Document [original]

Extreme and Fatigue Load Reducing Control

for Wind Turbines: A Model Predictive Control

Approach using Robust State Constraints

vorgelegt von

Dipl.-Ing.

Arne Körber

geb. in Berlin

von der Fakultät III - Prozesswissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

- Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuss

Vorsitzender: Prof. Dr. Matthias Kraume

Gutachter: Prof. Dr. Rudibert King

Gutachter: Prof. Dr. Po-Wen Cheng

Tag der wissenschaftlichen Aussprache: 28. April 2014

Berlin 2014

Abstract

The cost of energy for power generated using wind turbines is largely driven by the cost of the

wind turbines itself. As many turbine components are dimensioned by the mechanical loads, they

need to be able to withstand, reducing those loads through the use of smart control algorithms

directly impacts turbine costs. Model Predictive Control (MPC) is an advanced control technique

that originated in the process industries but is now used for a large variety of applications. This

thesis exemplarily demonstrates the benefits of MPC for wind turbine control and shows how MPC

can be employed to reduce both extreme and fatigue loading on wind turbines.

An analytical, nine-state, non-linear model of a wind turbine is developed and its parameters iden-

tified via closed-loop system identification. This model is used to design an Extended Kalman Filter

for state and wind speed estimation and a Model Predictive Controller. The Model Predictive Con-

troller is continuously linearized, i.e., the plant model is linearized and the optimization problem

is formulated as a quadratic program (QP) and solved numerically at each time step. The stability

of the linear MPC is ensured through the use of terminal cost and constraint terms in the problem

formulation. The controller contains both control and state constraints. Using the example of the

rotor overspeed constraint, it is exemplarily demonstrated how state constraints can be robustified

against uncertainties in future wind speeds. The developed Model Predictive Controller includes

the trajectory of future wind speeds explicitly in the problem formulation. This allows integration

of preview signals, for example measured using a turbine mounted Light Detection and Ranging

(LiDAR) device, in the controller without increasing complexity and retaining all MPC properties

such as optimality and constraint handling.

The performance of the MPC is evaluated in extensive simulations using the aero-elastic simulation

tool FAST and the NREL 5 MW reference turbine. Results show that a preview MPC controller sig-

nificantly outperforms both non-preview MPCs as well as a classically designed baseline controller

and significantly reduces extreme and fatigue loads. If no preview information is available, the

differences between MPC and the baseline controller are more gradual. However, it is also shown

that, due to the use of control and state constraints, the MPC provides benefits especially in non-

normal operating conditions such as extreme gusts or turbine fault situations where it reduces loads

or helps to keep the turbine online.

It is concluded that MPC is a natural fit for integrating preview signals into the turbine control

system. Even if no preview information is available, it offers significant benefits especially in special

scenarios. Future research on MPC for wind turbine applications should therefore also consider the

case where no LiDAR/ preview is available.

III

Kurzfassung

Die Energieerzeugungskosten für Strom aus Windenergie werden von den Kosten für die Wind-

kraftanlagen dominiert. Die meisten Anlagenkomponenten werden durch die mechanischen Belas-

tungen, denen sie widerstehen können, müssen dimensioniert Deshalb wirkt sich eine Reduktion

dieser Belastungen durch intelligente Anlagenregelungsstrategien direkt auf die Anlagenkosten

aus. Modellprädiktive Regelung (engl. Model Predictive Control/ MPC) ist ein fortgeschrittenes

Regelungsverfahren, das ursprünglich hauptsächlich in der Prozessindustrie zum Einsatz kam. In-

zwischen gibt es allerdings auch Anwendungen in einer Vielzahl anderer Bereiche. Diese Arbeit

zeigt exemplarisch, welche Vorteile MPC für die Regelung von Windkraftanlagen bietet und wie sie

verwendet werden kann, um sowohl Extrem- als auch Ermüdungslasten an Windkraftanlagen zu

reduzieren.

Ein analytisches, nichtlineares Modell einer Windkraftanlage mit neun Zuständen wird entwickelt,

wobei die Parameter des Modells anhand des Verhaltens des geschlossenen Regelkreises identi-

fiziert werden. Dieses Modell wird verwendet um einen Erweiterten Kalman-Filter zur Zustands-

und Störgrößenschätzung sowie einen Modellprädiktiven Regler zu entwickeln. Dieser Modell-

prädiktive Regler wird kontinuierlich literarisiert, d.h. in jedem Zeitschritt wird das Streckenmod-

ell neu linearisiert und der Reglerentwurf als Quadratisches Programm (QP) formuliert und nu-

merisch gelöst. Hierbei wird die Stabilität des Reglers durch die Verwendung von finalen Zustand-

skosten und -beschränkungen sichergestellt. Anhand des Beispiels der Überdrehzahl wird exem-

plarisch gezeigt, wie Zustandsbeschränkungen gegen Unsicherheiten in der Kenntnis der zukünfti-

gen Windgeschwindigkeiten robustifiziert werden können. Der entwickelte MPC berücksichtigt

dabei die zukünftigen Windgeschwindigkeiten explizit in der Reglerproblemstellung. Dadurch

können eventuelle vorhandene Vorhersagen der zukünftigen Windgeschwindigkeit zum Beispiel

von einem LiDAR (Light Detection and Ranging) System direkt verwendet werden, ohne die Kom-

plexität des Reglers zu erhöhen und unter Beibehaltung aller Vorteile der MPC Regelung wie zum

Beispiel der Optimalität und der Berücksichtigung von Beschränkungen.

Die Performance des MPC wird anhand des aero-elastischen Simulationsprograms FAST und der

NREL 5 MW Referenzwindkraftanlage untersucht. Die Resultate zeigen, dass ein MPC mit Kennt-

nis der zukünftigen Windgeschwindigkeit eine deutlich bessere Performance als sowohl ein MPC

ohne diese Kenntnisse als auch ein klassisch entworfener Referenzregler hat. Ohne Kenntnisse der

zukünftigen Windgeschwindigkeiten sind die Unterschiede zwischen MPC und dem Referenzregler

geringer. Allerdings zeigt sich, dass durch die Verwendung der Regel- und Zustandsbeschränkun-

gen, die MPC Regelung gerade in ”nicht normalen” Betriebsbedingungen wie zum Beispiel ex-

tremen Böen oder Anlagenfehlern Vorteile hat und in diesen Situation die Lasten reduzieren kann

bzw. dafür sorgt, dass die Anlage länger am Netz bleibt.

Daraus wird geschlossen, das MPC die natürliche Wahl für den Reglerentwurf ist, falls Informatio-

nen über zukünftige Windgeschwindigkeiten vorliegen. Selbst wenn solche Informationen nicht

verfügbar sind, bietet MPC immer noch große Vorteile in vielen Betriebsszenarien. Zukünftige

Forschung im Bereich der Anwendung von MPC zur Regelung von Windkraftanlagen sollte sich

deshalb nicht auf den Fall der Regelung unter der Annahme, dass ein LiDAR oder ähnliches Sys-

tem verfügbar ist, beschränken.

Foreword

This thesis is the result of my time as a doctoral student at the faculty of process sciences of

the Technische Universiät Berlin. Foremost, I would like to thank Professor Rudibert King

for agreeing to what started out as a vague research proposal several years ago and for all

the support I have received since then. Without his guidance, patience, and always helpful

feedback this project would not have been possible. Further, I would like to thank Professor

Po-Wen Cheng at the University of Stuttgart for agreeing to act as the second reviewer

and examiner on short notice. Finally, I would like to thank all colleagues, friends, and

collaborators that, through technical discussions, proof-reading or simply encouragement,

have contributed and helped me to complete this thesis.

VII

Contents

List of Abbreviations X

List of Symbols XII

List of Figures XV

List of Tables XVII

1. Introduction 1

2. Background 4

2.1. Control of Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1. Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2. Speed and Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.3. Load Reducing Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.4. ShutdownControl .............................. 15

2.1.5. FRTControl.................................. 16

2.2. System Simulation and Loads Analysis of Wind Turbines . . . . . . . . . . . . 17

2.2.1. Models..................................... 19

2.2.2. LoadCases .................................. 22

2.2.3. ExtremeAnalysis............................... 25

2.2.4. FatigueAnalysis ............................... 26

2.2.5. The Aero-elastic Simulation Tool FAST ................... 27

2.2.6. The NREL 5MW Reference Turbine . . . . . . . . . . . . . . . . . . . . 28

2.3. Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.1. Models and Problem Formulation . . . . . . . . . . . . . . . . . . . . . 30

2.3.2. LinearMPC.................................. 32

2.3.3. Stability .................................... 33

3. Related Work 36

3.1. FeedforwardControl................................. 36

3.1.1. Wind Speed Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.2. Wind Speed Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.3. Wind Speed Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.4. Preview Control for Individual Pitch Control . . . . . . . . . . . . . . . 39

3.1.5. Preview Control for Speed and Power Regulation . . . . . . . . . . . . 39

3.2. WindTurbineControl ................................ 40

3.2.1. MPC for Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2. Extreme Load Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

VIII

Contents

4. Turbine Model 44

4.1. Aerodynamics..................................... 46

4.2. StructuralDynamics ................................. 47

4.2.1. Rotor...................................... 47

4.2.2. Tower ..................................... 48

4.2.3. Blade Fore-Aft Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3. EffectiveWindSpeed................................. 52

4.4. ActuatorModels ................................... 53

4.5. State Space Formulation and Linearization . . . . . . . . . . . . . . . . . . . . 53

4.5.1. Discretization................................. 58

4.6. Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6.1. Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6.2. Structural Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . 60

4.7. Model Validation and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7.1. Open-Loop Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.7.2. Closed-Loop Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7.3. Frequency Domain Characteristics . . . . . . . . . . . . . . . . . . . . . 67

4.8. ModelingSummary.................................. 70

5. Controller Design 72

5.1. ExtendedKalmanFilter ............................... 72

5.1.1. Disturbance Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.2. Performance.................................. 76

5.2. Model Predictive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1. Tuning and Choice of Constraints . . . . . . . . . . . . . . . . . . . . . 79

5.2.2. Linearization ................................. 82

5.2.3. Stability .................................... 85

5.2.4. ControlEquations .............................. 88

5.2.5. Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3. Robust State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3.1. Additive Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3.2. Controller Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3.3. Unmeasured Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.4. BackupMode................................. 98

5.4. Algorithmic Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6. Results 101

6.1. SimulationSetup ...................................101

6.1.1. MPC Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.1.2. Baseline Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2. Normal Power Production Operation . . . . . . . . . . . . . . . . . . . . . . . 104

6.2.1. Overspeed Risk in NPP . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.2. Tuning.....................................110

6.3. Gusts..........................................113

6.3.1. Extreme Operating Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.2. Extreme Coherent Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Contents

6.4. FaultConditions....................................118

6.4.1. EmergencyStop................................118

6.4.2. Loss of Grid Connection during an Extreme Operating Gust . . . . . . 120

6.4.3. FaultRideThrough..............................123

7. Conclusion and Outlook 128

Bibliography 131

A. Determination of the Maximal Control Invariant Set 138

B. Design of a Baseline Controller 141

B.1. PitchController....................................141

B.2. Torque Controller and Switching Conditions . . . . . . . . . . . . . . . . . . . 143

B.3. TowerDamper.....................................144

List of Abbreviations

BEM Blade Element Momentum

CFD Computational Fluid Dynamics

CCU Converter Control Unit

DEL Damage Equivalent Load

DLL Dynamic Link Library

DOF Degree of Freedom

EKF Extended Kalman Filter

ECG Extreme Coherent Gust

EOG Extreme Operating Gust

FEM Finite Element Method

FRT Fault Ride Through

HH Hub Height

IEC International Electrotechnical Commission

IPC Individual Pitch Control

LIDAR Light Detection and Ranging

LPV Linear Parameter Varying

LTI Linear Time Invariant

LQR Linear Quadratic Regulator

LVRT Low Voltage Ride Through

MIMO Multi Input Multi Output

MPC Model Predictive Control(er)

NPP Normal Power Production

NREL National Renewable Energy Laboratory

PI Proportional-Integral(-Controller)

PLC Programmable Logic Controller

QP Quadratic Program

RMS Root Mean Square

SISO Single-Input-Single-Output

STD Standard Deviation

List of Symbols

ascaling constant for effect of blade tip movement on effective

wind speed

a1blade center of gravity ratio

Acinequality constraint matrix

Atc terminal inequality constraint matrix

CMtorque coeffcient

CTthrust coeffcient

dvector of disturbance inputs

Ddamping matrix

Fforce gain matrix

FAaerodynamic thrust

Fθpartial derivative of aerodynamic thrust with respect to θ

Fvpartial derivative of aerodynamic thrust with respect to v

Fωpartial derivative of aerodynamic thrust with respect to ω

FTlinearized aerodynamic thrust

Jrotor inertia

JCost function

kquadratic torque control law constant for variable speed opera-

tion

Kf a tower excitation parameter in the fore-aft direction

Kunconstrained gain matrix

Kstiffness matrix

Lprediction horizon

lgradial position of center of gravity of blade

laradial position of aerodynamic center of blade

Mmass matrix

MAaerodynamik torque

mbrotor mass

MGgenerator torque

MG,0 rated generator torque

MG,Cgenerator torque command

XII

MG,max maximum generator torque

Mθpartial derivative of aerodynamic torque with respect to θ

Mvpartial derivative of aerodynamic torque with respect to v

Mωpartial derivative of aerodynamic torque with respect to ω

nGgearbox Ratio

Pelectrical power

Ppitch actuator transfer function

PAaerodynamic power

P0rated power

piadditive noise for ith model output

Qoutput or state weights

Qmodel covariance (for EKF)

qi,jentries of Q

qiadditive noise for ith model state

Rrotor radius

Rcontrol output weights

Routput covariance (for EKF)

ri,jentries of R

sLaplace variable

ttime

tssampling time

uk|Lcontrol sequence

vfree stream wind speed

vestimated wind speed

veeffective wind speed

Vfterminal cost function

vrrated wind speed

v0linearization wind speed

xturbine model states

xaaugmented state vector for EKF

xblongitudinal position of blade tip

xtlongitudinal position of tower top

x0states at which system is linearized

ρair density

κterminal controller

ω0,bflapwise eigenfrequency of blades

ωp,0 eigenfrequency of pitch actuator system

ωangular velocity of the rotor

XIII

ωmax maximum rotor speed

ω0rated rotor speed

ωT,0 eigenfrequency of tower system

λtip speed ratio

θpitch angle

θ0steady state pitch angle

θCpitch setpoint

ζbrelative damping of blade flapwise bending

ζprelative damping of pitch actuator system

ζTrelative damping of tower system

XIV

List of Figures

2.1. Torque-SpeedCurve ................................. 6

2.2. PowerCoefficient................................... 8

2.3. Block diagrams for the two commonly used control laws for speed control in

the variable speed operating region . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4. Two baseline control schemes for the full load speed and power control prob-

lem ........................................... 11

2.5. PitchSensitivity.................................... 12

2.6. Individual Pitch Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7. CombinedController................................. 15

2.8. GenericLVRTcurve.................................. 17

2.9. Turbulence Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.10.GustShapes ...................................... 24

4.1. High-Level Controller Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2. Simplified turbine model used by the controller . . . . . . . . . . . . . . . . . 46

4.3. Simplifiedtowermodel ............................... 49

4.4. BladeModel...................................... 51

4.5. Mechanical Turbine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6. Torque and Thrust Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.7. Block diagram of setup for identification of the structural model parameters

ofthesimplifiedmodel................................ 61

4.8. Step Response for optimal fit of tower model . . . . . . . . . . . . . . . . . . . 62

4.9. Step Response for optimal fit of blade model . . . . . . . . . . . . . . . . . . . 63

4.10. Open-loop comparison between non-linear and linear models . . . . . . . . . 64

4.11. Closed-loop comparison between non-linear and linear models . . . . . . . . 65

4.12. Goodness of fit for simplified model . . . . . . . . . . . . . . . . . . . . . . . . 66

4.13. Bode Plots for the different models . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.14. Bode Plots for the different models . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.15. Poles and Zeros of the Turbine Model . . . . . . . . . . . . . . . . . . . . . . . 70

5.1. True and estimated wind speed a series of wind speed steps with a step size

of 1m/s......................................... 76

List of Figures

5.2. Estimator tracking performance for wind speed steps with a step size of 1m/s

starting between 12m/s(dark red) and 24 m/s(light red) . . . . . . . . . . . . . 77

5.3. Estimator tracking performance during an EOG50 gust starting at 13.8m/s. . 77

5.4. Estimator tracking performance with respect to the hub height (HH) wind

speed during turbulent wind conditions with a mean wind speed of 20m/s

and a turbulence intensity of 15% . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5. Steady state pitch angles as a function of wind speed . . . . . . . . . . . . . . 84

5.6. TerminalConstraint.................................. 87

5.7. x1-state trajectories (thin lines) from introduction of a measurement error at

different instants and bounding x1-trajectory (thick line) . . . . . . . . . . . . 94

5.8. x1-state trajectories (solid lines) from 200 random disturbance trajectories

and bounding trajectory (dashed line) . . . . . . . . . . . . . . . . . . . . . . . 95

5.9. ConstraintTightening ................................ 96

6.1. ComputationalLoad .................................103

6.2. BaselineController ..................................104

6.3. NPP - Operational Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4. NPP - Loads per Wind Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.5. NPP-LoadsSummary................................109

6.6. Rotor Speed Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . 110

6.7. Normal probability plots of rotor speed at different turbulence levels . . . . . 111

6.8. Tuning - Operational Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.9. Comparison of MPC and baseline controller during an Extreme Operating

Gust at a wind speed of 13.8m/swith a return period of 50 years . . . . . . . . 114

6.10. Comparison of MPC and baseline controller during an extreme coherent gust

at a wind speed of 13.8m/swith a return period of 50 years . . . . . . . . . . . 117

6.11. Comparison of MPC and simple shutdown procedure during grid loss event

at a wind speed of 17m/s...............................119

6.12. Maximum rotor speed and tower base bending moment for MPC and open-

loop pitch ramp shut-down control for wind speeds from 14m/sto 25m/s. . . 120

6.13. Comparison of MPC and baseline controller during an Extreme Operating

Gust at a wind speed of 13.8m/swith a return period of 1 year and occurrence

of a grid loss at the wind speed maximum . . . . . . . . . . . . . . . . . . . . . 121

6.14. Comparison of MPC and baseline control procedure during a grid fault event

at a wind speed of 13m/s...............................124

6.15. Maximum rotor speed and tower base bending moment during an FRT event

for state constrained and unconstrained MPC variants and baseline control

for wind speeds from 13m/sto 25m/s........................125

XVI

List of Tables

6.16. Comparison of MPC and baseline control procedures under turbulent condi-

tions with an average wind speed of 20m/sand a turbulence intensity of 15%

and an occurence of a grid fault every 30 seconds. . . . . . . . . . . . . . . . . 127

B.1. Gain Schedule for Baseline Pitch Controller . . . . . . . . . . . . . . . . . . . . 143

List of Tables

2.1. Main loadcases according to [39] that are relevant for the turbine control de-

sign. U:Ultimate(Extreme) Analysis, F: Fatigue Analysis . . . . . . . . . . . . 23

2.2. Key Properties of the NREL 5MW Reference Turbine [43] . . . . . . . . . . . . 29

4.1. Summary of turbine model states . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2. Summary of turbine model outputs . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3. Overview of turbine model parameters . . . . . . . . . . . . . . . . . . . . . . 59

6.1. Maximum values of rotor speed, pitch angle, and tower base moment during

EOG50 gust simulations starting at v=11.5 m/s,v=13.8 m/s, and v=25 m/s. 115

6.2. Maximum values of rotor speed, pitch angle, and tower base moment during

an Extreme Coherent Gust (ECG) starting at v=11.5 m/s............117

6.3. Maximum values of rotor speed, pitch angle, and tower base moment during

an EOG1 gust with a grid loss event occurring at the highest wind speed . . . 122

6.4. FRT performance of the different control methods in turbulent conditions . . 126

XVII

1. Introduction

Over the past decade, wind energy has evolved from a niche to a mainstream source of

energy in many countries. As part of this trend, there is an ever increasing pressure to

reduce the cost of energy generated by wind turbines. Wind energy is increasingly expected

to compete with energy from fossil fuel sources on a cost basis and in a subsidiary-free

environment. Unlike for most traditional sources of electric energy, the ”fuel” of a wind

turbine is free. The cost of electricity generated from wind turbines is therefore driven

largely by the cost of the wind turbine itself. Since there is little room for improvements

in turbine efficiency, wind turbine costs need to be reduced in order to drive their cost of

energy down.

A big contributor to the overall turbine cost are the costs for the structural, load bearing

components such as the tower, blades, and main shaft. These components are sized by

the mechanical load they need to be able to withstand. So any reduction in mechanical

loads will directly translate into a reduction in material costs for those components. There

is a strong interaction between the turbine control system and the loads the turbine expe-

riences, especially due to the slender structure and associated susceptibility to mechanical

oscillations as well as the generally challenging dynamics caused by the strong interaction

between the structural dynamics and the aerodynamics. On modern wind turbines, the

turbine control system is increasingly used to help mitigate some of these challenges and

thereby reduce loads. Generally, the larger the wind turbine, the more pronounced are the

dynamic challenges. At the same time, the benefits from any load reduction through use

of advanced control algorithms are also higher which is why there is an increasing need

for smart controller designs, especially for the very large offshore wind turbines which can

have power ratings exceeding 5MW and rotor diameters exceeding 120m.

There are two distinct types of mechanical loads: extreme and fatigue loads. Extreme loads

are loads that a given component needs to be able to withstand once; while fatigue loads are

accumulating over time and threaten to damage the turbine after several years of operation.

The design for each turbine component may be driven by either extreme loads, fatigue

loads, or a combination of both. Most of the existing research on wind turbine controls

focuses on reducing fatigue loads either through the addition of sensors to the turbine or by

smartly designing the algorithms for the existing control loops. For example, many modern

1. Introduction

wind turbines now have some means for sensing rotor imbalance loads and can correct

those by individually pitching each of the blades. As a result of the achieved reduction in

fatigue loads, the component design is driven more and more by extreme loads.

An important source of extreme loads are rare but extreme gusts occurring while the turbine

is in power production operation. One method that has been proposed to reduce those

gust loads is to install a forward looking, usually laser-based, sensor on the turbine that is

capable of measuring the wind speed several hundred meters ahead of the turbine, thus

giving the turbine enough time to adjust itself to a potential incoming gust.

Further extreme loads may occur during so-called fault events. If the turbine spins too fast

or if a fault in the electrical network it is connected to is detected, it will generally need to be

shut down and extreme loads can occur either prior or during the shut-down process. Any

control system that helps to avoid unnecessary faults or that improves the behavior during

the shut-down process will therefore also reduce turbine loads or contribute to turbine

availability.

Model Predictive Control (MPC) is a modern control method widely employed. While at

first it was mainly used in the process industry, it is now exploited in a large variety of con-

trol problems. Its basic principle is to use the plant model to predict the output trajectory

over a finite horizon at every sampling point given the current state and input informa-

tion. This information forms the basis to calculate the future control input trajectory as

the solution of an optimization problem. As this online optimization requires significant

computational power, MPC was initially employed for comparably slow processes where

update times in the range of minutes or even hours rather than seconds are required. How-

ever, with ever increasing computational power available in real time control systems, MPC

is now becoming feasible for much faster systems as well. With the clear trend towards in-

creasing wind turbine dimensions, the costs associated with using advanced controls meth-

ods, such as MPC, become smaller in relation to the potential benefits:

•Direct MIMO Formulation: The wind turbine problem is a multi-input-multi-output

problem (MIMO). Several actuators, such as torque and pitch, need to react on a mul-

titude of measured variables, such as speed and power, in a coordinated fashion.

MPC allows to directly design a single controller for the entire problem instead of

having to design individual controllers using decoupling or cascading strategies.

•Trade-offs: There are several conflicting controller objectives in wind turbine control.

For example, tight control of the rotor speed usually requires fast pitch actuation.

This, however, will lead to increased tower motion and loads. The controller will

need to be tuned to provide an acceptable trade-off between tower loads and speed

control. Such a tuning trade-off can be performed using most controllers, but MPC

allows a direct control over these kinds of trade-off and allows a tuning in terms of

the quantities that are relevant for the turbine design.

•Constraint Handling: Wind turbine control is inherently constrained. For example,

the pitch rates are usually limited by the maximum power of the pitch motors, while

the power electronics limit the maximum allowable generator torque rates. There are

also hard limits on some of the controlled variables such as generator speed. As too

high rotor speeds can damage the turbine, there are usually safety systems that will

shut down the entire turbine if the speed rises above a certain threshold. For any con-

troller, it is therefore crucial to maintain the speed below these trigger values. Using

MPC, these control and state constraints can be included in the problem formula-

tion and the controller will not lose optimality when operating on constraints. The

inclusion of the constraints is especially beneficial in situations where the turbines

routinely hit either the control or state constraints, such as gust or shut-down type

events.

•Preview Control: As described, one of the most promising candidates for improving

wind turbine controller performance with respect to extreme loads is the use of up-

stream wind speed measurements. Using MPC, this type of control, also known as

preview control, can be easily integrated in the feedback controller without adding

complexity, thus optimality and constraint handling are not lost when operating with

a combined feedback-feedforward controller.

This thesis explores these potential benefits of applying MPC to the wind turbine control

problem. It details how a Model Predictive Controller for the full load operation of a large,

variable speed, pitch-controlled wind turbine can be designed. It particularly focuses on

two of the MPC benefits: It exemplarily shows how state constraints can be included ro-

bustly in the MPC using the case of the overspeed trigger and analyzes the effect on the

turbine behavior. It further shows how preview measurements can be included directly in

the MPC formulation and its effects.

The thesis is organized as follows: Chapter two provides the necessary background infor-

mation on wind turbine control, system simulation, and MPC. Chapter three examines the

prior research that has been conducted in this area. In chapter four, a simplified turbine

model is developed and analyzed which is then used in chapter five as the basis for the ac-

tual design of the MPC. Finally, in chapter six, extensive simulation results comparing the

MPC with a traditionally designed baseline controller in various scenarios are presented.

2. Background

This chapter introduces the key concepts that are relevant for the controller design and the

simulation results that are presented in the later chapters. The first part provides a short

overview over the control system design of modern wind turbines and outlines some of

the design challenges and commonly used design techniques. The second part introduces

the system simulations of wind turbines that serve as the basis for the results presented

in chapter 6. Finally, Model Predictive Control and its mathematical formulation in state-

space is introduced.

2.1. Control of Wind Turbines

There are several fundamentally different turbine control concepts and in the early days

of the wind turbine industry, the main choice to be made when designing a wind turbine

controller was the control concept itself. Today however, almost all large turbines that are

being built are pitch controlled variable speed turbines. These wind turbines share two

distinct features which define their basic control strategy. Their electrical system allows a

variable speed operation and the torque demand at the generator can be directly controlled

through a frequency converter. Secondly, they can shed power by rotating the blades about

their longitudinal axis, also called pitching. While some textbooks (e.g., [26]) still treat

the fundamental design choices, the focus of current research on wind turbine controls has

shifted to the question of how to best control these variable speed, pitch controlled turbines.

With the choice of the basic control concept settled, this section details the control problem

for these turbines and introduces the basic strategies that are widely used and serve as a

baseline for much of the research on advanced wind turbine control methods.

2.1.1. Problem Description

The primary objective of wind turbine control is to maximize the energy that is extracted

from the wind while maintaining the turbine within its operational limits. These opera-

tional limits are, for example, the maximum electrical power that can be generated by the

2.1. Control of Wind Turbines

generator or the maximum rotor speed the drive train is designed for. As a secondary objec-

tive, wind turbine controllers are increasingly expected to also reduce mechanical loading

of the turbine.

The main actuators to achieve these goals are the pitch angles at each of the three blades

and the generator torque.

The blades can be pitched using either hydraulic or electric motors. While most older tur-

bines could only pitch all three blades collectively, newer turbines are capable of moving

them independently. By changing the pitch angle at a blade, the angle of attack of the air

flow and thereby also the generated lift and drag forces change. Pitch controlled turbines

generally pitch to feather. That means that in order to decrease the generated lift, the pitch

angle is increased resulting in a reduction of the angle of attack. The feather position is the

angle at which the angle of attack is reduced to zero and very low lift is produced.

The electrical system is designed so that the generator may run at rotational speeds that are

independent of the frequency of the grid the turbine is connected to. This is achieved by

either converting the entire power output from the generator to direct current (full conver-

sion) and then to the grid frequency or by using a generator setup that allows a variable

slip (doubly fed induction generator [87]). With these types of electrical architecture, the

generator torque is independent of the rotational speed of the generator as well. The con-

verter control unit (CCU) controls the current and voltage at the generator so that the air

gap torque is driven to a value commanded by the main turbine controller. While the con-

verter control strategies for wind turbine applications that are used in the CCU are a widely

researched topic as well [35, 12], for the turbine control problem the generator torque is the

main actuated variable and the underlying converter control is part of the respective actu-

ator model.

Since the same actuators are used to achieve both the primary and the secondary objective,

the two objectives cannot be treated independently and there are significant trade-offs to

be made. Nevertheless, there is generally a part of the overall controller that is mainly con-

cerned with the primary objective of controlling speed and power output of the turbine,

while performance with respect to the secondary objective is increased by adding addi-

tional control loops on top of the main speed and power controller.

2.1.2. Speed and Power Control

The basic speed and power controller is concerned with maximizing the power output

under certain constraints on both the stationary and transient rotor speed and generator

2. Background

torque values. These constraints lead to two distinct operating regions that are best de-

scribed in the torque-speed curve (figure 2.1). For a given turbine configuration, the max-

imum stationary power output is called rated power P0, and the corresponding generator

speed and torque values are called rated torque MG,0 and speed ω0.

Figure 2.1.: Torque Speed Curves for variable Speed Wind Turbines. Green: Partial load oper-

ating region for speed constrained turbine Blue: Partial load operating region for torque con-

strained turbine. Red: Range of potential full load operating region concepts

In the partial load operating region, the turbine is operating below the rated torque and

speed values. Therefore, the speed and torque can be freely varied in order to extract the

maximum energy out of the wind. At a certain wind speed, the energy in the wind will

be so large that the maximum extraction strategy would cause the turbine to run either

above the rated speed or torque value. This region, where the turbine has still not reached

rated power but cannot run at the aerodynamic optimum due to either the limit on speed

or power, is called the transition region. Whether a turbine is torque or speed constrained

depends on a number of factors such as the ratio of rated speed to rated torque and the ratio

of rotor area to rated power. The majority of wind turbines is speed constrained and some

overview articles do not even consider the torque constrained case at all ([3]). At a certain

wind speed the turbine will reach both rated torque and rated speed. This wind speed is

called the rated wind speed. The task of the controller for the so-called full load operating

region is now to keep the power output at its rated value P0at all wind speeds above this

rated wind speed.

2.1. Control of Wind Turbines

Rated speed and torque are stationary constraints, i.e., that turbine may not operate above

these values for an extended period of time. Due to transient variations in the incoming

wind field, the turbine will however operate above either of these limits for a short period

of time. There are also constraints on the instantaneous torque MG,max and speed ωmax.

While the maximum generator torque is a control constraint which can easily be honored,

the maximum speed is an output constraint that, if violated, would lead to an immediate

shut-down of the turbine. While these constraints on the instantaneous values are active in

all control regions, they are mainly relevant for control in the full load operating region as

here the torque and speed are closest to their respective short-term limits.

Variable Speed Operation The aerodynamic power generated by the rotor is given by:

PA=1

2ρπR2v3CP(λ,θ)(2.1)

where ρis the air density, Rthe rotor radius, vthe free-stream wind speed, and CPthe

dimensionless power coefficient. The power coefficient itself is a function of the pitch angle

θand the tip speed ratio:

λ=ωR

v. (2.2)

An example of the power coefficient as a function of pitch and tip speed ratio is shown in

figure 2.2. In variable speed operation the goal of the controller is to operate such that CP

is at its maximum CP,max(λopt,θopt). As the optimum pitch angle θopt does not depend on

the current wind speed, the pitch angle is simply held constant at this optimum value, also

called the fine pitch angle, throughout the entire variable speed operating region. The tip

speed ratio λis however a function of the rotor and wind speed. In order to operate at the

optimum tip speed ratio, the turbine needs to operate at a rotor speed which depends on

the current wind speed:

ωopt =λoptv

R. (2.3)

With the pitch angles held constant, the generator torque is the only remaining actuator that

is used to track this optimum rotor speed. In summary, in variable speed operation the con-

trol problem is a SISO tracking problem where the generator torque is used to control the

rotor speed so that it tracks a varying reference that depends on the wind speed. This track-

ing problem is complicated by the fact that on most turbines the current free-stream wind

speed vcannot be measured with enough accuracy to use it as an input to the controller.

