model predictive control of dfig-based wind power

University of Calgary

PRISM: University of Calgary's Digital Repository

Graduate Studies The Vault: Electronic Theses and Dissertations

2013-04-12

Model Predictive Control of DFIG-Based Wind Power

Generation Systems

Soliman, Mostafa

Soliman, M. (2013). Model Predictive Control of DFIG-Based Wind Power Generation Systems

(Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26967

http://hdl.handle.net/11023/601

doctoral thesis

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UNIVERSITY OF CALGARY

Model Predictive Control of DFIG-Based Wind Power Generation Systems

by

Mostafa Soliman

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

CALGARY, ALBERTA

APRIL, 2013

© Mostafa Soliman 2013

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Abstract

Novel control strategies that improve the cost effectiveness of wind energy conversion

systems are proposed in this thesis. The main focus is on grid-connected variable-speed variable-

pitch wind turbines equipped with doubly fed induction generators (DFIGs).

At the wind turbine control level, a multivariable control strategy based on model

predictive control techniques is proposed. The proposed strategy is formulated for the whole

operating region of the wind turbine, i.e., both partial and full load regimes. The pitch angle and

generator torque are controlled simultaneously to maximize energy capture, mitigate drive train

dynamic loads, and smooth the power generated while reducing the pitch actuator activity. This

has the effect of improving the efficiency and the power quality of the electrical power

generated, and increasing the life expectancy of the installation. Extensive simulation studies

show that the proposed control strategy provides superior performance when compared to

classical control strategies commonly used in the litterature.

For applications having fault tolerant control requirements, such as offshore wind farms,

a new wind turbine control strategy based on adaptive subspace predictive control is proposed. In

contrast with subspace predictive control algorithms previously proposed in the literature, the

proposed strategy ensures offset-free tracking. The effectiveness of the proposed strategy is

illustrated by simulating a wind turbine under normal operation and a fault in the hydraulic pitch

system.

Another control problem considered in this thesis is the design of the generator control

system to ensure fault ride through for DFIG-based wind turbines. This requirement is dictated

by recent grid codes, and it necessitates that the DFIG should be connected to the grid and

capable of providing reactive power support during large voltage dips. This is challenging for

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DFIG-based wind turbines due to their partially rated power converters. In this thesis, a novel

control strategy, based on using model predictive control and a dynamic series resistance

protection scheme, is proposed to ensure fault ride through requirement.

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Acknowledgements

First of all, I would like to thank Allah, the most gracious, the most merciful, for giving

me the patience and guidance to complete this work.

I would like to express my gratitude to my advisor, Prof. Om Malik, for his help and

supervision. I want to thank him for giving me the freedom in selecting my research topic, and

for continuously putting me on track when my research adventures drive me off the road.

I could not find words to express my gratitude and appreciation to my co-supervisor,

Prof. David Westwick, for his support and guidance throughout this project. His knowledge as

well as his kind and friendly personality have been always inspirational to me. Without his

constructive comments and insightful discussions, this dissertation would not have been possible.

The financial support received from the Alberta Ingenuity Fund (AIF) has been

instrumental in successfully completing this dissertation. I sincerely appreciate this support.

Warmest thanks are to my mother, Dr. NematAllah Rashad, who has always

overwhelmed me with her love, patience, and support. She made every effort to raise me as a

righteous person who appreciates the value of knowledge. I have always done my best to make

you, my dear mother, happy and proud of me. I hope that I achieved this goal.

My father, Prof. Hesham Soliman, has been always a role model for me. He showed me

the beauty of research and the joy of teaching. His dedication and passion for science have been

very influential in shaping my own carrer goals. Many thanks for my dear father for enlightening

the road for me and for his support.

My final warmest thanks go to my beloved wife, Dina Elsherbini, who was the source of

joy and love during this long journey. No words can express my gratitude and appreciation to her

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for her unbounded support and patience. You will always be my source of joy and the love of my

life.

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Dedication

To NematAllah, Dina, Laila, Arwa, and all my dear parents

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Table of Contents

Abstract ............................................................................................................................... ii Acknowledgements ............................................................................................................ iv Table of Contents .............................................................................................................. vii List of Tables .................................................................................................................... xii

List of Figures and Illustrations ....................................................................................... xiii List of Symbols, Abbreviations and Nomenclature ......................................................... xxi

INTRODUCTION ..................................................................................1 CHAPTER ONE:

1.1 Background ................................................................................................................4

1.1.1 Overview of wind energy conversion systems ..................................................4 1.1.2 Wind turbines classification ..............................................................................7

1.1.3 Wind turbines with doubly fed induction generators ........................................8 1.2 Wind turbines control ................................................................................................9

1.2.1 Control modes ...................................................................................................9 1.2.2 Control system hierarchy .................................................................................10 1.2.3 Control challenges ...........................................................................................12

1.2.3.1 Wind turbine control level .....................................................................13 1.2.3.2 Generator control level ..........................................................................15

1.3 Scope and objectives ................................................................................................17 1.4 Thesis outline ...........................................................................................................19

FUNDAMENTALS OF WIND ENERGY CONVERSION SYSTEMSCHAPTER TWO:

...................................................................................................................................21

2.1 The wind ..................................................................................................................21 2.1.1 The source of the wind ....................................................................................21 2.1.2 The power contained in the wind ....................................................................22

2.2 Wind turbine aerodynamics .....................................................................................23 2.2.1 Aerodynamics concepts ...................................................................................23

2.2.2 The power coefficient and the tip speed ratio .................................................29 2.3 Wind turbine control principles ...............................................................................31

2.3.1 Maximizing the power of a WECS .................................................................31 2.3.2 Limiting the power of a WECS .......................................................................33

2.3.2.1 Passive-stall control ...............................................................................33 2.3.2.2 Pitch control (pitch-to-feather) ..............................................................35

2.3.2.3 Active-stall control (pitch-to-stall) ........................................................35 2.4 Commercial wind turbine concepts .........................................................................36

2.4.1 Type 1 –Conventional fixed-speed SCIG ........................................................36

2.4.2 Type 2 – Wound rotor induction generator with variable external rotor resistance

..........................................................................................................................37 2.4.3 Type 3 – Doubly fed induction generator with partial-scale converter ...........39 2.4.4 Type 4− Permanent magnet synchronous generator with full-scale converter39

2.5 VSVP wind turbines ................................................................................................40 2.5.1 Advantages of VSVP wind turbines ................................................................40

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2.5.2 VSVP wind turbine power curve .....................................................................41

2.6 WECS control objectives .........................................................................................46

MODELING OF VARIABLE-SPEED VARIABLE-PITCH WIND CHAPTER THREE:

ENERGY CONVERSION SYSTEMS.....................................................................52 3.1 Overview of the WECS model ................................................................................52 3.2 Wind speed stochastic model ...................................................................................54

3.2.1 Wind speed at one fixed point .........................................................................54 3.2.1.1 Van der Hoven’s spectral model ............................................................54 3.2.1.2 Turbulence model ..................................................................................56 3.2.1.3 Overall model of the wind speed at one fixed point ..............................57

3.2.2 Wind speed experienced by the turbine blades ...............................................59 3.2.2.1 Wind shear effect ...................................................................................61

3.2.2.2 Tower shadow effect ..............................................................................63 3.2.2.3 Rotational turbulence .............................................................................64

3.2.3 Effective wind speed .......................................................................................65 3.3 Aerodynamics model ...............................................................................................70 3.4 Blade pitch system ...................................................................................................72

3.5 Drive train model .....................................................................................................73 3.6 Electric subsystem ...................................................................................................76

3.6.1 Wound rotor induction generator model .........................................................77 3.6.1.1 WRIG dynamic model ...........................................................................77 3.6.1.2 WRIG steady state model ......................................................................85

3.6.2 Modeling of the grid-side converter connection to the grid ............................88

3.6.3 Modeling the power converters .......................................................................90 3.6.4 Modeling the converter dc Link ......................................................................90 3.6.5 Modeling the generator control system ...........................................................91

3.7 Grid interonnection ..................................................................................................92 3.8 Modeling in per unit system ....................................................................................93

3.9 Overall WECS model ..............................................................................................94 3.10 Summary ................................................................................................................97

MODEL PREDICTIVE CONTROL .................................................98 CHAPTER FOUR:

4.1 Introduction ..............................................................................................................98 4.2 MPC ingredients ....................................................................................................102

4.2.1 Prediction model ............................................................................................103

4.2.2 Objective function .........................................................................................104 4.2.3 Constraints .....................................................................................................105 4.2.4 State estimation .............................................................................................106

4.3 MPC optimisation problem ....................................................................................106 4.3.1 Constructing the predictor .............................................................................107 4.3.2 Formulating the MPC optimization problem as a standard QP .....................108

4.4 Analysis of MPC controllers ..................................................................................111 4.4.1 Unconstrained MPC ......................................................................................111 4.4.2 Constrained MPC ..........................................................................................112

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4.5 Offset-free MPC ....................................................................................................115

4.6 Stability ..................................................................................................................119 4.7 Real-time MPC implementation ............................................................................123 4.8 Summary ................................................................................................................125

MULTIPLE MODEL MIMO PREDICTIVE CONTROL FOR CHAPTER FIVE:

VARIABLE-SPEED VARIABLE-PITCH WIND TURBINES ............................126

5.1 Control problem description ..................................................................................127 5.2 Simplified model of variable-speed variable-pitch wind turbines .........................130 5.3 Baseline wind turbine controller ............................................................................134 5.4 Proposed control strategy .......................................................................................136

5.4.1 Multiple model predictive control for variable-speed variable-pitch WECS 136 5.4.1.1 Prediction model bank .........................................................................137

5.4.1.2 Optimization problem ..........................................................................139 5.4.1.3 State estimation ....................................................................................141

5.4.1.4 Bumpless switching between different MPC controllers ....................142 5.4.2 MMPC controller design ...............................................................................142

5.4.2.1 MMPC weight selection ......................................................................142

5.4.2.2 Disturbance model selection ................................................................144

5.4.2.3 Partitioning the whole operating region into operating sub-regions146 5.5 MPPT algorithm ....................................................................................................149 5.6 Simulation results ..................................................................................................151

5.6.1 Simulation set-up ...........................................................................................151 5.6.2 Performance measures ...................................................................................152

5.6.3 MMPC design ................................................................................................153 5.6.4 PI baseline controller design .........................................................................161

5.6.5 Comparison of the MMPC and PI controllers - Deterministic wind speed ...164 5.6.6 Comparison of the MMPC and PI controllers - Stochastic wind speed ........167

5.7 Conclusions ............................................................................................................179

ADAPTIVE SUBSPACE PREDICTIVE CONTROL OF VARIABLE-CHAPTER SIX:

SPEED VARIABLE-PITCH WIND TURBINES ..................................................180

6.1 Introduction ............................................................................................................181 6.2 Review of subspace predictive control ..................................................................183

6.2.1 Subspace system identification ......................................................................183

6.2.2 Subspace predictive control ...........................................................................187

6.3 Offset-free subspace predictive control .................................................................190 6.3.1 Formulating the subspace predictor ...............................................................190 6.3.2 VARIX model identification .........................................................................195

6.3.2.1 Off-line Identification of ..................................................................195

6.3.2.2 On-line (recursive) identification of ................................................196 6.3.3 OFSPC algorithm ..........................................................................................198

6.3.4 Examples .......................................................................................................203 6.4 Application of OFSPC in wind turbine control .....................................................205

6.4.1 OFSPC controller design ...............................................................................205

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6.4.2 WECS model .................................................................................................207

6.5 Simulation results ..................................................................................................209 6.5.1 Performance comparison during normal WECS operation ...........................210 6.5.2 Performance comparison during a fault in the pitch actuator ........................216

6.6 Conclusions ............................................................................................................218

ENSURING FAULT RIDE THROUGH FOR DFIG-BASED WIND CHAPTER SEVEN:

TURBINES .............................................................................................................220 7.1 Introduction ............................................................................................................220 7.2 DFIG behavior under voltage dips .........................................................................224

7.2.1 DFIG behavior during normal operation .......................................................226

7.2.2 DFIG behavior under a voltage dip ...............................................................226 7.3 RSC protection schemes ........................................................................................228

7.3.1 Crowbar protection ........................................................................................228 7.3.2 Dynamic series resistance ..............................................................................229

7.4 Proposed control strategy based on MPC and DSR protection scheme ................230 7.4.1 RSC control design requirements ..................................................................230 7.4.2 Motivation for using MPC .............................................................................231

7.4.3 Overview of the proposed control strategy ...................................................232 7.4.4 Stator voltage space vector reference frame orientation ...............................234

7.4.5 MPC design ...................................................................................................235 7.4.5.1 Prediction model ..................................................................................236 7.4.5.2 MPC optimization problem .................................................................237

7.4.5.3 Possible MPC implementation approaches ..........................................239

7.4.6 Decision maker design ..................................................................................240 7.5 Simulation results ..................................................................................................242

7.5.1 Comparison of different MPC implementations ...........................................242

7.5.2 Evaluation of the proposed control strategy ..................................................243 7.5.2.1 Normal DFIG operation .......................................................................244

7.5.2.2 DFIG operation during voltage dips ....................................................244 7.5.2.3 Reactive power injection during a voltage dip ....................................260

7.5.2.4 Discussion ............................................................................................261 7.6 Conclusions ............................................................................................................263

CONCLUSIONS AND FUTURE WORK .....................................264 CHAPTER EIGHT:

8.1 Summary of contributions .....................................................................................264

8.2 Thesis outcomes .....................................................................................................268 8.3 Future work ............................................................................................................269

REFERENCES ................................................................................................................274

APPENDIX A: BASELINE WIND TURBINE GENERATOR CONTROLLER .........289 A.1. Vector control of the RSC ....................................................................................289 A.2. Vector control of the GSC ...................................................................................293

APPENDIX B: PER UNIT REPRESENTATION OF THE DFIG MODEL ..................296

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B.1. Base values ...........................................................................................................296

B.2. Per unit model of the drive train ..........................................................................297 B.3. Per unit model of the WRIG ................................................................................297 B.4. Per unit model of the GSC connection to the grid ...............................................298 B.5. Per unit model of the dc link ................................................................................298

APPENDIX C: REVIEW OF WIND TURBINE CONTROL SCHEMES .....................299

C.1. Partial load regime ...............................................................................................299 C.2. Full load regime ...................................................................................................302

APPENDIX D: MODEL PARAMETERS ......................................................................304 D.1. Wind turbine ........................................................................................................304

D.2. Pitch actuator .......................................................................................................304 D.3. Drive train ............................................................................................................304

D.4. DFIG system ........................................................................................................305 D.5. Generator control system parameters ...................................................................305

D.6. Wind speed simulator ..........................................................................................306 D.7. Power system model parameters ..........................................................................307

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List of Tables

Table 3.1 coefficients for 0, 1, …,4. ........................................................................ 71

Table 5.1 MMPC sub-regions and controller data ...................................................................... 159

Table 5.2 Low wind speeds statistics (no tower shadow and wind shear effects) ...................... 171

Table 5.3 Low wind speeds statistics (with tower shadow and wind shear effects) ................... 171

Table 5.4 Medium wind speeds statistics (no tower shadow and wind shear effects)................ 175

Table 5.5 Medium wind speeds statistics (with tower shadow and wind shear effects) ............ 175

Table 5.6 High wind speeds statistics (no tower shadow and wind shear effects) ..................... 178

Table 5.7 High wind speeds statistics (with tower shadow and wind shear effects) .................. 178

Table 6.1 OFSPC controller weights .......................................................................................... 209

Table 6.2 Low wind speeds statistics (all quantities are normalized to the OFSPC controller). 212

Table 6.3 Medium wind speeds statistics (all quantities are normalized to the OFSPC

controller). ........................................................................................................................... 214

Table 6.4 High wind speeds statistics (all quantities are normalized to the OFSPC controller). 216

Table 6.5 Performance comparison between OFSPC, CLSPC, MMPC1, MMPC2 and PI

controllers during a fault in the hydrolic pitch actuator. Bold and normal font show

quantities normalized to the OFSPC during faulty and normal operation, respectively. ... 218

Table 7.1 Different MPC implementation approaches. .............................................................. 242

Table 7.2 Computational time statistics for different MPCs. ..................................................... 243

Table 7.3 Comparison between the proposed MPC and baseline PI strategies during a small

voltage dip ( p.u.). .......................................................................................... 249

Table 7.4 Comparison between the proposed MPC with rotor- and stator-connected DSR, and

the baseline PI control strategies during a medium dip ( p.u.). ...................... 255

Table 7.5 Comparison between the proposed MPC with rotor- and stator- connected DSR,

and the baseline PI control strategies during a large dip ( p.u.). .................... 260

Table B.1 Selected base values. ................................................................................................. 296

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List of Figures and Illustrations

Figure 1.1 Evolution of grid-connected wind turbines in terms of power rating, hub height H,

and rotor diameter . The wind turbines are drawn to scale and they are compared with

the size of Boeing 747 and an American football field [2, 7]. ................................................ 3

Figure 1.2 A utility-scale horizontal axis wind turbine [12]. .......................................................... 6

Figure 1.3 Components of a horizontal axis wind turbine [12]. .................................................... 6

Figure 1.4 Wind turbine with a doubly fed induction generator.................................................... 9

Figure 1.5 Output power curve of a 1.5 MW wind turbine. In partial load region ( ), the WT power capture should be maximized, and in full load region ( ), the output power of the WT should be regulated at its rated value. ............................ 10

Figure 1.6 Control system hierarchy of a DFIG-based WECS. .................................................... 11

Figure 2.1 Wind flow over an airfoil section. .............................................................................. 24

Figure 2.2 Aerodynamic forces produced on an airfoil fixed in a stream of wind. ..................... 24

Figure 2.3 An airfoil section located at a distance from the hub. ............................................. 25

Figure 2.4 Aerodynamic forces produced on an airfoil section of a HAWT [13]. ...................... 25

Figure 2.5 Typical Lift and drag coefficients of an airfoil [12]. .................................................. 28

Figure 2.6 Typical of an airfoil [12]. ............................................................................... 28

Figure 2.7 Typical power coefficient variations of a HAWT [55]. ............................................. 30

Figure 2.8 Optimal regime characteristic (red) and wind turbine power and torque curves

(blue) at different wind speeds and shown on (a) the - plane and (b) the

- plane [13]. ................................................................................................................... 32

Figure 2.9 Aerodynamic stall at a rotor blade with fixed pitch angle at increasing wind speed

and constant rotor speed [15]. ............................................................................................... 34

Figure 2.10 Controlling the rotor power using pitch-to-feather and pitch-to-stall strategies

[15]. ....................................................................................................................................... 36

Figure 2.11 The four main grid-connected wind turbine concepts: (a) Type 1 - conventional

fixed-speed SCIG, (b) Type 2 - WRIG with variable external rotor resistance, (c) Type 3

- DFIG concept and (d) Type 4 – direct-drive PMSG with full-scale converter [17, 23]. .... 38

xiv

Figure 2.12 Typical power curve of a 1.5 MW VSVP wind turbine [65]. The solid line

shows the power curve, the dashed line shows the wind power , and the dotted line

shows the maximum power that can be extracted by the wind turbine ( ),

where . ......................................................................................................... 42

Figure 2.13 Variation of (a) the turbine speed, and (c) the pitch angle all operating wind

speeds. ................................................................................................................................... 43

Figure 2.14 Variation of (a) and (b) over all operating wind speeds. .................................. 44

Figure 2.15 Weibull probability distribution of the mean wind speed at a given site. ................ 45

Figure 2.16 Power density versus mean wind speed at a given site. ........................................... 45

Figure 3.1 Main subsystems of grid-connected DFIG-based wind turbine. ................................ 53

Figure 3.2 Block diagram of a grid-connected DFIG-based wind turbine. ................................. 53

Figure 3.3 Van der Hoven’s spectral model of the wind speed [12]. .......................................... 55

Figure 3.4 Non-stationary wind speed simulation at one point [12]............................................ 58

Figure 3.5 Non-stationary wind speed simulation: the low frequency component (black

solid) and the total wind speed at one point (blue dotted). ................................................... 59

Figure 3.6 Approximate von Karman’s spectrum for two different values of mean wind

speed: (a) turbulence PSDs and (b) shaping filter gains. ...................................................... 59

Figure 3.7 Spatial wind-speed distribution over the swept area of the turbine rotor. .................. 60

Figure 3.8 Wind shear and tower shadow effects [100]. ............................................................. 61

Figure 3.9 Torque oscillations due to the wind shear alone: (a) normalized individual blade

torque and (b) normalized wind turbine torque. ................................................................... 62

Figure 3.10 Torque oscillations due to the tower shadow alone: (a) normalized individual

blade torque and (b) normalized wind turbine torque. .......................................................... 64

Figure 3.11 Effective wind speed simulator [13]......................................................................... 66

Figure 3.12 Wind speed simulation: (a) wind speed at one point, (b) rotationally sampled

wind speed and (c) zoom on the rotationally sampled wind speed. ...................................... 69

Figure 3.13 Comparison of the shaping filters gains of the wind speed at one fixed point and

the rotationally sampled wind speed. .................................................................................... 69

Figure 3.14 Low and high wind speed simulations. .................................................................... 70

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Figure 3.15 Effective wind speed PSD comparison at low and high wind.................................. 70

Figure 3.16 - characteristics for different values of pitch angle [14, 52]. .......................... 72

Figure 3.17 Blade pitch system model [24]. ................................................................................ 73

Figure 3.18 Two mass model of the wind turbine drive train subsystem[32]. ............................ 75

Figure 3.19 Electrical connection diagram of a DFIG-based wind turbine [78]. ........................ 76

Figure 3.20 (a) Idealized three-phase, two-pole induction machine with concentric three

phase windings and (b) magnetic axes of the stator and rotor windings. ............................. 77

Figure 3.21 Relationship between abc reference frame and arbitrary rotating dq0 reference

frame. .................................................................................................................................... 81

Figure 3.22 Equivalent circuit of a WRIG [114]. ........................................................................ 86

Figure 3.23 GSC connection to the grid [26]. .............................................................................. 89

Figure 3.24 Converter dc link schematic. .................................................................................... 90

Figure 3.25 Generator controller. .................................................................................................. 92

Figure 4.1 MPC concept. ............................................................................................................ 100

Figure 4.2 Closed loop MPC control system. ............................................................................. 111

Figure 4.3 PI feedback control loop. ........................................................................................... 116

Figure 5.1 Nominal operating trajectory of a VSVP wind turbine. ............................................ 127

Figure 5.2 Variations of , and evaluated along the nominal WT operating trajectory

in Figure 5.1. ....................................................................................................................... 133

Figure 5.3 Bode magnitude plots of the WT model (5.11)-(5.13). Gray lines represent low

wind speeds (partial load) and black lines represent high wind speeds (full load). ............ 134

Figure 5.4 Classical control strategy using two PI controllers . ................................................. 135

Figure 5.5 Proposed control strategy using MMPC. .................................................................. 137

Figure 5.6 MPPT algorithm. ....................................................................................................... 150

Figure 5.7 Power system studied. ............................................................................................... 152

Figure 5.8 Response to a step change in wind speed from 6.5 to 7.5 m/s using ,

, and . ................................................................................................... 155

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Figure 5.9 Response to a step change in wind speed from 6.5 to 7.5 m/s using , , and . ..................................................................................................................... 156

Figure 5.10 Response to a step change in wind speed from 20 to 21 m/s using ,

, and . ....................................................................................................... 157


, and . ........................................................................................................ 158

Figure 5.12 Wind speed profile. ................................................................................................. 159

Figure 5.13 Performance comparison between SMPC and MMPC. .......................................... 160

Figure 5.14 Bode plots of at different wind speeds in the partial load regime. ..... 163

Figure 5.15 Bode plot of at different wind speeds in the full load regime. ........... 163

Figure 5.16 PI controller gains as functions of the mean wind speed. (a) partial load, and (b)

full load. .............................................................................................................................. 164

Figure 5.17 Response to a step change in wind speed from 6.5 to 7.5 m/s using the MMPC

and the PI control strategies. ............................................................................................... 165

Figure 5.18 Response to a positive step in wind speed from 10 to 11 m/s (left) and a negative

step in wind speed from 10 to 9 m/s (right) using the MMPC and the PI control

strategies. ............................................................................................................................ 166

Figure 5.19 Response to a step change in wind speed from 20 to 21 m/s using the MMPC and

the PI control strategies. (a) generator speed, (b) WTG output power, and (c) torsional

torque. ................................................................................................................................. 168

Figure 5.20 Response to a step change in wind speed from 20 to 21 m/s using the MMPC and

the PI control strategies. (a) generator torque, and (b) pitch angle. .................................... 169


Figure 5.22 Simulation results for low wind speeds. .................................................................. 170


Figure 5.24 MMPC switching signal. ......................................................................................... 172

Figure 5.25 Simulation results for medium wind speeds. (a) generator speed, (b) WTG output

power, and (c) torsional torque. .......................................................................................... 173

Figure 5.26 Simulation results for medium wind speeds. (a) Generator torque, (b) pitch angle.174

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Figure 5.27 Simulation results for medium wind speeds (zoomed from =15 to 50 s). ............. 175


Figure 5.29 MMPC switching signal. ......................................................................................... 176

Figure 5.30 Simulation results for high wind speeds. ................................................................. 177

Figure 6.1 Comparison between (a) classical MPC design, and (b) SPC design frameworks

using I/O data from the controlled plant. ............................................................................ 189

Figure 6.2 Histograms Markov Parameters identified using (a) and (b) (thick

black line shows the true parameter value). ........................................................................ 204

Figure 6.3 Open-loop system model. .......................................................................................... 205

Figure 6.4 Performance comparison between the CLSPC and OFSPC algorithms with three

values of = 0.95, 0.99 and 0.995 during a step input disturbance. .................................. 206

Figure 6.5 Proposed wind turbine FTC strategy based on OFSPC. ............................................ 207

Figure 6.6 Step response of the pitch actuator system during normal and faulty operation. ...... 209

Figure 6.7 Wind speed profile. ................................................................................................... 210

Figure 6.8 Simulation results for low wind speeds (zoomed from =300 to 360 s). .................. 211

Figure 6.9 Wind speed profile. ................................................................................................... 212

Figure 6.10 Simulation results for medium wind speeds (zoomed from =340 to 460 s). ......... 213


Figure 6.12 Simulation results for high wind speeds (zoomed from =300 to 330 s). ............... 215

Figure 6.13 Wind profile used during simulations of an abrupt fault in the pitch actuator. ....... 216

Figure 6.14 Comparison between the OFSPC (black), the CLSPC (red), and the classical PI

(blue) strategies after an abrupt drop in the hydraulic pressure. ......................................... 217

Figure 7.1 Three phase stator voltages (top) and rotor currents (bottom) for a terminal voltage

dip of 80% with no protection............................................................................................ 221

Figure 7.2 DFIG-based wind turbine with a crowbar for RSC protection [237]. ....................... 222

Figure 7.3 Fault Ride Through standard according to US grid codes [236]. .............................. 223

Figure 7.4 DFIG rotor equivalent circuit [43]. ........................................................................... 225

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Figure 7.5 DFIG-based wind turbine with (a) rotor-connected DSR [41] and (b) stator-

connected DSR [40] for RSC protection............................................................................. 230

Figure 7.6 Baseline control strategy. .......................................................................................... 231

Figure 7.7 Proposed RSC control strategy. ................................................................................. 233

Figure 7.8 Shaded regions show the polytopic approximation of the (a) rotor voltage

constraint in (7.36) and (b) rotor current constraint in (7.37). ............................................ 240

Figure 7.9 Decision maker block. ............................................................................................... 241

Figure 7.10 Tracking of step changes in the rotor currents’ set points using the proposed

MPC strategy. ..................................................................................................................... 245

Figure 7.11 Tracking of step changes in the rotor currents’ set points using the baseline PI

strategy. ............................................................................................................................... 245

Figure 7.12 Tracking of step changes in the generator torque and stator reactive power set

points using the proposed MPC strategy. ........................................................................... 245

Figure 7.13 Stator voltages during a small voltage dip ( p.u.). .............................. 246

Figure 7.14 DFIG response using the baseline PI strategy during a small voltage dip ( p.u.): (a) rotor voltages, (b) rotor currents, (c) generator torque, and (d) stator

reactive power. (dashed lines show maximum RSC limits) ............................................... 247

Figure 7.15 DFIG response using the proposed MPC strategy during a small voltage dip

( p.u.): (a) rotor voltages, (b) rotor currents, (c) generator torque, and (d)

stator reactive power. (dashed lines show maximum RSC limits) ..................................... 248

Figure 7.16 dc link voltage during a small voltage dip ( p.u.) using the baseline

PI (dashed) and the proposed MPC (solid) strategies. ........................................................ 249

Figure 7.17 Stator voltages during a medium voltage dip ( p.u.). .......................... 250

Figure 7.18 Crowbar activation signal during a medium voltage dip......................................... 250

Figure 7.19 Rotor-connected DSR activation signal during a medium voltage dip. .................. 250

Figure 7.20 Stator-connected DSR activation signal during a medium voltage dip. .................. 250

Figure 7.21 DFIG response using the baseline PI strategy during a medium voltage dip


stator reactive power. (dashed lines show maximum RSC limits) ..................................... 251

xix

Figure 7.22 DFIG response using the proposed MPC strategy with a rotor-connected DSR

during a medium voltage dip ( p.u.): (a) rotor voltages, (b) rotor currents,

(c) generator torque, and (d) stator reactive power. (dashed lines show maximum RSC

limits) .................................................................................................................................. 252

Figure 7.23 DFIG response using the proposed MPC strategy with a stator-connected DSR

during a medium voltage dip ( p.u.): (a) rotor voltages, (b) rotor currents,

(c) generator torque, and (d) stator reactive power. (dashed lines show maximum RSC

limits) .................................................................................................................................. 253

Figure 7.24 dc link voltage during a medium voltage dip ( p.u.) using the

baseline PI (dashed), the proposed MPC with rotor-connected DSR (dotted), and the

proposed MPC with stator-connected DSR (solid) strategies. ............................................ 254

Figure 7.25 Stator voltages during a large voltage dip ( p.u.). ............................... 256

Figure 7.26 Crowbar activation signal during a large voltage dip. ............................................. 256

Figure 7.27 Rotor-connected DSR activation signal during a large voltage dip. ....................... 256

Figure 7.28 Stator-connected DSR activation signal during a large voltage dip. ....................... 256

Figure 7.29 DFIG response using the baseline PI strategy during a large voltage dip ( p.u.): (a) rotor voltages, (b) rotor currents, (c) generator torque, and (d) stator

reactive power. (dashed lines show maximum RSC limits) ............................................... 257


during a large voltage dip ( p.u.): (a) rotor voltages, (b) rotor currents, (c)

generator torque, and (d) stator reactive power. (dashed lines show maximum RSC

limits) .................................................................................................................................. 258



generator torque, and (d) stator reactive power. (dashed lines show maximum RSC

limits) .................................................................................................................................. 259

Figure 7.32 dc link voltage during a large voltage dip ( p.u.) using the baseline

PI (dashed), the proposed MPC with rotor-connected DSR (dotted), and the proposed

MPC with stator-connected DSR (solid) strategies. ........................................................... 260

Figure 7.33 Stator voltages. ........................................................................................................ 261

Figure 7.34 Stator reactive power control during a voltage dip: (a) baseline PI strategy, (b)

proposed MPC strategy with a rotor-connected DSR, and (c) proposed MPC strategy

with a stator-connected DSR. ............................................................................................. 262

Figure A.1 Vector control structure of the RSC [25]. ................................................................. 293

xx

Figure A.2 Vector control structure of the GSC [26]. ................................................................ 295

Figure C.1 Control schemes used in the partial load regime. ..................................................... 299

Figure C.2 Optimal regime characteristic in the - plane. .................................................... 300

Figure C.3 Control schemes used in full load regime [13]. ........................................................ 303

xxi

List of Symbols, Abbreviations and Nomenclature

General Notation

Symbol Definition

the set of integers

the set of real numbers

the set of real-valued n-dimensional vectors

regular letters denote scalars

small bold letters denote vectors

capital bold letters denote matrices

vector formed by stacking , i.e.,

[

]

identity matrix

, zero matrix, zero matrix

the entry of the matrix

the row of the matrix

the column of the matrix

transpose of

trace of

rank of

xxii

maximum singular value of

Kronecker product

‖ ‖ 2-norm of the matrix , i.e., ‖ ‖

‖ ‖ Frobenius norm of the matrix , i.e., ‖ ‖ √

‖ ‖ 2-norm (Euclidean norm) of , i.e., ‖ ‖ √

‖ ‖ norm of weighted with the positive definite matrix , i.e.,

‖ ‖ √

[ ] Mathematical expectation of the random vector

maximum (minimum) limit of

value of at certain operating point

deviation of from its operating point value, i.e.,

backward difference of , i.e.,

derivative of , i.e.,

differential operator, i.e.,

set point value of

estimate of

base value of

rated value of

in per unit, i.e.,

| | magnitude of the complex number

real part of the complex number

xxiii

imaginary part of the complex number

conjugate of , i.e.,

Wind speed model parameters

wind speed signal at one fixed point [m/s]

low-frequency wind speed component [m/s]

turbulent wind speed component [m/s]

turbulence length scale [m]

turbulence intensity

power spectral density of the turbulent component

standard deviation of the turbulence

sampling interval of the low frequency component [s]

sampling interval of the turbulence component [s]

empirical wind shear exponent

hub height [m]

, spatial filter coefficients

tower radius

normal distance from the rotor to the tower center-line

Wind turbine model parameters

pitch angle [o]

pitch angle set point [o]

xxiv

time constant of the pitch system [s]

tip speed ratio

air density [kg/m3]

blade length of a wind turbine [m]

wind power [W]

harvested mechanical power of a wind turbine [W]

torque coefficient

power coefficient

low speed shaft (turbine rotor) torque [N m]

high speed shaft (generator elctromechanical) torque [N m]

turbine rotor rotational speed (speed of the turbine’s low speed

shaft) [rad/s]

generator rotational speed (speed of the turbine’s high speed

shaft) [rad/s]

generator inertia [kg m2]

wind turbine inertia [kg m2]

turbine inertia constant [s]

generator inertia constant [s]

shaft twist angle [rad]

drive train torsional torque [N m]

, shaft stiffness and damping coefficients

gear ratio

xxv

gear box efficiency

( ) base speed at the high (low) speed shaft [rad/s]

( ) base torque at the high (low) speed shaft [Nm]

rated wind speed (minimum wind speed at which the system

rated power is achieved) [m/s]

minimum wind speed at which the turbine speed reaches its rated

value, [m/s]

cut-in wind speed [m/s]

cut-out wind speed [m/s]

Doubly fed induction generator model parameters

, , current [A], voltage [V], flux linkage [Wb-t]

, , resistance [Ω], inductance [H], reactance [Ω]

, , active power [W], reactive power [var], apparent power [VA]

, , stator quantity, rotor quantity, grid-side converter quantity

, direct axis quantity, quadrature axis quantity

a quantity referred to the stationary reference frame

a quantity referred to the synchronously rotating reference frame

a quantity referred to the abc reference frame

, , 3-phase stator voltages [V]

, , 3-phase rotor voltages [V]

, , 3-phase stator currents [A]

xxvi

, , 3-phase rotor currents [A]

, , 3-phase stator flux linkages [Wb-t]

, , 3-phase rotor flux linkages [Wb-t]

, , 3-phase voltages at the terminals of the grid-side converter [V]

,

[

], [

]

,

[

], [

]

leakage inductance of the stator windings [H]

magnetizing inductance of the stator windings [H]

leakage inductance of the rotor windings [H]

magnetizing inductance of the rotor windings [H]

peak value of the stator-rotor mutual inductance [H]

magnetizing inductance on the stator side [H]

grid-side filter inductance [H]

induction machine leakage factor, i.e.,

stator resistance [Ω]

rotor resistance [Ω]

grid-side filter resistance [Ω]

position of the rotor a-axis with respect to the stator a-axis [elec.

rad]

xxvii

position of the synchronously rotating frame [elec. rad]

speed of the synchronously rotating frame [elec. rad/s]

generator rotor speed [elec. rad/s]

slip speed, i.e., [elec. rad/s]

slip, i.e.,

number of pole pairs


transferred between the grid-side converter and the grid


transferred between the wind turbine generator and the grid

copper losses in the rotor circuit [W]

voltage at the dc link [V]

, dc currents in the dc link [A]

per unit capacitance of the dc link

grid frequency [rad/s]

grid frequency [Hz]

root-mean-square (rms) Line to Line voltage [V]

synchronously rotating direct and quadrature axes

stationary direct and quadrature axes

coordinate transformation matrix

complex space vector

xxviii

complex time phasor

Grid model parameters

short circuit power at the point of common coupling (the fault

level), i.e. √ [VA]

short circuit current following during a three phase fault at the

point of common coupling [A]

equivalent Thevenin impedance of the grid [Ω]

equivalent Thevenin resistance of the grid [Ω]

equivalent Thevenin reactance of the grid [Ω]

voltage phasor at the wind turbine generator terminals

short circuit capacity ratio, i.e.,

grid impedance angle [deg]

short term flicker severity

Model predictive control

, , number of states, inputs and outputs

, , , , state vector, input vector, measurement vector, controlled outputs

vector, and reference vector

, , process noise, measurement noise, innovation process

An augmented vector containing the inputs and the outputs, i.e.,

xxix

[

]

extended observability matrix

, , prediction horizon, control horizon, past horizon

forgetting factor

Remark on Notation

Due to the multidisciplinary nature of the research conducted in this thesis, which

includes aerodynamics, mechanical, electrical and control engineering disciplines; and the

finiteness of the alphabet that can be used as symbols for different quantities, it was unavoidable

to use similar symbols for different quantities. In this research, is used to denote the wind

speed while and denote the stator and rotor voltages, respectively. Furthermore, is used

to denote the blade length of a wind turbine, , , denote the stator resistance, the rotor

resistance, and the Thevenin resistance of the grid, respectively, and denotes the reference

vector of a digital controller. Finally, is used to denote the sampling time of a digital

controller, while and denote the turbine and generator torques, respectively. Fortunately,

these quantities belong to different disciplines and the meaning of the symbol will be clear from

the context.

xxx

Abbreviations and Acronyms

ac Alternating current

BET Blade Element Theory

CLSPC Closed Loop Subspace Predictive Control

dc Direct current

DEL Damage Equivalent Load

DFIG Doubly Fed Induction Generator

DSR Dynamic Series Resistance

FDI Fault Detection and Isolation

GSC Grid-Side Converter

HAWT Horizontal Axis Wind Turbine

LQG Linear Quadratic Gaussian

LS Least Squares

MIMO Multiple-input Multiple-output

MOESP MIMO Output-Error State sPace algorithms

MPC Model Predictive Control

MMPC Multiple Model Predictive Control

OFSPC Offset-free Subspace Predictive Control

ORC Optimal Regime Characteristic

PCC Point of Common Coupling

PEM Prediction Error Method

PMSG Permanent Magnet Synchronous Generator

xxxi

PSD Power Spectral Density

QP Quadratic Programming

rms root-mean-square

RSC Rotor-Side Converter

SCIG Squirrel Cage Induction Generator

SCR Short Circuit capacity Ratio

SIM Subspace Identification Methods

STATCOM Static synchronous compensator

SISO Single-input Single-output

SPC Subspace Predictive Control

SVD Singular Value Decomposition

VARX Vector Auto Regressive with eXogenous input

VARIX Integrated VARX

VSVP Variable-Speed Variable-Pitch

WECS Wind Energy Conversion System

WRIG Wound Rotor Induction Generator

WT Wind Turbine

WTG Wind Turbine Generator

1

Introduction Chapter One:

Wind power is one of the fastest growing energy sources worldwide. With an average

annual increase in the installed wind energy capacity of 25% over the past five years [1, 2], and

with a current worldwide installed capacity of approximately 238,000 MW [1], wind energy

seems certain to play an important role in the world’s energy future.

The huge growth in wind power installations is driven by many factors. First, wind is a

clean energy source. With the increased worldwide concern about global warming and climate

change, wind energy is regarded as a part of the solution for these problems. Many studies

indicate that increasing the share of wind power compared to conventional power generation

using fossil fuels can significantly reduce carbon dioxide emissions from electricity generation

[3]. Another selling point of wind energy is its renewable nature. In contrast with fossil fuels,

wind energy will never get exhausted as it is used. One alerting indicator shown by recent studies

is that, with current production rates, the proven coal reserve is sufficient for only the next 120

years [4]. Finally, with increasing oil prices and the desire to secure alternative energy sources

to meet the continuously increasing electricity consumption, many governments are heavily

supporting the usage of wind power [5].

Currently, there are many countries that have succeeded in integrating a relatively large

amount of wind power generation within their electrical networks [6]. Most notably, Denmark

has 19% of its electric power generation capacity from wind. Other countries, such as Spain and

Germany, have a moderate wind power penetration level of approximately 10% [5]. There are

currently many ambitious plans in the USA and Europe to reach 20% wind contribution to

electricity supply by 2030 [5]. In the USA, it is reported that a total of more than 300,000 MW of

wind capacity should be installed by 2030 to reach this target [6].

2

Two main challenges are facing the achievement of large wind power penetration to the

grid. First, the cost of wind power generation must be reduced so that it can be competitive to

that of conventional power generation. Second, grid-connected wind turbines must meet grid

interconnection regulations, known as grid codes. These regulations are generally defined by grid

operators so that the integration of wind power does not deteriorate the reliability and the power

quality of the system.

There has been a significant evolution in the wind power generation technology during

the past three decades. In the early days of wind power grid integration, wind turbines were

small-sized (Figure 1.1), simple and self-regulated systems. These wind turbines operated at a

fixed speed dictated by the grid frequency and the stall effect was used to regulate their output

power during high winds. Because larger wind turbines have better energy-capture, the typical

size of grid-connected wind turbines has grown significantly as shown in Figure 1.1 [2].

Furthermore, with the technological advances and the reduction in costs of power electronic

drives and position actuators, most grid-connected wind turbines of today are equipped with

power converters and pitch angle servo systems. The power converter allows variable speed

operation of the wind turbine by decoupling its rotational speed from the constant grid

frequency. This allows the wind turbine to operate at maximum efficiency for a wide range of

wind speeds. The pitch servomechanism is used to rotate the wind turbines’ blades along their

longitudinal axes and to control the aerodynamics of the blades.

Advances in wind power generation technology have affected wind turbine control

systems. As wind turbines are growing in size, their structures tend to be more flexible. Since

wind turbines are operating under highly fluctuating wind profiles, the control system must be

designed to mitigate mechanical loads affecting the wind turbine components. Furthermore, the

3

widespread use of power converters and pitch actuators offers great flexibility and controllability

over the wind turbine behavior in terms of energy capture, active power and reactive power

control. However, this comes with increased complexity of the control system.

1985 1990 1995 2000 2005

Football Field

Boeing 747

H 43 m H 54 m H 80 m H 104 m H 114 m

100 kW

Ø 20 m

500 kW

Ø 40 m

800 kW

Ø 50 m

2000 kW

Ø 80 m

5000 kW

Ø 124 m

Figure 1.1 Evolution of grid-connected wind turbines in terms of power rating, hub height

H, and rotor diameter . The wind turbines are drawn to scale and they are compared with

the size of Boeing 747 and an American football field [2, 7].

Control systems play a very important role in modern wind energy conversion systems

(WECSs). A well-designed WECS control system can reduce the cost of wind energy generation

by:

maximizing the wind turbine’s generation efficiency and increasing its output energy

mitigating mechanical loads resulting in increased life of the installation

4

ensuring good power quality and meeting grid codes without needing extra electrical

equipment to enhance these aspects

Clearly, control systems have a direct impact on the cost of energy produced.

