Optimizing Blade Pitch Control of Wind Turbines with Preview Measurements of the Wind

Transcription

1 University of Colorado, Boulder CU Scholar Electrical, Computer & Energy Engineering Graduate Theses & Dissertations Electrical, Computer & Energy Engineering Spring Optimizing Blade Pitch Control of Wind Turbines with Preview Measurements of the Wind Fiona Dunne University of Colorado at Boulder, Follow this and additional works at: Part of the Power and Energy Commons Recommended Citation Dunne, Fiona, "Optimizing Blade Pitch Control of Wind Turbines with Preview Measurements of the Wind" (2016). Electrical, Computer & Energy Engineering Graduate Theses & Dissertations This Dissertation is brought to you for free and open access by Electrical, Computer & Energy Engineering at CU Scholar. It has been accepted for inclusion in Electrical, Computer & Energy Engineering Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact

2 Optimizing Blade Pitch Control of Wind Turbines with Preview Measurements of the Wind by F. Dunne B.S., University of California, Santa Barbara, 2006 M.S., University of Colorado Boulder, 2010 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical, Computer, and Energy Engineering 2016

3 This thesis entitled: Optimizing Blade Pitch Control of Wind Turbines with Preview Measurements of the Wind written by F. Dunne has been approved for the Department of Electrical, Computer, and Energy Engineering Prof. Lucy Y. Pao Dr. Alan Wright Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

4 Dunne, F. (Ph.D., Electrical Engineering) Optimizing Blade Pitch Control of Wind Turbines with Preview Measurements of the Wind Thesis directed by Prof. Lucy Y. Pao In above-rated wind speeds, the goal of a wind turbine blade pitch controller is to regulate rotor speed while minimizing structural loads and pitch actuation. This controller is typically feedback-only, relying on a generator speed measurement, and sometimes strain gauges and accelerometers. A preview measurement of the incoming wind speed (from a turbine-mounted lidar, for example) allows the addition of feedforward control, which enables improved performance compared to feedback-only control. The performance improvement depends both on the amount of preview time available in the wind speed measurement as well as the coherence (correlation as a function of frequency) between the wind measurement and the wind that is experienced by the turbine. This thesis shows how to design an optimal collective-pitch controller that takes both preview time and measurement coherence into account. Simulation results show significantly reduced pitch actuation, improved generator speed regulation, and reduced structural loads compared to several different baseline cases. In addition, linear-model-based results show how the benefit of preview depends on the preview time and measurement coherence. Effective lidar-based control also requires knowledge of the expected arrival time of the measured wind. Arrival time is the time it takes for the wind to travel from the measurement focus location to the turbine rotor. Arrival time is often assumed to be equal to the distance traveled divided by the average wind speed. This thesis, using field test data, studies deviations from this assumption. Control implementation across the full range of above-rated wind speeds can be achieved through gain scheduling. The effect of gain scheduling implementation on effective feedforward

5 iv and feedback gains is not straightforward. This thesis provides a detailed explanation of the effective gains resulting from two different gain-scheduling implementations. It includes an analysis of a simplified version of a nonlinear gain scheduling feedback loop as well as verification through simulation with a full nonlinear controller and turbine model. Additional topics covered in this thesis include a model-inverse-based analysis of the conditions under which lidar is beneficial, a breakdown of the maximum useful amount of preview time by its different uses, and a comparison of two lidar-based individual pitch controllers.

6 Dedication To my parents Mary Ellen and Denis Dunne.

7 vi Acknowledgements I would like to thank my advisor Prof. Lucy Pao for her tireless help, encouragement, and kindness and her thorough, insightful feedback; Dr. Alan Wright for his crucial advising and support; the rest of my dissertation committee Prof. Behrouz Touri, Prof. Dale Lawrence, Prof. Eric Frew, and Prof. Jason Marden for their generous time and feedback; Dr. David Schlipf, Dr. Eric Simley, and Dr. Jason Laks for their especially significant suggestions and discussions in guiding the direction of this research; the many anonymous peer reviewers who gave very helpful feedback on parts of this work; a fantastic group of labmates and visiting researchers including Dr. Hua Zhong, Marian Chaffe, Dr. Jeff Butterworth, Dr. Jason Laks, Dr. Eric Simley, Dr. Shervin Shajiee, Jake Aho, Andrew Buckspan, Dan Zalkind, Arnold Braker, Dr. David Schlipf, Floris Teeuwisse, and Bart Doekemeijer for both their friendship and their help and feedback on this research; Prof. Kathryn Johnson, Dr. Na Wang, Andy Scholbrock, Dr. Pieter Gebraad, Bonnie Jonkman, Neil Kelley, Dr. Rod Frehlich, Dr. Jason Jonkman, Dr. Paul Fleming, Jen Annoni, and many others for valuable advice and discussions at NREL meetings; all the friends, Ultimate players, and hiking and climbing partners who were a part of great fun and adventures outside of lab; and finally my husband Daniel Garrett for always being there to support and encourage me.

8 vii Contents Chapter 1 Introduction Motivation Background Introduction to Wind Turbine Control Lidar Measurements Rotor-Effective Wind Speed Measurement Coherence Contributions Related Work on Region 3 Blade Pitch Control Commercial Region 3 Blade Pitch Control Published Research on Region 3 Blade Pitch Control Differences Between this Thesis and Most-Closely-Related Work Organization Model-Inverse-Based Analysis Introduction Perfect Feedforward Imperfect Feedforward Conclusion

9 viii 3 Optimal blade pitch control with realistic preview wind measurements Introduction Design Methods Turbine Models Cost Function Augmented Plant for H 2 Synthesis Solving the H 2 Problem Baseline and Lowpass Filter Feedforward (LPF FF) Controllers NREL 5-MW Baseline Controller (PI FB) Lowpass Filter Feedforward (LPF FF) Controller Simulations Simulation Environment Simulation Implementation Considerations Simulation Results Linear-Model-Based Performance Predictions Conclusions and Future Work Preview Time Analysis How Preview Time is Used Actuator Lowpass Filtering Turbine Available Preview Time Lidar Measurements, Wind Turbine, and Wind Speed Estimator Windowed Cross-Covariances Filtering Lidar Measurements Using a Variable Time Delay Conclusions

10 ix 5 Comparison of Two Lidar-Based Independent Pitch Control Designs Introduction Simulated Turbine and Turbulent Inflow MW Turbine Model and Baseline Control Stochastic Turbulent Wind Field Simulator Controller Comparison Individual Pitch Feedforward Control Conclusions and Future Work General Conclusions and Future Recommendations 110 Bibliography 114 Appendix A Proof of Effect of Filters K, L, and H on Modeled Coherence and Power Spectra 119 A.1 Proof of Lemma A.2 Proof of Lemma B Analysis of Gain-Scheduling Implementation for the NREL 5-MW Turbine Blade Pitch Controller 123 B.1 Introduction B.2 Gain-scheduling Feedback Loop B.3 Multiply First Implementation B.4 Simulation Results B.5 Integrate First Implementation with New f(u) B.6 Conclusions C Effective Closed-Loop Gain Derivation for Integrate-First Implementation 138

11 D Gain Scheduling Implementation Details of the NREL 5-MW and Other Controllers 141 x

12 xi Tables Table 3.1 Properties of the NREL 5-MW turbine A list of names and properties of controllers discussed throughout Chapter Properties of the NREL CART2 turbine Data sets from 2012 CART2 field tests Mean values of t d (t) and T v : either uncorrected or corrected for induction zone Coherence bandwidth vs. method of variable time delay TurbSim boundary conditions Method notation. x, y, and z can each be either A, B, or (none) Features of feedforward controllers A & B Features of lidar configurations A & B

13 xii Figures Figure 1.1 Wind coordinate system and wind turbine actuator axes of rotation Sample steady-state power curve Turbine transfer functions from u, v, and w wind components Coherence bandwidth vs. number of lidar measurement samples per circle Block diagram showing linearized models of turbine, feedback, and feedforward PSDs without and with ideal feedforward to eliminate generator speed error Standard H 2 problem configuration Augmented plant with feedback and feedforward controllers Intermediate step in rearranging Fig. 3.3 into Fig Equivalent augmented plant to Fig Equivalent augmented plant to Fig. 3.3, fitting Hazell s framework Magnitude of the wind spectrum filter K Magnitude of L, corresponding H nok, and L 2 + H nok Frequency response of H 2 and PI feedback controllers, as individual parts Overall frequency response of PI plus H 2 feedback controllers, along with PI alone Frequency response of the feedforward part of an H 2 controller Impulse response of the feedforward part of an H 2 controller Block diagram of control implementation

14 xiii 3.14 Steady-state pitch angle versus steady-state wind speed, and linearization Performance metrics from simulation vs. PI FB Performance metrics from simulation vs. LPF FF* Performance metrics from simulation vs. H 2 FB, medpp Linear-model expectations vs. simulation results: gen speed error and pitch rate PSDs Linear-model expectations vs. simulation results for RMS generator speed error Linear-model expectations vs. simulation results for RMS pitch rate RMS gen speed error for a range of preview times and coherence bandwidths RMS pitch rate for a range of preview times and coherence bandwidths Required preview time Phase φ of actuator model Time delay τ φ of actuator model Preview time required for F with all DOFs on Preview time required for F with 5 DOFs on Block diagram of wind turbine control with lidar Time delay naming conventions Lidar measurement configuration Time series of wind measurement and estimate Time of peak cross-covariances Arrival time multipliers Time series including variably-time-delayed upstream measurement Coherence after using different variable time delays Coherence bandwidth vs cutoff frequency Feedforward control added to feedback control Impulse response of feedforward controller A Method AAA simulation results

15 xiv 5.4 Lidar configuration B pulsed lidar scanning pattern PSDs for method BBB under its original simulation conditions Simulated turbine loads using individual pitch controllers, 13 m/s Great Plains wind Simulated turbine loads using individual pitch controllers, 14 m/s Class A wind Bode plots of feedforward controllers A and B, excluding scheduled gains Time delay of feedforward controllers A and B Tower top fore-aft pitching moment as a function of frequency Blade root bending moment as a function of frequency Fore-aft nacelle acceleration as a function of frequency Blade pitch angle as a function of frequency Generator torque as a function of frequency B.1 Integrate-First implementation B.2 Nonlinear gain-scheduling loop. A simplified part of Fig. B.1, where d = / B.3 Steady-state pitch angle ( ) β o as a function of steady-state wind speed B.4 Effective closed-loop gain of Fig. B.2 from β 1 to β o when β 2 = B.5 Effective additional gain factor caused by the Integrate-First implementation B.6 Effective closed-loop gain of Fig. B B.7 Multiply-First implementation B.8 Blade pitch time series, simulation under different implementations B.9 Generator speed time series, simulation under different implementations B.10 Blade pitch PSD, simulation under different implementations B.11 Generator speed PSD, simulation under different implementations B.12 Effective closed-loop gain with new f(u) divided by desired closed-loop gain B.13 Blade pitch PSD, simulation with new f(u) implementation included B.14 Generator speed PSD, simulation with new f(u) implementation included

16 Chapter 1 Introduction Wind turbine control is typically feedback-only, relying on feedback measurements from a generator speed sensor, and sometimes strain gauges and accelerometers. Recently, however, it has become feasible to obtain measurements of the speed of the wind approaching a turbine by using a turbine-mounted Doppler lidar or other technology that remotely measures wind speed. This provides an estimate of the wind speed that will arrive at the turbine with a few seconds of preview. The topic of this thesis is incorporating this feedforward signal into the wind turbine s blade pitch control system in order to better counteract the effects of wind disturbances. The specific focus is improved performance in the above-rated wind speed operating region, where the control goals are rotor speed regulation, structural load reduction, and minimal pitch actuation. 1.1 Motivation The impact of this research depends on the percent reduction in the cost of wind energy due to improved control performance using lidar, the cost of a lidar relative to the cost of a wind turbine, and the size of the wind energy industry. This section provides some brief background on these factors. A turbine-mounted lidar can scan the incoming wind field and provide preview wind speed measurements a few seconds in advance [1]. This allows anticipatory blade pitch adjustments which can reduce structural loads compared to feedback alone. Structural load reduction leads to reduced cost of energy through either reduced material costs or increased turbine lifetime. A

17 2 small reduction in fatigue load often leads to a large increase in material lifetime. This is especially true for composite blade materials, where with an S-N (Wöhler) curve exponent of 10 [2], a 7% decrease in fatigue load leads to a doubling of number of cycles to failure. Therefore a relatively small reduction in structural load can have a relatively large impact on cost of energy. Coherent Doppler lidar has seen recent large improvements in cost, compactness, and reliability because of fiber-optic technology and related components developed for use in the telecommunications industry [3]. In addition, as wind turbines are becoming larger, a lidar becomes a smaller percentage of the cost of a turbine. In 2013, the average newly installed US wind turbine was rated at 1.87 MW, with a capacity-weighted average installed cost of $1,630/kW [4]. A Doppler lidar, at a cost of roughly $170,000, is then currently about 6% of the installed cost of today s average $3 million turbine. Finally, the potential impact of this work scales with the growth of the wind energy industry. The wind energy industry is growing rapidly, with wind power being the largest source of new United States electricity in 2014 [5]. Wind power currently generates 4% of U.S. electricity, and the U.S. Department of Energy has presented a goal of meeting 20% of U.S. electricity needs with wind by Globally, wind power is projected to deliver 5.3% of the electricity consumed worldwide by 2019 [6]. 1.2 Background This section contains some background information necessary to understanding this thesis. Topics include an introduction to wind turbine control, an explanation of lidar measurements, a discussion of rotor-effective wind speed, and a definition of measurement coherence.

18 Generator Torque w u v Nacelle Yaw Blade Pitch 3 Figure 1.1: Blue lines show the perpendicular directions of the u (downwind), v (transverse), and w (vertical) components of the wind when the turbine is yawed into the mean wind 4/43 direction. Red lines show the axes of rotation for the three different turbine actuators Introduction to Wind Turbine Control Actuators A modern multi-megawatt wind turbine relies on active control of the nacelle yaw, the generator torque, and the blade pitch angles. Fig. 1.1 shows the axis of rotation for each of these actuator types in red. Nacelle Yaw Nacelle yaw control keeps the wind turbine rotor directed into the wind. It operates slowly in order to minimize gyroscopic forces and avoids making continuous small corrections in order to avoid shock load cycles due to gear backlash [7]. A wind vane mounted on the turbine measures yaw error, and the turbine is moved in the correct direction only when the yaw error has been above some value for some period of time, often on the order of 5 degrees error for the past 10 minutes. A yaw brake is used whenever the yaw drive is not in motion. Generator Torque Generator torque actuation is achieved using the power electronics that connect the wind turbine generator to the utility grid. Generator torque response to torque commands is fast enough to be considered instantaneous relative to blade pitch control. The generator torque affects rotor speed according to τ a τ g = I ω (1.1)

19 4 where τ a is the aerodynamic torque applied to the rotor by the wind, τ g is the torque applied by the generator, I is the moment of inertia of the rotor, generator, and drivetrain, and ω is the rotor acceleration. Power production depends on generator torque according to P = τ g ω where P is the power produced by the generator and ω is the rotor speed. For simplicity, these two equations assume a gearbox ratio of 1 and omit drivetrain and electrical losses. Blade Pitch Blade pitch actuation is the rotation of an entire blade about its long axis. This changes the angle of attack between the airfoil and the wind it experiences, and therefore changes the lift and drag coefficients which determine the aerodynamic torque τ a as well as the thrust on the rotor. A pitch angle of 0 is often defined as the blade pitch angle that captures maximum power, with increasing values towards 90 increasingly shedding power, typically done in the pitch-to-feather direction for a more smooth and continuous reduction in power capture than the pitch-to-stall direction. For a 5-MW turbine, pitch actuation is possible up to a bandwidth of about 1 Hz, and this bandwidth generally decreases with increasing turbine size. Pitch actuation is typically achieved using one pitch motor for each blade, which can provide redundant braking capability [8]. This also enables the possibility of individual, rather than collective, blade pitch control, which may be used to compensate for non-uniform wind over the rotor plane, at the expense of increased pitch actuation. This thesis focuses mainly on collective pitch control, with one chapter on individual pitch control Control Regions Wind turbine control can be divided into the four control regions shown in Fig In Region 1, below the cut-in wind speed, the power production is zero because the wind speed is too low for operation to be worthwhile. In Region 2, the turbine is controlled to maximize power capture, typically achieved by holding the pitch angle constant at its optimal value of 0, and controlling generator torque based on generator speed feedback as τ g = kω 2 g, where k is a

