advanced engineering models for wind turbines …curranc/files/curran_crawford_phd.pdftitle:...

Advanced Engineering Models forWind Turbines with Application to

the Design of aConing Rotor Concept

Curran A. Crawford

Trinity CollegeDepartment of Engineering

This dissertation is submitted for the degree of

Doctor of Philosophy

October 2006

“Engineering is the science of economy, of conserving theenergy, kinetic and potential, provided and stored up bynature for the use of man. It is the business of engineeringto utilize this energy to the best advantage, so that theremay be the least possible waste.”

– Willard A. Smith (1908)

I, Curran Crawford, certify that this dissertation is the result of my ownwork and includes nothing which is the outcome of work done in collabora-tion except where specifically indicated in the text. The thesis has a totalword count of less than 65,000 and includes less than 150 figures.

This manuscript was prepared using the LATEX2e system implemented inMiKTeX and worked on in TeXnicCenter. The diagrams were prepared inOpenOffice.org Draw and the figures in Matlab and Excel.

The text is set in 11 pt font.

Copyright c© 2006 by Curran Crawford

Title: Advanced Engineering Models for Wind Turbines with Application to theDesign of a Coning Rotor Concept

Author: Curran A. CrawfordKeywords: Wind Turbine, Coning Rotor, BEM Theory, Rotor Optimization

AbstractThe core focus of this thesis is the development of engineering-level aerodynamic

and structural models, suitable for the design of highly flexible wind turbines withlarge coning and yaw angles. The enhanced models are integrated to enable op-timization of a coning rotor wind turbine concept. GH BLADEDTM is used as areference industry-standard code, to evaluate the validity of current design toolsapplied to highly flexible concepts.

The coning rotor concept combines the load shedding properties of flap-hingedblades with gross change in rotor area, via large coning angles and lengthened blades,to achieve increased energy capture at nominally constant system cost. Based onup-scaling the original detailed design work from the mid-1990’s, a comparison to amodern conventional machine indicates that the coning rotor remains a valid alter-nate technology track. Design theory is also used to justify the coning conceptualapproach.

The primary theoretical contribution of this work is an enhanced Blade ElementMomentum (BEM) method. Utilizing vortex theory to model induction, computa-tionally efficient corrections are derived that are key in more accurately predictingperformance for coned rotors. The theory is extended to include wake expansion,dynamic inflow, and yawed conditions, as well as considering centrifugal and radial-flow induced stall-delay. The theory is favourably validated against ComputationalFluid Dynamics (CFD) and experimental results for both real and idealized rotors.

BLADEDTM was to be modified with the enhanced BEM method for dynamicanalyses. To support these analyses, a beam sectional model and Finite ElementMethod (FEM) approach to the generalized centrifugally stiffened beam problemwere implemented. Ultimately, the linear structural theory in current codes pre-cluded accurate predictions at large flap angles. In lieu of a fully non-linear flexible-body simulation, a rigid-body dynamic model of the system was developed. Thecoupled aerodynamic and structural models were then used to analyse steady-stateand dynamic operation, including optimal control schedules.

Parametric optimization studies were used to examine the interplay between de-sign variables for the coning rotor, relative to a reference conventional machine.Increased blade length, shape and airfoil choice were found to be tightly coupled,yielding energy gains of 10–30% over conventional rotors. Airfoil choice and controlmechanism were found critical to limiting torque and thrust. The fundamental non-linear open-loop dynamics were also examined, including flap and edgewise dampingbehaviour. Low-Frequency Noise (LFN) was computed with a properly implementedphysics-based model, to quantify sensitivity to design and operational parameters.

The current work is a preliminary, but critical step, in proving the worth ofthe coning rotor. Controller design and an accurate flexible-body code will be re-quired for full load-set simulations, to affect detailed component design and costing.Ultimately, prototype testing will be needed to validate the complicated stallingbehaviour of the coning rotor.

AcknowledgementsI would like to begin by thanking Jim Platts and Dr. Bill O’Niel for their

supervision throughout the process. Without Jim’s open “what if?” at ourweekly meetings, none of the important questions would have been asked.

The support given by members of Garrad Hassan, in terms of time, con-versation and code (BLADEDTM), is greatly acknowledged. In particular,Peter Jamieson’s initial overview of the coning rotor concept was invalu-able. David Quarton’s generous offer of BLADED enabled comparison toan advanced, industry-level code. Ervin Bossanyi, Graeme McCann, TonyMercer, and Robert Rawlinson-Smith always responded quickly to e-mailqueries, and their efforts to integrate my changes into BLADED are greatlyappreciated.

Thomas Wulf and Andrew Streatfield at Rothe Erde provided valuableinsight into very large bearing design and applicabilty to wind turbines.

Robert Mikkelsen at the Technical University of Denmark provided valu-able feedback and data for the development of the BEM method.

Henk Polinder and Maxime Dubois from the Delft University of Technol-ogy provided data and insight into direct-drive generator costing.

Dr. Coton at Glasgow University generously provided the source code forHAWTDAWG and helped in attempting to modify it to suit the coningrotor.

The team at NREL in the US, especially Patrick Moriarty, Jason Jonkmanand Scott Schreck, made available the UAE dataset, as well as insight intotheir codes and noise prediction. Paul Veers and Jose Zayas at SandiaNational Labs also provided a North American perspective to wind energy.

Funding by the UK Commonwealth and Canadian NSERC scholarshipsis gratefully acknowledged.

On a more personal note, numerous people at Cambridge provided goodcompany during the course of the PhD, especially Thomas, Andy, andTamas. Carlos proved a good friend at our meetings on both sides ofthe Atlantic. I have also enjoyed spending time away from work with allthose bound together by Portugal Street and Canadian friends from “TheOther Place.” My family has, as ever, given me support and encouragementthroughout my time at Cambridge. Finally, Rina has shared the whole pro-cess with me, making the endeavour truly worthwhile.

Table of Contents

1 Introduction 11.1 Wind Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scoping the Coning Rotor . . . . . . . . . . . . . . . . . . . . . . . 21.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I Wind Energy in Context

2 System Considerations 92.1 The Resource . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Characterisation . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 The Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 System Constraints . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Supporting the Grid . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Decentralization . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 The End User . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Locale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Further Impacts . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Variations on a Theme 153.1 Chasing Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Design Theory . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 COE and CF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 To Lift or Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Lift-Drag Comparison . . . . . . . . . . . . . . . . . . . . . 213.3.3 Operational Conditions . . . . . . . . . . . . . . . . . . . . 21

3.4 VAWT or HAWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6.1 Goals and Methods . . . . . . . . . . . . . . . . . . . . . . . 263.6.2 Dynamic Considerations . . . . . . . . . . . . . . . . . . . . 29

3.7 Adapting Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 303.7.1 Dynamic Motion . . . . . . . . . . . . . . . . . . . . . . . . 303.7.2 Load Matching . . . . . . . . . . . . . . . . . . . . . . . . . 31

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3.7.3 Soft Approach . . . . . . . . . . . . . . . . . . . . . . . . . 313.7.4 Coning Rotor Adaptation . . . . . . . . . . . . . . . . . . . 32

3.8 Function to Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.8.1 Historical Context . . . . . . . . . . . . . . . . . . . . . . . 323.8.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.8.3 Pitch Control . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Coning Rotor Challenges and Opportunities . . . . . . . . . . . . . 36

II Analysis

4 Analytic Development 414.1 Industry Standard Comparison (BLADEDTM) . . . . . . . . . . . 424.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Coned Rotor Performance Metrics . . . . . . . . . . . . . . . . . . 454.4 Aerodynamic Modelling . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4.1 Back to Basics . . . . . . . . . . . . . . . . . . . . . . . . . 474.4.2 Flow Field Kinematics . . . . . . . . . . . . . . . . . . . . . 474.4.3 Momentum Balances . . . . . . . . . . . . . . . . . . . . . . 494.4.4 Blade Elements . . . . . . . . . . . . . . . . . . . . . . . . . 534.4.5 Wake Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 544.4.6 Relating Disc to Far Field . . . . . . . . . . . . . . . . . . . 574.4.7 Azimuthal Induced Velocity . . . . . . . . . . . . . . . . . . 584.4.8 Stall Delay and Centrifugal Pumping . . . . . . . . . . . . . 604.4.9 Spanwise Flow . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.10 Closed Equations . . . . . . . . . . . . . . . . . . . . . . . . 724.4.11 Expanding Wake . . . . . . . . . . . . . . . . . . . . . . . . 734.4.12 Additional Wake Geometries . . . . . . . . . . . . . . . . . 754.4.13 Aerodynamic Analytic Optimum . . . . . . . . . . . . . . . 764.4.14 Dynamic BEM . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Acoustic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.1 Aerodynamic Noise . . . . . . . . . . . . . . . . . . . . . . . 974.5.2 Modelling Approaches . . . . . . . . . . . . . . . . . . . . . 1004.5.3 Low-Frequency Noise . . . . . . . . . . . . . . . . . . . . . . 1034.5.4 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6 Structural Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.6.1 Sectional Modelling . . . . . . . . . . . . . . . . . . . . . . 1144.6.2 Kinetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.6.3 Dynamic Modelling . . . . . . . . . . . . . . . . . . . . . . 1244.6.4 Centrifugally Stiffened Beam . . . . . . . . . . . . . . . . . 130

4.7 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.7.1 Steady State Operation . . . . . . . . . . . . . . . . . . . . 1374.7.2 Dynamic Control . . . . . . . . . . . . . . . . . . . . . . . . 141

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4.8 Generator Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.8.1 Shear Stress Generator Model . . . . . . . . . . . . . . . . . 1424.8.2 Analytic Magnetic Circuit Generator Model . . . . . . . . . 143

4.9 Cost Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.9.1 Proxy Cost Metric . . . . . . . . . . . . . . . . . . . . . . . 1444.9.2 Magnetic Materials Cost . . . . . . . . . . . . . . . . . . . . 144

5 Validation 1475.1 Aerodynamic Validation . . . . . . . . . . . . . . . . . . . . . . . . 147

5.1.1 Uniformly Loaded Rotor . . . . . . . . . . . . . . . . . . . . 1485.1.2 Uniform Loading Results . . . . . . . . . . . . . . . . . . . 1495.1.3 Fundamental Model Deficiencies . . . . . . . . . . . . . . . 1535.1.4 Radial and Far-Field Flow . . . . . . . . . . . . . . . . . . . 1555.1.5 Tjaereborg Rotor . . . . . . . . . . . . . . . . . . . . . . . . 1565.1.6 NREL UAE . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.1.7 Dynamic Inflow . . . . . . . . . . . . . . . . . . . . . . . . . 1645.1.8 Yawed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.2 Acoustics Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.3 Structural Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.3.1 Sectional Properties . . . . . . . . . . . . . . . . . . . . . . 1715.3.2 Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.4 BLADEDTM Validation and Suitability . . . . . . . . . . . . . . . 1775.4.1 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.4.2 Steady State Operation . . . . . . . . . . . . . . . . . . . . 1785.4.3 Flapping Hinge . . . . . . . . . . . . . . . . . . . . . . . . . 1795.4.4 Mode Shape Modification by Coning . . . . . . . . . . . . . 181

5.5 Generator Model Validation . . . . . . . . . . . . . . . . . . . . . . 182

III Design

6 Rotor Optimization 1856.1 Initial Comparison and Updating . . . . . . . . . . . . . . . . . . . 185

6.1.1 Shifting Benchmarks . . . . . . . . . . . . . . . . . . . . . . 1856.1.2 Up-Scaling CONE-450 Results . . . . . . . . . . . . . . . . 1866.1.3 Parked Conditions . . . . . . . . . . . . . . . . . . . . . . . 187

6.2 COE and CF Performance . . . . . . . . . . . . . . . . . . . . . . . 1886.3 Aerodynamic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . 1906.4 Rotor Optimization Challenge . . . . . . . . . . . . . . . . . . . . . 191

6.4.1 Conventional Rotors . . . . . . . . . . . . . . . . . . . . . . 1916.4.2 Coning Rotor Optimization . . . . . . . . . . . . . . . . . . 1936.4.3 CONE-450 Approach . . . . . . . . . . . . . . . . . . . . . . 194

6.5 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

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6.5.1 Design Variables and Parameters . . . . . . . . . . . . . . . 1976.5.2 Optimizer Tuning . . . . . . . . . . . . . . . . . . . . . . . 1996.5.3 Stepwise Optimization Approach . . . . . . . . . . . . . . . 2016.5.4 Aerodynamic Optimum . . . . . . . . . . . . . . . . . . . . 2036.5.5 Blade Length . . . . . . . . . . . . . . . . . . . . . . . . . . 2036.5.6 Control Strategy Impacts . . . . . . . . . . . . . . . . . . . 2086.5.7 Aerodynamic Uncertainty . . . . . . . . . . . . . . . . . . . 211

7 Design Considerations 2177.1 Dynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.1.1 Fundamental Response . . . . . . . . . . . . . . . . . . . . . 2177.1.2 Aerodynamic Damping . . . . . . . . . . . . . . . . . . . . 222

7.2 Tower Thump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.2.1 Variation with Wind Speed . . . . . . . . . . . . . . . . . . 2307.2.2 Wake Profile Influence . . . . . . . . . . . . . . . . . . . . . 2317.2.3 Mitigation by Offset . . . . . . . . . . . . . . . . . . . . . . 2347.2.4 Control Strategy Effects . . . . . . . . . . . . . . . . . . . . 2377.2.5 Dynamic Motion . . . . . . . . . . . . . . . . . . . . . . . . 2387.2.6 Remaining Issues . . . . . . . . . . . . . . . . . . . . . . . . 239

8 Conclusions 2418.1 Analytical Contributions . . . . . . . . . . . . . . . . . . . . . . . . 2418.2 Design Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2428.3 The Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Bibliography 245

A Geometric Transformations A-1

B Cross Section Analysis B-1

C Reference Machine Specifications C-1C.1 Tjaereborg Machine . . . . . . . . . . . . . . . . . . . . . . . . . . C-1C.2 NREL UAE Phase IV . . . . . . . . . . . . . . . . . . . . . . . . . C-4C.3 1.5 MW Reference Machine (REF-1500) . . . . . . . . . . . . . . . C-7C.4 GH Demo Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . C-10C.5 CONE-450 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . C-12C.6 Coning Rotor Study . . . . . . . . . . . . . . . . . . . . . . . . . . C-12

D Historical Machines D-1

E Material Specifications E-1

iv

List of Figures1.1 Example “Danish concept” HAWT machines . . . . . . . . . . . . . . . 31.2 Example VAWT machines . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Coning rotor illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 CP and CF for translating airfoils utilizing lift and drag . . . . . . . . 223.2 Variation of CP and CF with translation direction θ for lift device . . 223.3 Blade mass trend with rotor diameter . . . . . . . . . . . . . . . . . . 253.4 Specific energy capture as a function of machine rating . . . . . . . . . 253.5 Power curve regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 Wind velocity and force vectors for various control strategies . . . . . 273.7 Power control in CP − λ and τ − Ω planes . . . . . . . . . . . . . . . . 283.8 Coning rotor schematic layout . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Kinematics setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 Kinematics at rotor for single stream-tube . . . . . . . . . . . . . . . . 484.4 Flow relative to airfoil section . . . . . . . . . . . . . . . . . . . . . . . 504.5 Thrust CT,model model behaviour . . . . . . . . . . . . . . . . . . . . . 524.6 Wake decomposition into rings and filaments . . . . . . . . . . . . . . 554.7 Vortex ring geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.8 Velocity field around vortex cylinder . . . . . . . . . . . . . . . . . . . 574.9 Axial, root and bound vortex filaments . . . . . . . . . . . . . . . . . . 594.10 Azimuthal induction along blade . . . . . . . . . . . . . . . . . . . . . 614.11 Skewed flow velocity vectors and boundary layer profiles . . . . . . . . 644.12 Effective sweep angle with coning for typical operating conditions . . . 654.13 Experimental 2D lift curves for varying yaw angle . . . . . . . . . . . 674.14 Wing section lift curves for 45 swept wing . . . . . . . . . . . . . . . 684.15 Stall delay behaviour for 2D sections in yawed flow . . . . . . . . . . . 684.16 Sectional data corrected for spanwise flow . . . . . . . . . . . . . . . . 714.17 Expanded wake profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.18 Skewed wake integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.19 Grids for ε table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.20 Skewed wake table transformation . . . . . . . . . . . . . . . . . . . . 824.21 Tower shadow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.22 Example total velocity tower shadow flow field . . . . . . . . . . . . . 864.23 Dynamic wind vector decomposition . . . . . . . . . . . . . . . . . . . 864.24 Dynamic flow relative to airfoil section . . . . . . . . . . . . . . . . . . 87

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4.25 Dynamic stall parameters . . . . . . . . . . . . . . . . . . . . . . . . . 904.26 Dynamic stall hysteresis loop for cyclic α . . . . . . . . . . . . . . . . 914.27 Section definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.28 Blade section triangulation . . . . . . . . . . . . . . . . . . . . . . . . 1174.29 Definition of section coordinate systems . . . . . . . . . . . . . . . . . 1184.30 Section layout for kinetostatics . . . . . . . . . . . . . . . . . . . . . . 1204.31 Kinetostatic offset vectors . . . . . . . . . . . . . . . . . . . . . . . . . 1204.32 Integrated loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.33 Beam element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.34 Strained fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.35 Element centrifugal force . . . . . . . . . . . . . . . . . . . . . . . . . 1344.36 Operating point solution flow chart . . . . . . . . . . . . . . . . . . . . 1394.37 Speed windows for optimal and limiting operation . . . . . . . . . . . 140

5.1 Uniformly loaded axial induction (β = 0, CT = 0.89) . . . . . . . . . 1495.2 Uniformly loaded axial induction as a function of loading CT (β = 0) 1505.3 Uniformly loaded axial induction (β = 20, CT = 0.89) . . . . . . . . . 1515.4 Uniformly loaded axial induction (β = −20, CT = 0.89) . . . . . . . . 1515.5 Uniformly loaded axial induction as a function of β for Bladed thrust

model (CT = 0.89, no wake expansion) . . . . . . . . . . . . . . . . . . 1525.6 Uniformly loaded radially induced velocity as a function of β (CT = 0.89) 1555.7 Tjaereborg axial induction (β = 0) . . . . . . . . . . . . . . . . . . . 1575.8 Tjaereborg axial induction (β = 20) . . . . . . . . . . . . . . . . . . . 1585.9 Tjaereborg axial induction (β = −20) . . . . . . . . . . . . . . . . . . 1595.10 Tjaereborg aerodynamic loading as a function of β . . . . . . . . . . . 1605.11 Power coefficient map for rotor operating points as a function of β . . 1615.12 NREL UAE Sequence S measurements and predictions . . . . . . . . . 1635.13 NREL UAE Sequence F measurements and predictions . . . . . . . . 1645.14 Tjaereborg root bending moments in yawed flow . . . . . . . . . . . . 1665.15 Simple isolated rotor acoustic prediction . . . . . . . . . . . . . . . . . 1685.16 SPL for upwind rotor with tower influence for varying azimuthal step

size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.17 Effect of tilt on section out-of-plane velocity . . . . . . . . . . . . . . . 1705.18 Effect of tilt on pressure signature for upwind rotor . . . . . . . . . . . 1705.19 Effect of tilt on downwind rotor SPL . . . . . . . . . . . . . . . . . . . 1715.20 Acoustic footprint of downwind rotor with tilt and shear . . . . . . . . 1715.21 Uniform blade Out-of-Plane (OP) rotating modes . . . . . . . . . . . . 1735.22 Uniform blade In-Plane (IP) rotating modes . . . . . . . . . . . . . . . 1745.23 Uniform blade OP rotating modes for teetered hub . . . . . . . . . . . 1745.24 Free-hub rotating uniform blade IP modes . . . . . . . . . . . . . . . . 1755.25 lhub = 0.5 free-hub rotating uniform blade with IP modes . . . . . . . 1755.26 Collective IP mode shapes for Demo blade . . . . . . . . . . . . . . . . 1775.27 BLADEDTM computed steady power curve for PTS . . . . . . . . . . 179

vi

6.1 Operational rotor thrust curves for upscaled machines relative to REF-1500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.2 Betz limit variable area rotor performance . . . . . . . . . . . . . . . . 1886.3 Industry machine performance . . . . . . . . . . . . . . . . . . . . . . 1896.4 Performance maps for actual tip radius . . . . . . . . . . . . . . . . . 1906.5 Performance maps for reference tip radius and control variable . . . . 1916.6 Chord and twist profile control . . . . . . . . . . . . . . . . . . . . . . 1976.7 Typical section layup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2006.8 Finite differencing prediction for CP as a function of solution tolerance

tol and step size h for chord and twist control points . . . . . . . . . . 2016.9 Power coefficient CP , annual energy yield Eann and total blade mass

variation with aerodynamic design tip speed λdesign,aero and cone angleβdesign,aero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6.10 Blade chord profiles for λdesign,aero = 6–8 . . . . . . . . . . . . . . . . 2056.11 Relative energy yield and maximum thrust variation with blade length

Stip and balance angle βbalance . . . . . . . . . . . . . . . . . . . . . . 2066.12 Energy yield relative to REF-1500 varying with mean wind speed and

hub height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.13 Operational curves for optimal rotors at βbalance = 30 . . . . . . . . . 2096.14 Generator cost variation with diameter and torque/speed levels . . . . 2106.15 CP − λ curves for β = 0 and varying airfoil set . . . . . . . . . . . . . 2116.16 Variation in operational curves with stall delay models for set II rotors 215

7.1 Flap angle solution expansion terms . . . . . . . . . . . . . . . . . . . 2187.2 Blade flapping without aerodynamic contribution . . . . . . . . . . . . 2197.3 Hub loads for blade flapping without aerodynamic contribution . . . . 2207.4 Dynamic situation with flapping blades and rotating imbalance . . . . 2217.5 Logarithmic decrement δd for flap hinge damping . . . . . . . . . . . . 2237.6 Damping constant variation with control strategy for 15% thick set I

airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2267.7 Damping constant variation with airfoil for 15% thick airfoils and PTS 2277.8 Variation of Angle of Attack (AOA) α and total twist angles over wind

speed range with airfoil choice (15% thick) for PTS . . . . . . . . . . . 2287.9 Damping constant variation with airfoil for 21% thick airfoils and PTS 2287.10 SPL variation with wind speed . . . . . . . . . . . . . . . . . . . . . . 2307.11 Acoustic spectrum variation with wind speed, including stall delay . . 2317.12 SPL variation with wake width . . . . . . . . . . . . . . . . . . . . . . 2327.13 Acoustic spectrum variation with wake width . . . . . . . . . . . . . . 2327.14 SPL variation with wake deficit . . . . . . . . . . . . . . . . . . . . . . 2337.15 Acoustic spectrum variation with wake deficit . . . . . . . . . . . . . . 2337.16 SPL variation with offset from tower . . . . . . . . . . . . . . . . . . . 2347.17 Acoustic spectrum variation with offset from tower . . . . . . . . . . . 2357.18 SPL variation with cone angle and observer azimuth angle . . . . . . . 236

vii

7.19 Acoustic spectrum variation with cone angle . . . . . . . . . . . . . . 2367.20 Footprint SPL changes with variation in cone angle . . . . . . . . . . . 2377.21 Rotor thrust variation over azimuth for control strategy choice . . . . 238

B.1 Section integral coordinate transform . . . . . . . . . . . . . . . . . . . B-1

C.1 Tjaereborg blade details . . . . . . . . . . . . . . . . . . . . . . . . . . C-2C.2 NREL UAE blade details . . . . . . . . . . . . . . . . . . . . . . . . . C-6C.3 REF-1500 details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8C.4 Demo machine details . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11C.5 CONE-450 details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-13C.6 Airfoil coefficients (NACA 63-4XX) . . . . . . . . . . . . . . . . . . . . C-14

viii

List of Tables2.1 IEC Wind Turbine Classes . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Axiomatic/TRIZ design method similarities . . . . . . . . . . . . . . . 19

4.1 Windowing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2 Active magnetic specific material costs . . . . . . . . . . . . . . . . . . 145

5.1 Uniformly loaded power coefficients (CT = 0.89) . . . . . . . . . . . . 1525.2 Upwind SPL predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.3 Uniform blade with long hub eigenfrequencies . . . . . . . . . . . . . . 1765.4 Collective IP eigenfrequencies for Demo blade . . . . . . . . . . . . . . 1775.5 Steady flap angle variation with actuator moment for Ω =50 RPM . . 181

6.1 Outer skin layup schedule . . . . . . . . . . . . . . . . . . . . . . . . . 2006.2 Web layup schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

7.1 SPL variation with control strategy above rated . . . . . . . . . . . . . 2387.2 SPL variation with dynamic motion and cone angle . . . . . . . . . . . 239

C.1 Tjaereborg parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1C.2 NREL UAE parameters . . . . . . . . . . . . . . . . . . . . . . . . . . C-5C.3 REF-1500 parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . C-7C.4 Demo machine parameters . . . . . . . . . . . . . . . . . . . . . . . . . C-10C.5 CONE-450 parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . C-12

D.1 Soft machine history . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2

E.1 Section layup material specifications . . . . . . . . . . . . . . . . . . . E-2

ix

List of AcronymsAMT Axial Momentum TheoryAOA Angle of AttackBEM Blade Element MomentumBOP Balance of PlantDV Design VariableCF Capacity FactorCFD Computational Fluid DynamicsCHP Combined Heat and PowerCOE Cost of EnergyCS Coordinate SystemDES Detached Eddy SimulationDFT Discrete Fourier TransformDNS Direct Numerical SimulationDOE Design of ExperimentsDOF Degrees of FreedomDP Design ParameterEOM Equations of MotionFD Finite DifferenceFEM Finite Element MethodFFT Fast Fourier TransformFR Functional RequirementFSS Fixed Speed StallFW-H Ffowcs Williams-HawkingsGDT Japanese General Design TheoryGH Garrad Hassan and PartnersHAWT Horizontal Axis Wind TurbineHVDC High Voltage Direct CurrentIP In-PlaneLE Leading EdgeLES Large Eddy SimulationLFN Low-Frequency NoiseLVRT Low Voltage Ride ThroughNA Neutral AxisNPV Net Present ValueNREL National Renewable Energy LaboratoryNS Navier-StokesODE Ordinary Differential Equation

xi

OP Out-of-PlanePDE Partial Differential EquationPDF Probability Density FunctionPMG Permanent Magnet GeneratorPTF Pitch to FinePTS Pitch to StallPV Process VariableRANS Reynolds-Averaged Navier-StokesSGSF Savitzky-Golay Smoothing FiltersSQP Sequential Quadratic ProgrammingSPL Sound Pressure LevelTE Trailing EdgeTRIZ Theory of Inventive Problem SolvingUAE Unsteady Aerodynamics ExperimentURANS Unsteady Reynolds-Averaged Navier-StokesVAWT Vertical Axis Wind TurbineVSS Variable Speed StallWTC Wind Turbine Company

xii

List of SymbolsGiven the range of equations covered, only the most generic and consistent areincluded here. The remaining symbols are defined in context as they are introducedand used.

English Symbols

a Axial induction factor

a′ Tangential induction factor

A Aspect ratio

B Number of blades

c Chord length

cd Sectional (2D) drag coefficient

cl Sectional (2D) lift coefficient

cm Sectional (2D) moment coefficient

CP Power coefficient P/(1/2ρV 3πD2

)CT Thrust coefficient T/

(1/2ρV 2πD2

)m Mass

P Power; Rotation frequency and its multiples e.g. 1P , 3P

p Pressure

r Radius

Re Reynolds number V l/ν

s, S Spanwise coordinate, blade length

T Thrust

V Velocity (wind speed if no subscript specified)

~w, ~u Induced velocities in momentum and vortex theories respectively

Greek Symbols

α Airfoil AOA

β Cone angle (positive downwind)

χ Aerodynamic skew angle

ε BEM correction factors; Strain

xiii

Γ Vortex filament strength

γ Vortex strength per unit area

γtot Total blade section twist angle (all pitch angles positive towards feather)

γtwist Blade section geometric twist angle

γset Blade fixed pitch offset (set angle)

γpitch Active blade pitch angle

Λ Sweep angle

λ Tip-speed ratio rΩ/V

µ Dynamic viscosity ρν

ν Kinematic viscosity µ/ρ

Ω Rotor rotation speed

ψt Nacelle tilt angle

ψyaw Nacelle yaw angle

ρ Density

σ Blade solidity Bc/2πr; Stress

Notation

x–y A set of 2D (or 3D) axes; Plane containing 2D axes

xi Element index i; where i is an axis label, indicates vector component in thatdirection

xn Time index n

A Matrix; also[a bc d

]x Column vector; also

ab

I Identity matrix

~x Geometric vector; also(x, y)

x Unit geometric vector

xiv

Chapter 1

Introduction

Energy lies at the heart of modern society, enabling everything from heating, light-

ing, computers, and food production through to manufacturing, medicine, and trans-

port. To properly appreciate the context of this thesis, consider that worldwide in

2003, wind power provided 0.5% of the 13.5% slice electricty represented of the

125 PWh consumed at end-use from all primary sources [1, 2]. A growing apprecia-

tion of the deleterious climatic effects of anthropogenic greenhouse gas emissions is

increasing regulatory pressure to mitigate or avoid CO2 release into the atmosphere

[3]. Simultaneously, many populous nations are rapidly aspiring to Western stan-

dards of living, intensifying the use of and competition for energy. The tractability

of tackling stationary energy use and supply, as opposed to transport fuels, and

in particular electricity production, has made this area of key public importance

[4]. However, it is important to bear in mind the overall scale of our society’s en-

ergy use, in all forms and with myriad correctable inefficiencies, when considering

enhancement of electricity production.

Ultimately, the highly competitive electricity market will select the most cost-

effective technology. Unfortunately, inclusion of externalities, e.g. a carbon tax,

unknown future decommissioning costs, as well as subsidies to all forms of energy,

creates an artificially skewed playing field. To compete in this economic environ-

ment, wind power must deliver power of grid-standard quality and at a competitive

price relative to “conventional” generation. It is in this overall context that this the-

sis seeks to develop engineering tools capable of properly analysing and optimizing

a potentially much more cost-effective wind turbine concept, the coning rotor.

1.1 Wind Power

Wind power has for the past 5 years increased in installed capacity by 20-30% each

year [2], indicating commercial viability with proper siting and regulatory frame-

work. The turbines are among the best energy sources in terms of energy payback

1

2 Chapter 1 Introduction

time1 (3-7 months [5]) and ratio2 (17–39 [6]). Once the turbines are built, their

impact is virtually nil, given proper design and consideration of local communities

and wildlife. These strong economic and sustainability credentials make wind power

attractive from multiple perspectives.

Wind turbine design has achieved a high level of sophistication, enabling pro-

duction of 120 m diameter, 5 MW machines, sited both on-shore and off-shore.

Progress from the 6 m diameter, 15 kW machines of the early 1980’s to today has

been achieved by technical and scale evolution of the “Danish concept”. Modern

examples are shown in Fig. 1.1, typified by rigid three-bladed rotors placed upwind

of the towers. In a modern machine, a gearbox transfers torque from the rotor to an

actively controlled generator, although some directly drive a low-speed generator.

Pitch actuators at the roots of the blades directly control the aerodynamic power

input to the rotor. Almost universally, a Horizontal Axis Wind Turbine (HAWT) is

chosen (i.e. the rotation axis of the rotor is near horizontal), although the Vertical

Axis Wind Turbine (VAWT) (examples shown in Fig. 1.2) with a vertical rotation

axis received much attention in the past, and may still find application at extremely

small and large scales.

Key modern development work is aimed at increased reliability, particularly for

off-shore sites, and load mitigation to reduce structural cost. The latter is being

tackled by more active control (in turn made possible by power electronics and fast

actuators) and refined structural designs and materials. It should be appreciated

that wind turbines, although sharing many engineering principles with aircraft de-

sign, are quite distinct in their objective (to deliver reliable energy) and therefore

demand more cost-sensitive solutions.

1.2 Scoping the Coning Rotor

The coning rotor was studied in the mid-1990’s as an alternative technology track

[7], potentially capable of delivering significant cost of energy reductions relative

to then state-of-the-art machines. The coning rotor builds on a lineage of flexible

machine designs, as opposed to the rigid “Danish concept”, in an attempt to both

capture more energy and achieve a lighter (cheaper) structure. The essential physical

features of the coning rotor, illustrated in Fig. 1.3, are hinges at the blade roots,

1Energy payback time represents the time taken to produce as much energy as used inproduction of the generating device.

2Energy payback ratio is the lifetime energy produced divided by energy used to producethe technology.

1.2 Scoping the Coning Rotor 3

(a) 1.3 MW Bonus machines at Lambriggwind farm

(b) REpower 5 MW HAWT being installedoffshore near Beatrice oil platform in the

North Sea (Image fromhttp://www.repower.de))

Figure 1.1 Example “Danish concept” HAWT machines

(a) 6 kW XC02 Quiet Revolution QR5VAWT (Image from

www.quietrevolution.co.uk)

(b) 300 kW Flowind Darrieus-type VAWT(Image from www.ecopowerusa.com)

Figure 1.2 Example VAWT machines


enabling the blades to flap downwind, thus sweeping out a cone. The rotor must

obviously be downwind of the tower. The hinges afford a steady load-matching

advantage by turning the blades from bending beams into tension members, as the

centrifugal and aerodynamic forces balance each other.

Figure 1.3 Coning rotor illustration

Dynamic movement of the blade may also further mitigate transitory loading.

Flap-hinged blades have long been known to afford this benefit [8–10]. The differen-

tiating features of the coning rotor are larger operational and parked coning angles.

In shut-down, extreme wind conditions, the blades are fully-coned to approximately

85. By avoiding extreme loading, the blades can be relatively longer, capturing

more energy.

Even with these positive aspects, flexible concepts have not achieved commer-

cial success, particularly at scales greater than a few hundred kilowatts. The root

difficulty has been in achieving accurate models, both aerodynamically and struc-

1.3 Thesis Outline 5

turally.1 These are required to adequately size components and synthesize conser-

vative operational profiles. The inherent flexibility of conventional “rigid” machines

is a difficulty in itself, without the further non-linearities of gross-coning. Failures

have been typically caused by dynamic behaviour that was not accurately predicted,

making a clear understanding of underlying assumptions important for all types of

wind turbines.

Numerous areas of study present themselves for further validating the coning

rotor concept. This thesis is limited to enhancing the engineering models that will

underpin any future development work. Without properly formulated tools, any

more detailed analysis and design is premature. In particular, the novel aspect of

the concept, the rotor itself, is of primary interest, with due consideration given to

the capabilities of other components. Furthermore, technical analysis is pursued,

rather than economic estimates.

Having developed the requisite analysis suite, a parametric study is made of the

concept, focusing on blade design and operational strategy. This step represents

the most optimistic view of performance benefit relative to traditional machines,

and must be completed in order to justify and motivate future work. This first

stage is required in any case as a base for future dynamic simulations and further

refinement of the design loads. Although the focus is on the coning rotor, the

developed theory and optimization procedure have relevance to current conventional

machines, operating at large yaw angles with increasingly flexible blades.

1.3 Thesis Outline

This thesis is divided into three distinct parts, following a natural progression from

problem definition to solution. Part I deals with the fundamental aspects of har-

vesting wind energy. Chapter 2 outlines the relevant characteristics of the wind

resource, together with the constraints placed on the wind energy converter by the

energy system being supplied. Chapter 3 provides a fundamental overview of possi-

ble functional and physical approaches, based on the set of constraints imposed by

the overall energy system. It also more clearly defines the operational and physical

aspects of the coning rotor, justifying the concept from a design theory perspective,

relative to other conceptual proposals.

1There have of course been cases in which elements of the machine were designed incor-rectly for financial or expediency reasons.


Part II presents the analysis tools required to explore highly flexible wind turbine

concepts, motivated by the analysis requirements created by the physical aspects

of the coning rotor in Chapter 3. Chapter 4 presents the theories that have been

developed and implemented in a unified analysis suite, ExcelBEM. In particular, a

novel Blade Element Momentum (BEM) method is presented, incorporating correc-

tions to more correctly model coned and yawed rotors in steady and unsteady flow.

The implementation of a Low-Frequency Noise (LFN) model is presented next. A

structural model for the blade sections is then developed, followed by dynamic Equa-

tions of Motion (EOM) and a centrifugally stiffened Finite Element Method (FEM)

beam model. Finally, an optimal control scheduling algorithm is described, followed

by a brief reference to a generator model and cost metrics. The work carried out

to validate the various theories, to the extent possible, is presented in Chapter 5.

The BEM method developments draw on the author’s published material [11]. The

remaining limits and shortcomings of the analytic methods are discussed, as is the

suitability of currently available industry codes. It should be noted at this point that

the reference code, BLADEDTM, is clearly capable of its intended use in simulating

conventional machines; it is the suitability for unconventional concepts such as the

coning rotor that is being explored.

Using the analytic tools developed in Part II, Part III explores the fundamentals of

designing a coning rotor, including aspects of the author’s published papers [12, 13].

Chapter 6 first up-scales the results of the original CONE-450 study to a modern

MW-scale machine. The aerodynamic characteristics of the coning rotor are then

discussed, with relevance to the optimization of rotors. A set of optimization and

parametric studies of coning rotors is then presented and discussed. The control

strategy, airfoil choice and blade length are primary foci. Chapter 7 then explores

some critical elements for the further design of the coning rotor. The fundamen-

tal dynamic response and aerodynamic damping are presented first, followed by a

parametric study of LFN trends with design parameters.

The thesis culminates with Chapter 8, by highlighting the main outcomes of this

work, covering both the theoretical contributions to wind turbines in general, and

the specific coning rotor concept study. The thesis concludes with suggestions for

future work to be carried out to address the outstanding issues and move the concept

forward towards reality.

Part I

Wind Energy in Context

Chapter 2

System Considerations

This chapter gives only the most brief of introductions to the source, transmission

and end-use of wind energy. The emphasis is on the salient points which must be

considered when proposing and analysing an alternative wind turbine concept. Nu-

merous authors inform the following sections, and provide more in-depth discussions

of the facets presented [14–19]. The resource itself is introduced in §2.1, followed by

transmission via the grid in §2.2 to the end user in §2.3.

2.1 The Resource

Wind energy is actually a form of solar energy, ultimately originating from the

nuclear reactions in the sun. Solar radiation differentially heats the atmosphere

and earth’s surface. This creates unbalanced pressure forces that drive the kinetic

motion of the air.

2.1.1 Scales

Wind energy is present at three spatial scales: global (e.g. trade winds); secondary

scales impacting the lower atmosphere (e.g. hurricanes, monsoon circulation); and

tertiary localized winds (e.g. land/sea breezes, valley/mountain winds, thunder-

storms, tornadoes). Temporal variation exists on an inter-annual, annual, diurnal

(day/night) and short term basis.

2.1.2 Characterisation

An atmospheric boundary layer exists above the surface of the earth, so that the

wind-speed grows with height z above reference height zref and wind-speed Vref :

V

Vref=(

z

zref

)α

(2.1.1)

The power-law exponent α is typically 1/7, varying with wind conditions and surface

roughness.

9

10 Chapter 2 System Considerations

The annual temporal variation is described well by a Weibull distribution with

Probability Density Function (PDF) f(V ):

f(V ) =dF

dV=

k

ckV k−1exp

(−(V

c

)k)

(2.1.2)

The shape factor k is in the range 1.8–2.2, and the scaling parameter c is used

to fit the mean wind-speed. The specific power content of the wind is 1/2ρV 3.

This cubic relationship with velocity V means that higher wind speeds deliver much

more energy than low speeds. Total energy yield from a turbine with power output

function P = P (V ) is computed from:

E = 8760

Vco∫Vci

P (V )f(V )dV (2.1.3)

where Vci and Vco are the lower and upper bounds of the wind turbine’s operational

wind speeds. The 8760 = 24 × 365 constant in the formula requires V in units of

m/s and P in kW to yield E in kWh, the standard energy “unit” in the electricity

industry.

More detailed temporal and spatial description of the wind-speed is provided by

the following [17]:

• Turbulence intensity• Probability density function• Autocorrelation function• Integral time• Power spectral density function

Offshore wind conditions are generally favourable compared to those on-shore. The

relatively smooth and unobstructed surface increases the shear rate α (i.e. higher

speeds at lower heights) and reduces turbulence levels.

2.1.3 Metrics

Potential installation sites are characterised by their annual mean wind-speed. The

standard at 10 m height ranges from Class I (0.0–4.4 m/s) through to Class VII

(7.0–9.4 m/s). Wind energy cost is typically quoted for Class VI sites. Bear in

mind that the V 3 relationship means that that a Class VII site has 50X more power

than a Class I site. Accurate knowledge of local conditions is critical for delivering

veracious energy yield predictions.

For the purposes of loads calculation, the IEC standard [20] specifies four classes

of wind speed shown in Table 2.1, defined at the hub height.

2.2 The Grid 11

Table 2.1 IEC Wind Turbine Classes (Adapted from [20])

ClassItem I II III IV

Reference wind speed Uref 50 42.5 37.5 30 (m/s)Annual average wind speed Uave 10 8.5 7.5 6 (m/s)50 yr return gust speed 1.4 Uref 70 59.5 52.5 42 (m/s)1 yr return gust speed 1.05 Uref 52.5 44.6 39.4 31.5 (m/s)

2.2 The Grid

The electricity grid interconnects generators and consumers, often across municipal,

state and national boundaries. It has evolved to facilitate large central generat-

ing stations exploiting economies of scale. New transmission technologies, such as

High Voltage Direct Current (HVDC) [21], are permitting increasing transmission

distances. A mix of generation technologies is used to constantly adjust power to

match the demand load. The time-constant of the generator, whether it be base-

load (e.g. coal, nuclear) or quick load-following (e.g. natural gas), determines the

relative economics and hence dispatch choice.

2.2.1 System Constraints

The market structure can play a dominant role in wind power’s economic success.

The closer the scheduling of supply is to actual dispatch, the more accurate the

forecast and hence system value. The Capacity Factor (CF) (see §3.2) of the machine

will also affect the efficient utilization of transmission resources. Geography has a

large impact as well, as transmitting electricity long distances is quite costly. Moving

offshore only increases this burden. Unfortunately, wind, as with many renewables,

has a low power density and is often best captured far from load centres.

Wind power is often ascribed a low value in displacing conventional plant, and

described as “intermittent.” While it is true that the wind does not blow all the time,

nationally distributed wind-power plants will tend to smooth output fluctuations.

It is more accurate to say that wind power is “variable,” but then so to is electricity

demand. Detailed studies and recent experience has indicated that up to a 20%

market share of electricity generation, little backup capacity is required [22].


2.2.2 Supporting the Grid

Wind turbines have been grid-connected successfully for many years [23]. In the

past, simple induction generators were used to passively soft-couple the rotor to

the grid frequency. At very low penetrations of wind power, and for smaller scale

machines, this approach was, and is quite effective.

For structural and power-capture reasons, machines are increasingly variable-

speed, further decoupling the grid and rotor. With increasing levels of wind plant

on the grid, regulators have toughened the requirements for connection. In par-

ticular, Low Voltage Ride Through (LVRT) is a requirement in many localities, as

is the ability to provide reactive power to actively control the power factor. This

has placed increasing demands on power electronics and generators to supply these

services to the grid, either at the individual machine or overall wind farm level.

2.2.3 Decentralization

There has been recent interest in returning to a distributed generation model, as

it was at the dawn of the electrified age in the early 1900’s. Savings accruing to

Combined Heat and Power (CHP), including the costs to some users associated with

network outages, have prompted fossil-fuel based machines of modest scale to be

installed. Embedding generation within the grid network of course poses its own set

of technical and regulatory challenges.

It has been suggested that given the inherently distributed nature of renewables, it

makes sense to pursue a distributed model [24]. At the same time, the scale effects of

wind turbines must be born in mind when considering small installations, including

the quality of the wind resource which grows with height (i.e. size of machine). Until

such time as a cost-effective energy storage solution (e.g. compressed air, flywheels,

batteries, hydrogen [25, 26]) is available, the grid will be required to redistribute

power to continually satisfy demand regardless of wind resource. Nevertheless, the

social engagement with the power system generated by embedded generation may

in itself be beneficial in terms of demand management.

2.3 The End User

The final consumers of electricity are not interested in energy per se, but rather the

services provided by that energy. The typical consumer does not engage with the

2.3 The End User 13

generation technology unless it either adversely affects them, or they are concerned

about the wider impacts of the technology (e.g. climate change).

2.3.1 Locale

As mentioned in §2.2.1, wind power is frequently generated close to a minority pop-

ulation to serve larger centres. Apart from the technical impact of remoteness, this

requires careful consideration of populations local to the machines, when the gener-

ated power is many hundreds or thousands of miles away [27, 28]. The involvement

of large companies in financing and installing wind parks has enabled the rapid de-

ployment seen in recent times. However, there is a danger of alienating consumers if

they are not properly consulted, requiring standards of practice in installation and

machine design to be upheld [29].

2.3.2 Further Impacts

Apart from the subjective impact of wind turbines and wind parks mentioned in

§2.3.1, a number of quantifiable concerns are often expressed. With proper machine

design and siting, these can be mitigated.

Avian Largely addressed by appropriate siting, a better understanding of

the visual [30] and audible [31] signals that birds use for navigation, and a

shift away from multi-element truss towers attractive for nesting.

Aviation Proper siting away from major flight-paths and markers on the

machines minimise the possibility of collision. MoD objections to a number

of projects on the grounds that national security is threatened by blocking

radar sites [32] may largely be alleviated by radar placement and even

stealth blades [33].

Land Use While it is often assumed to encompass the entire area of the

wind park [34], in reality it only consists of the tower bases, access roads

and any sub-stations that may be present. Animals are happy to graze

close to turbines and so farming can continue unabated.

Noise Noise was a large problem with early turbines, originating from the

blade aerodynamics (see §4.5.1) and mechanical sources in the nacelle. Na-

celle acoustic insulation, variable-speed machines (matching noise to back-

ground wind noise), and blade shaping (tip and Trailing Edge (TE)) have

largely alleviated these problems. Two-bladed machines tend to produce


more noise than three-bladed machines [35], and tower thump (see §7.2)

from downwind machines must be taken into account.

Visual Flicker Proper siting must avoid sunlight shining though the spin-

ning blades and casting a flickering shadow on nearby building occupants.

Chapter 3

Variations on a Theme

All wind turbines operate in the context set out in Chapter 2, with the goal to trans-

form energy from the wind into useful work. It is possible to propose myriad concepts

for achieving this overall function. Indeed, many researchers and practitioners have

done and continue to do so. Jamieson [36] gives a good overview, including var-

ious wind concentrators, as well as charged particle, airborne, sail-based [37] and

multi-rotor concepts.

To be efficient in selecting ideas for more detailed study and effort, it is possible to

utilize approaches from design theory to tease out the fundamentals of wind energy.

Section 3.1 provides a brief summary of this body of theory. The principles are then

applied to the justification of the coning rotor and similar concepts in §3.2 through

3.7, based on the functional requirements. The topology and operational principles

of the coning rotor are expounded in §3.8. As outlined in §3.9, the challenges to

obtaining the benefits of the coning rotor motivate the analysis and more detailed

design work presented in Parts II and III respectively.

3.1 Chasing Function

Pahl et al. [38] presents design as a four step process. Although presented as a

sequential process, in practice iteration between steps is carried out. This is either

required as a result of errors in previous steps, or as a beneficial part of the de-

sign process where the design process itself is sufficiently integrated and flexible to

accommodate it through design space exploration. The steps are:

1. Planning and clarifying the task

2. Conceptual design

3. Embodiment design

4. Detailed design

The task here is clear, to produce a concept that cost-effectively provides elec-

tricity from the wind. The second and third stages typically involve the creativity

15

16 Chapter 3 Variations on a Theme

of the designer. These steps are crucial, as they determine at a fundamental level

the effectiveness of the design. Unfortunately, they are also the most unquantifiable

and ill-defined steps. The final detailed design step is relatively well handled by the

application of quantitative engineering theory. To aid in the intermediate steps, a

number of generalized design theories are useful in analytically deriving the design,

by separating the functional and embodiment aspects of the design.

3.1.1 Design Theory

Engineering design can follow one of two approaches, intuitive or discursive, both of

which may be complemented by “conventional” methods [38]. Conventional meth-

ods include: literature searches, bio-mimicry, reverse engineering, analogies, and

model testing. Intuitive methods, such as brainstorming and other interactive ac-

tivities, rely on the unconscious flashes of inspiration. They are thus highly designer

dependent, and not useful in rigorous concept justification.

Discursive methods follow a set of deliberate procedures to analyse the design,

taking many forms from rigid and automated design catalogues [39], to knowledge-

capture tools [40], through to generic frameworks with abstracted solution spaces

[41]. The latter set of methods is useful here, to inform a generic discussion of

wind energy converters. In particular, Axiomatic design and TRIZ are the most

applicable amongst other possible options: bond-graphs [42], topological-spatial-

physical decomposition software [43], and CAD-based software [44]. The key to

both is separate consideration of the design in functional and physical spaces.

3.1.1.1 Axiomatic Design

Axiomatic design theory is an attempt to set down rigorous rules (axioms) governing

design [41, 45]. It does not set out specific steps to follow, but is rather a framework

to work in, consisting of two domains. The first is the functional domain, an abstract

space containing the decomposed functionality of the design, from top-level (convert

wind to electricity) through to minute detail (e.g. blade root connection). The

second is the physical space, containing the actual parts and assemblies required to

perform the specific functions.

Design activity consists of mapping Functional Requirements (FRs) in the abstract

space to Design Parameters (DPs) and Process Variables (PVs) in the physical space.

Starting with the top-level functionality, the FRs are sub-divided into 2–10’s of

subordinate levels. In parallel, a set of DPs and PVs are co-evolved to affect the

3.1 Chasing Function 17

FR’s [46]. Design constraints are not considered FRs, but impose limits in FRs, DPs

and Design Variables (DVs).

Two fundamental axioms are used (together with numerous corollaries) to inform

the choice of the best design. These may be stated in word and numerical form

as:

Independence Axiom The FRs must be satisfied independently; i.e. vari-

ation in a DP or PV must only affect one FR at a time. Note that physical

integration of parts is not precluded, as long as the functionality may be

properly controlled. Mathematically this is defined as:

FR = ADP (3.1.1a)

DP = B PV (3.1.1b)

C = A B (3.1.1c)

Aij =∂FRi

∂DPj(3.1.1d)

Bij =∂DPi

∂PVj(3.1.1e)

If the design is properly uncoupled, C will be diagonal or triangular.

Information Axiom Minimise the information content I; i.e. keep the de-

sign as simple as possible to achieve a high probability of success p:

I = log2

( range

tolerance

)= log2

1p

(3.1.2)

German design theory of the Workshop-Design-Konstruktion (WDK) school pro-

poses a level of abstraction between the physical and functional spaces: the organ

domain [47]. An organ is a collection of Wirk elements, each a point, line surface

or volume where a Wirkung (fulfilment of an FR) is performed. For example, for

a tabletop in the physical space, the top surface is a Wirk element performing the

function of holding up an object.

Japanese General Design Theory (GDT) also centres on a decompositional ap-

proach [47]. Taguchi et al. [48] espouses robust design, achieved by incorporating

the stochastic nature of the design to ensure that functionality is achieved in all

circumstances. This is akin to the information axiom.

3.1.1.2 TRIZ

Altshuller in Russia instigated an exhaustive patent search to extract principles

common to all innovation, independent of the specific application [49]. The Theory


of Inventive Problem Solving (known by its Russian acronym TRIZ) and the com-

putation tool ARIZ deriving from those efforts, is quite dogmatic and requires large

investments of time and user skill. However, the generic aspects of the method are

useful here.

Eight lines of “technical evolution” were identified: life cycle, dynamization, mul-

tiplication cycle, transition from macro to micro level, synchronization, scaling

up/down, uneven development of parts, and automation. Forty “inventive prin-

ciples” were found (e.g. segmentation, taking out, asymmetry), useful in solving a

“contradiction.” A contradiction in TRIZ is one of three types: administrative (e.g.

lower cost and higher performance), technical (e.g. improving one parameter at the

expense of another) or physical (e.g. requirement of multiple properties from same

material). Parameters here are mass, length, temperature, etc.

The concept of ideality figures prominently in the method, defined as:

Ideality =∑

Benefits∑Expenses +

∑Harms

(3.1.3)

Technical systems evolve towards higher ideality by utilizing external and internal

resources, the former frequently overlooked in a poor design (e.g. a refrigerator in a

cold environment obviating the need for a refrigeration cycle). At infinite ideality,

the mechanism disappears leaving only function.

3.1.1.3 Common Elements

An overarching theme common to many design theories is that of thinking in multiple

domains, an approach followed in later sections for the wind turbine. Abstraction of

function is found to be a powerful tool for breaking out of old ideas, by emphasizing

the generality of the essential principles involved [38]. By postponing visualization

of the means-to-an-end, the end itself can be concentrated upon, as it is the critical

denominator of success. A complimentary theme is a multi-level approach in all

domains, as the mind has limited capacity to effectively focus on multiple issues

simultaneously. The base functionality must be broken down into sub-functions

[38], analogous to parts and assemblies in the physical domain.

Implicit in a number of the methods is the concept of lean design, which can be

applied in three contexts: process, material and integration. Lean process refers to

the streamlining of the design process itself, of focusing attention on the core issues.

Material leanness is achieved by optimized form, using the minimal amount of the

appropriate material required for the intended purpose. Integration deals with part

3.2 COE and CF 19

count issues and the overall simplicity of the design. The idea of function sharing

has also been explored by Ulrich, Seering, Eppinger in their Functional Analysis

System Technique (FAST)/Value Analysis [47].

The second axiom of axiomatic design makes lean design explicit, as does the

concept of ideality in TRIZ. This is echoed by a number of other corollaries, elements

of axiomatic design and TRIZ respectively, compared in Table 3.1.

Table 3.1 Axiomatic/TRIZ design method similarities (Adapted from [50])

Axiomatic Design TRIZ

Corollary 2 Ideal Final ResultMinimise the number of FRs System imposes fee for realizing func-

tion, therefore minimize substance,energy and complexity

Corollary 3 Evolution Pattern 5Integration of physical parts Increased complexity followed by sim-

plificationCorollary 7 Evolution Line Mo-Bi-PolyUncoupled design with less Informa-tion

Guidelines for reducing complicationof a system

3.2 COE and CF

The primary function of a wind turbine is to efficiently (economically and techni-

cally) convert wind power to electricity.1 Two metrics are useful in this discussion,

Cost of Energy (COE) and Capacity Factor (CF):

COE =CostAmortized capital + BOP + OM

Energy CapturedActual

(3.2.1a)

CF =Energy capturedActual

Energy capturedTheoretical operation at rated power

(3.2.1b)

A related concept is availability, defined as the proportion of time the machine is

available to produce power (i.e. excluding down-time due to maintenance, etc.). CF

and availability are related but not synonymous. It is important to realize the wind

turbine availability is on-par with conventional plants. The CF of wind is much

lower, typically around 30%, and this is usually misconstrued that “wind turbines

only work 30% of the time.” This is false; the CF is merely a reflection of the

economic decision of rated power, which is a trade-off between low-wind energy

1Electricity production, rather than direct mechanical energy (e.g. traditional water-pumping) is the current focus.


capture and high-wind loading. Hypothetically, and extremely large rotor and very

small generator would yield a CF of 100%, assuming 100% availability.