In the past few years extensive research work has been done to develop effective wind

turbine control systems [2, 8-13]. The main challenge is the presence of a number of conflicting

control objectives in addition to the complexity of the wind energy system which involves

electrical, mechanical and aerodynamic subsystems. Currently, simple Proportional Integral

Derivative (PID)-based control strategies are typically implemented in most industrial wind

turbines [2, 14]. These strategies do not fully exploit the control capabilities of modern WECSs.

Broadly speaking, the research objective of this thesis is to develop new control strategies that

can effectively realize most of the control objectives of a modern WECS. More detailed thesis

objectives are stated in §1.3.

This chapter is organized as follows. Modern grid-connected wind turbines are

introduced in §1.1. Wind turbines control systems and challenges are described in §1.2. The

thesis scope and objectives are stated in §1.3, and §1.4 provides an outline of the thesis.

1.1 Background

1.1.1 Overview of wind energy conversion systems

A wind energy conversion system is a system that converts the kinetic energy contained

in the incoming air stream into electrical energy. This conversion occurs in two stages. The first

stage occurs at the wind turbine blades which convert the kinetic energy stored in the wind into

mechanical power. Then, in the second stage, an electrical generator converts the harvested

mechanical power into electricity.

5

A typical three blade Horizontal Axis Wind Turbine (HAWT) is shown in Figure 1.2 [12,

15], and its internal structure and major components are illustrated in Figure 1.3. A HAWT has

the following major components.

1. Wind turbine rotor

2. Nacelle

3. Tower

4. Yaw mechanism

The wind turbine rotor is composed of the wind turbine blades, where the aerodynamic

power conversion takes place, and the hub, where the blades and the low-speed shaft are

attached. Furthermore, most modern wind turbine rotors are equipped with pitch servos, inside

the hub, that rotate the blades along their longitudinal axes to control the aerodynamic behaviour

of the blades.

The nacelle of a HAWT contains the power transmission system, also known as the drive

train, the electric generator, control subsystems and some auxiliary elements such as cooling and

braking systems.

The drive train transmits the mechanical power captured by the rotor to the electric

generator. It consists of a low-speed shaft coupled to the rotating hub, a gear box (speed

multiplier) that increases the low rotational speed of the rotor to a higher speed suitable for the

electric generator, and a high-speed shaft driving the electric generator.

The electric generator converts the mechanical power transmitted by the drive train into

electric power supplied to the grid. Wound Rotor Induction Generators (WRIG), Squirrel Cage

Induction Generators (SCIG) and Permanent Magnet Synchronous Generators (PMSG) have

been used successfully with HAWTs.

6

Figure 1.2 A utility-scale horizontal axis wind turbine [12].

GeneratorGear box

brake

Low- speed

shaft

High- speed

shaft

Nacelle

Tower

Yaw mechanism

Blade

Hub

Rotor

Aanemometer

Wind

vane

Figure 1.3 Components of a horizontal axis wind turbine [12].

Rotor

Nacelle

Tower

Hub

Blade

7

Typically, an anemometer and a wind vane are installed on top of the nacelle to measure

the wind speed and wind direction at the hub height. It should be noted that the anemometer

measurements do not represent the actual wind speed seen by the rotor as they are distorted by

the rotor itself. However, they are commonly used in the general operation of the wind turbine

such as start-up and shutdown.

The tower structure holds the nacelle and the rotor at relatively large heights, typically

from 70-120 m [16], where the wind speeds are higher than near the ground. Tubular steel and

concrete towers are used with large wind turbines.

All modern HAWTs are equipped with a yaw mechanism which consists of electric

motors and gears that allow the nacelle to rotate around the tower axis to keep the rotor plane

perpendicular to the wind direction in order to maximize the power extraction.

1.1.2 Wind turbines classification

Large scale grid-connected wind turbines, with 1-5 MW power rating, can be

distinguished according to their operation, control principles and components. They can be

classified as [17]:

1. (i) Fixed-speed wind turbines, operating at constant speed very close to the synchronous

speed specified by the grid frequency (60 Hz); or (ii) variable-speed wind turbines, where

the turbine speed can be controlled to operate within a large range, both below and above

the synchronous speed.

2. (i) Fixed-pitch wind turbines where the the rotor blades are bolted to the hub at a certain

pitch angle that does not change dynamically; or (ii) variable-pitch wind turbines where

the blades pitch angle are controlled to limit the turbine power for high wind speeds.

8

3. Wind turbines having (i) a gearless drive train (direct-drive system) [18], where the

turbine rotor is coupled directly to the generator, or (ii) a conventional drive train

consisting of a low-speed shaft, a gear box, and a high-speed shaft.

4. Wind turbines equipped with (i) squirrel cage induction generators, (ii) wound rotor

induction generators, (iii) wound rotor synchronous generators, or (iv) multi-pole

permanent magnet synchronous generators.

5. Wind turbines that are (i) directly connected to the grid without a power electronic

converter, or (ii) connected to the grid via a power converter.

1.1.3 Wind turbines with doubly fed induction generators

Among the many possible wind turbine configurations, Variable-Speed Variable-Pitch

(VSVP) wind turbines are the most common. Especially, wind turbines equipped with Doubly

Fed Induction Generators (DFIGs) are currently the most used configuration for wind power

generation [19-21]. They represent approximately 50% of wind power installations worldwide

[19].

A DFIG-based WECS, shown in Figure 1.4, consists of a three-bladed wind turbine rotor

coupled to a WRIG through a gear box, a low-speed shaft and a high-speed shaft. The stator of

the WRIG is directly connected to the grid while the rotor winding is connected to the grid via a

partial scale power converter. The power converter consists of the Rotor-Side Converter (RSC),

a dc link and the Grid-Side Converter (GSC). The converter allows the interface between

variable frequency voltages at the DFIG rotor windings and constant frequency voltages at the

grid. Since the power converter is connected to the rotor circuit, only a portion of the DFIG

generated power flows through it. For this reason, the converter is typically rated at 30% of the

generator nominal power. The DFIG concept is attractive and popular from an economic point of

9

view because it offers variable-speed operation at a reduced cost of the partially rated power

converter. The DFIG power converters can also be controlled to perform reactive power

compensation [22]. Finally, most DFIG wind turbines are pitch-controlled [23]. This allows high

performance output power limitation during high wind speeds.

WRIG

ac/dc dc link

Grid

dc/ac

Gear Box

rotor

RSC GSC

Figure 1.4 Wind turbine with a doubly fed induction generator.

1.2 Wind turbines control

1.2.1 Control modes

A VSVP wind turbine (WT) has two operating regions with different control objectives,

namely the partial load regime and the full load regime, shown in Figure 1.5.

The partial load regime includes all wind speeds between the cut-in wind speed, ,

and the rated wind speed, , defined as the lowest wind speed at which the system rated

power is achieved. In this region, wind speeds have relatively low values and the power that can

be captured from the wind is less than the rated wind turbine power. Therefore, the main control

objective in this region is to maximize the energy capture. This is achieved by varying the

turbine rotational speed in proportion with the operating wind speed, and by fixing the blades

pitch angle at its optimum value.

When the wind speed is above and below the cut-out wind speed , the turbine is

operating in the full load regime. In this region, the available wind power is higher than the rated

10

wind turbine power. The main control objective is thus to regulate the wind turbine’s output

power and rotational speed at their rated values. This is achieved by adjusting the pitch angle of

the blades to shed excess input power.

Figure 1.5 Output power curve of a 1.5 MW wind turbine. In partial load region ( ), the WT power capture should be maximized, and in full load region ( ), the output power of the WT should be regulated at its rated value.

For wind speeds lower than , the available wind power is much smaller than the wind

turbine’s operational losses, and for wind speeds higher than , the available wind power is

much higher than the wind turbine’s design limits. In both situations, the wind turbine is shut

down.

1.2.2 Control system hierarchy

Due to large differences in the time scales of the electrical and mechanical dynamics (the

electrical dynamics are much faster than the mechanical ones) [22], the DFIG-based WECS has a

multilayer control structure as shown in Figure 1.6. Three control levels can be identified.

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

v, m/s

WT

ou

tpu

t p

ow

er,

MW

Partial Load Full Load

vci

vrat

vco

11

WECS

Optimization

Turbine

Controller

GSC

Controller

RSC

Controller

Generator

Controller

*WTGP

β*Tg

*

*dcV*

GCQ*sQ

WRIG

RSC

Grid

Gear Box

Rotor

GSC

vr vC

*rv *

Cv

*g

Figure 1.6 Control system hierarchy of a DFIG-based WECS.

WECS optimization level

This is the highest control level that is mainly responsible for operating the wind turbine

at its optimal operating trajectory. At this control level, a Maximum Power Point Tracking

(MPPT) algorithm [12] is typically implemented to calculate the generator speed set point, ,

so that the energy conversion efficiency is maximized in the partial load regime. In the full load

regime, this control level generates , and the wind turbine’s output power set point

, so that the generator speed and power are regulated at their rated values

and , respectively.

Turbine controller

This controller supervises the pitch control system and the generator controller. The task

of the turbine controller is to control the generator speed and output power to follow their desired

12

values dictated by the WECS optimization level. This is achieved by manipulating the pitch

angle set point, , and the generator torque set point, .

Generator controller

All generator electrical control loops, that are characterized by a much faster dynamic

response compared to the turbine control loops, are implemented at this control level. The

generator controller aims to control the generator torque and reactive power independently. It

consists of the RSC controller and the GSC controller.

The primary task of the RSC controller is to control the generator torque to quickly track

that is generated by the wind turbine control level. Another task is to control the reactive

power flow between the stator and the grid to follow a certain desired reference [24-26].

The GSC controller is used to regulate the dc link voltage at a certain rated value

. Maintaining a constant dc link voltage ensures that the active power is

appropriately transmitted between the generator rotor and the grid. A secondary task of the GSC

controller is to control the reactive power flow between the GSC and the grid to follow a certain

desired reference [22].

1.2.3 Control challenges

The main objective of a wind turbine’s control system is to maximize the cost-

effectiveness of wind power generation [12, 13]. Based on this goal, many partial control

objectives can be stated as follows:

Maximizing the energy capture

Reducing mechanical loads, especially the ones resulting from drive train torque

pulsations causing costly gearbox failures

13

Reducing the wind turbine’s down time by increasing the system’s robustness against

changes in wind turbine dynamics resulting from faults, wear, debris buildup on the

blades, or other causes.

Enhancing the power quality by smoothing the output power and reducing voltage

fluctuations at the point of interconnection to the grid

Ensuring compliance with recent grid codes and providing grid support during severe

network disturbances such as short circuits.

The achievement of all these objectives in the presence of a highly fluctuating input wind power

is not an easy task. By considering the multiple-input multiple-output (MIMO) nature of the

system, the system nonlinearity, and the presence of physical constraints on the system’s

variables, such as the ones on the pitch angle and output power, the control design task becomes

more challenging. In the following, the design challenges associated with the wind turbine and

generator control levels are detailed.

1.2.3.1 Wind turbine control level

In the partial load regime, the main control challenge is to design a controller that can

maximize the conversion efficiency while minimizing dynamic loads. It is stated in [9, 10, 27,

28] that tight tracking of the wind turbine optimal operating trajectory in the presence of

turbulent winds is usually associated with large generator torque variations, high mechanical

stresses, and severe fatigue loading. This can lead to premature and costly failures of critical

wind turbine components such as the gear box. In general, the control system should be designed

to achieve a suitable compromise between the energy maximization and load minimization

The main control challenge in the full load regime is to regulate the turbine output power

and speed in the presence of severe fluctuations in the turbine input power caused by erratic

14

variations in the wind speed. Input power fluctuations can lead to severe torque pulsations in the

drive train, and large fluctuations in the electric power supplied to the grid. These, in turn, can

cause a reduction in the WECS life time and the power quality.

When the wind turbine is operating at wind speeds around the rated value, the wind

turbine is said to operate in the transition region. In this region, the wind speed fluctuates around

the rated value, and the partial load and full load wind turbine controllers must be continuously

switched. Several studies indicate that undesirable drive train transient loads and power

overshoots can occur as a result of this switching [13]. For some turbines, it was found that the

maximum mechanical damage occurs during this transition [2].

Many control techniques have been proposed to control WECSs in the partial load regime

[10, 13, 22, 28-35]. The design of classical Proportional Integral (PI) controllers is described in

[10, 22, 29, 30]. To cope with the system nonlinearity and to allow a trade-off between energy

maximization and load reduction, the use of a gain-scheduled Linear Quadratic Gaussian (LQG)

controller is proposed in [28]. A gain-scheduled controller is suggested in [13] and [31]. The

use of nonlinear control methods including feedback-linearization and sliding-mode control are

suggested in [32, 33] and [35], respectively. In [34], adaptive control techniques are applied in

the design of wind turbine controllers.

Many papers focusing on the control of VSVP wind turbines operating in the full load

regime have appeared recently [10, 36-38]. Most of the work reported ignores the multivariable

nature of the problem [10, 37, 38]. Classical PI controllers are used in [10]. A PI controller in the

power control loop and an adaptive self-tuning regulator in the speed control loop are proposed

in [37]. A state feedback power regulator designed using pole placement techniques is suggested

in [38]. A multivariable gain-scheduled controller is proposed in [36].

15

Most of the work reported in the literature uses decentralized control structures, where

both the wind turbine power and speed are controlled independently. Furthermore, the design

method is typically described for either the partial load or the full load regime. It is discussed in

[13] and [36] that by recognizing the multivariable nature of the problem and designing a MIMO

controller, a much superior performance can be obtained as compared to the decentralized

approaches. To the knowledge of the author, the only work that provides a common framework

for designing a multivariable controller that can work in both partial and full load regimes is

found in [13] and [39], where a multivariable gain-scheduled controller is proposed. Finally,

it should be pointed out that no control method that systematically eliminates torque and power

overshoots occurring in the transition region is currently available.

1.2.3.2 Generator control level

Large penetration of wind power into the grid poses new challenging requirements on the

wind turbine’s generator control system. Currently, the generator controller is not only required

to provide high performance control over its active and reactive power generation, but it also

should meet the requirements imposed by grid codes. One of the most challenging grid codes

from the generator control perspective is Fault Ride-Through (FRT) capability, also known as

Low Voltage Ride-Through (LVRT) [11, 19, 20, 40, 41]. This requirement specifies the desired

behavior of wind turbines during and immediately after grid faults, in order to maintain good

dynamic performance and stability of the power network.

When an external grid fault occurs, a large voltage dip typically is produced across large

areas on the network. Large voltage dips at the WECS terminals lead to high currents flowing in

the generator and power converters. This might lead to the destruction of the converters unless

protective actions are taken [42-44]. This problem is very critical for DFIG-based wind turbines

16

because their power converters are partially rated at 30% of the rated generator power and thus

they are more sensitive to high currents compared to generators with fully rated converters.

A common approach used to protect the DFIG’s RSC is to disable the converter and to

divert the high rotor currents into a parallel resistive circuit known as crowbar. Once rotor

currents reach safe values, the crowbar is disconnected and the RSC is reactivated and the DFIG

resumes normal operation. According to recent grid codes, this approach is currently not

accepted. Most grid codes currently specify that for a wind turbine to be FRT capable, it must

remain connected to the grid during faults and be capable of injecting reactive power to support

the grid in recovering its rated voltage [45]. Clearly, when the crowbar approach is used, the

controllability of the DFIG is transiently lost when the RSC is deactivated. Even worse, during

this period, the DFIG acts as a conventional induction generator that consumes reactive power,

further deteriorating the voltage recovery process for the system.

A huge research effort has recently been undertaken in order to come up with new

control/protection strategies to ensure FRT capability for DFIGs [19, 20, 40, 41, 46-51]. One

approach is to modify the conventional PI-based RSC control algorithm [26] such that FRT is

achieved without using additional protection hardware. The use of fuzzy logic controllers to

control the rotor currents is suggested in [47]. Stator flux demagnetization using RSC control is

proposed in [46]. The addition of feed-forward compensation terms to the conventional

decoupling current controller is described in [48]. The use of these approaches can provide

successful FRT for DFIGs when the magnitude of the voltage dip is small or moderate [43].

It is shown in [43] that FRT cannot be met solely by the DFIG control when the voltage

dip is severe. This motivated the development of new converter protection strategies [20, 40, 41,

51] to be used instead of the classical crowbar protection such that the RSC is not disconnected

17

from the DFIG rotor and the control over the DFIG is maintained even during severe voltage

dips. In all these protection schemes, the conventional RSC control algorithm in [26] is used to

control the DFIG. Another solution for the FRT problem proposes the installation of a suitably

sized static synchronous compensator (STATCOM) at the wind farm to provide reactive power

support during grid faults [49, 50]. This approach is relatively expensive.

1.3 Scope and objectives

This thesis will focus on developing advanced control strategies that improve the cost-

effectiveness of grid-connected wind turbines. Utility-scale HAWTs with power ratings larger

than one MW are considered in this thesis. The focus is on the DFIG wind turbine configuration

because it is the most commonly used one today. However, it should be noted that all control

strategies developed in thesis are applicable to any VSVP wind turbine configuration.

The control strategies developed in this thesis correspond to the turbine control and

generator control levels shown in Figure 1.6. Standard MPPT techniques described in the

literature will be used with the developed control strategies to test the overall wind turbine

control system.

The thesis objectives can be summarized as follows.

Wind turbine control level

Develop a multivariable wind turbine control strategy that provides the desired WECS

performance over its whole operating region, i.e. partial load, transition and full load

regions.

In the partial load regime, the wind turbine controller should maximize the energy

capture without increasing mechanical loads in the drive train.

18

In the transition region, the control design method should systematically eliminate all

power and torque overshoots above the rated values. It should also allow smooth

transition between partial and full load operation.

In the full load regime, the control system should simultaneously control the generator

torque and blades pitch angle to smooth the output power, reduce voltage fluctuations at

the point of interconnection to the grid, and reduce mechanical loads in the drive train

without increasing the pitch activity.

The developed wind turbine control strategy should keep all critical WECS variables

within their safe operating limits. It should also provide good dynamic performance over

the whole operating wind speed range despite the nonlinear WECS dynamics and the

continuous variation in the wind speed.

The developed control strategy should be easily tuned to achieve the desired trade-off

between different conflicting objectives such as maximizing energy capture, minimizing

loads, reducing pitch activity, and reducing power and voltage fluctuations.

For applications requiring increased reliability such as offshore wind turbines, the control

system should be capable of adapting to changes in the wind turbine dynamics. This

“self-tuning” feature allows the controller to be resilient against unknown changes in the

system dynamics that might occur during WECS operation.

Generator control level

Develop a generator control strategy that allows high performance control over the

generator torque and the generator reactive power, as well as ensuring FRT requirement

according to recent grid codes.

19

1.4 Thesis outline

This thesis is organised in eight chapters followed by four appendices.

Chapter 2 represents a general introduction to wind energy conversion systems. It

explains how the wind power is converted into useful mechanical power by the wind turbine

rotor, and the main aerodynamic concepts behind this conversion. These concepts are used to

outline the main principles that are used in controlling a WECS. An overview of different

commercial wind turbine concepts is also provided. General wind turbine control objectives are

summarized in this chapter

Chapter 3 details the dynamic WECS model that is used in this thesis. A wind speed

simulator that emulates the effective wind speed seen by the rotor is described. The modeling of

different WECS subsystems including the aerodynamic, the pitch actuator, the drive train and the

electrical subsystems is explained. Finally, the modeling of the WECS grid interconnection is

described.

Chapter 4 gives an overview of Model-based Predictive Control (MPC) techniques. MPC

is the main tool that is used in this thesis to develop effective wind turbine control strategies.

This chapter summarizes main MPC concepts and some selected results that are relevant for this

work.

In Chapter 5 a novel multivariable control strategy based on MPC techniques for the

control of variable-speed variable-pitch wind turbines is proposed. The proposed control strategy

is described for the whole operating region of the wind turbine, i.e. both partial and full load

regimes.

A new adaptive predictive wind turbine control strategy is proposed in Chapter 6. The

proposed strategy uses a model predictive control algorithm with its predictor matrices

20

continuously updated using recursive subspace identification techniques. Offset-free tracking of

the proposed controller is proved. The application of this algorithm in the design of a wind

turbine control strategy is provided.

In Chapter 7, a novel RSC control strategy that ensures FRT for DFIG-based wind

turbines according to recent grid codes is proposed. The proposed strategy uses an MPC RSC

controller incorporating most of the DFIG’s constraints and a Dynamic Series Resistance (DSR)

protection scheme. Different MPC implementation alternatives are compared.

Chapter 8 summarizes the thesis contributions and offers suggestions for further

developments and improvements.

Appendix A provides the standard design method of the DFIG generator control system.

Appendix B describes the per unit representation of the DFIG model. Different wind turbine

control schemes and MPPT algorithms commonly used in the literature are reviewed in

Appendix C. Finally, Appendix D contains all WECS parameters that are used in all simulation

studies. These are the parameters of a 1.5 MW industrial General Electric (GE) wind turbine [14,

52].

21

Fundamentals of Wind Energy Conversion Systems Chapter Two:

To design an effective control system, it is important to understand the process to be

controlled. This chapter gives the necessary background for understanding the fundamentals of

WECSs. The source of power for a WECS, the wind, is described in §2.1. The basic

aerodynamics concepts behind the conversion of the wind power into mechanical power by the

wind turbine rotor are explained in §2.2. The main principles used for designing the wind turbine

control system are provided in §2.3. In §2.4, an overview of different commercial wind turbine

concepts is provided. VSVP wind turbines are reviewed in §2.5, and §2.6 outlines the main

objectives of a wind turbine control system.

2.1 The wind

2.1.1 The source of the wind

Wind energy is basically an indirect form of solar energy. The equatorial region receives

more radiation from the sun than do the polar regions. As the air at the equatorial region gets

warmer and lighter, it rises up into the atmosphere to a certain attitude and then spreads toward

the north and the south. This creates a low pressure region near the equator which attracts the

cooler air from the poles. The air flow circulating between the equator and the poles is diverted

by Coriolis forces resulting from the rotation of the earth. These large-scale air flows that are

found at higher altitudes in the atmosphere constitute the geostrophic winds, or more commonly

the global wind [4, 13]. It can be seen that the wind is basically generated due to the pressure

gradient resulting from the uneven solar heating of the earth’s surface.

Winds that flow near the surface, up to a height of 100 m, are known as local winds. The

velocity, direction and pattern of local winds are affected by many local factors such as the

surface roughness, and the presence of obstacles, seas, large lakes and mountains. For example,

22

the land in a region close to the sea gets heated faster than the sea surface during day time.

Winds, similar in nature to the geostrophic wind, flow from the sea toward the land. This is

known as the sea breeze. During the night, the process is reversed and the wind blows from the

land to the sea, which is called the land breeze [4].

Wind turbines capture the energy of the wind close to the ground. In general, the wind at

a given site near the ground results from the combination of the geostrophic and local winds.

2.1.2 The power contained in the wind

The kinetic energy stored in a stream of air with mass and moving with a speed can

be expressed as:

(2.1)

The wind power stored in this stream when flowing through an area is given by the kinetic

energy of the flowing air mass per unit time. That is,

(2.2)

where is the mass flow rate of the air.

The mass flow rate can be expressed using the air density and the wind speed as:

(2.3)

Hence, the wind power can be expressed as:

(2.4)

In the case of a HAWT with a rotor radius, , the wind power can be written as:

(2.5)

23

From (2.5), one should note that the wind power is proportional to:

the density of the air ( kg/m3 at 15

oC and normal pressure),

the area swept by the rotor (or the square of rotor radius for a HAWT), and

the cube of the wind velocity.

It is clear that the wind speed has the most prominent effect on the power contained in the wind.

2.2 Wind turbine aerodynamics

2.2.1 Aerodynamics concepts

The conversion of wind power into mechanical power, occurring at the rotor of a HAWT,

is based on the aerodynamic forces produced when wind flows around airfoil shaped blades.

How wind flow produces forces and torques on the wind turbine is described in this section.

The basic aerodynamic principles behind the force production on the wind turbine blades

are similar to the ones for the wings of airplanes [15]. To understand these principles it is useful

to start with a simple case. Assume an airfoil section of a blade is fixed in a stream of wind,

flowing with speed, , as shown in Figure 2.1. Due to the airfoil shape, the wind stream at the

top of the airfoil has to traverse a longer path than the air stream at the bottom, leading to a

difference in velocities. According to Bernoulli's principle, this results in a difference in pressure

between the two sides of the airfoil and a lift force is produced in the normal direction of the

wind flow. In addition, the wind has a dragging effect on the airfoil and it produces a drag force

in the direction of the wind. Figure 2.2 illustrates these forces.

For HAWTs, the Blade Element Theory (BET) is commonly used to derive expressions

for the forces and torques produced on the wind turbine, and the power captured by the rotor.

Here, the essential concepts of aerodynamic power conversion based on BET are explained. For

more details and practical aspects, see [12, 53].

24

Figure 2.1 Wind flow over an airfoil section.

Trailing edge

Leading

edgeChord

fL

fD

vWind

Angle of

attack

Figure 2.2 Aerodynamic forces produced on an airfoil fixed in a stream of wind.

The BET is based on dividing the blades of a HAWT into a number of transversal

sections along each blade. Each blade section has a thickness . It is assumed that the

geometrical and aerodynamic properties of each section of the blade are known. The main idea is

to calculate the lift and drag forces on each section and integrate over the blade span to obtain the

total force acting on the blade.

The airfoil blade section of a HAWT, located at a radius measured from the hub, is

shown in Figure 2.3. In contrast with the fixed airfoil section shown in Figure 2.1, the airfoil

section of a HAWT is rotating with a rotational speed, . Therefore, the lift and drag forces

acting on the blade element of a HAWT should be based on the wind speed as seen by the airfoil,

called relative wind speed, . This relative speed is given by the vector sum of the axial wind

velocity, , and the negative of the tangential velocity of the blade section, , as shown in

the left part of Figure 2.4 [53].

25

va

ωt r

ωt

rdr

Figure 2.3 An airfoil section located at a distance from the hub.

va

-ωt rvrel

vrel

Wind

ChordDirection

of blade

movement

fD

fL

fa

ft

βα

fnet

Plane of

rotation

ϕ

ϕ

Figure 2.4 Aerodynamic forces produced on an airfoil section of a HAWT [13].

Due to the airfoil shape of the blade element, a lift force that is normal to the the

relative wind, and a drag force that is in the direction of the relative wind are produced. Their

26

resultant force can be decomposed into a tangential component that acts in the direction

of rotation and an axial component that is normal to the rotor’s plane of rotation, as shown in

Figure 2.4. The tangential force develops the turbine torque that causes the rotation of the

rotor and it produces useful work. The axial component, known as the axial thrust force, acts on

the wind turbine structure; thus the rotor, tower and foundations must be designed to withstand

this force [12, 13, 53].

The aerodynamic forces acting on the airfoil section shown in Figure 2.4 can be

calculated as follows. The lift and drag forces per unit length are given by [13]:

,

,

(2.6)

(2.7)

where is the chord length, is the lift coefficient and is the drag coefficient. The local

angle of attack, , known also as the angle of incidence, is shown in Figure 2.4 and it is defined

as the angle between the relative wind speed vector and the chord line of the airfoil.

The angle of attack is the key variable determining the aerodynamic behavior of the

wind turbine and the efficiency of power conversion [12]. This angle should be differentiated

from the pitch angle that is defined as the angle between the chord line of the airfoil section

and the plane of rotation, see Figure 2.4. The pitch angle is a design parameter for the blade and

it can be changed during the operation of pitch-controlled wind turbines; while the angle of

attack is an aerodynamic parameter that depends on the pitch angle, the turbine rotational speed

and the wind speed. It can be seen from Figure 2.4 that the angle of attack decreases with

increasing the pitch angle, increasing the turbine rotational speed, or decreasing the wind speed.

27

From Figure 2.4, the torque producing force and the axial thrust force can be

computed by resolving and into their axial and tangential components as given by (2.8)-

(2.9), where is the angle between the relative wind speed and the rotor plane.

(2.8)

(2.9)

The torque per unit length produced on the airfoil section is given by:

(

) (2.10)

The turbine rotational torque and the total thrust force acting on the rotor can be computed by

integrating and over the rotor radius. The power captured by the turbine rotor is estimated

by multiplying the torque by . Finally, based on (2.10) and Figure 2.4, it should be pointed out

that the turbine aerodynamic torque depends on three main variables, namely, , and .

To increase the tangential force, turbine torque, and power capture, (2.8) and Figure 2.4

suggest that the lift force should be maximized and the drag force should be minimized. This can

be achieved by placing the airfoil at an optimum angle of attack at which the ratio is

maximized. Figure 2.5 shows typical curves for and of an airfoil section of a

HAWT [12], and the ratio is plotted as a function of in Figure 2.6. It can be seen that

for small values of , is linearly proportional with while is almost constant at very

small values. The lift reaches its maximum at a certain value for an airfoil ( in this

example). When the angle of attack increases above this critical value, the lift decreases steeply

28

while the drag increases with . This phenomenon is known as the stall effect and it is said that

the airfoil is stalled [12]. This behaviour results from the separation of airflow from the airfoil

that occurs when exceeds this critical value and the airflow is no longer laminar.

Figure 2.5 Typical Lift and drag coefficients of an airfoil [12].

Figure 2.6 Typical of an airfoil [12].

0 5 10 15 20 250

0.5

1

1.5

2

, o

CL

CD

0 5 10 15 20 250

10

20

30

40

50

, o

CL /

CD

Optimal angle of attack

Stall

Feathering

29

The stall region is emphasized in Figure 2.6. It should be noticed that, in this region, a

slight increase in the angle of attack results in a large decrease in the ratio, the turbine

torque and the captured power. In contrast with this region, for small values of , changes in the

angle of attack result in smoother changes in the ratio, the turbine torque and the captured

power.

2.2.2 The power coefficient and the tip speed ratio

The power coefficient and the tip speed ratio of a HAWT are defined as follows [17].

Definition 2.1: Power coefficient,

The power coefficient is the ratio between the mechanical power captured by the wind turbine

rotor and the theoretical power available in the wind, during steady state operation. That is,

(2.11)

Definition 2.2: Tip speed ratio,

The tip speed ratio is the ratio between the tangential velocity of the tip of a ind turbine blade

and the wind speed of the incoming stream. That is,

(2.12)

The power coefficient represents the efficiency of the rotor in converting the wind power

into mechanical power. Unfortunately, the power conversion process must involve some loss. As

the wind speed behind the turbine rotor cannot be zero, otherwise the wind flow would be

completely blocked by the rotor; thus the conversion efficiency must be less than unity. In fact, it

30

was shown that the power coefficient of any wind turbine cannot exceed a value of

.

This value is known as the Betz limit [15, 17]. Practically, the maximum power coefficients of

modern commercial HAWTs are around [15].

The power coefficient of a HAWT is usually described as a function of the tip speed ratio

and the pitch angle [23, 37, 54, 55]. Variation in the power coefficient of a typical HWAT,

as a function of the tip speed ratio and the pitch angle, is shown in Figure 2.7 . It can observed

that:

The power coefficient has a unique maximum which occurs when the pitch angle

and the tip speed ratio are at certain optimum values, and , respectively.

Increasing the pitch angle generally causes a decrease in the power coefficient. This

agrees with Figure 2.6, as increasing the pitch angle results in a decrease in the angle of

attack and ratio.

Figure 2.7 Typical power coefficient variations of a HAWT [55].

0

5

10

15 05

1015

20250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

CP(

,)

CP(

o,

o)

31

The power coefficient is commonly used to calculate the mechanical power captured by

the rotor and the turbine torque using (2.13), and (2.14), where the torque coefficient

is defined in (2.15).

(2.13)

(2.14)

(2.15)

2.3 Wind turbine control principles

From §2.2, many insights on the control system of a WECS can be drawn. The main control

principles of a WECS are summarized below.

2.3.1 Maximizing the power of a WECS

One of the most important control objectives of a WECS is to maximize its energy

capture. Figure 2.7 shows that to operate the WECS at its maximum power coefficient

(efficiency), , the following conditions should be satisfied.

The pitch angle should be fixed at its optimal value (typically very close to 0o).

The tip speed ratio should be fixed at its optimal value (typically between 6 and 8 for

three-bladed HAWT [15]). This is achieved by continuously varying the turbine speed

to match variations in the wind speed such that the ratio

is kept constant at .

Clearly, WECSs that are capable of operating at different speeds, known as variable-speed

WECSs, can operate at maximum efficiency over a wide range of wind speeds. On the other

hand, fixed-speed WECSs operate at maximum efficiency for only one value of the wind speed.

When the wind turbine is operating at , , and consequently ,

the wind turbine is said to be working at the Optimal Regime Characteristic (ORC) [12, 28]. At

32

the ORC, the turbine power and torque are given by (2.16) and (2.17), respectively, where is

defined in (2.18). Figure 2.8 shows the ORC in the and planes.

(2.16)

(2.17)

(2.18)

(a)

(b)

Figure 2.8 Optimal regime characteristic (red) and wind turbine power and torque curves

(blue) at different wind speeds and shown on (a) the plane and (b) the

plane [13].

0 0.5 1 1.5 2 2.50

200

400

600

800

1000

1200

1400

t, rad/s

4m/s

5m/s

6m/s

7m/s

8m/s

9m/s

10m/s

ORCPt,

KW

0 0.5 1 1.5 2 2.50

100

200

300

400

500

600

700

t, rad/s

4m/s5m/s

6m/s

7m/s

8m/s

9m/s

10m/s

ORCTt,

KN

m

33

Remark 2.1: Intuitively, if the wind turbine rotates at low speed compared to the wind speed,

most of the wind will flow through the rotor without interacting with the blades and thus without

energy transfer. On the other hand, if the wind turbine turns very quickly, the rotor will almost

act as a solid wall obstructing the wind flow and reducing the energy extraction. This is an

intuitive explanation on why there is certain optimal ratio between the rotor speed and the wind

speed at which the rotor efficiency is maximized [4].

2.3.2 Limiting the power of a WECS

At wind speeds higher than , the power that can be captured from the wind exceeds

the design limits of the wind turbine components and the rated power of the generator.

Therefore, the aerodynamic power must be reduced and limited to its rated value by reducing the

rotor efficiency and shedding excess power. This can be achieved by controlling the angle of

attack.

Figure 2.6 suggests two possible approaches to reduce the ratio and the power

conversion efficiency. The first approach, known as blade feathering, is based on decreasing the

angle of attack by increasing the turbine rotational speed and/or pitch angle. The second

approach, known as blade stall, involves increasing the angle of attack by decreasing the

rotational speed and/or pitch angle to low values [12].

2.3.2.1 Passive-stall control

This approach is the simplest, most robust and cheapest control method that can be used

to limit the power of a HAWT during high winds. It is typically used with fixed-speed fixed-pitch

wind turbines. No pitch actuator is used to adjust the blades’ pitch angle. The rotor blades are

designed such that the rotor power is self-regulated using the aerodynamic stall occurring with

increased wind speed and constant rotor speed.

34

Figure 2.9 illustrates the concept of the passive-stall method [15]. When the rotor speed

and the blade’s pitch angle are kept constant, the angle of attack increases with the increase in

the wind speed. Beyond a particular angle of attack, , air flow separation occurs and the blade

is stalled. This is accompanied with a reduction in the lift force and the ratio (see Figure

2.6); and consequently, the captured rotor power is reduced to a safe value.

β

α

v

-ωt r

vrel

α

v

-ωt r

vrel

operational

wind speed

plane of

rotation

Flow

separation

(stall)

β

high

wind speed

plane of

rotation

Figure 2.9 Aerodynamic stall at a rotor blade with fixed pitch angle at increasing wind

speed and constant rotor speed [15].

Despite its simplicity, passive-stall control has the following disadvantages [15, 23]:

The stall effect is characterized by a large increase in the drag coefficient (see Figure 2.5)

and drag forces. This yields increased thrust forces causing severe aerodynamic loads on

the wind turbine structure that should be accounted for in the wind turbine design.

The power regulation quality of the passive-stall control method is rather poor. This can

lead to poor power quality.

Blades with passive-stall cannot assist in the start-up or an emergency stop of the WT.

35

Due to these shortcomings, passive-stall control is not commonly used these days for utility-scale

wind turbines [13, 15].

2.3.2.2 Pitch control (pitch-to-feather)

In this approach, power limitation in high winds occurs by rotating the blades using a

pitch actuator towards the feathered position. It can be seen from Figure 2.10, that this results in

a decrease in the angle of attack. From Figure 2.5, decreasing the angle of attack decreases the

whereas remains low resulting in a decrease in the power conversion efficiency.

Compared with stall-based control methods, pitch-controlled wind turbines allow for

much steadier operation and better regulation of the turbine power and speed in high winds [4,

15, 23, 36]. Furthermore, the thrust forces produced during power limitation by pitch-to-feather

control method are significantly reduced. This is a significant advantage, as smaller aerodynamic

loads are applied to the wind turbine structure. This comes with the price of using a pitch

actuator that must be capable of rotating the blade over a relatively wide range of pitch angles

(typically within 0°-45°) and a more complicated control system. Despite that, almost all large

grid-connected wind turbines use the pitch-to-feather control method [13, 17, 23, 36].

2.3.2.3 Active-stall control (pitch-to-stall)

In this approach, rotor power is limited by pitching the blades in the opposite direction of

the pitch control case, thus reducing the pitch angle to even negative values[13, 15]. Figure 2.10

shows that this has the effect of increasing the angle of attack to large values, higher than ,

where the blades are forced to operate in the stall region. This method allows better power

regulation over the passive-stall method as the angle of attack can be continuously controlled to

effectively regulate the output. As a main attractive feature over the pitch-to-feather approach,

this method requires much smaller changes in the pitch angle and lower control effort to regulate

36

power. However, similar to the passive-stall method, significantly greater thrust loads are

produced on the wind turbine structure [15].

β

α

v

-ωt r

vrel

βα

v

-ωt r

vrel

β

α

v

-ωt r

vrel

Pitch-to-feather

Pitch-to-stall

feathered

position

operational

position

plane of

rotation

Flow

separation

(stall)

Figure 2.10 Controlling the rotor power using pitch-to-feather and pitch-to-stall strategies

[15].

2.4 Commercial wind turbine concepts

Despite the large number of possible wind turbine topologies described in §1.1.2, there

are four main wind turbine concepts, shown in Figure 2.11, that are used in power systems

applications. These concepts are described below [17, 23, 56, 57].

2.4.1 Type 1 –Conventional fixed-speed SCIG

This wind turbine concept, also known as the “Danish concept” [25, 57], can be

considered as the first generation of grid-connected wind turbines [17]. The main configuration

of a Type 1 WECS is shown in Figure 2.11 (a). It consists of a three-bladed HAWT that drives a

SCIG through a multi-stage gear box, a low-speed shaft and a high-speed shaft. The SCIG is

directly connected to the grid via a transformer. Since a SCIG always consumes reactive power,

37

this configuration uses a capacitor bank for reactive power compensation. A soft starter is also

used to allow a smoother start-up of the SCIG. Although passive-stall control is commonly used

with this concept, pitch control and active stall control have also been applied [23].

This concept is characterized by its almost fixed-speed operation. Due to the direct

connection of the SCIG to the grid, the SCIG operates only in a narrow range around the

synchronous speed, , where is the grid frequency (60 Hz) and is the number

of pole pairs of the generator. Typically, the rotational speed is from 0% to 1% above the

synchronous speed.

Although this concept is relatively simple, robust and cheap, its fixed-speed operation has

severe drawbacks compared to variable-speed wind turbines as explained in §2.5.1 [58]. For that

reason, Type 1 WECS concept has witnessed a severe decline in the last decade in comparison

with variable-speed WECSs [13, 57, 58].

2.4.2 Type 2 – Wound rotor induction generator with variable external rotor resistance

This concept is similar to Type 1 WECSs except that a WRIG is used instead of the

SCIG. The rotor windings of the WRIG are accessible and they are connected to an additional

variable rotor resistance that is adjusted by an optically controlled power converter. The

resistance and the converter are mounted on the rotor shaft, thus eliminating the need for slip

rings which are costly and require maintenance. This concept uses the pitch control method for

power limitation. The connection of a Type 2 WECS is shown Figure 2.11 (b) [57, 58].

Type 2 WECSs are characterized by their limited variable-speed operation. The insertion

of controllable external resistance to the rotor varies the rotor effective resistance and the shape

of torque-speed characteristic of the IG. Consequently, the speed of the wind turbine can be

controlled from 0% to 10% above the synchronous speed [56].

38

SCIG

Grid

Soft

starterCapacitor

bank

Gear Box

(a)

Wind

turbine

ac/dc

WRIG

Grid

Resistor

Soft

starter Capacitor

bankWind

turbine

Gear Box

(b)

WRIG

ac/dc dc link

Grid

dc/ac

Gear Box

Wind

turbine(c)

PMSG

ac/dc dc link

Grid

dc/ac

N

S

Wind

turbine(d)

Figure 2.11 The four main grid-connected wind turbine concepts: (a) Type 1 - conventional

fixed-speed SCIG, (b) Type 2 - WRIG with variable external rotor resistance, (c) Type 3 -

DFIG concept and (d) Type 4 – direct-drive PMSG with full-scale converter [17, 23].

39

2.4.3 Type 3 – Doubly fed induction generator with partial-scale converter

The main components of a DFIG-based wind turbine are described in §1.1.3. The use of

partially rated power converters in the DFIG concept allows limited variable-speed operation of

the wind turbine. The generator speed can be controlled by the partially rated converter to

operate in a wider range compared with Type 2 WECSs, depending on the size of the converter.

Typically, the speed range is from -40% to +30% around the synchronous speed [25, 58].

Fortunately, this range is convenient for wind energy applications since wind turbines operate in

a limited range of wind speeds as shown in Figure 1.5. Furthermore, reactive power

compensation is achieved by controlling the power converter [22]. All DFIG wind turbines use

the pitch control method for power limitation [23].

2.4.4 Type 4− Permanent magnet synchronous generator with full-scale converter

In this configuration, the generator is connected to the grid through a full-scale power

converter as shown in Figure 2.11 (d). Recent advances in, together with lower costs for, power

electronics make it feasible to use a power converter with the same rating as the wind turbine

[57]. Typically, PMSGs and the pitch control method are used in this configuration [58]. The

PMSG is designed to have a large number of poles so that it can rotate at low speeds similar to

the turbine rotor. Consequently, the PMSG is connected directly to the turbine rotor and the

gearbox is omitted in this concept. This is considered to be one of the main advantages of this

configuration since the gear box is the component that is most prone to failures and its removal

increases the system reliability. The price is an increased diameter, weight and volume of the

generator to achieve a large number of poles and torque [59].

40

Type 4 wind turbines have a full variable-speed capability. The generator speed is

completely decoupled from the grid frequency due to the use of the full-scale converter. The

converter is used to control the generator speed and reactive power independently [60, 61].

2.5 VSVP wind turbines

2.5.1 Advantages of VSVP wind turbines

VSVP wind turbines (Type 3 and 4) are currently the most used configurations in power

systems applications due to their control flexibility [25]. The main advantages of these

configurations over the fixed-speed concept (Type 1) are [57, 59, 62, 63] :

They allow higher conversion efficiency. Due to the variable-speed capability, the turbine

speed is continuously adjusted to track all wind speed variations so that the power

coefficient is kept at its maximum value throughout the partial load regime.

