UNIVERSITY OF CALGARY. Data-Driven Modeling of Wind Turbine Structural Dynamics and Its Application. to Wind Speed Estimation. Vahid Saberi Nasrabad

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1 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

2 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

3 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

4 Dedicated to my parents for their endless love & support

5 Table of Contents Abstract ii Acknowledgements iii Table of Contents v List of Tables vii List of Figures viii List of Symbols xi 1 Introduction Wind Power Wind Turbine System Wind Turbine Aerodynamics Wind Turbine Control System Effective Wind Speed Estimation Challenges Wind Shear Tower Shadow Turbulence Effective Wind Speed Estimation literature review Dynamics of Wind Turbine Rotor Speed Dynamics of Wind Turbine Tower Deflection Wind Speed Estimation Based on Wind Turbine Tower Deflection Contributions Organization of the Thesis Wind Turbine Aerodynamics Modelling Blade Element Momentum Theory Momentum Theory Blade Element Theory Corrections of Blade Element Momentum Theory Tip-Loss and Hub-Loss Glauert Correction Blade Element Momentum Theory Iterative Procedure for BEM Method BEM Validation System Identification of Wind Turbine Tower Tower Fore-aft Motion Dynamics Hammerstein System System Identification Notation Subspaces Projection State-space Subspace System Identification Wind Turbine Tower System Identification Practical Aspects of Wind Turbine Tower System Identification Control System Design Data Collection for System Identification v

6 3.5.3 Input-output Data Measurement for System Identification System Identification Evaluation Estimation Thrust Force Estimator Design Kalman Filter Design Residual Sequence Estimation Thrust Force Estimation Thrust Force Estimation Summary Effective Wind Speed Estimation Using Neural Network Neural Network Training Numerical Results and Verification Results and Discussion Verification Method Wind Turbine Simulation Tools Turbulent Wind Field Specifications Simulation Results Summary and Suggestions Summary Suggestions Bibliography A Airfoil Data B Subspace Identification vi

7 List of Tables 3.1 Model validation using modal parameters of wind turbine tower in terms of natural frequency f n and damping ratio ζ Wind field and simulation conditions Mean value of effective thrust force estimation error Comparison of wind speeds mean value A.1 Blades are splitted to 15 elemnts considering a constant twist angle, chord length and cross-section geometry along each of them. The distance of the blade elemnts center from the hub and the airfoil numbers are provided in the table A.2 The lift and drag coefficient data of the airfoil #1 based on angle of attack. 110 A.3 The lift and drag coefficient data of the airfoil #2 based on angle of attack. 111 A.4 The lift and drag coefficient data of the airfoil #3 based on angle of attack. 112 A.5 The lift and drag coefficient data of the airfoil #4 based on angle of attack. 113 vii

8 List of Figures and Illustrations 1.1 Global annual installed wind capacity (Source: Ref. [51]) Global cumulative installed wind capacity (Source: Ref. [51]) Wind energy price in different countries in 2010 and 2014 (Information retrieved from Ref. [31]) Wind turbine system: diffirent parts of a horizontal axis wind turbine (HAWT) are shown. The wind speed creates aerodynamic torque T a and thrust force F T on the blades. The gearbox transmits the torque to generator and increases the angular velocity from rotor speed Ω to generator shaft speed ω. The generator torque T g and blade pitch angle β are two control inputs to the system. The yaw control system changes the direction of the wind turbine towards the highest wind speed Torque coefficient C Q and power coefficient C p curves: highly nonlinear changes of torque and power coefficients is depicted versus variation of the tip speed ratio λ and pitch angle β. As shown, maximum power coefficient C pmax is associated with optimal tip speed ratio (λ = λ 0 ) and β 0 0 [10] Ideal power curve and control system operational regions. In Region I, wind turbine ideally can capture all avaiable energy in the wind if we maitain the optimal tip speed ratio by controlling the rotor speed. In Region III, the generator torque and power are saturated and we discard the excessive energy in 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 Wind speed field [10] Tower shadow causes purturbations in aerodynamic torque by reducing the torque when blades pass the tower in each revolution. For a three bladed wind turbine it causes purturbations with frequency of 3P [10] Wind speed estimation method using wind turbine tower deflection modelling. By solving the inverse problem for tower structural model we can find the estimated thrust force F ˆ T from tower deflection measurement Def ˆ Then the estimated wind speed Û can be calculated by solving the invesre problem for aerodynamic model The control volume considered for wind turbine to apply the conservation of linera momentum. In this figure U is the flow velocity which is shown in four different cross-sections Annular stream tube. In this figure the flow velocity on rotor plane and downstream are shown as a function of free stream velocity U and axial induction factor a The air flow finds a tangential velocity component passing the rotor plane which is proporional to tangential induction factor á To use blade element theory we split the blade into several blade elements operating independently with no aerodynamical interaction. The chord length c and twist angle θ T may be different for each blade element viii

