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2016-01-27

Data-Driven Modeling of Wind Turbine Structural

Dynamics and Its Application to Wind Speed

Estimation

Saberi Nasrabad, Vahid

Saberi Nasrabad, V. (2016). Data-Driven Modeling of Wind Turbine Structural Dynamics and Its

Application to Wind Speed Estimation (Unpublished master's thesis). University of Calgary,

Calgary, AB. doi:10.11575/PRISM/25518

http://hdl.handle.net/11023/2788

master thesis

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UNIVERSITY OF CALGARY

Data-Driven Modeling of Wind Turbine Structural Dynamics and Its Application

to Wind Speed Estimation

by

Vahid Saberi Nasrabad

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING

CALGARY, ALBERTA

January, 2016

c© Vahid Saberi Nasrabad 2016

Abstract

In wind turbine control systems, the wind speed measurement is used in order to derive

the optimal shaft speed for achieving the Maximum Power Point Tracking (MPPT) and to

adjust the pitch angle optimally for protecting the turbine from excessive loading. In this

thesis, a tower deflection based effective wind speed estimation method is proposed. The

tower dynamics is identified using subspace system identification method. To estimate the

effective wind speed, an online model-based aerodynamic thrust force estimator is designed

and implemented using Kalman filter and recursive least square algorithm. The estimated

aerodynamic thrust force is used as an input to a neural network estimator to solve the

inverse aerodynamic thrust force equation and estimate the effective wind speed. Finally,

the simulation results for effective wind speed estimation for a turbulent wind field are

presented and an evaluation method based on correlation coefficient is used to validate the

results.

ii

Acknowledgements

I would like to express my sincerest gratitude and thanks to my supervisor, Dr. Qiao Sun,

for her continuous support, guidance and helpful insights during my research. I would also

like to thank her for being so nice, supportive and caring to her students and helping me to

gain industrial experience. Besides, I would like to thank my thesis committee: Dr. David

Wood, Dr. David Westwick and Dr. Abdulmajeed Mohamad for their great questions in my

defense and their insightful comments.

My very special thanks go to lovely Haleh for her great support, suggestions, grammatical

editing and helping me to structure and write this thesis.

I would also like to thank my great groupmates, Dayuan, Graeme and Ehsan, for their

helpful suggestions. Also, many thanks go to my dear roommates Hossein and Omid who

supported me during my studies.

Finally, I wish to give my deepest gratitude to my dear parents and my beloved brother,

Navid, for their unconditional love, support and encouragement.

iii

Dedicated to my parents for their endless love & support

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Wind Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Wind Turbine System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Wind Turbine Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Wind Turbine Control System . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Effective Wind Speed Estimation Challenges . . . . . . . . . . . . . . . . . . 111.3.1 Wind Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Tower Shadow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.3 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Effective Wind Speed Estimation literature review . . . . . . . . . . . . . . . 141.4.1 Dynamics of Wind Turbine Rotor Speed . . . . . . . . . . . . . . . . 161.4.2 Dynamics of Wind Turbine Tower Deflection . . . . . . . . . . . . . . 18

1.5 Wind Speed Estimation Based on Wind Turbine Tower Deflection . . . . . . 201.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Wind Turbine Aerodynamics Modelling . . . . . . . . . . . . . . . . . . . . . 24

2.0.1 Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . 242.1 Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Blade Element Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Corrections of Blade Element Momentum Theory . . . . . . . . . . . . . . . 32

2.3.1 Tip-Loss and Hub-Loss . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.2 Glauert Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . 332.5 Iterative Procedure for BEM Method . . . . . . . . . . . . . . . . . . . . . . 342.6 BEM Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 System Identification of Wind Turbine Tower . . . . . . . . . . . . . . . . . 423.1 Tower Fore-aft Motion Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Hammerstein System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.2 Subspaces Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.3 State-space Subspace System Identification . . . . . . . . . . . . . . . 52

3.4 Wind Turbine Tower System Identification . . . . . . . . . . . . . . . . . . . 563.5 Practical Aspects of Wind Turbine Tower System Identification . . . . . . . 58

3.5.1 Control System Design . . . . . . . . . . . . . . . . . . . . . . . . . 583.5.2 Data Collection for System Identification . . . . . . . . . . . . . . . . 59

v

3.5.3 Input-output Data Measurement for System Identification . . . . . . 613.6 System Identification Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 634 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1 Thrust Force Estimator Design . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Kalman Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.2 Residual Sequence Estimation . . . . . . . . . . . . . . . . . . . . . . 734.1.3 Thrust Force Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 754.1.4 Thrust Force Estimation Summary . . . . . . . . . . . . . . . . . . . 78

4.2 Effective Wind Speed Estimation Using Neural Network . . . . . . . . . . . 804.2.1 Neural Network Training . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Numerical Results and Verification . . . . . . . . . . . . . . . . . . . . . . . 825 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1 Verification Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.1.1 Wind Turbine Simulation Tools . . . . . . . . . . . . . . . . . . . . . 855.2 Turbulent Wind Field Specifications . . . . . . . . . . . . . . . . . . . . . . . 855.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 Summary and Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2 Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A Airfoil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110B Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

vi

List of Tables

3.1 Model validation using modal parameters of wind turbine tower in terms ofnatural frequency fn and damping ratio ζ . . . . . . . . . . . . . . . . . . . 67

5.1 Wind field and simulation conditions . . . . . . . . . . . . . . . . . . . . . . 895.2 Mean value of effective thrust force estimation error . . . . . . . . . . . . . . 895.3 Comparison of wind speeds mean value . . . . . . . . . . . . . . . . . . . . . 89

A.1 Blades are splitted to 15 elemnts considering a constant twist angle, chordlength and cross-section geometry along each of them. The distance of theblade elemnts center from the hub and the airfoil numbers are provided in thetable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.2 The lift and drag coefficient data of the airfoil #1 based on angle of attack . 110A.3 The lift and drag coefficient data of the airfoil #2 based on angle of attack . 111A.4 The lift and drag coefficient data of the airfoil #3 based on angle of attack . 112A.5 The lift and drag coefficient data of the airfoil #4 based on angle of attack . 113

vii

List of Figures and Illustrations

1.1 Global annual installed wind capacity (Source: Ref. [51]) . . . . . . . . . . . 31.2 Global cumulative installed wind capacity (Source: Ref. [51]) . . . . . . . . . 31.3 Wind energy price in different countries in 2010 and 2014 (Information re-

trieved from Ref. [31]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Wind turbine system: diffirent parts of a horizontal axis wind turbine (HAWT)

are shown. The wind speed creates aerodynamic torque Ta and thrust forceFT on the blades. The gearbox transmits the torque to generator and in-creases the angular velocity from rotor speed Ω to generator shaft speed ω.The generator torque Tg and blade pitch angle β are two control inputs to thesystem. The yaw control system changes the direction of the wind turbinetowards the highest wind speed. . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Torque coefficient CQ and power coefficient Cp curves: highly nonlinear changesof torque and power coefficients is depicted versus variation of the tip speedratio λ and pitch angle β. As shown, maximum power coefficient Cpmax isassociated with optimal tip speed ratio (λ = λ0) and β0 ' 0 [10] . . . . . . . 7

1.6 Ideal power curve and control system operational regions. In Region I, windturbine ideally can capture all avaiable energy in the wind if we maitain theoptimal tip speed ratio by controlling the rotor speed. In Region III, thegenerator torque and power are saturated and we discard the excessive energyin the wind by controlling the pitch angle to avoid damage to wind turbine.In Region II, a smooth transition between Regions I and III is desired. . . . 9

1.7 Wind speed field [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.8 Tower shadow causes purturbations in aerodynamic torque by reducing the

torque when blades pass the tower in each revolution. For a three bladed windturbine it causes purturbations with frequency of 3P [10] . . . . . . . . . . . 14

1.9 Wind speed estimation method using wind turbine tower deflection modelling.By solving the inverse problem for tower structural model we can find theestimated thrust force FT from tower deflection measurement ˆDef Then theestimated wind speed U can be calculated by solving the invesre problem foraerodynamic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 The control volume considered for wind turbine to apply the conservation oflinera momentum. In this figure U is the flow velocity which is shown in fourdifferent cross-sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Annular stream tube. In this figure the flow velocity on rotor plane and down-stream are shown as a function of free stream velocity U and axial inductionfactor a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 The air flow finds a tangential velocity component passing the rotor planewhich is proporional to tangential induction factor a . . . . . . . . . . . . . . 29

2.4 To use blade element theory we split the blade into several blade elementsoperating independently with no aerodynamical interaction. The chord lengthc and twist angle θT may be different for each blade element. . . . . . . . . . 30

viii

2.5 Cross-section of a blade element where dFL is the lift force, dFD is the dragforce. The drag and lift forces are projected into two components, one is thetangential force dFT which lies in the rotor plane and causes the torque andthe other is normal to the rotor plane dFN and causes the thrust force. . . . 31

2.6 BEM iterative algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7 Verification of the BEM method result with the FAST output which uses

the generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitchangle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrustforce in frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.8 Verification of the BEM method result with the FAST output which usesthe generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitchangle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrustforce in frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.9 Verification of the BEM method result with the FAST output which usesthe generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitchangle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrustforce in frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1 Wind turbine dynamic system is described using Hammerstein structure wherethe Wind Turbine Aerodynamics is considered as the static nonlinearity andthe Tower Dynamics is considered as a LTI system. The entire dynamicsystem of wind turbine is shown under closed-loop control; however, open-loop system identification method still can be used for identification of TowerDynamics because there is no feedback from tower fore-aft deflection DFA toaerodynamic thrust force FT . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Orthogonal Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Oblique Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Diagonal elements of S matrix in singular value decomposition (SVD) of Oh

matrix which is used to determine the order of the model . . . . . . . . . . 573.5 The control system of the wind turbine is shown which consists of two parts,

namely, generator torque control and blade pitch angle control. As shown,the random binary signals are added in order to make enough excitation inthe system to collect informative data . . . . . . . . . . . . . . . . . . . . . . 61

3.6 Input data set used for evaluation of wind turbine tower identified model. Thepitch angle is adjusted by the control system to maitain the rotor speed atrated values (12rpm) a) input wind speed profile b) pitch angle experienced bythe wind turbine blades c) rotor speed d) aerodynamic thrust force associatedwith the presented wind speed, pitch angle and rotor speed. The thrust forcecalculated by BEM is compared with the FAST output. . . . . . . . . . . . . 64

3.7 Wind turbine tower deflection associated with the wind speed, pitch angle androtor speed condition presented in Figure 3.6. The output of the identifiedmodel is compared with FAST output. a) time domain and b) frequencydomain representation of tower deflection data . . . . . . . . . . . . . . . . . 65

4.1 a) Thrust force estimator validation. b) Wind speed estimator validation . . 83

ix

5.1 Spatial distribution of wind speed in a turbulent wind field at four differenttime instants. Wind turbine hub is located in the center of the circle and thecircle with R = 35m shows rotor plane of the wind turbine. The color barshows the wind speed (m/s). (a) t = 50s (b) t = 55s (c) t = 60s (d) t = 65s 86

5.2 Spatial distribution of the mean wind speed in a turbulent wind filed over1000s. Wind turbine hub is located in the center of the circle and the circlewith R = 35m shows rotor plane of the wind turbine. The color bar showsthe wind speed (m/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3 Correlation coefficient of the turbulent wind field with the hub-height windspeed. Wind turbine hub is located in the center of the circle and the circlewith R = 35m shows rotor plane of the wind turbine. The color bar showsdimetionless values of the correlation coefficient. . . . . . . . . . . . . . . . 87

5.4 Comparison of estimated values of effective thrust force and FAST thrust forceoutput for test #1 (top) and test #2 (bottom). . . . . . . . . . . . . . . . . 90

5.5 Effective wind speed and hub-height wind speed comparison for test #1 (top)and test #2 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.6 Correlation coefficient of the turbulent wind field with the effective wind speedfor test #1 (top) and test #2 (bottom). Wind turbine hub is located in thecenter of the circle and the circle with R = 35m shows the rotor plane ofwind turbine. The color bar shows the dimensionless values of the correlationcoefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.7 Frequency domain representation of hub-height and effective wind speeds(test #1). Effective wind speed has some higher frequency components atf1 ' 0.4Hz, f2 ' 0.6Hz and f3 ' 1.2Hz which are not present in the hub-height wind speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.8 Frequency domain representation of hub-height and effective wind speeds(test #2). Effective wind speed has some higher frequency components atf1 ' 0.4Hz, f2 ' 0.6Hz and f3 ' 1.2Hz which are not present in the hub-height wind speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.9 Rotor speed for test #2 which is maintained around Ω = 12rpm using pitchcontrol system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

x

List of Symbols and Acronyms

Symbol Definition

ANFIS Adaptive Neuro Fuzzy Inference Systems

ANN Artificial Neural Network

BEM Blade Element Momentum

BPANN Back-Propagation Artificial Neural Network

ELM Extreme Learning Machine

ESN Echo State Network

FAST Fatigue, Aerodynamics, Structures, and Turbulence

FBG Fiber Bragg Grating

GRBFN Gaussian Radial Basis Function Network

HAWT Horizontal Axis Wind Turbine

LIDAR Light Detection And Ranging

LPV Linear, Parameter-Varying

LTI Linear Time-Invariant

MLPNN Multi-Layer Perceptron Neural Network

MPPT Maximum Power Point Tracking

NREL National Renewable Energy Laboratory

PEM Prediction Error Method

SODAR Sound Detection And Ranging

SVD Singular Value Decomposition

SVR Support Vector Regression

TI Turbulence Intensity

VSVP Variable-Speed Variable-Pitch

4SID State-Space Subspace System Identification

xi

Symbol Definition

' Approximately equal to

= Equal to

6= Unequal to

∈ Element of

Rn×n n× n space

E[] Expected value

R The real numbers

Rn n space

A/B Orthogonal projection of row space A on row space B

A⊥ The space perpendicular to row space A

A/B

C Oblique projection of row space A along B on C

A−1 Inverse of matrix A

AT Transpose of matrix A

A† MoorePenrose pseudoinverse of matrix A

Cov(A,B) Covariance of A and B

xii

Chapter 1

Introduction

1.1 Wind Power

Using wind energy is one of the fastest growing, most sustainable and cleanest ways to

produce electricity. It is renewable and produces no greenhouse gases which makes it a

viable alternative for fossil fuels. By the end of 2014, the global wind electricity-generating

capacity increased to 369,597 MW from 197,943 MW in 2010. This means that in four

years the world wind industry grew by 87%. The Figures 1.1 and 1.2 show that global

wind installation rate has increased very rapidly in recent years such that 2014 was a record-

breaking year for the wind industry with more than 51 GW installed in a single year, bringing

the global total close to 370 GW [51].

2014 was also a record year for wind energy development projects in Canada. Canada

with about 1.9 GW annual wind power installation, was among ten leading countries in

2014. Also, with about 9.7 GW wind power capacity, Canada ranks 7th in the world based

on the cumulative wind power capacity and China with about 114.6 GW has the highest

cumulative wind capacity [51]. The distribution of wind power installation in Canada shows

that Alberta with total installation of about 1.5 GW (reported in June 2015) has the third

highest capacity among the provinces [20].

Although the wind energy is renewable and does not have negative environmental im-

pacts, to take a higher share of the energy market it should compete with fossil fuels in terms

of the energy cost. The advances in wind turbines technologies, enabling us to build very

large wind turbines with variable rotor speed, has decreased the energy cost of wind energy

significantly in the past decades. Although wind energy cost defers from country to country

depending on the utilized technology and region, the global cost has significantly dropped

1

from 2010 to 2014 as in Figure 1.3.

Design and implementation of large wind turbines plays an important role in cost re-

duction. Using a large wind turbine can reduce the building and installation capital cost

significantly compared to when we use several small wind turbines. Also, having a small

number of wind turbines can reduce the maintenance cost. The service life of wind turbines

is the other main factor affecting the wind energy cost. Wind speed estimation can reduce

the fatigue loads on the wind turbine and increase its operating life which will lead to cheaper

wind energy. Wind speed estimation is one of the topics that will be discussed in detail in

this thesis.

1.2 Wind Turbine System

As it is shown in Figure 1.4, a horizontal axis wind turbine (HAWT) consists of different

subsystems which can be organized in four main categories, namely, the aerodynamic, me-

chanical, electrical and control subsystems. The aerodynamic subsystem converts the wind

energy to mechanical energy. The mechanical subsystem caries out two main jobs: transmit-

ting the torque from rotor to generator and supporting the wind turbine system in height

against the thrust force. The first job is carried out by the drive-train which includes the

gearbox and transmitting shafts and the second one is done by the tower structure and foun-

dation. The electrical subsystem consists of generator and transformer which converts the

mechanical energy to electrical power with a desired voltage and frequency. Finally, the role

of control subsystem is to adjust the torque and thrust force by changing the pitch angle

and generator torque. The control subsystem has a sensor to measure the wind speed and

includes a hydraulic or electromechanical system to change the pitch angle of the blades.

As it is shown in Figure 1.4 when the wind blows through wind turbine blades, it applies

torque and thrust force on the wind turbine. The thrust force causes deflection in blades and

tower of the wind turbine and the torque drives the wind turbine rotor. The gearbox included

2

1530 2520 3440 37606500 7270 8133 8207

1153114701

20286

26952

38478 3898940943

44929

35692

51473

0

10,000

20,000

30,000

40,000

50,000

60,000

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

MW

Figure 1.1: Global annual installed wind capacity (Source: Ref. [51])

7600 10200 13600 17400 23900 31100 39431 47620 5909173949

93901120715

159079

197943

238435

283132

318644

369597

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

MW

Figure 1.2: Global cumulative installed wind capacity (Source: Ref. [51])

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2010 2014

20

14

USD

/kW

h

Year

France Japan Australia US California US non-California Germany China

Figure 1.3: Wind energy price in different countries in 2010 and 2014 (Information retrievedfrom Ref. [31])

3

Generator GearboxgT

Yawing

Tower

aTTF

Nacclle

R

Figure 1.4: Wind turbine system: diffirent parts of a horizontal axis wind turbine (HAWT)are shown. The wind speed creates aerodynamic torque Ta and thrust force FT on the blades.The gearbox transmits the torque to generator and increases the angular velocity from rotorspeed Ω to generator shaft speed ω. The generator torque Tg and blade pitch angle β aretwo control inputs to the system. The yaw control system changes the direction of the windturbine towards the highest wind speed.

in drive-train increases the rotational speed of low-speed shaft Ω to rotational speed of the

high-speed shaft (generator shaft) ω and transmits the aerodynamic torque to the generator.

Finally, the generator converts the mechanical energy to electricity.