Variable-speed operation reduces drive train torque fluctuations as the variations in the

wind are absorbed by changes in the generator speed and its kinetic energy. The variable-

speed operation creates an “elasticity” that mitigates the loads affecting the wind turbine.

Variable-speed operation allows the smoothing of the power supplied to the grid. By

varying the wind turbine speed, the system inertia acts as an energy buffer between the

highly fluctuating input power and the power supplied to the grid. During positive wind

gusts, excess input power can be transiently stored as kinetic energy in the turbine. This

process is reversed during an abrupt decrease in the wind speed. Smoothing the wind

turbine output power results in less voltage fluctuations and better power quality.

The use of power converters in Type 3 and Type 4 WECSs allows reactive power control.

Consequently, both configurations can be used to support the grid voltage during large

41

voltage dips. This differs from Type 1 WECSs that cannot generate reactive power to

support the grid voltage [57].

Both the DFIG concept and the direct-drive PMSG concept possess their relative

advantages and disadvantages [57, 59]. The DFIG concept offers smaller converter size, cost and

losses compared the direct-drive PSMG concept. It also uses smaller and cheaper inverter filters,

and it typically results in less harmonic distortion. On the other hand, the direct-drive PSMG

concept does not use gearboxes. This results in a higher reliability and availability of the wind

turbine and a reduction in the mechanical power losses, the weight and maintenance costs. It also

does not use slip rings. Furthermore, Type 4 WTs have lower copper losses as they have no field

windings. Finally, the direct-drive PSMG concept is less sensitive to grid faults since the power

converters are full-scale. Despite their competition, the DFIG concept is still dominating the

market in the wind turbine industry especially for large grid-connected wind turbines [57, 59].

For that reason, in the rest of the thesis, the DFIG system will be used as the benchmark system

that is used to validate and illustrate the effectiveness of the developed control strategies.

2.5.2 VSVP wind turbine power curve

The power curve of a wind turbine is a plot of its output power versus the wind

speed . The power curve is one of the most important characterizations of a wind turbine as it

reveals its operation states and power production capabilities [12, 13, 64].

A typical power curve of a 1.5 MW VSVP wind turbine is shown in Figure 2.12. Clearly,

the wind turbine operates within a specific wind speed range between the cut-in wind speed, ,

and the cut-out wind speed, . To realize the power curve shown in Figure 2.12, the VSVP

wind turbine should be controlled as follows.

42

Figure 2.12 Typical power curve of a 1.5 MW VSVP wind turbine [65]. The solid line

shows the power curve, the dashed line shows the wind power , and the dotted line

shows the maximum power that can be extracted by the wind turbine ( ), where

.

In the partial load regime, , to operate the wind turbine at its maximum

efficiency, thus maximizing the energy capture, the turbine speed must vary in proportion to the

wind speed to maintain , and the pitch angle must be fixed at its optimum value (typically

). This is shown in region I.1 in Figure 2.13. At a certain wind speed , typically less that

, the turbine speed reaches its rated value, , and cannot be increased any further.

Consequently, the wind turbine is operated at constant speed for as shown in

region I.2 in Figure 2.13. In general, the rotational speed is limited to prevent high acoustic noise

emission and to keep centrifugal forces within safe values [13].

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

v, m/s

Po

we

r, M

W

PWTG

Pw

CP,max

* Pw

PartialLoad

vrat

vci

vco

Full Load

43

(a)

(b)

Figure 2.13 Variation of (a) the turbine speed, and (c) the pitch angle all operating wind

speeds.

In the full load regime, , the wind turbine output power and rotational

speed should be kept at their rated values by increasing the pitch angle of the blades and

shedding excess input wind power. This is shown in Region II in Figure 2.13.

Figure 2.14 shows the evolution of and over all operating wind speeds. It can be

seen that the wind turbine is operated at its maximum efficiency only in Region I.1.

0 5 10 15 20 25 308

10

12

14

16

18

20

22

v, m/s

t,

rpm


III.2I.1

vco

vci

vrat

v,rat

0 5 10 15 20 25 300

5

10

15

20

25

v, m/s

,

o

Full Load

III.2I.1

vci

v,rat v

ratv

co

Partial Load

44

(a)

(b)

Figure 2.14 Variation of (a) and (b) over all operating wind speeds.

Having an initial look at the wind turbine power curve, it may appear unreasonable to

limit the power capture of the wind turbine in the full load regime and to operate it at low

efficiency as shown in Region II in Figure 2.14. From the energy generation perspective, it is

more appealing to continue operating the wind turbine on the curve shown in Figure

2.12 and to generate more power in the full load regime. However, this turns out to be less

economical [4, 13].

This can be explained by looking at the probability distribution of the average wind speed

at a given site, , that is commonly described by the Weibull distribution, such as the one

shown in Figure 2.15 [13]. The wind speed distribution reveals that, despite their high power

0 5 10 15 20 25 302

3

4

5

6

7

8

9

v, m/s

III.2I.1

vci

v,rat v

ratv

co


0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

v, m/s

CP

Full LoadPartial Load

III.2I.1

vci

v,rat

vrat

vco

45

content, large wind speeds rarely occur. More insight can be gained by drawing the wind power

density, defined in (2.19), as shown in Figure 2.16. It is easy to observe that available wind

energy during high wind speeds ( ) is relatively low. To extract this little amount of energy,

it is required to construct the turbine to withstand large mechanical stresses resulting from high

wind speeds and to use a huge generator with a very large power rating. That is why stopping the

wind turbine during high winds ( ) is much more economical than designing the system to

operate at such conditions. For similar reasons, in the full load regime, the turbine is designed to

limit the power to its rated value which is much less than the power that can be captured.

(2.19)

Figure 2.15 Weibull probability distribution of the mean wind speed at a given site.

Figure 2.16 Power density versus mean wind speed at a given site.

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

vm

, m/s

p( v

m )

0 5 10 15 20 25 300

10

20

30

40

vm

, m/s

Pow

er

density,

W/m

2

vcov

civ

rat

46

2.6 WECS control objectives

This section summarizes the main control objectives of a WECS. In Chapters 5-7,

advanced control strategies are developed to achieve most of these objectives.

2.6.1 Maximizing the energy capture

One of the most important control objectives is to maximize the generated power

supplied to the grid. This increases the profitability of a wind power generation plant. This

objective should be realized while respecting all safe operational limits of the WECS, such as

rated power of the wind turbine, rated rotational speed, and pitch actuator limits.

2.6.2 Reducing mechanical loads

Wind turbines are large and flexible structures interacting with the wind. Due to the high

wind speed variability, wind turbines’ components are exposed to different dynamical loads.

Reducing these loads increases the wind turbine lifetime and the cumulative energy produced

during the wind turbines’ service [32, 39, 66].

Dynamic loads affecting the wind turbine can be classified into transient and cyclic loads

[13, 15]. Transient loads result from temporary wind speed variations, such as wind gusts. On the

other hand, cyclic loads have a periodic nature and they result from the rotation of the blades in a

non-uniform wind speed field. Due to the flexibility of the wind turbine structure, dynamic loads

can excite poorly damped modes of the system, leading to excessive vibrations in the tower,

blades, and drive train. These oscillations can fatigue the wind turbines’ components, resulting in

premature failure and a reduction in the lifetime of the installation.

Wind turbine control systems can play an important role in damping vibrational modes,

and reducing transient loads, especially when the wind turbine operates near the rated wind

speed (transition region) [9].

47

2.6.3 Increasing system reliability

The increasing use of offshore wind turbines poses new challenges on WT control

systems in terms of reliability [67-69]. Due to high unscheduled maintenance costs in the

offshore environment, it is desirable to design control systems that are resilient to unknown

changes in the WT dynamics resulting from faults, wear, debris buildup, or other causes. This

can reduce wind turbines’ down-time and maintenance cost, and increase electricity production.

2.6.4 Enhancing power quality and ensuring compliance with grid codes

Large penetration of wind power into the grid, with plans of reaching 20% penetration in

the US and Europe [5, 70], has a significant impact on the power quality and stability of the grid

[44, 71]. The use of power electronic converters and the injection of highly variable wind power

affect the voltage magnitude and its waveform (harmonic content) at the point of common

coupling (PCC) between the wind farm and the electric network [44]. Furthermore, to maintain

network stability in the presence of large wind power generation, it is extremely desirable to

design wind farms to provide reactive power control, frequency control and fault ride-through

capability, like conventional power plants [72, 73].

In response to the fast wind power development, regulations for interconnecting wind

turbines to the grid have been specified and continuously updated during the last two decades

[11, 73]. These regulations are specified by grid operators to allow continuous integration of

large amounts of wind power while maintaining the stability and security of the network. Grid

codes differ from country to country and they represent the mandatory minimum technical

requirements that a wind farm should fulfil to be connected to the grid.

By taking power quality and grid code issues into account during wind turbine controller

design, the control system can make best use of the existing wind turbines’ hardware and their

48

power electronic converters to ensure compliance with grid codes. This can eliminate the need to

invest in extra equipment such as Static Var Compensators (SVC), STATCOMs, or additional

transmission lines that are typically used to enhance power quality and stability [13, 44].

The most relevant regulations, namely, voltage flicker, harmonics, fault ride-through,

reactive power/voltage control, and active power/frequency control are briefly discussed below.

2.6.4.1 Flicker

Flicker is defined as [74]:

“an impression of unsteadiness of visual sensation induced by a

light stimulus, whose luminance or spectral distribution fluctuates

with time”.

Flicker can cause consumers annoyance, and, thus it is desirable to reduce its effects.

Large WECS output power fluctuations, resulting from wind speed variations, cause

voltage flicker. These power fluctuations yield load flow changes within the grid, leading to

voltage fluctuations at the PCC. Unfortunately, these variations, especially the ones occurring at

the 3P frequency (see §3.2.2), may fall in the range of human eye sensitivity causing consumers’

annoyance [75-77].

One commonly used measure of flicker emissions is the short-term flicker severity

coefficient, . The flickermeter described in IEC 61000-4-15 [77] is used to calculate based

on voltage measurements at the PCC over a ten-minute period. In general, the short-term flicker

severity should be kept below a certain limit which is specified in power quality standards.

The magnitude of voltage variations and flicker emissions at the PCC depend on

numerous factors [44, 71, 74]:

the ‘strength’ of the WECS connection to the grid, that is typically quantified by

the Short Circuit capacity Ratio (SCR) defined in (2.20), where is the short

49

circuit apparent power at the PCC without contribution from the WECS, is

the rated apparent power of the wind generation system, is the rated line-to-

line voltage at the PCC, and is the Thevenin impedance of the grid at the PCC

| |⁄

(2.20)

the grid impedance angle defined in (2.21), where and are the

resistive and reactive components of

(

) (2.21)

amount of active and reactive power exchanged between the WECS and the grid.

Typically, wind farm locations are in sparsely populated areas with high average wind

speeds. Long transmission and distribution lines are normally required to connect wind farms to

the grid. As a result, short circuit levels at the wind farms are generally low, making them weak

electrical systems and more prone to flicker problems [78].

Flicker mitigation can be realized by either controlling the WECS reactive

power/terminal voltage, or by smoothing the WECS output power and damping power

oscillations. Clearly, a well-designed control system can effectively reduce flicker emissions and

enhance power quality [13].

2.6.4.2 Harmonics

Variable-speed wind turbines used today are equipped with power electronic converters.

These electronic converters produce high order harmonic currents that distort the voltage

waveform in the neighboring busses. Similar to flicker emissions, power quality standards

50

enforce that harmonic voltages and the total harmonic distortion of the voltage must be kept

below certain limits [12, 79].

Harmonics can be reduced by using appropriate harmonic filters or advanced pulse width

modulation techniques [11, 12]. Since the control system is mainly concerned with the

fundamental frequency components, harmonic reduction is considered out of the scope of this

thesis and it will not be considered any further.

2.6.4.3 Fault ride-through

As described in §1.2.3, the Fault ride-through requirement describes the desired dynamic

behavior of the wind turbine during and immediately after external network faults. This

requirement can be addressed within the wind turbine generator control system. The FRT

problem is considered in Chapter 8.

2.6.4.4 Reactive power/voltage control

Voltage variations at the PCC occur as a result of variations in the WT’s output power

with the mean wind speed, and other load flow variations within the network. In general, there

are stringent requirements on the extent to which bus voltages can be allowed to deviate from

their nominal value. Consequently, control of reactive power flow is typically implemented to

control the voltage without affecting power generation [13, 44, 64, 79].

Use of power electronic converters with variable-speed wind turbines (Type 3 and 4)

offers significant control flexibility over fixed-speed ones. In addition to enabling variable-speed

operation of the WT, the power converters can be used to control the reactive power exchange

between the WECS and the grid. This eliminates the need for auxiliary devices (such as capacitor

banks and SVCs) that are typically used for reactive power control. The control can be

implemented in the form of a reactive power, power factor, or terminal voltage control loop [44].

51

2.6.4.5 Active power/ frequency control

In any electrical power system, the active power generated should balance the loads in

real time. Any mismatch between the generation and consumption causes a deviation of the

system frequency. With the continued increase in wind generation, a potential concern for grid

operators is the capability of wind farms to provide dynamic frequency support [11, 73].

In the case synchronous generators connected to the grid, when the grid frequency

reduces from its nominal value, due for example to a sudden load increase, kinetic energy is

taken from the rotating mass, thus reducing the frequency decline. Variable-speed wind turbines

do not provide this “inertial response” as their speed is decoupled from the grid frequency by the

power converters [80, 81]. Research is currently being undertaken to make use of the wind

turbine inertia and its stored kinetic energy to dynamically provide frequency regulation using

fast WT active power control [80-82].

Typically, reactive power/voltage and active power/frequency control are implemented at

the wind farm control level. The overall control performance of the wind farm highly depends on

the control capabilities of its individual wind turbines [13]. The design of high performance

controllers controlling the active and reactive power of individual wind turbines is one of the

objectives considered in this thesis.

After discussing the most relevant WECS control objectives in this section, it should be

pointed out that many of these objectives are conflicting. For example maximizing the energy

capture is usually associated with large mechanical loads. A trade-off between different control

objectives must be decided by the control system designer [10, 27, 28, 33, 83]. For that reason, it

is desirable that any proposed wind turbine control strategy can be easily tuned to achieve these

trade-offs.

52

Modeling of Variable-Speed Variable-Pitch Wind Energy Conversion Chapter Three:

Systems

For the design, testing and assessment of WT control systems, it is essential to have a

reliable model representing the WECS dynamics. The model should include not only the WECS

dynamics, but also the external systems with which the WECS interacts. Basically, the WECS

interacts with the wind speed and the power system to which the WECS is interconnected. This

chapter will focus on the modeling of VSVP DFIG-based wind turbines. An overview of WECS

modeling is given in §3.1. The main driving signal of the WECS, the wind speed, is modeled in

§3.2. The WECS is modeled as several interconnected subsystems. The aerodynamics, the pitch

actuator, the drive train and the electrical subsystems of the WECS are modeled in §3.3-§3.6,

respectively. Modeling of the WECS grid interconnection is provided in §3.7. WECS modeling

using per unit system is described in §3.8, and §3.9 describes the overall model of the system.

3.1 Overview of the WECS model

The modeling details of VSVP WECSs vary depending on the objective of the study. In

the literature, research focusing on the WECS mechanical performance requirements tends to

overlook the electrical details [9, 31, 34, 37-39, 83-90]; while the research conducted by authors

with an electrical background tends to simplify the aerodynamics and the mechanical subsystems

[25, 26, 91-97]. In this thesis, the objective is to develop control strategies that optimize the

energy capture, reduce the drive train loads and enhance the power quality (in terms of reduced

voltage flicker level). Therefore, the aerodynamics, the mechanical and electric subsystems will

be modeled in sufficient detail.

A model of a DFIG-based WECS can be structured as several interconnected subsystems,

as shown in Figures 3.1-3.2 [13]. The aerodynamics subsystem represents the transformation of

53

kinetic energy stored in the wind into useful aerodynamic torque acting on the rotor. The pitch

actuator subsystem models the pitch servo system that rotates the blades along their longitudinal

axes. The drive-train subsystem represents the mechanical parts that transfer the power captured

by the rotor to the generator shaft. The electrical subsystem describes the electric generator, the

power electronic converters and the generator control system that converts the harvested

mechanical power into electric power supplied to the grid.

WRIG

AC/DC DC link

Grid

AC/DC

Gear Box

Rotor

Wind

Aerodynamics

+ Pitch actuatorDrive Train

Electrical Subsystem

(Generator + Converters + Generator Controller)

smoothing

inductor

WECS

Figure 3.1 Main subsystems of grid-connected DFIG-based wind turbine.

Pitch

system

Aero

dynamics

Generator

Controller

Generator+

Convetrer

Drive

Train

β* β

v

Tg

Tt

ωg

ωt

PWTG

ωg

Tg*

vWTG

Electrical Subsystem

Wind speed

simulator

Power system

Grid

feedback

iWTG

WECS

Figure 3.2 Block diagram of a grid-connected DFIG-based wind turbine.

54

The WECS interacts with the wind through the wind speed signal , and with the power

system grid through the Wind Turbine Generator (WTG) terminal voltage and the currents

flowing between the WTG and the grid . In Figure 3.2, the input signals are the generator

torque set point and the pitch angle set point . The measured outputs are, typically, the

generator speed, and the electric power generated by the WTG and supplied to the grid,

.

3.2 Wind speed stochastic model

Wind is a very complex process. Its magnitude and direction vary in space and time. It is

affected by many factors such as atmospheric conditions, the surface roughness, the presence of

obstacles and the altitude of the wind, to name a few. For this reason, the wind speed is typically

modeled as a non-stationary stochastic process [13, 53, 91, 98-100].

This section starts by providing a model for the wind speed at one fixed point in space

(§3.2.1). Since the turbine blades rotate in a non-uniform wind speed field, the wind speed

experienced by the blades can differ significantly from the wind speed at one fixed point. This is

discussed in §3.2.2. Based on §3.2.1 and §3.2.2, a model of the effective wind speed that can be

used in wind turbines simulations is provided in §3.2.3.

3.2.1 Wind speed at one fixed point

3.2.1.1 Van der Hoven’s spectral model

An early attempt to characterize the stochastic variation of the wind speed is Van der

Hoven’s spectral model shown in Figure 3.3 [12, 13, 98, 101]. This model is considered to be

one of the best known references for large band wind speed modeling. The model shows the

product of the wind speed power spectral density and the frequency over a range of

frequencies from 0.0007 to 900 cycles/h (over 6 decades).

55

Figure 3.3 Van der Hoven’s spectral model of the wind speed [12].

Van der Hoven’s spectral model reveals the kinetic energy distribution of the wind in the

frequency domain. It can be seen from Figure 3.3, that the spectrum has two peaks occurring at

approximately 0.01 cycles/h (4 days period) and 50 cycles/h (1 min period). Furthermore, the

spectrum shows that the kinetic energy of the wind is concentrated in two disjoint frequency

ranges that are separated by an energy gap between periods of 10 min to 2 h. This suggests

modeling the wind speed signal at one fixed point, , as the superposition of two components

[12, 13]:

(3.1)

where is the low-frequency component describing the long-term, slow variations of the

wind; and is the turbulence component, describing the high frequency, fast variations. The

turbulence component is typically modeled as a zero mean random process while the low-

10-3

10-2

10-1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

f, cycles/hour

fSv

v(f

), m

2/s

2

Energygap

10 min2 h4 days40 days 5 s Period

Turbulant component

Low-frequency component

1 h

56

frequency component is regarded as a constant equal to the mean wind speed when viewed at the

turbulence time scale (typically, 10 min time window) [12, 13].

3.2.1.2 Turbulence model

The turbulence component is modeled as a zero mean random process and it is

characterized by its power spectrum. One of the basic models that is used in modeling the

turbulence is the von Karman power spectrum in (3.2), where is the Power Spectral Density

(PSD) of the turbulence component, is the frequency in rad/s, is the turbulence intensity and

is the turbulence length scale [12, 13, 98].

( (

)

)

⁄ (3.2)

The von Karman power spectrum is characterized by two parameters: the turbulence

length scale and the turbulence intensity defined in (3.3), where is the standard

deviation of the turbulence. Both parameters can be determined empirically for a specific site, or

they can be adopted according to standards [12, 13, 98]. Typically, for WECSs, and are in

the ranges of 100-330 m and 0.1-0.2, respectively [13].

√∫

(3.3)

The von Karman spectrum (3.2) shows that the turbulence component characteristics are

dependent on the low-frequency component . In particular, (3.3) shows that the standard

deviation of the turbulence is proportional to .

57

3.2.1.3 Overall model of the wind speed at one fixed point

A wind speed model representing both the turbulence component (short-term wind speed

variation), and the low-frequency component (medium- and long-term wind speed evolution), is

developed in [98]. The model combines the low and medium frequency part of Van der Hoven’s

model to generate the low-frequency component, and the von Karman model to generate the

non-stationary turbulence component. The superposition of both components allows large-band

modeling of the wind speed. This model uses two time scales in simulating the wind speed. The

slow frequency component is sampled with a sampling period that is taken in the minutes

range (3-10 min) while the sampling time of , , is in the seconds range (0.1-2 s).

To generate the low-frequency component , the wind PSD , based on Van

der Hoven’s spectrum, is sampled at discrete frequencies , , in the spectral

range representing the medium- and long-term wind speed evolution [98]. In [98], this range is

assumed as [

] [

] (see Figure 3.3) and 10 samples per decade are used.

The low-frequency is calculated using (3.4), where is given in (3.5), [ ] is a

randomly generated number using a uniform distribution, and is the average wind speed

calculated on a time horizon much greater than ⁄ .

∑

(3.4)

√

( ) (3.5)

To simulate the turbulence component in a fast and efficient way, it was suggested in [98]

to filter a unit variance white noise by the rational shaping filter in (3.6) with and

given in (3.7) [13, 98]. The colored noise output of has a unit variance; and consequently,

58

it should be multiplied by to adjust the standard deviation of the colored noise to . It was

shown in [98] that this approach can accurately approximate the irrational PSD in (3.2) over a

large range of frequencies. The parameters and of the filter remain constant along

the sampling interval as long as is constant. They are updated at each sampling period

according to the current value of .

(3.6)

√

(

)

(3.7)

The procedure proposed in [98] to simulate the wind speed at one point is illustrated in

the block diagram representation in Figure 3.4. A wind speed simulation is carried out for about

12 h using this model and is shown in Figure 3.5. The model clearly captures the variations in the

and the non-stationary behaviour of the turbulence and its dependence on . The PSDs and

the shaping filter gains of the turbulence component corresponding to mean wind speeds of 6 and

15 m/s are shown in Figure 3.6 (a) and (b), respectively. The general behaviour is that the

turbulence bandwidth and power increase with increasing mean wind speed.

Calculate vm(t) using (3.4)

Calculate TF and

using (3.7)It

Σ

Time varying Filter

vm(t)

e(t)White noise

vt(t)

v1p(t)

)(FH

Evaluated each Tsm

vm(t)

TF , σvtFK

FK

Figure 3.4 Non-stationary wind speed simulation at one point [12].

59

Figure 3.5 Non-stationary wind speed simulation: the low frequency component (black

solid) and the total wind speed at one point (blue dotted).

Figure 3.6 Approximate von Karman’s spectrum for two different values of mean wind

speed: (a) turbulence PSDs and (b) shaping filter gains.

3.2.2 Wind speed experienced by the turbine blades

The wind turbine blades rotate in a three-dimensional wind speed field. As the rotor

swept area can be well beyond 1,000 m2 for utility-scale wind turbines, the wind speed spatial

distribution over this area is far from being uniform [13, 15, 102]. This subsection uses a

2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85 2.9 2.95 3

x 105

0

5

10

15

20

25

time, s

win

d s

pee

d,

m/s

vm

v1p

= vm

+ vt

10-3

10-2

10-1

100

101

0

10

20

30

40

50

60

, rad/s

(a)

S v tv t()

vm

= 6 m/s

vm

= 15 m/s

10-3

10-2

10-1

100

101

-30

-20

-10

0

10

20

30

, rad/s

(b)

Shap

ing f

ilte

r gai

n

v

m = 6 m/s

vm

= 15 m/s

60

qualitative approach to explain different factors causing this non-uniformity and its effect on the

shape of the turbine torque produced.

Figure 3.7 shows a typical wind speed distribution over the rotor swept area at a certain

instant of time. In general, the magnitude of both the mean wind speed and the turbulence

component can vary significantly from one point to another in the rotor swept area.

Consequently, a blade element rotating in this wind field can experience wind speed variations

that differ significantly from wind speed variations observed at one fixed point. This is known as

the rotational sampling effect [12, 13, 102].

Figure 3.7 Spatial wind-speed distribution over the swept area of the turbine rotor.

The wind speed variations experienced by a rotating blade element can be classified as

deterministic or stochastic variations [13]. Wind shear and tower shadow are the main causes of

deterministic variations. Spatial variations in the turbulence component at different points over

the rotor swept area are the cause of the stochastic variations.

61

3.2.2.1 Wind shear effect

The Wind shear effect expresses the increase in the wind speed with height. One common

expression for this variation is (3.8), where is the empirical wind shear exponent, is the

mean wind speed at height , is the mean wind speed at the hub height, and is the hub

height [100, 103, 104]. Typical wind speed variations with the height are shown in the left part

of Figure 3.8.

(

)

(3.8)

ϴ

Tower shadow

Wind shear gradient

Hei

ght

ϴ

Blade element (br)

Figure 3.8 Wind shear and tower shadow effects [100].

To understand the wind shear effect on the shape of the turbine torque, assume the blade element

shown in Figure 3.8, located at distance from the hub, performs one complete revolution.

Clearly, the height of will vary sinusoidally with the blade angle , i.e. .

This leads to a cyclic variation in the wind speed experienced by . It can be seen from Figure

3.8 that the maximum wind speed is experienced by when it is located at the maximum

height, i.e. when the blade is upward ( ; and the minimum occurs when the blade is

62

downward ( . Since the torque produced on is proportional to the square of the wind

speed (see (2.10)), all cyclic variations in the wind speed are translated into cyclic variations in

the torque produced [13].

Typical torque oscillations produced on the wind turbine’s individual blades are shown in

Figure 3.9 (a). The torque is normalized to the torque produced assuming a uniform wind speed

equal to . It is clear that the torque has an almost sinusoidal waveform with a frequency equal

to the turbine rotational speed . This frequency is known as the 1P frequency [100]. The total

turbine torque produced on the rotor, shown in Figure 3.9 (b), is the sum of the torque produced

by each blade. Since the angle between two blades of a three-bladed wind turbine is

, the

fundamental frequency (1P) is cancelled out in the sum and the turbine torque is dominated by

the third harmonic frequency that is three times the turbine rotational speed. This frequency is

known as the 3P frequency [100, 103, 104]. It should be noted that the magnitude of fluctuations

in the turbine torque are significantly reduced when compared to the individual blades’ torques.

Figure 3.9 Torque oscillations due to the wind shear alone: (a) normalized individual blade

torque and (b) normalized wind turbine torque.

0 45 90 135 180 225 270 315 36095

100

105

Bla

de

torq

ue,

%

(a)

0 45 90 135 180 225 270 315 36099.1

99.2

99.3

99.4

Tu

rbin

e to

rque,

%

, o

(b)

blade 1 blade 2 blade 3

63

Cyclic blades and turbine torques yield cyclic loads on the wind turbine components that

may lead to fatigue and reduction in the wind turbine lifetime. Furthermore, torque fluctuations

are transformed into power fluctuations supplied to the grid reducing the power quality.

However, it should be noted that due to the averaging effect of the rotor, the wind shear

aerodynamic loads on the blades are significantly attenuated when propagated down the hub, the

drive train and the generator[13].

3.2.2.2 Tower shadow effect

The tower shadow effect expresses the deficit occurring in the mean wind speed in an

influence zone in front of the tower of a HAWT (Figure 3.8). Typically, the tower has a

cylindrical shape and it represents an obstacle to the airflow. The presence of the tower causes

the deviation of the air streamlines around the tower, resulting in a decrease in the magnitude of

the axial wind speed in a small region in front of the tower [13, 103].

The tower shadow causes a periodic drop in the blade torque each time the blade passes

through the tower shadow region in front of the tower. Typical torque variations produced at the

blades and at the rotor, resulting from the tower shadow effect, are illustrated in Figure 3.10 (a)

and (b), respectively [103]. It is clear that the individual blade torque is periodic with frequency

1P and the total turbine torque is periodic with the frequency at which the blades pass by the

tower, i.e. the 3P frequency [103]. Since the turbine torque is not sinusoidal, its spectrum

consists of impulses at integer multiples of the 3P frequency.

Similar to wind shear, the tower shadow induced torque oscillations cause cyclic loads on

the wind turbine components and power fluctuations to be injected to the grid. In general, the

tower shadow effect dominates that of the wind shear [13, 53, 103]. This can be seen by

64

comparing the magnitude of the tower shadow induced torque oscillations ( ) in Figure 3.10

(b) and the wind shear induced ones ( ) in Figure 3.9 (b).

Figure 3.10 Torque oscillations due to the tower shadow alone: (a) normalized individual

blade torque and (b) normalized wind turbine torque.

3.2.2.3 Rotational turbulence

The turbulence experienced by a rotating blade element, known also as the rotational

turbulence, can differ significantly from the turbulence observed at one fixed point [12, 13, 91,

100, 104]. The difference depends on many factors such as the distance of the blade element to

the hub, the rotational speed of the rotor and turbulence bandwidth. To understand the concept,

assume the extreme case, where the turbulence temporal variations are extremely slow compared

to the rotor speed, i.e. the turbulence variation is almost frozen in time [13]. When a blade

element rotates in a time-invariant wind speed distribution such as the one in Figure 3.7, the

blade element observes the same wind speed variations at each revolution. This results in a

periodic wind speed variation experienced by the blade element and the resulting power

spectrum will consist of impulses at integer multiple of the 1P frequency. In reality, the wind

speed is not frozen in time and the turbulence variations are typically much slower than the

turbine rotational speed. Consequently, the turbulence variation observed by a blade element is

not periodic and there exist slight variations from one period to another. Therefore, the spectral

0 45 90 135 180 225 270 315 36090

95

100

105

Bla

de

torq

ue,

%

(a)

0 45 90 135 180 225 270 315 36090

95

100

105

Tu

rbin

e to

rque,

%

, o

(b)

blade 1 blade 2 blade 3

65

peaks at integers of the 1P frequency leak over all frequencies. The rotational turbulence is

reflected on the total turbine torque produced in the form of fluctuations occurring at integers of

the 3P frequency. Furthermore, these fluctuations are transferred to the grid in the form of power

fluctuations which can lead to voltage flicker problems.

Another phenomenon which can be associated with the spatial variation of the turbulence

is known as spatial filtering [12, 91, 100]. The turbulence component at one point of the rotor

swept area is correlated with the turbulence at neighboring points. As different points get further

apart, their turbulence components become more and more uncorrelated. Due to the averaging

effect of the rotor (the turbine torque is the sum of the torque produced by each blade), the

turbulence induced turbine torque is smoother than that produced at each blade. In fact, the rotor

can be regarded as a low pass filter suppressing the high frequency content of the turbulence

spectrum at one fixed point.

3.2.3 Effective wind speed

The blade element theory described in §2.2 can be used to compute the total aerodynamic

torque produced on the wind turbine rotor [100]. In this approach, the rotor swept area is divided

into sectors and the wind speeds at a grid of points in each sector are assumed to be known. At

each computation step, the magnitude of the incident wind speed at any blade element is

determined based on the current rotor position; and the torque produced on the blade element and

the total turbine torque are calculated as explained in §2.2.1. However, this approach requires

large computational effort and results in slow simulations [12, 100].

Another approach that is computationally more efficient is based on using a fictitious

wind speed, known as the effective wind speed , which is equivalent in some sense to the three-

dimensional wind speed field [12, 13, 91, 100, 105]. The equivalence here is in the sense that the

66

turbine torque calculated by applying the effective wind speed to the power coefficient

and the scalar torque equation in (2.14) is the same, or has the same PSD, as the torque obtained

using the BET and a three-dimensional wind field approach [13, 100].

Figure 3.11 shows a block diagram of the wind speed simulator that is used in this thesis

to generate the effective wind speed [13]. First, a time series of the wind speed signal at one

fixed point (at the hub height) is generated using the model in §3.2.3 (Figure 3.4). Both the mean

wind speed and the turbulence component at the hub height are adjusted by shaping filters and

additional terms to account for the wind turbine blades interaction with the wind speed

distribution over the swept area (§3.2.2).

Fixed point wind

speed model

(Fig. 3.4)

HSF (s) HRSF (s)

Rotational sampling model

Σ

∫

Wind

Shear

Tower

Shadow

3

Σ

vm

vt

v1p

ωt

vm vmω3P

vEffective

wind speed

vWS

vTS

vm

ϴ

Figure 3.11 Effective wind speed simulator [13].

The turbulence component is modified by two series filters: the spatial filter and

the rotational sampling filter , as shown in Figure 3.11. The spatial filter is used to

attenuate the high-frequency components of the turbulence to represent the spatial filtering

67

property of the rotor. A typical transfer function of is given in (3.9), where ,

and is the mean wind speed at the hub height [12]. The rotational sampling filter

is used to represent the rotational sampling of the wind speed by the rotor. In the

literature, the rotational sampling transfer function in (3.10) is proposed, where and

is the damping factor which depends on the mean wind speed [91]. This filter amplifies

those components with frequencies close to the blade passing frequency (3P). At other

frequencies, this filter has an almost unity gain. Similar filters can be used to amplify the

spectrum at higher harmonics.

√

(√ √ ) (

√

)

(3.9)

( )( ) (3.10)

To account for wind shear and tower shadow effects in the effective wind speed model,

two additional terms are added to the mean wind speed at the hub height [103]. The first term

, given in (3.11), represents the wind shear effect. The second term , representing the

tower shadow effect, is given in (3.12) where is the tower radius, is the normal distance from

the rotor to the tower center-line,

and is defined in (3.13). It was shown

in [103] that these terms are equivalent to the actual wind shear and tower shadow effects, under

mild assumptions.

(

(

)

(

)

) (3.11)

68

∑(

(

)

)

∑( (

)

)

(3.12)

(3.13)

A 5 min simulation of the effective wind speed using the model in Figure 3.11 is shown

in Figure 3.12. The mean wind speed is 10 m/s. The parameters are taken as: min,

s, m, , m, (no tower shadow effect), m and

. Simulations show a significant difference between the wind speed at one fixed point

and the effective wind speed . The difference is due to rotational sampling. Oscillations

occurring at 3P are clearly observed by zooming on the effective wind speed as shown in Figure

3.12 (c). The filter gains used with the wind speed at one fixed point and the effective wind

speed are compared in Figure 3.13.

Low and high wind speed simulations at mean wind speeds of 6 and 15 m/s, respectively,

are performed using the model in Figure 3.11. Results are shown in Figure 3.14. The non-

stationary behavior of the effective wind speed and its dependence on the mean wind speed can

be observed. This is confirmed by comparing the PSD estimates of the effective wind speed at

low and high wind speeds as shown in Figure 3.15.

69

Figure 3.12 Wind speed simulation: (a) wind speed at one point, (b) rotationally sampled

wind speed and (c) zoom on the rotationally sampled wind speed.

Figure 3.13 Comparison of the shaping filters gains of the wind speed at one fixed point

and the rotationally sampled wind speed.

0 100 200 3007

8

9

10

11

12

13

14

time, s

(a)

v 1p

, m

/s

0 100 200 3007

8

9

10

11

12

13

14

time, s

(b)

v, m

/s

90 91 92 93 94 95 96 97 98 99 10010.7

10.8

10.9

11

11.1

11.2

11.3

time, s

(c)

v, m

/s

10-3

10-2

10-1

100

101

102

103

-140

-120

-100

-80

-60

-40

-20

0

20

40

, rad/s

Shap

ing f

ilte

r gai

n

Wind speed at one point

Rotationally sampled wind speed

70

Figure 3.14 Low and high wind speed simulations.

Figure 3.15 Effective wind speed PSD comparison at low and high wind.

3.3 Aerodynamics model

Using the effective wind speed approach described in §3.2.3, the aerodynamic model is

greatly simplified to the scalar turbine power and torque equations in (3.14) and (3.15),

respectively. Here, is the air density, is the rotor radius, is the turbine speed, is the

effective wind speed, is the tip speed ratio, is the pitch angle, and is the power coefficient.

The aerodynamic subsystem in Figure 3.2 receives , and as inputs to the model and

calculates the turbine torque produced using (3.15).

0 100 200 300 400 500 6000

5

10

15

20

time, s

Eff

ecti

ve

win

d s

pee

d,

m/s

High wind speed(v

m=15 m/s)

Low wind speed(v

m= 6m/s)

10-2

10-1

100

101

-80

-60

-40

-20

0

20

40

Frequency, Hz

Pow

er/f

requen

cy,

dB

/Hz 3P frequencies

Low wind speed(v

m=6 m/s)

High wind speed(v

m=15 m/s)

71

(3.14)

(3.15)

The power coefficient is a characteristic of the wind turbine. Look-up tables and

nonlinear functions are typically used to model the power coefficient of a wind turbine as a

function of and [37, 39, 54]. In [14, 52], the power coefficient of a 1.5 MW General Electric

(GE) industrial wind turbine is modeled using (3.16), where the coefficients for 0,1,…,

4 are listed in Table 3.1.

∑∑

(3.16)

Table 3.1 coefficients for 0, 1, …,4.

-4.1909e-1 2.1808e-1 -1.2406e-2 -1.3365e-4 1.1524e-5

-6.7606e-2 6.0405e-2 -1.3934e-2 1.0683e-3 -2.3895e-5

1.5727e-2 -1.0996e-2 2.1495e-3 -1.4855e-4 2.7937e-6

-8.6018e-4 5.7051e-4 -1.0479e-4 5.9924e-6 -8.9194e-8

1.4787e-5 -9.4839e-6 1.6167e-6 -7.1535e-8 4.9686e-10

Another widely used equation that is used to model is given by [54]:

(

)

(3.17)

(3.18)

Here, (3.16) and Table 3.1 will be used in all simulation studies presented in Chapter 5-7.

The characteristics for different values of pitch angles are shown in Figure 3.16. The

72

maximum power coefficient is 0.48 and it is achieved at the optimal values and

.

Figure 3.16 characteristics for different values of pitch angle [14, 52].

3.4 Blade pitch system

Most modern grid-connected wind turbines are equipped with blade pitch mechanisms to

limit the rotor power and speed during high winds and to brake the rotor aerodynamically during

emergency situations. This is achieved by rotating the wind turbine blades along their

longitudinal axes to a certain desired pitch angle. The pitch mechanism is actuated using

hydraulic drives or electrical motors [15]. A closed loop blade positioning system is used to

ensure precise control of the blades pitch angles.

The closed loop pitch system can be modeled using a simple first-order dynamic system

[24, 106]. However, the model should include the physical limits on operating range of the pitch

angle and of the pitch angle rate that exist in all blade pitch actuation systems. Typically, the

pitch angle takes values between to and the pitch angle rate is limited to [14, 52].

3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.1

0.2

0.3

0.4

0.5

= 0

= 2

= 4

= 6

= 10

= 15

= 20

CP

CP,max

0

73

The modeling equations of the blade pitch system are given in (3.19)-(3.21), where is the time

constant of the pitch system and is the maximum (minimum) limit of •. A block

diagram of the pitch system model is shown in Figure 3.17.

(3.19)

(3.20)

(3.21)

∑

p

1

s

1

p

1

*

Figure 3.17 Blade pitch system model [24].

3.5 Drive train model

The drive train of a wind turbine is the system that transmits the mechanical power

produced at the rotor to the high-speed shaft driving the electric generator. The drive train system

encompasses all rotating parts including the rotor, the gearbox, the low-speed shaft and high-

speed shaft, and the generator.

Typically, the drive train contains flexible components such as the low-speed shaft and

the hub with blades [107-109]. Due to the softness in the drive train and the propagation of the

wind speed fluctuations to the drive train, drive train torsional oscillations and mechanical

vibrations occur. This can lead to excessive loading and reduction of wind turbine life time if

their effect is not appropriately mitigated by the wind turbine control system. Furthermore, these

oscillations go through the generator shaft and appear as power and current oscillations injected

74

into the grid. Many studies have been carried out and they concluded that accurate modeling of

the low frequency modes of the drive train dynamics is essential for wind turbine control systems

design, and the assessment of the WECS performance and power quality [22, 106, 110-112].

The most common way to model the drive train is to treat the system as a finite number

of rigid bodies connected together by springs defined by their damping and stiffness coefficients.

Different numbers of masses have been suggested to model the wind turbine drive train. A six

mass model representing the three blades, the hub, the gearbox and the generator is described in

[110]. Simpler models consisting of three masses and two masses have also been proposed [32,

110]. Different drive train models are compared in [108, 110] and it was concluded that the drive

train dynamics can be modeled with a reasonable accuracy using a two-mass model. The two-

mass model allows for the representation of the dominant resonance mode in the drive train and

the assessment of transient torques stressing the drive train. These results are confirmed in [27,

91, 108, 112]. For that reason, a two-mass model is used here to represent the drive train.

A two-mass model of the drive train is illustrated in Figure 3.18. The rotor inertia is

driven by the turbine torque at speed , and the generator inertia is driven by the generator

electromagnetic torque at speed . The parameters and are the shaft stiffness and

damping coefficients rendered at the high-speed shaft, respectively. These parameters define the

flexible coupling between the two inertias. From Figure 3.18, the drive train modeling equations

in (3.22) can be derived where is the internal drive train torsional torque defined in (3.23),

is the twist angle in mechanical rad/s, is the gear ratio, and is the gearbox efficiency

[12, 32]. Referring to Figure 3.2, the inputs of the drive train subsystem model are and , the

outputs are and and the model has three internal states that are governed by (3.22).

75

ωt

Tg

Tt

Ks

Bs

NM,ηM

NM ωt Jg

Jt

ηTt

NM

ωg

Figure 3.18 Two mass model of the wind turbine drive train subsystem[32].

(

)

(3.22)

(3.23)

Remark 3.1: The flexible shaft shown in Figure 3.18 does not necessarily represent one of the

physical shafts of the wind turbine. It represents the most flexible part in the wind turbine and the

2-mass model represents the fundamental resonance frequency which may occur in either the

rotor or the transmission. Consequently, the rotor inertia and the generator inertia do not

necessarily represent the actual physical inertias of the rotor and the generator.

76

3.6 Electric subsystem

Figure 3.19 illustrates the electrical connection of a DFIG-based wind turbine [26, 78,

113]. The stator windings of the WRIG are connected directly to the gird or via a transformer

(not shown). The rotor windings are connected to the grid via a partial scale ac-ac power

converter, consisting of a RSC, dc link and GSC. The GSC converts the constant frequency ac

voltage of the grid to a constant dc voltage across the capacitor at the dc link. The RSC

transforms the dc voltage at the dc link to an ac voltage with adjustable frequency and magnitude

that is applied to the rotor terminals. Slip rings must be used to connect the rotor windings

rotating at speed to the converter which is fixed inside the nacelle.

WRIG

Tg

HSS

Slip rings

Rotor side

Converter

(RSC)

Grid side

Converter

(GSC)

dc link

Power

grid

vsabc

vrabcvCabc

vWTGabc

iWTG

abc

irabc

isabc

iCabc

I1I2

Vdc

Smoothing

inductor

Figure 3.19 Electrical connection diagram of a DFIG-based wind turbine [78].