9 2.5 Cross-section of a blade element where df L is the lift force, df D is the drag force. The drag and lift forces are projected into two components, one is the tangential force df T which lies in the rotor plane and causes the torque and the other is normal to the rotor plane df N and causes the thrust force BEM iterative algorithm Verification of the BEM method result with the FAST output which uses the generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrust force in frequency domain Verification of the BEM method result with the FAST output which uses the generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrust force in frequency domain Verification of the BEM method result with the FAST output which uses the generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrust force in frequency domain Wind turbine dynamic system is described using Hammerstein structure where the Wind Turbine Aerodynamics is considered as the static nonlinearity and the Tower Dynamics is considered as a LTI system. The entire dynamic system of wind turbine is shown under closed-loop control; however, openloop system identification method still can be used for identification of Tower Dynamics because there is no feedback from tower fore-aft deflection D F A to aerodynamic thrust force F T Orthogonal Projection Oblique Projection Diagonal elements of S matrix in singular value decomposition (SVD) of O h matrix which is used to determine the order of the model 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 in the system to collect informative data Input data set used for evaluation of wind turbine tower identified model. The pitch 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 blades c) rotor speed d) aerodynamic thrust force associated with the presented wind speed, pitch angle and rotor speed. The thrust force calculated by BEM is compared with the FAST output Wind turbine tower deflection associated with the wind speed, pitch angle and rotor speed condition presented in Figure 3.6. The output of the identified model is compared with FAST output. a) time domain and b) frequency domain representation of tower deflection data a) Thrust force estimator validation. b) Wind speed estimator validation.. 83 ix

10 5.1 Spatial distribution of wind speed in a turbulent wind field at four different time instants. Wind turbine hub is located in the center of the circle and the circle with R = 35m shows 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 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 shows rotor plane of the wind turbine. The color bar shows the wind speed (m/s) 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 shows rotor plane of the wind turbine. The color bar shows dimetionless values of the correlation coefficient Comparison of estimated values of effective thrust force and FAST thrust force output for test #1 (top) and test #2 (bottom) Effective wind speed and hub-height wind speed comparison for test #1 (top) and test #2 (bottom) Correlation coefficient of the turbulent wind field with the effective wind speed for test #1 (top) and test #2 (bottom). Wind turbine hub is located in the center of the circle and the circle with R = 35m shows the rotor plane of wind turbine. The color bar shows the dimensionless values of the correlation coefficient Frequency domain representation of hub-height and effective wind speeds( test #1). Effective wind speed has some higher frequency components at f 1 0.4Hz, f 2 0.6Hz and f 3 1.2Hz which are not present in the hubheight wind speed Frequency domain representation of hub-height and effective wind speeds( test #2). Effective wind speed has some higher frequency components at f 1 0.4Hz, f 2 0.6Hz and f 3 1.2Hz which are not present in the hubheight wind speed Rotor speed for test #2 which is maintained around Ω = 12rpm using pitch control system x