There are two main approaches that are used for designing a controller for this problem.

2. Background

Figure 2.2.: Power coefficient CPas a function of pitch angle θand tip speed ratio λ

The first one uses the specific nature of the problem in order to design a tracking con-

troller that does not require knowledge of the current optimum rotor speed: In stationary

conditions and neglecting the mechanical and electrical losses the power generated at the

generator

PE=MGωnG, (2.4)

where nGis the gearbox ratio, is equal to the aerodynamic power produced at the rotor

MGωnG=1

2ρπR2v3CP(λ,θ)(2.5)

which can be solved for the generator torque

MG=1

2ρπ R5

ω3R3

|{z}

λ−3

CP(λ,θ)ω2. (2.6)

Now, assuming the turbine operates at its aerodynamic optimum

CP(λ,θ) = CP,max,λ=λopt (2.7)

2.1. Control of Wind Turbines

equation (2.6) becomes:

MG=1

2ρπ R5

nGλ3

opt

CP,max

| {z }

ω2(2.8)

where all constants are lumped into kand the only remaining variable is the rotor speed ω

which is measurable. This nonlinear control law MG=kω2has been derived here using

stationary considerations only. It can, however, be shown that it will also drive the rotor

speed towards its optimum if the rotor speed is not at its optimum and therefore will lead

to tracking of the rotor speed reference [42]. One disadvantage of this control strategy is

that, due to the lack of tuning parameters, there is no control over how tightly the reference

signal ωopt is tracked.

The second approach relies on the wind speed estimator that is used on many modern wind

turbines. The wind speed estimator uses a turbine model and the available measurements

to compute an estimated wind speed ˆ

vthat approximates the actual wind speed:

v≈v. (2.9)

See sections 3.1.2 and 5.1 for more details on wind speed estimators. With ˆ

vavailable, the

reference rotor speed can be explicitly calculated:

ωset =λopt ˆ

R. (2.10)

Now the speed can be controlled by designing a controller −C(s)1that drives the control

error to zero:

MG=−C(s)ωerr with ωerr =ωset −ω. (2.11)

The regulator C(s)is commonly designed as a PI controller, but other designs are also possi-

ble. Unlike in the MG=kω2approach, it is possible to influence the tracking performance

through the design of the regulator. Figure 2.3 shows block diagrams of these two com-

monly used control approaches for variable speed operation.

Transition Region In the transition region, also called upper partial load operation, the

control strategy depends mainly on whether the turbine is speed or torque constrained. In

both cases the control objective is to maintain operation as close as possible to the optimal

1As an increasing generator torque, will generally cause the rotor speed to decrease the controller needs to

have a negative gain. Here, and in the pitch controller covered later, the the factor −1 is pulled out so that

C(s)has a positive gain.

2. Background

Wind Turbine

kM G=G

(a) MG=kω2control law

Wind Turbine

vopt

set

=set

err

)(sC G

−

(b) wind estimator based control scheme

Figure 2.3.: Block diagrams for the two commonly used control laws for speed control in the

variable speed operating region

curve that is followed in the variable speed operation but to stay within the speed or torque

limits. Therefore, in the speed constrained case, when rated speed is reached, the speed is

held constant by controlling the torque accordingly. This can be achieved using a PI-type

or similar controller that regulates the difference between the current rotor speed and the

rated speed to zero. Due to variations in the wind speed, the torque required to maintain

rated speed and thereby also the power output vary in this operating regime. If the power

increases above the rated power value the turbine will transition to full load operation.

Similarly, if the torque drops below the kω2curve the turbine will transition to the variable

speed operation that was previously described. The described approach for transition re-

gion control is similar to the estimator approach for variable speed tracking and can also be

realized easily by simply using the rated speed as an upper bound for the calculated speed

set-point.

In the torque constrained case, the torque is simply held constant at its rated value and

the speed is allowed to vary freely. The turbine is essentially uncontrolled in this region.

Similar to the speed constrained case, switching to the other operating regions is performed

if the kω2curve or respectively the constant power curve in the torque speed plane are hit.

Full Load Operation In full load operation there is more power available in the wind than

can be used by the turbine. The turbine has to shed power by pitching the blades. The

main goal of the controller is to regulate both power and turbine speed to their respective

set-points with the wind acting as a disturbance. As both collective pitch and the generator

torque are used to achieve these goals, the control problem becomes a 2x2 MIMO distur-

bance rejection problem. This 2x2 control problem is often simplified by either maintaining

the generator torque constant at the rated value (constant torque control) or by controlling

2.1. Control of Wind Turbines

it so that the power output is constant (constant power control):

MG=(MG,0 : constant torque control

nGω: constant power control (2.12)

In either case, the power regulation performance is then directly linked to the speed control

performance. If the turbine is controlled to run at rated speed, for both these torque con-

trol laws the power output will be rated power and regulation of the rotor speed to rated

speed translates into regulation of power to rated power. The remaining control problem,

using the pitch to control speed, is again a SISO problem. Obviously, the choice of constant

power control will give the best performance with respect to the power regulation control

objective. Yet, it will also make speed regulation more difficult as for rotor speeds above the

set-point the generator torque is decreased which further speeds up the rotor. It has been

shown that a hybrid strategy that lies somewhere in between these two extremes might

give better overall performance [85].

Wind Turbine

)(sC

−

0,G

err

(a) constant torque

Wind Turbine

)(sC

−

(

)

GG nPM 0

err

(b) constant power

Figure 2.4.: Two baseline control schemes for the full load speed and power control problem

The aerodynamic torque generated by the rotor is

MA=1

2ρπR3v2CM(λ,θ)(2.13)

where the CMis the torque coefficient. Equation (2.13) shows that the controller needs to

counter any increase in wind speed vby reducing the torque coefficient through a change in

pitch angle θ. In stationary conditions the aerodynamic torque needs to be equal to the rated

torque at the generator for all wind speeds above rated wind speed. The pitch controller

Cθ(s)is also typically designed as a PI-based controller that acts on the speed error:

θ=−Cθ(s)(ω−ω0). (2.14)

2. Background

The partial derivative of the aerodynamic torque with respect to the pitch

∂MA

∂θ =1

2ρπR3v2∂CM

∂θ (2.15)

is also called the pitch sensitivity. Especially due to the quadratic influence of the wind

speed, there is a significant change in this sensitivity over the full load operating region.

For typical turbines, the pitch sensitivity near the cut-out wind speed2is about five times

as large as near rated wind speed as shown in figure 2.5. This nonlinearity needs to be

5 10 15 20 25

−2

−1

v [m/s]

normalized pitch sensitivity

Figure 2.5.: Pitch Sensitivity normalized to sensitivity at rated wind speed for an example tur-

bine with vr=11 m/s

accounted for when designing the pitch controller for the full load operating region. This

is typically done by using a gain scheduling approach: The gains are adjusted as a function

of the wind speed, for example using the inverse of the pitch sensitivity [66]. If no wind

speed estimator is used and the wind speed is not available this scheduling is usually done

as a function of the pitch angle.

2.1.3. Load Reducing Control

There are several additions that are commonly added on top of the basic speed and power

controller which are designed to reduce mechanical loads acting on the turbine.

Drive Train Damper Unlike the flapwise or out-of-plane motion of the blades, the edge-

wise oscillations modes have very little aerodynamic damping and thus also very little

2The cut-out wind speed is defined as the wind speed above which the turbine is not allowed to operate

continuously. If the wind speed exceeds this limit for a certain period of time (e.g., 10 minutes), the turbine

is powered down by pitching the blades to a completely feathered position.

2.1. Control of Wind Turbines

overall damping. These edgewise motions are also coupled with the drive train dynamics

which are arising from the flexible coupling of the generator inertia with the low speed

shaft. One of the most problematic modes is the so-called collective edge mode where all

three blades oscillate in phase against the generator inertia [26]. To add extra damping to

this mode, a drive train damper that uses torque actuation based on the measured genera-

tor speed is commonly used. The torque output from the drive train damper is then added

to the generator torque command from the main speed and power controller. This drive

train damper is often designed to be a bandpass or high pass filter in order to not interfere

with the main speed regulation which occurs at lower frequencies than these types of drive

train oscillations [3].

Tower Feedback A tower damper or tower feedback controller uses the measured tower

top acceleration along the direction of the rotor axis (longitudinal) to generate a pitch offset

that is added to the pitch command from the speed controller [4]. Using a simple one degree

of freedom model for tower motion

xt+2ζTωT,0 ˙

xt+ω2

T,0xt=Kf aFA(ω,v,θ)(2.16)

where xtis the longitudinal tower top position and FAis the aerodynamic thrust force acting

on the system, and linearizing the thrust at a fixed operating point

xt+2ζTωT,0 ˙

xt+ω2

T,0xt=Kf a

∂FA

∂vδv+Kf a

∂FA

∂θ δθ (2.17)

it can be seen that a simple controller consisting only of integral feedback

δθ =−KI

xtdt(2.18)

would suffice to add damping to the tower motion:

xt+2ζTωT,0 +KIKf a

∂FA

∂θ ˙

xt+ω2

T,0xt=Kf a

∂FA

∂vδv. (2.19)

Instead of a pure integral controller the tower damper is often implemented as a PI con-

troller to provide an additional degree of freedom to account for phase loss, for example

from the pitch actuator dynamics or necessary filtering on the tower acceleration signal.

The tower damping control has been treated here as independent of the pitch control for

speed regulation. However, since they use a common actuator, there is a significant amount

of cross-coupling. Any pitch actuation to regulate speed will also excite the tower motion;

2. Background

while any attempt to dampen the tower motion will also impact the rotor speed. There-

fore, proper attention has to be paid during the design of these controllers to limit these

interactions [63, 64].

Individual Pitch Control On many modern wind turbines the three blades can be pitched

individually. This capability can be used to reduce loads cased by the asymmetric loading

of the rotor. For example, due to the wind shear, the wind speed a blade pointing up is

subjected to is usually greater than the wind speed for a blade pointing down. This creates

a nodding moment that would bend the main shaft upwards as each of the blades generates

more thrust while pointing upward than when pointing downwards. This effect can be

countered by increasing the pitch angle for each blade a little bit while pointing up and

decreasing it while pointing down. Although many different designs for this individual

pitch control (IPC) exist, many follow the basic ideas as outlined by Bossanyi [4, 5]:

D-Regulator

Q-Regulator

Inverse D-Q

transformation

∆

D-Q

transformation

D-Moment

Q-Moment

Blade root out-of-

plane moments

Figure 2.6.: Basic individual pitch control scheme as shown in [5]

The controller is acting and designed in the fixed (non-rotating) coordinate frame. The

nodding (D) and yawing (Q) moment are either measured at the main shaft or derived

from load measurements at the blade roots. For both the D and the Q axis a controller is

designed to regulate the respective moment to zero by generating a fixed frame pitch offset

for each axis. The actual pitch offsets to be added to the collective pitch command are then

calculated by transforming the fixed frame pitch angles back into the rotating coordinate

frame. The design of the D- and Q-regulators is often based on a simple PI controller, but

needs to include extensive filtering in order not to react on loads that cannot be countered

by once-per-revolution (1P) pitching, such as the load caused by the tower dam effect.

The IPC as described here constrains the pitch offsets to a mean of zero. Thus, there is very

limited interaction with the control loops that have the collective pitch angle as an output

and the individual control problem can be treated as decoupled [27]. There might however

be some benefit in relaxing this constraint and designing the individual and collective pitch

controllers in a unified framework [22].

2.1. Control of Wind Turbines

Figure 2.7 shows the controller structure resulting from the combination of the speed and

power controller with the load reducing control loops described here.

Individual

Pitch

Controller

Speed and

Power

Controller

Drive Train

Damper

Tower

Damper

Blade root load

measurements

Generator

Speed

Tower Top

Acceleration

Generator

Speed

Pitch Angle Blade 1

Pitch Angle Blade 2

Pitch Angle Blade 3

Generator Torque

Command

Figure 2.7.: General control structure with the most commonly used load reduction features

2.1.4. Shutdown Control

The control strategy including the load reduction features that are described in the previous

sections are only active when the turbine is operating to produce power. Traditionally, most

of the research and development has been focused on this controller for normal power

production (NPP) operation. However, there are also situations where the turbine needs

to be controlled dynamically while it is not producing power, e.g., the start-up and the

shutdown process. In the context of load reducing control, the start-up is of relatively low

importance since load levels during start-up are low. The loads the turbine encounters

during a shut-down, however, can contribute significantly to the overall turbine design

and lifetime. A shutdown procedure may be initiated either by an operator request or by a

fault that is detected by the turbine controller. Such faults may for example be related to the

wind conditions such as the wind speed being above the design wind speed of the turbine,

the wind direction exceeding a predefined limit, or to the turbine operation such as the

generator speed approaching a safety critical level. Once a fault is triggered, the turbine will

be brought to a stand-still using a shutdown controller. Depending on the kind of scenario

2. Background

or error message, different shut-down controllers are used. For example, if a regular stop

of the turbine is commanded, the stopping procedure will generally be more gentle than in

case of an emergency stop where the objective is to bring the turbine to a halt as quickly

as possible. Although more elaborate strategies for such an emergency stop are possible,

the most common control scheme, which is also used as a baseline here, simply pitches the

blades towards the feathered position at a fixed rate without any feedback of rotor speed

or other measured signals. The pitch rate during the shut-down process depends on the

capabilities of the pitch system but is in normally somewhere between 5deg/sand 15deg/s.

2.1.5. FRT Control

In the past, whenever there was a disturbance in the electrical grid the wind turbine was

connected to, the turbine would simply be shut down. With the increasing penetration of

wind energy in many grids, however, this is no longer practical as turbines disconnecting

due to grid disturbance would further weaken the grid and could cause even more sys-

tems to trip. More specifically, today, in many grids wind turbines are required to stay

connected to the grid for a certain amount of time if there is a drop in voltage. This is gen-

erally referred to as Fault Ride Through (FRT) capability. The exact type of the requirement

is defined by the grid operator and varies between the various grids [40]. Nevertheless,

most requirements are similar to the generic curve shown in figure 2.8 for a so-called Low

Voltage Ride Through (LVRT) requirement. The grid voltage drops suddenly, potentially

stays flat at a low level for a short amount of time, and then increases slowly. Of course,

grid disturbances usually do not have this exact shape; a curve like this defines the most

severe disturbance the turbine needs to be able to ride through: As long as the grid voltage

is above the specified curve the turbine must stay connected to the grid. LVRT is mainly

a challenge for the converter control unit and grid interconnection. A significant amount

of research has been performed on the design of the control algorithms for the electrical

system (e.g., [14, 78, 121]) for these scenarios. However, LVRT and other grid events also

challenge the main turbine controller. In many of these grid events, the turbine generators

are not able to produce any torque for up to a few seconds because no power can be fed

to the grid [76, 20, 21, 74]. Due to the loss of counter torque at the generator, the rotor will

speed up threatening to cause an overspeed fault which would in turn lead to an emer-

gency shut-down of the turbine. The loss of counter torque will need to be countered by

increasing the pitch angle at the blades. Although this pitch control problem is very similar

in structure to pitch control in normal operation, the plant dynamics might be very differ-

ent since the turbine is operating far off its normal operating curve and therefore requires a

specifically tailored pitch controller.

2.2. System Simulation and Loads Analysis of Wind Turbines

Voltage [%]

Time

Turbine must stay

connected

Turbine may trip

100

Time of

event

Figure 2.8.: Generic LVRT requirement curve [40]

2.2. System Simulation and Loads Analysis of Wind Turbines

The design of modern wind turbines relies heavily on simulation methods. All mechani-

cal components need to be dimensioned so that they can withstand all loads the turbine is

likely to encounter and to last at least a specified lifetime, usually 20 years. The main driver

for the mechanical loading of the turbine are the aerodynamic forces acting on the turbine

generated by blades. These aerodynamic forces are a function of the local flow conditions

on the turbine blades and are thus highly dependent on the incoming wind field. The wind

is stochastic in nature and varies both on short (e.g., seconds) and longer time scales (hours

to months). The short term fluctuations are mainly caused by local phenomena and the

landscape surrounding the turbine. These short time fluctuations in wind speed are su-

perimposed on the slower phenomena caused by more global effects such as diurnal or

seasonal variations [10, 26]. The design of a wind turbine needs to consider a set of wind

conditions that is representative for both typical and extreme conditions, such as storms

and gusts, which the turbine may experience at its site of erection. Next to variations in the

wind field, turbine loading is also influenced by the way it is operated. For example, if the

turbine is in an idling state, the loading will be very different from when it is producing full

power. Finally, a failure in one of the components should not endanger the whole turbine

and the various component failure modes also need to be considered. Overall, the combi-

nation of wind conditions, operating modes, and potential turbine or component failures,

2. Background

leads to a very large number of scenarios that need to be taken into account. The design

assessment of a typical modern wind turbine can easily require more than 1000 single sce-

narios to be analyzed. Such a large number of scenarios can only effectively be evaluated

through the use of automated simulation tools. Hence, the design of wind turbines relies

heavily on simulation tools that model the overall aero-elastic system behavior and dy-

namics. Common tools that are used include for example BLADED [6], FAST [44], or FLEX

[84].

For several reasons, the design of wind turbine controllers also relies heavily on the same

simulation tools as the loads simulations: With the turbine operating in such a large variety

of conditions the controller needs to able to handle all these conditions. Firstly, with no

wind tunnels available to test full size, modern wind turbines, the wind as the main driving

factor for the turbine dynamics is largely out of control and one has to rely on favorable

conditions occurring at the test site which can easily be prohibitive. Secondly, not only can

the wind field the turbine is subjected to not be controlled, it is also very difficult to measure

the full wind field. With the uncertainty on wind speed, a valid conclusion on the control

performance can only be made by measuring the turbine performance over an extended

period of time and using statistics to reduce the variability. Finally, if the design of a turbine

is to be certified by an external party, the simulations have to follow certain guidelines or

standards such as the IEC regulations on the design of wind turbines [39]. Even without

an external certification, the conditions and load cases defined in these regulations have

become the widely accepted standard for wind turbine design. A significant goal of the

design of controllers for wind turbines is the reduction of mechanical loads acting on the

turbine. The mechanical loads are assessed through the use of standardized aero-elastic

simulation tools and conditions. Therefore, any wind turbine control design focused on

load reduction needs to be assessed in terms of its impact on the same loads simulations

that define the turbine design and ultimately the turbine cost. Subsequently, the majority

of all research on control of wind turbines that has been performed uses simulation studies

only. Even in the few existing studies where advanced controllers have been tested on real

turbines (e.g., [118, 119, 42, 49]), the actual design was performed in simulation and the

field test is only used to confirm that the field performance matches the predictions from

simulations for a very limited range of conditions.

The following sections will provide an overview of the key features of the loads simulations

and outline their impact on the control design.

2.2. System Simulation and Loads Analysis of Wind Turbines

2.2.1. Models

In order to model the dynamic behavior of a wind turbine, models for several components

need to be combined: An aerodynamic model is used to calculate the forces and moments

acting on the rotor blades that are caused by the local flow conditions. A mechanical model

then models the response of the main structural components such as main shaft, blades, and

tower to the aerodynamic loading. The turbine controller will react on measured turbine

states by sending commands to the actuators such as the generator system or the pitch

motors, and the behavior of these actuators needs to be modeled as well.

For all of these subsystems, there exist general purpose, high fidelity modeling methods

such as CFD or FEM-based tools. However, due to the described large number of simula-

tions that need to be evaluated in the turbine design process, the computational speed of

the simulation tools is, even given modern computing power, still of high importance and

model fidelity for these system level simulation is kept at a minimum. Further, because

of the unique combination of aerodynamic, structural, and electro-mechanical modeling,

wind turbine system simulations are generally performed using tools that have been specif-

ically created for wind turbine simulations. Although there is some variance in the exact

modeling technique that is used, most of the commonly used codes use similar modeling

approaches in principal [86]. Therefore, here, only the main modeling approaches that are

implemented in the commonly used tools are introduced.

It should further be noted that the loads evaluation process for wind turbines usually uses

two levels of analysis. In the first step, the so-called system simulation, the overall behav-

ior of the entire wind turbine system is simulated. These simulations use the described low

fidelity, highly specialized system simulation tools. The results from these simulations are

the mechanical loads each component will be subjected to. Additionally, for each subsys-

tem, there are often additional simulation methods and tools to perform a more detailed

analysis. As an example, the main frame of a turbine is generally not modeled as a flexible

body in the system simulation as the dynamic coupling between the structural modes of

the main frame and the overall turbine modes is low. Instead, the main frame is assumed

to be a rigid body. The result from the system simulation would then be the loads acting

on the mainframe. As a next step, these loads could then be used as inputs, to a more

detailed simulation, e.g., using 3D-FEM, of the mainframe dynamics only. For the control

design, these higher frequency dynamics generally do not have to be considered since the

controller mainly interacts with the low-frequency, turbine level modes. Therefore, only the

models typically used in the system simulations are introduced here.

2. Background

Aerodynamics The aerodynamics are generally modeled using Blade Element Momen-

tum (BEM) Theory [10]. BEM is a quasi-steady approach where it is assumed that the

aerodynamic forces acting on each section of the blades can be calculated purely based on

the geometry of that section. All 3D effects, i.e., air flow along the span of the blade, are

ignored. Under this assumption the lift FLand drag FDforces acting on each blade section

are then calculated based on the local flow velocity c3and the lift and drag coefficients cl

and cd

FL=cl(α)ρ

2c2bstsFD=cd(α)ρ

2c2bsts. (2.20)

where ρis the air density and bsand tsare the blade section and chord length respectively.

The lift and drag coefficients are a function of the local angle of attack αwhich itself is a

function of the free stream wind speed, the rotational speed of the rotor, and the local in-

duction factor4. The functional relationships of the lift and drag coefficients, also called

profile polars, are generally determined either using wind tunnel tests or in CFD simula-

tions. It should be noted here that because of the quadratic influence of the wind speed and

the lift and drag coefficients, the aerodynamic model is highly nonlinear.

This basic BEM model is then augmented with several additional semi-empirical models to

improve accuracy: A tip-loss model modifies equation (2.20) so that the blade sections near

the tip produce less lift because of the flow along the span of the blade and around the tip.

A dynamic stall model [62] modifies the profile polars in case the blade section is rotating

about its center axis, for example due to pitch motion or torsional vibrations of the blade.

The described BEM model is quasi-stationary, i.e., the assumed local flow conditions and

resulting aerodynamic forces only depend on the current conditions and would change

immediately with any change in the wind field or blade movement. In reality, the fluid

flow needs some time to adjust to the new situation and the change is not instantaneous.

Therefore, a dynamic wake model is used that models the fact that the flow conditions

do not change instantaneously by adding some dynamics, usually in the form of low-pass

behavior, to the calculation of the local induction factor.

The output of the aerodynamic calculations are the distributed forces along each of the

blades. These forces then act as the external forcing on the structural dynamics of the tur-

bine.

3The local velocity depends largely on the circumferential speed of the particular blade section and the free

stream wind speed and should not be confused with the latter

4The induction factor is the factor by which the free stream wind speed has been slowed down when it reaches

the rotor plane due to the rotor extracting kinetic energy.

2.2. System Simulation and Loads Analysis of Wind Turbines

Structural Dynamics In most tools, the structural model is a combination of rigid and

flexible bodies. Usually, only the blades and the tower are modeled as being flexible while

all other components are rigid. In order to reduce the computational complexity, most codes

provide some means for a modal reduction of the flexible bodies. The blade and tower

motion is assumed to be limited to a superposition of the first few eigenmodes. In most

cases, only the first two modes are considered, which can be justified by the higher modes

being difficult to detect in actual measurements and having a negligible energy content. It

should be noted that the modal decomposition and reduction is performed for each axis

individually: E.g., each of the blades may have two modal degrees of freedom for each of

flapwise and edgewise deflection as well as torsion. Combining these modal degrees of

freedom with the rigid body degrees of freedom results in an overall, coupled mechanical

system with between 20 and 100 degrees of freedom.

The mechanical system of a modern turbine has a very low inherent damping. Especially

for structural motion along the direction of the wind speed, most of the damping is pro-

vided by the aerodynamics: Any movement in the wind direction reduces the relative wind

speed at the blades and thereby reduces the thrust produced by the rotor while movement

towards the wind has the opposite effect. This aerodynamic damping of the fore-aft motion

is only one example of an effect arising from the coupling of structural and aerodynamics.

These types of aero-elastic phenomena are important for many aspects of wind turbine dy-

namics and therefore need to be considered specifically not only in the system simulation

but also in the control design.

Actuators The three principal actuators yaw, pitch, and generator each consist of sev-

eral subsystems and underlying control loops and may have complex dynamics. While a

detailed modeling of these systems in the overall system level simulation tools can be ben-

eficial for the design of the actuators, it is not necessarily required for the overall system

and controller design.

The output of the controller is a commanded generator torque. The converter control unit

(CCU) then uses the capabilities of the converter to produce a current in the generator that

produces a torque that matches the desired value. The dynamics of this underlying elec-

trical control loop are at least one order of magnitude faster than the mechanical dynamics

of the system. Therefore, they are not included explicitly in the simulation of the overall

system, but instead are typically represented by a small communication delay and time con-

stant between the torque demanded by the generator and the actual air-gap torque acting

on the mechanical system.

The pitch system receives a desired pitch angle for each blade from the controller. It then

needs to control the pitch motors to drive the blades in the desired positions. Due to the

2. Background

large inertia of the blades, there is a significant delay between a blade pitch command and

the blade actually arriving in the desired position. This delay cannot be ignored in the sys-

tem simulations. These dynamics are largely dominated by the blade and motor inertia,

the capabilities of the pitch system, and the tuning of the underlying control loops while

the torque caused by the aerodynamic loading of the blades only has a small impact. Many

simulation tools simply assume that all effects that depend on the current state of the tur-

bine can be ignored and lump the entire pitch system dynamics into one SISO model that

has the commanded pitch angle as its only input and the actual pitch angle as output. Often

the pitch system is modeled as a linear second oder system with low pass characteristics

plus additional rate limits on the pitch angle that represent the power limits of the pitch

motors.

The yaw system of a typical turbine is significantly slower than the pitch system and the

structural dynamics. Since the yaw rates are so slow that there is very little interaction

with the rest of the system behavior, it is often not considered at all. Within one typical ten

minute time series, the yaw is simply assumed to be fixed. If yawing is to be considered

it can be modeled similar to the pitch system but with significantly lower bandwidth and

rate limits.

2.2.2. Load Cases

A full loads analysis requires consideration of a number of predefined scenarios, for exam-

ple representing an operational state or failure mode. Each scenario needs to be evaluated

over a large range of external conditions such as mean wind speed, turbulence, and start-

ing values. Such a group of simulations is called a load case. The different load cases that

are to be considered are for example defined in the IEC guidelines [39]. Not all of the load

cases are relevant for the control design, either because in this load case the controller does

not affect the loads that occur, or because they are generally not driving the turbine design.

For example, during storms most turbines are in an idling mode where the pitch angles

are simply kept at a fixed value and the turbine is not actively controlled. Table 2.1 lists the

subset of load cases where the controller design has the most impact on the turbine loads.

Normal Power Production NPP is the state in which the turbine will operate for the vast

majority of the time. The NPP simulations emulate the case where the turbine is connected

to the grid and produces power and need to capture the typical conditions the turbine will

be subjected to. These simulations are performed with a turbulent wind field. For a given

single simulation, the timeseries of the wind field is a superposition of a constant mean

wind speed and a stochastic turbulence with a mean of zero. To capture the low frequency

2.2. System Simulation and Loads Analysis of Wind Turbines

Load Case Description Main Varying Factors Relevant Controller Type of Loads

Analysis

1.1

Power Production Oper-

ation in turbulent condi-

tions without fault

Mean Wind Speed,

Turbulence NPP U/F

1.3 Extreme Coherent Gust

and Direction Change

Wind Speed, Gust

Amplitude, Direction

Change

NPP, Shutdown U

1.5 Extreme Operating Gust

(1-year) and grid loss

Wind Speed, Gust

Amplitude, Timing

of Grid Loss

NPP, Shutdown U

1.6 Extreme Operating Gust

(50-year)

Wind Speed, Gust

Amplitude NPP, (Shutdown) U

1.9 Extreme Coherent Gust Wind Speed, Gust

Amplitude NPP, (Shutdown) U

5.1 Emergency Stop Wind Speed Shutdown U/F

Table 2.1.: Main loadcases according to [39] that are relevant for the turbine control design.

U:Ultimate(Extreme) Analysis, F: Fatigue Analysis

changes in wind speed, simulations are performed at all wind speeds between the cut-in

and the cut-out wind speed, usually in steps of 1m/s. Generally, the turbulence intensity is

a function of the mean wind speed. In the turbine design process generic turbulence levels

such as defined by the IEC classes are typically used. See figure 2.9 for the turbulence as a

function of mean wind speed for turbulence classes A and B. The turbulent component of

the wind speed time series is created using a turbulence model such as the Kaimal or Mann

[26] models. Generation of the turbulent wind requires some form of random number gen-

erator. In order to reduce the statistical variation, the turbine behavior at each wind speed

is simulated with different starting values (”seeds”) for the random number generator used

in the creation of the turbulence.

0 5 10 15 20 25

v [m/s]

Ti [%]

Figure 2.9.: Turbulence intensity as a function of mean wind speed for turbulence classes A and

B according to [39]

The turbine controller should be designed so that it can handle all situations that might

2. Background

arise during normal power production. Therefore, in load case 1.1 the turbine is assumed

to be operating without fault and only the NPP-controller is of relevance.

Gusts Next to the conditions the turbine would typically encounter, extreme wind condi-

tions that the turbine may only experience in rare situations also need to be considered. For

example load case 1.6 describes a gust that is so strong that on average it occurs only once

in fifty years while load case 1.3 describes the case where a so-called Extreme Coherent

Gust (ECG) gust coincides with an extreme change in wind direction. These simulations

are performed using deterministic wind profiles, i.e., the wind is not stochastic but follows

a predefined gust shape and only the amplitude and starting wind speed are varied. Figure

2.10 shows the wind speeds for an Extreme Operating Gusts (EOG) which are used in load

case 1.5 and 1.6 and the Extreme Coherent Gust (ECG) used in load case 1.3 and 1.9.

The turbine will enter into these events in NPP mode. Therefore, the NPP controller will

define the behavior during the gust. However, due to these gust representing extreme

conditions, one or more of the safety thresholds might be triggered. For example, in load

case 1.6 the gust can easily be so strong that the generator speed rises above the overspeed

trigger level even with a functioning NPP controller and the turbine would shut down. In

this case, the control strategy used for shutting down the turbine also has an impact on the

loads occurring during that load case.

0 5 10 15

time [s]

v [m/s]

EOG1

EOG50

ECG

Figure 2.10.: Extreme Operating Gusts with a 1-year (EOG1) and 50-year (EOG50) return period

and Extreme Coherent Gust (ECG) according to [39] at a wind speed of 11 m/sfor a turbine with

a rotor diameter of 126m

Grid loss Extreme events are not only caused by the wind conditions. They can also be

caused by failures in the turbine itself. Most importantly, faults in the electrical system

need to be considered. These can be caused by failures of electrical components such as the

2.2. System Simulation and Loads Analysis of Wind Turbines

frequency converters, but can also arise as a result of the connection to the power grid. For

example if the grid frequency deviates by more than a specified amount from its base value,

most turbines will be disconnected from the grid. These types of events are characterized

by an immediate disconnection of the electrical system and resultant loss of counter torque

at the generator.

Load case 5.1 simulates an emergency stop. If an emergency shut-down is triggered, the

turbine will also disconnect the generator and the behavior is very similar to a grid loss.

While these simulations are performed at multiple wind speeds, the wind speed during

each of these simulations is constant. Without any gust or turbulence, the NPP controller

that is active during the beginning of the simulation has no impact and this load case is

only important for the design of the shut-down controller that is used once the fault is

triggered.