20 Power (%) Rotor Speed (rpm) Blade Pitch (deg) Wind Speed (m/s) Figure 1.2: A sample steady-state power curve, with corresponding steady-state rotor speeds and blade pitch angles. Wind speeds are below cut-in in Region 1, below-rated in Region 2, above-rated in Region 3, and above cut-out in Region 4. Transition Regions 1.5 and 2.5 are omitted. This thesis focuses on blade pitch control in Region 3 (highlighted). constant that depends on the turbine and the air density, and ω g is the generator speed, which is approximately equal to the rotor speed ω times the gearbox ratio. This nonlinear torque feedback law regulates the rotor speed to the aerodynamic optimal value for the current wind speed. In Region 3, above the rated wind speed, the turbine is controlled to regulate rotor speed and power to their rated values primarily by using blade pitch control. Pitch control in Region 3 is often gainscheduled proportional-integral (PI) control of generator speed error. Region 3 generator torque is either held constant at its rated value or set to be inversely proportional to generator speed, maintaining constant power. In Region 4, above the cut-out wind speed, the turbine is shut down because the wind speed is too high. Not shown in Fig. 1.2 are Regions 1.5 and 2.5, which allow gradual transition between operating regions Turbine Models The National Renewable Energy Laboratory (NREL) 5-MW turbine model [9] is a publiclyavailable and very commonly used turbine model for control systems and other turbine design research. This thesis mainly uses the NREL 5-MW turbine model. In addition, this thesis discusses lidar field test data taken from the NREL two-bladed Controls Advanced Research Turbine

21 6 (CART2), which is a smaller 600-kW wind turbine used for controls field testing at NREL Lidar Measurements Coherent Doppler lidar remotely measures wind speed by emitting a laser, often infrared, which is reflected by aerosols, and by measuring the Doppler shift of the return signal to determine wind speed. The resulting wind speed measurement is a spatially-averaged line-of-sight wind speed centered at some chosen focus distance. The line-of-sight wind speed components are spatially averaged along the line of sight, weighted by a Gaussian (for pulsed lidar) or Lorentzian (for continuous-wave lidar) range-weighting function with peak centered approximately at the focus distance [10, 11]. The turbine is most affected by changes in the u-component of the wind: the component of the wind vector in line with the mean wind direction, which is pointing directly at the turbine rotor, assuming no yaw error. Fig. 1.1 shows the directions of the perpendicular wind components u, v, and w in blue. The effect of the u component in comparison to that of the v and w components is shown in Fig. 1.3, where for a linear model of the NREL 5-MW turbine, the u-component of the wind, at most frequencies, has at least ten times the effect on the turbine as the v (transverse) and w (vertical) components. To obtain an estimate of u-component of the wind, it must be reconstructed from the lidar line-of-sight wind speed, which is often not directly in line with the u-component. The lidar is mounted on top of a turbine nacelle or in the turbine hub, and one or multiple beams are typically directed to sample the incoming wind field at distributed locations representative of the wind approaching the entire rotor area. For example, the lidar might scan a circle with a half-cone angle of 15 or 30 degrees. The u-component can be reconstructed based on the line-of-sight measurement angle and the assumption that the v and w wind components are zero. The true v and w components contribute to u-component-estimate error, which increases as the lidar measurement angle increases. This is just one of several factors that affect the coherence between the rotor-effective wind speed predicted using lidar measurements and the rotor-effective wind speed experienced by the turbine.

22 7 Magnitude (abs) To: Gen speed error (rpm) To: RootMycC (kn m) To: RootMycV (kn m) To: RootMycH (kn m) Bode Diagram From: u (m/s) From: v (m/s) From: w (m/s) Frequency (Hz) Figure 1.3: Magnitudes of transfer functions from uniform u (downwind), v (transverse), and w (vertical) wind components, as deviations from the operating point, to generator speed error and the out-of-plane root bending moments of the three blades, as deviations from the operating point, presented in collective (RootMycC), vertical (RootMycV), and horizontal (RootMycH) multi-blade coordinate transformation (MBC) components, for the NREL 5-MW turbine model with baseline feedback controller (Multiply-First Implementation, to be discussed further in Appendix B), linearized in 13 m/s wind.

23 Rotor-Effective Wind Speed The rotor-effective wind speed is the single uniform wind speed that has the same effect on the turbine as the full 3D vector field, in terms of a particular turbine parameter: usually either aerodynamic torque or aerodynamic thrust. The rotor-effective wind speed experienced by the turbine can be determined by using a wind speed estimator, essentially using the entire turbine as an anemometer [12]. One relatively simple rotor-effective wind speed estimation method is the torque-balance method, in which the first step is to solve for the aerodynamic torque according to equation (1.1), given generator torque and rotor acceleration. Then a lookup table based on the turbine model is used to determine the current wind speed given the current aerodynamic torque, blade pitch angle, and rotor speed. Many improved versions of wind speed estimators have also been used, for example, a Kalman filter that accounts for tower and blade bending [13, 11]. In simulation, rotor-effective wind speed can also be estimated by using a weighted average of the u-components of the wind speeds over the rotor disc. The specific shape of the weighting function depends on the mean wind speed and whether aerodynamic torque or thrust is of interest, but even a simple average over the rotor disc is sometimes used as a sufficient estimate. The conversion from individual lidar samples of line-of-sight wind speeds to a prediction of rotor-effective wind speed depends on the scan pattern. For a simple circular scan, the rotoreffective wind speed prediction can be a running average over the past full circle of measurements, after correcting from line-of-sight velocities to u-components Measurement Coherence Measurement coherence is the correlation as a function of frequency between the rotoreffective wind speed predicted using lidar measurements and the rotor-effective wind speed experienced by the turbine. More formally, we define measurement coherence as the magnitude squared

24 9 coherence (γ 2 am) as a function of frequency (f): γ 2 am(f) = S am(f) 2 S aa (f)s mm (f) (1.2) where S am (f) is the cross power spectral density between the rotor-effective wind speed predicted using lidar measurements, w m (measured wind), and the rotor-effective wind speed experienced by the turbine, w a (actual wind). The wind experienced by the turbine is estimated as described in Section S mm (f) and S aa (f) are, respectively, the individual power spectral densities of each signal. Magnitude squared coherence can range from zero to one, where zero means no correlation between the measured and actual wind, and one means perfect correlation. In other words, a coherence of one means there is some linear time-invariant (LTI) transfer function between the two signals, and a coherence of less than one means some noise, disturbances, or nonlinearities are present between the two signals. This coherence function between the actual and measured wind is affected by wind evolution (which increases with focus distance), u-component estimation error (Section 1.2.2) due to the line-of-sight angle (which decreases with focus distance, given a fixed scan radius), the turbine induction zone (the wind in front of the turbine is slowed and redirected because the turbine is capturing power), and the use of limited lidar measurement locations to represent the effective wind speed over the entire rotor plane (lidar spatial averaging is usually beneficial). The overall effect of these error sources is generally strongest at high frequencies, resulting in a measurement coherence function that drops from close to one at low frequencies down to zero at high frequencies, with the exact shape, DC value, and bandwidth depending on the measurement configuration and wind conditions. The overall quality of measurement coherence is sometimes summarized by the coherence bandwidth, which gives an approximate frequency up to which preview measurements can be generally trusted to represent the effective wind speed experienced by the rotor. Coherence bandwidth is sometimes defined as the frequency at which the magnitude squared coherence drops below 0.5. In other cases, it is defined as the cutoff frequency of the lowpass filter whose magnitude squared

25 10 best fits the magnitude squared coherence. Coherence bandwidth may be expressed in temporal frequency (Hz or rad/s) or spatial frequency (rad/m), where the average wind speed is the conversion factor between the two notations. As average wind speed changes, coherence bandwidth often remains more constant when expressed as a spatial frequency than when expressed as a temporal frequency A Coherence Bandwidth Example For a circular scan pattern, one factor affecting the measurement coherence is the number of measurement samples per circle. Fig. 1.4 shows that coherence bandwidth generally increases as number of samples per circle increases. It also shows that using a non-integer number of samples per circle results in increased coherence bandwidth compared to using an integer number of samples per circle, assuming a fixed scan rate of one circle per second. This is because sampling at varying azimuth angles better distributes the measurement points throughout the structures in the wind. (Note that this data is based on lidar simulations where the lidar model instantaneously takes each measurement. Instead, with realistic non-instantaneous measurements, if the lidar were to continue scanning without pausing to take each measurement, a smearing effect would be introduced, partially overriding the benefit of using a non-integer number of samples.) 1.3 Contributions The focus of this work is in improving wind turbine blade pitch control performance in aboverated wind speeds using lidar-based wind speed measurements, with the control goals of rotor speed regulation, structural load reduction, and minimal pitch actuation. The specific research questions addressed are: How can an optimal controller be designed while accounting for measurement coherence and preview time in the design process? When does lidar allow improved performance, and by how much?

26 Coherence Bandwidth (rad/m) # of Measurement Points Per Circle Figure 1.4: Coherence bandwidth vs. number of lidar measurement samples per circle, for a simulated lidar scanning a circle 80 m upwind of the NREL 5-MW turbine in 18 m/s wind. The coherence bandwidth plotted is the pole location of the first-order lowpass filter, with DC gain 1, whose magnitude squared best fits the magnitude squared measurement coherence resulting from the simulation. Lidar scan rate: 1 second per circle. Scan radius: 73% span. (Data points are limited to measurement sample rates that correspond to multiples of the s simulation time step.)

27 How much preview time is useful to the feedforward controller, and how much is provided by the lidar? 12 The primary contributions of this work are: (1) An optimal control design process that incorporates lidar measurements and directly accounts for measurement coherence and available preview time. By optimal control, we mean a control design that minimizes some cost function, allowing tradeoffs between different outputs. Simulation results show significantly improved Region 3 performance, with tradeoffs between performance measures being easily tunable through the choice of weights. For example, one implementation on the NREL 5-MW turbine in 18 m/s wind with 17% turbulence intensity results in reductions of 48.3% in RMS generator speed error, 43.2% in pitch rate standard deviation, 7.7% in blade root out-of-plane damage equivalent load, 18.4% in tower base fore-aft damage equivalent load, and 3.3% in shaft torque damage equivalent load, compared to the baseline PI feedback only case. (2) Quantification of how performance improvements due to lidar depend on measurement coherence and preview time. Increases in both measurement coherence and preview time improve performance up to a point of diminishing returns. A computationallyinexpensive method to quantify these performance improvements using linear model-based predictions is presented and validated using simulation results. (3) An improved understanding of the amount of preview time that is useful for blade pitch control, broken down by different sources of delay. The major factors that determine the maximum amount of useful preview time, including pitch actuator delay and delay introduced by lowpass filtering, are explained. (4) An improved understanding of the amount of preview time provided by the lidar and an analysis of how to best variably-time-delay the wind preview, given a controller that expects a constant preview time. This analysis is based on

28 13 lidar and turbine data provided by NREL after field experiments carried out on a 600-kW wind turbine known as the two-bladed Controls Advanced Research Turbine (CART2) at NREL. Additional contributions include: (5) An analysis of often-overlooked implementation details. A gain scheduling study shows that a change in the order of multiplication with a gain-scheduling factor significantly changes the effective gains of the nonlinear closed loop. In addition, after correcting for these effective gain changes, one PI feedback-only control implementation has significant load reduction compared to another. Additional implementation details studied include a method to turn lidar-based control on and off smoothly and an integrator anti-windup implementation that functions effectively even in the presence of a non-zero-mean feedforward signal. (6) A lidar-based individual pitch controller comparison. Two different lidar-based individual pitch control designs created by two different researchers are studied, with the differences broken down by feedforward controller, feedback controller, and lidar configuration. 1.4 Related Work on Region 3 Blade Pitch Control Commercial Region 3 Blade Pitch Control Little information is available on commercially-implemented control strategies because these are usually proprietary. However, the most common commercial Region 3 blade pitch control strategy appears to be collective-pitch gain-scheduled PI control. Some manufacturers may also implement individual-pitch gain-scheduled PI control. Both of these strategies have been field tested [14].

29 Published Research on Region 3 Blade Pitch Control Feedback-Only In the research community, a wide variety of feedback-only Region 3 pitch control methods have been studied. These include model predictive control (MPC) [15, 16, 17], l 1 optimal control [18], state-space methods including Linear-Quadratic Regulator (LQR) and disturbanceaccommodating control (DAC) [19, 20], Linear-Quadratic-Gaussian (LQG) control [21], periodic control [22], adaptive control [23, 24], H control [25], a classical/lqg/h comparison [26], sliding-mode control design via H theory [27], feedback linearization [28], and LPV H 2 /H control [29] Wind Speed Preview Included Control Design Recently and often simultaneously with this dissertation research, many Region 3 pitch control methods that include feedforward of a wind speed preview have also been studied. The majority assume perfect measurements in the original design process, although many are later simulated with imperfect measurements and re-tuned based on simulation, or designed assuming perfect measurements and then augmented with a prefilter to filter out the uncorrelated high frequency portion of the wind measurements. In the following list, the designs where measurement coherence is instead directly considered in the design process are marked with a *. The first known lidar-based control study [3] replaces state estimates of vertical wind shear in a DAC controller with measured values from a simulated lidar, resulting in reduced blade root loads. *Successful field testing [30, 31, 32] of lidar-based collective pitch control has been implemented. *Reference [33] discusses continuous-time H 2 model-matching for preview feedforward control to achieve tower fore-aft oscillation reduction. This strategy accounts for measurement

30 15 distortion in the control design process. Two different methods of fatigue load reduction are discussed in [34]. One is an adaptive feedforward controller based on a filtered-x recursive least squares (FX-RLS) algorithm, and the other is a combined feedforward/feedback linear-quadratic (LQ) preview control augmentation to the existing feedback controller. In [35], model-inverse control and preview designed using Tomizuka s H 2 Preview Control method [36] are studied. Both of these design processes are based on an assumption of perfect measurements. A genetic-algorithm control design method* based on simulation results is also studied. Lidar for extreme event control, rather than fatigue load reduction, is discussed in [37]. In [38], a lidar-based H 2 controller is tested with simulated wind evolution. This H 2 control design is optimal assuming a white noise wind spectrum and no measurement error. The thesis [39] contains further discussion of H 2 preview design and considers additive white measurement noise, but not noise that varies as a function of frequency. Reference [39] also discusses MPC using a wind speed measurement and a DAC approach. MPC with wind measurement is also discussed in [40], [41], [42], and [43]. Reference [44] uses spinner-mounted lidar for individual pitch control to reduce once-perrevolution (1P) fatigue loads. Reference [45] discusses bounds on achievable performance, by using linear programming for a given wind profile, and also discusses H control. Wind and Lidar Modeling and Scanning In addition to control design research, research related to Region 3 blade pitch control with a wind speed preview also includes wind and lidar modeling and scan patterns.

31 Reference [1] is the first known experimental implementation of a spinner-mounted lidar. Various scan patterns were implemented to capture wind speed measurements. 16 Reference [10] discusses lidar measurements for wind turbine control, showing how lidar configuration, including measurement range, angular offset, and measurement noise affect measurement error. This study excludes wind evolution. Reference [46] discusses a coherence model for wind evolution, based on data from large eddy simulations. Reference [47] describes the design of a minimum mean-square error prefilter based on lidar measurement and rotor-effective wind speed statistics. This prefilter is ideal for augmenting a controller that is designed assuming perfect lidar measurements. Reference [48] discusses the effects of the induction zone on lidar measurements. Reference [49] discusses a model for the correlation between lidar measurement and rotoreffective wind speed, based on the Kaimal wind spectrum and accounting for measurement configuration, lidar spatial averaging, and wind evolution. Reference [32] discusses wind field reconstruction, lidar modeling, and experimental implementations. Reference [50] discusses a variety of scan patterns, provides rules of thumb for obtaining lidar measurements representative of the entire rotor plane, and discusses both collective and individual pitch control strategies Differences Between this Thesis and Most-Closely-Related Work Most of the controllers above are designed assuming perfect measurements, although many are simulated and sometimes tuned in imperfect conditions. Instead, this work directly accounts for measurement coherence, in addition to available preview time, in an optimal control design process. Two other control designs mentioned above also directly include measurement coherence.