11 List of Symbols and Acronyms Symbol ANFIS ANN BEM BPANN ELM ESN FAST FBG GRBFN HAWT LIDAR LPV LTI MLPNN MPPT NREL PEM SODAR SVD SVR TI VSVP 4SID Definition Adaptive Neuro Fuzzy Inference Systems Artificial Neural Network Blade Element Momentum Back-Propagation Artificial Neural Network Extreme Learning Machine Echo State Network Fatigue, Aerodynamics, Structures, and Turbulence Fiber Bragg Grating Gaussian Radial Basis Function Network Horizontal Axis Wind Turbine Light Detection And Ranging Linear, Parameter-Varying Linear Time-Invariant Multi-Layer Perceptron Neural Network Maximum Power Point Tracking National Renewable Energy Laboratory Prediction Error Method Sound Detection And Ranging Singular Value Decomposition Support Vector Regression Turbulence Intensity Variable-Speed Variable-Pitch State-Space Subspace System Identification xi

12 Symbol Definition Approximately equal to = Equal to R n n E[] R R n A/B A A/ C B A 1 A T A Cov(A, B) Unequal to Element of n n space Expected value The real numbers n space Orthogonal projection of row space A on row space B The space perpendicular to row space A Oblique projection of row space A along B on C Inverse of matrix A Transpose of matrix A MoorePenrose pseudoinverse of matrix A Covariance of A and B xii

13 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 M W from 197,943 M W in 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 recordbreaking year for the wind industry with more than 51 GW installed in a single year, bringing the global total close to 370 GW [51] 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 Also, with about 9.7 GW wind power capacity, Canada ranks 7 th in the world based on the cumulative wind power capacity and China with about 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 impacts, 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

14 from 2010 to 2014 as in Figure 1.3. Design and implementation of large wind turbines plays an important role in cost reduction. 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, mechanical, electrical and control subsystems. The aerodynamic subsystem converts the wind energy to mechanical energy. The mechanical subsystem caries out two main jobs: transmitting 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 foundation. 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

15 2014 USD/kWh 60,000 50,000 40,000 MW ,000 20,000 10, Figure 1.1: Global annual installed wind capacity (Source: Ref. [51]) 400, , , , , , ,000 50,000 0 MW Figure 1.2: Global cumulative installed wind capacity (Source: Ref. [51]) France Japan Australia US California US non-california Germany China Year Figure 1.3: Wind energy price in different countries in 2010 and 2014 (Information retrieved from Ref. [31]) 3

16 Nacclle T a T g Generator Gearbox Yawing F T R Tower Figure 1.4: Wind turbine system: diffirent parts of a horizontal axis wind turbine (HAWT) are shown. The wind speed creates aerodynamic torque T a and thrust force F T on the blades. The gearbox transmits the torque to generator and increases the angular velocity from rotor speed Ω to generator shaft speed ω. The generator torque T g and blade pitch angle β are two control inputs to the system. The yaw control system changes the direction of the wind turbine 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

17 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 Using BEM (see Chapter 2), the total thrust force F T (1.1) and torque F Q 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 C T and torque C Q coefficients which are a function of pitch angle β and tip speed ratio λ. F T = 1 2 ρπr2 C T (λ, β)u 2 (1.2) F Q = 1 2 ρπr3 C Q (λ, β)u 2 (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 C P : P = 1 2 ρπr2 C P (λ, β)u 3 (1.4) 5

18 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 C P and C Q with respect to deviations of λ and β. The maximum point for wind turbine power happens in a pitch angle close to zero (β 0 0) and the optimal tip speed ratio (λ = λ 0 ). 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 (C p = C pmax ) 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 (C p C pmax ). The control strategies used in wind turbine to regulate the aerodynamic power will be discussed in the following section 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 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

19 20 2 The Wind and Wind Turbines The torque and power coefficients are of special interest for control purposes. Figure 2.8 shows typical variations of C Q and C P with the tip-speedratio and the pitch angle deviation. In the case of fixed-pitch wind turbines, C Q and C P vary only with λ, sinceβ = 0 naturally. So, with some abuse of notation we will write C Q (λ) andc P (λ) todenotec Q (λ, 0) and C P (λ, 0), respectively. Figure 2.9 depicts typical coefficients C Q (λ) andc P (λ) offixedpitch turbines in two-dimensional graphs. CQ β (a) 5 λ C P max C P (λ o,β o) CP β (b) 5 λ Fig Typical variations of (a) C Q and (b) C P for a variable-pitch wind turbine Figure 1.5: Torque coefficient C Q and power coefficient C p curves: highly nonlinear changes of torque and power coefficients is depicted versus variation of the tip speed ratio λ and pitch angle β. As shown, maximum power coefficient C pmax is associated with optimal tip speed ratio (λ = λ 0 ) and β 0 0 [10] 7