In old generations of wind turbines, rotor speed was fixed and governed by the constant

frequency of the grid. However, advances in power electronics of the wind turbines have

insulated the wind turbine rotor speed from the constant frequency of the grid. Therefore,

in new generations of wind turbines, the rotor speed is variable which gives the wind turbine

control system flexibility to capture the maximum energy and minimize the loads. Most of

the large wind turbines are variable-speed variable-pitch (VSVP) where the control system

regulates the rotor speed and the captured energy of wind turbine by altering the generator

torque and the blades pitch angle. In the following sections, the control strategy of VSVP

wind turbines will be discussed and explanations will be given about the importance of wind

speed estimation for the performance of wind turbines.

4

1.2.1 Wind Turbine Aerodynamics

Here an introduction is given about the aerodynamic power generation in wind turbine rotor

based on the Blade Element Momentum (BEM) Theory [28] and how it can be used to

maximize power generation. BEM is a wind turbine aerodynamic modeling method which

calculates the aerodynamic torque and thrust force by splitting wind turbine blades to several

aerodynamically independent units, called blade elements, and using conservation of the

linear and angular momentum. The aerodynamics of wind turbine will be discussed in detail

in Chapter 2.

The main goal of aerodynamic modeling of wind turbine is to calculate the aerodynamic

torque, thrust force, and power which can be used for control of wind turbine and wind

speed estimation.

The wind turbine thrust force, torque and power depend on the wind speed U , blade

pitch angle β, and tip speed ratio λ. where λ is defined as the ratio of blade tip speed ΩR

to wind speed U :

λ =ΩR

U(1.1)

Using BEM (see Chapter 2), the total thrust force FT and torque FQ exerted on the

blade can be obtained by integrating Equations 2.34 and 2.35 respectively along the blade

span. The result can be expressed in terms of non-dimensional thrust CT and torque CQ

coefficients which are a function of pitch angle β and tip speed ratio λ.

FT =1

2ρπR2CT (λ, β)U2 (1.2)

FQ =1

2ρπR3CQ(λ, β)U2 (1.3)

Where ρ is the air density and R is the blade radius. Also, wind turbine power can be

expressed similarly using the non-dimensional power coefficient CP :

P =1

2ρπR2CP (λ, β)U3 (1.4)

5

Among the parameters influencing wind turbine aerodynamics, wind speed can not be

controlled and we can only control the tip speed ratio λ and pitch angle β. Tip speed ratio

λ can be controlled by changing the rotor angular speed and blade pitch angle β can be

adjusted using the pitch angle servo. Figure 1.5 shows the variations of CP and CQ with

respect to deviations of λ and β. The maximum point for wind turbine power happens in

a pitch angle close to zero (β0 ' 0) and the optimal tip speed ratio (λ = λ0). In low wind

speeds, where the available energy in the wind is less than the wind turbine capacity, the

system should track the optimum point (Cp = Cpmax) to capture the maximum amount of

energy. However, in high wind speeds where the available energy in the wind is higher than

the wind turbine capacity, the system should just capture a portion of the energy in the

wind which is equal to the wind turbine capacity (Cp ≺ Cpmax).

The control strategies used in wind turbine to regulate the aerodynamic power will be

discussed in the following section.

1.2.2 Wind Turbine Control System

For efficient power generation in wind turbine systems, the control system is necessary. The

control system decides what should be the rotational speed of wind turbine in different wind

speeds and keeps the wind turbine in designed operational range to avoid the damages to the

system. In this section the control approach for variable-speed variable-pitch wind turbines

will be discussed and the importance of wind speed measurement for performance of wind

turbine control system will be explained.

1.2.2.1 Wind Turbine Operational Regions

We can determine different operational regions for a wind turbine based on the available

wind speed and the wind turbine power capacity. Control objective of the wind turbine is

different in these regions and wind turbine should behave differently in each region in order

to maximize the power, reduce damages and maintenance cost.

6

20 2 The Wind and Wind Turbines

The torque and power coefficients are of special interest for control pur-poses. Figure 2.8 shows typical variations of CQ and CP with the tip-speed-ratio and the pitch angle deviation. In the case of fixed-pitch wind turbines,CQ and CP vary only with λ, since β = 0 naturally. So, with some abuseof notation we will write CQ(λ) and CP (λ) to denote CQ(λ, 0) and CP (λ, 0),respectively. Figure 2.9 depicts typical coefficients CQ(λ) and CP (λ) of fixed-pitch turbines in two-dimensional graphs.

5

10

150

10

20

30

0

0.02

0.04

0.06

0.08

0.1C

Q

λβ

(a)

5

10

150

10

20

30

0

0.1

0.2

0.3

0.4

0.5

CP

λβ

CPmax ≡ CP (λo, βo)

(b)

Fig. 2.8. Typical variations of (a) CQ and (b) CP for a variable-pitch wind turbineFigure 1.5: Torque coefficient CQ and power coefficient Cp curves: highly nonlinear changesof torque and power coefficients is depicted versus variation of the tip speed ratio λ and pitchangle β. As shown, maximum power coefficient Cpmax is associated with optimal tip speedratio (λ = λ0) and β0 ' 0 [10]

7

Wind turbines have an operational range starting from cut-in wind speed (Umin) to cut-

out wind speed (Umax). Beyond this range, the wind turbine does not operate because the

wind speed is either too low in which the available energy in the wind does not justify the

operational cost of the wind turbine, or too high which may damage the wind turbine system.

Within the operational range, the available energy of wind varies from Pmin = P (λ, β, Umin)

to Pmax = P (λ, β, Umax) which is a large range as the power is proportional to third power

of wind speed P ∝ U3 (see Equation 1.4). By increasing the capacity of wind turbine, the

capital cost will increase; therefore, it is important to find the optimal power capacity of

wind turbine called rated power.

As shown in Figure 1.6, the operational range of wind turbine can be split into two main

regions based on the rated wind speed, the wind speed associated with the rated power.

Having wind speeds below the rated wind speed Ur, marked as Region I in Figure 1.6 , the

wind turbine ideally can capture the maximum available energy in the wind. However, if

the wind speed falls within Ur and Umax, marked as Region III in Figure 1.6, the available

energy in wind is higher than the wind turbine capacity; therefore, the wind turbine should

discard part of the energy and only capture the constant amount of energy (equal to rated

power) in order to avoid the damages the wind turbine. Region II in Figure 1.6 shows a

transition between Regions I and III.

1.2.2.2 Wind Turbine Control Approach

Based on the operational regions and type of the wind turbine, the control system has

different duties. Most of the large and modern wind turbines are variable-speed variable-

pitch. In this type of wind turbines, rotor speed is independent of power grid and can be

changed. Also, the blades’ pitch angle is adjustable.

For variable-speed variable-pitch wind turbines, control system duties in different regions

are defined as follows:

In Region I, the control system should maintain the optimal tip-speed ratio λopt by

8

Figure 1.6: Ideal power curve and control system operational regions. In Region I, windturbine ideally can capture all avaiable energy in the wind if we maitain the optimal tipspeed ratio by controlling the rotor speed. In Region III, the generator torque and powerare saturated and we discard the excessive energy in the wind by controlling the pitch angleto avoid damage to wind turbine. In Region II, a smooth transition between Regions I andIII is desired.

controlling rotor speed according to wind speed variations. Based on Equation 1.1, when

the wind speed changes, the control system should change the rotor speed to Ωopt = Uλopt/R

according to new wind speed. Therefore, accurate estimation of effective wind speed will

result in higher amount of energy captured by the wind turbine. In this region, control

system keeps the pitch angle around zero and uses the generator torque as control input to

regulate the rotor speed to track the maximum power.

In Region III, the control system should maintain the rated power and keep the rotor

speed in its rated value by adjusting the pitch angle to discard the wind energy exceeding

the rated power. To keep the rotor speed in its rated value and avoid over speeding, the wind

turbine has two main control inputs: generator torque and pitch angle. In Region III, the

generator torque is saturated and has reached its rated limit and can not be increased any

more to regulate the rotor speed. Therefore, by feathering the pitch angle, control system

reduces the aerodynamic torque to keep the rotor speed in its rated value. In Figure 1.5, we

can see the effect of pitch angle change on aerodynamic torque and power, showing that we

can discard the excessive portion of wind energy by increasing the pitch angle.

9

The objectives of pitch angle control system in Region III can be summarized as follows

[72, 37]:

• Preventing aerodynamic power to exceed the rated power by discarding the

excess power in wind. Pitch control maintains the rotor speed at its rated

value by keeping the aerodynamic torque at the rated value.

• Smoothing the output power and increasing the power quality. Wind speed has

rapidly varying turbulent components in addition to its slowly varying mean

value. The pitch control system should continuously take the fluctuations into

account and alter the pitch angle accordingly to avoid fluctuations in power.

• Decreasing fatigue loads on the wind turbine components. For instance, gear-

box fatigue is caused by stressing of the gearbox teeth in response to torque

overloads and generator fatigue is because of thermal stressing of the genera-

tor windings caused by rotor speeds over the rated value. The pitch control

system should alleviate the fatigue loads by taking appropriate action against

external disturbances.

Achieving these goals depends on the ability of the pitch control system in monitoring

fluctuations in wind speed and adapting the pitch angle in order to avoid fluctuations appear

in the output power and to reduce fatigue damages.

A wind turbine experiences a three-dimensional wind field in which wind speed varies in

both temporal and spatial manners. Wind speed variation in rotor plane is because of the

turbulence, tower shadow, wind shear, localized wind gusts, etc. As rotor sweeps the rotor

plane, it experiences different wind speeds causing fluctuations in the aerodynamic torque

and thrust force, leading to some damaging loads. Therefore, effective wind speed estimation

for pitch angle control system plays an important role in reducing the damages by enabling

the control system to cancel fluctuations. Any inaccuracy or delay in wind speed estimation,

10

however, will result in damaging loads on wind turbine components which may reduce the

service life of a wind turbine and increase maintenance costs.

Finally, the aim of the control system in Region II is to provide smooth transition between

Region I and Region II and prevent the excessive loads and vibrations which may damage

the system. In some wind turbines, this region does not exist and the parabolic curve of

Region I, governed by the third order polynomial of Equation 1.4, is directly connected to

the constant power in Region III.

1.3 Effective Wind Speed Estimation Challenges

As discussed in previous sections, the aim of control algorithms in new generations of wind

turbines is to track the maximum power point in wind speeds less than rated wind speed

and to avoid over-speeding in wind speeds greater than rated wind speed. The former can

be achieved by adjusting the shaft speed using torque control and the latter can be done

by adapting pitch angle using pitch control. Since wind turbine operates in varying wind

speeds, wind speed measurement or estimation can improve the performance of wind turbine

control system.

A more accurate wind speed estimation will increase the control system performance.

Generally, an anemometer is used for wind speed measurement. Anemometer is not accurate

and reliable and has some limitations. It provides wind speed information in a single point

only which does not represent the effective wind speed for the whole rotor plane. Also,

anemometer is usually mounted on nacelle and measures the wind speed affected by wind

turbine which is not the original wind speed experienced by the blades. The effective wind

speed is the spatial average of wind speed over rotor plane.

The model used in aerodynamic modeling of the wind turbine considers wind speed to be

uniform in the area swept by wind turbine blades; however, in reality the wind speed varies

from point to point in rotor plane. Particularly, for large wind turbines, the area swept by

11

blades is very huge which makes wind speed variations very significant. Therefore, wind

turbine blade experiences different values of wind speed as it rotates. This issue introduces a

cyclic perturbation in blade thrust force and torque with frequency of wind turbine rotational

speed (1P ). For a wind turbine with N number of blades, the total thrust force and torque

will contain perturbations with frequency of NP and its multiplies. The main causes for wind

speed variations in the rotor plane and perturbation in the aerodynamic force and torque

are wind shear, tower shadow and turbulence. A brief introduction about these effects is

given in the following section using reference [10].

1.3.1 Wind Shear

The mean value of wind speed increases as we go further from the ground. If we consider a

blade element which is in the distance r from the hub, the height of the blade element can

be determined as h+ rcos(ψ) where h is the height of tower (shown in Figure 1.7).

When the blade is upwards (ψ = 0), the height of blade element is h + r and when the

blade is downwards (ψ = π), the height is h− r. As the mean value of wind speed is bigger

in higher heights, the blade element experiences a cyclic change in the mean value of the

wind speed which results in cyclic loads on the blade. For each blade of the wind turbine,

this variation happens with a frequency of 1P . However there is a phase difference between

the perturbation of different blades. For example, the phase difference for a wind turbine

with three blades is 23π. When we average the perturbations coming from three blades, as

they have a phase difference of 23π, the 1P components of torque and thrust force will be

cancelled out and only the 3P components will affect the system [10].

1.3.2 Tower Shadow

The horizontal-axis wind turbines have two common configurations, namely, upwind and

downwind in which the rotor is upstream and downstream of the tower respectively. Most of

the modern wind turbines are upwind. In either case, the tower is an obstacle in front of the

12

Figure 1.7: Wind speed field [10]

wind flow and changes the air streamlines and decreases the wind speed. This effect is called

tower shadow. As the tower is present only in the lower half of the rotor plane, the tower

shadow effect is not present in the upper half. When the wind turbine blades pass the tower,

they experience a reduction in torque and thrust force. As it is shown in Figure 1.8, in a

wind turbine with three blades, the tower shadow causes a perturbation with the frequency

of 3P and its multiples.

1.3.3 Turbulence

Wind shear and tower shadow are components which have a deterministic nature. However,

there are some stochastic variations in wind speed during time, called turbulence. Turbu-

lence causes perturbations in thrust force and torque of the wind turbine and it presents

fluctuations in thrust force and torque with frequency of 1P and its harmonics.

The turbulence cyclic perturbation effect is more significant in blade elements far from

the hub. That is because these elements produce the most part of torque and thrust force.

As blades travel a larger distance in each revolution, they encounter less correlated stochastic

variations [10].

13

Figure 1.8: Tower shadow causes purturbations in aerodynamic torque by reducing thetorque when blades pass the tower in each revolution. For a three bladed wind turbine itcauses purturbations with frequency of 3P [10]

1.4 Effective Wind Speed Estimation literature review

Effective wind speed estimation has drawn considerable attention among researchers. Several

methods are proposed in literature to estimate effective wind speed without using anemome-

ter [49, 57, 8, 9, 32, 38, 71, 58, 2, 1, 47, 41, 54, 5, 48, 45, 69].

Different types of neural networks with different training data have been used for wind

speed estimation [32]. For instance, multi-layer perceptron neural network (MLPNN) [38],

Soft sensor based support vector machine [71], Gaussian radial basis function network

(GRBFN) [58], Support vector regression (SVR) [2, 1], Echo state network (ESN) [47], Ex-

treme learning machine (ELM) [45, 69], and Adaptive neuro fuzzy inference systems (ANFIS)

[41, 54] are some of the methods that are proposed for neural network based estimation of

wind speed. The ideas behind all of these methods is to use the inverse power characteristic

function. Given the mechanical power of wind turbine, rotor speed, and the pitch angle, the

wind speed can be obtained by solving the inverse power characteristic function. All of these

estimation methods use neural networks to solve the nonlinear inverse power charactristic

function. The differences are in types of neural networks and methods of training they use.

For example, in [48], the mechanical power is estimated from electrical power by modelling

14

the generator and drive-train. The wind speed is then estimated with the information of

shaft speed, pitch angle, and mechanical power by using a back-propagation artificial neural

network (BPANN).

The polynomial based methods for wind speed estimation are similar to neural network

based methods. The only difference is that these methods utilize root-finding algorithms

instead of neural networks for solving the inverse charactristic function. For instance, in[57,

8, 9], the iterative methods like Newton Raphson or Bisection method are utilized to solve

the power characteristic function expressed as a polynomial in terms of tip speed ratio.

In addition to neural network and polynomial based methods, some other techniques are

reported in the literature for wind speed estimation such as Kalman filter based methods

[15, 13, 17, 70, 12, 16, 18, 46, 14], linear and non-linear observer based method [29, 27, 53]

, differentiation based methods, lookup table based method [44]. In all of these techniques,

first the aerodynamic torque is estimated and then the inverse torque characteristic function

is solved for wind speed estimation. In other words, despite the previous methods in which

wind speed estimation was based on power charactristic function, in these methods torque

charactristic function is used. For example, in [15, 17], the Kalman filter is used for estimation

of states such as rotor speed and aerodynamic torque. Then, Newton Raphson method is

used to solve the inverse torque characteristic function in order to obtain wind speed.

In summary, the reviewed methods use the dynamic behaviour of wind turbine rotational

system —which includes rotor, drive-train, and generator— for wind speed estimation. The

main drawback of using this approach is that it has a large time constant which makes it

too slow for estimating the rapidly fluctuating wind speed components. The extremely high

inertia of wind turbine rotor, drive-train, and generator causes time delay in wind turbine

rotational system [23, 43]. The second problem with this approach is that the rotational

system of wind turbine behaves like a narrow low-pass filter, filtering out high-frequency

components of the wind profile such as turbulence, tower shadow, wind shear, wind gusts,

15

etc. [21]. In order to estimate the wind speed with the highest fidelity, i.e., lowest time

delay and with complete frequency content, we need a subsystem of wind turbine with quick

response and wider passband. In this thesis, we propose using of wind turbine tower dynamics

for wind speed estimation. As it is shown in Section 1.4.2, tower dynamic behaves like an

under-damped second order system which has a quick response time and a wide passband

covering the desired frequency range, providing wind speed information with lower delay and

higher fidelity.

In the following sections, dynamics of wind turbine rotor system and tower system will be

presented and their dynamic behavior will be studied to compare their suitability for wind

speed estimation.

1.4.1 Dynamics of Wind Turbine Rotor Speed

The rotor speed dynamics is given in the following equations [5]:

Jrωr +Brωr = Tr − Ttr (1.5)

Jgωg +Bgωg = Ttg − Tg (1.6)

Ttrωr = Ttgωg (1.7)

where Jr are Jg the rotor and generator inertia, Tr is the wind generated torque in the rotor,

Tg is the generator torque, Ttr is the torque in rotor side of the gearbox, Ttg is the torque

in generator side of the gearbox, ωr is the angular velocity of the rotor, ωg is the angular

velocity of the generator shaft, Br and Bg are the viscus friction coefficient of the rotor and

generator respectively. The relation between angular velocity of the rotor ωr and angular

velocity of the generator is given by the gearbox ratio γ:

γ =ωgωr

(1.8)

Using Equations 1.5, 1.6 and 1.7 it is obtained:

Jω +Bω = Tr − γTg (1.9)

16

with

J = Jr + γ2Jg (1.10)

B = Br + γ2Bg (1.11)

The Equation 1.9 can be transformed to the following form:

ω(s)

∆T (s)= K

1

τs+ 1(1.12)

Which is a first order low–pass filter, where τ = JB

is the time constant of the filter,

K = 1B

is the gain of the filter, s is the Laplace transform variable and ∆T = Tr − γTg is

the difference between wind generated torque and generator torque which drives the system

as input.