32 17 The first is the collective pitch feedforward control design discussed in [30, 31, 32]. We use a very similar control design as a baseline for comparison in Chapter 3, where it is referred to as lowpass filter feedforward (LPF FF). In contrast to LPF FF control, this dissertation research uses an optimal control strategy that allows tradeoffs between different goals, includes a model of wind turbine dynamics, and allows optimal design of both the feedback and feedforward parts of the controller. The second control design that directly includes measurement coherence is [33]. Using a different approach to [33], this dissertation research addresses the feedback design with the same optimal control method as the combined feedforward/feedback design and hence allows a consistent evaluation of the benefit of feedforward. In addition, the results in this thesis are for discrete-time rather than continuous-time design, control blade pitch angle directly rather than controlling a power reference, and allow modeling of coherence functions with non-unity DC values. Other controllers designed assuming perfect measurements can be corrected by augmentation with a minimum-mean-square-error prefilter as in [47]. However, this prefilter is designed given some available preview time, where the amount available to the prefilter depends on the amount used by the feedforward controller. Instead, the current dissertation treats the prefilter and controller as one whole system, essentially solving the two problems together for optimal sharing of preview time. Many studies that look at the amount of preview time that is useful to a feedforward controller quote values in the fraction of a second range [39, 45]. This dissertation research instead finds that several seconds of preview are useful and comes to this conclusion by accounting for the delay introduced by lowpass filtering to remove the uncorrelated high frequency portion of the wind measurements, and also by including a pitch actuator model. A method to predict arrival time of lidar measurements is also introduced in [42]. The work in this thesis, however, provides additional insights through the use of real experimental data.

33 Organization This thesis is organized as follows. Chapter 2 gives a model-inverse-based analysis of disturbance feedforward control. This single-output linear analysis provides initial insight into the question of when lidar allows improved performance, while accounting for measurement uncertainty. Chapter 3 covers an H 2 optimal control design process that includes measurement coherence and preview time, along with simulation results, as described in Contribution 1. It also includes linearmodel-based predictions as described in Contribution 2, and some of the implementation details described in Contribution 5. Chapter 4 provides a preview time analysis. The first section provides a breakdown of how preview time is used, as described in Contribution 3, and the second section uses CART2 field experiment data to provide an analysis of available preview time and an explanation of how to best variably-time-delay preview measurements, as described in Contribution 4. Chapter 5 presents an individual pitch controller comparison, as described in Contribution 6. Each chapter contains its own detailed conclusions, and Chapter 6 contains general conclusions and future recommendations. Appendix A provides a proof addressing the modeled coherence and power spectra. Appendix B explains new results in gain scheduling implementation, as described in Contribution 5. Appendices C and D provide further details on gain scheduling implementation. Chapter 3 has partially appeared in [51]. Section 4.1 has partially appeared in [52]. Section 4.2 has partially appeared in [53]. Chapter 5 has partially appeared in [52]. Parts of Chapter 3 and Appendix B have also been submitted for publication.

34 Chapter 2 Model-Inverse-Based Analysis 2.1 Introduction One possible method for lidar-based wind turbine blade pitch control is to use model-inverse feedforward control [54]. This method involves inverting the linear plant model for some chosen output. Except when additional prefiltering is used, model-inverse control assumes that a perfect measurement is available. This is a valid assumption for reference-feedforward model-inverse control because the reference is usually perfectly known. However, blade pitch control using wind measurements is instead a disturbance-feedforward control problem, and this wind disturbance is not perfectly known. In this chapter, we analyze the results of a model-inverse blade pitch controller that is designed under the assumption of perfect measurements but is then subjected to error in wind disturbance measurements. 2.2 Perfect Feedforward We study the linearized system in Figure 2.1, where we have added feedforward control in an attempt to reduce the effect of the wind disturbance on some output y. In this section, we assume perfect feedforward control: F = T 1 yβ T yw, where T yβ is the closed-loop transfer function from β F to y, and T yw is the closed-loop transfer function from w a to y. We call this choice of F perfect feedforward control because it results in y=0 when combined with perfect wind measurements (w m = w a ). Solving for y without perfect measurements, we find:

35 20 F wm wa T C + βf P Ω y Figure 2.1: Block diagram showing linearized models of the wind turbine P, the feedback controller C, and the feedforward controller F. The labeled signals are the differences from the linearization operating point: measured wind speed w m, actual wind speed w a, feedforward pitch command β F, generator speed Ω, and some output of interest y. For example, y may be equal to Ω, to a blade bending moment, or to a tower bending moment. Each of these signals, excluding Ω, may be a vector of three values when considering individual pitch control, or they may each be a single value when considering collective pitch control. T is the closed-loop model containing the feedback controller and the turbine. y = T yw w a + T yβ F w m y = T yw w a + T yβ ( T 1 yβ T yw)w m y = T yw (w a w m ) Similarly, with no feedforward, we can solve that y = T yw w a. Therefore, at any given frequency, perfect feedforward is better than no feedforward at reducing y(f) whenever w a (f) w m (f) < w a (f). That is, perfect feedforward is helpful whenever the norm of the error in measurement is less than the norm of the actual wind. There is an exception at DC, where an integrator in the feedback loop makes T yw (0) = 0, and therefore y(0) = 0 regardless of what is done with feedforward. 2.3 Imperfect Feedforward Perfect feedforward (F = T 1 yβ T yw) will never be possible because of errors in plant modeling and because T yβ often can only be inverted approximately, either because y is a vector of outputs, or, even if y is a single output, because of non-minimum-phase zeros in the plant model. When non-minimum-phase zeros are present, approximate inversion methods include the non-causal

36 series expansion, nonminimum-phase zeros ignore, zero-phase-error tracking controller, and zeromagnitude-error tracking controller methods [54]. In this section, we study Figure 2.1 with imperfect feedforward control: F = T 1 yβ T yw + ɛ. Then solving for y: 21 y = T yw w a + T yβ F w m y = T yw w a + T yβ ( T 1 yβ T yw + ɛ)w m y = T yw (w a w m ) + T yβ ɛw m Again, with no feedforward, y = T yw w a. Therefore, imperfect feedforward is helpful whenever T yw (f)(w a (f) w m (f)) + T yβ (f)ɛ(f)w m (f) < T yw (f)w a (f) Now the error in the output y is the sum of two terms: one due to measurement error and another due to the error in F. It is possible but unlikely that these two sources of error would cancel each other out. To reduce y, we need a balance between accurate measurements and small measurements, where the tradeoff depends on the sizes of ɛ, T yw, and T yβ, and small and sizes refer to the magnitudes or norms of the signals and transfer functions. Because errors increase with frequency, this suggests that the value used for w m should be a lowpass-filtered version of what was actually measured. 2.4 Conclusion When a feedforward controller is designed assuming perfect measurements, it improves performance at a given frequency only when the norm of the measurement error, combined with the error due to an imperfect model-inverse, is less than the norm of the wind disturbance. Otherwise feedforward is harmful compared to feedback alone. However, the wind measurements can be lowpass filtered before being input into an ideal model-inverse controller. Simley [47] has extended the work in this chapter to the design of an optimal prefilter for the wind measurements; using this prefilter, feedforward is never harmful. Combining a model-inverse controller with a prefilter has limitations: only a single output can be effectively minimized using model-inverse control, and

37 when the turbine model is non-minimum phase, either infinite preview time or an approximation is required to create the model-inverse controller. 22

38 Chapter 3 Optimal blade pitch control with realistic preview wind measurements 3.1 Introduction H 2 optimal control allows the designer to make tradeoffs between multiple objectives. In above-rated wind speeds, the wind turbine control goals are rotor speed regulation, structural load minimization, and minimal pitch actuation. Improvements in meeting these goals result in increased turbine lifetime and therefore reduced cost of energy. The benefit of a lidar sensor in achieving these control goals is a function both of preview time, which is approximately determined by the lidar focus distance divided by wind speed, and of measurement coherence: the correlation as a function of frequency between the wind measurement and the wind that is actually experienced by the turbine. This chapter explains an H 2 optimal control design method that accounts for measurement coherence and preview time in the design process. It also includes simulation results in addition to linear-model-based results, and the linear-model-based results are validated through simulation. The combined feedforward/feedback H 2 controller we design is wrapped around the standard proportional-integral (PI) feedback controller, rather than directly applied to the turbine. This means that there is an H 2 feedback part that is added in parallel to the standard PI feedback, and there is also an added H 2 feedforward part. By wrapping the combined feedforward/feedback H 2 controller around the standard PI feedback controller, we facilitate smoothly turning off the lidar-based part of the control, returning to PI-control when necessary. Keeping the PI controller included also means that the augmented turbine model used in the H 2 control design process is

39 24 less uncertain because it contains feedback. In addition, adding H 2 feedback allows greater control freedom than using PI feedback alone; the differences in operation between these two feedback parts are shown in Section We use Hazell s method of solving for the optimal controller [55], taking advantage of the chain of delays in the problem structure to reduce the size of the Riccati equations we need to solve. This makes the solution faster, more numerically accurate, and easier to implement than standard H 2 optimal control. For a range of different preview times, coherence bandwidths, and cost function weights, we solve for an optimal H 2 combined feedback/feedforward wind turbine blade pitch controller whose output is to be added to that of the baseline PI feedback controller. We present simulation results for a small number of cases and linear model-based results for a larger number of cases. This chapter is organized as follows. In Section 3.2, we describe our design methods, including the turbine models, the cost function, the augmented plant used for H 2 synthesis, and details on solving the H 2 problem. In Section 3.3, we describe the baseline PI feedback controller for the 5- MW turbine and also discuss a lowpass filter feedforward controller used for comparing results. In Section 3.4, we explain the simulation environment and implementation considerations and present simulation results. We show linear-model-based results in Section 3.5, and finally we summarize conclusions and describe future work in Section Design Methods Turbine Models Simulation Turbine Model Simulations are performed using NREL s FAST code [56], an aeroelastic wind turbine simulation code, using the land-based version of the NREL 5-MW turbine model [9]. Table 3.1 lists properties of the turbine. All 16 available degrees of freedom are turned on in simulation. The standard turbine model does not include a pitch-actuator model, so we have added appropriate

40 Table 3.1: Properties of the NREL 5-MW turbine. 25 Hub Height (m) 90 Rotor Radius (m) 63 Rotor Diameter (m) 126 Rotor Orientation Upwind Number of Blades 3 Rated Wind Speed (m/s) 11.4 Rated Rotor Speed (rpm) 12.1 Rated Generator Speed (rpm) Gearbox Ratio 97 dynamics to represent a pitch actuator (a second-order lowpass filter [57] with a cutoff frequency of 1 Hz and a damping ratio of 0.7) because without a pitch actuator model, there is no delay between pitch command and pitch actuation, and the benefit of preview is not as apparent. In simulation, dynamic inflow is turned on, whereas in linearization, dynamic inflow is not an option Linear Turbine Model The turbine model is linearized using FAST, with reduced degrees of freedom: only the generator, first tower fore-aft mode, and first flapwise blade mode are turned on for linearization because these are the modes that have a noticeable effect on the value of the turbine transfer functions at the frequencies (below about 0.4 Hz) where collective pitch actuation is to be used. Using a limited set of degrees of freedom reduces the complexity of solving for the optimal controller. The turbine is linearized about a wind speed operating point of interest (18 m/s) within the range of above-rated wind speed operation, which occurs from 11.4 m/s to 25 m/s. After linearization, the unobservable generator degree of freedom is removed from the linear turbine model (the plant), and the plant is converted to discrete time using the sample-and-hold method with a sample rate of 10 Hz. The plant is also augmented with a linearization of the turbine model s standard Region 3 constant-power torque feedback controller [9] and the actuator model described above. The linearized torque controller is τ = ( Nm/rpm) Ω

41 26 where τ is generator torque in Nm (as a deviation from the operating point) and Ω is generator speed error in rpm. The linearization and augmentation result in an overall linear turbine model P. P has two outputs: y and Ω, and two inputs: w and β. The vector y is the cost to be minimized and is described in Section The other 3 signals are scalars: Ω is the generator speed error in rpm and is used for feedback control, w is the rotor-effective wind speed in m/s (as a deviation from the operating point), and β is the collective blade pitch command in degrees (as a deviation from the operating point). For example, at the 18 m/s operating point, the turbine model transfer function from w to Ω is P Ωw = (z )(z 0.118)(z z ) (z )(z z )(z z ) Sampling time T s : 1/10 s Cost Function The cost function minimized in the H 2 control design process is { 1 N ( J = E lim Ω i 2 + α 2 pitchrate N N β 2 ( ) i + α 2 H2 pitchaccel βh2 2)} (3.1) i i=0 or the sum of the variances of the components of the augmented linear turbine model s cost output y. In this study, we use y = Ω (rpm) β α pitchrate (rad/s) β H2 α H2 pitchaccel (rad/s 2 ) The cost function includes the generator speed error Ω, the overall pitch rate β, and the H 2 pitch acceleration β H2 ; along with scalar weights α pitchrate and α H2 pitchaccel, which allow tuning of relative penalties between the three outputs of interest. Generator speed error is penalized because improved generator speed regulation is itself a control goal, and because improved generator speed regulation generally also indirectly leads to

42 27 structural load reduction, as shown in Fig In Fig. 3.1, when the ideal feedforward controller is used to eliminate generator speed error, it also significantly reduces blade root moment, tower base fore-aft moment, and pitch rate at low frequencies. However, it significantly increases tower bending and pitch rate at the 0.33 Hz tower resonant frequency. For simplicity, direct penalization of structural loads is not included in our cost output. Our design framework does allow for the possibility of direct penalization of structural loads, but leaving out direct penalization of structural loads results in more consistent valid solutions to the numerical optimizations. The overall pitch rate is penalized in order to preserve the life of the pitch actuators. This also indirectly leads to reduced tower base fore-aft moment at the resonant frequency, compared to penalizing generator speed error alone. The H 2 pitch acceleration output is the second derivative of the pitch command output of the H 2 part of the controller. In our design, this pitch command output is added to the pitch command output of the baseline PI feedback controller, and together the two commands sum to form an overall pitch command β. The H 2 pitch acceleration is penalized to minimize high frequency actuation from the H 2 part of the controller because of model uncertainty and unmodeled dynamics at high frequencies. The weight α pitchrate allows tuning of the tradeoff between generator speed error and overall pitch rate. For reference, we found that the α pitchrate value for which the baseline PI controller is optimal is in 18 m/s wind. In Section 3.4.3, Simulation Results, we call this value of α pitchrate the medium pitch penalty. The penalty on the second derivative of the H 2 controller output (H 2 pitch acceleration) is automatically tuned through an iterative simulation process that updates α H2 pitchaccel in order to approximately minimize the first two terms in the cost function in simulation: N ( Ω i 2 + αpitchrate 2 β ) 2 i (3.2) i=0 After α H2 pitchaccel is updated, the iterative process then designs a new controller using this new penalty, simulates the turbine with the new controller, and calculates the performance metric (3.2)

43 28 Gen speed error (rpm) RootMyc (knm) TwrBsMyt (knm) e+007 4e+007 2e e+010 2e+010 1e+010 Pitch rate (deg/s) Linear Model Prediction of Power Spectral Densities (Power/Hz) no F F designed for 0 gen speed error Frequency (Hz) Figure 3.1: Linear-model predictions of power spectral densities of generator speed error, blade root out-of-plane moment (RootMyc), tower base fore-aft moment (TwrBsMyt), and pitch rate for the NREL 5-MW turbine, with all 16 degrees of freedom, linearized at 18 m/s, assuming a rotoraveraged Kaimal wind spectrum with class A turbulence. Blue represents feedback-only control (no F ), and green represents feedback control augmented with an ideal feedforward control (F ) designed to perfectly eliminate generator speed error.