The COE numerator is dominated by machine and balance of plant (BOP) capital

cost, to finance a structure capable of sustaining the applied loading required to

convert and transmit power from the wind through to the electrical connection

point. The denominator represents the amount of energy captured over the economic

lifetime of the machine. At each wind speed V , power coefficient CP and area

A = π/4D2 at that point, the power captured is:

P = 1/8ρV 3CPπD2 (3.2.2)

Note that CP is defined relative to the rotor area. Another common misconception

is that for a CP of say, 0.5, a turbine is only capturing 50% of the available wind.

In fact, this is a technical measure, fundamentally limited to the “Betz limit” of16/27 = 0.593. It is akin to stating that a heat engine is only 15% efficient, relative

to a Carnot efficiency of say 30%.

Both numerator and denominator of COE are affected by the control strategy,

influencing both the loads and energy picked up from the wind. This is in turn

constrained by the level of compliance in the drive train (i.e. fixed or variable speed)

and aerodynamic control (e.g. pitch actuation, active yaw, etc.). Some added cost

(e.g. for a coning mechanism) is justified on the basis that the rotor cost is only

approximately 10-20% of total cost, so that increase in energy capture can give an

overall reduction in COE. Malcolm and Hansen [51, p. 29] found a 10% rotor cost

change led to a 1% to 1.5% COE change.

The typical FR of a wind turbine is a low delivered COE. The CF of a machine is

typically seen as an artefact of that design process and the wind regime on-site. With

remote load-centres, for example offshore or in remote areas, it may be beneficial to

consider a high CF as a FR in its own right. This would better utilize the expensive

transmission cabling and infrastructure.

3.3 To Lift or Drag

The second-level choice for the functionality in a wind energy technology is to base

it on either lift or drag forces. As will be seen in §3.3.2, the choice is critical.

3.3 To Lift or Drag 21

3.3.1 Governing Equations

In the most general case, a device of area A (frontal area for pure drag device,

planform area for lift) travels at velocity ~Vd through wind speed ~V , both relative

to fixed ground. The two vectors subtend an angle θ, and their magnitude ratio is

δ = Vd/V . The lift CL and drag CD coefficients are the fully 3D values for the device.

The energy extracting force F in the direction of motion is non-dimensionalized as:

CF =F

1/2ρV 2A

= [cos θ (CL sinβ + CD cosβ) + sin θ (CL cosβ − CD sinβ)] ·[1− 2δ cos θ + δ2

]tanβ =

δ sin θ1− δ cos θ

(3.3.1)

The power coefficient is then simply CP = δCF .

3.3.2 Lift-Drag Comparison

A simple caparison is shown in Fig. 3.1, taking CD = 2.0 and θ = 0 for the drag

device, and θ = 90, CL = 0.8 and CD = 0.1 for the lift device. These values

are assumed constant throughout (i.e. trimmed to maintain the same Angle of

Attack (AOA)). The drag device can never move faster than the wind, limiting its

power capture. In fact, its prime functional advantage is a better CF for δ < 0.2.

It is also possible to construct much simpler physical devices utilizing drag, so for

high-force/low-speed applications these may have an advantage. Otherwise, the

drag device is severely limited, especially if energy capture is the objective.

Figure 3.2 shows the effect of the next decision, translation angle θ. Angles around

θ = 90 have the highest peak force and power coefficients. This is fortuitous, con-

sidering that the HAWT discussed next in §3.4 operates at θ = 90. The alternative

is either a VAWT, or a device the translates on rails at some angle to the wind. The

latter involves considerable physical infrastructure.

3.3.3 Operational Conditions

Key non-dimensional parameters for aerodynamic performance are the Reynolds

number Re = ρV l/µ, and some measure of surface roughness. Both characterise the

conditions in which the boundary layer develops. With higher Re more energy is

fed into the boundary layer, to overcome the surface roughness attempting to retard

the flow. Leaving aside the laminar separation bubbles seen at very low Re (around


0

1

2

3

4

5

6

7

8

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0δ

C P

0.0

0.5

1.0

1.5

2.0

C F

Lift C_PDrag C_PLift C_FDrag C_F

Figure 3.1 CP and CF for translating airfoils utilizing lift and drag

0123456789

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0δ

C P

0.00.20.40.60.81.01.21.41.61.8

C F

C_P 20

C_F 20

C_F 80C_F 100

C_F 90

C_P 100

C_P 90

C_P 80

Figure 3.2 Variation of CP and CF with translation direction θ (deg) forlift device

3.4 VAWT or HAWT 23

1x104), higher Re conditions and smoother surfaces tend to permit higher cl,max by

avoiding separation.

Critical to lift devices, the cl curve in stall may vary dramatically with Re and

roughness, for the same airfoil. To set wind turbines in aerodynamic context with

other applications, the wing sections on a large jet transport aircraft operate around

1–10x107. A small wind turbine blade would see Re around 1–10x105, and a large

turbine around 1–10x106. The range that wind turbines operate in, is the same

range over which large differences in cl behaviour appear. It is therefore essential

that airfoil data is obtained for the operational Re and surface roughness.

3.4 VAWT or HAWT

The fundamental momentum balance derived in §4.4.3 predicts equal performance

for a VAWT and HAWT (see Figs. 1.1 and 1.2). The limiting case for a VAWT is

a gyromill, having vertical straight blades with continuous pitching to only shed

vorticity (change lift) as the blades transit directly upstream/downstream. VAWTs

have certain physical advantages including: generator location at ground-level; abil-

ity to operate in any wind direction; and no cyclic gravity loading. Mitigating

against these are: cyclical aerodynamic loading (this led to fatigue failure of early

aluminium blades); operation in the wake of the tower and other blades; usually

close to ground loosing wind shear benefit. For structural reasons, the blades are

usually formed in a troposkein shape (see Fig. 1.2(b)) so that aerodynamically, the

blades are constantly changing lift, even in ideal conditions. This fundamentally

limits their CP , even with complicated pitching systems.

For all of these reasons, VAWTs have been commercially abandoned, for the most

part. The exceptions are a resurgent interest at small and very large scale. The

former may have benefits over a HAWT in built-up areas, with rapidly varying yaw

angle.1 The latter is predicated on avoiding the cyclic gravity loads that are starting

to drive the design of very large HAWT blades.2

3.5 Scale

The ideal scale of machine is very hard to derive analytically. At the most basic level,

COE would be expected to rise with scale, according to the “square-cube” law which

1E.g. Quietrevolution from XCO22E.g. Aerogenerator from Wind Power Ltd.


states that energy capture increases with diameter squared D2, while the volume

of material (cost) increases as D3. This is of course overly simplistic, for a number

of reasons, including falling installation and maintenance costs with fewer overall

machines, and improving wind resource with tower height. Some authors have used

component-wise scaling laws to arrive at curves predicting optimum machine size [16,

18], while others have pursued more detailed design study on components including

blades [52] and balance-of-plant [53]. Jamieson [36] found that a multi-rotor concept

would improve the area/volume relationship.

Coulomb and Neuhoff [54] have studied the cost progression of machines, before

vendors ceased disclosing list prices, from the perspective of learning curves. A

technology typically exhibits cost reduction from the experience gained over time.

In this case of wind turbines, when analysing cost data with size it was found

important to include wind shear exposing machines to higher wind-speeds as they

grow. With these considerations, learning has dropped costs by 12.7% with each

doubling in installed capacity. Based solely on machine cost, 400-500 kW machines

were found to be optimal, although this excluded Balance of Plant (BOP) factors.

In general, optimal size predictions depend on myriad assumptions that are diffi-

cult to prove in the absence of real experience. Moving offshore changes the equa-

tion, shifting the cost centre away from the turbine to BOP. There is some finite

upper-limit on machine size, as the machine cost certainly exceeds the D2 exponen-

tial progression. Griffin [55] found in a scaling study a Dx cost exponent of 2.9,

whereas the commercial average is 2.4. The largest influence was design condition

(see Table 2.1) making tailored machine/rotor design increasingly important.

The TRIZ technical evolution trend of scaling up is clearly evident in the wind

industry. Onshore the limit is practically around 2 MW, owing to transportation re-

strictions. Offshore it is less clear, with 5 MW prototypes currently being installed.

Changing materials complicate trend analysis of blade weight (a proxy cost met-

ric), shown in Fig. 3.3 compiled from vendor data sheets. Vestas is increasing the

use of carbon fibre to obtain stiffness, while LM evolves their ‘standard’ polyester-

glass blades. High-performance materials (e.g. carbon fibre) may deliver technical

performance, however cost performance can be adversely affected if aerospace type

composites are employed (Vestas is transitioning to pultruded carbon fibre-wood

composite blades to avoid this). The averaged curve fit indicates a cost exponent

with D of 2.1, for this data set.

Figure 3.4 shows historical data compiled from WindStats magazine for a range

3.5 Scale 25

y = 0.7911x2.1055

0

5000

10000

15000

20000

25000

20 40 60 80 100 120 140Rotor Diameter (m)

Per B

lade M

ass (

kg)

LMVestasPNE (Multibrid)EnerconSiemensEcotecnia

Figure 3.3 Blade mass trend with rotor diameter

of machines comprising full data sets over the year. No offshore data is included,

as the wind resource would tend to skew general trends. There is large data scatter

and some outlying data sets, with less variance in summer months possibly owing to

less persistent storm activity. The seasonal variation is evident in the seasonal trend

lines, with winter winds higher than those in summer. There is a general increase

in average energy capture with machine size, but it is fairly gradual and only in

evident in winter and spring.

050

100150200250300350400

0 500 1000 1500 2000Rating (kW)

Spec

ific E

nergy

Cap

ture (

kWh/m

2 )

AutumnWinterSpringSummer

Figure 3.4 Specific energy capture as a function of machine rating

For the purposes of this thesis, a 1.5 MWe machine is used as a target scale.

This size is representative of on-shore machines currently being installed. Data for


the REF-1500, a conventional machine (see §C.3) was also available for this size

machine.

3.6 Control Strategy

Wind turbine control synthesis is divided into two sequential stages: scheduling and

controller design [56]. The latter provides the control loops and gains using control

theory [57], but is subordinate to the first task of specifying the control strategy

and targets. Functionally, this is a complex and critical step (as will be shown in

Chapter 6). The second step is therefore left to previous [56, 58, 59] and future

work.

3.6.1 Goals and Methods

All wind turbines have two competing goals, to capture energy while avoiding loads.

Figure 3.5 shows the three control regions: Region I below Vci, where the wind

turbine is parked; Region II to optimally track the maximum power extraction

point; and Region III above Vr to track the peak power of the generator, up to a

maximum Vco where the rotor is parked. The choices of Vci and Vr are economic,

as there is low energy content at low wind speeds (see §3.2). Likewise, Vco avoids

extreme loads on the machine, at speeds that have very high energy content but

very low frequency of occurrence.

Figure 3.5 Power curve regions

The mechanism of control is important to both loads and energy capture. In

Region II, variable speed operation permits optimal energy capture operation. This

is usually done at fixed pitch angle, ideally with maximal capture area. Region III

3.6 Control Strategy 27

must limit rotor power to that of the generator maximum, though one of a number

of mechanisms. Yaw1 and coning both affect the gross capture area, via Eq. (3.2.2).

More conventional control methods for Region III, in order of decreasing usage are:

Pitch to Fine (PTF), Fixed Speed Stall (FSS), Pitch to Stall (PTS), and Variable

Speed Stall (VSS). Each alters CP , either by changing the relative velocity vector

(FSS, VSS) or AOA via pitch angle (PTF, PTS), as shown in Fig. 3.6. Notice that

each maintains the same product of torque-producing force vector and rotational

velocity rΩ, to produce the same power. The resulting thrust force also varies

between strategies.

Figure 3.6 Wind velocity and force vectors for various control strategies

From this list of choices, modern machines use almost exclusively PTF. A coning

rotor combines coning control with VSS or PTS. The Proven machine (see Table D.1)

adopts an even more complicated use of PTS coupled to rotation speed, and PTF

coupled to cone angle. Adaptive flexible coupling between flexible blades (in bending

or speed-linked extension) and torsion have been proposed by a number of authors

[60, 61]. Veers et al. [62] provides a good overview of the static possibilities, and

highlights the dynamic stability bounds and strict manufacturing tolerances required

to practically execute such a strategy.

Figure 3.7(a) shows the conventional control strategies considered from the non-

dimensional rotor perspective, with associated CP loss mechanisms either side of

the peak CP,opt. Pitching strategies alter the CP − λ curve directly, while VSS and

FSS both move along a nominally constant curve.2 Note that FSS, VSS and PTS

all operate in the left-hand stalling portion of the CP − λ curve, while PTF avoids

1E.g. Gamma 60 machine [16, p. 357]2Re effects modify the curve somewhat.


stalling. The impacts of coning on this control map are discussed in §4.3, 4.7 and

6.3.

(a) CP − λ (b) τ − Ω

Figure 3.7 Power control in CP − λ and τ − Ω planes

Figure 3.7(b) presents the control schedule as it is actually implemented. The wind

speed cannot be accurately measured, so it is necessary to indirectly control based

on an aerodynamic1 torque estimate τ and rotor speed Ω. The generator will impose

lower Ωmin and upper Ωmax bounds. After Vci, the rotor tracks from A to B in low

winds. With sufficient wind, the optimal CP,opt of the rotor is tracked by maintaining

λopt. This may be altered somewhat by the speed-dependent inefficiencies in the

drivetrain/generator. At some point C the upper speed limit is reached, sometimes

“prematurely” to smoothly transition between Region II and Region III controllers.

Pitching strategies PTF and PTS maintain a constant torque at Dpitch, while VSS

must increase torque to reduce Ω and stall the rotor towards point Dstall.

Additional control inputs reduce somewhat the burden of analysis fidelity and

afford performance enhancements. In particular, the stall behaviour of the rotor

is subject to considerable uncertainty between prediction and measurement [64].

Provision of pitch action enables on-line tuning to account for modelling errors, and

adjustment for air density variation and blade soiling. Active control of power in

Region III (i.e. not FSS) removes the prime arguments for low-lift airfoils [65, 66],

enabling the use of high-lift airfoils for more cost-effective blade designs.

1Controlling on generator torque is sub-optimal, as the rotor inertia is involved ratherthan energy-producing aerodynamic torque [63].

3.6 Control Strategy 29

3.6.2 Dynamic Considerations

Although this work focuses on the steady-state facets of the control problem, a

number of dynamic considerations must be kept in mind. All variable-speed concepts

attempt to maintain optimal power capture in Region II. The slope either side of

the CP peak in Fig. 3.7(a) effects the ability of the control system to accomplish

this objective in unsteady winds. If the peak is sharp (usually a steep stalling front

to the left), sharp drop-offs in power will occur since the rotor speed response is

limited by inertia. In particular, VSS rotors require a sharp peak to limit power in

Region III, making optimal low-wind operation a competing design objective.

Mercer [58], Hoffmann [59] and Burton et al. [16, p. 476] all discuss the fundamen-

tally poor power regulation of VSS concepts in Region III. VSS is better than FSS,

in that steady-state control can maintain rated power and actively transition from

the optimal cl/cd point to a stalled point. However, to do this, excess torque must

be applied not only in steady-state (torque must rise to maintain power P = τΩ),

but also dynamic torque to alter the inertia of the rotor (via τdyn = IdΩ/dt). This

lag provides greatly reduced power quality above rated. The negative slope of the

Dpitch–Dstall VSS control objective in Fig. 3.7(b) is unstable and demands a com-

pensating controller. New grid demands such as LVRT (see §2.2.2) also make direct

control of rotor input power, via pitch action, increasingly a control requirement.

PTF has been widely adopted because with fast actuators, it directly and quickly

limits input rotor power. It is usually preferred over VSS because of the difficulty

in analysing stalled rotors. The only large turbine to use PTS is the V-82,1 the

perceived advantages being smaller/quicker pitch actions and reduced load variation

associated with gust-slicing [16, p. 355]. The latter occurs because PTF operates in

the linear-regime where much larger cl changes can occur with AOA than in stall.

PTF may also interact with tower vibrations in the following sequence: pitch action,

thrust decreases, tower moves upwind, relative velocity increase, more pitch action.

PTS operates in an opposite sense, reducing fatigue loading while increasing mean

loads, as discussed in §6.1.2.

Practically, experience2 is that PTS does reduce fatigue loading relative to PTF, if

the ill-damped edgewise vibrations found in large blades are adequately controlled

1V82 1.65 MW machine (previously NM-82), Vestas website, http://www.vestas.com/pdf/produkter/2006/V82_UK.pdf, July, 2006

2Personal communication, Tomas Vronsky at Vestas Wind Systems A/S

http://www.vestas.com/pdf/produkter/2006/V82_UK.pdf

http://www.vestas.com/pdf/produkter/2006/V82_UK.pdf


(see §7.1.2.2). However, extreme loads are higher in extreme yaw error situations.

In turn, Raben et al. [67] found higher blade flap fatigue for PTF versus FSS.

3.7 Adapting Structures

Structures incorporating Degrees of Freedom (DOF) are able to undergo conforma-

tional change. Properly designed, these changes are capable of reducing the applied

loading and resulting structural stresses developed in the structure. The following

sections explore the application of this concept to wind turbine design.

3.7.1 Dynamic Motion

Since Putnam’s original flapping blade machine in the 1940’s [8], motion of the

blades has been proposed to alleviate dynamic system loads. Indeed, teetering

of two-bladed machines is almost essential for viable performance [15, 68], in the

same way as hinges are required for successful helicopter designs [69]. A number of

researchers have examined two and three-bladed machines with individual discrete

flapping hinges [7, 51, 66, 70–73] or combined flexible hinging and teetering [74, 75].

However, only the Carter machine has achieved any widespread deployment in the

past.

The functioning of a flexible structure is best explained by the fundamental dy-

namic equation of any structure, be it connected by discrete or flexible elements:

[M ] x+ [Caero] x+ [Cstructure] x+ [K]x = F (t) (3.7.1)

where [M ], [Caero], [Cstructure], and [K] are the mass, aerodynamic and structural

damping and stiffness matrices, x the generalized displacements and F the applied

forces.

In a stiff structure, displacements are relatively small; therefore forces are reacted

almost exclusively by the stiffness, which must be large. This has the effect that the

ultimate stress/strain capabilities of the material are underutilized and the structure

may be overly heavy and expensive. One way around this is to use light high-

modulus materials, such as carbon fibre composites, but this drives up the cost when

used in large quantities. In a compliant structure, the velocities and accelerations

are non-negligible, and the displacements are also larger. As a result, the stiffness

requirement is reduced, since the forces may be reduced by modifying the airflow.

What forces are imparted, will be reacted more by the mass and damping of the

3.7 Adapting Structures 31

structure, rather than stiffness. The trend towards dynamization is one line of

technical evolution in TRIZ [49].

Aerodynamic blade loading may be tailored with soft structures, most notably in

the blades themselves, such as bend-twist coupling [60], flapping flex-beams [9], or

by discrete flapping/teetering hinges [71]. All act to dynamically change the angle of

attack at the blade sections by configurational change, so that negative feedback of

load is achieved. Discrete hinges near the root of the blades avoid the complexity and

stringent manufacturing tolerances of flexible blades or hinges. Kelley et al. [76] in

examining the Wind Eagle (a derivative of the Carter machine), and Quarton [75]

from monitoring of a Carter 200/300, have highlighted the loads seen in practice

from imbalance in flexibility and mass between blades in a flexible approach.

3.7.2 Load Matching

Any variably coned rotor also benefits from a static matching of thrust and cen-

trifugal loads. The steady out-of-plane bending moment carried along the blade is

reduced, ideally leaving only an axial tension load. Lighter, cheaper blades then feed

back to reduced edgewise gravity-dominated loads, further reducing blade weight

and cost. Recent commercial efforts [70] and research studies [71, 77] have high-

lighted the effectiveness of flapping blades in this respect. Obviously, any design

with significant flapping or coning must have a downwind orientation in order to

avoid tower strike. Kelley et al. [76] has noted that the Wind Eagle has reduced

loading at high winds relative to conventional rotors, but the reverse in low winds.

Evidently the bending and coning at high winds alleviates loading in high winds,

but suffers from a lack of pre-cone (from centrifugal opening) in low winds.

3.7.3 Soft Approach

The concept of an adaptable machine has been pursued in a number of machines

and design approaches. In addition to the rotor itself, other aspects of the machine

may adopt a stiff or soft approach. For example, the fundamental mode of the tower

may be tuned relative to the blade passing frequency BΩ:

ftower > BΩ stiff-stiff (3.7.2a)

Ω < ftower < BΩ stiff-soft (3.7.2b)

ftower < Ω soft-soft (3.7.2c)


where B is the number of blades. Softer designs yield lower mass (cost) towers.

The generator may also incorporate flexibility, from direct-connected synchronous,

to standard induction, variable resistance (slip) induction and doubly-fed induction

through to fully-variable synchronous generators.

3.7.4 Coning Rotor Adaptation

The coning rotor concept at its heart employs flap-hinged blades, thereby inherently

benefiting from the static and dynamic load alleviation just mentioned. The coning

rotor is further differentiated from a flapping rotor by two other operational char-

acteristics. Firstly, extending the range of coning angles (gross coning up to 85)

avoids storm loading when shut down. Secondly, by utilizing relatively long blades,

the COE’s denominator is increased by enhanced energy capture in partial load

conditions. The longer blades are made possible by coning to appreciable angles

(20–35) as rated power is reached, to have similar loading to conventional rotors at

that load-critical point.

This creates an adaptive rotor with large area in low winds and smaller area

in high winds. As the limits of aerodynamic performance (power coefficient CP )

are reached by conventional designs, this adaptive rotor acts on the power/energy

Eq. (3.2.2) in the most direct way, via the capture area π/4D2.

Conventional PTF machine developments towards individual pitch control [78, 79]

are aimed at load mitigation. With reduced operational loading, longer blades are

possible [80]. The goal is the same as the coning rotor, to maximize energy capture

from a larger rotor area. In contrast to coning rotors though, conventional rotors

will always remain susceptible to 3D turbulence-induced limit loading while shut-

down. The parasitic power loss and bearing life effects associated with aggressive

continuous pitch actuation schemes have also yet to be quantified.

3.8 Function to Form

Having examined the functional aspects of an ideal wind turbine, evolution of the

concept in the physical domain is explored. These choices lead to the concept

explored in more detail in Part III.

3.8.1 Historical Context

Adaptable machines are typically much more difficult to design, given their dynamic

nature. Indeed, it is partially the increased design challenge, associated with a lack

3.8 Function to Form 33

of adequate design tools, that has limited the success of a soft approach. Past

experience is important, and is garnered here from a review of past machines in

Table D.1, to yield the following principles:

Blade Count The proclivity for two blades is a testament to the fact that

soft designs require by their very nature more advanced analysis and design

to be successful operationally. When a soft design approach is adopted, the

more detailed analysis also seems to typically lead the designer towards a

two bladed configuration. This is presumably for reasons of economy and

simplicity when designing systems for load alleviation.

Blade Articulation The only machine not employing articulation is a 3-

bladed one. This implies a clear requirement for the blades to teeter/flap

so as to alleviate aero and structural loads, as previously discussed in this

section.

Coning Coning is usually used in a fixed sense, relying on the inherent

changes in effective coning angle resulting from blade flexibility. The Risø

Soft Rotor and Carter machines separated the coning and teetering func-

tions. Both blade roots incorporated flex-beams to provide stiffness to a

pivoted joint. The MS-4 machine also employed flex-beams which turned

out to be difficult components to design . The WTC machine employs

independent hydraulically-damped discrete hinges. The Cone-450 concept

machine was to use a central hydraulic cylinder to collectively cone the

three blades, with dampers incorporated into the control links. It was the

only one to use gross coning to adjust capture area for power control.

Blade Flexibility The Hutter-Allgaier machine opted for a low-solidity ro-

tor, so that the slender fibreglass blades were quite flexible; as a result the

prototype suffered from flutter. The Risø Soft Cone concept used no shear

webs so as to be lightweight and offer the possibility of adaptable airfoil

geometry. A common theme is to consider the bending moment supported

by the blade. In order to flex (in bending) to any large degree requires

large bending moments, which in itself is not desirable from a stress point

of view.

Downwind Orientation The majority of machines use a downwind config-

uration, although not universally, due to potential tower-shadow problems.

Scale Early machines attempted to jump into the multi-megawatt scale in

order to prove themselves on a conventional utility scale. This approach


was an almost universal and unmitigated failure. The more successful ma-

chines to date have been much smaller. This is also echoed by the evolution

of the Danish concept that has matured by growing in size, building on past

experience.

3.8.2 Topology

The nominal layout of the rotor under consideration in this study is shown in Fig. 3.8.

In the CONE-450 original work [66], the blades were hinged on a space-frame with

the hinge axis significantly away from the rotor axis. The lightweight carbon fibre

blades under consideration required this configuration to reduce the aerodynamic

hinge moment and obtain reasonable free-coning angles (≈ 30). A more conven-

tional compact cast hub is preferable from a complexity perspective, with conven-

tional blade materials and mass tuning.

Figure 3.8 Coning rotor schematic layout (Only symmetric half of nacelle andinboard part of one blade shown)

A central hydraulic actuator was envisaged for the CONE-450, with links out to

the blade roots that extended inboard of the hinge points, to impose equal moments

about the flapping hinge. The basic operational principle was for collective coning

action, with the actuator active only during very light winds to hold the rotor more

open with minimal bending moment, and during fully coned shut-down. The rest

of the time the actuator was simply a means of applying damping (alleviating the

loss of aerodynamic damping in stall) with relief values to assure free-coning from

just below rated.

3.8 Function to Form 35

In the original coning work, the rigid links from the central actuator to the blades

resulted in excessive dynamic overturning moments at the tower head. This was

determined to be a result of the proximity of the 3P frequency range and the first

blade flapwise mode (this is a fundamental result, see §4.6.3.2). To alleviate this

loading, flexible damped links were incorporated into the model, solving the problem

in the simulations. The physical implementation of the dampers was never discussed

in the reports. Experimentation with blade flexibility was found effective in reducing

blade loads, but individual flapping was required to alleviate overturning moments.

In the current work, the focus is on three independent actuators for three blades.

This is a reflection of the need for some individual blade motion found in previous

work, and for a cleaner practical implementation. A more conventional compact

hub with the actuators acting outboard of the hinge line on moderate weight, mass

tuned blades is also adopted. A more integrated design approach is also being

pursued, to closely couple a Permanent Magnet Generator (PMG) into a simple

rotor, for enhanced reliability and cost effectiveness. This approach is being pursued

on conventional machines with “standard” generator designs [81, 82] and also using

extremely large-diameter bearings for the generator [83].

The two-bladed Wind Turbine Company (WTC) [84] prototype machines use in-

dependent dampers on individual flap hinges, achieving a type of teeter. Pierce

[77] investigated a machine with an actuator between two otherwise freely flapping

blades. Both employ PTF (see §3.8.3) and therefore maintain small cone angles. A

big problem with conventional teetering and these concepts is hitting stops, which

re-introduces large loads. The coning rotor operates well away from any stops in a

range of coning from 5 to 85, instead of the WTC -5 to 15. The failure of the

WTC prototype from tower strike, presumably after stop impact, highlighted this

risk.

A three-bladed rotor is used in the present work, for a number of reasons:

• Public acceptance and aesthetic studies have indicated this preference [36]

• Lower optimal tip-speed ratios for on-shore siting issues

• Aerodynamic performance loss for two-bladed rotors is significant [51]

• Two blades are not necessarily cheaper than three when size (solidity) and

loads are accounted for1

• Cyclic rotating imbalance-type hub loads [71] should be less in a three-bladed

rotationally symmetric flapping rotor

1Personal communication with Peter Jamieson, Garrad Hassan and Partners (GH) 2004


The imbalance loads were not mentioned as significant in the original coning rotor

work but were in the two-bladed flapping study [71]. The latter study also suggested

high tip-speeds to maintain a relatively flat rotor (to keep energy capture high) with

the potential for attendant dynamic instability resulting from the low-solidity rotors

required. Note that the coning rotor uses longer blades and hinge moment bias to

effect even greater energy capture than a completely planar rotor.

3.8.3 Pitch Control

Chapter 6 will discuss the requirement for pitch control of some kind on a coning

rotor. The CONE-450 used fixed pitch and VSS. The attraction of VSS is elimination

of a set of actuators and the possibility of integrating the hinges into the blades

without requiring a circular root.1 Pitchable tips could be used instead, as is done

for tip-brakes on some FSS machines. However, the mechanisms are difficult to

integrate structurally and can pose maintenance issues being located at the blade

extremities.

An additional method of pitch control is possible by passively coupling pitch angle

γ to flap angle β by inclining the hinge axis by an angle δ3 in the plane normal to

the rotor rotation axis. The non-linear relation is:

∆γ = ∆β cosβ tan δ3 (3.8.1)

Helicopters and teetered rotors use +δ3 to pitch towards feather with flap angle,

reducing loads.

The coning rotor must positively cone with rising winds, so +δ3 is inappropriate.

The rotor must stall (either VSS or PTS) to move away from the tower. While −δ3would achieve this objective, dynamically it would exacerbate stall instabilities. At

larger cone angles, gross “in-plane” (azimuthal) movement of the blade axis would

further complicate matters (rotating imbalance, aerodynamic feedback). In any

case, most pitch action occurs near β = 0, while it is only required at larger β near

and above rated power.

3.9 Coning Rotor Challenges and Opportunities

The preceding sections of this chapter have outlined the rationalization for the im-

portant functional and physical elements of the coning rotor. Using the decom-

1Another concept is an inner member with flap hinge allowing rotation of an outer shell,e.g. MS-4, WTC [85]

3.9 Coning Rotor Challenges and Opportunities 37

position approach of design theory, minimum COE has been identified as the core

function of a wind turbine. Based on fundamental arguments, a lift-based, HAWT

of around 1.5 MWe has been identified as the optimal basic approach. The qualita-

tive steady state and dynamic ramifications of control strategy have been discussed,

as have the fundamental reasons for adopting stall-limited (VSS or PTS) high-wind

operation of the coning rotor. The notion of an adaptive machine to tailor and

mitigate loading, both in steady state and dynamically, has been identified as a key

driver for load reduction.

The key physical aspects of the coning rotor are flapping hinges at the blade roots,

allowing the blades to sweep out a cone and to park in the streamwise direction in

high winds. This configuration affords reductions in both parked high-wind loads

and operational blade bending moments. The coning rotor exploits these load reduc-

tions by employing relatively longer blades, with nominally constant cost, to yield

lower COE relative to conventional machines. Although advanced “conventional”

PTF machines (for example with independent pitch actuation) may also increase

blade length by reducing loads, the non-flapping blade roots of these concepts fun-

damentally remain more susceptible to high-wind parked loads. They also must

resist larger steady bending moments during operation. The coning rotor there-

fore appears a viable alternate approach worthy of more detailed consideration in

Part III, given the potentially large benefit in reduced COE.

As a conceptual approach to extracting energy from the wind, the coning rotor

shares many elements with conventional machines. The analysis tools required will

therefore be similar to those currently used and validated, but with important differ-

ences, as covered in Part II. The primary area of uncertainty is in the aerodynamics,

which is already complex in stalled unconed rotors. The theory on which almost

all wind turbine design tools are founded, BEM, requires modification to handle the

geometry of the coning rotor, as detailed in §4.4. BEM theory is extended here,

rather than employing a higher order model, to facilitate dynamic simulations and

optimization on a design relevant time-scale.

The inclusion of non-linearities in the structural model (frequently linearised for

conventional machines) are also found to be important when considering a coning

rotor. The required downwind orientation of the coning rotor may lead to LFN,

a negative effect not encountered with conventional machines but which must be

mitigated for downwind machines. Finally, the solution of the optimization and

control problem is complicated by the presence of the flap hinges, requiring modified


approaches and due consideration of the attached generator.

Motivated by the qualitative potential benefits of the coning rotor, the remainder

of this thesis is focused on developing, validating, and applying design tools to

examine the quantitative performance of such a rotor. The ultimate question to be

answered is the permissible increase in blade length, which will in turn determine

the COE advantage of the concept. In turn, this question may only be answered

with any certainty after building confidence in the design tools employed, with due

appreciation for any shortcomings in the implemented analysis methods.

Part II

Analysis

Chapter 4

Analytic Development

This chapter develops the theory validated in Chapter 5 and used later in Part III

for design studies. The presentation is motivated by the unique configuration of

the coning rotor described in Chapter 3, which violates some of the assumptions in

standard models, most principally aerodynamic (§4.4) and structures (§4.6) related.

Most of the models are implemented in a common software program, ExcelBEM,

to facilitate later work.1 Throughout this work, emphasis is placed on extend-

ing engineering-level models, rather than more complex full-field simulations, to

facilitate rapid optimization and for incorporation into dynamic simulations while

retaining reasonable runtimes.

A number of analytical models are required, principally a novel BEM method ca-

pable of properly analysing a coned rotor (§4.4), equally applicable to other highly

flexible concepts and extensible to yawed rotors and dynamic inflow (§4.4.14). The

implementation of a low-frequency acoustic model is presented next in §4.5, required

for evaluating the “tower thump” noise associated with downwind rotors. Structural

modelling methods are presented next in §4.6, including sectional property compu-

tations (§4.6.1), loads integration (§4.6.2), dynamic EOM for the flap-hinged rotor

(§4.6.3), and derivation of a FEM approach to the centrifugally stiffened beam prob-

lem in §4.6.4. Using these components, steady-state and dynamic control scheduling

methods are presented in §4.7. Finally, a generator (§4.8) and cost model (§4.9) are

presented. Before any of these models are presented, an industry-standard reference

code is introduced (§4.1), in addition to an overall coordinate system (§4.2) and

performance metrics for the coning rotor (§4.3).

1The name ExcelBEM derives from the fact that Microsoft Excel is used as a graphicaluser interface, with sheets to define the blades, structure, etc. Matlab m-files and compiledC functions are used for computations and interaction with Excel.

41

42 Chapter 4 Analytic Development

4.1 Industry Standard Comparison (BLADEDTM)

Modern, industry standard, fully aeroelastic time-domain wind turbine simulation

programs are quite complex. They have been developed over many years and certi-

fied by regulatory bodies. A large quantity of data must be properly input describing

the machine, operation and conditions, including the blades, drive train, tower, con-

trollers, and wind/wave inflow. Wind turbines are inherently flexible structures,

making accurate coupling between aerodynamic loading and structural deforma-

tion critical for accurate performance prediction. These codes therefore adopt some

modal or discretized description of the flexible components of the system. Taken to-

gether, developing a full commercial-grade code is a large undertaking, well beyond

the scope of an individual thesis.

The BLADEDTM software from GH is used throughout this thesis as a compar-

ison industry-level code. BLADEDTM is a program frequently used in industry

for turbine design and certification, incorporating many useful features for detailed

analysis and design. Preprocessor modules can create turbulent wind histories and

compute modal properties. Steady-state aerodynamic and power curve analyses are

possible, as are dynamic, fully aero-elastic computations including offshore wave

loading and controllers. Post-processors are available for data reduction and visu-

alization, including extreme and fatigue load computations, and state-space model

linearisation.

4.2 Coordinate Systems

Prior to embarking on topical development, a common Coordinate System (CS) is

developed to describe the machine. A set of DOF are shown in Fig. 4.1 that define

the position and orientation of the following components:

Tower Top-head translational (xn, yn) and rotational (ψnod, ψroll, ψyaw) mo-

tion, in that order

Nacelle Offsets from the yaw-axis (o, h, l)

Hub Shaft tilt ψt about y′′ and rotation θ about x′′ (positive rotation clock-

wise when looking downwind), and offset Rhinge of the coning hinge axis

Blade Cone angle β (positive coning downwind), distance from coning hinge

to blade root shub and section distance along blade axis s.

All section twist and pitch angles are about the zaero axis, positive towards feather.

These DOF are sufficient for the current purposes which focus on rigid-body motion,

4.2 Coordinate Systems 43

but which contain enough generality that extension to flexible components should

be straight forward.

(a) Earth-tower-nacelle

Figure 4.1 Coordinate systems

It is important to point out a further simplification that is made with respect to

any flexibility of the blades, and the blade pre-bend defined in Fig. 4.29, both of

which move the section chordline off the pitch axis (zaero axis in Fig. 4.1(d)). For

the purposes of aerodynamics, any off-axis position is ignored, as it will be small

compared with other dimensions. Any structural velocity, however, resulting from

flexibility, is included in the aerodynamics. The structural calculations in §4.6.1.4

take into account the pre-bend, as do the load calculations in §4.6.2, as any mass

offset from the pitch axis will have an important effect.

The various CSs associated with the components are also detailed in Fig. 4.1.

Throughout the following development, the transformation matrices defined in Ap-

pendix A are used to relate CSs. The most general transformation from blade


(b) Nacelle-shaft (downwind) (c) Nacelle-shaft (upwind)

(d) Shaft-blade

Figure 4.1 Coordinate systems (cont.)

aerodynamic axes xaero–yaero–zaero to earth (fixed) xe–ye–ze is:

xyz1

e

= T (xn, yn, ht) R y(ψnod) R x(ψroll) R z(ψyaw) T (0, l, h) . . .

R y(ψt) R x(θ) T (0, 0, Rhinge) R y(β)

00

shub + s1

(4.2.1)

For anti-clockwise rotor rotation, the sign of θ is negative in Eq. (4.2.1). Also,

for downwind rotors ψt is negative, while for upwind rotors o is negative. Unless

otherwise stated, the tower is assumed rigid throughout this thesis (i.e. xn = yn =

ψnod = ψroll = 0).

4.3 Coned Rotor Performance Metrics 45

4.3 Coned Rotor Performance Metrics

A variable-area rotor introduces an ambiguity into the conventional definitions for

tip speed ratio λ, thrust CT and power CP coefficients. Each may be based on either

the actual R = Ractual (real, coned) or reference (unconed) tip radius R = Rref .

For a coning rotor of blade length S, they are defined as:

Ractual = Rhinge + S sin(β) (4.3.1a)

Rref = Rhinge + S (4.3.1b)

λ =ΩRU

(4.3.1c)

CP =P

1/2ρU3πR2(4.3.1d)

CT =T

1/2ρU2πR2(4.3.1e)

4.4 Aerodynamic Modelling

Clearly, aerodynamics lies at the heart of energy extraction from the wind. To

properly design a blade, accurate predictions of the aerodynamic forces and derived

quantities (e.g. rotor power, torque) are required, in both steady and dynamic condi-

tions. Numerous tools have been developed based on three basic formulations: Blade

Element Momentum (BEM), potential flow (vortex filaments/lifting-line), and full-

field solutions of the Navier-Stokes (NS) equations. Advanced aerodynamic analysis

techniques for wind turbines are in principle available to handle the aerodynamics

of the coning rotor concept, including lifting-line [86, 87], Navier-Stokes actuator

line and disk [88, 89] and full-field simulations [90].

It should be noted at this point that an initial attempt was made by the author

to modify a computationally efficient1 vortex code provided by Dr. Coton [87], to

handle the coned rotor. This approach was abandoned for two reasons. The first was

that even in its unmodified original form, the analysis of a single steady case took

on the order of minutes to compute. This rendered it unsuitable for later design

work. Secondly, it became apparent that the underlying equations would require

extensive reformulation to account for a coned rotor geometry.

1This formulation splits the wake into a near and far part, with prescribed wake geometry.This formulation involved the least possible computations, principally evaluations of someform of the Biot-Savart law (see §4.4.5).


In any case, practical design codes retain, at their core, implementations of BEM

theory. The BEM approach is much less computationally demanding than any other

approach, a key requirement for analyses taking into account inherent and important

coupled elastic behaviour, on a time scale relevant for design iterations. It has of

course been acknowledged that more advanced aerodynamic models are required for

better understanding of many wind turbine phenomena [91], but currently available

resources prevent these approaches from being beneficial in design practice and

optimization.

Bearing in mind these constraints and discussions of the inherent contradictions

in BEM [92], it has nevertheless proven to be quite adept at adequately predicting

performance for standard rotors. Highly coned rotors violate some of the underlying

assumptions of standard BEM theory, although on closer examination and with the

proper corrections, not as badly as might be expected. In light of BEM’s relevance

to wind turbine design, its underlying assumptions and history are elucidated in

§4.4.1 and used to correct the model in later sections, to make it applicable to the

analysis of coning rotor concepts.

The revised BEM method is first derived for the steady case to illustrate the

fundamental concepts. The reader is led from a description of the flow field in

§4.4.2, through momentum equations in §4.4.3, then to blade element forces in §4.4.4,

before introduction to the derived correction factors in §4.4.5 through §4.4.7. The

postulate is that the critical limitation of BEM is the usual assumed relation between

disc and far wake induction. Proper consideration of the relative placement of the

wake, and inclusion of radially induced velocity, using potential flow theory, yields

the corrections to the BEM method. The validation presented in §5.1 demonstrates

that the main limitation of BEM theory in application to coned and yawed rotors is

in the assumed wake position, not the planar disc nor the independence of stream-

tube assumptions discussed later. A model for centrifugal pumping is selected in

§4.4.8, before synthesizing a spanwise flow stall delay model in §4.4.9. Finally,

§4.4.10 presents the final corrected BEM equations, with additional wake geometries

in §4.4.11 and §4.4.12. Building on the same fundamental insights, the revised BEM

method is extended in §4.4.14 to a time-domain formulation handling dynamic and

yawed inflow.

4.4 Aerodynamic Modelling 47

4.4.1 Back to Basics

The lineage of BEM theory dates back to Rankine [93] and Froude [94], who before

1900 developed uniform actuator disc models. These were based on linear momen-

tum balances, and were extended by Betz [95] in the 1920’s to include wake rota-

tion. The idea of using blade elements to predict forces on the blades was originally

developed by Froude [96] and Drzewiecki [97]. Glauert [98, 99] provides a useful

discussion of the marrying of momentum theory to blade element theory by 1940

and the subsequent history of accounting for finite numbers of blades, which has

remained virtually unchanged in modern BEM. It is this 60 year old presentation

which has lent the insight required to extend BEM to account for coned rotors.

The approach taken is somewhat analogous to recent extensions for skewed rotors

[100], arising from work done in the 1940’s and 1950’s on helicopter rotors [101–103].

In order to properly derive a modified BEM theory appropriate for coned rotors, it

is necessary to go back and consider separately the momentum and blade element

components. This can only be done after properly defining the flow field through

the rotor.

4.4.2 Flow Field Kinematics

In keeping with the standard set-up for BEM, the axisymmetric flow through a gen-

eralized coned rotor is shown in Fig. 4.2, from far up-stream conditions at station 0,

passing through the rotor at station 1, to far down-stream at station 2.1 Inviscid

flow is assumed throughout, except for the thin shear layer at the boundary of the

wake. The flow is implicitly assumed to mix back to the free stream velocity in the

very far wake downstream of station 2. The details of the far wake field will be

discussed further in §4.4.11 as they relate to modelling at high induction factors.

The incident flow at station 0 is taken as constant at all radii. The effects of inflow

turbulence, yaw and wind shear are all ignored in the present steady, axisymmetric

analysis until §4.4.14. The downstream velocity varies radially between stream-

tubes, as a function of the calculated loading. Both far-field velocities, 2D vectors~V0 and ~V2 in (r, z), are taken to have zero radial component, the former’s magnitude

being an input variable and the latter an output of the simulation. The situation

at the rotor for a single stream-tube is expanded in Fig. 4.3. The radial component

of the flow (normally neglected) is shown, together with the aggregate rotor/blade

1Note that this aerodynamic coordinate system differs from §4.2. The equations wereoriginally derived for comparison to Mikkelsen [89] who used the CS used in this section.


Figure 4.2 Kinematics setup

generated forces acting on the flow through the individual stream-tube. The forces

are either reactions of the lift and drag forces on the blades, or hypothetical idealized

pressure forces. For a real rotor imparting azimuthal forces as well, there is also an

azimuthal velocity component at the disc wθ and wθ,2 in the far field, both pointing

into the page.

Figure 4.3 Kinematics at rotor for single stream-tube

Note that the radial velocity component modifies Vn only for coned rotors, and

affects the predicted velocity relative to the 2D airfoil. Critically, this is a description

of the generalized velocity components of the true net flow field through the rotor.

Mikkelsen [89] presented a BEM method (in addition to the CFD results discussed

in §5.1) which accounted for the radial velocity component, but as will be shown

later, erroneously assumed wn to be parallel to ~Fn. The proposed rationalization

was that BEM governing equations only consider axial momentum, and there could

therefore be no induced velocity parallel to the rotor blades. In fact, there is radial

flow and it is simply a modelling deficiency that the radial flow is conventionally


unaccounted for.

In the present method, the induced velocity components ~w are the orthogonal in-

crements from the free stream velocity at the rotor, proportioned via the flow model

to give the true net flow vector at the disc. The net flow field can be rationalized by

momentum considerations to be the result of the effects of the local loading of the

disc (retardation of the flow in the stream-tube, which must result in flow expan-

sion and hence radial flow), or as the induced effect of the vorticity bound on the

blades and trailed in the wake. The latter approach will later prove quantitatively

expedient in §4.4.5, yielding induced velocities ~u.1

The standard relations between the axial a and tangential a′ induction factors

and induced velocities wz and wθ are retained in the present analysis for convenient

comparison to the standard BEM equations:

wz = aV0 (4.4.1a)

wθ = a′rΩ (4.4.1b)

Note that these are the induced velocities at the disc, station 1 in Fig. 4.2.

4.4.3 Momentum Balances

Momentum theory was the basis of Froude’s early analysis of a uniformly loaded

disc. Later, the disc was divided into annular tubes, that afford radially varying

induction at the disc, when pressure and friction effects between stream-tubes are

neglected.2 Pressure drop in the wake owing to rotation is also ignored in assuming a

pressure of p0 at station 2 [16, p. 62]. Assuming wake rotation to be small compared

to Ω also greatly simplifies the following momentum equations, as shown by Glauert

[98]. All of these common simplifications are adopted here.

4.4.3.1 Axial Momentum

An axial momentum balance for a control volume enclosing a single stream-tube

between station 0 and station 2 yields:

BFz = −(V2 − V0

)∆m (4.4.2)

∆m = ρ [(V0 − wz) cosβ − wr sinβ] 2πr∆s (4.4.3)

1Note that for clarity ~w is used in the momentum equations, and ~u is used in referenceto the results of vortex theory.

2A variable area/expanding stream-tube will have non-zero integrated pressure forces onthe streamwise sides, although Mikkelsen [89] showed these to be negligible.


Note that the mass flow is calculated as the dot product of ~V1 and area normal

vector∣∣∣ ~dA∣∣∣ = 2πr∆s. The standard BEM formulation assumes equivalently wr = 0

or stream-tube cross-sectional area of 2πr∆s cosβ perpendicular to the total flow at

station 1 (V0−wz). Both formulations overestimate the mass flow which is relatively

reduced for a non-zero β, due to the radial inclination of the flow.

Figure 4.4 Flow relative to airfoil section

4.4.3.2 Tangential Momentum

The tangential velocity imparted to the flow as a torque reaction is calculated from

a tangential momentum balance:

BrFθ = ∆mr2wθ,2 (4.4.4)

The notation wθ refers to azimuthal velocity at the disc and wθ,2 azimuthal veloc-

ities downstream. As with the rest of the method development, the basic theory

essentially ignores wake expansion and assumes an infinite number of blades. Con-

sidering the reference frame fixed to the blade section shown in Fig. 4.4, the average

tangential velocity wθ through the rotor is half the tangential exit velocity 2wθ,

since there is no tangential velocity upstream of the rotor. With no wake expansion

(r1 = r2), conservation of angular momentum yields:

2wθr21 = wθ,2r

22 ⇒ wθ,2 = 2wθ (4.4.5)


4.4.3.3 Finite Length Blades

A finite number of blades B will produce distinct helical vortex sheets all moving

downstream with the same velocity, rather than a continuous volume of vorticity.

For finite numbers of blades, Prandtl [98] developed a simplified wake structure

model idealized as semi-infinite discs moving downstream at speed V2, spacing the

idealized discs at the pitch distance of the helical sheets. The average velocities in

the wake V2 and wθ,2 are then predicted from complex-variable flow analysis around

the discs, in the form of a correction factor F to the far-field velocities:

F =2π

cos−1

(exp

[−B

2R− rr sin(φ)

])(4.4.6a)

wz,2 = Fwz,2 (4.4.6b)

wθ,2 = Fwθ,2 (4.4.6c)

This accounts for the average momentum transferred to the bulk fluid relative to

the uniform flow produced by an infinite number of blades. Thought of another

way, the velocity at the blades (the velocity of the wake sheets) is higher than the

average flow between the sheets near the edges.

Some researchers have therefore also applied F to the induced velocity at sta-

tion 1 in the mass flow term (as a circumferentially averaged quantity). This is not

convention1[104, p.55] or done here, because Prandtl’s derivation requires an infinite

series of discs which is not a good approximation in the near wake.

Prantl’s model should strictly use the helix angle at the tip, φtip. For the coned

rotor, φ should be the wake helix angle in a plane tangent to the rotor axis, not

normal to the blade pitch axis. Again, this detail is lost in the overall approximation,

so φ at each section normal is used.

4.4.3.4 High Induction

The BEM model breaks down for high a values (occurring equivalently at high λ

and low V0 for fixed Ω). In reality, unsteady turbulent mixing occurs with the flow

outside the bounding streamline downstream of the rotor, to satisfy mass continuity

(i.e. the velocity cannot be zero in the wake as predicted by basic momentum theory).

To account for this effect, the stream-tube axial momentum balances are modified

by empirically derived thrust coefficients CT,model, determined experimentally from

aggregate values for complete rotors.

1BLADEDTM uses a not aF in the mass flow term. Note that some authors may alsoinvert the definitions of a and aF by defining a as the averaged quantity.


Three existing models (referred to later as thrust models) that have been fitted

to experimental data are used here, when the induction factor in the plane of the

rotor tip adisc exceeds ac:

CT,Bladed (ac = 0.3539) = 0.5998 + 0.61adisc + 0.78a2disc (4.4.7a)

CT,Glauert (ac = 0.4) =(adisc − 0.143)2 − 0.0203

0.6427+ 0.889 (4.4.7b)

CT,Spera (ac = 0.2) = 4(a2

c + (1− 2ac) adisc

)(4.4.7c)

The first CT,Bladed model, implemented in BLADEDTM, is quite similar to the

Glauert model [17]. Results presented later obtained from runs of the commer-

cial code are labelled ‘BLADED’. The same thrust model is used internally in the

implementation of the current work, for which results are labelled ‘CT Bladed’, for

comparison to the ‘CT Spera’ results using the Spera model [15]. The behaviour

of the three models is compared in Fig. 4.5, plotted with the original data Glauert

used to derive his model [15], which will become relevant later.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a

CT

Momentum BLADED

Glauert Spera

NACA MR No L5009 NACA TN 221

BRC R&M 885

Figure 4.5 Thrust CT,model model behaviour

Again, for the unconed rotor these models should strictly use adisc = aF not

adisc = a to account for the average velocity at the disc, as the data was gathered

based on an average velocity. As in §4.4.3.3, this is again not done to follow conven-

tion [16, p. 93]. However, one must be careful in comparing data as some authors

choose a and others aF .


4.4.4 Blade Elements

Momentum theory alone cannot predict the forces imparted by a blade with par-

ticular airfoil cross-section profiles. However, using the predicted induced velocities

at the rotor, 2D airfoil data can be used to calculate the loading imparted to the

stream-tube in the momentum equations. Prior to an understanding of the induced

velocities produced from 3D wake structures, there was great debate as to the flow

conditions at the rotor suitable for determining the aerodynamic forces [98]. Blade

element methods which ignored any inflow effects (i.e. assumed unaltered V0 velocity

at the blades) were used with apparently heuristically derived airfoil datasets, each

differing in the aspect ratio of the blade, in a misguided attempt to account for the

actual 3D flow effects. Lifting line methods can now of course explicitly account for

the velocities induced at the blade by the vorticity trailed downstream.

The closure of this problem is left to the next section. For now, the generalized

flow relative to a blade section shown in Fig. 4.4 is considered. Spanwise flow cannot

be directly modelled by BEM. However, it will serve to further force the un-modelled

3D boundary layer and may exacerbate stall-delay effects from spanwise boundary

layer migration [105] (see §4.4.9).

The aerodynamic forces fz and fθ per unit length of blade are predicted from

2D airfoil test data (lift/drag coefficients cl, cd) dependent on AOA α and Reynolds

number Re:

fz = cnρV 2

rel

2c cosβ (4.4.8a)

fθ = cθρV 2

rel

2c (4.4.8b)

cn = cl cosφ+ cd sinφ (4.4.8c)

cθ = cl sinφ− cd cosφ (4.4.8d)

The total angle of the section γtot is found from:

γtot = γtwist + γset + γpitch (4.4.9)

with γtwist the blade twist, γset the fixed pitch offset, and γpitch any active pitch

angle. The AOA is then α = φ− γtot.

Lindenburg [104, p. 57] notes that the effective camberline of an airfoil section on

a coned rotor will be modified, due to the curvature of the flow relative to the airfoil.

The effects may be included by referencing the AOA to the 3/4c point on the section

and modifying the AOA–cm relation. This effect is ignored here however, given the

spatial resolution of the BEM method.


4.4.5 Wake Analysis

The momentum equations must be closed by relating the far field velocities to those

at the disc. As previously mentioned, the detailed flow at the disc is required to

compute the aerodynamic forces developed by the section. This is normally done

via Bernoulli’s equation, relating the difference in static pressure just before and

after the rotor (equivalent to Fn/dA) to the upstream and downstream velocities.

This again neglects wake rotation and expansion, but more importantly is the root

of an inability to predict coning effects. Essentially, as has been noted in full flow

field studies [88, 106], standard BEM behaves as if the calculations are done in a

radial plane through the blade tip. In essence they are, since information about the

relative stream-wise position of the stations along the blade is absent.

An alternative approach to closing the equations is to examine the wake structure

in more detail using potential flow theory. It is of course possible to dispense with the

momentum equations altogether, by modelling the blades and wake as bound and

free vortex filaments to determine the flow at the rotor (bound vortex), but this is

overly computationally intensive for design purposes. To exploit the computational

efficiency of BEM, analogous with fundamental wake studies [107], yawed turbines

[100] and helicopter rotors [102], the helical vortex sheets shed into the wake can be

decomposed into rings and axial filaments extending infinitely far downstream.

4.4.5.1 Wake Decomposition

The simplest case is obtained by restricting attention to the uniformly loaded rotor

(constant pressure exerted on the flow by the disc) with an infinite number of blades.

Vorticity is continuously shed only at the blade tips to form a single sheet of vorticity,

as shown in Fig. 4.6. The bound vortices on the disc also merge into a single axial

root vortex joining from downstream at the blade roots. Note that in general as the

loading along the blade changes for a real rotor, vortex rings and filaments will be

shed to fill the entire downstream volume with helical sheets.