Remark 3.2: The word “rotor” is used in two different contexts. The “turbine rotor” is used to

refer to the hub and blades assembly of a wind turbine, while the “generator rotor” refers to the

generator parts rotating inside the generator casing.

77

A model of the electrical system should include the WRIG, the GSC connection to the

grid, the power converters, the dc link, and the generator control system. This is detailed below.

3.6.1 Wound rotor induction generator model

3.6.1.1 WRIG dynamic model

A WRIG has a cylindrical rotor rotating inside a stationary stator core. Both the stator

and the rotor cores are made with laminated ferromagnetic sheets and they are separated by a

uniform air-gap (see Figure 3.20 (a)) [114]. Three-phase symmetrical stator and rotor windings

are embedded in the stator and rotor slots that are uniformly distributed over the stator inner and

rotor outer circumferences, respectively. The magnetic axes of the three-phase symmetrical

winding (either the stator or the rotor) are displaced in space by ⁄ electrical radians as

shown in Figure 3.20 (b).

x

x

x

x

x

x

as1

as2

bs1

bs2

cs1

cs2 ar1

ar2

cr1

br2

cr2

br1

as-axis

a r-ax

is

ϴr

ωr

Stator

Generator

rotor

as

bs

cs

ar

cr

br as-axis

a r-ax

is

ϴr

ωr

Air

gap

(a) (b) Figure 3.20 (a) Idealized three-phase, two-pole induction machine with concentric three

phase windings and (b) magnetic axes of the stator and rotor windings.

It is a common practice in electric machines analysis to express angles in electrical rad,

and rotational speeds in electrical rad/s. The relation between electrical and mechanical angles is

78

given by (3.24), where is the number of pole pairs of the machine. In this section, the

generator rotor speed in electrical rad/s is denoted by and it is related to by (3.25).

[ ] [ ] (3.24)

[ ] [ ] (3.25)

To derive the WRIG model, it is common to assume that [115, 116]: 1) Hysteresis and

eddy current losses are negligible. 2) Magnetic hysteresis and saturation are negligible. 3) Both

stator and rotor have symmetrical windings. 4) Stray capacitances of the windings are negligible.

5) The effects of the stator and rotor slots are negligible. 6) Mutual inductances between the

stator and the rotor vary as the cosine of the rotor angle.

Based on the assumptions above and following the motor convention, the voltage

equations of the magnetically coupled stator coils (as, bs and cs), and rotor coils (ar, br and cr )

shown in Figure 3.20 (b) can be written as [114]:

(3.26)

where , and denote voltages, currents and flux linkages, respectively. The stator and rotor

resistances are denoted by and , respectively.The subscripts ( , , ) refer to quantities

corresponding to the three phase stator coils (as, bs, cs) while the subscripts ( , , ) refer to

quantities corresponding to the three phase rotor coils (ar, br, cr).

The stator and rotor voltages can be written compactly using matrix notation as:

79

(3.27)

where the notation [ ] and

[ ] is used with

representing currents, voltages, or flux linkages.

Furthermore, the flux linkages of the stator and rotor windings can be written as:

[

] [

] [

] (3.28)

where the sub-matrices and denote the stator-to-stator and rotor-to-rotor winding

inductances. Due to the air-gap uniformity, both matrices are made up of constants and they are

defined in (3.29), where ( ) is the leakage inductance of the stator (rotor) windings and

( ) is the magnetizing inductance of the stator (rotor) windings.

[

]

[

]

(3.29)

The sub-matrix containing the stator-to-rotor mutual inductances depends on the angular

position of the rotor . Since these mutual inductances vary sinusoidally with , can be

written as in (3.30), where is the peak value of the stator-to-rotor mutual inductance.

[

⁄ ⁄

⁄ ⁄

⁄ ⁄

] (3.30)

The modeling equations of the WRIG in terms of the phase variables are (3.27)-(3.30).

Due to the dependence of stator-to-rotor mutual inductances on , the magnitude of these

80

inductances vary with time when the rotor rotates. This leads to a dynamic model with time

varying coefficients that can be difficult to analyze and simulate.

The standard approach to cancel out the time variation of the stator-to-rotor mutual

inductances is to use what is known as dq0 variable transformation [114, 116, 117]. The use of

the dq0 transformation was proposed by R. H. Park in the 1920s to develop dynamic models for

synchronous machines. He proposed a change of variables which, in effect, replaces the three

phase stator variables with another set of variables associated with two orthogonal fictitious

windings rotating at synchronous speed. In the new coordinate frame, known as the

synchronously rotating reference frame, it was shown that the time variations in the inductances

are eliminated. This theory was further extended to model induction machines and other rotating

machines.

For induction machine modeling, different dq0 reference frames can be used depending

on the purpose of study. The two common reference frames that are used are the stationary and

the synchronously rotating reference frames [114, 117]. In the stationary reference frame, the

and axes are attached to the stator with the axis aligned to the magnetic axis of the stator a-

phase winding. In the synchronously rotating reference frame, the and axes rotate at the

same speed of the WRIG rotating magnetic field, i.e. at synchronous speed .

Consequently, at steady state, all machine variables appear steady when referred to this reference

frame. For that reason, the WRIG model in the ( , ) reference frame is extremely useful for

linear analysis and control synthesis.

Here, a general model of the WRIG at an arbitrary rotating reference frame rotating at

speed as shown in Figure 3.21 is derived. Based on this model, WRIG modeling equations at

the stationary and synchronously rotating reference frames can be easily obtained [78, 114].

81

as-axis

ar-axis

ϴr

ωr

d sk

qsk

dk-axis

d rk

qrk

ωk

bs

cs cr

br

ϴk

qk-axis

Figure 3.21 Relationship between abc reference frame and arbitrary rotating dq0

reference frame.

The fictitious stator (d, q, 0) variables are obtained from the (a, b, c) phase variables

using a linear time varying orthogonal transformation (similar to Park’s transformation) defined

as [78]:

[

]

⏟

√

[

⁄ ⁄

⁄ ⁄

√ ⁄ √ ⁄ √ ⁄

]

⏟

[

]⏟

(3.31)

where is used to represent current, voltage, or flux linkage, and is the angle of the rotating

-axis with respect to the stator a-phase axis as shown in Figure 3.21. The corresponding

inverse transformation is:

[

] √

[

√ ⁄

⁄ ⁄ √ ⁄

⁄ ⁄ √ ⁄

]

⏟

[

] (3.32)

These transformations can be written compactly using matrix notation as:

82

(3.33)

Referring to Figure 3.21, it can be seen that the rotor quantities can be transformed onto

the same dq0 frame by using the same transformation matrices in (3.31)-(3.32) with the angle

replaced by as shown in (3.34).

(3.34)

Remark 3.3: The transformation matrix is orthogonal. This selection ensures that the

transformation is power invariant [78, 115].

Remark 3.4: Since induction machines are generally operated under balanced conditions, the

zero sequence components and

are typically zero and they are not considered in most of

the analysis [78].

Using (3.33)-(3.34), the voltage equation (3.27) and the flux linkage equation (3.28) can

be written in the arbitrary rotating reference frame as [26, 78, 114]:

(3.35)

where

83

(3.36)

(3.37)

Furthermore, it can be shown that the WRIG electromagnetic torque , the stator active power

, the stator reactive power , the rotor active power , and the rotor reactive power are,

respectively, given by:

(

)

(3.38)

(3.39)

(3.40)

(3.41)

(3.42)

Remark 3.5: In (3.35)-(3.42), all rotor quantities and parameters are referred to the stator.

Finally, the WRIG modeling equations in an arbitrary rotating ( , ) reference frame

are (3.35)-(3.42). The model receives the stator voltages, specified by the grid, and the rotor

voltage, determined by the RSC, as inputs to the model. The outputs of the model are the

electromagnetic torque, the stator active and reactive power, and the rotor active and reactive

power. The model has four internal states as can be seen from (3.35).

Another common representation of the WRIG model in (3.35) can be obtained using

complex space vectors defined as [114]:

(3.43)

84

Using space vector notation, the WRIG voltage equations (3.35) can be compactly written as:

(3.44)

where

(3.45)

It should be noted that these equations are mainly used for steady state analysis of the WRIG.

This is typically done by replacing by in (3.44), where is the frequency of the forcing

inputs. The stator active and reactive power can be calculated using (3.46), where, for a complex

number , is the real part of , is the imaginary part of and is the conjugate of .

Similar expressions of the rotor active and reactive power can be used.

(3.46)

Based on the WRIG modeling equations (3.35)-(3.46) in the arbitrary rotating reference

frame, the following models are easily derived.

The WRIG model in the stationary reference frame is obtained by setting in

(3.33)-(3.34), and setting and replacing all superscripts with in (3.35)-(3.46).

The WRIG model in the synchronously rotating reference frame is obtained by replacing

by in (3.33)-(3.34), and setting and replacing all

superscripts with in (3.35)-(3.46).

The WRIG model in the rotor reference frame is obtained by replacing by in

(3.33)-(3.34), and setting and replacing all superscripts with in (3.35)-

(3.46).

85

As the WRIG model in the synchronously rotating reference frame is mostly used in this

thesis, the superscript will often be dropped for this reference frame only to simplify notation.

3.6.1.2 WRIG steady state model

Under steady state and balanced operation of the WRIG, the three phase stator voltages

and currents can be expressed as:

√

√

√

√

√

√

(3.47)

where and are the root-mean-square (rms) of the stator voltage and currents, respectively.

The stator voltage and current time phasors denoted by and , respectively, are defined as:

(3.48)

From (3.33), the stator abc-phase voltages and currents in (3.47) can be transformed into a

synchronously rotating frame yielding

√

√

√

√ (3.49)

Using (3.43), (3.49) and (3.48), the following relation between space vectors and time phasors

can be established at steady state [114]:

√ √

√ √

(3.50)

Similar analysis can be performed to the rotor abc-phase voltages and currents yielding

√ √

(3.51)

It can be seen from (3.49)-(3.51) that the stator and rotor variables are constant when referred to

a synchronously rotating frame. Consequently, the steady state model is derived from (3.44) by

setting yielding [114]

86

(3.52)

where the slip

represents the relative speed between the rotor and the synchronously

rotating magnetic field. Using (3.50)-(3.52), the WRIG steady state model is given by (3.53)-

(3.54). By defining , (3.54) can be written as (3.55). Based on (3.53)-(3.55),

the equivalent circuit of WRIG is shown Figure 3.22.

(3.53)

(

) (3.54)

(

) (

) (3.55)

rs jXls

jXm

RrjXlrsI~

mI~

rI~

rV~

sV~

mV~

+ ++

Rr(1-s)

s(1-s)

srV~

Figure 3.22 Equivalent circuit of a WRIG [114].

The active and reactive power of the stator and the rotor can be calculated using (3.56),

where is the phase difference between the voltage and current time phasors.

(3.56)

87

By multiplying (3.55) by and considering the real part of the equation, (3.57) is obtained,

where is the air gap power, |

| is the copper losses in the rotor

circuit, and is the power injected to the rotor.

(

) (

) (3.57)

Using the equivalent circuit in Figure 3.22, it is easy to verify the power balance equation in

(3.58), where is the mechanical power at the generator shaft.

(3.58)

Using (3.57)-(3.58), (3.59) can be deduced [78]. In general is imposed by the prime mover

and it has a negative sign due to the motoring sign convention used. Furthermore, is much

smaller than and . Consequently, (3.59) shows that the power injected to the rotor, , can

be used to adjust the slip and speed of the WRIG.

(3.59)

In general, depending on the injected rotor power, the WRIG has two operating modes:

1. Sub-synchronous generating mode [78]

In this mode the WRIG rotates at speeds lower than the synchronous speed ( ).

This mode is achieved by injecting active power to the rotor from the grid. Since is negative

and is positive for the typical slip range of of a DFIG, (3.59) shows that

is positive when . A positive slip implies that the WRIG is running at a speed

smaller than the synchronous speed.

88

2. Super-synchronous generating mode [78]

In this mode the WRIG rotates at speeds higher than the synchronous speed ( ).

This mode is achieved by generating active power from the rotor and injecting it to the grid.

Since is negative and is positive for the typical slip range of of a

DFIG, (3.59) shows that is negative when . A negative slip implies that the WRIG is

running at a speed higher than the synchronous speed.

From the previous steady state analysis, the following comments are in place:

The magnitude and direction of the rotor power determines the rotational speed of the

DFIG and its operating mode (sub-synchronous or super-synchronous mode).

The power flow between the rotor and the RSC can be controlled by controlling the

magnitude and phase of the rotor injected voltage . The desired frequency, magnitude

and phase of the rotor voltage is obtained by controlling the RSC.

By neglecting the rotor and stator copper losses, one can see that

(from

(3.59)) and and therefore [78],

(3.60)

This implies that the rotor power is only a fraction of the stator power.

As the typical DFIG slip range is , the power converters should be rated at around

of the stator rated power. This partial rating of the converters represent the main

economic advantage of the DFIG configuration [57].

3.6.2 Modeling of the grid-side converter connection to the grid

Figure 3.23 shows the schematic of the GSC connection to the grid. The inductor filter is

represented by a resistance and an inductance . The voltage equations across the inductor

89

are given by (3.61), where [ ] is the three phase WTG

terminal voltage vector, [ ] is the three phase GSC terminal voltage

vector and [ ]

is the vector representing the three phase currents flowing

between the GSC and the grid [26, 106].

(3.61)

Voltage equations in (3.61) can be transformed into a synchronously rotating reference by

transforming the abc-phase voltages and currents using where is defined in (3.31).

The transformed voltage equations are

(3.62)

GSCdc linkGrid

vcC

I1I2

Vdc

Smoothing

inductor filter

vbC

vaCva,WTG

vb,WTG

vc,WTG

iaC

ibC

icC

Rf

Rf

Rf

Lf

Lf

Lf

PC,QC PGC,QGC

Figure 3.23 GSC connection to the grid [26].

The active power and reactive power transferred to or from the grid, as shown in

Figure 3.23, is given in (3.63)-(3.64), and the active power of the GSC is calculated using (3.65).

(3.63)

(3.64)

(3.65)

90

The modeling equations of the GSC connection to the grid are (3.62)-(3.65). The model receives

the WTG voltages , specified by the grid, and the GSC terminal voltage

, determined

by the GSC, as inputs to the model. The outputs are , an . The model has two internal

states as can be seen from (3.62).

3.6.3 Modeling the power converters

Due to the high switching frequency of the power electronics, the dynamics of the power

converters are much faster than the dynamics of the rest of the system. For that reason, the

converter dynamics can be neglected [106, 111, 118, 119]. For control purposes, it will be

assumed that the converters are able to instantaneously follow the reference values of the

voltages and

calculated by the generator control system. Therefore, and , and

and

will be used interchangeably. Furthermore, since the study of the harmonics produced by the

converters is out of the scope of this thesis, only fundamental frequency components are

assumed.

3.6.4 Modeling the converter dc Link

Figure 3.24 shows the schematic of the converter dc link and the power flow through the GSC,

the dc link and the RSC.

GSCdc link vCabc

vrabc

irabc iCabc

Vdc

I1I2

RSC

PCP1P2Pr

Figure 3.24 Converter dc link schematic.

91

The charging and discharging of the capacitor at the dc link can be described by [106, 119]

(3.66)

In general, the RSC and the GSC converter losses are small (around 1% of rated output

power). By neglecting converter losses, it is easy to see that and (see Figure

3.24). Therefore, the dynamics of the dc link are modeled by (3.67) where, and are

calculated using (3.65) and (3.41), respectively.

(3.67)

3.6.5 Modeling the generator control system

As shown in Figure 1.6, the WECS control system is composed of three control levels. In

order to design and test control strategies for the wind turbine control level, the generator control

system should be considered and modeled as a part of the controlled plant.

Vector control is the standard method used in the design of wind turbines’ generators

controllers [11, 22, 23]. For the DFIG system in Figure 3.19, the method relies on representing

all the system variables in a synchronously rotating orthogonal d-q coordinate system. In this

reference frame, decoupled control of the active and reactive power of the DFIG can be

implemented. This yields much faster dynamic responses compared to scalar control methods.

The price is that a more complicated estimation should be implemented to transform the

machines’ variables from the a-b-c to d-q coordinate system.

The block diagram representation of the DFIG generator controller is shown in Figure

3.25. All design details of the vector control strategy for DFIGs are provided in Appendix A. The

RSC consists of four Proportional Integral (PI) controllers and cross-coupling compensation

terms and

as shown in Figure 3.25. This control structure allows decoupled control of

92

the generator torque and the stator reactive power . The GSC controller has a similar

structure to the RSC and it performs decoupled control of the dc link voltage and the reactive

power exchange between the GSC and the grid, . For more details about the generator

controller, see Appendix A.

PI PI

PI PI

PI PI

PI PI

Rotor Side Converter Controller

Grid Side Converter Controller

Generator Controller

Tg*

Tg

iqr*

iqr

vqr*

idr*

idr

vdr*

Qs

Qs*

Vdc*

QGC*

QGC

Vdc

idC*

idC

vdC*

iqC*

iqC

vqC*

ccqrv

ccdrv

cc

qCv

cc

dCv

Figure 3.25 Generator controller.

3.7 Grid interonnection

Modeling of different power system components and interconnections is a complex task.

Depending on the purpose of study, various models with different degrees of complexity can be

used to represent a power system. Power systems modeling is treated in detail in [64, 114, 116].

Here, the main concepts are outlined.

The general modeling framework, known as the quasi-sinusoidal (or phasor)

approximation of power system dynamics, is adopted here [64, 114, 116]. In this framework, the

transmission network is modeled by a set of steady state algebraic equations, while other

dynamic elements such as synchronous generators, wind turbines, and induction motors are

93

modeled using detailed dynamical models. The resulting model is a set of differential-algebraic

equations of the form [120]:

(3.68)

(3.69)

where (3.68) represents the dynamics of the power system dynamic elements; and (3.69)

represents the power network equations. Here and are extended vectors consisting of sub-

vectors related to the modeled elements of the power system. In (3.69), is a vector containing

complex phasors of the currents injected into network, a vector containing complex phasors of

the phase to neutral bus voltages, and is the network nodal admittance matrix. All and

components must be referred to the same synchronously rotating reference frame. In multi-

machines simulation with an infinite bus, its voltage phasor can be conveniently chosen as the

reference phasor for the angles of other bus voltages. If there is no infinite bus, the d or q axis of

one of the generators reference frames may be chosen as the reference axis instead [114].

Following the approach described above, the DFIG-based wind turbine interaction with

the grid can be simulated by solving the network equation to calculate the terminal voltage of the

DFIG. This voltage is used to calculate the currents injected by the DFIG using the DFIG

dynamical equation. This procedure is continuously repeated during the whole simulation.

3.8 Modeling in per unit system

Modeling using per unit quantities is a common practice in power systems. In the per unit

system, all variables are normalized with respect to user defined base values that are usually the

nominal (rated) values. Here, all quantities in per unit are denoted by an overbar and they are

defined as follows [116].

94

Definition 3.1: Quantity in per unit,

(3.70)

The use of per unit quantities offers several advantages such as:

1. The values of equipment parameters expressed in per unit stay in a fairly narrow and

known ranges irrespective of equipments ratings.

2. Calculations using the per unit system are simpler. In the presence of transformers, there

is no need to render quantities at the high or low voltage side.

3. It is more useful to express electrical quantities as percentages of their rated values.

The base values of the per unit system used here are chosen to match the generator rated

values. They are summarized in Table B.1 in Appendix B. According to this selection, the per

unit system representation of the modeling equations (3.22)-(3.23), (3.35)-(3.42), (3.62)-(3.65)

and (3.67) can be easily obtained, as summarized in Appendix B.

3.9 Overall WECS model

By combining the dynamical equations (3.19), (3.22), (3.35), (3.62) and (3.67) that

governs the WECS subsystems with the dynamics of the generator controller in Figure 3.25, the

overall WECS dynamics can be described by:

(3.71)

where [

] is the state vector, is defined in (3.72),

and are vectors containing the states of the four PI regulators of the RSC and GSC

controllers in Figure 3.25. In (3.71), [ ] is a vector containing

95

all inputs affecting the WECS. The nonlinear function can be easily obtained

from (3.19), (3.22), (3.35), (3.62), (3.67) and Figure 3.25.

[ ] (3.72)

Finding an accurate operating point for the WECS (3.71) is important for initializing the

model during simulation studies. This eliminates any initial transients that might result from

using wrong initial conditions. Furthermore, the knowledge of an operating point allows

performing linear analysis on the nonlinear system (3.71).

The calculation of an operating point for a grid-connected wind turbine is not straight-

forward. It requires the solution of the set of nonlinear algebraic equations (3.19), (3.22), (3.35),

(3.62) and (3.67) with

set to zero. Furthermore, the calculation depends on the operating

region of the WECS. In the full load regime, the WT operation is characterized by the following:

The DFIG output power is equal to the rated value,

The turbine speed is equal to the rated value,

A wind speed value satisfying must be picked in order to uniquely define

the operating point.

On the other hand, in the partial load regime (Region I.1 in Figure 2.13), the following

conditions must be satisfied:

The tip speed ratio is equal to it optimum value,

The pitch angle is fixed at zero,

A DFIG output power satisfying must be picked in order to uniquely

define the operating point.

96

In both the partial load and full load cases, the WTG output power is known. Furthermore,

the WTG reactive power, , is known since both and are controlled by

the generator controller. The knowledge of and allows performing load flow analysis

for the power system under study, thus calculating the values of time phasors of the terminal

voltage, , and WTG current . The d-q stator voltages can be easily calculated from

as described in §3.6.1.2. After this step, the nonlinear algebraic equations (3.19), (3.22),

(3.35), (3.62) and (3.67) with

set to zero must be solved simultaneously to find all initial

values of the WECS variables. Nonlinear equation solvers must be supplied with initial values

for the unknowns that should be close to the solution. Procedure 3.1, proposes a method to find

these values. This procedure is based on neglecting the losses in the DFIG system and using the

WRIG equivalent circuit in Figure 3.22.

Procedure 3.1 (Quick initialization of the WECS variables)

Step 0: if the WT is operating at partial load, calculate the turbine speed and generator slip using

√

(see 2.16 and let )

(see (3.25))

Step 1: calculate the power flow in the DFIG as follows

(neglect losses of the DFIG)

(see (3.59))

(see (3.60))

(neglect losses in power converter and the GSC filter)

Step 2: calculate the stator and GSC apparent power using

97

,

Step 3: calculate stator and GSC current phasors using

(

)

, (

)

Step 4: calculate the GSC terminal voltage phasor using

( )

Step 5: use the equivalent circuit in Figure 3.22 to calculate the following

,

,

, and

(

)

Step 6: calculate all d-q variable corresponding to , , , and , and calculate stator and

rotor flux linkages using (3.36).

Step 7: if the WT is operating at full load, calculate by solving the nonlinear equation

(

)

3.10 Summary

In this chapter, the modeling of VSVP DFIG-based wind turbines is detailed. The model

includes relevant aerodynamic and mechanical aspects such as rotational sampling, wind shear,

tower shadow, and drive train resonance, as well as the electrical aspects such as the WRIG, dc

link, and GSC link dynamics. The initialization of the model is discussed and a quick procedure

that approximately calculates WECS operating points is proposed. The model will be used in

Chapter 5-7 to test and simulate the WECS with the developed control strategies. This allows

performance assessment of the developed methods in terms of energy maximization, mechanical

load reduction, and voltage flicker mitigation.

98

Model Predictive Control Chapter Four:

Model-based Predictive Control is the main tool that is used in this thesis to develop

effective wind turbine control strategies. This chapter reviews main MPC concepts and some

selected results that are relevant for this work. For a more complete coverage of MPC theory, see

[121-125]. The basic MPC idea is introduced in §4.1. The ingredients that are used to construct

an MPC algorithm are described in §4.2. In §4.3, the MPC control policy is formulated as an

optimization problem to be solved at each sampling instant. The main properties of MPC

controllers are described in §4.4-§4.6. A discussion on different alternatives that can be used to

implement a real-time MPC controller is given in §4.7.

4.1 Introduction

Model-based predictive control is the only one among all advanced control techniques

(more advanced than PID controllers) which has been tremendously successful in industrial

applications in recent decades [125-127]. More than 4600 MPC applications in petrochemicals,

automotive, aerospace, and food industries are reported in [127]. The main reasons for this

success are [123, 125]:

It can directly take into account physical constraints on the inputs, outputs and states of

the controlled plant.

It allows the plant to operate near its limits. This typically leads to more profitable

operation of the plant.

It can be applied to multivariable (MIMO) systems in a systematic way. In contrast, the

design of PID controllers for such systems is usually very difficult.

The inclusion of feed-forward from measurable disturbances can be easily implemented

within the MPC formulation.

99

The main idea of MPC algorithms is easy to understand; and the tuning of MPC

controllers is relatively intuitive and easy.

It can handle complex control objectives, allowing the user to perform required trade-offs

between different performance requirements.

The basic idea behind MPC can be explained as follows [128, 129]. First, a dynamic

model of the controlled system and the constraints on the system variables must be known. At

each sampling instant, the dynamic model is used to predict the future system behavior within a

predefined prediction horizon. The optimal control sequence is calculated by solving a

constrained optimization problem that includes the system constraints and a performance index

that reflects the system performance. The first input in the optimal sequence is then sent to the

system while the rest is discarded. The entire calculation is repeated at subsequent control

intervals with shifted prediction horizons and using the current measurements of the system. The

continuous shift in the prediction horizon is known as the receding horizon concept. The idea is

illustrated in Figure 4.1.

It should be noted that MPC resembles, in some sense, human behaviour. An individual

usually selects his/her current decision that will lead to the best predicted outcome within a

certain prediction horizon. For example, in the process of driving, the driver considers the visible

distance of the road to anticipate for any potential dangers and to take the best possible decisions

during driving without violating constraints (speed limits, street borders, etc…). The visible zone

of the road is continuously receding as the car is moving and the decisions are updated based on

the most recent information.

100

Output

Input

Time

Time

Constraints

Reference

t0

Prediction

Horizon

t0+1 t0+Np

(a) MPC at time t0

t0+Np+1

Output

Input

Time

Time

Constraints

t0

Prediction

Horizon

t0+1 t0+2

(b) MPC at time t0+1

t0+Np+1

Figure 4.1 MPC concept.

All practical control systems have an associated set of constraints [123, 130]. Input

constraints typically arise from actuator limitations such as position and slew rate limits of a

valve or a servomechanism. On the other hand, output constraints typically arise from safety and

design considerations such as maximum limits on the temperature of an oven or the speed of a

generator. Typically, the most profitable operation of the plant occurs by “pushing the system

101

hard” and operating the system near some of these constraints. Therefore, constraints cannot be

ignored if one is seeking high performance [121, 123, 125].

The main salient feature that distinguishes MPC from other control methods is the

explicit inclusion of system constraints in the controller formulation [126, 128, 129]. This feature

allows the MPC to anticipate and prevent future constraint violations while optimizing the

system performance. Other control methods deal with system constraints using basically two

approaches [123, 126]. In the first approach, a (linear) controller is designed first after ignoring

all systems constraints. After the design is completed, constraints are dealt with using ad-hoc

fixes or other systematic approaches such as integrator wind-up [131]. In the second approach,

the controller is “detuned” and the system is operated far from the constraints, and thus

constraint violations are avoided [123]. Clearly, both approaches are not as effective as MPC that

systematically deals with system constraints from the very first beginning of the controller

design [123, 125].

MPC controllers have a nonlinear behavior [121, 123, 125, 132]. Due to the MPC

awareness of the constraints, the controller can react very differently when positive and negative

disturbances with the same amplitude are applied to a system operating near an output constraint.

In one of these cases, the disturbance will push the output toward the constraint and the MPC

will react aggressively to prevent constraint violation irrespective of any performance loss. In the

other case, the disturbance is moving the system far from the constraint and thus the MPC will

react in a relaxed way. Clearly, this behavior can never be achieved using any linear controller.

Many variants of MPC algorithms with different names and acronyms exist [125, 127,

133]. The first versions of MPC algorithms are the Model Predictive Heuristic Control (MPHC)

proposed in [134], and the Dynamic Matrix Control (DMC) proposed in [135]. Since then (early

102

1980s), many MPC algorithms have been proposed, including Generalized Predictive Control

(GPC) [136, 137], Quadratic Dynamic Matrix Control (QDMC) [138], and Receding Horizon

Control (RHC) [122]. These algorithms mainly differ in the type of model and objective function

used. However, all of them share the essential features of predictive control: an explicit internal

model, the receding horizon idea, and the computation of the control signal by solving an

optimization problem.

MPC algorithms require online solution of a quadratic program (QP) during each

sampling interval. Compared with PID controllers, MPC is a computationally demanding

algorithm. This initially restricted the application of MPC to petrochemicals applications with

slow dynamics and large sampling periods [127, 133]. Currently, with the tremendous increase in

computers’ speeds and advances in computational algorithms [139-144], there is widespread use

of MPC technology in a wide variety of application areas with relatively fast dynamics,

including aerospace [145, 146], automotive [147-149] and power converters [150-152]

applications, to name a few. It is reported in [140] that an MPC algorithm involving 12 states, 3

controls, and a horizon of 30 time steps (which entails solving a QP with 450 variables and 1284

constraints) can be solved in about 5 ms. Clearly, MPC can now be applied in control problems

approaching kilohertz sample rates.

4.2 MPC ingredients

For any MPC algorithm, the main components that must be specified by the designer are

the prediction model, performance index, system constraints, and state estimator. Details of these

components are described below [124, 125].

103

4.2.1 Prediction model

A model of the plant is essential to predict the future behavior of the system.

Conceptually, any model including finite impulse response, step response, transfer function,

linear state space, or nonlinear models can be used. In Chapter 5-7, linear discrete-time state

space models are used. This selection makes the predictions linear (affine) with the inputs, thus

facilitating the optimization problem. Furthermore, it is well known that state space models are

best suited for multivariable system [125].

The model considered is described by (4.1)-(4.3), where is the state vector,

is the input, is the unmeasured disturbance, is the measurable

disturbance, is the measured output, is the controlled output. In general, the

variables and overlap and in some cases they will be the same. In such cases the model is

described by (4.1)-(4.2) and will denote both the measurements and the controlled outputs.

(4.1)

(4.2)

(4.3)

The fictitious disturbance is used to represent the effect of unmeasured plant

disturbances and model uncertainties. It is modeled as the output of the linear time invariant

system (4.4)-(4.5), where is the disturbance state vector. The eigenvalues of the matrix

are chosen inside and/or on the unit circle.

(4.4)

(4.5)

The model (4.1)-(4.5) can be written compactly as

104

(4.6)

(4.7)

(4.8)

where

[

], [

], [

], [

]

[ ], [ ]

(4.9)

4.2.2 Objective function

The objective function to be minimized by the MPC controller is given by (4.10) [124,

125]. Here, , , is the i-step ahead prediction of based on measurements up to

time ; i.e. based on output measurements up to and knowledge of the inputs only up to

, since the current input has not yet been determined. In (4.10), is the

prediction horizon, is the control horizon, and is the set of all positive integers. It is

assumed that . The control move is defined as . The symbol

‖ ‖ denotes a weighted norm.

∑‖ ‖

∑ ‖ ‖

(4.10)

The weighing matrix is used to penalize the deviations of the future

controlled output from the desired reference . On the other hand, the weighing matrix

is used to penalize changes in the input vector. Typically these matrices have a diagonal

structure and their diagonal elements are tuned to achieve a desired trade-off between tracking

performance of different control variables, and the activity of each control input. Other tuning

parameters that affect the controller performance are and . By increasing their values,

105

MPC tends to perform as an infinite horizon optimal controller, typically leading to better

performance and stability properties [121, 124, 125].

4.2.3 Constraints

Here, the linear constraints on the inputs increments, inputs, and controlled outputs

(4.11)-(4.13) are considered. In general, the set is a polytope defined by

, and the sets and are polytopes defined similarly.

, (4.11)

, (4.12)

, (4.13)

One common case found in applications is when the system variables are limited between

maximum and minimum values. In this case, the constraints can be expressed in (4.14)-(4.16),

where denotes the maximum (minimum) dynamical limit of •.

, (4.14)

, (4.15)

, (4.16)

It is common to classify constraints into hard and soft ones [124, 125]. The hard

constraints are the ones that can never be violated such as the maximum and minimum limits of

a valve opening. On the other hand, soft constraints are the ones that are typically imposed on

the controlled outputs. These constraints can be violated in the presence of extreme disturbances

on the system.

106

4.2.4 State estimation

To predict the future outputs of the system using the model (4.1)-(4.5), the current state

of the plant and the disturbance model must be known. Typically, not all these states

are measurable and an estimator should be used to reconstruct them. Here, the estimator in

(4.17)-(4.18) is used, where | and | denote the estimate and one-step ahead

prediction of based on measurements up to time , respectively. The observer gain

can be designed using pole placement or Kalman filtering techniques [153, 154].

| | ( | ) (4.17)

| | (4.18)

4.3 MPC optimisation problem

In this section the MPC optimization problem is formulated as described in [125]1. Using

the ingredients in §4.2, the MPC optimization problem that is solved online at each sampling

instant can be expressed as in (4.19)-(4.27). Since the constraints are linear and the objective

function is quadratic, this optimization problem can be cast as a standard QP. This will be

outlined in the rest of this section.

∑‖ ‖

∑ ‖ ‖

(4.19)

, (4.20)

, (4.21)

[ | | ] , (4.22)

1 The notation used in this section is different from the one in [125].

107

, (4.23)

, (4.24)

, (4.25)

, (4.26)

, (4.27)

4.3.1 Constructing the predictor

To write (4.19)-(4.27) as a standard QP, the first step is to eliminate the equality

constraints in (4.20)-(4.21) representing the prediction model. This is done by expressing the

future output predictions as linear (affine) functions of the inputs, i.e. constructing the predictor.

Using (4.20)-(4.21), the prediction is given by (4.28). Thus, the vector containing

future output predictions can be expressed as in (4.29).

∑

[ ]

(4.28)

[

] [

]

⏟

[

]

⏟

[

]

[

]

⏟

[

]

(4.29)

To simplify (4.29), the stacked vector notation is used as follows. Let , then the

stacked vector is defined as

108

[ ] (4.30)

Using this notation, the predictor can be written as (4.31) where , , and are

defined in (4.29).

(4.31)

By observing that ∑ , (4.32) can be obtained, where

and are defined in (4.33).

(4.32)

[

], [

] (4.33)

Using (4.24), and (4.31)-(4.32), the predictor can be written in (4.34), where

and are defined in (4.35).

(4.34)

(4.35)

4.3.2 Formulating the MPC optimization problem as a standard QP

Based on the predictor (4.34) and using the stacked vector notation, an equivalent form of

(4.19)-(4.27) can be given by (4.36)-(4.41).

‖

‖

‖ ‖

(4.36)

(4.37)

[ | | ] (4.38)

109

(4.39)

(4.40)

(4.41)

Here, and are defined in (4.42), ,

, [

] and [

]

. Similar definitions are used for , , and , .

[

], [

] (4.42)

Using (4.36)-(4.41), it is straightforward to write the MPC optimization problem as the

QP

(4.43)

(4.44)

where

is the optimal solution of the QP, is a constant that is irrelevant to the QP

solution, and , , , , and are defined in (4.45)-(4.49), respectively.

(4.45)

(4.46)

(4.47)

[

]

(4.48)

110

[

]

(4.49)

If future information about the reference and the measured disturbance is available, then

it should be used in (4.47) and (4.49), respectively. However, if such information is not

available, one can assume that the reference and the measurable disturbances are constant within

the prediction horizon, and equal to their current values and , respectively. In that case,

and

can be set to (4.50) [125]. This assumption will be adopted here.

[

] , [

] . (4.50)

The optimization problem is strictly convex since for any and .

Consequently, the problem has a unique global minimum. The QP has decision variables

and linear constraints.

The MPC controller solves the QP (4.43)-(4.44) at each sampling instant. Only the first

optimal control increment is implemented. The MPC control law can be written in (4.51).

Figure 4.2 represents a closed loop system with an MPC controller.

[ ]

(4.51)

111

Plant

Estimator

MPC

z-1

r

v d

y

u

x

dx

Figure 4.2 Closed loop MPC control system.

4.4 Analysis of MPC controllers

The explicit MPC control law that defines as a function of the states, the reference,

and the measureable disturbance is hidden by the optimization formulation (4.43) and (4.51).

Knowing the MPC control law is useful in analysing the performance and stability of the system.

This topic is considered in [123, 125, 143, 155] and the main results are summarized in this

section.

4.4.1 Unconstrained MPC

In this subsection, the simple unconstrained MPC controller is described. Assume that

there are no constraints in the MPC formulation or that the constraints (4.44) are inactive at the

optimal solution of the QP (4.43)-(4.44). In both cases, the optimal solution is given by (4.52).

(4.52)

Using (4.47) and (4.50), it is straightforward to show that the unconstrained MPC control

law is given by [125]

(4.53)

112

where

[ ] [

]

[ ]

[ ]

[ ] [

]

(4.54)

It can be seen that the unconstrained MPC is basically an observer-based linear time-

invariant state feedback controller. Consequently, all linear analysis tools can be used to analyse

unconstrained MPC controllers. In particular, stability analysis can be carried out based on the

closed loop poles, and frequency response analysis.

4.4.2 Constrained MPC

To derive the explicit MPC control law for the general constrained case, it should be

noticed that both and in (4.43)-(4.44) depend linearly on a parameter vector defined in

(4.55). The other QP matrices and are constants. To emphasize that, the QP (4.43)-(4.44)

is rewritten in (4.56)-(4.57), where the time argument is omitted to simplify notation, and ,

and are constant matrices that are easily obtained from (4.43)-(4.44).

[ ] (4.55)

(4.56)

(4.57)

113

Clearly, the optimal solution of (4.56)-(4.57),

, is dependent on . This type of

optimization problem is known as a multi-parametric quadratic program (mp-QP) [155]. Its

closed-form solution is the function

, where is the set

of all for which (4.57) is feasible. The function

is the explicit MPC control law.

The optimality conditions on the optimizer can be explored to derive the explicit MPC

control law as follows [155]. Assume that for a certain feasible parameter vector , the QP

(4.56)-(4.57) is solved and let

be the optimal solution. Therefore,

satisfies the

Karush–Kuhn–Tucker (KKT) conditions [156]

(

)

(4.58)

(4.59)

(4.60)

(4.61)

where is a vector of Lagrange multipliers, and is the number of

inequality constraints in (4.57). Assume that the active constraints at

are known and

described by (4.62), where , and are formed by the rows of , and corresponding

to these active constraints, respectively. Let and denote the Lagrange multipliers

corresponding to active and inactive constraints, respectively. Thus, (4.63) and (4.64) can be

obtained from (4.58)-(4.59) and (4.61).

(4.62)

(4.63)

(4.64)

114

Assume that the rows of are linearly independent, then (4.62) and (4.64) can be

simultaneously solved yielding (4.65). Clearly,

and are both affine functions of the

parameters vector .

[

] [

]

([

] [ ])

[

] [

] [

]

(4.65)

It can be concluded from (4.65) that, for a fixed set of active constraints, the solution of the mp-

QP, and the explicit MPC control law are affine functions of [143, 155].

The largest set of parameters for which the set of active constraints in (4.62) is the

optimal active set and for which

in (4.65) is the optimizer can be specified from (4.57) and

(4.63) as in (4.66) [143, 155]. From (4.65), it can be seen that the set (4.66) is indeed a polytope

in the parameter space .

(4.66)

In principle, one can perform the previous calculations starting from any feasible

parameter vector . This will reveal the MPC control law within a certain polyhedral partition

corresponding to the current set of active constraints. This procedure is successively repeated and

new polyhedral partitions and their associated MPC control laws are explored at each step. This

procedure is stopped when the whole set of parameters of interest has been explored and thus the

explicit MPC controller has been identified. Many algorithms were suggested implementing this

idea [143, 155, 157]. The following theorem summarizes the discussion.

115

Theorem 4.1 [155]: Consider the MPC control law (4.51), where

is the optimizer

of the multi-parametric quadratic program (4.56)-(4.57). Then, the explicit MPC control law is a

continuous piecewise affine function of over polyhedra, i.e. the MPC control law is given by

(4.67)

where the polyhedral sets form a polyhedral partition of a given

feasible polyhedral set of parameters of interest.

Proof: see [155].

Remark 4.1: It is discussed in [155] that the number of polyhedral partitions in (4.67) can grow

exponentially with the number of constraints of the QP. That is why the calculation of the

explicit MPC (4.67) is only limited to small sized problems with small numbers of constraints

and prediction horizons.

Remark 4.2: The MPC controller in (4.67) is indeed nonlinear.

4.5 Offset-free MPC

To motivate the concepts described in this section, consider a PI controller in a feedback

control loop as shown in Figure 4.3. By assuming that the closed loop system is asymptotically

stable, the controller integral action guarantees that the output must be equal to the desired

reference at steady state. This property is ensured irrespective of the plant dynamics, and the

presence of unknown piecewise constant disturbances affecting the plant. It is said that the PI

controller guarantees offset-free tracking [158].

116

Plantr yu

di do

dp

+

-

KP

KI ∫

Σ

PI controller

+

+

+

+

Figure 4.3 PI feedback control loop.

Definition 4.1: Offset-free tracking [121, 158, 159]

Offset-free tracking is a property that ensures that the controlled variables converge to their set

point values without offset at steady state. Mathematically, this is expressed as

as . (4.68)

Definition 4.2: Offset-free controller

An offset-free controller is a controller that guarantees offset-free tracking.

Obviously, it is desirable that MPC controllers guarantee offset-free tracking.

Unfortunately, this property is not automatically satisfied by the MPC formulation described in

§4.3 and special considerations must be taken to ensure this property. This problem is studied in

[158-160] and the main results are summarized below.

In this section, it is assumed that the controlled outputs are linear combinations of the

measurements and they are less than or equal to the number of measurements as shown in (4.69).

This assumption is required to guarantee offset-free tracking of .

, (4.69)

One general framework that is used to compensate for unmeasured disturbances is to

augment the process model to include a model of the disturbance. Then, an observer is used to

117

estimate this disturbance based on the output measurements. Finally, the controller is designed to

(ideally) eliminate the effect of the disturbance on the controlled outputs [121, 153, 161]. Here,

the objective is to ensure offset-free tracking in the presence of piecewise constant disturbances.

Therefore, it is natural to use integrated white noise (4.70)-(4.71) as the disturbance model (4.4)-

(4.5) of the MPC formulation [158-160, 162]. This is achieved by setting , and as shown

in (4.72). This choice is also consistent with the internal model principle introduced in [163].

(4.70)

(4.71)

, and

(4.72)

Using this selection, the augmented system model used for the state estimator design is

given by

[

] [

] [

] [

] [

] (4.73)

[ ] [

] (4.74)

Through the choice of and , one can design the integrated disturbances to be

applied at the plant input, output, state, or a combination of these alternatives. The only

restriction is that the augmented system must be detectable. The detectability of the augmented

system (4.73)-(4.74) can be checked using Lemma 5.1 [158].

Lemma 4.1 [158]: The augmented system (4.73)-(4.74) is detectable if and only if the non-

augmented system is detectable, and the following condition holds

[

] (4.75)

118

As described in [124, 158-160], the use of the disturbance model in (4.70)-(4.71) is not

sufficient to guarantee offset-free tracking using an MPC controller. The general conditions that

are required to design an offset-free MPC controller are [124]:

1. The objective function used in the MPC formulation must be well-posed in the sense that

the minimum of must be consistent with zero tracking errors.