20 Wind turbines have an operational range starting from cut-in wind speed (U min ) to cutout wind speed (U max ). 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 P min = P (λ, β, U min ) to P max = P (λ, β, U max ) which is a large range as the power is proportional to third power of wind speed P U 3 (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 U r, 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 U r and U max, 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 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 variablepitch. 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 λ opt by 8

21 Figure 1.6: Ideal power curve and control system operational regions. In Region I, wind turbine ideally can capture all avaiable energy in the wind if we maitain the optimal tip speed ratio by controlling the rotor speed. In Region III, the generator torque and power are saturated and we discard the excessive energy in 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. 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 Ω opt = Uλ opt /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

22 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, gearbox fatigue is caused by stressing of the gearbox teeth in response to torque overloads and generator fatigue is because of thermal stressing of the generator 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

23 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

24 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 N P 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] 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 2 π. When we average the perturbations coming from three blades, as 3 they have a phase difference of 2 π, the 1P components of torque and thrust force will be 3 cancelled out and only the 3P components will affect the system [10] 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

25 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 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. Turbulence 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

26 Figure 1.8: Tower shadow causes purturbations in aerodynamic torque by reducing the torque when blades pass the tower in each revolution. For a three bladed wind turbine it causes 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 anemometer [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], Extreme 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

27 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

28 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 Dynamics of Wind Turbine Rotor Speed The rotor speed dynamics is given in the following equations [5]: J r ω r + B r ω r = T r T tr (1.5) J g ω g + B g ω g = T tg T g (1.6) T tr ω r = T tg ω g (1.7) where J r are J g the rotor and generator inertia, T r is the wind generated torque in the rotor, T g is the generator torque, T tr is the torque in rotor side of the gearbox, T tg is the torque in generator side of the gearbox, ω r is the angular velocity of the rotor, ω g is the angular velocity of the generator shaft, B r and B g are the viscus friction coefficient of the rotor and generator respectively. The relation between angular velocity of the rotor ω r and angular velocity of the generator is given by the gearbox ratio γ: Using Equations 1.5, 1.6 and 1.7 it is obtained: γ = ω g ω r (1.8) J ω + Bω = T r γt g (1.9) 16

29 with J = J r + γ 2 J g (1.10) B = B r + γ 2 B g (1.11) The Equation 1.9 can be transformed to the following form: ω(s) T (s) = K 1 τs + 1 Which is a first order low pass filter, where τ = J B (1.12) is the time constant of the filter, K = 1 B is the gain of the filter, s is the Laplace transform variable and T = T r γt g 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 = kg.m 2 and the time constant the cut off frequency are approximately τ = 30s and f c = 0.03Hz [35]. This characteristic of wind turbine rotor speed dynamics has different influences on behaviour 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

30 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 (P captured = P electrical ) [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 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 f c = 0.03Hz is very slow because of huge 18

31 inertia of the rotor [35]. On the other hand, wind turbine tower deflection is lightly damped and has a cut-off frequency of f c = 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 f c = 0.03 and tower dynamics f c = 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

32 1.5 Wind Speed Estimation Based on Wind Turbine Tower Deflection In this thesis, effective wind speed estimation based on wind turbine tower deflection measurement 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 Hammerstein 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 aerodynamics 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 crosssections inertia, materials properties, etc. Since there is uncertainty in defining these physical properties, we need a systematic method to identify the model parameters. System identification techniques can be used to identify the accurate model parameters using input-output 20