Load case 1.5 combines a deterministic gust with a grid fault. The gust has the same ”Mex-

ican Hat” shape as the one used in load case 1.6, but a smaller amplitude as a return period

of only one year needs to be considered. This gust coincides with a grid loss that happens

at a point in time during the gust. The relevance to the controller design is similar to load

case 1.6 as the NPP controller is used initially and then switches to a shutdown controller.

However, unlike in load case 1.6, the turbine will always shut-down independently of how

the NPP controller performed during the first part of the gust.

2.2.3. Extreme Analysis

The analysis of extreme or ultimate loads is concerned with dimensioning the turbine com-

ponents so that they can withstand the maximum load the turbine is likely to incur during

its lifetime. Here, the component is required to only withstand the load once and all ef-

fects that are a result of multiple applications of a load, such as crack growth, are ignored.

Subsequently, the magnitude that is considered in the extreme load analysis is computed

by taking the maximum over all considered load cases and time series. This maximum is

computed independently for each point and load of interest5and the maxima will gener-

ally occur in different load cases. For example, the maximum tower base bending moment

will usually occur in different conditions than the maximum main shaft torque.

Generally, all load cases including NPP need to be taken into account in the extreme analy-

sis. On the other hand, the maximum load on most sensors usually does not occur during

normal power production, but rather in one of the gust, fault, or storm load cases. There-

fore, one important aim of the wind turbine controller design is to reduce the load peaks

especially under these gust and fault conditions. The reduction of extreme loads that occur

5often called a sensor in the loads analysis context

2. Background

during normal power production loading is less relevant as these loads are often not driv-

ing the turbine design: For example, if the maximum load for a given sensor that occurred

during load case 1.1 is significantly below the load in load case 1.6, any reduction of this

load in load case 1.1 through a modified controls strategy will not bring the extreme load

for this sensor down. The sensor is then said to be dimensioned by load case 1.6.

2.2.4. Fatigue Analysis

Unlike the extreme analysis, the fatigue analysis is concerned with the effect of applying a

load repeatedly. Most mechanical components on a wind turbine are exposed to a complex

sequence of loads. These loads are mainly caused by variations in the wind and also har-

monic excitation effects caused by the rotation of the rotor. For example the slight difference

in the weight of the three blades that is always present will cause a revolving load on the

main shaft with the amplitude and frequency depending on the rotor speed. The entire set

of load cycles the turbine will encounter during its lifetime is called the loads collective. For

wind turbines, it consists of load cycles covering a large range of magnitudes and number

of occurrences.

In order to facilitate the analysis of fatigue loads, the entire loads collective for each sensor

that the turbine will encounter during its lifetime is combined into a damage equivalent load

(DEL). The DEL is the load that, if applied with a specific frequency, would cause the same

damage to the component as the loads collective. So a complex load consisting of cycles

with a variable amplitude and frequency is transformed into an equivalent load with a

fixed amplitude and frequency.

For a given time series, first the loads collective, the number nand amplitudes σiof all

cycles needs to be determined. This is done using a cycle counting method such as Rain-

Flow-Counting [95]. The total damage Dcaused by this loads collective is computed using

the linear damage accumulation hypotheses according to Palmgren-Miner (e.g., [113]): The

total damage caused by this loads collective is

∑

i=1

σm

i, (2.21)

where mis the material-dependent slope of the Wöhler curve. The DEL σDEL is now calcu-

lated such that, if applied for Nre f cycles it leads to the same overall damage:

Nre f σm

del =D. (2.22)

2.2. System Simulation and Loads Analysis of Wind Turbines

The number of reference cycles for a time series with length tts is chosen so that it corre-

sponds to a reference frequency fre f :

Nre f =fre f tts, (2.23)

where the reference frequency is a frequency in the same order of magnitude as the frequen-

cies most often seen in the loads collective. For wind turbines, this is usually the frequency

of the rotor revolution during operation at rated speed.

The overall DEL for each sensor of a wind turbine is a combination of the loading at dif-

ferent operating conditions. Therefore, for each time series ithat is considered, the DELs

σ(i)are calculated. The overall DEL is then obtained by combining the single DELs ac-

cording to the occurrence probability p(i)of this time series and using the linear damage

accumulation hypotheses:

σdel,tot =m

r∑

p(i)σ(i)m. (2.24)

Although some event load cases such as emergency stops (load case 5.1) are included in the

fatigue analysis, their contribution to the overall fatigue is low due to their low share of the

overall lifetime. Thus, in order to impact the fatigue loads using controls, the focus needs

to be on the normal turbine operation as simulated in load case 1.1 and therefore also on

the NPP-controller.

2.2.5. The Aero-elastic Simulation Tool FAST

FAST (Fatigue, Aerodynamics, Structures, and Turbulence) [44] is a simulation code for the

aero-elastic simulation of wind turbines that was developed by the National Renewable

Energy Laboratory (NREL) of the United States starting in the mid-nineties. Unlike similar

tools such as BLADED or FLEX, FAST is available free of charge and its source code is pub-

lished. This has lead to FAST probably being the most commonly used tool for aero-elastic

wind turbine simulations, especially for academic research.

Using FAST the behavior of two- or three-bladed on- or offshore wind turbines in a up- or

down-wind configuration can be studied.

Strictly speaking, the aerodynamic modelling is not a part of FAST itself. Instead, the aero-

dynamic code AERODYN [57] is used. However, since AERODYN is now fully integrated

in FAST, this distinction is generally not made. AERODYN models the aerodynamics of

horizontal-axis wind turbines using the Blade Element Momentum Theory (BEM) (e.g.,

[26, 10]). In addition to the pure BEM, corrections for tip and hub losses, skewed wakes,

2. Background

dynamic stall, and tower shadow are also implemented [75]. Next to deterministic wind

speed time series defined at hub height only, AERODYN also has the ability to use fully

three-dimensional turbulent wind fields.

The structural model in FAST consists of both flexible and rigid bodies. The tower, drive-

train, and blades are assumed to be flexible while all other components are modeled as

being rigid. The flexible bodies are included in the multi-body formulation using a modal

formulation. For a three-bladed turbine up two 24 degrees-of-freedom (DOF) are available.

Of these 24 DOF, four belong to the tower bending (2 each longitudinal and lateral) and

nine to blade bending (2 flap-wise modes and one edge-wise mode for each blade) while the

remaining DOF belong to the support platform (6 DOF), generator and rotor speed (2 DOF),

yaw flexibility (1 DOF), and rotor tail furl (2 DOF)6. Which of these DOF actually needs to be

used depends heavily on the specific turbine configuration that is being analyzed, and FAST

therefore provides functionality to selectively activate or deactivate individual DOF.

FAST also includes several simple models for the behavior of the generator and simple con-

trol algorithm for speed and power control. It is also possible to include custom generator

models and controller codes using either a user-written dynamic link library (dll) or by run-

ning the program in SIMULINK using the provided interface and modeling the controller

and generator there. Further, no model of the pitch actuators is provided and, if required,

this also needs to be added as a dll or included in the SIMULINK model.

2.2.6. The NREL 5MW Reference Turbine

The National Renewable Energy Laboratory (NREL) has developed and published a model

of a 5MW reference turbine [43]. Although this turbine has not and will not be built, it has

been designed to represent typical wind turbines in the class of large offshore turbines and

is also heavily based on existing 5MW turbines such as the REPower 5M [108]. This ref-

erence model has been developed specifically to act as a baseline for various wind turbine

related studies and to give wind turbine researchers access to a full turbine model without

restrictions. It has been used in hundreds of studies focused on wind turbine aerodynamics,

system dynamics, and controls research.

The NREL 5MW reference turbine is a three-bladed upwind turbine with a variable-speed,

active-pitch control system. As such, its configuration matches the majority of modern on-

and offshore turbines. It has a rotor with a 126m diameter and a hub height of 90m. This is

a typical size of modern commercial offshore turbines and somewhat larger than turbines

6Windmill Furling is a method where the tail of a wind mill is connected to the turbine using a hinge that is

angled back from the vertical which causes the turbine to automatically turn out of the wind once a certain

wind load threshold is exceeded [109]. Furling is generally not used on large wind turbines.

2.3. Model Predictive Control

that are currently deployed at onshore locations. It is however likely that in the future

also onshore turbines will reach this size. Table 2.2 summarizes the key properties of this

reference turbine configuration.

Rating 5 MW

Rotor Orientation, Configuration Upwind, 3 Blades

Control Variable Speed, Collective Pitch

Drivetrain High Speed, Multiple-Stage Gearbox

Rotor, Hub Diameter 126 m, 3 m

Hub Height 90 m

Cut-In, Rated, Cut-Out Wind Speed 3 m/s, 11.4 m/s, 25 m/s

Cut-In, Rated Rotor Speed 6.9 rpm, 12.1 rpm

Rated Tip Speed 80 m/s

Overhang, Shaft Tilt, Precone 5 m, 5o, 2.5o

Rotor Mass 110000 kg

Nacelle Mass 240000 kg

Tower Mass 347460 kg

Table 2.2.: Key Properties of the NREL 5MW Reference Turbine [43]

The NREL 5MW reference turbine has been chosen as the turbine model that is used through-

out this study. This ensures maximum comparability to the majority of the wind turbine

controls literature that also uses this turbine. It allows a direct comparison and reproduc-

tion of the results. Here, the onshore version of the reference turbine is used, as it is ex-

pected that all results related to the use of MPC for turbine control are applicable to both

onshore and non-floating offshore turbines without modifications so that including the off-

shore conditions only increases the complexity of the necessary system simulations without

providing additional insight.

2.3. Model Predictive Control

Model Predictive Control is a control method that has been used in industrial applications

for several decades now [89]. Its basic idea is to use a process model to explicitly predict

the outputs of the plant and then use these predictions to calculate a sequence of future

control inputs that optimizes the future plant outputs. In general, only the first part of this

optimal control sequence is actually applied to the plant and the entire optimization is re-

peated in the next control interval. MPC is closely linked to many optimal control methods

such as LQR control. As in optimal control, instead of a control law, a controller objective

based on the resultant performance is specified. In most classical optimal control methods,

this control problem is solved analytically and off-line, yielding an optimal control law that

only needs to be applied to the process. Contrary to that, using MPC, the optimization is

2. Background

generally performed online using explicit numerical optimization routines. The online so-

lution allows incorporation of control and process constraints, such as actuator limitations

or safety critical plant outputs, directly in the control formulation. As these constraints

are found in many control applications and are difficult to handle using most other control

techniques, it is especially this feature that has contributed to the increasing use of MPC.

On the other hand, this online optimization requires significant processing power which is

why the use of MPC has initially been limited to comparably ”slow” plants such at those

found in the process industries. With the increasing computing power and corresponding

drop in prices of modern controls hardware, MPC is now being increasingly used also for

faster applications such as walking robots (e.g., [105]) or combustion engines (e.g., [106]).

This section introduces the key concepts of MPC that are relevant for the wind turbine

controller design in chapter 5.2. It follows mainly the text books by Maciejowski [70] and

Rawlings and Mayne [91]. There exists a large body of literature that deals with both the

practical and the more theoretical aspects of MPC. Here, the properties and formulations

are only introduced on a level of detail required for the understanding of the specific wind

turbine controller design. For a more mathematically rigorous presentation of the material,

the respective literature should be consulted.

2.3.1. Models and Problem Formulation

As a model based control method, MPC relies on a process model. While in the early stages

impulse response models were often used, now almost the entirety of MPC applications

and research is based on discrete state space models

xk+1=f(xk,uk,dk)

yk=h(xk,uk,dk)(2.25)

where xk,uk,dk, and ykare the vectors of states, controlled inputs, disturbance inputs, and

plant outputs respectively. At any point in time, the goal of the MPC is to determine a

sequence of future control moves. The controller only determines the future control inputs

for a finite number of time steps, the so-called control horizon L. The controller needs to

determine all control inputs

uk,uk+1, . . . , uk+L. (2.26)

2.3. Model Predictive Control

The following notation for sequences of vectors is introduced: If the current time step is

indicated by k, then y(k)is the value of yat time step kand the vector ym|Nis defined as:

ym|N(k) = yT

myT

m+1. . . yT

m+NT. (2.27)

To simplify the notation, the explicit dependence on the time step is dropped whenever the

vector sequence starts at the current time step which is generally denoted by k:

yk|N≡yk|N(k). (2.28)

At time step k, the controller determines uk|L. This sequence of control moves is determined

by minimizing a cost function Vxk|L,uk|L(k). Only the current and future control inputs

uk|Lare the decision variables here and the future states xk+1|Ldepend on the control and

disturbance inputs so that the system dynamics (2.25) act as a constraint. Here and in

the following, only the regulation problem, i.e., driving the state vector to the origin is

considered. If instead of the origin the states were to be driven to a reference trajectory rk|L,

the objective function would also be a function of this reference trajectory. In addition to

the system dynamics constraint, there can also be any number and type of constraint on the

control inputs and states so that all values of xand uneed to be within the sets Xand U,

respectively. The optimization problem that is solved at every time step is thus:

minuk|LVxk|L,uk|L

subject to

xk+1=f(xk,uk,dk)

x(i+1)∈X,u(i)∈Ufor all i=k. . . L. (2.29)

An MPC controller solves this optimization problem numerically and online. However, the

optimization problem in this general form as stated in (2.29) can be extremely challenging to

solve as it combines a nonlinear objective function with nonlinear constraints. Not only can

it easily lead to computation times that can be prohibitive for an online implementation, the

general form of the problem makes it difficult to ensure the chosen algorithm actually finds

the solution. Therefore, almost all MPC applications place some conditions on the structure

of the objective function, the system dynamics, and the control and state constraints with

the most common type being linear MPC.

2. Background

2.3.2. Linear MPC

Linear MPC derives its name from the use of linear state space models

xk+1=Axk+Buk+Edk

yk=Cxk+Duk+Fdk(2.30)

and constraints

Ecuk|L≤ec

Fcxk+1|L≤fc. (2.31)

The objective function is chosen to be quadratic in the control and state trajectories:

Vxk|L,uk|L=uT

k|LRuk|L+xT

k+1|LQxk+1|L. (2.32)

Because of the linear system dynamics, the state trajectory depends linearly on the control

uk|Land disturbance dk|Ltrajectories and the current state x(k):

xk+1|L=Apx(k) + Bpuk|L+Epdk|L(2.33)

where prediction matrices Ap,Bp, and Epare given by appropriately ”stacking” the system

matrices from (2.30):

Ap=











Bp=





B 0 . . . 0

AB B . . . 0

.....

AL−1B AL−2B. . . B







Ep=





E 0 . . . 0

AE E . . . 0

.....

AL−1E AL−2E. . . E







(2.34)

Substituting (2.33) in the objective function (2.32) and constraints the optimization problem

becomes:

minuk|LuT

k|LR+BpTQBpuk|L+2uT

k|LBpTQApx(k) + Epdk|L+

Apx(k) + Epdk|LTQApx(k) + Epdk|L

subject to

Ecuk|L≤ec

FcBpuk|L≤fc−FcApx(k)−FcEpdk|L. (2.35)

2.3. Model Predictive Control

As x(k)and dk|Lare constant, this optimization problem is clearly quadratic in uk|Lwith

linear constraints. This class of optimization problems is called a Quadratic Program (QP)

for which there exist numerous efficient and fast numerical solution methods. If further

Rand Qare chosen appropriately, the problem is also convex. This simplifies solving the

optimization problem since in convex optimization any local minimum is also the global

minimum avoiding the issue of the algorithm not converging to the global minimum. It is

precisely because of these benefits that the majority of MPC applications use linear MPC.

2.3.3. Stability

As with any control design technique, designing a Model Predictive Controller should re-

sult in a closed loop system that is stable. Even with linear MPC, however, analyzing and

ensuring stability is significantly more difficult than for most other controllers.

A system is called input-to-state stable if for any type of bounded input all states will re-

main bounded [103]. Using this criterion, stability is essentially defined on an infinite time

horizon as for a finite horizon usually any signal is bounded. Yet, the MPC formulation

introduced in the previous sections, only defines a controller for a finite prediction hori-

zon. So in order to analyze the stability of the controller the system behavior needs to be

analyzed on the infinite horizon even if the controller is only defined on a finite horizon.

In essence, according to Mayne et al. [71], using MPC will lead to a stable system if the

following criteria are met:

1. The objective function Vplaces a cost on the states for the infinite horizon and not

only the prediction horizon, and the contribution of each time step to the overall cost,

the so-called stage-wise cost function, is positive definite.

2. This objective function for the infinite horizon is bounded and the additional cost

incurred by including an additional time step in the infinite horizon objective function

decreases7.

3. It is ensured that the state and control constrained are not violated on the infinite

horizon also.

In practice, this is achieved by making an assumption for how the controller behaves for all

time steps past the control horizon:

u(i) = κ(x(i)) for i>L. (2.36)

7This requirement is essentially derived from using the cost associated with the stage-wise infinite horizon as

a Lyapunov function.

2. Background

This assumption has lead to the term dual-mode control as now the infinite horizon is split

into two parts each using a different controller: The finite horizon where the controlled

inputs are determined from solving the constrained optimization problem and the remain-

der of the infinite horizon where the controlled inputs are given by the control law κ, the

so-called terminal controller. It should be noted that due to the moving horizon nature of

MPC, the terminal controller is never actually applied to the plant; it is only an assump-

tion that is made to better handle the stability considerations. This terminal controller is

assumed to start controlling the system once the final state x(L)of the prediction horizon is

reached. If the terminal controller is designed to stabilize the plant, then the cost associated

with bringing the system from the terminal state to a rest using this controller is bounded.

This cost, which depends only on the terminal state, is termed the terminal cost Vf(x(L)). If

now this terminal cost is added to the objective function, the first and second requirement

are already fulfilled. The third requirement is generally fulfilled by placing an additional

constraint on the final state x(L)∈X.

How exactly the terminal constraint, controller, and associated cost will need to be set to

ensure stability depends heavily on the specific type of plant and the state and control

constraints that are used. Rawlings and Mayne [91] list these various requirements with

only those relevant for the control design at hand being discussed further here.

To motivate the argumentation, first the simplest possible problem is considered. For a

linear plant that is asymptotically stable and has only control constraints but no state con-

straints, the choice of terminal controller becomes an easy one: It can simply be assumed

that no controller is used after the prediction horizon

u(i) = 0 for i>L. (2.37)

As the plant is asymptotically stable, it will move towards the origin even without any

active control and the associated terminal cost is bounded. Further, the third requirement

is always met as there are no state constraints and without a terminal controller there is no

risk of violating the control constraints.

The linearized plant model that is derived in chapter 4, however, is only marginally stable

and further also does have state constraints. Because of the only marginally stable plant,

the system will not return to the origin if no terminal controller is used and the terminal

cost will not be bounded. Thus, a terminal controller is required. This terminal controller

is commonly designed as a Linear Quadratic Regulator (LQR) which will lead to a linear

control law κ(x(i)) = −KLQRx(i). If the cost function for the LQR controller is

∞

∑

i=L+1

x(i)TQLQRx(i) + u(i)TRLQRu(i)(2.38)

2.3. Model Predictive Control

then the control matrix KLQR can be calculated by solving the discrete algebraic Riccati

equation:

P=−ATPB(BTPB +RLQR)−1(ATPB)T+ATPA +QLQR (2.39)

for Pand setting

KLQR =BTPB +RLQR−1BTPAT. (2.40)

Designing the terminal controller as a LQR controller has the benefit that it directly provides

the terminal cost associated with the controller as

Vf(x(L)) = 1

2x(L)TPx(L). (2.41)

Of course, any other choice for a stabilizing linear control law is also possible as long as

the terminal cost defined via (2.41) is calculated by solving the associated discrete algebraic

Riccati equation. With the control law and terminal cost defined the first and second con-

dition for stability are again fulfilled. Now it only needs to be ensured that by using this

terminal controller, the system can be brought to the origin without violating either the con-

trol or state constraints. This is generally ensured by choosing a terminal constraint that is

a subset of the maximal control admissible set. The initial state of a linear system is control

admissible if using this state as the initial condition will not result in violation of either the

state or control constraints at any point in time. The maximal control admissible set is the

set of all state vectors that meet this condition. The maximal control admissible set for lin-

ear systems under linear state and control constraints can for example be calculated using

Algorithm 3.1 from Gilbert and Tan [28] which is also reproduced in appendix A.

3. Related Work

As stated in the introductory chapter, one of the key benefits MPC has to offer to the wind

turbine control problem is the ability to include feedforward control action in the closed

loop controller in an integrated framework. This chapter explores some of the previous

research that has been conducted on using feedforward control and MPC for wind tur-

bines. First, wind speed measurements and feedforward control for wind turbines are ex-

amined for applications that do not rely on MPC. Then previous studies applying MPC,

with and without a preview component, to wind turbines are introduced. Finally, one fur-

ther promising application of MPC to wind turbine control is control during special events

such as shut-down procedures and an overview of existing methods for this control prob-

lem is given.

3.1. Feedforward Control

Classical wind turbine speed and power control relies on measurements of the generator

speed only [3]. From a control systems point of view, the wind speed fluctuations act as

a disturbance which the closed-loop controller needs to reject. However, it is also possible

to include control action based on this disturbance directly. This is commonly known as

disturbance feedforward control. Feedforward control can be very effective in improving

control performance especially if the disturbance can be measured. Further improvements

are possible if the disturbance can not only be measured the instant it occurs, but also if it

is known some time before it arrives at the plant1. Usually, feedforward is implemented

as an additional block on top of the feedback control loop. However, it is also possible

to incorporate feedforward characteristics directly into the main closed loop controller for

example using Model Predictive Control.

Generally, for the use of feedforward two somewhat separate issues have to be solved: The

first is the question of how suitable disturbance (wind speed) information can be obtained?

The second is how can this information best be used to improve control performance?

1e.g, upstream measurements in process applications

3.1. Feedforward Control

All large modern wind turbines are equipped with an anemometer. It is usually located

on top of the nacelle behind the rotor. As the wind speed is not uniform over the entire

rotor plane, the one point measurement from the anemometer is usually not considered

to provide sufficient information about the wind speed driving the turbine dynamics and

as such cannot be used directly for turbine control. Therefore, the necessary wind speed

information has to be obtained from other sensors.

3.1.1. Wind Speed Measurement

In the last few years, Light Detection and Ranging (LIDAR) [114] has emerged as one of

the most commonly used measurement principles for wind speed measurements for wind

turbines. Especially the possibility to measure the wind speed at a distance of several hun-

dred meters and being able to measure not just at one point in space but to scan a surface or

even volume have been proven to be very valuable for wind energy applications. While at

first LIDAR has been mostly considered for an improved accuracy site assessment or power

curve validation (e.g., [1, 16]), there are now also several studies that specifically focus on

LIDAR measurements as potential controller inputs.

Harris et al. [34] study the potential of turbine mounted LIDAR. Three different setups

based on commercially available systems are studied: Two hub mounted system with-

out internal scanning that rely on the turbine rotation for conical scanning and a nacelle

mounted system with internal scanning. The systems studied there are capable of mea-

suring at a distance ranging from 40 m to approximately 200 m. While the short ranges

are probably not sufficient for preview control, they can still be used for yaw and shear

estimation and the longer ranges studied seem more than sufficient to be used for preview

control. The study concludes that turbine mounted LIDAR might be economically feasible

even if only used for shear regulation via individual pitch feedforward as in this study.

As part of the RAVE project, LIDAR was also tested extensively for variety of applications

including turbine mounted LIDAR for control applications [93, 92].

Mikkelsen et al. [73] describes the setup of a test facility in Denmark where a 3D turbulent

wind field is measured using several ground based LIDAR systems and already envisions

a spinner based LIDAR for control applications. Mikkelsen further reports the first test re-

sults from such a spinner mounted forward looking LIDAR [72]. Their system was capable

of measuring the oncoming wind field at a distance of 100 m in front of the rotor with a full

scan of the measurement cone taking less than one second. Next to the magnitude of the

oncoming wind field, the system presented can also measure the instantaneous yaw error

which is proposed as an input for both yaw control and individual pitch control.

3. Related Work

Simley et al. [102] further analyse the suitability of hub-mounted spinning LIDAR systems

as control inputs and quantify the minimum expected measurement errors. If no wind

evolution is assumed to occur between wind speed measurement upstream of the turbine

and the actual wind field hitting the turbine, the smallest measurement errors for the out-

board rotor sections are expected to be at a distance of between 150 m and 200 m in front

of the turbine which coincides with the approximate distance required for collective pitch

preview control.

3.1.2. Wind Speed Estimation

While the anemometer signal is not sufficiently capturing the effective wind speed, it is

possible to derive the effective wind speed from other available signals. This can be seen

as using the entire rotor as an anemometer. There are numerous studies that use some

form of effective wind speed estimation (see Østergaard [81] for an overview), but most

of these techniques rely on measuring the rotor speed or acceleration which together with

the commanded generator torque allows an estimation of the aerodynamic torque and sub-

sequently the wind speed if the aerodynamic characteristics of the rotor are known. This

type of effective wind speed estimate could also be used to add a feedforward component

to the speed control setup and has been suggested in several studies [112, 80, 111, 110].

However, since the wind speed estimate relies on the generator speed, most of the benefits

from feedforward, that is the ability to react before the effect of the disturbance is felt at the

output of the plant, is lost. It is in fact doubtful whether distinguishing between feedback

and feedforward is appropriate at all for situations such as this one where the disturbance

is estimated based on the output of the plant that is to be controlled.

Alternatively, it has also been proposed to estimate the local wind speed at each blade and

use this as an input for a feedforward component in the individual pitch controller [101].

This type of estimation usually relies on additional load measurements such as main shaft

or blade root bending moments sensors.

However, more commonly than for feedforward control, the estimated wind speed is used

for improved gain scheduling of linear controllers and improved λopt-tracking in partial

load operation (e.g., [80, 8] and others).

3.1.3. Wind Speed Prediction

As an alternative to actual wind speed measurement for preview applications, it has also

been suggested to use wind speed prediction. The estimated effective wind speed or the

wind speed measured in the rotor plane is used as a basis for predicting speed ahead over

3.1. Feedforward Control

the required interval. In general, the accuracy of such a prediction is expected to be much

lower than actual measurements. However, since no additional hardware is required any

prediction that beats the consistency assumption of the wind speed remaining fixed at its

current value that is either implicitly or explicitly made in almost all turbine controllers,

would improve the control performance. Methods that have been used include autoregres-

sive wind models ”trained” on recorded wind time series [79] or filtering of the wind data

according to its known frequency content in combination with linear [94] or polynomial

[15] extrapolation.

3.1.4. Preview Control for Individual Pitch Control

Laks et al. [61] study the use of feedforward from measurement of the tip wind speeds

to reduce blade flap loads. They find that if exact measurements of local wind speeds are

available, blade loads can be reduced significantly as long as no pitch rate constraints are

considered. As the required pitch rates are fairly high and above what most pitch systems

are capable of, they conclude that in order to realize these benefits on a realistic system, the

individual pitch controller needs to account for rate constraints and also requires knowl-

edge of local wind speeds ahead in time. Finally, in a further study [60] such an individual

pitch feedforward controller with preview and explicit bounding of the pitch rates via an

LMI formulation is considered. One main result of this study is that the required preview

times depend on the allowable pitch rates and that for typical pitch rates around 0.5 seconds

of preview time are required.

3.1.5. Preview Control for Speed and Power Regulation

Schlipf and Kühn [98, 96] study the use of preview control2for collective pitch control under

the assumption of perfect wind speed measurement. They use static system inversion in

combination with a predefined time shift as the feedforward component. The mapping of

pitch angle to wind speed in stationary conditions is inverted and used as a wind speed de-

pendent feedforward gain. The time shift that gives the best performance was determined

to be around 1 second. As such a controller would lead to very aggressive pitching for most

types of wind speed fluctuations, the incoming effective wind speed signal is low-pass fil-

tered before being passed to the described feedforward structure. Significant performance

improvements in both turbulent time series as well as in gust type events are reported.

Laks et al. [60] also propose a collective pitch preview controller in a simulation study. Their

controller is based on the inversion of the gain scheduled transfer function from wind to

2called ”predictive disturbance compensation” in this study

3. Related Work

generator speed with three different realizations of such a controller studied. The wind in-

put to the feedforward controller was an approximate disk average 90 meters ahead of the

turbine calculated from 5 measurement points. Similar to the approach by Schlipf, the in-

put to the feedforward controller had to be heavily low pass filtered. They further reported

that their approach for collective pitch feedforward has not resulted in a performance im-

provement possibly due to the low-pass effectively canceling all controller dynamics or the

wind speed measurement not being representative of the true effective wind speed. In a

follow-up study [19], several methods to improve the system inversion approach are ex-

amined with some improvement in the control performance. One of the limiting factors

stated in this study is the SISO nature of the inverse transfer function approach and the

missing possibility to trade the conflicting controller objectives such as speed regulation,

pitch actuator usage, and mechanical loads off against each other.

Wang et. al. [115] also address the design of a collective pitch feedforward controller that is

added on top of an existing feedback control algorithm. Two different algorithm are stud-

ied: The first is a dynamic system inversion approach that, similar to the earlier results by

Laks et. al. [60], does not provide a significant improvement in performance. The second

algorithm is an adaptive FIR-filter based algorithm. A recursive least squares algorithm is

used to continuously adjust the feedforward path controller based on the speed tracking

performance in the feedback path. This approach leads to significant improvement in per-

formance showing that the design of the feedforward controller does need to consider the

feedback controller as well.

Finally, in the summer of 2012, Schlipf et. al. performed the first reported field test of a LI-

DAR enabled feedforward controller [97]. They used the static system inversion approach

with a fixed time shift to design a feedforward controller for the CART2 600 kW turbine

at NREL and could show a reduction in the rotor speed deviation and tower loads at low

frequency for a small measurement period.

3.2. Wind Turbine Control

State-feedback was among the earliest methods used for the wind turbine control problem.

While pole placement has been used [41, 117], it is usually hard to specify the pole loca-

tion required for the wind turbine control objectives. The more common approach is to

formulate the problem as an optimal control problem either in LQ [30, 88, 82, 77, 107] or in

H∞form [27, 30, 13]. The key benefit of using optimal control design techniques was that

they provide ”management level tuning knobs” [31, 32] that link the controller design and

tuning directly to the performance objectives such as load reduction and speed tracking

performance.

3.2. Wind Turbine Control

Most of these controllers face the same challenge as using linear MPC: They are based on

an LTI system description of the plant dynamics and can therefore only be designed for a

single operating point. In order to use them, they have to be gain-scheduled or interpolated

between the different operating points. Another option is the use of a single sufficiently

robust controller for the whole operating region. Although it is possible to find a single

stabilizing controller, this will usually lead to sub-optimal performance in all operating

points except the nominal operating point. Instead of an LTI formulation, both Lescher et.

al. [68, 69, 67] and Østergaard et. al. [80, 83] use LPV-methods to design a robust controller.

Through the use of one system description for all operating points, the gain-scheduling is

here done implicitly. Both studies showed a significant reduction in loads compared to LQ

and PI controllers.

3.2.1. MPC for Wind Turbines

There are several studies investigating the potential application of Model Predictive Control

to wind turbines for both non-preview and preview applications.

Henriksen [36] describes the development of a linear MPC for a wind turbine. One linear

MPC was developed for each operating region and tested for both floating and land based

turbines. Although the nonlinearity of the wind turbine has not been taken into account

when the controller was designed, the study shows a single linear non-robust MPC con-

troller to be stable for the entire full load operating range. The study also demonstrates

how off-set free tracking of the desired rotor speed can be achieved in MPC even with the

wind acting as a persistent disturbance through disturbance estimation and origin shift-

ing. This MPC application is later [37] expanded to include soft state constraints and a

re-linearization scheme. Henriksen also states that the implementation of a hard constraint

on the rotor speed state is difficult due to potential infeasiblity of the control problem, an

issue that is addressed at length in this thesis.

Botasso et al. [7] use an adaptive controller that includes some elements from nonlinear

MPC but does not include any constraints in the problem formulation. Using the results

obtained with this controller, they conclude that for nominal operating conditions it is hard

to beat a well-tuned classical controller.

Kumar and Stol [54] present a scheduled Model Predictive Controller for full load opera-

tion. The controller consists of three separate linear MPCs, each similar to the controller

presented by Henriksen and based on a turbine model linearized at a different wind speed,

that are run in parallel. Using the estimate of the effective wind speed obtained from an

EKF, the output from each of these controllers is weighted according to the proximity to the

respective linearization wind speeds. The final controller output is then the sum of the three

3. Related Work

weighted controller outputs. The performance of the scheduled MPC is compared against a

single linear MPC and a scheduled PI controller. The results show a slightly higher perfor-

mance of the scheduled MPC over the two other variants. The authors further argue that an

additional benefit of the scheduled MPC is the ability to tune each controller individually

so that the response at different wind speeds can be tailored to specific requirements.