The Biot-Savart law is used to compute the induced velocities ~u for a vortex

filament of strength Γ and length d~l with vector ~r to the point of interest:

d~u =Γ4π

d~l × ~r|~r|3

(4.4.10)

Using this formula, it is shown later in §4.4.7 that the axial filaments on the wake

cylinder and at the root, together with the bound vortices on the disc, induce only

second-order velocities. As in other works, the streamwise (axial) [100] and radial


Figure 4.6 Wake decomposition into rings and filaments

induction effects are of primary importance here. The swirl is of course treated

in the conventional manner, by the azimuthal momentum equation. Provision for

further treatment is provided by the inclusion of εθ in the final set of equations. The

correction factor is defined from the vortex filament results as:

εθ =uθ,2

2uθ

∣∣∣∣vortexsheet

(4.4.11a)

wθ,2 = 2a′Ωrεθ (4.4.11b)

In any case, swirl is irrelevant to the solution of the uniformly loaded disc, since

only the axial momentum equation is solved. The axial and azimuthal equations

are uncoupled and the latter uniformly zero, due to a lack of azimuthally applied

pressure.

For a real rotor, the loading (including an azimuthal loading) will of course depend

on φ, since the lift and drag depend on the angle of attack which is determined by

φ together with the blade twist and pitch. Again using lifting line theory, it can be

shown that the bound vortices on the disc do not contribute to the induced velocities

on the disc in the case of an unconed rotor. For a coned rotor, the axial and bound

filaments only induce azimuthal velocity, as for the unconed rotor. The net effect is

that the azimuthal induction is changed less than 1% over the length of the blade

for a coned rotor relative to an unconed rotor (see §4.4.7 for further details). In

summary, inclusion of the swirl correction factor for coning εθ will have zero effect

for a uniformly loaded rotor and a second order effect for a real rotor.

4.4.5.2 Evaluation of Correction Factors

Focusing back on just the vortex cylinder comprising the wake, the induced velocity

from the cylindrical vortex sheet in Fig. 4.7 at point(x0, 0, z0

)is found from inte-

gration of Eq. (4.4.10) over the surface of the cylinder with vortex strength per unit


area γ = Γ/dzRdθ:

~u = 2

π∫0

γR

4π

∞∫z0

z cos θi+ z sin θj + (R− x0 cos θ) k(x2

0 − 2Rx0 cos θ +R2 + z2 + (δcR)2) 3

2

dzdθ (4.4.12)

The point is x0 units radially off the centreline and z0 units upwind from the end of

the vortex tube. The inner z portion of the integral is easily evaluated, however the

azimuthal component requires either elliptical integral functions or as is done here,

numerical integration. The z limits of integration may be set to ±∞ to evaluate

the induced velocity in the far wake. The vortex core addition to the formula, δcR,

avoids a discontinuity as the sheet is approached.1 The core size δcR is taken as

0.1%R, with minimal effect on the results below this value.

Figure 4.7 Vortex ring geometry

Correction factors εz and εr are calculated using the induced velocities from

Eq. (4.4.12), computed for each section radius at a point on the coned disc and

in the far wake:

εz =uz,2

2uz


(4.4.13a)

εr =ur

uz


(4.4.13b)

The division operations cancel the vortex sheet strengths γ in Eqn. Eq. (4.4.12),

which can be arbitrary here. It should be possible to bypass the use of the momentum

1This is a numerical artefact of the mathematical model. Viscosity has important effectsnear the centre of the vortex, leading to various models to describe the induced velocitiesas the core is approached [108].


equations, by continuing the vortex analysis and computing the sheet strengths,

rather than the induction factors. This method would amount to a prescribed wake

vortex approach, and was not explored here as the purpose was to provide small

corrections to well known and trusted BEM codes.

Figure 4.8 shows the vortex sheet predictions for a unit radius rotor operating at

a = 1/3. Only a single cylindrical vortex is shed at the tip . The vectors show the

induced velocity from the vortex sheet, the grey streamlines the overall flow, and

the ellipses centred on the tip are isolines of the induced radial velocity. An unconed

rotor is shown, but the same diagram will exist for a unity radius rotor with coning

(i.e. a longer blade). The axial correction factor εz is greater than unity for points

upstream of the tip-plane, unity at the plane and less than unity downstream. The

radial correction factor εr is always greater than zero and bounded to approximately

4 near the wake sheet (depending on vortex core size).

-2.0-1.5-1.0 -0.50.00.5 1.00.0

0.5

1.0

1.5

2.0

z

r

Student Version of MATLAB

Figure 4.8 Velocity field around vortex cylinder

4.4.6 Relating Disc to Far Field

Using the vortex filament results (εz and εr) at each section along the blade, the far

field velocity may now be related to that at the disc for closure of the momentum

equations and definition of the induced velocities. The following equations define


the induced velocities in terms of εz and εr:

wz,2 = 2wzεz

= 2aV0εz (4.4.14a)

wr = wzεr (4.4.14b)

For an unconed rotor, the vortex calculations yield a unity axial correction factor

εz. This is consistent with the usual assumed result from Bernoulli’s equation,

wz,2 = 2wz.

The inclusion of a radial component in the flow modifies the velocity normal to

the blade in calculating the lift and drag coefficients. The normal velocity seen by

the blade section can now be defined using the radial correction factor:

V1 = V0

((1− a)2 + (aεr)

2)1/2

(4.4.15a)

Vn = V1 cos (β + δ) (4.4.15b)

δ = arctan(aεr

1− a

)(4.4.15c)

4.4.7 Azimuthal Induced Velocity

The axial filaments on the wake cylinder sheet, the root streamwise vortex, and the

bound vorticity on the blades shown in Fig. 4.9 are responsible for all azimuthally

induced velocities. Qualitatively, the velocity vector increments at the blade in-

duced by the filaments can be worked out graphically, using the right-hand rule to

determine the directionality of the induced velocities. Considering each of the three

contributors in turn, it can be demonstrated that the net streamwise and radial

induced velocities are zero for each set, independent of cone angle. The azimuthal

contribution, however, is non-zero for the wake and root filaments for an unconed

rotor. Additionally, the bound vortices have a net contribution for a coned rotor.

The error associated with ignoring the details of the azimuthal induction can be

quantified using the Biot-Savart law Eq. (4.4.10). For any straight vortex filament

of strength Γ, defining ~r1 and ~r2 as vectors from the tail and head of the filament

to the point of interest respectively, the induced velocity may be calculated from:

~u =Γ4π

(|~r1|+ |~r2|) (~r1 × ~r2)|~r1| |~r2| (|~r1| |~r2|+ ~r1 · ~r2) + δ2

(4.4.16)

The δ factor again ensures that the formula converges near the vortex core.

In order to quantify the induced velocities, the magnitude of Γ must be determined

for a given operating condition consisting of free-stream velocity V0, axial induction


Figure 4.9 Axial, root and bound vortex filaments

a and tip-speed ratio λ, as done by Burton et al. [16]. Considering the wake cylinder

opened flat, over one revolution of the wake the sum Γs of all the trailing vortex

filaments will be distributed over a distance 2πRsinφt perpendicular to the filaments.

For an infinite number of blades, this gives an areal density of vortex strength of:

γw =Γs

2πRsinφt(4.4.17)

The angle φt is the flow angle at the tip, which from geometry may be determined

from:

tanφt =1− a

λ(1− a′t)(4.4.18)

The vortex density γ normal to the streamwise direction is a component of γw from

geometry, γ = γw cosφt. As will be shown later, in the far wake this induces a

uniform axial velocity of γk. This induction is equal to 2aV0 from classical theory,

yielding:

Γs =4πV 2

0 a(1− a)Ω(a+ a′t)

(4.4.19)

where a′t is the tangential induction factor at the tip. For the unconed rotor, only

the root vortex of strength Γ = Γs contributes to the tangential velocity, so by

definition of a′t and Eq. (4.4.16):

a′t =Γ

4πR2Ω(4.4.20)


Combining Eqs. (4.4.19) and (4.4.20) we obtain a quadratic for a′t:

a′t2 + a′t −

a(1− a)λ2

= 0 (4.4.21)

This may be solved for a′t for the given flow condition, and then substituted into

Eq. (4.4.19) to yield Γ.

Given Γ and using Eq. (4.4.16), the contributions to the net induced velocity at

any point in the flow may be determined. The tangential induction factor at points

along the blade at radius r can be computed from:

a′t =uθR

rV0λ(4.4.22)

where uθ is azimuthal induced velocity and R the tip radius.

The results shown in Fig. 4.10 are for a typical case of a = 0.3 and λ = 6, with

δ = 0.001R. 100 blades are used for the numerical calculations. The tip radius is

the same for the coned and unconed cases. For these results, the above relations for

Γ are assumed valid for a coned rotor. As will be shown, Eq. (4.4.20) is minimally

in error due to the additional influence of the bound vortices on a coned rotor.

As expected, only the root vortex has any influence at the disc for β = 0. The

dip at the tip is due to the vortex core model. For the downwind coned rotor, the

bound vortex has an influence in opposition to that created by the root vortex.

However, the influence of the root vortex is increased, as the blade now “sees” more

of the vortex. For the upwind coned rotor, the effects are reversed. The net effect

in both cases is that the azimuthal induction is changed less than 1% between the

coned and unconed cases. These results lends credence to the decision to ignore the

details of the azimuthal induction in the coning BEM theory.

4.4.8 Stall Delay and Centrifugal Pumping

It has long been recognized that propeller and wind turbine blades behave as if

sections near the root of the blades have delayed stall, to higher cl (up to double)

and αstall, than would be predicted by 2D section data. It has also been noted that

a much more benign and opposite effect appears to exist near the tips. Lindenburg

[104, §4.2.2] provided an excellent review of the relevant studies that have examined

this effect. As the effect has only been observed on rotating blades, it is variously

referred to as “centrifugal pumping”, “spanwise pumping” and “radial pumping”.

Section 4.4.8.1 first gives an overview of the experimental challenges in studying stall

delay, followed by a comparison of the various theories that have been proposed to


0 0.2 0.4 0.6 0.8 1

-1

0

1

2

3

4

r/R

a’

bnd

cyl

root

total

(a) β = 0

0 0.2 0.4 0.6 0.8 1

-1

0

1

2

3

4

r/R

a’

(b) β = 40

0 0.2 0.4 0.6 0.8 1

-1

0

1

2

3

4

r/R

a’

(c) β = −40

Figure 4.10 Azimuthal induction along blade

explain the phenomenon in §4.4.8.2. Based on this background, §4.4.8.3 selects and

presents the best existing model for use in this thesis.

4.4.8.1 Experimental Investigation

Post-processing experimental data to validate the occurrence of the phenomenon is

complicated [109]. Measurements of aggregate shaft torque/power mask the details

of any spanwise variation. A common method of investigation is to employ pressure

taps at sections along the blade. Unfortunately, the spatially resolution is fairly

coarse, and the angle of attack must be computed in some manner for decomposing


the normal cn and tangent ct force coefficients into conventional aerodynamic cl and

cd ones.

To avoid the latter ambiguity, some data reduction (and models in §4.4.8.2)

present the data in terms of cn [104]. Others make use of full vortex analysis mod-

els to back-out the angles of attack and thereby decompose the measured pressures

into lift/drag coefficients. This has produced conflicting results, particularly as some

studies note an increase in cd while others find a decrease. There is no doubt, how-

ever, that the effect is real and particularly pronounced at the root.

4.4.8.2 Mechanisms for Centrifugal Pumping

Stall delay is clearly a boundary layer effect, whereby gross separation is delayed and

stall avoided. Based on this, various researchers have employed the NS equations

in an attempt to tease out the underlying mechanism of stall delay. The earliest

collection of efforts [110] reduced the equations to their laminar boundary layer

form. In fact, at its most reduced, no stall delay is predicted for c << r. More

recent efforts, as detailed and expanded on by Corten [105], have emphasised the

use of order-of-magnitude analysis for eliminating terms in the NS equations. This

reference should be consulted for more information on the following discussion.

Snel [111] employed this technique [112], embedding the equations into X-Foil1 to

solve for enhanced lift characteristics. The theoretical examination of the bound-

ary layer equations predicted a trend with (c/r)2/3, however the calculations found

a relation of (c/r)2. Corten [105] disagreed with a number of assumptions in the

reduction, fundamentally whether the use of the boundary layer equations was jus-

tified in stall. A revised analysis was framed in a cylindrical CS rotating with the

blade, without the boundary layer assumption, but neglecting viscosity (reduction

to the Euler equations). Emphasis was placed on a separation point existing on the

blade surface, so that the separated boundary layer is thick and velocities negligible.

The final result is a prediction that stall delay effects are proportional to c/r.

At issue between the two methods is the underlying mechanism. In the former,

it is a centrifugal pressure gradient owing to the rotation, which sets up a radial

flow. In turn, the radial flow creates a positive chordwise pressure gradient, owing

to Coriolis acceleration in the rotating reference frame. This reduces the adverse

pressure gradient and hence delays stall. Corten [105] on the other hand, finds that

12D coupled inviscid-viscous boundary layer code by Dr. Drela at MIT


radial convection is the underlying mechanism. A related phenomenon predicted by

the latter approach is a helical flow outwards in the separated region near the TE.

It seems somewhat duplicitous to simultaneously assume a separated region of

flow and delay of stall, since separation is the mechanism of airfoil stall. If a stall

region extends from the TE, then a drag increment should be more evident in the

results. This would arise both from a chordwise pressure differential, and energy

input to the large volume of separated flow. Ultimate resolution of the theoretical

discrepancy will depend on more experimental data becoming available.

4.4.8.3 Modelling Centrifugal Pumping

In an attempt to match analysis to experimental data for the National Renewable

Energy Laboratory (NREL) Unsteady Aerodynamics Experiment (UAE) experiment

(see §C.2), Lindenburg [104] compared the available models based on the preceding

analyses. A new model was also proposed by Lindenburg [113], based on the sep-

aration point at the TE, together with a set of assumptions about fully-developed

spanwise flow at the TE creating a Coriolis acceleration. The lift increment is then

taken as proportional to the chordwise extent of the separation region. The other

models propose modifications to cl, cd and cn in various forms.

Based on the accuracy of the predictions in that report, relative to the experimen-

tal data, the modified form of Snel’s model is used in the current work. As outlined

in §4.4.8.2, the Snel model also appears the most physically based. It is expressed

as a cl increment from the static non-rotating coefficient cl,non−rot:

cl,rot = cl,non−rot + 3.1 cos2 φ( cr

)2(2π(α− α0)− cl,non−rot) (4.4.23)

The factor of 3.1 was obtained empirically. The cos2 φ local tip-speed ratio λr

dependency was introduced after the original derivation, to enable analysis during

start-up and shut-down. BLADEDTM can only accept static corrections to airfoil

tables computed a priori, so this model cannot be used in BLADED runs. A number

of additional features of the model implementation are given in the following list:

• The maximum lift increment is limited so that cl,rot never exceeds 2π(α−α0),

the linear lift curve.

• A linear reduction between r = 0.75 . . . 0.8 to zero limits the spanwise extent

of the model to be consistent with experiments.1

1The linear reduction (rather than a sharp cut-off) is required to avoid numerical diffi-culties in optimization as sections pass the cut-off radius under DV manipulation.


• The model is only applied for α > α0. It is also linearly tapered to zero

over 30 < α < 50. This ensures stall occurs at some point over that range,

consistent with experimental evidence.

4.4.9 Spanwise Flow

By virtue of its geometry, a coning rotor will encounter a velocity component Vs

along the spanwise axis, as shown in Fig. 4.11. Any rotor operating in yaw or flow

non-parallel to the shaft will also experience spanwise flow. The effective sweep

angle Λ over the blade is shown in Fig. 4.12, for a range of coning angles β at

zero yaw. Two operating points are shown, one representative of optimal operation

(Fig. 4.12(a)) and the other power limiting (Fig. 4.12(b)).

Figure 4.11 Skewed flow velocity vectors and boundary layer profiles

The induced velocities (axial and radial) have been computed using the correction

factors developed in §4.4.5.2. The induction factor corrections have been computed

for an infinite number of uniformly loaded blades, so the only limit on induced

velocity close to the tip is the vortex core size. Consequently the effective sweep

angle is found to rise sharply towards the tips. In reality, a finite number of blades

will reduce this large increase seen over the last 5% of the radius. Due to the flow

expansion predicted by the vortex model, even an unconed blade will experience

some degree of spanwise flow. In this case though, the angles are quite small, even


0.2 0.4 0.6 0.8 10

20

400

10

20

30

40

r

4

6

2

8101 2

β (deg)

Λ (de

g)


(a) a = 1/3, λ = 7

0.2 0.4 0.6 0.8 10

2040

0

10

20

30

40

r

1 08

12141 6

6

18

β (deg)

Λ (de

g)


(b) a = 2/3, λ = 3

Figure 4.12 Effective sweep angle with coning for typical operating conditions

in the stalling condition Fig. 4.12(b), and so would not have much of an effect based

on the experimental evidence to follow. Conversely, the increasingly large Λ seen

over the blade length, particularily towards the root as the cone angle β increases,

can be expected to influence the stalling behaviour of the rotor.

This section first examines the common assumptions regarding spanwise flow in

§4.4.9.1, before referencing experimental evidence in §4.4.9.2 to show that the com-

mon assumptions break down in stall. To deal with the conditions of the coned rotor,

a model is developed in §4.4.9.3 that is used to modify the 2D airfoil characteristics

to account for spanwise flow.

4.4.9.1 Independence Principle

Almost universally, the spanwise flow component is ignored when computing the

aerodynamic properties of the section. It is assumed that the lift and drag of the

section may be predicted solely from Vrel and α, computed as V cos Λ. This ap-

proach is justified by the independence principle [114], also applied to swept wings

and helicopter rotors, and which may be rationalized by the following thought ex-

periment.

Consider an infinitely long wing (unbounded zaero), of constant cross section,

immersed in a frictionless flow. If the flow is at right angles to the wing axis (Λ = 0),

then the flow around the wing is uniquely determined by the cross-sectional profile

normal to the spanwise axis. If the wing is now assumed to translate along its own

axis, the lack of friction means that the external flow is unaltered from the original


condition. It is also equivalent to an effective yaw angle Λ being experienced by the

wing. The thought experiment of course breaks down when the viscosity of the flow

becomes important, i.e. when the boundary layer is no-longer thin and separation

(stall) begins to occur. The laminar boundary layer profiles ([see 110]) before stall

are illustrated in the upper right corner of Fig. 4.11, with a greatly exaggerated

dimension normal to the airfoil surface.

Simple wing sweep theory simply relates angle of attack αΛ, lift curve slope clαΛ

and lift coefficient cl,Λ for a section in the streamwise direction, to the analogous

parameters in the section normal to the span axis, α, clα and cl:

αΛ = α cos Λ (4.4.24a)

clαΛ = clα cos Λ (4.4.24b)

cl,Λ = cl cos2 Λ (4.4.24c)

These formulae can be derived from the geometry of Fig. 4.11, decomposing the

rotation vector α lying along the spanwise axis into a component normal to the free-

stream, and recognising that c s = cΛ cos Λ b/ cos Λ = cΛb, i.e. the non-dimensional

areas are equal for both un-swept (chord c, length s) and swept (chord cΛ, span b

perpendicular to free-stream) definitions.

4.4.9.2 Experimental Evidence

Experimentally, the validity of the independence principle has been confirmed for

real wings, at least before 2D results would predict stall. In the set of swept wing test

data to follow, sweep angle and yaw angle (in an aircraft sense) Λ are synonymous

with each other. Figure 4.13 shows a typical set of 2D lift curve data (i.e. for sections

perpendicular to spanwise axis), derived from a 3D yawed wing test. Another set

of 2D lift data shown in Fig. 4.14 has been measured from pressure taps on two

swept wings, one without twist and employing a symmetric airfoil, and the other

with twist and a non-symmetric airfoil [115]. The various lift curves are labelled by

the section percentage location along the span. The data is somewhat complicated

by the induced velocities from the finite wing length.1 The effect is to alter the

linear lift-curve slope differentially along the span, as well as shift the zero-lift AOA

for the non-symmetric wing.

1Secondarily, a fuselage was also present at the root of the wing, and the pressure taps atthe most inboard and outboard locations were orientated in the free-stream direction. Therest of the mid-span pressure taps were located along chords perpendicular to the sweptwing axis. The tests were also carried out for Re ≈ 108.


0°15°30°

35°40° 45°

60°

75°

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0 10 20 30 40 50AOA (deg)

c l

Figure 4.13 Experimental 2D lift curves for varying yaw angle (data from Harris[110])

Experimental 2D lift curves for Λ = 0, shown as the dashed lines in Fig. 4.14,

where used to compute estimated lift curves based on the simple sweep theory given

in Eq. (4.4.24). Note that AOA is the overall wing angle of attack for this dataset,

not the sectional angle of attack α. A Weissigner lifting-line method was then used

to compute linear lift-curve slopes which were then used to adjust the 2D curves.

The predictions are not shown in Fig. 4.14 for simplicity, but suffice it to say that

the linear portions matched well, but the predicted stall behaviour was akin to the

2D curve in terms of stall AOA and cl,max. The measured data therefore shows

significant stall delay, increasing in effect inboard.

Based on data such as this, it has been found that a yawed wing will follow

the 2D linear lift curve well past the 2D un-swept stall angle and lift coefficient

cl,max(Λ = 0), delaying stall to much higher lift coefficients cl,max(Λ) and stall

AOAs. Data points taken from a number of studies for cl,max and αstall are given in

Fig. 4.15 illustrating this behaviour.1

The proposed mechanism for the observed stall delay is a thinning and energization

of the boundary layer by the spanwise flow. The boundary layer is therefore able

to continue to navigate the adverse pressure gradient without separation for much

higher 2D AOAs α. Furthermore, it has been found experimentally that the 2D drag

and moment coefficient curves are basically unaffected by Λ. Of course, energy is

1The additional data in Fig. 4.15(a) for the NACA study was obtained from full-wingCL data, scaled using Eq. (4.4.24c).


0.167

0.3830.5450.815

0.9240.707

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 10 20AOA (deg)

c l

2D

(a) Plain wing

0.1670.383

0.5450.707

0.815

0.924

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

-5 5 15 25

AOA (deg)

c l 2D

(b) Twisted wing

Figure 4.14 Wing section lift curves for 45 swept wing (data from [115])

being put into the boundary layer to avoid 2D stall, and so increased drag is present,

but in the spanwise direction.

1.01.21.41.61.82.02.22.42.62.83.0

0 10 20 30 40 50 60 70Sweep angle (deg)

c l,max( Λ

)/cl,m

ax Λ

=0

HarrisHarris 2NACAcos^1.5cos^1cos^0.75

(a) Maximum lift

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70Sweep angle (deg)

AOA s

tall

(b) Stall angle

Figure 4.15 Stall delay behaviour for 2D sections in yawed flow (data from[110, 116])

It should be noted that this spanwise drag component is quite important for heli-

copter rotors [110, 117], since it translates into an additional power requirement for

forward flight. Helicopter blades experience continually varying yaw with azimuth


angle, owing to the relative component of the free-stream in forward flight (greater

angles towards advancing side, [see 110, Figure 2]).

In aircraft applications [118, pg. 252], stall is generally avoided, so only the linear

theory (below stall) is presented. Further comments are usually directed towards

the mitigation of tip stall (washout, vanes), resulting from outboard migration of

the boundary layer on swept wings, which precipitate dangerous pitch-up and loss

of roll control. This migration results from a spanwise pressure gradient developed

between sections normal to the free-stream [116], since the airfoils in those planes

are displaced relative to each other in a streamwise sense. Assuming the pressure

profiles are identical, there will therefore be a pressure differential between sections.

Another important effect with relevance to aircraft is vortex lift. As noted by

Furlong and McHugh [116] in an overview report of spanwise flow, much larger climprovements are found for airfoils with sharp Leading Edges (LEs), which stall

from the LE, rather than TE for airfoils with more rounded LE. The mechanism is

a stable vortex [118] created over the upper surface, near the LE. The lift increase

associated with this flow structure was included as a safety factor in the original

CONE-450 study [66] for parked calculations. It is not clear whether or not such

enhanced vortex lift would in fact develop, given the “negative sweep” angle formed

by the thinning chordwise distribution.1 No prediction of effects from spanwise flow

in operation was included in the original study.

For wind turbines, it is the enhanced lift and delayed stall effects that are critical.

Predictions in yaw above stall remain within the realm of experiment, defying an

easy analytic solution [110]. Boundary layer theory can be applied to the chordwise-

spanwise coupled problem, but is only valid to the lower limit of stall initiation2 and

available in closed form only for laminar flow. Most data obtained for aircraft

design has looked at wings with relatively low aspect ratio (average A = 5) [116].

More experimental effort to explore these effects for the very high aspect ratio wind

turbine blades (approaching A = 25) is required for validation. It is expected that

the effects may be even more pronounced, given the larger span absent of tip-effects,

over which the spanwise flow can develop.

1This is opposite to the conventional delta wing that increases in span (wind turbineblade chord) in the flow direction.

2The same is true for predicting the stall delay effects of centrifugal pumping discussedin §4.4.8.


4.4.9.3 Spanwise Flow Model

Below the point of 2D stall, sectional data may be used in the BEM method with

some confidence. There is clearly no fully satisfying analytic method available for

predicting the 2D properties beyond the 2D stall point when spanwise flow is present.

In the face of this uncertainty, Harris [110] originally proposed a simple approach,

beginning with the assumption that a section in the streamwise direction will stall

at approximately the same αΛ,stall as a 2D section without sweep αstall. This was

rationalized from transforming Fig. 4.13 to a free-stream definition, and noting that

the clΛ,max values were somewhat close (0.85 . . . 0.95 up to Λ = 45) and the lift

curves linear to 11 . . . 12. From Fig. 4.15(a), curves were fit through the single

data point marked with a triangle (1/ cos0.75 Λ and 1/ cos1 relations). The first was

postulated as relationship for cl,max(Λ) for turbulent conditions, and the latter for

laminar conditions, both being somewhat consistent with the original assumption.

The theory is then left for further investigation, and based on the sparse dataset in

that study the curve fits were not overly accurate.

Moreover, what is required is some prediction of the airfoil behaviour over the

complete range of α found in a BEM simulation. To that end, new curve fits for

1/ cos1.5 Λ are shown in Fig. 4.15, for both lift coefficient and 2D stall angle. This

relation fits the available data better, even if no better theoretical basis can be

offered. By scaling both cl and α by the same ratio, the linear portion of the lift

curve is left un-altered. This is important, as it is known that performance in this

region is in fact unaltered by spanwise flow. The scaling for spanwise flow is therefore

implemented as given by Eq. (4.4.25):

α(Λ) = α0 + (α− α0)sf (4.4.25a)

cl(Λ) = clsf (4.4.25b)

sf =r

Rtip+(

1− r

Rtip

)cos−n Λ (4.4.25c)

The exponent n is taken as 1.5 based on the experimental data presented in §4.4.9.2.

The effect is tapered based on radius towards zero at the tip.

Using the experimental data for Λ = 0 given in Fig. 4.13 and Eq. (4.4.25),

corrected data up to Λ = 45 is shown as dotted lines in Fig. 4.16. The full set

of experimental data from Λ = 0 is also shown as solid lines. The scaling seems

reasonable, at least up to 30. Beyond this angle, the delay in stall angle prediction

degrades, although cl,max is predicted well. A much more involved model would be


required for a better prediction, but with such a limited dataset would be hard to

synthesize in a generic and computationally simple manner.

0°15°

30°

35°40°

45°

0.0

0.4

0.8

1.2

1.6

2.0

0 5 10 15 20 25 30AOA (deg)

c l

Figure 4.16 Sectional data corrected for spanwise flow

With no direct experiments available, it is unclear whether it is valid to simultane-

ously apply spanwise flow models for both sweep effects and the centrifugal pumping

discussed in §4.4.8. Both produce similar effects, one from a direct velocity compo-

nent and the other owing to a rotating reference frame. Since the underlying mech-

anisms both energize the boundary layer, albeit from different sources, it stands to

reason that there is at least some degree of synergy and additive effect. Further-

more, it is conservative from both an overall force and power limiting capability

perspectives to analyse the effect of considering both simultaneously.

4.4.9.4 Dynamic Spanwise Flow

It is important to remember that helicopter rotors and wind turbine blades expe-

riencing yawed flow will experience dynamically changing sweep Λ and AOA (see

§4.4.14). These dynamically changing flow conditions can be expected to deviate

from the steady-state behaviour just discussed [117]. Based on some experimental

evidence, Leishman [119] concludes that the normal force coefficient cn for a swept

wing in oscillating pitch does not attain a larger maximum, but does produce a

higher average value with the eventual dynamic stall delayed to larger AOAs, com-

pared to an un-swept wing. The net effect is an increase in rotor thrust with sweep

angle.


Leishman [117] concludes that although applying the static corrections just dis-

cussed is predicated on different fundamental mechanisms, the net result is represen-

tative of the experimentally observed behaviour. Fortuitously, purely coned rotors

in a uniform un-yawed flow experience a constant sweep angle and so the assumption

of constant sweep angle is more accurate. Of course, the spanwise velocity will vary

along the span and is thus different than the swept wing case.

4.4.10 Closed Equations

Combining the developments of previous sections, the momentum-derived thrust

coefficient CT,mom (from Eq. (4.4.2)) and the coefficient obtained from blade element

forces, CT,loc may be derived:

CT,mom = 4aFεz [(1− a)− aεr tanβ] (4.4.26a)

CT,loc = σcnVn

V 20 sin2 φ

(4.4.26b)

= σcncos2(β + δ)

[(1− a)2 + (aεr)2

]sin2 φ

(4.4.26c)

The manipulation from Eq. (4.4.26b) to Eq. (4.4.26c) is possible by vector geometry

visible in Fig. 4.4.

Together with the tangential momentum balance Eq. (4.4.4), the final set of equa-

tions is:

anew =c1(1 + c2) + c3

1 + c1(4.4.27a)

c1 =σcn

sin2 φ

H

4Fεzcos2(β + δ) (4.4.27b)

c2 =(aεr)

2

1− a(4.4.27c)

c3 =a2εr tanβ(1− a)

(4.4.27d)

σ =Bc

2πr(4.4.27e)

H =

1 adisc ≤ ac

εz4a(1− a)− 4a2εr tanβ

CT,modeladisc > ac

(4.4.27f)

a′new =σcθ4F

V 2rel

V0 [(1− anew) cosβ − anewεr sinβ] Ωrεθ(4.4.28)

The set of equations is now closed and allows an iterative solution to be obtained

using fixed-point iteration to determine anew = f(a, a′) and a′new = f(a, anew, a′)


from the current estimates a and a′.1 To handle parked and very low induction

cases, a and a′ are blended to zero over local tip speed ratio λ(r) = 1.5 . . . 0.5, after

limiting a and a′ to −1 . . . 1.

The deviations from the standard BEM are in the εz and εr terms calculated

a priori with vortex theory, and the inclusion of proper induced velocities in the

various equations. As mentioned before, the details of swirl as predicted by vortex

theory are ignored in the present work, so εθ is set to 1.

Some authors [16] argue that the cd term should be zero in computing the induc-

tion factors. The rationale is that as only pressure drag contributes to the velocity

deficit in the overall flow, the velocity deficit owing to shear on the airfoil is confined

to the wake sheet. This argument breaks down for the stalled airfoil, and so cd is

generally included in modern codes and ExcelBEM.

The velocity in the tip-plane adisc = εza derives from the vortex results for the

wake geometry discussed thus far, and is assumed valid in the modifications con-

sidered in the following sections. The induction in the tip-plane is used as the

criteria, rather than the induction on the blade, to be consistent with the definition

of the original experimental data. As the CT,model equations no longer intersect

the momentum-derived CT,mom function curve, a smooth blending function over

a = ±0.05 around ac is used to transition between the momentum and experimental

CT models.

It was found in some cases (equivalently high a, CT or λ) that convergence of the

fixed point iteration was problematic. This was found to be a result of Eq. (4.4.14),

which ties radial to axial induction and in these specific cases computed unrealisti-

cally large flow angles δ. The flow angle will never physically reach extremely large

angles, as the vortex wake state will be entered, instead of the ever expanding wake

diameter predicted by a simple non-recirculating mass flux analysis. For this reason

and to aid convergence, δ has therefore been limited to ±60 by modifying εr, a

condition only ever found near the tip in any case.

4.4.11 Expanding Wake

In reality, any induction must expand the flow through the rotor to satisfy continuity.

Figure 4.8 demonstrates that the expansion is non-negligible for the common Betz

limit operating point (a = 1/3) and standard assumed wake geometry. The vortex

1Moves are limited to half that predicted by Eqs. (4.4.27) and (4.4.28) to facilitateconvergence.


wake tube should strictly follow the tip streamline, since free vortices must strictly

convect with the flow. To approximate this condition, alternate induced velocity

equations were derived for a cone shaped vortex sheet (i.e. radius R a linear function

of z in Eqn. Eq. (4.4.12)). The wake could then be approximated as a series of N

cones with the mid-side of each cone (dots in figure) aligned with the local flow,

as shown in Fig. 4.17. The coincident edges of the cones are positioned in a cosine

distribution downstream, to put more points in the high-curvature near-field region,

joining a cylinder wake at 5Rtip. The strength of the sheets γi are constant over

each individual cone.

Figure 4.17 Expanded wake profile

The wake is initialized to have constant radius Ri = Rtip and γi = γcyl. After each

BEM solution, each wake segment is updated sequentially downstream by comput-

ing the induced velocity at the mid-point and modifying the slope of the segment

to align it with the flow. After all segments are updated, εz and εr are recomputed

based on the deformed wake geometry and a new BEM solution is obtained. The

convergence condition was a movement of the wake end-points between wake itera-

tions of less than 0.001Rtip. Convergence was obtained after approximately 10–20

wake iterations for the cases presented in §5.1.

The wake sheet strength γcyl of the far-wake cylinder is determined from mass

continuity and vortex theory. The former is an integration of the mass flow in

Eq. (4.4.3) and the latter the result that at the centre of an infinitely long cylinder,

the induced velocity is uniform across the cross-section and equal to the vortex sheet

strength. Rearranging this relationship yields:

γcyl = V0 −m/ρ

πR2far

(4.4.29)


The wake strength is not strictly constant (as assumed in the classic result wz,2 =

2wz) for either the cylindrical or conical wake geometries, since it must convect with

the flow. Note that γ is an area density of vorticity, varying only in the streamwise

direction due to the ‘bunching up’ of vortex filaments as they slow down. The

vorticity density must therefore increase, so that γ = γ(Vz). Circumferentially, the

vortex lines simply stretch as the local wake radius increases, changing the core size,

but maintaining the same path integral (circulation) around them. The strength of

each wake segment γi is then simply:

γi = γcylVz,far

Vz,i(4.4.30)

The wake length used (2.5 D) is a commonly used value in wind farm wake

models [120], as the point at which expansion is considered complete. Variation

of the length had minimal effect on the results compared in §5.1 for the expanded

(Ex) and unexpanded (Unex) wake geometries. Likewise, the results are shown for

N = 10 wake segments, as increasing N had little effect on accuracy.

4.4.12 Additional Wake Geometries

Two additional wake geometries are developed for comparison. For any real blade,

the loading along the blades varies; therefore, vorticity must be continually shed

along the span. To crudely model this, additional stream-tubes joining points along

the blade from downstream are considered. The circumferential direction of the

vorticity of these sheets is opposite to the one shed at the tip to reflect this direc-

tionality. The first geometry is similar to that in [100], consisting of a single tube

at some fractional radius towards the root. Here the tube is shed at 20% of the tip

radius, approximately where the cuff of the blade profile greatly reduces the lift on

the blade (large change in vorticity). The second wake geometry consists of equal

strength vorticity tubes joining the blade at 10 equally spaced radial stations, with

the cumulative vorticity shed at the tip.

In all cases, a straight cylindrical sheet is assumed. The correction factor εz is

calculated from the vortex system at each radial position. It would be possible to

include a second induction factor iteration loop, to adjust the vortex sheet strengths

according to fed back changes in loading along the blade using the multiple vortex

tube model. This would entail dramatically more computational effort and is not

pursued here.


4.4.13 Aerodynamic Analytic Optimum

An analytic optimum blade shape (chord and twist distributions) can be derived

from the BEM equations. To define the optimum, the airfoil must be selected,

in particular the AOA αclcd,max for maximum lift-to-drag cl/cd ratio. An analytic

optimum is only possible for fixed values of λ and β. For conventional variable-

speed rotors operating at a fixed λ, the optimum is well-defined by the analytic

solution. For fixed-speed and coning rotors, which sweep through λ and β, the

analytic optimum can only serve as an initial blade design. This is also true if the

airfoil varies along the blade, producing non-smooth geometry. Likewise, for a real

blade, structural and other considerations (circular blade root) will require further

optimization to produce a smooth and realistic blade shape. This problem is treated

in §6.4.

To derive an analytic optimum, it is assumed that cd = 0 and F = F (λ, µ) in

Eq. (4.4.6a) for convenience. It will turn out that CT is below the critical point, so

the equations equating momentum and element forces can be non-dimensionalized

as:

BV 2rel

RV 20

ccl cosφ cosβ = [(1− a) cosβ − aεr sinβ] 8πFaεzµ (4.4.31a)

BV 2rel

RV 20

ccl sinφ = [(1− a) cosβ − aεr sinβ] 8πFa′λµ2 (4.4.31b)

for axial and azimuthal momentum respectively, using λ = ΩR/V0 and µ = r/R to

follow Burton et al. [16]. Likewise with Eq. (4.4.6a):

F =2π

cos−1

exp

−B2

1− µµ

√1 +

(3λµ2

)2 (4.4.32)

where it is assumed from later calculations that a = 1/3. Next, the two balance

equations are divided and combined with the geometric formula for tanφ to relate

a and a′:

a′ =((1− a)− aεr tanβ) aεz

λ2µ2(1 + a′)(4.4.33)

To find maximal power output, the torque must be maximized, so the stationary

point of the right-hand-side of Eq. (4.4.31b) is found from:

∂Cθ

∂a= 0 = a′ (− cosβ − εr sinβ) +

∂a′

∂a[(1− a) cosβ − aεr sinβ] (4.4.34)

Using Eq. (4.4.33) with (1 + a′) ∼= 1 in the above yields:

a =13

(1 + εr tanβ) (4.4.35)


which agrees with other analyses for β = 0 [16].

Finally, a “blade parameter” σclλ may be found from the blade element half of

Eq. (4.4.31b):

σclλ =4Fa′λ2µ2√

(λµ (1 + a′))2 + ((1− a) cosβ − aεr sinβ)2(4.4.36)

The blade parameter may be thought of as a non-dimensional chord.

Given a set of operating conditions (λ, β), section location µ, and best cl at

αclcd,max, Eq. (4.4.35), Eq. (4.4.33) and Eq. (4.4.36) define the optimal blade shape

with twist angle from:

γtwist = φ− αclcd,max (4.4.37)

The new set of optimum equations include the correction factors of the revised BEM

theory, indicating a rotor optimized for non-zero β will differ from that optimized

for a planar rotor.

4.4.14 Dynamic BEM

In reality, wind turbines never operate in a uniform inflow. Wind shear, tower

shadow, temporal variation, and skewed flow all alter the conditions at the blades

and in the wake. To account for these effects, five main elements must be considered:

• The induction effects of a skewed wake

• Proper projections of the incident velocities

• The 2D airfoil properties must include dynamic AOA changes

• The transport equations’ method of dealing with yaw

• Some accounting of the time-delay associated with temporal changes in induc-

tion

These are treated sequentially in the following sections, to derive a unified BEM

applicable to a generalized yawed, coned rotor in unsteady flow. A final solution

iteration scheme is outlined last. Note that the following derivation leads to dynamic

equations that do not require azimuthal iteration [16] or assumed induction [100]

for yawed rotors. The current method thus contrasts to previous treatments of the

subject that are not able to compute a general unsteady solution.

4.4.14.1 Skewed Wake Analysis

The wake may be skewed relative to the rotor shaft of the rotor by the aggregate

effects of structural deflection (yaw, tilt, etc.) and the free-stream wind velocity.


The wake vortices may be decomposed as before in Fig. 4.6, in this case assuming

the plane of the ring elements remains parallel to the tip-plane plane of the rotor

as they migrate downstream. Extending the analysis of §4.4.7 to include skew, it is

found that the axial filaments will induce both azimuthal and axial velocities that

vary with skew and cone angles. For β = 0, the root filament is responsible [16],

while for β 6= 0 the bound vortices also contribute. Nevertheless, the axial filaments

are again ignored by setting εθ = 1, assuming again second order effects [100]. As the

blades sweep through a yawed flow, the AOA and hence lift and shed vortex strength

will continually change with azimuth. The assumption adopted here, as before, that

the constant strength tip-wake cylinder dominates, is therefore not strictly valid.

The simplification from varying strength helical sheets to tip cylinder is however

consistent with the previous analysis.

The integral for the skewed wake induction is set up with reference to Fig. 4.18.

The coordinate system is the shaft-aligned x′′–y′′–z′′ axes from Fig. 4.1(d). Point ~p

is the point of interest, located a vector ~r away from the integration point on the

vortex cylinder. The wake is assumed to develop as a skewed elliptical cylinder of

strength γ and constant radius R. The vortex rings lie in planes parallel to the

y′′–z′′ plane with their common centreline at a skew angle χ from the x′′ axis. The

wake axis tracks the point(xw, yw, zw

), related to the tangent of the skew angles

csy and csz projected in the x′′–y′′ and x′′–z′′ planes respectively:

yw = csyxw (4.4.38a)

zw = cszxw (4.4.38b)

The Biot-Savart integral of Eq. (4.4.10) can be evaluated by first defining the

following quantities:

d~l = ~rring × idθ (4.4.39a)

= R(0, cos θ,− sin θ

)dθ (4.4.39b)

~r = ~p−[R(0, sin θ, cos θ

)+(xw, yw, zw

)](4.4.39c)

=(px − xw, py −R sin θ − yw, pz −R cos θ − zw

)(4.4.39d)

=(rx, ry, rz

)(4.4.39e)

d~l × ~r = R(rz cos θ + ry sin θ,−rx sin θ,−rx cos θ

)dθ (4.4.39f)

|~r|3 = (r2x + r2y + r2z + (δcR)2)3/2 (4.4.39g)


Figure 4.18 Skewed wake integral

Equation (4.4.39g) adds the same cut-off model as used previously in Eq. (4.4.12).

The final integral that must be computed for the wake from xw = 0 . . .∞ is:

~u =γ

4π

2π∫0

∞∫0

d~l × ~r|~r|3

dxw (4.4.40)

When this integral is expanded, dxw (as a result of Eq. (4.4.38), yw and zw are a

function of xw) and dθ become the variables of integration. As before, the axial dxw

portion of the integral may be analytically solved, while the dθ part of the integral

is solved numerically.

4.4.14.2 Correction Factors

The induced velocity ~u now has 3 components(ux, uy, uz

). Evaluating at points

f(~p)→ ~u and f(xw = 1000R)→ ~ufar (at same radial distance from skew axis as ~p)

yields the following correction factor relations:1

εz =ux,far

2ux(4.4.41a)

εuy = −uy

ux(4.4.41b)

εuz = −uz

ux(4.4.41c)

1Note that the z and x axes are swapped for the skewed wake formulation relative tothe simple coned formulation in §4.4.5.2, to be consistent with the global CS of §4.2. Thecorrection factor εz definition is kept as before.


Note that the previous result that |~ufar| = −γ is not correct for the skewed wake.

For ~p =(xp,−r cos θ, r sin θ

), the radial and azimuthal velocity components are:

ur = −uy sin θ + uz cos θ (4.4.42a)

uθ = uy cos θ + uz sin θ (4.4.42b)

The previous radial factor εr (see Eq. (4.4.13)) is then:

εr =ur

ux(4.4.43)

A new azimuthal correction factor εθz (owing to the vortex rings, not the axial

filaments as for εθ):

εθz =uθ

ux(4.4.44)

These correction factors are used later, as in §4.4.6, to relate the induction a at

the section to the far-field induction, and to the radially and tangentially induced

velocity components.

To reduce the on-line computational burden of evaluating the integrals at each

time-step, a 4D table of axial, radial and azimuthal induced velocities was computed.

The points are referenced by [χtab, rtab, xtab, θtab], as shown in Fig. 4.19. Utilizing

symmetry, the table need only cover χtab = 0 . . . π/2, rtab = 0 . . . 1 (R = 1), θtab =

−π/2 . . . π/2 and xtab = − tanβmax . . . tanβmax. The angle βmax is set to 60, and

is the maximal cone angle with Rhinge = 0 the table can completely cover (outside

the axial range the edge value is used). The maximal value of χtab is set to 80as a

practical limit, and rtab is bounded by 0.98 to avoid numerical difficulties with the

vortex singularity.1 The points are distributed using cosine spacing, to place more

near the rotor tip axially and near θ ± π/2 in the regions of maximum ~u gradient.

The table is computed with the skew axis in the x′′–y′′ plane.

In order to use the table, the coordinates of the actual points on the blade sections

~ps must be transformed into the table coordinate system. As shown in Fig. 4.20,

the skew axis in general lies in a plane rotated κ = arctan csz/csy around the x′′

(shaft) axis of Fig. 4.1(d). The skew axis is assumed to pass through the shaft

axis a distance xw0 away from the y′′–z′′ plane, corresponding to the tip-plane.2

Using the azimuthal angle θs and local radius rs, the table may be entered with the

1An excessive numerical quadrature tolerance was required to accurately compute ~u pastthis radius.

2The tip-plane for each blade is computed independently, varying with β.


-0.5 0 0.50

0.2

0.4

0.6

0.8

y

z

(a) y–z grid points

-1 0 10

0.5

x

r

(b) x–r grid points

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

x

y

(c) Skew axis (χ) range

Figure 4.19 Grids for ε table

following coordinates:

xtab = s sinβ − xw0 (4.4.45a)

rtab = Rhinge + s cosβ (4.4.45b)

θtab = θs − κ (4.4.45c)

χtab = arctan(√

c2sz + c2sy

)(4.4.45d)

Note that κ is signed, allowing the table to only cover positive χ values. The angle

θtab is bounded to±π by adding/subtracting 2π as appropriate, and finally computed

as:

θtab =

θtab − π/2 ≤ θtab ≤ π/2

π − θtab θtab > π/2

− π − θtab θtab < −π/2

(4.4.46)

Finally, the ε factors are computed from the interpolated induced velocities. The εθz

is the only non-symmetric component, requiring a change of sign for |θtab > π/2|.


Figure 4.20 Skewed wake table transformation

The axial factor εz is practically bounded to ±10. Storing ~u instead of ε was prefer-

able for accurate interpolation. As ux → 0, εz → ∞, making interpolation on a

relatively coarse grid of ε relatively less accurate. The other factors, εχ1 and εχ2,

introduced later in §4.4.14.6, are computed and stored directly as they avoid these

numerical issues.

4.4.14.3 Wind Vectors

The aerodynamic velocity vectors for the fully dynamic set of equations must be

carefully composed from their components, as only a subset are affected by the in-

duction factors. The total velocity seen by the blade section is given by Eq. (4.4.47),

including the induced velocities ~Vind, wind ~Vwind, rotational velocity ~Vrot = rΩ, and

structural velocity ~Vstr owing to the dynamic motion of the section. These vectors

are all in the global earth CS of Fig. 4.1(a).

~Vtot = ~Vind + ~Vwind + ~Vrot + ~Vstr (4.4.47)

The induced velocities are left to §4.4.14.4. The wind velocity given in Eq. (4.4.48)

is composed of the turbulent wind at the section location, ~Vturb(t, x, y, z), modified

by the velocity increments owing to wind shear ∆~Vshear, tower shadow ∆~Vshadow,


and velocity deficits or turbulence from up-stream turbines ~Vupwind turb.

~Vwind = ~Vturb(t, x, y, z) + ∆~Vshear + ∆~Vshadow + ∆~Vupwind turb (4.4.48)

The upwind turbine contribution ∆~Vupwind turb is not explored further in the current

work.

The fully turbulent wind ~Vturb(t, x, y, z) is in general obtained from a probability

function [121]. A fully 3D block of wind vectors is synthesized over the plane of the

rotor, for a region of space extending in the streamwise direction a length equal to the

mean wind speed times total simulation time. This wind field then passes through

the rotor over the run of the simulation, defining interpolated velocity vectors for

any position in the field corresponding to the coned and rotating blade sections

(or tower sections). The presence of the rotor is assumed to have no effect on this

volume of velocities.

The wind shear increment ∆~Vshear is defined according to the standard expo-

nential model of Eqs. (2.1.1) and (4.4.49). Only the components of wind in the

xe–ye plane of Fig. 4.1(a) (i.e. parallel to ground plane) are scaled, according to

∆~Vshear(x, y) = (1− fws)~Vturb(t, x, y) with h = z in Eq. (4.4.49).

V

Vref=(

h

href

)αws

= fws (4.4.49)

The tower shadow model accounts for the retardation (or acceleration) of the

flow due to the presence of the tower. The model smoothly blends a potential flow

model upstream of the tower [122], with a wake deficit model downstream [123–125].

To apply the model, the wind vector ~Vturb,tow(t, xtow, ytow, ztow) at the centreline of

the tower at time t and height z is found. Again, the model is only applied to

the velocity components in a plane parallel to the ground plane. The local angle

θ is therefore found from arctan(Vturb,tow,y/Vturb,tow,x) with the wind magnitude

Vw,tow =√V 2

turb,tow,y + V 2turb,tow,x. The point of interest (i.e.

(x, y)

position of the

blade section) is translated to the (x′–y′) CS with R z(−θ).

Dropping the primes on the transformed coordinate notation, the velocity incre-

ment is calculated from:

∆~Vshadow = Vw,towfv(ui+ vj) (4.4.50)


Figure 4.21 Tower shadow model

The velocity increments u and v are computed separately upwind(uu, vu

):

uu = −R2t (x

2 − y2)r4

+cd2π

Rtx

r2(4.4.51a)

vu = −2R2

txy

r4+cd2π

Rty

r2(4.4.51b)

r =√x2 + y2 (4.4.51c)

and downwind(ud, vd

):

ud =

−∆√x

cos2(πyw

)|y| ≤ w/2

0 |y| > w/2(4.4.52a)

vd = 0 (4.4.52b)

x =x

2lwRt(4.4.52c)

w = 2wwRt

√x (4.4.52d)

d =r

R2t

(4.4.52e)

The upwind equations are the solution for inviscid flow around a cylinder, with a

drag correction term. The downwind equations are empirical, based on a cosine curve

distribution for the velocity deficit. The wake deficit ∆ is non-dimensionalized by

the free-stream. The wake width ww and downstream distance lw define the location

and extent of the specified deficit, both non-dimensionalized by the tower diameter

2Rt. An alternate formulation for ud that only depends on cd (with less investigative


flexibility) has been proposed [122]:

ud =

−cd√d

cos2(

πy

2Rt

√d

)|y| ≤

√d

0 |y| >√d

(4.4.53a)

d =

√r2

R2t

(4.4.53b)

Although not used directly here, this formulation provides an estimate of ww and

∆, based on cd.

The upwind and downwind models are smoothly blended in the x′–y′ plane using

a cosine function:

u =

uu x ≤ 0

fauu + (1− fa)ud 0 < x ≤ Rt

ud x > Rt

(4.4.54a)

v =

vu x ≤ 0

favu 0 < x ≤ Rt

0 x > Rt

(4.4.54b)

fa =12

(cos(πx

Rt

)+ 1)

(4.4.54c)

An example complete flow field is shown in Fig. 4.22, for typical parameters (see

§5.2). The tower cylinder is superimposed on the field. Note that the inclusion

of a drag-dependent term in the inviscid upstream formulation produces a stagna-

tion region upstream of the tower. The influence of the tower on upwind rotors

is typically assumed minimal, while it is recognized as critical to downwind rotors.

Figure 4.22 shows that within Rt of the tower surface, the flow disturbance is of

similar magnitude, indicating its significance to all rotor orientations.

It is also necessary to blend the tower wake model smoothly from the top of the

tower.1 For points above the tower top, the blend can either be azimuthal or vertical;

the latter was chosen and implemented as:

fv =

1 z ≤ ztop

12

(cos(π(z − ztop)lblend

)+ 1)

ztop < z ≤ ztop + lblend

0 z > ztop + lblend

(4.4.55)

The blending length lblend is taken as the diameter of the tower at the tower top.

1It was discovered that BLADEDTM does not do this blending which resulted in severederivatives when performing noise calculations (see §4.5.3).


x’

y’

-3 -2 -1 0 1 2 3 4 5 60

1

2

3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Figure 4.22 Example total velocity tower shadow flow field√

(1 + u)2 + v2

(Rt = 1, cd = 1, ∆ = 0.3, ww = 2.5, lw = 3.25)

4.4.14.4 Wind Vector Decomposition

Having defined the various velocity components, a number of components can be

computed with reference to Fig. 4.23. The first is the out-of-plane velocity Vw,OP =~Vwind ·OP . This vector corresponds to V0 in the steady BEM formulation in Fig. 4.2.

It is effectively the shaft-aligned free-stream velocity, normal to the unconed plane

of the rotor.

Figure 4.23 Dynamic wind vector decomposition

The normal, tangential and spanwise sectional velocities(Vw,n, Vw,t, Vw,s

)are com-

puted as the dot product of ~Vwind with n, t and s respectively. These are the wind

velocities that the section sees without any induction. The computation of the


structural velocity term ~Vstr is defined later in §4.6.3 and decomposed to normal

and tangential components Vstr,n and Vstr,t respectively.

The total velocity that the 2D section sees, ~Vrel is defined with reference to

Fig. 4.24 and Eq. (4.4.56). The aerodynamic force vectors are the same as in Fig. 4.4,

however the axes and velocity vectors are modified. The induced components Vind,n

and Vind,t shown in Fig. 4.24 are components of the axial, radial and tangential

induction velocities. These represent the proper induced velocities, acting on the

out-of-plane VOP and in-plane rΩ component velocities respectively. The distinction

between out-of-plane/in-plane and normal/tangential velocities, and which veloci-

ties are “induced” constitutes one source of discrepancy between analysis codes.1

Vn = Vw,n − aVw,OP (cosβ + εr sinβ) + Vstr,n (4.4.56a)

Vθ = Vw,t + rΩ(1 + a′

)+ aVw,OP εθz + Vstr,t (4.4.56b)

Vs = Vw,s − aVw,OP (sinβ − εr cosβ) + Vstr,s (4.4.56c)

Figure 4.24 Dynamic flow relative to airfoil section (velocity vectors scaled forclarity, not indicative of relative magnitudes)

The final element is the computation of χ, the generalized skew angle. The~Vturb,tt(t, x, y, z) vector used for computing the wake skew vector is determined at

1BLADEDTM computes induction based on total section velocity, whereas NREL’s Aero-Dyn is similar to the present derivation acting only on the wind vectors.


the centreline of the tower-top, with wind shear but without tower shadow. It is

assumed that the wake convects downstream in-line with this vector. In reality, the

wake will self-induct and tend to increase the skew angle [16], especially near the

rotor. Thought of another way, the thrust force (directed normal to the rotor) on

the flow is directed upwind and in the direction of aerodynamic yaw γyaw, thereby

increasing the skew angle.