2. The output predictions are unbiased at steady state.

The objective function (4.19) or (4.36), with , satisfies the first condition because

occurs when

and . The second condition can be

ensured by applying Lemma 4.2 [158].

Lemma 4.2 [158]: By choosing and designing an asymptotically stable observer for

(4.73)-(4.74), (4.76) is satisfied, where , | , and

| are the steady state output measurements, process state and disturbance state

estimates (one-step predictions).

(4.76)

Proof [158]: From (4.72), (4.9) and (4.17)-(4.18), the observer equations can be written as

[ |

| ] [

] [ |

| ]

[ | | ]

[

] [

]

(4.77)

where

[

] [

]. (4.78)

At steady state, the second block row of (4.77) reduces to

119

[ ] (4.79)

If is full rank, then (4.76) must be satisfied. The fact that is full rank follows directly from

the stability of the observer (4.77)-(4.78) as detailed in [158].

Extended mathematical proofs and analysis of offset-free tracking MPC can be reviewed

in [158, 160]. The main design requirements for the MPC described in §4.3 to satisfy offset-free

tracking can be summarized as:

The MPC disturbance model is taken as integrated white noise model such as in (4.70)-

(4.71)

The number of integrated white noise disturbances is equal to the number of

measurements

(4.80)

and are chosen such that the augmented system (4.73)-(4.74) is detectable

Remark 4.3: Intuitively, it might appear to be sufficient to use integrated disturbances to

ensure offset-free tracking of . However, in cases where , the matrix , with

dimension , will indeed have a nonzero null space, and thus there might be an offset

between and .

4.6 Stability

MPC has a reputation for usually giving closed-loop stability [126]. It is interesting to

know that MPC technology had been successfully and widely used in industrial applications for

more than 15 years before any theoretical result concerning its stability appeared [127, 130, 133].

During this period, posteriori stability checks were done by performing linear analysis on the

unconstrained MPC control law or by using simulations. Despite that, the development of MPC

120

stability theory succeeded in providing a solid theoretical basis for MPC and in enhancing the

understanding of MPC properties [124].

Proving stability of MPC algorithms is a nontrivial task. The main difficulty is that MPC

controllers optimize the system performance over a finite future horizon, while stability is a

characterization of the system over an infinite future horizon [123]. To resolve this conflict, the

MPC should be formulated for an infinite horizon so that stability can be guaranteed. This is

done by setting and in (4.19). However, a problem immediately arises in this

context. The tractability of the MPC optimization (4.19)-(4.27) is lost as the optimization

problem now has an infinite number of decision variables. Many approaches have been proposed

in the literature to solve this dilemma. One of them is outlined below [132]. To simplify the

discussion, the linear time-invariant discrete-time system (4.81) with constraints (4.82)-(4.83) is

considered. The control objective is to regulate the state of the system to the origin. Finally, it is

assumed the system states are measurable and that .

(4.81)

, (4.82)

, (4.83)

Dual mode control is one of the main ideas used to convert infinite horizon problems into

finite horizon ones that are suitable for MPC implementation [124]. The general idea of dual

mode control is to use two control laws, one is applied when the system is far from the

equilibrium point, and the other is used when the system is close to the equilibrium. In the MPC

context, the control law in the second control mode, known as the terminal control law, is

predetermined and is typically a simple state feedback controller. In contrast, the control inputs

121

in the first mode are left free and taken as optimization variables. Using this approach, the

number of decision variables is finite and thus the optimisation is tractable.

Based on the dual mode idea, a natural choice of the terminal control law is the

unconstrained LQR

(4.84)

where

(4.85)

and is the solution of the algebraic Riccati equation (4.86).

(4.86)

Furthermore, let be an invariant set for the system in which

the constraints are satisfied. This means that

, for (4.87)

There exist many algorithms that can be used to calculate the set [124, 164]

With these ingredients, consider the infinite horizon optimization problem (4.88)-(4.93).

∑

(4.88)

, (4.89)

, (4.90)

, (4.91)

, (4.92)

, (4.93)

122

The use of the terminal constraint in (4.92) ensures that the constraints will not be violated by

applying the terminal control law (4.93) for . Also, the objective function in (4.88) can be

written as , where ∑ [ ] and

∑ [ ] . From optimal

control theory, it can be seen that and thus (4.88)-(4.93) is

equivalent to (4.94)-(4.98). Clearly, the problem has been converted into a finite horizon one that

is suitable for MPC implementation. The associated MPC control law is (4.99). Now, it remains

to verify that this controller yields a stable control law. This is shown in Theorem 4.2.

∑[ ]

(4.94)

, (4.95)

, (4.96)

, (4.97)

, (4.98)

[ ] (4.99)

Theorem 4.2 [123, 132]: Consider the closed loop system formed by the system (4.81) that is

controlled by the MPC control law (4.94)-(4.99). Assume that and

;

and that and are chosen as in (4.86) and (4.87), respectively. Then, the origin of the

closed loop system is asymptotically stable.

Proof: See [123, 132].

The main idea of the proof of Theorem 4.2 is to show that the condition (4.100) is

satisfied, where is the optimal cost of (4.94)-(4.98). This condition implies that is non-

123

increasing along the closed-loop system trajectory. Since is bounded below by 0, the sequence

converges and consequently, and as follows

from (4.100) and the positive definiteness of and .

( ) ( ) (4.100)

In summary, there are three main ingredients to ensure stability of an MPC controller

[123, 132]. One should select a stabilizing terminal control law. A terminal constraint set that is

invariant and admissible under the terminal control law must be added to the finite horizon

formulation. Finally, a terminal state penalty must be used in the objective function of the finite

horizon optimisation to account for the infinite cost associated with the terminal control law.

It is important to note that the additional terminal constraint introduced to guarantee

stability renders the solution of the QP more difficult, possibly leading to feasibility problems

during real-time implementation [133]. It is shown in [165] that closed-loop stability can be

maintained without including the terminal constraint set in the MPC formulation. This is done by

selecting a sufficiently large prediction horizon. This comes with the price of increased

computational requirements.

4.7 Real-time MPC implementation

To implement the MPC controller described in §4.3 in real-time, the QP (4.43)-(4.51)

must be solved at each sampling time. This can be done using the two approaches described

below.

Online QP solver

In this approach, numerical optimization techniques are used to solve the QP online. The most

common QP solvers used in MPC problems are based on active set methods [166]. These

algorithms have guaranteed convergence. Interior point methods are also starting to be used in

124

MPC problems with very promising results in terms of computational speeds [139, 140, 142].

The details of these algorithms can be reviewed in [141, 166, 167].

Explicit MPC implementation

As described in §4.4.2, the QP (4.56)-(4.57) can be solved offline for a given set of parameters of

interest, and the corresponding explicit MPC control law (4.67) can be determined [143, 155,

157]. Once (4.67) has been calculated, the MPC is implemented online as a lookup table of

affine state feedback controllers. At each time step, a lookup table search for the current

parameter vector is done, followed by an affine control law evaluation. For small MPC

problems, with relatively few states and constraints, this approach can lead to very fast

computations. However, as pointed out in §4.4.2, when the problem size increases, the number of

polyhedral partitions in (4.67) can be huge. This can make the lookup table search time-

consuming, leading to slower calculations compared to online QP solvers [168].

Another important issue related to real-time implementation is the insurance of online

feasibility of the MPC optimization problem. The presence of constraints in (4.44) might render

the QP infeasible. Obviously, this is not acceptable in real-time implemetations because in that

case there is no calculated control input to be applied to the system. The most common approach

to prevent this problem is to make the output constraints soft. This is done by introducing slack

variables which are kept small by introducing a corresponding penalty term in the objective

function. Specifically, this is done by using (4.101) instead of (4.27), and (4.102) instead of

(4.19) in the MPC formulation, where is a slack variable, is used to penalize the amount

of constraint violation, and [ ] [125].

, (4.101)

125

∑‖ ‖

∑ ‖ ‖

(4.102)

4.8 Summary

This chapter reviewed the main MPC techniques, properties, and results. The standard

MPC formulation based on discrete time state space models is described. Offset-free tracking

and stability properties of the MPC controller are discussed. Different alternatives for

implementing MPC controllers in real-time are outlined.

There are many features that make MPC an effective control solution to the wind turbine

control problem. The most important ones are:

MPC is a multivariable controller that can fully utilize the control capabilities of the wind

turbine system.

MPC algorithms can directly take into account wind turbines’ constraints.

MPC is based on optimal control techniques. This is useful in achieving desired trade-

offs between different competing control objectives.

Novel wind turbine control strategies that are based on MPC are developed in In Chapter 5 and

6. In Chapter 7, MPC techniques are used to design a RSC control strategy that guarantees FRT

requirement for DFIGs.

126

Multiple Model MIMO Predictive Control for Variable-Speed Variable-Chapter Five:

Pitch Wind Turbines

A multivariable control strategy based on model predictive control techniques for the

control of variable-speed variable-pitch wind turbines is proposed in this chapter. The proposed

control strategy is described for the whole operating region of the wind turbine, i.e. both partial

and full load regimes. Pitch angle and generator torque are controlled simultaneously to

maximize energy capture, mitigate drive train loads and smooth the power generated while

reducing the pitch actuator activity. This has the effect of improving the efficiency and the power

quality of the electrical power generated, and increasing life expectancy of the installation.

Furthermore, safe and acceptable operation of the system is guaranteed by incorporating all

relevant constraints on the physical variables of the WECS in the controller design. In order to

cope with nonlinearities in the WECS and continuous variations in the operating point, a

multiple model predictive controller is suggested which provides desired performance

throughout the whole operating region.

This chapter is based on [169-171] and it is organized as follows. The wind turbine

control problem is introduced in §5.1. A simplified wind turbine model that is used by the

proposed MPC strategy is described in §5.2. A baseline wind turbine control strategy that is

commonly used in industrial wind turbines is described in §5.3. The proposed MPC control

strategy is detailed in §5.4. The MPPT algorithm that is used with the proposed wind turbine

control strategy is described in §5.5. Simulation results demonstrating the effectiveness of the

proposed strategy are given in §5.6, and §5.7 concludes the chapter.

127

5.1 Control problem description

As described in §2.5.2, a VSVP wind turbine has two operating regions, with different

control objectives, as shown in Figures 2.12-2.13 and redrawn in Figure 5.1 for convenience

[13].

Figure 5.1 Nominal operating trajectory of a VSVP wind turbine.

The partial load regime controller should maximize the wind turbine energy capture. This

is done by continuously adjusting the turbine speed according to (5.1), and thus is

maintained at its optimal value . By also fixing the blades’ pitch angle at its optimal value ,

the power conversion efficiency is maximized. Due to the continuous wind speed variation, it

can be seen that the reference turbine speed should be time-varying as given by (5.1). Therefore,

the partial load control problem is basically a servo (tracking) control problem.

(5.1)

In the full load regime, the main control objective is to regulate both the output power

and the generator speed at their rated values and , respectively. This

0 5 10 15 20 25 300

0.5

1

1.5

2

v, m/s

Pt,

MW

vci


vr v

co

0 5 10 15 20 25 300

10

20

30

v, m/s

, o

t, rpm

Full LoadPartial Load

vci

vr v

co

128

should be achieved in the presence of a severely fluctuating disturbance, the wind. Clearly, the

full load control problem is a MIMO regulator control problem.

The regulation of the WECS variables can be achieved by manipulating the pitch angle

set point and/or the generator torque set point . As shown in (3.14), can be regulated by

changing that can be interpreted as a variable gain controlled by and . Consequently,

manipulating the pitch angle results in deviations in the power extracted by the wind turbine and,

indirectly, induces deviations in the turbine speed via the drive train dynamics. Similarly, the

generator torque can affect the turbine speed through the drive train dynamics and can be used to

control the power extracted by the wind turbine by controlling . The design of a multivariable

controller that can harmonize the use of both the pitch and torque control can significantly

enhance the transient response of the system and reduce the pitch actuator activity [13, 36].

Clearly, the control objectives and schemes that are used in the partial and full load

regimes are different. The issue of how to combine the controllers designed separately for low

and high wind speeds is important. Most of the work reported in the literature consists in

switching between both controllers. Undesirable frequent switching between controllers may

occur when wind speed fluctuates around its rated value, which can lead to large transient loads

and power fluctuations [13]. Furthermore, when the WECS operates in the partial load regime

and near the rated wind speed, large output power and drive train torque overshoots can occur.

This occurs when a positive wind gust acts on the rotor while the pitch actuator is deactivated

(fixed at ) as dictated by the partial load control strategy.

The control goals associated with different wind turbine operating regions can be

summarized as follows.

129

Partial load regime:

Maximizing energy capture by tightly tracking the ORC

Reducing mechanical loads by damping drive train oscillations. This is important due to

the prohibitive cost of replacement of failed gearboxes [66, 172].

Full load regime:

Smoothing the wind turbine’s output power and reducing flicker emissions

Reducing drive train torsional torque fluctuations and mechanical loads

Reducing the pitch activity

Transition region:

Ensuring smooth transition between the partial and full load controllers, and eliminating

power and drive train torque overshoots

In addition to the different challenges specific to wind turbine operating regions

discussed in §1.2.3.1, the following challenges should be considered to design effective wind

turbine controllers. The wind turbine is a nonlinear system with a continuously varying operating

point, depending on the mean wind speed. Therefore, the control system should cope with such

variations and provide good performance over all operating wind speeds. Another challenge is

the presence of cyclic aerodynamic torque variations at triple the rotational speed of the wind

turbine blades. These 3P frequency fluctuations are the result of rotational sampling, wind shear

and tower shadow effects as described in §3.2.2. Cyclic torque variations can increase dynamic

loads and the voltage flicker severity [13, 74]. Finally, the control system should realize all the

control objectives while keeping system variables within their safe operating limits. This is

important to avoid unnecessarily stopping of the wind turbine during normal operating

conditions because of, for example, generator over-speed or overheating of electrical

130

components [39]. Furthermore, as described in [124, 125], ignoring these limits during controller

design can lead to severe degradation in the performance of the closed loop system. The relevant

physical wind turbine constraints are summarized in (5.2)-(5.6), where denote the

maximum (minimum) dynamical limit of •.

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

The work presented here is motivated by the desire to develop an overall MIMO control

strategy that can work in both partial and full load regimes. As reviewed in §1.2.3.1, most of the

control strategies in the literature use decentralized control structures to control the wind turbine.

These approaches do not exploit the full control capabilities of the multivariable system and

generally result in an inferior performance as compared to MIMO controllers [13, 36]. In this

chapter, a new MIMO control strategy based on MPC techniques is proposed to control variable-

speed variable-pitch WECSs in both partial and full load regimes. The DFIG configuration is

used to verify the proposed control strategy as it is the most popular type used today [19]. The

proposed strategy is a multivariable method that uses the full capability of the system to obtain

the desired performance in the whole WECS operating region, while keeping the system

variables within safe limits.

5.2 Simplified model of variable-speed variable-pitch wind turbines

The dynamic model of a variable-speed variable-pitch WECS is detailed in Chapter 3. It

consists of several interconnected subsystems as shown in Figure 3.2.

131

For controller design purposes, it is important to use simple models that capture the

relevant dynamics of the system. Using the overall WECS model in (3.71), gives rise to a high-

order model that includes 19 state variables. Fortunately, the dynamics of the electrical

subsystem are much faster than the turbine dynamics. Furthermore, the generator torque control

can be decoupled from reactive power control. For these reasons, simple models can be used to

represent the dynamics of the electrical subsystem shown in Figure 3.2. Here, the first order

model (5.7)-(5.8) is used, where , and are the generator torque, time constant and

efficiency, respectively. Similar simplifications are commonly used in the literature [39].

(5.7)

(5.8)

Despite this simplification, the resulting overall WECS model described by (3.19)-(3.22)

and (5.7)-(5.8) is nonlinear. The main nonlinearity is due to the nonlinear aerodynamic torque

expression in (3.15). Linearizing the turbine torque equation in (3.15) yields

(5.9)

|(

),

|(

),

|(

). (5.10)

where denotes the deviation of the variable from its operating point value

denoted by . Since the steady state values and are dependent on , as shown in

the nominal operating trajectory of the wind turbine in Figure 5.1, the WECS operating point is

completely defined by , which is equal to the operating mean wind speed [13]. From

132

(3.19)-(3.22) and (5.7)-(5.9), a linear state space model representing the WECS dynamics at

certain operating wind speed is given by

(5.11)

where

[

(

)

]

(5.12)

[

]

[

]

[

] (5.13)

Here, [ ] is the state vector,

[ ] is the control input, and [ ] is the measured output.

The variations of the coefficients , and , and the Bode magnitude plots of the

model (5.11)-(5.13) over the WT nominal operating trajectory in Figure 5.1 are shown in Figures

5.2-5.3, respectively. One salient feature that can be observed from Figure 5.3 is the presence of

large open loop gains occurring around a particular resonant frequency. This is due to the

presence of lightly damped complex poles in the drive train dynamics system (3.22). It can be

shown that the resonant frequency is given by (5.14). In general, this resonance can cause a

133

reduction in the closed loop stability margins and performance. Finally, the model (5.10)-(5.13)

also shows that system is MIMO and that the system dynamics vary when the mean wind speed

varies.

√

(

)

[ ⁄ ] (5.14)

Figure 5.2 Variations of , and evaluated along the nominal WT operating

trajectory in Figure 5.1.

0 5 10 15 20 25 30-3000

-2000

-1000

0

v, m/s

T

t/

t, K

Nm

.s/r

ad

0 5 10 15 20 25 300

100

200

300

v, m/s

T

t/ v,

KN

.s

0 5 10 15 20 25 30-300

-200

-100

0

v, m/s

T

t/

, K

Nm

/ o

134

Figure 5.3 Bode magnitude plots of the WT model (5.11)-(5.13). Gray lines represent low

wind speeds (partial load) and black lines represent high wind speeds (full load).

5.3 Baseline wind turbine controller

In this section, the baseline WECS control strategy that is widely used by industrial wind

turbines is described. This controller is used for comparison with advanced controllers proposed

in Chapters 5-6. Further details about this control strategy and other control strategies commonly

found in the literature are provided in Appendix C.

The baseline turbine controller is shown in Figure 5.4 [39]. The main part of the

controller is a set of two PI controllers which regulate the generator speed in the partial and full

load regimes, respectively. In partial load operation, is fixed at zero and is manipulated by

the first PI controller so that the generator speed tracks the desired generator speed set point

. In the full load regime,

is fixed at its rated value, , while is manipulated by the

135

second PI controller to regulate the generator speed at its rated value. In order to take into

account variations in the aerodynamics, these PI controllers are generally gain-scheduled.

Furthermore, bumpless switching between the partial and the full load control configurations is

implemented.

v

PI

WECS

βd

Tg*

ωg

ωg*

Scheduling

Signal

MPPT

PI

0o

Tg,rat

Full/partial

load

+

-

Figure 5.4 Classical control strategy using two PI controllers .

The classical PI wind turbine strategy has many drawbacks. First, in the partial load

regime, the tuning of PI controllers to achieve the desired trade-off between energy

maximization and reliability demands in terms of mechanical loads in the drive train is not easy

[12]. Furthermore, when the system is operating near the rated wind speed, the partial load

control structure focuses on controlling the generator speed only irrespective of the generator

power. Due to wind speed fluctuations, significant power and drive train torsional torque

overshoots can occur [13]. Finally, in the full load regime, using the pitch actuator alone to

regulate the generator speed can cause large pitch activity and severe power fluctuations. This in

turns reduces the life time of the equipment and deteriorates the quality of the power produced.

136

5.4 Proposed control strategy

5.4.1 Multiple model predictive control for variable-speed variable-pitch WECS

The use of linear model predictive control with a nonlinear plant, such as the wind

turbine model in Chapter 3 in which the operating point is continuously changing, can lead to

degradation in the closed loop performance [121]. There has been extensive research effort to

extend the applicability of MPC to nonlinear systems [173]. One of the most straightforward

approaches is to use Multiple Model Predictive Control (MMPC) [174-179].

The proposed wind turbine control strategy based on MMPC is shown in Figure 5.5. The

main idea is to use a multivariable MPC controller at the turbine control level to control the

WECS behavior by simultaneously manipulating both and . WECS constraints, such as

limits on the pitch angle magnitude, pitch angle rate, the generated power and the turbine speed,

are explicitly incorporated into the MPC controller. To cope with WECS nonlinearities and the

continuous variation in the operating point, the whole operating region of the WECS is divided

into operating sub-regions with linearized models that adequately represent the local

system dynamics within each sub-region. A linear MPC controller based on each model is

designed. Finally, a criterion by which the control system switches from one controller to another

as operating conditions change is defined. This approach is known in the literature as MMPC

[174-179].

The main components of the proposed MMPC strategy are the prediction model bank, the

optimization problem formulation, the state estimator and a model switching criterion [124, 125,

128, 180]. These are detailed in the rest of this section.

137

v

MPC WECS

β*

Tg*

PWTG

ωg

ωg*

MMPC

Estimator

Bank

Model

Bank

)|(ˆ tti

x

Ai, Bi, Cc

i, Dcui

Scheduling

Signal

WECS

Optimization

)|(ˆ ttid

x

*WTGP

Figure 5.5 Proposed control strategy using MMPC.

5.4.1.1 Prediction model bank

A model bank (5.15), consisting of linearized models that represent the WECS

dynamics in the whole operating region shown in Figure 5.1 must be available.

, (5.15)

Here, the superscript is used to indicate the index of the model used.

For the case of VSVP wind turbines, the control input vector , the state vector ,

the measurement vector , and the controlled output vector of model in (5.15) at a

sampling instant are defined in (5.16).

[

]

[ ]

[

]

[

]

(5.16)

138

The controlled output vector is chosen as the measurement vector augmented by the

derivative of the internal shaft torsional torque . Controlling the measurement vector allows

tracking/regulation of the wind turbine speed and power. The use of the unmeasured signal

in

as in (5.16) and the inclusion of this signal in the objective function of the

optimization problem allow damping high frequency oscillations in the drive train torsional

torque. This approach is commonly used in optimal control [161].

The fictitious unmeasured disturbance is used to represent the effect of

actual unmeasured disturbances, such as the wind, and it is modeled as the output of the system

(5.17).

(5.17)

The discrete linear models in (5.15) can be obtained by discretizing (5.11) at

different mean wind speeds, with , representing the whole wind speed operating

range. At certain operating wind speed ,The matrices , , ,

, and in (5.15) are

computed using (5.18)-(5.19), where is the sampling period, and and denote the

third row in and , respectively.

( ) ∫ (

)

( ) (5.18)

[

( )

] [

] (5.19)

Combining (5.15) and (5.17), the augmented prediction model bank used in the MPC

formulation is given by

139

[

]⏟

[

]⏟

[

]⏟

[

]

⏟

[

]⏟

[

]⏟

, . (5.20)

5.4.1.2 Optimization problem

Assuming knowledge of the estimates of the plant states | and disturbance states

| , given the data up to time , the MMPC controller solves the quadratic optimization

problem given by:

∑‖ ‖

∑ ‖ ‖

∑ ‖ ‖

(5.21)

, (5.22)

[ | | ], (5.23)

, (5.24)

, (5.25)

, (5.26)

, (5.27)

Here, is the reference vector defined as [

] , is defined as

, and is the value of the input vector at operating point , i.e.

[ ]

. The vectors , , and are defined in (5.28). From (5.16), it is easy

140

to see that the input constraints (5.2)-(5.3) and output constraints (5.5)-(5.6) are guaranteed by

(5.26) and (5.27), respectively. Furthermore, for sufficiently small sampling time, the pitch angle

rate constraint (5.4) can be approximated by (5.25).

[ ]

[ ]

[

]

(5.28)

The weight matrices , and are defined in (5.29). The base values for the speed

, torque , power and pitch angle are given in Appendix B. It should be noted that

the form in (5.29) is used to normalize the physical variables penalized in (5.21) to their base

values. This facilitates the tuning of the weights ,

, ,

, and

, and make the process

insensitive to the units of the physical variables that are penalized in the objective function

(5.21).

[

]

, [

], [

] (5.29)

Finally, the MMPC controller is given by (5.30), where

is the solution of (5.21)-

(5.27).

[ ]

(5.30)

Using the MPC formulation in (5.21)-(5.27), there are six weights, ,

, ,

, and

, that can be tuned to reach the desired compromise between different control objectives. The

weights and

penalize regulation/tracking errors in the generator speed and power,

141

respectively. The weight penalizes the magnitude of torsional torque rate of variation. This

parameter can be tuned to achieve the desired damping of oscillations in the drive train torsional

torque. The weights and

penalize high control activity in the generator torque and pitch

angle set points, respectively. Finally, the weight is used to penalize the magnitude of the

pitch angle. Details of weight selection for partial and full load regimes are provided in §5.4.2.

5.4.1.3 State estimation

As described in Chapter 4, the value of the state or its estimate is required to implement

the MPC algorithm. Here, the state estimates are computed using an observer bank consisting of

state observers designed for the model (5.31). The process noise and the

measurement noise are zero mean Gaussian white noise with symmetric positive

definite covariance matrices and , respectively.

(5.31)

The observer bank is given by

| | [ | ] (5.32)

| | . (5.33)

The Kalman gains, , are computed using

( )

(5.34)

where is the unique symmetric positive definite solution of the discrete

algebraic Riccati equation

( )

(5.35)

142

5.4.1.4 Bumpless switching between different MPC controllers

At any sampling instant, there is only one MPC controller that is active; and its

optimization problem is solved to determine the value of the control signal. The switching

between different MPCs is based on the value of a scheduling signal as shown in Figure 5.5. In

the case of a WECS, the scheduling signal can be the generator speed and pitch angle or an

average wind speed estimate which can be determined online by filtering the wind speed

measured by an anemometer located at the wind turbine [31, 39].

Due to wind speed fluctuations, it is important to ensure bumpless switching between

different MPC controllers. This is ensured by (i) using an MMPC algorithm that calculates only

the control increments, and , and (ii) by continuously updating the internal state

of all estimators in the estimator bank based on the current control and measurement signals at

each sampling instant. This reduces the transients in the state estimates when switching between

MCP controllers.

5.4.2 MMPC controller design

5.4.2.1 MMPC weight selection

In the proposed MMPC strategy, the whole operating region is partitioned into sub-

regions. The selection of the weights ,

, ,

, and

for each sub-region depends on the

current wind turbine operating mode.

Partial load regime

For sub-regions corresponding to partial load operation, the weights and

in (5.21)

should be set to large values to force the pitch angle set point to be fixed at zero. This disables

the activation of the pitch angle in the partial load regime. Since the objective is to track the

generator speed set point, the weight should be set to zero while the weights

and in

143

(5.21) should be selected to achieve the desired trade-off between energy maximization and drive

train transient load minimization. The desired damping of oscillations in the drive train torsional

torque can be achieved by tuning .

Remark 5.1: Another way that can be used to fix the pitch angle at zero is to set

in (5.28). However, the penalty approach adopted here allows the pitch angle to be only used in

preventing the violation of WECS constraints that might result from sudden extreme

disturbances affecting the system.

Transition region

For sub-regions corresponding to partial load operation near the rated wind speed, the

weights ,

, and

should be selected similar to those used in sub-regions corresponding

to partial load operation. However, the weights and

should be reduced in comparison with

sub-regions corresponding to partial load operation, and should be replaced by

in (5.28). This introduces the constraint (5.36) instead of (5.6) in the MPC formulation.

The use of the constraint (5.36) allows the pitch system to be only activated when required to

prevent the power from exceeding its rated value when the wind speed fluctuates near the rated

wind speed. This results in eliminating all power and drive train torsional torque overshoots

above the rated values that might occur in the transition region. This advantage is a consequence

of the multivariable formulation and the flexibility provided by the MPC controller.

, (5.36)

Full load regime

For sub-regions corresponding to the full load regime, the weight should be set to zero

to allow the pitch angle to take any required value. The weights ,

, and

are tuned to

144

achieve the desired trade-off between generator speed regulation, power smoothing, drive train

transient loads reduction and pitch angle activity, respectively.

5.4.2.2 Disturbance model selection

The disturbance model in (5.17) is chosen to guarantee offset-free tracking for the WECS

[158, 160, 162]. Following the guidelines in §4.5 [158, 160, 162], two integrated white noise

unmeasured disturbances, , entering at the inputs are assumed. This is achieved by

using (5.37).

,

, , and (5.37)

According to results in [158, 160, 162], the condition in (4.69) must also be satisfied to

guarantee offset-free tracking. This condition requires that the controlled output used in the MPC

objective function is a linear combination of the measurements. This requirements is not satisfied

in the formulation (5.21)-(5.27) due to the presence of a term containing the unmeasured

controlled output, , in (5.16). Despite that, the next theorem shows that at steady state, the

objective function of the MPC controller used here is equivalent to the one in [162] and

therefore, offset-free tracking is still guaranteed for the proposed MPC strategy.

Theorem 5.1: Consider a WECS controlled by the MMPC in (5.21)-(5.30). If the closed loop

system is stable and constraints are not active at steady state, then and

asymptotically.

Proof: The main difference between the MPC controller (5.21)-(5.30) and the one described in

[162] is the presence of an additional term in the objective function (5.21) that depends on the

unmeasured controlled output . This proof shows that this additional term is forced to zero at

steady state, and thus results in [162] guarantee that the MPC (5.21)-(5.30) ensures offset-free

tracking. To simplify the notation, the superscript in (5.15)-(5.33) is dropped.

145

First, it should be noticed that the system (5.11) is a non-integrating process and both

and are invertible. By assumption, when the closed loop system reaches steady state, it

follows from the observer equations in (5.32)-(5.33) and the disturbance model matrices (5.37)

that [162]

(5.38)

(5.39)

where , , and are the steady state values of the measurement vector, control input,

plant state estimate and disturbance state estimate. Substituting (5.39) and (5.37) in the

prediction model equations (5.18)-(5.20), (5.40) is obtained.

, . (5.40)

Substituting (5.39) in (5.40) gives:

(5.41)

Since is invertible, in (5.18) can be written as

(5.42)

Substituting in (5.18) and in (5.42) in (5.41), (5.43) is obtained.

( )

(5.43)

Since [ ] from (5.13), it can be concluded that at steady state

and, therefore, the objective function (5.21) is equivalent to the one used in [162].

146

5.4.2.3 Partitioning the whole operating region into operating sub-regions

The proposed MMPC control strategy is based on partitioning the whole operating region

into operating sub-regions as shown in (5.44), where sub-regions

are defined in (5.45). This is done to approximate the nonlinear WECS dynamics by local

linear models.

[ ]

(5.44)

[ , [ , …, [ ]

where

(5.45)

Special care must be taken when partitioning the whole operating region. In general,

increasing the number of partitions and, consequently, reducing the range of each sub-region will

enhance the linear approximation and the prediction accuracy of the linear model. This comes at

the cost of increasing the controller complexity and the computational burden.

To obtain the partitions in (5.45), the following simple algorithm is proposed. The

algorithm assumes that a fine grid of linear models (5.11) linearized at different wind speeds

, where the increment

and is chosen to be a

sufficiently large positive integer, is available. It can be seen that this grid of wind speeds covers

the whole operating region. These models can be discretized using (5.18)-(5.19) to obtain

discrete time models of the form (5.15). The basic idea is to partition the whole operating region

such that the distance between different prediction models in the same sub-region is bounded by

a user defined value, .

147

The distance between different models is based here on the mismatch between their

predictors. Based on the development in §4.3.1, it is straightforward to show that the predictor

for the model (5.18)-(5.20) and (5.37), linearized at is given by

(5.46)

where

[

], (5.47)

[

] , (5.48)

[

] ,

[

∑

]

, (5.49)

[

]

. (5.50)

For a given , the difference between predictions obtained using two different models

that are linearized at two different wind speeds and

is given by (5.51).

(5.51)

The norm of the prediction mismatch between those two models is written in (5.52), where

‖ ‖ √ and is the maximum singular value of . It can be seen from (5.52) that the

prediction mismatch between both models is kept small when the value of is small.

Based on this discussion, Algorithm 5.1 is used to generate the sub-regions in (5.45) that

partition the whole WECS operating region.

148

‖

‖ ‖ ‖

‖ ‖ ‖ ‖

‖ ‖

(5.52)

Algorithm 5.1 (Partitioning the whole WECS operating region):

Input: , , and parameters ( and ) of model (5.15) linearized at

with

.

Output: sub-regions where

Initialization: , , , calculate using (5.47)-(5.50)

1. For do

2. calculate from and using (5.47)-(5.50);

3. if ( )

, then [

, , , endif;

4.

5. endfor

Using the above algorithm and applying the triangle inequality for matrices, it can be

inferred that for any ,

[

) and , inequality (5.53) is

satisfied. This shows that Algorithm 5.1 guarantees that the distance between different prediction

models within the same sub-region is bounded by , and thus any of these models can be picked

as a candidate model that represents the other models in this sub-region.

‖ ‖ ‖ ‖

‖ ‖ ‖ ‖

(5.53)

149

Remark 5.2: Selecting large values of in the Algorithm 5.1 means that it is acceptable to have

a large prediction mismatch within the same sub-region. In this case, it should be expected that

the algorithm will produce a small number of partitions with large ranges. Reducing the value of

increases the number of partitions calculated by the proposed algorithm and enhances the

linear approximation within each sub-region.

5.5 MPPT algorithm

In order to test and evaluate the proposed MMPC control strategy, a MPPT algorithm

should be implemented to calculate the generator speed set point during partial load

operation. Many MPPT algorithms have been proposed in the literature [12]. The most

straightforward approach is to use the wind speed signal or its estimate to calculate using

[1], [6], [12], and [13]:

(5.54)

The main short-coming of this approach is that it requires the knowledge of the effective wind

speed experienced by the rotor blades. This fictitious speed cannot be measured and it is very

difficult to estimate.

The MPPT algorithm that is used in this thesis is shown in Figure 5.6. An experimental

validation of this approach on a DFIG test bench is provided in [29]. This approach does not

require knowledge of the effective wind speed; and is calculated using (5.55), where is an

estimate of the turbine torque. It can be seen from (2.17) that (5.55) enforces the operation of the

wind turbine at the ORC. For further description about commonly used MPPT algorithms, see

Appendix C.

150

√ ⁄ (5.55)

NM

opt

t

k

T*g

Estimator

tTg

Figure 5.6 MPPT algorithm.

The turbine torque estimator described in [32] is used. In this work, a dynamic torque

estimator that is based on a two-mass model representation of the drive train is used. From the

drive train model in (3.22), turbine torque estimation can be regarded as an unknown input

estimation problem. To convert the problem into a state estimation problem, a piecewise constant

turbine torque model is used. Thus, the augmented drive train model can be written in (5.56)-

(5.57), where the process noise and the measurement noise are zero mean white noises with

variances and

, respectively. Kalman filtering techniques can be used to generate the state

estimates , , and of the model (5.56)-(5.57).

[

]

[

(

)

]

[

]

[

]

[

] (5.56)

[ ] [

] (5.57)

151

Remark 5.3: By neglecting the drive train dynamics, the turbine torque can be estimated using

the simple relation . However, it was shown in [181] that this approach results in poor

tracking of the ORC and that much superior performance is obtained using a dynamic torque

estimator.

Remark 5.4: Other dynamic torque estimators using single mass drive train models are reported

in [10, 182].

Remark 5.5: Knowing , and at any time instant, an estimate of the wind speed can be

calculated by numerically solving the nonlinear equation (5.58) using Newton method [32, 181].

(

) (5.58)

5.6 Simulation results

5.6.1 Simulation set-up

In this section, the performance of the proposed control strategy is compared with the

classical gain-scheduled PI control strategy described in §5.3. For a fair comparison, the same

MPPT algorithm, described in §5.5, and the same generator control system described in

Appendix A, are used with both controllers in all simulations.

The system under study, shown in Figure 5.7, assumes that a 1.5 MW DFIG wind turbine

is connected to a radial distribution system [64]. All power system parameters are provided in

Appendix D. The SCR at the PCC, defined in (2.20), is 13.3 MVA and the short circuit

impedance angle, defined in (2.21), is 61 . In all simulations presented in this section, the

designed controllers are tested on the nonlinear model described in Chapter 3, with data given in

Appendix D. These parameters correspond to a 1.5 MW industrial GE wind turbine [14, 52]. The

152

nonlinear WECS model is implemented using SIMULINK® and the other power system

compenents are simulated using SimPowerSystemsTM

[183]. The speed, torque and power

signals have been normalized based on the per unit system described in Appendix B.

1 2 3 4 5

6 7 89

T12

L6

T47 T48

T59

PCC

DFIG

WECSL7

L9

Grid

T36

Figure 5.7 Power system studied.

5.6.2 Performance measures

In order to compare different control strategies, there must be some performance

indicators that reflect the performance of the closed loop system. In the partial load regime,

controller tracking performance is measured using the average power produced during the entire

simulation. To measure the flicker emission, a digital flickermeter is implemented in

MATLAB®, based on the IEC 61000-4-15 standard [77, 184], to calculate the short-term flicker

severity at the PCC bus shown in Figure 5.7. To compare the pitch activity in full load and

transition regions, the standard deviation of the pitch rate will be used.

The general approach to assess fatigue loads is to perform fatigue life prediction of

structural components subjected to random stresses. For a uniform sinusoidal stress load, the

expected fatigue life can be determined from the so-called S-N curve that relates the range of a

153

cyclic stress with the number of such stress cycles to failure [66, 185]. For random loads, the

‘rainflow’ counting algorithm is typically used to translate a spectrum of varying stress into sets

of reversing stress cycles of constant magnitude. Once these stress cycles have been determined,

the Palmgren-Miner linear damage rule can be used to estimate the fatigue life of the equipment

[66, 172, 185, 186].

For wind turbine control system evaluation in terms of fatigue, it is common to calculate

the damage equivalent load (DEL) [66]. The DEL is defined as the amplitude of a sinusoidal

load of constant frequency which produces the same damage as the original load. The DEL can

be calculated by performing the ‘rainflow’ counting algorithm to calculate the number of load

cycles from time domain simulation results. The MATLAB toolbox developed in [185] is used

for this purpose. Assume a load history with different stress levels with ranges , for

. Denote the number of load cycles corresponding to each these stress levels by .

The DEL can be calculated using [66]:

(∑

)

(5.59)

where is the duration of the load history, and is the Wohler coefficient of the material

under stress.

5.6.3 MMPC design

The tuning of the proposed MMPC controller according to the guidelines in §5.4.2 is

illustrated in this section. Different case studies with different weight settings are used in this

subsection to demonstrate the trade-offs between the competing objectives.

154

Partial load regime

An MPC controller is designed based on the WECS model (5.15) linearized at an

operating wind speed of 6.5 m/s. The effect of the weight is studied by simulating the

closed loop WECS system with different values of taken as 0, 0.05, and 0.1. In all three cases,

the other weights are fixed at , , , , and . The sampling

time and prediction horizons are chosen as

, , . (5.60)

Simulation results for a unit step wind speed are shown in Figure 5.8. It can be seen that

increasing the weight results in a better damping of drive train torsional oscillations.

The trade-off between fast tracking of the generator speed and lower generator torque

activity can be achieved by adjusting the ratio between and . This is illustrated by

simulating the closed loop WECS system using different values of taken as 1, 4, and 8. The

other weights are fixed at , , , , and . Simulation

results in Figure 5.9 show that increasing compared to results in slower response speed and

smoother torque variations.

Full load regime

An MPC controller is designed based on the WECS model (5.15) linearized at an

operating wind speed of 20 m/s. Using the multivariable MPC formulation, it is easy to

achieve certain desired trade-off between generator speed and output power regulation

performance. This is obtained by tuning the ratio between and . To show that, the closed

loop WECS system is simulated with different values of taken as 1, 10, and 30. In all three

cases, the other weights are fixed at , , , and . Simulation

155

results are shown in Figure 5.10. It can be seen that increasing compared to results in a

better regulation in the generator speed control loop and performance degradation in the power

control loop.


, and .

0 2 4 6 8 100.8

0.85

0.9

0.95

1

time, s

g,

p.u

.

Q3 = 0 Q

3 = 0.05 Q

3 = 0.1

g*

0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

time, s

Ttw

, p.u

.

Q3 = 0 Q

3 = 0.05 Q

3 = 0.1

0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

time, s

Tg,

p.u

.

Q

3 = 0 Q

3 = 0.05 Q

3 = 0.1

156


, and .

Trade-off between the closed loop response speed and the pitch activity is achieved by

tuning . Figure 5.11 shows the step response of the closed loop WECS system with three

different values of taken as 1, 30, and 60. The other controller weights are taken as ,

, , , and . It can be seen that increasing results in less pitch

activity and slower response speed.

0 2 4 6 8 100.8

0.85

0.9

0.95

1

time, s

g,

p.u

.

r1 = 1 r

1 = 4 r

1 = 8

g*

0 2 4 6 8 100

0.1

0.2

0.3

0.4

time, s

Ttw

, p.u

.

r1 = 1 r

1 = 4 r

1 = 8

0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

time, s

Tg,

p.u

.

r1 = 1 r

1 = 4 r

1 = 8

157


, and .

1 2 3 4 5 61.19

1.2

1.21

1.22

time, s

g,

p.u

.

q1 = 1 q

1 = 10 q

1 = 30

g*

1 2 3 4 5 60.88

0.9

0.92

0.94

time, s

PW

TG

, p.u

.

q1 = 1 q

1 = 10 q

1 = 30 P

WTG*

1 2 3 4 5 60.72

0.74

0.76

0.78

time, s

Tg,

p.u

.

q1 = 1 q

1 = 10 q

1 = 30

1 2 3 4 5 616.5

17

17.5

18

18.5

time, s

,

o

q

1 = 1 q

1 = 10 q

1 = 30

158


, and .

Based on §5.4.2 and the above discussion, the design of the MMPC controller is

described as follows. Using the Algorithm 5.1 with 0.5 and 0.1, the whole operating

region, with 4 m/s and 26 m/s, is partitioned into six operating sub-regions defined

in Table 5.1. For each of these sub-regions, a model (5.15) linearized at in Table 5.1 is

calculated. Using these models, six MPC controllers are designed following the guidelines in

0 2 4 6 8 10

1.19

1.2

1.21

1.22

1.23

time, s

g,

p.u

.

r2 = 1 r

2 = 30 r

2 = 60

g*

0 2 4 6 8 100.88

0.9

0.92

0.94

time, s

PW

TG

, p.u

.

r2 = 1 r

2 = 30 r

2 = 60 P

WTG*

0 2 4 6 8 1016.5

17

17.5

18

18.5

time, s

,

o

r2 = 1 r

2 = 30 r

2 = 60

159

§5.4.2.1, resulting in the settings listed in Table 5.1. The MMPC sampling time and prediction

horizons are chosen as in (5.60).

Table 5.1 MMPC sub-regions and controller data

, m/s

Par

tial

load

1 [4, 8.9) 6.5 2 0 0.05 1 1000 1000

2 [8.9, 11) 10 2 0 0.05 1 4 0.25

Full

load

3 [11,13.5) 12.5 2 2.5 0.05 1 6 0

4 [13.5, 18.3) 16 1.5 2 0.05 1 6 0

5 [18.3, 21.8) 20 1.5 2 0.05 1 6 0

6 [21.8, 26] 24 1.5 2 0.05 1 6 0

Remark 5.6: It was found that the MMPC calculations required less than 25 ms (about 50% of

the sampling time) in simulations carried on a 1.66 GHz dual core PC.

To test the performance of the MMPC controller over the full load regime, the staircase

wind speed signal in Figure 5.12 is applied to the WECS. The closed loop system is simulated

with the MMPC in Table 5.1 and with a Single model MPC (SMPC) that corresponds to in

Table 5.1. Simulation results are shown in Figure 5.13.