Friis et al. [25] propose to use linear MPC to design a repetitive controller for individual

pitch control. Unlike the previous MPC applications, the wind speed disturbance states,

which are estimated using a Kalman Filter, are included directly in the MPC formulation

and thereby adding a feedforward component to the overall control scheme. This study

further includes both output and state constraints, although no details are given with re-

spect to how the state constraints are handled in the presence of unknown disturbances or

estimation errors. In the proposed scheme, stability is achieved using a sufficiently long

prediction horizon. The results show a significant reduction of vibration levels for several

components.

Körber and King [50] first show that that preview information can be easily integrated into a

linear MPC for the full load operating region. Under the assumption of perfect wind speed

measurements five seconds ahead of the turbine, large reductions in both fatigue and ex-

treme loads were observed. If no preview information was available, a small performance

improvement of the MPC over a classically-designed baseline controller remained. In a

follow-up study [51], the controller was expanded to also cover partial load operation and

several schemes to handle the wind turbine nonlinearity were compared with two main

results: Firstly, preview MPC can increase the tracking of the optimum tip speed ratio in

partial load drastically but, for the examined turbine, this does not lead to any significant

increase in power production. Secondly, using a scheduled or continuously linearized MPC

did not result in a significant loss of performance compared to a fully nonlinear MPC.

Spencer et al. [104] use a linear MPC to design a combined speed and power controller

and individual pitch controller that assumes perfect measurements of the wind field up

to five seconds before it arrives at the turbine to be available. It is found that large load

reductions are possible and also that the benefit of preview-MPC is mainly seen in the full

load operating region.

Laks et al. [59] as well as as Schlipf et al. [99] use MPC to design a preview controller

where instead of assuming perfect measurement of the wind speed, a more realistic model

for the preview measurements is assumed. Both studies conclude that even in the presence

of imperfect wind speed measurement, large reductions in both extreme and fatigue loads

are possible.

Finally, many of the concepts introduced and evaluated in this thesis, such as the robust

3.2. Wind Turbine Control

handling of the over-speed constraint and the linearization scheme, were introduced and

published by Körber and King [52] in 2013.

To the knowledge of the author no one has tested a Model Predictive Controller on an actual

turbine so far.

3.2.2. Extreme Load Control

Given their importance for the overall wind turbine design, very little academic research

has been performed on the design of wind turbine controllers that reduce extreme loads.

Of the studies that do consider the non-NPP load cases, most deal with handling gust

cases (e.g., load cases 1.3, 1.6, 1.93). However, gust cases are mostly considered when

designing controllers that make use of additional sensing means, such as the previously

discussed LIDAR-based preview controllers or increased actuation capability through the

use of smart rotors [55].

Kanev and van Engelen [47] present a method to improve controller performance during

extreme gust without additional sensing or actuation for a regular pitch controlled variable

speed wind turbine. Their concept relies on an EKF-based wind condition observer that

is used to detect extreme events. Once such an event is detected, a special extreme event

controller is activated that is designed to prevent the turbine from causing an overspeed

fault. Control during these events essentially consists of pitching out the blades at the

maximum rate and holding the generator torque constant. One reported down-side of this

open-loop scheme is that it can cause unnecessarily high tower loads because the pitch

is controlled at the maximum possible rate and not at the rate necessary to prevent the

overspeed. Furthermore, because the controller now has two discrete operating modes,

toggling between modes can occur and a hysteresis is required. Both of these down-sides

are addressed when using the MPC developed in this thesis for the same control problem

of overspeed avoidance during gust events.

While the idea of maintaining closed-loop control of the rotor speed during emergency

stops is almost as old as the wind turbine industry [53], no academic studies investigating

the design of such a shut-down controller are known to the author.

3see table 2.1 in section 2.2.2

4. Turbine Model

The Model Predictive Controller relies on the full state and disturbance information being

available at every time step. While all modern turbines are equipped with an anemometer,

which is usually mounted on the nacelle, the signal from such an anemometer is usually

not considered to be suitable for turbine control for several reasons: Firstly, the anemome-

ter only measures at one point of the rotor plane. The wind speed can vary significantly

across the area swept by the rotor and measurements taken at a single point in this rotor

plane are not necessarily representative of the rotor area effective wind speed. Secondly,

the anemometer is usually placed behind the blades so that all disturbance created by the

passing blades will be visible in the signal. Finally, by extracting energy from the wind, the

turbine causes the wind to slow down significantly. The controller however would require

the free stream wind speed which can only be measured at a significant distance ahead

of the turbine. While numerous methods for state and disturbance estimation exist, an Ex-

tended Kalman Filter is chosen here as it is the most straight-forward and widely employed

method used for wind speed estimation on wind turbines (see section 3.1.2).

Thus, as shown in figure 4.1, the full controller consists of the EKF based estimator and the

Model Predictive Controller.

Figure 4.1.: High-Level Controller Structure

Both the EKF and the MPC are model based techniques: They rely on an internal model of

the plant for which the states are to be estimated and which is to be controlled. In the first

section of this chapter, the dynamic model and its parameters, used by both the estimator

and controller, are discussed.

As outlined in section 2.2, normally complex models with highly coupled structural and

aerodynamic components, as implemented in specialized software like BLADED, FLEX5,

or FAST, are used to evaluate turbine dynamics. These models are too complex to be im-

plemented directly in an MPC scheme. Instead, most control designs for both MPC and

non-MPC approaches use models that are based on the models used in the system simula-

tions but are significantly reduced in complexity.

There are two main distinct approaches to generate models that are suitable for the control

design which are frequently used:

•Some of the commonly used simulation tools (especially BLADED and FAST) provide

routines that allow an automatic generation of linear state space models for a given

operating point. The resulting linear models will generally still be fairly large and

may consist of several hundred states, yet can be reduced using model order reduc-

tion techniques. The main advantages of this approach are the ease at which models

can be generated and the fact that all dynamic interactions are included. In the MPC

for wind turbines context, this approach has for example been used by Kumar and

Stol [54] and Spencer [104].

•The second approach is to use a first principle model of the turbine and identify its

parameters from the aero-elastic code using system identification routines. By choos-

ing to ignore certain interactions that are deemed not to be relevant for the control

problem at hand, models with just a few states are possible using this approach. An-

other advantage is that using first principle models makes it is easier to include the

nonlinearities of the model explicitly and thus allows more flexibility in handling the

nonlinearities in the control design. First principle modeling is the technique that has

been used by Henriksen [36, 37] or Friis et al. [25].

In this study, the second approach is chosen because of the smaller size of the resulting

model and explicit inclusion of nonlinearities. How to design scheduled MPC based on

linear models at different operating points is outlined in [54]. The remainder of this section

describes and analyses the simplified first principle model that is used in the control design.

One benefit of using a first principle model is that the model can be specifically tailored

to the needs of the control problem to be solved. The model presented in this chapter

has been chosen to allow a design of an MPC for the speed, power, and tower vibration

4. Turbine Model

control problem. It would however require some extensions if other control problems such

as individual pitch control or drive train damping are to be considered.

This simplified model consists of linear structural dynamics and actuator models coupled

with a nonlinear aerodynamic model as illustrated in figure 4.2.

Figure 4.2.: Simplified turbine model used by the controller

The following sections will describe each of the sub models and explain how its parameters

are identified from the full aero-elastic simulation tool.

4.1. Aerodynamics

The output of the aerodynamic model are the aerodynamic torque and thrust generated by

the rotor. These are mainly dependent on the current local inflow conditions at each section

of the blades. Instead of treating each blade section individually, as would be done us-

ing Blade Element Momentum (BEM) theory, and calculating the local aerodynamic forces,

only the resulting torque and thrust generated by the entire rotor is considered. The aero-

dynamic torque MAand thrust FAare defined in terms of the torque and thrust coefficients

CMand CT[26]

MA=1

2ρπR3v2

eCM(λ,θ)

FA=1

2ρπR2v2

eCT(λ,θ). (4.1)

where veis the effective wind speed, ρthe air density and Rthe rotor radius. The effective

wind speed veis a modification of the free stream wind speed vthat was used in section

2.1.2 to define the aerodynamic characteristics of the wind turbine that also includes the

effect of the structural turbine motion on the aerodynamics. It will be properly defined in

section 4.3. The coefficients themselves are a function of the current collective pitch angle θ

4.2. Structural Dynamics

and the tip speed ratio λ:

λ=ωR

. (4.2)

4.2. Structural Dynamics

Only three degrees of freedom are considered to model the structural dynamics of the tur-

bine. The first describes the rotation of the rotor, consisting of the three blades and all drive

train components about its principal axis of rotation. The other two model the fore-aft dy-

namics of the tower and rotor, respectively.

4.2.1. Rotor

The rotor model lumps all rotating components into a single degree of freedom. On a

geared turbine, the generator and so-called high speed shaft runs at a higher rotational

speed ωGthan the main shaft, hub, and attached rotor blades. Here, it is assumed that both

sides are rigidly coupled through the gearbox and any potential flexibility is ignored.

ωG=nGω(4.3)

By ignoring the drive train flexibility, the generator speed ωGdoes not have to be included

as a separate degree of freedom since it can always be expressed in terms of the rotor speed

ω. Although on most turbine the generator speed and not the rotor speed is measured and

used as a controller input, in the following all equations are expressed in terms of the rotor

speed and it is understood that rotor and generator speed can be used interchangeably.

It should be noted that some modern wind turbines do not use a gearbox. The drive train

model introduced here however remains applicable even for these so-called direct-drive

turbines since a turbine without a gearbox is essentially a turbine with a gearbox ratio of

nG=1.

It should further be noted that some studies on wind turbine control design (e.g., [8]) do

include flexibility in the drive train and model the rotor as a two-mass system. However,

this is necessary only if the drive train damping problem is to be considered. As the drive

train damping functionality is not to be included in the MPC, modeling the drive train

flexibility is not required and the additional degree of freedom is omitted in order to keep

the model as small as possible.

4. Turbine Model

The inertias about the principal axis of rotation for the three blades JB, all rotating compo-

nents on the slow speed shaft JSS and high speed shaft, including the generator, JHS are

combined into a single inertia J. Since this effective inertia is associated with the rotational

speed on the slow speed side of the gearbox, the gearbox ratio needs to be taken into ac-

count when including the high speed side inertia:

J=3JB+JSS +n2

GJHS. (4.4)

The generator torque MGand aerodynamic torque MAare the moments acting on this ro-

tational drive train model:

J˙

ω=MA(θ,ve,ω)−nGMG. (4.5)

Equation (4.5) represents the simplest model that can be used to design a speed controller

for a wind turbine with MGand θbeing the controlled variables, vethe disturbance, and

ωand P=ωMGthe plant outputs that are to be controlled. It has, however, been shown

that especially for large turbine structural dynamics of the tower and blades do have a

significant impact on even the pure speed control problem [65]. Hence, the pure drive train

model is augmented with tower and blade fore-aft models. The impact of the blade and

tower dynamics on the rotor speed dynamics are further analyzed later in this section.

4.2.2. Tower

The aerodynamic thrust causes movement of the wind turbine head mainly in the direction

of the wind speed, the so-called fore-aft direction. This fore-aft motion of the tower is

modeled as a single mass oscillator as shown in figure 4.3. Using this single degree of

freedom model for the tower motion essentially treats the entire tower as a linear spring

and ignores the higher modes of the flexible tower. However, due to the low energy content,

the impact of the higher modes is comparatively small and can generally be ignored in the

control design.

Instead of using the mass m, stiffness k, and damping properties dthe resulting equation

of motion is directly given in terms of the natural frequency ωT,0, damping ratio ζT, and

forcing gain Kf a

xt+2ζTωT,0 ˙

xt+ω2

T,0xt=Kf aω2

T,0FA. (4.6)

Under certain circumstances, the machine head of a wind turbine may also sway sideways.

This so-called side-side tower oscillation is significantly smaller then the fore-aft movement

4.2. Structural Dynamics

(

)

eA vF

Figure 4.3.: Simplified tower model

and, as it is perpendicular to the wind direction, has very little interaction with the turbine

aerodynamics. It is therefore ignored here.

4.2.3. Blade Fore-Aft Motion

The bending motion of a flexible wind turbine blade is usually decomposed into two prin-

cipal axis of deflection. If the blades have not been pitched, ”edgewise” deflection takes

place in the plane of rotation while ”flapwise” motion describes the motion out of the plane

of rotation.

Depending on the control problem at hand, different blade models need to be included

in the overall turbine model. While there are some phenomena like controls induced blade

flutter that are a result of the dynamic coupling between structural dynamics, aerodynamics

and controls that require detailed nonlinear models of the blades like the one developed by

Kallesoe [46], most control designs only require rudimentary blade models to be included

in the turbine model.

For the design of a drive train damper, the edgewise motion of the three blades should

be included as it is heavily coupled with the drive train dynamics, while the design of an

individual pitch controller requires a model of the individual motion of each blade in the

flapwise direction. Since the tower fore-aft motion is coupled with the flapwise motion of

the blades, the flapwise motion should also be included in a structural model for speed

control and tower damping [65].

Geyler and Caselitz [27] augment their single degree-of-freedom tower model with a sin-

gle degree-of-freedom model for the flapwise deflection of each of the blades. They then

transform the three individual blade tip deflections in the flapwise direction xb1−xb3into

4. Turbine Model

a collective tip deflection xb, which describes the average tip flapwise deflection, as well as

the sine xbs and cosine xbc components of the deviations from the collective deflection.







xbs

xbc





=





1 sin(ψ)cos(ψ)

1 sin(ψ+2

3π)cos(ψ+2

3π)

1 sin(ψ+4

3π)cos(ψ+4

3π)







−1





xb1

xb2

xb3





(4.7)

where ψis the rotor azimuth.

Here, since individual pitch control is not treated, only the collective flap motion xbis con-

sidered and the asymmetric components xbs and xbc are ignored in order to reduce model

complexity. The equation of motion for this collective flap deflection is derived using a sin-

gle blade, but it is implied that all three blades behave identically and are lumped into this

one degree-of-freedom.

Figure 4.4 shows the simple model. The blade is modeled as a single stiff body with rota-

tional inertia Jand mass mbconnected to the tower via the reactive force R, elasticity k, and

damping d. The position of the tower itself is xt. The motion can be described via the tip

deflection of the blade xbwhich is measured in the reference frame with its origin at the

tower top and is thus not an inertial coordinate system. A deflection angle

ϕ=arctan xb

l(4.8)

is introduced and it is immediately assumed that the deflections angles are small so that

the assumption

ϕ≈xb

l(4.9)

can be made. It is important to mention that this angle has no physical meaning since on

the real turbine the blade is not stiff and straight but flexible and curved and the bending

angle at the hub will always be zero. For the same reason, the scaling factor a1has to be

introduced which describes the translatory motion of the center of gravity with respect to

the tip deflection

xcog =a1xb. (4.10)

In order to derive the equations of motion D’Alembert’s principle of inertial forces is em-

ployed where the inertial forces caused by the translatory accelerations ¨

xtand ¨

xband the

angular acceleration ¨

ϕare included in the free body diagram as if they were external forces.

The system is then treated as a static system with ∑F=0 and ∑M=0.

4.2. Structural Dynamics





tb xm 

bb xam 

wind direction

Figure 4.4.: Blade Model

The equation of motion for this system is now derived by taking the balance of moments

about the blade root:

−J¨

ϕ−kϕ−d˙

ϕ−mba1lg¨

xb+laFA−mblg¨

xt=0 (4.11)

which can be written as

xb+2ω0,bζb˙

xb+ω2

0,bxb=kbω2

0,b(FA−mb

xt)(4.12)

using

ω2

0,b=k

J+mblga1l

ζb=dω0,b

kb=lal

k. (4.13)

The force acting on the tower Rbecomes:

R=FA−mba1¨

xb−mb¨

|{z}

already included in tower model

. (4.14)

where the inertial force of the blades mb¨

xtwith respect to the the tower motion is already

included in the mass of the tower system (4.6). Equation (4.12) can be combined with the

tower dynamics (4.6) to form the two mass oscillator shown in figure 4.5. The combined

4. Turbine Model

Figure 4.5.: Mechanical Turbine Model

equations in motion can be written in matrix notation:

"1Kf aω2

T,0mba1

lakbω2

0,b1#

| {z }

xb!+"2ωT,0ζT0

0 2ω0,bζb#

| {z }

xb!

+"ω2

T,0 0

0ω2

0,b#

| {z }

xb!="Kf aω2

T,0

kbω2

0,b#

| {z }

FA(4.15)

M ¨

xb!+D ˙

xb!+K xt

xb!=FFA(4.16)

4.3. Effective Wind Speed

The wind speed ”seen” by the rotor will change with tower and blade motion. If the turbine

moves towards the wind, the relative wind speed at the rotor is increased by the tower top

velocity ˙

xt. Similarly, blade deflection in the flapwise direction will also effect the relative

wind speed at the blades. Therefore, the effective wind speed veis defined as the rotor

plane average of the free stream wind speed vcorrected for tower and blade motion

ve=v−˙

xt−a˙

xb. (4.17)

If the blade is deflected with ˙

xbin the flapwise direction, only the tip will see a change

in relative wind velocity of ˙

xb; all other parts of the blade will move at a lower velocity.

4.4. Actuator Models

Therefore the scaling constant ais introduced to express the total change in effective wind

speed as a fraction of the blade tip out-of-plane velocity. If the blade were actually stiff,

awould be equal to the location of the aerodynamic center relative to the blade length

a=la

4.4. Actuator Models

The pitch system is modeled as a linear second order system relating the commanded pitch

angle θCto the actual pitch angle θ:

θ+2ωp,0ζp˙

θ+ω2

p,0θ=ω2

p,0θC. (4.18)

The electrical dynamics are considered to be significantly faster than structural dynamics,

so it is assumed that the actual generator torque MGdirectly follows the torque set-point

MG,C:

MG=MG,C. (4.19)

For the MPC, the constraints on the commanded actuator rates play an important role. In

order to allow treating these rate constraints as simple bounds on the controlled variables,

equations (4.18) and (4.19) are differentiated with respect to time

...

θ+2ωp,0ζp¨

θ+ω2

p,0 ˙

θ=ω2

p,0 ˙

θC

MG=˙

MG,C(4.20)

and the commanded pitch ˙

θCand torque ˙

MG,Crates are seen as the controller inputs to the

model.

4.5. State Space Formulation and Linearization

The turbine model, consisting of equations (4.20), (4.17), (4.16), (4.5), and (4.1), is now com-

bined into a single state space model by defining the states shown in table 4.1. Defining the

state vector

x=ω θ ˙

θ¨

θxtxb˙

xt˙

xbMGT, (4.21)

4. Turbine Model

x1Rotor Speed ω

x2Collective Pitch Position θ

x3Collective Pitch Rate ˙

x4Collective Pitch Acceleration ¨

x5Long. Tower Position xt

x6Collective Flap Position xb

x7Long. Tower Velocity ˙

x8Collective Flap Velocity ˙

x9Generator Torque MG

Table 4.1.: Summary of turbine model states

a vector of controlled inputs

u=˙

MG,C˙

θCT, (4.22)

and vector of disturbance inputs

d=v(4.23)

the turbine model can be represented in the standard form dx

dt =f(x,u,d)with the follow-

ing state evolution equations

x1=1

J(MA(x1,x2,x7,x8,v)−nGx9)

x2=x3

x3=x4

x4=−ω2

p,0x3−2ωp,0ζpx4+ω2

p,0 ˙

θC

x5=x7

x6=x8

x8!=M−1 FFA(x1,x2,x7,x8,v)−K x5

x6!−D x7

x8!!

x9=˙

MG,C. (4.24)

y1Rotor Speed ω

y2Turbine Power P

y3Collective Pitch Position θ

y4Generator Torque MG

y5Long. Tower Velocity ˙

y6Collective Flap Velocity ˙

y7Long. Tower Acceleration ¨

Table 4.2.: Summary of turbine model outputs

4.5. State Space Formulation and Linearization

Table 4.2 lists all quantities that are seen as turbine outputs. These are combined in the

system output vector yand the output equation is also given in its standard form y=

h(x,u,d)when ˙

x7is replaced by (4.24):

y1=x1

y2=x1nGx9

y3=x2

y4=x9

y5=x7

y6=x8

y7=˙

x7. (4.25)

Only a subset of these outputs is measurable. The distinction between measurable and

unmeasurable outputs will need to be taken into account when designing the state and

disturbance estimator (see section 5.1) where only measurable outputs are considered and

the designing of the actual controller (see section 5.2) where a full state controller can have

its performance specified based on both measurable and unmeasurable outputs. Here, only

the quantities which are commonly measured on modern wind turbines are assumed to

be measurable: rotor speed, power output, pitch angle, generator torque, and tower top

acceleration. This defines the vector of measurable outputs z:

z=y1y2y3y4y7T(4.26)

The design for both the controller and the state and disturbance estimator relies on lin-

earizing the turbine model at various operating points. In the described model the entire

nonlinearity is contained in the aerodynamics. The structural and actuator models are fully

linear. Therefore, in order to linearize the system dynamics only the aerodynamics need to

be linearized which can easily be performed analytically.

As the torque and thrust coefficients CMand CTin (4.1) depend on the pitch angle and

tip speed ratio, which in turn depends on the effective wind speed and rotor speed, the

aerodynamic torque and thrust depend nonlinearly on effective wind speed ve, rotor speed

4. Turbine Model

ω, and pitch angle θ. The partial derivatives with respect to these variables are given by:

MA=1

2ρπR3v2

eCM(λ,θ)

∂MA

∂ω =1

2ρπR4ve

∂CM

∂λ

∂MA

∂θ =1

2ρπR3v2

∂CM

∂θ

∂MA

∂ve

=ρπR3veCM−1

2ρπR4∂CM

∂λ ω(4.27)

and

FA=1

2ρπR2v2

eCT(λ,θ)

∂FA

∂ω =1

2ρπR3ve

∂CT

∂λ

∂FA

∂θ =1

2ρπR2v2

∂CT

∂θ

∂FA

∂ve

=ρπR2veCM−1

2ρπR3∂CT

∂λ ω. (4.28)

Using these partial derivatives, the entire system can be represented in the standard form

of a linear state space system:

x(t) = Ax(t) + Bu(t) + Ed(t)

y(t) = Cx(t) + Du(t) + Fd(t). (4.29)

The matrices of this linear state-space model are given by the partial derivatives of the sys-

tem and output equations with respect to the state, output and disturbance vectors. As the

the system is linear with respect to the control inputs u, the linearized system only depends

4.5. State Space Formulation and Linearization

on the chosen state vector x0and wind speed v0at which the system is linearized.

A=∂f

∂xx0,v0







JMω1

JMθ0 0 0 0 −1

JMv−1

JaMv−nG

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 −ω2

p,0 −2ωp,0ζp0 0 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

ft1Fωft1Fθ0 0 −kt11 −kt12 −dt11 −ft1Fv−dt12 −ft1Fv0

ft2Fωft2Fθ0 0 −kt21 −kt22 −dt21 −a ft2Fv−dt22 −a ft2Fv0

0 0 0 0 0 0 0 0 0







B=∂f

∂ux0,v0

="0 0 0 0 0 0 0 0 1

0 0 0 ω2

p,0 00000#T

E=∂f

∂dx0,v0

=h1

JMv00000ft1Fvft2Fv0iT

C=∂h

∂xx0,v0







1 0 0 0 0 0 0 0 0

Mg,0nG0 0 0 0 0 0 0 ωr,0nG

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

ft1Fωft1Fθ0 0 −kt11 −kt12 −dt11 −ft1Fv−dt12 −ft1Fv0







D=∂h

∂ux0,v0

="000000000

000000000#T

F=∂h

∂dx0,v0

=h000000ft1Fv0 0iT(4.30)

where ktij,dtij, and fti are the entries of the transformed matrices

Kt=M−1K Dt=M−1D Ft=M−1F. (4.31)

4. Turbine Model

and the terms

Mv=∂MA

∂ve

,Mθ=∂MA

∂θ ,Mω=∂MA

∂ω and

Fv=∂FA

∂ve

,Fθ=∂FA

∂θ ,Fω=∂FA

∂ω (4.32)

denote the partial derivatives of the aerodynamic torque MAand thrust FAas defined in

(4.27) and (4.28). These terms are the only terms that will change depending on at which

linearization point, defined by x0and v0, the system is linearized. All entries in the system

matrices that are a function of the linearization point are marked in red, while the black

terms are invariant with respect to the linearization point. If further the linearization points

are restricted to lie on the stationary operating curve of the turbine, the linear system matri-

ces only depend on the free stream wind speed as in stationary conditions all state values

are functions of this wind speed.

If, instead of the full output vector, only the measurable outputs are to be considered, the

linearized output equations becomes

z(t) = Cmx(t) + Fmd(t)(4.33)

with Cmand Fmdetermined by appropriately reducing Cand F.

4.5.1. Discretization

The model that was derived and linearized in this section is in continuous time. For the

purpose of implementing the MPC controller, it is preferable to use a discretized version of

the model equations. Therefore, the linearized model equations are discretized using the

zero order hold method [17]:

"A[B E]

0 0 #disc

=exp ts·"A[B E]

0 0 #cont!

Cdisc =Ccont

Fdisc =Fcont

where tsis the sample time. For ease of notation, the distinction between continuous and

discrete formulation is omitted in the following, where it is clear from context which matri-

ces are used.

4.6. Parameter Identification

Overall, the model described in this section has 17 parameters. Besides the air density,

which is an environmental parameter, they all describe the aerodynamic or mechanical

properties of the wind turbine. Some parameters, like the rotor radius or inertia, are known

turbine characteristics, while some others are not readily available as the aero-elastic sim-

ulation tool models that aspect differently. These parameters are determined by running

special system identification simulations and fitting the model parameters appropriately.

Table 4.3 lists all model parameters, its values, and the source of these values.

Parameter Identification Value

JOverall Inertia of Rotor 4.05×107kgm2

RTurbine Definition 63m

nGTurbine Definition 97

CM(λ,θ)from WT_PERF Fig. 4.6

CT(λ,θ)from WT_PERF Fig. 4.6

ωT,0 Optimal fit of tower top acceleration to wind step input 0.324 Hz

ζTOptimal fit of tower top acceleration to wind step input 0.0133

Kf a Optimal fit of tower top acceleration to wind step input 1.72×10−7

ωp,0 Parameter of actuator model 0.955 Hz

ζpParameter of actuator model 0.7

ω0,beigenfrequency of first flap mode 0.7 Hz

ζbstructural damping of first flap mode 0.0047

mbthree times mass of individual blade 17740 kg

kbfitted to static deflection of blade tip with respect to change in thrust 2.2×10−6

laoptimal fit of response to wind speed step 0.558

a1optimal fit of response to wind speed step 0.470

aoptimal fit of response to wind speed step 0.399

Table 4.3.: Overview of turbine model parameters

4.6.1. Aerodynamic Coefficients

The aerodynamic coefficients CMand CTare generated using the tool WT_PERF [9]. The

data is generated on a grid of pitch angles and tip speed ratios with a step size of 0.1deg

and 0.25, respectively. The matrices of partial derivatives with respect to λand θare created

by differentiating CMand CTnumerically. Figure 4.6 shows the resulting aerodynamic coef-

ficients. Here, the coefficients are shown only over a relatively small range near the optimal

operating point of the rotor. For simulations, however, the coefficients are determined over

a range that covers all potential operating points.

4. Turbine Model

θ [deg]

0.072247

0.062814

0.05338

0.03923

0.025079

0.010929

−0.0079388

−2 0 2 4 6 8 10

(a) Torque Coefficient

θ [deg]

0.65163

0.98705

1.3225

1.6579

1.9933

2.3288

2.6642

2.9996

−2 0 2 4 6 8 10

(b) Thrust Coefficient

Figure 4.6.: Torque and thrust coefficients CMand CTas functions of pitch angle θand tip speed

ratio λ

4.6.2. Structural Model Parameters

The structural model of the blade and tower in the simplified turbine model has only two

degrees of freedom. As such, it is much simpler than the model implemented in FAST

and most of its parameters cannot be derived directly from the FAST parameters. Instead,

these parameters are determined using closed-loop system identification: The system, in

closed-loop with a PI pitch controller, is subjected to a step change in wind speed and the

responses of the generator speed and tower acceleration are observed. The simulation is

performed in parallel with both FAST and the simplified model, and the goal of the system

identification is to minimize the difference between the responses of the two models. See

figure 4.7 for a schematic of this setup. During these simulations, the generator torque is

held constant at the rated torque value and the wind speed is chosen to change from 13m/s

to 14m/s. A wind speed slightly above rated speed is chosen for the identification, as this

operating range is the most critical for the control design and it is desirable to achieve the

best accuracy for the simplified model at these wind speeds. In section 4.7, the validity of

the model at different wind speeds is further examined.

The actual parameter identification is performed in two steps: First, only those parameters

related to the tower are identified and the blades are assumed to be stiff. In a second step,

the blade parameters are identified with the tower parameters fixed from the first identifi-

cation step.

In the first step, only the tower parameter ωT,0,ζT, and Kf a are determined. The blades are

assumed to be rigid by deactivating the respective degrees-of-freedom in FAST and by us-

ing mb=0 in the simplified model. The tower parameters are determined so that the root

4.6. Parameter Identification

PI-

Controller

−

0,G

Simplified

Model

PI-

Controller

−

0,G

AST

Simplified

Model

−

res

−



rest



Figure 4.7.: Block diagram of setup for identification of the structural model parameters of the

simplified model

mean square value of the residual of the tower top acceleration ¨

xt,res is minimized (least

squares fit) where the numerical optimization is performed numerically using the Nelder-

Mead Simplex algorithm [56]. See figure 4.8 for a comparison of the FAST response and the

response of the fitted tower model. The identified tower eigenfrequency of ωT,0 =0.323Hz

matches the frequency of 0.324 Hz that is reported for the FAST model in [43] almost ex-

actly.

Of the parameters related to the blade dynamics, ω0,b,ζb,mb, and kbare set directly based

on the blade properties reported in [43]. The remaining blade model parameters are iden-

tified in a similar fashion as the tower models. While for the tower model identification no

blade dynamics were assumed, the blade model identification already assumes the tower

model identified in the previous step to be fixed for the simplified model and all degrees-

of-freedom to be active in FAST. In order to provide sufficient excitation to the blades, the

system is simulated with more aggressive controller gains for the PI-controller so that the

blade motion is clearly visible in the generator speed signal. Unlike the tower top acceler-

ation, the blade tip deflection is generally not measured directly. While an accurate tower

model is required since one goal of the controller design is to reduce tower vibrations, the

blade model is included in the overall model because of the impact of blade motion on the

aerodynamics. Further, due to the simplifications described in section 4.2, it is difficult to

compare the states of the blade model directly to states of the full order, nonlinear model

in FAST. Therefore, the parameters of the blade model are identified not by fitting the re-

sponse of the tip deflection, but by fitting the generator speed response and the objective

4. Turbine Model

0 20 40 60 80

12.5

13.5

14.5

t[s]

v[m/s]

(a) Wind Speed

0 20 40 60 80

−1

−0.5

0.5

1.5

2.5

t[s]

θ[rad]

(b) Pitch Angle

0 20 40 60 80

−0.03

−0.02

−0.01

0.01

0.02

0.03

t[s]

ω−ω0[rad/s]

0 20 40 60 80

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

¨xt

t[s]

(d) Tower Top Acceleration

Figure 4.8.: Response to step changes in wind speed from simulation with FAST (blue) com-

pared against the fitted response (green) of the simplified model for the first step of the identi-

fication of the structural model

for the numerical optimization problem becomes:

min

la,a1,a

(RMS(¨

xt,res)·RMS(ωres)). (4.34)

Figure 4.9 shows the resulting fit.

4.7. Model Validation and Analysis

This section examines the simplified turbine model that is developed in the previous sec-

tion. The main focus is to show that the simplified model accurately captures the relevant

4.7. Model Validation and Analysis

0 20 40 60 80

−0.03

−0.02

−0.01

0.01

0.02

0.03

t[s]

ω−ω0[rad/s]

(a) Rotor Speed

0 20 40 60 80

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

¨xt

t[s]

(b) Tower Top Acceleration

Figure 4.9.: Response to step changes in wind speed from simulation with FAST (blue) com-

pared against the fitted response (green) of the simplified model for the second step of the

identification of the structural model

dynamics of the full, nonlinear model and to highlight key characteristics of this model

that are relevant for the subsequent control design. The model is examined in both time

and frequency domain.