Burton et al. [16] provides a formula approximating the “proper” skew angle χcorr

derived from vortex theory:

χcorr = (0.6a+ 1)χyaw (4.4.57)

where χyaw is the geometric skew angle without induction. This formula assumes

that the wake convects far downstream along the velocity vector at the centre of

the disk, which is also not strictly true. It also implies a dependency on a requiring

further solution iteration. For these reasons, no correction for wake self-induction is

made to χ in the current formulation.

4.4.14.5 Unsteady Sectional Aerodynamics

There are two basic approaches to modelling the 2D aerodynamics of an airfoil in

dynamic situations. The simplest method is to adopt the quasi-steady model, which

assumes that the airfoil behaves according to steady 2D lift curves. More advanced

theories recognize that changes in lift and drag do not develop instantaneously.

There is therefore some lag introduced in the tracking of the steady AOA-cl/cd/cmcurves. Two DOF are important: pitching and heaving. Both dynamically alter the

angle of attack and the latter the incident velocity as well.

In order for the aerodynamic force system to change, the flow around the airfoil

must change over a finite period of time. The effects are separated into lift-dependent

(circulatory) and added-mass (non-circulatory) effects. The former is further differ-

entiated into an inner and outer problem, owing to vorticity on the airfoil and in

the wake respectively [126].

Outside of the linear lift regime, dynamic stall behaviour has been observed ex-

perimentally. Governed by the rapidity of AOA variation, the airfoil can achieve

much higher lift coefficients (approaching the linear 2πα) well past AOAs for which

stall would have occurred in a steady flow. This is possible because the boundary

layer only changes relatively slowly, so that separation (leading to stall) takes time

to develop. Eventually the airfoil does progress to full-stall, and does so in a much


more violent manner (and over a smaller AOA range) than in the steady case. Once

the airfoil has fully stalled, a hysteresis effect is observed, wherein the fully stalled

condition is maintained as the AOA is reduced. The dynamic airfoil behaviour only

returns to the steady characteristic once the airfoil is well back into the linear regime.

The nature of the stall (from LE or TE) is largely a function of thickness.

A number of models have been proposed to account for the dynamic stall phe-

nomenon [89, 126, 127]. The model adopted here was chosen for its relative sim-

plicity and for comparison with Computational Fluid Dynamics (CFD) results [89]

in §5.1. Accurate stall prediction is difficult even for much more advanced models

[126], which also incorporate better dynamic effects below stall.

The present method, presented by Mikkelsen [89], only considers changes in cl,

and essentially maps the airfoil between two curves as the AOA α changes. The

upper curve is the linear one, defined by:

cl,lin = 2π(α− α0) (4.4.58)

and the lower curve is the fully-separated curve:

cl,sep = Gcl,st (4.4.59a)

G = (1−A0) +A0

2[1 + tanh (B0 (|α− α0| − α1))] (4.4.59b)

where the suggested constant values are A0 = 0.3 . . . 0.5, B0 = 5 . . . 15 and α1 is the

first stall angle of the steady lift curve data defined by cl,st(α). The zero lift angle

of the airfoil is α0. The tanh function smoothly blends the assumed fully-separated

cl,sep curve1 to the cl,st curve (G→ 1 away from α = α1). The three separate curves

are given in Fig. 4.25 for an example airfoil, with the parameters A0 and B0 varied

for cl,sep. The mid values for the fit constants (0.3 and 10) are used in the remainder

of the current work.

The movement between the curves is determined by a fractional function fst(α):

fst(α) =cl,st(α)− cl,sep(α)cl,lin(α)− cl,sep(α)

(4.4.60)

Clearly, this equation defines the steady cl,st curve for fst = 0.5. For each time step

i, the value of fst(α) is computed with Eq. (4.4.60). A simple lag equation is then

used to compute the fraction f ist to use for the current step:

f ist = f i−1

st +[fst(α)− f i−1

st

] ∆tτdyn stall

(4.4.61)

1This curve is a rough approximation, with approximately half the cl,st curve slope nearα0 and joining the cl,st curve around 30.


-40 -30 -20 -10 0 10 20 30 40

-0.5

0

0.5

1

1.5

α (deg)

cl c

l,lin

cl,st

cl,sep,mid

cl,sep,low

cl,sep,high

Figure 4.25 Dynamic stall parameters (low to high values for A0 and B0 are[0.3, 0.4, 0.5] and [5, 10, 15] respectively)

where ∆t is the time since the last computation, and τdyn stall the lag time step.

The suggested value of 4c/Vrel is used, changing locally with chord c but using a

fixed average Vrel of 30 m/s.1 Finally, the current lift coefficient is computed by

interpolation:

cl(α) = f istcl,lin(α) + (1− f i

st)cl,sep(α) (4.4.62)

To test the implementation, an example hysteresis loop was run, is shown in

Fig. 4.26. The model displays a smooth behaviour and predicts a lift increment into

stall, as desired.

4.4.14.6 Governing Equations

Implicit in the following discussion is the continued validity of the stream tube

independence assumption and to varying degrees azimuthal averaged effects. For the

yawed rotor, these assumptions are even less rigorous than the steady, un-yawed case.

They are first adopted pragmatically and the results compared to other predictions

and measurement in §5.1 to ascertain the valdity of the assumptions.

There are three possible departure points for deriving the momentum equations,

based again on considering averaged conditions over an idealized uniform disc [16].

The first, termed Axial Momentum Theory (AMT), assumes the thrust (pressure)

force of the disc creates average induction a at the disc and 2a in the far-field, both

1It was found that varying Vrel with operating condition led to numerical difficulties.


4 6 8 10 12 14 16 18 20 22

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

α (deg)

cl

cl,lin

cl,st

cl,sep

cl

Figure 4.26 Dynamic stall hysteresis loop for cyclic α (α = [5 . . . 20] at 1 Hz)

perpendicular to the disc. The axial momentum equation is formulated in the same

perpendicular direction, ignoring unbalanced pressure forces that also contribute to

the wake skew.

Glauert proposed the second theory, by comparing the autogiro downwash system

to that of a highly yawed wind turbine rotor. The key difference to AMT is that the

transport velocity through the disc area is the total velocity at the disc, not that

perpendicular to the disc. This allows a component of the thrust force to create an

effective lift force on the rotor, analogous to a wing or autogiro rotor at an AOA.

Burton et al. [16] proposed a third model based on averaged vortex theory. The

induced velocities at rotor centre and downstream are used in the Bernoulli equation

to derive the thrust equation for the disc.

It is observed that AMT will predict power most accurately (lowest CP (Λ)), and

Glauert the thrust (highest CP (Λ)) [16]. Notice that Glauert attributes some lift

force to thrust, and as this is perpendicular to the velocity, cannot extract work

from the flow. The vortex theory produces results intermediate to these theories for

CP . BLADEDTM and most other codes uses AMT, and so this theory is employed

here with those caveats.


The axial (shaft-aligned, x′′–y′′–z′′ CS of Fig. 4.1) momentum equation is derived

as before in Eq. (4.4.2), with V0 replaced by Vw,OP :

∆(momentum) = −(V2 − Vw,OP

)∆m (4.4.63a)

= (2FaεzVw,OP )∆m (4.4.63b)

∆m = ρ

∆A∫0

~V1 · d ~A (4.4.63c)

The definition of d ~A varies with the AMT or Glauert assumptions:

d ~AAMT = dA(cosβi+ sinβ

(sin θj − cos θk

))(4.4.64a)

d ~AGlauert = dA~V1∣∣∣~V1

∣∣∣ (4.4.64b)

The transport velocity at the section ~V1 includes the section wind velocity ~Vwind =(Vw,x, Vw,y, Vw,z

)and the induced velocities. Adopting AMT, the mass-flow term

becomes, with dA = dsrdθ and θ from Fig. 4.1(d):

∆m = ρVw,OP

∆s∫0

2π∫0

cosβ (1− a) +

sinβ((

Vw,y

Vw,OP+ εuya

)sin θ −

(Vw,z

Vw,OP+ εuza

)cos θ

)rdsdθ (4.4.65)

The tilde over a and ε are used to emphasize that these quantities vary during

integration with θ. Since the induction in the wake afar = 2FaεzVw,OP is constant

over θ, the local a and a may be related as εza = εza. Now εz = f(θ), whereas

(εz, a) 6= f(θ) in the velocity change term of Eq. (4.4.63a). For a temporally and

spatially uniform inflow (assumed approximately valid in dynamic flow as well),

Vw,y and Vw,z will also be invariant in θ. Performing the integrals while noting the

components that evaluate to zero over θ = 0 . . . 2π yields:

∆m = ρVw,OP r∆s2π [cosβ (1− aεzεχ1)− aεχ2 sinβ] (4.4.66)

where

εχ1 =12π

2π∫0

1εzdθ (4.4.67a)

εχ2 =12π

2π∫0

(εuz

εzcos θ − εuy

εzsin θ

)dθ =

12π

2π∫0

εrεzdθ (4.4.67b)


These factors are evaluated with trapezoidal integration from the table quantities

stored in §4.4.14.2. For χ = 0, these equations reduce to the un-yawed ones in

§4.4.10.

As before in §4.4.3.4, empirical CT,model values are used for adisc = aεz > ac to

introduce anH factor into the momentum equation for CT,mom, non-dimensionalized

by 1/2ρV 2w,OP 2πr∆s cosβ. The blade element CT,loc equation remains unchanged

as:

CT,loc =Bc

2πrcz

cosβV 2

rel

V 2w,OP

= σcnV 2

rel

V 2w,OP

(4.4.68)

It must be assumed that CT,loc applies to the whole annulus, even though it will

vary with θ as the conditions at the blade change (i.e. ~Vrel). A steady code could

iterate on this to achieve uniform aεz over θ by using the averaged element forces,

however a dynamic code does not have this luxury.

Temporally varying inflow can be handled in five ways [16, 89, 125, 128, 129], in

increasing order of complexity:

Frozen wake This method assumes that the wake is frozen after the first

time step. The induced velocities aVw,OP and a′rΩ do not change there-

after.

Equilibrium wake This method assumes that the wake changes instanta-

neously with loading changes. The solution is computed anew from the

steady equations, at every time step.

Dynamic inflow The most realistic assumption is that the wake will evolve

with some time-lag relative to loading changes.

Vortex filaments The wake is evolved as time-dependent vortex strengths

and paths.

CFD The full-field simulation of the Euler or NS equations simply include

temporal effects when implemented in their unsteady forms.

Notice that it is the induced velocities aVw,OP , not induction factors a, that change.

The dynamic inflow is preferred here, for comparison to BLADEDTM. The solu-

tions may either be formulated generally by including multiple assumed functions

for the pressure distribution on the disc, or in a reduced form containing terms

for aggregate thrust and moment. The former uses acceleration potential methods

based on asymptotic solutions of the Euler equations to model the flow. Wake rota-

tion is not explicitly included in the theory, modifications are required for the high a

levels of wind turbines, and coning cannot be accounted for [122, 130]. BLADEDTM

adopts the latter, most primary and simple assumption, that only thrust is affected.


Essentially, the forces applied by the blades on the flow must not only balance the

steady momentum change of the flow, but also apply a force to temporally change

the induction velocity of the flow. This is handled by an additional term in the

momentum equation:

CT = CT,mom + CT,dyn (4.4.69)

where

CT,dyn =mA

d(wz,far/2)dt

1/2ρV 2w,OPdA

(4.4.70)

The far-field induction term wz,far/2 = aVw,OP εz is used to emphasize that it is the

far wake induction that changes. BLADEDTM and standard theories can only use

the value at the blade aVw,OP , as no information is available about the change in

induction from relative wake position with θ. Neglecting this subtlety would tend

to over-predict the dynamic inflow changes.

From potential theory, the appropriate added mass term mA for a disc of radius

R is 8/3ρR3 [16, 125]. Therefore, for an annulus from r1 to r2:

CT,dyn = kd (aVw,OP εz)

dt(4.4.71a)

k =163π

r32 − r31r22 − r21

(4.4.71b)

BLADEDTM simply sets k = Rtip.1 Each blade is computed independently (i.e.

different β, a solutions, etc.) assuming the equations valid for the entire annulus.

To initialize the dynamic inflow equations, and for the frozen and equilibrium

wake cases, the axial momentum equation is rearranged for fixed-point iteration as:

anew = f(a) =c1

1− ac2(4.4.72a)

c1 = σcnV 2

rel

V 2w,OP

H

4Fεz+ a2εχ2εz tanβ (4.4.72b)

c2 = εzεχ1 (4.4.72c)

The geometric relations used to derive Eq. (4.4.26c) are no longer applicable, how-

ever this turns out to be fortuitous. In fixed-point iteration, it the iterate equation

with minimal f ′(a) that is the fastest [131]. The choice of iterate equation is con-

strained by the condition that |f ′(a)| < 1 to converge. Equation (4.4.72) violates

1The difference was found to be minimal, and Eq. (4.4.71b) is used here.


this near a = 1/c2 where f(a) → ∞. Solutions above this limit are not viable in

any case, so the updates are limited as:

anew =

1/c2 − ξ c2a− 1 < ξ or anew > 1/c2 − ξ

anew anew ≤ 1/c2 − ξ(4.4.73)

The offset ξ is set to 0.01 to avoid the discontinuity. It was evident from the decreased

solution iterations required that f ′(a) for this formulation is smaller than that of

Eq. (4.4.27).

The tangential equation uses the dynamic mass-flow term of Eq. (4.4.66) in

Eq. (4.4.4) to yield:

a′new = σcθ4F

V 2rel

εθrΩVw,OP (cosβ (1− aεzεχ1)− aεχ2εz sinβ)(4.4.74)

Including the added mass term in CT for the dynamic inflow model, the axial

iterate equation is:

d (aVw,OP )dt

=1Kεz

(σcnV

2rel

H

F− 4εzV 2

w,OP

(a (1− ac2)− a2εzεχ2 tanβ

))(4.4.75)

A simple Euler scheme is used to advance the solution and obtain the fixed-point

iteration equation for the dynamic wake:

anew =[(aVw,OP )old,1 +

d (aVw,OP )dt

(t− told,1)]V −1

w,OP (4.4.76)

The estimate of anew is bounded as in Eq. (4.4.73). New solutions are only com-

puted when t− told,1 exceeds a set “dynamic wake time step”, typically 0.02 s for a

large machine [125]. The new value of aVw,OP is then stored as (aVw,OP )old,1, after

updating the previous value of (aVw,OP )old,2 ← (aVw,OP )old,1. The induction factor

is finally linearly interpolated based on t between the values at told,1 and told,2, so

that the wake evolves smoothly over the next wake time step, thereby introducing

aerodynamic lag. The tangential update equation is unchanged from Eq. (4.4.74),

but is lagged from its dependency on Eq. (4.4.75).

4.4.14.7 Computational Procedure

For clarity, the overall ordering of computations in the general case is given in the

following procedural list. All steps are carried out for each section on each blade in

turn, with coupling to the structural model of §4.6.3.

1. Structural positions and velocities are obtained either deterministically, or

computed as part of the time-marching solution of structural equations (see

§4.6.3).


2. The wind vectors, as described in §4.4.14.3 and §4.4.14.4 are computed based

on the position and orientation of the blade sections.3. Induction factors a and a′ are initialized according to the unsteady induction

model:

Initial Timestep/New Dynamic Solution a = 0.3 and a′ = 0

Frozen Previous values

Equilibrium Default values

Dynamic If (t − told,1) ≥ τdyn, a new dynamic solution is computed.

If (t− told,1) < τdyn, the induction values are interpolated:

fdyn =t− told,1

told,1 − told,2(4.4.77a)

a =(aV0)old,2 + fdyn((aV0)old,1 − (aV0)old,2)

Vw,OP(4.4.77b)

a′ =(a′rΩ)old,2 + fdyn((a′rΩ)old,1 − (a′rΩ)old,2)

rΩ(4.4.77c)

The induced velocities (aVw,OP )old,1 and (a′rΩ)old,1 are the most re-

cent estimates computed at time told,1. (aVw,OP )old,2 and (a′rΩ)old,2

are the previous estimates computed at time told,2. The interpolation

function fdyn therefore allows the new induction to fully develop over

the time span τdyn, introducing the aerodynamic lag.4. The full dynamic flow velocities are computed from the relations in §4.4.14.4

and the induction factors.5. The aerodynamic forces are computed, including the effects of stall delay (see

§4.4.8 and 4.4.9) and dynamic stall (see §4.4.14.5). The tip-loss factor F and

turbulent wake H parameters are also computed.6. Induction factors are updated if required. This is done in all cases for the

first time-step and for the equilibrium wake model using Eq. (4.4.72). The

dynamic wake model uses Eq. (4.4.75) for all subsequent time-steps when

new inductions factors must be computed ((t − told,1) ≥ τdyn). Steps 4–6 are

repeated to convergence of the induction factors.7. The various aerodynamic values are recorded and loads computed (see §4.6.2).

The dynamic stall fractions fst are updated, as are the dynamic wake induced

velocities (aVw,OP )old,i and (a′rΩ)old,i.

4.5 Acoustic Modelling

The noise created by wind turbines can be separated into two main areas: aerody-

namic and structural. Structural noise can arise from the following sources [132]:

4.5 Acoustic Modelling 97

• Transmission (gear meshing, bearings)

• Coolant systems

• Electrical components

Structural noise on modern wind turbines has largely been eliminated by careful

design of components, acoustic treatment of the nacelle, facilitated by relatively well

understood physical mechanisms. The focus of the noise analysis presented currently

is therefore on aerodynamically generated noise. Section 4.5.1 first introduces the

metrics and mechanisms of noise. Next, §4.5.2 overviews the modelling approaches

and mathematical background of noise prediction. Section 4.5.3 then introduces

LFN, the primary mechanism of concern for the coning rotor. Fortuitously, theory is

available in the literature to model LFN from first principles, presented in §4.5.3.2.

The nuances of building the theory into an accurate prediction code are however not

present in the literature in a unified manner, and so are discussed last in §4.5.3.4

and 4.5.4.

4.5.1 Aerodynamic Noise

At a fundamental level, aerodynamic sounds (noise if an undesired sound) are acous-

tic waves travelling through the air. Physically, sound is equivalently a pressure or

density (condensations and rarefactions) wave travelling at a characteristic speed

c0, depending on the ambient fluid temperature and composition (343 m/s at 20 C

and 1 atm for air). Humans experience sound as pressure fluctuations acting on

the ear drum. The frequency of the wave is experienced as tone, and the amplitude

as volume level. Since the ear is a dynamic mechanical system, it has a defined

frequency/amplitude response that influences the perception of sound.

4.5.1.1 Intensity

To account for the human response to noise, Sound Pressure Level (SPL) Lp is

measured on a logarithmic scale:

Lp = 10 log10

p2rms

p2ref

= 20 log10

prms

pref(4.5.1)

where pref is a standardized pressure of 20 µPa corresponding to the limits of human

hearing. The units of Lp are decibels (dB). The sound power level LP and intensity


LI may also be defined:

LP = 10 log10

P

Pref(4.5.2a)

LI = 10 log10

I

Iref(4.5.2b)

where Pref is 10−12 W and Iref is 10−12 Wm−2. The power is a constant function

of the acoustic source, while the pressure and intensity both decrease with distance

r from the source. In free-space, I is proportional to p2rms; LI and Lp will decay as

r2 as the sound is spread over a spherical surface centred at the source.

4.5.1.2 Frequency Spectrum

At the most basic level of characterization, SPL may be computed by sampling the

entire frequency spectrum. To account for the frequency response of the human ear,

three weighted sounds levels (A, B, C) are used. Each defines a filter function over

the range of human hearing (20 Hz–20 kHz) to attenuate sound at the high and

low ends of this range. Commonly referred to as dBA sound levels, this denotes a

sampling with the A filter suitable for most situations. The B and C weightings are

more appropriate for low-frequency sounds [132].

Two methods are used to resolve specific frequency content of sound, octave and

Fast Fourier Transform (FFT) analysis. Octave analysis divides the frequency spec-

trum into bins defined by:

fc,i+1

fc,i= 21/n (4.5.3a)

fl,i = 2−(1/2n)fc,i (4.5.3b)

fu,i = 2(1/2n)fc,i (4.5.3c)

where fc, fl and fu are centre, lower and upper frequencies for each bin i. By

international standard, 125 Hz is a centre frequency (the others may be derived

from the above equations). Most analysis is done for n equal to 1 or 3, termed 1/1

and 1/3 octaves respectively. In practice, sound meters use multiple analogue filters

tuned to these frequency ranges to sample the sound and obtain Lp for each bin.

To obtain finer frequency resolution, FFT analysis can be performed on data

discretely sampled from the continuous pressure signal (see §4.5.4). It is possible to

convert from FFT results to octave bins Lp,bin by summing the FFT Lp,j in each bin:

Lp,bin = 10 log10

b∑j=a

10Lp,j/10 (4.5.4)


where a and b are the lower and upper indices of the FFT results in that bin. Note

that is it the sound intensity I that must be summed, not the Lp values. The above

equation is only valid for incoherent (different frequency and phase) noise sources.

4.5.1.3 Noise Mechanisms

The minimisation of aerodynamic noise from wind turbines remains a challenge,

but by careful design can be minimised. The first step is to identify the individual

aerodynamic noise mechanisms [133]:

TE - Turbulent Boundary Layer The turbulent boundary flowing over

the trailing edge of an airfoil will radiate noise from both suction and

pressure surface TEs.

Separated Flow Progressive stall from the TE increases the boundary layer

thickness and size of coherent eddies passing over the TE. In full stall, noise

is radiated from the entire suction surface as as result of these large eddies

and can dominate the TE source.

TE - Laminar Boundary Layer This tonal noise is created by amplifica-

tion of Tollmien-Schlicting waves via vortices shed at the TE, into coherent

vortex shedding. The Reynolds number must be small enough for laminar

flow to develop and so this source is more important for small machines.

Blunt TE Depending on geometry, especially for thick and blunt TEs, co-

herent vortices are shed creating tonal noise. The phenomenon is analogous

to the von Karman vortex street behind bluff bodies.

Turbulent Inflow The eddies present in the turbulent inflow incident on the

rotor interact with the LE of the airfoils to radiate noise. The magnitude

of the noise is larger when the ratio of turbulence length scale to LE radius

is large, the former ranging in size from 1 mm to 100 m.

Tip Vortex Unlike all of the other noise sources which are predominantly

2D phenomena, this origins of this source are 3D. The interaction be-

tween the shed tip-vortex and the blade surfaces create an important high-

frequency noise source. The strength of the vortex is determined by the

shape of the spanwise load distribution and can be mitigated from this

perspective. Also, it has been found that parabolic tips, and others with

cut-away TEs reduce tip noise. The former tailors the spanwise loading,

while the latter also moves the tip surfaces upwind of the tip-vortex spiral.


Low-Frequency Noise (LFN) Described further in §4.5.3, this source is

typically at or below the threshold of human hearing, but nevertheless is

important as it is “felt”. LFN is created by the blades sweeping out a

volume of air and also fluctuating sectional aerodynamic forces acting on

the fluid and moving relative to the observer. The primary source of LFN is

from the blade passing through the tower wake creating rapidly fluctuating

forces.

The noise heard by an observer will be modified by two factors, directionality

and propagation. Directionality modifies the sound perceived by the observer owing

to the relative position (directivity of the sound) and motion of the sound source,

the time-lag due to propagation from spatially distributed sources, and convection

of the noise with the free-stream. The propagation of the sound from source to

observer will also introduce artefacts owing to attenuation, reflection off terrain and

buildings, and the non-uniform properties of the atmosphere.

All but the last of these noise sources can be alleviated by clean airfoils, sharp

trailing edges and tip-shape selection. Operationally, stalled rotors and those with

high tip-speeds produce relatively more noise. The latter is a design choice, typically

limited to 70 m/s on-shore. Off-shore however, where noise is less of a constraint,

tip-speeds may increase substantially in the future, leading to cost-reduction through

reduction of torque/forces and hence material usage. LFN will be discussed further

in §4.5.3.

4.5.2 Modelling Approaches

Three classes of model may be used to model aerodynamic noise [132]. The first

Class I models are empirically based on measurements from existing machines. They

typically are parametrised in terms of generic quantities such as rotor diameter, rated

power and tip speed to give an overall dBA sound level. They can give an overall

indication of noise production, but not detailed spectral information or accurate

quantitative predictions.

Class II models are the current state-of-the-art, such as those in the NREL FAST

code [134], and rely on more detailed information about the machine. They consider

the noise mechanisms separately to synthesize the overall acoustic signature. These

models are based on fundamental theory, either in terms of deterministic forces or

scaling laws from non-dimensional 2D airfoil analyses. They are also supported and

validated by experimental work [135]. Both Class II and III models are capable of


providing time-domain predictions of acoustic pressure, and hence frequency content

and overall noise levels.

Class III models rely on detailed description of the geometry within the con-

text of a full-field time-domain simulation. These approaches typically solve the

NS equations in either compressible or incompressible form (see [136] for a dis-

cussion of the relative merits). The predictions must be accurate for separated

flow, so Reynolds-Averaged Navier-Stokes (RANS) (steady solution) and Unsteady

Reynolds-Averaged Navier-Stokes (URANS) (excessive numerical dissipation) are in-

adequate; Large Eddy Simulation (LES), Detached Eddy Simulation (DES) or Direct

Numerical Simulation (DNS) simulations are required. Clearly the underlying NS

solution is quite expensive [137], and due to cost is not used to any great extent

in practice. This can be reduced somewhat by employing the acoustic analogy dis-

cussed next.

4.5.2.1 Acoustic Analogies

Mathematically, acoustic sound is governed by the NS equations. Together with the

continuity equation, these equations fully describe the generation and prorogation

of acoustic signals, i.e. aerodynamic noise. Unfortunately, the full set of equations

requires fine spatial and temporal resolution for accurate predictions. This precludes

the use of fine grids extending from a wind turbine to observer. To overcome this

limitation, Lighthill developed his “acoustic analogy” [138]. Without delving into

the mathematical details, it is essentially a re-arrangement and combination of the

continuity and momentum equation resulting in a single compact equation:

∂2ρ

∂t2− c20∇2ρ =

∂2Tij

∂xi∂xj(4.5.5)

where Tij , the Lighthill stress tensor is:

Tij = ρuiuj + (p− ρc20)δij + σij (4.5.6)

where δij is the Kronecker delta function, xi and ui are position and velocity com-

ponents, and σij are the viscous stresses (usually ignored as second order). This

equation is a hyperbolic partial differential describing a wave travelling at speed

c0 in a quiescent medium (left-half), forced by source terms (right-half). This in-

homogeneous wave equation (acoustics) is the “analogy“ to the full fluid-dynamics

equations.

Equation (4.5.5) can be solved for the small disturbance density ρ(~x, t)′ = ρ(~x, t)−ρ0, and hence pressure via the isentropic relation p′ = a2

0ρ′ at the observer position


~x. The final solution involves an integral of the stress tensor through a volume

bounded by a fixed surface enclosing the noise generating regions. It also includes

an integral of pressure over the same bounding surface.

Using this mathematical result, the process of computing noise is separated into

two distinct tasks. First, a detailed computation is made of the flow conditions

within a volume bounding the important noise generating mechanisms. Using these

results, the far-field noise signature is then obtained by integrating the detailed

solution, as just described, at each time-step. The flow-field conditions are therefore

know a priori for the second step.

Lighthill originally developed the theory for jet noise, using a stationary bound-

ary. Ffowcs Williams-Hawkings extended the analogy to include moving boundary

surfaces, introducing two additional terms into the solution [136]:

∇2p′(~x, t) =∂

∂t[ρ0vn + ρ(un − vn)] δ(f)−

∂

∂xi[∆Pijnj + ρui(un − vn)] δ(f)+

∂2

∂xi∂xj[TijH(f)] (4.5.7)

where un and vn are the fluid and boundary surface velocities normal to the boundary

surface f = 0, δ(f) is the Dirac delta function, and ∆Pij is the compressive stress

tensor jump across the surface. The addition of a moving boundary allows the

bounding surface to be a real (on the blade surface) or an imaginary surface located

in the flow field but conveniently co-rotating with the blades.

4.5.2.2 Mathematical and Physical Noise Sources

The three source terms in Eq. (4.5.7) are mathematically monopole, dipole and

quadrupole elementary acoustic sources [132]. The abstract and physical nature of

each source, and examples for each (in brackets) are as follows:

Monopole A point source with radial symmetry (radially pulsating sphere

surface); fluctuating sources of mass (siren); moving volumes (propeller

blades)

Dipole Fluctuating and/or translating forces exerted on fluid (oscillating

sphere); varying lift (airfoil angle of attack change); moving lift force (ro-

tating blade)

Quadrupole A pair of dipoles (small deforming sphere); turbulent flows,

fluctuating Reynolds stresses (eddies from stalled flow over wing)


4.5.2.3 Advanced Analysis

Although the Ffowcs Williams-Hawkings (FW-H) equations are conveniently com-

pact, they assume sound radiation into an unbounded domain. Wind turbines are

typically placed in a complex environment. To perform a detailed simulation of

atmospheric absorption and terrain effects, the linearised Euler equations may be

used with clever computation of the Green’s function [136]. However, this level of

complexity is beyond the scope of the current work.

4.5.3 Low-Frequency Noise

This section presents the relevance (§4.5.3.1), theory (§4.5.3.2 and 4.5.3.3), and

implementation nuances (§4.5.3.4) for analysing the LFN of the coning rotor.

4.5.3.1 Relevance

A noise study was commissioned during the course of the original work on the coning

rotor concept [139]. That study only included two noise mechanisms, turbulent

inflow and TE - turbulent boundary layer interaction. Mention was also made of

using cut-away TEs tips to limit tip-noise and sharp trailing edges (< 3 mm) to

avoid vortex shedding noise. The overall findings were that the coning rotor concept

did not create noise levels much in excess of a conventional machine. The modest

increase that was observed, derived mainly from the increased TE length of the

relatively longer blades. The variable-speed coning rotor presumably also better

matched noise production to background levels compared to the constant speed

reference machine.

Missing from this earlier study was a treatment of LFN. While this noise source

does not have a large component of A-weighted noise levels, it is very important

for public acceptance. This is based on practical experience in both the US with

the MOD-1 [15, 140] and in Sweden with the WECS Maglarp machine [141]. Both

were large (vØ80 m rotors) prototype downwind machines with 2 bladed rotors.

The MOD-1 had a heavy-weight truss-type tower with 4 major structural elements

creating a downstream wake with 2–4 distinct velocity deficits (depending on yaw)

and a broad wake. The rotor was also completely rigid. The WECS machine had a

more conventional cylindrical tower. Both experimental efforts generated complaints

from neighbours due to the characteristic “tower thump” (LFN described in §4.5.1)

heard (and felt) at the blade passing frequency.


Clearly then, LFN is an important aspect of the coning rotor design, as the rotor

must be downwind of the tower. Owing to the fact that LFN can travel long distances

before being attenuated, it will be important for both on-shore and near to medium-

offshore concepts. LFN is not necessarily a fundamental constraint on the concept,

but must be carefully considered if the coning rotor is to be publicly acceptable.

Some comfort may be taken from other downwind concepts that have been built

and did not suffer unduly from LFN: Hutter’s machines of the 1950/60s; the MS-4

machine in the UK; and the Wind Turbine Company’s prototypes.

4.5.3.2 Theory

Utilizing a number of simplifying assumptions, LFN can be treated relatively easily

using a development of Eq. (4.5.7). This development is outlined by Wagner et al.

[132, p. 98], citing detailed work by Succi and Farassat and is described here. For

wind turbines operating at low Mach numbers, the driving mechanisms of LFN are

monopole and dipole sources (see §4.5.2.2). The monopole source is created by the

moving volume of the blades as they move through the air. This can be visualized

as the blade temporarily displacing a volume of air in a given region as the blade

sweeps through it, and it termed the thickness noise. The dipole source can be

divided into two distinct mechanisms. Both are created by the aerodynamic forces

(lift and drag) developed by the blade, vectorially opposite to those experienced by

the blade, as they are exerted on the fluid. The first source is created by a constant

magnitude force moving relative to the observer and is termed near-field noise. The

second source is developed by a temporally varying force vector (magnitude and

direction) and termed far-field noise.

The solutions to Eq. (4.5.7) typically contain inconvenient spatial derivatives that

may be re-cast as time derivatives. They also contain integrals over a bounding

volume and surface. The volume integral may be ignored, as no quadrupole sources

(turbulence) are important for LFN. The surface integrals may be conveniently con-

sidered as acoustically compact for frequencies below 50 Hz. Using this assumption,

the monopole and dipole sources may be evaluated as discrete sources on the blade,

rather than requiring integration over the actual blade surface. This approach is

commensurate with the BEM method (see §4.4), which provides position and load-

ing output for sections along the blade. It is assumed here that the forces are located

on the pitch axes of the blades.


The final formulas governing the time dependent pressures created at observer

location ~x produced by the three sources are as follows:

pthk,i(τ, ~yi) =ρV0

4π

[1

r(1−Mr)∂

∂τ

(1

1−Mr

∂

∂τ

(1

1−Mr

))](4.5.8a)

pnear,i(τ, ~yi) =14π

[1

r2(1−Mr)2

(~ri · ~fi

1− ~Mi · ~Mi

1−Mr− ~fi · ~Mi

)](4.5.8b)

pfar,i(τ, ~yi) =14π

[1

c0r(1−Mr)2

(~ri ·

∂ ~fi

∂τ+

~ri · ~fi

1−Mr

(~ri ·

∂ ~Mi

∂τ

))](4.5.8c)

These equations are applied at each spanwise element of each blade, using the time-

dependent aerodynamic force vector ~fi in units of force per unit-length of blade.

Likewise, the sectional volume V0, thickness pthk,i(τ, ~yi), near pnear,i(τ, ~yi), and far

pfar,i(τ, ~yi) pressures are per unit span (denoted by the overbar). The position of

each blade section is ~yi(τ). Note that all input quantities are known a priori from

the BEM simulation for all i = 1 . . . ns stations on ib = 1 . . . B blades.

The other parameters in Eq. (4.5.8) are the unit vector from source to observer

ri and the distance between source and observer ri:

ri = |~x− ~yi| (4.5.9a)

ri =~x− ~yi

ri(4.5.9b)

the source Mach vector and its derivative:

~Mi =1c0

∂~yi

∂τ(4.5.10a)

∂ ~Mi

∂τ=

1c0

∂2~yi

∂τ2(4.5.10b)

and the relative Mach number:

Mr = ri · ~Mi (4.5.11)

For simplicity, the sectional volume is computed with the following formula:

V0 =π

2tpc+

t

3(1− p)c (4.5.12)

where t is the section thickness, c the chord length and p ∈ 0 . . . 1 a parameter

determining the fractional chord wise location of maximum thickness (taken as 0.35

throughout this work). This formula approximates the airfoil as a half-ellipse for

the first pc length of blade, and a triangle for the remaining length.


Returning to Eq. (4.5.8), the three components are first evaluated for each section

in retarded time τ (the original simulation time corresponding to the time when the

noise is generated). The observer time ti (time at which the pressure disturbance

reaches the observer) may then be computed for each segment from:

ti(τ) = τ +ri(τ)c0

(4.5.13)

Implicit in this equation is a disregarding of the free-stream velocity, which will

modify the transmission velocity from source to observer. For wind turbines, this

velocity differential is less than 0.1c0 and is typically ignored. The pressures in

retarded time psrc,i(τ) are therefore converted to observer pressures psrc,i(t).

The next step is to properly assemble the pressure signals from each source and

location i. This is done by sorting each pressure psrc,i(t) according to arrival time at

the observer. Practically, this is accomplished in the implementation used for this

thesis by linearly interpolating each psrc,i(t) value to a common tobs time computed

as:

t0obs = max [min [ti(0)]] (4.5.14)

tn+1 = tn + dtobs (4.5.15)

t ≤ min [max [ti(τmax)]] (4.5.16)

The choice of dtobs will become important for proper FFT analysis (see §4.5.4).

The penultimate step is to integrate over the span of the blade(s) to obtain physical

pressures psrc,j(t, ~yj) from pressure per unit span psrc,i(t, ~yi):

psrc,j(t, ~yj) = (sj+1 − sj)(psrc,j+1 − psrc,j) j ∈ 1 . . . ns − 1 (4.5.17)

These pressure sources therefore effectively originate at the mid-point of the spanwise

segments located at ~yj(τ). The final step to compute the overall pressure time-

history at ~x is to sum the contributions from each source and panel:

p(~x, t) =B∑

ib=1

ns−1∑j=1

pthk,j(t, ~yj) + pnear,j(t, ~yj) + pfar,j(t, ~yj) (4.5.18)

4.5.3.3 Simplified Theory

It should be noted that an even more simplified approach is possible [132]. First,

the acoustically compact approach is carried one step further, by assuming that the

forces on the blades can be lumped at one spanwise point (e.g. 72–75% radius). The


loading is then assumed to move in a circle, any dynamic blade motion is ignored,

and the rotor is assumed to rotate at constant speed.

Using these additional assumptions, an analytic formula for the nth harmonic of

RMS pressure at the observer location may be derived. The formula depends only

on the Fourier coefficients of overall rotor thrust and torque (derived over one rotor

rotation period). It has been applied in the study of noise from rotating forces

(Gutin propeller noise) and wind turbines [141]. The current work uses the more

accurate and complete formula of Eq. (4.5.8).

4.5.3.4 Nuances

Equation (4.5.8) is relatively compact and straight-forward to implement, but care

must be taken with the input data. Specifically, as the equations are dependent on

first and second order derivatives of position, loading, and derived quantities, the

numerical procedure for obtaining these derivatives becomes important. The time-

step used in the original simulation must also be sufficiently small to accurately

determine the derivatives, as well as for issues associated with post-processing (see

§4.5.4).

Ordinarily, derivatives are avoided if possible in data-gathering. Data is gath-

ered for derivative values directly (e.g. acceleration) and then integrated to yield

lower-order quantities (e.g. velocity and position). This procedure avoids the noise

amplification associated with computing numerical derivatives of data.1

When only zeroth degree data is available, as is the case for Eq. (4.5.8), vari-

ous finite-difference formulas with varying degree f (n), location of the centre point,

number of included points, and error O(h) may be derived from either interpolating

polynomials or Taylor series expansion. Typical formulae for the first and second

derivatives of a function f(x) with error terms are [142]:

f (1)(xi) =f(xi+1)− f(xi−1)

2h+

16h2f (3)(ξ) (4.5.19a)

f (2)(xi) =f(xi−1)− 2f(xi) + f(xi+1)

h2− 1

12h2f (4)(ξ) (4.5.19b)

Unfortunately, sampled data invariably contains some noise component, either phys-

ical artefacts of sampling a real system, or numerical noise in simulations. Moreover,

data samples only possesses a certain level of accuracy. Consequently, these formu-

las may depend on very small differences between large numbers, that may in fact

be noise rather than signal. This leads to cancellation and rounding errors and in

1It does of course introduce the new problem of integrator drift error.


turn to highly inaccurate derivative estimates. In application to Eq. (4.5.8a), the

problem is even worse owing to a double-use of first derivative formulas.

One approach to this problem is to construct a moving average estimate of the

derivative [143]. Using linear regression, the sampled data f(x) = y(x) may be

assumed to follow a linear relationship y = mx + b over a limited interval. For

derivative computation, only the slopem is important, yielding the following formula

for the derivative:

mi =n (∑xiyi)− (

∑xi) (

∑yi)

n(∑

x2i

)− (∑xi)

2 (4.5.20)

where mi is the derivative estimate at data point (xi, yi). The summations are

carried out over n data points around xi.

In fact, Eq. (4.5.20) is a type of low-pass digital filter, more generally defined by:

fi =nR∑

n=−nL

cnf(xi+n) (4.5.21)

where fi is the smoothed estimate, the cn a set of weights, applied to a set of points

nL and nR to the left and right of the current data point xi [144, Chap. 14.8].

In the case of simple data-smoothing, the most basic constant coefficient would be

cn = 1/(nL + nR + 1) (moving window averaging). For constant or linearly varying

data, this would produce an exact result, but for higher order variation would be in

error. Mathematically, depending on the weights, the various moments1 of the data

are preserved or not.

To analyse chemical spectra, Savitzky and Golay [145] developed a set of filters

referred to as Savitzky-Golay Smoothing Filters (SGSF)2 that preserve the higher

moments up to some degree. To build the filter, a polynomial function of a given

order is fit to the data, in a least-squares optimal sense. The parameters for the fit

include: the total number of points to use (ns = nL +nR + 1); centring of the other

data points relative to the current one (i.e. nL 6= nR potentially for data near start

and end of sampling); and the degree of the polynomial n. Since a least-squares fit

is a linear operation, the “fit” procedure only has to be done once and thereafter

applied to a moving set of data as a digital filter.

The mathematical details of the filter derivation are left to the interested reader.

The implementation of the filter weights used in the current work was obtained

1Moment of order e µe defined as µe =∑

i xeif(xi). The zeroth moment is the area, 1st

moment location in x, etc.2Also referred to as a least-squares smoothing and differentiating filter.


as the Matlab function sgsdf_gram_poly.1 The code came pre-validated against

other data [146], implemented in an efficient recursive algorithm and containing

parameters for polynomial degree, derivative degree, and number and centring of

data points.

The polynomial interpolation approach readily yields a set of weights that directly

lead to the derivatives of the sampled data. The order of the derivatives s must of

course be less than or equal to n and a sufficient number of data points used. The

choice of of n and ns effects the characteristics of the smoothing. More points and

higher polynomial degree better preserve data peaks, but at the expense of decreased

noise rejection. Based on guidance given by Press et al. [144] and parametric varia-

tion, the SGSF filters used to evaluate the acoustic equations use n = 3 and ns = 20.

One-sided filters are used for the first and last ns data points.

4.5.4 Post-Processing

The final output of the analysis procedure outlined above is a pressure-time history

at an observer position. In order to evaluate the noise characteristics, the pressure

signal p(~x, t) is transferred to the frequency domain using Fast Fourier Transform

(FFT) analysis. This topic is covered in the literature [147, 148], but elucidation

of the nuances involved can be hard to find [149]. In order to obtain accurate

quantitative results, there are a number of aspects to consider: proper units, sample

rate, windowing, and binning.

4.5.4.1 FFT Units

The Fast Fourier Transform Xk is the discretely-sampled (and computationally ef-

ficient) analogue of the continuous-time Fourier transform X(jω), defined here by

the following equations:

X(jω) =

+∞∫−∞

x(t)e−jωtdt (4.5.22a)

Xk =N∑

i=1

xie− 2πj

N(i−1)(k−1) (4.5.22b)

where j is here√−1, k = 1 . . . N . The xi are the N sampled points, uniform spaced

in time at dt = 1/fs, corresponding to a sampling frequency of fs. Essentially, the

1Source code from Jianwen Luo, PhD Candidate, Department of Biomedical Engineer-ing, Tsinghua University, Beijing 100084, P. R. China, e-mail: [email protected],[email protected]


FFT computes the frequency components of a signal, returning the amplitude and

phase of each component as a complex quantity.1

The theory derives from the Fourier series, whereby any periodic signal may be

decomposed into an infinite summation of sines and cosines of increasing frequency,

plus a constant term. The Fourier transform (integral) extends this to non-periodic

signals. The Discrete Fourier Transform (DFT) replaces the continuous integrals

with discrete summations. The FFT is simply an efficient way of computing the

DFT. In order for the FFT algorithm to work, N must be a power of two N = 2n

(typically 256, 1024).

The FFT may be thought of as frequency bins, with each Xk acting as a filter of

bandwidth ∆f . The achievable frequency resolution and range both depend on the

original sample frequency and number of sample points according to:

∆f =fs

N(4.5.23a)

fmax =fs

2−∆f (4.5.23b)

Matlab’s fft function was used here, which implements the above equations and

returns a two-sided spectrum2 of complex values. To convert from the computed

spectrum values to amplitude Ak for the positive frequency components, the follow-

ing relation is used:

Ak, peak =

|X1|N

k = 1

2|Xk|N

k = 2 . . . N/2(4.5.24)

Since the FFT amplitudes represent sinusoidal components of the signal, the RMS

values of the (non-zero frequency) components may then be found from:

Ak, RMS =Ak, peak√

2k > 1 (4.5.25)

The power spectrum is defined as:

Sk = A2k, RMS (4.5.26)

This allows definition of the signal frequency components on the Decibel scale as:

dB = 10 log10

Sk

Sref= 20 log10

Ak, RMS

Aref(4.5.27)

Additionally, phase information is available from the complex vectors Xk.1Note that some canned functions may return the power spectrum, not amplitude spec-

trum.2Both ± frequencies are returned, zero frequency component first followed by positive

then negative frequency components.


4.5.4.2 Sample Rate and Aliasing

The sample rate fs is important not only for determining the spectral range in

Eq. (4.5.23), but also in accurate decomposition of the signal. It turns out [148] that

a digitally sampled signal, where fs < 2f∗, will suffer from an effect called aliasing.

The quantity f∗ is the Nyquist frequency, and defines the permissible lower bound

on the sampling frequency, given a maximum frequency content f in the sampled

signal. As an explanation, the digital sampling is not occurring frequently enough

over the period of the signal 1/f . The peaks and troughs of the signal are sampled

at points spaced too far apart which then appear as an entirely different frequency

waveform. This in effect creates a false signal at an alias frequency fa = ±(f −mfs)

if f ≥ fs/2, for m = 1, 2, . . ..

In physical measurements, this problem is avoided by employing a high sample

frequency. If the analog signal cannot be assured to have frequency components

below fs/2, anti-aliasing filters are used before the analog-to-digital converter to

clean the signal. This approach is not possible for numerical simulated systems,

such as the current LFN computations. There are essentially three types of numerical

simulation that may be encountered in the present context:

Partial Differential Equations (PDEs) Equations1 solved using a field

method (e.g. finite difference/element) to yield solutions in position and

time. The temporal discretization method must satisfy von Neumann sta-

bility criteria, and the spatial discretization the Courant limit. An example

is the solution of the NS equations.

Ordinary Differential Equations (ODEs) Solved to yield positional and

force quantities over time. The numerical integration schemes (e.g. Runge-

Kutta) must satisfy the von Neumann stability criteria. Examples include

the usual coupled dynamic equations for a BEM-structural simulation of

a flexible wind turbine, or prescribed airfoil section (or blade) motion to

solve dynamic aerodynamic equations (e.g dynamic inflow, dynamic stall).

In the latter case, the time-step may be discretionary (i.e. stability easily

satisfied) but still requires a sufficiently small time-step to attain reasonable

accuracy.2

1Hyperbolic or parabolic in this context.2E.g. simple first-order Euler equation used for dynamic inflow equations has poor

accuracy due to local truncation error.


Prescribed Time A deterministic system of equations with known analytic

solution for all t is sampled at discrete time intervals.

In each of the cases, satisfaction of the stability criteria, or sufficient prescribed

temporal resolution, will ensure that the computational sample frequency fs is suf-

ficiently fast to capture the frequencies for the relevant phenomenon. The Nyquist

criteria will therefore be satisfied, as no higher frequency component can exist in

the signal, as it is not present in the numerical simulation.

4.5.4.3 Windowing

Implicit in the FFT algorithm is that the signal is sampled for an integral number

of periods, and that the signal is infinitely repeated in time. Unless the sampling

is triggered to be synchronous with the process (enabling capture of an integral

number of cycles), or the signal decays to zero over the sample time, this condition

will never be true. This sampling-introduced artefact is termed spectral leakage.

In order to mitigate these effects, the sampled data is windowed [150]. A window

is a digital filter that modifies the tail regions of the data sample, gradually reducing

them to zero. This avoids sampling non-integral numbers of periods. The defini-

tions and characteristics of a number of common windowing functions are given in

Table 4.1, where fi = (i− 1)/(N − 1) is the fractional position of each data point i

in the sample.

Table 4.1 Windowing functions

WindowPoint

WeightCoherent

Gain

NoisePower

Bandwidth

Worst-CaseAmplitude

(dB)

Uniform 1 1.0 1.00 3.92Hann 1−cos 2πfi

2 0.50 1.50 1.42Hamming 0.54− 0.46 cos 2πfi 0.54 1.36 1.75

It is actually impossible not to use a window; without specifically applying one, the

Uniform window is applied. To understand the artefacts introduced by windowing,

it is useful to consider the frequency spectrum of the window. Every window has a

peak centre lobe (0 dB) with side lobes of decreasing magnitude (towards −∞ dB).

The centre lobes are located at the frequency components of the true signal content.

The application of the window to the signal can be viewed as a convolution of the

signal frequencies with the frequency characteristic of the window. As an example, a

signal with a single frequency, sampled for integral number of periods, will produce


an exact FFT result. If sampled for a non-integral number of cycles, the centre lobe

of the window is shifted away from the true signal content. The side lobes therefore

appear in the FFT results. Viewed another way, the FFT process views the convolved

spectrum through a set of slits at the frequencies defined by Eq. (4.5.23); this is called

the “picket fence” effect.

The width and magnitudes of the lobes determine the spreading of frequency con-

tent between adjacent Xk, important for resolving closely spaced frequency content.

However, the minimum width of the centre lobe is limited by increased spectral

leakage as the energy content of the side lobes increases with decreasing main lobe

width. The Hann window is used here, as it has good frequency resolution with

minimal spectral leakage.

Once a window is applied, it introduces two effects. The first is to reduce the

overall amplitudes of the signal, obvious from the applying the point weights of

Table 4.1 (all < 1). To account for this, the finalXk are divided by the coherent gain

of the window. The second effect is that the windows spread the frequency content,

effectively increasing the bandwidth ∆f of each FFT bin. The factor increase is given

in Table 4.1 as the noise power bandwidth. For estimating the true peak frequency

for a component of a non-continuous spectrum,1 a frequency-weighted average of

power over ±3 adjacent peaks can be used. The true peak amplitude is then the

sum of the adjacent powers, divided by the noise power bandwidth.

4.5.4.4 Ensemble Averaging

To improve accuracy, multiple FFTs may be performed on a given signal, if the

signal’s length is long enough (N < Ns). In these cases, sections of signal, possible

overlapping from 0–100%, are ensemble average to yield a better estimate of the Xk.

The last sample may be padded with zeros to fill a complete set of N samples. Of

course, since the starting point of each sample is different, and in general not con-

temporaneous with an integral number of periods, the phase information embodied

in the Xk is lost.

4.5.4.5 Binning

Deriving octave binned frequency spectra from FFT results is accomplished with

Eq. (4.5.4). Doing so provides a more coarse, but concise and standardized, repre-

sentation of the noise. The total overall noise level may also me computed by the

1This procedure is not valid for continuous spectra


same formula, either for the entire spectrum or a specific sub-set. The steady X0

component is disregarded throughout these calculations.

4.6 Structural Modelling

A wind turbine is an inherently aeroelastic structure, especially for large scale multi-

MW machines. There is therefore a need for an accurate model of the structure for

combination with the aerodynamic models discussed in §4.4. In this section, sequen-

tial aspects of the structural modelling are developed: a general beam-section model

(§4.6.1); distributed load integration (§4.6.2); a rigid-body model of an indepen-

dently hinged rotor (§4.6.3); and finally a FEM method for the blade modes(§4.6.4).

Only the EOM for the rigid rotor are derived here, leaving flexible body dynamics

to the commercial code (BLADEDTM) with proper modes from the FEM method.

Commensurate with the scope of this thesis, the primary emphasis is on the

modelling of the rotor (blades). It is recognized that the overall dynamics of the

complete machine, including drive-train (shaft(s), transmission and generator), yaw

mechanism and tower are all critically important for a final complete analysis in

future studies.

4.6.1 Sectional Modelling

Wind turbine blades (and towers) are typically treated as linear beams. This is

standard practice, even though it is arguable in some cases whether blade cross-

sections are well approximated as slender beams [151]. The primary quantities of

interest for beam theory are the mass/unit length mps and area moments of inertia

I, which must be computed for sections made up of many composite layers. The

approach adopted here is similar to the one the author has used in the past [152].

The definition of the composite layup is different however, obtained here from direct

user-entry rather than being CAD-based. The section properties are used in later

derivations as inputs to describe the dynamic loads and motion of the blades.

4.6.1.1 Layup Definition

For each airfoil defined for a blade, two input sheets are required in ExcelBEM. The

first contains the aerodynamic properties, and the other the airfoil profiles. These

profiles are defined at various percent thicknesses covering those used in the blade

for (linear) interpolation. Internally, the profiles are interpolated when loaded to

yield profiles with a uniform number and location of chordwise spacing, to speed

4.6 Structural Modelling 115

computation. The same chordwise cosine spacing is used for the upper and lower

surfaces, which are input separately to accommodate a range of data sources.

The layup of blade sections is defined on a separate sheet. Every station may be

defined, or only a subset which is kept constant for intermediate sections. The input

sheet for the definition of an example single section is shown in Fig. 4.27, and the

resulting section shown in the left-half of Fig. 4.28.

1 Upper Surfacec (%) t (m) Mat'l

0.00 0.0005 Gelcoat0.0004 Random mat0.0009 CDB3400.0050 Balsa0.0009 CDB340

0.18 0.0005 Gelcoat0.0004 Random mat0.0009 CDB3400.0100 Spar cap mix0.0009 CDB340

0.53 0.0005 Gelcoat0.0004 Random mat0.0009 CDB3400.0100 Balsa0.0009 CDB340

0.85 0.00051 Gelcoat0.00038 Random mat0.00089 CDB340

0.99 0.0005 Gelcoat0.0004 Random mat0.002 TE spline

0.0009 CDB340

Lower Surfacec (%) t (m) Mat'l

0.00 0.0005 Gelcoat0.0004 Random mat

0.00089 CDB3400.005 Balsa

0.00089 CDB3400.12 0.00051 Gelcoat

0.0004 Random mat0.00089 CDB340

0.01 Spar cap mix0.0009 CDB340

0.47 0.00051 Gelcoat0.0004 Random mat

0.00089 CDB3400.01 Balsa

0.0009 CDB3400.85 0.0005 Gelcoat

0.0004 Random mat0.0009 CDB340

Web(s)c (%) t (m) Mat'l0.18:0.12 0.0015 CDB340

0.0100 Balsa0.0015 CDB340

0.53:0.47 0.0015 CDB3400.0100 Balsa0.0015 CDB340

Figure 4.27 Section definition

The definition is split into three, one for each of the upper and lower surfaces, and

a third to define the webs. For the upper and lower surfaces, fractional chordwise

section divisions are input, starting with 0. The definition is assumed to continue

to the TE, at c(%) = 1, so a minimum of one chordwise region must be defined


for both skins. These define the locations of lay-up transitions along the skins. In

this example, there are 5 lay-up definitions for the LE, spar cap, balsa-core TE,

skinned TE and finally a TE spline. The last two columns define the lay-up for

each section, defined by layer thickness and material. The material definitions are

named and defined on a separate sheet (density, moduli, cost, etc.). The web section

may be blank, or contain an unlimited number of shear web definitions. To allow

for angled webs, the chordwise locations may be defined as colon separted pairs,

defining the fraction chordwise locations of the web intersection with the upper and

lower surfaces respectively. The layup defintion for the web(s) is identical to the

other surfaces. It is assumed that the thickness of the web projects towards the TE.