Figure 5.12 Wind speed profile.

0 20 40 60 80 10010

15

20

25

30

time, s

v,

m/s

160

Figure 5.13 Performance comparison between SMPC and MMPC.

0 20 40 60 80 1001.19

1.2

1.21

1.22

1.23

time, s

g,

p.u

.

SMPC MMPC g*

0 20 40 60 80 1000.88

0.89

0.9

0.91

time, s

PW

TG

, p.u

.

SMPC MMPC PWTG*

0 20 40 60 80 1000.65

0.7

0.75

0.8

time, s

Ttw

, p.u

.

SMPC MMPC

0 20 40 60 80 100

-4

-2

0

2

4

6

time, s

d

/dt,

o/s

SMPC MMPC

161

It can be seen from Figure 5.13 that the performance of the SMPC deteriorates

significantly when the system is operating at wind speeds far from the one at which the SMPC is

designed. On the other hand, the MMPC provides the desired performance over a wide range of

operating wind speeds.

5.6.4 PI baseline controller design

As described in §5.3, the classical wind turbine control strategy consists of two speed

control loops. The first loop is active during partial load operation where is adjusted by the

partial load controller to allow good tracking of the generator speed set point. In the full

load regime, the second loop is active and is adjusted by to regulate the generator

speed at its rated value. Both controllers can be designed using single-input single-output (SISO)

design techniques on the transfer functions , from to , and , from to ,

respectively.

The controllers and are tuned using an off-the-shelf MATLAB routine

‘pidtune'. This routine calculates the PI controller gains based on user specified performance

requirements. Here, a desired gain crossover frequency and phase margin are specified. The

crossover frequency is selected such that the closed loop system has a similar response speed to

the one obtained using the MMPC controller, and the phase margin is used to ensure certain

desired stability margin.

In the partial load regime, the WECS model is linearized at five different operating wind

speeds ranging from 4 m/s to 10 m/s. A PI controller (5.61) is designed for each model to

achieve a crossover frequency of 1.25 rad/s and a phase margin of 60 . The Bode plots of

and the PI gains for the five PI controllers are shown in Figure 5.14 and Figure

5.16 (a), respectively.

162

(5.61)

In the full load regime, the WECS model is linearized at seven different operating wind

speeds ranging from 11 to 26 m/s. Due to resonance effects, the ‘pidtune’ function failed in

finding a PI controller that achieves the desired phase margin. To solve that issue, a notch filter

is used in series with the PI controller resulting in of the form (5.62), where ,

and is defined in (5.14). The notch filter attenuates the open loop gain at the

resonance frequency and helps in mitigating resonance effects. This solution is commonly used

in the literature [187, 188]. With of the form (5.62), seven different PI controllers are

designed to achieve a crossover frequency of 2 rad/s and a phase margin of 50 . The bode plots

of are plotted in Figure 5.15. The gains and in (5.62) are plotted in Figure

5.16 (b).

(

) (

) (5.62)

Figure 5.16 shows that PI controller gains are practically constant in the partial load

regime, while the gains of the full load controller vary significantly with the operating wind

speed. Therefore, a gain scheduling scheme is applied to the full load controller. Here, a simple

lookup table with linear interpolation is used to determine the controller gains as a function of

the mean wind speed. An anti-windup strategy and a bumpless switching between the partial and

full load controllers are implemented [131].

163

Figure 5.14 Bode plots of at different wind speeds in the partial load regime.

Figure 5.15 Bode plot of at different wind speeds in the full load regime.

-50

0

50

100

From Tg* To g

Magnitu

de (

dB

)

Bode Diagram

Frequency (rad/s)

10-2

10-1

100

101

102

-180

-90

0

90

Phase (

deg)

Bode Diagram

Frequency (rad/s)

-150

-100

-50

0

50

100

From: Betasp To: Out(1)

Magnitu

de (

dB

)

10-2

10-1

100

101

102

-450

-360

-270

-180

-90

Phase (

deg)

164

(a)

(b)

Figure 5.16 PI controller gains as functions of the mean wind speed. (a) partial load, and

(b) full load.

5.6.5 Comparison of the MMPC and PI controllers - Deterministic wind speed

The MMPC controller designed in §5.6.3 is compared with the gain-scheduled PI

controller designed in §5.6.4 using a deterministic wind speed profile. The following cases are

considered.

Partial load with variable-speed operation (low wind speed)

The closed WECS is simulated using a unit step change in the wind speed from 6.5 to 7.5

m/s. The performance of the MMPC and the PI controllers is compared in Figure 5.17. It can be

seen that both controllers provide similar tracking performance of the generator speed set point.

However, the MMPC controller outperforms the PI controller in damping torsional torque

oscillations.

Partial load operation at near rated wind speed (medium wind speed)

The closed loop WECS is simulated using a positive step change (from 10 to 11 m/s) and

a negative step change (from 10 to 9 m/s) in the wind speed. The performance of the MMPC and

the PI controllers are compared in Figure 5.18.

4 6 8 100

2

4

6

8

10

v, m/s

PI

contr

oller

gain

spartial load

-Kp

-Ki

10 15 20 250

2

4

6

8

10

v, m/s

PI

contr

oller

gain

s

full load

-Kp

-Ki

165

Figure 5.17 Response to a step change in wind speed from 6.5 to 7.5 m/s using the MMPC

and the PI control strategies.

2 3 4 5 6 7 8 9 100.85

0.9

0.95

1

1.05

time, s

g,

p.u

.

MPC

PI

g*

2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

time, s

PW

TG

, p.u

.

MPC

PI

2 3 4 5 6 7 8 9 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

time, s

Ttw

, p.u

.

MPC

PI

2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

time, s

Tg,

p.u

.

MPC

PI

166

Figure 5.18 Response to a positive step in wind speed from 10 to 11 m/s (left) and a negative

step in wind speed from 10 to 9 m/s (right) using the MMPC and the PI control strategies.

0 2 4 6 8 101.195

1.2

1.205

1.21

1.215

1.22

time, s

g,

p.u

.

MPC

PI

g*

0 2 4 6 8 101.185

1.19

1.195

1.2

1.205

1.21

time, s

g,

p.u

.

MPC

PI

g*

0 2 4 6 8 10

0.65

0.7

0.75

0.8

0.85

0.9

0.95

time, s

PW

TG

, p.u

.

MPC

PI

0 2 4 6 8 100.45

0.5

0.55

0.6

0.65

0.7

time, s

PW

TG

, p.u

.

MPC

PI

0 2 4 6 8 100.55

0.6

0.65

0.7

0.75

0.8

time, s

Tg,

p.u

.

MPC

PI

0 2 4 6 8 10

0.35

0.4

0.45

0.5

0.55

0.6

0.65

time, s

Tg,

p.u

.

MPC

PI

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

time, s

,

o

MPC

PI

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

time, s

,

o

MPC

PI

167

The nonlinear behavior of the MPC controller is clearly observed as the controller

behavior for a positive step change in the wind speed is totally different from the negative one. In

the first case, the pitch actuator is activated to prevent the power from exceeding its rated value

as dictated by the constraint (5.36). On the other hand, when the wind speed is pushing the

system away from violating this constraint, very little activity can be observed in the pitch angle.

This behavior allows the removal of any power or torque overshoots above their rated values. On

the other hand, a power overshoot above the rated value can be seen in the PI controller case in

the case of positive wind speed step.

Full load operation (High wind speed)

The closed loop WECS is simulated using a unit step change in the wind speed from 20

to 21 m/s. The performance of the MMPC and the PI controllers is compared in Figures 5.19-

5.20. The MMPC controller outperforms the PI controller in damping torsional torque

oscillations and reducing output power fluctuations. A slight reduction in the pitch activity can

also be observed. However, these improvements are achieved with more fluctuations in the

generator speed.

5.6.6 Comparison of the MMPC and PI controllers - Stochastic wind speed

The MMPC controller designed in §5.6.3 is compared with the gain-scheduled PI

controller designed in §5.6.4 using a stochastic wind speed profile. The stochastic wind speed

model described in §3.2.3 is used. The following cases are considered.


A simulation of ten minutes of partial load WECS operation was performed. The mean

wind speed is 6.5 m/s, the turbulence intensity is 12%. The tower shadow and wind shear effects

are ignored during simulations. All other wind speed model parameters are given in Appendix D.

168

The wind profile is shown in Figure 5.21. A portion of sixty seconds of simulation results is

shown in Figure 5.22.

(a)

(b)

(c)

Figure 5.19 Response to a step change in wind speed from 20 to 21 m/s using the MMPC

and the PI control strategies. (a) generator speed, (b) WTG output power, and (c) torsional

torque.

0 2 4 6 8 101.195

1.2

1.205

1.21

1.215

1.22

time, s

g,

p.u

.

MPC

PI

g*

0 2 4 6 8 100.89

0.895

0.9

0.905

0.91

0.915

time, s

PW

TG

, p.u

.

MPC

PI

PWTG*

0 2 4 6 8 100.72

0.74

0.76

0.78

0.8

0.82

time, s

Ttw

, p.u

.

MPC

PI

169

(a)

(b)

Figure 5.20 Response to a step change in wind speed from 20 to 21 m/s using the MMPC

and the PI control strategies. (a) generator torque, and (b) pitch angle.


0 2 4 6 8 100.73

0.74

0.75

0.76

0.77

time, s

Tg,

p.u

.

MPC

PI

0 2 4 6 8 1016.5

17

17.5

18

18.5

time, s

,

o

MPC

PI

0 100 200 300 400 500 6004

6

8

10

time, s

v,

m/s

170

Figure 5.22 Simulation results for low wind speeds.

400 410 420 430 440 450 4600.6

0.7

0.8

time, s

g,

p.u

.

MPC

PI

g*

400 410 420 430 440 450 460

0

0.1

0.2

0.3

time, s

PW

TG

, p.u

.

MPC

PI

400 410 420 430 440 450 4600

0.1

0.2

0.3

0.4

time, s

Ttw

, p.u

.

MPC

PI

400 410 420 430 440 450 460

0

0.1

0.2

0.3

0.4

time, s

Tg,

p.u

.

MPC

PI

171

It can be seen that both the MMPC and the classical controller provide similar tracking

performance of the generator speed. Calculating the average power in both simulations, as shown

in Table 5.2, reveals that both controllers have similar power production. Furthermore, it can be

observed in Figure 5.21 that and fluctuate regardless of the type of control strategy

used. This is due to the continuous variation in to track the MPP. However, the amplitudes of

these oscillations decrease when the MMPC control strategy is used. Table 5.2 indicates that the

MMPC strategy provides a reduction of 33% in the flicker emission and 11% in drive train loads

in comparison to the classical strategy.

Table 5.2 Low wind speeds statistics (no tower shadow and wind shear effects)

Quantity MPC PI MPC/PI

AVG( ) 0.20 0.20 1

0.08 0.12 0.67

DEL 0.17 0.19 0.89

The whole simulation is repeated with the tower shadow and wind shear effects included.

The tower radius is taken as 1.5 m and the wind shear exponent is set to 0.2. Statistics in Table

5.3 indicate that the MMPC strategy provides a reduction of 24% in the flicker emission and

18% in drive train loads in comparison to the classical strategy.

Table 5.3 Low wind speeds statistics (with tower shadow and wind shear effects)


AVG( ) 0.2 0.2 1

0.10 0.13 0.76

DEL 0.18 0.22 0.82

172


A simulation of ten minutes of partial load WECS operation near the rated wind speed

(transition region) was performed. The mean wind speed is 10 m/s. The tower shadow and wind

shear effects are ignored during simulations. The wind profile is shown in Figure 5.23. The

switching signal of the MMPC and the WECS variables are shown in Figures 5.24-5.26.


Figure 5.24 MMPC switching signal.

It can be seen that power and drive train torque overshoots occur when using the classical

control strategy. This is clearly shown in Figure 5.27, where a portion of 35 s of the simulation is

shown. The MMPC eliminated these overshoots and the generator power and speed stay within

the rated values given in Appendix D. Statistics given in Table 5.4 confirm these observations. It

0 100 200 300 400 500 6006

8

10

12

14

time, s

v,

m/s

0 100 200 300 400 500 6000

2

4

6

time, s

Sw

itchin

g S

ignal

173

can be seen in the third column in Table 5.4 that the MMPC controller reduces the flicker

emission, the DEL, the pitch activity, the maximum drive train torque, and the maximum power

by about 21%, 15%, 34%, 6%, and 8%, respectively, when compared to the classical PI strategy.

(a)

(b)

(c)

Figure 5.25 Simulation results for medium wind speeds. (a) generator speed, (b) WTG

output power, and (c) torsional torque.

0 100 200 300 400 500 6001

1.1

1.2

time, s

g,

p.u

.

MPC

PI

g*

0 100 200 300 400 500 6000.2

0.4

0.6

0.8

1

time, s

PW

TG

, p.u

.

MPC

PI

0 100 200 300 400 500 600

0.4

0.6

0.8

time, s

Ttw

, p.u

.

MPC

PI

174

(a)

(b)

Figure 5.26 Simulation results for medium wind speeds. (a) Generator torque, (b) pitch

angle.


The tower radius is taken as 1.5 m and the wind shear exponent is set to . Statistics in Table

5.5 confirms the previous findings.


A simulation of ten minutes of full load WECS operation at an average wind speed of 20

m/s was performed. The tower shadow and wind shear effects are ignored during simulations.

The wind profile is shown in Figure 5.28. The switching signal of the MMPC and the WECS

variables are shown in Figures 5.29-5.30.

0 100 200 300 400 500 600

0.4

0.6

0.8

time, s

Tg,

p.u

.

MPC

PI

0 100 200 300 400 500 6000

2

4

6

8

time, s

,

o

MPC

PI

175

Figure 5.27 Simulation results for medium wind speeds (zoomed from =15 to 50 s).

Table 5.4 Medium wind speeds statistics (no tower shadow and wind shear effects)


AVG( ) 0.73 0.73 1

0.11 0.14 0.79

DEL 0.22 0.26 0.85

STD( ) 0.35 0.53 0.66

Max( ) 0.76 0.81 0.94

Max( ) 0.90 0.98 0.92

Table 5.5 Medium wind speeds statistics (with tower shadow and wind shear effects)

15 20 25 30 35 40 45 500.7

0.8

0.9

1

time, s

PW

TG

, p.u

.

MPC

PI

15 20 25 30 35 40 45 50

0.65

0.7

0.75

0.8

0.85

time, s

Ttw

, p.u

.

MPC

PI

15 20 25 30 35 40 45 500

1

2

3

4

5

time, s

,

o

MPC

PI

176


AVG( ) 0.72 0.72 1

0.12 0.15 0.80

DEL 0.23 0.26 0.88

STD( ) 0.35 0.57 0.61

Max( ) 0.78 0.81 0.96

Max( ) 0.90 0.98 0.92


Figure 5.29 MMPC switching signal.

0 100 200 300 400 500 60010

15

20

25

30

time, s

v,

m/s

0 100 200 300 400 500 6000

2

4

6

time, s

Sw

itchin

g S

ignal

177

Figure 5.30 Simulation results for high wind speeds.

0 100 200 300 400 500 600

1.14

1.16

1.18

1.2

1.22

1.24

time, s

g,

p.u

.

MPC PI g*

0 100 200 300 400 500 6000.87

0.88

0.89

0.9

0.91

0.92

time, s

PW

TG

, p.u

.

MPC PI PWTG*

0 100 200 300 400 500 6000.72

0.74

0.76

0.78

0.8

time, s

Tg,

p.u

.

MPC

PI

0 100 200 300 400 500 6005

10

15

20

25

time, s

,

o

MPC

PI

178

It can be observed that when using the MMPC controller, the power fluctuations are

significantly reduced. The price of this reduction is an increase in the generator speed

fluctuations in comparison with the classical control strategy. Table 5.6 shows that about 81%

reduction in power fluctuations, 40% reduction in the flicker emission, and 22% reduction in the

pitch activity are obtained when using the MMPC controller. The DELs obtained using both

controllers are approximately equal.


The tower radius is taken as 1.5 m and the wind shear exponent is set to 0.2. Statistics in Table

5.7 confirm the previous findings.

Table 5.6 High wind speeds statistics (no tower shadow and wind shear effects)


STD( ) 0.0127 0.0079 1.60

STD( ) 0.0011 0.0059 0.19

0.0136 0.0228 0.60

DEL 0.0692 0.0701 0.99

STD( ) 0.9015 1.1545 0.78

Table 5.7 High wind speeds statistics (with tower shadow and wind shear effects)


STD( ) 0.0128 0.0080 1.60

STD( ) 0.0025 0.0060 0.42

0.0165 0.0294 0.56

DEL 0.0848 0.0918 0.92

STD( ) 0.9411 1.2075 0.78

179

5.7 Conclusions

A multivariable control strategy based on MPC techniques is proposed to control

variable-speed variable-pitch WECSs over their full operating ranges. In the partial load regime,

the MMPC controller can be designed to provide the desired tradeoff between energy

maximization and reduction of the drive train torsional torque. Near the rated wind speed, power

and drive train torsional torque overshoots are eliminated and flicker emissions can be

significantly reduced. In the full load regime, the MMPC controller uses both the pitch angle and

the generator torque to regulate the generator power and speed. This reduces pitch activity,

smoothes the generated power and reduces flicker emissions. Furthermore, the MMPC controller

provides the desired WECS performance while keeping the system variables within safe

operating limits. Performance of the MMPC controller is compared with the classical gain-

scheduled PI control strategy. Extensive simulation results show superiority of the proposed

strategy over the whole operating region of the WECS.

180

Adaptive Subspace Predictive Control of Variable-Speed Variable-Pitch Chapter Six:

Wind Turbines

A new adaptive predictive wind turbine control strategy is proposed in this chapter. The

main difference between this strategy and the MMPC control strategy proposed in Chapter 5 is

the incorporation of a recursive subspace identification algorithm that continuously updates the

plant model used by the MPC controller. The motivation for this adaptive strategy is:

Grid integration of offshore wind farms is dramatically increasing all over the world. Due

to the high maintenance costs in the offshore environment, it is important to design wind

turbine controllers that provide robustness and resiliency against variations in the wind

turbine dynamics that might result from faults or other causes.

The use of accurate models in MPC algorithms enhances the prediction quality and the

overall control system performance. The MMPC controller in Chapter 5 uses a piecewise

constant disturbance model to account for the wind speed disturbance signal. By using an

online identification method, a better stochastic model of the wind speed disturbance can

be used and better performance should be expected.

However, the drawbacks of using an adaptive strategy compared to the MMPC strategy are

increased algorithmic complexity and computational requirements.

In this chapter, a new wind turbine strategy based on adaptive Subspace Predictive

Control (SPC) is proposed. The proposed strategy uses a model predictive control algorithm with

its predictor matrices continuously updated using recursive subspace identification techniques. In

contrast with SPC algorithms previously proposed in the literature, the proposed strategy

includes integral action, and consequently offers better disturbance rejection and better

181

performance. The effectiveness of the proposed strategy is illustrated using the 1.5 MW DFIG-

based WECS described in Chapters 3 and 5.

This chapter is based on [189] and it is organized as follows. An introduction and a

review of SPC methods are provided in §6.1-6.2. The proposed adaptive SPC algorithm is

formulated in §6.3. Application of this algorithm in the design of a wind turbine control strategy

is provided in §6.4. Simulation results are given in §6.5, and §6.6 concludes the chapter.

6.1 Introduction

The use of offshore wind farms is continuously increasing worldwide. This is driven by

the availability of higher mean wind speeds and less turbulence at offshore sites compared to

onshore ones [67]. However, offshore wind farms are more expensive than onshore ones.

The increasing use of offshore wind turbines poses new challenges on wind turbine

control systems in terms of reliability and performance. Primarily, the control system should

ensure good performance of the system in terms of energy production maximization and load

reduction during normal operation of the wind turbine. Furthermore, due to high unscheduled

maintenance costs in the offshore environment, it is desirable to design fault-tolerant control

systems that can allow the operation of the wind turbine and the production of electricity

between the occurrence of a fault and the next scheduled service.

The design of wind turbine fault-tolerant controllers (FTCs) has started to receive

increasing attention these days. A wind turbine FTC that is based on control theory and

linear parameter varying models is proposed in [68, 69]. The design of passive FTC and active

FTC is considered. The active FTC uses a Fault Detection and Isolation (FDI) system that is able

to detect faults and, based on this information, the controller is reconfigured (adapted) according

to the current fault. In the passive FTC case, no fault information is available online and robust

182

control techniques are used to design the controller to be resilient against a certain set of faults.

Both approaches are compared in [68] in the case of a fault in the hydraulic pitch system, and in

this case the active FTC significantly outperformed the passive FTC. This is due to the

conservatism obtained by designing a single passive FTC that ensures system robustness in

presence of large variations in the system dynamics. This conservatism is avoided in the design

of active FTCs that are capable to adapt to different fault scenarios.

Recently, SPC has been successfully employed in many applications, including fault

tolerant control [190, 191]. SPC is a collection of algorithms [191-194] that combines a subspace

predictor, identified using subspace identification techniques, with an MPC control law. In these

algorithms, explicit knowledge of the state space model parameters is not required. In fact, the

identification step is completed once the output predictor, to be used by the MPC controller, is

estimated from Input/Output (I/O) data. That is why these algorithms are considered as ‘model-

free’ or ‘data-driven’ ones.

SPC offers several advantages that make it a good candidate for FTC applications. First,

subspace identification can be recursively implemented online allowing for the adaptation of the

SPC controller to changes in the system dynamics. Furthermore, as argued by [195], MPC has

implicit fault tolerant control capabilities in cases of actuator redundancies. Finally, SPC is very

suitable for multivariable constrained control problems.

A new FTC strategy for variable-speed variable-pitch wind turbines is proposed in this

chapter. The contributions of the chapter can be stated as follows. First, a new adaptive Offset-

Free Subspace Predictive Control (OFSPC) algorithm that is based on the framework of [191] is

proposed. In contrast with the SPC algorithm of [191], the proposed OFSPC algorithm

systematically includes integral action in the SPC controller. Consequently, offset-free

183

tracking/regulation is guaranteed under piecewise constant disturbances affecting the system.

The second contribution of this chapter is to use the OFSPC algorithm in the design of a wind

turbine control strategy. The effectiveness of the proposed controller will be assessed during

normal WECS operation. Furthermore, a fault in the hydraulic pitch actuator [68] is considered

to illustrate the FTC capabilities of the proposed controller.

6.2 Review of subspace predictive control

6.2.1 Subspace system identification

Subspace Identification Methods (SIMs) are a set of identification algorithms originally

developed to identify the parameters of state space models [196, 197]. These algorithms were

initially proposed in the literature in the early 1990s, and they have enjoyed significant

development in both theory and practice [198, 199]. They have been successfully used in

different applications areas such as power systems [200-202], biomedical systems [203, 204],

and mechanical applications [205, 206].

SIMs offer an attractive alternative to prediction error methods (PEMs) for the following

reasons [154, 207]:

SIMs estimate state-space model parameters without requiring the solution of nonlinear

optimization problems that are typically encountered when using PEMs.

Subspace identification methods are based on robust numerical algorithms such as the

QR decomposition and the Singular Value Decomposition (SVD).

SIMs are effective with multivariable MIMO systems as well as SISO ones.

Many subspace identification algorithms have been proposed in the literature. The most

common ones are MOESP [208-211], N4SID algorithms [197, 212], and Canonical Variate

Analysis (CVA) [213, 214]. All these algorithms can be interpreted as singular value

184

decomposition of a weighted matrix [215]. Generally, this matrix is obtained from a QR

decomposition of a block-Hankel matrix constructed from the input and output measurements of

the system. Here, it is not the objective to give a comprehensive review of these algorithms.

Instead, the main algorithmic steps of one of the SIMs [216] is outlined below as it is useful in

understanding SPC algorithms described in this chapter.

Consider a system described by the innovation form model

(6.1)

(6.2)

where is the state vector, is the input, is the measured output, is

zero mean white noise (innovations) with covariance .

Define the past and future output block-Hankel matrices as:

[

]

[

]

(6.3)

The subscripts of and denote, respectively, the index in the left upper entry of the

block-Hankel matrix, the number of block-rows and the number of block-columns. Similar

definitions are used to denote , , , and . Based on these definitions and (6.1)-(6.2), it

is straightforward to deduce the following subspace matrix equations

(6.4)

(6.5)

185

where

[

]

,

[

]

, (6.6)

[

]

, (6.7)

[ ]

[ ]

(6.8)

(6.9)

In subspace identification algorithms, must be chosen to be larger than the system order

, and must be chosen to be much larger than to obtain good estimates of the model

parameters.

It is shown in [216, 217] that as , (6.10) is obtained, where is defined in (6.11)

and is a constant matrix that depends on the model parameters. Consequently, (6.4) can be

approximated by (6.12) for sufficiently large .

(6.10)

[

] (6.11)

(6.12)

The subspace algorithm in [216] now can be summarized in the following steps [192, 217].

Step 1: Estimate and by solving the Least Squares (LS) problem

‖ (

)‖

(6.13)

186

where the Frobenius norm ‖ ‖ . This step can be interpreted in the context of

performing an orthogonal projection of the row space of into the row space spanned by

and . The LS solution of (6.13) can be obtained in a numerically robust way by performing an

RQ-factorization as detailed in [216, 217].

Step 2: perform the SVD

[ ] [

] [

] (6.14)

Ideally, should be equal to zero and the dimension of determines the system order .

However, due to the presence of noise, does not need to be zero and the system order is

determined by inspecting the number of significant singular values. Once the SVD is performed,

and can be calculated using (6.15).

⁄

⁄

(6.15)

Step 3: In the last step, the state space matrices , , , , , and are

calculated using and/or . Many approaches exist to do this step [197]. One approach is to

estimate and using (6.16), where denotes the pseudo-inverse of . The rest of the

model parameters can be calculated from the inputs, outputs, and the estimates and . See

[154, 196] for details.

(6.16)

187

6.2.2 Subspace predictive control

Extensive research has been done to combine subspace identification techniques with

MPC to form an adaptive control strategy [190-194, 218-224]. This is motivated by the fact that

both tools are very effective with multivariable systems. Two approaches that can be identified

in the literature fall in the following two categories:

Indirect adaptive approach

Direct adaptive approach

In the indirect adaptive approach [218, 221-223], the state space model matrices ( , , ,

and ) are estimated online by a recursive subspace identification algorithm, and a conventional

MPC algorithm is used based on the identified model. One of the main bottlenecks for this

approach is the recursification of the SVD step as the SVD is computationally expensive to

update. This problem is solved in [225], where the signal subspace-tracking was interpreted as

the solution of an unconstrained minimization problem. Furthermore, the Projection

Approximation Subspace Tracking (PAST) algorithm was developed to update the subspace

estimate using recursive least squares techniques. The application of this algorithm in recursive

subspace identification problems is proposed in [226, 227]. Another challenge for this indirect

approach is the estimation of the current state. As described in Chapter 4, in order to accurately

predict the outputs using state space models, both the model parameters and the initial state must

be known. Recursive subspace identification does not provide estimates for the current state.

This problem was pointed out in [200], where a heuristic approach based on continuously

redesigning an observer using pole placement techniques is used.

188

The other approach is known in the literature as subspace predictive control [190-194,

220, 224]. In this approach, the state space model parameters are not explicitly identified online.

Instead, the predictor matrices are identified and used directly by the MPC controller.

SPC was initially proposed in [192]. In this work, it was observed that the matrices

and in (6.13) that are calculated in the first step of the subspace identification algorithm can

be used directly to formulate the predictor used by MPC algorithms. Therefore, once these

matrices have been calculated, the identification procedure can be stopped without performing

the SVD and calculating the model parameters , , , and . Furthermore, the online update of

these matrices can be done easily allowing the use of SPC for slowly varying systems. Finally, it

was shown that the resulting unconstrained SPC controller is equivalent to an LQG controller

under some mild assumptions.

The SPC design approach in [192] can be compared with the classical MPC design

approach as shown in Figure 6.1, where both controllers are designed from I/O data. It can be

seen that SPC controller is directly calculated from I/O data with much fewer steps and without

requiring knowledge of the state space model parameters. That is why in the literature, SPC is

considered as a model-free or data driven controller.

Many enhancements to the SPC proposed in [192] can be found in the literature. SPC was

extended in [193, 224] for the design of SPC controllers. It was shown in [228] that open

loop subspace identification algorithms lead to biased predictors when using closed loop data.

This can be seen from (6.12)-(6.13), where the input must be uncorrelated with so

that the LS solution in (6.13) gives consistent estimates. This condition is generally not satisfied

in closed loop systems. Consequently, SPC algorithms based on open loop SIMs can result in

poor performance when implemented in adaptive control applications. Motivated by that, a

189

Closed-Loop SPC (CLSPC) algorithm is proposed in [191] where closed-loop subspace

identification [229, 230] and Vector Auto Regressive with eXogenous input (VARX) modeling

[229] are used. However, this algorithm does not include ‘integral action’ in the controller

formulation. This can lead to poor performance when rejecting piecewise constant disturbances

[231]. For that reason, a new offset-free SPC algorithm that combines closed loop subspace

identification and includes integral action is proposed in §6.3.

I/O Data

Subspace matrices

Lw and Gu

Extended

observability matrix

A, B, C, D

State estimator Predictor

MPC controller

QR-factorization

SVD

LS

Kalman filter

design

I/O Data

Subspace (Predictor)

matrices Lw and Gu

MPC controller

QR-factorization

(a) (b)

Subspace

Identification

Predictor

calculation

Figure 6.1 Comparison between (a) classical MPC design, and (b) SPC design frameworks

using I/O data from the controlled plant.

190

6.3 Offset-free subspace predictive control

The proposed OFSPC algorithm is described in this section. The proposed algorithm

systematically includes integral action, and it is formulated in a stochastic closed-loop setting

allowing for online adaptation of the controller. The formulation of the subspace predictor is

provided in §6.3.1. Off-line and on-line identification of the predictor are described in §6.3.2. In

§6.3.3, the OFSPC algorithm is described. The effectiveness of the proposed strategy is

illustrated by examples in §6.3.4.

6.3.1 Formulating the subspace predictor

Here, the subspace predictor is formulated based on a state space model described by

(6.17)

(6.18)

(6.19)

where is the state vector, is the input, is the measured output, is a

zero mean white noise process and is the backward difference operator defined by

where is the backward shift operator. In contrast with the MMPC formulation in Chapter 5,

no discrimination is made here between the controlled outputs and the measurements, i.e.

.

The key difference between the proposed OFSPC algorithm and the CLSPC of [191] is

the use of the model (6.17)-(6.19), where is an integrated white noise process. The CLSPC

algorithm uses the standard state space model in innovation form described by (6.1)-(6.2).

According to [158, 160], in order to ensure zero steady-state error between the outputs and the

set points in the presence of piecewise constant disturbances and references, the noise model

191

should include integrated white noise disturbances. This condition is clearly not satisfied by the

model used in [191]. Consequently, the use of CLSPC may result in poor rejection of piecewise

constant disturbances. The model (6.17)-(6.19) was previously used in [194, 217] in an open-

loop non-adaptive setting.

To derive the subspace predictor that can be identified from closed-loop data, similar

ideas proposed in [191, 229] are used for the model (6.17)-(6.19). It is convenient to write the

model (6.17)-(6.19) in the equivalent form (6.20)-(6.21), where , ,

with , is defined in (6.22), and and are defined in (6.23). In the following, it will

be assumed that is stable.

(6.20)

(6.21)

[ ] (6.22)

, [ ] (6.23)

Based on (6.20), can be expressed as

[ ] (6.24)

where is a predefined past horizon. Since is assumed stable, the term in

(6.24) can be made arbitrarily small by selecting sufficiently large. Substituting (6.24) in (6.21)

and ignoring the term, is given by (6.25), where is defined in

(6.26). Following the same naming convention used in [229], the model (6.25) is considered to

be an Integrated VARX (VARIX) model.

(6.25)

192

[ | | | ]

[ | | | ]

(6.26)

Remark 6.1: If the matrix or its estimate is known, the Markov parameters , and

can be determined by appropriately partitioning as given in (6.26).

Using (6.25)-(6.26), the vector formed by stacking

is given by

(6.27)

where

[

] [

] (6.28)

[

] (6.29)

From the definitions of and in (6.22)-(6.23), (6.27) can be written in (6.30), where and

are defined similar to in (6.29) with replaced by and , respectively.

(6.30)

Let denote the -step ahead optimal predicted output increment given the measured

outputs and the known inputs . Based on (6.30), the vector

formed by stacking ( ) is given by

(6.31)

193

that can be expanded as

[

]

[

( )]

[

] [

]

[

( )]

(6.32)

Since and are lower triangular block Toeplitz matrices, (6.32) can be solved

efficiently, without inverting , using the forward substitution algorithm yielding

[

]

[

] (6.33)

where and can be recursively calculated for using (6.34)-

(6.35) [191].

∑ ( ) , (6.34)

( ) ∑ ( ) , (6.35)

Since is known, the first block row in (6.33) can be deleted. Furthermore, by

observing that does not depend on ( ), can be written as in

(6.36), where and are defined in (6.37)

(6.36)

[

] [

] (6.37)

194

By observing that ∑ , the output predictor

can be

related to using (6.38), where and are defined in (6.39).

(6.38)

[

] [

] (6.39)

Finally, by using (6.36) and (6.38), the output predictor in (6.40) can be obtained. The predictor

(6.40) allows the calculation of the output predictions within a prediction horizon from ,

, and .

(6.40)

Algorithm 6.1 below describes a procedure to calculate and of the predictor (6.40)

given the matrix . It should be noted that Algorithm 6.1 does not require the explicit knowledge

of the state-space model parameters , , and .

Algorithm 6.1 (Calculation of and given the matrix )

Input:

Output: and

1. Extract the Markov parameters , and from by

appropriately partitioning as given in (6.26).

2. Construct for as defined in (6.28).

3. Calculate and , using (6.34)-(6.35).

4. Construct and as defined in (6.37).

195

In §6.3.2, an algorithm is described to consistently estimate from I/O data collected

from an open-loop or closed-loop experiment. Once this estimate is available, consistent

estimates of and can be calculated using Algorithm 6.1.

6.3.2 VARIX model identification

6.3.2.1 Off-line Identification of

Assume the input and output measurements

, where , are available for the identification of in (6.25). It should be

noted that (6.25) can be regarded as a linear regression and therefore, can be estimated using

the Least Squares method.

To calculate the LS estimate , it is convenient to define , and [ as in

(6.41)-(6.43), respectively. The subscript indicates the time index of the last column of and

, and the subscript [ indicates the range of time indices of the last column of [ .

[ ] (6.41)

[ ] (6.42)

[ [ ]

[

]

(6.43)

Using (6.41)-(6.43) and (6.25), the LS estimate can be calculated using (6.44), where

denotes the pseudo-inverse of .

‖ [ ‖

[

(6.44)

196

An elegant solution of the LS problem (6.44) can be obtained by performing the RQ-

factorization in (6.45), from which can be calculated using (6.46) [226].

[ [

] [

] [

] (6.45)

(6.46)

Let [ ] denote the mathematical expectation. By assuming that is a quasi-stationary second

order ergodic process, and letting , it can be seen that

[

[ ] is always satisfied whether the system is open-loop, or closed-loop

under the assumption of a causal feedback controller. Therefore, the LS estimate in (6.44) is

consistent if either open-loop or closed-loop data is used.

Remark 6.2: is a lower triangular matrix and, therefore, can be calculated efficiently

without requiring the inversion of by solving the upper triangular system of equations

using the backward substitution algorithm.

6.3.2.2 On-line (recursive) identification of

For an adaptive SPC formulation, the estimate should be continuously updated using

the most recent measurements from the system. Furthermore, to account for (slow) time

variations in the system dynamics, one approach that is adopted in this chapter is to solve the

weighted LS problem (6.47) instead of (6.44), where the forgetting factor [ ] is used to

discount older data.

∑ ‖ ‖

(6.47)

197

A recursive solution to the LS problem of the form (6.47) is suggested in [226]. The idea

is to update and of the RQ-factorization in (6.45) at time based on their estimates at

. The concept is briefly reviewed as follows. To simplify the discussion, let . Assume

that the RQ factorization in (6.45) at is known, i.e. , and

are known. At time , a new measurement vector and a new stacked input-output vector

are available. From (6.41) and (6.43), the data matrix at time can be written as

[ [

] [

[

|

]

[

|

]

⏟

[

]

(6.48)

(6.49)

Instead of performing a new RQ-factorization for the whole data matrix (6.48), it is more

efficient to complete the RQ-factorization in (6.49) by annihilating the last column of .

In the general case, where [ ], it is shown that and can be updated by

applying the Givens rotation algorithm to find an orthogonal matrix such that (6.50) is

satisfied, with a lower triangular matrix and [190, 219]. Once and have

been updated, can be updated using (6.46). The procedure for updating and using the

Givens rotation algorithm is summarized in Algorithm 6.2 [190, 226].

[√

√ |

] [

|

] (6.50)

Algorithm 6.2 [190, 226] (Updating and )

Input: , , and

Output:

198

1. [√

√ ]

2. For

3. ,

4. √ , ⁄ , ⁄

5. [

]

6. End For

7.

8.

By performing Algorithm 6.2, it can be seen that the elements of the last column of in

step 1 are successively annihilated at each iteration. Furthermore, it should be noted that

Algorithm 6.2 does not perform the full RQ update. In fact, the Givens rotation procedure is

stopped once and have been updated. These matrices are the only matrices required for

calculating using (6.46). That is why the matrix does not appear in Algorithm 6.2.

Similarly, the matrices and are not used in Algorithm 6.2.

6.3.3 OFSPC algorithm

The MPC formulation considered here is based on the quadratic objective function (6.51),

where is the prediction horizon, is the control horizon, and ‖ ‖ .

Here, it is assumed that . Furthermore, it is assumed that physical limitations on the

plant can be modeled by the constraints (6.52), where denotes the maximum

(minimum) limit of •.

199

∑‖ ‖

∑ ‖ ‖

(6.51)

(6.52)

To formulate the MPC optimization problem, the output predictions for

should be available. Here, the subspace predictor (6.40) derived in §6.3.1 is used

for this task. In MPC, it is common to set for .

Consequently, the predictor (6.40) can be written as in (6.53), where and are defined in

(6.54).

(6.53)

(6.54)

It is convenient to define and as in (6.55), and

[

] and [

] . Similar definitions are

used for , , and , .

[

] [

] (6.55)

Based on (6.51)-(6.55), it is straightforward to write the MPC optimization problem as the QP

(6.56)

subject to:

(6.57)

where is a constant that is irrelevant to the QP solution,

200

, , (6.58)

, (6.59)

=

[

]

, =

[

]

, (6.60)

and and are defined similarly to and in (6.39), respectively,

with replaced by .

The OFSPC controller solves the QP (6.56)-(6.57) at each sampling instant. Only the first

optimal control increment is implemented. The MPC control law can be written in (6.61).

[ ]

(6.61)

The following proposition provides an analytic expression of the OFSPC control law

(6.61) when the inequality (6.57) is inactive. This is referred to as the unconstrained OFSPC.

Then, the conditions required to ensure offset-free tracking of the SPC are provided in Theorem

6.1.

Proposition 6.1: The unconstrained OFSPC control law is given by (6.62), where

and are defined in (6.63)-(6.64).

( ) (6.62)

[ ] (6.63)

[ ] (6.64)

201

Proof: Since in (6.58) is a positive definite matrix, the unique unconstrained optimal solution

of (6.56) is given by

. Using (6.58)-(6.59) and by taking

, (6.62)-(6.64) can be verified.

Theorem 6.1: Consider a system controlled by the OFSPC algorithm described by (6.56)-(6.57)

and (6.61), and the reference as . Assume that the closed-loop system reaches

steady state with the input and output constraints (6.57) inactive. If the gain matrix in (6.63)

is full column rank ( , then there is zero offset between the output and the reference,

i.e. where is the system output at steady state.

Proof: At steady state, the input and output constraints are not active, by assumption, and,

therefore, the MPC control law is (6.62). Furthermore, , where is the

steady state input, and . Consequently, at steady state, (6.62) implies

. If , then has empty null space and the unique solution of

is implying that the controller ensures offset free tracking of the reference.

The unconstrained CLSP developed in [191] is given by (6.65), where and are

defined in [191]. By comparing (6.65) and (6.62), it is easy to see that the CLSPC controller

structure does not contain integral action. Consequently, disturbance rejection is mainly achieved

by the adaptation of the model parameters. On the other hand, the proposed OFSPC algorithm

has integral action. This ensures effective rejection of piecewise constant disturbances.

(6.65)

Remark 6.3: For a SISO system, where , the unconstrained OFSPC in (6.62) reduces

to (6.66), where is the unit shift operator and and are defined in (6.67). The

202

controller in (6.66) is an output feedback controller with order and it includes a term

proportional to the integral of the error.

( ) (6.66)

∑

∑

(6.67)

Finally, the OFSPC algorithm that combines the predictor of §6.3.1, the identification

algorithm in §6.3.2, and the MPC controller (6.56)-(6.61) is summarized in Algorithm 6.3.

Algorithm 6.3 (OFSPC algorithm)

Input: , and

Output:

Predictor estimation

1. Calculate and .

2. Calculate and in (6.50) using Algorithm 6.2.

3. Solve

for using the backward substitution algorithm.

4. Use Algorithm 6.1 to compute , from .

5. Calculate and using (6.54).

MPC control calculation

6. Calculate , , and using (6.59)-(6.58).

7. Solve the QP (6.56)-(6.57) and apply the input (6.61).

203

Remark 6.4: If the objective function (6.68) is used by the OFSPC controller instead of (6.51), it

is easy to show that Algorithm 6.3 can be still applied. The only modification is to use and

as defined in (6.69) instead of (6.58), where is defined in (6.70).

∑‖ ‖

∑ ‖ ‖

∑ ‖ ‖

(6.68)

( )

(6.69)

[

] (6.70)

6.3.4 Examples

Example 1: Effect of the past horizon

In the OFSPC formulation in §6.3.1, the term in (6.24) is ignored. This

generally causes a bias in the estimates of the Markov parameters in (6.26). This example shows

that this bias can be made arbitrarily small by increasing the past horizon .

Consider the system (6.17)-(6.19), with parameters (6.71)-(6.72). The system is simulated

for 1000 samples using uncorrelated white noise signals applied at and with variances 1 and

0.1, respectively. Monte Carlo simulations are conducted by repeating the experiment 500 times.

The I/O data from each experiment is used to estimate as described in §6.3.2.1 using two

values of and . In both cases, is set to 5. The histograms of the first four

Markov parameters ( , , , are shown in Figure 6.2. One can see that the

estimates are biased for while the bias is almost eliminated when is used.

[

], [

], [

] (6.71)

[ ] (6.72)

204

(a)

(b)

Figure 6.2 Histograms Markov Parameters identified using (a) and (b) (thick

black line shows the true parameter value).