In order to motivate the choice of turbine model, three different stages of the simplified

model are considered. The first model is the most basic model that could be used to design

a speed controller for a wind turbine. It consists of just the aerodynamic model (4.1) and

the simple 1-DoF drive train model (4.5). In the second step, this model is augmented with

the flexible tower model (4.6) while the model of the blade flap-wise motion is only added

in the third step.

4.7.1. Open-Loop Comparison

At first, the simplified model is compared against the full, nonlinear model in pure open-

loop. The system is subjected to both step and pulse inputs in wind speed and pitch angle

and the responses of the different models are shown in figure 4.10.

It can be seen that for the steps the response essentially follows a first order behavior and

that the pure drive train model captures the dynamics well. There is very little improve-

ment of the fit between full and simplified model by adding the tower and blade models to

the drive train model. For the responses to the pulse inputs, the overall dynamics are still

well captured by the pure drive train model, but there are significant higher order dynamics

on top of the first order behavior.

4. Turbine Model

0 10 20 30 40 50

−0,015

−0,01

−0,005

0,005

t[s]

ω−ω0[rad/s]

(a) pitch pulse

0 10 20 30 40 50

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0.02

t[s]

ω−ω0[rad/s]

(b) pitch step

0 10 20 30 40 50

−0.005

0.005

0.01

0.015

0.02

0.025

t[s]

ω−ω0[rad/s]

0 10 20 30 40 50

−0.05

0.05

0.1

0.15

0.2

0.25

0.3

t[s]

ω−ω0[rad/s]

FAST

Simpl. Model, Twr+Bld stiff

Simpl. Model, Twr flexible, Bld stiff

Simpl. Model, Twr+Bld flexible

(d) wind step

Figure 4.10.: Comparison of the full non-linear model from FAST with the simple linear model

for unit step inputs and unit pulse inputs with a 1sduration.

4.7.2. Closed-Loop Comparison

The open-loop comparison shows that the match between simplified model and the full

model depends on the type of excitation the system is subjected to. In order to compare

the behavior for situations that are more representative for the actual turbine operation, the

systems are compared in a closed-loop configuration. The system is simulated in full load

with the generator torque command held constant and the pitch command controlled by a

PI controller. The gains of the PI pitch controller have been chosen so that for the simple

drive train model the resulting closed-loop transfer function has a unity damping. Figure

4.11 shows the response of the generator speed to a step in wind speed for two levels of

controller aggressiveness.

4.7. Model Validation and Analysis

0 10 20 30 40 50

−0.005

0.005

0.01

0.015

0.02

0.025

0.03

t[s]

ω−ω0[rad/s]

FAST

Simpl. Model, Twr+Bld stiff

Simpl. Model, Twr flexible, Bld stiff

Simpl. Model, Twr+Bld flexible

(a) slow controller

0 10 20 30 40 50

−0.005

0.005

0.01

0.015

0.02

0.025

t[s]

ω−ω0[rad/s]

(b) aggressive controller

Figure 4.11.: PI control closed-loop comparison of the full non-linear model from FAST with

the simple linear model for a wind step from 13 m/s to 14 m/s at different levels of controller

actuation

Again, at slow pitch action the pure drive train model already captures the dynamics quite

well and the larger models do not provide a large improvement. At higher frequencies,

however, it can be seen that the model that includes the flexible rotor captures the dynam-

ics significantly better than the simpler models. As a result, it can be concluded that it is

possible to use the simple drive train model only to design a wind turbine controller. How-

ever, the range of controller strategies for which such a model provides an adequate match

is significantly smaller than if the tower and blade interactions are included. Especially for

fast controller actuation, i.e., high closed-loop eigenfrequecies, the interaction of the pitch

controller with the blade and tower motion becomes more significant. The more aggressive

a controller is designed to regulate the turbine speed, the higher the interaction with the

tower and blade motion will be. Therefore, ignoring the tower and blade dynamics in the

controller design confines the possible closed loop behavior to a limited frequency range

and reduces the potential control performance.

In order to assess whether the identified model also represents the model implemented in

FAST accurately at other operating points, the goodness of fit value χis introduced. It is

defined as the root mean square of the residual divided by the root mean square of the

FAST-simulation over the step event:

χω=RMS (ωres)

RMS (ωFAST −ω0)χ¨

xt=RMS (¨

xt,res)

RMS (¨

xt,FAST). (4.35)

A value of χ=0 therefore means that the simplified model matches the behavior of FAST

perfectly, while a value of χ=1 would mean that none of the FAST behavior is represented

4. Turbine Model

in the simplified model.

12 14 16 18 20 22 24

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

v [m/s]

χω

Twr+Bld stiff

Twr flexible, Bld stiff

Twr+Bld flexible

(a) Rotor Speed

12 14 16 18 20 22 24

0.1

0.2

0.3

0.4

0.5

0.6

0.7

v [m/s]

χ¨xt

(b) Tower Top Acceleration

Figure 4.12.: Goodness of fit for responses to wind speed step inputs at different wind speeds

Figure 4.12 shows the values of χfor responses to a 1m/s step change in wind speed start-

ing at wind speeds between 12m/s and 25 m/s. Several conclusions can be drawn from

this figure:

a. As was already visible in figure 4.11 b), the addition of the structural model to the

simple aerodynamic and rotor model improves the fit of the rotor speed significantly.

b. The necessity of including the structural model in order to provide a good fit of the

rotor speed decreases with wind speed. This is due to the reduced excitation of the

structural states at high wind speeds because the higher pitch angles cause the turbine

to run at lower thrust levels.

c. If the model that considers the flexible tower and blade is used, the goodness of the

fit for the rotor speed is slightly worse at wind speeds above approximately 18m/s

compared to the fit at wind speeds close to rated. This is most likely caused by the

higher stiffness of the blades in the out-of-plane orientation at high wind speeds due

to running at increased pitch angles. However, the decrease is only small and the

overall fit remains very good with only a few percent mismatch even at high wind

speeds.

d. For the tower top acceleration, the match of the simplified model is generally lower

than for the rotor speed. Overall the majority of the behavior is still captured.

e. Similar to the rotor speed fit, the agreement of the tower top accelerations between

FAST and the simplified model that includes flexible blades, decreases with increasing

4.7. Model Validation and Analysis

wind speeds which, again, is due to ignoring the changes in the blade dynamics with

increasing pitch angles.

f. Including the flexible rotor model leads to better capturing of the system dynamics,

also for the tower top acceleration.

It should be noted that in the simulation of the non-linear model, many effects that lead

to harmonic disturbances, like blade misalignment or mass offsets and most importantly

the aerodynamic effect of the tower on the passing blade, have been omitted. It remains to

be investigated whether including these effects in a model for control design improves the

controller performance.

4.7.3. Frequency Domain Characteristics

In order to evaluate the simplified model in the frequency domain, the model is linearized

according to section 4.5 at different operating points of the turbine each uniquely defined

by the wind speed.

Figure 4.13 compares the resulting bode diagrams from the different models for the key

input/output pairs. Again, the analysis is performed for the intermediate stages of the

model development as well.

The torque rate command to generator speed transfer function Figure 4.13 a) shows an

almost exactly linear magnitude drop with 40dB per decade, as would be expected from

the combination of the integration of the torque command (4.20) and the principal equation

of motion for the rotor (4.5). The addition of the tower and blade models has only a minor

effect. Further, from Figure 4.13 b) it becomes apparent that only very little excitation to the

tower motion is caused by the generator torque commands even though they are coupled.

Figure 4.13 c) shows a significant reduction in the magnitude of the pitch to rotor speed

transfer function at the tower eigenfrequency. At this frequency, the interaction between

the rotor speed and tower motion becomes so strong that any pitch controller designed to

regulate rotor speed essentially has a ”blind spot” because the impact of pitching the blades

on rotor speed is small while at the same time the excitation of the tower motion as seen

in Figure 4.13 d) is at its maximum. Corresponding to this ”blind spot” is a large drop

of phase at the tower eigenfrequency in the pitch to rotor speed transfer function. If no

special precautions are taken, it is therefore usually inferred that the control performance

is limited to that frequency [64, 63]. Although including the blade model does not change

the amplitude plot at frequencies below the tower frequency, it adds an additional phase

4. Turbine Model

−180

−160

−140

−120

−100

Magnitude [dB]

0.1 1 5

−45

Frequency [Hz]

Phase [Deg]

Twr+Bld stiff

Twr flexible, Bld stiff

Twr+Bld flexible

(a) Torque Rate Command to Rotor Speed

−200

−150

−100

Magnitude [dB]

0.1 1 5

−360

−180

180

Frequency [Hz]

Phase [Deg]

(b) Torque Rate Command to Tower Top Acceleration

−80

−60

−40

−20

Magnitude [dB]

0.1 1 5

−720

−540

−360

−180

Frequency [Hz]

Phase [Deg]

−60

−40

−20

Magnitude [dB]

0.1 1 5

−720

−540

−360

−180

Frequency [Hz]

Phase [Deg]

(d) Pitch to Tower Top Acceleration

Figure 4.13.: Bode plots for the transfer functions from pitch command to rotor speed for the

different models linearized at v=14m/s

loss in the frequency range below 0.2 Hz which is where most of speed control takes place1.

The analysis of the turbine model in the frequency domain confirms the observation made

in the time domain that including the tower/blade model is not only crucial for accurately

capturing tower dynamics, but also for the rotor speed.

So far the plant model has only been considered at one wind speed. Since the torque coeffi-

cients Mω,Mv, and Mθ, as well as the thrust coefficients Fω,Fv, and Fθ, vary depending on

the linearization point, the linearization will result in different transfer functions at different

wind speeds. Figure 4.14 shows the bode plot of Ω

sΘC(s)and s2Xt

sΘC(s)over the range of wind

speeds in which the turbine operates pitch controlled. The low frequency gain in the pitch

1As speed control, at least in full load operation, is a disturbance rejection problem, the relevant frequency

range is determined by the frequency content of the wind disturbance which is difficult to accurately spec-

ify. However, most wind models show a rapidly decreasing power spectral density of the wind above

approximately 0.1 Hz [26]

4.7. Model Validation and Analysis

to rotor speed transfer function varies by as much as 10dB. This nonlinearity needs to be

accounted for in the controller design and is the main reason why the majority of wind tur-

bine pitch controllers are either gain scheduled linear controllers or nonlinear controllers. It

is further important to note that at the lower wind speeds of the full load operating region,

the system is a non-minimum phase system, easily identifiable by the large phase drop at

the tower and blade eigenfrequencies. With higher wind speeds, this effect becomes less

pronounced and at a certain point, the system becomes minimum phase and the phase

changes abruptly. Leithead and Dominguez [65] describe this effect and its implications for

the control design in detail and also report similar bode plots for the pitch to speed transfer

function. The pitch to tower acceleration transfer function in Figure 4.14 b) shows much

less variance between the different wind speeds, which explains why most tower damper

designs are generally not gain scheduled and instead use one linear controller for the entire

full load operating range [4].

−80

−60

−40

−20

Magnitude [dB]

0.1 1 5

−720

−540

−360

−180

Frequency [Hz]

Phase [Deg]

increasing

wind speed

(a) Pitch to Rotor Speed

−40

−20

Magnitude [dB]

0.1 1 5

−540

−360

−180

Frequency [Hz]

Phase [Deg]

increasing

wind speed

(b) Pitch to Tower Top Acceleration

Figure 4.14.: Bode plots for the transfer functions from pitch command to generator speed and

tower top acceleration at wind speeds between 13m/sand 25 m/s

Figure 4.15 shows the locations of the poles and zeros of the resulting transfer function from

pitch command to rotor speed which, as outlined, is the most critical input/output path.

At both the component eigenfrequencies of the tower ωT,0 and blades ω0,b, there is a set

of zeros which moves with wind speed. At 13 m/sboth the zero associated with the tower

and the zero associated with the blade is in the right half plane causing the system to be

non-minimum phase and leading to the large phase drop at these frequencies observed in

Figure 4.13. With increasing wind speed, these zeros move towards the left half plane with

first the tower zero crossing the imaginary axis in between 16m/sand 19m/sand then the

blade zero between 23m/sand 25m/s. These crossings of the imaginary axis correspond to

the sudden changes in phase already shown in Figure 4.14 a).

4. Turbine Model

Pole−Zero Map

Real Axis (seconds−1)

Imaginary Axis (seconds−1)

−5 −4 −3 −2 −1 0 1

−5

−4

−3

−2

−1

0.72

0.86

0.96

0.10.220.320.440.58

0.72

0.86

0.96

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.10.220.320.440.58

Pole associated

with pitch actuator

Pole associated

with blade

Pole/Zero

associated

with drive train

Zero associated

with blade

Pole/Zero

associated

with tower

Increasing

Wind Speed

Figure 4.15.: Poles and zeros for the pitch to rotor speed transfer function at wind speeds be-

tween 13m/sand 25 m/s

4.8. Modeling Summary

In this section, a simplified model of the wind turbine has been developed using first prin-

ciple modeling techniques. The developed model is used as the basis for both the wind

speed and state estimator and the Model Predictive Controller that are introduced in the

next chapter. The model consists of a nonlinear aerodynamic model coupled with linear

structural and actuator models. Using first principle modeling allows tailoring of the model

fidelity, expressed chiefly in the chosen number of states, specifically to the control prob-

lem at hand: The design of a MIMO controller for speed and power regulation and tower

damping. This will generally lead to smaller models than those derived from the automatic

linearization routines of the aero-elastic tools which can be essential especially for MPC

applications.

The analysis of the model in the time and frequency domain showed that adding a struc-

tural model of the blade and tower motion to the simple drive train model, which is the

4.8. Modeling Summary

simplest model that can be used for the design of a pitch controller (e.g., [8, 33, 43, 54]), im-

proves the match between the full model as represented by the aero-elastic simulation tool

and the simple model for control design significantly. It is exactly this interaction between

the aerodynamics and the structural dynamics that makes this control problem challenging

by introducing right hand plane zeros and the associated control bandwidth restrictions.

Moreover, this behavior varies significantly with wind speed and it is therefore crucial for

any controller operating near those limits to accurately know and account for these differ-

ences caused by the nonlinearity of the plant model.

5. Controller Design

This chapter describes how first the simplified wind turbine model that was discussed in

the previous chapter is used to design a wind speed and turbine state estimator and pro-

vides details on its performance. It then details how a continuously linearized Model Pre-

dictive Controller that makes use of this information can be designed and introduces a

method to robustify the state constraints against unmeasured disturbances.

5.1. Extended Kalman Filter

The MPC is designed as a full state controller, i.e., the controller requires the current value

of all model states xto compute the controller output u. Not all states of the turbine model

introduced in the previous section are measurable. Therefore, the state vector needs to be

estimated based on the available measurements using an estimator. Further, as described

previously, on most wind turbines it is not possible to measure the wind speed accurately

enough in order to use that signal in the turbine controller and it needs to be estimated as

well. In a prior study [50], wind speed estimation and state estimation were performed in

two separate estimators. However, since both estimation processes essentially require the

same model, sample times, and need to handle the same nonlinearity caused by the turbine

aerodynamics, they are combined into a single estimator here. This section first describes

the generic Kalman Filter process that is used for the estimator design, then details how

the estimator is designed based on the system model derived in the previous section, and

finally examines its performance.

The wind speed and state estimator is designed as a standard Extended Kalman Filter [116].

An EKF uses a stochastic system model

xk=f(xk−1,uk−1) + qk−1

zk=h(xk) + pk(5.1)

to reconstruct the state vector xkbased on the measurements zkwhere the random variables

qkand pkrepresent the noise on the system and outputs respectively. Assuming that qkand

pkare linear additive disturbances as in equation (5.1), is an assumption that is commonly

5.1. Extended Kalman Filter

made but that would not be required. Here, only linear disturbances are considered for the

description of the EKF. Its general idea is to linearize the nonlinear system model (5.1) at

each time step about the current estimated state vector ˆ

xk. The estimator is then designed

to minimize the trace of the covariance of the estimation error

Pk=E(ekeT

k)with: ek=˜

xk−xk(5.2)

under the assumption that qkand pkare Gaussian white noise with covariances Qand R

respectively.

The estimator that results from solving this minimization problem can be described in terms

of two major steps that are performed at each time step: A time update where the system

behavior is predicted ahead based on the current state estimate and a measurement up-

date where the prediction is corrected by comparing the predicted system outputs with the

actual measurements.

Without any further derivation, the resulting equations for both steps are stated here [116]:

Time update:

x−

k=f(ˆ

xk−1,uk−1)

P−

k=Ao,kPk−1AT

o,k+Qk−1(5.3)

Measurement update:

Kk=P−

kHT

k(Ho,kP−

kHT

o,k+Rk)−1

xk=ˆ

x−

k+Kk(zk−h(ˆ

x−

k))

Pk= (I−KkHo,k)P−

k(5.4)

with:

Ao,k=∂f

∂x(ˆ

xk−1,uk−1)

Ho,k=∂h

∂x(ˆ

xk). (5.5)

5.1.1. Disturbance Estimation

As stated before, the EKF needs to serve two purposes: It estimates the state and distur-

bance information that is not measurable. In order to facilitate the disturbance estimation,

the equations of the wind turbine model are rewritten to include the wind speed as a state

in the model. A state vector for the process to be observed xais defined as the state vector

5. Controller Design

of the original model augmented with a disturbance state:

xa= x

v!(5.6)

The wind speed is now not treated as an input to the model, but is instead assumed to be

driven by the system noise process ˙

q10:

x10 =˙

q10 (5.7)

The EKF system model (5.1) and its associated system noise qkis defined in a discrete for-

mulation while the wind turbine system model (4.24) was derived in a continuous formula-

tion in the previous chapter. Therefore, the augmented system model under the assumption

of linear additive system qknoise is stated here in continuous time but it is understood that

the properties of the system noise qkare specified with respect to the discrete formulation.

Using the augmented state vector, assuming only the measurable outputs of (4.26), and un-

der the assumption of linear additive system qand measurement noise pthe process to be

estimated becomes:

x1=1

J(MA(x1,x2,x7,x8,x10)−nGx9)+˙

x2=x3+˙

x3=x4+˙

x4=−ω2

p,0x3−2ωp,0ζpx4+ω2

p,0 ˙

θC+˙

x5=x7+˙

x6=x8+˙

x8!=M−1 FFA(x1,x2,x7,x8,x10)−K x5

x6!−D x7

x8!!+ ˙

q8!

x9=˙

MG,C+˙

x10 =˙

q10

z1=x1+p1

z2=x1nGx9+p2

z3=x2+p3

z4=x9+p4

z7=˙

x7+p5(5.8)

The EKF estimation process requires linearization of the system and output equations for

each time step about the current estimated state vector. The corresponding matrices Ao,k

5.1. Extended Kalman Filter

and Ho,kfollow directly from the process model (5.8) similar to (4.30):

Ao=∂f

∂xˆ

="A E

00#

Ho=∂h

∂xˆ







1000000

0100000

0010000

0001000

0000001







"C F

00#(5.9)

Again, the system model is stated here in the continuous form but and the process needs to

be discretized using (4.34) for an algorithmic implementation according to (5.3) and (5.4).

Choice of Noise Covariances With the process model for the estimation defined, only the

assumed covariances Rand Qremain to be set. The real process noise can practically not

be determined. The measurement noise levels could be specified in terms of the accuracy of

the used sensors. However, even those do not have the assumed white noise characteristic.

In the end, the values of Rand Qare often not corresponding to the real noise levels of

the process, but are used as knobs to tune the estimator performance according to specific

performance requirements. This tuning generally involves a trade-off between removing

the measurement noise from the estimated signals via setting Rto large values and limiting

the time lag in the estimation via using a large Q. In the wind turbine context, this trade-off

has to be mainly evaluated in terms of the wind speed estimation as the process uncertainty

due to the unknown wind speeds is several orders of magnitude larger than the uncertainty

in the aerodynamic and structural models and the measurement noise. As such, the main

tuning knobs are the entries qi,jand ri,jin Qand Rthat correspond to ˙

q10 and p1: Increasing

r1,1 leads to the estimator not ”believing” the measured rotor speed and thus only slowly

adjusting the estimated wind speed if the rotor speed changes. On the other hand, large

values of q10,10 mean that the estimator will assume that any change in rotor speed is due

to a change in wind speed and the estimator will adjust its wind speed estimate quickly if

the rotor speed changes. This will, however, cause unmodelled effects, such as the tower

shadow, to incorrectly show up in the estimated wind speed.

5. Controller Design

Here, the covariance matrices are chosen to be diagonal matrices with entries:

q1,1 =1.0 ×10−1q2,2 =3.8 ×102q3,3 =3.8 ×102q4,4 =3.8 ×102q5,5 =1.0 ×102

q6,6 =1.0 ×104q7,7 =1.0 ×102q8,8 =1.0 ×104q9,9 =0q10,10 =1.4 ×104

r1,1 =3.1 r2,2 =0r3,3 =1.0 ×10−3r4,4 =0r5,5 =3.0 ×10−2(5.10)

5.1.2. Performance

It was discussed in the previous section that the performance of the wind speed estimator

is driven by the choice of covariance matrices. To motivate the choice of estimator tuning

values, its performance is examined using FAST simulations. The full wind turbine model

is simulated in FAST using a simple baseline controller, and the estimator as described in

the previous sections is implemented in SIMULINK. The simulation setup and the baseline

controller will be discussed in more detail also in the next chapter.

The first simulation is a series of step changes in wind speed ranging from 10 m/sto 25m/s

with a step size of 1m/s. Figure 5.1 shows this series of steps and the resulting estimated

wind speed from the EKF. On a high level, it can be seen that the estimated wind speed

0 100 200 300 400 500 600

t[s]

v[m/s]

truth

estimated

Figure 5.1.: True and estimated wind speed a series of wind speed steps with a step size of 1m/s

clearly tracks the true wind speed. At low wind speeds, in steady-state, the estimated

wind speed matches the true wind speed almost exactly. Only at wind speeds of 22m/s

and higher, there is a clearly visible steady state offset with the estimator overestimating

the wind speed by close to 0.5m/sat the highest wind speed. As the steady-state estimator

performance is mainly driven by the stored aerodynamic properties, this difference is likely

due to slightly different aerodynamic models used for the generation of the CMtables (in

WT_PERF) and for the simulation (AERODYN) at low tip speed ratios or high pitch angles.

5.1. Extended Kalman Filter

0 5 10 15

−0.5

0.5

1.5

∆ v[m/s]

t[s]

truth

estimated

increasing wind speed

Figure 5.2.: Estimator tracking performance for wind speed steps with a step size of 1m/sstart-

ing between 12m/s(dark red) and 24m/s(light red)

In this figure, it can also be observed that there is some time delay and overshooting in

the estimated wind. To examine these tracking dynamics in more detail, figure 5.2 shows

a zoom at the step responses where several steps are co-plotted by removing the constant

component of the wind speed. The estimator tracks the true wind speed with an approxi-

mate time constant of 0.5s and an overshoot of up to 20 %. Generally, this tracking perfor-

mance can be influenced by the choice of assumed estimator noise covariances. For exam-

ple, it is easily possible to tune the estimator so that a step change in free stream wind speed

is tracked without any overshoot by using a lower value for the assumed model noise q10,10

associated with the wind speed state. However, such a tuning would also lead to a signif-

icant increase of the tracking time constant. It was found that the MPC presented in the

next section can easily handle some overshoot or a slightly oscillatory estimator response

but would be heavily affected by a sluggish tracking performance.

0 5 10 15 20 25

t[s]

v[m/s]

truth

estimated

Figure 5.3.: Estimator tracking performance during an EOG50 gust starting at 13.8m/s

5. Controller Design

Except for the steady state offsets, the tracking behavior is very similar between wind

speeds showing that the estimator correctly accounts for the plant nonlinearity. However,

under more realistic conditions, the wind speed never changes in steps. To assess the track-

ing behavior in conditions that are relevant for the wind turbine design, figure 5.3 shows the

tracking performance during an extreme operating gust with a return period of 50 years,

which is representative of the most extreme wind speed changes a wind turbine will en-

counter in power producing operation. The shape of the estimated wind speed is similar to

the actual ”mexican hat” gust shape, especially during the early part of the gust. Only start-

ing at around t=14 s is there a notably bigger difference and more oscillatory response.

This is likely caused by the excitation of unmodelled structural states by the strong gust.

Similar to the step responses, for the early portion of the gust, there is a tracking time con-

stant of around 0.5s and a slightly reduced amplitude. Generally, the difference between

the true and estimated wind speed never exceeds 2m/sand is much smaller for most of

the gust. The implications of the estimator mismatch on the ability of the MPC to correctly

handle state constraints is discussed at length in section 5.3.

0 100 200 300 400 500 600

t[s]

v[m/s]

HH wind speed

estimated

Figure 5.4.: Estimator tracking performance with respect to the hub height (HH) wind speed

during turbulent wind conditions with a mean wind speed of 20m/sand a turbulence intensity

of 15%

Finally, figure 5.4 shows the result of the wind speed estimation under turbulent conditions

for a 10 minute time series with a mean wind speed of 20m/sand a turbulence intensity

of 15%. Similar to the deterministic simulations, much of the high frequency content is

removed by the wind speed estimator. In the turbulent case, not all of this is due to the

low-pass type behaviour of the filter itself. In turbulent conditions, the wind speed is not

uniform across the rotor plane so there is not one single wind speed to compare the es-

timator against. The estimator will estimate the rotor area effective wind speed, i.e., the

uniform wind speed that would have the same effect on the rotor speed as the current tur-

bulent wind field, and it is clear that due to this averging effect the effective wind speed

5.2. Model Predictive Controller

will be much smoother than the single wind speed the estimator is compared to here. Nev-

ertheless, it can be seen that the estimator correctly tracks the lower frequencxy changes in

the wind speed.

5.2. Model Predictive Controller

At every time step, the controller calculates a sequence of future controller outputs uk|Lby

solving an optimization problem of which only the first controller output is then applied

to the plant u(k) = uk. For simplicity, it is assumed that no time delay appears when com-

puting uk. Otherwise, uk+1would be the first variable determined in the optimization step.

The objective function for the minimization problem is the quadratic sum of the controller

outputs uk|Land deviations of the plant outputs yk+1|Lfrom a given reference trajectory

rk+1|Lover the prediction horizon L. The sums are weighted using the matrices Qand R:

minuk|LJk=(yk+1|L−rk+1|L)TQ(yk+1|L−rk+1|L)

+uT

k|LRuk|L+Vf(xL+1)(5.11)

The term Vfis an additional cost penalizing the final state. It should be noted that, because

u= ( ˙

MG,C˙

θC)Tcontains the commanded actuator rates, this choice of objective function

will lead to u= (0 0)Tin steady state even if r6=0. The optimization problem (5.11) is

subject to the system dynamics

x=f(x,u,v)

y=h(x,u,v), (5.12)

bounds on the controller commands

umin

k|L≤uk|L≤umax

k|L, (5.13)

and state and terminal constraints

Acxk+1|L≤bc

AtcxL≤btc (5.14)

5.2.1. Tuning and Choice of Constraints

The MPC is tuned through the choice of weighting matrices Qand R. For example, large

values of Rmean the controller strongly penalizes controller actuation and will only uses

5. Controller Design

the actuators cautiously. On the other hand, if Ris small compared to Q, the controller

places a higher importance on regulating the outputs than on careful use of the actuators

and will act aggressively. By selecting the values for Qand R, the wind turbine designer can

therefore tune the controller response according to the design needs. It is not only possible

to trade-off actuator expenditure against regulation performance. By changing the entries

of Q, the tracking performance of the outputs can also be traded-off against each other.

For example, if q1,1, the entry of Qcorresponding to the rotor speed, is small compared to

the tower top acceleration weight q7,7, the controller will place more emphasis on reducing

tower oscillations than on tightly tracking the generator speed. Here, the following values

have been chosen:

Q=





Q1. . . 0

.....

0. . . Q1





R=





R1. . . 0

.....

0. . . R1







Q1=





q1,1 . . . 0

.....

0 . . . q7,7





R1="r1,1 0

0r2,2#

with

q1,1 =1.5 ×103q2,2 =7.2 ×10−6q3,3 =0q4,4 =0q5,5 =1.5 ×102q6,6 =0q7,7 =0

r1,1 =1.0 ×10−6r2,2 =3.0 ×103(5.15)

The entry corresponding corresponding to the pitch rate ˙

θ,q3,3, is set to zero because ex-

cessive pitch movement can also be penalized via the penalty on the commanded pitch

rate r2,2. A generator torque penalty q4,4 is not required as only high torque rates but not

the torque itself have a negative impact on the system and the torque rate is penalized via

r1,1. The tower top acceleration is not penalized via q7,7 as it was found that using the ac-

celeration penalty does not offer any additional benefit on top of the tower top velocity

penalty q5,5. Finally, the blade tip speed penalty q6,6 is not used due to the limitations of the

collective flap blade model. The effect of the choice on the control performance is further

discussed in section 6.2.2.

The generic constraint formulation (5.13) and (5.14), allows for a wide range of potential

constraints. Here, however, only the following constraints are used. The rates for the both

generator torque actuation as well as the pitch actuation are limited:

|˙

θC| ≤ 10 deg/s

|˙

MG,C| ≤ MG,0

10 1/s. (5.16)

5.2. Model Predictive Controller

Two state state constraints are used. The rotor speed is constraint to be below 10% above

the rated speed

x1≤11

10ω0(5.17)

Placing this hard constraint on the rotor speed has severe implications for the feasibility of

the control problem that are specifically addressed in section 5.3. Secondly, the pitch angle

is limited to not be below 1deg.

x2≥1deg (5.18)

1deg is roughly the optimal pitch angle for variable speed operation. This constraint is

necessary when operating near the rated wind speed. Without it, the turbine might be

pitching towards stall in case of low wind speeds.

Although the focus of the designed controller is on the full load operating region, it needs

to be able to handle at least upper partial load operation as in turbulent conditions the

turbine will have to operate below rated wind speed for short periods of time, even if the

average wind speed is above rated. This partial load controller is implemented through

simply changing the tuning weights and constraints whenever the turbine is in partial load

operation. In partial load operation, the turbine is not supposed to use the pitch to control

the turbine. Therefore, the pitch rate is constrained to zero:

|˙

θC|=0. (5.19)

With the pitch not available, the turbine can no longer track the rotor speed and power

output independently. Therefore, the power regulation objective is deactivated by setting

the corresponding weight to zero:

q2,2 =0. (5.20)

It should be noted that this type of partial load controller is capable of handling both lower

and upper partial load operation. The only difference between the two is that in upper

partial load operation the rotor speed set-point is constant, while in lower partial load the

set-point would be a function of the estimated wind speed (see section 2.1.2). However,

for the purposes of evaluating the wind turbine performance near and above rated wind

speed, the lower partial load controller does not need to be considered any further.

Switching Conditions The turbine switches from full load operation to partial load opera-

tion where the modified constraints (5.19) and weights (5.20) are used when the pitch angle

5. Controller Design

is less than or equal to the minimum pitch angle of θ=1 deg and the power is below rated

power. The turbine switches from partial load operation to full load if the power is higher

than rated power.

5.2.2. Linearization

The state and output equations in the optimization problem, as given by equations (4.24)

and (4.25), are nonlinear functions resulting in the optimization problem being nonlinear.

There are several options to handle nonlinear plant models in a model predictive control

setup:

•Robustness: If the linear MPC is designed with enough robustness, it is possible to

use a single linear MPC for an entire operating range of the plant (e.g., full load op-

eration). This approach was demonstrated by Henriksen for both the partial load

operation and the full load operation [36].

•Scheduling: Similar to classical gain scheduling, the plant model used for setting up

the MPC-problem can be obtained by linearization not only at one but at several sta-

tionary operating points. The most straight forward implementation of this strategy

is to run several controllers in parallel with the output being a weighted sum of the

outputs from the single controllers depending on the current operating point. This

method was employed using three linearization points by Kumar and Stol [54]. If

only a few linearization points are used, this method can be seen as accounting for

the slow changes due to variations in mean wind speed, but neglects the fast turbu-

lence induced changes in plant dynamics.