33 U Tower System D e f Mesurement Tower System Model Uˆ Aerodynamics FˆT Structural Dynamics (LTI) ˆ Def 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 thrust force F ˆ T from tower deflection measurement Def ˆ Then the estimated wind speed Û can be calculated 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

34 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

35 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

36 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 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 Section 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

37 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 conservation 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 element 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: F T = ṁ(u 1 U 4 ) = (ρa 2 U 2 )(U 1 U 4 ) momentum conservation (2.1) ṁ = (ρau) 1 = (ρau) 2 = (ρau) 4 mass conservation (2.2) Where, F T 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 ṁ is the mass flow rate. 25

38 Stream tube U1 U 2 U3 U 4 Rotor Disk Figure 2.1: The control volume considered for wind turbine to apply the conservation of linera momentum. In this figure U is the flow velocity which is shown in four different cross-sections. Rotor Disk Stream tube dr U U 1 a U 12 a r R Rotor Disk Figure 2.2: Annular stream tube. In this figure the flow velocity on rotor plane and downstream 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: F T = A 2 (p 2 p 3 ) (2.3) In which p 2 and p 3 are the air pressures before and after the wind turbine rotor. 26

39 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: p ρu 2 1 = p ρu 2 2 upstream of the rotor (2.4) p ρu 2 3 = p ρu 2 4 downstream of the rotor (2.5) Assuming that the far upstream and far downstream pressures are equal (p 1 = p 4 ) and there is no axial velocity changes across the rotor plane (U 2 = U 3 ), Equations 2.4 and 2.5 can be combined as: By substituting Equation 2.6 into Equation 2.3 we can obtain: p 2 p 3 = 1 2 ρ(u 2 1 U 2 4 ) (2.6) F T = 1 2 A 2ρ(U 2 1 U 2 4 ) (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: U 2 = (U 1 + U 4 ) 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 = (U 1 U 2 ) U 1 (2.9) Then the rotor plane and the downstream wind velocities can be defined as a function of upstream wind velocity and induction factor: U 2 = U 1 (1 a) (2.10) U 4 = U 1 (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: df T = 1 2 ρ(u 2 1 U 2 4 )(2πrdr) = 4πρU 2 (1 a)ardr (2.12) 27

40 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 2Ωrá where á 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)(2ωár)r = 4πr 3 ρuω(1 a)ádr (2.13) With this last equation we finish deriving relations for the thrust force df T 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

41 r Figure 2.3: The air flow finds a tangential velocity component passing the rotor plane which is proporional to tangential induction factor á Lift force df L : The component of aerodynamic force that is perpendicular to the relative wind speed Drag force df D : The component of aerodynamic force that is parallel to the relative wind speed Section pitch angle θ p : The angle between the chord line and the plane of rotation Blade pitch angle θ p,0 : The pitch angle at the tip of the blade Section twist angle θ T : The angle between the section chord line and the chord line at the blade tip. It can be expressed as θ T = θ p θ p,0 Angle of attack α : The angle between the relative wind speed and the chord line of the airfoil defined as: α = ϕ θ T θ p,0 (2.14) 29

42 dr c T R r Figure 2.4: To use blade element theory we split the blade into several blade elements operating independently with no aerodynamical interaction. The chord length c and twist angle θ T 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 = θ p + α As it is shown in Figure 2.5, the relative wind speed U rel has a component in the rotor plane Ωr(1 + á) and a component perpendicular to the rotor plane U(1 a). The perpendicular component has two parts, Ωr is due to the rotation of the blade and Ωrá 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 (df D ) and lift (df L ) forces can be determined using the drag (C d ) and lift (C l ) coefficients as: 1 df L = C l 2 ρu relcdr 2 (2.15) 1 df D = C d 2 ρu relcdr 2 (2.16) 30

43 df N df L Chord line P,0 df T df D Plane of rotation P T U(1 a) U rel r (1 a ) Figure 2.5: Cross-section of a blade element where df L is the lift force, df D is the drag force. The drag and lift forces are projected into two components, one is the tangential force df T which lies in the rotor plane and causes the torque and the other is normal to the rotor plane df N and causes the thrust force. where c is the chord length and U rel is the relative wind speed defined as: U rel = U(1 a) sin(ϕ) (2.17) Using 2.5 the normal and tangential forces can be defined as: df N = df L cos(ϕ) + df D sin(ϕ) (2.18) df T = df L sin(ϕ) df D cos(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 = BdF N (2.20) dq = BrdF T (2.21) 31