4.6.1.2 Section Computations

Using the layup of the section’s structure, together with the airfoil profiles, the

relevant quantities may be computed as follows:

1. The relevant airfoil profiles are interpolated and scaled to the proper chord

length. For thicknesses outside the defined range, simple scaling of thickness

is used.1

2. The normals ni shown in Fig. 4.28 are computed for each point on the airfoil

surfaces using:

~ni =12

yi−1 + yi+1

xi+1 − xi−1

(4.6.1a)

ni,upper =~ni

|~ni|(4.6.1b)

ni,lower = − ~ni

|~ni|(4.6.1c)

where(x, y)

are the airfoil profile coordinates.

3. Points and normals are linearly interpolated to place points exactly on chord-

wise layup transition points.

4. The upper and lower surfaces are processed by defining each layup thickness

block as two triangles, and processing each triangle. The derivation of section

properties for the triangular elements is included in Appendix B. The com-

putation proceeds along each interval of the surfaces, and through the layup

thickness along the normals.

5. The web is processed in a similar manner, using normals defined by the(x, y)

coordinates of the web termination points on the airfoil surfaces.1More properly, the camberline should be extracted from the thickness profile, the thick-

ness profile scaled and then recombined with the camberline [52].


The formulas for overall section properties (~xCG, mps, EI, etc.) from the triangular

element aggregatrations are also included in Appendix B.

Figure 4.28 Blade section triangulation

4.6.1.3 Common Confusions and Assumptions

It is important at this point to clarify two issues, as they are commonly misconstrued.

The first is that the torsional constant GJe is not in general the same as the shear

modulus weighted polar moment of area, computed here as GJ about ~xEA. They

are equal only for axisymmetric circular bars [151]. The computation of the effective

polar moment of area Je is more involved than computing J , utilizing either thin

shell theory (shear flow) or more generally Prantl’s membrane analogy. These two

quantities are however often used interchangeably, but may be in significant (non-

conservative) error, especially near the root [153]. Related to this is the definition

of the shear centre, which is not in general co-located with the Neutral Axis (NA)

(~xEA), although it is commonly assumed so. Likewise, the centre-of-mass (~xCG) and

NA are not in general co-located (the circle and cross in Fig. 4.28 respectively).

In any case, the pure torsional modes of blades are not typically used in wind

turbine analysis. Notwithstanding a wealth of research [61, 154], there has never

been any evidence of flutter1 in practice. Flutter can be avoided in any case by

ensuring the section ~xCG lies between the shear centre and the LE. Deliberate flap-

twist coupling has also been investigated, and so provision for input of these coupling

values is provided in commercial codes (e.g. BLADEDTM). The investigations are

usually performed in terms of either an arbitrary coupling coefficient [61], or derived

from more detailed FEM analysis [155].

Another confusion surrounds the definition of flapwise and edgewise/chordwise,

the relation to Out-of-Plane (OP) and In-Plane (IP) and structural twist [80]. This

confusion also extends into aerodynamic modelling. To clarify the definitions, a

1Flutter is caused by the coupling of bending and torsional modes


general blade section with various coordinate systems is shown in Fig. 4.29. The

section is viewed looking in inboard along the blade (i.e. towards −z direction).

The blade is translating to the right of the page, and the wind vector is nominally

up the page, modified by the cosine of the cone angle.

Figure 4.29 Definition of section coordinate systems

At the LE is the(x, y)

CS used for structural computations.1 The next coordi-

nate system is on the chordline, at the star symbol c1/4 from the LE. This is the

point where aerodynamic coefficients (cl, cd, cm) are defined. The next coordinate

system is lcx away from the LE on the chordline. The inclined set of vectors define

the flapwise and edgewise bending moment directions. The horizontal-vertical axis

located pb (pre-bend) off the chordline defines the pitch axis location. All other anal-

ysis (aerodynamic and structural) compute forces decomposed into the (xaero, yaero)

directions.2 These are the OP and IP directions, for β = 0.

4.6.1.4 Stress/Strain Computation

Wind turbine blades are usually governed structurally by direct normal stresses,

rather than shear forces; the latter are therefore ignored in the stress calculations.

The location of maximum stress (or strain) is usually checked at a number of points

around the periphery of the section. In general, given combined loading, the curved

1Standard airfoil coordinates take the LE as(1, 0)

and TE as(0, 0), so these must be

reversed in x here.2The aerodynamic axes usually fall on the chordline (pb = 0). The pre-bend is included

here to be comparable to BLADEDTM load results.


geometry and variable layup material properties (both stiffness and strength allow-

ables), the location of extreme stress is not known a priori.

Therefore, the current analysis checks all grid-points computed in §4.6.1.2, using

the following formula:

ε =Fz

EA+yp(MxEIy +MyEIxy)− xp(MxEIxy +MyEIx)

EIxEIy − EI2xy

(4.6.2)

The strain formulation is used to accommodate heterogeneous stiffnesses of the

layers. The stress is of course found from σ = Eε using the layer’s Young’s modulus

E, although composites are frequently characterised in terms of strain rather than

stress. The loads (Fz, Mx and My) must be translated from their definition on pitch

axis (from other calculations) to xEA. The test points(xp, yp

)are defined in the(

x, y)

LE coordinate system of Fig. 4.29 as vectors from xEA to the test point in the

layup.

Fatigue calculations (e.g. rainflow counting) may proceed from the resulting stress

or stain time-history, in the case of dynamic loading. Although fatigue tends to

dominate many wind turbine components, static loads are used throughout this

thesis to provide initial structural designs for later dynamic simulations of fatigue

loading. Skin buckling could also be incorporated in the current framework, and

should also be checked in a final design [153, 156].

4.6.2 Kinetostatics

The key elements of aerodynamic force prediction and the mass properties of the

blades have been developed in §4.4 and §4.6.1.2 respectively. The current task is

twofold: to compute the moments required for equilibrium operation (see §4.7.1) and

power production, and secondly to compute the loading in the blade for dynamic

operation (§4.6.3) and as input for structural load calculations (§4.6.1.4). The loads

are computed on the pitch axis (zaero in Fig. 4.1(d)), for each section on the blade

lying in the xaero–yaero plane. Three sources of loading are considered, aerodynamic

(~faero, ~maero), gravitational (~fgrav, ~mgrav) and inertial (~facc, ~macc), comprising the

total forces ~f and moments ~m per unit length. The loading is developed with

reference to Fig. 4.30, which is consistent with Fig. 4.29. D’Alembert’s method is

used to handle the inertial accelerations.


Figure 4.30 Section layout for kinetostatics

Figure 4.31 Kinetostatic offset vectors

The lengths in Fig. 4.31 define the position of P , which represents either the

aerodynamic (da) or mass centre (dm).

da = c (pA − 1/4)

dm = c (pA − CG)

lx = −pb − d sin γtot

ly = −d cos γtot

(4.6.3)

where lx and ly apply for either the aerodynamic or mass centre with subscript a or

m respectively. The applied forces will then generate moments according to:

~moff = (lxi+ ly j)× ~f (4.6.4)

from this offset. The sectional mass is assumed concentrated at the CG, so no

direct moments are produced from gravitational and inertial forces, only the ~moff

moments. Point masses are included as increments to the defined mass per unit

length on the blade (mps), computed to preserve the 1st moment about the flap

hinge, by distributing the mass over two adjacent sections. The sectional forces and


moments (per unit span) are now outlined in §4.6.2.1 through 4.6.2.3, followed by

the integrated quantities in §4.6.2.4.

4.6.2.1 Aerodynamic Loading

The aerodynamic forces follow directly from the aerodynamic characteristics pre-

sented in §4.4.4. It only remains to decompose the vectors appropriately and include

the effect of the offset section in the moment calculations. The 2D aerodynamic ap-

plied forces are:

q = 1/2ρV 2relc (4.6.5a)

fx,aero = q (cl cosφ+ cd sinφ) (4.6.5b)

fy,aero = −q (cl sinφ− cd cosφ) (4.6.5c)

~faero = fx,aeroi+ fy,aeroj (4.6.5d)

As discussed in §4.4.9, any shear force normal to the section, owing to the viscosity

of the spanwise flow, is considered insignificant as a direct force.

The pure moment contribution is found from:

~maero = (qccm) k (4.6.6)

which will be in addition to the moment found from Eq. (4.6.4).

4.6.2.2 Gravitational Loading

The gravitational force on the section is easily found from ~fgrav,e = −mpsgk. In

order to compute the loading in the section CS, the unit vectors uaero,i describing

the axes xaero–yaero–zaero are first found from:

uaero,i = R z(yaw) R y(ψt) R x(θ) R y(β)ui (4.6.7)

where ui are the unit vectors i, j, k, in the column notation of Appendix A. The

section loading along each axis is then:

~fgrav,i = ~fgrav,e · uaero,i (4.6.8)

4.6.2.3 Inertial Loading

The inertial loading is determined by the acceleration vector from ~facc = −mps~a.

The determination of the acceleration vector ~a is found from the kinetics of the


active DOF discussed in §4.6.3, the azimuth θ and cone angle β. The position vector

x of the section (on the pitch axis) relative to the centre of the hub (i.e. in x′′–y′′–z′′

CS of Fig. 4.1(b)) is found from expanding the following:

x = R x(θ) T (0, 0, Rhinge) R y(β)

lx,m

ly,m

s′

1

(4.6.9)

where s′ = shub + s. The full form of this equation, and the derivatives x and x

are not included here for brevity. A set of reduced and easily verifiable equations is

obtained with lx = ly = Rhinge = 0:

~x = s′β cosβi+

(β sin θ sinβ − θ cos θ cosβ

)j−(

θ sin θ cosβ − β cos θ sinβ)k

(4.6.10a)

~x = s′(β cosβ − β2 sinβi

)i+(

θ cos θ cosβ + θ2 sin θ cosβ + 2θβ cos θ sinβ + β2 sin θ cosβ + β sin θ sinβ)j+(

−θ sin θ cosβ − θ2 cos θ cosβ + 2θβ sin θ sinβ − β2 cos θ cosβ − β cos θ sinβ)k

(4.6.10b)

Equation (4.6.10a) contains the velocity terms owing to azimuthal rotation and

flapping, decomposed onto the coordinate axes. Equation (4.6.10b) contains the

direct acceleration motions in azimuth and flap, the Coriolis accelerations associated

with the rotating reference frames (θβ), and the centrifugal acceleration terms (θ2

and β2), again decomposed through θ and β.

4.6.2.4 Integrated Loading

The loading from each source may now be integrated to yield shear loads and bend-

ing moments along the blade, for each section. Each contribution is computed

separately, as the dynamic model in §4.6.3 requires only the integrated aerody-

namic hinge moments. The properties of the blade are assumed to vary linearly

between sections, as are the sectional forces. Figure 4.32 illustrates the blade as a

beam, where section i is of interest, located si from the blade root and extending

by ∆s = (si+1 − si) outboard to section si+1. The sectional applied forces ~f and

moments ~m per unit length in §4.6.2.1, 4.6.2.2 and 4.6.2.3 are assumed to vary lin-

early with s. Both shear and moment loading are integrated from tip (constraint

and force free) to hub.


Figure 4.32 Integrated loading

The orthogonal axial and shear forces ~Vi is easily obtained from integration over

the section:

~Vi = ~Vi+1 +~fi + ~fi+1

2∆s (4.6.11)

The moment components of ~Mi are also found by integration and from outboard

shear forces as:

Mi,x = Mi+1,x +(mi+1,x +mi,x

2− Vi+1,y

)∆s−

(fi,y

6+fi+1,y

3

)∆s2 (4.6.12a)

Mi,y = Mi+1,y +(mi+1,y +mi,y

2+ Vi+1,x

)∆s+

(fi,x

6+fi+1,x

3

)∆s2 (4.6.12b)

Mi,z = Mi+1,z +(mi+1,z +mi,z

2

)∆s (4.6.12c)

4.6.2.5 Transformed Loading

The final step is to transform the shear and bending load from the most inboard

section to the flap hinge and hub centre. The important hinge loads are found from:

Mx,hinge = M1,x − shubV1,y (4.6.13a)

My,hinge = M1,y + shubV1,x (4.6.13b)

Mz,hinge = M1,z (4.6.13c)

My,hinge is the critical coning moment, either summed over all sources, or just the

aerodynamic forces (for §4.6.3). The hub loads in the x′′–y′′–z′′ CS of Fig. 4.1(c)


are:

Mx,hub = Mx,hinge cosβ +Mz,hinge sinβ −RhingeV1,y (4.6.14a)

Vx,hub = V1,x cosβ + V1,z sinβ (4.6.14b)

Summing the hub loads for each blade, the driving torque is τ =∑

B Mx,hub, power

P = τΩ and the rotor thrust T =∑

B Vx,hub.

The moments and shears for each section can also be easily transformed into the

flapwise and edgewise directions:

Vflap = Vx cos γtot − Vy sin γtot (4.6.15a)

Vedge = −Vx sin γtot − Vy cos γtot (4.6.15b)

Mflap = Mx sin γtot +My cos γtot (4.6.15c)

Medge = Mx cos γtot −My sin γtot (4.6.15d)

4.6.3 Dynamic Modelling

Wind turbines are described by multiple DOF, both rigid and flexible (blades and

tower). The EOM may be derived by numerous methods, all deriving fundamen-

tally from a Newtonian force or energy approach (Hamilton’s principle). Newtonian

methods tend to be more laborious, as they require explicit vectorial consideration

of each loading mechanism. Energy-based methods permit more general DOF and a

scalar formulation to realize more compact EOM.

To reduce the DOF to be included at run-time, modal representations of the

flexible structures are typically used. An alternative is an FEM model, either as beam

elements or lumped mass/force fields. Both serve to reduce the partial differential

equations to ordinary differential equations amenable to solution, the modal method

being preferred for compactness.

The modal approaches use either true normal modes, or approximate modes con-

sistent with imposed boundary conditions and with reasonably accurate mode shapes

(e.g. §4.6.4) [69]. In the former case, the PDEs are directly converted to ODEs by

substitution of the modes (usually up to second or third modes). In the latter case,

a Galerkin method may be used to reduce the PDE from a Newtonian formulation.

Alternatively, Lagrange’s equation (Eq. (4.6.16)) or Kane’s method may be used to

directly synthesize the EOM in terms of assumed modes.

The EOM may be further reduced by substituting the flexible component mass

and stiffness (including rotational stiffness) terms with modal shape and frequency


terms.1 This is only correct for normal mode shapes (i.e. no coupling of modes).

Even then, the rotation speed Ω must be constant and the stiffness linear (it will

be non-linear for large continuous bending or flapping deflections). Some authors

adopt this simplification, assuming the last non-linear considerations small, but as

will be seen in §5.4.3, this can lead to large errors.

In most cases, flapwise/edgewise mode shapes are used, so that with twist in-

plane/out-of-plane motion is coupled but the modes themselves are normal and

invariant with pitch angle (to first order). Others, for example BLADEDTM, use

in-plane and out-of-plane assumed mode shapes, which vary with pitch angle.

4.6.3.1 Equations of Motion

A full and detailed accounting of the all DOF in Fig. 4.1 is quite involved, especially

when flexible elements are incorporated. Numerous couplings are introduced by the

various rotating and translating reference frames, yielding obtuse sets of equations.

The post-processing of loads (see §4.6.2) further burdens the process of writing a

full simulation code.

BLADEDTM was therefore targeted as an eventual platform for full, flexible body

simulations. Unfortunately, commercial time constraints have hindered implemen-

tation of the aerodynamic refinements appropriate to the coning rotor. Moreover,

as illuminated in §5.4.3, it appears the structural model internal to BLADEDTM

will require modification as well. In short, a full simulation has had to be relegated

to future work (see §8.3).

Here, only a rigid body set of EOM is developed, with individual blade coning βj

and rotor azimuth θ as the DOF.2 This will allow examination of the fundamental

flapping motion of the coning rotor. It is also the same number of DOF adopted by

other authors studying flapping blades [77]. Lagrange’s equations are used to define

the EOM [69]. The method followed is similar to that in the original CONE-450

report [66], but including more discussion of non-linearities, gravity, tilt angle, and

independent flapping.

The total kinetic T and potential U energy of the system are obtained from

kinematic analysis of the various DOF xi. The generalized forces on the system are

1The terms of the homogeneous PDEs used to find the modes are also present in the fullnon-homogeneous PDEs (see [69, p. 387]).

2Note that the blade-dependent azimuth angle θj is offset from the rotor azimuth angleθ by (j − 1)2π/B.


contained in Fi (forces for linear DOF and moments for rotational DOF).

L = T − U (4.6.16a)∂

∂t

(∂L

∂xi

)− ∂L

∂xi= Fi (4.6.16b)

The resulting equations are equivalent to those obtained by kinetostatic analysis

(Newton’s Law) or Kane’s equations. Lagrange’s approach was chosen as it is quite

straightforward to apply and generates reasonable compact equations, although any

internal forces must be obtained by post-processing.

Applying the method to the reduced set of DOF used for dynamic analysis, the

following expressions are obtained for T and U :

T =12IH θ

2 +B∑

j=1

[12Irj θ

2j +

12Iββ

2j

](4.6.17a)

U =B∑

j=1

Mbghj (4.6.17b)

The blade moment of inertia about the hinge axis Iβ, blade mass Mb, and centre

of mass distance sCG are assumed constant and equal for all blades (no imbalance).

It is also implicit in the dynamics formula that the masses are on the pitch axis

(pb = 0 and pA = CG in Fig. 4.30). IH is the inertial moment of the hub about the

rotation axis. The moment of inertia about the centre of mass ICG of an individual

blade yields Iβ = ICG + Mbs2CG. The moment of inertia of the blades about the

shaft varies with cone angle βj as:

Irj =

S∫s0

r2dm = R2hMb + 2RhMbsCG cosβj + Iβ cos2 βj (4.6.18)

The potential energy term U needs a height hj , defined relative to the hub centre:

hj = sCG (− sin(ψt) sinβj + cos(ψt) cos θj cosβj) +Rh cos(ψt) cos θj (4.6.19)

The tilt angle ψt here is positive for an upwind rotor, and negative for a downwind

rotor, as defined in Fig. 4.1(b).

The final set of 1 +B equations is derived as:IH +B∑

j=1

Irj

θ −B∑

j=1

[θβj (2RhMbsCG sinβj + Iβ sin 2βj)

]=

Qaero −Qr −B∑

j=1

[Mbg (Rh cos(ψt) sin θj + sCG cos(ψt) sin θj cosβj)] (4.6.20a)


Iββj +HactDβj + (2RhMbsCG sinβj + Iβ sin 2βj)θ2

2=

Haero −Hact +MbgsCG (cos(ψt) cos θj sinβj + sin(ψt) cosβj)

for j = 1 . . . B (4.6.20b)

Equation (4.6.20a) represents the azimuthal DOF and Eq. (4.6.20b) the B inde-

pendent (in general) flapping DOF. The aerodynamic moments, Qaero and Haero

about the rotor and flap axes respectively, are dependent on the state variables[θ, θ, θ, βj , βj β

], thereby introducing spring and damping action dependent on the

aerodynamics. A damping element HactD and actuator moment Hact are also in-

troduced. The second term in Eq. (4.6.20a) is a damper developed as a result of

flapping motion, which in the rotating reference frame of the rotor is manifest as a

torque from Coriolis acceleration. The moment Qr is the reaction torque from the

generator, assumed to transmit its torque through an infinitely stiff structure.1 The

second term in Eq. (4.6.20b) contributes the centrifugal stiffening of the rotor. In

both equations, the gravity terms introduce cyclical forces with π/2 phasing between

the DOF.

Equation (4.6.20) is first rewritten in state-space form. The model is then solved

with Matlab’s ode45 solver, implementing a 4th or 5th order Runge-Kutta method

with variable step size. Without a controller, θ is set to zero for a constant θ

rotation speed. The state variables are time-stepped according to the internal solver

algorithm. The aerodynamic calculations are only run at specified time steps, after

the EOM equations have converged to sufficient accuracy based on the previous

aerodynamic values, using the OutputFcn facility of ode45. The time steps are

specified in terms of azimuth angle, ranging from 0.5–5.

4.6.3.2 Linearised Equations of Motion

Equation (4.6.20) represents the full non-linear set of equations that describe the

system. A degree of further insight may be gained by linearising Eq. (4.6.20b) about

βj = 0, taking ψt = 0, ignoring gravity and hinge moments (both aerodynamic and

actuator):

β +(

1 +RhMbsCG

Iβ

)θ2βj = 0 (4.6.21)

Recall that the standard definitions of critical damping cc, damping ratio ζ, natural

frequency ωn and damped natural frequency ωnd for an under-damped and un-forced1If a flexible shaft(s) is used with a gearbox, this additional compliance can be very

important, however a closely coupled direct-drive machine will be less sensitive to this as-sumption.


dynamic system described by displacement x are:

mx+ cx+ kx = 0 (4.6.22a)

ωn =

√k

m(4.6.22b)

cc = 2mωn (4.6.22c)

ζ =c

cc(4.6.22d)

ωnd = ωn

√1− ζ2 (4.6.22e)

A non-dimensional number relevant to the current discussion is the Locke number.

As the change in aerodynamic force is proportional to clα, the ratio of aerodynamic

to inertial forces is termed the Locke number and defined as:

Lk =ρcclαR

4

Iβ(4.6.23)

Finally, examination of Eq. (4.6.21) yields the following insights:

• The centrifugal stiffening effect is increased with Rh, which is a partial reason

for why the CONE-450 employed a space-frame, since it employed very light

carbon fibre blades.

• Lk may also be defined to include the full stiffness term in Eq. (4.6.21) and is

a metric of equilibrium coning angle.

• The un-damped natural frequency of the system is found to be

ωn = θ

√1 +

RhMbsCG

Iβ(4.6.24)

indicating that Rh again increases the stiffness and hence frequency.

4.6.3.3 System Damping

The sources of damping in the system are the embedded hinge damper, HactD, active

damping imposed by active control of Hact, and any aerodynamic damping that may

be present in Haero (developed by relative AOA changes with flapping motion β). In

the absence of any direct hinge damping, aerodynamic damping becomes critical (see

§7.1.2.1). In the absence of an active actuator moment, the damping of the system

may be estimated by further assuming that Haero = Haero

(θ, β, V0 − seff β cosβ

)[66]. The length seff along the blade is an effective moment arm where the aero-

dynamic forces are assumed to act. This accounts for aerodynamic flapping mo-

ments created by β, without requiring a separate map of Haero with β. A value


of seff = 0.7S has been found appropriate [66]. With this linearising assumption,

Eq. (4.6.21) can be expanded to include aerodynamic forces as:

βj +1Iβ

[HactD + seff

∂Haero

∂V0

]βj + ω2

nβj = 0 (4.6.25)

by incorporating

∆Haero =∂Haero

∂θ∆θ +

∂Haero

∂β∆βj +

∂Haero

∂V0

(∆V0 − seff βj

)(4.6.26)

and taking the steady-state case where ∆θ = ∆V0 = 0 and ignoring the added

stiffness term ∂Haero∂β in ωn. The damping ratio is then computed as:

ζ =HactD + seff

∂Haero∂V0

2Iβωn(4.6.27)

To achieve a certain minimum positive damping factor ζ, the actuator damping must

be positive to balance any negatively sloping aerodynamic hinge moment curve. This

is typically associated with negative Lk (negative clα) found in stalled conditions.

The non-linear damping of the system may be studied using the full simulation

code as well. In this case, damping may be computed via measurement of the loga-

rithmic decrement [157, p. 137] in free vibration from a non-equilibrium start point.

Other possible approaches include utilizing the Hilbert transform [158], general-

ized system identification methods [125], or work equivalence1 [159], however these

methods are more complex than is required here.

The logarithmic decrement δd is based on the following equation:

δd =1N

lnx0

xN=

2πζ√1− ζ2

(4.6.28)

which derives from the analytic solution to Eq. (4.6.22a), where x0 and xN are

peak positive displacements N cycles apart. Critically, this equation depends on an

equilibrium value of xinf = 0. The offset can be explicitly removed from the data (if

known) [158], or dealt with implicitly by computing:

yi = ln |xi − xi+1| (4.6.29)

where xi are now theNp extrema (positive and negative), located in the displacement

time history with a peak-finding algorithm. A good estimate of δd is then obtained as

δd = −2m, where m is the least-squares fit to the yi points at indices i = 1 . . . Np−1.

Finally the damping ratio ζ is obtained from Eq. (4.6.28) and damping constant from

Eq. (4.6.22), if ωn and m are known for the mode.1The work equivalence method is quite useful in determining the damping of individual

flexible-body modes, but requires accurate ωn.


4.6.4 Centrifugally Stiffened Beam

In order to include flapping in BLADEDTM which incorporates flexible blades, a

method was required to compute the mode shapes of hinged blades. BLADEDTM

takes as input a defined number of in-plane and out-of-plane modes, described

by their mode shapes, natural frequencies (stationary and rotating), and assumed

damping factors for each mode. The modes are computed by transforming the

structural stiffness through the section twist and pitch angles, using Eq. (B.8).

There is no built-in facility in BLADEDTM for computing freely hinged modes.

The approach taken to this problem is to model the blades with an FEM repre-

sentation. The centrifugal stiffening and linear variation of beam properties are

somewhat non-standard and are therefore developed here. BLADEDTM uses some

form of iterative technique [157] on the beam PDE for conventional fixed-root blades.

4.6.4.1 Finite Element Formulation

The specific approach taken to this derivation is the minimization of potential energy

πp = U+Ω, following the general approach and notation of Logan [160, p. 56] where

standard beam equations are presented (constant properties, without centrifugal

stiffening). A beam element is shown in Fig. 4.33. Linear theory is used throughout

this derivation, both in displacements and material properties (i.e. σ = Eε).

Figure 4.33 Beam element coordinate system (x–y), displacement (v) and nodaldisplacements (d), rotations (φ), forces (f) and moments (m)

The shape functions define the displacement v as:

v = N d =[N1N2N3N4

]d1y

φ1

d2y

φ2

(4.6.30a)


N1 =1L3

(2x3 − 3x2L+ L3

)N2 =

1L3

(x3L− 2x2L2 + xL3

)N3 =

1L3

(−2x3 + 3x2L

)N4 =

1L3

(x3L− x2L2

) (4.6.30b)

The strain may be defined from its definition and the deformed geometry of the

beam with axial extension u (see Logan [160]):

u = −y dvdx

(4.6.31a)

εx(x, y) =du

dx= −y d

2v

dx2(4.6.31b)

Using the shape functions N , the strain is found as:

εx(x, y) = −yB d (4.6.32)

where B is the second derivative of N with respect to x:

B =1L3

[12x− 6L, 6xL− 4L2,−12x+ 6L, 6xL− 2L2

](4.6.33)

The stress is simply defined as:

σx = D εx = E(x, y)εx (4.6.34)

The strain energy U is found to be:

U =∫∫∫

V

12σxεxdV (4.6.35)

by neglecting shear stress (i.e. Euler-Bernoulli beam, not Timoshenko beam). Direct

axial strain (owing to displacement u) is also ignored in this derivation. Substituting

the above constituent quantities yields:

U =12

L∫0

∫∫A

E(x, y)y2dT

B T B dAdxd =12

L∫0

EI(x)dT

B T B dxd (4.6.36)

where the usual integral for the moment of inertia is used:

EI(x) =∫∫A

E(x, y)y2dA (4.6.37)

and the product EI in general varies over the length of the beam:

EI(x) = (EI)1 +(EI)2 − (EI)1

Lx (4.6.38)


Evaluating Eq. (4.6.36) and differentiating for each DOF of the four beam variables

and setting each equal to zero (minimum energy condition), yields the elemental

stiffness matrix k stiff :

k stiff =

L∫0

EI(x) B T B dx (4.6.39)

the expanded matrix form of which is omitted here for brevity.

It was found that using constant EI values (average of end section values) pro-

duced significantly different results to this exact equation. The same was found for

the mass matrix, defined “consistently”1 as:

m =∫∫∫

V

ρN T N dV (4.6.40a)

=

L∫0

mps(x) N T N dx (4.6.40b)

mps(x) = mps,1 +mps,2 −mps,1

Lx (4.6.40c)

Equation (4.6.40a) may be obtained from virtual work or D’Alembert’s principle

(see Logan [160] for the derivation).

An additional term is required to account for “centrifugal stiffening”, essentially

increased stiffness, k cent. Two approaches are possible to account for this: body

forces (Ω term) or pre-stress (U term). The latter is presented here, as it is more

general (e.g. vibrating tensioned string).

The starting point is to consider a general strained fibre in the beam, as shown

in Fig. 4.34. As the fibre is strained from length dS to ds = ((dS + du)2 + dv2)1/2,

the strain developed is [161, § 3.3]:

εx =ds− dSdS

=ds

dS− 1

=

(1 + 2

du

dS+(du

dS

)2

+(dv

dS

)2)1/2

− 1(4.6.41)

In keeping with linear theory, dS ≈ dx, and using the first three terms of the

binomial expansion of the (. . .)1/2 term above yields:

εx =du

dx+

12

((du

dx

)2

+(dv

dx

)2)

(4.6.42)

1A lumped-mass formulation was found inadequately accurate.


Figure 4.34 Strained fibre

The origin of Eq. (4.6.35) is now relevant:

U =∫∫∫

V

εx∫0

EεxdεxdV (4.6.43)

obtained from the work of internal forces on a differential volume dU = σxdεxdV

(see [160]), substituting σx = Eεx, and integrating over the volume. Substituting

the first term of Eq. (4.6.42) into the above yields Eq. (4.6.35). The second term in

Eq. (4.6.42) is ignored as higher order, leaving the third non-linear v displacement

term. This latter term may be integrated by noting that it is a constant (6= f(εx))

to yield:

U =12

∫∫∫V

σx

(dv

dx

)2

dV (4.6.44)

Equation (4.6.44) is the strain energy for the beam bending problem, owing to pre-

stress σx. With an assumed uniform stress over the cross-sections, this reduces

to:

U =12

L∫0

F

(dv

dx

)2

dx (4.6.45)

where F is the force on the cross-section. The equation may be vectorized as:

U =12

L∫0

F d T N Tx N x d dx (4.6.46)

where N x is the first derivative with respect to x of the shape functions N (see

Eq. (4.6.30a)). The centrifugal “stiffness matrix” kcent is obtained by taking the

derivative as before:

k cent =

L∫0

F N Tx N xdx (4.6.47)


It only remains to define the section force F that varies over the blade length. As-

suming that external forces perfectly balance the centrifugal forces in the y direction,

the centrifugal axial force is found with reference to Fig. 4.35 as:

F = F2 + Ω2 cosβ

L∫x

mpsrdx (4.6.48)

where F2 is the aggregate of the outboard forces and mps varies as in Eq. (4.6.40c).

The radius r = Rh +(s1 + x) cosβ depends on the hub, as Rh = Rhinge for fixed and

outboard flapping hinges, and Rh = Rhinge cosβ for centrally flapping and teetered

roots. The final integrated matrix form of Eq. (4.6.47) is quite large and therefore

omitted.

Figure 4.35 Element centrifugal force

4.6.4.2 Assembly and Master-Slave Method

The overall stiffness matrix (K ) and mass matrix (M ) are assembled by super-

position (direct stiffness method [160]) of the the various element matrices ( k stiff ,

k cent and m ), by matching DOF for each blade section element and blade.1 A

central hub node (with displacement and rotation) is also included. Note that all

rotational DOF φ are the same direction, and the displacements are normal to the

un-deformed blade axes (i.e. not using general rotated elements).

The Master-Slave method [162] is used to couple the blade root nodes to the

central nodes. The only exception is for a hub with known stiffness, in which case

the hub stiffness matrix is simply assembled as above. The general method begins by

partitioning the matrix in master nodes u m, slave nodes u s, and “uncommitted”

1No symmetry is in general present to allow reduction in DOF (in-plane symmetric/anti-symmetric modes, teetering.


nodes u u: k uu k um k us

k Tum k mm k ms

k Tus k T

ms k ss

u u

u m

u s

=

f u

f m

f s

(4.6.49)

The constraint equation prescriptively couples DOF with infinite stiffness:

A m u m + A s u s = g (4.6.50)

Rearranging, and solving for u s defines the matrix T :

u s = −A−1s A m u m + A−1

s g = T u m + g (4.6.51)

which finally yields a reduced system of equations:[k uu k um T

T T k Tum T T k mm T

] [u u

u m

]=[

f u − k us g

f m − k ms g

](4.6.52)

In this application, a mass matrix is also required, g is zero and no external forces

are present, giving:

M ¨u + K u = 0 (4.6.53)

The T matrix is augmented from the general formulation, to map the complete set

of original DOF to a reduced set u containing u u and u m.

u = T u (4.6.54a)

M = T T M T (4.6.54b)

K = T T K T (4.6.54c)

Two couplings are used between the hub and blade root nodes to define T :

• Rigid link of length L v1φ1

v2φ2

=

1 00 11 L0 1

v1φ1

(4.6.55)

• Hinge at end of rigid link of length Lv1φ1

v2

=

1 00 11 L

v1φ1

(4.6.56)

A global T matrix is assembled in the same manner as the mass and stiffness

matrices by superposition, starting with T = I .


4.6.4.3 Boundary Conditions

The boundary conditions and coupling are specific to the hub. In all cases the

displacement of the hub node is zero. For OP modes:

Rigid No rotation of hub and rigid coupling of blade roots to hub node.

Central Hinge No rotation of hub and hinged coupling of blade roots to

hub node.

Hinge at Rhinge No rotation of hub, no translation of blade roots.

Teetered Rigid connection between blade roots and hub node.

For IP modes:

Brake Applied (Fixed) No rotation of hub and rigid coupling of blade

roots to hub node.

Free-Hub Rigid connection between blade roots and hub node.

4.6.4.4 Modal Solution

The mode shapes are obtained from the eigenvalues (λn) and vectors (x n) re-

turned from Matlab’s eig function. This solves Eq. (4.6.53), by assuming a so-

lution form of the form u = x exp(iλnt) to yield the standard eigenvalue problem

λn M x n = K x n. Before solution, Eq. (4.6.53) is further reduced by removing

zeroed DOF. After solution, the modes are sorted from lowest to highest frequency,

the displacements normalized to a maximum of 1 for the largest tip deflection. The

free-body (zero frequency azimuthal rotation) in-plane mode for the free-hub is also

discarded.

4.7 Control

This section describes the solution of the wind turbine control problem. Section 4.7.1

presents the derivation and implementation in ExcelBEM of an optimal steady-state

control scheduling algorithm, complete with equilibrium coning of the coning rotor.

Section 4.7.2 then provides a brief introduction to the synthesis of a dynamic con-

troller, including extraction of a suitable dynamic model for the task. The detailed

execution of the task is left for future work, as only constant speed rotors and fixed

applied hinge moments are considered in this thesis.

4.7 Control 137

4.7.1 Steady State Operation

BEM programs typically solve for steady state conditions with a fixed geometry and

flow conditions. The solution iterations are confined to the aerodynamics (a and a′),

possibly including yaw correction factors which require azimuthally averaged and it-

erated values. When searching for an operating point defined in the torque/speed

plane, another level of iteration is required, since there is no closed-form solution

to the operating condition problem. The wake iterations presented in §4.4.11 also

require iteration in a loop outside the induction factors, as does free-hinging opera-

tion.

BLADEDTM includes steady-state power curve calculations, but does not include

provision for a flap hinge in this mode. The definition of the control is also different,

requiring the same torque–speed relationships used as target schedules in dynamic

simulations. These must be provided by the user as either look-up tables or constants

defining a generic curve.

In the present work, steady solutions are required for variable geometry (equilib-

rium coning), as well as optimal energy capture and power limiting control schedules.

The latter are used to define operational target control schedules over the complete

wind speed range. Defined schedules with wind speed for applied actuator hinge

moment, cone angle, and rotor speed may also be defined. The latter two may be

used either as initial conditions for the optimum and limiting schemes, or as fully-

defined control schedules. The challenge of optimal operation for the coning rotor

is discussed further in §6.4.2.

4.7.1.1 Cone Angle Equilibrium

The aerodynamic loading on the blade will change with cone angle, as will the

centrifugal stiffening forces, creating a non-linear system. BLADEDTM does not

have a capability within its steady calculations to model this effect. Only static

pre-cone may be specified. Alternatively, dynamic simulations may be run to steady

state.

In ExcelBEM, a secant method [131] is run as an iterative loop outside the induc-

tion factor solution. The following equation is solved for the net moment about the

hinge:

Maero(β) +Mcentrifugal(β) +Mactuator = 0 (4.7.1)


The algorithm is initialized with the value of β in memory and a second cone angle

10% larger (or 2 for zero intial cone angle). A solution termination tolerance of

0.1 is used on β between iterates, together with solution lagging after 25 iterations.

4.7.1.2 Optimal Energy Capture and Limiting Operation

The operating point algorithm computes either deterministic performance over the

wind speed range, or an optimal control schedule (Region II) with or without limiting

(Region III). The overall flow of these calculations is shown in Fig. 4.36. Switches

for limiting (“Limiting?”) and optimization (“Optimizing?”) specify the output

quantity for iteration f(xi) at iterate xi. The f(xi) are either power or torque,

produced aerodynamically or electrically (including parasitic losses). Optimizing is

only done in xi = Ω; limiting may be with xi = Ω or xi = γpitch. The “Run BEM”

steps include the β iterations of §4.7.1.1, if a free hinge is defined.

Each of the scheduling steps is only done if a schedule is defined; otherwise nominal

values are used. The unconed λ is limited to less than 15 at Ωmax by limiting

Ω, in addition to user-input generator rotation speed bounds. This was done to

avoid numerical problems associated with free hinging and negative power (propeller

mode).

The limiting and optimization iterations are converged to 0.5% change in the

limit and nominal values respectively; the change in variables (Ω or γpitch) must be

less than 0.001Ωmin and 0.1 respectively. Robust search algorithms are used to

guarantee proper solutions in what is a highly non-linear problem.

Optimal solutions for Ω are found with a modified Golden Section search algorithm

[163]. The algorithm is initialized with bounding values Ωmin and Ωmax. The

function f(xi) in general has multiple local minima, so at each step of the algorithm

the f values on the bounds (Ωmin,Ωmax) are compared to the f(x1) and f(x2) values

on the interior of the standard algorithm. Contraction of the search domain then

occurs towards the absolute minimum, which may occur on the bounds.

For optimization, the pitch angle is assumed saturated at its upper or lower limit

for PTS and PTF control respectively, over the Region II range below rated. Fig-

ure 4.37 illustrates the speed windows ∆Ω defining the operational envelope of the

rotor, bounded by the limits on rotation speed, for two wind speeds. Point A rep-

resents the upper speed limit at a low wind speed, and B the optimal solution for

that wind speed.

4.7 Control 139

Figure 4.36 Operating point solution flow chart

The limiting iterations are done with a bounded secant method, rather than the

slower Regula Falsi method [131]. The algorithm is bounded for VSS rotors by

Ωmin . . .Ωmax and by γpitch,min . . . γpitch,max for PTS and PTF. The starting point

x1 is Ωmax for VSS, γpitch,max for PTS and γpitch,min for PTF. An initial line search is

performed in the variable to locate x2 where f(x1)f(x2) < 0, or until the other bound

is reached, by stepping towards the opposite bound. This line search is necessary to

accommodate all potential shapes of power/torque curves (e.g. negative slopes). If

the other bound is reached, the algorithm is terminated and limiting is impossible.

Otherwise, the modified secant algorithm is used with x1 and x2 as starting points.


Figure 4.37 Speed windows for optimal and limiting operation

At each step, the xi step is bounded to the variable limits.

The feedback path from optimized solution to limiting operates only on Ω. This

step is required in some cases when the speed window is straddling the peak of the

power curve, as shown in Fig. 4.37 by point C. The wind speed is large enough

in these cases that the dimensional power (or torque) exceeds the limit value. In

these cases, both D and E are valid solutions, as they both correspond to the CP

for limiting.

The correct choice depends on the operating strategy. For VSS, Point E is an

appropriate choice, since the stall half of the curve is to be tracked past rated power.

Tracking D would yield lower drive-train torque, but would make a controller very

hard to design. Eventually, a drastic speed reduction would be required at higher

wind speeds to move from the drag-limited right half of the CP curve to the left stall-

limited range. For the pitch controlled rotor, the choice is taken to track point D,

since the rotor can eventually use pitch when required to limit. This is only one

possible choice; ideally point C’ would be located, but this would require a more

costly multi-dimensional optimization. In general, these are pathological cases in any

event, and will only be encountered if the upper rotor speed limit is unrealistically

high. In reality, there is usually some constant speed mode near rated to smooth

controller mode switching and/or limit tip-speed.

4.8 Generator Modelling 141

4.7.2 Dynamic Control

The purpose and time-frame of the current project has precluded any detailed work

on a dynamic controller for the coning rotor. Connor et al. [164] detailed the con-

troller design for the CONE-450 to effect the steady-state control reference targets

developed along the lines of §4.7.1. A relatively standard PI-type was used, with

estimators for aerodynamic torque and consideration of control-mode switching.

The design of the controller was carried out by linearising the EOM for the 2

DOF system, in that case a single central actuator (i.e. β flapping) and rotational θ

DOF. A similar approach may be taken on the decoupled flapping concept. Equa-

tion (4.6.20) is first fully linearised around an operating point, taking due care of

the equation parameters that are functions of the linearising parameters (θ, θ, βj

and βj). By then taking the Laplace transform of the equations, the constant terms

of the linearisation disappear, and a set of 1 +B coupled equations result.

The equations are expressed in perturbation variables of the linearising param-

eters. They contain parameters dependent on the mass properties of the system,

and the gradients of the aerodynamic driving (θ direction) and coning (β direc-

tion) torques. The latter gradients are determined numerically from maps of the

aerodynamic performance, computed at the linearisation point. Using the standard

representation of the system thus developed, controller synthesis techniques may

then be applied for a set of operating points from start-up through shut-down. In

the original work [66], Bode plot transfer function shaping was used.

An alternative approach would be to numerically linearise the system directly from

a full dynamic simulation, including any flexible body modes. This is a relatively new

approach for wind turbines, and may be carried out in BLADEDTM. The output is a

state-space model described by internal state variables and control outputs. If pitch

control is to be considered in a revised coning rotor design, this approach may prove

expedient, and will certainly be more accurate. Of course, a whole host of controller

synthesis methods may then be applied (e.g. loop shaping, PID, state-space), as

they have been for standard wind turbines.

4.8 Generator Modelling

Generator analysis, similar to the aerodynamic and structural analyses presented

earlier, can be conducted at various levels. The focus here is on the magnetic

circuit, lying at the heart of the generator and driving the cost. The most detailed


analysis required for final design work includes FEM simulation of the magnetic flux

and temperatures through the rotor and stator, over the operating conditions of the

machine.

Clearly, this level of analysis is well outside the requirements of the current work.

Instead, what is required is a mix of a very simple model using basic parameters, and

a more involved parametric analytic model. Both will be required in Part III to place

bounds on the generator capacity, so that together with the costs given in §4.9.2, the

generator requirement does not become unrealistic. To that end, the following two

models are considered. Section 4.8.1 computes representative values for the simplest

shear stress model, while §4.8.2 briefly introduces a parametric model developed by

Dubois [165].

4.8.1 Shear Stress Generator Model

For electrical machines with low rotation speed, the design is driven by the torque

requirement, rather than the machine power rating [166]. This is because the mag-

netic fields must be quite strong to develop sufficient torque according to:

τgen =Pgen

Ωgen(4.8.1)

The magnetic forces developed by the interaction of magnetic fields (created by the

magnets and energised windings) manifest themselves as an effective shear stress

σag in the air gap between rotor and stator. The shear stress is itself fundamentally

limited by the magnetic flux density that can be accommodated in the iron core and

magnets. Considerations of rotor and stator geometry and thermal management

serve to further limit the maximum value that σag can attain.

The thermal management in particular has a large effect. Concepts employing free

convection are quite limited relative to actively cooled designs. Of course, there are

trade-offs in complexity and additional cost of the cooling system. Once the general

concept of the generator is determined, σag can be used to estimate an overall size:

τgen = 2πRlσag (4.8.2)

of air-gap radius R and length l.

Backing out the value of σag for a freely cooled design, the Vensys 1.2 MW [81]

direct drive machine, yields approximately 14.5 kPa. For comparison, an actively

cooled 1.5 MW design uses a value of 42.1 kPa [167].

4.9 Cost Modelling 143

4.8.2 Analytic Magnetic Circuit Generator Model

The equations governing magnetic fields may be integrated over the cross-sectional

area and average path lengths of a magnetic device, to yield a system of equations

analogous to electric circuits [168, 169]. The basic elements are the magnet MMF

and winding nI “motive forces”, reluctance R, and flux Φ, equivalent to the voltage

V , resistance R, and current I of an electric circuit.

Dubois [165] has constructed this type of model for sizing of a radial-flux generator.

That model has been implemented in a spreadsheet analysis by the current author,

to further investigate parametric variation of the generator design parameters in

Part III. The details of the model can be gleaned from the original thesis. The

only modification to the model that has been made is to change the dimensioning

variables to describe an outer rotor (magnets) and inner stator (windings).

For optimization purposes, a diameter D and rotation speed Ω are specified as

parameters, and efficiency η as a constraint (95%). The DVs are then pole pitch τp,

current density J , no-load flux density in the air-gap Bg, tooth/slot length ratio in

the stator bt/bs, an the ratio of stator length to diameter Krad. All parameters are

given realistic limits. Constraints are also included for field saturation and short-

circuit currents. An additional constraint for σag < 25 kPa as also imposed in the

present work. Using Excel’s solver, it has been possible to optimize for a specified

power or torque requirement.

4.9 Cost Modelling

Obviously an accurate measure of the cost of any new concept must be developed

and compared to the existing state-of-the-art in order to definitively demonstrate an

economic advantage. Cost models vary greatly for various scoping studies, ranging

from simple functions of diameter and power, through aggregated mass dependent

cost functions for individual components, and ultimately vendor tenders for fully-

defined components. Mass dependent models [16] typically incorporate a fixed and

variable component:

c = cref

((1− µ) + µ

v

vref

)(4.9.1)

where cref is the cost of the reference machine/component and µ the percentage of

the cost assumed to vary with metric v (e.g. rotor area πR2, mass m).

The refinement of the cost model should be commensurate with the level of engi-

neering analysis and stage in the design process. The level of detail of the cost model


will also dictate the confidence that can be placed in any optimization results. The

CONE-450 report [66] presented a detailed accounting from manufacturers of the

costs of major components, especially those specific to the coning rotor. This level

of detailed cost is appropriate at a highly refined, engineering study stage. Given

the resources and focus of the current work, this level of cost detail is inappropri-

ate and beyond the current scope. Ultimately, any revised coning concept must

be accounted for in a similar manner to the original study. However, this activity

is better tackled by a commercial design study. The current approach is a proxy

cost metric, overviewed in §4.9.1, and more quantifiable estimates for the magnetic

material costs (§4.9.2).

4.9.1 Proxy Cost Metric

In Part III, the strictly technical merits of the concept are of primary focus. For

the most part, cost is only accounted for by a proxy metric, material mass. This

approach effectively sets µ = 1 and v = m in Eq. (4.9.1). Even cost estimates

for bulk quantities of material can vary widely, depended on market conditions

and international exchange rates, making a fundamental comparison difficult. The

attempt here is therefore to examine the performance trade-offs of various design

choices, leaving to more detailed future studies the task of determining economic

optimality, in current market conditions.

This approach is viable, as the rotor blades are the primary component under

consideration, so that conventional materials and techniques can be compared on a

like-for-like basis. This should make mass a fairly consistent and reliable measure

of cost-effectiveness when comparing the coning rotor to a standard rotor.

As already mentioned, parts such as the coning actuators and hub are unique to

the concept, and hence will require more detailed work in the future. Based on the

original work, they should not drive the cost of the concept to any great extend,

although reliability may be of concern.

4.9.2 Magnetic Materials Cost

One component that has been examined in more detail is the generator, for reasons

that will become clear in Chapter 6. For a generator, the value of µ in Eq. (4.9.1)

will be quite high for the active materials used. Some notion of the installed costs of

these materials has therefore been collated in Table 4.2. The costs include both raw

material and assembly into a generator. They clearly cover a large range, confused

4.9 Cost Modelling 145

further by currency conversion, temporal supply/demand variation affecting cost,

and the details of the generator design (winding layout, fabrication techniques).

The magnet costs are representative of rare-earth materials (NdFeB most common).

Table 4.2 Active magnetic specific material costs

Item Dubois [170] (e/kg) Polindera(e/kg) WindPACT [167] ($US/kg)

Copper 6 10–25 13Iron 6 3 5Magnet 40 20 50a Data from personal communication with Dr. Polinder, Delft University of

Technology, Laboratory of Electrical Power Processing, Delft, The Nether-lands, September 2004

Chapter 5

Validation

In order to have confidence in the analysis methods of Chapter 4 for use in the design

work of Part III, they must be validated to the extent possible. The aerodynamic

models are validated first in §5.1, against externally available CFD and experimental

results, the former for both idealized and real geometry. The acoustic model is

checked next in §5.2, followed by the structural sectional and beam models in §5.3.

In §5.4, the outputs of BLADEDTM are investigated to ascertain the modifications

required to properly analyse the coning rotor. Finally, a brief verification of the

generator model is made in §5.5.

5.1 Aerodynamic Validation

Aerodynamic validation is done for three increasingly complex cases, for both coned

and unconed rotors. First, a uniformly loaded theoretical rotor is examined numeri-

cally (§5.1.1–§5.1.4), to exclude blade shape and airfoil effects. Secondly, numerical

simulations are compared for a real rotor in §5.1.5, to include airfoil effects. Wind

tunnel test data for a large rotor is then compared to ExcelBEM predictions in

§5.1.6. The aerodynamic behaviour in dynamic inflow conditions is then compared

to BLADEDTM in §5.1.7. Finally, numerical predictions are compared for a yawed

rotor in §5.1.8.

It should be noted that experimental results for wind turbines are quite sparse, and

the NREL UAE in §C.2 is virtually the de facto standard comparison dataset. The

primary problem in turbine testing, as opposed to say aircraft, is the requirement for

large models to simultaneously match Re, λ and geometric similarity. Atmospheric

testing is troubled by the non-uniform inflow that cannot be input exactly into

simulations for comparison. Very large wind tunnels are therefore required, doubly

so since the wake must freely expand, demanding tunnels much larger than the rotor.

The results in this section are identified by the following labels and unless other-

wise noted, indicate results from the modified BEM method of §4.4 (ExcelBEM):

147

148 Chapter 5 Validation

• “CT Bladed” and “CT Spera” indicate the thrust model used in ExcelBEM

(see §4.4.3).

• “Unex” and “Ex” indicates unexpanded and expanded wake geometries in

ExcelBEM (see §4.4.11). “Unex” is implicit unless otherwise stated.

• “Madsen” and “Mikkelsen” indicate CFD study results.

• “BLADED” indicates BLADEDTM code results.

5.1.1 Uniformly Loaded Rotor

The classic uniformly loaded rotor (with no tangential loading fθ) provides a valu-

able check on the modified BEM method. At low induction factors with an infinite

number of blades, the vortex model geometry should be an almost exact representa-

tion of the flow conditions. With constant loading, vorticity is only shed at the tips.

No swirl is present for the purely axially loaded rotor (i.e. a′ irrelevant). No airfoil

data is used, avoiding any ambiguity associated with stall delay, etc. Standard BEM

formulations predict a constant induction a along the blade, regardless of cone angle

β or CT . The power and thrust coefficients are also constant when referenced to the

coned disc area.

The numerical results from two full-field CFD studies were used for compari-

son1: Madsen and Rasmussen [106] used an axisymmetric Navier-Stokes model with

volumetric body forces representing the blade loading; Mikkelsen et al. [88] and

[89] adopted a vorticity-swirl-streamline model, in both axisymmetric and fully 3D

actuator line formulations, but only the axisymmetric portion of the work is rele-

vant here. Madsen’s study investigated only downwind coning on straight, winglet

equipped and curved blades. Mikkelsen only looked at straight rotors, but covered

upwind and downwind coning, as well as real rotor performance, to be considered

in §5.1.5.

The unit area axial loading fu,z of the disc is specified according to the desired

thrust coefficient CT . For the applied loading to be perpendicular to the coned rotor

surface, the normal unit loading fu,n at the set coning angle is found from:

fu,z = fu,n = CT12ρV 2

0 (5.1.1)

Power output is calculated from:

P = −R∫

0

~V1 · ~fu,n2πrds (5.1.2)

1Note that the CFD studies define cone angle in the opposite sense.

5.1 Aerodynamic Validation 149

Numerically this equation is evaluated between each blade section assuming linear

variation in velocities (induction factors, ε, etc.).

5.1.2 Uniform Loading Results

The unconed rotor is examined first in §5.1.2.1, followed by the coned rotor in

§5.1.2.2.

5.1.2.1 Unconed Rotor

The most basic case of the unconed rotor is presented first in Fig. 5.1. The current

method with no wake expansion predicts uniform induction, as per the standard

theory. With the inclusion of wake expansion however, the correct trend in induction

(increasing towards tip) is predicted. The thrust model is also observed to have a

large effect on the results, with the Spera model giving lower a as expected from

Fig. 4.5. Note that Mikkelsen’s CFD results deviate from Madsen’s, illustrating that

even a full-field method is subject to considerable uncertainties in this simplest of

cases.

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

az

CT_Bladed, Unex

CT_Spera, Unex

CT_Bladed, Ex

CT_Spera, Ex

Madsen

Mikkelsen

Figure 5.1 Uniformly loaded axial induction (β = 0, CT = 0.89)

At low induction factors, the CFD method, like BEM without expanded wake

geometry (the light grey horizontal lines in Fig. 5.2), predicts a uniform induction.

As CT rises from 0.4 to 1.0 in Fig. 5.2, the induction factor near the tip of the disc

grows in both the CFD results and the corrected BEM method with wake expansion.


Again, this is predicted qualitatively by the vortex model, as outboard sections see

more influence from the proximal shape of the sheet. Near the centre of the disc,

the induced velocity is reduced as the wake moves outboard, away from the disc.

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.0 0.2 0.4 0.6 0.8 1.0

r/Rtip

az

CT = 0.4, CT_Bladed0.4 Mikkelsen0.8 CT_Bladed0.8 Mikkelsen0.89 CT_Bladed0.89 Mikkelsen1.0 CT_Bladed1.0 Mikkelsen

Figure 5.2 Uniformly loaded axial induction as a function of loading CT (β = 0)

5.1.2.2 Coned Rotor

The present formulation is able to predict the correct induction factor trend for both

upwind (−β) and downwind coning (+β), as shown in Figs. 5.3 and 5.4 respectively.

The variation over a wider range of cone angles excluding wake expansion is shown in

Fig. 5.5. The downwind results of primary interest are better predicted, for reasons

elicited later.

The somewhat unintuitive result that the CT Spera induction factors are higher

at the centreline than the CT Bladed ones in this case owes to the adisc indexing

of the the thrust models.1 Induction factors normal to the blade an follow similar

trends to the axial ones a = az and are not discussed further.

Returning to the flow field picture in Fig. 4.8, for positive coning, the disc is

sitting in a region of reduced influence from the wake cylinder, while the reverse is

true for negative coning angles. In order to balance the momentum equations in

1Referring to Fig. 4.5, for the same CT , the Spera model has a lower adisc. This is stilltrue in the downwind case, from the definition of adisc.