•Continuous Linearization: In order to avoid some of the limitations of the scheduling

approach, the model and resulting MPC formulation could be updated at every or

close to every time step. This would allow for also accounting for the (fast) turbulence

induced variations in operating point.

•Nonlinear MPC: In both the scheduling and the continuous linearization approach,

a single linear model is used for predicting the output trajectory over the prediction

horizon and calculating the optimal control trajectory. In the case of nonlinear MPC,

the nonlinear model is used for the output trajectory calculation and the gradient

is calculated along this trajectory. As the problem is no longer quadratic and needs

numerical integration of the model equations, a much higher computational burden

is involved.

While a large body of literature dealing with Nonlinear Model Predictive Control (NMPC)

exists, and even application to comparably fast systems seems viable (e.g., [120]), using

5.2. Model Predictive Controller

NMPC still increases complexity and computational burden over linear MPC significantly

and it needs to be evaluated carefully whether using NMPC actually provides the perfor-

mance improvement that would warrant the additional complexity. The different methods

for linearization have been evaluated in a previous study [51]. The main result was that

for turbulent simulations with a given mean wind speed and typical turbulence intensities

using NMPC provided almost no observable benefit over continuous linearization or even

linearization about the stationary operating point corresponding to the mean wind speed.

However, as described in section 2.1.2, there is a significant variation in plant behavior if

variations in mean wind speed are also considered. Therefore, the approach that is chosen

here is a hybrid of the scheduling and the continuous linearization methods: The system

is linearized at every time step. The linearization is however not performed at the current

state and input vectors. Instead, the system is linearized at the stationary operating point

corresponding to the current estimated wind speed. This approach avoids having to run

several controllers in parallel as in the scheduling approach while still handling even fast

changes in wind speed. Compared to the continuous linearization, where the linearization

point is the current state vector instead of the steady state value, the approach will always

have the origin in state space as its set-point which significantly simplifies the control for-

mulation and stability assessment.

At each time step, a linear state space model

δ˙

x=Akδx+Bδu+Ekδd

δy=Ckδx+Fkδd(5.21)

is derived by linearizing the equations (4.24) and (4.25). The index kis used for the matrices

A,C,E, and Fto indicate that they will change with every time step. The linearization

is performed about the steady state values xss corresponding to the current estimate or

measurement of the free stream wind speed v:

x0=xss(v). (5.22)

Offset free tracking is assumed for the rotor speed and electrical power. Therefore, the

steady state value for the rotor speed is simply the set-point

xss,1 =ωss =ω0(5.23)

while the steady state generator torque is simply the corresponding rated torque value

MG,0, which can be calculated based on rated power:

xss,9 =MG,0 =P0

ω0nG

. (5.24)

5. Controller Design

With rated rotor speed and generator torque known, there is a pitch angle θ0that balances

the aerodynamic torque with the generator torque. This pitch angle is obtained using a

precomputed mapping Θof steady state pitch angles as a function of corresponding wind

speeds:

θ0=Θ(v). (5.25)

The function Θis calculated by numerically solving P0/ω0=MA(v,ω0,θ)for θat all

0 5 10 15 20 25

v [m/s]

θ0 [deg]

Figure 5.5.: Steady state pitch angles as a function of wind speed

relevant values of v. Fig. 5.5 shows the resultant pitch angles as a function of wind speed.

With θ0(v)known, the steady state value of the aerodynamic thrust can be calculated, which

in turn is used to calculate steady-state blade and tower positions:

xt,ss

xb,ss!=K−1FFA(v,ω0,θ0(v)). (5.26)

All remaining states clearly have steady state values of zero. In summary, the steady state

vector for a given wind speed is defined as:

xss(v) =







ω0

Θ(v)

K−1FFA(v,ω0,θ0(v))

ω0nG







. (5.27)

For ease of notation, the difference between the current state vector and the linearization

point δx(t) = x(t)−x0(v)is simply called xwherever it is clear from context that the

deviations are used. Further, while the model has been derived in continuous form, it

5.2. Model Predictive Controller

is implemented in a discrete state space formulation, and the system matrices A,B,E,

and Fneed to be discretized accordingly. Therefore, in the following, the system model is

simply

xk+1=Akxk+Buk+Ekdk

yk=Ckxk+Fkdk. (5.28)

and it is implied that the input, state, and output vectors are the deviations from the lin-

earization point, and that the discrete system matrices were obtained by discretizing the

system model derived in this section using the zero order hold method.

5.2.3. Stability

As discussed in section 2.3.3, it is a well-known property of a linear MPC that stability can

be guaranteed independently of the choice of prediction horizon by choosing an appropri-

ate terminal cost Vf(xL+1)and constraint Atc and btc [91]. On the other hand, stability can

also be achieved using a sufficiently large prediction horizon which also has been used in

many applications of MPC for wind turbines. On the other hand, in order to achieve guar-

anteed stability using large prediction horizons, generally a prediction horizon at least in

the order of the settling time of the plant, is required [71]. Especially if the tower dynamics

are considered, as in the present study, this is a severe limitation since the natural frequency

of the tower is usually below 0.5 Hz and, combined with the low damping, this would lead

to prediction horizons that are above the horizons typically considered.

As stated by Mayne and Rawlings [91], stability can be achieved if the terminal cost is

chosen to reflect the infinite horizon cost of bringing the system to rest from the terminal

state of the prediction horizon using a terminal controller, and the terminal constraint set

is chosen appropriately. The terminal constraint set is a control invariant set, meaning that

for no state vector in this set the constraints are violated while bringing the system to rest

with the terminal controller.

It should be noted that choosing the terminal cost and constraint like this will only lead

to stability for the linear system. The system considered in this study, however, is nonlin-

ear, so that generally stability of the closed loop system will only be ensured in a region

around the linearization point. In order to ensure stability over a wider range of lineariza-

tion points, the variations in plant and controller behavior need to be considered explicitly,

for example using parametric uncertainties. This, however, is not seen as a severe restric-

tion, as the nonlinearity, while not negligible when considering the full operating envelope,

is fairly ”tame” in the proximity of the operating points and thus leads to large regions of

attraction.

5. Controller Design

A linear quadratic regulator (LQR) is chosen as the terminal controller. The infinite horizon

cost associated with this controller is 1

2xT

L+1PkxL+1where Pkcan be calculated by solving

the discrete algebraic Riccati equation

Pk=−AT

kPkBk(BT

kPBk+R)−1(AT

kPkBk)T

+AT

kPkAk+Qk,x(5.29)

with Qk,xdefined so that the cost placed on the states is equivalent to the cost placed on the

system outputs within the prediction horizon:

Qk,x=CTQ1C. (5.30)

As the system matrices Akand BKchange over time, the Riccati equation (5.29) is solved

for Pkat every time step and the terminal cost is set to:

Vf(xL+1) = 1

2xT

L+1PkxL+1. (5.31)

The terminal constraint is chosen to be a subset of the maximum control invariant set. A

subset chosen as the maximum set is defined by a large number of inequalities which make

the implementation impractical. Also, due to the nonlinearity of the plant model, the max-

imum admissible set varies with operating point and the terminal constraints would need

to be updated at every time step. Instead, the terminal constraint set is chosen to be a

hyper cuboid in the 9-dimensional state space which is a subset of the maximum output

admissible sets for all wind speeds. As an example, figure 5.6 shows the maximum out-

put admissible sets at four different wind speeds and the chosen terminal constraint set

in the x1-x2(speed vs. pitch angle) and the x1-x9(speed vs. generator torque) slices of the

state space where the maximum control invariant set is calculated using the algorithm from

Gilbert and Tan [28] which is reproduced in appendix A.

For the calculation of the terminal constraint, the constraint placed on the pitch state is ig-

nored. This is done because the constraint is not a true constraint where a violation would

lead to some form of system failure. Instead, the pitch angle constraint is used to help

the transition between full load and partial load operation. Specifically, violation of the

minimum pitch angle constraint is not a problem for the system, but merely one necessary

condition for switching from full load to partial load operation. Therefore, only the maxi-

mum rotor speed and the bounds on the commanded torque and pitch rates are considered

here.

Looking at the x1-x2slice, it can be seen that at low pitch angles lower rotor speed are re-

quired. This is intuitively understandable considering that the combination of low pitch

5.2. Model Predictive Controller

(a) x1-x2slice (b) x1-x9slice

Figure 5.6.: x1-x2and x1-x9slices of the selected terminal constraint (black) and maximum ad-

missible sets (colored) at four different wind speeds. The dashed blue line marks the rotor

speed state constraint while the dashed green line in the x1-x9slice marks the constant power

line.

angle and high rotor speed clearly leads to the highest overspeed risk. The maximum ad-

missible set has a fairly similar shape between the different shown wind speeds. However,

at high wind speeds, slightly higher rotor speeds can be allowed at non-negative pitch an-

gles. This is due to the higher sensitivity of the aerodynamic torque with respect to pitch

changes that allows the controller to bring the speed down faster with the same maximum

pitch rate. At the same time, because of this higher sensitivity, lower pitch angles also

pose more of a challenge to the system. Subsequently, the maximum admissible set extends

further into the negative pitch angles at lower wind speeds.

Another interesting observation can be made when looking at the x1-x9slice: The maxi-

mum admissible set seems to be almost centered around the constant power (dashed green)

line. This is caused by the very restrictive torque rate bound. If the power is too far above

(high rotor speed and high generator torque) or below (low rotor speed and low genera-

tor torque), then the terminal controller will aggressively use the torque to maintain rated

power and thereby violate the torque rate bound. Unlike in the x1-x2slice, there are only

small differences between the different wind speeds showing the power control problem is

only remotely affected by the non-linear aerodynamics.

Finally, the selected terminal constraint appears to be much smaller than the shown maxi-

mum admissible sets. However, for each of those plots all seven other states are assumed

to be zero. The potential variations in the other states will also need to be considered, i.e.,

the maximum admissible set in the x1-x2slice will be significantly smaller if all potential

variations in the other states are covered as well. For the same reason, a terminal constraint

5. Controller Design

needs to be defined for all nine states even if a state constraint is only used for the rotor

speed. For example, a very large tower terminal velocity can easily result in controller ac-

tuation above the rate limits and high rotor speeds. Therefore, it needs to be ensured that

the rotor velocity is within a certain bound at the terminal state even if no explicit state

constraint is used on that state. The chosen terminal constraint set is:

|x1| ≤ 30 RPM

|x2| ≤ 1 deg |x3| ≤ 0.2 deg/s

|x4| ≤ 0.1 deg/s2|x5| ≤ 0.15 m |x6| ≤ 0.15 m

|x7| ≤ 0.1 m/s|x8| ≤ 0.1 m/s|x9| ≤ 100 Nm (5.32)

It should be noted that, as the linearized system description is used, the terminal constraint

set is centered around the origin of the linearized state space. The terminal constraint (5.32)

thus defines the maximum deviations at the terminal state from the steady state operating

point given by (5.27).

The maximum admissible set would need to be calculated at an infinite number of op-

erating points but here the assumptions is made that, if the chosen set is a subset of the

maximum set at a number of operating points (here, all integer valued wind speeds), this

holds for all operating points.

5.2.4. Control Equations

In order to solve the controller optimization problem ((5.11)-(5.14)), the system dynamics

constraint is eliminated from the problem formulation by expressing the outputs yk+1|Las a

function of the controller inputs uk|Lwhich are the decision variables for the optimization.

For ease of implementation, a slightly different formulation than that used in section 2.3 is

employed here.

The trajectory of future states and outputs can be seen as the sum of the unforced state and

output trajectories xf,k|Land pk|Land the effect of the future control inputs

xk+1|L=xf,k|L+Bpuk|L

yk+1|L=pk|L+CpBpuk|L. (5.33)

The unforced state and output trajectories are a function of the current state and the trajec-

tory of future disturbance inputs

xf,k|L=Apx(k) + Epvk|L

pk|L=Cpxf,k|L(5.34)

5.2. Model Predictive Controller

with the prediction matrices for both the unforced and forced responses defined as [18,

70]

Ap=











Bp=





Bk0. . . 0

AkBkBk. . . 0

.....

AL−1

kBkAL−2

kBk. . . Bk







Cp=











Ep=





Ek0. . . 0

AkEkEk. . . 0

.....

AL−1

kEkAL−2

kEk. . . Ek







. (5.35)

Finally, the terminal state xL+1also needs to be expressed as a function of the current state

and future control and disturbance inputs

xL+1=xf,L+1+BpLuk|L

xf,L+1=ApLx(k) + EpLvk|L(5.36)

where the matrices ApL,BpL, and EpL are defined accordingly

ApL =hAL+1i

BpL =hALB AL−1B. . . Bi

EpL =hALE AL−1E. . . Ei.

(5.37)

Substituting the prediction model, (5.33) and (5.34), and the terminal cost (5.31) in the ob-

jective function (5.11) gives:

Jk=uT

k|LBpTCpTQCpBp+BpLTPkBpL +Ruk|L

+2uT

k|LBpTCpTQpk|L+BpLTPkxf,L+1

+pT

k|LQpk+1|L+xT

f,L+1Pxf,L+1. (5.38)

Similar to the objective function, the constraints are also expressed as functions of the un-

5. Controller Design

forced response and the control action:

umin

k|L≤uk|L≤umax

k|L

AcBpuk|L≤bc−Acxf,k|L

AtcBpLuk|L≤btc −Atcxf,L+1. (5.39)

Equations (5.38) and (5.39) now define a quadratic problem (QP) in uk|L.

5.2.5. Feedforward Control

By considering the future disturbance inputs explicitly when calculating the unforced out-

put trajectory (5.34), the described controller includes a disturbance feedforward compo-

nent in the feedback scheme. Generally, the trajectory of the future disturbance inputs is

not known and only an estimate of the current effective wind speed ˆ

vis available. In this

case, the assumption is made that the wind speed will remain constant over the entire pre-

diction horizon, effectively causing the terms in the prediction model (5.34) that depend on

the disturbance to vanish, as the model has been linearized at the current estimated wind

speed.

However, if upwind information, e.g., from LIDAR measurements, is available and used

as the trajectory of future disturbance inputs vk|L, the controller will consider the effect of

future wind speed changes. In the wind turbine controls context, this type of control is

often termed preview control.

Next to measurements, the controller can also use predictions of the future wind speeds, for

example from a wind prediction model as described in section 3.1.3. In this case, this type

of controller is not a true disturbance feedforward as the disturbance itself, and potentially

also the predicted wind speed, is only derived from the same measurements also used for

the feedback control. Nevertheless, this type of controller will improve the performance

over a pure feedback controller as the model information of how disturbances act on the

controlled outputs is taken into account explicitly.

Most feedforward schemes combine a feedforward controller with a feedback controller,

where the feedback controller has to ”handle” all remaining disturbances that have not been

canceled by the feedforward due to modeling or measurement errors. Due to the described

explicit output prediction using disturbance information in the control formulation, this

MPC scheme, however, combines both feedback and feedforward in a single controller.

This has the advantage that optimality and constraint handling can be maintained for the

overall controller.

5.3. Robust State Constraints

Most studies investigating the use of MPC for wind turbine control only consider control

constraints. These are typically representing hardware limitations on the actuators, such

as maximum torque and pitch rates. The main benefit of including those in the problem

formulation explicitly is that a controller, which does not account for these limitations and

simply saturates the actuated variables, will lose optimality in case it was designed using

some form of optimal control design method, and also that stability cannot be guaran-

teed.

There are, however, also state constraints in the wind turbine control problem. Most no-

tably, the generator may not exceed a certain safety critical threshold running under load.

Therefore, most turbines will disconnect the generator and initiate a braking procedure if

this threshold is reached. In order to avoid these shut-downs, the controller should ensure

that the threshold is never exceeded during operation, which is an inequality constraint on

the rotor speed state:

x1≤ωmax, load. (5.40)

A constraint like (5.40) could be implemented directly in the described MPC setup similar

to the control constraints. Unlike the control constraint, however, it cannot be guaranteed

that the constraint is not violated in case of modeling errors or unknown present and fu-

ture disturbances. For example, if the turbine is subjected to a step like gust that would,

if the state constraint was omitted, lead to a rotor speed above the threshold, then includ-

ing the constraint in the optimization will result in a trajectory that will hit the constraint

exactly. Now, if either the current wind speed or future wind speed at any given point in

the prediction horizon is even slightly higher than what was assumed by the controller, the

generator speed will increase above the threshold value and violate the constraint. If the

future wind speed is assumed to remain constant over the prediction horizon, it can clearly

be seen that due to the stochastic nature of the wind, this condition will frequently be met

and the constraint will be violated. Even if the future wind speeds are measured (see sec-

tion 5.2.5), there will always be measurement errors causing the controller to transgress the

constraint.

In order to make the use of state constraints viable, the controller needs to be robustified

so that as long as certain assumptions about the disturbance are met, the state and control

constraints will not be violated in the presence of unknown disturbances.

The method that is used to robustify the controller is based on the tube-based robust con-

troller using an additive disturbance model as described by Rawlings and Mayne [91]; al-

though it does not include all of its elements.

5. Controller Design

Here, only robust handling of the overspeed limit is considered, but the method can be

expanded to include any linear state constraint. Furthermore, so far only robustification

against unknown disturbances is considered, while modeling and estimation errors are ig-

nored. This can be justified by the fact that the uncertainty on the future wind is several

orders of magnitude larger than the modeling uncertainty and completely dominates the

overall system behavior. If modeling uncertainty, which usually takes the form of a para-

metric uncertainty, were to be included, it could however be transferred into an equivalent

additive disturbance as demonstrated by Rawlings and Mayne.

The basic concept of the tube-based robust controller relies on the assumption that an un-

known but bounded disturbance is acting on the plant. While the actual disturbance is

unknown to the controller and cannot be included in the prediction model, it is possible to

calculate or approximate the maximum effect this disturbance can have on the predicted

output trajectory based on the bounds of the disturbance. Based on this maximum effect,

the control problem can then be modified accordingly such that the constraints are honored

for all possible disturbance sequences.

Definition: In this context, a vector defined over the prediction horizon being smaller than

another vector

ym|N<zm|N(5.41)

is defined as each entry being smaller than the corresponding entry

ym+i<zm+i∀i=0...N. (5.42)

If ym+iand zm+iare vectors themselves, this holds for every entry.

5.3.1. Additive Disturbance

An additive disturbance is assumed to be acting on the plant. As described, only the mis-

match between assumed and actual wind speed is considered as a disturbance so that the

system model becomes

xk+1=Akxk+Bkuk+Ekdk+Ekvd,k

yk=Ckxk(5.43)

with vdas the unmeasured wind speed. This unmeasured wind speed encompasses both

the measurement error as well as undetected future changes in wind speed at any point

5.3. Robust State Constraints

within the prediction horizon. Similar to the future wind speeds, a vector of future un-

known wind speeds is defined:

vd,k|L=vd,kvd,k+1. . . vd,k+LT. (5.44)

Due to the unmeasured disturbance, the actual state trajectory will differ from the nomi-

nal trajectory as assumed by the controller. The real trajectory is the sum of the unforced

response of the system xf,k|L, which is a function of the current state and known distur-

bance trajectory, the effect of the controller outputs uk|L, and the response to the unknown

disturbances xd,k|L

xk+1|L=xf,k|L+Bpuk|L

| {z }

xnom

k+1|L

+xd,k|L(5.45)

where the state trajectory due to the unknown disturbance is

xd,k|L=Epvd,k|L. (5.46)

Calculating the disturbance trajectory as in (5.46) ignores the effect of the controller as it

assumes the controller will not ”see” and react on the effect of the disturbance. Due to its

receding horizon nature, the Model Predictive Controller will however react on the state

deviations caused by the unmeasured disturbance. The effect of closed loop control is in-

cluded by employing a modified prediction matrix EK,p

xd,k|L=EK,pvd,k|L. (5.47)

which is obtained by modifying (5.35) to use a state matrix which includes the effect of

feedback

AK=A−BK (5.48)

where Kis the feedback matrix corresponding to the unconstrained MPC. As the distur-

bance trajectory vd,k|Lis unknown, it is also not possible to calculate the resulting distur-

bance state trajectory xd,k|L. It is, however, possible to approximate bounds on these distur-

bance state trajectories. Here, the assumption is made that the disturbance trajectories are

bounded by a maximum measurement error:

|vd,k| ≤ vmax

d(5.49)

and that the maximum state trajectory is caused by the maximum measurement error being

introduced at any time during the prediction horizon and then remaining constant over the

5. Controller Design

20 40 60 80 100

0.02

0.04

0.06

0.08

0.1

k|L

Figure 5.7.: x1-state trajectories (thin lines) from introduction of a measurement error at differ-

ent instants and bounding x1-trajectory (thick line)

rest of the prediction horizon:

xmax

d,k+1|L=max EK,pvj

d,k|Lj=0...L.

d,k+i=





0i<j

vmax

di≥j. (5.50)

The resulting assumed bounding state trajectory for the state x1is shown in Fig. 5.7. This

assumption can be motivated by the choice of a state constraint on only the rotor speed and

knowledge that the wind speed to rotor speed input/output characteristic follows what

is essentially a first order low-pass behavior. If different state constraints are considered,

approximating the maximum disturbance trajectories becomes more complex. One way to

approximate the maximum disturbance trajectories is using Monte-Carlo simulations: The

system behavior is simulated for a large number of potential disturbance trajectories and

the maximum trajectory is chosen to encompass all or most of the simulated trajectories.

This is illustrated in Fig. 5.8, where 200 x1-trajectories obtained from simulations with

200 randomly chosen disturbance trajectories are plotted, and it can clearly be seen that

all state trajectories fall within the assumed bounds. For the generation of the disturbance

trajectories, it was only assumed that (5.49) holds. Generally, the more conditions are placed

on potential disturbance trajectories, such as rate limits or assumed frequency content, the

tighter the bounding state trajectories can be chosen, which will be essential for placing

constraints on the more oscillatory states such as tower motion. Under this assumption,

it can easily be seen that all possible disturbance trajectories will fall within the assumed

lower and upper bounds:

−xmax

d,k+1|L≤xd,k+1|L≤xmax

d,k+1|L. (5.51)

5.3. Robust State Constraints

10 20 30 40 50

−0.08

−0.06

−0.04

−0.02

0.02

0.04

0.06

0.08

k|L

Figure 5.8.: x1-state trajectories (solid lines) from 200 random disturbance trajectories and

bounding trajectory (dashed line)

This maximum disturbance trajectory then also defines the maximum difference between

nominal and actual state trajectories:

xk+1|L=xnom

k+1|L+xd,k|L

xk+1|L<xnom

k+1|L+xmax

d,k+1|L

xk+1|L>xnom

k+1|L−xmax

d,k+1|L. (5.52)

The maximum disturbance trajectory can be seen as defining an outer bounding tube on

the set of all possible state trajectories. This tube is centered on the nominal trajectory as

illustrated in Fig. 5.9.

5.3.2. Controller Modification

The main goal of the robust controller is to ensure that the state constraint is not violated in

the presence of unknown disturbances. In other words, it needs to be ensured that the ac-

tual trajectory xk+1|Ldoes not violate the constraint for any possible sequence of unknown

disturbances:

xk+1|L=xnom

k+1|L+xd,k|L<xmax. (5.53)

However, since in the calculation of the nominal trajectory only the measured disturbances

are considered, it will only be ensured that

xnom

k+1|L<xmax. (5.54)

5. Controller Design

5 10 15 20

−0.02

−0.01

0.01

0.02

0.03

0.04

0.05

0.06

k|L

xmax

xnom

k+1|L

xnom

k+1|L+xmax

d,k+1|L

(a) Without Constraint Tightening

5 10 15 20

−0.02

−0.01

0.01

0.02

0.03

0.04

0.05

0.06

k|L

xnom

k+1|L+xmax

d,k+1|L

xnom

k+1|L

xmax

xmax −xmax

d,k+1|L

(b) With Constraint Tightening

Figure 5.9.: Illustration of the effect of tube based constraint tightening on the predicted state

trajectory.

It can clearly be seen that if the constraints for the nominal control problem are tightened

to xmax −xmax

d,k+1|Lthen:

xnom

k+1|L<xmax −xmax

d,k+1|L

⇒xk+1|L=xnom

k+1|L+xd,k+1|L<xmax (5.55)

as xd,k+1|L<xmax

d,k+1|Lfor all possible disturbance sequences vd,k|L. This is further illustrated

in Fig. 5.9. In the left plot no constraint tightening is used. While the nominal trajectory

does not cross the bound on the state, parts of the tube of actual state trajectories lie outside

of the allowed state values. In the figure on the right the constraints are tightened. This

causes the nominal trajectory to have a certain distance to the state bound, such that the

entire tube of possible real trajectories lies within the range of allowed values. When calcu-

lating the disturbance state trajectories according to (5.47), the assumption was made that

the controller acts on the effect of the unmeasured disturbance. This, however, is only pos-

sible if the additional control action caused by the unmeasured disturbance does not cause

the controller to operate at the control constraints. Therefore, not only the state constraints,

but also the control constraints for the nominal control problem need to be tightened so

that, even in the case of the maximum disturbance state trajectory occurring, the overall

control action will be within the controller limits.

umin

k+1|L+Kxmax

d,k+1|L≤uk+1|L≤umax

k+1|L−Kxmax

d,k+1|L(5.56)

This can be seen as requiring the controller to keep a certain control reserve, which is to

be used to handle unforeseen disturbances. How much control reserve is used depends on

5.3. Robust State Constraints

the assumed bounds: The better the measurements of current and potentially future wind

speed, the more the controller is actually allowed to assume operation near its limits when

calculating the optimal control trajectory. It should be noted that the constraint on the first

time step ukis not affected. Since only this first time step will actually be applied to the

plant, this type of constraint tightening does not restrict the controllers ability to operate

at its limits. It only becomes more cautious with respect to potential disturbances at future

points in time.

5.3.3. Unmeasured Disturbance

The choice of bounds on the unmeasured disturbance needs to cover both the measure-

ment/estimation error and potentially the error introduced by assuming the wind speed

will remain constant at its current value over the prediction horizon in the non-preview

case. Generally, it is not possible to actually determine the upper bounds on the unmea-

sured disturbance across all operating ranges and wind conditions. Moreover, it is most

likely too conservative to actually use this maximum unmeasured disturbance as this would

result in a controller which, even under normal operating conditions, will react too cau-

tiously and might not provide the best performance. Instead, the values specifying the

assumed unmeasured disturbance should be seen as an additional tuning parameter. They

implicitly determine the safety distance the controller maintains to the actual hard limits

on the state values. For example, using large values, the probability of an overspeed fault

occurring can be reduced, but at the same time due to the large ”safety zone”, the controller

has to keep the rotor speed excursions in a smaller range, which requires more pitch action

and increases tower loads.

Here, the unmeasured disturbance is assumed to be vmax

d=3.5m/sin the non-preview case.

This is in the same order of magnitude as the estimation error in the extreme gust cases

(e.g., figure 5.3) but less than the worst case change in wind speed over any 5s prediction

horizon during these extreme gust events. Nevertheless, this choice has proven to provide

a good performance in the load cases that are discussed in chapter 6. The assumption

that the unmeasured disturbance acts as a step input is likely conservative with respect to

the actual unmeasured disturbance trajectories explaining why lower than actual assumed

disturbances already provide an acceptable performance.

This value is too high only at wind speeds near rated wind speed. Due to the low sensitivity

of the aerodynamic moment to pitch movements, choosing a value this high for the unmea-

sured disturbance leads to the controller constantly operating near or at the robustified state

constraint. Therefore, at rated wind speed an unmeasured disturbance of vmax

d=1.5m/sis

used and the value is increased linearly with the wind speed so that at 13 m/sthe full value

5. Controller Design

is reached:

vmax

d=









1.5m/sv<vr

3.5m/sv>13m/s

1.5m/s+2m/s·v−vr

13m/s−vrelse

. (5.57)

Whenever preview control is considered, a fixed value of vmax

d=0.5m/sis used.

5.3.4. Backup Mode

If state constraints are used, there is a possibility that the QP problem is infeasible, i.e., no

solution that does not violate any constraint exists. For the presented controller, the most

likely cause for infeasibility would be an incorrect assumption on the unmeasured distur-

bance as discussed in the previous section. If a gust is stronger than the value of the unmea-

sured disturbance that was assumed, the controller may not be able to contain the turbine

states within its limits without violating the controller actuation bounds. Another poten-

tial cause for infeasibility is mismatch between the turbine model in the controller and the

actual plant leading to incorrect disturbance tubes. Further, one event that routinely needs

to be considered in the wind turbine design is a partial or complete loss of counter torque

at the generator. As this event is not included in the calculation of the disturbance tube,

it can also not be guaranteed that it is possible to stay below the overspeed limit with the

available actuator limits. In many cases, the model mismatch or torque loss uncertainty are

not a severe challenge to the controller as they are also implicitly covered by the allowance

for unmeasured disturbances. Only if they occur together with extreme changes in wind

speed, such as in load case 1.5 where a grid loss is assumed to happen during an extreme

operating gust with a one year return period, will they actually lead to infeasibility.

Infeasibility means that the controller does not ”see” any way of maintaining operation

within the given limits and thus, that there is a potential safety hazard. If a classical wind

turbine controller were used and a safety hazard was detected, the supervisory control

system would switch the controller from a power production mode to shut-down mode,

which in many cases means the turbine is brought to a stop by pitching out the blades at a

fixed pitch rate. Analogously, if in the MPC case the problem is infeasible the commanded

and generator torque pitch rates are set to the maximum:

uk= ˙

Mmax

G,C

θmax

c!(5.58)

5.4. Algorithmic Implementation

Unlike in the case of a shut-down controller, this backup mode is not used until the turbine

has come to a rest; it is only used for one time step. At each time step, there is a new

attempt at solving the QP problem and only if it is still not feasible, the backup mode is

used. As a result, if the controller is within safe operating limits again, it will immediately

resume normal operation. If not, the turbine will shut down completely. It should be noted

that the choice of backup mode is clearly driven by the nature of the state constraint that is

considered. The backup mode is simply the controller action that reduces the rotor speed

as fast as possible and therefore minimizes the severity of the constraint violation. If, e.g.,

a tower deflection constraint would be considered, there is no such clearly defined backup

mode and a more elaborate scheme for handling infeasibility is required.

5.4. Algorithmic Implementation

To summarize the controller design, the steps that need to be performed at each controller

cycle are stated here:

1. Perform measurement and time update of EKF according to equations (5.4) and (5.3)

using the current turbine output measurements (4.26) and the state estimate of the

last time step.

2. Determine the steady state operating point corresponding to the current wind speed

estimate according to (5.27) and define the difference between the current state esti-

mate and the linearization point as new state vector.

3. Update the partial derivatives with respect to aerodynamic torque and thrust in the

continuous, linear state space model (4.30) according to the current wind speed esti-

mate.

4. Discretize the state space model according to (4.34)

5. Solve the discrete Algebraic Ricatti Equation (2.39) to get the infinite horizon cost Pk

and the terminal controller KLQR (2.40).

6. Built up MPC prediction matrices (2.34).

7. If no preview control is considered, set vk|Lequal to the current wind speed estimate.

8. Calculate the unforced output trajectory according to (5.34).

9. Calculate disturbance tubes using the current linear system model and terminal con-

troller KLQR according to (5.50) and using the chosen values for the unmeasured dis-

turbance vmax

5. Controller Design

10. Calculate the control and state constraints according to the current linearization point

and the calculated disturbance tubes (5.55) and (5.56) and update the constraints (5.39)

accordingly.

11. Solve QP problem (5.38)+(5.39) for uk|L.

12. If no feasible solution exists, set ukaccording to backup controller (5.58).

13. Send the first entry ukin the computed optimal control sequence uk|Lto the plant.

100

6. Results

This chapter examines the performance of the controller design that was presented in the

previous chapter. After a quick look at some of the implementation details and introduction

of a reference, baseline controller, it discusses results from simulations closely resembling

those load cases that are also considered when designing an entire wind turbine.

6.1. Simulation Setup

All simulations are run in SIMULINK. The aero-elastic simulation tool FAST (see section

2.2.5) that is used to simulate the plant, i.e., the wind turbine, is included in the SIMULINK

model via the provided S-function interface. For all simulations that require turbulent wind

fields, the wind time series are generated using the Kaimal model [45] with TURBSIM [48]

according to the IEC regulations [39]. FAST does not include any actuator models so those

are added directly in SIMULINK: The pitch actuator model consists of a second order trans-

fer function (4.18) with the same parameters as stated in table 4.3 and rate limitation to

±10deg/sfor each of the three blades. The generator torque actuator model is a simple one

time step delay operator.