In large wind turbines, the large inertia of the rotor causes the time constant τ to be

huge. For example, for a 5 MW wind turbine, the total inertia is about J = 4 × 107kg .m2

and the time constant the cut–off frequency are approximately τ = 30s and fc = 0.03Hz

[35].

This characteristic of wind turbine rotor speed dynamics has different influences on be-

haviour of the control system and wind speed estimator. In Region I, the slow transient

response of the rotor speed dynamics causes a time delay in tracking the optimal tip speed

ratio. In fact, when the wind speed varies, it takes a long time for torque control system

to change the rotor angular speed from one optimal speed to the next optimal speed. It

means that the wind turbine operates in a non-optimal rotational speed for a while which

deteriorates the maximum power point tracking (MPPT) and decreases the average power

output [22].

On the other hand, the inertia of the rotor behaves like an inductor in an electrical circuit

which can increase the wind turbine power quality, smooth the power output and filter out

the wind speed fluctuations [43]. However, the large time delay and low cut–off frequency

of this system is not desired for wind speed estimation. For instance, in Region III, where

17

wind speed is higher than the rated value, the control system should ideally maintain the

output power at rated level and avoid over speeding of the wind turbine rotor. If so, we

can capture the maximum power and prevent the damages to the wind turbine components.

To attain this goal, the pitch angle controller should react to the wind speed variations fast

enough such that acceleration and deceleration of the rotor is zero and the power captured

from wind is equal to the electrical power (Pcaptured = Pelectrical) [43].

In wind speed estimation based on rotor dynamics, the measurement of angular velocity

is used for wind speed estimation. Therefore, when the wind turbine is subject to a change

in wind speed, the estimation occurs only after rotor speed has responded to the variations.

Since the rotor speed response is very slow, the pitch controller will react when the wind

speed variation has already influenced the system. Hence, the wind turbine experiences

large overloads and rotor speeds over the rated value which lead to damages in wind turbine

components. Using a system with faster response time for wind speed estimation can alleviate

this problem.

In order to increase wind speed estimation performance, we propose using wind turbine

tower system for this purpose. Tower deflection of wind turbine provides a wider passband

and a quicker response time which makes it suitable for wind speed estimation. In the

following sections, dynamic behavior of wind turbine tower deflection will be discussed and

the proposed method for wind speed estimation based on tower deflection will be discussed.

1.4.2 Dynamics of Wind Turbine Tower Deflection

Modeling of wind turbine tower deflection is discussed in detail in Chapter 4 of this thesis.

Wind turbine tower system is an under-damped second order system considering the first

vibration mode. Comparing the cut-off frequency of the wind turbine tower deflection with

the rotor dynamics system, we see that tower dynamics has a relatively wider passband. For

example, for a 5MW wind turbine with tower height of 87.6m and rotor diameter of 126m,

rotor speed dynamics with cut-off frequency of fc = 0.03Hz is very slow because of huge

18

inertia of the rotor [35]. On the other hand, wind turbine tower deflection is lightly damped

and has a cut-off frequency of fc = 0.32Hz which provides a wider passband and a quicker

response time for wind speed estimation.

Using the tower dynamic system for wind speed estimation we can estimate wind speed

fluctuations with higher frequencies and provide the effective wind speed with a lower time

delay. This can help us capture the maximum power and avoid damages to the system by

improving the control system performance in both maximum power point tracking (Region

I) and pitch angle control (Region III).

For example, rated rotor speed for a 5MW wind turbine is 12.1rpm [35] which means

the wind turbine blades experience new values of wind speed with frequency of f = 0.2Hz

sweeping the rotor plane. Comparing this rotational frequency with cut-off frequency of

rotor dynamics fc = 0.03 and tower dynamics fc = 0.32Hz , we can see that wind speed

variations lay into the bassband of the tower dynamic system; however, the rotor dynamic

system kills the faster components of wind speed and only estimates the mean value.

The downside of wind speed estimation using wind turbine tower dynamics is that the

system is lightly damped around the natural frequency which may cause instability and may

introduce error to wind speed estimation results. However, in wind turbine control system

design, rotor speeds close to structural natural frequencies of the system are avoided. For

example, for certain values of wind speed in Region I the optimal rotor speed is close to

wind turbine structural natural frequency; however, these rotor speeds are avoided because

of the trade-off between maximum energy capturing and vibration and fatigue load reduction.

Moreover, in wind turbine pitch control system, the operational frequency of pitch actuator

is designed to be far from the wind turbine structural natural frequencies.

19

1.5 Wind Speed Estimation Based on Wind Turbine Tower Deflection

In this thesis, effective wind speed estimation based on wind turbine tower deflection mea-

surement is proposed. The idea behind this method is to use wind turbine itself as an

anemometer by using its dynamic behavior to estimate wind speed.

Since wind turbine tower deflection is caused by the effective wind speed, by creating

a proper model for wind turbine tower dynamics and solving the inverse problem we can

estimate the effective wind speed. To build a model for wind turbine tower deflection, we

need to model both aerodynamics and structural dynamics. Since the aerodynamics of the

wind turbine is highly nonlinear, it is hard to create a nonlinear model describing structural

dynamics and aerodynamics all together. Moreover, solving the inverse problem for such a

nonlinear model is difficult.

To alleviate this problems, wind turbine tower dynamics is described using a Hammer-

stein structure [64] where we separate the model to a static nonlinearity part, which is

aerodynamics of the wind turbine, followed by a linear time invariant (LTI) part, which is

the tower structural model.

We first use the Blade Element Momentum (BEM) Theory [28] to model the aerodynam-

ics of the wind turbine. We calculate thrust force of the wind turbine having wind speed,

rotor speed and blades pitch angle using BEM. Then, the aerodynamic thrust force is used

as an input to tower structural dynamics model which is assumed linear time invariant (LTI)

in this work.

The structural dynamics of the wind turbine tower is modeled using EulerBernoulli beam

theory [7] and assumed modes method [50] is applied to obtain the state-space model. This

method of modeling results in a model with different physical parameters such as cross-

sections inertia, materials properties, etc. Since there is uncertainty in defining these physical

properties, we need a systematic method to identify the model parameters. System identifi-

cation techniques can be used to identify the accurate model parameters using input-output

20

Tower System Mesurement

Tower System Model

Aerodynamics Structural

Dynamics (LTI) ˆTF ˆDef

D e fU

U

Figure 1.9: Wind speed estimation method using wind turbine tower deflection modelling.By solving the inverse problem for tower structural model we can find the estimated thrustforce FT from tower deflection measurement ˆDef Then the estimated wind speed U can becalculated by solving the invesre problem for aerodynamic model.

data of the system. Using Hammerstein structure enables us to use the well established

linear system identification methods. In this thesis, we use subspace system identification

method [61] to identify the state space matrices for wind turbine tower deflection model.

After obtaining the wind turbine tower deflection model, we can use it to design an

effective wind speed estimator. For this purpose, we need to measure the tower deflection

and solve the inverse problem to estimate the wind speed as shown in Figure 1.9. Since we

split the model into two parts using Hammerstein structure, we solve the inverse problem in

two steps: first for the structural sub-model and then for the aerodynamics.

By solving the inverse model for structural part we can estimate the aerodynamic thrust

from the tower deflection measurement. Finally, we use the estimated thrust force from the

first step to solve the second inverse problem in the aerodynamics of the wind turbine to

estimate the effective wind speed.

In this thesis, a model-based estimation method based on Kalman filter [25] and Recursive

Least Squares [40] method is presented which estimates the thrust force. Also, an Artificial

Neural Network [66] is used to solve the aerodynamic inverse problem and estimate the

21

effective wind speed from the thrust force.

In chapter 5, some numerical results and simulations will be presented for effective wind

speed estimation using the proposed method to show its performance.

1.6 Contributions

As discussed in Section 1.4, huge amount of research has been done on improving wind speed

estimation. In all of the reviewed publications, the main focus is on reducing the time delay

and computational effort by improving the methods of neural network training and input

estimation. Estimating wind speed based on wind turbine rotor dynamics is the part that

most of these methods have in common. In this work, dynamic behavior of wind turbine

rotor speed is discussed and it is shown that the slow dynamic response of wind turbine

rotor system causes basic limitations for wind speed estimation. Also, wind turbine tower

dynamics is proposed as an alternative to use in wind speed estimation.

Moreover, to prove the feasibility of the proposed approach, wind turbine tower dynamics

is modeled and identified and a wind speed estimation method based on wind turbine tower

deflection is developed and implemented. To show the performance of the proposed method,

wind speed is estimated for different cases using FAST wind turbine simulator. Also, for

turbulent wind fields, the effective wind speed is estimated and the correlation of the effective

wind speed with wind speed in different points of the field is presented which shows higher

correlation inside the rotor plane. The main contributions of this work are summarized as

follows:

• Developing a novel method for effective wind speed estimation based on the wind turbine

tower dynamic behavior

• Studying the effect of inertia on time delay in estimation and proposing a method to use

the vibration of the wind turbine tower in effective wind speed estimation

22

• Developing a model for wind turbine tower structural dynamics and model parameter

identification

• Evaluation of wind speed estimation method based on wind turbine tower deflection and

showing its capability in effective wind speed estimation

1.7 Organization of the Thesis

• Chapter 1: A comprehensive overview of wind turbine systems and a literature review of

wind speed estimation for wind turbine control is presented.

• Chapter 2: The aerodynamics of the wind turbine is modelled using the Blade Element

Momentum (BEM) theory and the iterative algorithm of BEM method is implemented

and verified to be used for calculation of aerodynamic thrust force and torque.

• Chapter 3: A dynamic model for tower deflection of the wind turbine is developed and the

subspace system identification method is implemented to identify the model parameters.

The identified parameters are validated using different methods.

• Chapter 4: Wind turbine tower structural dynamics is used to estimate the effective thrust

force from tower deflection measurement. To solve the inverse problem, Kalman filter and

recursive mean squares method are used. Also, the aerodynamic inverse problem is solved

using neural network to determine the effective wind speed from the estimated thrust

force.

• Chapter 5: The wind speed estimation method developed in this thesis is used for effective

wind speed estimation in turbulent wind fields and some numerical results are provided

to show the performance of our method.

• Chapter 6: The summary of the thesis along with suggestions for future work are presented.

23

Chapter 2

Wind Turbine Aerodynamics Modelling

In this chapter, the Blade Element Momentum theory is presented and implemented for

modelling the aerodynamic thrust force and torque of wind turbine. A brief introduction

is given for Momentum theory and Blade Element theory and the iterative procedure for

calculating the aerodynamic thrust force and torque is presented. At the end, some results

are presented and compared with the FAST software outputs for verification.

2.0.1 Blade Element Momentum Theory

Blade Element Momentum (BEM) theory (attributed to Betz and Glauert [28]) is one of

the most commonly used methods for aerodynamic calculations and aero-elastic modelling

of wind turbine in steady state condition. This method is computationally cheap and very

fast and if we provide reliable airfoil data for lift and drag coefficients, it yields accurate

results [30]. The BEM method calculates the induced velocities assuming a quasi-steady

condition in which the wake is in equilibrium with the aerodynamic loads. However, this

assumption is not valid for unsteady flow of variable-pitch wind turbines which is the focus

of this thesis. On the other hand, as it is shown by M.O.L Hansen et al. [30] that the result

of BEM method has a close agreement with experimental results in which the pitch angle

changes suddenly. Also, this method is used widely by researchers and has gained enormous

popularity for wind turbine identification, control and wind speed estimation applications

where the calculations are real-time and accurate result is required [56]. Moreover, in Sec-

tion 2.6 the results of the BEM theory is compared with results of FAST software, which is

accredited by the most of the academic research papers and is verified by experimental data.

The comparison shows the accuracy of the BEM theory results.

There are some other methods based on the Euler and Navier-Stokes equations which

24

use more physics and less experimental data than the BEM method [52, 30]. These methods

are useful for more complicated cases, e.g. when analysing the interaction of several wind

turbines in a wind farm [30].

BEM consists of two different theories, namely, momentum theory and blade element

theory. Momentum theory considers an annular stream tube as control volume. It uses con-

servation of linear and angular momentum to obtain the axial and angular induced velocities

and to derive the forces on the wind turbine blade. The second part of BEM, the blade el-

ement theory, divides the blade into several annular elements operating independently in

local aerodynamic condition assuming that there is no interaction between blade elements.

The blade elements are considered as two-dimensional airfoils whose aerodynamic forces are

defined as a function of blade geometry using the drag and lift coefficients. To obtain the

total aerodynamic forces and moments experienced by the wind turbine, one should integrate

the elemental forces along the blade span. In the following sections, these two theories will

be combined to calculate the aerodynamic torque and thrust force on a wind turbine.

2.1 Momentum Theory

Assuming a steady state incompressible flow with no fractional drag, the linear momentum

conservation law can be applied for the control volume shown in Figure 2.1 to obtain the

thrust force as follows:

FT = m(U1 − U4) = (ρA2U2)(U1 − U4) momentum conservation (2.1)

m = (ρAU)1 = (ρAU)2 = (ρAU)4 mass conservation (2.2)

Where, FT is the thrust force, U and A are the velocity of the air flow and the area of the

cross-section in different places as it is shown in Figure 2.1, ρ is the air density and m is the

mass flow rate.

25

1U 2U 3U 4U

Rotor Disk

Stream tube

Figure 2.1: The control volume considered for wind turbine to apply the conservation oflinera momentum. In this figure U is the flow velocity which is shown in four differentcross-sections.

1U a 1 2U a

Rotor Disk

Stream tube

R

dr

Rotor Disk

U r

Figure 2.2: Annular stream tube. In this figure the flow velocity on rotor plane and down-stream are shown as a function of free stream velocity U and axial induction factor a.

The thrust force also can be obtained from summation of the forces on each side of the

rotor plane as follows:

FT = A2(p2 − p3) (2.3)

In which p2 and p3 are the air pressures before and after the wind turbine rotor.

26

No work is done on the system on either side of the rotor plane and the rotor is the

only element which does work on the air particles; therefore, the Bernoulli equation can be

utilized in two control volumes before and after the rotor plane:

p1 +1

2ρU2

1 = p2 +1

2ρU2

2 upstream of the rotor (2.4)

p3 +1

2ρU2

3 = p4 +1

2ρU2

4 downstream of the rotor (2.5)

Assuming that the far upstream and far downstream pressures are equal (p1 = p4) and there

is no axial velocity changes across the rotor plane (U2 = U3), Equations 2.4 and 2.5 can be

combined as:

p2 − p3 =1

2ρ(U2

1 − U24 ) (2.6)

By substituting Equation 2.6 into Equation 2.3 we can obtain:

FT =1

2A2ρ(U2

1 − U24 ) (2.7)

Now comparing the value of thrust force from Equations 2.1 and 2.7, wind speed at the rotor

plane can be calculated by:

U2 =(U1 + U4)

2(2.8)

The axial induction factor, a, can be defined as the fractional decrease between the free

stream and the rotor plane wind velocities:

a =(U1 − U2)

U1

(2.9)

Then the rotor plane and the downstream wind velocities can be defined as a function of

upstream wind velocity and induction factor:

U2 = U1(1− a) (2.10)

U4 = U1(1− 2a) (2.11)

Now, substituting Equations 2.10 and 2.11 into Equation 2.7 the thrust force on an

annular cross-section can be expressed as:

dFT =1

2ρ(U2

1 − U24 )(2πrdr) = 4πρU2(1− a)ardr (2.12)

27

In which r indicates the radius of the annular stream tube cross-section in rotor plane as it

is shown in Figure 2.2.

In addition to the thrust force, the wind flow passing through the wind turbine exerts

a torque on the rotor. The rotor also imposes a reaction torque on the air particles. The

reaction torque causes the wind flow to rotate in the opposite direction to the rotor. In

other words, as the wind flow passes the rotor plane, its axial velocity remains constant and

it gains an tangential velocity component (Figure 2.3). The tangential velocity is defined as

2Ωra where a is the tangential induction factor. It should be noted that the kinetic energy

associated with the extra velocity component is provided by the reduction in static pressure.

Now a relation for rotor torque can be obtained by applying the conservation of angular

momentum on the system [19]:

torque = rate of change of angular momentum

= mass flow rate× change of tangential velocity× radius

Then:

dQ = ρ(2πrdr)U(1− a)(2Ωar)r = 4πr3ρUΩ(1− a)adr (2.13)

With this last equation we finish deriving relations for the thrust force dFT and torquedQ

using the momentum theory.

2.2 Blade Element Theory

If we consider the blade as a two-dimensional airfoil which is divided into several sections

(Figure 2.4) where there is no interaction between these sections, then we can express the

forces acting on the blade elements as a function of drag and lift coefficients and the angle of

attack. The aerodynamic concepts and parameters which are used in this section are defined

before deriving the force equations:

28

r

Figure 2.3: The air flow finds a tangential velocity component passing the rotor plane whichis proporional to tangential induction factor a

• Lift force dFL : The component of aerodynamic force that is perpendicular to

the relative wind speed

• Drag force dFD : The component of aerodynamic force that is parallel to the

relative wind speed

• Section pitch angle θp: The angle between the chord line and the plane of

rotation

• Blade pitch angle θp,0: The pitch angle at the tip of the blade

• Section twist angle θT : The angle between the section chord line and the chord

line at the blade tip. It can be expressed as θT = θp − θp,0

• Angle of attack α : The angle between the relative wind speed and the chord

line of the airfoil defined as:

α = ϕ− θT − θp,0 (2.14)

29

T

R

r

c

dr

Figure 2.4: To use blade element theory we split the blade into several blade elementsoperating independently with no aerodynamical interaction. The chord length c and twistangle θT may be different for each blade element.

• Angle of relative wind speed ϕ : The angle between the relative wind speed

and the rotor plane which is equal to ϕ = θp + α

As it is shown in Figure 2.5, the relative wind speed Urel has a component in the rotor

plane Ωr(1 + a) and a component perpendicular to the rotor plane U(1 − a). The perpen-

dicular component has two parts, Ωr is due to the rotation of the blade and Ωra is induced

by the conversion of angular momentum (wake rotation).