44 for the current iteration. The process is repeated until the performance metric (3.2) improvement 29 has been less than 0.5% for each of the past three iterations. This automatic tuning is done individually for each controller. Ideally, in order to show the benefit of lidar, we would like to use the same cost function in designing both the combined feedforward/feedback controller and the feedback-only controller to which it is compared, but in practice, α H2 pitchaccel will differ for each individual controller even when α pitchrate is the same. However, the two controllers can still be designed with the same overall objective: minimize (3.2) in simulation over the set of optimal controllers designed using (3.1) for all α H2 pitchaccel. As an alternative to H 2 control, our control design framework also allows for implementation of H control, which instead minimizes the largest possible amplification over all frequencies of a sinusoidal input, and allows for robustness guarantees. Through use of a simple switch, our design framework can solve either type of optimal controller for our linear augmented plant model. However, we have not yet fine-tuned the cost function, uncertainty-modeling, and implementation consideration changes necessary for good simulation results in the H control case, so further detail on H control is outside the scope of this thesis Augmented Plant for H 2 Synthesis Initial Augmented Plant The standard H 2 problem configuration is shown in Fig Given the augmented plant P H2, the controller K H2 is solved to minimize the H 2 norm of the closed-loop transfer function from u 1 to y 1 : T y1 u 1 2. This is equivalent to minimizing the 2-norm of the output vector y 1 for a zero-mean Gaussian white noise input u 1. Fig. 3.3 shows the augmented plant that models our turbine with imperfect preview wind measurements, with blocks grouped to show how they fit into the standard problem configuration. In Fig. 3.3, everything outlined in red corresponds to P H2 in Fig. 3.2, and the remaining feedforward controller H 2 FF and feedback controller H 2 FB in Fig. 3.3 correspond to K H2 in Fig. 3.2.

45 u 1 u 2 P H2 K H2 y 2 y 1 30 Figure 3.2: Standard H 2 problem configuration Corresponding input and output signals are highlighted in matching colors and defined below. u 1 = n 1 n 2 ; u 2 = β H2 ; y 1 = y ; y 2 = w m The linear turbine model, P, is described in Section ; its output y contains the weighted outputs to be minimized as described in Section 3.2.2, and its output Ω is generator speed error, used for feedback. PI FB is the linearized model of the baseline gain-scheduled PI feedback controller. The actual wind w a is represented by white noise n 2 multiplied by the wind spectrum K. The signal w a arrives at the turbine after a delay equal to the amount of preview time, delay = z τ P /T s, where τ P is the preview time and T s = 1/10 s is the sample time. In this study, we look at a range of preview times from 0 to 10 seconds in linear modeling, and in simulation focus on a preview time of seconds, determined by the wind speed and choice of lidar scan pattern. The measured wind, w m, is the signal that is measured by the lidar and is input to the feedforward controller. In this model, it is represented by w m = Lw a +Hn 1 where L is typically a lowpass filter, H is typically a highpass filter, and n 1 is white noise. L and H are designed to make the coherence between w a and w m equal to a desired coherence, and to make the power spectrums of the two signals match: S aa = S mm. On a real turbine, the coherence between the actual and measured wind can be measured, using an observer to determine the actual wind. We can then set up our model s w a and w m to match this desired coherence. The power spectrum of both signals can also be measured on a real turbine. The actual and measured wind tend to have similar power spectrums, and we set them equal for simplicity in our model. This is reasonable because the difference in power spectrums tends to affect Ω

46 31 n 1 n 2 wm H + L K wa K H2 H 2 FB H 2 FF + Ω β H2 + PI FB β delay w P P H2 y Figure 3.3: Augmented plant with feedback and feedforward controllers. Color coding corresponds with Fig. 3.2.

47 32 n 1 n 2 wm K L H + wa delay w Figure 3.4: Intermediate step in rearranging top part of Fig. 3.3 into Fig Color coding corresponds with Fig only the resulting controller, not the cost. For example, say S mm = 4S aa, and we correctly model the coherence, but we incorrectly assume S mm = S aa while solving for an H 2 optimal controller and cost. The resulting cost is still correct, and the resulting feedforward controller simply needs Saa to be multiplied by = 1 Saa to be corrected. The caveat is that if varies as a function S mm 2 S mm of frequency, it will have a nonzero phase delay, and therefore some additional preview time may be required in order to maintain causality while correcting the feedforward controller Rearranged Augmented Plant The augmented plant as shown in Fig. 3.3 does not yet meet the requirements for H 2 synthesis [58]. Informally, two of the requirements are: Each measurement output (each component of y 2 in Fig. 3.2) should have direct feedthrough n from a disturbance input (a component 1 of u n 2 1 ). Each actuator input (u 2 ) should K have direct feedthrough H to a cost output (a component of y 1 ). wm L delay + Direct feedthrough means that an output s value at the current sample time is affected (at least partially) by an input s value at the current sample w a (delayed) time. The second condition is met because the pitch rate is included in our cost, because there is direct feedthrough from the H 2

48 H2 2 2 H2 H2 Figure 3.5: Equivalent augmented plant to Fig. 3.3, rearranged to a format more suitable for H 2 synthesis. Color coding corresponds with Fig. 3.2.

49 34 pitch command β H2 to the pitch command β and because, through careful choice of discrete-time implementation of the pitch actuator model, there is direct feedthrough from pitch command β to pitch rate. The position of the delay block in Fig. 3.3 is the main roadblock in meeting the first condition because it prevents any direct feedthrough from a white noise disturbance n 1 or n 2 to the generator speed measurement Ω. We therefore rearrange the augmented plant in two steps as follows. First, note that magnitude squared coherence is commutative, and the H, L, and K filters have been chosen to give a desired coherence between w a and w m and make S aa = S mm. Therefore, from Fig. 3.3, we can flip the top half of the block diagram as shown in Fig. 3.4 so that the signal that once reached w a now reaches w m, and vice versa. By swapping the inputs to w a and w m, we have changed neither the coherence between the two signals nor the fact that S aa = S mm. Therefore this new arrangement in Fig. 3.4 is equivalent for our purposes. (Note that while we use S aa = S mm for simplicity, it is not a requirement. The details for modeling a non-unity ratio S aa /S mm are mentioned in Section ) Starting with Fig. 3.4, the second step is to move the delay block, which is now at the output of a summing junction, to each input side of the summing junction. This is simply using the distributive property of multiplication over addition. This results in a delay block after the L filter and another delay block after the H filter. Any delay on the H side of the summing junction can be ignored because the input to H is white noise, resulting in Fig Fig. 3.5 is equivalent for our purposes to Fig. 3.3, and the delay block is no longer in the path from n 2 to the generator speed output Ω. The two augmented plants are equivalent but not exactly equal because they match in magnitude, and in coherence between w a and w m, but the phase from the white noise inputs to the augmented plant outputs differs between the two. This phase difference does not matter because the goal is only to minimize the closed-loop magnitude, with no constraints on the closed-loop phase. In addition to fitting into the standard H 2 problem configuration, our rearranged augmented plant also now fits into the framework of the generalized regulator problem with both previewable

50 35 and non-previewable disturbances solved by Hazell [55]. Hazell s method recognizes that when a chain of delays is a specific part of the augmented plant structure, then the order of the delay block (the number of samples of delay) does not need to be included in the dimension of the two Discrete Algebraic Riccati Equations (DAREs) to be solved. Instead of the high-dimension full-information DARE, the same result can be achieved using a reduced-dimension DARE, along with a discrete Lyapunov equation and a Stein equation. Instead of the high-dimension output-feedback DARE, the same result can be achieved using a reduced-dimension DARE and by recognizing that the states of the delay chain can be perfectly reconstructed by the controller. Fig. 3.6 contains essentially the same rearranged augmented plant as shown in Fig. 3.5, but with four outlined blocks corresponding to the four blocks of Hazell s framework (Figure 1.5 within reference [55]). The dimension of each reduced-order DARE is the order of the system to be controlled block Filters K, L, and H Fig. 3.7 shows the magnitude of our filter K. It models magnitude of the rotor-averaged wind speed from simulation with an IEC Kaimal spectrum [59], assuming Class A turbulence (high turbulence) and the normal turbulence model (NTM), with a mean wind speed of 18 m/s. The rotor-averaged wind speed at each time step is calculated as w a = ( 19 i=1 w CP (i)f A (i) 36 j=1 v ) 1 u(i, j) where w a is the rotor-averaged wind speed, w a = w a 18 m/s is the actual wind experienced by the turbine (as a deviation from the operating point) for use in coherence calculations, i is a radius index, j is an azimuth index, w CP (i) is a weight representing the relative contribution to power coefficient at that radius, f A (i) is the fraction of the swept area represented by that radius point, and v u (i, j) is the u-component of the wind speed at point i,j. Rotor-averaging gives a more accurate representation of the frequency content of the rotor-effective wind disturbance than directly using the Kaimal spectrum, which is representative of the wind at a single point in space. The Kaimal spectrum is one of the simplest of many spectral models of the wind. All models have

51 H2 Figure 3.6: Equivalent augmented plant to Fig. 3.3, rearranged to a format more suitable for H 2 synthesis, as in Fig. 3.5, but with blocks outlined to correspond with those in Hazell s framework [55]. (The delay and L blocks are also swapped compared to Fig. 3.5.) Magnitude Frequency (Hz) Figure 3.7: Magnitude of the filter K at 18 m/s operating point, obtained using rotor-average wind speed from simulation with Class A normal turbulence model (NTM) [59].

52 37 lowpass filter characteristics, with more energy in the wind at lower frequencies. The phase of our filter K does not matter because the input to K is white noise. To determine this low-order filter K with magnitude matching the magnitude obtained from simulation, we use log-chebychev magnitude design to fit a minimum-phase state-space model (Matlab [60] function fitmagfrd()), achieving a close match with two poles and two zeros. We require the same number of zeros as poles in the fit in order to preserve direct feedthrough. After obtaining this state-space model, we also artificially increased the value of the direct feedthrough term (the D matrix) in order to further increase the likelihood of finding a valid solution to the Riccati equations. This artificial increase in the direct feedthrough term is responsible for the rise in the magnitude of K with frequency at frequencies above 1 Hz, as seen in Fig. 3.7, but it does not significantly affect the magnitude of K at the lower frequencies (those below about 0.4 Hz) where the H 2 part of the controller is active. L and H are determined as follows: L = coh(f) (3.3) L = 0 (3.4) H = K 2 (1 L 2 ) (3.5) H = irrelevant (3.6) where coh(f) is the desired measurement coherence as defined in (1.2). This choice of magnitudes for L and H ensures that the coherence between w a and w m equals the desired measurement coherence (coh(f)) and that S aa = S mm as proven in Appendix A. (For S aa /S mm 1, in order to leave coherence unchanged while correctly modeling the ratio of S aa /S mm, H should remain unchanged from above based on the original K and L, the L above should be multiplied by S aa /S mm, and K should be multiplied by S mm /S aa.) The choice of phase for L in (3.4) ensures that the average phase between the two signals is zero, which is typically true for real measurements when we have accounted for the correct amount of preview time. In this study, our desired magnitude squared coherence is the magnitude squared of a scaled

53 38 Butterworth lowpass filter (our L). This filter is determined by three design parameters: (1) We choose a filter order of 1 because this best fits the magnitude squared coherence found in simulation. (2) We find solutions for a range of different coherence bandwidths, which is our name for the cutoff frequency of the filter (the frequency where the magnitude response of the filter is 1/2 of its DC value). For various lidar trajectories studied by Schlipf for the NREL 5- MW turbine, achievable coherence bandwidths ranged from 0.03 rad/m to 0.07 rad/m [32]. Our best-fit coherence bandwidth for our simulation with an 18 m/s IEC wind field and a modeled lidar, as described in Section 3.4, is Hz, or rad/m, approximately equal to that of Schlipf s optimal lidar trajectory (2π Hz/(18 m/s) = rad/m). This is the highest coherence bandwidth we would expect to see with current experimental lidar configurations, and it is at the top of the achievable range in part because wind evolution and the induction zone are not modeled in our simulations. (3) A scaling factor χ 1 is included because otherwise the control design process expects almost perfect coherence at low frequencies, and therefore almost eliminates feedback control because it expects that feedforward can almost completely take over. However, feedback is still important because of model uncertainty. We found a scaling factor of χ = 0.95 to work well. For a filter order of 1, coherence bandwidth Hz, and χ = 0.95: L = (0.95)( )(z + 1) ; T s = 1/10 s. (3.7) z This choice of L satisfies L = coh(f) but does not satisfy L = 0. It is impossible for a causal filter to have both this desired magnitude and a phase of zero. To approximately correct for the phase delay in L, we first convert its phase delay to a time delay: t L (f) = φ L(f)(in degrees) 360 degrees 1 f

54 where t L (f) is the time delay of L and φ L (f) is the phase of L. The time delay of L is relatively constant as a function of frequency at all frequencies below the bandwidth, and drops as frequency 39 increases above the bandwidth. We make the approximation t L (f) = t L (f low ), where f low is a very low frequency relative to the bandwidth. We can then subtract this constant value of time delay, t L (f low ), from the delay block that precedes the L filter in the augmented plant (Fig. 3.6). Then although the phase of L does not satisfy (3.4), the phase across both L and the delay block together is approximately correct. At higher frequencies, the amount we subtract from the delay block is an overapproximation, so our model shows slightly less high-frequency delay, and therefore less high-frequency preview available than there really is. However, we do correctly model the low coherence values at higher frequencies, so the resulting controller would be unable to make much use of additional high-frequency preview time. This small error in modeling available preview time is especially not a concern for preview times above about 6 seconds, because as we will see in Section 3.5, the cost quickly flattens out as a function of preview time. A filter H that meets the magnitude requirement of equation (3.5) is H = K H nok, where H nok is a filter which satisfies L 2 + H nok 2 = 1. In the case where L has a scaling factor of χ = 1, H nok is simply the highpass Butterworth filter with the same order and cutoff frequency as L. With a non-unity scaling factor χ, an H nok with a very good approximation to the required magnitude can be determined as follows: H nok = 1 χ 2 L ω2 /L ω1 where L ω1 and L ω2 are lowpass Butterworth filters of order n where n is the order of L, with DC gains of 1, and with cutoff frequencies ω 1 and ω 2 respectively, where ω 1 < ω 2. The cutoff frequency ω 2 is set to match the cutoff frequency of L, so L ω2 = L/χ. The cutoff frequency ω 1 can then be found by noting that the ratio between ω 2 and ω 1 determines the amount by which H nok rises as frequency goes from DC to infinity. H nok must rise overall by a factor of 1 1 χ 2 because its DC magnitude is 1 χ 2, and it must have unity magnitude as frequency approaches infinity because L = 0 as frequency approaches infinity. The overall increase of H nok will be 20n db per

55 40 1 ω ω L H nok L 2 + H nok Frequency (Hz) Figure 3.8: Magnitude of example L from equation (3.7), along with corresponding H nok and L 2 + H nok 2. Cutoff frequency ω 1 is labeled on H nok, and cutoff frequency ω 2 is labeled on both L and H nok. ( ) ω2 decade between ω 1 and ω 2, where the number of decades between ω 1 and ω 2 is log 10. An ω 1 increase of 20n db is equivalent to being multiplied by a factor of 10 n. In equation form, we then have ( ) ω2 1 log = 10 1 χ 2 (10n ) ω 1 This equation can be rearranged and simplified to solve for the cutoff frequency ω 1 : ( ) 1 ( ) ω 1 = ω 2 1 χ 2 n This completes the design of H nok. Fig. 3.8 shows L and H nok for the example L from equation (3.7). Their magnitudes squared sum to almost exactly one, as required Solving the H 2 Problem The rearranged augmented plant described in Section and shown in Fig. 3.6 still does not quite meet the requirements of H 2 synthesis because the turbine model P does not have direct feedthrough from its disturbance input (wind) to its measurement output (generator speed). Since our turbine model transfer function has one fewer zero than pole, we simply multiply its disturbance

56 41 input by z to add one more zero, and then add one more sample of delay (z 1 ) to the delay block so that the augmented plant is still equivalent. In the special case of infinite coherence bandwidth with a scaling factor of χ = 1, the filter L = 1 and H = 0, and therefore, once again we do not meet a requirement of H 2 synthesis because there is no direct feedthrough from a disturbance input to each measurement output: as shown in Fig. 3.6, the measurement outputs are w m and Ω, and when H = 0, this requirement is not met for the measurement output Ω. Our solution to this problem, in this infinite coherence bandwidth case only, is to simply delete the measurement output Ω, leaving only the measurement output w m. This means we are using no feedback control, only feedforward control. Fortunately, in this case, the measurement Ω is redundant, and anything we could have done with feedback control can also be done with feedforward control because the assumption in this perfect feedforward case is that all inputs to the plant model and the plant model itself are perfectly known, therefore the outputs of the plant model are perfectly known without measurement. We use Hazell s method [55] to solve for the minimum-h 2 -cost controller. Both standard H 2 synthesis and Hazell s method require the solution of two discrete algebraic Riccati equations (DAREs). Hazell s method is relevant for H 2 (or H ) synthesis in problems where the augmented plant P H2 contains a significant amount of delay, which often appears in modeling a preview of a reference or disturbance signal. The method is an improvement upon standard H 2 synthesis for these problems because it allows the size of the DAREs to be reduced to the order of the plant and weighting functions, excluding the order of the delay. Two additional techniques, in addition to using Hazell s method, are used to minimize the occurrence of numerical problems that can prevent accurate solution of the two DAREs. The first is to use a balanced realization (Matlab [60] function balreal()) on the augmented plant model before doing the H 2 control synthesis. The second is multiplying the output vector y by a scalar. Sometimes the algorithm cannot solve for the controller that minimizes y, but can solve the equivalent problem of minimizing 10y, for example. We automatically loop through increasing factors until a valid solution is returned.

