objectives and constraints for wind turbine optimizationobjectives and constraints for windturbine...
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Objectives and Constraints for WindTurbine Optimization
S.Andrew Ning
National Renewable Energy Laboratory
January, 2013
NREL is a national laboratory of the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.
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Overview
1. Introduction 2. Methodology (aerodynamics,
structures, cost, reference model, optimization)
3. Maximum Annual Energy Production (AEP)
4. Minimum m/AEP 5. Minimum Cost of Energy (COE) 6. Conclusions
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Introduction
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WindTurbine Design
Optimization and Uncertainty Quantification
Aerodynamic Performance
Structural Design
Control Strategy
Site Selection
Farm Layout
Capital Costs
Maintenance Costs
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WindTurbine Optimization
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Objectives
• Max P/min Mb
• Max AEP • Min COE
Design Vars
• Blade shape • Rotor/nacelle • Turbine
Fidelity
• Analytic • Aeroelastic • 3D CFD
Optimization
• Gradient • Direct search • Multi-level
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WindTurbine Optimization
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Objectives
• Max P/min Mb
• Max AEP • Min COE
Design Vars
• Blade shape • Rotor/nacelle • Turbine
Fidelity
• Analytic • Aeroelastic • 3D CFD
Optimization
• Gradient • Direct search • Multi-level
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Methodology
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Model Development
1. Capture fundamental trade-offs (physics-based) 2. Execute rapidly (simple physics) 3. Robust convergence (reliable gradients)
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Aerodynamics
• Blade-element momentum theory
‣ hub and tip losses, high-induction factor correction, inclusion of drag
‣ 2-dimensional cubic splines for lift and drag coefficient (angle of attack, Reynolds number)
• Drivetrain losses incorporated in power curve • Region 2.5 when max rotation speed reached • Rayleigh distribution with 10 m/s mean wind
speed (Class I turbine) • Availability and array loss factors
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�
Blade Element Momentum
⌦r(1 + a 0)
plane of rotation
U1(1 - a)W
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a0)
a)1-U1(1-
⌦r(1 + a
�
Blade Element Momentum
plane of rotation
⌦r(1 +
U1(1 -W
a)a)
1 + a 0)0)
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�
Blade Element Momentum
⌦r(1 + a 0)
plane of rotation
U1(1 - a)W
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Blade Element Momentum
sin ¢ cos ¢f(¢) = - = 0
1 - a(¢) Ar(1 + a0(¢))
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Blade Element Momentum
S.Andrew Ning, “A Simple Solution Method for the Blade Element Momentum Equations with Guaranteed Convergence,” Wind Energy, to appear. CCBlade
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Figure 1: Example of composites layup at a typical blade section. Principal material direction of laminas, shown in brown, is skewed with respect to the blade axis. This causes bend-twist coupling as evident at the blade tip.
igure 2: Example of spar-cap-type layup of composites at a blade section F
rn
t
LD�
S ply angle
principal materialdirection
Composite Sectional Analysis
Sector 3 Sector 1
AA BB
Sector 2
Laminas schedule at AA Laminas schedule at BB
PreComp NATIONAL RENEWABLE ENERGY LABORATORY 15
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Beam Finite Element Analysis
1 Z 1
i (⌘)f00Kij = [EI](⌘)f 00 j (⌘)d⌘
L30 pBEAM
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Panel Buckling
b
⇣ ⇡ ⌘2 Z
E(⌧ )⌧ 2
Ncr = 3.6 d⌧2b 1� ⌫(⌧ )
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Fatigue (gravity-loads)
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Cost Model
• Based on NREL Cost and Scaling Model • Replaced blade mass/cost estimate • Replaced tower mass estimate • New balance-of-station model
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Reference Geometry
• NREL 5-MW reference design • Sandia National Laboratories
initial layup • Parameterized for optimization
purposes
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Design Variables
Description Name # of Vars
chord distribution {c} 5
twist distribution {θ} 4
spar cap thickness distribution {t} 3
tip speed ratio in region 2 λ 1
rotor diameter D 1
machine rating rating 1
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ultimate tensile strain
ultimate compressive strain
spar cap buckling
tip deflection at rated speed
blade natural frequency
fatigue at blade root due to gravity loads
fatigue at blade root due to gravity loads
maximum tip speed (imposed directly in analysis)
Constraints
minimize x
subject to
J(x)
(r
f rm✏
50
i
)/✏
ult < 1, i = 1, . . . , N
(r
f rm✏
50
i
)/✏
ult > -1, i = 1, . . . , N
(✏
50
j rf - ✏
cr
)/✏
ult > 0, j = 1, . . . ,M
6/6
0 < 1.1
!