Now the equations for thrust and torque can be derived based on the definitions and

Figure 2.5. The drag (dFD) and lift (dFL) forces can be determined using the drag (Cd) and

lift (Cl) coefficients as:

dFL = Cl1

2ρU2

relcdr (2.15)

dFD = Cd1

2ρU2

relcdr (2.16)

30

Chord line

Plane of rotation

(1 )U a

PT

,0P

r (1 )a

relU

TdFDdF

LdF

NdF

Figure 2.5: Cross-section of a blade element where dFL is the lift force, dFD is the dragforce. The drag and lift forces are projected into two components, one is the tangential forcedFT which lies in the rotor plane and causes the torque and the other is normal to the rotorplane dFN and causes the thrust force.

where c is the chord length and Urel is the relative wind speed defined as:

Urel =U(1− a)

sin(ϕ)(2.17)

Using 2.5 the normal and tangential forces can be defined as:

dFN = dFLcos(ϕ) + dFDsin(ϕ) (2.18)

dFT = dFLsin(ϕ)− dFDcos(ϕ) (2.19)

For a blade element at a distance r and a rotor with number of blades of B, the thrust and

torque can be determined as follows:

dT = BdFN (2.20)

dQ = BrdFT (2.21)

31

therefore

dT = B1

2ρU2

rel(Clcos(ϕ) + Cdsin(ϕ))cdr (2.22)

dQ = B1

2ρU2

rel(Clsin(ϕ)− Cdcos(ϕ))crdr (2.23)

The angle of relative wind speed ϕ can be related to axial (a) and tangential (a) induction

factors as:

ϕ = tan−1(U(1− a)

Ωr(1 + a)) (2.24)

2.3 Corrections of Blade Element Momentum Theory

Blade Element Momentum (BEM) Theory has some limitations in practical applications. In

this section some corrections are presented to compensate the limitations of the BEM.

2.3.1 Tip-Loss and Hub-Loss

One of the things that is not considered in BEM and influences the induced velocity is the

helical shape vortices in the wake created by the blade tips. The vortices close to the hub of

the rotor also have a similar effect and influence the induced velocities. These two effects are

called tip-loss and hub-loss effects and some corrections should be made in BEM to address

these effects. By applying the following correction factor F to the momentum part of the

BEM, one can modify it for practical applications [42]:

F = FtipFhub (2.25)

where Ftip and Ftip are tip-loss and hub-loss factors defined as:

Ftip =2

πcos−1e−

B(R−r)2rsinϕ (2.26)

Fhub =2

πcos−1e−

B(r−Rhub)

2rsinϕ (2.27)

In which B is number of blades, R is the blade radius, r is the radial distance of blade

element from center of rotor plane,Rhub is the radius of hub, and ϕ is the inflow angle. Then

32

using the correction factor the modified thrust and torque will be:

dT = 4πρU2(1− a)aFrdr (2.28)

dQ = 4πr3ρUΩ(1− a)aFdr (2.29)

2.3.2 Glauert Correction

According to Equation 2.11, the BEM predicts that for a = 0.5 the wind velocity in the

far wake U4 will be zero and the flow will come to stop. Also, for a > 0.5 the far wake

wind velocity will be negative and flow reversal will happen. This situation violates the

assumptions of the BEM theory and for a > 0.5, the behaviour of the flow will be different

from what BEM predicts. To compensate for this effect in BEM, an empirical relation for

turbulent wake state ( a > 0.5 ) is presented by Glauret. According to the modified Glauert

relation, for CT > 0.96F , the following new axial induction factor should be used instead of

the axial induction factor calculated by the BEM [42]

a =18F − 20− 3

√CT (50− 36F ) + 12F (3F − 4)

36F − 50(2.30)

where F is defined by Equation 2.25 and CT is the thrust coefficient of the blade element

and can be obtained using the following equation:

CT =σ(1− a)2(Clcosϕ+ Cdsinϕ)

sin2ϕ(2.31)

2.4 Blade Element Momentum Theory

In Appendix A the momentum theory is used to derive the thrust force and torque as follows:

dT = 4πρU2(1− a)aFrdr (2.32)

dQ = 4πr3ρUΩ(1− a)aFdr (2.33)

33

also the blade element theory is used to find:

dT = B1

2ρU2

rel(Clcos(ϕ) + Cdsin(ϕ))cdr (2.34)

dQ = B1

2ρU2

rel(Clsin(ϕ)− Cdcos(ϕ))crdr (2.35)

Equating dT and dQ from these two sets of equations, we will find:

a =

[1 +

4Fsin2(ϕ)

σ(Clcos(ϕ) + Cdsin(ϕ))

]−1

(2.36)

a =

[− 1 +

4Fsin(ϕ)cos(ϕ)

σ(Clsin(ϕ)− Cdcos(ϕ))

]−1

(2.37)

where σ is local solidity defined as:

σ =Bc

2πr(2.38)

2.5 Iterative Procedure for BEM Method

In order to calculate the thrust force along the blade, the thrust force of each blade element

should be calculated first using the BEM iterative algorithm. Then the total thrust force

can be obtained by summation of blade elements thrust forces. The main inputs to BEM

iterative algorithm are pitch angle β and tip speed ratio at blade element location defined

as:

λr =U

Ωr(2.39)

The thrust force calculation using BEM method is summarized here and illustrated in Fig-

ure 2.6:

1. Start the iteration with the initial estimations [42]:

a =1

4

[2 + πλrσ −

√4− 4πλrσ + πλ2

rσ(8β + πσ)

](2.40)

a = 0 (2.41)

34

2. Calculate the inflow angle:

ϕ = tan−1((1− a)

λr(1 + a)) (2.42)

3. Calculate correction factor F :

F = FtipFhub (2.43)

Ftip =2

πcos−1e−

B(R−r)2rsinϕ (2.44)

Fhub =2

πcos−1e−

B(r−Rhub)

2rsinϕ (2.45)

4. Calculate the thrust coefficient for the element using the following equation:

CT =σ(1− a)2(Clcosϕ+ Cdsinϕ)

sin2ϕ(2.46)

5. If CT 0.96F , the element is highly loaded therefore, use the modified Glauert

correction instead of standard BEM theory to calculate the axial induction

factor as:

a =18F − 20− 3

√CT (50− 36F ) + 12F (3F − 4)

36F − 50(2.47)

6. If CT 0.96F , use the BEM theory to determine the axial induction factor:

a =

[1 +

4Fsin2(ϕ)

σ(Clcos(ϕ) + Cdsin(ϕ))

]−1

(2.48)

7. Calculate the tangential induction factor using BEM theory:

a =

[− 1 +

4Fsin(ϕ)cos(ϕ)

σ(Clsin(ϕ)− Cdcos(ϕ))

]−1

(2.49)

8. Repeat steps 2 to 6 until the axial induction factor converges.

9. Calculate the thrust force of the blade element using the following equation:

dT = B1

2ρU2

rel(Clcos(ϕ) + Cdsin(ϕ))cdr (2.50)

10. Repeat the iterative algorithm for all of the blade elements and then find the

total thrust force by summation of the blade elements thrust forces.

35

Initialize a and a fromEquations 2.8 and 2.9

1. Calculate ϕ from Eq. 2.242. Calculate F from Eq. 2.25

Calculate CT (Equa-tion 2.31) by readingCl and Cd from Ta-

bles A.2, A.3, A.4, or A.5

CT 0.96F

Calculate a and afrom Equations 2.30and 2.37 respectively

Calculate a and afrom Equations 2.36and 2.37 respectively

| anew − aold |≺ ε

Calculate thrust force fromEquation 2.22

no

yes

no

yes

Figure 2.6: BEM iterative algorithm

36

2.6 BEM Validation

In this section the BEM method is used to simulate the thrust force for a given wind speed

profile, pitch angle and angular speed of rotor. As the real wind turbine was not availble

to do experiment in this work the output of BEM method is compared and verified using

the FAST [34] which is a wind turbine simulation package developed by National Renewable

Energy Laboratory (NREL). The simulations are for a 1.5 MW wind turbine and the blades

are divided to 15 blade elements for simulation. Four different airfoils are used in the blades

whose lift and drag coefficients are provided in Tables A.2, A.3, A.4, and A.5. The chord

length, twist angle and geometry of the cross-section are assumed to be constant along each

blade element as it is shown in Table A.1. As it is shown in Figures 2.7, 2.8 and 2.9, thrust

force is simulated for different wind speed profiles. As the wind speeds are higher than the

rated wind speed, the pitch control system is used to regulate the rotor angular speed and

keep it at its rated value. The control system adjusts the pitch angle according to the wind

speed variations as it is shown in the Figures 2.7, 2.8 and 2.9. The BEM theory is used to

simulate the thrust force for the provided data of the variable wind speed and pitch angle

and the results are studied in time and frequency domain.

37

0 200 400 600 800 100012

14

16

18

20

22

24

26

28

Time (s)

Win

d S

peed

(m

/s)

(a)

0 200 400 600 800 1000−2

−1

0

1

2

3

4

5

6

7

Time (s)

Pitc

h A

ngle

(de

gree

)

(b)

0 200 400 600 800 10000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

5

Time (s)

Thr

ust F

orce

(N

)

Model (BEM)FAST

(c)

10−3

10−2

10−1

100

101

102

0

20

40

60

80

100

120

140

160

f (Hz)

|FT(f

)|

Model (BEM)FAST

(d)

Figure 2.7: Verification of the BEM method result with the FAST output which uses thegeneralized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerody-namics thrust force in time domain (d) Aerodynamic thrust force in frequency domain

38

0 200 400 600 800 100010

12

14

16

18

20

22

24

26

Time (s)

Win

d S

peed

(m

/s)

(a)

0 200 400 600 800 1000−1

0

1

2

3

4

5

6

Time (s)

Pitc

h A

ngle

(de

gree

)

(b)

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

5

Time (s)

Thr

ust F

orce

(N

)

Model (BEM)FAST

(c)

10−3

10−2

10−1

100

101

102

0

50

100

150

200

250

f (Hz)

|FT(f

)|

ModelFAST

(d)

Figure 2.8: Verification of the BEM method result with the FAST output which uses thegeneralized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerody-namics thrust force in time domain (d) Aerodynamic thrust force in frequency domain

39

0 200 400 600 800 10008

10

12

14

16

18

20

22

24

26

Time (s)

Win

d S

peed

(m

/s)

(a)

0 200 400 600 800 1000−2

−1

0

1

2

3

4

5

6

7

Time (s)

Pitc

h A

ngle

(de

gree

)

(b)

0 200 400 600 800 10000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

5

Time (s)

Thr

ust F

orce

(N

)

ModelFAST

(c)

10−3

10−2

10−1

100

101

102

0

5

10

15

20

25

30

35

40

45

50

f (Hz)

|FT1(

f)|

ModelFAST

(d)

Figure 2.9: Verification of the BEM method result with the FAST output which uses thegeneralized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerody-namics thrust force in time domain (d) Aerodynamic thrust force in frequency domain

40

The simulation result of the BEM theory shows close agreement with the FAST outputs.

Comparing the results in time domain shows a small error between the outputs of BEM

theory and the FAST. By comparing the results in frequency domain this small difference

between the results can be justified. As it is shown in the Figures A.2, A.3, A.4, and A.5

there is a close match between the BEM theory and FAST in low frequencies. However, in

high frequencies the difference between the BEM output and the FAST is more noticeable.

Also, in certain frequencies there is a significant difference between the results. Comparing

these frequencies in different simulations shows that the frequencies are same for all of the

cases regardless of the inputs. Considering the structural vibration modes of wind turbine

we can see that these frequencies are related to the natural frequencies of the blades flapwise

vibration and tower vibration. The vibration of the blades and tower affect the relative wind

speed and consequently influence the thrust force exerted on the wind turbine. However,

it should be noted that the error introduced to the simulations due to the blade and tower

vibration is not because of using BEM theory. If we take into account the blade and tower

vibration in BEM simulations, the results will be closer to the FAST output. However,

for some practical reasons, the vibrations are not considered in simulations. First of all,

considering the blades vibration has some practical issues in real cases applications. For

example as the blades are rotating, the blades deflection measurement will be very difficult

and noisy. Besides, as the results provided in Chapter 3 shows, the calculated thrust force

using BEM theory is accurate enough for our application in this thesis which is the system

identification of the wind turbine tower deflection.

In summary, the thrust force modeling using the BEM theory provides aerodynamic

thrust force results with reasonable accuracy and computational cost for our application

which will be used for wind turbine tower system identification.

41

Chapter 3

System Identification of Wind Turbine Tower

Motivated by the fact that the model parameter estimation of dynamic systems is not feasible

using first principals modelling, in this chapter a subspace system identification framework is

presented for modelling and parameter estimation of wind turbine tower dynamics. Subspace

system identification technique uses input-output data of system to build a data-driven

model. This chapter is organized as follows. First, the dynamic model of the wind turbine

tower fore-aft motion is presented and then the subspace system identification is used as a

compliment to obtain accurate estimation of the model parameters. At the end, the identified

model is tested and evaluated using different sets of data.

3.1 Tower Fore-aft Motion Dynamics

In order to estimate the effective wind speed based on the wind turbine tower deflection,

we should first model the wind turbine tower dynamics. In this section a physical model

for wind turbine tower deflection is presented which describes the wind turbine tower as a

cantilever beam subjected to the aerodynamic thrust force. Then a discrete-time state-space

description of the model will be presented to be used in the following sections for system

identification of the wind turbine tower deflection.

To start the modelling of wind turbine tower deflection, consider the equation of motion

of a beam under distributed transverse force is given by:

∂2

∂x2

[EI(x)

∂2w(x, t)

∂x2

]+ ρA(x)

∂2w(x, t)

∂t2= f(x, t) (3.1)

where w is the lateral deflection of blade, f(x, t) is the transverse force, E represents the

elasticity modules, I is the area moment of inertia of the cross section of the beam, ρ is the

42

linear density of the beam and A is the area of the cross section of the beam. The boundary

conditions of the tower are clamped-free at two ends and at t = 0 we have w(x, 0) = 0 and

w(x, 0) = 0 . Using the modal analysis the solution of Equation 3.1 is assumed to be a

linear combination of the normal modes of the beam as:

w(x, t) =∞∑i=1

Wi(x)qi(t) (3.2)

where qi(t) are the generalized coordinates and Wi are the normal modes found by solving

the following eigenvalue equation using the boundary conditions:

d2

dx2

[EI(x)

d2Wi(x, t)

dx2

]− ρA(x)ω2

iWi(x) = 0 (3.3)

where ωi is ith natural frequency of the system. The normal modes Wi and Wj are orthogonal

if i 6= j and hold the following relation which is called the orthogonality relation (see Ref. [50]

for the proof):

∫ l

0

ρA(x)Wi(x)Wj(x)dx =

0 i 6= j

1 i = j(3.4)

Now using Equations 3.2 and 3.3, Equation 3.1 can be expressed as:

ρA(x)∞∑i=1

ω2iWi(x)qi(t) + ρA(x)

∞∑i=1

Wi(x)d2qi(t)

dt2= f(x, t) (3.5)

Multiplying Equation 3.5 by Wj and integrating from 0 to l results in:

∞∑i=1

qi(t)

∫ l

0

ρA(x)ω2iWi(x)Wj(x)dx

+∞∑i=1

d2qi(t)

dt2

∫ l

0

ρA(x)Wi(x)Wj(x)dx =

∫ l

0

Wj(x)f(x, t)dx (3.6)

in which l is the length of the blade. Using the orthogonality condition (Equation 3.4),

Equation 3.6 can be reduced to:

d2qi(t)

dt2+ ω2

i qi(t) = Qi(t), i = 1, 2, . . . (3.7)

43

where Qi(t) is the generalized force corresponding to the ith mode given by:

Qi(t) =

∫ l

0

Wi(x)f(x, t)dx, i = 1, 2, . . . (3.8)

Considering the thrust force to be exerted on the tip of the wind turbine tower and assuming

the aerodynamic forces exerted to other parts of the tower to be small (because of the small

area) the generalized force can be reduced to:

Qi(t) = Wi(l)f(l, t), i = 1, 2, . . . (3.9)

Also using the Equation 3.2 and considering finite number of modes, the tip deflection of

the wind turbine tower can be determined by:

w(l, t) =n∑i=1

Wi(l)qi(t) (3.10)

Therefore, using the Equations 3.7, 3.9 and 3.10, the tip deflection of wind turbine tower

can be expressed by state-space model as follows:

x(t) = Ax(t) + Bu(t) (3.11)

y(t) = Cx(t) (3.12)

where y(t) is the tower deflection, u(t) = f(l, t) is the thrust force and x(t) is the states

vector. Considering the time step of T = ∆t the discretized state-space model will be:

x(k + 1) = Ax(k) +Bu(k) (3.13)

y(k) = Cx(k) (3.14)

in which

A = eA∆t (3.15)

B =

∫ (k+1)∆t

k∆t

eA[(k+1)∆t−τ ]Bdτ (3.16)

x(k) = [x1(k), x2(k), . . .]T (3.17)

44

The discretized state-space model is suitable for wind speed estimator design. However,

the A, B and C matrices can not be determined accurately using the presented method. The

reason is that these matrices contain several physical parameters of the system which cannot

be measured or estimated accurately. In Section 3.3 the system identification technique will

be used to determine the model parameters precisely.

3.2 Hammerstein System

In this section, system identification method for modelling of wind turbine tower deflection

is described and the dynamic system of the wind turbine is illustrated to justify the use of

system identification technique. As explained in previous chapters, wind speed is the main

input to the wind turbine system which we have no control on. Additional to wind speed,

there are two more inputs to wind turbine system called control inputs: generator torque

and blades pitch angle ,which we can use the control system to decide about the values of

these inputs.

As illustrated in previous chapters, wind turbine dynamic behaviour is nonlinear. In

order to model the wind turbine dynamic system, we should deal with nonlinearities which

are mainly because of the aerodynamic subsystem.

Using nonlinear system identification methods is one of the ways to identify the model

for wind turbine system. However, nonlinear system identification techniques are not mature

and well-developed to be used for complicated systems like wind turbine dynamics. Some of

nonlinear system identification methods are only tested on simulated systems and are not

applicable in real life system identification problems [11]. The other popular method for

dealing with nonlinear systems is to linearise the system and consider it as a linear time

invariant (LTI) system around a constant operating point. However, using this method for

wind turbine has some challenges. First of all, in wind turbines the operating point varies

contentiously because of the varying wind field or changing pitch angle. On the other hand,

45

Rotational

Dynamics

Tower Dynamics (LTI)

2

2

, ,

, ,

T

Q

F U

F U

Wind turbine mechanics

R

U

Controller

U

TF

QF

gT

FAD

SSD

Wind Turbine Dynamics

Wind Turbine Aerodynamics

Figure 3.1: Wind turbine dynamic system is described using Hammerstein structure wherethe Wind Turbine Aerodynamics is considered as the static nonlinearity and the TowerDynamics is considered as a LTI system. The entire dynamic system of wind turbine isshown under closed-loop control; however, open-loop system identification method still canbe used for identification of Tower Dynamics because there is no feedback from tower fore-aftdeflection DFA to aerodynamic thrust force FT .

in order to identify the LTI model of wind turbine, we need to collect the input-output data

of the system around a fixed operating point. However, it is difficult to ensure that the wind

turbine is operating around a constant operating point during the measurement.