0.1 0.2 0.3 0.40

50

100

0 0.1 0.2 0.30

50

100

0.13 0.14 0.15 0.16 0.170

50

100

0.13 0.14 0.15 0.16 0.170

50

100

0.1 0.15 0.2 0.25 0.3 0.35 0.40

20

40

60

80

100

120

-0.1 0 0.1 0.2 0.30

20

40

60

80

100

120

0.1 0.11 0.12 0.13 0.14 0.150

20

40

60

80

100

0.12 0.13 0.14 0.15 0.16 0.170

20

40

60

80

100

205

Example 2: Comparison between the OFSPC and CLSPC [191] algorithms

The model in Figure 6.3 with

[191], is used to compare the performance of the

proposed OFSPC and the CLSPC [191] when a unit step disturbance is applied at = 140 s

and is white noise with variance 10-4

. The controller parameters are = = 5, = 6, = 2,

= 1, and the sampling time = 0.5 s. Both controllers are simulated with three different

values of = 0.95, 0.99 and 0.995. Results are shown in Figure 6.4. It can be seen that, in

general, the CLSPC does not perform well under piecewise constant disturbances. Furthermore,

the disturbance rejection speed is sensitive to . This is due to the fact that the CLSPC does not

have integral action. Therefore, the disturbance rejection is mainly achieved by the adaptation of

the model parameters. These drawbacks are eliminated by using the proposed OFSPC.

G(s)∑ ∑u

d e

y

Figure 6.3 Open-loop system model.

6.4 Application of OFSPC in wind turbine control

6.4.1 OFSPC controller design

The proposed wind turbine control strategy based on OFSPC (Algorithm 6.3) is shown in

Figure 6.5. For the wind turbine case, [ ] , [ ] , and

[

] . The objective function to be minimized by the OFSPC is

∑‖ ‖

∑ ‖ ‖

∑ ‖ ‖

(6.73)

where

206

[

], [

], [

]. (6.74)

(a) = 0.95

(b) = 0.99

(c) = 0.995

Figure 6.4 Performance comparison between the CLSPC and OFSPC algorithms with

three values of = 0.95, 0.99 and 0.995 during a step input disturbance.

100 110 120 130 140 150 160 170 180 190 200-1.5

-1

-0.5

0

0.5

1

1.5

r

CLSPC

OFSPC

Step

disturbance

100 110 120 130 140 150 160 170 180 190 200-1.5

-1

-0.5

0

0.5

1

1.5

r

CLSPC

OFSPC

Step

disturbance

100 110 120 130 140 150 160 170 180 190 200-1.5

-1

-0.5

0

0.5

1

1.5

time, s

r

CLSPC

OFSPC

Step

disturbance

207

Here, three sets of weights that correspond to the partial load, transition and full load

regions must be chosen by the designer. It should be noted that the weight used by the MMPC

does not appear in the OFSPC formulation. This is a result of the OFSPC predictor formulation

that does not rely on the internal state of the system. Consequently, direct tuning of the torsional

torque damping cannot be done in the OFSPC case. Despite that, the DEL will be calculated in

all simulation studies to assess drive train loads resulting from torsional oscillations. Finally, the

wind turbine constraints that are incorporated in the OFSPC formulation are the same as the ones

of the MMPC controller and they are given in (5.24)-(5.26).

Remark 6.5: If a FDI is available in the wind turbine, it can be integrated easily with the

proposed wind turbine FTC strategy based on OFSCP shown in Figure 6.5. In that case, the FDI

can be used to speed up the adaptation process by feeding the predictor estimation block with an

initial model that corresponds to the fault detected.

Wind

Turbine

β*

Tg*PWTG

ωg

PWTG*

ωg*

Predictor

Estimation

MPC

OFSPC

Lz, S

v

Figure 6.5 Proposed wind turbine FTC strategy based on OFSPC.

6.4.2 WECS model

The WECS model described in Chapter 3 and used in Chapter 5 is considered here. The

parameters of the WECS are detailed in Appendix D.

208

To illustrate the FTC capabilities of the OFSPC controller, the pitch system model in §3.4

is modified to account for a fault in the hydraulic actuator. The model described in [68] is

considered. It is assumed that the wind turbine has a hydraulic pitch actuator, and the fault

considered is a drop in the hydraulic pressure. According to [68], the pitch system is modeled by

the second order system (6.75), where is the damping factor and is the natural frequency.

The hydraulic pressure drop results in changes to the dynamics of the pitch actuator. This is

accounted for by considering and as functions of the fault parameter [ ] as given in

(6.76)-(6.77). According to (6.76)-(6.77), the damping factor and the natural frequency are equal

to their nominal values and , when ; and they are equal to their values at low

pressure and when .

( ) ( ) ( )

( ) (6.75)

where

( ) ( )

(6.76)

( ) ( ) ( ) (6.77)

The parameters , , and are given in (6.78) [68] and the step response of the pitch

system is compared during normal and faulty operation in Figure 6.6.

,

,

(6.78)

As described in §3.4, the pitch model should also include saturation blocks to represent

the physical limits on the pitch angle operating range and the pitch angle rate.

209

Figure 6.6 Step response of the pitch actuator system during normal and faulty operation.


The same simulation setup in §5.6.1 is used to verify the proposed OFSPC control

strategy. Here five wind turbine controllers are compared

OFSPC (Figure 6.5): This controller implements Algorithm 6.3. The controller

parameters are given in Table 6.1 and (6.79)-(6.80).

CLSPC: This controller implements the CLSPC algorithm described in [191]. The

controller parameters are given in Table 6.1 and (6.79)-(6.80).

MMPC1: This controller is based on the MMPC strategy in Chapter 5 with the controller

parameters in Table 6.1 and (6.79).

MMPC2: This is the MMPC controller designed in Chapter 5.

PI: This is the gain-scheduled PI controller designed in Chapter 5.

Table 6.1 OFSPC controller weights

Region , m/s

Partial load [4, 9) 1 2 0 1 1000 1000

Transition [9, 11) 2 2 0 1 4 0.25

Full load [11,26) 3 2 2.5 1 6 0

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

,

o

Normal

Fault

210

= 25, = 10, = 100 ms (6.79)

= 25, = 0.998 (6.80)

6.5.1 Performance comparison during normal WECS operation

The controllers are compared using a stochastic wind speed signal generated using the

model in §3.2.3 when the WECS is operating in partial load, transition region, and full load.

Within simulations, the tower shadow and wind shear effects are ignored. Furthermore, the speed

and power measurements are corrupted by additive zero-mean white noise with standard

deviation of 0.0158 rad/s, and 10-2

KW, respectively [232].


A simulation of ten minutes of partial load WECS operation was performed. The mean wind

speed is 6.5 m/s, the turbulence intensity is 12%. All other wind speed model parameters are

given in Appendix D. The wind profile is shown in Figure 6.7. A portion of sixty seconds of

simulation results for the OFSPC, CLSPC, and PI controllers is shown in Figure 6.8. The

average power and the DEL are compared for all five controllers in Table 6.2.


0 100 200 300 400 500 6005

6

7

8

9

time, s

v,

m/s

211

Figure 6.8 Simulation results for low wind speeds (zoomed from =300 to 360 s).

300 310 320 330 340 350 3600.75

0.8

0.85

0.9

0.95

time, s

g,

p.u

.

OFSPC

CLSPC

PI

g*

300 310 320 330 340 350 3600

0.1

0.2

0.3

0.4

time, s

PW

TG

, p.u

.

OFSPC

CLSPC

PI

300 310 320 330 340 350 3600

0.1

0.2

0.3

0.4

time, s

Ttw

, p.u

.

OFSPC

CLSPC

PI

300 310 320 330 340 350 3600

0.1

0.2

0.3

0.4

time, s

Tg,

p.u

.

OFSPC

CLSPC

PI

212

Table 6.2 Low wind speeds statistics (all quantities are normalized to the OFSPC

controller).

Quantity OFSPC CLSPC MMPC1 MMPC2 PI

AVG( ) 1 1 1 1 1

DEL 1 1 1.07 1.02 1.31

It can be seen that all controllers provide good tracking of the generator reference speed,

and they all have similar power production. However, the OFSPC and CLSPC controllers have

less drive train loads compared to MMPC and PI control strategies.


A simulation of ten minutes of partial load WECS operation near the rated wind speed (transition

region) was performed. The mean wind speed is 10 m/s. The wind profile is shown in Figure 6.9.

A portion of two minutes of simulation results for the OFSPC, CLSPC, and PI controllers is

shown in Figure 6.10. Statistics corresponding to all five controllers are given in Table 6.3.

Similar to the partial load case, all controllers provide similar power production. However, the

OFSPC provides significant improvement in DEL compared to the CLSPC and the PI

controllers. Results also show that the performance of the MMPC controllers is very similar to

the OFSPC.


0 100 200 300 400 500 6006

8

10

12

14

time, s

v,

m/s

213

Figure 6.10 Simulation results for medium wind speeds (zoomed from =340 to 460 s).

340 360 380 400 420 440 4601.05

1.1

1.15

1.2

1.25

time, s

g,

p.u

.

OFSPC

CLSPC

PI

g*

340 360 380 400 420 440 460

0.2

0.4

0.6

0.8

1

time, s

PW

TG

, p.u

.

OFSPC

CLSPC

PI

340 360 380 400 420 440 460

0.2

0.4

0.6

0.8

time, s

Ttw

, p.u

.

OFSPC

CLSPC

PI

340 360 380 400 420 440 4600

2

4

6

8

time, s

,

o

OFSPC

CLSPC

PI

214

Table 6.3 Medium wind speeds statistics (all quantities are normalized to the OFSPC

controller).


AVG( ) 1 1 1 1 1

DEL 1 1.1742 1.01 0.99 1.11

STD( ) 1 1 0.96 0.99 1.32

Max( ) 1 1 1.01 1.01 1.06

Max( ) 1 1 1 1 1.08


A simulation of ten minutes of full load operation at an average wind speed of 20 m/s

was performed. The wind profile is shown in Figure 6.11. A portion of thirty seconds of

simulation results for the OFSPC, CLSPC, and PI controllers is shown in Figure 6.12. Statistics

corresponding to all five controllers are given in Table 6.4. It can be observed that the OFSPC

provides a significant reduction in power fluctuations compared to the CLSPC and the other

controllers. Furthermore, the OFSPC provides better performance than the MMPC and PI

controllers in terms of generator speed fluctuations, drive train loads, and pitch activity.


0 100 200 300 400 500 60014

16

18

20

22

24

26

time, s

v,

m/s

215

Figure 6.12 Simulation results for high wind speeds (zoomed from =300 to 330 s).

300 305 310 315 320 325 330

1.17

1.18

1.19

1.2

1.21

1.22

time, s

g,

p.u

.

OFSPC CLSPC PI g*

300 305 310 315 320 325 3300.87

0.88

0.89

0.9

0.91

0.92

time, s

PW

TG

, p.u

.

OFSPC CLSPC PI PWTG*

300 305 310 315 320 325 330

0.7

0.75

0.8

time, s

Ttw

, p.u

.

OFSPC CLSPC PI

300 305 310 315 320 325 33012

14

16

18

time, s

,

o

OFSPC CLSPC PI

216

Table 6.4 High wind speeds statistics (all quantities are normalized to the OFSPC

controller).


STD( ) 1 0.99 3.44 2.67 1.68

STD( ) 1 1.27 3.85 4.35 13.28

DEL 1 1.01 1.44 1.38 1.39

STD( ) 1 0.97 1 1.10 1.39

6.5.2 Performance comparison during a fault in the pitch actuator

The wind turbine is simulated at an average wind speed of 20 m/s with an abrupt drop in

the hydraulic pressure of the pitch actuator occurring at = 200 s. The pitch actuator model in

§6.4.2 is used. The wind profile is shown in Figure 6.13. Simulation results for the OFSPC,

CLSPC and PI controllers are shown in Figure 6.14. It can be seen that the OFSPC and the

CLSPC succeeded in tolerating the fault with an acceptable adaptation speed. On the other hand,

large oscillations are produced after the fault when using the PI controller. These oscillations can

cause fatigue loading and a reduction of the wind turbine lifetime.

Figure 6.13 Wind profile used during simulations of an abrupt fault in the pitch actuator.

190 200 210 220 230 240 250 26019

20

21

22

23

24

25

time, s

v,

m/s

217

Figure 6.14 Comparison between the OFSPC (black), the CLSPC (red), and the classical PI

(blue) strategies after an abrupt drop in the hydraulic pressure.

190 200 210 220 230 240 250 2601.16

1.18

1.2

1.22

1.24

time, s

g,

p.u

.

abrupt

fault

190 200 210 220 230 240 250 2600.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

time, s

PW

TG

, p.u

.

abrupt

fault

190 200 210 220 230 240 250 26014

16

18

20

22

time, s

,

o

abrupt

fault

218

To compare the performance of the OFSPC, CLSPC, MMPC1, MMPC2, and PI

controllers during normal and faulty operation, a 10 min simulation is conducted in each case for

the same wind profile shown in Figure 6.11. Performance measures are calculated and compared

in Table 6.5. It can be seen that the PI controller performance deteriorates significantly during

faulty operation. This is expected since the PI controller is not designed to operate in such

conditions. In contrast, the OFSPC maintains good performance during normal and faulty

conditions and provides a better performance than the CLSPC.

Table 6.5 Performance comparison between OFSPC, CLSPC, MMPC1, MMPC2 and PI

controllers during a fault in the hydrolic pitch actuator. Bold and normal font show

quantities normalized to the OFSPC during faulty and normal operation, respectively.

Controller STD( ) STD( ) DEL STD( )

OFSPC 1 (1.93) 1 (1.71) 1 (1.25) 1 (1.12)

CLSPC 0.99 (1.91) 1.22 (2.09) 1.02 (1.27) 0.97 (1.09)

MMPC1 2.24 (4.34) 2.71 (4.64) 1.49 (1.86) 1.11 (1.24)

MMPC2 1.92 (3.71) 2.68 (4.60) 1.44 (1.81) 1.13 (1.28)

PI 1.51 (2.93) 13.34 (22.88) 1.97 (2.46) 1.63 (1.84)

6.6 Conclusions

An adaptive subspace predictive control algorithm, the OFSPC, is proposed to control

variable-speed variable-pitch wind turbines. In contrast with previously developed SPC

algorithms, the OFSPC includes integral action, and consequently, better rejection of piecewise

constant disturbances is guaranteed. The OFSPC strategy is compared with a classical PI strategy

and a closed-loop SPC strategy in controlling a 1.5 MW wind turbine under normal operation

and low hydraulic pressure fault in the pitch actuator. Simulation results show the superiority of

OFSPC in tolerating faults and offering good performance during normal and faulty operation.

219

This is due to the offset-free formulation of the OFSPC and its capability to adapt to variations in

the wind turbine dynamics.

220

Ensuring Fault Ride Through for DFIG-based Wind Turbines Chapter Seven:

A novel RSC control strategy that ensures Fault Ride Through (FRT) for DFIG-based

wind turbines according to recent grid codes is proposed in this chapter. During large voltage

dips, large currents are induced in the rotor that can destroy the RSC. The common approach

used to protect the RSC, known as the crowbar, is to disable the converter and to dissipate the

rotor power into a resistor bank. This approach is currently not accepted by most grid codes

which dictate that grid-connected wind turbines should remain connected to the grid during

severe faults with full control over their active and reactive power. The proposed strategy

ensures FRT for DFIG grid-connected wind turbines without using a crowbar, and therefore

meets recent grid code requirements. This is achieved by using an MPC controller incorporating

most of the DFIG’s constraints and a dynamic series resistance protection scheme.

This chapter is based on [233] and it is organized as follows. The FRT problem

associated with DFIG wind turbines is introduced in §7.1. Analysis of the DFIG behavior during

voltage dips is described in §7.2. Different RSC protection schemes are reviewed in §7.3. The

proposed MPC control solution is described in §7.4. Simulation results illustrating the

effectiveness of the developed method are given in §7.5, and §7.6 concludes the chapter.

7.1 Introduction

Wind turbines equipped with DFIGs are currently the most used configuration for wind

power generation [19-21]. With its partially rated power converters (typically 20-30% of the

system rated power), variable-speed operation of the wind turbine is provided at low cost in

comparison with wind turbine generators with fully rated converters.

The main drawback of DFIGs is that they are very sensitive to abrupt changes in their

terminal voltage [21, 43, 234]. When an external grid fault occurs, a large voltage dip is

221

produced at the DFIG terminals and large currents flow in both the DFIG’s stator and rotor

windings. Figure 7.1 illustrates the response of the DFIG in §3.6 with the generator controller in

Appendix A for an 80% stator voltage dip without using any protection. It can be observed that

the rotor current reaches large values that can lead to the destruction of the RSC [235].

Figure 7.1 Three phase stator voltages (top) and rotor currents (bottom) for a terminal

voltage dip of 80% with no protection.

The desired behavior for wind turbines during voltage dips, as specified by grid codes,

has faced dramatic changes during the past two decades [19, 236]. Initially, at the early stage of

wind power grid integration, it was acceptable to disconnect wind turbines from the grid during

faults to protect them from high currents. With the continuous increase in wind power capacity,

many countries specified grid codes such that wind turbines must remain connected to the grid

during faults [44]. This is motivated by the fact that disconnecting a large amount of wind power

0.1 0.2 0.3 0.4 0.5 0.6 0.7

-1

-0.5

0

0.5

1

time,s

v s,a

bc,

pu

0.1 0.2 0.3 0.4 0.5 0.6 0.7-3

-2

-1

0

1

2

3

4

time,s

i r,abc,

pu

3.1 pu peak rotor current

222

generation can deteriorate the stability of the grid. Wind turbine manufacturers reacted by

designing a protection strategy known as active crowbar [234, 237-239]. The main idea is to

disconnect the RSC during grid faults when rotor currents reach high values and divert them to

the crowbar, as shown in Figure 7.2. Once the fault is cleared, the crowbar is deactivated, the

RSC is reconnected and the DFIG resumes normal operation.

WRIG

RSC GSCGear

box

Wind

turbine

GSC

Controller

RSC

Controller

DC link

Generator

Controller

Filter

Tg*

vr* vC

*

*dcV*

GCQ*sQ

Grid

ucrow

Crowbar

Figure 7.2 DFIG-based wind turbine with a crowbar for RSC protection [237].

The use of active crowbar protection has many drawbacks [19, 40, 41, 51, 235]. First,

when the crowbar is active, the DFIG behaves like a conventional induction generator,

consuming more reactive power, resulting in voltage stability deterioration. Furthermore, during

this period, the RSC is disconnected and thus the control of the DFIG active and reactive power

is temporarily lost.

Quite recently, grid codes were modified such that for a wind turbine to be FRT capable,

it must not only remain connected to the grid during faults, but also control over its active and

reactive power should be maintained [44, 45, 240]. Generally, FRT capability is specified by a

region, see Figure 7.3, in which the wind turbine must not trip under symmetrical faults at the

223

point of interconnection. Different countries have adopted different FRT curves similar to the

one shown in Figure 7.3 [19, 236].

0.15 pu

Time, s

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2 2.5 3 3.5

Wind Plant

May Trip

0.9 puWind Plant

Must Not Trip

Vd

ip

Test

Fault

Vo

ltag

e at

th

e p

oin

t o

f

inte

rco

nn

ecti

on

, p

u

0.625 s

Beginning of voltage dip

Figure 7.3 Fault Ride Through standard according to US grid codes [236].

According to recent grid codes requirements, the use of active crowbar is not acceptable.

Many control/protection solutions have been proposed in literature in order to ensure FRT for

DFIGs [19, 20, 40, 41, 46-51]. As described in §1.2.3.2, many authors suggested modifying the

conventional RSC control algorithm [26] in order to account for stator voltage variation and

realize good disturbance rejection properties for the DFIG [46-48]. However, it is shown in [20]

that FRT cannot be met solely by the DFIG control when the voltage dip is severe and it

suggested using a combination of flux demagnetization and crowbar protection. A combined

converter protection scheme based on using a rotor-connected DSR and a crowbar is proposed in

[41]. The advantage achieved in [20, 41] is that the crowbar operation is limited to very short

periods of time. Despite this improvement, the RSC control is temporarily lost during crowbar

activation. The use of a stator-connected DSR to protect the RSC during severe voltage dips was

recently proposed in [40, 51].

224

A novel control strategy that ensures FRT for wind turbines with DFIGs is proposed in

this chapter. The proposed strategy uses a multivariable MPC controller to control the RSC and a

DSR to protect the RSC during severe grid faults. The proposed strategy does not require the

disconnection of the RSC and meets recent grid codes’ requirements.

7.2 DFIG behavior under voltage dips

The DFIG dynamic behavior and the factors affecting the rotor currents magnitudes

during three-phase voltage dips are studied in [43]. The main results are summarized in this

section. It is assumed that the rotor speed remains constant during electrical transients.

Furthermore, it is assumed that the DFIG is in steady state before the voltage dip occurs.

From the induction machine model in (3.51), the stator voltage at the stator reference

frame, and the rotor voltage at the rotor reference frame are governed by (7.1)- (7.2), where the

notation is defined in (3.43) and

.

, (7.1)

. (7.2)

From (3.45), the stator current in the stator reference frame can be written in terms of the rotor

current and stator flux linkage vectors as in (7.3). Similarly, (3.45) can be used to express the

rotor flux linkage in the rotor reference frame as in (7.4), where is the leakage factor.

(

) (7.3)

(7.4)

Substituting (7.3) in (7.1), and (7.4) in (7.2), the stator and rotor dynamical equations (7.5)-(7.6)

are obtained, where is defined in (7.7) [43].

225

(

)

, (7.5)

(7.6)

(7.7)

In (7.6), the term represents the voltage drop in both the rotor resistance and

the transient inductance ; and the term represents the rotor internal voltage induced due to

the stator flux variations as seen by the rotor. This voltage is produced at the rotor terminals if

the rotor is open-circuit [43].

An equivalent circuit of the rotor dynamics can be drawn, Figure 7.4, based on (7.6). It

can be concluded that the magnitude of the rotor currents is dependent on:

rotor resistance and transient inductance

internal rotor voltage

rotor terminal voltage controlled by the RSC

In general, the magnitude of is limited by the RSC capability, and and are small [43].

Therefore, it should be expected that if significantly exceeds the voltage limits of the RSC,

then large rotor currents are produced.

rri

rrv

rre

rR rL

Rotor

Side

Converter

Figure 7.4 DFIG rotor equivalent circuit [43].

226

7.2.1 DFIG behavior during normal operation

Consider the DFIG system with the three-phase balanced stator voltages given by (3.47),

where is the nominal stator line-to-neutral voltage, and is the grid frequency. Assuming

steady state operation, it can be shown that the stator voltage space vector is given by (7.8),

where is defined in (3.48) [43]. The stator flux linkage can be determined from (7.8) and (7.5)

by neglecting and setting , as given in (7.9). It can be seen that both and

are

vectors rotating steadily with a constant speed .

√

(7.8)

√

(7.9)

From (7.9) and (7.7), the stator flux linkage and the rotor internal voltage space vectors in the

rotor reference frame are given by (7.10)-(7.11), where

is the slip.

√

(7.10)

(

√ ) (7.11)

For DFIG systems, and the magnitude of the rotor internal voltage can be approximated

by (7.12). It can be concluded that the magnitude of the rotor internal voltage is proportional to

the slip during normal operation [43].

| | | |√ (7.12)

7.2.2 DFIG behavior under a voltage dip

Consider first a full voltage dip occurring at the stator terminals at . This might be

caused by a bolted three-phase fault occurring at the DFIG terminals. In that case the stator

227

voltage space vector is for . By neglecting the term

in (7.5), the stator

dynamics are governed by

(

)

, (7.13)

The solution of (7.13) with initial condition

is given by (7.14), where

. In contrast with (7.9), the natural stator flux component (7.14) appearing during a full

voltage dip is a fixed vector. Its amplitude decreases exponentially to zero according to the stator

time constant . The relative speed between this flux component and the rotor is the rotor speed

.

√

, (7.14)

From (7.14) and (7.7), the rotor internal voltage after a full voltage dip is given by (7.15).

√

(

)

, (7.15)

Typically, ⁄ is much smaller than and . Therefore, the maximum magnitude of

the rotor internal voltage, occurring at , can be approximated by (7.16) [43]. This shows

that the rotor internal voltage is proportional to . This value is much larger than the value

in (7.12) obtained during normal operation for a typical DFIG slip range .

| | √ (7.16)

In the more general case, grid faults cause partial voltage dips such as the one shown by

the dotted line in Figure 7.3, where denotes the dip depth. In that case, it is shown in [43]

that the stator flux linkage is given by (7.17), where and

are the stator natural and forced

228

flux components, respectively. Furthermore, the rotor internal voltage at can be

approximated by (7.18) [43].

√

⏟

( )√

⏟

(7.17)

| | ( ( ))√ (7.18)

Typically, the term ( ) in (7.18) is small as it is proportional to the slip; and the

magnitude of the rotor internal voltage is dominated by the term √ . This shows

that the magnitude of the rotor internal voltage during a partial voltage dip is proportional to the

dip depth and the p.u. rotor speed .

It can be concluded from the analysis that the amplitude of the voltage induced in the

rotor at the first moment of the dip is similar to the stator rated voltage instead of the small

percentage induced in normal operation [43]. The overvoltage caused by the dip notably exceeds

the maximum voltage that can be produced by the RSC. Therefore, high rotor currents are likely

to appear, that can destroy the RSC and the dc link unless protective action is taken [21, 43, 237].

7.3 RSC protection schemes

7.3.1 Crowbar protection

Currently, active crowbar protection [234, 237], shown in Figure 7.2, is the most

employed protection scheme for DFIGs [19]. An active crowbar consists of a resistor bank that

can be connected and disconnected from the rotor windings. The crowbar activation is based on

two signals that are continuously monitored, the rotor currents and the DC link voltage. If any of

these signals reaches a certain maximum limit, the RSC is blocked and the crowbar circuit is

activated short circuiting the rotor winding through the resistor bank. Consequently, the high

229

rotor currents flow through the crowbar protecting the converter. Once safe operation of the RSC

is detected, the crowbar is deactivated and the RSC is reactivated resuming normal operation of

the DFIG. Two criteria are proposed to decide that the RSC can operate safely. It is proposed in

[234, 237] to deactivate the crowbar once the stator voltage has recovered and is above a certain

safe level. Another approach [20] is based on estimating the natural flux component, and normal

operation is resumed once the natural flux magnitude is below certain threshold. This value is

chosen in [20] as the maximum natural flux magnitude that can be cancelled by the rotor current

without violating the RSC current limits.

7.3.2 Dynamic series resistance

Recently, new protection configurations based on using a dynamic series resistance

inserted into the rotor or the stator windings have been proposed [40, 41, 51]. This protection

scheme was originally proposed for Type 1 wind turbines in [241].

The use of a rotor-connected DSR, as shown in Figure 7.5 (a), is proposed in [41]. During

normal operation, the power electronic switch is on and the series resistance is bypassed. Once a

fault occurs and rotor currents increase above the maximum allowable value of the RSC, the

switch is turned off and the resistance is inserted in series with the rotor windings and the

RSC, limiting the rotor current. It can be seen from Figure 7.4 that by increasing the rotor

effective resistance using a sufficiently large , and by using an appropriate RSC control

strategy, the rotor currents can be kept within acceptable limits without requiring the

disconnection of the RSC.

A similar idea is proposed in [40, 51]. The main difference is that the DSR is installed

between the stator windings and the grid as shown in Figure 7.5 (b). The insertion of the stator-

connected DSR during low voltage dips limits the stator and rotor currents, allows faster

230

damping of the stator natural flux, and limits the oscillations of DFIG transient response [40].

However, compared to the rotor-connected DSR approach, the power electronic switches used in

this configuration have higher ratings, and are thus more expensive, as they are connected to the

stator windings.

WRIG

RSC GSCGear

box

Wind

turbine

GSC

Controller

RSC

Controller

DC link

Generator

Controller

Filter

Tg*

vr*

vC*

*dcV*

GCQ*sQ

Grid

DSR

WRIG

RSC GSCGear

box

Wind

turbine

DC linkFilter

Grid

DSR

uDSR

GSC

Controller

RSC

Controller

Generator

Controller

Tg*

vr*

vC*

*dcV*

GCQ*sQ

uDSR

(a)

(b)

Figure 7.5 DFIG-based wind turbine with (a) rotor-connected DSR [41] and (b) stator-

connected DSR [40] for RSC protection.

7.4 Proposed control strategy based on MPC and DSR protection scheme

7.4.1 RSC control design requirements

There are two main design requirements that must be addressed in any proposed RSC

control scheme, namely:

231

During normal DFIG operation, the RSC controller should ensure good tracking of the

generator torque and reactive power set points.

During voltage dips, the RSC should remain connected to the rotor and the control

system should ensure fast rejection of stator voltage variations.

It is useful at this point to assess the baseline RSC controller described in Appendix A in

terms of both requirements. A simplified representation of this strategy is shown in Figure 7.6

for convenience. As described in Appendix A, the RSC controller can be designed to guarantee

fast decoupled control of and during normal DFIG operation, and thus meet the first

requirement. However, its performance significantly deteriorates during voltage dips [47]. The

reason is that its design is based on the assumption that the stator voltage level is constant and

the stator flux linkage space vector is rotating steadily at synchronous speed. Clearly, these

assumptions are invalid during voltage dips as described in §7.2.2.

PI PI

PI PI

RSC Controller

Tgiqr

vqr

idr

vdr

Qs

DecouplerCurrent ControllerTg&Qs Controller

Induction

machine

ucrow

vqs

vds

feedback

ccqrv

ccdrv

'qrv

'drv

*gT

*dri*

sQ

*qri

Decision

Maker

iqridr,

sn

Figure 7.6 Baseline control strategy.

7.4.2 Motivation for using MPC

The use of MPC techniques described in Chapter 4 [125, 128] for RSC control offers

many advantages over commonly used PI-based control strategies [19, 20, 40, 41, 46, 51, 234,

232

237]. MPC techniques can explicitly handle RSC current and voltage constraints. Furthermore,

feed-forward compensation of measurable disturbances, such as the stator voltage, can be

achieved effectively by including the measurable disturbances in the prediction model. This can

ensure fast rejection of stator voltage dips without waiting for the rotor currents to reach high

values. In addition, an MPC controller can be easily reconfigured by changing the prediction

model used. This feature is useful since the dynamics of the controlled system (WRIG) change

when the DSR is switched on and off. Finally, MPC is a multivariable control strategy which is

suitable for MIMO systems such as the WRIG.

7.4.3 Overview of the proposed control strategy

The proposed control strategy for RSC control is shown in Figure 7.7. The proposed

strategy differs from the conventional one, shown in Figure 7.6, in that the PI current controllers

and the cross-coupling compensation are replaced with a multivariable MPC current controller.

Furthermore, the MPC uses the stator voltage measurements, and as measurable

disturbances. This allows the RSC controller to react immediately after a voltage dip to prevent

future current and voltage constraint violations. Finally, the stator and rotor currents are

estimated in a synchronously rotating reference frame aligned with the stator voltage space

vector instead of the stator flux linkage space vector. The reason for this choice is explained in

§7.4.4.

The binary signal in Figure 7.7 is used to switch the DSR switch on and off, and it

is defined in (7.19). This signal is fed to the MPC controller to determine the suitable prediction

model of the WRIG.

(7.19)

233

PI

PI

RSC Controller

Tg

iqr

vqr

idr

vdr

Qs

Tg&Qs Controller

Induction

machine

Decision

Maker

uDSR

vqs

vds

feedback

iqridr,

*gT

*dri

*sQ

*qri

MPC

sn

iqsids

Figure 7.7 Proposed RSC control strategy.

The main idea of the proposed strategy is based on using an MPC controller with two

switching internal models to control the RSC. The first model represents the WRIG without any

external resistance added to the rotor or the stator. This model is used during normal DFIG

operation when the DSR switch is on (closed). The second model represents the DFIG dynamics

with the DSR resistance inserted into the rotor or the stator winding depending on the chosen

DSR protection scheme. This model is used during severe voltage dips, when the DSR switch is

off (opened), and thus the effective rotor or stator resistance is increased to limit the rotor

currents.

The combination of the proposed MPC controller and the DSR protection does not

require the disconnection of the RSC, and thus the DFIG behavior is controlled even during

severe voltage dips. The design of the MPC and decision maker blocks shown in Figure 7.7 is

detailed in the rest of this section.

234

Remark 7.1: It is more intuitive to use one multivariable MPC controller that directly controls

the generator torque and reactive power instead of the two-level control structure in Figure 7.7.

However, the controlled output variables and in that case are nonlinear functions of the

WRIG states as shown in (3.38) and (3.40). This renders the MPC optimization problem non-

convex and hard to solve at the fast sampling rate that is used in RSC control.

7.4.4 Stator voltage space vector reference frame orientation

An essential requirement for an effective MPC controller is to have an accurate model

representing the dynamics of the controlled system. In order to use the model (A.1)-(A.11) in the

MPC design, all machine variables must be estimated in a synchronously rotating reference

frame. The reference frame aligned with the stator flux space vector is chosen in the baseline

vector control strategy described in Appendix A. This choice is valid during normal operation as

the stator flux space vector is rotating at synchronous speed, and in this reference frame

decoupled control of the torque and reactive power can be realized [26]. However, the analysis in

§7.2 reveals that this property does not hold during voltage dips. To resolve that issue and to

allow the use of the model (A.1)-(A.11), the stator voltage space vector reference frame

orientation is used in the proposed strategy instead of the stator flux orientation used in

Appendix A.

Estimation of the WRIG variables in the stator voltage oriented reference frame is

described as follows. First the stator voltage angle and magnitude can be calculated from the

a-b-c stator voltages using (A.41) and (A.42). Then, the d-q-0 stator and rotor currents are

estimated using (7.20) and (7.21), respectively, where is defined in (3.31). The generator

torque and stator reactive power can be calculated using (7.22) and (7.23). These estimated

235

variables are used by the proposed control system in Figure 7.7. Finally and , calculated

by the proposed strategy, can be transformed to , , and using (7.24).

(7.20)

(7.21)

(7.22)

(7.23)

(7.24)

Now it remains to check that decoupled control of and can be achieved during

normal operation by controlling the rotor currents in the chosen reference frame. This follows by

observing that and is constant during normal operation. Therefore, and

follow from (A.1) and (A.2) by neglecting the stator resistance. Based on these

observations, (7.25) and (7.26) can be easily deduced from (A.5)-(A.6) and (7.22)-(7.23).

Equations (7.25) and (7.26) show that, under stator voltage orientation, decoupled control of

and can be achieved by controlling and , respectively.

(7.25)

( ) (7.26)

7.4.5 MPC design

The main components of the RSC MPC controller shown in Figure 7.7 are described in

this section. It is assumed that the states of the WRIG are accurately calculated as described in

§7.4.4 and thus a state observer is not used. The MPC controller formulation is detailed for the

236

case of a rotor-connected DSR protection scheme. The formulation for the stator-connected DSR

protection follows similar lines.

7.4.5.1 Prediction model

The WRIG model in a stator voltage oriented reference frame rotating at synchronous

speed is given in (7.27)-(7.29), where [ ] is the control input,

[ ] is the state vector, [ ] is the measurable

disturbance and [ ] is the controlled output. This model is obtained from

(3.35) by eliminating the stator and rotor flux linkages using (3.36). The term is

introduced in (7.30) to reflect the fact that the value of the effective rotor resistance is either

equal to if the DSR switch is on, or if the DSR switch is off.

(7.27)

(7.28)

(7.29)

[

( )

( )

( )

( )

]

, (7.30)

[

]

,

[

]

, (7.31)

[ ]. (7.32)

237

Two discrete time models (7.33) can be obtained by discretizing (7.27)-(7.29). The one

corresponding to , denoted by , is used by the MPC during normal DFIG operation.

The other one corresponding to , denoted by , is used when the DSR is inserted in

series with the rotor. The model (7.33) is a special case of the model (4.1)-(4.3), and it is

obtained by setting , , , and .

, for (7.33)

Remark 7.2: The model of the WRIG in the case of stator-connected DSR is the same as (7.27)-

(7.29) with replaced by , and .

7.4.5.2 MPC optimization problem

Good tracking performance of the rotor currents while keeping the magnitudes of the

rotor voltages and currents within safe limits is achieved by solving the optimization problem

(7.34)-(7.38) at each sampling time by the MPC controller. Since the prediction model is linear

and the constraints (7.36)-(7.37) are quadratic, the problem (7.34)-(7.38) is a Quadratically

Constrained Quadratic Program (QCQP) [167]. This is a convex optimization problem that can

be solved effectively using interior point algorithms [167].

∑‖ ‖

∑ ‖ ‖

(7.34)

prediction model equations in (7.33) (7.35)

‖ ‖

, (7.36)

238

‖ ‖

, (7.37)

. (7.38)

Here, , , and denote the prediction horizon, the control horizon, the

maximum allowable RSC voltage and current, respectively. The weights and are defined

in (7.39). The control move is defined as and ‖ ‖ √ is the

Euclidian norm of the vector .

[

],

[

] (7.39)

The slack variable is used in (7.37) to soften the rotor current magnitude constraint.

However, constraint violations are penalized by a large penalty in the objective function

represented by the term , where is a user defined large positive scalar The use of soft

output constraints in the MPC formulation (7.34)-(7.38) prevents running into infeasibility

problems during real time control.

The quadratic constraints (7.36)-(7.37) ensure that the instantaneous values of the rotor

currents and voltages are kept below and . This can be shown as follows. From the

definitions of and

and since for balanced rotor voltages, (7.40) is obtained.

Using (7.40) and (7.24), and from the orthogonally of , (7.41) and (7.42) are obtained.

Consequently, the constraint (7.36) is equivalent to (7.43). For a three-phase balanced sinusoidal

rotor voltage, (7.43) implies that the maximum rotor voltage is less than .

‖ ‖ ‖

‖

‖ ‖

‖ ‖

(7.40)

(7.41)

(7.42)

239

(7.43)

7.4.5.3 Possible MPC implementation approaches

There are many approaches that can be used for implementing the MPC controller

previously discussed. The first approach is to use a Semi-Definite Program (SDP) solver [167,

242] to solve the QCQP in (7.34)-(7.38). This approach will be referred by . Another

alternative is to approximate the quadratic constraints in (7.36) and (7.37) by the two polytopes

shown in Figure 7.8, with vertices defined in (7.44) where , and denote the number of

polytope vertices, the real part and the imaginary part of a complex number, respectively. In that

case the QPQC in (7.34)-(7.38) can be approximated by the QP in (7.45)-(7.49), where is a

vector entirely composed of s and , , , and can be

easily derived from the polytope vertices in (7.44).

√

[ (

) (

)]

√

[ (

) (

)]

, for (7.44)

∑‖ ‖

∑ ‖ ‖

(7.45)

prediction model equations in (7.33) (7.46)

, (7.47)

, (7.48)

. (7.49)

Remark 7.3: Increasing improves the approximation accuracy. However, this also increases

the number of linear constraints and the complexity of the QP solved.

240

dri

qri

max,2

3rI

ci,1

ci,2ci,3

ci,4

ci,5 ci,6

drv

qrv

max,2

3rV

cv,1

cv,2cv,3

cv,4

cv,5 cv,6

(b)(a)

Figure 7.8 Shaded regions show the polytopic approximation of the (a) rotor voltage

constraint in (7.36) and (b) rotor current constraint in (7.37).

As described in §4.7, there are two approaches that can be used to implement the MPC

controller (7.45)-(7.49) in real time. The first approach relies on using a QP solver to solve the

QP online at each sampling time. This approach will be referred by . The other approach

relies on calculating the explicit MPC control law offline and to implement the MPC online as a

lookup table of affine state feedback controllers. This approach will be referred by . All

three approaches will be compared in §7.5.

7.4.6 Decision maker design

The main function of the decision maker block in Figure 7.7 is to decide whether the

DSR should be connected or not. Figure 7.9 shows the logic implemented in the decision maker.

The DSR protection is triggered if one of the following conditions occurs:

the rotor currents exceed the maximum allowable limit of the RSC, or

the stator natural flux component is higher than certain threshold value .

241

The DSR protection is deactivated when both of the above conditions cease to be present. The

“Hold” block in Figure 7.9 is used to ensure that once the DSR is triggered, it will be activated

for at least a period of time defined by . This is used to prevent unnecessary high

frequency switching of the DSR. The parameter is selected as the maximum stator

natural flux magnitude that can be cancelled by the rotor current without violating the RSC

current limits [20]. This value is given by (7.50).

(√

) (7.50)

Typically, the following sequence of events occurs when the DFIG is exposed to a severe

voltage dip. At the moment of the dip, a large stator natural flux component appears causing an

increase in the rotor currents. Once the rotor currents exceed the RSC current limit, the DSR is

connected limiting the rotor currents. During that period, the DFIG is demagnetized. Once the

stator natural flux component is below , the DSR is removed from the rotor or stator

winding and normal operation is resumed.

>

22qrdr ii

OR

sn

>

idr

iqr

sn uDSR

max,2/3 rI

threshold

OR

Hold

THOLD

Figure 7.9 Decision maker block.

242


7.5.1 Comparison of different MPC implementations

The objective is to compare the computational speed of , and ,

described in §7.4.5.3. For any of these algorithms to be implementable in controlling the RSC,

the algorithm’s computational time must be sufficiently smaller than the sampling period used,

which is typically in the order of milliseconds.

The , and controllers are designed and tested using MATLAB

and SIMULINK®. The controller is implemented using YALMIP modeling language

[243] and the SEDUMI optimization solver [242] is used to solve the QCQP (7.34)-(7.38)

online. The controller uses the QPC solver [244] to solve the QP (7.45)-(7.49) online.

The is designed and implemented using the MPT toolbox [245]. In both the and

, polytopes with eight vertices are used to approximate the quadratic norm constraints as

shown in Figure 7.8. All three MPC controllers’ data are summarized in Table 7.1.

Table 7.1 Different MPC implementation approaches.

, 8, 4 8, 4 3, 2

Complexity 12 quadratic constraints 96 linear constraints 22000 polytopic

regions in

Solver SeDumi QPC MPT

The 1.5 MW DFIG system modeled in Chapter 3, with parameters given in Appendix D,

is simulated with all three RSC controllers for 10 seconds with a sampling period of 1 ms and

random changes in the rotor current set points and the stator voltages. Simulations are carried on

two dual core PCs. The first PC, C1, is 1.66 GHz with 2 MB cache while the second one, C2 is

243

3.16 GHz with 6 MB cache. Table 7.2 provides an idea about the computational time required by

each MPC algorithm. It can be seen that the use of is suitable for RSC control at fast

sampling rates. This controller is used in the rest of simulations provided in this section.

Table 7.2 Computational time statistics for different MPCs.

CPU time statistics C1 (1.66 GHz, 2 MB) C2 (3.16 GHz, 6 MB)

Max time (ms) 510 0.55 376.9 176.8 0.16 138.1

AVG time (ms) 218.4 0.17 49.8 70.6 0.067 16.2

% (time<1ms) 0 100 60 0 100 70

7.5.2 Evaluation of the proposed control strategy

In this subsection, the following RSC control/protection strategies are compared:

baseline PI strategy in Figure 7.6 with an active crowbar protection scheme

proposed MPC strategy in Figure 7.7 with a rotor-connected DSR protection scheme

proposed MPC strategy in Figure 7.7 with a stator-connected DSR protection scheme

These control strategies are used to control the RSC of the 1.5 MW DFIG-based wind

turbine described in Chapter 3. The values of the crowbar resistance in Figure 7.2, the rotor-

connected DSR in Figure 7.5 (a), and the stator-connected DSR in Figure 7.5 (b) are all equal to

0.75 p.u. The crowbar and DSR protection schemes are operated according to the logic in Figure

7.9.

The DFIG system used in simulations is operating at full load at 20 m/s with a rotor

rotational speed of 1.2 p.u. The RSC current limit, , and voltage limit, , are taken as

1.5 p.u. and 0.5 p.u., respectively [20, 46].