44 therefore dt = B 1 2 ρu 2 rel(c l cos(ϕ) + C d sin(ϕ))cdr (2.22) dq = B 1 2 ρu 2 rel(c l sin(ϕ) C d cos(ϕ))crdr (2.23) The angle of relative wind speed ϕ can be related to axial (a) and tangential (á) induction factors as: ϕ = tan 1 U(1 a) ( Ωr(1 + á) ) (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 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 = F tip F hub (2.25) where F tip and F tip are tip-loss and hub-loss factors defined as: F tip = 2 π cos 1 e B(R r) 2rsinϕ (2.26) F hub = 2 π cos 1 e B(r R hub ) 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,r hub is the radius of hub, and ϕ is the inflow angle. Then 32

45 using the correction factor the modified thrust and torque will be: dt = 4πρU 2 (1 a)af rdr (2.28) dq = 4πr 3 ρuω(1 a)áf dr (2.29) Glauert Correction According to Equation 2.11, the BEM predicts that for a = 0.5 the wind velocity in the far wake U 4 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 C T > 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 C T (50 36F ) + 12F (3F 4) 36F 50 (2.30) where F is defined by Equation 2.25 and C T is the thrust coefficient of the blade element and can be obtained using the following equation: C T = σ(1 a)2 (C l cosϕ + C d sinϕ) sin 2 ϕ (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 = 4πρU 2 (1 a)af rdr (2.32) dq = 4πr 3 ρuω(1 a)áf dr (2.33) 33

46 also the blade element theory is used to find: dt = B 1 2 ρu 2 rel(c l cos(ϕ) + C d sin(ϕ))cdr (2.34) dq = B 1 2 ρu 2 rel(c l sin(ϕ) C d cos(ϕ))crdr (2.35) Equating dt and dq from these two sets of equations, we will find: a = [ 1 + [ á = 1 + ] 4F sin 2 1 (ϕ) (2.36) σ(c l cos(ϕ) + C d sin(ϕ)) ] 1 (2.37) 4F sin(ϕ)cos(ϕ) σ(c l sin(ϕ) C d cos(ϕ)) 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: λ r = U Ωr (2.39) The thrust force calculation using BEM method is summarized here and illustrated in Figure 2.6: 1. Start the iteration with the initial estimations [42]: a = 1 4 [ 2 + πλ r σ ] 4 4πλ r σ + πλ 2 r σ(8β + π σ) (2.40) á = 0 (2.41) 34

47 2. Calculate the inflow angle: 3. Calculate correction factor F : ϕ = tan 1 (1 a) ( λ r (1 + á) ) (2.42) F = F tip F hub (2.43) F tip = 2 π cos 1 e B(R r) 2rsinϕ (2.44) F hub = 2 π cos 1 e B(r R hub ) 2rsinϕ (2.45) 4. Calculate the thrust coefficient for the element using the following equation: C T = σ(1 a)2 (C l cosϕ + C d sinϕ) sin 2 ϕ (2.46) 5. If C T 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 C T (50 36F ) + 12F (3F 4) 36F 50 (2.47) 6. If C T 0.96F, use the BEM theory to determine the axial induction factor: [ ] 4F sin 2 1 (ϕ) a = 1 + (2.48) σ(c l cos(ϕ) + C d sin(ϕ)) 7. Calculate the tangential induction factor using BEM theory: [ ] 1 4F sin(ϕ)cos(ϕ) á = 1 + (2.49) σ(c l sin(ϕ) C d cos(ϕ)) 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 = B 1 2 ρu 2 rel(c l cos(ϕ) + C d sin(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