Modelling and control of wind turbines can also be performed using the methods devel-

oped for linear, parameter-varying (LPV) systems [62]. Considering the wind turbine as a

LPV system, the Gain Scheduling control techniques can be used to control the wind turbine.

However, using LPV models for control of wind turbine has some limitations. Wind speed

is one of the scheduling parameters of LPV models which is very hard to measure and it will

cause error in model parameters scheduling. In addition, LPV model of wind turbine is not

useful for our application because we need the model to estimate the wind speed, whereas

LPV model needs wind speed as an input to select the model parameters.

46

To overcome the indicated problems, identifying a relatively simple and globally valid

model, which is valid over entire operating domain, for wind turbine seems to be necessary.

As the nonlinearities of the wind turbine dynamic system are dominated by the aerodynamic

part, by splitting the system into two parts we can consider the dynamic model of wind tur-

bine as a so-called Hammerstein system. The Hammerstein structure describes the systems

that consist of a static nonlinearity in series with a LTI dynamic part. As in Figure 3.1,

dynamic system of wind turbine tower is represented as a Hammerstein system where the

nonlinear mapping of aerodynamic thrust coefficient is considered as a static nonlinearity

followed by a LTI dynamic system of tower fore-aft motion. By doing so, we can use the

well-established LTI system identification techniques to identify a model for wind turbine.

The other advantage of this method is that we are not restricted to measure the input-output

data around a fixed operating point, therefore, we can record longer sequences of data for

system identification. Also, the identified system will be globally valid in the operating

domain of wind turbine which is vital for wind speed estimation.

In summary, we use the Blade Element Momentum Theory, presented in Chapter 2, to

calculate the aerodynamic thrust force. Now the static nonlinearity associated with the

aerodynamic part is excluded from the dynamic system and the aerodynamic thrust force is

considered as an input to LTI dynamic model. Then, we use the subspace system identifi-

cation method to identify the LTI dynamic model of wind turbine tower fore-aft motion.

3.3 System Identification

Physical modelling of the dynamic systems can provide us with a great insight about the

behaviour of the system especially in the design stage; however, it is hard to find the value

of the model parameters accurately using this approach. Also, for specific applications like

control and estimation, the simple models which can capture the main characteristics of the

system and can be processed real-time are required.

47

The input-output data of real system, are a great source of information for building

a model. The mathematical techniques which enable us to build a dynamic model from

observed input-output data are called system identification. Using system identification

method we can find the model parameters regardless of the type of wind turbine tower.

Some of the possible tower types for wind turbines are monopole, guyed, sectional lattice

and tabular lattice [68].

For linear time invariant (LTI) systems, there are two main system identification tech-

niques, namely, prediction error method (PEM) and subspace identification method. In

prediction error the optimization methods are used to minimize the prediction error of the

model and find the model parameters. However, the subspace system identification uses the

linear algebra techniques to determine the model parameters.

In this section a subspace system identification framework is presented. The subspace

system identification method has some advantages over prediction error identification meth-

ods in practical cases[26]. For example, they are fast and they avoid the difficulties of finding

a model parametrization and performing a non-convex gradient based optimization which

may lead to numerical issues and local minima [60]. In this work State-space subspace sys-

tem identification (4SID) method [61] is implemented and will be applied to identify the

dynamic model of wind turbine tower fore-aft motion.

3.3.1 Notation

The state-space representation of discrete-time linear time invariant systems subject to mea-

surement and/or process noise can be described as follows:

xk+1 = Axk +Buk + wk (3.18)

yk = Cxk +Duk + vk (3.19)

with A ∈ Rn×n, B ∈ Rn×nu ,C ∈ Rny×n. The vectors xk ∈ Rn, uk ∈ Rnu , yk ∈ Rny ,

wk ∈ Rn, and vk ∈ Rny are the state vector, input signal, and output signal, process noise,

48

and measurement noise respectively.

For subspace identification method it is more suitable to describe the system using the

following equivalent state space model of combined deterministic-stochastic system in an

innovation form:

xk+1 = Axk +Buk +Kek (3.20)

yk = Cxk +Duk + ek (3.21)

where K ∈ Rn×ny is the Kalman gain and ek ∈ Rny is an unknown innovation with covari-

ance matrix E[ekekT ] = Re. In order to use linear algebra mathematical tools for matrices

identification, we need to write the state space model in the form of a single linear matrix

equation. Before doing that, some notations and mathematical techniques are required to

be introduced.

The input uk for k ∈ (0, 1, . . . , i+ h+ j − 2) can be arranged into block Hankel matrices

as follows:

(UpUf

) =

u0 u1 . . . uj−1

u1 u2 . . . uj...

.... . .

...

ui−1 ui . . . ui+j−2

ui ui+1 . . . ui+j−1

ui+1 ui+2 . . . ui+j...

.... . .

...

ui+h−1 ui+h . . . ui+h+j−2

= (U+p

U−f) =

u0 u1 . . . uj−1

u1 u2 . . . uj...

.... . .

...

ui−1 ui . . . ui+j−2

ui ui+1 . . . ui+j−1

ui+1 ui+2 . . . ui+j...

.... . .

...

ui+h−1 ui+h . . . ui+h+j−2

(3.22)

where Up is the matrix of past inputs and Uf is the matrix of future inputs. In the same

way, the output signal yk and noise signal ek can be arranged in Yf , Yp, Ef , and Ep. Also

49

the past input-output data can be arranged in Wp and W+p as follows:

Wp =

Up

Yp

(3.23)

W+p =

U+p

Y +p

(3.24)

In addition, the block-Toeplitz matrices Hdk and Hs

k associated with the deterministic and

stochastic subsystems will be defined according to:

Hdk =

D 0 0 . . . 0

CB D 0 . . . 0

CAB CB D . . ....

......

.... . .

...

CAk−2B CAk−3B CAk−4B . . . D

(3.25)

Hsk =

I 0 0 . . . 0

CK I 0 . . . 0

CAK CK I . . ....

......

.... . .

...

CAk−2K CAk−3K CAk−4K . . . I

(3.26)

The extended observability matrix Γk is defined as:

Γk =

C

CA

. . .

CAk−1

(3.27)

50

System state sequence also can be arranged in a matrix form as follow:

Xp = (x0, x1, . . . , xj−1) (3.28)

Xf = (xi, x1, . . . , xi+j−1) (3.29)

3.3.2 Subspaces Projection

Subspaces projection will be used as the main mathematical tool in the process of parameters

identification. In this section, a brief overview of orthogonal and oblique projections is

provided.

3.3.2.1 Orthogonal Projection

The row space A can be projected orthogonally on the row space B as shown in Figure 3.2

which will decompose it to two terms with orthogonal row spaces as follows:

A = A/B + A/B⊥ (3.30)

The orthogonal projection can be done using LQ decomposition: B

A

= LQ =

L11 0

L21 L22

Q1

Q2

(3.31)

Then, we have:

A/B = L21Q1

A/B⊥ = L22Q2

3.3.2.2 Oblique Projection

As shown in Figure 3.3 the oblique projection can be used to decompose the row space A to

three components as follows:

A = A/B

C + A/C

B + A/

B

C

⊥

(3.32)

51

/A B

/A B

B

B

A

Figure 3.2: Orthogonal Projection

In which A/B

C is oblique projection of the row space A along the row space of B on the row

space of C and

B

C

is the joint row space of B and C. The LQ decomposition can be

used to find the oblique projection:B

C

A

= LQ =

L11 0 0

L21 L22 0

L31 L32 L33

Q1

Q2

Q3

(3.33)

Then, the orthogonal projection can be obtained as:

A/B

C = L32L−122 C = L32L

−122

(L21 L22

) Q1

Q2

(3.34)

3.3.3 State-space Subspace System Identification

The starting point to performe the state-space subspace system identification is to combine

the recursive state space innovation model into a single linear matrix equation by recursively

substituting Equation 3.20 into Equation 3.21:

Yf = ΓhXf +HdhUf +Hs

hEf (3.35)

52

/C

A B

/C

A BC

B

A

/B

AC

Figure 3.3: Oblique Projection

If we can determine the row space of states sequence matrix XF , it would be enough to find

the state space model parameters A, B, C and D by solving the following equation using

least squares method: Xi+1

Yi

=

A B

C D

Xi

Ui

(3.36)

The reason for sufficiency of row space is that the state space model for a system is not

unique and its parameters can change by defining different sates for the system. However,

the row space of Xf is an invariant subspace regardless of the state space basis we use to

span it. Therefore, knowing the values of inputs Ui and outputs Yi in Equation 3.36, we can

determine state space model matrices only by knowing the row space of Xf rather than the

exact numerical value of Xf . The first step to find the row space of Xf is determining ΓhXf

by eliminating HdhUf and Hs

hEf from right hand side of Equation 3.35 which will be done

using oblique projection of subspaces. After that, singular value decomposition (SVD) can

be used to split ΓhXf to row space of Xf and column space of Γh.

3.3.3.1 Obtaining ΓhXf From Input-Output Data

It can be proven that Yf lies in the joint row space of Uf and Wp and also that the Ef is

perpendicular to it [61]. With these two assumptions and using Equation 3.35, the oblique

53

projection of Yf along the Uf on Wp will give us the ΓhXf :

Oh = Yf /Uf

Wp = ΓhXf (3.37)

3.3.3.2 Splitting Oh = ΓhXf

The singular value decomposition (SVD) can be used to split the Oh to observability matrix

Γh and states sequence matrix Xf . In this process, the order of the model n can also be

determined by separating the diagonal singular values matrix S to two parts in which S1 is

a diagonal n×n matrix and contains the significant singular values and S2 contains the rest

of the singular values:

Oh =

(U1 U2

) S1 0

0 S2

V T

1

V T2

(3.38)

Γh = U1S1/21 (3.39)

Xf = S1/21 V T

1 (3.40)

3.3.3.3 Estimating State Space Model Matrices

Now we can use the row space of Xf to estimate A, B, C and D matrices. The state

sequences, input and output, are related to state space model matrices by the following

relation: Xi+1

Yi

=

A B

C D

Xi

Ui

+

ρw

ρv

(3.41)

In which the inputs Ui and the outputs Yi are available and the state sequences Xi and Xi+1

are estimated from the previous steps of subspace system identification:

Xi =

(xi . . . xi+j−1

)(3.42)

Xi+1 =

(xi+1 . . . xi+j

)(3.43)

Ui =

(ui . . . ui+j−1

)(3.44)

Yi =

(yi . . . yi+j−1

)(3.45)

54

Therefore, Equation 3.41 can be solved using the least squares as follows to obtain the

state space model parameters: A B

C D

=

Xi+1

Yi

Xi

Ui

†

(3.46)

The A and C matrices can also be determined using column space of Γh. The matrix

C can be read easily from the first block row of Γh and the matrix A can be obtained by

shifting the structure of Γh. If we obtain the Γh and Γh by removing the first and last block

rows respectively from Γh:

Γh =

C

CA

. . .

CAh−2

(3.47)

Γh =

CA

CA2

. . .

CAh−1

(3.48)

Then the matrix A can be obtained by solving the following equation using the least

square method:

ΓhA = Γh (3.49)

3.3.3.4 Subspace System Identification Algorithm Summary

The complete algorithm of subspace system identification is summarized here [61]:

1. Arrange the input-output data into Hankel signal matrices Up, Uf , Yp, Yf , U+p ,

U−f , Y +p , Y −f , Wp and W+

p . (see Section 3.3.1)

55

2. Calculate the oblique projections (see Section 3.3.2.2)

Oi = Yf /Uf

Wp = ΓiXi (3.50)

Oi+1 = Y −f /U−f

W+p = Γi−1Xi+1 (3.51)

3. Calculate singular value decomposition (SVD) of Oh and determine the order

of the system by picking significant singular values S1 (see Section 3.3.3.2)

Oh = USV T =

(U1 U2

) S1 0

0 S2

V T

1

V T2

(3.52)

4. Determine Γh and Γh as follows: (see Sections 3.3.3.2 and 3.3.3.3)

Γh = U1S1/21 (3.53)

Γh = Γh(1 : h− 1, :) (3.54)

5. Determine state sequences

Xi = Γ†iOi (3.55)

Xi+1 = Γi†Oi+1 (3.56)

6. Estimate the model parameters (see Section 3.3.3.3) A B

C D

=

Xi+1

Yi

Xi

Ui

†

(3.57)

3.4 Wind Turbine Tower System Identification

Using the knowledge of wind speed U , rotor speed Ω, and pitch angle β, the aerodynamic

thrust force can be calculated using the BEM (see Chapter 2) as:

FT (k) =1

2ρπR3CT (λ(k), β(k))U2(k) (3.58)

56

1 2 3 4 5 6 7 80

100

200

300

400

500

600

700

Order

Sin

gula

r V

alue

Figure 3.4: Diagonal elements of S matrix in singular value decomposition (SVD) of Oh

matrix which is used to determine the order of the model

where, k is time step, λ is tip speed ratio and CT is the thrust coefficient calculated by BEM

algorithm. Now we can consider thrust force as an input to tower dynamic system which is

assumed to be Linear Time Invariant (LTI) as discussed in Section 3.2. If we describe the

dynamic model of wind turbine tower deflection using the innovation state-space structure

(Equation 3.20), we will have:

yk = DFA(k) (3.59)

uk = FT (k) (3.60)

where DFA(k) and FT (k) are wind turbine tower fore-aft deflection and aerodynamic thrust

force respectively as shown in 3.1.

The diagonal elements of the matrix S in Equation 3.52 are plotted in Figure 3.4 which

shows that the first four singular values are significant. Therefore, we can choose order of

the model using this figure which shows S1 should be a 4× 4 matrix.

Studying the residual values of the identification shows that the noise can be considered

zero mean and it has a small variance; however, it is not a white noise. The reason is that

the random noise in the wind speed goes through the static nonlineariaty which causes the

input noise to the structural model to be coloured noise.

57

3.5 Practical Aspects of Wind Turbine Tower System Identification

In this section, practical aspects of wind turbine tower identification such as control system

design, data collection for system identification, and wind speed estimation will be discussed.

3.5.1 Control System Design

In order to identify the wind turbine tower system dynamic model, we need to collect input-

output data for the system in the desired operating condition. Wind turbine systems are

operating in a closed-loop system in which the wind speed acts like a disturbance to the

system and the generator torque and blades pitch angle are used as control inputs to keep

the wind turbine in a desired operating point. To identify the wind turbine tower dynamic

system, one should use the data measured under closed-loop control. Therefore, designing a

control system is required for system identification. In this thesis, adaptive control is designed

and implemented to control the wind turbine. The data used for system identification is

collected under closed-loop operation of wind turbine.

It should be noted that using open-loop subspace system identification methods for iden-

tifying the systems that are operating under closed-looped control may lead to a biased

estimation of model parameters. The reason for this is that the formulation used for open-

loop system identification assumes that the input signal uk to the system is uncorrelated

with the past noise process ek. Using closed-loop data for system identification violates the

assumption:

EukeTj 6= 0 for j < k (3.61)

In other words, when we have closed-loop control the feedback from output of system is

used to correct the input to the system which causes the past process noise to enter to input

of the system. This in turn results in an input signal which is correlated with the process

noise.

In subspace system identification algorithm, past input-output data is used as the re-

58

gressor to eliminate the noise in the matrix equation assuming that the past input-output

data is uncorrelated with the noise. Using closed-loop data for system identification, this

assumption will not be valid any more which may lead to biased parameter estimation.

However, despite of the fact that wind turbine operates under closed-loop control, we

still can use subspace system identification technique for identifying the model parameters

of the wind turbine tower. It can be seen in Figure 3.1 that tower dynamics of wind turbine

does not have any feedback loop. Therefore, considering thrust force FT as the input and

tower fore-aft deflection DFA as the output, we can employ open loop system identification

methods to identify the wind turbine tower dynamic model.

3.5.2 Data Collection for System Identification

In order to identify the dynamic model of wind turbine tower deflection successfully, informa-

tive data should be used. Collecting informative data requires careful design of experiments.

For system identification experiment design, the condition in which data is collected is

the most important thing that we should decide about. As the model might be biased to the

operating domain, the experimental condition should resemble the condition in which the

model will be used. Therefore, the data for identification of wind turbine tower deflection

should be collected in operational region of wind turbine. For example, for a 5MW wind

turbine, the operational range is defined as the wind speeds between cut-in and cut-out

wind speeds (3m/s to 25m/s), the rotor speeds between cut-in and rated values (6.9rpm to

12.1rpm) and the pitch angles between defined limits (0 to 90) [35].

There are some considerations for selecting the type of the input which used in the

experiment. To have informative data, which allows us to discriminate between different

models, the input must be exciting.

Having a closed loop control system decreases the information of the data by relating

the input and output specially when the reference control input is time-invariant. This issue

should be considered in data collection for wind turbine tower system identification as it

59

operates under closed-loop system.

In wind turbine system, the controller regulates the aerodynamic torque by changing the

tip-speed ratio or the pitch angle. In Region I (as defined in Section 1.2.2), the reference

control input r(t) = ω0(t) is time-variant and changes with variations in wind speed to

maintain the optimal value of the tip speed ratio λ. However, in Region III, the reference

control input r(t) = ω0 is equal to the rated rotor speed which is constant. Ideally the pitch

controller will regulate the aerodynamic torque to stay constant. Although the aerodynamic

torque is not directly related to the wind turbine tower system, it is correlated with the

aerodynamic thrust force which is the input to the wind turbine tower system. As a result,

the torque feedback control system with time-invariant reference input in Region III might

affect the thrust force input to the wind turbine tower system to be non-exciting and there

is a chance to collect non-informative data.

To avoid this problem, one should inject perturbations into the controller reference signals

to excite the system and collect the data which is informative enough for system identifica-

tion. Figure 3.5 shows that the pitch angle β is perturbed by adding binary random signals

of ±0.5.

The primary input which excites the wind turbine system is the wind speed variations.

As we do not have control on wind speed variations, we can not guarantee that the wind

speed excitation will be enough for collecting informative data during the experiment.

In Region III, the aim of the controller is to cancel the effect of the wind speed variations

on aerodynamic torque, therefore, the regulator may reduce level of the excitation of thrust

force even for the case that the wind speed is exciting.