0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

az

CT_Bladed, Unex

CT_Spera, Unex

CT_Bladed, Ex

CT_Spera, Ex

Madsen

Mikkelsen

Figure 5.3 Uniformly loaded axial induction (β = 20, CT = 0.89)

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

az

CT_Bladed, Unex

CT_Spera, Unex

CT_Bladed, Ex

CT_Spera, Ex

Mikkelsen

Figure 5.4 Uniformly loaded axial induction (β = −20, CT = 0.89)


0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

az

-40-30-20-10010203040

Figure 5.5 Uniformly loaded axial induction as a function of β (deg) for Bladedthrust model (CT = 0.89, no wake expansion)

light of greater than and less than unity εz respectively, the induction factors must

be inversely modified.

The aggregate power coefficients (based on coned area) are compared next in

Table 5.1. At CT = 0.89, classic BEM theory predicts a power coefficient of 0.593,

the Betz limit. The CFD models and modified BEM all predict reasonably close to

this value for the straight and downwind coned rotor in all cases. The invariance of

CP found here and in the CFD results is explained by the wake structure, which for

the uniformly loaded rotor is determined uniquely by the thrust level. The attendant

wake structure results in identical power outputs for equally sized rotors, when a

control volume bounded by the wake and far-field upstream/downstream boundaries

is considered.

Table 5.1 Uniformly loaded power coefficients (CT = 0.89)

CPβ (deg)

0 20 -20

Mikkelsen 0.601 0.601 0.604Madsen 0.573 0.571 -CT Bladed, Unex 0.593 0.593 0.592CT Spera, Unex 0.619 0.618 0.620CT Bladed, Ex 0.586 0.588 0.586CT Spera, Ex 0.616 0.615 0.616


Within like thrust and wake expansion models, the variation of CP with β is at

the level of numerical accuracy for the modified BEM method. The Betz limit was

of course derived from the momentum thrust equation, which from Fig. 4.5 may

be slightly low at this induction level. This effect provides some explanation for

Mikkelsen’s and the CT Spera cases’ exceedance of the Betz limit.

5.1.3 Fundamental Model Deficiencies

The results obtained demonstrate that it is too simplistic to dismiss BEM as invalid

because of the stream-tube independence assumption or planar disc derivation. It is

in fact the classic Bernoulli derived relation between disc induced velocity and far-

field velocity that is presumptuous, once a radial flow component is included and the

mass flow term correctly derived. In fact, Glauert discussed the induced velocities

of the shed vortex wake as the key link between momentum and blade element

theory [98]. For convenience, the Bernoulli rationalisation is usually presented, but

this approach bypasses the core understanding of the physical situation required to

account for coning.

Clearly, there are still deficiencies in the present method, even for the most basic

uniformly loaded case. As stated, the method is strictly only valid for lightly loaded

conditions, implying no expansion of the wake. Based on the results, inclusion of this

geometrical effect is capable of resolving the bulk of the disparity between reality and

the standard BEM method. Leaving aside problems with 2D airfoil predictions, such

as stall delay, present for real rotors (and not present for the uniformly loaded rotor),

the key model uncertainty is in the thrust model, as evidenced by the variation in

the results with the chosen model. The deviation is much worse for rotors with large

a, encountered more in the upwind case than the downwind one.

It is instructive to consider the origin of the breakdown in the momentum thrust

model. For a lightly loaded rotor, the wake is confined to a thin shear layer, well

modelled by a thin sheet of vorticity. As the loading increases, the layer thickness

will grow and follow the tip streamline (i.e. expand), but is still represented well as a

vortex sheet. At still higher loadings, the wake thickness will grow even larger before

finally filling the entire wake space as unsteady large-scale eddies. This is analogous

to inviscid airfoil modelling, which predicts ever increasing lift, by neglecting the

large separated region at high angles of attack. Even strongly coupling a viscous

boundary larger to an inviscid outer flow is only valid up to a limit slightly beyond

stall.


The CFD results were obtained by inviscid steady-state formulations. As a field

method, it is capable of resolving the thickening of the wake shear layer, and so

correctly predicts the induced velocity at the disc. Mikkelsen does report however,

that as loading is increased past CT = 0.89 and for low wind speeds (both high

a conditions), steady-state convergence is increasingly hard to obtain, eventually

becoming impossible. The unsteady flow is also mentioned as an upper constraint

on λ in Nygaard’s Euler flow solution [171]. The wake state of a uniformly loaded

rotor has been studied specifically using a time-marching full-field scheme to track

the development of the flow field from initial uniform conditions to steady state

[172]. It was found that the turbulent wake state is in fact an unstable transition

between the windmill brake state and a vortex ring or propeller brake state flow,

characterized by a recirculation region through and upstream of the rotor.

Methods based solely on vortex filaments, using the Biot-Savart equation to eval-

uate induced velocities, have been tried to correct for high tip speed conditions

[128, 173–175]. Numerical convergence problems appear and in any case they are

steady formulations that are only capable of accounting for wake expansion, as is

done here. The pitch of the wake filaments is considered, but in one approach de-

generates to the momentum thrust equation [174]. Yet another study assumes all

turbulent mixing to occur in the very far wake [176], which must occur for the flow to

return to free-stream, patched onto the conventional up-stream-downstream inviscid

analysis.

The thrust models are therefore an attempt to capture unsteady viscous behaviour

within the confines of a steady inviscid method. Both the Spera and Bladed models

are fits to the experimental data points in Fig. 4.5, which are most likely unsteady

conditions as evidenced by their scatter and CFD results [172]. BEM’s implicit

stream-tube independence assumption is therefore hard pressed to deal with either

the unsteady mixing or steady-state recirculation flow fields.

The upwind coned case is less accurate because of geometry. As the wake sheet

thickens, the meanline moves towards the centreline, and for the upwind case to-

wards the blade. This in turn increases εz which is below unity in this case, and

hence reduces the predicted induction. This effect can be seen by comparing the

unexpanded and expanded wake cases in Fig. 5.4; in the latter the wake is further

from the centre of the rotor and so εz is larger and a smaller. The effect can also be

thought of as a result of the stream-tube independence assumption, since turbulent

mixing will bring energized flow in from the tip towards the core of the flow. This


will have a non-uniform effect on the inboard sections of an upwind coned rotor as

the flow mixes in downstream of the rotor tip towards the root sections.

The straight and downwind coned cases are relatively insensitive in a inboard

because of the relative position of wake and blade. Further prediction of this effect

is beyond the scope of this work, as the downwind case is of primary interest, but

is clearly required for a better understanding and prediction of high a performance.

5.1.4 Radial and Far-Field Flow

The vortex model confirms the CFD results that the distribution of radial velocity

along the blade is nearly independent of cone angle. Figure 5.6 shows the pre-

dicted radial velocity for the CT Bladed expanded wake. Although the magnitude

is marginally and uniformly higher than the CFD results, the variation between

cases is negligible.

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

vr/V0

-20 CT_Bladed, Ex

-20 Mikkelsen

0 CT_Bladed, Ex

0 Mikkelsen

20 CT_Bladed, Ex

20 Mikkelsen

Figure 5.6 Uniformly loaded radially induced velocity as a function of β (deg)(CT = 0.89)

This result is best explained by reference to Fig. 4.8, noting the radially induced

velocity isocurves. As they are centred about the tip, and a wake will be shed from

that point in the coned and unconed configurations, the radial component is almost

constant in a circle about the tip. The result is only slightly skewed by the fact

that a constant tip radius requires a longer blade for a coned rotor. The isocurves

will therefore cross the blade at modified points. Notice that Mikkelsen’s BEM


method radially induced velocity vector (normal to blade) is in the wrong direction

for upwind coning, since the radially induced velocity is in fact outboard in all cases.

The CFD results predicted uniform induction across the far-wake, inside the limit-

ing stream-tube. This is consistent with the vortex model which predicts a uniform

induction downstream away from the leading edge of the sheet (analogous to the

magnetic field in an infinite solenoid).

5.1.5 Tjaereborg Rotor

The modified BEM formulation is immediately applicable to real rotor geometries,

unlike the more qualitative results presented by Chaney et al. [100] for an assumed

unmodified induction factor. Mikkelsen’s CFD study included results for the Tjaere-

borg rotor, which is considered presently (see Appendix C.1 for machine details).

The root of the blade is at 1.46 m radius; however, the CFD results were presented

to the centreline. No 3D stall delay effects (see §4.4.8 and 4.4.9) were included in

the CFD study, so spanwise flow effects were not included in the ExcelBEM runs

either. The induction factors shown are the product of tip loss factor F and calcu-

lated a value, to show the average induction. All single operating point results are

for 10 m/s free-stream with a rotor speed of 22.1 RPM.

5.1.5.1 Unconed Rotor

The induction factor distribution for the unconed rotor is shown in Fig. 5.7. The

effect of the thrust model is again evident, lowering the curve for the Spera model

results. No results including wake expansion corrections are shown,1 as the inclusion

of tip loss factor F mitigates the tip-effects seen in the uniformly loaded case, by

forcing aF to zero at the tip. The Bladed thrust model results are almost identical to

those from BLADEDTM, since radial flow does not affect either the axial momentum

balance or flow relative to the section (i.e. aerodynamic force calculation) for an

unconed rotor.

5.1.5.2 Coned Rotor

The predicted induction factors are examined first in §5.1.5.3, as these are good in-

dicators of the efficacy of the aerodynamic model. Loading predictions are presented

second in §5.1.5.4 and are of ultimate interest in performance prediction for design.

1Almost identical results to the unexpanded case were obtained by including wake ex-pansion.


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

az

CT_Spera

CT_Bladed

Mikkelsen

BLADED

Figure 5.7 Tjaereborg axial induction (β = 0)

5.1.5.3 Induction Results

The downwind coning case is examined first in Fig. 5.8. As the radial velocity

component becomes important, the BLADEDTM results are much higher than all

of the other results. The modified BEM results for both Spera and Bladed wake

models track the magnitude and trend of Mikkelsen’s CFD results quite well. As

ac is lower for the Spera thrust model, it predicts lower a values, compared to the

Bladed thrust model that produces results closer to the CFD ones, especially over

the outer portion of the blade.

The alternate wake geometries (see §4.4.12) provide a marginally better prediction

at mid-span, but overall degrade the accuracy of the results. Results for the single

inboard model are not shown inboard of 0.2Rtip, as it predicts zero induction (εz = 0)

for r < 0.2Rtip. Obviously the chosen vorticity distribution for the multiple vortex

model is arbitrary; iteration on the BEM calculated loading back into the shed

vorticity calculations would have doubtlessly improved the results. Unfortunately,

that iteration would detract from the simplicity of the BEM equations.

Figure 5.9 shows the upwind coning predictions for the various models. The Spera

thrust model again predicts lower a than the Bladed thrust model, as the induction

factors are above ac in both models. This is particularly evident near the tip where

aF rises for the Bladed thrust model, since the values of a become quite large as


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

az

CT_SperaCT_BladedMikkelsenBLADEDCT_Bladed, SingleCT_Bladed, Mult

Figure 5.8 Tjaereborg axial induction (β = 20)

F → 0. The single vortex wake geometry moderately improves the results near to

the attachment point at 20%Rtip. Otherwise, both alternate wake geometries again

generally degrade the results. The modified BEM method does again follow the

right trend, outperforming the BLADEDTM results in the sense that BLADEDTM is

invariant with the coning direction. The radial component becomes more important

in the real rotor case, as the disc loading must be predicted from the velocity relative

to the blade, creating a double dependence on the radial velocity.

The predictions for β < 0 are generally poorer than those downwind. This may

be explained by reference to Fig. 4.8, noting that a cylindrical vortex sheet does

not produce much influence upwind and at greater radii than the sheet’s own outer

radius. For the downwind coned rotor, this means that most of the rotor is sitting

in a region of reduced sensitivity to the sheets, due to the variable loading along

the blade length. In the upwind coned case however, the entire blade is inboard

and behind the forward lip of the sheets shed by the outer portions of the blade.

The cumulative influence is therefore much greater and felt increasingly towards the

root, as evidenced by the increasing deviation between BEM and CFD results in

Fig. 5.9 towards the root.


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

az

CT_SperaCT_BladedMikkelsenBLADEDCT_Bladed, SingleCT_Bladed, Mult

Figure 5.9 Tjaereborg axial induction (β = −20)

5.1.5.4 Loading Results

The aerodynamic loading on the blade is shown in Fig. 5.10. For the unconed case,

the CFD and BEM results agree quite well. The BLADEDTM and CT Bladed results

are almost identical for the unconed rotor, demonstrating recovery of the standard

BEM results at β = 0. When coned downwind, the modified BEM method tracks

the CFD results fairly well, under-predicting around r/Rtip = 0.8. For upwind

coning the loading is uniformly under-predicted. BLADEDTM predicts the same

distribution for ±β, under predicting the modified BEM method and CFD in all

cases for r/Rtip < 0.9. In terms of integrated loading at s = 1.46 m (the blade

root location), for β = 20 the shear force predicted by BLADEDTM is 6.6% below

the modified BEM results and bending moment 4.9% too low. For β = −20, the

loading under-predictions are 3.8% and 5.2% respectively.

The other main aggregate performance metric is the power coefficient CP (based

on coned radius), shown in Fig. 5.11. The figure is fully non-dimensional, but

obtained from the same operating points as used in the CFD study, which set the

average pitch angle of the blade as a function of wind speed. With no coning, the

CFD results are higher but in relative agreement. The shift with downwind coning

is similar for both CFD and modified BEM method, given the initial discrepancy

at β = 0. The upwind coning prediction is much less well predicted by the current


0

500

1000

1500

2000

2500

3000

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

F x, a

ero (

N/m)

-20 Mikkelsen-20 CT_Bladed0 Mikkelsen0 CT_Bladed20 Bladed

0

500

1000

1500

2000

2500

3000

0.0 0.2 0.4 0.6 0.8 1.0r/Rtip

F x, a

ero (

N/m)

0 Mikkelsen0 CT_Bladed0 BLADED20 Mikkelsen20 CT_Bladed20 Bladed

Figure 5.10 Tjaereborg aerodynamic loading as a function of β (deg)

method, due to the reasons previously discussed.

BLADEDTM is of course only capable of a single prediction for an absolute coning

angle. The shift in CP is much larger than either modified BEM or CFD in both

cases. The discrepancy at β = 0 for both BEM methods presumably relates to the

thrust model, which is absent from the CFD predictions. This is evidenced by the

increasing agreement between CFD and BEM towards lower λ, as BEM transitions

from the empirical thrust curve to the momentum derived one.

The modifications considered in this section all relate to a more accurate treatment

of induction at the rotor, and can therefore be expected to have reduced significance

at low λ (low induction). This is shown in Fig. 5.11, but note that the corrections

are substantial down to λ = 5, corresponding to a V0 of 14 m/s for the Tjaereborg

machine. This covers the critical Region II energy capture wind speed range when

the rotor is tracking optimal CP up to rated power. A coning rotor derives its energy

capture advantage in this region, so accurate prediction of power over this range is

important. For the coning rotor, the accurate prediction of root bending moment is

also important, for determining operational coning angle and loading.

5.1.6 NREL UAE

The UAE dataset provides a unique repository of data collected from a large-scale,

controlled wind tunnel rotor test. Appendix C.2 gives the background and relevant


0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

4 6 8 10 12λ

C P

-20 Mikkelsen-20 CT_Bladed0 Mikkelsen0 CT_Bladed20 Bladed

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

4 6 8 10 12λ

C P

0 Mikkelsen0 CT_Bladed0 BLADED20 Mikkelsen20 CT_Bladed20 Bladed

Figure 5.11 Power coefficient map for rotor operating points as a function ofβ (deg)

technical details of the UAE dataset. Two test sequences are of immediate interest

in relation to exploring coning rotor aerodynamics, from among the numerous data

campaigns available. The first is Sequence S, the upwind baseline case with no probes

installed on the baseline blade. The blade tip pitch was 3, teeter was locked out

with rigid links, and yaw angle 0. The second case, Sequence F, is for a downwind

rigid rotor coned to 18. Sequence S has data available from 5–25 m/s and F from

10–20 m/s.1 Sequence C was the downwind baseline case, but was not examined

here because it had a rigid cone angle of only 3.4.

5.1.6.1 NREL UAE Baseline

Before the results of the wind tunnel tests were made public, a competition was held

to predict the results of the various test sequences. The predictions were made solely

on the geometric and operational parameters of the turbine, and a number of airfoil

datasets for the blade airfoils were provided. A variety of codes were used from BEM

to full CFD. It turned out that even for the most simple, steady case, Sequence S,

1Hand et al. [177, p. 17] states that “excessive inertial loading prevented operation atlower wind speeds”. Upon clarification with NREL, this comment refers to an excessivecentrifugal force along the blade that is not counterbalanced by the thrust loads until higherwind speeds.


the predictions varied greatly from around stall to 25 m/s [178]. Only one full CFD

method was able to predict the aerodynamic behaviour with any accuracy.1

Since then, numerous authors have analysed the data [104, 109], focusing on the

stall-delay phenomenon. Gerber et al. [179] have used a vortex lattice method to

determine the flow conditions at the blade sections (principally AOA), to then back

out the lift and drag properties from the pressure tap measurements. It appears that

a complicated stall region is developed on the blade, with the mid-section isolated

by strong vortex “stall fences” inboard and outboard. All this is to say that all but

the most advanced CFD codes have significant trouble predicting this set of data.

5.1.6.2 Sequence S

Figure 5.12 presents predictions and measurements of the power curve and normal

force coefficients (cn) for Sequence S. Note the error bars for a few points on the

measured data. Even in the controlled conditions of the wind tunnel, there was

significant variation of power with azimuth angle.

The results are fairly typical of other BEM predictions [104], matching very well

before stall occurs. The BLADEDTM and ExcelBEM results without stall de-

lay (“Plain”) are in exact agreement, and under-predict power drastically in the

stalled region. Including stall delay in ExcelBEM (“w SD”) improves the prediction

markedly, but still with significant deviation from the measured results. The cnplots (obtained from pressure tap data) clearly illustrate the delay in stall of the

sections. While the inboard sections under-predict cn with the stall delay model,

the outboard ones over-predict the power, leading to the over-prediction and under-

predictions seen respectively in lower and upper parts the power curve.

5.1.6.3 Sequence F

Bearing in mind the difficulties of predicting even the most simple case, the re-

sults for the coned Sequence F are given in Fig. 5.13. The first point to note is

the generally low induction factors in Fig. 5.13(b), even at low wind speeds. This

means that unfortunately, the results are relatively insensitive to the aerodynamic

coning corrections. The error bars on the measurement results are also much larger,

presumably owing to the greater influence of the coned rotor inertia on the torque

strain gauge measurements.

1The machine and blades were quite rigid, so structural response had only a second-ordereffect on the results.


0

2

4

6

8

10

12

5 10 15 20 25Wind speed (m/s)

Aero

powe

r (kW

)

MeasuredBLADEDPlainw SD

(a) Power curve

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25Wind Speed (m/s)

C n

MeasuredExcel BEM

(b) 0.30r/R

0.0

0.5

1.0

1.5

5 10 15 20 25Wind Speed (m/s)

C n

MeasuredExcel BEM

(c) 0.63r/R

0.0

0.5

1.0

1.5

5 10 15 20 25Wind Speed (m/s)

C n

MeasuredExcel BEM

(d) 0.95r/R

Figure 5.12 NREL UAE Sequence S measurements and predictions (cn datainclude centrifugal stall delay)

The BLADEDTM and “Plain” results are quite similar, the difference owing to

the geometrics of the two solution algorithms. Addition of the centrifugal pumping

stall delay effects (with SD) improves the predictions markedly, especially around

rated. Adding the coning correction factors εz and εr (“w C” and “w C & SD”)

only slightly modifies the results without and with stall delay. Finally, including

sweep stall delay effects as well (“w C & SD & S”) improves agreement with the

measurements. Ultimately, experimental results at higher induction factors, and

hopefully less complicated stall behaviour, are required to better compare analytic


0

2

4

6

8

10

12

5 10 15 20Wind speed (m/s)

Aero

powe

r (kW

)

MeasuredBLADEDPlainw Cw SDw C & SDw C & SD & S

(a) Power curve

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.1 0.3 0.5 0.7 0.9r/R

a

5 67 89 10

(b) Induction factors for low windspeeds (m/s) with coning corrections and

stall delay

Figure 5.13 NREL UAE Sequence F measurements and predictions

predictions.

5.1.7 Dynamic Inflow

ExcelBEM in unsteady flow has been validated against BLADEDTM, to the ex-

tend possible (i.e. without coning or yaw). First, a simple blade of uniform chord

and twist and airfoils with 2πα lift curve were run with dynamic inflow solution

method. No dynamic stall model was turned on. The difference in results between

BLADEDTM and ExcelBEM was extremely small, and related to the solution toler-

ance on a.

Next, the same blade profile but with LS1-GH airfoils (from §C.3) was run to

obtain results for both equilibrium and dynamic wake solutions. Again no stall delay

model was used, but wind shear was added. Again, the results were in agreement,

but required a solution tolerance on a of 10−5 for exact agreement.

Finally, the dynamic stall models were turned on in both BLADEDTM and Ex-

celBEM. As expected, the results were not exactly identical, as expected from the

difference in dynamic stall models. Both models did produce lift hysteresis loops.

The BLADEDTM model exhibited this feature well below stall as well, since it is

more advanced and includes unstalled unsteady aerodynamic effects.


5.1.8 Yawed Flow

Mikkelsen [89] also compared his CFD results to actual test data for the real Tjaere-

borg machine §C.1. The datasets available were azimuthally binned “flapwise”

bending moments (properly My) at the blade root (r = 1.46 m). The CFD re-

sults, “Mikkelsen” in Fig. 5.14, are compared against the experimental “Exp”,

BLADEDTM “BLADED” and predictions from ExcelBEM. The CFD results (and

of course “Exp”) include a flexible structural model, while the rest do not. The

dynamic inflow model of §4.4.14.6 is used by default, except for the steady (equi-

librium wake) formulation for the “Corr Stdy” results. The “Corr” results include

the ε terms from vortex theory, while the “NoCorr” results use the default values

that transform the equations to the standard ones in BLADEDTM.1 The wind shear

exponents α and hub-height wind speed V used (based on experimental data) are

noted in the figures.

The nominally un-yawed case in Fig. 5.14(a) shows good agreement between the

various BEM formulations. There is a relative offset of the tower effect in all cases

past θ = 180 in the CFD and Exp results. This effect, and the lower peak moments

for ψyaw = −3, are most likely related to the structural flexibility that is un-

modelled in the BEM results.

Noting this un-yawed discrepancy, the other yawed cases are generally in good

agreement with the Exp and CFD results, when the correction factors are included.

The reduction in the moment decrease through the tower wake of Fig. 5.14(b) is

likely a structural modelling effect. The prediction improvements for the revised

BEM method are most noticeable in the peaks near θ = 90 and 270 where the

wake influence is greatest.

In general, it appears that the corrected steady formulation is a closer match to the

comparison data, relative to the dynamic inflow formulation. This is understandable,

given that the flow is in fact steady. Even so, the dynamic formulation artificially

introduces a time-lag, with the artefact that the changing blade forces and induction

factors with azimuth are accelerating the flow. In fact, they should be merely varying

with local flow conditions created by a steady-state wake.

1The formulations are not entirely equivalent, owing to the different treatments of in-duced and structural velocities in the iterative solution.


0 90 180 270 360500

550

600

650

700

750

Azimuth (deg)

My (

kN

m)

(a) ψyaw = −3, V=8.6 m/s, α = 0.17

0 90 180 270 360400

450

500

550

600

650

700

Azimuth (deg)

My (

kN

m)

(b) ψyaw = 32, V=8.5 m/s, α = 0.31

0 90 180 270 360200

250

300

350

400

450

500

550

600

650

Azimuth (deg)

My (

kN

m)

(c) ψyaw = −51, V=8.3 m/s, α = 0.27

0 90 180 270 360150

200

250

300

350

400

450

500

550

Azimuth (deg)

My (

kN

m)

(d) ψyaw = 54, V=7.8 m/s, α = 0.30

700Exp

Mikkelsen

BLADED

600 NoCorr

Corr

Corr Stdy

Figure 5.14 Tjaereborg root bending moments in yawed flow

5.2 Acoustics Validation 167

5.2 Acoustics Validation

Validation of the acoustics model was hampered by a lack of experimental or compu-

tational comparison data. Only qualitative comparison has been possible, in terms

of approximate magnitude and frequency response, as reference machine data was

not available. Fortunately both sets of results are for large, downwind machines.

Wagner et al. [132] provides figures for the WTS-4 turbine (2 MW in 12 m/s wind),

both frequency response and directivity patterns, including detailed and simplified

models and experimental measurements. Sounds levels were found to be up to

80 dB for an observer 91.5 m downwind at ground level, with a peak near 5 Hz

experimentally and roll-off to 65 dB at 20 Hz. The predictions followed fairly closely,

except below 2 Hz where instead of continuing to decrease, it spiked to 95 dB.

Spatially, lobes were found up-wind and downwind for an observer 200 m away from

the tower base.

Barman et al. [141] examined the Maglarp 3 MW machine, and again presented

experimental and predictive results, both for actual pressure sound level. The ob-

server location was 0.12 km downwind at ground-level. The experimental results

showed pressure spikes of ±1 Pa about a mean, whereas predictions were ±1.6–

±3 Pa. Both were associated with passage through the tower wake. The predicted

SPL peaks were at 75 dB at 10 Hz, decreasing to 35 dB at 50 Hz and 60 dB at 1.6 Hz

either side of the peak.

As comparison, results for the REF-1500 (see §C.3) were computed for the up-

wind configuration, using ExcelBEM. The observer in all of the following is 100 m

downwind in a wind of 11 m/s (rated power, but below pitch action). The rotor

speed is 20 RPM, conveniently giving a 3P frequency of 1 Hz. The SPLs given are

for the summed octave bands to 50 Hz.1 Post-processing is done for re-sampled data

at 1000 Hz with a 4096-point FFT.

Shaft tilt, wind shear, and tower influence are excluded to start. All forms of

stall delay are turned off, and dynamic wake modelling is used. Figure 5.15 presents

pressure and SPL for the isolated rotor. Even without tower influence, it is clear that

the moving forces produce a pressure signature at the observer location. The primary

source is the near-field, as expected from the original equations (see §4.5.3.2). As

will be seen throughout, the thickness noise is negligible.

1This excludes the zero, near-zero and high frequency components, the former of whichis not noise and the latter inaccurately predicted by the current method.


0 2 4 6-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

tObs

(s)

p (

Pa)

thk

near

far

total

(a) Pressure

1.6 3.1 6.3 12.5-100

-50

0

50

100

f (Hz)

SP

L (

dB

)

0 5 101 2 3 4 6 7 8 9 9 -100

-50

0

50

100

f (Hz)

SP

L (

dB

)(b) SPL

Figure 5.15 Simple isolated rotor acoustic prediction

Figure 5.16 shows the effect of including the tower influence (without tilt). The

tower cd is 0.3 (above Recrit). The frequency content is clearly centred at the

blade passing frequency. Three cases are shown for increasing refinement of the

aerodynamic simulation time-step, governed by the azimuthal step in degrees. It is

clear from Fig. 5.16 that to capture 9P content and above, 0.5 steps are required

(at least near the tower; this could be relaxed away from the tower). The same

accuracy in not evidently required for the overall SPL, as shown in Table 5.2. A 0.5

step is used in the remainder of the results.

Including the tilt (5) moves the blades away from the tower, reducing the SPL by

9 dB. The SPL still appears quite high, but bear in mind the observer is only 100 m

away, and as shown in Fig. 5.17, the assumed pressure around the tower causes quite

large velocity fluctuations even for this upwind case. With wind shear, and then

cd = 0 for the tower, the SPL is reduced further.

Figure 5.18 illustrates the observer pressure signature, which typically exhibits

peaks at the 3P frequency. It also shows the peak reduction by tilt. Adding wind

shear has fairly minimal effect on either spectral content or SPL.

With the rotor downwind and taking lw = 3.25, ∆ = .3 and ww = 2.5 as typical

for a cylindrical tower [87], the SPL is 88.4 dB without tilt and 85.3 dB with tilt.

Figure 5.19 shows the energy content is shifted to somewhat higher frequency (au-

dibility) compared to the upwind case. These are only mildly higher than the data

presented in the other quoted studies. It is likely that the current “test” rotor is

5.3 Structural Validation 169

1.6 3.1 6.3 12.5 25 0

25

50

75

100

f (Hz)

SP

L (

dB

)

0 2 4 6 8 10 12 14 16 18 200

25

50

75

100

f (Hz)

SP

L (

dB

)

2

1

0.5

Figure 5.16 SPL for upwind rotor with tower influence for varying azimuthalstep size (deg)

Table 5.2 Upwind SPL predictions

Case SPL (dB)

Isolated rotor 17.3Tower influence 2 step 77.4

1 step 81.70.5 step 83.1Tilt 73.9Tilt and wind shear 73.0Tilt and wind shear and cd = 0 72.6

proportionally closer to the tower.

The acoustic footprint of the rotor is given in Fig. 5.20. The shift in the cross-

stream direction is consistent with the comparison data set [132] and is caused by the

rotor advancing/retreating from the observer. The lobes are caused by the dominant

forces being aligned with the free-stream. No account is made of acoustic reflectivity

from either the ground or tower, which would further modify the footprint.

5.3 Structural Validation

The cross-sectional model is validated first in §5.3.1, followed by the FEM beam

model in §5.3.2.


-40-20

0 20

40

40

60

80

100

120

7

8

9

10

11

yz

Vop (

m/s

)

(a) No tilt

-40-20

0 20

40

40

60

80

100

120

9

9.5

10

10.5

11

yz

Vop (

m/s

)(b) With 5 tilt

Figure 5.17 Effect of tilt on section out-of-plane velocity for upwind rotor (globalcoordinate system)

0 2 4 6-1.5

-1

-0.5

0

0.5

1

tObs

(s)

p (

Pa

)

thk

near

far

total

(a) No tilt

0 2 4 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

tObs

(s)

p (

Pa

)

thk

near

far

total

(b) With 5 tilt

Figure 5.18 Effect of tilt on pressure signature for upwind rotor


0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

f (Hz)

SP

L (

dB

)No tilt

Tilt

Figure 5.19 Effect of tilt on downwind rotor SPL

0°

15°

30°

45°

60°

75°90°105°

120°

135°

150°

165°

±180°

-165°

-150°

-135°

-120°

-105° -90° -75°

-60°

-45°

-30°

-15°

708090

Observer azimuth (0° downwind)

Figure 5.20 Acoustic footprint (dB) of downwind rotor with tilt and shear

5.3.1 Sectional Properties

The sectional computations were verified against analytic formulas for a thin ring,

box section and I-beam, for all stiffness and mass properties. The analytic formulae

were obtained and derived from the standard analysis methods (parallel-axis the-

orem, etc.) presented by Ugural and Fenster [151]. For the circle, refinement of

the element size (arc-length) showed asymptotic convergence to the analytic value

(below 0.2% at for 100 chordwise sections). The other shapes matched identically.


5.3.2 Beam Model

The beam model has been checked against BLADEDTM’s modal predictions, for all

but the flapping boundary condition, which is not part of BLADEDTM’s capability.

Four test cases are considered: uniform (unity) mass, stiffness, and flexible blade

length L = 1 (§5.3.2.1); linearly varying mass and stiffness (both 1.1–0.1, root–

tip) with L = 1 (§5.3.2.2); GH Demo blade (§5.3.2.3); and the CONE-450 blade

(§5.3.2.4). Two and three bladed rotors were checked, as were clamped and free-

rotation for the in-plane modes.

In the figures to follow, the FEM results are given by dots, those from BLADEDTM

by crosses. The modal frequencies λn are given as e.g. 4 Hz (5 Hz) for FEM

(BLADEDTM) results. Where it is stated that the two sets of results are same,

what is meant is that the modal basis vectors are equivalent. BLADEDTM solves

for modes that typically have motion of all blades simultaneously. The FEM method

typically produced modes in which one blade is displaced and the others have no

displacements (for clamped root conditions). When the frequencies are identical,

the modal basis vectors describe the same eigenspace.

5.3.2.1 Uniform Blade

For reference, the OP and IP modes for a rotor with three uniform blades, with freely

rotating hub, are given in Fig. 5.21 and Fig. 5.22. The results have no flap hinge and

lhub = 0,1 and the rotation speed Ω is 100 RPM. As expected, the OP modes show

the blades completely un-coupled. The mode shapes and frequencies are identical

to those from BLADEDTM. Figure 5.23 shows the modes with a teetered hub.2 The

rigid-body modes associated with teeter are clearly present. The static and rotating

mode shapes are identical in all cases, and the rotating modes are higher frequency,

as expected. With flapping hinges, the FEM results agree with Eq. (4.6.24).

The IP modes however showed some discrepancies. For two (three) bladed rotors

with freely rotating hub, every second (third) mode differed in frequency and shape,

as shown in Fig. 5.24. The BLADEDTM results for this test case deviate from the

FEM results for the collective IP mode shapes. Double curvature, as evident in

1Note that lhub = Rhinge in the vernacular of §4.6.4 for an un-hinged blade, representingthe offset of the blade root from the rotation axis.

2The differences between the two sub figures owe to the coordinate definitions;BLADEDTM uses a positive displacement for upwind motion of all blades, whereas the FEMformulation couples the rotation of the hub node in teeter, so that positive displacement isupwind for one blade and downwind for the other.


0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#1: 1.86 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#2: 1.86 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#3: 1.86 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#4: 5.50 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#5: 5.50 Hz

rd

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#6: 5.50 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#7: 12.06 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#8: 12.06 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#9: 12.06 Hz

r

d

Figure 5.21 Uniform blade OP rotating modes

the FEM results, must exist for these modes, owing to the free hub.1 As shown in

Fig. 5.22, there should be sets of B − 1 reactionless modes and a single collective

mode.

Increasing lhub to 0.5, the OP modes still match. The IP modes shown in Fig. 5.25

again exhibit the slope error just discussed. BLADEDTM gives a zero displacement

at the root. As the hub is free to rotate, the offset should yield some displacement

at the root (vroot = φhublhub for small angles). Further investigation, by increasing

lhub to 49, gave the results in Table 5.3. The mode shapes are not shown, however

the BLADEDTM results were found to have some minimal displacement at the root

for both OP and IP modes. The IP modes were also again different for the collective

modes.

The discrepancies in the BLADEDTM IP results seem to indicate some finite

stiffness is associated with the hub. The internal details of the BLADEDTM method

were not available, so the cause could not be definitively determined. As will be

1This may also be justified by drawing the BLADEDTM modes as a rotor. It is thenobvious that a moment would have to be present at the hub for the deflections predicted byBLADEDTM to occur.


0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#1: 1.86 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#2: 1.86 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#3: 4.85 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#4: 5.50 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#5: 5.50 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#6: 10.59 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#7: 12.06 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#8: 12.06 Hz

r

d

0 0.2 0.4 0.6 0.8 1 -1

-0.5

0

0.5

1

#9: 19.35 Hz

rd

Figure 5.22 Uniform blade IP rotating modes

0 0.2 0.4 0.6 0.8 1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

No

rma

lized

dis

pla

cem

en

t

r/R

B1:1

B2:1

B1,2:2

B1:3

B2:3

(a) BLADEDTM

0 0.2 0.4 0.6 0.8 1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

No

rma

lized

dis

pla

cem

en

t

r/R

B1,2:1

B1:2

B2:2B1,2:3

(b) FEM

Figure 5.23 Uniform blade OP rotating modes for teetered hub (Bladenumber:mode number)


0 0.2 0.4 0.6 0.8 1 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.8

Norm

aliz

ed

dis

pla

cem

ent

r/R

(a) Mode 3, 4.85 Hz (1.86 Hz)

0 0.2 0.4 0.6 0.8 1 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Norm

aliz

ed

dis

pla

cem

ent

r/R

(b) Mode 6, 10.59 Hz (5.50 Hz)

Figure 5.24 Free-hub rotating uniform blade IP modes, FEM (BLADEDTM)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.6

Norm

aliz

ed d

isp

lace

ment

r/R

(a) Mode 3, 4.01 Hz (2.35 Hz)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.8

Norm

aliz

ed d

isp

lace

ment

r/R

(b) Mode 6, 8.94 Hz (6.47 Hz)

Figure 5.25 lhub = 0.5 free-hub rotating uniform blade with IP modes, FEM(BLADEDTM)


Table 5.3 Uniform blade with long hub eigenfrequencies (lhub = 49)

Static (Hz) Rotating (Hz)Mode # BLADEDTM FEM BLADEDTM FEM

OP 1 0.56 0.56 2.64 14.242 0.56 0.56 2.64 14.243 0.56 0.56 2.64 14.24

IP 1 0.56 0.56 2.64 14.242 0.56 0.56 2.64 14.243 0.57 0.90 9.71 22.58

shown for realistic blade properties in §5.3.2.3 and §5.3.2.4, the BLADEDTM results

do not exhibit this error. The present test case is therefore a pathological example

that with very small blade mass, stiffness and length drastically alters the usual

hub–blade stiffness relationship. The BLADEDTM method is a pragmatic one, and

is valid for real blades. The OP frequencies do match in all cases, which is curious,

considering the finite deflection at the root for the BLADEDTM results.

5.3.2.2 TaperedBlade

The tapered blade was compared for both lhub = 0 and 0.5. Overall, the results

matched BLADEDTM, with the same discrepancies discussed for the uniform blade.

5.3.2.3 DEMO Blade

The inclusion of generally varying stiffness, mass and twist, as well as collective

pitch and modest offset, tests all facets of the FEM method. Comparison with

BLADEDTM results at 18 RPM, for both a rigid 3-bladed rotor and teetered 2-

bladed rotor, showed virtually identical results. There were small differences in

λn for higher modes (>= 6 for 3 bladed rotor), presumably owing to the different

numerical approaches.

Figure 5.26 shows the third, sixth and ninth collective IP modes, and Table 5.4

list the associated λn.

In contrast to the uniform and tapered blade, for a real geometry the BLADEDTM

and FEM results are in very close agreement for the collective modes. It is likely

some stiffness value is used for the hub in BLADEDTM, producing odd results only

for unrealistic pathological test cases.

5.4 BLADEDTM Validation and Suitability 177

0 10 20 30 40-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

r (m)

Norm

aliz

ed

dis

pla

cem

ent

3

6

9

(a) BLADEDTM

0 10 20 30 40-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

r (m)

Norm

aliz

ed

dis

pla

cem

ent

3

6

9

(b) FEM

Figure 5.26 Collective IP mode shapes for Demo blade

Table 5.4 Collective IP eigenfrequencies for Demo blade

Static (Hz) Rotating (Hz)Mode # BLADEDTM FEM BLADEDTM FEM

3 3.10 3.11 3.17 3.176 8.32 8.37 8.38 8.439 16.52 16.67 16.57 16.72

5.3.2.4 CONE-450 Blade

The IP modes again match as per the DEMO blade. No BLADEDTM solution is

available for the OP modes, due to the flap hinge. The integrated blade properties for

this blade enable analytic computation of the first three flap modes (rigid body) from

Eq. (4.6.24). Testing at Ω = 25 RPM, and with the properties in §C.5, Eq. (4.6.24)

predicts 0.456 Hz, which is exactly equal to the first three frequencies from the FEM

code, corresponding the three independent rigid-body flap modes.

5.4 BLADEDTM Validation and Suitability

BLADEDTM is commercial software, with full Germanischer Lloyd certification and

a development period of 20 years. To achieve that status, it has been validated

against numerous machines and used successfully in many projects. Throughout this

thesis, the code has been stretched to allow analysis of the coning rotor. It should be

noted that for the original CONE-450 work, Jamieson [66] used a custom version of


the code, differing from the current version (v3.65) primarily in parked aerodynamics

and the structural model. This section deals with the modifications of the current

version to handle the coning rotor concept. The areas of discrepancy found were

aerodynamics (§5.4.1), steady-state solution including coning and VSS (§5.4.2), and

the dynamic structural model (§5.4.3). To reiterate, the issues discussed in this

section pertain to the applicability of BLADEDTM to the coning rotor, and are not

meant to indicate unsuitability for conventional machines.

5.4.1 Aerodynamics

Efforts towards implementing the aerodynamic refinements developed in §4.4 in

BLADEDTM have been made. Unfortunately, to date the work is incomplete, de-

spite the best efforts of all parties. In addition to implementing the induction-factor

corrections, it was discovered well into the project that a number of other aerody-

namic modelling features will be required:

• Smooth blending of tower shadow at tower top (see §4.4.14.3)

• Some spanwise flow model should be included (see §4.4.9)

• A condition-dependent stall-delay model is required (see §4.4.8)

The issues in the following sections further added to this list of outstanding is-

sues, and precluded the development of a full enhanced implementation within

BLADEDTM in the available time-scale.

5.4.2 Steady State Operation

For the purposes of design optimization (Part III), steady-state operation points

are required. Unfortunately, no direct coning equilibrium calculation is available

within BLADEDTM, akin to that developed in §4.7.1.1. Likewise, optimal operation

calculations, as presented in §4.7.1.2, are not a feature of BLADEDTM. It would

of course be possible to bypass these by utilizing the dynamic calculations with a

self-tuning controller, but at increased computational effort.

It was also found that the steady-power curve calculation in BLADEDTM, while

able to iterate on pitch angle or rotor speed for limiting power §4.7.1.2, in some

cases produced non-smooth power curves.1 An example is shown in Fig. 5.27 for

PTS operation of the REF-1500. This behaviour would have proven problematic for

the optimization studies which require accurate estimates for changes in objective

1Interestingly the rated electrical power is constant, however the shaft (aerodynamic)power is not. The latter is presumably the determinant of the pitch/speed schedule.


0.00.20.40.60.81.01.21.41.61.8

0.0 5.0 10.0 15.0 20.0 25.0Windspeed (m/s)

Powe

r (MW

)

-7-6-5-4-3-2-10

Pitch

angle

(deg

)

Shaft power Electrical power Pitch angle

Figure 5.27 BLADEDTM computed steady power curve for PTS

function. Based on the author’s experience with ExcelBEM, solving accurately for

power limiting with either PTS or VSS requires extremely tight termination criteria

for the iteration algorithm. This owes to the steep derivatives of power/torque with

control variable, when the rotor is in stall. Unfortunately, the solution tolerances

for this calculation in BLADEDTM are not accessible to the user.

5.4.3 Flapping Hinge

BLADEDTM was modified to feed flap angles βj and flap velocities βj to an exter-

nally implemented controller. A switch was also added to turn on/off the built-in

flap restraints pre-existing in the code. The restraints are modelled as individual

hydraulic cylinders attached at the pin-end to a point offset from the rotor axis,

and the rod-end to a pin on the blade outboard of the hinge axis.1 The physical

geometry and hydraulic parameters (including end-stops and accumulator pressure)

are used to compute an equivalent hinge-moment that is applied to the flap hinge in

the simulation. For generality, a facility was added for the current project to allow

the external controller to impose an arbitrary hinge moment. This allows modelling

of a generalized actuator, either passive or with active control.

To model a flapping hinge in BLADEDTM, the mode shapes and natural frequen-

cies must be provided from the FEM predictions developed in §4.6.4. To exclude

1This modelling was added to analyse the WTC prototype.


any complicating structural effects, the uniform blade from §5.3.2.1 was used with

Rhinge = 0. Only the first flap mode was input to BLADEDTM, known to be rigid

body rotation about the hinge. Testing of the structural simulation, in isolation from

the aerodynamics, was possible by defining airfoils with zero lift and drag. Gravity

was also turned off with g = 0. A simple external controller was defined to apply a

constant moment plus some damping to each blade. Analytic solutions are available

from Eq. (4.6.20b) (non-linear) and Eq. (4.6.21) (linear) by setting βj = βj = 0.1

Using this approach, the blade was modelling in ExcelBEM using 5 sections, in

both Steady and Dynamic modes, the later with critical damping applied. The

results were compared to BLADEDTM models with 5, 25 and 100 equally-spaced

sections. The results are given in Table 5.5, which yield the follow insights:

• The analytic results show that assuming linearity artificially stiffens the rotor,

becoming important around 10.

• ExcelBEM solutions all match the non-linear analytic solution. The slight nu-

merical discrepancy in the Steady results owes to the accuracy of the algorithm

in §4.7.1.1.2

• BLADEDTM results follow quite closely the analytical linear predictions.

• The BLADEDTM predictions do not converge to either the linear or non-linear

analytic solutions with increasing numbers of sections.

The last point suggests another effect, other than just a linearity assumption, is

present in BLADEDTM. Based on the operational profile of the coning rotor (up

to β =≈ 40), the “virtual flap stiffness” present in the BLADEDTM model means

that equilibrium β may be in error by approximately 30%. The power, thrust and

other predictions will also be affected by approximately the cosine of this error level.

The source of the discrepancy is evident from examination of the BLADEDTM

theory manual [125]. As discussed in §4.6.3, BLADEDTM simplifies the EOM by

assuming a known modal frequency, even though the modes used are not normal. In

the case of gross-coning, the modal frequencies are altered (as the centrifugal stiffen-

ing changes with β), and so the embedded EOM adhere to a linearising assumption

that is increasingly in error with β. The original work [66] appears to have used only

mode shapes, not frequencies, so those predictions should be accurate in coning.

1From the solution to these equations is also apparent that two solutions for βj arepossible for a given Hact, either side of ±45.

2The solution is obtained much faster with the Steady method however, which is why itis preferred during optimization over running the dynamic simulation to steady-state.


Table 5.5 Steady flap angle variation with actuator moment for Ω =50 RPM

β (deg)Hact (Nm)

Model 100 200 300 350 400

Analytic linear 6.88 13.75 20.63 24.06 27.50Analytic non-linear 6.94 14.34 23.03 28.57 36.87ExcelBEM Steady 6.88 14.30 23.01 28.57 36.85ExcelBEM Dynamic 6.94 14.34 23.03 28.57 36.87BLADEDTM 5 6.67 13.33 20.00 23.34 26.67BLADEDTM 25 6.87 13.74 20.61 24.04 27.48BLADEDTM 100 6.74 13.48 20.22 23.60 26.97

5.4.4 Mode Shape Modification by Coning

The pre-stress present in the blade owing to centrifugal loading will vary with cone

angle, approximately as cos2 β. In general, the aerodynamic and inertial forces

will not precisely align with the axis of the blade, but as assumed in the deriva-

tion of §4.6.4.1, some insight is gained by ignoring these effects. The analytic fre-

quency of the free-body flapping mode may be computed with the non-linear form

of Eq. (4.6.24), linearized about β0:

ωn = θ

√cos(2β0) +

RhMbsCG

Iβcos(β0) (5.4.1)

Here again, ωn is imaginary for β0 > 45 (for Rh = 0), indicating instability.

The FEM results do not agree with Eq. (5.4.1) for the fundamental flap mode,

predicting higher natural frequencies as β0 is increased. The higher OP and IP

modes are almost identical in shape and frequency to the unconed case. The reason

for the discrepancy is the difference in boundary conditions between the analytic

and FEM equations. Equation (5.4.1) assumes a hinge moment is applied to create

the equilibrium condition, whereas the FEM method assumes forces applied along

the blade exactly balance the transverse centrifugal force component. To harmonize

the two approaches would require including the equilibrium bending moment stress

in Eq. (4.6.45), rather than assuming plain stress. This extension is left as future

work, as the mode shapes are relatively unaffected by the assumption of β0 = 0.

Moreover, as just discussed in §5.4.3, modal frequencies cannot be used directly in

the dynamic simulation of the coned rotor.


5.5 Generator Model Validation

Dubois [165] presented 14 optimized design variables (τp, J , Bg, bt/bs and Krad)

for a range of parameters (diameters D and efficiency η). The Solver built into

Excel has been used to conduct multi-parameter optimization of the spreadsheet

model with the same parameters and constraints as the original thesis. Comparison

of the original results with the current ones indicate that the model is consistently

implemented.

Part III

Design

Chapter 6

Rotor Optimization

Part II focused on developing tools capable of analysing the coning rotor. In addi-

tion to analytical differences, compared to a conventional machine the operational

concept of the coning rotor demands a somewhat different and more integrated de-

sign approach. This chapter first uses the results of the original CONE-450 study to

predict performance relative to modern machines, in terms of load (§6.1) and energy

capture (§6.2). The generic aerodynamic behaviour of the coning rotor is examined

next in §6.3. Section 6.4 then elucidates the challenges inherent in optimizing the

coning rotor. Finally, a parametric study of the DVs of the coning rotor is presented

in §6.5.

6.1 Initial Comparison and Updating

The present section first contextualizes the original coning rotor work (§6.1.1), before

properly scaling up the loading results from that study in §6.1.2. Justification is then

given in §6.1.3 for leaving the revisiting of parked conditions to future work.

6.1.1 Shifting Benchmarks

In the mid-1990’s, when the original coning rotor work was done, the benchmark

comparison machine was a 450 kW, constant-speed, FSS regulated machine (REF-

450) [7]. Technical evolution means that an appropriate benchmark is now a 1.5 MW

variable-speed pitch regulated machine.1 This evolution has been by incremental

refinements in design and analysis within the wind turbine community, spurred on by

cost and performance enhancements of actuators, materials, and power electronics.

The two main shifts relevant to the present comparison have been to substantial

variable speed operation and increasingly fast, individual blade pitch control.

11.5 MW was chosen as generic machine data was available for this size and is around theaverage capacity actually installed on land, notwithstanding the industry drive to capacitiesexceeding 4 MW for offshore sites where economies of size are still unclear.

185

186 Chapter 6 Rotor Optimization

6.1.2 Up-Scaling CONE-450 Results

As an initial scoping exercise, the representative operating curves from the original

450 kW machine (CONE-450) and REF-450 dynamic simulations [36] were scaled

up to a 1.5 MW machine (CONE-450 US and REF-450 US). The loads shown

in Fig. 6.1 were obtained by power scaling based on the reference rotor diameter

Dref , in turn scaled proportionally to the square root of the rated power. These

are compared against the loads computed for a generic 1.5 MW pitch controlled

machine (REF-1500, see §C.3) using BLADEDTM.

0

50

100

150

200

250

0 5 10 15 20 25

Windspeed (m/s)

Thrust (kN)

REF-1500

CONE-450 US

REF-450 US

Figure 6.1 Operational rotor thrust curves for upscaled machines relative toREF-1500

The results indicate that the overall loading for the coning concept at a larger

scale is reasonable, and highlights the differences between the pitch control and

active stall/coning strategies. The design of the original CONE-450 (see §6.4.3) was

based on maintaining the same maximum thrust level as the REF-450, occurring

for the CONE-450 at rated and for the REF-450 at cut-out (25 m/s). The CONE-

450 US attains the same maximum thrust, but stays at a relatively high thrust

after rated relative to the REF-1500, as stalling imparts more force than pitching

to feather. It can be expected that the parked extreme loading case, discussed

further in §6.1.3, remains a non design-driving condition for the upscaled coning

rotor. Fatigue loading comparisons must be deferred to more detailed time-domain

analysis in the future.

6.1 Initial Comparison and Updating 187

The original energy yield results included not only a larger rotor for the CONE-

450, but also an increase in tower height from 35 m to 50 m. The tower height

increase was enabled primarily by equal tower bottom overturning moment when

parked (i.e. coning rotor thrust less severe in parked extreme conditions). The

energy yield advantages, relative to other conventional machines (250–600 kW),

varied from -15–64%, depending on machine rating and mean wind speed. The

CONE-450 was superior in low winds (5-6 m/s mean), but lost this advantage in

high winds (9–10 m/s), paricularily for machines of equal tower height and higher

rating. The net competitive advantage (in terms of COE) was positive in all cases,

ranging from 0.2–62%. It was noted that the CONE-450 was designed for true

Class I conditions, whereas the reference machines were not, and so the CONE-450

would in fact be more cost competitive than indicated by the figures. Against the

reference machine in normal UK conditions, the CONE-450 had an 18% Net Present

Value (NPV) advantage without taller tower.

6.1.3 Parked Conditions

A novel aspect of the coning rotor is the parking strategy (fully coned to ≈ 85).

Jamieson [66] examined this condition in some detail, including fully orthotropic

turbulence and various delta-wing stall delay models (see §4.4.9.2).1 The results

indicated that parking did not drive the design of the CONE-450. Based on this,

the present work does not explore in detail this aspect of the coning rotor.

A primary concern might be the overhang weight associated with a larger scale

machine. For the blades discussed later in §6.5 (CONE-1500), deterministic steady

blade root bending moments are ≈500 kNm owing to gravity. The blade structure

design loads2 are ≈1500 kNm, indicating a large static factor of safety. The REF-

1500 experiences a maximum of 2107 kNm overturning moment at the yaw bearing.

The ≈ 3 × 500 = 1500 kNm equivalent moment for the CONE-1500 is therefore

not unreasonable (the extreme moment from the CONE-450 US is 913 kNm when

parked, versus 3512 kNm in extreme operation).

Further validation of the 1.5 MW scale concept must revisit this issue to verify the

assumption that parked loads do not dominate. The models must include full tur-

bulence, and ideally wind-tunnel testing to verify aerodynamic loading predictions

for the fully-coned blades experiencing extensive spanwise flow.

1The IEC standard of the day only specified steady winds for the parked case.2Obtained as the root moments to hold the blades open at rated conditions.


6.2 COE and CF Performance

The interrelation between COE and CF, and their use as metrics in wind turbine

optimization were introduced in §3.2. Figure 6.2 shows the ideal performance of a

variable area rotor, able to operate at CP = 0.48 with full diameter up to rated, and

1.5 MW above. Performance is shown over a range of diameters (Ø70–100 m) for

a nominal site with k = 2.2 in Fig. 6.2(a). Three mean wind speeds are compared,

based on a realistic cost exponent with diameter (see §3.5) of 2.4 [15], relative to

the baseline Ø70 m rotor diameter. Based on this simple analysis, COE rises quite

quickly with CF, but less so for low-wind sites.

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8CF

COE

V=6V=8V=10

(a) CF dependency (k = 2.2)

0.20

0.25

0.30

0.35

0.40

0.45

0.50

70 80 90 100D (m)

CF

2000

3000

4000

5000

6000

Energ

y Yiel

d (MW

h)

k=1.7k=2.2k=2.5

E

CF

(b) k dependency (Vmean=6 m/s)

Figure 6.2 Betz limit variable area rotor performance

The effect of k is shown in Fig. 6.2(b), across the range of diameters. As expected,

CF and energy E both rise with diameter, while the k effect is rather subtle. Low k

factors (broad f curve) reduce both dependencies, and visa versa.

The design choice of rating and rotor size varies between designers. Indeed, mul-

tiple rotors of varying diameter may be offered for the same nominal machine (i.e.

generator, tower, etc.), based on operation in different wind classes (see §2.1.3). To

examine the impact of these design decisions, a number of power curves for real

1.2–2 MW machines were obtained from product literature.

Performance is compared for k = 2.29 and Vmean = 8.24 m/s, based on a typical

US site [15], in Fig. 6.3. Tower height is assumed constant. Figure 6.3(a) shows the

PDFs of energy capture for the machines relative to the wind speed profile, ideal

6.2 COE and CF Performance 189

Betz limit (over entire wind speed range), and a scaled power curve for the CONE-

450 US. The ideal capture is clearly weighted to higher V than both the wind and

actual machine capture. The machine profiles are clustered quite closely around

11 m/s, with the CONE-450 US attaining the highest peak (tightest distribution).