48 Initialize a and á from Equations 2.8 and Calculate ϕ from Eq Calculate F from Eq Calculate C T (Equation 2.31) by reading C l and C d from Tables A.2, A.3, A.4, or A.5 Calculate a and á from Equations 2.30 and 2.37 respectively yes C T 0.96F no Calculate a and á from Equations 2.36 and 2.37 respectively a new a old ε no yes Calculate thrust force from Equation 2.22 Figure 2.6: BEM iterative algorithm 36

49 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

50 Wind Speed (m/s) Pitch Angle (degree) Time (s) (a) Time (s) (b) 2.2 x Model (BEM) FAST Model (BEM) FAST Thrust Force (N) F T (f) Time (s) (c) f (Hz) (d) Figure 2.7: Verification of the BEM method result with the FAST output which uses the generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrust force in frequency domain 38

51 26 6 Wind Speed (m/s) Pitch Angle (degree) Time (s) (a) Time (s) (b) 2 x Model (BEM) FAST 250 Model FAST Thrust Force (N) Time (s) (c) F T (f) f (Hz) (d) Figure 2.8: Verification of the BEM method result with the FAST output which uses the generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrust force in frequency domain 39

52 Wind Speed (m/s) Pitch Angle (degree) Time (s) (a) Time (s) (b) 2.2 x Model FAST Model FAST Thrust Force (N) F T 1(f) Time (s) (c) f (Hz) (d) Figure 2.9: Verification of the BEM method result with the FAST output which uses the generalized-dynamic-wake model for simulation (a) Wind speed (b) Pitch angle (c) Aerodynamics thrust force in time domain (d) Aerodynamic thrust force in frequency domain 40

53 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

54 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 EI(x) 2 w(x, t) + ρa(x) 2 w(x, t) = f(x, t) (3.1) x 2 x 2 t 2 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

55 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 ẇ(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) = W i (x)q i (t) (3.2) i=1 where q i (t) are the generalized coordinates and W i are the normal modes found by solving the following eigenvalue equation using the boundary conditions: [ ] d 2 EI(x) d2 W i (x, t) ρa(x)ω 2 dx 2 dx 2 i W i (x) = 0 (3.3) where ω i is ith natural frequency of the system. The normal modes W i and W j are orthogonal if i j and hold the following relation which is called the orthogonality relation (see Ref. [50] for the proof): l 0 { 0 i j ρa(x)w i (x)w j (x)dx = 1 i = j (3.4) Now using Equations 3.2 and 3.3, Equation 3.1 can be expressed as: ρa(x) ωi 2 W i (x)q i (t) + ρa(x) W i (x) d2 q i (t) = f(x, t) (3.5) dt 2 i=1 Multiplying Equation 3.5 by W j and integrating from 0 to l results in: i=1 l q i (t) ρa(x)ωi 2 W i (x)w j (x)dx 0 d 2 q i (t) + dt 2 i=1 l 0 i=1 ρa(x)w i (x)w j (x)dx = l 0 W j (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: d 2 q i (t) dt 2 + ω 2 i q i (t) = Q i (t), i = 1, 2,... (3.7) 43

56 where Q i (t) is the generalized force corresponding to the ith mode given by: Q i (t) = l 0 W i (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: Q i (t) = W i (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 W i (l)q i (t) (3.10) i=1 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: ẋ(t) = Áx(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 B = (k+1) t k t A = eá t (3.15) eá[(k+1) t τ] Bdτ (3.16) x(k) = [x 1 (k), x 2 (k),...] T (3.17) 44

57 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

58 R U Wind Turbine Dynamics U Wind Turbine Aerodynamics F F T Q,, U,, U 2 2 F Q T g FT Wind turbine mechanics Rotational Dynamics Tower Dynamics (LTI) D FA D SS Controller Figure 3.1: Wind turbine dynamic system is described using Hammerstein structure where the Wind Turbine Aerodynamics is considered as the static nonlinearity and the Tower Dynamics is considered as a LTI system. The entire dynamic system of wind turbine is shown under closed-loop control; however, open-loop system identification method still can be used for identification of Tower Dynamics because there is no feedback from tower fore-aft deflection D F A to aerodynamic thrust force F T. 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 developed 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