1
/(3⌦
rated
) > 1.1
0
root-gravity
/S
f < 1
0
root-gravity
/S
f > -1
V
tip < V
tip
max
ultimate tensile strength ultimate compressive strength spar cap buckling tip deflection at rated blade natural frequency fatigue at blade root (gravity loads) fatigue at blade root (gravity loads) maximum tip speed
cset(x) < 0
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Maximum Annual Energy Production
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Why a single-discipline objective?
1. No structural model 2. No cost model 3. Organizational structure 4. Computational limitations
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Maximize AEP at Fixed Mass
maximize AEP (x)
with respect to x = {{c}, {✓}, �}subject to cset < 0
mblade = mc
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Maximize AEP at Fixed Mass
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AEP First
maximize AEP (x)
with respect to x = {{c}, {✓}, A}subject to V
tip < V
tip
max
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i
aero
< i
aero0
S
plan
< S
plan0
maximize AEP (x)
with respect to x = {{c}, {✓}, }subject to V
tip
< V
tip
max
AEP First
maximize AEP (x)
with respect to
subject to
x = {{c}, {✓}, A}V
tip < V
tip
max
i
aero < i
aero0
✓
iaero ⌘
Z M
b
t dr
◆
S
plan < S
plan0
(
root < (
root0
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minimize m(x)
with respect to x = {t}subject to c
set
(x) < 0
maximize AEP (x)
with respect to x = {{c}, {✓}, }subject to V
tip
< V
tip
max
i
aero
< i
aero0
S
plan
< S
plan0
J
root
< J
root0
AEP First
maximize AEP (x)
with respect to x = {{c}, {✓},A}subject to V
tip < V
tip
max
i
aero < i
aero0
S
plan < S
plan0
(
root < (
root0
minimize m(x)
with respect to x = {t}subject to c
set
(x) < 0
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mass 1st min COE
Comparison Between Methods
AEP mass COE
0
0.5
% c
hang
e
-0.5
-1
-1.5 AEP 1st
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Mass First minimize
with respect to
subject to
maximize
with respect to
subject to
minimize
with respect to
subject to
m(x)
x = {{c}, {t}}
cset(x) < 0
AEP (x)
x = {{✓}, �}V
tip < V
tip
max
m(x)
x = {{c}, {t}}
cset(x) < 0
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min COE
Comparison Between Methods
AEP mass COE
-1
-0.5
0
0.5
% c
hang
e
-1.5 AEP 1st mass 1st
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Minimize COE
minimize COE(x)
with respect to x = {{c}, {✓}, {t},�}subject to cset(x) < 0
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Comparison Between Methods
AEP mass COE
-1
-0.5
0
0.5
% c
hang
e
-1.5 AEP first mass first min COE
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Conclusions
1. Similar aerodynamic performance can be achieved with feasible designs with very different masses.
2. Sequential aero/structural optimization is significantly inferior to metrics that combine aerodynamic and structural performance.
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Minimum Turbine Mass / AEP
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Cost of Energy FCR (TCC + BOS) + O&M mturbine
COE = ⇡ AEP AEP
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Vary Rotor Diameter
minimize COE(x; D) or m(x; D)/AEP (x; D)
with respect to x = {{c}, {✓}, {t}, �}
subject to cset(x) < 0
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Maximize AEP at Fixed Mass
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Tower Contributions to Mass/Cost mass cost
53%33%
14% 17%
24%
9%
38%
12%
rotor nacelle tower rotor nacelle tower bos o&m
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Maximize AEP at Fixed Mass
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Maximize AEP at Fixed Mass
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Fixed Mass m
fixed = mblades + m
other
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Fixed Mass
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Fixed Mass
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Conclusions
1. m/AEP can work well at a fixed diameter but is often misleading for variable diameter optimization
2. Problem must be constructed carefully to prevent over-incentivizing the optimizer to reduce tower mass
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Minimum Cost of Energy
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Robust Optimization
Wind Power Class Wind Speed (50m) 3 6.4
4 7.0
5 7.5
6 8.0
7 11.9
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Robust Optimization
minimize < COE(x; V hub) >
where V hub ⇠ U(6.4, 11.9) with respect to x = {{c}, {✓}, {t},�, D, rating}subject to cset(x) < 0
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Robust Design
Vhub
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Conclusions
1. Optimization under uncertainty is important given the stochastic nature of the problem
2. Fidelity of the cost model can dramatically affect results
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Conclusions
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Conclusions
1. Sequential (or single-discipline) optimization is significantly inferior as compared to integrated metrics
2. m/AEP can be a useful metric at a fixed diameter if tower mass is handled carefully
3. High-fidelity cost modeling and inclusion of uncertainty are important considerations
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Acknowledgements
• Rick Damiani
• Pat Moriarty
• Katherine Dykes
• George Scott
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ThankYou