57 Sample H 2 Controllers The frequency and impulse responses of two sample H 2 controllers are shown in Figs. 3.9, 3.10, 3.11, and The controllers, as labeled in the legends, are summarized in Table 3.2, which lists the parameters and explains the abbreviated names of the controllers discussed throughout Chapter 3. In Fig. 3.9, Feedback Parts, the feedback parts of the two H 2 controllers have approximately the same magnitude at low frequencies as the PI feedback controller to which they are added in parallel, but they have the opposite phase, meaning they are canceling much of the low frequency action of the PI feedback control. The peak at about 0.32 Hz corresponds to the first tower mode. This low-frequency magnitude reduction compared to PI alone can be seen in Fig. 3.10, Feedback Overall. In addition, H 2 FF/FB, medpp control has a low-frequency magnitude reduction in its overall feedback compared to H 2 FB, medpp. This is expected because when the feedforward controller is compensating for the wind disturbances at low frequencies, it frees the feedback controller to shift towards disturbance rejection at higher frequencies, since the feedback disturbance rejection across all frequencies is limited by the Bode sensitivity integral [61]. This low-frequency magnitude reduction for H 2 FF/FB, medpp feedback is also consistent with other work [30, 50] that has shown reductions in PI gains to be beneficial when using lidar-based feedforward control. The frequency response of the feedforward part of the combined H 2 feedforward/feedback controller, shown in Fig. 3.11, has a similar bandwidth to the coherence bandwidth used in the design, but also contains peaks at the tower frequency and its harmonics. The impulse response of this same feedforward part is shown in Fig This is the output of the linear feedforward block when its input is an impulse at time zero. The blue line indicates the s preview time, and the first s of this response represents preview actuation. In addition, the negative initial response indicates that the feedforward part of the controller is nonminimum phase.

58 Table 3.2: A list of names and properties of controllers discussed throughout Chapter 3 43 Controller Name Mean Wind Speed (m/s) Preview Time (s) Coherence Bandwidth (Hz) χ α pitchrate α H2 pitchaccel PI FB 18 LPF FF(*) H 2 FF/FB, medpp H 2 FF/FB, lowpp H 2 FF/FB, hipp H 2 FB, medpp PI FB Baseline proportional-integral feedback only LPF FF(*) Lowpass filter feedforward H 2 FF/FB Combined H 2 feedforward/feedback H 2 FB H 2 feedback only medpp with medium pitch penalty (medium α pitchrate ) lowpp with low pitch penalty (low α pitchrate ) hipp with high pitch penalty (high α pitchrate ) *: When * is included, the cutoff frequency is the transfer function estimate bandwidth rather than the coherence bandwidth, as described in Section : Except in Section 3.5, where a range of values is specified. : This amount of preview time is available, but the controller delays the signal, effectively using less preview time. H 2 Feedback Part of H 2 Controllers; and PI Feedback Magnitude (db) H 2 FB, medpp H 2 FF/FB, medpp PI FB Phase (deg) Frequency (Hz) Figure 3.9: Frequency response of H 2 and PI feedback controllers, as individual parts (before addition in parallel). (For H 2 FF/FB, only its feedback portion is shown here.) Input: generator speed error (Ω) (rpm). Output: pitch (β) (rad).

59 44 Overall (H 2 plus PI) Feedback Part of H 2 Controllers Magnitude (db) H 2 FB, medpp H 2 FF/FB, medpp no H 2, PI FB only 90 Phase (deg) Frequency (Hz) Figure 3.10: Overall frequency response of PI plus H 2 feedback controllers, along with PI alone. Input: generator speed error (Ω) (rpm). Output: pitch (β) (rad). Feedforward Part of H 2 FF/FB Controller Magnitude (db) Phase (deg) H 2 FF/FB, medpp Frequency (Hz) Figure 3.11: Frequency response of the feedforward part of an H 2 combined feedforward/feedback controller. Input: measured wind (w m ) (m/s). Output: pitch (β) (rad).

60 45 8 x 10 5 Impulse Response of Feedforward Part of H 2 FF/FB Controller H 2 FF/FB, medpp 6 Amplitude Time (sec) Figure 3.12: Impulse response of the feedforward part of an H 2 combined feedforward/feedback controller. Input: measured wind (w m ) (m/s). Output: pitch (β) (rad).

61 generator speed (rpm) + _ Ω rpm (desired speed) X 0 switch if sat 0 sat 3.37 K P H 2 FB K I + b + + a _ + lookup table H 2 FF saturation 0 to 90 wm + _ 18 m/s rate limit 8 /s wind speed measurement (m/s) actuator model gain scheduling blade pitch 46 Figure 3.13: Block diagram of control implementation in simulation. The baseline gain-scheduled proportional-integral feedback (PI FB) controller is represented by everything not colored blue. Blue blocks and lines represent the H 2 control additions, so that the overall block diagram represents the combined H 2 feedforward/feedback controller. 3.3 Baseline and Lowpass Filter Feedforward (LPF FF) Controllers NREL 5-MW Baseline Controller (PI FB) Our baseline controller is the standard 5-MW turbine collective-pitch feedback controller, a gain-scheduled PI controller [9], with three changes from the original referenced controller. This controller is represented in Fig by all parts of the block diagram that are not colored blue. Other aspects of this figure will be explained later in this section, and in the upcoming Section The first change from the original referenced controller [9] is that because the gain-scheduling factor is based on blade pitch angle, our addition of an actuator model introduces the design choice of whether to use the signal from before or after the actuator model. We chose to use the signal after the actuator model so that the gain scheduling is based on the actual pitch angle rather than the pitch command, but the two options appear to produce very similar simulation results. Second, the gain-scheduling multiplication happens before the proportional-integral (PI) control blocks. The original gain-scheduling order, with multiplication after the PI blocks, has unintended consequences, effectively reducing control gains from their design values. Our change in multiplication order preserves the originally-designed control gains and also allows easier implemen-

62 47 tation when adding H 2 feedback in parallel. A detailed analysis of gain-scheduling implementation is contained in Appendix B. Third, we changed the integrator anti-windup method to allow fully-functional operation even when feedforward is added. The original anti-windup method limits the integrator state to be within minimum and maximum values, with the lower limit being zero. With feedback alone, and a lower pitch angle saturation limit of zero, this correctly works to prevent the integrator from accumulating a negative value in below-rated wind speeds where the blade pitch is saturated at zero and the generator speed is below the setpoint. However, if a feedforward signal with a positive average offset value is contributing to the blade pitch command, the average value of the integrator is reduced. The integrator will often need to accumulate a negative value in order to perform its normal function. The original anti-windup method prevents this because of the lower limit of zero on the integrator state. Our new anti-windup method does not limit the integrator state. Instead, we use the purple switch block shown in the lower-left of Fig This switch holds the generator speed error input to the integrator at zero whenever the pitch command saturation block is saturated. This replacement of the input with zero prevents windup. Because this method does not place any limits on the integrator state, the integrator can now operate at negative values when necessary Lowpass Filter Feedforward (LPF FF) Controller We compare our H 2 controller results not only to the baseline PI controller, but also to a lowpass filter feedforward (LPF FF) controller which is similar to a design that has been successfully field tested [30]. Our LPF FF controller has three components: time delay First, the input to LPF FF, w m, is delayed by the preview time τ P minus the time delays of the lowpass filter and pitch actuator, where w m is the lidar measurement before subtraction of the operating point (w m = w m + 18 m/s), as described in Section lowpass filter Second, the delayed measurement is lowpass filtered using a cutoff frequency equal

63 48 to: the coherence bandwidth, in linear-model results, where we assume for simplicity that the power spectrums of actual and measured wind are equal. the transfer function estimate bandwidth, in simulation, where for improved performance we do not assume equal power spectrums. This refers to the transfer function estimate between measured and actual wind. In this case we will use the name LPF FF*. lookup table Finally, the delayed-and-filtered measurement is input to a lookup table which maps steady-state wind speed to steady-state pitch angle. The output of LPF FF is the resulting feedforward pitch command β H2, which is added to that of the baseline PI feedback controller (PI FB). 3.4 Simulations Simulation Environment Simulations are performed using NREL s FAST code [56], an aeroelastic wind turbine simulation code, augmented with a continuous-wave Doppler lidar model [10]. We use a simulation sample rate of 80 Hz, where the controller is resampled from 10 Hz to 80 Hz using the Tustin approximation. We use a lidar sample rate of 40 Hz, and a lidar scan rate of 1 second per circle. We scan a circle using a half-cone angle of 30, a radius of 46.2 m (73.3% span), and a distance of 80 m ahead of the turbine rotor. The measured wind signal w m used as input to the feedforward controller is w m = wm 18 m/s, where wm is the running average of the past full circle of measurements corrected for line-of-sight angle as in [10] and 18 m/s is the operating point. The wind field used in simulations is an IEC wind field of 1 hour duration, with 18 m/s average wind speed and 17% turbulence intensity. The first 100 seconds of data are discarded before calculation of performance measures to allow time for start-up transients to settle.

64 Simulation Implementation Considerations Gain Scheduling the H 2 Controller In this turbulent 18 m/s wind field, we use H 2 control designs based upon a linear turbine model, linearized at 18 m/s. These linear designs require gain scheduling in order to operate effectively throughout the large range of wind speeds encountered in the wind field, which has an instantaneous hub-height minimum of 7.4 m/s and maximum of 29.1 m/s. We found the best results by separately scheduling the feedforward and feedback parts of the H 2 controller, as shown in Fig. 3.13, because this allows the preview (feedforward) part to be scheduled based on the previewed wind speed while the feedback part maintains the same scheduling as the PI feedback, based on the non-previewed current pitch angle. The H 2 feedforward scheduling uses a lookup table from: f βl (w SS ) to: f β (w SS ) where f βl gives the linearized steady-state pitch angles for a given steady-state wind speed and f β gives the actual steady-state pitch angles for a given steady-state wind speed, each plotted versus wind speed in Fig. 3.14, w SS is a vector of steady-state wind speeds, and is the steady-state pitch angle in radians at the 18 m/s wind speed operating point. The H 2 feedback scheduling is accomplished because its input is the generator speed error that has already been multiplied by the gain-scheduling output. It is important that the same gain-scheduling signal is used for both the PI and the H 2 feedback parts because the H 2 feedback design partially cancels out the PI feedback, and the gain scheduling needs to match for this to happen accurately. We also multiply the H 2 input signal by 1/0.297 = 3.37 to correct the scaling to 1 at the 18 m/s operating point, where the gain-scheduling factor has value 0.297, because the H 2 feedback block was designed based on an unscaled generator speed error input.

65 50 Steady State Pitch Angle (rad) Actual Linearized at 18 m/s Operating Point Steady State Wind Speed (m/s) Figure 3.14: Steady-state pitch angle versus steady-state wind speed, along with the curve s linearization at the 18 m/s operating point Anti-Windup for the H 2 Controller Additionally, the anti-windup implementation for the two feedback parts matches. The signal that is the input to the PI feedback integrator is the same signal used as input to the H 2 feedback, as shown in Fig. 3.13, so whenever the scheduled generator speed error is switched to zero because of pitch saturation, it affects both feedback controllers in the same way. This should prevent integrator windup in the H 2 feedback in addition to the PI feedback, allowing a smooth transition between below-rated and above-rated operation Turning the H 2 Controller On and Off Smoothly Finally, lidar may not always be available, and the lidar-based part of the control may need to be turned off. Both feedforward and feedback parts of the H 2 controller should be switched off together because the feedback part is designed to operate together with the feedforward part and is not optimal operating alone. Smooth and instantaneous switching between PI-only and PI-plus-H 2 control operation should be achievable as follows. Let a(k) be the value of the signal at point a in Fig at time sample k, b(k) be the value of the signal at point b in Fig at time sample k, and ξ(k) be the internal state of the integrator at time sample k (assuming a forward-euler

66 51 integrator implementation where the state equals the output). When turning off the H 2 part of the controller at time sample k, set a(k) = a(k + 1) = a(k + 2) = 0, and ξ(k) = ξ(k) + a(k 1)/K I, where ξ(k) is the value the integrator would normally have under standard operation without this addition, and K I is the integral gain shown in Fig This results in no step change in the total control command a + b because a(k) + b(k) = b(k) = b(k) + a(k 1), where b(k) is the value b(k) would normally have without the addition to the integrator state. When turning on the H 2 part of the controller at time k, the H 2 control states should be initialized based on their steady-state values for the current wind speed, and the initial H 2 control output value, a(k), should be scaled and subtracted from the internal state of the integrator at time k: ξ(k) = ξ(k) a(k)/k I. Again, by following these steps, there should be no step change in the total control command a + b Simulation Results Fig shows a summary of simulation results for simulation conditions described in Section 3.4.1, relative to the baseline PI control. All controllers shown are an overall improvement compared to the baseline PI control, with the exception of the H 2 FF/FB, hipp controller increasing generator speed error. The medium-pitch-penalty H 2 feedback-only controller (H 2 FB, medpp) results in only a small performance improvement over PI FB, but for controller with feedforward included, the performance improvements are substantial. For example, the medium-pitch-penalty H 2 combined feedforward/feedback controller (H 2 FF/FB, medpp) results in reductions of 51.3% in RMS generator speed error, 39.0% in pitch rate standard deviation, 7.5% in blade root out-ofplane damage equivalent load, 16.9% in tower base fore-aft damage equivalent load, and 4.3% in shaft torque damage equivalent load, compared to the baseline PI feedback only case. In addition, the tradeoff between pitch rate and generator speed error can be easily controlled by tuning the pitch rate penalty α pitchrate, as shown by the differences between the lowpp, medpp, and hipp results. Blade root and tower base loads tend to decrease with pitch rate standard deviation, while shaft torque loads tend to decrease with generator speed error. Damage equivalent loads were calculated using rainflow counting, assuming an S-N-curve