In order to make the experiments cost effective and avoid redoing them, we can use per-

turbation in the reference signals to excite the system. In this work we have used the binary

signals as the perturbation. It should be noted that using random binary signals for iden-

tifying the systems with static nonlinarity may lead to ignorance of the static nonlinearity.

60

Wind Turbine System

Torque Controller

Pitch Controller

U

rated

opt

gT

Figure 3.5: The control system of the wind turbine is shown which consists of two parts,namely, generator torque control and blade pitch angle control. As shown, the random binarysignals are added in order to make enough excitation in the system to collect informativedata

That is because when the binary signal goes through static nonlinearity the result still will

be a binary signal. Here we use random binary signals only for perturbation and the static

nonlinearity is already modelled using BEM theory.

For system identification of the wind turbine tower deflection, a measurement sequence

of 1000s (15 min) is used. The sampling rate of 100Hz is used which is in the order of 10

times higher than the bandwidth of the system as advised [39]. The sampling rate of the

acquisition system could be governed by the control system or be a fixed value. In these

cases, the anti-aliasing filter should be used to downsample the data.

3.5.3 Input-output Data Measurement for System Identification

To identify the wind turbine tower dynamics, the tower deflection, rotor speed and wind

speed should be measured. For wind speed measurement the remote sensing techniques such

as LIDAR (Light Detection And Ranging) [55] and SODAR (Sound Detection And Ranging)

61

[36] can be used. In these techniques we use the sound or laser to detect the velocity of the

air stream in the upstream of the wind turbine. The working principle of these techniques

is similar to radar in which we use the Doppler effect. For example, the LIDAR measures

the wind speed by transmitting a laser beam into atmosphere boundary layer and measuring

the scattered radiation received back at the instrument [36]. The SODAR uses sound wave

instead of laser beam.

The tower deflection of wind turbine can be measured using an array of strain gauges or

accelerometers. A successful example of tower deflection measurement is given in [4] where

the arrayed Fiber Bragg Grating (FBG) sensors are employed to accurately measure the

strain along the wind turbine tower. Displacement-Strain Transformation (DST) is used

to obtain the tower deflection from strain measurements. The DST matrix is found using

modal approach by doing Finite Element (FE) analysis of wind turbine tower structure.

It should be mentioned that the changes in yaw angle of wind turbine changes the di-

rection of the fore-aft and side-side tower deflection which can cause error in wind turbine

tower deflection measurement. This problem can be alleviated by installing enough number

of strain gauges around the tower circumference or building higher order DST matrix.

The other thing that should be measured for system identification of wind turbine tower

is the rotor speed Ω. In commercial wind turbines, the sensor for measuring the rotational

speed of low-speed shaft is not provided. However, the rotational speed of the high-speed

shaft (generator side) is measured. A simple method is to ignore the rotational dynamics

of the drive-train and only consider the rigid body motion of gearbox. Using this method,

the rotational speed of low-speed shaft can be obtained from rotational speed of high-speed

shaft using the gearbox ratio. However, a lumped mass model for rotational dynamics of

wind turbine drive-train along with Kalman filter can be used to estimate the rotor speed

of wind turbine [17, 65].

62

3.6 System Identification Evaluation

In this section, the developed model for wind turbine tower deflection will be validated. The

identified model will be applied on some new data sets, which are independent from the

data set that we used for identification, to examine its performance in predicting the system

outputs and simulating the dynamic behavior of the system.

As the real wind turbine was not availble to do experiment in this work, wind turbine

tower model is identified and validated for the WP 1.5MW wind turbine using the simulation

data provided by FAST [34] which is a certified wind turbine simulation package developed

by National Renewable Energy Laboratory (NREL).

63

0 100 200 300 400 500 600 700 800 900 100010

12

14

16

18

20

22

24

26

28

Time (s)

Win

d S

peed

(m

/s)

(a)

0 100 200 300 400 500 600 700 800 900 1000−2

0

2

4

6

8

10

Time (s)

Pitc

h A

ngle

(de

gree

)

(b)

0 100 200 300 400 500 600 700 800 900 100011.7

11.8

11.9

12

12.1

12.2

12.3

12.4

Time (s)

Rot

or S

peed

(rp

m)

(c)

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Thr

ust F

orce

(kN

)

FASTBEM

(d)

Figure 3.6: Input data set used for evaluation of wind turbine tower identified model. Thepitch angle is adjusted by the control system to maitain the rotor speed at rated values(12rpm) a) input wind speed profile b) pitch angle experienced by the wind turbine bladesc) rotor speed d) aerodynamic thrust force associated with the presented wind speed, pitchangle and rotor speed. The thrust force calculated by BEM is compared with the FASToutput.

64

0 100 200 300 400 500 600 700 800 900 1000−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (s)

Tow

er D

efle

ctio

n (m

)

FASTIdentified Model

(a)

10−3

10−2

10−1

100

101

102

0

0.002

0.004

0.006

0.008

0.01

0.012

f (Hz)

|Tow

er D

efle

ctio

n (f

)|

FASTIdentified Model

(b)

Figure 3.7: Wind turbine tower deflection associated with the wind speed, pitch angle androtor speed condition presented in Figure 3.6. The output of the identified model is com-pared with FAST output. a) time domain and b) frequency domain representation of towerdeflection data

65

The input-output data are collected under closed-loop pitch angle control system. Fig-

ure 3.6 shows an example of wind speed profile and associated pitch angle variations where

the rotor speed is maintained at rated value (12rpm here) by controlling the pitch angle of

the blades. Also, the BEM theory which we implemented in Chapter 2 is used to calculate

the thrust force for the given wind speed, pitch angle and rotor speed data. In Figure 3.6,

the result of thrust force calculation is compared with the output of FAST which is simulated

by dynamic-wake theory [42].

To validate the model of wind turbine tower deflection we used the thrust force data in

Figure 3.6 as an input to the identified model and compared tower deflection result with

FAST output. The time domain and frequency domain representation of the results are

presented in Figure 3.7 which shows a close match.

In this validation, we involve the error caused by the inaccuracy of aerodynamic model as

well by using calculated thrust force as the input to the model. However, the purpose of the

aerodynamic thrust force calculation in this work is to identify wind turbine tower structural

dynamics and it will not be used for wind speed estimation. Therefore, we can exclude the

error associated with aerodynamic model and only evaluate the identified structural model.

To verify the accuracy of identified model alone we used aerodynamic thrust force calculated

by FAST as the input to identified model. The variance-accounted-for (VAF) [63] is used as

a quality measure for the identified model which is defined as:

V AF = max0, (1− var(y − y)

var(y))× 100% (3.62)

where y is the measured output and y is the output predicted by the model. For different

wind conditions we calculated VAF value by considering the FAST thrust output as the

input to model. The result was V AF ' 97% which shows the quality of identified model.

The other way to evaluate the identified model is to extract physical parameters of wind

turbine tower structural dynamics from identified model. In Table 3.1, we have compared

the natural frequencies and damping coefficients of the wind turbine tower with the values

66

FAST Identified Modelfn (Hz) ζ fn (Hz) ζ

1st Mode 0.4068 2.89e− 2 0.3804 3.99e− 22ed Mode 2.7301 3.26e− 2 2.6420 3e− 1

Table 3.1: Model validation using modal parameters of wind turbine tower in terms of naturalfrequency fn and damping ratio ζ

obtained from FAST which shows a good agreement indicating that the identified model

describes the dynamic behavior of the system.

In summary, evaluation of the identified model for wind turbine tower dynamics shows

the validity of the model to be used for wind speed estimation. In the following chapters, we

will use the identified tower dynamics model to develop a wind speed estimation algorithm;

therefore, the quality of the model can be verified by evaluating its capability to estimate

the wind speed as well.

67

Chapter 4

Estimation

In previous chapters, we built a model for wind turbine tower deflection and used system

identification technique to estimate the model parameters. The developed model’s inputs

are wind speed profile, rotor speed and blades pitch angle and its output is the tower fore-aft

deflection. In this chapter, we discuss our method to solve the inverse problem to estimate

the effective wind speed from tower deflection measurements.

As the model consists of two parts, namely aerodynamic static nonlinearity and tower

structural dynamics, the inverse solution to the problem will also have two separate parts.

First, we need to solve the state-space model inversely and estimate the thrust force from

the tower deflection measurements. Second, having the thrust force, we should solve the

aerodynamic model inversely to estimate the wind speed.

The first part of the model is a linear time invariant (LTI) state space structure and

we can use Kalman Filter to estimate the states and the input from the output. An online

recursive thrust force estimator based on Kalman filter is presented in this chapter. The

second part of the model is a static nonlinearity governed by the blades aerodynamics.

In Chapter 2, we used BEM theory to calculate the thrust force from wind speed in the

modelling of wind turbine tower dynamics. However, the BEM model cannot be solved

inversely as it is an iterative algorithm. In this thesis, neural network will be used to solve

the nonlinear aerodynamic relation and estimate the wind speed from aerodynamic thrust

force.

68

4.1 Thrust Force Estimator Design

In this section, we present an online algorithm to estimate the system input from output

of the system. This type of problems are categorized as inverse problems [3] and are very

common in engineering and science areas. For example, in heat transfer problems, we can

solve the inverse problem to estimate the input, e.g. heat flux in boundaries, by measuring

the output which is the temperature in different points.

The method used in this work is adopted from an inverse heat transfer problem [59]

and will be used here to estimate the aerodynamic thrust force from the tower deflection

measurement. Since the wind turbine control system requires online estimation of wind

speed, the thrust force estimator must use recursive estimation algorithm to provide online

estimation.

The input estimation method, presented in following sections, consists of two parts. In

the first part we estimate the innovation residual sequences using Kalman filter. We define

innovation residual in Kalman filter design as the deference between output estimated by the

model and the measured output. In this method we define the innovation residuals with and

without the effect of the input which results in an equation that we can solve it to estimate

the unknown input. In the second part we solve the equation using recursive lest squares

method to estimate the unknown input from innovation residual sequences.

4.1.1 Kalman Filter Design

We use the tower deflection measurement to estimate the thrust force by solving the inverse

problem. To solve the inverse problem we use the innovation residual which is defined as the

difference between measured output and model estimated output. For calculating the model

estimated output we need to first estimate the states of the system. Kalman filter is used

to estimate the states of the system and the innovation residual which will be introduced in

this section.

69

In some control and estimation applications we need to estimate the states of the dynamic

system (x vector in Equation 3.13). Since we usually do not have access to states of the

system to measure them directly or the states measurement is difficult we estimate the states

of the system using the dynamic model of the system.

As you can see in Equation 3.13, states vector x can be estimated using the states vector

in previous time step and the input. Also, using Equation 3.14 we can estimate the states of

the system by measuring the output. The idea behind Kalman filter is to design an optimum

state estimator by combining both of these sources: model estimation and measurement.

When we have two different predictions for states of the dynamic system Kalman fil-

ter considers the weighted average of predictions as the optimum prediction. In averaging

process, predictions with smaller uncertainty values are more trusted by Kalman filter and

are given larger weights where the measure of uncertainty for different predictions is their

covariances.

The Kalman gain plays the role of the weights in weighted average. In other words, it

defines which prediction is more trusted, whether model estimation or the measurement.

Lower values of Kalman gain mean that the prediction made by the model is more trusted

than the measurement. On the other hand, higher values of Kalman gain mean that the

measurement is more trusted than the model estimation.

In this section, a discrete-time Kalman filter is presented for use in system states and

residual sequence estimation.

In order to use Kalman filter for residual sequence estimation, the dynamic system model

is required. In previous chapters, we described the dynamic system model of the wind turbine

tower as a discrete-time state-space model which has the following general form:

x(k + 1) = Ax(k) +B[F (k) + w(k)] (4.1)

z(k) = Hx(k) + v(k) (4.2)

70

where A and B are the state space matrices, w(k) is the Gaussian process noise sequence with

zero mean and covariance of Qk, H is the measurement matrix, x(k) is the states sequence,

z(k) is the tower deflection measurement sequence, F is the thrust force input to the system,

and v(k) is the Gaussian measurement noise with zero mean and covariance of Rk. xk|k−1 is

defined as state estimation using dynamic model by considering the previous measurements

up to t = k − 1.

xk|k−1 = Axk−1|k−1 +BF (k − 1) (4.3)

Also, xk|k is defined as the estimation of the state sequence xk given the observations up

to t = k. xk|k can be obtained by correcting the model estimation (xk|k−1) using the Kalman

gain (Kk) as follows:

xk|k = xk|k−1 +Kkyk (4.4)

where yk is the innovation residual defined as follows:

yk = zk −Hkxk|k−1 (4.5)

The optimum Kalman filter gain can be derived by minimizing the mean-square error for

xk − xk|k which will result in:

Kk = Pk|k−1HTS−1

k (4.6)

where Sk = Cov(yk) is the covariance of innovation and Pk|k−1 = Cov(xk − xk|k−1) is the

covariance of state estimate. If we expand Sk and Pk|k−1, we reach the following expressions:

Pk|k−1 = APk−1|k−1AT +BQBT (4.7)

Sk = HPk|k−1HT +R (4.8)

In the process of estimating the system states, we should find Kalman gain in each time

step. This requires to update the covariance of states. For an optimum Kalman gain, the

following recursive equation can be used to update the covariance of states in each time

step[67]:

Pk|k = (I −KkHk)Pk|k−1 (4.9)

71

It should be noted that this simple recursive equation is only valid for optimum Kalman

gain. For other gain values, a more complex equation should be used.

4.1.1.1 Kalman Filter Summary

State estimation using discrite-time Kalman filter can be summarized as follows. First, we

use the state space model of the system to estimate the states using the states at previous

time step:

x(k + 1) = Ax(k) +BF (k) (4.10)

Pk|k−1 = APk−1|k−1AT +BQBT (4.11)

where Pk|k−1 is the predicted covariance of the state estimate at t = k. Next step is to correct

the estimation using the measurement zk:

yk = zk −Hkxk|k−1 (4.12)

Sk = HPk|k−1HT +R (4.13)

Kk = Pk|k−1HTS−1

k (4.14)

xk|k = xk|k−1 +Kkyk (4.15)

Pk|k = (I −KkHk)Pk|k−1 (4.16)

where yk is the innovation residual, Sk is innovation covariance, Kk is the optimal Kalman

gain, xk|k is the corrected (updated) state estimate and Pk|k is the covariance of the updated

state estimate.

4.1.1.2 Kalman Filter Parameter Tuning

In order to implement the Kalman filter, the measurement noise covariance R and process

noise covariance Q should be measured or estimated. Better tuning the values of R and Q

Kalman filter will provide better state estimation.

Measuring the measurement noise covariance is practical because we always can do offline

measurement and collect some observation samples to calculate the measurement noise co-

72

variance R. However, determining the process noise covariance Q is more difficult in general

because we usually do not have access to observation of the states which we estimate or we

do not know the model uncertainties. Subspace system identification technique, which we

implemented in Chapter 3, can be used for process noise covariance Q estimation. From

Equation 3.41, process noise covariance can be estimated as Q = E[ρwρwT ].

When real values of Q and R are constant, Kalman gain Kk and estimation covariance

Pk will converge to a costant value and remain constant after some iterations. If this is the

case, Q and R can be pre-determined by running the Kalman filter offline [67]. However,

as model uncertainty, disturbance and sensor noise are not constant in different situations,

usually this is not the case and Q and R do not remain constant.

4.1.2 Residual Sequence Estimation

So far we used Kalman filter to estimate the states of the system and the innovation residual.

Now, we are going to use the estimation of innovation residual sequence with and without

the effect of input for estimating the unknown input Fk.

The innovation residual with effect of input is defined as zk = yk . Using Equations 4.5

and 4.3, we have:

zk = z(k)−Hxk|k−1 = z(k)−HAxk−1|k−1 −HBFk−1 (4.17)

where xk−1|k−1 is the Kalman filter estimation of the system states which can be rewritten

by combining Equations 4.4, 4.5 and 4.3 as follows:

xk|k = (I −KkHk)(Axk−1|k−1 +BFk−1) +Kkzk (4.18)

The innovation residual without the effect of the input zk can be defined by removing the

effect of input F (k) from Equation 4.17:

zk = z(k)−Hxk|k−1 = z(k)−HAxk−1|k−1 (4.19)

73

where xk|k is the estimation of the system states without the effect of input Fk which can be

defined by removing the effect of input F (k) from Equation 4.18 as follows:

xk|k = (I −KkHk)Axk−1|k−1 +Kkzk (4.20)

Unknown input can be estimated by defining ∆zk = zk − zk which is the difference

between innovation residuals with effect of the input zk and without effect of the input zk.

Therefore, by subtracting Equation 4.17 from Equation 4.19 and defining ∆xk = xk|k − xk|k

we have:

zk − zk = HA∆xk +HBFk−1 (4.21)

By subtracting Equation 4.18 from Equation 4.20 and assuming the input to be zero

before the time step k = n (Fk = 0 for k ≺ n) and to be constant from k = n to k = n + l

(Fk = Constant over interval k = n, n+ 1, . . . , n+ l) we can express ∆xk = MkBFk−1 where

Mk can be determined from the following recursive relation:

Mk =

(I −KkHk)(AMk−1 + I) n ≺ k n+ l

0 k ≺ n(4.22)

Finally, substituting ∆xk = MkBFk−1 into Equation 4.21 leads to the following equation

which will be used for unknown input Fk estimation:

zk =

zk k n

zk + ΛkFk−1 n ≺ k n+ l(4.23)

where

Λk = H(AMk−1 + I)B (4.24)

For input Fk, estimation Equation 4.23 can be arranged in the following matrix form by

assuming Fk to be constant over interval n ≺ k n+ l:

Y (N) = Φ(N)F + ε(N) (4.25)

74

where

Y (N) =

(z(n+ 1) z(n+ 2) . . . z(n+ l)

)T(4.26)

Φ(N) =

HB

H(AM(n+ 1) + I)B

...

H(AM(n+ l − 1) + I)B

(4.27)

ε(N) =

(z(n+ 1) z(n+ 2) . . . z(n+ l)

)T(4.28)

In Equation 4.25, ε(N) can be treated as a disturbance with the following covariance

matrix:

Σ(N) =

s(n+ 1) 0 . . . 0

0 s(n+ 1) . . . 0

......

. . ....

0 0 . . . s(n+ l)

(4.29)

where s(n) is the covariance of zk = yk and can be determined from Equation 4.8. It should

be noted that Σ(N) is a positive definite matrix because its diagonal elements all are positive

defined by Equation 4.8 .

Now we should solve Equation 4.25 to estimate the unknown thrust force F . In the fol-

lowing section, the least square method will be presented to solve this equation and estimate

the thrust force.