244

7.5.2.1 Normal DFIG operation

The main tuning parameters of the MPC controller (7.45)-(7.49) are the weights and

. These weights are chosen to give approximately the same tracking speed of rotor current set

points obtained using the baseline PI rotor current controllers with parameters in Appendix D.

The values of and are given in (7.51) and other parameters are given in (7.52).

[

], [

], for (7.51)

, , , (7.52)

The DFIG responses using the MPC and the baseline PI current controllers under step

changes in the rotor current set points are shown in Figure 7.10 and Figure 7.11, respectively. It

can be seen that the MPC current controller provides fast decoupled control of the rotor direct

and quadrature axis currents with a settling time of 10 ms.

The torque and reactive power controllers in Figure 7.7 have the same parameters of the

corresponding PI controllers of the baseline PI strategy in Figure 7.6. These parameters are given

in Appendix D. The response of the DFIG system with the proposed MPC strategy under step

changes in the generator torque and reactive power set points is shown in Figure 7.12. It can be

seen that the MPC current controller provides fast decoupled tracking of the generator torque and

reactive power set points, and thus meets the first design requirement described in §7.4.1.

7.5.2.2 DFIG operation during voltage dips

The objective in this section is to evaluate the performance of the proposed MPC strategy

during stator voltage dips. A test fault shown in dotted line in Figure 7.3, with duration of 300

ms, is applied at 0.2 s at the DFIG terminals. Three case studies, with different voltage dip

magnitudes, , are considered. In these studies, the stator reactive power set point is fixed at 0.

245

Figure 7.10 Tracking of step changes in the rotor currents’ set points using the proposed

MPC strategy.

Figure 7.11 Tracking of step changes in the rotor currents’ set points using the baseline PI

strategy.

Figure 7.12 Tracking of step changes in the generator torque and stator reactive power set

points using the proposed MPC strategy.

0.05 0.1 0.15 0.20.75

0.8

0.85

0.9

(a)

i dr,

pu

0.05 0.1 0.15 0.2

-0.35

-0.3

-0.25

-0.2

time, s

(b)

i qr,

pu

iqr

iqr*

idr

idr*

0.05 0.1 0.15 0.2

0.35

0.4

0.45

0.5

(a)

i dr,

pu

0.05 0.1 0.15 0.20.75

0.8

0.85

0.9

time, s

(b)

i qr,

pu

iqr

iqr*

idr

idr*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.75

0.8

0.85

0.9

time, s

Tg,

pu

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.05

0.1

0.15

time, s

Qs,

pu

Tg

Tg*

Qs

Qs*

246

Case 1: small voltage dip ( p.u.)

The three-phase stator voltage is shown in Figure 7.13. In this case, neither the crowbar

nor the DSR are triggered by the decision maker. The DFIG responses using the baseline PI and

the proposed MPC strategies are shown in Figures 7.14 and 7.15, respectively. It can be seen that

the proposed strategy offers much faster rejection of the grid disturbances. Furthermore, the

oscillations in the torque and stator reactive power are effectively reduced. This typically results

in a reduction in drive train transient loads produced during grid faults. It should be noted that

this performance improvement is obtained without any performance degradation of the DFIG

during normal operation as shown in §7.5.2.1.

The dc link voltage is one of the DFIG variables that are indirectly affected by the

performance of the RSC controller. As described in Appendix A, this signal is regulated at its

nominal value using the GSC controller. However, the active power required by the rotor is

transferred through the dc link. Therefore, any perturbation in the rotor active power affects the

dc link voltage. Figure 7.16 shows the dc link voltage obtained during simulations. The proposed

MPC strategy provides a relatively smaller impact on the dc link voltage compared to the

baseline PI strategy.

Figure 7.13 Stator voltages during a small voltage dip ( p.u.).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

-0.5

0

0.5

1

time,s

v s,a

bc,

pu

247

Figure 7.14 DFIG response using the baseline PI strategy during a small voltage dip


stator reactive power. (dashed lines show maximum RSC limits)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.8

-0.4

0

0.4

0.8

time, s

(d)

Qs,

pu

248

Figure 7.15 DFIG response using the proposed MPC strategy during a small voltage dip



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.8

-0.4

0

0.4

0.8

time, s

(d)

Qs,

pu

249

Figure 7.16 dc link voltage during a small voltage dip ( p.u.) using the baseline

PI (dashed) and the proposed MPC (solid) strategies.

The performance of the proposed and baseline PI control strategies are compared in terms

of maximum , the duration for which the error in the stator reactive power is greater

than 5%, the maximum drive train torsional torque, and the maximum dc link voltage in Table

7.3. Results show significant performance improvement by using the proposed MPC strategy.

Table 7.3 Comparison between the proposed MPC and baseline PI strategies during a

small voltage dip ( p.u.).

Quantity MPC PI

Max( , p.u. 0.11 0.76

, s 0.3 1.32

Max( , p.u. 0.96 1.75

Max , p.u. 1.07 1.11

Case 2: medium voltage dip ( p.u.)

The three-phase stator voltage is shown in Figure 7.17. In this case, the crowbar, the

rotor-connected DSR, and the stator-connected DSR are triggered during simulations. Their

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.85

0.9

0.95

1

1.05

1.1

time, s

Vdc,

pu

PI

MPC

250

activation signals are shown in Figures 7.18-7.20. The DFIG responses using the baseline PI

strategy, the proposed MPC strategy with rotor-connected DSR, and the proposed MPC strategy

with stator-connected DSR are shown in Figures 7.21-7.23, respectively.

Figure 7.17 Stator voltages during a medium voltage dip ( p.u.).

Figure 7.18 Crowbar activation signal during a medium voltage dip.

Figure 7.19 Rotor-connected DSR activation signal during a medium voltage dip.

Figure 7.20 Stator-connected DSR activation signal during a medium voltage dip.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

-0.5

0

0.5

1

time,s

v s,a

bc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

time,s

cro

wbar

Sig

nal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

time,s

DS

R S

ignal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

time,s

DS

R S

ignal

251

Figure 7.21 DFIG response using the baseline PI strategy during a medium voltage dip



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

2

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

time, s

(d)

Qs,

pu

252


during a medium voltage dip ( p.u.): (a) rotor voltages, (b) rotor currents, (c)

generator torque, and (d) stator reactive power. (dashed lines show maximum RSC limits)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

time, s

(d)

Qs,

pu

253


during a medium voltage dip ( p.u.): (a) rotor voltages, (b) rotor currents, (c)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.3

-0.2

-0.1

0

0.1

0.2

0.3

time, s

(d)

Qs,

pu

254

It can be seen that the proposed MPC strategy provides fast rejection of the grid voltage

disturbances and less oscillations in the generator torque and reactive power compared to the

baseline PI strategy. In particular, the proposed MPC strategy with a stator-connected DSR

provides the best performance among all three control/protection strategies. It should be noticed

also that the MPC control strategy is able to keep the rotor currents and voltages within

allowable limits of the RSC (Figures 7.22-7.23 (a) and (b)). This allows the RSC to be always

connected to the rotor controlling the DFIG behavior.

Figure 7.24 shows the dc link voltage obtained during simulations. It can be seen that

voltage reaches a large value when the baseline PI strategy is used. This is due to the large

amount of rotor power produced that charges the dc link capacitor. Typically, the dc link voltage

is protected using a parallel dc-chopper circuit that is triggered to dissipate excess rotor power,

thus keeping the dc voltage level within safe limits. On the other hand, the proposed MPC

strategy (in particular the one with a stator-connected DSR) provides a much smaller impact on

the dc link voltage. Table 7.4 compares the performance of all three control strategies.

Figure 7.24 dc link voltage during a medium voltage dip ( p.u.) using the

baseline PI (dashed), the proposed MPC with rotor-connected DSR (dotted), and the

proposed MPC with stator-connected DSR (solid) strategies.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.8

0.9

1

1.1

1.2

1.3

1.4

time, s

Vdc,

pu

PI

MPC-DSRr

MPC-DSRs

255

Table 7.4 Comparison between the proposed MPC with rotor- and stator-connected DSR,

and the baseline PI control strategies during a medium dip ( p.u.).

Quantity MPC-DSRr MPC-DSRs PI

Max( , p.u. 0.26 0.18 1.1

, s 1.32 0.16 1.41

Max( , p.u. 0.81 0.76 2

Max , p.u. 1.16 1.04 1.42

Case 3: large voltage dip ( p.u.)

The three-phase stator voltage is shown in Figure 7.25. In this case, the crowbar, the

rotor-connected DSR, and the stator-connected DSR are triggered during simulations. Their

activation signals are shown in Figures 7.26-7.28. The DFIG responses using the baseline PI

strategy, the proposed MPC strategy with rotor-connected DSR, and the proposed MPC strategy

with stator-connected DSR are shown in Figures 7.29-7.31, respectively.

It can be seen from Figures 7.29-7.31 that fast rejection of the grid voltage disturbances is

achieved using the proposed MPC strategy. In particular, the proposed MPC strategy with a

stator-connected DSR provides the best performance among all three control/protection

strategies. Furthermore, the MPC controller maintained the rotor currents and voltages within

allowable limits of the RSC, and therefore, the RSC is always connected to the rotor and control

over the DFIG is maintained. The dc link voltage is shown in Figure 7.32. Clearly, the proposed

MPC strategy with stator-connected DSR has the minimum impact on the dc link voltage. Table

7.5 compares the performance of all three control strategies. Results confirm the superiority of

the MPC strategy with stator-connected DSR.

256

Figure 7.25 Stator voltages during a large voltage dip ( p.u.).

Figure 7.26 Crowbar activation signal during a large voltage dip.

Figure 7.27 Rotor-connected DSR activation signal during a large voltage dip.

Figure 7.28 Stator-connected DSR activation signal during a large voltage dip.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

-0.5

0

0.5

1

time,s

v s,a

bc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

time,s

cro

wbar

Sig

nal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

time, s

DS

R S

ignal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

time,s

DS

R S

ignal

257

Figure 7.29 DFIG response using the baseline PI strategy during a large voltage dip



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

1.5

2

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

time, s

(d)

Qs,

pu

258




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1.5

-1

-0.5

0

0.5

1

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

time, s

(d)

Qs,

pu

259




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

0.4

0.6

(a)

v r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

0

1

(b)

i r,abc,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

(c)

Tg,

pu

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.2

0

0.2

0.4

0.6

time, s

(d)

Qs,

pu

260

Figure 7.32 dc link voltage during a large voltage dip ( p.u.) using the baseline

PI (dashed), the proposed MPC with rotor-connected DSR (dotted), and the proposed MPC

with stator-connected DSR (solid) strategies.

Table 7.5 Comparison between the proposed MPC with rotor- and stator- connected DSR,

and the baseline PI control strategies during a large dip ( p.u.).

Quantity MPC-DSRr MPC-DSRs PI

Max( , p.u. 0.84 0.18 1.4

, s 1.4 0.2 1.53

Max( , p.u. 1.6 0.75 2

Max , p.u. 1.3 1.16 1.62

7.5.2.3 Reactive power injection during a voltage dip

In this case study, the voltage profile in Figure 7.33 is applied at the DFIG terminals.

This represents a voltage dip with a dip depth of 0.85 p.u. and 600 ms duration. During the dip,

the stator reactive power set point is changed to command the DFIG to supply reactive power to

the grid. This is a typical situation, where electric generators are required to support the grid

voltage recovery by injecting reactive power. The objective here is to evaluate the capability of

each control/protection strategy in following these reactive power commands.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

1.2

1.4

1.6

time, s

Vdc,

pu

PI

MPC-DSRr

MPC-DSRs

261

Figure 7.33 Stator voltages.

Figure 7.34 shows and obtained using all three control/protection strategies. It can

be seen that baseline PI strategy gives very poor tracking of the reactive power set point. On the

other hand, the proposed MPC strategy with stator-connected DSR provides a fast tracking of the

stator set point even during severe voltage dips. It can be seen from Figure 7.34 (c) that the DFIG

is capable of supplying the desired stator reactive power after approximately 50 ms from the start

of the dip.

7.5.2.4 Discussion

From simulation results provided in this section, it can be concluded that the proposed

MPC strategy significantly outperforms the baseline PI strategy in terms of rejecting grid

disturbances. This is due to the inclusion of RSC current and voltage constraints and a feed-

forward from the measurable disturbance (the stator voltage) in the controller formulation.

Simulations also show that the use of stator-connected DSR protection scheme with the

proposed MPC strategy provides the following advantages compared to the baseline PI strategy:

small oscillations in the generator torque and stator reactive power

small fluctuations in the dc link voltage

fast tracking of stator reactive power set points during voltage dips

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

-0.5

0

0.5

1

time,s

v s,a

bc,

pu

262

Short activation time of the DSR

Figure 7.34 Stator reactive power control during a voltage dip: (a) baseline PI strategy, (b)

proposed MPC strategy with a rotor-connected DSR, and (c) proposed MPC strategy with

a stator-connected DSR.

The main reason for this superior performance is that increasing the stator resistance

reduces the stator time constant

. This allows a faster decay of the stator flux component

which is the main cause of oscillations in the DFIG variables. This property is not provided by

the rotor-connected DSR scheme. For that reason, although the rotor-connected DSR succeeded

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

(a)

Qs,

pu

Qs

Qs*

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

(b)

Qs,

pu

Qs

Qs*

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.4

-0.2

0

0.2

time, s

(c)

Qs,

pu

Qs

Qs*

263

in limiting the rotor current, its dynamic performance during large voltage dips is not as

satisfactory as the one obtained using a stator-connected DSR.

7.6 Conclusions

A novel control strategy that ensures FRT requirement for wind turbines with DFIGs

according to recent grid codes is proposed. This strategy uses a multivariable MPC controller for

controlling the RSC. Limits on the RSC currents and voltages are explicitly incorporated in the

controller. To limit the RSC current during severe grid faults without disconnecting the RSC, a

DSR protection scheme is used. Both rotor- and stator-connected DSR schemes are compared.

Simulation results show that the proposed MPC strategy with a stator-connected DSR provides

fast rejection of grid disturbances and better damping of generator torque and reactive power

oscillations when compared with the baseline PI strategy with crowbar protection scheme.

Alternative implementations of the MPC algorithm are compared in terms of computational

speed in order to evaluate the feasibility of the proposed strategy for DFIG control at fast

sampling rates.

264

Conclusions and future work Chapter Eight:

Advanced model predictive control techniques are used in this thesis to develop effective

control solutions for the following problems existing in the wind turbine literature.

Problem 1 (turbine control level): Designing a control strategy that provides energy

efficiency maximization, mechanical dynamic load minimization, voltage flicker

reduction, and resiliency against faults and other changes in system dynamics.

Problem 2 (generator control level): Designing an effective control strategy that ensures

good reference tracking during normal DFIG operation as well as the realization of fault

ride through requirement during grid faults according to recent grid codes.

The main thesis contributions are summarized in §8.1.

8.1 Summary of contributions

1. New multiple model MIMO predictive control strategy for controlling variable-speed

variable-pitch wind turbines over their full operating regions.

The proposed strategy is a multivariable one, where the generator torque and blade pitch

angles are simultaneously controlled to achieve the desired wind turbine control performance.

This formulation fully exploits the control capability of the system as compared to SISO PID-

based controllers typically used in industrial wind turbines. The proposed MPC controller uses

multiple models that are scheduled according to the operating wind speed to provide good

dynamic performance of the system over the whole operating wind speed range. Furthermore, all

WECS operating constraints are explicitly incorporated in the controller formulation to ensure

safe operation of the WECS.

The proposed MMPC is an optimization-based control algorithm. This allows the

designer to easily achieve the desired trade-off between different conflicting objectives such as

265

energy efficiency maximization, mechanical load minimization, power smoothing, and flicker

reduction. Furthermore, the proposed controller provides a flexible and easy-to-tune framework

for controlling the WECS in its whole operating region, i.e. the partial load, transition, and full

load regimes. Specific contributions for each of these regions are summarized below.

1.1. Partial load regime

The standard MPC algorithm is modified to include an additional term in the objective

function that allows desired damping of drive train torsional torque oscillations. Offset-free

tracking is proved for the modified MPC algorithm. This formulation allows the designer to

easily assign the desired trade-off between fast tracking of the generator speed set point,

damping torsional torque oscillations, and low generator torque activity.

1.2. Transition region

A novel control solution is proposed for controlling the wind turbine in the transition

region. The proposed control strategy eliminates power and torsional torque overshoots that are

typically encountered when using classical switching strategies between partial and full load PI-

based controllers. This is achieved by: a) including a constraint forcing the generator power to be

less than its rated value in the MPC formulation, and b) using a multivariable MPC formulation

where the pitch angle can be adjusted in conjunction with the generator torque to shed excess

input power during wind gusts.

1.3. Full load regime

A MIMO MMPC controller that uses the pitch angle and the generator torque to regulate

the generator power and speed at their rated values is proposed. The MMPC controller is tuned

to allow larger generator speed variations compared to output power variations. This allows the

wind turbine inertia to be treated as an energy buffer between the highly fluctuating input wind

266

power and the output electric power. Consequently, the output power is smoothed and less

voltage flicker is obtained. At the same time, the speed and output power are kept below their

dynamical limits by putting constraints on their values in the MPC formulation. This, for

example, can prevent the wind turbine from over-speeding during normal wind conditions. The

proposed formulation typically results in smoother output power, reduction in flicker emissions,

and lower pitch activity compared with SISO or decentralized control approaches typically used

in the literature.

2. New adaptive offset-free subspace predictive control algorithm.

A new adaptive subspace predictive control algorithm, the OFSPC, is developed in this

thesis based on the framework described in [191]. In contrast with previously developed SPC

algorithms, the OFSPC systematically includes integral action in the controller formulation.

Consequently, better rejection of piecewise constant disturbances is provided. Furthermore, the

recursive identification algorithm used by the OFSPC algorithm provides consistent estimates of

the predictor parameters using open or closed loop data. This allows its use in adaptive control

applications.

3. Application of offset-free subspace predictive control in DFIG-based wind turbines

control

The OFSPC algorithm is applied in designing an adaptive wind turbine control system.

The integral action provided by the OFSPC allows rejecting slow variations in the mean wind

speed that are disturbing the system.

The proposed OFSPC wind turbine control strategy inherits the control advantages of the

MMPC strategy. In addition to that, it offers the following two features:

267

Adaptation of the OFSPC controller provides resiliency against changes in the system

dynamics, and fault tolerance capabilities. These features are particularly important for

offshore wind turbines, where unscheduled maintenance costs are high.

The OFSPC uses a deterministic/stochastic model that is continuously updated online.

This allows better predictions of the system outputs in the presence of stochastic wind

speed fluctuations. This typically results in a better control performance of the predictive

control algorithm.

On the other hand, the OFSPC has the following shortcomings compared to MMPC.

It has higher computational requirements. The OFSPC algorithm implements the

updating and the calculation of the predictor at each sampling time, in addition to solving

the MPC QP step that is only required by the MMPC.

It has less tuning flexibility. In particular, the MMPC formulation allows easier tuning of

the damping of torsional torque oscillations.

Additional parameters related to the identification algorithm, such as the past horizon

and the forgetting factor , must be chosen by the designer. This might not be an easy

task. In general, poor selection of these parameters might lead to a poor or even unstable

performance.

In this thesis, the OFSPC controller is simulated with a 1.5 MW wind turbine nonlinear

model under normal operation and low hydraulic pressure fault in the pitch actuator. Simulation

results show that the OFSPC is capable of tolerating faults and offering good performance during

normal and faulty operation.

4. Novel control strategy that ensures FRT requirement for wind turbines with DFIGs

according to recent grid codes.

268

A new multivariable MPC controller for controlling the RSC is proposed. The RSC

current and voltage limits and a feed-forward term from the stator voltage disturbance are

incorporated in the controller. During small voltage dips, where the rotor currents do not reach

the maximum allowable RSC limit, the proposed strategy shows a significant improvement in the

damping of generator torque and reactive power oscillations when compared with the classical

PI-based vector control strategy.

To limit the RSC current during severe grid faults without disconnecting the RSC, a DSR

protection scheme is used. Both rotor- and stator-connected DSR schemes are compared.

Simulation results show that the proposed strategy with a stator-connected DSR provides fast

rejection of grid disturbances and faster control over the reactive power injected to the grid when

compared with the classical RSC strategy.

Three different implementation that use explicit, QP, and QPQC MPC formulations are

compared in terms of computational speed. Simulations indicate that the MPC with online QP

solver can be implemented in much less than 1 ms.

8.2 Thesis outcomes

The following publications are the outcome of the research conducted in this thesis:

1. M. Soliman, O. P. Malik, and D. T. Westwick, "Multiple Model Predictive Control for

Wind Turbines With Doubly Fed Induction Generators," IEEE Transactions on

Sustainable Energy, vol. 2, pp. 215-25, 2011.

2. M. Soliman, O. P. Malik, and D. T. Westwick, "Multiple model multiple-input multiple-

output predictive control for variable speed variable pitch wind energy conversion

systems," IET Renewable Power Generation, vol. 5, pp. 124-136, 2011.

269

3. M. Soliman, D. Westwick, and O. P. Malik, "Identification of Heffron-Phillips model

parameters for synchronous generators operating in closed loop," IET Generation,

Transmission and Distribution, vol. 2, pp. 530-541, 2008.

4. M. Soliman, O. P. Malik, and D. T. Westwick, "Fault Tolerant Control of Variable-Speed

Variable-Pitch Wind Turbines: a Subspace Predictive Control Approach," in Proceedings

of the 16th IFAC Symposium on System Identification - SYSID 2012, Brussels, Belgium,

2012.

5. M. Soliman, O. P. Malik, and D. T. Westwick, "Ensuring Fault Ride Through for Wind

Turbines with Doubly Fed Induction Generator: a Model Predictive Control Approach,"

in Proceedings of the 18th IFAC World Congress, Milano (Italy), 2011.

6. M. Soliman, O. P. Malik, and D. Westwick, "Multiple Model MIMO Predictive Control

for Variable Speed Variable Pitch Wind Turbines," in Proceedings of the 2010 American

Control Conference - ACC 2010, Baltimore, MD, USA, 2010.

The following paper is under preparation.

7. M. Soliman, O. P. Malik, and D. T. Westwick, "Adaptive Predictive Control of Variable-

Speed Variable-Pitch Wind Turbines," To be submitted to IET Control Theory &

Applications.

8.3 Future work

There are still many questions and research problems that need to be tackled. They are

summarized as follows.

1. Experimental validation

All control strategies developed in this thesis have been tested on a simulation model that

captures the relevant electrical, mechanical and aerodynamic aspects. The model parameters

270

used correspond to the an actual 1.5 MW industrial GE wind turbine [14, 52]. Despite that, it is

important to verify these control strategies on an actual wind turbine or on a small-scale test

bench that emulates the actual system.

2. Use of advanced wind speed sensors and actuators

Extensive research is currently being undertaken to develop advanced sensors and

actuators to control large scale wind turbines [2]. This is driven by the desire to decrease

mechanical loads affecting wind turbines structures that are continuously increasing in size. One

of the promising technologies is based on light detection and ranging sensors, known as lidars [2,

7, 8]. The use of lidars can allow the measurement of the wind speed and direction at different

points in the rotor plane. Feed-forward from these measurements can be used in pitch and torque

control to improve disturbance rejection and performance.

Another technology that is currently under investigation is to use multiple pitch actuators

at each blade to allow for different pitch angles at different radial positions along the blades

[246]. This can be used to mitigate loads resulting from the blades rotation in a non-uniform

wind speed field.

One of the advantages of the control strategies proposed in this thesis is that they are

formulated for MIMO systems. Therefore, the design framework will not differ by incorporating

more measurements and actuators. However, to be able to incorporate these advanced sensing

and actuator technologies, there is still much work that needs to be done in the modelling and

analysis of these devices and their effect on the WECS.

3. Global stability of the MMPC strategy

The stability of the proposed MMPC wind turbine control strategy is verified through

extensive simulation studies. An interesting research problem that requires further investigation

271

is: is it possible to prove global stability of the nonlinear WECS system controlled by locally

designed MPC controllers over the whole operating region?

4. Use of the nonlinear MPC

The MMPC strategy proposed in Chapter 6 uses local linear models to represent the

WECS dynamics over the whole operating strategy. This formulation allows formulating the

MPC optimization problem as a QP that can be efficiently solved online at fast sampling rates.

Another route is to use a discretized nonlinear model representing the overall WECS dynamics.

This model can give better predictions of the output especially when the WECS is significantly

perturbed and is operating far from the nominal trajectory. However, the use of a nonlinear

model will lead to a nonlinear optimization problem that is hard to solve online. This is the main

barrier for using nonlinear MPC. There has been extensive research effort toward efficient

nonlinear MPC algorithms [247]. However, it seems that there is much research work that needs

to be undertaken in this direction.

5. OFSPC versus CLSPC

In Chapter 7, it is shown that the OFSPC controller offers better steady state control

performance in comparison with the CLSPC controller. This is due to the inclusion of integral

action in the OFSPC algorithm. However, the inclusion of integrators in the controller generally

results in deterioration in the transient performance of the closed loop system. On the other hand,

absence of integral action in an adaptive control strategy results in biased model parameters

when the system is exposed to constant disturbances. This also results in deterioration in the

transient performance.

In all simulation studies conducted here, performance with OFSPC is better than with

CLSPC. However, it seems that further analytical studies of the transient performance of both

272

control strategies should be conducted to have a better understanding of the behavior of both

controllers.

6. Distributed MPC theory and control system coordination between the GSC and the RSC

In Chapter 8, the RSC was designed using MPC to provide FRT capabilities for DFIG-

based wind turbines. No advanced control strategy was suggested to control the GSC. Both the

RSC and GSC controllers operate in a decentralized fashion where each controller is not aware

of the actions of the other. In general, GSC regulates the dc link voltage and is set to be reactive

power neutral. This control approach does not exploit the full control capability of the DFIG

system. A better FRT performance can be obtained by designing a single multivariable MPC

controller that simultaneously controls the RSC and the GSC such that:

The reactive power desired by the grid is met via the GSC and the stator terminals of the

DFIG

The RSC and GSC voltage and current limits are not exceeded

The dc link voltage is not exceeded

However, due to the nonlinear dynamics of the DFIG, the formulation of such MPC

controller will lead to a nonlinear optimization problem that is hard to solve at fast sampling

rates.

Another promising solution that is worth exploration is to use the recently developed

distributed MPC theory [121] to design cooperative RSC and GSC controllers. This allows both

controllers to cooperate together to provide a better FRT performance compared to the

decentralized control approach.

7. Design of the control/protection system

273

To the knowledge of the author, there is no systematic way to select the value of the

crowbar resistance used in the DFIG protection. In general, this value is selected to be

sufficiently large to limit the rotor currents. In general, the proposed RSC control performance

during severe voltage dips is dependent on the crowbar resistance value. The selection of the

crowbar resistance that will lead to the best control performance is worth studying.

274

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289

APPENDIX A: BASELINE WIND TURBINE GENERATOR CONTROLLER

A.1. Vector control of the RSC

The main objective of the RSC controller is to control generator torque and stator

reactive power to follow certain desired set points and

, respectively. The design of the

RSC controller using vector control is explained below [25, 26, 92, 93, 96].

From (3.35)-(3.42), the WRIG modeling equations can be written in a synchronously

rotating reference frame with angle as shown in (A.1)-(A.11). The superscript is suppressed

to simplify notation.

Voltage equations:

(A.1)

(A.2)

(A.3)

(A.4)

Flux linkage equations:

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

Generator torque and stator reactive power equations:

( ) (A.10)

(A.11)

290

Since the stator is connected to the grid, it can be assumed that the three phase stator

voltages have constant amplitude, frequency, and phase such as the ones in (3.47). It can be seen

from (3.49) that both and are constants. Now, by choosing the synchronously rotating

reference frame with its d-axis aligned with the stator flux linkage space vector, (A.12) is

obtained.

(A.12)

Since has a practically negligible value and due to the frame orientation, it

can be seen from (A.2) that can be well approximated using (A.13), and hence, its value does

not change with time. Using this fact, (A.14) can be obtained from (A.1).

(A.13)

(A.14)

By substituting (A.12) in (A.5)-(A.6), the stator currents can be related to the rotor

currents using (A.15)-(A.16). These equations show that variations in rotor currents and

are directly reflected on their corresponding stator currents and , respectively.

Consequently, the stator currents can be controlled by controlling the rotor currents.

(A.15)

(A.16)

From (A.10)-(A.16), the generator torque and stator reactive power can be written as (A.17) and

(A.18), respectively.

(A.17)

291

(A.18)

Equations (A.17) and (A.18) clearly show that, under stator flux orientation, independent

(decoupled) control of the generator torque and the stator reactive power can be achieved

by controlling and , respectively. In what follows, design of the dynamic rotor current

control loops is described.

Substituting (A.7)-(A.8) and (A.15)-(A.16) in the direct axis rotor voltage equation (A.3) yields

(A.19), where and

is the leakage factor.

(

) (A.19)

By neglecting the time variations in , (A.19) can be written as:

(A.20)

Applying similar arguments to the q-axis rotor voltage equation, (A.21) is obtained.

(

) (A.21)

As discussed in §3.6.3, the power converter dynamics is extremely fast and it can be assumed

that , and

, where and

are the desired rotor voltages set points

calculated by the RSC controller.

To cancel out coupling terms in (A.20)-(A.21), and

can be synthesized using

(A.22)-(A.23), where and

are cross-coupling compensation terms, and and

are

dynamic control terms generated by two PI controllers controlling the d- and q-axis rotor

currents. Clearly, by substituting (A.20)-(A.21), the rotor currents dynamics are governed by

292

(A.24)-(A.25), and hence simple PI tuning methods can be used to design the d- and q-axis rotor

currents regulators.

(A.22)

(A.23)

(A.24)

(A.25)

The key to the implementation of the stator flux oriented control is to transform all a-b-c

current and voltage measurements into a synchronously rotating reference frame aligned with the

stator flux space linkage space vector. This is achieved by estimating the instantaneous angle of

the stator flux linkage space vector, . To do that, the first step is to calculate the stator and

rotor currents in the stationary reference frame using (A.26)-(A.27), where the transformation

matrix is defined in (3.31). Then, the stator flux linkage d-q components in the stationary

reference frame are calculated using (A.28). Finally, and can be estimated using (A.29).

The vector control structure of the RSC is shown in Figure A.1.

[

] [

] (A.26)

[

] [

] (A.27)

(A.28)

293

√

(A.29)

TT(θe-θr) PWM

PI

PI

PI

PI

T(0)

T(-θr)

encoder

T(θe-θr)

Calculation

of θe and λds

Decoupler

Calculation

of Qs and Tg T(θe)

iar, ibr, icr

θr

θe-θr

θe

ias, ibs, ics

vas, vbs, vcs vds, vqs

ids, iqs

λdssqs

sds ii ,

TgQs

*gT

*sQ

idr

iqr

*dri

*qri

*arv*brv*crv

*drv

*qrv

'drv

'qrv

sqr

sdr ii ,

ccqrv

ccdrv

+

-+

-

+

-

+

-

+

+

+

+

+ -

RSC

GSC

Grid

WRIG

Figure A.1 Vector control structure of the RSC [25].

A.2. Vector control of the GSC

The main objective of the RSC controller is to regulate the dc link voltage, , and the

reactive power flow between the grid and the GSC, , at certain desired set points and

, respectively.

Vector control of the GSC is implemented in a synchronously rotating reference frame

with its d-axis aligned with the stator (grid) voltage space vector [26, 96]. Based on this

orientation, and by assuming balanced supply voltage, (A.30) and (A.31) are obtained, where

is the rms stator line-to-neutral voltage.

294

(A.30)

√ (A.31)

By substituting (A.30)-(A.31) in (3.63)-(3.64), and by noting that and

the GSC active and reactive power exchanged with the grid can be written as in

(A.32)-(A.33), respectively. Furthermore, by neglecting losses in the inductor filter, (3.67) can

be written as (A.34).

√ (A.32)

√ (A.33)

√ (A.34)

Equations (A.33) and (A.34) show that, under stator voltage orientation, independent

(decoupled) control of the dc link Voltage and the reactive power interchanged with the grid

can be achieved by controlling and , respectively. In what follows, the design of

dynamic rotor current control loops is described.

From (3.62) and (A.30)-(A.31), the GSC connection to the grid is modeled by:

(A.35)

(A.36)

By synthesizing and

as in (A.37)-(A.38), (A.39)-(A.40) are obtained from (A.35)-

(A.36), respectively. The terms and

are the outputs of two PI controllers that are used to

control the d- and q-axis rotor currents. These PI controllers can be easily designed based on the

modified dynamics in (A.39)-(A.40).

(A.37)

295

(A.38)

(A.39)

(A.40)

To calculate the variables in a synchronously rotating frame aligned with the stator

voltage, the instantaneous angle of the stator voltage space vector must be estimated. This can be

achieved using (A.41)-(A.42).

[

] [

] (A.41)

√

(A.42)

TT(θe) PWM

PI

PI

PI

PI

T(θe)

Decoupler

θe

iaC, ibC, icC

vas, vbs, vcs

sdsv

QGC

*GCQ

*dcV

*dCi

*qCi

*aCv*bCv*cCv

*dCv

*qCv

'dCv

'qCv

ccqCv

ccdCv

+

-+

-

+

-

+

-

+

+

+

+

RSC

GSC

Grid

WRIG

T(0)Calculation

of θe and vds

Calculation

of QGC sqsv

dsv

idC

iqC

Vdc

Figure A.2 Vector control structure of the GSC [26].

296

APPENDIX B: PER UNIT REPRESENTATION OF THE DFIG MODEL

B.1. Base values

Table B.1 Selected base values.

Base quantity Selection

Base Power, Rated apparent power of the generator, i.e.

[VA]

Base frequency, Grid frequency (synchronous speed in elec. rad/s), i.e.

[rad/s]

Base stator voltage and rotor

voltage in stator units,

√ rated rms phase voltage , i.e.

√ [V]

Base stator current and rotor

current in stator units,

√ rated rms phase current , i.e.

√

[A]

Base impedance, ⁄ [ ]

Base flux linkage, ⁄ [Wb-t]

Base inductance, ⁄ [H]

Base HSS speed, Synchronous speed rendered at the HSS, i.e.

[mech. rad/s]

Base LSS speed, Synchronous speed rendered at the LSS, i.e.

[mech. rad/s]

Base HSS torque,

[N m]

Base LSS torque,

[N m]

Base dc link voltage, rated dc link voltage, i.e.

[V]

Base pitch angle, Maximum pitch angle, i.e.

[o]

297

B.2. Per unit model of the drive train

( )

(

)

(B.1)

(B.2)

where

[ ⁄ ]

[ ⁄ ] (B.3)

[ ]

[ ] (B.4)

[ ] [ ] (B.5)

B.3. Per unit model of the WRIG

(B.6)

(B.7)

298

(B.8)

(

) (B.9)

(B.10)

(B.11)

(B.12)

(B.13)

B.4. Per unit model of the GSC connection to the grid

(B.14)

(B.15)

(B.16)

B.5. Per unit model of the dc link

(B.17)

where

(

) ⁄ [ ] (B.18)

299

APPENDIX C: REVIEW OF WIND TURBINE CONTROL SCHEMES

C.1. Partial load regime

There are two main control schemes that are usually used in the partial load regime,

namely, Indirect Speed Control (ISC), Figure C.1(a), and Direct Speed Control (DSC), Figure

C.1(b) [13, 26, 29].

WECS*

gT

β*=0

g

(•)2

WTGP

3

M

o

N

k

v

(a) ISC (torque-mode control)

WTGP

WECSSpeed

Controller

*

gT

β*=0

g

*

g +

-

v

(b) DSC (speed-mode control)

Figure C.1 Control schemes used in the partial load regime.

In the ISC scheme, known also as torque-mode control, is directly calculated from

using

. This selection makes the generator torque, when referred to the LSS,

correspond to the ORC shown in Figure C.2. It can be shown that this control scheme guarantees

operation at the ORC at steady state [9, 34]. To illustrate that, assume that the WECS is

operating at point ‘A’ in Figure C.2 and the wind increases from 6 to 8 m/s (point ‘B’). The

300

accelerating torque, being the difference between the turbine torque (point ‘B’) and the generator

torque (point ‘A’) causes the DFIG to accelerate. Eventually, the machine will reach the point

‘C’ where there is no accelerating torque.

Figure C.2 Optimal regime characteristic in the plane.

In the DSC scheme (Figure C.1(b)), known also as speed control mode, an explicit speed

feedback control loop is used. The generator speed set point must be calculated to ensure the

operation of the WECS at the ORC. Many MPPT algorithms proposed in the literature realize

this objective. The most important ones are summarized below.

MPPT using estimated wind speed [32, 39]

The most straightforward approach is to use the wind speed signal or its estimate to

calculate using (C.1). In general, it is impossible to measure the effective wind speed. The

one point wind speed measured by an anemometer is totally different from the effective wind

speed. Furthermore, the anemometer is located on the nacelle and therefore, the measurement

does not represent the speed of the wind in front of the rotor. Therefore, a wind speed estimator

0 0.5 1 1.5 2 2.50

100

200

300

400

500

600

700

t, rad/s

4m/s

5m/s

6m/s

7m/s

8m/s

9m/s

10m/s

ORC

Tt,

KN

m B

A

C

301

must be used in this approach. The estimation of the effective wind speed based on WECS

measurements is discussed in [32, 181].

(C.1)

MPPT using measured (estimated) turbine torque [29]

In this approach the wind speed measurement is avoided. Instead, is calculated using

(C.2), where is an estimate of the turbine torque. Many turbine torque estimators have been

proposed in the literature [29].

√ ⁄ (C.2)

MPPT using a gradient ascent search approach [12]

This technique continuously tracks the MPP using the property that the curve has

a single smooth maximum point. In this approach,

is estimated online based on the power

and rotational speed measurements. Then, is incrementally increased (decreased) if

is

positive (negative). The search stops once the gradient is zero. The main difficulty in this

approach is on how to obtain a good estimate for

in the presence of noise and measurements

that are perturbed by the wind turbulence.

Despite its simplicity and smooth response, the ISC scheme is characterized by a very

slow response, especially with large wind turbines having big rotor inertia [13, 29, 32]. In the

presence of turbulent winds, the large rotor inertia prevents it from tracking fast wind speed

variations. Consequently, the wind turbine operates most of the time far from the ORC. In

contrast, the use of a dynamic speed controller in the DSC allows the closed loop bandwidth of

302

the system to be increased and the generator torque can be manipulated to allow faster tracking

of wind speed variations [29].

C.2. Full load regime

There are many control schemes that can be used to achieve power and speed regulation

for the WECS during full load operation, as shown in Figure C.3.

The simplest approach is to fix at its rated value, , and to regulate the generator

speed by manipulating using a SISO speed control loop [13, 39]. Using this approach,

( , where is the DFIG efficiency) is indirectly regulated at its rated value. The

design of the controller is relatively easy as it can be done using SISO design techniques.

However, this control scheme does not fully exploit the control capabilities of the system as the

generator torque is not dynamically used in the control. Consequently, large pitch angle activity

and power fluctuations can occur [13].

Another control scheme, shown in Figure C.3(b), is based on using a decentralized

structure, where two separate SISO controllers are designed to regulate the generator speed and

power independently [32, 37, 248]. This approach has many disadvantages. First, designing these

two controllers is a difficult task due to the presence of interaction between these two control

loops. Second, in cases of slow pitch actuators, the generator speed is kept almost constant using

the fast generator torque control, and consequently all wind power fluctuations are directly

transmitted to the grid. Furthermore, large torsional torque variations typically occur in the drive

train.

To fully exploit the control capability of the system, a multivariable MIMO controller

(Figure C.3(c)) can be used [13, 39]. In this scheme, the generator output power and speed are

controlled by simultaneously manipulating and . As argued in [13], this approach results in

303

much superior performance compared to the decentralized control structure in terms of power

smoothing, flicker, dynamic loads, and pitch actuator activity.

ratg ,WTGP

WECS

ratgg TT ,*

β*

g

*g

+

-

v

Speed

Controller

(a) SISO speed control loop

*WTGP

ratWTGP ,

ratg ,

WTGP

WECS

Speed

Controller

*gT

β*

g*g

+

-

v

Power

Controller

+-

(b) Decentralized control

*WTGP

ratWTGP ,

ratg ,

WTGP

WECS*gT

β*

g*g

+

-

v

+-

Multi-

variable

controller

(c) Multivariable control

Figure C.3 Control schemes used in full load regime [13].

304

APPENDIX D: MODEL PARAMETERS

D.1. Wind turbine

Parameter Value

System rated power, 1.5 MW

Rated turbine speed, 2.2 rad/s (21 rpm or 1.2 p.u.)

Max turbine speed, 2.4 rad/s (22.8 rpm or 1.3 p.u.)

Blade radius, 35.5 m

Max power coefficient, 0.48

Optimal tip speed ratio, 8.6

cut-in speed / cut-out speed 4 / 26 m/s

rated wind speed, 11 m/s

9 m/s

Air density, 1.225 kg/m3

Number of blades 3

D.2. Pitch actuator

Parameter Value

Pitch system time constant, 0.1 s

Min/Max pitch angle, / 0/45 o

Min/Max pitch rate, / -10/10 o/s

D.3. Drive train

Parameter Value

Gearbox ratio, 68.5

Gearbox efficiency, 1

Turbine inertia constant, 3 s

Generator inertia constant, 0.5 s

305

Shaft stiffness, 0.5 p.u./elec. rad

Shaft damping, 0.5 p.u./elec. rad

D.4. DFIG system

Parameter Value

Rated apparent power, 1.5/0.9 MVA

Rated line to line voltage, √ 575 V

Rated frequency, 60 Hz

Pole pairs, 3

Stator resistance, 0.00706 p.u.

Stator leakage inductance, 0.171 p.u.

Rotor resistance, 0.005 p.u.

Rotor leakage inductance, 0.156 p.u.

Magnetizing inductance, 2.9 p.u.

Grid-side filter resistance, 0.0015 p.u.

Grid-side filter inductance, 0.15 p.u.

dc link capacitance, 1000 μF

Rated dc link voltage, 1200 V

D.5. Generator control system parameters

All generator PI regulators have the transfer function in (D.1).

(D.1)

The parameters and are as follows.

306

RSC PI regulators parameters

Regulator

Rotor d-axis current regulator 0.3 1.8

Rotor q-axis current regulator 0.3 1.8

Generator torque regulator 0.44 160

Stator reactive power regulator 0.74 186

GSC PI regulators parameters

Regulator

GSC d-axis current regulator 0.3 1.8

GSC q-axis current regulator 0.3 1.8

dc link voltage regulator 4.7 26

GSC reactive power regulator 4.7 26

D.6. Wind speed simulator

Parameter Value

Turbulence length scale, 150 m

Turbulence intensity, 0.12

0.55 s

1.3 s

Tower height, 80 m

Tower radius, 1.5 m

Normal distance from the rotor to the tower center-line, 4.5 m

Wind shear exponent, 0.2

Anemometer time constant, 0.5 s

307

D.7. Power system model parameters

Transmission Lines

Resistance = , Reactance = , Susceptance = ,

Length =

Loads (L6, L7, L9)

0.15 + j0.147 MVA

Transformers

[MVA] [KV] [KV] [%] [KW]

25 110 15 11 110

2 15 0.575 6 13.58

, , 0.63 15 0.21 6 6

: rated apparent power

( ): rated voltage of the high (low) voltage side

: Nominal short circuit voltage

: copper losses at rated power

Grid

Short Circuit Level (SCL) = 30 MVA

X/R ratio = 2

model predictive control of dfig-based wind power

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