67 52 slope m = 10 for blade material and m = 3 for tower and shaft materials [2]. When m = 10, a 7% decrease in DEL corresponds to a doubling (1/ = 2.1) in lifetime, and when m = 3, a 21% decrease in DEL corresponds to a doubling (1/ = 2.0) in lifetime. Fig shows the same simulation results as Fig but now shown relative to the LPF FF* control. The H 2 FF/FB controller can be tuned such that performance is improved in all five metrics compared to LPF FF*. For example H 2 FF/FB, medpp results in reductions of 0.9% in RMS generator speed error, 27.3% in pitch rate standard deviation, 1.2% in blade root out-ofplane damage equivalent load, 4.7% in tower base fore-aft damage equivalent load, and 0.9% in shaft torque damage equivalent load, compared to the LPF FF* case. Fig also shows the same simulation results as Fig but now shown relative to the H 2 FB, medpp control. The H 2 FF/FB, medpp results in reductions of 50.6% in RMS generator speed error, 31.4% in pitch rate standard deviation, 5.1% in blade root out-of-plane damage equivalent load, 16.1% in tower base fore-aft damage equivalent load, and 3.2% in shaft torque damage equivalent load, compared to the H 2 FB, medpp case Comparison to Expected Linear-Model-Based Results Fig shows power spectral densities of generator speed error and pitch rate, both asexpected based on the linear model used for control design and as-recorded from simulation. General trends in the linear case based on reduced-order linear modeling carry over to the simulation case based on high-order nonlinear modeling. Linear-model-based results give helpful predictions of expected percent reductions in RMS generator speed error and RMS pitch rate that should be observed in simulation. The square root of the area under each curve in Fig is equal to the standard deviation of that signal, which also equals the RMS value when the mean is zero. Linear-model-based predictions of RMS generator speed error, along with the corresponding simulation results of RMS generator speed error, are shown in Fig. 3.19, and linear-model-based predictions of RMS pitch rate, along with the corresponding simulation results of RMS pitch rate, are shown in Fig The results are

68 53 normalized to PI FBonly RMS(gen speed error) std(pitch rate) mean blade root OOP DEL tower base FA DEL shaft torque DEL LPF FF* H FF/FB, H FF/FB, H FF/FB, lowpp medpp hipp H 2 FB, medpp Figure 3.15: Performance metrics (OOP: out-of-plane, DEL: damage equivalent load, FA: fore-aft), normalized so all baseline PI feedback bars would equal 1 if displayed, for five controllers listed in Table RMS(gen speed error) std(pitch rate) mean blade root OOP DEL tower base FA DEL shaft torque DEL normalized to LPF FF* H 2 FF/FB, H 2 FF/FB, H 2 FF/FB, lowpp medpp hipp H 2 FB, medpp PI FB Figure 3.16: Performance metrics (OOP: out-of-plane, DEL: damage equivalent load, FA: fore-aft), normalized so all LPF FF* bars would equal 1 if displayed, for five controllers listed in Table 3.2.

69 54 normalized to H 2 FB, medpp RMS(gen speed error) std(pitch rate) mean blade root OOP DEL tower base FA DEL shaft torque DEL H 2 FF/FB, H 2 FF/FB, H 2 FF/FB, lowpp medpp hipp PI FB LPF FF* Figure 3.17: Performance metrics (OOP: out-of-plane, DEL: damage equivalent load, FA: fore-aft), normalized so all H 2 FB, medpp bars would equal 1 if displayed, for five controllers listed in Table 3.2. Power Spectral Density (Power/Hz) To: gen speed error (rpm) To: pitch rate (deg/s) Linear Model Simulation PI FB LPF FF* H 2 FB, medpp H 2 FF/FB, medpp Frequency (Hz) Frequency (Hz) Figure 3.18: Linear-model expectations and simulation results of generator speed error and pitch rate power spectral densities, for four controllers listed in Table 3.2.

70 Simulation RMS Generator Speed Error, in % of Baseline PI 140 LPF FF* 120 H FB, medpp H 2 FF/FB, medpp H 2 FF/FB, lowpp H 2 FF/FB, hipp Linear Expectation Figure 3.19: Linear-model expectations vs. simulation results for RMS generator speed error, as percentages of baseline PI feedback-only (PI FB), for controllers listed in Table 3.2. Best-fit line equation: y = 0.93x well-correlated, all falling along roughly the same line. The trends of Figs and 3.20 can be used to map linear-model-based expectations to a better prediction of actual simulation results. For example, if the linear model predicts an H 2 feedforward/feedback controller to have 40% of the RMS pitch rate of the baseline PI feedback controller, we can use the line fit through the points in Fig. 3.20, and determine that the simulation result is likely to show this H 2 feedforward/feedback controller to actually have 50% of the RMS pitch rate of the baseline PI feedback controller ( = 50). 3.5 Linear-Model-Based Performance Predictions Linear-model-based performance predictions are computationally-inexpensive compared to simulation, allowing quick study over a wide range of design parameters. The predictions in this section should be referenced to Section to determine how these predictions generally translate to actual simulation results, and the results should be considered most trustworthy when the preview time and coherence bandwidth are close to the simulation conditions of Hz coherence bandwidth and seconds of preview. Limited simulations with varied coherence bandwidths

71 RMS Pitch Rate, in % of Baseline PI 100 Simulation LPF FF* H 2 FB, medpp H 2 FF/FB, medpp H 2 FF/FB, lowpp H 2 FF/FB, hipp Linear Expectation Figure 3.20: Linear-model expectations vs. simulation results for RMS pitch rate, as percentages of baseline PI feedback-only (PI FB), for controllers listed in Table 3.2. Best-fit line equation: y = 0.89x

72 57 and preview times have shown similar trends to these linear-model-based predictions. Fig shows linear-model-based expectations of RMS generator speed error, and Fig shows linear-model-based expectations of RMS pitch rate, for a range of coherence bandwidths and preview times. LPF FF control and H 2 control have relatively similar generator speed error predictions, but H 2 control has much lower pitch rate predictions. LPF FF results improve with increasing coherence bandwidth but are unaffected by increasing preview time. Fig and Fig quantify the H 2 control improvements possible as preview time increases from 0 to 10 seconds, and as coherence bandwidth increases from 0 to infinity, where 0 represents no feedforward and infinity represents perfect measurements. Both increasing coherence bandwidth and increasing preview time reach a point of diminishing returns. The maximum useful preview time is the preview time above which RMS generator speed error and pitch rate remain flat. The maximum useful preview time increases as coherence bandwidth decreases because at lower coherence bandwidths, more lowpass filtering is required, and this introduces more delay, which can be compensated with preview time [52]. However, even with infinite coherence bandwidth, which does not require lowpass filtering to minimize measurement error, up to 3 seconds of preview time is useful because of our pitch rate and H 2 pitch acceleration penalties. The zero coherence bandwidth result for the LPF FF controller is the same as PI FB. The zero coherence bandwidth result for the H 2 controller shows only a small improvement in generator speed error compared to PI FB, but pitch rate is significantly improved compared to PI FB. For coherence bandwidths other than zero and infinity, there are no results below some minimum preview time. This is because our design process involves subtracting the delay (t L (f low )) of the lowpass filter (L) from the delay block as explained in Section Before this subtraction, the amount of delay in the delay block is equal to the amount of preview time. If t L (f low ) is greater than the amount of preview time, the subtraction results in a negative delay, which is not implementable. The overall cost must theoretically be monotonically decreasing with increasing preview time and with increasing coherence bandwidth, because there is only increased information. However,

73 58 the generator speed error and pitch rate, being individual components of the cost function, can and do individually violate this monotonicity, as long as an increased cost in one of these output components is balanced by a corresponding decreased cost in another, including β H2, which is not displayed here. 3.6 Conclusions and Future Work H 2 optimal control design, within the limits of our modeling assumptions and cost functions, allows a comparison of the best performance possible with lidar versus the best performance possible without lidar, while using a realistic model of wind measurements in the design process by including a known coherence model and preview time. In simulation, H 2 optimal combined feedforward/feedback control significantly reduces structural loads, pitch rate, and generator speed error compared to three other types of control: baseline PI feedback only, H 2 optimal feedback only, and lowpass filter feedforward control. For example, the medium-pitch-penalty H 2 combined feedforward/feedback controller, on the NREL 5-MW turbine in 18 m/s wind with 17% turbulence intensity, results in reductions of 51.3% in RMS generator speed error, 39.0% in pitch rate standard deviation, 7.5% in blade root out-of-plane damage equivalent load, 16.9% in tower base fore-aft damage equivalent load, and 4.3% in shaft torque damage equivalent load, compared to the baseline PI feedback only case; and reductions of 0.9% in RMS generator speed error, 27.3% in pitch rate standard deviation, 1.2% in blade root out-of-plane damage equivalent load, 4.7% in tower base fore-aft damage equivalent load, and 0.9% in shaft torque damage equivalent load, compared to the LPF FF* case. In addition, tradeoffs between outputs can be easily tuned through the choice of weights. Linear-model-based results are also presented alongside simulation results for comparison and validation, and additional linear-model-based results are presented to show how the benefit of lidar is expected to vary across a range of coherence bandwidths and preview times. Suggested future work includes improving gain scheduling by solving for linear H 2 controllers at multiple operating points and transitioning between them, field testing, and using the same control design techniques to study the benefit of lidar for individual pitch control. This chapter

74 59 18 m/s Linear Model Prediction: RMS Generator Speed Error normalized to PI FBonly Coherence Bandwidth (Hz) Inf Preview Time (s) Figure 3.21: 18 m/s linear-model expectations of RMS generator speed error, normalized so that the expected RMS generator speed error of baseline PI feedback-only equals 1, for a range of preview times and coherence bandwidths, for lowpass filter feedforward (LPF FF) control in dashed lines and H 2 feedforward/feedback control with medium pitch penalty (H 2 FF/FB, medpp) in solid lines. 18 m/s Linear Model Prediction: RMS Pitch Rate normalized to PI FBonly Coherence Bandwidth (Hz) Inf Preview Time (s) Figure 3.22: 18 m/s linear-model expectations of RMS pitch rate, normalized so that the expected RMS pitch rate of baseline PI feedback-only equals 1, for a range of preview times and coherence bandwidths, for lowpass filter feedforward (LPF FF) control in dashed lines and H 2 feedforward/feedback control with medium pitch penalty (H 2 FF/FB, medpp) in solid lines.

75 60 has discussed collective pitch control, where each blade receives the same pitch command based on rotor-averaged wind speed measurements and generator speed error. Lidar-based individual pitch control could be implemented using additional cyclic pitch commands, allowing the turbine to react to previews of changing wind shear or yaw error. Chapter 5 discusses individual pitch control, but does not use this H 2 optimal design strategy.

76 Chapter 4 Preview Time Analysis 4.1 How Preview Time is Used The maximum preview time that is useful for feedforward control depends on the particular choice of cost function or output to be minimized. For the simplest case where minimizing only generator speed error is of interest, the maximum useful preview time can be broken down into a sum of three values. The first is determined by the actuator. The second is determined by the lowpass filtering which is necessary because the measured wind does not exactly match the actual wind experienced by the turbine, especially at high frequencies. The third is determined by the turbine itself. Figure 4.1 shows estimates for these three values for the NREL 5-MW turbine, along with the approximate expected preview time available from the lidar, assuming a fixed focus distance 80 m upwind, a 0.5 s delay due to the scan pattern, a coherence bandwidth fixed at 0.06 rad/m in spatial frequency regardless of wind speed, and using Taylor s frozen turbulence hypothesis [62, 63], which assumes that the frozen wind field is marched toward to turbine at a constant average wind speed. The preview time estimates shown for the actuator, lowpass filter, and turbine are explained in the following subsections Actuator Our actuator is modeled by a second-order lowpass filter with a natural frequency of 1 Hz and a damping ratio of 0.7. Figures 4.2 and 4.3 show how phase and time delay vary with frequency for this model. At frequencies below 0.2 Hz, the time delay remains at about seconds. Therefore

77 62 Required Preview Time (s) For: Available from Lidar Turbine + Actuator + LPF Turbine + Actuator Turbine Wind Speed (m/s) Figure 4.1: Cumulative required preview times to compensate the delay introduced by the turbine, actuator, and lowpass filter. Available preview time from lidar also shown, with 0.5 s scan pattern delay included. Assumptions include a fixed 80 m upwind focus distance, fixed 0.06 rad/m coherence bandwidth, Taylors frozen turbulence hypothesis, and generator speed error minimization only.

78 0 63 phase φ (deg) frequency f (Hz) Figure 4.2: Phase φ of second-order actuator model (natural frequency of 1 Hz and damping ratio of 0.7) seconds of preview time is required to compensate for this actuator delay Lowpass Filtering At high frequencies, lidar-based wind preview measurements are not an accurate representation of the rotor-effective wind speed experienced by the turbine. Therefore some lowpass filtering must be included in any feedforward controller that relies on these measurements. The ideal prefilter is S am (f)/s mm (f), where S am (f) is the cross power spectral density between the measured wind w m and the actual wind w a experienced by the turbine, and S mm (f) is the power spectral density of w m. This ideal prefilter has been referred to both as the transfer function estimate from w m to w a [32] and as the minimum mean squared error Wiener smoothing filter for estimating w a given w m [47]. The magnitude of this ideal prefilter is equal to the square root of the magnitude squared coherence, normalized by the ratio between the power spectra of w a and w m : S am (f)/s mm (f) = γ 2 am S aa S mm. If S aa = S mm, the magnitude squared of the prefilter is simply equal to the measurement coherence. Therefore the coherence bandwidth is a rough estimate of the cutoff frequency of the ideal prefilter. The more that is filtered out by this lowpass prefilter, the greater the delay it introduces, and therefore the greater the maximum useful preview time: Doubling the order of the lowpass filter

79 time delay τ φ (s) frequency f (Hz) Figure 4.3: Time delay τ φ of second-order actuator model (natural frequency of 1 Hz damping ratio of 0.7). τ φ = φ/360 /f

80 65 (LPF) doubles the amount of useful preview time corresponding to this filter because it doubles the phase delay. Doubling the cutoff frequency (the location of the LPF poles) cuts this useful preview time in half because the majority of the phase delay shifts to higher frequencies. For this example we assume a measurement coherence that is best fit by a first-order LPF with a cutoff of 0.06 rad/m in spatial frequency, and we assume that this spatial frequency will remain approximately constant as the wind speed increases. Multiplication with wind speed gives the cutoff frequency in Hz, and because that cutoff frequency increases with wind speed, the delay introduced by the filter decreases with wind speed. The resulting delay is plotted in Figure 4.1 as the LPF component Turbine The third value affecting maximum useful preview time appears because for some output y, there is often more phase delay in the Blade-Pitch-to-y transfer function than there is in the Wind-Speed-to-y transfer function of the wind turbine. This difference in phase delay is equal to the phase advance of the ideal model-inverse feedforward controller F = T 1 yβ T yw, where T yβ is the closed-loop (feedback included) transfer function from blade pitch to output, and T yw is the closed-loop transfer function from wind speed to output, as in Fig For ease of calculations in this section, we let y be the generator speed output Ω, and we use collective pitch feedforward control. Then the equation above simplifies to F = P 1 yβ P yw, where P is the open-loop plant, and the feedback controllers do not matter. This is because the torque and collective pitch feedback controllers C both take their input from generator speed, and since the feedforward is designed to reduce generator speed error to zero, these torque and collective pitch feedback controllers C drop out of the equation for F : T yβ = (1 P yβ C) 1 P yβ T yw = (1 P yβ C) 1 P yw when y = Ω when y = Ω T 1 yβ T yw = P 1 yβ P yw Any individual pitch feedback control also does not matter because it does not affect the average