4.1.3 Thrust Force Estimation

In previous section, we found a linear matrix equation which should be solved to obtain the

unknown thrust force F . In Equation 4.25, we use l number of measurements to estimate

F which means the system is over-determined. The advantage of over-determined system

for this case is that it reduces the effect of disturbance ε(N) by considering extra number of

measurements and filtering the data.

75

The least square method can be used for solving over-determined problems. Consider

the over-determined linear model presented in Equation 4.25 where Y (N) is a N × 1 vector

containing N measurements, Φ(N) is a N × n matrix, ε(N) is a N × 1 vector containing

measurement noise components and finally F is a n × 1 vector containing the unknown

inputs to the system which should be estimated. In our case, we have n = 1 because F only

contains the unknown thrust force; therefore, number of measurements is higher than the

number of unknown parameters (N n) and the system is over determined. If we consider

F to be the estimated parameter, then we can define the vector of estimation residuals as

follows:

e = Y (N)− Y (N) = Y (N)− Φ(N)F (4.30)

The idea behind the least square method is to minimize the square of the estimation

error by minimizing the following objective function:

S = eTWe = (Y (N)− Φ(N)F )TW (Y (N)− Φ(N)F ) (4.31)

where W is a N×N weighting matrix which must be chosen symmetric and positive definite

[40]. By solving the optimization problem we can determine the estimated value of F as

follows:

F = [Φ(N)TWΦ(N)]−1Φ(N)TWY (N) (4.32)

From this equation, we can obtain the estimated thrust force using N measurements.

The weighting matrix inspired by Equation 4.29 can be selected as follows:

Σ(N)−1 =

s−1(n+ 1)γl−1 0 . . . 0

0 s−1(n+ 2)γl−2 . . . 0

......

. . ....

0 0 . . . s−1(n+ l)

(4.33)

where 0 ≺ γ 1 is the fading factor which causes the recent measurements to be weighted

more heavily than the less recent ones. Using Σ(N)−1 as the weighting matrix, the estimated

76

thrust force will be:

F = [Φ(N)TΣ(N)−1Φ(N)]−1Φ(N)TΣ(N)−1Y (N) (4.34)

Now, suppose that we want to include an additional measurement z(N+1) and use N+1

measurements for estimating F instead of N measurements. For the additional measurement

we have:

z(N + 1) = Λ(n+ 1)F + z(N + 1) (4.35)

To estimate the thrust force with additional measurement, we solve the following system

which can be obtained by combining Equation 4.35 and Equation 4.25:

Y (N + 1) = Φ(N + 1)F + ε(N + 1) (4.36)

where

Y (N + 1) =

Y (N)

z(N + 1)

(4.37)

Φ(N + 1) =

Φ(N)

Λ(N + 1)

(4.38)

ε(N + 1) =

ε(N)

z(N + 1)

(4.39)

Now, least square method (Equation 4.34) can be applied to estimate the thrust force

with additional measurement:

ˆF (N + 1) = [Φ(N + 1)TΣ(N + 1)−1Φ(N + 1)]−1Φ(N + 1)TΣ(N + 1)−1Y (N + 1) (4.40)

where

Σ−1(N + 1) =

γΣ−1(N) 0

0 s−1(N + 1)

(4.41)

in which s(N + 1) can be determined from Equation 4.8.

77

If we want to repeat this procedure for each new measurement, it will be very expensive

in terms of computational cost specially if we want to use it for real time estimation of F .

Recursive least square method can alleviate this problem by utilizing the previous estimation

of F (N) for estimating the new F (N+1). Using recursive least square method for estimating

Fk+1 leads to the following result (also see Appendix B):

F (N + 1) = F (N) +KRLS(N + 1)[z(N + 1)− Λ(N + 1)F (N)] (4.42)

where KRLS(N + 1) and PRLS(N + 1) are defined as:

KRLS(N + 1) = γ−1PRLS(N)ΛT (N + 1)[Λ(N + 1)γ−1PRLS(N)ΛT (N + 1) + s(N + 1)]−1

(4.43)

PRLS(N + 1) = [I −KRLS(N + 1)Λ(N + 1)]γ−1PRLS(N) (4.44)

Then, we can use Equation 4.42 for recursive estimation of thrust force using the wind

turbine tower deflection measurement.

4.1.4 Thrust Force Estimation Summary

In previous sections, we presented a recursive method to estimate the aerodynamic thrust

force Fk using wind turbine tower deflection measurement zk. As a summary, the recursive

thrust force estimation algorithm explained in previous sections can be performed using the

following steps:

First, we use Kalman filter to estimate the innovation residual sequence. Kalman filter

78

equations, presented in Section 4.1.1, can be summarized as follows:

xk|k−1 = Axk−1|k−1 (4.45)

Pk|k−1 = APk−1|k−1AT +BQBT (4.46)

Sk = HPk|k−1HT +R (4.47)

Kk = Pk|k−1HTS−1

k (4.48)

Pk|k = (I −KkHk)Pk|k−1 (4.49)

zk = z(k)−Hxk|k−1 (4.50)

xk|k = xk|k−1 +Kkzk (4.51)

where A and B are state space model matrices, H is the measurement matrix, Q and R are

covariances of process and measurement noises respectively, Pk|k−1 is the predicted states

covariance, Pk|k is the updated states covariance, z(k) is the tower deflection measurement

sequence, Kk is the Kalman gain, zk is the innovation residual without effect of the input,

xk|k−1 is the predicted state vector without effect of input, and xk|k is the updated state

vector without effect of input.

Then, we use recursive least squares method to estimate the unknown input thrust force

Fk from the innovation residual which we estimated in the previous step. The recursive least

square equations for input thrust force estimation are summarized as:

Λk = H(AMk−1 + I)B (4.52)

Mk = (I −KkHk)(AMk−1 + I) (4.53)

KRLS(k) = γ−1PRLS(k − 1)ΛT (k)[Λ(k)γ−1PRLS(k − 1)ΛT (k) + s(k)]−1 (4.54)

PRLS(k) = [I −KRLS(k)Λ(k)]γ−1PRLS(k − 1) (4.55)

F (k) = F (k − 1) +KRLS(k)[z(k)− Λ(k)F (k − 1)] (4.56)

where F (k) is the estimated thrust force.

79

4.2 Effective Wind Speed Estimation Using Neural Network

In previous sections, we estimated the aerodynamic thrust force using wind turbine tower

deflection measurement. The aim of this section is to estimate the effective wind speed using

estimated thrust force. For this purpose, we use Neural Networks.

Effective wind speed estimation based on the tower deflection measurement requires

solving two inverse problems; one is to estimate the thrust force from tower deflection and the

other is to estimate the effective wind speed from estimated aerodynamic thrust force. In the

first problem, we are involved with solving the state-space model of the wind turbine tower

inversely. Since the tower model is linear, in previous sections we presented an algorithm

based on Kalman filter and recursive least square to estimate the aerodynamic thrust force

from wind turbine tower deflection measurement. In the second problem, we should solve

the aerodynamic thrust force model developed using BEM theory. The main challenge is

that this model is nonlinear and iterative. Therefore, we will use neural network to solve

this inverse problem and estimate the effective wind speed from estimated thrust force.

Neural network is a powerful tool to identify and simulate the underlying relationships in

a set of data. Having a set of data, we can train a neural network to learn the relationship

between inputs and outputs. Training a neural network involves solving an optimization

problem to find the optimal neural network parameters and to map the inputs to outputs.

After training, neural network can be applied to a new set of inputs to estimate the associated

outputs.

Neural network can be trained and applied for solving either forward or inverse problems.

In the following section, a neural network will be trained to estimate the effective wind speed

from aerodynamic thrust force by solving the inverse problem.

80

4.2.1 Neural Network Training

In previous sections, we used wind turbine tower deflection measurements to estimate the

aerodynamic thrust force FT . From Equation 1.2, given the estimated aerodynamic thrust

force, the blade pitch angle β and the rotor speed Ω, we can solve the inverse thrust force

function to estimate the wind speed. Artificial Neural Network (ANN) has been used suc-

cessfully for solving nonlinear inverse problems [48]. If we train an ANN with the input

vector of thrust force, rotor speed and pitch angle [FT ,Ω, β] and the output of wind speed

U , we can solve Equation 1.2 inversely and estimate the wind speed from it. However, some

manipulations in Equation 1.2 can improve the performance of wind speed estimation. By

substituting wind speed U from Equation 1.1, we can rewrite Equation 1.2 as:

2FTρπR4Ω

=CT (β, λ)

λ2(4.57)

The left hand side of Equation 4.57 can be calculated providing the information of thrust

force FT and rotor speed Ω. We can use the input vector of [ 2FT

ρφR4Ω, β] and the output of

tip-speed ratio λ, for training the ANN. Using this approach, we can calculate the estimated

tip-speed ratio λ from ANN which can be used for estimation of the effective wind speed

U = RΩ

λ.

Solving Equation 4.57 for wind speed estimation has some advantages over solving Equa-

tion 1.2 [69]. Using Equation 4.57, we use two inputs for training the ANN while using

Equation 1.2 we use three inputs for training the ANN. Having smaller number of inputs to

ANN results in requiring less number of hidden-layers in training the ANN. Also, having the

same number of hidden-layers, the ANN with less number of inputs has a better computa-

tional performance. Moreover, in Equation 1.2, we have the air density inside the mapping

function. Therefore, if the air density changes from the situation we collected the training

data, we should train the neural network again. Using Equation 4.57, the air density is

inside input vector and the neural network is independent of the air density. It should be

noted that the air density varies due to changes in height and temperature which can cause

81

significant change in wind turbine power generation [68].

In this work, MATLAB neural network toolbox [24] is used for training a neural network

with five hidden-layers, two inputs [ 2FT

ρπR4Ω, β] and one output λ where the effective wind

speed is determined as U = RΩ

λ. In the following section, some numerical results will be

presented to verify the performance of the neural network effective wind speed estimator.

4.3 Numerical Results and Verification

In this section, the developed method for wind speed estimation will be validated. As the

real wind turbine was not availble to do experiment in this work, wind speed estimation is

performed for the WP 1.5MW wind turbine using the simulation data provided by FAST

[34] which is a certified wind turbine simulation package developed by National Renewable

Energy Laboratory (NREL).

In Figure 4.1, the wind speed and aerodynamic thrust force estimated by the developed

method is compared to the input wind speed profile to the wind turbine and the thrust force

output of FAST.

The presented results demonstrate the capability of the developed method for wind speed

estimation. The wind profile used for evaluation in this section is spatially constant in rotor

plane and are only varying during the time. In the following chapter, wind speed estimation

method developed in this thesis will be used for estimating the effective wind speed in fully

turbulent wind fields where the wind profile varies temporally and spatially over the rotor

plane.

82

0 100 200 300 400 500 600 700 800 900 1000−100

0

100

200

300

400

500

600

700

Time (s)

Thr

ust F

orce

(kN

)

EstimatedFAST

(a)

0 200 400 600 800 100014

16

18

20

22

24

26

28

30

Time (s)

Win

d S

peed

(m

/s)

Estimated Wind Speed Wind Speed

(b)

Figure 4.1: a) Thrust force estimator validation. b) Wind speed estimator validation

83

Chapter 5

Results and Discussion

In previous chapters, we developed a model for wind turbine tower deflection and presented

a method to solve the inverse problem to estimate the effective wind speed from the wind

turbine tower deflection. In this chapter, we will use the developed wind speed estimation

algorithm to estimate the effective wind speed in turbulent wind conditions. Some numerical

results for effective wind speed estimation will be presented and the performance of the

estimator will be tested and verified.

5.1 Verification Method

In Chapter 4, we applied our developed wind speed estimation method to estimate wind

speed for a uniform wind field with no spatial variations. In other words, the wind speed

was varying uniformly over rotor plane. To verify results for such a case, we can compare

the result of estimation with the input wind speed. In this section, we are going to apply the

wind speed estimation method for estimating effective wind speed for a turbulent wind field

where the wind speed varies both temporally and spatially over rotor plane. The effective

wind speed is expected to be a weighted average of the wind speeds in different points of the

wind filed.

In this section, a verification method based on correlation coefficient is proposed to eval-

uate the estimated effective wind speed. Considering σA and Cov(A,B) to be the standard

deviation of A and covariance of A and B respectively, the correlation coefficient is defined

as follows:

ρ(A,B) =Cov(A,B)

σAσB(5.1)

84

which is the measure of dependency between two signals. It varies from 0 to 1 and higher

values of correlation coefficient show higher dependency between the signals.

To evaluate the wind speed estimation method, we determine correlation coefficient of

the estimated effective wind speed with wind speeds at different points of rotor plane. We

expect to see higher values of correlation coefficient inside the rotor plane.

5.1.1 Wind Turbine Simulation Tools

In order to verify the effective wind speed estimation method developed in this thesis, we

need high fidelity wind turbine simulation tools. Recently, some accredited simulation tools

are developed by National Renewable Energies Laboratory (NREL) which are popular among

researchers. FAST (Fatigue, Aerodynamics, Structures, and Turbulence) [34] is the software

package which will be used to simulate a wind turbine in different conditions and with

different wind speed profiles. TurbSim [33] is another product of NREL which can be used

for turbulent wind field simulation.

In the following sections, some simulations are performed using TurbSim and FAST to

verify the performance of the effective wind speed estimation method. A 1.5MW wind

turbine is simulated using FAST in different turbulent wind conditions generated by TurSim

and the proposed evaluation method is used to validate the results.

5.2 Turbulent Wind Field Specifications

Simulation results for effective wind speed estimation of turbulent wind conditions are pre-

sented in this section. In turbulent wind fields, wind speed changes spatially and temporally

over rotor plane. For example, in Figure 5.1 four snapshots of a turbulent wind field are

shown illustrating that wind speed is time varying and has a nonuniform distribution over

a field. Furthermore, variation of wind speed mean value over a wind field is presented in

Figure 5.2 which shows the mean value of wind speed increases as we go further from the

85

ground. For the 120m × 120m turbulent wind field shown in Figure 5.2, the mean value of

wind speed changes from 14.5m/s at the bottom to 20m/s at the top and mean value of

wind speed varies about 3m/s from top to bottom of rotor plane which is shown by a circle

with radius R = 35m. This variation in wind speed mean value is because of wind shear

effect discussed in Section 1.3.

Wind Field Width (m)

Win

d F

ield

Hei

ght (

m)

0 20 40 60 80 100 1200

20

40

60

80

100

120

6

8

10

12

14

16

18

20

22

(a)

Wind Field Width (m)

Win

d F

ield

Hei

ght (

m)

0 20 40 60 80 100 1200

20

40

60

80

100

120

11

12

13

14

15

16

17

18

19

20

21

(b)

Wind Field Width (m)

Win

d F

ield

Hei

ght (

m)

0 20 40 60 80 100 1200

20

40

60

80

100

120

12

14

16

18

20

22

24

(c)

Wind Field Width (m)

Win

d F

ield

Hei

ght (

m)

0 20 40 60 80 100 1200

20

40

60

80

100

120

14

16

18

20

22

24

26

(d)

Figure 5.1: Spatial distribution of wind speed in a turbulent wind field at four different timeinstants. Wind turbine hub is located in the center of the circle and the circle with R = 35mshows rotor plane of the wind turbine. The color bar shows the wind speed (m/s). (a)t = 50s (b) t = 55s (c) t = 60s (d) t = 65s

86

Wind Field Width (m)

Win

d F

ield

Hei

ght (

m)

0 20 40 60 80 100 1200

20

40

60

80

100

120

14.5

15

15.5

16

16.5

17

17.5

18

18.5

19

19.5

20

Figure 5.2: Spatial distribution of the mean wind speed in a turbulent wind filed over 1000s.Wind turbine hub is located in the center of the circle and the circle with R = 35m showsrotor plane of the wind turbine. The color bar shows the wind speed (m/s).

Wind Field Width (m)

Win

d F

ield

Hei

ght (

m)

0 20 40 60 80 100 1200

20

40

60

80

100

120

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5.3: Correlation coefficient of the turbulent wind field with the hub-height wind speed.Wind turbine hub is located in the center of the circle and the circle with R = 35m showsrotor plane of the wind turbine. The color bar shows dimetionless values of the correlationcoefficient.

87

The strength of turbulence in a wind field is measured by Turbulence Intensity (TI)

defined as follows [6]:

TI =σVV× 100% (5.2)

where σV is the standard deviation of the wind field and V is the mean wind speed. For

example, for wind field presented in Figures 5.1 and 5.2, we have TI = 16% which is a

relatively high turbulence intensity.

To show how wind speed values are correlated over the wind field we determine the

correlation coefficient of wind speeds at different points with the hub-height wind speed.

The results are presented in Figure 5.3 which shows wind speed varies significantly over the

wind field and wind speeds at the points located far from each other have low correlation.

Since the turbulent wind field is defined using a 16 × 16 grid and there is no grid point

located exactly at the hub location, we have averaged the wind speed in four closest grid

points to determine hub-height wind speed. This is why correlation coefficient value is not

1 even in the center of the circle.

In the following section, our wind speed estimation method will be applied to estimate the

effective wind speed for two different turbulent wind fields and the results will be presented.

5.3 Simulation Results

In this section, we use our wind speed estimation method to estimate effective wind speed for

two different turbulent wind fields. Simulation conditions for the wind fields are presented

in Table 5.1.

In order to verify the effective thrust force estimation results, we have compared the

estimated values for thrust force with thrust force output of FAST. The results of comparisons

for test #1 and test #2 are presented in Figure 5.4. We can see that test #1 shows more

fluctuations in both estimated thrust force and FAST output. The reason is that for test

#1 we have a higher value of TI and higher wind speed sampling frequency which cause

88

test # 1 2TI 16% 12%

mean wind speed 18m/s 25m/swind field size 120m× 120m 120m× 120m

rotor plane diameter 70m 70mwind sampling time step 0.01s 5s

simulation period 1000s 1000ssimulation time step 0.01s 0.01s

Table 5.1: Wind field and simulation conditions

test # 1 2thrust force estimation error (mean) 13.8% 7.3%

Table 5.2: Mean value of effective thrust force estimation error

more fluctuations in thrust force. The estimation error for effective thrust force is presented

in Table 5.2 showing that estimation error for test #1 is higher than test #2. The reason

is that, wind speed profile in test #1 has greater amount of high frequency content falling

beyond the bandwidth of tower dynamics. This shows that the thrust force estimator has a

good performance even for highly turbulent wind conditions and its performance gets better

for wind fields with lower turbulence intensity.