The following signals and their respective FAST sensor names are assumed to be measurable

and available to the controller:

•The rotor speed RotSpeed

•The pitch angle1BldPitch1

•The tower top acceleration in the longitudinal direction YawBrTAxp

Loads are evaluated for the three principal components tower, main shaft, and blades at

these three sensor locations:

•Tower base bending moment in the fore-aft direction TMyt

•Main shaft torque RotTorq

1as no individual pitch control is used the pitch angles for all three blades are always identical and the pitch

angle at blade 1 can be used as the collective pitch angle

101

6. Results

•Out-of-plane bending moment at the blade root for blade 1 RootMyc1

Fatigue loads at these locations are calculated as damage equivalent loads (see section 2.2.4)

which are calculated in a postprocessing step after the actual time series simulation using

the WAFO toolbox [113] for MATLAB with a reference cycle frequency of 1Hz. A material

slope of m=4 corresponding to steel is used for the tower and main shaft while for the

blades, which are made of composite materials, a slope of m=10 is applicable.

6.1.1. MPC Implementation

The MPC that is described in the previous chapter is implemented using an interpreted

MATLAB script that is called from the SIMULINK environment. QPOASES [23] is used as the

underlying QP-solver which is called once in every time step. The initial condition for the

QP-solver is the uk|L=0 at every time step and the option to ”hot-start” the solver that is

provided in QPOASES is not used.

Both the MPC and the estimator are chosen to run at a sample rate of 10Hz and the ”mea-

surements” taken from FAST are down-sampled accordingly. A prediction horizon of 5s

is used so that l=50 and, as there are two controller outputs, the QP-problem has the

dimension 100.

FAST does not provide a convenient sensor for the rotor plane effective wind speed which

needs to be assumed measurable if preview control is simulated. Therefore, the estimated

wind speed that is calculated via the estimator is used as a surrogate for the wind speed

preview measurements. Whenever preview control is considered, the time series of the

estimated wind speed, corrected for estimator delay, from a prior run using the same wind

field is made available to the controller for the full prediction horizon. Especially the high

frequency components of the wind field are attenuated by the wind speed estimation so

that this assumption is conservative as it assumes imperfect preview measurements. It has

been shown [100, 96, 99] that by using LIDAR measurements, it is not possible to correctly

measure the high frequency components of the wind field ahead in time due to the limited

spatial resolution of most LIDAR systems and the fact that these frequencies are generally

related to smaller structures in the turbulent wind field that tend to disperse before they

arrive at the turbine. Hence, the choice of using the estimated wind speed as the preview

signal, although made because of lack of suitable alternative, might be a more realistic

assumption, than using the true rotor plane effective wind speed.

One important aspect for any MPC application is always the computational load. In the cur-

rent implementation, it takes about 0.37 s of CPU time to simulate 1 second of real time on a

standard commercial PC with a clock frequency of 2.5 GHz. This simulation time includes

102

6.1. Simulation Setup

the computational load that is required to run FAST as well as the MATLAB/SIMULINK over-

head. For a real controller implementation, only the controller portion of the simulation

setup would need to be implemented on a real-time system such as a Programmable Logic

Controller (PLC) and the time required to run the plant model, in this case FAST, is there-

fore of no concern. Figure 6.1 shows the distribution of the computational load for each of

the major steps in the MPC algorithm. It can be seen that a significant amount of time is

required for setting up the equations and constraints, while solving the actual QP-problem

only requires around 10% of the total computational load. Considering that especially the

non-QP steps are in no way optimized for speed2and that they could be made significantly

faster, an implementation on a real-time system does not seem impossible.

FAST+Overhead (60%) Determination of

Linearization Point (1%)

Building of MPC

matrices (24%)

DARE solution (4%)

Constraint

Robustification (4%)

QP solution (4%)

EKF (3%)

Figure 6.1.: Distribution of the computational load for each controller time step

6.1.2. Baseline Controller

In order to better analyze the behavior of the Model Predictive Controller, its response for

the various scenarios that are evaluated is always compared against a baseline controller.

This baseline controller is designed to be a representation of a typical, classically designed

controller that can be found on modern wind turbines. As the performance evaluation is

performed only in the above rate, pitch controlled operating region, only the controller for

full load operation is described here. Similarly to the MPC, the baseline controller will also

be in partial load operation temporarily, even if the mean wind speed is above rated wind

speed. The partial load controller and the switching conditions between partial load and

full load are described in appendix B.2.

The torque controller is chosen to follow the constant power strategy with the torque control

law accordingly given by (2.12). The pitch controller is designed as a gain-scheduled PI

2Unlike the QP-solver they also run in the MATLAB interpreter mode which is slow compared to compiled

code

103

6. Results

controller which controls the rotor speed. Details on the tuning and gain scheduling of this

pitch controller can be found in appendix B.

The EKF, which is used in the MPC setup to estimate the turbine states and wind speed,

assumes the tower top acceleration to be measurable. In order to allow a performance

comparison that is as fair as possible, the baseline controller also needs to make use of

the tower acceleration signal. Therefore, a tower damper is added that uses the measured

tower top acceleration to generate a pitch offset that is added to the output from the speed

controller. The tower damper is also designed as a PI-controller and details about its tuning

can also be found in appendix B.

Since the MPC only considers collective pitch action, an individual pitch controller is not

included in the baseline control design. The same applies to a potential Drive Train Damper

which was excluded in the MPC problem formulation. Figure 6.2 shows the final baseline

controls structure.

−



(

)

GG nPM 0

Torque Controller

Pitch Controller

Tower Damper

Figure 6.2.: Block diagram of baseline controller in full load operation

One event that is considered in the following simulation results is the so-called grid loss.

In case of a grid loss, the turbine is no longer capable of feeding power to the grid so the

torque command needs to be set to zero. The baseline controller then stops the turbine

by pitching towards a pitch angle of 90deg with a fixed pitch rate of 10 deg/s. The tower

damper is deactivated during this shut-down process.

6.2. Normal Power Production Operation

First, the behavior of the MPC is examined under Normal Power Production Operation

(NPP) that would typically be run during the design assessment of any wind turbine. The

turbine behavior is simulated during multiple time series each with a 10 minute duration

and varying wind conditions. The wind speed for each time series consists of a constant

104

6.2. Normal Power Production Operation

mean wind speed and a superimposed turbulent component with zero mean speed where

the turbulence intensity is a function of the mean wind speed for the given time series.

Here, mean wind speeds from 12m/sto 25m/sin steps of 1m/sare considered and a turbu-

lence distribution according to the IEC turbulence class ”B” (figure 2.9) is assumed. Wind

speeds below 12m/sare not evaluated as, due to the rated wind speed of the given tur-

bine being above 11m/s, the turbine will spent most of the time in partial load operation

during these time series. While the MPC developed in the previous chapter is able to han-

dle partial load operation, the presented MPC has been designed with a focus on full load

operation and it has been shown earlier that the benefits of MPC in partial load are neg-

ligible [51, 104] so that this operating range is excluded from the analysis. At each mean

wind speed, two seeds, i.e., two different realizations of the turbulence generated using a

random number generator, are used. Any further variation that is sometimes considered

during a loads assessment such as upflow or yaw misalignment is ignored so that in total

the NPP set consists of 28 individual time series.

Four different controllers are considered for the analysis: The first is the MPC described

in the previous chapter without any preview information available, i.e., it is assumed that

the current estimate of the free stream wind speed is valid throughout the prediction hori-

zon. For this first controller, the constraint on the rotor speed state is further deactivated so

that only the control constraints and the constraint on the pitch angle are active. The sec-

ond controller is identical to the first controller except that here the rotor speed constraint,

including its robustification, is active. The third controller is also the MPC with the rotor

speed constraint, but preview information is assumed to be available and the assumption

on unmeasured disturbance is modified according to section 5.3.3. Finally, the fourth con-

troller is the classically designed baseline controller that was introduced in section 6.1.2.

Figures 6.3 and 6.4 show the key operational metrics and resulting loads per wind speed.

Whenever a standard deviation (STD) is calculated, it is averaged between the two sim-

ulations at the same wind speed. For the maxima (MAX), the higher value of the two

simulations is used while for the damage equivalent loads (DEL), the two time series are

combined using the Palmgren-Miner Rule (see section 2.2.4). All signals are normalized:

The rotor speed ωand the generator torque MGare normalized to their respective rated

values. The tower base bending moment in the fore-aft direction TMyt, the main shaft

torque RotTorq, and the out-of-plane bending moment at the root of blade 1 RootMyc1are

all normalized to the stationary values corresponding to the mean wind speed of the given

time series. Similarly, the pitch angle and rate are normalized to pitch angle necessary to

achieve steady state at the given mean wind speed and rated torque.

The first observation can be made by looking at the maximum and standard deviation of

the rotor speed in figure 6.3 (a) and (b): All four controllers clearly manage to track the

105

6. Results

12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.01

0.02

0.03

0.04

mean(v)[m/s]

std(ω/ω0)

(a) Rotor Speed - STD

12 13 14 15 16 17 18 19 20 21 22 23 24 25

1.05

1.1

1.15

mean(v)[m/s]

max(ω/ω0)

(b) Rotor Speed - MAX

12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.05

0.1

0.15

0.2

mean(v)[m/s]

std(MG/MG,0)

12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.05

0.1

0.15

0.2

mean(v)[m/s]

std(θ/θ0)

MPC no state constr

MPC + state constr

Prev. MPC + state constr

Baseline

(d) Pitch Rate - STD

Figure 6.3.: Key operational metrics from NPP simulations with a mean wind speed ranging

from 12m/sto 25m/swith two seeds per wind speed and using IEC B turbulence

rotor speed set-point tightly with standard deviation of the tracking error only being a few

percent of the set-point. Except for the baseline controller at 24 m/sand 25 m/s, all controllers

further manage to keep the rotor speed below its limit at all times. Generally, the baseline

controller provides a level of speed tracking performance in the same order of magnitude

as the two non-preview MPCs. However, with the baseline controller, the speed tracking

performance shows greater variability over the different wind speeds with the standard

deviation of the rotor speed being significantly higher at the high end of the wind speed

range, while with the MPCs, the speed tracking performance is flatter across wind speeds.

This is due to the simple gain scheduling approach that is used for the baseline controller

which, for example, does not account for the variation in the sensitivity of the aerodynamic

moment with respect to the wind speed. Looking at the speed tracking performance, it

can further be noted that under the simulated conditions, the rotor speed constraint is not

necessary. Even with the first controller, where this constraint is deactivated, the rotor

speed is never above the limit. However, as the results with and without this constraint are

not numerically identical for some wind speeds, there must be multiple situations where

the state-constrained controller ”sees” an overspeed threat and therefore reacts differently

than the unconstrained controller. The effect of the rotor speed constraint in NPP operation

is further analyzed in section 6.2.1.

106

6.2. Normal Power Production Operation

12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.2

0.4

0.6

0.8

mean(v)[m/s]

DEL(TMyt/TMyt,0)

(a) Tower Base Moment - DEL

12 13 14 15 16 17 18 19 20 21 22 23 24 25

1.5

2.5

mean(v)[m/s]

max(TMyt/TMyt,0)

(b) Tower Base Moment - MAX

12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.1

0.2

0.3

0.4

mean(v)[m/s]

DEL(RotTorq/RotTorq

12 13 14 15 16 17 18 19 20 21 22 23 24 25

1.1

1.2

1.3

1.4

1.5

mean(v)[m/s]

max(RotTorq/RotTorq

(d) Main Shaft Torque - MAX

12 13 14 15 16 17 18 19 20 21 22 23 24 25

mean(v)[m/s]

DEL(RootMyc1/RootMyc10

)

(e) Blade Root Flap - DEL

12 13 14 15 16 17 18 19 20 21 22 23 24 25

mean(v)[m/s]

max(RootMyc1/RootMyc10

)

MPC no state constr

MPC + state constr

Prev. MPC + state constr

Baseline

(f) Blade Root Flap - MAX

Figure 6.4.: Maximum and Damage Equivalent Loads from NPP simulations with a mean wind

speed ranging from 12m/sto 25m/swith two seeds per wind speed and using IEC B turbulence

Looking at the distribution of maximum and damage equivalent loads for the range of

wind speeds in figure 6.4, it becomes clear that the two non-preview MPCs provide a level

of performance that differs only slightly from the baseline controller results; although slight

improvement can be seen in the fatigue and extreme loads at most wind speeds. Only the

tower fatigue loads at wind speeds above approximately 20 m/sincrease through the use of

the MPC. However, in this range of wind speeds the difference in speed tracking perfor-

mance is also the greatest. Within certain limitations, it is generally possible to trade-off

speed tracking against tower fatigue loads [50] through the choice of controller tuning, i.e.,

some speed control performance may be sacrificed to reduce tower loads. Section 6.2.2 will

107

6. Results

further examine this property of the full load speed control problem. The slight improve-

ment in blade flap and tower DELs at low wind speeds, where the speed control perfor-

mance is on a similar level, can be explained by the fact that the MPC design does include

knowledge of the blade dynamics and blade-tower interaction explicitly through the used

state-space model while the baseline controller does not.

Interestingly, the two non-preview MPCs seem to perform worse than the baseline con-

troller at 12m/swith respect to containing rotor speed excursions. They require more pitch

actuation, visible in the increased standard deviation of the pitch rate ˙

θ, but still lead to

a higher maximum rotor speed. Here, the issue is that at 12 m/sthe controller will switch

frequently between full load operation and partial load operation. The switching behav-

ior is not included in the MPC problem formulation, i.e., for the calculation of the optimal

control trajectory the controller will assume to stay in the operational mode in which it is

currently operating, even if in reality, there is a mode switch in the next controller time step.

This causes the controller to switch sub-optimally during these manoeuvres and diminishes

controller performance.

Figure 6.5 shows the extreme and fatigue loads obtained from combining all 28 time series

of the NPP set. For the extreme loads, the highest load that occurs at the specific component

over all time series determines the overall maximum load, while for the fatigue loads, the

DELs for the individual time series are combined into a single DEL under the assumption

of an IEC wind class 2 distribution of mean wind speeds. These results show an overall

tower and main shaft fatigue loads reduction of 5% and a blade root fatigue reduction

of 2% for the MPCs with and without the state constraint compared to the baseline level.

Similar to the fatigue loads, the maximum tower and main shaft loads are also decreased by

around 4%. Neither of the two non-preview MPC is able to deliver extreme blade root loads

below the baseline level though. Nevertheless, the extreme blade root load is reduced by

the addition of the state constraint, indicating that the event where the highest blade load

occurs is one where the constraint is active. Overall, with fatigue and exteme loads only

slightly below the baseline level, the results for the two non-preview MPCs are in line with

Botasso’s result that it is hard to significantly beat a well tuned classical controller under

nominal conditions [7].

Compared to the three other controllers, the preview controller shows a significantly im-

proved performance. It is able to cut the rotor speed deviation at least to a third of the

non-preview level with less pitch and generator torque activity. The resulting extreme and

fatigue loads are lower at all wind speeds for all three main components. The biggest loads

reductions are seen at the tower base where the maximum load is reduced by 6% while

damage equivalent load is down by just over 30%. Of course, this loads reduction is only

applicable to the full load operating region, but considering that approximatly 70% of all

108

6.2. Normal Power Production Operation

tower shaft blade

100

110

relative load level [%]

(a) Fatigue

tower shaft blade

100

110

relative load level [%]

MPC no state constr

MPC + state constr

Prev. MPC + state constr

Baseline

(b) Extreme

Figure 6.5.: Combined extreme and fatigue loads for the entire full load operating range from

12m/sto 25 m/s. Results are normalized to the baseline controller level.

tower fatigue is caused in full load and assuming there is no change to partial load oper-

ation, the overall reduction in tower DEL is still likely in the range of 20%, which would

more than double the lifetime of the tower.

6.2.1. Overspeed Risk in NPP

The results presented in the previous section show that except for the baseline controller at

very high wind speeds, there is no real risk of triggering an overspeed fault. This is not very

surprising as any controller which is not capable of maintaining operation would be poorly

designed. This, however, also means that under these conditions the constraint on the rotor

speed state and its robustification would not be necessary. To analyze the behavior and ef-

fectiveness of this constraint, first the distribution of rotor speeds within one time series is

examined. Figure 6.6 shows the probability of occurrence of a particular rotor speed com-

puted based on one of the time series from the NPP set at a mean wind speed of 20m/s. The

distributions resemble a normal distribution for all four controllers. The distribution ob-

tained with the preview controller is significantly more centered around the mean value. A

result which is expected considering the significantly lower standard deviation. The three

distributions for the other controllers are all very similar. As no overspeed violations were

observed for any controller during this 10 minute time series, the frequency of occurrence is

zero for ω>1.1ω0for all controllers. Nevertheless, it is clear that the shape and variance of

the distribution directly drives the probability of triggering an overspeed fault. Especially

if the distributions are considered to be normal, the average number of overspeed faults,

e.g., trips per month, could be calculated for a given wind speed and turbulence intensity

based on the standard deviation of the rotor speed. Further, if the rotor speed tracking

is tightened by a more aggressive tuning of the controller, the trip probability will be re-

duced. Unfortunately, there is always a trade-off involved in tuning the controller. Tighter

109

6. Results

speed tracking generally also leads to higher tower loads. A relationship which is further

examined in the next section.

0.9 0.95 1 1.05 1.1

0.05

0.1

0.15

0.2

ω/ω0

Probability

MPC no state constr

MPC + state constr

Prev. MPC

Baseline

Figure 6.6.: Relative frequency of occurrence of the normalized rotor speed from an NPP simu-

lation at a mean wind speed of 20m/susing a bin width of 0.005 for f racωω0

In order to take a deeper look at the influence of the rotor speed constraint on the overspeed

risk, the same time series is also simulated with a considerably higher3turbulence intensity

of Ti=21%. Figure 6.7 compares the normal probability plots at both turbulence intensities.

The first observation is that while the curves are reasonably linear near the center, they

depart from the normal curve at both ends for all controllers. The distributions are clearly

thin-tailed and normal distribution cannot be assumed in the range near the maximum

rotor speed. At the higher turbulence intensity, the slope of the curve is lower, which in

the case of the baseline controller and unconstrained MPC causes it to cross the overspeed

level. While at Ti=15%, the upper tails for the three non-preview controllers are almost

identical, at Ti=21%, the constrained MPC differs from the other two at high rotor speeds.

Its curve bends upward earlier and thus never crosses the limit.

In summary, the addition of the rotor speed constraint helps to avoid overspeed trips

caused by high turbulence by modifying the controller behavior at high rotor speeds, i.e.,

the high tail of the probability distribution, only. In a certain range, it can avoid having to

make the overall controller more aggressive.

6.2.2. Tuning

One of the claimed benefits of MPC is the direct control over the performance trade-offs

that need to be considered for the controller tuning. To demonstrate the usefulness of this

3Using Turbulence class B, the turbulence intensity at 20 m/swas 15%

110

6.2. Normal Power Production Operation

0.85 0.9 0.95 1 1.05 1.1

0.0001

0.05

0.1

0.25

0.5

0.75

0.9

0.95

0.99

ω/ω0

Probability

(a) Ti=15%

0.85 0.9 0.95 1 1.05 1.1

0.0001

0.05

0.1

0.25

0.5

0.75

0.9

0.95

0.99

ω/ω0

Probability

MPC no state constr

MPC + state constr

Prev. MPC

Baseline

(b) Ti=21%

Figure 6.7.: Normal probability plot of the rotor speed from a NPP simulation at a mean wind

speed of 20m/sfor two different turbulence intensities

MPC feature for wind turbine control, a single NPP time series is simulated for a range

of potential tuning settings. Here, the weight placed on the rotor speed state (the first

entry of Q1) is chosen to exemplarily show the effect of the tuning weights and is varied

between 1.5 ×102and 1.5 ×104using logarithmic spacing. Figure 6.8 shows the effect of

these different settings on the controller performance for key metrics.

The first observation is that increasing the weight placed on the rotor speed increases the

rotor speed tracking performance as expected. For all three MPC variants, a higher weight

leads to a lower standard deviation. This comes at the cost of significantly higher pitch ac-

tivity and somewhat higher tower fatigue loads confirming the three way trade-off between

speed control, tower loads, and pitch activity that was already reported in [50]. The impact

on tower fatigue loads seems to be smaller than on pitch rate and rotor speed control, but

considering that a material slope of m=4 is used to calculate these DELs, the approximate

difference of 10% between the lowest and highest load for each controller still corresponds4

to a 50% increase in accumulated component damage and corresponding lifetime reduction

by one third. Given the relativly small differences in performance that were observed in the

previous sections, it is likely that it is also possible to tune the baseline controller to provide

a performance similar to the different tuning points. However, since there is no clear link

between gains and performance such a tuning is much more cumbersome in the baseline

case compared to the ”tuning knobs” the MPC, as well as most other optimal control for-

mulations, provide the turbine designer with. For the same reason, having this easy and

direct access to the design trade-offs is also a pre-requisite for any kind of site-specific or

self-adjusting control design.

41.14≈1.5

111

6. Results

103104

0.01

0.02

0.03

0.04

0.05

Q(1,1)

std(ωr/ω0)

(a) Rotor Speed - STD

103104

1.05

1.1

1.15

Q(1,1)

max(ωr/ω0)

(b) Rotor Speed - MAX

103104

0.3

0.4

0.5

0.6

Q(1,1)

DEL(TMyt/TMyt,0)

103104

0.02

0.03

0.04

Q(1,1)

std(θ/θ0)

MPC no state constr

MPC + state constr

Prev. MPC + state constr

(d) Pitch Rate - STD

Figure 6.8.: Key operational metrics from NPP simulations with a mean wind speed of 20m/s

and 15% turbulence intensity for different MPC tuning weights placed on the rotor speed state.

The dashed red line marks the chosen setting that is considered in all other simulations of this

chapter.

Secondly, the MPCs with and without the state constraint behave very similar except at

low rotor speed weight settings. Without the state constraint, the overspeed threshold is

clearly violated for all settings below 1.0 ×104while the constrained controller manages to

stay within the limit. The effect on the standard deviation of the rotor speed and the tower

loads is only seen in the most extreme cases. The addition of the speed constraint can be

seen as decoupling the overspeed behavior from the general tuning problem. Only at very

low weights placed on the rotor speed, the tracking performance will become so poor that

the speed constraint will be active frequently and drive the tower loads and pitch activity

up.

Finally, the preview controller provides a significantly improved performance for all con-

sidered tuning options and for all four performance metrics. This should be considered

when designing any type of preview controller, MPC based or not, as by modifying the

tuning settings some of the performance increase in, e.g., overspeed prevention can be sac-

rificed for further loads reductions. Here, the big benefit of MPC is that with the addition

of the feedforward component, the feedforward and feedback components of the controller

are still treated and tuned together. Further, some of the proposed non-MPC preview con-

112

6.3. Gusts

trollers, such as the static system inversion approach [98], do not provide any means for

tuning the feedforward path controller as they are designed purely with the objective of

canceling the disturbance impact on the rotor speed.

6.3. Gusts

Next to operation in turbulent conditions, which represent normal power production oper-

ation, the performance of the controller during deterministic gust load cases is also crucial

for the turbine design. Here, the main objective of the controller is to reduce the maximum

load that occurs during the event as well as potentially keep the turbine within its operat-

ing limit so that it does not need to be shut down. The most important gusts that have to

be considered during the design are the Extreme Operating Gust (EOG) and the Extreme

Coherent Gust (ECG) as prescribed by the IEC guidelines [39].

6.3.1. Extreme Operating Gust

Load case 1.6 of the IEC guidelines requires simulation of an EOG with a return period of

50 years. Here the simulation is performed at rated wind speed, 120% of rated wind speed,

and the cut-out wind speed or v=11.5 m/s,v=13.8 m/s, and v=25 m/s, respectivly. The

magnitude of the gust is based on the turbulence class B assumption that was also used in

the NPP simulations. Figure 6.9 shows the resulting time series for the v=13.8m/scase

while table 6.1 compares the resulting maximum values for all three wind speeds.

It can be seen that both the unconstrained MPC and the baseline controller lead to violation

of the overspeed constraint, while the constrained MPC manages to keep the rotor speed

just below the limit. If preview control is used, the turbine will start pitching much earlier

and the speed excursion is greatly reduced so that there is no threat of an overspeed fault.

This type of behavior matches what has been found in other studies that applied preview

control to the main speed regulation problem (e.g., [50, 51, 98]).

When comparing the maximum resultant tower base bending moments, several observa-

tions can be made: As expected, the preview controller also reduces the load on the tower.

The two non-preview MPC have almost identical maximum tower loads as the maximum

occurs when the controller just starts to ”see” the state constraint and the controllers begin

to differ in their behavior. So in this case, using a state constraint does not increase the load

although the controller reacts more aggressively.

Further observations can also be made from comparing the results for the different wind

speeds given in table 6.1:

113

6. Results

0 10 20 30 40 50

0.8

0.85

0.9

0.95

1.05

1.1

1.15

1.2

t[s]

ω/ω0

MPC no state constr

MPC + state constr

Prev. MPC + state constr

Baseline

(a) Rotor Speed

0 10 20 30 40 50

0.5

1.5

2.5

t[s]

θ/θ0

(b) Pitch

0 10 20 30 40 50

−2

−1

t[s]

TMyt/TMyt,0

0 10 20 30 40 50

t[s]

v[m/s]

(d) Wind Speed

Figure 6.9.: Comparison of MPC and baseline controller during an Extreme Operating Gust at

a wind speed of 13.8m/swith a return period of 50 years

•The preview MPC provides superior performance across all three wind speeds with

the resultant rotor speed excursions and tower base bending moments being the

lowest of all four controllers at all simulated wind speeds. At v=11.5 m/sand

v=13.8m/s, the resulting tower base bending moment is at least 10% below the three

other variants while at v=25 m/sthe benefit is even higher. Considering that load

case 1.6 is often the load case that drives the design of the tower, this is a significant

reduction in loads that in some cases may directly translate into reduced costs for the

tower.

•With the addition of the state constraint to the MPC, the maximum rotor speed stays

below the assumed overspeed trigger value at all three wind speeds even though in

the unconstrained case there is a significant exceedance. That means in these scenar-

114

6.3. Gusts

max ω

ω0max θ

θ0max TMyt

TMyt,0 

Baseline 1.221 5.68 2.16

unconstrained MPC 1.124 5.11 1.88

constrained MPC 1.071 5.13 1.81

constrained MPC + preview 1.015 8.89 1.55

(a) v=11.5m/s

max ω

ω0max θ

θ0max TMyt

TMyt,0 

Baseline 1.218 2.25 2.83

unconstrained MPC 1.160 2.15 2.25

constrained MPC 1.080 2.36 2.21

constrained MPC + preview 1.008 1.76 1.99

(b) v=13.8m/s

max ω

ω0max θ

θ0max TMyt

TMyt,0 

Baseline 1.319 1.32 3.98

unconstrained MPC 1.185 1.42 3.10

constrained MPC 1.099 1.58 3.80

constrained MPC + preview 1.051 1.48 2.33

Table 6.1.: Maximum values of rotor speed, pitch angle, and tower base moment during EOG50

gust simulations starting at v=11.5 m/s,v=13.8 m/s, and v=25m/s

ios, the behavior of the controller is mainly driven by the state constraints and less by

trying to minimize the cost function. The controller is operating at its constraints.

•Even at v=11.5 m/s, the state constraint is necessary to prevent the overspeed fault.

However, it can be seen that with the current choice of tuning parameters and unmea-

sured disturbance assumption the controller is overly conservative. At ω

ω0=1.071,

the maximum rotor speed is still significantly below the limit of 1.1, but the constraint

was clearly active as the unconstrained case shows a different result. This behavior

can be explained with the low sensitivity of the aerodynamic torque with respect to

changes in pitch angle near rated wind speed (compare figure 2.5). Due to the low

sensitivity, the controller would need to pitch the blades out more in case of an un-

measured disturbance and the controller therefore keeps more control reserve via the

tube formulation.

•At v=11.5m/s, using the preview controller results in the highest pitch excursion.

This is due to the turbine briefly going into partial load operation during the early

part of the gust. In the current design, the MPC is not allowed to use the pitch, i.e.,

switch to full load operation, for the entire duration of the prediction horizon even if

the previewed wind speed is above rated wind speed. Therefore, once the turbine is

back to full load operation it needs to pitch especially fast. This is a unique situation

for the preview controller near rated wind speed. The behavior during these events

115

6. Results

could be improved by an improved set-point scheduling in the transition region [58].

•At v=25m/s, it can be observed that adding the state constrained increases the max-

imum resultant tower base bending moment significantly over the non-constrained

case. This is due to the very fast pitch motion and corresponding change in thrust that

the constrained controller requires to maintain rotor speed below the limit. In case the

tower load at this wind speed is of greater concern than the gust-ride-through feature

of the MPC, it might be necessary to relax the constraint and include the maximum

tower load as a further state constraint.

•At v=25m/s, even using the preview controller leads to a very significant speed

excursion. This is mainly due to the assumption that the preview controller has the

wind speed that the estimator would observe available ahead in time and not the true

wind speed. As shown in section 5.1.2, the estimator will attenuate the gust amplitude

somewhat and this attenuation will cause imperfect gust ride-through, especially at

high wind speeds where the assumed gust amplitudes are also the highest. For the

controller, this means that some unmeasured disturbance needs to be assumed also in

the preview case. How much exactly will be needed depends directly on the perfor-

mance of the measurement device that is used to generate the preview measurements

and can only be evaluated using a detailed sensor model [100].

6.3.2. Extreme Coherent Gust

Figure 6.10 shows the resulting time series from an Extreme Coherent Gust simulation start-

ing at rated wind speed corresponding to load case 1.9 of the IEC guidelines and table 6.2

lists the resulting maximum values. These results echo what was also observed for the

Extreme Operating Gust: If preview information is available, the turbine is able to ride

through the gust without much trouble. For this type of gust, neither the rotor speed nor

the tower base bending moment increase above the respective stationary values at all. If no

preview information is available, then the MPC without the state constraint behaves simi-

lar to the baseline controller: The rotor speed rises well above the assumed maximum rotor

speed and the turbine would have triggered an overspeed fault. If the speed constraint

is activated, the turbine starts to pitch aggressively once it ”sees” the overspeed threat at

approximately t=8s and manages to contain the speed below the limit. This, however,

comes at the cost of a strong excitation of the tower which the controller then needs to

dampen out through rapid pitch motions which are clearly visible between t=10 s and

t=20s in figure 6.10 (b).

It can be concluded that for gust events, as has been shown earlier, preview MPC provides

a significant performance increase. Even without preview measurements, MPC has signif-

116

6.3. Gusts

0 5 10 15 20 25

0.8

0.85

0.9

0.95

1.05

1.1

1.15

1.2

t[s]

ω/ω0

MPC no state constr

MPC + state constr

Prev. MPC + state constr

Baseline

(a) Rotor Speed

0 5 10 15 20 25

t[s]

θ/θ0

(b) Pitch

0 5 10 15 20 25

−0.5

0.5

1.5

t[s]

TMyt/TMyt,0

0 5 10 15 20 25

t[s]

v[m/s]

(d) Wind Speed

Figure 6.10.: Comparison of MPC and baseline controller during an extreme coherent gust at a

wind speed of 13.8m/swith a return period of 50 years

max ω

ω0max θ

θ0max TMyt

TMyt,0 

Baseline 1.191 8.57 1.24

unconstrained MPC 1.170 8.76 1.20

constrained MPC 1.097 8.76 1.20

constrained MPC + preview 1.028 8.44 1.00

Table 6.2.: Maximum values of rotor speed, pitch angle, and tower base moment during an

Extreme Coherent Gust (ECG) starting at v=11.5m/s

icant benefits as with the use of state constraints unnecessary shut-downs can be avoided.

Due to its multi-variable nature and optimal control formulation, it also provides improved

performance when compared to a simple controller, even if state constraints can be ig-

nored.