81 66 Figure 4.4: Preview time required for F with all available degrees of freedom (DOFs) turned on in linearization. Above 0.5 Hz, time values begin to vary much more widely, but we do not intend to use frequencies any higher for feedforward control. blade pitch, which is what we are adding to with this collective pitch feedforward control. To find the preview time required to implement F in the above configuration, we first linearize the open loop turbine model at 7 different wind speeds from 12 m/s to 24 m/s and, at each wind speed, solve for F = P 1 yβ P yw. For each F, we then take the phase advance in degrees (φ) at various frequencies (f) in Hz and convert it to time advance ( τ φ ) in seconds, according to τ φ = φ/360 /f The results are shown in Figures 4.4 and 4.5. Figure 4.4 shows the results when all available degrees of freedom are turned on in linearization. The preview time required varies with wind speed, but stays fairly constant over the frequencies of interest. Figure 4.5 shows results when only 5 degrees of freedom (DOFs) are turned on in linearization: first flapwise blade mode ( 3 blades), generator, and drivetrain rotational-flexibility. Because Figures 4.4 and 4.5 are very similar, we can conclude that the required preview time depends mainly on some or all of only these 5 DOFs. A third set of linearizations was also created with only two DOFs: generator and drivetrain rotational-flexibility. The resulting required preview time was almost exactly zero at all frequencies and at all wind speeds! This shows that the first

82 67 Figure 4.5: Preview time required for F with 5 DOFs turned on in linearization: first flapwise blade mode ( 3 blades), generator, and drivetrain rotational-flexibility. Above 0.5 Hz, time values begin to vary much more widely, but we do not intend to use frequencies any higher for feedforward control. flapwise blade mode is the main reason why the phase delay from Blade-Pitch-to-Generator-Speed is different than the phase delay from Wind-Speed-to-Generator-Speed. In other words, the amount of preview time required by the turbine depends mainly on blade flexibility. It is also interesting to note that when blade flap DOF is turned on, a nonminimum phase zero appears in the Blade-Pitch-to-Generator-Speed transfer function, but not in the Wind-Speedto-Generator-Speed transfer function. It is unlikely to be a coincidence that this nonminimum phase zero and the difference in phase delay appear only under the same circumstances. Preview time can also be used for model-inverse-based control when the plant has non-minimum-phase (unstable) zeros. Non-minimum-phase zeros often appear in plants with non-collocated sensors and actuators, and they appear in the linearized 5-MW model when blade flexibility degrees of freedom are turned on. In this case, the ideal model-inverse feedforward controller is unstable, but it can be approximated by a stable, non-causal controller. Various approximation methods exist, including the non-causal series expansion [64, 65]. Unlike model-inverse-based control, H 2 control synthesis does not address non-mimimum-phase zeros directly, but their presence in the plant does affect the resulting H 2 optimal controller and the performance it can achieve. A smaller amount of preview time than a plotted required value is still useful because it

83 allows a better approximation to the ideal F than would be possible without any preview time. In addition, required turbine preview time may increase greatly from the less than 0.3 sec shown 68 here if we focus on other outputs besides generator speed. A controller that is designed only to regulate generator speed is not ideal for reducing loads, although it should help reduce them somewhat. Previous work done using preview control [35] suggests a turbine required preview time of 2 seconds when only penalizing pitch rate and generator speed error, and 20 seconds when tower sway is also penalized. However, other work [66] shows that the preview time required to minimize loads depends on wind speed and is at most 1 second. Future work should further investigate the required turbine preview time when designing for minimizing pitch actuation and blade root and tower loads as well as generator speed regulation.

84 Available Preview Time So far in this thesis we have assumed a fixed, known amount of preview time, according to Taylor s frozen turbulence hypothesis [62, 63], and used the FAST simulation environment which does not model wind evolution, but instead models a frozen 3D wind field marching toward the turbine at a constant average wind speed. However, in real-world operation, the wind field evolves, and, given a fixed lidar focus distance, the preview time is always changing and not precisely known. Fig. 4.6 shows an example block diagram of combined feedforward/feedback wind turbine control using a lidar, now with additional blocks included to analyze and manage the real-world variation in available preview time. We use v u (t) to refer to the upstream estimate of the approaching rotor-effective wind speed. Throughout this section, the rotor-effective wind speed is estimated based on line-of-sight lidar measurements distributed over a vertical plane upstream of and parallel to the turbine rotor plane, by using the running average of the past full circle of measurements corrected for line-of-sight angle as in [10], while assuming that the turbine is perfectly aligned with the wind. We use v r (t) to refer to the wind experienced at the turbine rotor, as determined by a wind speed estimator, which essentially uses the turbine as an anemometer. The relationship between v u (t) and v r (t) can be characterized by their coherence (correlation as a function of frequency) and the time delay between them. This section focuses on the time delay between the two signals using data from field tests on a 600-kW wind turbine known as the two-bladed Controls Advanced Research Turbine (CART2) [67], a research wind turbine that is used to test advanced control algorithms at NREL. Arrival time t a (t) is the time it takes for the wind to travel from the measurement location to the turbine rotor. Preview time t p (t) and time delay t d (t) are closely related to t a (t), as shown visually in Fig Given one of these three timing signals, the others can be easily determined. Compared to the wind at the measurement location, the upstream v u (t) is delayed by one-half the time it takes to complete one scan pattern because v u (t) always depends on the most recent complete scan, which includes measurements taken one-half scan pattern ago on average. The most

85 70 LPF(v u (t)) Low-pass Filter v u (t) Variable Time Delay Estimate Rotor- Effective Wind Speed over Scan Pattern Blade Pitch Combined and/or Generator Feedforward/ Torque Commands Feedback Controller Wind at Measurement Location (Ahead of Turbine) Lidar Delay and Evolution Wind Turbine Wind Speed Estimator v r (t) Wind at Rotor Outputs Figure 4.6: Block diagram of wind turbine control using a turbine-mounted lidar under real-world conditions with varying preview time. Physical devices and processes are shown in blue, and control system components are shown in green. It is likely that a different low-pass filter (sometimes called a prefilter) is also contained in the feedforward part of the Combined Feedforward/Feedback Controller.

86 Table 4.1: Properties of the NREL CART2 turbine. 71 Rated Power (kw) 600 Rotor Diameter (m) 42.7 Number of Blades 2 recent complete scan is used to provide a representation of the wind across the full rotor disk. There are mixed conclusions in the available literature on the importance of accurate timing predictions. One study involving a collective pitch feedforward controller [57] shows (in its Figure 9) that errors of more than 5 s are needed before the combined feedforward/feedback controller performance is worse than that of feedback alone. Another study [42] states that lead or lag errors in the wind speed measurement, which is fed to the controller, severely reduce the performance of the controller. The second study uses peak time of cross-covariance between v u (t) and v r (t) to predict timing, but it does not use real-world data for these signals. In this section, we analyze timing by using real-world data. We show how quickly timing can vary, and we show how helpful it would be to correctly predict it if that were possible, rather than assuming it stays constant for a constant average wind speed. In Section 4.2.1, we describe the lidar measurements and how we obtain v u (t) and low-pass filter it into LPF(v u (t)). We also describe the wind turbine and the wind speed estimator used to obtain v r (t). In Section 4.2.2, we describe taking windowed cross-covariances between LPF(v u (t)) and v r (t) to find t d (t), the time delay between them. In Section 4.2.3, we pass the signal v u (t) through the varying time delay t d (t), and show that this provides an improved preview measurement for use in a time-invariant controller. Finally, in Section 4.2.4, we summarize conclusions Lidar Measurements, Wind Turbine, and Wind Speed Estimator Lidar measurements were obtained from CART2 field tests [30] at NREL. Table 4.1 contains the CART2 turbine parameters. We use the four data sets shown in Table 4.2, downsampled to a rate of 40 Hz. These are

87 72 Arrival time t a (t) Preview time t p (t) Wind at measurement location v u (t) Time delay t d (t) LPF(v u (t)) Wind at rotor v r (t) d s, Delay due to scan time d LPF, Delay due to low-pass filtering of v u (t) d r, Delay due to rotor effective wind speed estimation Figure 4.7: Visual representation of possible introduced time delays and the naming conventions used in Section 4.2. Delays in blue are constant in time, and red indicates functions of time that vary with the evolving wind field. In Section 4.2, d LP F = d r = 0, and d s = 0.67s. We use the word timing to refer in general to all three signals t a (t), t p (t), and t d (t).

88 Table 4.2: Data sets from 2012 CART2 field tests 73 Set Date Start End Length Mean Time Time (mm:ss) Wind :43:09 22:18:09 35: m/s :39:29 16:35:42 56: m/s :41:57 15:11:57 30: m/s :29:22 16:10:37 41: m/s the four longest continuous data sets available from the 2012 CART2 field tests, excluding data taken before torque calibration, before hard target problems were solved, when lidar measurements were not available, and when the turbine was not operating in a region suitable for the wind speed estimator to produce accurate results. Set 1 had several periods of up to 4 s long of missing lidar measurements; we linearly interpolated to replace missing data. This did not cause any noticeable problems, and although we often plot data from Set 1 alone, we also provide a summary of results for each of the four data sets. The lidar used in the field tests is a modified Windcube [68], a pulsed lidar that scans a circular pattern at five focus distances, as shown in Fig Although the lidar can achieve other scan patterns, the circular pattern was chosen because it was expected to provide measurements well correlated to the wind experienced by the turbine. We chose to use data from the first focus distance of one rotor diameter (42.7 m) upwind of the lidar, because these measurements were best correlated to the turbine. The lidar was mounted on the wind turbine nacelle at a distance of 1.66 m downwind of the hub (rotor center). Thus the measurement location is 42.7 m 1.66 m = 41.0 m ahead of the rotor. Each circle of six measurement points was scanned in 1.33 s, yielding an average scan time delay (Fig. 4.7) of d s = 1.33/2 s = 0.67 s. The lidar measures the wind component along its line of sight. Each measurement is adjusted using the lidar line-of-sight angle to find the wind speed in the x-direction (directly downwind toward the turbine), under the assumption that the y and z components of the wind are zero. A running average over the past six adjusted measurements, from the first measurement plane only, was provided as an estimate of the x-component of the rotor-

89 z (m) x (m) y (m) Figure 4.8: Lidar measurement configuration. The lidar is shown as a white square on the nacelle of the turbine. The black dots represent the locations of wind speed measurements used in Section 4.2. The white dots represent locations where wind speed measurements were also recorded. average wind speed. This is our v u (t), which is shown in Fig Fig. 4.9 also contains the low-pass filtered version of v u (t), LPF(v u (t)). This includes low-pass filtering with a cutoff frequency of Hz and notches at once-per-revolution (1P), twice-perrevolution (2P), and the drivetrain torsion frequency (0.695 Hz, 1.39 Hz, and 3.36 Hz, respectively). These particular low-pass and notch filters do not introduce any delays because zero-phase filtering (filtfilt() in MATLAB [60]) is used, which is possible because the filtering is not done in real time. Essentially, the filtering is done both forward in time and backward in time so that any effects of the filter on phase are canceled out. Together these filters minimize the magnitude of v u (t) at frequencies at which it may be correlated to v r (t) because of tower and lidar motion rather than independent measurements of wind speed. The wind speed estimator uses the torque balance method, which assumes τ r τ g = J Ω (4.1) where τ r is the torque applied by the wind to the rotor, τ g is the torque applied at the generator (multiplied by the gearbox ratio), J is the drivetrain moment of inertia (as observed from the rotor side of the gearbox), and Ω is the rotor acceleration. Wind speed estimators that neglect rotor acceleration may introduce a time delay because of the inertial response of the rotor. After τ r is solved for using the above equation, a lookup table is used to determine the rotor-

90 v u (t) LPF(v u (t)) v r (t) Wind speed (m/s) Time (s) Figure 4.9: Upstream measurement (v u (t)), low-pass filtered v u (t), and rotor estimate (v r (t)). Five-minute portion from Set 1 data.

91 effective wind speed for the given τ r, blade pitch angle β, and rotor speed Ω, with an adjustment for air density ρ. These variables are related as follows: 76 τ r = ρπr2 C P (λ, β)v 3 r(t) 2Ω (4.2) where R is the rotor radius, C P is the power coefficient (fraction of wind power captured), and λ is the tip speed ratio (ΩR/v r (t)). Pitch, torque, and rotor speed signals are low-pass filtered and notch filtered (at 1P, 2P, and the drivetrain torsion frequency) using zero-phase filtering before being used in the estimator. The result of the lookup table is the rotor-effective wind speed v r (t) shown in Fig Windowed Cross-Covariances The time delay t d (t) between LPF(v u (t)) and v r (t) can be estimated from the times of peak windowed cross-covariances between LPF(v u (t)) and v r (t). However, simply shifting a window across the data and recording each time of peak cross-covariance results in a t d (t) that contains sudden jumps to unrealistic values. Three main steps can reduce these occurrences. First, we choose a Gaussian window with a large enough window size. Second, we detrend (subtract the best-fit line from) the data within each window before applying the cross-covariance function. Third, we enforce a rate limit on t d (t), along with a lower limit of 0 s. These steps yield the blue t d (t) curves shown in Fig We show results for three window sizes for comparison, with standard deviations of σ = 30 s, 15 s, and 60 s, and total window sizes of 180 s, 90 s, and 360 s, respectively. Throughout the remainder of Section 4.2, we use only the t d (t) resulting from the σ = 30 s window. This choice of window size is a trade-off because smaller sizes give better time resolution and tend to lead to the greatest increases in coherence bandwidth (when t d (t) is used as described in Section 4.2.3); however smaller sizes are also more likely to produce non-physical results (which can be impossible to predict) and contain instances of near-zero preview times. A minimum of 1 s to 2 s of preview would be preferable if it were available in real time for control. Based on Taylor s frozen turbulence hypothesis [62, 63], t a (t) is typically assumed to be equal

92 Time of peak cross covariance (t d (t)) σ=30 s Time of peak cross covariance (t (t)) σ=15 s d Time of peak cross covariance (t d (t)) σ=60 s Expected t d (t) (T LPF(vu (t)) ) Expected t d (t) (T vr (t) ) Time (s) Window center time (s) Figure 4.10: Time (t d (t)) of peaks of windowed cross-covariances of LPF(v u (t)) with v r (t). σ describes Gaussian window size; t d (t) with σ = 30s is used throughout this chapter and can be assumed when not specified; plots using the other two sizes are presented here for comparison. This plot also includes two versions of T v as in (4.3). Set 1 data used.

93 78 to D/v, where D is the distance traveled from the measurement location to the rotor and v is the average wind speed. Fig. 4.10, in addition to showing t d (t), also shows the expected t d (t) computed as T v = D/v d s (4.3) where D = 41 m, d s = 0.67 s, T stands for Taylor, and the subscript v varies depending on what choice of v we use in the equation. We use either LPF(v u (t)) or v r (t) for v in Fig At approximately 2,000 s, there is a large spike in t d (t). This corresponds to a data set portion when the lidar measurements are relatively inaccurate because of low wind speeds. In this portion, the wind speed as determined by the turbine-based estimator is below 6 m/s; whereas the lower limit of a possible lidar measurement is 6 m/s as a result of the technique used to eliminate data due to accidental lidar sensing of hard targets. In the remaining data (before 1,900 s) in Fig. 4.10, t d (t) on average matches its expected value T v relatively well, as shown in Table 4.3. It is likely that induction zone effects (slowing of wind as it approaches the rotor) are responsible for t d (t) being longer than expected on average. Theoretically, the induction zone velocity U is characterized by U U = ( 1 a[1 + ξ(1 + ξ 2 ) 1/2 ]) 1 (4.4) where U is the undisturbed velocity and ξ = x/r, where x is the distance from the rotor (negative upwind) and R is the rotor radius [69]. The axial induction factor a is approximately the optimal value 1/3 in below-rated wind speeds because the turbine is extracting as much power as possible from the wind. This is when we expect the induction-zone slowdown effect to be strongest. As wind speed increases above rated, the axial induction factor decreases, and the slowdown effect diminishes. The CART2 has a rated wind speed of approximately 13 m/s, and we have little data above this speed. Integrating (4.4) with respect to ξ and then multiplying by U/U /ξ gives the arrival time multipliers M shown in Fig To calculate a theoretical arrival time t a (t) that accounts for the

94 Table 4.3: Mean values of t d (t) and T v : either uncorrected or corrected for induction zone 79 *Final 110 s excluded Set Mean Mean Mean t d (t) [s] T [s] LPF(vu(t)) T v r(t) [s] Uncorrected Corrected Uncorrected Corrected 1* induction zone, use t a (t) T aylor,corrected = (D/v)M (4.5) where D is the measurement distance and v is the wind speed at the measurement location. Our measurement distance results in x/r = 1.92, and we assume a = 1/3, corresponding to an M of This means t a (t) should be 12% greater than the value we calculated without accounting for the induction zone. Table 4.3 shows the uncorrected and corrected versions of T, where LPF(vu(t)) the corrected version is (D/v)M d s. It also shows the uncorrected and corrected versions of T vr(t), where the corrected version is (D/v)M d s, where M results from integrating (4.4) with respect to ξ and then multiplying by 1/ξ. The multiplication factor U/U is not included in M because v r (t) is assumed to be a delayed estimate of U instead of U. In both cases, the induction zone correction improves the match with t d (t). On shorter time scales, t d (t) often differs from its expected value T v by several seconds, as shown in Fig Timing does not stay constant for a constant average wind speed. This may be in part because of the evolution of the wind field, and also the 3D nature of the wind field in which the wind speed at a given location does not always match the speed of travel of the turbulent airflow structure carrying that wind speed. The timing may also be affected by wind shear and wind components in the y and z directions, which exist although we assume them to be zero. In this section, our goal is not to improve the modeling to predict timing, but to use the time lag found in post-processing to estimate how much room for improvement exists in the current model.