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 5 10 15 20 25U (m/s)

f V, f E

ann (

s/m)

Wind Ideal V80 Vensys 62Bonus 2MW NM 64-1500 NM 72-2000 CONE-450 US

(a) Wind speed profile

0.35

0.40

0.45

0.50

0.25 0.30 0.35 0.40 0.45 0.50P/A (kW/m2)

CF

stall control

(b) Power density

4.0

4.5

5.0

5.5

6.0

6.5

0.30 0.35 0.40 0.45 0.50CF

D2.4 /E

(m2.4 /M

Wh)

0.20

0.22

0.24

0.26

0.28

0.30

0.32

P/E (k

W/MW

h)D/EP/E

VENSYS

(c) Cost metrics

Figure 6.3 Industry machine performance

Figure 6.3(b) shows that CF is in fact a linear function of the ratio of generator

power to rotor area, as expected. The one PTS machine deviates somewhat from the

overall trend, with lower CF. List prices are unavailable, but Fig. 6.3(c) compares


two proxy COE metrics: a cost function D2.4/E assuming cost to vary with diameter

(see §3.5), and the ratio of installed power to energy captured P/E. The former

metric results show costs rising with CF, mirroring earlier results in §3.2. The Ven-

sys machine evidently achieves a superior power curve relative to the conventional

machines, while the CONE-450 US would have a high cost if the D2.4 relationship

is not altered for the coning rotor. The P/E ratio shows that installed generator

power relative to rotor size is an almost linear function with CF. This mirrors the

CF–P/A relationship and suggests capture area A is a dominant factor.

6.3 Aerodynamic Behaviour

It is instructive to examine the non-dimensionalized aerodynamic behaviour of a

coning rotor, before considering its optimization. Figure 6.4 shows the performance

maps, non-dimensionalized by real rotor area, of the REF-1500 over a range of β

and λ. The optimal CP location moves in λ with β, and in this case CP,max rises

with β. These trends are generally present in other rotors examined, however the

direction of shift varies. Thrust behaviour is also seen to change with β.

0

20

40

600 2

4 6

8 10

0

0.1

0.2

0.3

0.4

0.5

0.5

λ

0.4

0.3

0.2

0.1

β (deg)

CP

(a) CP

0 10

2030

4050

0 2

4 6

8 10

0

0.2

0.4

0.6

0.8

1 1

0.8

λ

0.6

0.4

0.2

β (deg)

CT

(b) CT

Figure 6.4 Performance maps for actual tip radius

Figure 6.5 shows a set of curves for the REF-1500 non-dimensionalized by the

reference tip radius, for a range of cone and pitch angles. The intent is to illustrate

the extent of control possible based solely on coning versus a pitch-controlled rotor

(fine or stalling pitch). Power control is clearly more direct with pitch control, both

6.4 Rotor Optimization Challenge 191

in terms of CP reduction with angle and the ability to change pitch directly instead

of via an indirect speed-stall-cone relationship. The extremely fast decrease in CP

for PTS is also evident.

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

λ

CP

0

50

45

40

(a) Coning angle β (γ = 0)

2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

λ

CP

-10

-5

0

5

10

15

(b) Pitch angle γ (β = 0)

Figure 6.5 Performance maps for reference tip radius and control variable (β orγpitch in 5increments)

6.4 Rotor Optimization Challenge

Attention now shifts to refining the coning rotor design by considering a viable

optimization process. The procedure for a conventional rotors is outlined first in

§6.4.1, followed by the challenges introduced by the coning rotor in §6.4.2 and the

approach to the problem taken by previous investigators in §6.4.3.

6.4.1 Conventional Rotors

The modern variable-speed rotor is typically first optimized aerodynamically for a

specific tip speed ratio λopt [51]. The optimal operational speed profile is unambigu-

ously defined in both the rotor speed–wind speed (Ω–U) and torque–rotor speed

τ–Ω planes, and can be relatively easily computed from a knowledge of the non-

dimensional power curve and the equations in Fig. 3.7(b).

Treating airfoil choice (both shape and percent thickness) as parameters in the

optimization, the equations in §4.4.13 prescribe the chord and twist DV distributions

for the optimal blade. The overall scale of the machine will modify the results


somewhat, owing to Re variation, but otherwise the design is non-dimensional. The

blade length is guided by experience and an estimate of CP in Eq. (3.2.2) to yield

the rated power desired for the machine (influenced by the results of §6.2).

After this first-pass, the blade must be further refined in a number of ways. The

most apparent is the requirement to smooth the chord and twist profiles. Airfoil

data is not smoothly varying between thicknesses, so Eq. (4.4.36) will not yield

smooth profiles. Additionally, constraints are imposed on the geometry by functional

(a circular root for the pitch bearing) and manufacturing (maximum twist angle)

requirements. These changes may be incorporated by simply smoothing the results

of the analytic equations, or by numerically optimizing the profiles directly. FSS

rotors, with constantly varying λ, must be optimized numerically across the wind

speed range.

Dynamic load simulations are required to certify a design, and also to optimize

an internal structure. Paradoxically, those loads are unavailable without a defined

structure, as the mass and stiffness properties are required for the simulations. This

is usually handled by designing for some simplified, steady load cases. Malcolm

and Hansen [51], like others [180], uses extreme bending loads to size the spar-

cap thickness at 4 spanwise locations, with interpolation for intermediate sections.

Griffin [55] does likewise for the flap bending loads (yielding spar cap thickness), with

combined maximum torque and gravity fatigue loading for edgewise loads (to size a

TE spline at 0.95c). Tip deflection in a 50 year extreme gust is also checked to limit

minimum thickness. Some type of numerical optimization may be easily used at this

stage, either as interpolation functions for spar cap thickness as in the referenced

studies, or direct numerical optimization. The dynamic simulations are then run,

the real limit and fatigue loads computed, and the resulting strains checked. Some

manual refinement of thicknesses, structural concept, and/or materials is then made

to satisfy the material limitations.

Aerodynamic optimality is usually the highest priority for the blade design, given

that the rotor is only 10–15% of the total cost, while improvment of energy capture

more directly impacts COE. However, the interplay between aerodynamic and struc-

tural considerations in fact defines a pareto front, albeit heavily weighted towards

the aerodynamics (CP ). The main competing factors are:

• High performance airfoils (high lift and lift/drag ratio) yield small chords,

adversely reducing section thickness and affecting structural properties [181].

• FSS rotors require limited lift in the tip region to control power and ensure

6.4 Rotor Optimization Challenge 193

smooth stall behaviour to alleviate edgewise vibrations (see §7.1.2.2) [65].

Giguere et al. [65] concluded in an optimization study that COE was best

for higher-lift airfoils, as material use was reduced by limiting the 50 year gust

load with smaller chords. These are obviously conflicting requirements.• Small airfoil percent thicknesses limit both the structural area and height in

a section. Higher strength materials (e.g. carbon fibre) can accommodate

these constraints, with associated cost. Lower modulus materials (e.g. wood

or glass/polypropylene) struggle to fit enough material into the section, with

reduced structural efficiency as the thickness grows towards the elastic axis.• The airfoil shape itself influences the section stiffness. Boxier sections place

more material away from the elastic axis providing efficient use of material,

but possibly to the detriment of aerodynamic properties.• The aerodynamic optimum typically waists the chord around mid-span, a crit-

ical area for fatigue requiring good structural properties (i.e. larger chords and

thicknesses).

These effects may be incorporated into a coupled structural-aerodynamic optimiza-

tion, but more typically are manually iterated.

6.4.2 Coning Rotor Optimization

Unfortunately, the inherent complexity of the coning rotor is manifest when opti-

mization is attempted. The coning rotor inherently ties β to Ω via the rotor inertia

and aerodynamic hinge moment, so that below rated the optimal operation is not

simply defined by λopt. The aerodynamic and structural designs are more tightly

coupled for the coning rotor, requiring more integration of the optimization process.

In addition to the blade profile DVs discussed in §6.4.1, some inclusion of operational

behaviour must be included either explicitly or implicitly.

One approach is to track the locus of CP,max over β, following the ridge of the 3D

CP –λ–β surface in Fig. 6.4 (of course the surface must be computed for each design

iteration). This can be used to define a τ–Ω map in the below rated condition, with

the same speed limits and above rated operation as for the non-coning rotor. The

performance of the rotor can then be determined over a range of wind speeds in an

iterative manner.

This approach has a number of drawbacks, including: difficulty modifying the

curve directly to satisfy constraints (e.g. tip-speed for noise) since the control curve

is implicitly defined via the 3D performance map; optimality is not assured, as coned

diameter also affects power, not just a max CP condition; and many iterations are


required for both definition of the control curve and solution for operating conditions.

Additionally, although a simple non-dimensional inertia is readily derivable for a

centrally-hinged rotor, the inclusion of the hinge offset complicates the definition

(requiring some relationship of mass and first/second moments of inertia). Above

rated, another optimization exercise is required in the non-dimensional space to

define a constant power curve. For a PTS rotor the space grows to four dimensions

with the inclusion of pitch.

Returning to a dimensional approach, real control strategies are posed in the τ–Ω

plane, as wind speed cannot be directly measured. For the purposes of steady-

state optimization, control may be defined in the Ω-U plane, to avoid the required

iterations to find steady-state torque and speed.1 A naive approach is therefore

to treat the entire curve as a function of a set of design variables, from cut-in to

cut-out wind speed. This overly complicates the problem, which is better defined by

limits on rotor (and/or tip) speed, some DVs defining the nominally rated portion

of the power curve, and iteration on rotor speed (or pitch angle) to limit power to

rated (see §4.7.1.2). The optimizer can then control the below-rated schedule, with

implicit power control above rated. For a non-coning rotor, only two DV’s, Ω at

cut-in and dΩ/dU , are required to fully define the constant λ below rated optimal

operation.

Of course, using only two variables will ignore any drive train inefficiencies that

may, in general, be non-linear and coupled with Ω via the transmission, and Re

effects. With β variation as well, the optimal track in the Ω-U plane will certainly

no longer be linear. Therefore, the most generalized approach is to remove the

definition of Ω from the list of DVs. Speed optimization is then handled as a sub-

problem for each iterate, using the methods in §4.7.1.2. Analogous to using splines

to describe the blade profile (see §6.5.1.1), constraints and optimality conditions are

implicitly included in the formulation rather than the optimization process itself.

6.4.3 CONE-450 Approach

The CONE-450 concept was developed roughly by the following steps [66]. No stall

delay or coning corrections were used in the original study. Vortex-lift corrections

(based on delta-wing theory) were used in the coned-parked cases, with full 3D wind

loading.

1The τ–Ω map can be easily constructed once the operating profile over wind speed areknown.

6.5 Parametric Study 195

1. Rotor size was determined by first picking a rated wind speed (Vr=12 m/s)

and rated cone angle (25). The CT of the REF-450 at this speed was then

assumed equal to the CT of the coned rotor at rated, and used to compute the

rotor diameter for equal maximum thrust as the REF-450.

2. Chord and twist profiles were optimized on a section-wise basis for λ = 8.5

and β = 0.

3. Rotor speed schedules were computed for maximal energy capture, given a

prescribed β schedule. Peaks near rated were removed.

4. The chord profiles were linearised and twist adjusted to accommodate hinge-

line and avoid excessive energy capture degradation.

5. Steady blade loading was computed at 11 m/s with 200 kNm hinge moment

applied. Blade structure is then computed to survive the static forces, and

mass-tuned to achieve 25cone angle at Vr.

6. Final speed schedules were computed with free-coning and 60 kNm hinge mo-

ment between 5–9 m/s.

7. Detailed design of components and controller.

8. Iteration on blade structure and component design, based on survival of com-

puted dynamic loads. The blade structure was designed with steady loads,

eventually based on a steady 560 kNm applied hinge moment, to survive the

dynamic fatigue and limit loading.

6.5 Parametric Study

The analytic methods developed in Chapter 4 (BEM, power curve solution, dy-

namic simulations), in their integrated form, are unfortunately not readily amenable

to gradient-based optimization methods (e.g. Sequential Quadratic Programming

(SQP)). Unlike a coupled FEM-CFD code, complex-step [182], auto-differentiation,

and adjoint methods are unable to provide “easy” and accurate derivative informa-

tion. Finite-differencing techniques are expensive, and were found to suffer from

code solution granularities. For example, it was found that the BEM code must be

run to < 10−6 precision (see §6.5.2.2), while the power curve must be computed to

less than ≈0.2 m/s step-size resolution. Each optimizer step requires N calculations

to compute the gradient (for N DVs), plus a line-search step for SQP, making this

approach quite time-consuming when the gradients are computed accurately for the

complete coupled system.


Of course, heuristic algorithms (e.g. genetic algorithms [65, 183], simulated an-

nealing,1 particle swarm, random walk) avoid gradient computations, but substitute

many more direct objective and constraint calculations for each iterate. Tuning of

the algorithms is also difficult, often requiring close to a full-factorial search after

tuning! They are of course more able to avoid local minima than gradient methods,

but at great expense. Simplex methods [72] can be more efficient, and also avoid

the need for derivatives, but are not adept in incorporating general constraints.

Another approach is to simplify the design space. Reduced order modelling [184],

including basis functions, response surfaces, and Kriging, approximate the true de-

sign space. The optimizer then operates in this new space, which is much less

expensive to evaluate and usually smoother. The main difficulty in these methods

is obtaining a sufficient fidelity and accuracy of the reduced design space. Com-

putational expense typically limits the number of DV to ≈ 5. Similarly, variable

fidelity methods [185] switch between high and low-order codes (if available), using

the former to update corrections to the latter in which the optimizer operates.

The coning rotor design space is described by the following DVs: chord, twist, air-

foil (thickness & shape), material structure, and operating parameters. Given the

size and complexity of the design-space described by these DVs, a parametric study

approach (Design of Experiments (DOE)) is adopted here, in common with other

authors [186], rather than pure optimization of a final solution. Gradient-based (ef-

ficient) optimization is used where appropriate, but exploration of the tradespace is

considered more relevant at this stage than simply applying the optimization meth-

ods just outlined. The function used is Matlab’s fmincon, a refined SQP algorithm.

The specific DOE approach adopted is essentially to perform full-factorial stud-

ies on sub-sets of the problem. The author has tried other DOE methods in the

past, such as Latin Hypercube and orthogonal arrays, and based on this experience

has found the current approach more elucidatory. The primary areas explored in

the following sections are: the influence of coning on optimum aerodynamic profile

(§6.5.4); best-case energy advantage of the coning rotor over conventional designs

(i.e. blade length increase, §6.5.5); control strategy ramifications (§6.5.6); and the

influence of aerodynamic uncertainty (§6.5.7). A common element to all these areas

is the impact of airfoil selection. Before this presentation, the specific choice of DV

and parameters is given in §6.5.1, numerical optimization issues in §6.5.2, and the

optimization approach adopted here to deal with the sub-problems in §6.5.3.

1Adaptive simulated annealing (ASA) by Lester Ingber from http://www.ingber.com/

http://www.ingber.com/


6.5.1 Design Variables and Parameters

The coning rotor is assumed to mount onto the same generator/tower as the REF-

1500. No tower height advantage is included in the present results, as the parked

loading of modern rotors is more benign than the comparison rotor used by Jamieson

[66], and is less likely to afford such large increases. The results are therefore only

reflective of changes in the rotor itself. Hinge radius Rhinge has been chosen as 2 m

to achieve a realistic hub and blade-root structure and to accommodate actuators

capable of maximum coning deflection, positioned downwind of the rotor. The

control schedule is treated implicitly as outlined in §6.4.2. The control method

(VSS, PTS, PTF) is treated as a parameter.

6.5.1.1 Shape Profiles

The blade shape profiles are controlled to provide the realistic geometry shown in

Fig. 6.6, thereby avoiding the smoothing steps mentioned in §6.4.1. Bezier splines

are chosen over cubic splines to define chord and twist profiles, as they posses the

convenient convex-hull property [187], thereby avoiding oscillations in the profile.

The deCasteljau algorithm is used for computation of the polynomial coefficients.

The spanwise locations of all control points are fixed, except for the first twist point,

while the DVs prescribe the chord and twist at those points. The actual section

locations move with the profile itself (not control polygon), to place a section at

maximum chord and with cosine spacing towards the tip.

Figure 6.6 Chord and twist profile control

A 7th order Bezier curve is used to define the chord definition. The outer 4

points at s = [0.15 0.4 0.7 0.95] of blade span S are active DVs for chord. The


blade shapes incorporate a Ø2.2 m circular root (fixing the first spline point), rather

than the blended root possible with a long flap hinge, in case a pitch actuator

is required. The second and third point’s chords are equal to the the first and

fourth point’s respectively. The outer 5% of the blade has a parabolic tip shape, for

noise mitigation. This also produceds a low optimization sensitivity, desireable as

Prantl’s F factor reduces the solution sensitivity in this area. The chord formulation

allows the cuff (maximum chord cmax at sc,max) location to move spanwise implicitly,

without requiring another DV. It was found that chord is best bounded by DV

limits (0. . . 1.5× 3.8/stip m)1 and monotonic linear constraint on the DV to ensure

reducing chord with s. An alternate to the linear constraint is to define the 5th and

6th control points relative to the 4th and 7th points. This adversely affected the

optimizer sentivities to these DVs.

Twist is controlled by a 4th order Bezier curve and 4 active DVs. The first control

point is placed at sc,max (not 0.15s), and constant twist is defined inboard of sc,max.

Twist is constrained by DV bounds of -10–25. The airfoil thickness is tapered

linearly from 100% at the root to 30% thickness at sc,max, and then linearly to 15%

at the tip. The pitch axis tapers from 50% at the root to 30% at sc,max, and is

uniform outboard.

The final blade shape thus implicitly incorporates smooth geometry and man-

ufacturing constraints. Optimization over the entire power curve is facilitated by

this formulation, as sectional optimization is only effective for a single prescribed

operating condition. Global shape control also aids the optimizer by equalizing the

sensitivity between DVs. Simultaneously, a limited number of DVs are required,

rather than DVs to control each section.

6.5.1.2 Airfoils

Airfoil choice is treated as a parameter, rather than a discrete DV, avoiding mixed

discrete-continuous optimization issues. As will become apparent in §6.5.6 and 6.5.7,

the proximity of the optimum operation point (α for (cl/cd)max) to αstall, and the

behaviour after stalling, is critically important for blades incorporating particular

airfoils.

As a parametric study, the actual airfoil data used here is incidental; three sources

have been used to define three airfoil families for this study (sets I, II, III). Each

family defines a complete range of thickness, and covers a range of characteristics,1The 1.5 factor is required to achieve a cmax of 3.8 m, since the DV are spline control

points of a bounding polygon.


to highlight the influence of airfoil choice. In a real-world blade design, the opera-

tional Re and blade soiling levels must be carefully matched to the test data used

in an aerodynamic design, to achieve the desired performance characteristics. Sec-

tions 3.3.3 and 5.1.6 have both highlighted the widely varying data obtainable from

different sources, for ostensibly the same airfoil, at different Re.

The root airfoil for all three sets is a circle with a cd of 0.3. Set I uses LS1

data from GH (same as REF-1500 in §C.3), defined for 30%, 21%, 17%, and 13%

thicknesses. Set II uses the NACA 63-4XX data given in §C.6 to define a second

airfoil set over 21%–30%. Finally, set III uses the FX66-17AII-182 data from §C.5

for 17% thickness, combined with the 21%–100% data of the NACA 63-4XX data.

For thicknesses less than the covered range, the thinnest airfoil in the set is used.

These airfoils were chosen to be representative of increasingly sharply stalling and

high-performance (high cl/cd) airfoils.

The source data covers a range of Re that do not completely cover the opera-

tional range of Re, but is sufficient for its purposes here. To reiterate, the data is

illustrative, as each airfoil designation will produce differing performance with oper-

ating conditions. Centrifugal stall delay (§4.4.8) is included in the following studies.

Spanwise flow (§4.4.9) is inactive until §6.5.7.

6.5.1.3 Internal Structural Layup

A conventional glass-epoxy layup is used for this study. Future refinements will likely

include wood/bamboo-carbon-epoxy structural concepts. The layup and material

values (see Table E.1) of the section are the same as that used by Griffin [55].

Figure 6.7 shows a typical section. The layups for the outer skins and webs are

given in Table 6.1 and Table 6.2 respectively. The webs join the ends of the spar

caps. The relative placement of elastic (shear) and mass centres discussed in §4.6.1.3

was not considered here. A static load safety factor of 1.5 and cumulative material

safety factor of 2.94 (representing the standard combination of factors [55]) are used

to check all computed points in the cross-section against strain allowables.

6.5.2 Optimizer Tuning

Optimization is not simply a “black-box” solution. It requires a number of nuances

in its application, including proper DV scaling (§6.5.2.1) and set-up (§6.5.2.2).


-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-0.1

0

0.1

0.2

0.3

0.4


Figure 6.7 Typical section layup

Table 6.1 Outer skin layup schedule (Adapted from [55])

Layer Chordwise Location Material Thickness

1 0.0–1.0 Gel coat 0.51 mm2 0.0–1.0 Random mat 0.38 mm3 0.0–1.0 CDB340 0.89 mm4 0.0–0.15 Balsa 0.005c

0.15–0.5 Spar cap mixture Variable0.5–0.85 Balsa 0.01c

5 0.0–1.0 CDB340 0.89 mm

Table 6.2 Web layup schedule (Adapted from [55])

Layer Material Thickness

1 CDB340 1 mm2 Balsa 0.01c3 CDB340 1 mm

6.5.2.1 DV Scaling

Linear scaling of all DVs as per Eq. (6.5.1) is used throughout the present results.

This yields gradients of the same order for each design variable and well-conditioned

Hessian matrices [184]. The diagonal D matrix and vector c are calculated so that

the scaled design vector y ∈ [−1 . . . 1] corresponding to the limits on the dimensional

design vector x. Scaling based on the condition number of the Hessian matrix (by

scaling to unity the diagonal entries of the Hessian [188]) was experimented with,

but found to have minimal effect on the results. The Hessian was already well

conditioned with the existing scaling, proving to be a red herring, once the issues of

solution accuracy in the following section were addressed.

x = D y + c (6.5.1)


6.5.2.2 Finite Differencing

Anderson [72] stated that the BEM equations are not amenable to gradient-based

optimization methods, due to the inaccuracy of the fixed-point iteration solution.

Selig and Tangler [189] also make reference to difficulties in selecting appropriate

finite differencing step-sizes. A closer examination reveals that the finite differencing

step is quite sensitive, but tractable, by judicious selection of the solution tolerance

tol and differencing step-size h. A simple first-order forward difference formula is

used throughout:df(x)dx

=f(x+ h)− f(x)

h+O(h) (6.5.2)

Figure 6.8 shows the effect of the BEM solution tolerance for the axial and tan-

gential induction factors on the computed Finite Difference (FD) approximation for

chord and twist DVs. A tolerance of 1e-10 has been used to avoid solution tolerance

errors. Above and below step sizes of 1e-6 and 1e-12, truncation and round-off error

occur respectively. A manual step-size of 1e-6 has been used for the FD computa-

tions, in deference to the adaptive algorithm in fmincon.

10-14

10-12

10-10

10-8

10-6

10-4

10-2

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

dC

P/d

x

h

(a) tol = 1e-4

10-14

10-12

10-10

10-8

10-6

10-4

10-2

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

dC

P/d

x

h

(b) tol = 1e-10

Figure 6.8 Finite differencing prediction for CP as a function of solutiontolerance tol and step size h for chord and twist control points

6.5.3 Stepwise Optimization Approach

A sequential approach is followed for determining the optimum blade profile, struc-

ture, and operating schedule, given the fixed parameters examined in the following


sections. The chord and twist optimization steps are done at a prescribed cone angle

βdesign,aero and (real) tip speed ratio λdesign,areo:

1. The equations of §4.4.13 are used to compute initial chord and twist values for

each blade section.

2. A fast non-linear least-squares fit optimization determines the chord and twist

DVs of §6.5.1.1 that best match the analytic values over 0.15S–0.95S (i.e. ac-

tively controlled section of the blade). This is done using Matlab’s lsqnonlin

function.

3. The SQP algorithm is used to finalize the blade profile with the bounds in

given in §6.5.1.1, with CP as the objective at Ω = Ωmax .

With the resulting finalized aerodynamic design, flapwise and edgewise bending

loads are computed for a prescribed βbalance. These loads are used to numerically

optimize (with SQP) the symmetric spar cap thicknesses for each section on the

blade. Additional structural design variables were experimented with, including

independent spar cap thickness, variable chordwise extent of the spar caps (both

symmetric and with slanted webs allowing alignment of loading with section twist),

and materials (wood, carbon). Weight reductions were certainly possible with more

DVs, but the static design loads would have to be supplemented by operational

fatigue simulations to verify the feasibility of the results in practice. These results

did not affect the following remarks on the coning rotor, and are therefore excluded

for brevity.

The structural implications of the airfoils (in terms of structural shape) are of

course implicitly included. Recent efforts to develop “flatback” airfoils to maximize

structural efficiency have taken the airfoil shape DV to an extreme, with encouraging

results [180, 190].

Finally, the blade is mass balanced, with a point mass at the tip, so that free-

hinging to βbalance is achieved at Vrated=11 m/s and maximum rotation speed Ωmax.

The optimal operational schedule is computed, with a 400 kNm hinge moment ap-

plied from 5–8 m/s to an otherwise free hinge. This is based on scaling the CONE-

450 60 kNm low-wind applied moment. The directly applied moment was found to

be increasingly ineffective at higher wind speeds, requiring control either via pitch

(aerodynamic moment) or Ω (aerodynamic and centrifugal moments). A 7.5 m/s

hub-height wind (Vmean) is used for steady energy yield calculations, with shape

factor k = 2.2.


6.5.4 Aerodynamic Optimum

Some initial design point is required for the coning rotor. Rather than picking one

design condition, as in §6.4.3, a range of blade designs is considered using airfoil

set I. Section 4.4.13 predicts that the optimal blade shape should vary with the

βdesign,aero of the aerodynamic design point. As with conventional rotors, the design

(real) tip speed ratio λdesign,aero also has a large effect on performance. For this

study, a 40.15 m blade is used (same as REF-1500), βbalance = 25 and Ω is bounded

to 10–18 RPM.

Figure 6.9 shows the results of the optimization. Figure 6.9(a) shows the familiar

trend of increasing CP with λdesign,aero, owing primarily to reduced rotational mo-

mentum losses. Evidently increasing βdesign,aero also increases the efficiency of the

rotor, at the design point.

Figure 6.9(b) also shows higher λdesign,aero rotors to be more efficient in total

energy capture. As the blades are all mass balanced to equal βbalance, the variation

with βdesign,aero is explained with reference to Fig. 6.10. The higher βdesign,aero are

more slender, mirroring the effect of increasing λdesign,aero. The aggregate result is

an increase in λdesign,aero with βdesign,aero for the isoperformance lines of Fig. 6.9(b).

More subtle effects are also present, such as the off-design performance differences

(β 6= βdesign,aero) and different thrust behaviour affecting equilibrium coning angle

over the power curves, away from βbalance.

Figure 6.9(c) illustrates the blade mass variation with design variables. The in-

creasingly slender profiles with increasing λdesign,aero lead to lighter blades. The

variation with βdesign,aero is somewhat more complex. It appears that for larger

βdesign,aero, blade mass grows more rapidly at lower λdesign,aero, with the converse

true at higher λdesign,aero, relative to βdesign,aero = 0.

6.5.5 Blade Length

At the heart of the coning rotor energy advantage is an increase in blade length.

This increase cannot be unbounded, and is grossly limited by the tolerable increase

in rotor thrust at rated. To explore the effects of varying blade length, this section

varies blade length Stip and βbalance. The variation of βbalance effectively varies the

rated capture area, thereby altering the maximum thrust. Based on the results in

§6.5.4, βdesign,aero is chosen as 0 and λdesign,aero as 8.0. The bounds on Ω are varied

with Stip to achieve unconed tip speeds of 45–80 m/s as proxy noise constraints and

to introduce peak shaving.


0.4820.484

0.486

0.4880.49

0.492

0.494

0.496

0.4

96

0.4

98

0.498

0.5

0.5

0.502

0.5

04

0.5

06

0.5

08

0.5

10.5

12

0.5

14

βdesign, aero

(deg)

λdesig

n, aero

(re

al)

0 10 20 306

6.5

7

7.5

8

(a) CP (real) at aerodynamic design point

9898.5

98.599 99

99.5 99.5

100100

100.5

100.5

101

101

101.5

βdesign, aero

(deg)

λdesig

n, aero

(re

al)

0 10 20 306

6.5

7

7.5

8

(b) Eann (relative to ensemble mean, %)

9294

96

96

9898

98100

100

102

102

104

104

106108

110

βdesign, aero

(deg)

λdesig

n, aero

(re

al)

0 10 20 306

6.5

7

7.5

8

(c) Blade mass (relative to ensemble mean, %)

Figure 6.9 Power coefficient CP , annual energy yield Eann and total blade massvariation with aerodynamic design tip speed λdesign,aero and cone

angle βdesign,aero


0 5 10 15 20 25 30 35 400

1

2

3

4

s (m)

c (

m) Increasing β

design,aero

Figure 6.10 Blade chord profiles for λdesign,aero = 6–8

Figure 6.11 shows the overlapped contours of energy yield and maximum thrust;

both are relative to the values for the REF-1500. Note that as Prated is constant,

CF will follow the same trends (see §3.2 and §6.2). For all airfoils, the limit on Eann

increases with βbalance; the blade length must also increase to maintain Eann. The

blades weights obtained using the stated design condition are comparable to the

REF-1500.

The unbounded energy capture performance of all airfoils and control strategies

is fairly similar with variations in Stip and βbalance. The achievable energy increase

is, however, markedly different. The PTS rotors are self-limited by higher thrust

levels, relative to the VSS rotors. The difference in stall behaviour of the airfoils is

also manifest by an increase in tolerable Eann with the sharpness of stall. Modifying

the thrust bounds either side of 100% produces relatively similar increments in Eann

between the various rotors.

The energy yields of the rotors relative to the REF-1500 range from 0–30%, which

at the upper-end is similar to the original 29.9% for the CONE-450/REF-450 com-

parison at 7.5 m/s (with tower height advantage). Figure 6.12 compares the REF-

1500 power curve to those of the set II VSS and PTS rotors considered next in §6.5.6

(chosen for their equal maximum thrusts). A shape factor of k = 2.2 was used, and

Vmean is referenced to a nominal hhub=84 m. Figure 6.12(a) illustrates the generic

non-linear trends of decreasing Eann with Vmean. The coning rotors maintain an

energy advantage across the range, and all rotors benefit from hhub increase relative

to the nominal 84 m. The superior below rated power curve of Fig. 6.13 for the VSS

rotor is evident in the Eann results.

Figure 6.12(b) shows the coning rotor performance relative to the REF-1500 at

equal Vmean and constant hhub=84 m. Again, increasing hhub is clearly beneficial,


8590

95100105110115

120125130

135140145

150

βbalance

(deg)

Stip (

m)

8590

95100105110115

20 25 30 35 4032

34

36

38

40

42

44

46

48

50

(a) I PTS

859095100105110

115120

125130135

140

145

150

βbalance

(deg)

Stip (

m)

859095100

10511

0115

20 25 30 35 4032

34

36

38

40

42

44

46

48

50

(b) I VSS

9095100

105110115120125

130

135

140

145150

βbalance

(deg)

Stip (

m)

859095

100105110

115

20 25 30 35 4032

34

36

38

40

42

44

46

48

50

(c) II PTS

9095100105110115

120125130

135

140

145

150

βbalance

(deg)

Stip (

m)

8590

95100105110

115

20 25 30 35 4032

34

36

38

40

42

44

46

48

50

(d) II VSS

8590

95100105110115

120125130

135

140

145

150

βbalance

(deg)

Stip (

m)

8590

95100105110

115

20 25 30 35 4032

34

36

38

40

42

44

46

48

50

(e) III PTS

859095100

105

110115120125130

135

140

145

150

βbalance

(deg)

Stip (

m)

859095

100105

110115

20 25 30 35 4032

34

36

38

40

42

44

46

48

50

(f) III VSS

Figure 6.11 Relative energy yield and maximum thrust variation with bladelength Stip and balance angle βbalance ( energy yield; thrust)


4 5 6 7 8 9 10 11 120

50

100

150

200

Vmean

Eann/E

ann,r

ef (

%)

(a) Eann relative to REF-1500 at 7.5 m/s and hhub=84 m

4 5 6 7 8 9 10 11 12100

120

140

160

180

200

Vmean

Rela

tive

Eann (

%)

(b) Eann relative to REF-1500 at equal Vmean and hhub=84 m

Figure 6.12 Energy yield relative to REF-1500 varying with mean wind speedVmean and hub height hhub ( 1.0hhub, 1.2hhub, 1.4hhub;

REF-1500, VSS, PTS)


more so for lower wind speeds. The relative impacts of the coning rotor and tower

height effects are illustrated by the curves for the REF-1500, which only benefit from

hhub increase. The low wind advantage of the coning rotors also mirror the results

of Jamieson [66]. Without definitive cost metrics, no precise answer can be gleaned

for overall COE of the coning rotor. As all rotors have equal installed generator

capacity, the CF will clearly benefit for a coning rotor, as discussed in §6.2. The

slope dEann/dVmean is largest around 7.5 m/s, indicating the cost trade-offs are most

acute in this range.

6.5.6 Control Strategy Impacts

The dynamic implications of the control strategy have been outlined in §3.6.2. Con-

tinuing to concentrate on the steady-state operational profile, Fig. 6.13 shows curves

for rotors with similar thrust and equal βbalance, selected from Fig. 6.11. The blade

lengths are 38 m, with the exception of 40 m for the set II and III VSS rotors.

Although the PTS rotors in §6.5.5 cannot employ rotors as large as the VSS ones,

additional constraints are imposed by the generator. A problem synonymous with

VSS is the excess torque requirement to slow the rotor into stall [58, 59]. The large

torque increment after the speed limit has been reached is undesirable from the

generator perspective. Unlike a constant-speed stall controlled rotor, which must

accept power loss above-rated from passive stalling as the CP –λ curve is traversed,

the active stall-controlled rotor can move over the map to initiate stall. The added

expense of this strategy is a generator to provide the excess torque.

Using the PMG model of §4.8, Fig. 6.14 illustrates the cost variation of active

material required to achieve the extreme and rated torque/speeds of Fig. 6.13(f),

optimized for minimum cost/torque. The reduction in cost with generator diameter

Dgen is quite apparent for a PMG design. For the set III rotor (800–1025 kNm),

the torque increase leads to a 40% cost increase, based on equal Dgen. The set II

rotor (800–1120 kNm) leads to a 60% increase. This clearly represents an important

design consideration. Recall that dynamic overloads are tolerable by both generator

and power electronics; at discussion here are the steady torques that must be reacted

to maintain control steady-state control.

The excess torque requirement and high-wind behaviour are both directly at-

tributable to the choice of airfoil. More specifically, the proximity of the angles of

attack (α) for maximum lift-to-drag ratio αcl/cd,max and stall αstall are key, as is the

slope of the lift curve after stall. Experimentation with rotors designed with three


0 5 10 15 20 250

200

400

600

800

1000

1200

1400

1600

V (m/s)

Pow

er

(kW

)

I PTS

I VSS

II PTS

II VSS

III PTS

III VSS

(a) Power

0 5 10 15 20 250

50

100

150

200

250

V (m/s)

Th

rust

(kN

)

(b) Thrust

0 5 10 15 20 255

10

15

20

25

30

35

40

V (m/s)

β (

deg)

(c) β

0 5 10 15 20 250

200

400

600

800

1000

1200

V (m/s)

To

rque

(kN

m)

(d) Torque

0 5 10 15 20 250

2

4

6

8

10

12

14

V (m/s)

λ

(e) λ

10 15 200

200

400

600

800

1000

1200

Ω (RPM)

Torq

ue (

kN

m)

(f) τ − Ω

Figure 6.13 Operational curves for optimal rotors at βbalance = 30


€ 35,000

€ 45,000

€ 55,000

€ 65,000

€ 75,000

€ 85,000

€ 95,000

5.0 5.2 5.4 5.6 5.8 6.0Outer diameter (m)

Cost

(Euro

)

800kNm@18 [email protected]@14.8RPM

Figure 6.14 Generator cost variation with diameter and torque/speed levels

airfoil sets (I, II, III) has highlighted the common experience [36, 58, 59] that airfoil

curve characteristics are key to the steady rotor performance curves. Figure 6.15

shows the CP map for the VSS rotors just examined. Note that the data includes

stall-delay, heterogeneous blade lengths and blended airfoils, somewhat complicat-

ing a simple comparison between the 2D coefficients in Appendix C. The aggregate

effects of the airfoil behaviour are manifest in the “peakiness” of the CP,max point

of the curves, and the slope variation −dCP /dλ after stall.

The set II airfoils have the greatest difference between αcl/cd,max and αstall (≈10)

and the 17% set III the least (≈1.8). This characteristic affects the stall half of the

curve to the left of CPmax, where the set III rotor moves quickly into stall as the

airfoils reach stall, with minimal angle of attack change. The 17% set III airfoil is also

the most peaky in terms of lift-to-drag ratio change with AOA. This means that for

the optimized rotors operating at αcl/cd,max the performance is rapidly degraded off-

design. The set I rotor requires large movement in λ near rated, but limits extreme

torque by the sharp −dCP /dλ around λ = 3. The stall of the rotor is in fact too

pronounced at low λ, leading to the power drop-off observed in Fig. 6.13(a) above

V=23 m/s. It has been possible to optimize the twist to reduce the excess torque

requirement for the VSS rotors, but at the expense of below rated performance.

There are two other main differences between VSS and PTS rotors: the coning

angle above rated is less for the PTS rotors and the thrust levels higher. Both


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 4 8 12λ

C PIIIIII

Figure 6.15 CP − λ curves for β = 0 and varying airfoil set

effects arise from maintaining a constant Ω above rated. This rotationally stiffens the

rotor, keeping it more open. In turn, the capture area is larger, increasing the thrust

somewhat. More dominating, the faster rotation speed for the PTS rotors means

that the inflow angle is reduced relative to the VSS rotors and the relative velocity

at the blade higher. By examining the local flow and resulting forces, it is found

that even though the lift/drag coefficients are similar, the flow properties result in a

higher stream wise force component and smaller torque-inducing component for the

PTS rotors. Evidently the centrifugal stiffening force, proportional to Ω2, dominates

the Ω proportional thrust force in determining the equilibrium cone angle.

6.5.7 Aerodynamic Uncertainty

When interpreting design and optimization predications, it behoves the designer to

appreciate the limitations and potential pitfalls of the modelling tools employed. In

the case of the coning rotor, the primary area of modelling uncertainty is in the

aerodynamics. Many of the issues are not unique to this concept, but extend to any

wind turbine, especially those operating in stall and at high yaw angles. The issues

have been discussed in various sections of Part II, but are unified here to inform

future designers.

The issues may usefully be divided into an “outer” problem of induction and “in-

ner” problem of 2D section aerodynamics. In the BEM model, these two aspects


are coupled by the velocity components resolved parallel to each blade section. Full-

field CFD methods fully couple the two problems by resolving the flow with fine scale

grids near the blade surfaces, linked to coarser meshes extending well downstream

into the wake. Due to this tighter coupling and more fundamental approach, CFD

methods are theoretically better placed to avoid the issues discussed below. Practi-

cally speaking however, they remain too expensive for application to wind turbine

design work, necessitating the use of BEM theory with certain caveats. Issues with

the outer problem are typically manifest at lower wind speeds, and are important

for energy capture (Region II). Inner problem issues appear especially during stall,

when power limiting (Region III).

The outer problem is essential that of determining the proper induced velocities

which modify the incident flow as it approaches each blade section and in the wake.

The origin of the induced velocities is the wake shed by the blades. It includes the

following aspects:

Flow at the Disc The induced flow at the disc contains axial, radial and

azimuthal components. The proper prediction of all three components is

important (see §4.4.2).

Far Downstream For BEM, the velocities seen at each section on the disc

must be properly related to those in the far wake (see §4.4.6) to balance

the momentum equations.

The modified BEM method in this thesis includes both of these considerations,

producing improved aerodynamic and loading predictions relative to standard the-

ory (see §5.1.2). However, within the confines of a computationally appropriate BEM

method, only the dominant tip vortex tube may be considered (i.e. without further

iterations on more complicated wake structures). This ignores the variation in span-

wise loading on a real rotor which creates multiple vortex tubes, further modifying

the spanwise variation of induction. A number of other approximations are also

necessary:

3D Tip/Hub Losses The loading at the hub and tips of the blades must

trend to zero. Physically, this is an artefact of a finite number of finite A

blades creating distinct vortex sheets. BEM theory models this effect in

the momentum equations as averaged induction factors via the Prandtl F

factor (see §4.4.3.3).


Wake Expansion The trailing wake vorticity must strictly freely follow the

streamlines. As induction increases, the wake must expand to satisfy conti-

nuity. In turn, the induced velocity at the disc is modified by the modified

strength and location of the vortex sheets (see §4.4.11).

High Induction The momentum equations are fundamentally in error, even

with wake expansion, at very high induction factors, as they violate conti-

nuity. The thrust models of §4.4.3.4 attempt to overcome this deficiency,

but owing to the fundamental nature of the problem are unable to capture

the recirculating stream tubes.

Yawed Flow Akin to a coned rotor, a yawed rotor experiences varying in-

duction with azimuth owing to the relative position of the wake.

Wake expansion is better handled by the new BEM method (see §5.1.2.1), however

very high induction factor conditions remain quite challenging to predict for both

coned and unconed rotors. Yaw prediction has been improved with current method

(see §5.1.8), but again includes some error because a further iteration of the solution

for each azimuthal blade position is not possible in the general dynamic case.

The inner problem consists of predicting lift (cl), drag (cd) and moment (cm)

2D coefficients based on the velocity seen at the blade (as shown in Figs. 4.4 and

Fig. 4.11). Even with perfect resolution of the outer problem, the 2D sectional

problem is complicated by the following aspects:

Large AOA Measurements and numerical predictions are fairly accurate in

attached-flow regions (small AOA below stall). Design critical points occur

when the airfoils are stalled, even out to ±180, which is far outside the

standard test range for an airfoil. Fortunately, airfoils behave similarly

past a critical AOA of approximately 50 enabling extension of available

data (see Appendix C). Even below stall, drag prediction in particular can

lead to multiple datasets, for ostensibly the same airfoil [177].

Operational Changes Surface modifications during operation (e.g. soiling,

ice formation) will modify the 2D characteristics.

Re Effects Wind turbines operate in the Re range over which airfoils un-

dergo large performance changes, requiring data in the proper regime.

Spanwise Flow The independence principle (see §4.4.9) is generally valid

below stall. For stalling rotors, large modifications to the 2D characteristics

may occur owing to boundary layer transport. The models used here are

only a first approximation to this condition, which should receive more


detailed verification. In particular, the root and tip regions will tend to

undergo opposite changes (higher and lower maximum lift respectively) as

the boundary layer migrates outboard.

Centrifugal Pumping In addition to the spanwise flow stall delay (primar-

ily important for coned rotors), centrifugal pumping (see §4.4.8) will modify

lift/drag properties. The fundamental mechanisms remain misunderstood

and only partially modelled.

Dynamic Flow The flow around a 2D section behaves differently in steady-

state than dynamically. The flow responds with some lag, both in stalled

and unstalled conditions. The general case, including spanwise flow, may

only be treated with approximate models, which must be tuned to the

particular airfoil section employed.

These individual modelling difficulties ultimately manifest themselves in the ag-

gregated performance predictions. It is well recognized that stall prediction is very

difficult for any rotor (i.e. even unconed) [91]. In fact, the verification of above

rated performance for a real machine revealed a 17% under-prediction of power [58].

Surface roughness effects will drastically change stall behavior, although that should

be favorable from a safety perspective, as power and torque can only decrease; from

an energy capture perspective, it is of course detrimental.

Any form of stall delay to higher AOA (see §4.4.8 and 4.4.9) will act to decrease

safety margins. This is true for thrust, as well as the generator torque critical for

the VSS rotors. The modelling difficulties evident in §5.1.6 are a pathological worst-

case above stall, given the simplified geometry used in the study relative to most

commercial blades, and represent and upper bound on aerodynamic uncertainty.

Based on these results, power predictions may be erroneous by up to 30–60%, varying

with the stall delay model used.

To provide some quantitative indication of the aggregate modelling effects on the

coning rotor predictions, the operational curves for the set II rotors of §6.5.6 are

compared in Fig. 6.16. For the VSS rotor τ is increased by adding the stall delay

models. The extrema value is approximately 4% too low without any model. The

PTS rotor predictions are in error in β, in this case by 4% in extreme value. In both

cases, the spanwise flow effect is small around rated, but approximately equal to the

centrifugal pumping effect at Vco.

Adoption of a PTS strategy would provide a measure of design safety on the

difficult stalled-flow predictions. Ample control would be available from the pitch


0

200

400

600

800

1000

1200

0 5 10 15 20 25V (m/s)

τ (kN

m)

No CorrCPCP & SF

(a) VSS

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25V (m/s)

β (de

g) No CorrCPCP & SF

(b) PTS

Figure 6.16 Variation in operational curves with stall delay models for set IIrotors (CP = Centrifugal pumping, SF = spanwise flow)

system to compensate for any real world shortcomings of the model of the system.

The dynamic controller design should take this possibility into account by adopting

a robust approach.

It is difficult to place quantitative bounds on the aerodynamic uncertainty present

in aerodynamic predictions for wind turbines. Comparison of the modified BEM

method to CFD results in §5.1 did yield bounds relative to a more accurate method,

providing some indication of the level of error to be expected. This section has high-

lighted the various error sources, their domains of influence, and their conservative

or non-conservative effects on design predictions. Ultimately however, further exper-

imental results are required against which to valid the predictive models, in order to

obtain more quantitative bounds. Until such time as these results become available,

the designer is forced to heed past experience and the insight offered in this section

to synthesize robust designs that are relatively insensitive to the modelling errors

which may be present.

Chapter 7

Design Considerations

Using the initial rotor designs presented in Chapter 6, future work must extend

the analysis into unsteady simulations, to further refine the design. This chapter

highlights a number of design considerations that become critical to the time-domain

performance of the coning rotor. The first set concern the dynamics of the system

(§7.1). The second area of consideration is the low-frequency acoustic profile of

the coning rotor (§7.2) that must be accounted for and mitigated in the layout and

operation of any potential machine.

7.1 Dynamic Simulation

This section presents an exploration of the fundamental dynamic aspects of the

coning rotor. In particular, the non-linear flapping response to the various forcing

functions is discussed in §7.1.1, followed by aerodynamic damping of the flap and

edgewise modes in §7.1.2.1 and 7.1.2.2 respectively. To proceed further will require

full dynamic simulations of the rotor, including flexible blades and tower, drive-

train, yaw mechanism, and a viable controller. The synergistic modal interactions

of the various components will need to be checked on a Campbell diagram and/or

by examination of a linearised state-space model. The results of fully non-linear

time-domain simulations must then be used to further refine and verify the design.

Hopefully the basic insight offered in this section will clarify the execution of these

future tasks.

7.1.1 Fundamental Response

The intent of this section is to explore the fundamental non-linear effects that will

influence the loading of the coning rotor, without complicating the picture with

controllers or stochastic aerodynamic forces. Manwell et al. [17] presented the work

of Eggleston and Stoddard [14] for a linearised model about β0 = 0, incorporating

very simplified (linear) aerodynamics, steady yaw motion, and same type of hinge

217

218 Chapter 7 Design Considerations

offset, similar to the presentation in §4.6.3.1. Instead of applied damping or an

active moment, a linear spring was used to tune the response to simulate the first

flap mode of a flexible blade. The linearised equations were analytically examined,

using a solution expansion of constants, cosines and sines of the rotor azimuth angle

with fixed rotation speed.

The linearised equations of motion developed in §4.6.3.2 yielded basic insight into

the fundamental frequencies and critical damping (§4.6.3.3) of the coning rotor. In

this section, the non-linear EOM of §4.6.3.1 are examined numerically, using data

from the REF-1500, with the inclusion of tilt angle, unsteady aerodynamics and

non-linear cone angles. Constant hinge moment is used to bias the steady cone

angle, and a constant rotor speed Ω is used for a downwind rotor. Figure 7.1 is

useful for visualizing the motions as constant, cosine and sine responses.1

(a) Constant +β0 (b) Cosine +βc at θ = 0 (c) Sine +βs at θ = π/2

Figure 7.1 Flap angle solution expansion terms

To start with, the aerodynamic contribution is omitted. The gravity term in

Eq. (4.6.20b) imposes a cosine varying loading, and a constant term, both biased

by the tilt angle ψt. With ψt = 0, gravity is purely a cosine input. The analytic

solution yields a “cyclic sharing” term, dependent on the stiffness of the hinge spring,

determining the relative amount of cosine and sine response of the flap angle to the

input. It is found that in general, including the other forcing functions discussed

next, that stiff blades respond with little phase-lag to the input, while freely flapping

blades exhibit a π/2 phase lag. Therefore, gravity causes a cosine response in a stiff

rotor, and a sine response for a freely hinged rotor.

A critical damping coefficient HactD for ζ = 1 may be computed from Eq. (4.6.27),

and Hact for a steady 10 flap angle from Eq. (4.6.20b). Releasing from β = 0, the

10azimuth is defined here for the 1st blade vertically upwards, whereas Manwell et al.[17] define it vertically downwards.

7.1 Dynamic Simulation 219

system quickly reaches the example cyclic responses shown in Fig. 7.2. For ψt = 0

in Fig. 7.2(a), the blade angle phasing is as expected 120, and blade 1 has maximal

β at 1/4 rev showing a pure sine response to gravity. Changing to ψt = 5, as

shown in Fig. 7.2(b), the mean cone angle and flapping magnitude is reduced, but

the phasing remains the same.

2 2.5 3 9.5

10

10.5

Rotor revolutions

β (

deg

)

1

2

3

(a) ψt = 0

2 2.5 3

9.2

9.4

9.6

9.8

10

Rotor revolutions

β (

de

g)

1

2

3

(b) ψt = 5

Figure 7.2 Blade flapping without aerodynamic contribution

The aerodynamic loading is now added to the equations. In the absence of gravity

and wind shear, a trivial result of both the analytic and numerical approaches is

that the rotor operates at a steady cone angle. Hinge offset and positive resisting

moments/springs reduce the angle. As expected, the aerodynamic moment balances

the inertial centrifugal force and imposed root moments. As noted in §5.4.3, the

EOM admit two steady-state solutions, symmetric about β = 45. Dynamically, for

β > 45, the system is unstable; as β increases, the moment arm of the centrifugal

force increases, but this is overcome by the decreasing centrifugal force magnitude,

so β continues to decrease.

Wind shear is a sine input, and therefore the response for the coning rotor is

primarily a cosine one.1 Combined with gravity and tilt, both of these cyclic forcing

functions therefore conspire to develop a real or “effective”2 yaw angle for the rotor.

Manwell et al. [17] have shown the effect to be worse for faster-spinning and softer1An active hinge moment with some component proportional to β will effectively stiffen

the rotor and produce some sine response.2I.e. if held in fixed in yaw, the rotor tips will not sweep out a circle centred on the rotor

axis.


(no hinge spring) rotors. This has implications for a free-yaw design in which the

developed cross-flow on the rotor creates a sine input, owing to the effects of ad-

vancing/retreating blades. The sine and cosine inputs will balance at some steady

yaw angle. Operation in yaw, either “effective” or real, will also reduce the power

capture somewhat.

Removing the aerodynamic contribution again, with ψt = 0 and yaw angle held

at 0, the hub loads are shown in Fig. 7.3(a), in fixed coordinates (i.e. x′′–y′′–z′′

of Fig. 4.1(b), not rotating with blades, ), with the same hinge actuator moment

and critical damping as earlier. The main force produced is a steady gravity force

in the −z′′ direction, from total blade weight. There is also a very small mean −y′′

component, and the x′′ component is zero (no aerodynamic thrust present). Both

y′′ and z′′ components have a small 3P component (B = 3), π/2 out of phase (not

visible at the figure scale).

0 1 2 3 4-16

-14

-12

-10

-8

-6

-4

-2

0

2x 10

4

Rotor revolutions

Forc

e (

N)

Fx

Fy

Fz

(a) Forces

0 1 2 3 4-0.5

0

0.5

1

1.5

2

2.5

3x 10

5

Rotor revolutions

Mom

en

t (N

m)

Mx

My

Mz

(b) Moments

Figure 7.3 Hub loads for blade flapping without aerodynamic contribution(non-rotating CS)

This behaviour may be explained with reference to Fig. 7.4. Cyclic flapping is

superimposed on a mean cone angle (from constant Hact), creating blade paths

(equivalently tip paths or blade centres of mass) that are effectively yawed from the

mean path by βy as the blades pass through the x′′–y′′ plane. The net centre of mass

of the rotor (averaging all blades) therefore tracks a circle with frequency BP , offset

in the +y′′ direction from the x′′ axis. This effectively creates a rotating imbalance

force Fy(θ), only present for a rotor operating at mean non-zero β.


Figure 7.4 Dynamic situation with flapping blades and rotating imbalance

Lagging by π/2, the maximum flap velocity β develops as the blades pass through

the x′′–z′′ plane. Operating about a non-zero mean β, the sinβ (radial) component

of the flap velocity is non-negligible, and so a Coriolis force is created.1 The net

effect of both responses is the y′′ and z′′ cyclic forcing in the hub fixed CS, the

former with a negative mean. Both were observed to increase in amplitude with

fewer blades and larger mean cone angle.

Helicopter rotors are somewhat similar, in that the tip-path plane is tilted relative

to the shaft axis, to direct thrust forward and sideways for flight control. The effects

are smaller though, as the blades are much lighter. In addition, as the mean cone

angle is small, both the sinβ component of flap velocity and magnitude of centre

of mass movement are smaller. Two-bladed teetered rotors can obviate the rotor

imbalance force with an under-slung rotor, by placing the pivot axis on the rotation

axis, at the pre-coned rotor centre of mass. The transmission of Coriolis forces to

the hub are eliminated with lead-lag hinges in fully-articulated rotors.

The hub moments in Fig. 7.3(b) show Mx increasing then decreasing as the blades

flap towards equilibrium. This illustrates the rotor torque moment developed with

collective coning motion, to conserve rotational momentum. In this case Ω is held

1Note that the Coriolis force ~Fc = −2m~ω × ~vr is simply a manifestation of conservationof angular momentum and vice versa.


fixed, but with variable speed may be used to reduce the moment magnitude. All

moments have a small BP alternating component, owing to the effects just discussed.

Again, as with helicopters, the hinge offset Rhinge imparts additional moments to

the hub from the moment of the forces at the hinge location, relative to the hub

centre.

7.1.2 Aerodynamic Damping

The motion of a body through a fluid produces forces proportional to the velocity

of movement. It is the work of this aerodynamic force component(~F (~V ) · ~V ) that

gives rise to either positive or negative damping. In the latter case, vibrations

are reinforced, as the force component is in the same direction as the velocity. If

unchecked, either fatigue life is reduced, or catastrophic divergent oscillations may

develop. Aerodynamic damping is relevant to both discrete DOF (i.e. the rigid

blade’s flap hinge) as well as the flexible modes of the blades [113].