117

6. Results

It should be noted that using state constrained MPC cannot guarantee avoiding overspeed

faults for all gust situations. A gust can be so strong that no feasible solution exists; i.e.,

even running at the control constraints cannot prevent the rotor from transgressing the

constraint. Additionally, as stated in section 5.3.3, especially during gust situations, it is

not possible to include the entire future wind speed in the assumed unmeasured distur-

bance for the tube calculation. Even if perfect state constraint handling is not possible with

the presented tube-based MPC, it will still greatly reduce the overspeed probability and

increase overall controller performance, as is also evident from the performance evaluation

under turbulent conditions. The values that were chosen here for the unmeasured distur-

bance resulted in the state-constrained MPC being able to not trigger an overspeed fault

even in the event of a gust that has a return period of 50 years so that the residual over-

speed risk can be considered as small. For a real turbine, the risk of an overspeed due to

fault modes like temporary loss of communication between the turbine controller and the

pitch system will be several orders of magnitude larger than the risk linked to the controller

not being able to handle a certain gust scenario. Nevertheless, it remains an interesting

probabilistic design challenge to calculate the real life overspeed trigger probabilities with

the controller design and tuning.

6.4. Fault Conditions

6.4.1. Emergency Stop

As outlined in section 2.2.2, one of the fault scenarios that is commonly considered in the

design of wind turbines is load case 5.1 according to the IEC 61400 [39] guidelines: An

emergency stop of the turbine with immediate and complete disconnection of the generator.

In case of such an event, most turbines will pitch the blades out at a fixed rate to bring the

turbine to a stop using some kind of backup power source, e.g., batteries. With the backup

power available, it is however also possible to maintain closed-loop control of the turbine.

In order to demonstrate how state-constraint MPC can be used in a shut-down event, a

grid loss is simulated once with the MPC and once with a baseline strategy. This baseline

controller simply increases the pitch angles of all three blades using a fixed rate of 10 deg/s.

In the MPC case, once the grid-loss is detected, the generator torque state, the commanded

torque rate bounds, and the weights placed on the power output are set to zero. The con-

troller will now try to maintain rated speed without the torque from the generator and will

therefore start increasing the pitch angle immediately. As unconstrained MPC is not able to

handle this scenario, it is excluded from the comparison. Further, since the wind speed, at

least under IEC design conditions, does not change during this event, preview control does

118

6.4. Fault Conditions

0 5 10 15 20

0.7

0.8

0.9

1.1

t[s]

ω/ω0

(a) Rotor Speed

0 5 10 15 20

0.5

1.5

t[s]

θ/θ0

(b) Pitch

0 5 10 15 20

−4

−3

−2

−1

t[s]

TMyt/TMyt,0

0 5 10 15 20

0.2

0.4

0.6

0.8

1.2

1.4

t[s]

MG/MG,0

MPC + state constr

Baseline

(d) Generator Torque

Figure 6.11.: Comparison of MPC and simple shutdown procedure during grid loss event at a

wind speed of 17m/s

not have an effect, so that preview and non-preview MPC are identical and only a single

state-constrained MPC is considered here. The set-point for the rotor speed is held constant

during the entire event. Unlike in the simple open-loop scheme, the rotor speed is main-

tained near its rated value and the turbine is not coming to a stop. Stopping the turbine can

however easily be achieved by ramping down the speed set-point after the initial transient

has died out.

As can be seen in figure 6.11, using the MPC, the maximum rotor speed during the event

increases slightly above the level obtained with the simple shut-down, but remains below

the chosen constraint. On the other hand, the fast increase in pitch angle in the simple

shutdown case causes strong tower vibration with little damping, while when the MPC is

used, the tower base bending moment does not increase above its stationary value at all.

119

6. Results

Figure 6.12 shows the maximum rotor speed and tower base bending moment from grid

loss simulations over the entire range of full load wind speeds. It can be seen that what was

observed in Figure 6.11 is also true at other wind speeds: Using MPC during the shut-down

increases the maximum rotor speed slightly, but still manages to hold it below the rotor

speed constraint of ωmax =1.1ω0while the maximum tower load is reduced significantly.

14 16 18 20 22 24

1.04

1.05

1.06

1.07

1.08

1.09

v[m/s]

max(ω)/ω0

(a) Rotor Speed

14 16 18 20 22 24

1.5

2.5

3.5

v[m/s]

max(abs(TMyt))/TMyt,0

MPC + state constr

Baseline

(b) Tower Base Bending Moment

Figure 6.12.: Maximum rotor speed and tower base bending moment for MPC and open-loop

pitch ramp shut-down control for wind speeds from 14m/sto 25 m/s

Not all of this performance gain can be attributed to the use of MPC alone. Most of it is

probably due to the use of closed-loop control instead of the simple open-loop scheme.

However, the main aim of such a closed-loop shutdown controller is to maintain the rotor

speed and possibly additional states, such as the tower deflection, below safety critical

levels for a large range of operating conditions and fault scenarios (e.g., different levels of

available torque), which is difficult to achieve using non-MPC controller designs.

6.4.2. Loss of Grid Connection during an Extreme Operating Gust

Load case 1.5 simulates the loss of the grid connection and thus counter torque at the gener-

ator during an extreme operating gust. To account for the reduced probability of these two

events occurring at the same time, a return period of only year and corresponding lower

amplitude has to be considered for the EOG instead of the 50 years that were considered

in load case 1.6 (see figure 2.10). As part of a full turbine loads assessment, different times

of occurrence of the grid loss with respect to the gust would need to be considered. Here,

the intent is only to highlight the controller behavior in this scenario and not to compute

detailed loads so that the analysis is limited to the case where the grid loss occurs exactly in

the instant where the gust reaches its maximum wind speed. The controller reaction to the

120

6.4. Fault Conditions

grid loss is the same as described in the previous section: The torque command is set to zero

for all controllers. The baseline controller pitches the blades to the feathered position at a

fixed rate while the MPC tries to maintain the rotor speed at its set-point, but is constrained

to use only pitch actuation instead of pitch and torque.

Similar to the load case 1.6, the analysis is performed at rated wind speed, 120% of rated

wind speed, and the cut-out wind speed. Figure 6.13 shows the resulting time series for the

v=13.8m/scase, while table 6.3 shows the relevant maximum values for all three wind

speeds.

0 5 10 15 20 25 30 35

0.8

0.85

0.9

0.95

1.05

1.1

1.15

1.2

t[s]

ω/ω0

MPC no state constr

MPC + state constr

Prev. MPC + state constr

Baseline

(a) Rotor Speed

0 5 10 15 20 25 30 35

t[s]

θ/θ0

(b) Pitch

0 5 10 15 20 25 30 35

−4

−3

−2

−1

t[s]

TMyt/TMyt,0

0 5 10 15 20 25 30 35

0.2

0.4

0.6

0.8

1.2

1.4

t[s]

MG/MG,0

(d) Generator Torque

Figure 6.13.: Comparison of MPC and baseline controller during an Extreme Operating Gust at

a wind speed of 13.8m/swith a return period of 1 year and occurrence of a grid loss at the wind

speed maximum

The first important observation is that at all three wind speeds only the preview controller

is able to maintain operation below the overspeed level. Even the constrained MPC fails to

121

6. Results

max ω

ω0max θ

θ0max TMyt

TMyt,0 

Baseline 1.207 30.48 2.00

unconstrained MPC 1.183 6.47 1.98

constrained MPC 1.139 8.27 1.93

constrained MPC + preview 1.041 7.24 2.17

(a) v=11.5m/s

max ω

ω0max θ

θ0max TMyt

TMyt,0 

Baseline 1.206 10.26 2.70

unconstrained MPC 1.206 2.48 2.53

constrained MPC 1.138 2.91 2.50

constrained MPC + preview 1.082 1.99 1.45

(b) v=13.8m/s

max ω

ω0max θ

θ0max TMyt

TMyt,0 

Baseline 1.223 3.79 4.54

unconstrained MPC 1.320 1.51 3.68

constrained MPC 1.218 1.78 3.98

constrained MPC + preview 1.083 1.57 1.70

Table 6.3.: Maximum values of rotor speed, pitch angle, and tower base moment during an

EOG1 gust with a grid loss event occurring at the highest wind speed

limit the rotor speed as the scenario of a sudden loss of counter torque is not included in

the robustification of the state constraint and most of the control margin the controller had

has been ”used up” by the gust. In figure 6.13 b) the constrained MPC can be clearly seen

as being in Backup Mode between approximately t=10 s and t=12 s where the rotor speed

is above the limit and the pitch is increased at a fixed rate similar to the baseline controller.

Unlike the baseline controller, however, as soon as the rotor speed has been reduced below

the constraint, closed loop operation is resumed.

Similar to the pure grid loss simulations, it can be seen that maintaining closed loop control

and thus also tower damping, significantly helps in bringing the tower to a rest. In all of

the MPC cases, the tower base bending moment only has to go through two full cycles

before being at rest, while in the baseline case with its open-loop braking procedure, the

tower oscillation only has the low damping that the fore-aft mode naturally has at high

pitch angles.

In the preview case, most of the speed excursion coming from the gust can be canceled by

the controller by pitching ahead in time so that there is enough controller reserve to handle

the loss of counter-torque at t=10.3s and a potential rotor speed fault is avoided. While

at first, prevention of an overspeed might not seem necessary in this event as the turbine is

already in fault mode and does not produce any power, there are certain scenarios where it

is beneficial to keep the turbine running even though it is not producing power at least for

122

6.4. Fault Conditions

a short period of time. This behavior is analyzed in more detail in the next section. Finally,

the sub-optimal behavior of the preview controller for gusts near rated wind speed that

was observed in load case 1.6 can be observed here as well: The preview controller reacts

very aggressively during the gust at 11.5m/scausing a significant pitch excursion and tower

load.

6.4.3. Fault Ride Through

The loadcases 5.1 and 1.5 that were discussed in the previous sections simulate an emer-

gency stop with disconnection of the generator. This emergency stop can be caused by a

number of reasons with one of the most important being a so-called grid loss: Due to a

disturbance in the electrical network the turbine is connected to, the turbine needs to be

disconnected from the grid. There are, however, also grid disturbances that especially new,

modern wind turbines are expected to run through without triggering a fault. A feature

which is generally called Fault Ride Through (FRT). The types of grid disturbance a tur-

bine needs to run through vary heavily depending on the specific grid conditions. Running

through these faults is mainly a challenge that needs to be dealt with when choosing and

designing the system architecture of the electrical system and the control algorithms for the

CCU.

Nevertheless, for many potential grid faults and electrical system designs, the turbine

might need to run at a reduced level of generator torque or even no generator torque for a

short period of time and due to the sudden loss of counter torque, the entire rotor will speed

up. At this point, it is no longer only a CCU control issue but a turbine-level control issue

as well. As outlined by Ramtharan et al. [90], the sudden loss of counter torque can easily

cause the rotor speed to increase above its trip limit, trigger an overspeed fault, and lead to

a complete shut-down of the turbine which would be a violation of the FRT requirement.

The main goal of the turbine controller during these types of events is therefore to maintain

safe operation without triggering a fault even if the generator torque is not or only partially

available as an actuator to the turbine control system.

In order to demonstrate the usefulness of MPC during these types of events, the turbine

behavior is simulated during a fictitious grid event based on the generic Low Voltage Ride

Through (LVRT) curve introduced in figure 2.8: The grid voltage drops suddenly to 0% of

nominal. After 2.5 seconds, it starts to increase linearly up to about 75%. At that level, it

remains constant for another 2 seconds until it finally jumps back to 100%. For simplicity,

this voltage curve is directly translated into available generator torque. It should be noted

that the actual electrical behaviour is much more complicated and that in most cases, the

generator torque is not proportional to the voltage at the grid side converter. Nevertheless,

123

6. Results

no grid voltage generally means that no power can be transmitted into the grid and that, if

the turbine would continue to generate power, it would, for example, overload the DC link

on a full conversion turbine [90]. Therefore, using the voltage curve as a torque limitations

is an, although not fully realistic, reasonable choice to model this behaviour.

0 5 10 15 20 25

0.9

0.95

1.05

1.1

1.15

t[s]

ω/ω0

(a) Rotor Speed

0 5 10 15 20 25

0.5

1.5

2.5

t[s]

θ/θ0

(b) Pitch

0 5 10 15 20 25

−1

−0.5

0.5

1.5

t[s]

TMyt/TMyt,0

0 5 10 15 20 25

0.2

0.4

0.6

0.8

1.2

1.4

t[s]

MG/MG,0

MPC no state constr

MPC + state constr

Baseline

(d) Generator Torque

Figure 6.14.: Comparison of MPC and baseline control procedure during a grid fault event at a

wind speed of 13m/s

In the MPC case, as soon as the available torque drops below 100%, the torque rate in the

controller is constrained to be zero and the torque command from the MPC is replaced

with the maximum available torque. The turbine controller now only uses the pitch an-

gle as an actuator. Torque control authority is essentially ceded to the CCU. The turbine

level controller however is ”aware” of the current generator torque5and includes it in its

predictions.

5e.g., it is fed that information from the CCU

124

6.4. Fault Conditions

Similar to the MPC case, in the baseline case, the torque controller is replaced directly by

the maximum torque level according to the grid fault definition described above.

Figure 6.14 shows the behavior for the MPC variants and the baseline controller during

such an event with a constant wind speed of 13m/s. Although all three controllers behave

fairly similarly, it can be seen that using the baseline controller, the speed increases above

the overspeed trigger level and, had the respective supervision been modeled, would cause

an overspeed shut-down. There is also a slight difference between the MPC with the rotor

speed state constraint and the one without as due to the robustification, the constrained

MPC sees an overspeed threat and reacts more aggressively. The performance of the MPCs

is better than the baseline controller in this scenario because right at the time of the drop

in available torque, they ”see” the upcoming rise in generator speed and start to increase

the pitch angle while the baseline feedback controller needs to wait for the rotor speed

to actually increase. In a way, the MPC in the given formulation includes a feedforward

component from the available torque.

12 14 16 18 20 22 24

1.06

1.08

1.1

1.12

1.14

1.16

1.18

v[m/s]

max(ω)/ω0

(a) Rotor Speed

12 14 16 18 20 22 24

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

v[m/s]

max(abs(TMyt))/TMyt,0

MPC no state constr

MPC + state constr

Baseline

(b) Tower Base Bending Moment

Figure 6.15.: Maximum rotor speed and tower base bending moment during an FRT event for

state constrained and unconstrained MPC variants and baseline control for wind speeds from

13m/sto 25 m/s

In order to more systematically examine the performance, the same event is simulated at

all wind speeds from 12 m/sto 25 m/sand the resulting maximum rotor speed levels are

shown in figure 6.15 a). Using the baseline controller, the rotor speed increases above the

limit only at the lower wind speeds up to 15m/s. At higher wind speeds, the sensitivity of

the aerodynamic moment to changes in pitch is so high that even the baseline controller

manages to reduce the aerodynamic moment fast enough. At high wind speeds, due to the

faster pitch actuation, the tower bottom loads actually increase with the use of the MPCs.

Compared to other load cases, however, they are still small.

125

6. Results

Controller No. of max rotor

speed violations (out

of 20)

average maximum

rotor speed per event

maximum rotor

speed for all events

MPC no state constraint 2 1.071 1.114

MPC + state constraint 0 1.067 1.096

Preview MPC + state constraint 0 1.069 1.082

Baseline 10 1.094 1.136

Table 6.4.: FRT performance of the different control methods in turbulent conditions: Results

from a 600 second simulation with a FRT event triggered every 30 seconds (20 events total) and

an average wind speed of 20m/sand a turbulence intensity of 15%.

In reality, the wind speed is almost never constant. So even if, as shown in figure 6.15, there

is no threat of causing an overspeed fault during an FRT using the baseline controller at

high wind speeds, the FRT could be triggered while a gust is hitting the turbine and the

combination of turbulence and FRT could bring the speed above the limit. To show the

FRT performance of the MPC while the wind speed is varying, the turbine is simulated in

turbulent conditions and the FRT is triggered repeatably. Figure 6.16 shows the resulting

time series from a 10 minute simulation at an average wind speed of 20 m/sand a turbulence

intensity of 15% where the same grid event that was used previously is triggered every 30

seconds for a total of 20 FRT manoeuvres. Because the wind speed is now varying, MPC

with preview is also considered here.

As shown in table 6.4, even though in static conditions the baseline controller did not cause

rotor speeds above the overspeed threshold, in turbulent conditions an overspeed fault

would have been triggered in 10 out of 20 events. The unconstrained MPC is also no longer

capable of maintaining operation below the limit in all cases. Yet, due to the faster reaction

it only causes two transgressions. Finally, both the preview and non-preview constrained

MPC end up at a maximum rotor speed on a similar level as in the case with static wind

conditions showing that adding the constraint helps avoiding some of the variation due to

turbulence.

In summary, state-constrained MPC can provide improved Fault Ride Through perfor-

mance when compared to the baseline controller. Similar to the discussion on the closed-

loop braking procedure, it is most likely possible to achieve similar performance also by

modifying the classically designed baseline controller. For example, the pitch controller

could be made more aggressive whenever a grid fault is detected. However, especially for

real-life faults, a large number of potential fault modes and operating conditions need to

be considered and treating all these cases using specifically designed controllers can eas-

ily become prohibitive. On the other hand, using MPC one common framework for both

NPP and fault conditions can be used as, via the explicit prediction and control and rate

constraints, many of these scenarios can be directly included in the problem formulation.

In the end, almost all fault scenarios require the controller to run through an event without

126

6.4. Fault Conditions

0 100 200 300 400 500 600

0.9

0.95

1.05

1.1

1.15

t[s]

ω/ω0

MPC no state constr

MPC + state constr

Prev. MPC + state constr

Baseline

(a) Rotor Speed

0 100 200 300 400 500 600

0.5

1.5

t[s]

MG/MG,0

(b) Generator Torque

0 100 200 300 400 500 600

t[s]

v[m/s]

Figure 6.16.: Comparison of MPC and baseline control procedures under turbulent conditions

with an average wind speed of 20m/sand a turbulence intensity of 15% and an occurence of a

grid fault every 30 seconds.

leaving the operational limits of the turbine by using limited available control capability in

a somehow optimal manner.

127

7. Conclusion and Outlook

The application of the Model Predictive Control technique to the wind turbine control prob-

lem has been studied through one exemplary control design for a full load torque and col-

lective pitch controller. The controller has been evaluated through extensive aero-elastic

system simulations for various scenarios similar to those that are typically considered dur-

ing the design of a wind turbine.

The presented controller, although not formulated and implemented in a speed optimized

manner, is already capable of running in real time so that an implementation on an ac-

tual turbine does not seem unfeasible. The comparably low computational burden was

achieved through the use of a plant model that is specifically tailored to the control prob-

lem at hand and which has, especially compared to the models that are generated by some

of the aero-elastic simulation tools, only few states. Computational complexity is reduced

further through the use of the presented linearization scheme that avoids any type of true

nonlinear MPC formulation. One major step that would be required for full implementa-

tion is the extension of the MPC to also cover the entire partial load region. However, as

has also been shown for the upper partial load operating region, extension of the presented

concept to the entire wind turbine operation is straightforward and has only been excluded

here as earlier studies [51] indicated that there is little benefit.

It has been shown that MPC is a natural fit for integrating knowledge about the future wind

speed, for example from LIDAR measurements, in the control formulation. MPC is espe-

cially suitable for preview control, as it allows combining the feedback and the feedforward

controller in a single controller so that all the other features of MPC, such as constraint han-

dling and optimal control formulation, are applicable to the combined controller. While

the vast majority of all current MPC research for wind turbines focuses on using MPC to

design a LIDAR enabled preview controller, here both preview and non-preview control

are considered.

Much of the previous research on the application of MPC to wind turbines only included

constraints on the control variables, but not on the states. In this study, it was detailed that

including state constraints requires robustification of the controller due to the future wind

speed acting as a large unknown disturbance on the system. One method for robustification

based on the tube-based robust control approach by Rawlings and Mayne [91] has been

128

applied here and was used to include the overspeed limit of the turbine explicitly in the

control formulation.

The results showed only small benefits of the non-preview MPC when compared to a base-

line controller under normal, turbulent operating conditions. Bigger performance improve-

ments were found especially in the special scenarios such as the gust and fault load cases

where very significant extreme load reductions were observed. Especially the use of MPC

to control the turbine during shut-down and grid failure events has proven to be an appli-

cation with significant potential for loads reduction.

The inclusion of an inequality constraint on the rotor speed state using the presented robus-

tification scheme results in a significantly lower risk of the turbine creating an overspeed

fault. In fact, across all considered simulations, only the combination of a full grid loss,

which is not considered in the robustification scheme, and an extreme gust caused a viola-

tion of the overspeed limit if the state constraint was used. It was further shown that the

addition of the overspeed constraint helps in separating the overspeed avoidance objective

from the general tuning problem of the controller to a certain extent.

If the upwind wind speed information is available, it can be included without any further

controller modification. Even with the non-perfect preview wind speed information that

was considered in the simulations, the control performance improved drastically compared

to the non-preview controllers. Overall, tower extreme and fatigue loads were reduced by

around 5% and 30%, respectively, with at the same time reduced pitch activity and almost

completly eliminated overspeed risk. Even main shaft and blade loads were reduced, even

though they were not explicitly considered. It would however be foolish to attribute all

these gains to the use of MPC alone. Simply having the preview information available

drives most of the benefit that can be obtained by using the preview MPC. The performance

difference between a non-MPC preview controller and a MPC preview controller is likely to

be only of the same order of magnitude as the difference observed between the non-preview

MPC and the baseline controller.

Many of the examined benefits of MPC do not necessarily require MPC. For example, pre-

view control can be implemented using system inversion approaches, better overspeed

handling can be achieved by switching to a dedicated overspeed controller mode as sug-

gested by Kanev and van Engelen [47], and a closed-loop shut-down controller can likely

also be designed using classical methods. On the other hand, all of this adds complexity

to the controller design. Every new controller module or independent controller for a very

specific task will need to be specifically designed and likely have its own tuning param-

eters. Given the high number of different scenarios, operating points, system constraints

this approach of ”add-on” features can easily become prohibitive with numerous hard-to-

evaluate interactions between the various functions.

129

7. Conclusion and Outlook

By using state constraints and appropriately modifying the controller objective, MPC is

able to handle not only normal turbine operation but also special scenarios such as extreme

gusts or grid-loss cases. This is a first step towards fault tolerant control of wind turbines

where a single MPC is used to handle the large variety of conditions and scenarios a wind

turbine needs to be able to handle. This type of controller would not only focus on reducing

the mechanical loading of the turbine in normal operation, but also on minimizing the loads

during special events. By including the probability of certain faults occurring directly in the

control design via the selection of state constraints, it can further contribute to the overall

turbine reliability by keeping the turbine online longer and reducing unnecessary faults.

The presented approach for robustification of the state constraint essentially treats the un-

measured disturbance as a tuning parameter that is not connected to the true changes in

future wind speed. While this has proven to be a powerful method to improve the over-

speed performance, it also means the parameter needs to be set by ”trial and error” and

there is no way of automatically adjusting it based on, e.g., site turbulence or estimator

tuning. Particularly the estimator performance has proven to have a big impact on the per-

formance of the presented robustification scheme and careful tuning was required to mini-

mize the wind speed estimation error. For further research in this area, rethinking how the

disturbance tubes are generated and developing an approach that is less conservative than

the currently used step input approach will likely also have a big impact on the accuracy

and usefulness of the presented method.

One of the biggest benefits of MPC is the ability to also address extreme and not only the

fatigue loads that most controller design for wind turbines try to reduce. Using the pre-

sented methods, extreme loads reductions were achieved especially in the gust and fault

load cases. The next logical step would be to include the loads directly in the MPC formu-

lation in the form of another state constraint. The most obvious candidate to be included

next is the maximum tower base bending moment. As this load is closely related to the

tower top deflection, which is already a state in the current model, this constraint could be

included without any update to the plant model. Further, while the collective pitch and

torque controller can generally be treated as decoupled from a potential individual pitch

controller, integrating them in the MPC framework would allow implementation of a state

constraint on the blade tip deflection whenever a blade passes in front of the tower which

is a key blade design driving criterion that is relevant especially in some of the gust and

fault load cases.

Overall, it was shown that MPC offers numerous benefits to the wind turbine control prob-

lem. Some of which were examined in detail in this thesis. Future studies on MPC for wind

turbines should therefore not be limited to preview control, but also explore the benefits

MPC has to offer even if no preview control is used.

130

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137

Appendix A. Determination of the Maximal Control Invariant Set

Appendix A.

Determination of the Maximal Control Invariant

Set

In order to design a Model Predictive Controller that is stable, it is often necessary to con-

strain the final state so that it lies within an output admissible set. A set of state vectors

is admissible if using any vector within the set as an initial condition for a constrained dy-

namic system will result in all future future states lying within the constraint set. Therefore,

if the system is unconstrained every state vector is admissible. The term output admissible

is used for the more general case where the constraint set is not only defined on the sys-

tem states but also on further outputs. The maximal output admissible set is the set of all

state vectors that are output admissible. Gilbert and Tan [28] provide a method for the de-

termination of the maximal output admissible set. For a discrete linear state space system

without inputs

xk+1=Axk

yk=Cxk(A.1)

which is subjected to a set of sscalar linear output constraints

iy≤eii=1...s, (A.2)

where Eiis a vector and eia scalar, the maximum output admissible set can be calculated

using the following algorithm:

1. Set k=0.

138

2. Solve the following optimization problem:

J∗

i=maxxEiCAk+1x

s.t.







EjCx

EjCAx

EjCA2x

EjCAkx







≤













for all j=1...s(A.3)

for all values i=1...s. If J∗

i<eifor all i=1...sthen set k∗=kand exit algorithm.

3. Set k=k+1 and go to step 2.

With k∗determined, the maximum output admissible set O∞is described by stimes k∗

scalar inequalities:

O∞=nx∈Rn:EiCAkx<ei,i=1...s,k=0...k∗o(A.4)

The optimization problems defined in (A.3) are linear optimization problems which can be

solved very efficiently even for the large number of inequality constraints that are likely to

occur here using simplex based linear program solvers such as the COIN Linear Program

code [24].

In many MPC applications, it cannot be assumed that the plant is operated uncontrolled,

i.e., without any inputs, and an output admissible set for controlled operation is required.

In this case, the system model is:

xk+1=Axk+Buk

yk=Cxk+Duk(A.5)

Further, constraints are usually not only placed on the outputs but also on the controlled

variables:

Fiu≤fii=1...sc(A.6)

If a linear controller u=−Kxis used then by defining new system matrices

A∗=A−BK

C∗="C−DK

−K#(A.7)

139

Appendix A. Determination of the Maximal Control Invariant Set

and output vector

y∗= y

u!(A.8)

the system can be brought into the format of (A.1) again

xk+1=A∗xk

y∗

k=C∗xk. (A.9)

As the control inputs uare now part of the modified output vector y∗, the control constraints

(A.6) can be treated as output constraints. Using these definitions, it can be seen that the

problem of calculating a maximum admissible set for controlled operation is equivalent to

calculating the set for the uncontrolled case that was treated previously.

140

Appendix B.

Design of a Baseline Controller

B.1. Pitch Controller

The pitch controller is designed as a PI controller that sets the pitch command θcbased on

the difference of the rotor speed ωfrom a reference rotor speed ωref

θc=Kp+1

sKinG(ω−ωref)(B.1)

where Kpand Kiare the gains of the controller. The gearbox ratio nGis included here as

instead of the rotor speed the generator speed is the measured turbine output. During the

full load operation of a wind turbine the reference speed will simply be the rated rotor

speed ωref =ω0. However, for the purpose of this analysis, the reference is assumed to be

variable.

The gains of the PI controller are selected based on the fundamental drive train equation

(4.5):

J˙

ω=MA(θ,ve,ω)−nGMG. (B.2)

The generator torque, pitch angle, and wind speed can all be seen as inputs to this dynamic

system. Now, linearizing the aerodynamic moment gives the following transfer function

from pitch angle to rotor speed:

θ(s) =

∂MA

∂θ

Js −∂MA

∂ω

. (B.3)

By combining equations (B.1) and (B.3) a closed-loop transfer function from the speed set-

141

Appendix B. Design of a Baseline Controller

point ωref to the actual rotor speed is calculated:

ωref (s) =

∂MA

∂θ

JKps+KinG

s2+−∂MA

∂θ KpnG−∂MA

∂ω 

| {z }

2ζcl ωcl

s+−∂MA

∂θ KinG

| {z }

ω2

. (B.4)

Using (B.4) at a given operating point with known ∂MA

∂θ and ∂MA

∂ω , the gains for the PI-

controller Kpand Kiwill follow directly from the choice of closed-loop poles in terms of

ωcl and ζcl.

In the present case, the PI controller is designed at a wind speed of 20m/sand ωcl and ζcl

are chosen so that the speed control level is roughly similar to the MPC controller resulting

in poles characterized by

ωcl =0.48rad/sζcl =0.85 (B.5)

and corresponding PI gains

Kp=0.003rad/rad

sKi=0.0015rad/rad. (B.6)

The pitch controller further needs to be gain-scheduled to account for the differences in the

system dynamics due to the nonlinear aerodynamics. Here, the gains are adjusted so that

the terms Ki·∂MA

∂θ and Kp·∂MA

∂θ are constant. It can be seen in equation (B.4) that this choice

of gain scheduling corresponds to a constant value of ωcl. Since ∂MA

∂ω is small compared to

Kp, the damping value ζcl is also almost constant. Figure B.1 (a) shows the pitch sensitivity

∂MA

∂θ as a function of wind speed for the steady operating curve. As, on a wind turbine

without a wind speed estimator, the wind speed is difficult to measure reliably, the gain

schedule is however not implemented as a function of wind speed but as a function of the

pitch angle instead. Figure B.1 (b) shows the resulting gain multiplier, based on nominal PI

gains at a wind speed of 20m/s, directly calculated from the aerodynamic properties as well

as the smoothed curve which is implemented in the controller.

The controller is further implemented with a rate limitation max(|˙

θc|)≤9deg/sand a con-

ditional integrator to prevent integrator wind-up.

142

B.2. Torque Controller and Switching Conditions

12 14 16 18 20 22 24

−10

−8

−6

−4

−2

0x 107

v[m/s]

Mθ[Nm/rad]

(a) Pitch Sensitivity

5 10 15 20

0.5

1.5

2.5

3.5

θ[deg]

Gain Multiplier

raw

Gain Schedule

(b) Gain Schedule

Figure B.1.: Stationary torque-to-pitch sensitivity as a function of wind speed and resulting gain

schedule for baseline PI pitch controller

B.2. Torque Controller and Switching Conditions

In partial load operation, the turbine speed is controlled via torque actuation, while the

pitch angle is fixed at θ=1 deg. Only a torque controller for the upper partial load region

(the vertical line in the torque-speed-curve of figure 2.1) is implemented and just like the

pitch controller, the torque controller tries to maintain the rated rotor speed. A controller

for variable speed control in lower partial load operation is not implemented. Similar to

the pitch controller, the torque controller is designed as a PI-controller:

MG,c=Kp,trq +1

sKi,trq(ω−ωref)(B.7)

Here, no gain scheduling is used and the gains are chosen to be

Kp,trq =1200Nm/(rad/s)Ki,trq =600 Nm/rad (B.8)

corresponding roughly to closed-loop poles characterized by

ωcl =0.4rad/sζcl =0.45 (B.9)

using similar considerations as for the pitch controller.

Switching Conditions The turbine switches from full load operation, where the pitch is

controlling the speed and torque is used to maintain constant power, to partial load opera-

tion, where the pitch is held constant and the torque is controlling the speed when the pitch

143

Appendix B. Design of a Baseline Controller

angle is less than or equal to the fine pitch angle of θ=1 deg and the power is below rated

power. The turbine switches from partial load operation to full load if the power is higher

than rated power. In both cases, the integrator for the pitch and torque PI-controllers are

reset to the current pitch and torque values to allow for a smooth transition between the

two operating regions.

B.3. Tower Damper

The tower damper is designed as a PI controller

θc,TFA =Kp,TFA +1

sKi,TFA¨

xt(B.10)

acting directly on the measured tower top acceleration in the longitudinal (along the wind)

direction. The chosen gains are

Kp,TFA =0.2degs2/mKi,TFA =1degs/m(B.11)

The choice of an integral gain significantly larger than the proportional gain is motivated

by the considerations presented in section 2.1.3, where it was shown that integral control

can be seen as adding additional damping to the tower. The small proportional feedback is

added in addition to account for the phase delay caused by the pitch actuator. To prevent

the tower damper from interacting too much with the regulation of the rotor speed, the out-

put of the tower damper is limited to |θc,TFA| ≤ 1 deg and implemented using conditional

integration to prevent integrator wind-up as was also done for the main pitch controller.

The tower damper is deactivated whenever the turbine is not in full load operation.

144