Also, the estimated effective wind speed is compared to hub-height wind speed and the

results are presented in Figures 5.5. The mean value of the effective wind speed is expected

to be close to the mean value of wind speeds over the rotor plane. For the simulated cases,

these two values are also close to mean value of hub-height wind speed as shown in Table 5.3.

test # 1 2effective wind speed 18.54m/s 24.7

wind speed over rotor plane 18.7m/s 24.4hub-height wind speed 18.36 25.31m/s

Table 5.3: Comparison of wind speeds mean value

89

0 200 400 600 800 100060

80

100

120

140

160

180

200

220

Time (s)

Thr

ust F

orce

(kN

)

FASTEstimated

0 200 400 600 800 1000100

150

200

250

300

Time (s)

Thr

ust F

orce

(kN

)

EstimatedFAST

Figure 5.4: Comparison of estimated values of effective thrust force and FAST thrust forceoutput for test #1 (top) and test #2 (bottom).

90

0 200 400 600 800 100010

15

20

25

30

Time (s)

Win

d S

peed

(m

/s)

EffectiveHub−height

0 200 400 600 800 100015

20

25

30

35

Time (s)

Win

d S

peed

(m

/s)

EffectiveHub−height

Figure 5.5: Effective wind speed and hub-height wind speed comparison for test #1 (top)and test #2 (bottom).

Moreover, in Section 5.1, we discussed evaluation of the effective wind speed using cor-

relation coefficient. In Figure 5.6, the correlation of effective wind speeds with wind speeds

over the wind field is presented. As you can see, the correlation coefficients are higher for

test #2 where we have lower turbulence intensity. Also, from Figure 5.6, we have a higher

value of correlation coefficient for wind speeds inside the rotor plane.

Based on the definition of effective wind speed which is the spatial average of wind speeds

91

over rotor plane, we can conclude that the presented results meet the main purpose of this

research which is development of a tower deflection based effective wind speed estimation

method.

Finally, as we discussed in Section 1.3, effective wind speed has some components addi-

tional to original upstream wind speed which are created due to contribution of wind turbine

itself. For example, wind turbine blades experience different wind speed values sweeping the

rotor plane, causing fluctuations in effective wind speed with rotational frequency of wind

turbine P and more significantly its multiples such as 3P (for three-bladed wind turbines).

Also, wind turbine blades vibration contributes to effective wind speed by changing the

relative wind speed. To investigate these effects, we can study the frequency domain repre-

sentation of the effective wind speed.

The frequency domain comparison presented in Figures 5.7 and 5.8 shows that the effec-

tive wind speed and hub-height wind speed have similar frequency content at low frequency

range. However, effective wind speed for both test #1 and test #2 has some higher fre-

quency components which are not present in the hub-height wind speed. Furthermore, the

additional components have the same frequency of f1 ' 0.4Hz, f2 ' 0.6Hz and f3 ' 1.2Hz

for test #1 and test #2.

92

0 20 40 60 80 100 1200

20

40

60

80

100

120

0.2

0.25

0.3

0.35

0.4

0.45

0 20 40 60 80 100 1200

20

40

60

80

100

120

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Figure 5.6: Correlation coefficient of the turbulent wind field with the effective wind speedfor test #1 (top) and test #2 (bottom). Wind turbine hub is located in the center of thecircle and the circle with R = 35m shows the rotor plane of wind turbine. The color barshows the dimensionless values of the correlation coefficient.

93

10−3

10−2

10−1

100

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

f (Hz)

|Win

d S

peed

(f)

|

Hub−height Wind Speed

10−3

10−2

10−1

100

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f (Hz)

|Win

d S

peed

(f)

|

Effective Wind Speed

Figure 5.7: Frequency domain representation of hub-height and effective wind speeds( test#1). Effective wind speed has some higher frequency components at f1 ' 0.4Hz, f2 ' 0.6Hzand f3 ' 1.2Hz which are not present in the hub-height wind speed.

94

10−3

10−2

10−1

100

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

f (Hz)

|Dis

plac

emen

t(f)

|

Hub−height Wind Speed

10−3

10−2

10−1

100

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f (Hz)

|Dis

plac

emen

t(f)

|

Effective Wind Speed

Figure 5.8: Frequency domain representation of hub-height and effective wind speeds( test#2). Effective wind speed has some higher frequency components at f1 ' 0.4Hz, f2 ' 0.6Hzand f3 ' 1.2Hz which are not present in the hub-height wind speed.

95

0 200 400 600 800 100011

11.5

12

12.5

13

Time (s)

Rot

or S

peed

(rp

m)

Figure 5.9: Rotor speed for test #2 which is maintained around Ω = 12rpm using pitchcontrol system.

Modal analysis of the wind turbine structural dynamics shows that these additional

high frequency components happen in natural frequency of wind turbine blades f ' 1.2Hz

and tower f ' 0.4Hz. Furthermore, the rotor speed is maintained at Ω = 12rpm in our

simulations using the designed pitch control system (see Figure 5.9). Consequently, the

rotational frequency is P = 0.2Hz showing the frequency peak of f = 0.6Hz is associated

with 3P fluctuations caused by turbulence, wind shear, etc. as in Section 1.3.

In summary, the presented results in this chapter show that our tower deflection based

wind speed estimation method can predict the effective thrust force and wind speed success-

fully for turbulent wind fields.

96

Chapter 6

Summary and Suggestions

6.1 Summary

The fast-growing wind power industry demands large wind turbines to reduce the production

cost and compete with other sources of energy. More flexible structure of large wind turbines

has increased the complexity of control system design. Modern wind turbines require a more

sophisticated control system to reduce the loads on the wind turbine as well as capture the

maximum energy. Designing such a control system calls for more accurate information about

wind speed.

On the other hand, wind speed estimation is more difficult for large wind turbines as the

rotor blades experience large spatial changes in wind speed sweeping the rotor plane. Also,

wind speed measurement using anemometer provides very limited information because of

large size of rotor plane and the fact that it can measure the wind speed only in a single

point downstream of the rotor plane. This issue is addressed by researchers and several wind

speed estimation methods are presented in the literature. Majority of the presented methods

estimate wind speed based on rotor dynamics of the wind turbine and mostly are focused on

improving the estimation algorithm and reducing the computational cost. The aspect that

is less studied in the literature is the suitability of wind turbine rotor dynamics for wind

speed estimation.

In this thesis, we showed the limitations of wind turbine rotor dynamics for wind speed

estimation. Also, we proposed an effective wind speed estimation method based on wind

turbine tower deflection and showed its capability in estimation of effective wind speed in

turbulent wind fields.

In Chapter 1 of this thesis, a comprehensive overview of wind speed estimation methods

97

was presented and limitations caused by huge inertia of wind turbine for wind speed esti-

mation based on rotor system was studied. Also, a wind speed estimation method based on

wind turbine tower dynamics was proposed.

To design the wind speed estimator based on wind turbine tower deflection, first we

developed a model for wind turbine tower deflection. For this purpose, in Chapter 2, wind

turbine aerodynamics was modeled using Blade Element Momentum (BEM) theory. The

iterative algorithm of BEM calculates the aerodynamic thrust force providing the information

of wind speed profile, rotor speed and pitch angle. Finally, simulated results for aerodynamic

thrust force was presented to verify the model.

In Chapter 3, we used Euler-Bernoulli beam theory to model wind turbine tower deflec-

tion. Then, by considering the aerodynamic thrust force as an input to tower deflection

model, subspace system identification technique was used to identify a linear time invariant

(LTI) model for wind turbine tower deflection. The identified model was verified using modal

parameters of the system and simulated results.

In Chapter 4, the main goal was to solve the inverse problem for the wind turbine tower

deflection model and estimate the wind speed from wind turbine tower deflection. To achieve

this, two inverse problems were solved. The first inverse problem was solved for tower LTI

model of wind turbine tower deflection where we used Kalman filter along with recursive least

squares method to estimate the thrust force from tower deflection measurement. The second

inverse problem was solved for aerodynamics model to estimate wind speed from thrust force.

Since aerodynamics model was highly nonlinear, we used nueral network to solve the inverse

problem. Also, some simulation results were presented to verify the implemented methods.

In Chapter 5, we used the wind speed estimation method developed in this thesis to

estimate the effective wind speed for turbulent wind fields. An evaluation method based on

correlation coefficient was developed to verify the results of effective wind speed estimation.

The simulation was performed for a 1.5MW wind turbine with rotor radius of R = 70m

98

which is subject to a 120m × 120m turbulent wind field. Our results showed that the

developed method can estimate effective wind speed which is a spatial average of wind speed

over rotor plane. The result of verification shows that effective wind speed has a higher

correlation with wind speeds inside the rotor plane. These results meet the main purpose

of this research which is development of a wind speed estimation method based on wind

turbine tower deflection.

6.2 Suggestions

In this thesis, we used an evaluation method based on correlation coefficient to verify the

results. The evaluations show the capability of the developed method in estimating effective

wind speed for turbulent wind fields. However, as the eventual goal of effective wind speed

estimation is improving the performance of control system, it is important to have an eval-

uation in this regard as well. One of the possible future works is to develop an evaluation

method based on performance of the control system.

Furthermore, we compared rotor dynamics based and tower dynamics based wind speed

estimation methods by studying their dynamic behaviors. For future work, it will be useful

to implement a wind speed estimation method based on rotor dynamics and compare the

results with the presented results in this thesis.

The focus of this research was on wind speed estimation based on wind turbine tower

dynamics. However, wind turbine blades structural dynamics can also be used for wind

speed estimation which requires further studying. One of the possible future directions is to

use wind turbine blades deflection for wind speed estimation. This method can especially be

useful for individual pitch control systems where the pitch angle is controlled independently

for each blade and wind speed estimation for each blade is required.

Finally, in this thesis, we used simple back propagation method for the neural network.

The performance of wind speed estimation can be improved using higher performance map-

99

ping tools such as extreme learning machine (ELM).

100

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Appendix A

Airfoil Data

RNodes AeroTwst DRNodes Chord Airfoil #2.85833 11.10 2.21667 1.949 15.07500 11.10 2.21667 2.269 27.29167 11.10 2.21667 2.589 29.50833 10.41 2.21667 2.743 211.72500 8.38 2.21667 2.578 213.94167 6.35 2.21667 2.412 216.15833 4.33 2.21667 2.247 218.37500 2.85 2.21667 2.082 320.59167 2.22 2.21667 1.916 322.80833 1.58 2.21667 1.751 325.02500 0.95 2.21667 1.585 327.24167 0.53 2.21667 1.427 329.45833 0.38 2.21667 1.278 331.67500 0.23 2.21667 1.129 433.89167 0.08 2.21667 0.980 4

Table A.1: Blades are splitted to 15 elemnts considering a constant twist angle, chord lengthand cross-section geometry along each of them. The distance of the blade elemnts centerfrom the hub and the airfoil numbers are provided in the table.

Angle of attack Cl Cd-180.0 0.0 0.5

0.0 0.0 0.5180.0 0.0 0.5

Table A.2: The lift and drag coefficient data of the airfoil #1 based on angle of attack

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Angle of attack Cl Cd-180.00 -0.170 0.0200-170.00 0.640 0.0500-160.00 0.840 0.3100-150.00 1.080 0.6200-140.00 1.150 0.9600-130.00 1.090 1.3000-120.00 0.880 1.5200-110.00 0.600 1.6600-100.00 0.310 1.7600-90.00 0.000 1.8000-80.00 -0.310 1.7600-70.00 -0.600 1.6600-60.00 -0.880 1.5200-50.00 -1.090 1.3000-40.00 -1.150 0.9600-30.00 -1.080 0.6200-20.00 -0.840 0.3100-10.00 -0.640 0.0144-8.00 -0.480 0.0124-6.00 -0.090 0.0082-5.00 0.020 0.0082-4.00 0.130 0.0082-3.00 0.240 0.0082-2.00 0.350 0.0086-1.00 0.460 0.00860.00 0.570 0.00871.00 0.670 0.00882.00 0.780 0.0090

Angle of attack Cl Cd3.00 0.890 0.00934.00 0.990 0.00965.00 1.100 0.00996.00 1.200 0.01037.00 1.310 0.01088.00 1.410 0.01139.00 1.510 0.011810.00 1.560 0.019411.00 1.610 0.022112.00 1.650 0.024513.00 1.650 0.026914.00 1.630 0.029615.00 1.620 0.052030.00 1.080 0.620040.00 1.150 0.960050.00 1.090 1.300060.00 0.880 1.520070.00 0.600 1.660080.00 0.310 1.760090.00 0.000 1.8000100.00 -0.310 1.7600110.00 -0.600 1.6600120.00 -0.880 1.5200130.00 -1.090 1.3000140.00 -1.150 0.9600150.00 -1.080 0.6200160.00 -0.840 0.3100170.00 -0.640 0.0500180.00 -0.170 0.0200

Table A.3: The lift and drag coefficient data of the airfoil #2 based on angle of attack

111

Angle of attack Cl Cd-180.00 -0.170 0.0200-170.00 0.640 0.0500-160.00 0.840 0.3100-150.00 1.080 0.6200-140.00 1.150 0.9600-130.00 1.090 1.3000-120.00 0.880 1.5200-110.00 0.600 1.6600-100.00 0.310 1.7600-90.00 0.000 1.8000-80.00 -0.310 1.7600-70.00 -0.600 1.6600-60.00 -0.880 1.5200-50.00 -1.090 1.3000-40.00 -1.150 0.9600-30.00 -1.080 0.6200-20.00 -0.840 0.3100-10.00 -0.640 0.0144-8.00 -0.480 0.0124-6.00 0.010 0.0074-5.00 0.120 0.0075-4.00 0.230 0.0077-3.00 0.340 0.0078-2.00 0.440 0.0080-1.00 0.550 0.00820.00 0.660 0.00841.00 0.770 0.00862.00 0.880 0.0089

Angle of attack Cl Cd3.00 0.980 0.00914.00 1.090 0.00955.00 1.200 0.00986.00 1.300 0.01027.00 1.410 0.01078.00 1.490 0.01559.00 1.580 0.017910.00 1.660 0.020311.00 1.680 0.025012.00 1.700 0.027313.00 1.700 0.029714.00 1.680 0.032415.00 1.660 0.052030.00 1.080 0.620040.00 1.150 0.960050.00 1.090 1.300060.00 0.880 1.520070.00 0.600 1.660080.00 0.310 1.760090.00 0.000 1.8000100.00 -0.310 1.7600110.00 -0.600 1.6600120.00 -0.880 1.5200130.00 -1.090 1.3000140.00 -1.150 0.9600150.00 -1.080 0.6200160.00 -0.840 0.3100170.00 -0.640 0.0500180.00 -0.170 0.0200

Table A.4: The lift and drag coefficient data of the airfoil #3 based on angle of attack

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Angle of attack Cl Cd-180.00 -0.170 0.0200-170.00 0.640 0.0500-160.00 0.840 0.3100-150.00 1.080 0.6200-140.00 1.150 0.9600-130.00 1.090 1.3000-120.00 0.880 1.5200-110.00 0.600 1.6600-100.00 0.310 1.7600-90.00 0.000 1.8000-80.00 -0.310 1.7600-70.00 -0.600 1.6600-60.00 -0.880 1.5200-50.00 -1.090 1.3000-40.00 -1.150 0.9600-30.00 -1.080 0.6200-20.00 -0.840 0.3100-10.00 -0.640 0.0144-8.00 -0.480 0.0124-6.00 0.060 0.0092-5.00 0.170 0.0082-4.00 0.280 0.0067-3.00 0.390 0.0068-2.00 0.500 0.0069-1.00 0.600 0.00700.00 0.710 0.00721.00 0.820 0.00742.00 0.930 0.0076

Angle of attack Cl Cd3.00 1.040 0.00784.00 1.140 0.00825.00 1.250 0.00876.00 1.350 0.01047.00 1.440 0.01468.00 1.530 0.01849.00 1.630 0.020010.00 1.650 0.021911.00 1.670 0.023912.00 1.680 0.026213.00 1.670 0.028814.00 1.650 0.031615.00 1.630 0.052030.00 1.080 0.620040.00 1.150 0.960050.00 1.090 1.300060.00 0.880 1.520070.00 0.600 1.660080.00 0.310 1.760090.00 0.000 1.8000100.00 -0.310 1.7600110.00 -0.600 1.6600120.00 -0.880 1.5200130.00 -1.090 1.3000140.00 -1.150 0.9600150.00 -1.080 0.6200160.00 -0.840 0.3100170.00 -0.640 0.0500180.00 -0.170 0.0200

Table A.5: The lift and drag coefficient data of the airfoil #4 based on angle of attack

113

Appendix B

Subspace Identification

By substituting Equations 4.37, 4.38 and 4.41 into Equation 4.40 we have:

ˆF (N + 1) =[γΦ(N)TΣ−1(N)Φ(N) + Λ(N + 1)T s−1(N)Λ(N + 1)]−1

×[γΦ(N)TΣ−1(N)Y (N) + Λ(N + 1)T s−1(N + 1)z(N + 1)]

(B.1)

In Equation 4.40, we can expand the inverse of the matrix by defining PRLS as:

PRLS(N + 1) = [Φ(N + 1)TΣ(N + 1)−1Φ(N + 1)]−1

= [γP−1RLS(N) + Λ(N + 1)T s−1(N)Λ(N + 1)]−1

(B.2)

Using the matrix inversion lemma [40], PRLS(N + 1) can be written as:

PRLS(N + 1) = γ−1PRLS(N)− γ−1PRLS(N)ΛT (N + 1)

×[Λ(N + 1)γ−1PRLS(N)ΛT (N + 1) + s(N + 1)]−1

×Λ(N + 1)γ−1PRLS(N)

(B.3)

Substituting PRLS(N + 1) from Equation B.3 into Equation B.1 and using Equation 4.34:

F (N + 1) =F (N) + γ−1PRLS(N)ΛT (N + 1)s−1(N + 1)z(N + 1)

− γ−1PRLS(N)ΛT (N + 1)

× [Λ(N + 1)γ−1PRLS(N)ΛT (N + 1) + s(N + 1)]−1Λ(N + 1)

× [F (N) + γ−1PRLS(N)ΛT (N + 1)s−1(N + 1)z(N + 1)]

(B.4)

with some simplifications, Equation B.4 can be written as follows:

F (N + 1) = F (N) +KRLS(N + 1)[z(N + 1)− Λ(N + 1)F (N)] (B.5)

where KRLS(N + 1) is:

KRLS(N+1) = γ−1PRLS(N)ΛT (N+1)[Λ(N+1)γ−1PRLS(N)ΛT (N+1)+s(N+1)]−1 (B.6)

114

Also using the definition of KRLS(N + 1), Equation B.3 can be simplified as follows:

PRLS(N + 1) = [I −KRLS(N + 1)Λ(N + 1)]γ−1PRLS(N) (B.7)

Then, Equation B.5 can be used for thrust force input estimation.

115