7.1.2.1 Hinge Flapping Damping

The gross motion of the blade is flapping about its hinge axis. Jamieson [66] found

that tower-top loading with rigid coning links was excessive (natural flap frequency

and rotation speed too close), so damped links were envisaged instead. These links,

and the collective actuator both also provided damping to compensate for the loss

of aerodynamic damping on the stalled rotor. If inadequately damped, this motion

can lead to destructive fatigue loading on various components. The flap motion is

particularly prone to difficulty, as the natural flap frequency is always close to the

fundamental rotor speed (see Eq. (4.6.24)). The CONE-450 experience of resonant

excitement is therefore an intrinsic issue to the coning rotor.

Provision for adequate passive damping in a hydraulic coning actuator mechanism

may be readily engineered. A safety factor over the critical damping factor should

avoid excessive response. Alternatively, active feedback control may be used to

better tune the system response over the wind speed range. The level of aerodynamic

damping loss will vary with the rotor characteristics.

Figure 7.5 shows the logarithmic decrement δd (see §4.6.3.3) for the rotors consid-

ered in §6.5.6, obtained from the aerodynamically damped oscillation after release

from β = 0. The results include uniform wind, no gravity, dynamic and centrifugal

stall-delay models, and are computed at the optimal design operating points varying

with V . For V below those shown, the system was over-damped.


10 15 20 25-1

0

1

2

3

4

5

6

V (m/s)

δd

I

II

III

(a) VSS

10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

V (m/s)

δd

(b) PTS

Figure 7.5 Logarithmic decrement δd for flap hinge damping

All rotors experience their lowest levels of damping near rated. Airfoils with sharp

stall (set I) experience lower levels of damping. It appears that the PTS rotors are

better damped at higher V , whereas the reverse is true near rated.

7.1.2.2 Flexible Body Damping

The problems of aerodynamic damping are present in all rotor concepts employing

some type of stall control (FSS, VSS or PTS). The industry shift to PTF as machines

grew past 1 MW was at least partially motivated by the inherent technical difficulty

associated with stall, both from a prediction point of view and inherent physical

challenges. During the period of transition from FSS to PTS, Petersen et al. [159]1

presented an excellent analysis of the fundamental issues associated with the effects

of aerodynamic stall on the modal damping of flexible blades.

Concentrating on FSS machines, equations were derived and results presented

on a sectional and whole blade-mode basis. The damping coefficients in flapwise,

edgewise and coupled directions (from integrated flexibility over twisted structural

and geometric axes) were examined in terms of aerofoil lift/drag characteristics.2

The influence in terms of overall power and thrust curves of the section and blade

1Later repeated by Burton et al. [16, §7.1.9]2Edgewise and flapwise are “true” in this context, parallel and normal to the section

chord respectively.


were also considered. The main results with applicability to the current concept

may be summarized as:

• A negative power/wind speed curve creates negative damping. For an actively

controlled rotor, this may be avoided.

• Sharply negative cl −α slopes after stall are detrimental. Dynamic stall plays

a large role in modifying the behaviour.

• Low-lift (smoothly stalling) airfoils are preferred outboard, to minimise the

contribution of negative lift after stall. The mode shapes weight the contri-

bution to damping as radius squared, making outboard sections more critical

aerodynamically.

• Structural-pitch towards feather is favourable for edgewise vibration, while

minimally detrimental to flapwise vibration. The mechanism is a modification

of the sectional direction of motion to increase coupling of the edgewise mode

to the well-damped unstalled flapwise mode.

• Highly cambered airfoils, with α0 near 0 are preferable to minimise geometric

and structural pitch towards stall, thereby aligning the edgewise direction

favourably towards feather.

• Changing the structural or geometric collective pitch angle have essentially

the same effect. Therefore, with PTS, edgewise damping can be expected to

decrease.

• Increased chord increases negative aerodynamic damping, as well as increases

stiffness and structural damping. A compromise is therefore required, with

relatively large chord inboard where aerodynamic damping is less critical but

edgewise stiffness is greatly enhanced.

• Edgewise mode coupling to yaw and nod modes (i.e. close to blade edgewise

frequencies) should be avoided, by making the shaft/nacelle structure as stiff

as possible.

Petersen et al. [159] also derived isolated section damping coefficients, by linearis-

ing the variation in aerodynamic force FR about operating point Vn and rΩ. The

equations were derived assuming zero induction factor, and with linearised aerody-

namics including structural motion (i.e. relative aerodynamic velocities) in the θ

and n directions vs,θ and vs,n. The forces in these two directions then have steady

and damping (proportional to vs) components:

FR =FR

θ

FRn

=FR

θ0

FRn0

−

− ∂F Rθ

∂(rΩ)

∂F Rθ

∂Vn

− ∂F Rn

∂(rΩ)∂F R

n∂Vn

vs,θ

vs,n

= FR

0 −[cRθθ cRθn

cRnθ cRnn

]vs (7.1.1)


The resulting equations yield a full damping matrix with these two DOF, with the

signs accounting for direction of motion. Without repeating the derivation here, but

expressing the damping constants in the coordinate system notation of Fig. 4.4, the

damping coefficients for an isolated section are:

cRθθ(r, Vn) =12cρrΩVrel

[2r2Ω2 + V 2

n

rΩcd − Vn

∂cd∂α− V cl +

V 2

rΩ∂cl∂α

](7.1.2a)

cRθn(r, Vn) =12cρrΩVrel

[−Vncd − rΩ

∂cd∂α

+r2Ω2 + 2V 2

n

rΩcl + Vn

∂cl∂α

](7.1.2b)

cRnθ(r, Vn) =12cρrΩVrel

[−Vncd −

V 2n

rΩ∂cd∂α− 2r2Ω2 + V 2

n

rΩcl + Vn

∂cl∂α

](7.1.2c)

cRnn(r, Vn) =12cρrΩVrel

[2V 2

n + r2Ω2

rΩcd + Vn

∂cd∂α

+ Vncl + rΩ∂cl∂α

](7.1.2d)

A real section will move in a coupled direction, so the damping constants are

transformed here to an xB axis with:

cBx = cos2(θs)cRθθ − cos(θs) sin(θs)(cRθn − cRnθ

)+ sin2(θs)cRnn (7.1.3)

where θs is the angle measured clockwise from θ, the same as AOA α.1

The sectional power and thrust per unit span are computed with:

P ′ =12ρcrΩVrel (Vncl − rΩcd) (7.1.4a)

F ′ =12ρcΩVrel (rΩcl + Vncd) (7.1.4b)

Using these equations, a comparison between airfoils and control strategies can

be made at the fundamental sectional level. A section radius r of 30 m is used, and

local speed ratio λr = rΩ/Vn of 6 below rated. A nominal chord of 1 m is used for

the set I airfoils, and the other airfoils chords are scaled to achieve the same P ′ at

11 m/s. The airfoils are operated at their respective maximum cl/cd points below

rated. The airfoils have static twist γtw to achieve this, and are actively pitched by

γop above rated. All twist angles are positive towards feather. The section damping

cBx angle θs is expressed with γstr (structural twist relative to chordline) as:

θs = γstr + γop + γtw (7.1.5)

Edgewise and flapwise vibration directions are nominally 90 apart, so varying γstr =

−90 . . . 90 covers both. With the integrated influence of inboard sections, edgewise

vibration is approximately in the range of γstr = 0 . . . 20.

1This is opposite to Petersen et al. [159].


Figure 7.6 shows the damping achieved for the set I airfoil. All control strategies

achieve the same damping below rated, as expected. Above rated, the controls

utilizing stall clearly suffer from a loss of damping, particularly in the flap direction.

The PTS rotor is particularly affected, as a large pitch well into stall is required. In

addition, active pitch re-aligns the vibration direction in a detrimental way.

510

1520

25

-90-60-300306090-100

0

100

200

300

V (m/s)γstr

(deg)

cxB (

Ns/m

2)

(a) PTF

510

1520

25

-90-60-30 0306090

-1200

-1000

-800

-600

-400

-200

0

200

400

V (m/s)γstr

(deg)

cxB (

Ns/m

2)

(b) PTS

5

10

15

20

25

-90-60

-300

3060

90

-800

-600

-400

-200

0

200

400

V (m/s)γstr

(deg)

cxB (

Ns/m

2)

(c) VSS

5 10 15 20 25-15

-10

-5

0

5

10

15

20

25

V (m/s)

γ op (

deg

) or

Ω (

RP

M)

PTF

PTS

VSS

(d) Control variables

Figure 7.6 Damping constant variation with control strategy for 15% thick set Iairfoil (Ns/m2)

Additional airfoils from set II and III are compared in Fig. 7.7, for PTS control.

The 5 m/s case is quite similar and uniformly well damped in all directions. Moving

into stall at 15–25 m/s, the behaviour is a somewhat complicated mix of factors.


Firstly, the set III has the smallest chord (and set II the largest), increasing damping

(see earlier list [159]). Secondly, the pitch action shown in Fig. 7.8(a) is largest for

the set II airfoil. The vibration direction is therefore more negative, an overall

detriment to damping. Finally, and most importantly, note that the stall drop-off in

cl is roughly comparable for the set II and III airfoils, and both are much less severe

than set I, over the range of AOA shown in Fig. 7.8(b). This is the dominant factor

in the highly negative levels of damping for the set I airfoils. The set II and III

airfoils experience similar levels of damping, owing to their similar stall behaviour,

shifted in γstr from the overall pitch angle γop + γtw.

-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

200

γstr

(deg)

cxB (

Ns/m

2)

5

15

25

I

II

III

Figure 7.7 Damping constant variation with airfoil for 15% thick airfoils andPTS ( 5 m/s, 15 m/s, 25 m/s; · I, II, III)

Figure 7.9 shows damping data for a thicker set of airfoils.1 The set I damping

is clearly improved across the entire range of V and γstr, owing to the smoother

stall. The set II/III damping is reduced somewhat relative to the 15% airfoils. This

is explained by the second stall peak in the thin airfoil data, relative to the more

negatively sloping cl curve for the 21% thick data. Apparently the operation at

relatively constant γop + γtw = −8 (above rated) for the set II/III airfoils has little

positive damping effect, relative to the -18 angle of the 15% thick airfoil.

Overall, it can be seen that there are multiple competing effects that influence

damping, making airfoil choice and blade design all the more complicated. In the

original CONE-450 study [66], no edgewise mode shapes were included, only rigid

flapping and a flexible flap mode. Any potential edgewise damping issues would1Set III airfoil data is the same as set II data at this thickness.


5 10 15 20 25-18

-16

-14

-12

-10

-8

-6

-4

-2

0

V (m/s)

γtw

+ γ

op (

de

g)

I

II

III

(a) Total twist γ

5 10 15 20 250

5

10

15

20

25

30

35

V (m/s)

α (

deg)

(b) AOA α

Figure 7.8 Variation of AOA α and total twist angles over wind speed range withairfoil choice (15% thick) for PTS

-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

γstr

(deg)

cxB (

Ns/m

2)

5

15

25

I

II & III

Figure 7.9 Damping constant variation with airfoil for 21% thick airfoils andPTS ( 5 m/s, 15 m/s, 25 m/s; · I, II, III)

7.2 Tower Thump 229

therefore not have been manifest in the results. The unavailability of a suitably

modified version of BLADEDTM prohibited more detailed examination of these ef-

fects for a complete blade, which should be revisited in future work.

7.2 Tower Thump

Lowson and Lowson [139] concluded that the CONE-450 would produce only a min-

imal noise increment over an equally-rated VSS machine. Within the confines of the

analysis method, increased trailing-edge-length was the primary culprit. Tip-noise

can be avoided by proper design, and is now relatively well understood and funda-

mentally limited by tip-speed (<70 m/s). As discussed in §4.5.3.1, an important

effect not originally considered is “tower-thump”, present for any downwind rotor

passing through a tower wake. This noise has been problematic in the past for

downwind machines [191]. The model of §4.5.3 is used in this section to paramet-

rically examine the effects of wind speed (§7.2.1), wake profile (§7.2.2), offset from

the tower (§7.2.3), control strategy (§7.2.4), and rigid-body flap motion (§7.2.5).

Section 7.2.6 gives a brief outline of future directions for acoustic investigation.

It is not possible to draw absolute conclusions from the current investigations.

Rather, a quantitative indication of the effect of design parameters is sought. A

complete design would be required, as would a strict measure of acceptability, for

objective quantification in terms of maximum acceptable SPL. Some indications

were given in §5.2; Barman et al. [141], who measured real data for a downwind

turbine that caused LFN disturbance to neighbours, found pressure peaks of ±1.4–

1.6 Pa, corresponding to ≈ 75 dB peaks around 10 Hz (25 RPM rotor speed).

ExcelBEM predictions presented in §5.2 for an up-wind rotor (for which LFN is not

a practical consideration), indicate that 72 dB is an acceptable upper-limit on LFN

(at 11 m/s and 100 m observer distance).1 Attenuation with distance is minimal for

low-frequency noise, so observer distance is not a influencing factor in this analysis.

This will be true even for nearby offshore machines.

The REF-1500 rotor is used as a baseline, with the following configuration unless

otherwise stated:

• The observer is 100 m downwind of the tower (0 azimuth), at ground level.

• The nominal wind speed is 11 m/s, below the rated power of the machine but

at maximum rotor speed.

1Noise is measured on a logarithmic scale, so the nominal 12–15 dB difference in §5.2between upwind and downwind is substantial.


• Wind shear is included (α = 0.15).

• The tower wake profile has wake deficit ∆ 0.3 and width w 2.5D, both defined

3.25D downwind of the tower, typical values for the wake from a cylinder

Wang and Coton [87].

• No tilt angle is included.

• No dynamic motion, only fixed cone angles (0 nominal) and rotor speeds

(20 RPM nominal).

• Dynamic stall delay (referred to below as simply stall delay) is turned off.1

• Centrifugal pumping and spanwise flow corrections are turned off throughout.

• The dynamic inflow model is used for induction calculation.

7.2.1 Variation with Wind Speed

Barman et al. [141] presents non-dimensional wake profile data over a range of

wind speeds for the velocity deficit behind a tower, which are fairly invariant with

respect to free stream velocity, given the error-bounds. Based on this, the SPL can

be expected to grow with wind speed. To test this, predictions were made over

the operational profile of the REF-1500, including variable speed and pitch angle

(for the nominal control schedule, without stall delay). The results are shown in

Figs. 7.10 and 7.11.

5 10 15 20 2575

80

85

90

95

Windspeed (m/s)

SP

L (

dB

)

Stall delay

No stall delay

Figure 7.10 SPL variation with wind speed

These results confirm that the noise grows, at least until rated power. With pitch

action, the forces remain relatively constant, and therefore the SPL only continues

1Effects are only important above rated.

7.2 Tower Thump 231

0 5 10 15 200

10

20

30

40

50

60

70

80

f (Hz)

SP

L (

dB

)

5

8

15

20

25

Figure 7.11 Acoustic spectrum variation with wind speed (m/s), including stalldelay

to grow with increasing absolute velocity deficit. This is counterbalanced against

reduced overall force with pitch-to-fine, resulting in fairly constant LFN above rated.

Including stall delay effects is shown to have an important effect, since the force

magnitudes are larger, as are the transients in negative stall (for 20–25 m/s).

7.2.2 Wake Profile Influence

The tower design will influence the noise significantly. A thin, circular tower pro-

duces a deep, sharp wake, while a multi-element (truss) tower would create a less-

severe but broader wake. Another, possibly costly, option would be a streamlined

tower, able to rotate with the rotor in yaw either as a complete unit or by means

of a fairing around a circular tower. The influence of both are examined below.

The rotor is unconed in both cases operating at 20 RPM. The wake parameters are

defined 3.25D downwind.

7.2.2.1 Wake Width

The SPL trend with wake width w (expressed as multiples of tower diameter D),

at constant wake deficit ∆ = 0.3 is shown in Fig. 7.12. Even though the total

deficit remains constant, the larger time-period means that lift variations are more

sedate. Hence, less acoustical noise is produced. Figure 7.13 shows that the acoustic

energy is essentially constant for low-order harmonics, but falls off at higher (audible)

frequencies with increasing wake width.


2.5 3 3.5 487.2

87.4

87.6

87.8

88

88.2

88.4

88.6

w

SP

L (

dB

)

Figure 7.12 SPL variation with wake width w

0 5 10 15 200

20

40

60

80

100

f (Hz)

SP

L (

dB

)

2.5

3

3.5

4

Figure 7.13 Acoustic spectrum variation with wake width w

It is worth noting that the wake is assumed uniform. The smoothed, broader wake

from a small-element truss tower would mimic this. More substantial members of a

tripod-type structure would not, but would create more sharp individual wakes.1

7.2.2.2 Wake Deficit

Holding the wake width w fixed at 2.5D, Fig. 7.14 shows that as expected, SPL

grows with wake deficit ∆. It appears that the growth is non-linear, decreasing in

1This was a problem on the US MOD-1 machine [15].

7.2 Tower Thump 233

additive effect as ∆ is increased. Examining the detailed spectrum in Fig. 7.15,

it appears that the higher harmonics grow more quickly that the lower ones. The

steady pressure remains virtually constant, as the average forces are virtually un-

altered with ∆. The quick growth with ∆, especially of the more audible higher-

frequency components makes minimization of this parameter (at the blade location)

quite important.

0.2 0.4 0.6 0.884

86

88

90

92

94

96

∆

SP

L (

dB

)

Figure 7.14 SPL variation with wake deficit ∆

0 5 10 15 2020

30

40

50

60

70

80

90

f (Hz)

SP

L (

dB

)

0.2

0.4

0.6

0.8

Figure 7.15 Acoustic spectrum variation with wake deficit ∆


7.2.3 Mitigation by Offset

Structurally, the rotor hub is most efficiently supported near that tower centre.

Acoustically, the rotor should be placed as far downwind as possible. One option is

an elongated hub, which has been the common solution: Carter [75], MS-4 [9, 192],

Proven [193], and WTC [70]. The generator in all cases is overhung on the upwind

end of the nacelle, and a shaft transmits torque from the rotor. All but the small

Proven machines use gearboxes with induction generators. Klinger et al. [81] and

Versteegh [82] present the Vensys and Zephyros direct-drive machines respectively,

both of which have conventional upwind rotors. The generators are located on the

same side as the rotor, and use efficient compact-hub construction.

In order to avoid a long shaft for a PMG coning rotor, the acoustic effects must

be quantified. The CONE-450 with space-frame hub located the hinge axes ≈3 m

downwind of the yaw axis, while the direct-drive concept proposed in that report

had almost zero offset to the yaw axis [66]. Obviously tower diameter will directly

affect any strategy, in turn influenced by static and fatigue loading.

7.2.3.1 Offset from Yaw Axis

Figure 7.16 shows the SPL as the hub offset o is varied by its ratio with tower-top

diameter. It is clear this is an effective parameter in mitigating tower interaction

noise. Even at 7D however, the SPL is above the 72 dB that is the nominal reduction

target.

1 2 3 4 5 6 778

80

82

84

86

88

90

Ratio of offset to tower top diameter

SP

L (

dB

)

Figure 7.16 SPL variation with offset from tower o/Dtowertop

7.2 Tower Thump 235

Examining the spectrum in Fig. 7.17 reveals that increasing the offset has a non-

linear effect on SPL. The 3P SPL is hardly effected, while the mid-range around

10 Hz is most effected. At higher harmonics, above offsets of 3D the SPL changes

minimally, but is strongly dependent below this offset.

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80

f (Hz)

SP

L (

dB

)

1

2

3

4

5

6

7

Figure 7.17 Acoustic spectrum variation with offset from tower o/Dtowertop

7.2.3.2 Cone Angle

In order to obtain a fair comparison of coned configurations, the aerodynamic power

of the rotor is kept constant. This is achieved by scaling the blade sections and chords

by a constant multiple. The true tip-speed-ratio λ is kept constant, so that tip noise

would remain relatively unaltered, even if the TE noise would grow somewhat with

blade length. The coning hinge is at the outboard edge of the hub (1.75 m) and the

nominal 3.3 m offset from the yaw axis is used.

With this set of scaled blades, the acoustic variation in SPL, shown in Fig. 7.18,

is found to be even more effective than linear offset in reducing SPL, at least for the

0 observer azimuth (directly downwind). In fact, 10 of cone is equivalent to 3D

(8.5 m) of offset in this case. The outer portion of the blades, responsible for the

bulk of noise, moves fairly quickly away from the tower with β. The spectrum of the

noise in Fig. 7.19 shows that noise is reduced quite strongly across the entire range.

Both lower (infra sound) and higher audible sounds are mitigated more effectively

than with simple offset.


0 10 20 30 4060

65

70

75

80

85

90

Cone angle (deg)

SP

L (

dB

)0

90

180

270

Figure 7.18 SPL variation with cone angle β (deg) and observer azimuth angle(deg)

0 5 10 15 200

10

20

30

40

50

60

70

80

f (Hz)

SP

L (

dB

)

0

10

20

30

40

Figure 7.19 Acoustic spectrum variation with cone angle β (deg)

7.2 Tower Thump 237

With coning, the location and orientation of the acoustic sources changes more

than with simple offset. The acoustic footprint for the rest of the variations consid-

ered in this section is relatively invariant in profile. Figure 7.20 makes it obvious

however that the presence of coning alters not only the magnitude of the SPL, but

also the spatial lobes. With increasing cone angle, the upwind lobe is much larger

than the downwind one. This means that although the nominal downwind observer

considered here benefits, more investigation is warranted on the directionality of the

LFN. The trends for other observer locations in Fig. 7.18 show that the effects are

very non-linear, and should be considered in an acoustic evaluation of the coning

rotor.

Observer azimuth (0° downwind)

0°

15°

30°

45°

60°

75°90°105°

120°

135°

150°

165°

±180°

-165°

-150°

-135°

-120°

-105° -90° -75°

-60°

-45°

-30°

-15°

5060708090

Figure 7.20 Footprint SPL changes with variation in cone angle (β: 0 20

40)

7.2.4 Control Strategy Effects

Above rated power, the control strategy is found to have a strong effect on the SPL,

as shown in Table 7.1. Both stalling strategies (PTS and VSS) exhibit reduced noise

relative to the PTF machine (as mentioned in §3.8.3, PTF is un-viable in any case

for the coning rotor). Although the steady forces are higher (see §6.5.6), Fig. 7.21

shows that as each blade passes through the tower wake, the change in force is

actually much larger for the PTF concept. Acoustic noise is generated by changes

in pressures, so the larger variations increase the overall SPL. It is also evident

that stall delay effects are important in modifying the response, altering both the

magnitude and azimuthal profile.


Table 7.1 SPL variation with control strategy above rated

Without stall delay With stall delayWindspeed (m/s) PTF PTS VSS PTF PTS VSS

15 90.8 84.3 83.2 90.5 86.1 84.820 92.0 80.7 79.6 93.2 84.0 82.925 91.0 82.7 82.3 94.6 84.7 84.4

0.45 0.5 0.55-80

-60

-40

-20

0

20

Rotor revolutions

T -

Tm

ean (

kN

)

PTF

PTF s

PTS

PTS s

VSS

VSS s

Figure 7.21 Rotor thrust variation over azimuth for control strategy choice(20 m/s; ‘s’ indicates stall delay model included)

7.2.5 Dynamic Motion

Allowing the blades to freely flap has two potentially opposite effects. The first is

that cyclic inputs (e.g. wind shear, gravity), by creating cyclic β variation, increase

the noise by the motion of the acoustic forces. On the other hand, dynamic motion

of the blades, if fast enough, will reduce the transient aerodynamic forces and hence

noise.

Using the same lengthened blades and rotation speeds as in §7.2.3.2, and retaining

the nominal section weights in §C.3, steady simulations were then run at each cone

angle and the root bending moment noted. This moment was then applied in the

dynamic simulations to attain roughly the same cone angle.1 The results given next

include dynamic stall delay, wind shear and gravity.

Given the observer azimuth dependency found earlier in §7.2.3.2, Table 7.2 gives

the SPL results for the same range of angles. There is little variation found by

1There is a slight β offset owing to the cyclic inputs.

7.2 Tower Thump 239

including flap motion, especially for larger cone angles. Examination of the flap

angle histories revealed no discernible structural response to the tower wake. The

aerodynamic forces were therefore relatively un-altered, although extremely flexible

blades (i.e. include modal response) may change this result.

Table 7.2 SPL variation with dynamic motion and cone angle

Cone angle β (deg) Observer azimuth (deg)0 90 180 270

Without flap motion0 88.4 74.8 88.2 74.7

10 83.2 64.8 85.0 75.640 68.5 71.6 80.0 76.5

With flap motion0 87.9 75.8 87.6 75.1

10 83.2 66.2 85.0 75.940 68.7 71.5 80.2 76.5

Apparently, the cyclic flap motion of the blades has fairly minimal effect on the

acoustics. The SPL is slightly reduced upwind/downwind of the rotor, and slightly

increased SPL to the sides. These trends were observed across the frequency spec-

trum.

7.2.6 Remaining Issues

The primary requirement is for more experimental data to validate the acoustic

model. Although it is physics-based with minimal assumptions, the input tower wake

profile is of critical importance. Reynolds number has a large effect on the profile,

as does tower diameter and configuration (e.g. truss, multi-element or streamlined

towers). Additionally, no account is made of the modification of the wake by the

passing blades. In combination with experiment, it has been postulated that this

effect widens the wake [87]. Terrain modification of the LFN should also be investi-

gated. In particular, mountainous terrain may channel the noise. Propagation over

water should also be looked at for offshore locations.

Conventional noise mechanisms should also be quantified. Modern PTF machines

operate away from stall. A comparison gross stall noise should be carried out, to

compare a stalling strategy to PTF in terms of noise production. The original study

[139] did not include this comparison, as both the REF-450 and CONE-450 operated

in stall.


Finally, any final design must take mitigation of tower noise into account and strive

to mitigate the effects. The specific variable rotor speed schedule and turbulent

winds should be checked for their effect on the noise profile of the machine.

Chapter 8

Conclusions

Proposing and advancing an alternate wind turbine concept is a multi-step process,

with extremely broad scope, for which this thesis can only begin to tackle fundamen-

tal considerations. Part I began in Chapter 2 with an exposition of the contextual

placement and requirements of a wind turbine. Chapter 3 then approached the

problem from a functional design perspective. The basic trade-offs in design choices

were used to justify the coning rotor concept and describe the unique features of

the concept in topological and operational terms. Its relationship to conventional

and past unconventional designs was examined, with a view to exposing the critical

elements requiring analysis. Throughout the remainder of the thesis, reference was

made to a standard code, BLADEDTM, to identify the areas in conventional models

requiring alteration to analyse the coning rotor concept.

8.1 Analytical Contributions

Part II focused first in Chapter 4 on developing the analytical tools necessary to

properly design a coning rotor. Section 4.4 presented a novel BEM method devel-

oped by returning to the underlying assumptions and derivation of the theory. By

considering the wake structure with vortex filaments, a new BEM method was for-

mulated that more properly handles coned and yawed rotors in steady and dynamic

flow. Importantly, the low-cost computational structure of the BEM method was

retained. It is readily implementable in a fully dynamic, industry-level code in-

corporating aero-elastic coupling, at negligible computational cost. In §5.1, it was

demonstrated that the root of errors in the BEM method lies primarily not in the

planar disc assumption, but rather in a deficient accounting for the relative posi-

tion of the wake. The validity of the results has been demonstrated for uniformly

loaded and real rotors, with coning and yaw, by comparison with more accurate

CFD models, wind tunnel and field test results. Very high induction factor and

stalled conditions, however, remain difficult to predict with great certainty.

241

242 Chapter 8 Conclusions

Building on the aerodynamic analysis, the equations governing low-frequency

acoustics were presented in §4.5. Although the equations have been developed by

other authors, a number of important nuances in their implementation are pre-

sented. No quantitative comparative results were available, however parametric

variation produced reasonable predictions from what is an accurate physics-based

model.

Structural modelling tools were developed in §4.6, beginning with a highly flexible

cross-sectional property analysis. The equations governing distributed loads in the

blades were then developed. Next, the dynamic equations of motion for the rigid-

body coning rotor were shown to lead to analytic expressions for flap frequency and

damping. Finally, an FEM approach was used to derive a centrifugally stiffened beam

element with varying mass and stiffness properties, assembled to predict the modes

of a complete rotor with varying boundary conditions. Analytic and numerical

results from BLADEDTM were used for comparison.

Combining these elements, the computation of an optimal steady-state control

schedule was described, taking into account the equilibrium cone angle and applied

hinge moment bias. Based on these integrated analyses, the primary assumptions

requiring modification in BLADEDTM were identified. The BEM model was origi-

nally thought to be the primary difficulty, however it became evident that there are

additional linearity assumptions in the structural model of BLADEDTM that are

violated by the coning rotor.

Simplified PMG models were introduced, based on the driving magnetic materials

required. Together with cost estimates for those components, an estimation of the

cost-torque sensitivity of the generator was enabled.

8.2 Design Refinements

Part III used the analytic tools of Part II to affect a study of the DVs and parameters

determining the topology and operation of the coning rotor. In Chapter 6, results

from the original study on the coning rotor were up-scaled to a modern equivalent

machine in §6.1. A favourable comparison was made on a loading and energy capture

basis. The COE/CF and non-dimensional aerodynamic performance of a coning rotor

was then examined in §6.2 and 6.3. The challenge of optimizing a coning rotor, with

its inherent aerodynamic-structural integration, was then contrasted to that of a

conventional rotor in §6.4. Based on this complexity, and as a precursor step to

a full engineering design including dynamic simulations, a parametric study was

8.3 The Next Steps 243

made of blade shapes, airfoil families, and control strategies in §6.5. The results

illustrate the bounds placed on enhanced energy capture of the coning rotor by

steady loading and generator requirements. The importance of airfoil selection was

also underlined. The coning rotor range of energy capture advantage relative to a

conventional reference was found to be 10-30% at equivalent tower height.

Chapter 7 surveyed the areas of critical importance and uniqueness to the con-

ing rotor. The first was the dynamic response of a flap-hinged rotor (§7.1), in the

presence of aerodynamic, inertial and gravitational loading. The non-linear effects

(aerodynamic and structural) were found to be important for the coning rotor in

terms of modelling and cyclic loading, operating at non-zero cone angle, away from

the standard linearised conditions. The aerodynamic damping inherent to the sys-

tem was presented next, starting with the rigid-body flap mode which may enter

regions of negative damping. In the absence of a flexible-body simulation code,

sectional equations were used to examine the impacts of airfoil choice and control

method on the edgewise damping mode. For a stall-controlled rotor, this mode

tends to be unstable due to a lack of aerodynamic damping, and may require com-

pensation. Section 7.2 parametrically then explored the low-frequency acoustics of

the coning rotor. The effect of tower wake profile, operation and relative position

were quantified, to determine the boundaries placed on the coning rotor to avoid

this noise source, critical to downwind machines.

8.3 The Next Steps

As is clear from the length of this thesis, numerous technical areas must be properly

integrated to advance the coning rotor concept. The work presented here is only a

first step to improve the underlying theory and provide a starting point for further

design. The next immediate step is to modify BLADEDTM to incorporate the BEM

method and any structural modelling changes required. Additional experimental

test data for coned rotor configurations should also be obtained, to further refine

the aerodynamic modelling.

To further advance the actual design of the concept will require a full set of

dynamic simulations (to an established standard, e.g. IEC-14000) using that code.

This will of course require a controller, based initially on a rotor design from this

thesis. The controller and simulations will also have to revisit parked conditions

and mode-switching, over the entire operational profile. Working with the dynamic

simulations, the design can be further refined, to definitively size and cost a concept

244 Chapter 8 Conclusions

machine. Jamieson [66] also concluded that the dynamic response of the system was

the critical element of the design requiring further investigation.

In particular, if a PMG is to be employed, a more detailed design of the generator

will be required. The detailed design of the coning actuators and hub will also require

careful attention, heeding the lessons of difficulties encountered by previous flapping-

blade concepts. Assuming the final paper concept design demonstrates merit over

conventional machines, the subsequent step should be a prototype machine of modest

scale, to test the predictive accuracy of the analysis suite and design methods. Based

on any required refinements, the concept may then be scaled up and implemented

on a commercial scale.

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Appendices

Appendix A

Geometric Transformations

A common notation is used for the 4x4 geometric transformation matrices used

throughout. The geometric translation matrix is defined as:

T (∆x,∆y,∆z) =

1 0 0 ∆x0 1 0 ∆y0 0 1 ∆z0 0 0 1

(A.1)

Matrices defining a rotation θ about the x, y and z axes are defined respectively as:

R x(θ) =

1 0 0 00 C −S 00 S C 00 0 0 1

(A.2a)

R y(θ) =

C −S 0 0S C 0 00 0 1 00 0 0 1

(A.2b)

R z(θ) =

C 0 S 00 1 0 0−S 0 C 00 0 0 1

(A.2c)

where S = cos θ and C = cos θ.

To transform a coordinate pointx, y, z, 1

T through a rotation α about the x

axis and then β about the z axis, the matrices must be premultiplied in reverse

transformation order as:

x′ = R z(β) R x(α)

xyz1

(A.3)

A vector ~x may be transformed by setting the fourth element of the vector to

zero. Note that translations then have zero effect.

A-1

Appendix B

Cross Section Analysis

This appendix describes the computation of cross-sectional properties for a general-

ized composite beam section. It has been adapted from the author’s previous work

[152].

The basic idea is to map the coordinates of the triangle from the real space(x, y)

to a convenient unit vector space(u, v), as shown in B.1.

Figure B.1 Section integral coordinate transform

The mapping function ϕ may be defined as:

~x = ϕ~u =[Ax By

Ay By

]uv

(B.1)

for an arbitrary vector ~x in(x, y)

to ~u in(u, v), both lying in the triangular inte-

gration region.

The local and absolute centroid may be computed from:

~xc,local =~A+ ~B

3(B.2a)

~xc = ~O + ~xc,local (B.2b)

B-1

The area Atri of the element, and an elemental area dA may be computed from:

Atri = 1/2∣∣∣ ~A× ~B

∣∣∣ (B.3a)

dA = |d~x× d~y| = |a0a3 − a1a2| dudv = βdudv (B.3b)

The first (S) and second (I) moments of area for triangular domain, computed in

the real domain, may then be derived from the above transformation and quantities

as:

Sx =∫ydA = β

(Ay

6+By

6

)+Atri (~xc,y − ~xc,local,y) (B.4a)

Sy =∫xdA = β

(Ax

6+Bx

6

)+Atri (~xc,x − ~xc,local,x) (B.4b)

Ixx =∫ ∫

y2dA = β

(AyBy

12+A2

y

12+B2

y

12

)+Atri

(~x2

c,y − ~x2c,local,y

)(B.4c)

Iyy =∫ ∫

x2dA = β

(AxBx

12+A2

x

12+B2

x

12

)+Atri

(~x2

c,x − ~x2c,local,x

)(B.4d)

Ixy =∫ ∫

xydA = β

(AxBy +BxAy

24+AxAy

12+BxBy

12

)+ (B.4e)

Atri (~xc,x~xc,y − ~xc,local,x~xc,local,y) (B.4f)

Jz =∫ ∫

(x2 + y2)dA = Ixx + Iyy (B.4g)

The formulas are applied to each triangular element, and their contributions aggre-

gated as modulus (EA, ES, EI, GJ) and density (ρA, ρS, ρJ) weighted sectional

quantities. The over-tilde indicate final aggregate quantities. The final sectional

mass properties are then found from:

mps = ρA (B.5a)

~xCG = [ρSy , ρSx]/mps (B.5b)

Izz = ρJ −mps |~xCG|2 (B.5c)

where mps is the mass/unit-span of the section and the stiffness properties from:

~xEA = [ESy , ESx]/EA (B.6a)

EIx = EIx − EAx2EA,y (B.6b)

EIy = EIy − EAx2EA,x (B.6c)

EIxy = EIxy − EAxEA,xxEA,y (B.6d)

GJ = GJ − GA |~xEA|2 (B.6e)

B-2

The angle α and magnitude of the principle axes may be determined from:

α = 1/2 arctan2EIxy

EIy − EIx(B.7a)

EIprin,1 = EIx cos2 α+ EIy sin2 α− 2EIxy sin(α) cos(α) (B.7b)

EIprin,2 = EIx sin2 α+ EIy cos2 α− 2EIxy sin(α) cos(α) (B.7c)

The coordinate definition and nature of blade profiles means that the first principle

axis is invariably chordwise, and the second edgewise.

Given the total twist of the section γ, including pitch set angle, active pitch angle,

and section twist, the OP and IP bending stiffnesses may be computed as:

EIOP =EIx + EIy

2+EIx + EIy

2 cos 2γ− EIxy sin 2γ (B.8a)

EIIP =EIx + EIy

2− EIx + EIy

2 cos 2γ+ EIxy sin 2γ (B.8b)

B-3

Appendix C

Reference Machine Specifications

This appendix contains the details for the reference machines used in this thesis.

Where required, airfoil data has been extended in AOA using NREL’s AirfoilPrep

worksheet [194], which implements Viterna’s method [109] for post-stall, with a max

cd of 1.29 and an A of 10.

C.1 Tjaereborg Machine

The Tjaereborg machine was an operational turbine in Esbjerg, Denmark. It was

built by ELSAM in 1987, and decommissioned in 2001. The relevant data for the

rotor, as supplied by Mikkelsen [89] and used in his thesis, is given in Table C.1 and

Fig. C.1.

Table C.1 Tjaereborg parameters

Parameter Value

Rated power 2 MWRotor diameter 61 mRotor position UpwindHub length (Rhinge) 1.46 mNominal cone angle 0

Pitch set angle (γpitch,set) 0.5

Fixed rotation speed (Ω) 22.1 RPMNacelle tilt 3

Nacelle offset (o) 6.81 mHub height 61 mTower station height(diameter) 0.0 (7.25) m

28.0 (4.75) m56.0 (4.25) m

Assumed tower wake properties ∆ = 0.3ww = 2.50lw = 3.25

C-1

0

2

4

6

8

10

12

0 5 10 15 20 25 30Distance from root (m)

Chord

(m), t

wist (d

eg)

0

5

10

15

20

25

30

35

Thick

ness

(%)

ChordTwistThk

(a) Blade geometry parameterspitch

02468

1012141618202224

0 4 8 12 16 20 24V (m/s)

Pitch

(deg

)

(b) Operating profile

Figure C.1 Tjaereborg blade details

C-2

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-20 0 20 40 60AOA (deg)

c l 1215182124

(c) Airfoil lift coefficient (NACA 4412)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-20 0 20 40 60AOA (deg)

c d

1215182124

(d) Airfoil drag coefficient (NACA 4412)

Figure C.1 Tjaereborg blade details (cont.)

C-3

C.2 NREL UAE Phase IV

The UAE was a long-running program conducted by the National Renewable Energy

Laboratory (NREL) to provide experimental data to validate analysis tools. The

first phases of the program consisted of field trials, but due to atmospheric turbulence

creating inflow anomalies, a controlled wind tunnel experiment was conducted in

phase VI. The wind tunnel at the National Aeronautics and Space Administration

(NASA) Ames Research Centre was used to enable testing of a large scale machine,

thereby avoiding aerodynamic scaling difficulties. The NASA Ames tunnel test

section is 24.4 m x 36.6 m, allowing a rotor of nominal 10 m diameter to be tested

without wake blockage, wind shear or large-scale inflow turbulence.

The set-up for the test itself is well documented [113, 177, 178, 195, 196], so

only a brief description is included here and in Table C.2. The UAE machine is a

heavily modified Grumman Wind Stream-33 built to allow flexible configuration. It

employs a 2-bladed rotor that can be rigid, teetering or flapping and operate upwind

or downwind with full-blade independent pitch control. The nominal blade has a

tip radius of 5.029m, and can be fitted with extended or smoke tips. The drive-train

has a gearbox and constant-speed induction generator, but may be controlled via

power electronics. Telemetry provision is quite extensive, including instrumentation

for loads (blade, hub, drive train and tower strain gauges/load cells/tunnel balance)

and configuration (pitch, azimuth, yaw angles, etc.). The blades themselves are

instrumented with pressure taps and pitot-tube probes at fixed locations, and the

tunnel itself has various inflow measurements (wind speed, temperature, pressure

taps, etc.).

The data sets available are pre-corrected for the influence of the overhung rotor

weight on the shaft strain gauge measurements used to compute power. An im-

portant note when using the raw data is to correct the power reading for dynamic

imbalance of the rotor. Imbalance in the telemetry boom led to rotating imbalance

and in turn a cyclic error in the power [177, p. 75].

The blade data shown in Fig. C.2 for the 5.029 m radius rotor (i.e. without

extended tip). Hand et al. [177] provides tabular data for the extended-tip blade

that must be interpolated to derive these profiles. The pitch setting γpitch,tip of the

experiments refers to the out-of-plane angle at the (short) blade tip. The blade pitch

setting γpitch,set is therefore:

γpitch,set = γpitch,tip − γtwist,tip (C.1)

C-4

The data provided by NREL has a fairly large range in cl and cd1, at different

Re and from testing facilities. The airfoil dataset used here is the S809 OSU data

[177] with AOA data to 26, for best Re fit among the various sets available. The

cd curve was smoothed around ±10 and extended beyond the provided data range

using AirfoilPrep.

Table C.2 NREL UAE parameters

Parameter Value

Rated power 12 kWRotor diameter 11 mHub length (Rhinge) 0.508 mFixed rotation speed (Ω) 72 RPM

1Wake deficit and surface pressure integrations are both used for cd

C-5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Distance from root (m)

Chord

(m)

-20

0

20

40

60

80

100

120

Twist

(deg

), thic

knes

s (%)

ChordTwistThk

(a) Blade geometry parameters

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-20 0 20 40 60AOA (deg)

c l, c

d

clcd

(b) S809 airfoil data

Figure C.2 NREL UAE blade details

C-6

C.3 1.5 MW Reference Machine (REF-1500)

The REF-1500 was supplied by GH as a representative model of a current state-of-

the-art MW class machine. It has a custom controller for variable speed operation

with pitch control (PTF). The relevant parameters for the machine are given in

Table C.3, and the blade details in Fig. C.3. The tower is linearly tapered and rotor

upwind of the tower.

The airfoil data has been modified slightly around α = 16 for the high Re number

data to avoid extremely sharp stall in the GH data. A 100% thick airfoil (circle)

has also been defined, with zero lift and constant cd = 0.3 for a cylinder at Re >1e6

White [197, p. 298].1

Table C.3 REF-1500 parameters

Parameter Value

Rated electrical power 1.5 MWNo-load constant power loss 15 kWGenerator efficiency 95 %Rotor diameter 70 mHub length (Rhinge) 1.75 mPitch set angle (γpitch,set) 1

Rotation speed (Ω) 14.4–21 RPMHub height 84 mHub offset 3.3 mNominal shaft tilt 5

Tower height 82 mTower top/bottom diameter 2.823/5.663 m

1The original data used cd = 1.0.

C-7

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 5 10 15 20 25 30 35Distance from root (m)

Chord

(m)

0

20

40

60

80

100

120

Twist

(deg

), thic

knes

s (%)

ChordTwistThk


1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10


EI flap

, EI ed

ge (N

m2 )

10

100

1000

10000

Mass

/unit l

ength

(kg/m

), pitc

h ax

is (%

c)

EIflapEIedgempspitch axis

(b) Blade stiffness and mass parameters

Figure C.3 REF-1500 details

C-8

0

5

10

15

20

25

30

0 5 10 15 20 25V (m/s)

RPM,

pitch

angle

(deg

) RPMPitch angle

(c) Operating profile

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-20 0 20 40 60AOA (deg)

c l, c

d

13 cl17 cl21 cl13 cd17 cd21 cd

(d) Airfoil coefficients for Re = 2e6 (LS1)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-20 0 20 40 60AOA (deg)

c l, c

d

13 cl17 cl21 cl13 cd17 cd21 cd

(e) Airfoil coefficients for Re = 4e6 (LS1)

Figure C.3 REF-1500 details (cont.)

C-9

C.4 GH Demo Machine

The GH Demo Machine is supplied with BLADEDTM as an example set of data.

The data relevant to this thesis are given in Table C.4 and Fig. C.4.

Table C.4 Demo machine parameters

Parameter Value

Rated power 2.2 MWRotor diameter 80 mHub length (Rhinge) 1.25 m

C-10

0

2

4

6

8

10

12

14


Chord

(m), t

wist (d

eg)

0

20

40

60

80

100

120

Thick

ness

(%)

ChordTwistThk


1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10


EI flap

, EI ed

ge (N

m2 )

10

100

1000

10000

Mass

/unit l

ength

(kg/m

), pitc

h ax

is (%

c)

EIflapEIedgempspitch axis

(b) Blade stiffness and mass parameters

Figure C.4 Demo machine details

C-11

C.5 CONE-450 Concept

The details for the CONE-450 final design are given in Table C.5 and Fig. C.5 [66].

The airfoil coefficients in Fig. C.5(d) were not included in the original report, but

are taken from another GH report using same airfoil [58]. This data is used for the

“FX” blade sets in Part III.

Table C.5 CONE-450 parameters

Parameter Value

Rated power 450 kWRotor diameter 23.166 m (unconed)Hub length (Rhinge) 2.0 mBlade mass 641 kgFirst mass moment (about hinge) 4509 kgmSecond mass moment (about hinge) 45108 kgm2

C.6 Coning Rotor Study

The NACA 63(2)-4XX airfoil used in Part III is taken from the Risø airfoil library

[198] experimental data, at an Re of 1.5e6. The 17%–21% data is from Abbott, and

the 30% data from Velux. Some data has been smoothed to avoid sharp spikes, as

shown in Fig. C.6.

C-12

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12 14 16 18 20 22Distance from root (m)

Chord

(m)

0

5

10

15

20

25

Twist

(deg

), thic

knes

s (%)

ChordTwistThk


1.E+00

1.E+01

1.E+02

0 2 4 6 8 10 12 14 16 18 20 22Distance from root (m)

Mass

/unit l

ength

(kg/m

)

(b) Blade mass parameter

Figure C.5 CONE-450 details

C-13

RPM

0

5

10

15

20

25

30

35

0 5 10 15 20 25V (m/s)

RPM

(c) Operating profile

-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-20 0 20 40 60AOA (deg)

c l, c

d

cl

cd

(d) Airfoil coefficients (FX66-17AII-182)

Figure C.5 CONE-450 details (cont.)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-20 0 20 40 60AOA (deg)

c l, c

d

15 cl18 cl21 cl30 cl15 cd18 cd21 cd30 cd

Figure C.6 Airfoil coefficients (NACA 63-4XX)

C-14

Appendix D

Historical Machines

Table D.1 has been compiled from a number of sources [9, 15, 18, 66, 74, 193].

D-1

Table

D.1

Soft

mac

hine

hist

ory

Mac

hine

Date

Rating(kW)

Diameter(m)

#Blades

Orientationa

Yawb

Hubc

PowerControld

Coning

Towere

Smit

h-P

utna

mat

Gra

ndpa

’sK

nob1

1941

1250

53.3

2D

AIF

FSP

-3 /

6Tr

Hut

ter-

Allg

aier

219

60s

100

342

D?

TFSP

7T

GM

arti

nR

yle

(Cam

brid

ge)3

1980

s≈

5≈

52

DP

IFS

0–1

5T

Boe

ing

Mod

-24

1980

s25

0091

.42

UA

TP

SP-

TK

KRV

Swed

en/H

amilt

onSt

anda

rd(U

S)W

TS-

4519

80s

4000

782

DA

Td

FSP

?T

Car

ter6

1980

s25

020

2D

PFB

S>

0T

Boe

ing

Mod

-5B

719

8732

0097

.52

UA

TP

SP-

Ta

Ori

enta

tion

:D

=do

wnw

ind,

U=

upw

ind

bY

aw:

A=

acti

ve,P

=Pas

sive

cH

ub:

IF=

indi

vidu

alfla

ppin

g,T

=te

eter

ing,

Td

=te

eter

ing

wit

hd3

,T

p=

teet

erpi

tch

coup

ling,

FB

=fle

x-be

am,R

=ri

gid

dPow

erC

ontr

ol:

FSP

=fu

ll-sp

anpi

tch,

PSP

=pa

rtia

l-sp

anpi

tch,

Y=

yaw

cont

rol,

S=

stal

l,C

=co

nee

Tow

er:

Tr

=tr

uss,

T=

tubu

lar,

TG

=gu

yed

tubu

lar

1Fla

ppin

ghi

nges

adop

ted

toco

mba

tgy

rosc

opic

load

ing

whe

nya

win

g2

Low

solid

ity,

high

aspe

ctbl

ades

led

toflu

tter

prob

lem

s3

Onl

yri

gid

stop

son

flapp

ing

blad

es;Tap

ered

untw

iste

dbl

ades

4Fir

stso

ftde

sign

usin

gex

peri

ence

and

sim

ulat

ion;

Hig

her

than

expe

cted

cycl

iclo

adin

gfr

omin

adeq

uate

win

dm

odel

s5

Soft

tow

er;Sp

ring

mou

nted

gear

box

6Fle

xibl

esp

arin

stea

dof

flap

hing

es;N

extge

nera

tion

mac

hine

:“C

onst

ant-

coni

ng”

inst

ead

ofpi

tch-

flap

coup

ling

topr

ovid

esm

ooth

erpo

wer

,P

TS

&va

riab

lesp

eed

7Fir

stva

riab

lesp

eed

mac

hine

usin

gcy

cloc

onve

rter

D-2

Table

D.1

Soft

mac

hine

hist

ory

(con

t.)

Mac

hine

Date

Rating(kW)

Diamter(m)

#Blades

Orientationa

Yawb

Hubc

PowerControld

Coning

Towere

Ris

øTes

tM

achi

ne1

1991

3012

2D

PT

S5

TG

amm

a60

219

9220

00?

2U

AT

Y-

TC

ON

E-4

503

1994

450

443

DP

RC

,S

0/8

0T

GP

rove

n419

900.

6–15

2.55

–93

DP

IFS,

C5

–45

TM

S4-6

005

1997

600

41/4

53/

2D

PFB

FSP

,S,C

8st

atic

T1

Hig

hly

inst

rum

ente

dte

stm

achi

new

ith

vari

able

spri

ngs

and

dam

pers

used

onth

ete

eter

DO

F;E

xper

imen

tsco

ncer

ned

wit

hfr

eeya

wpe

rfor

man

cean

dre

quir

edfr

ee-t

eete

ran

gle

requ

irem

ents

2Tee

teri

nghu

bpe

rmit

shi

ghya

wra

tes

faci

litat

ing

relia

nce

onya

wco

ntro

lfor

pow

erlim

itin

g;B

road

-ran

gesp

eed

cont

rol;

Soft

tow

er;

Low

mod

ulus

blad

em

ater

ial

3H

ydra

ulic

cent

ral

actu

ator

toco

ntro

lco

llect

ive

coni

ngan

gle;

Bla

des

indi

vidu

ally

hing

ed,

but

esse

ntia

llyri

gid

due

toco

llect

ive

coni

ngm

echa

nism

4R

ange

ofm

achi

nesi

zes;

Con

stan

t-sp

eed

dire

ct-d

rive

gene

rato

rs;

Zee

bede

flexi

ble

incl

ined

hing

e(δ

3)

allo

ws

coni

ngof

blad

es,

com

bini

ngce

ntri

fuga

lfo

rce

tow

ards

feat

her

and

pitc

hw

ith

cone

angl

eto

war

dsst

all

5Fle

x-be

ams

allo

w50

%lo

adre

duct

ion

inhi

ghw

inds

;fu

ll-sp

anpi

tch

tow

ards

stal

l;fix

ed-s

peed

;co

unte

r-ba

lanc

edna

celle

,la

tera

llyoff

set

from

tow

erto

bala

nce

thru

stan

dro

tor

torq

ueco

mpo

nent

alon

gto

wer

from

tilt

edna

celle

(dow

nwin

den

dup

);se

lf-er

ecti

ng;

2-bl

aded

desi

gnfo

roff

shor

e;Ø

45m

desi

gnfo

rC

lass

IIsi

tes

D-3

Table

D.1

Soft

mac

hine

hist

ory

(con

t.)

Mac

hine

Date

Rating(kW)

Diamter(m)

#Blades

Orientationa

Yawb

Hubc

PowerControld

Coning

Towere

WT

C1

1998

250

402

DA

IFFSP

-5 /

15

TG

Soft

-Rot

orC

once

pt(R

isø)

219

9915

122

DP

T/I

FFSP

0/9

0T

GW

indfl

ow50

0320

0350

033

2U

AT

pFSP

-T

Ver

gnet

420

0327

530

2D

PT

FSP

>0

TG

1

1H

ydra

ulic

dam

pers

for

sem

i-pa

ssiv

eya

w;H

ydra

ulic

ally

dam

ped,

inde

pend

ent

blad

efla

ppin

g2

Tee

teri

nghu

bw

ith

stiff

ness

;Bla

des

not

rigi

dly

atta

ched

tote

eter

edhu

b,bu

tin

divi

dual

lysu

ppor

ted

byhi

nge

axis

incl

ined

togi

ve8

flap/

edge

coup

ling.

Aco

mpo

site

flex-

beam

acro

ssth

ehi

nge

axis

then

prov

ides

indi

vidu

albl

ade

flapp

ing

stiff

ness

;N

osh

ear

web

sw

ere

used

inth

ebl

ades

apar

tfr

omth

efle

x-be

am;B

lade

pitc

hing

isun

clea

r,bu

tap

pear

sto

envi

sage

das

15

wit

hac

tive

stal

lco

ntro

lin

futu

reve

rsio

ns;N

acel

leti

ltfle

xibi

lity

ispr

ovid

ed3

Tor

que

limit

ing

gear

box;

Nov

elpi

tch

teet

erco

uplin

g4

Tilt

dow

nfo

rex

trem

est

orm

s(e

.g.

hurr

ican

es)

D-4

Appendix E

Material Specifications

These material values are taken from Griffin [55], and are a result of test data and

laminate calculations without safety factors. Unless otherwise stated, the materials

are glass fibres in an epoxy matrix.

E-1

Table

E.1

Sect

ion

layu

pm

ater

ialsp

ecifi

cati

ons

Nam

eD

ensi

tyE

last

icTor

sion

alPoi

sson

’sU

ltim

ate

Stra

inM

odul

usM

odul

usR

atio

Ten

sile

Com

pres

sive

ρE

Gν

ε ut

ε uc

(kg/

m3)

(GPa)

(GPa)

--

-

CD

B34

0a17

0024

.24.

970.

390.

022

0.01

05A

260b

1700

31.0

3.52

0.31

0.02

20.

0105

Spar

cap

mix

c17

0027

.14.

700.

370.

022

0.01

05R

ando

mm

at16

709.

653.

860.

300.

022

0.01

05G

elco

at12

303.

441.

380.

300.

022

0.01

05B

alsa

144

2.07

0.14

00.

220.

022

0.01

05a

Tri

axia

lfa

bric

,25

%:2

5%:5

0%di

stri

buti

onof

+45

:-4

5:0 fi

bres

bU

niax

ialfa

bric

cA

lter

nati

ngla

yers

ofC

DB

340

and

A26

0gi

ving

70%

unia

xial

fibre

sby

mas

s

E-2

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