95 Arrival Time Multiplier M a=1/6 a=1/3 a=1/ x/r Figure 4.11: Arrival time multipliers M for use in (4.5). x is the measurement distance from the rotor (positive downwind), R is the rotor radius, and a is the axial induction factor Filtering Lidar Measurements Using a Variable Time Delay A controller incorporating lidar measurements, for example the controller used in the field testing [30], can be designed for some given preview time and some given correlation or transfer function estimate. The controller may be adapted as arrival time and correlation change, but so far this has been done on a timescale of a few minutes. In Fig t d (t) (σ = 30 s) jumps from 3 s to 6 s in only 68 s, for example. One way to address varying arrival time is to delay the measured signal v u (t) by varying amounts before sending it to the controller. This essentially uses interpolation to stretch and shrink various parts of v u (t) in time. The amount of varying delay that should be used on v u (t) is equal to the preview time t p (t) minus the constant amount of preview expected by the controller. For this method to be successful, two requirements must be met by the preview time t p (t). First, t p (t) must always be greater than or equal to the amount of preview required by the controller. This is required for real-time operation because it is impossible to use negative delay. Second, let represent taking the differences between adjacent elements in a time series, where t = 1/40 s because of the 40 Hz sample rate. Then the slope t p (t)/ t (and thus t a (t)/ t and t d (t)/ t as well) must always be greater than 1. If t p (t)/ t = 1, two different wind speeds would be expected to arrive at the same time, and below a slope of 1, wind speeds would be expected

96 81 to arrive in a swapped order than when they were measured. In situations when arrival time has a slope 1, a possible solution is to overwrite the older measurement with the more recently measured wind speed, which is arriving at the same time or sooner. With a small amount of zooming in, t d (t) in Fig appears to meet the requirement of t d (t)/ t > 1. However, upon zooming in enough to see individual time samples, we see that the time of peak cross-covariance is discretized at the sample rate (40 Hz). This results in steep steps, many with slope equal to 1. To solve this problem, we filter this data with a boxcar filter of 11 samples in length, which, for this data, provides just enough smoothing to keep t d (t)/ t > 1. The boxcar filter is centered so that no time delay is introduced. After smoothing the data, Fig was created by stretching and shrinking LPF(v u (t)) from Fig. 4.9 according to t d (t) from Fig Because of this, it sometimes lags and sometimes leads the original LPF(v u (t)), and this results in a better correlation at low frequencies to v r (t). We are most interested in improving low-frequency correlation because the low frequencies contain the most power in the wind and have the most effect on the turbine. Fig shows coherence (correlation as a function of frequency) using three methods of variable time delay. T LP F0.003 (v u(t)) means that the wind speed v used in (4.3) is the lidar measurement filtered with a first-order low-pass filter with a cutoff of Hz. This is most similar to the method used in the field tests: the time delay is updated slowly, on the order of minutes, based on a filtered version of the measured wind speed. Using T vu(t), the time delay is updated instantaneously, with no filter on the measured wind speed. Surprisingly, this improves coherence compared to T LP F0.003 (v u(t)), even though by using a time delay with such high frequency variations, we are subjecting the signal to a swapped order of data points due to slopes being below 1 as previously described. Fig shows that there is a general trend toward improved coherence bandwidth as the low-pass filtering cutoff frequency is increased. However, the minimum available preview time also decreases as the low-pass filtering cutoff frequency is increased, therefore low-pass filtering should not be completely neglected; instead some filtering should be employed to ensure availability of a minimum preview time for control. Using t d (t) for the time delay gives

97 LPF(v u (t)) LPF(v u (t)) w/ variable time delay v r (t) 14 Wind speed (m/s) Time (s) Figure 4.12: Low-pass filtered upstream measurement LPF(v u (t)), LPF(v u (t)) stretched and shrunk by filtering using the variable time delay t d (t) plotted in Fig. 4.10, and rotor estimate (v r (t)). Fiveminute portion from Set 1 data.

98 Magnitude mscohere( v r (t), v u (t) w/ variable time delay according to: ) T LPF0.003 (v u (t)) T vu (t) t d (t) Frequency (Hz) Figure 4.13: Magnitude squared coherence between turbine estimate v r (t) and lidar measurement v u (t) with each of the three different methods of variable time delay listed in the legend. Set 1 data. the best coherence bandwidth. Although creating the signal t d (t) in real time is not possible, this method estimates an upper limit of coherence bandwidth improvement that is possible to achieve through improved knowledge of arrival time. Table 4.4 shows the results from these three methods on the four data sets in terms of coherence bandwidth. We define coherence bandwidth as the pole location of the first-order low-pass filter whose magnitude squared best fits the magnitude squared coherence. On average across all four data sets, our results show a 26% increase in coherence bandwidth when going from using T LP F0.003 (v u(t)) to t d (t), and a 13% increase in coherence bandwidth when going from using T vu(t) to t d (t). These increases in coherence bandwidth, for example, would allow a pitch controller to provide an improvement in its combined goal of generator speed regulation and minimal pitch actuation as quantified in Fig and Fig Conclusions Using data from field tests on NREL s CART2 wind turbine, we have shown how the arrival time of the lidar measurements varies, we have filtered the measured wind speed signal using a variable time delay, and we have found an upper limit on the improvement that can be obtained

99 coherence bandwidth (Hz) Set 1 data 0.03 Set 2 data Set 3 data Set 4 data LPF cutoff frequency (Hz) Figure 4.14: Coherence bandwidth between turbine estimate v r (t) and lidar measurement v u (t) with variable time delay using T LP Fx-axis(v u(t)) (the wind speed v used in (4.3) is the lidar measurement filtered with a first-order low-pass filter with cutoff frequency shown on the x-axis). The dashed lines represent no filtering (infinite cutoff frequency). Table 4.4: Coherence bandwidths [Hz] Set % % T LP F0.003 (v u(t)) T vu(t) t d (t) Mean Coherence bandwidths [Hz] between turbine estimate v r (t) and lidar measurement v u (t) with three methods of variable time delay.

100 85 through better prediction of arrival time. The data show that we can improve the prediction of average arrival time by using an induction zone correction when using Taylor s frozen turbulence hypothesis. This allows a good prediction of average arrival time, but arrival time can temporarily deviate significantly above or below this average value. A method for accurate real-time prediction of these quick variations in arrival time is outside the scope of this work. Instead, we used post-processing to obtain these variations in arrival time from CART2 field test data. Knowing these variations in advance, compared to a method of predicting arrival time similar to that used in the field tests, would have increased coherence bandwidth between measured and rotor-estimated wind by 26% on average, and therefore could have improved control performance as quantified in Chapter 3. This work sets an upper limit of possible performance improvement because real-time prediction at best will be no more accurate than what can be achieved in post-processing.

101 Chapter 5 Comparison of Two Lidar-Based Independent Pitch Control Designs 5.1 Introduction This chapter discusses two different methods [35, 66] for implementing lidar-based individual pitch control that were previously designed under separate studies. Both of the two control designs in this chapter are feedforward controllers that are intended to be added on to a standard feedback controller as shown in Figure 5.1. Many design features differ between the two methods including lidar scan patterns, and feedback and feedforward control strategies. The goal of this chapter is to determine which features are most important in achieving good load reduction and which choices should be made for each feature. In this chapter, Section 5.2 describes the turbine model, baseline controller, and simulation conditions. In Section 5.3, we describe and compare the two controllers. Section 5.4 explores details on individual pitch (IP) feedforward control. Finally, Section 5.5 outlines conclusions and future work. 5.2 Simulated Turbine and Turbulent Inflow MW Turbine Model and Baseline Control Simulations are performed using a full non-linear turbine model, the NREL 5-MW reference turbine [9], in the FAST [56] software code. All 16 available degrees of freedom (DOFs) are turned on in simulations. For all simulations, a 2nd-order pitch actuator model has been added to the

102 Wind Speed Preview LIDAR Wind (ahead of turbine) 87 FEEDFORWARD Pitch Commands (3 blades) DELAY/ EVOLUTION TURBINE Outputs FEEDBACK Figure 5.1: Feedforward control added to feedback control 5-MW turbine model, with a natural frequency of 1 Hz and a damping ratio of 0.7 [57]. The standard collective-pitch feedback controller is a gain-scheduled PI control [9]. An individual pitch (IP) feedback-only controller was also designed [8], and we will use this as our baseline controller. In addition to generator speed, the controller inputs are the three out-of-plane blade root bending moments, and the rotor azimuth, which is used for the multi-blade coordinate [70] (MBC) (or d-q axis) transformation into horizontal and vertical blade root bending components due to horizontal and vertical wind shear. The horizontal and vertical components are each controlled with PI controllers. The collective component is controlled with the same PI feedback control as the standard collective-pitch controller. An inverse MBC transformation transforms the outputs of the three PI controllers into three individual blade pitch commands Stochastic Turbulent Wind Field Simulator The NREL TurbSim [71, 72] stochastic full-field inflow simulator is used to provide realistic wind fields for the turbine simulations. Most of the simulations described below are based on extensive observations taken in the high-plains environment of Southeast Colorado that now has a large operating wind farm. The Great Plains (GP-LLJ) spectral model available in TurbSim is used to simulate wind conditions present at this site. The boundary conditions for the TurbSim simulator are shown in Table 5.1. These values

103 Table 5.1: TurbSim Boundary Conditions for 90m Hub Height, 5-MW Turbine, Great Plains (Lamar, Colorado) Inflow Simulations. u hub hub-height mean wind speed, Ri T L vertical stability parameter, α D vertical power law shear exponent, u D mean friction velocity (shearing stress) over the rotor disk, coh struct coherent structures, n.a. jet instead of power law. Ensemble ID u hub (m/s) Ri T L α D u D (m/s) Coh Struct? Jet Height (m) AR N no jet - pwr law AR N no jet - pwr law AR N no jet - pwr law AR n.a Y 90 AR n.a Y are derived from the averages of subpopulations (e.g., AR1) of actual measured wind conditions associated with 13 m/s (above-rated) hub-height mean wind speeds. The (y-z) grid encompassing the turbine rotor disk contains points of three orthogonal wind components with a sample rate of 20/second and a total record length of 630 seconds. The first 99 seconds of data from each simulation are discarded before calculating any performance measures to allow initial transients to settle out. Thirty-one different realizations of each subpopulation were created, each 630 seconds long. In addition to the wind fields shown here, we also mention below the use of a 14 m/s average, class A wind field. This field is also 630 seconds long and created using TurbSim, but has a greater turbulence intensity (18%), and uses the IEC Kaimal Normal Turbulence Model (NTM) [59] spectral model with a Class A turbulence level. 5.3 Controller Comparison We compare two previously developed feedforward control methods, which we refer to as feedforward controller A [35] and feedforward controller B [66]. Each feedforward controller was designed for use in combination with its own individual pitch feedback controller. Both individual pitch feedback controllers are very similar, both following the description above in Section 5.2.1, but there are slight differences between the two, including differing gain-scheduling implementations. (See Appendix B for more on gain-scheduling implementations.) We will refer to these as feedback

104 Table 5.2: Method notation. x, y, and z can each be either A, B, or (none). 89 x y z Method xyz feedforward controller feedback controller lidar configuration controllers A [35] and B [66]. The two feedforward controllers were also each developed with their own lidar configuration, which we refer to as lidar configuration A [35, 73] and lidar configuration B [66]. To refer to different combinations of feedforward controller, feedback controller, and lidar configuration, we will use method xyz as shown in Table 5.2. Only a few of all possible combinations will be discussed. A summary of the differences between feedforward controllers is shown in Table 5.3, and a summary of the differences between lidar configurations is shown in Table 5.4. Feedforward controller A [35] uses a finite-impulse-response (FIR) design, with 5 seconds of preview. Its FIR filter coefficients were originally chosen heuristically. They were then optimized by using a genetic algorithm, trying thousands of variations, and converging on the set of coefficients with the best performance. Performance was based on fatigue load reduction (blade and tower damage equivalent loads and nacelle accelerations), RMS pitch rate, peak rotor speed, and average power achieved in a simulation of the nonlinear turbine, with all 16 DOFs, in above rated wind conditions, as described in [35]. The impulse response of feedforward controller A is shown in Figure 5.2. For input to feedforward controller A, three rotating hub-mounted continuous-wave lidar measurements [73] are taken 65 m ahead of the turbine, one measurement ahead of each blade, at about 75% span. We will refer to this wind measurement scheme as lidar configuration A. The three Table 5.3: Features of feedforward controllers A & B Feature Feedforward Controller A Feedforward Controller B Controller Design FIR + wind-to-pitch lookup LPF + wind-to-pitch lookup Individual Pitch Control? Yes (individual) Yes (cyclic)

105 90 Table 5.4: Features of lidar configurations A & B Feature Lidar Configuration A Lidar Configuration B Lidar Type CW Pulsed Lidar Sample Rate (Hz) 80 5 Measurement Locations 3 points, 1 ahead of 5 circles, 12 points each each blade at 75% span Measurement Distance (m) to 174 Preview Time Used (s) 5 varies Convert Measurements to Mean, No Yes Horizontal, and Vertical Shear? Assume Perfect Alignment w/ Wind? Yes Yes Figure 5.2: Impulse response of feedforward controller A

106 91 blades feedforward signals are controlled separately, each using its own wind speed measurement. Wind speed measurements are sampled at 80 Hz, matching the simulation rate and controller update rate for convenience. However, the higher frequencies of the wind speed measurements are not used, since the FIR filter acts like a lowpass filter, with a cutoff frequency of approximately 0.09 Hz. Each measurement is separately filtered by the FIR filter described above, which has a DC gain of 1. In series with each FIR filter is a lookup table from steady-state wind speed to blade pitch. Results of this design (method AAA) are shown in Figure 5.3. Results are also shown here for a collective pitch variation, where the three wind speed measurements are averaged together, and the same average is fed in to each blade s identical feedforward control channel. Feedforward controller B uses a pulsed lidar model, sweeping a circle in 2.4 seconds, with 12 points at each of five different distances, as shown in Figure 5.4. This trajectory has been implemented with a lidar system developed and installed on the nacelle of a 5-MW turbine [68]. In the simulation, effects such as obstruction of the laser beam by the blades, volume measurement, and mechanical constraints of the scanner from real experimental data were considered to obtain realistic measurements. For instance, the same loss of about 30% of points could be observed in the simulation and in the measurements due to obstruction by the moving blades. These lidar measurements are then reduced to three components: [ ] T d 0HV = v 0 δ H δ V, (5.1) where v 0 is the horizontal hub-height wind speed and δ H and δ V are the horizontal and vertical shear, respectively. These components are found by using a least squares method on the past 12 measurements (the past full circle). We will refer to this wind measurement scheme as lidar configuration B. Lidar configuration B is more realistic than lidar configuration A, and it also appears to have a higher bandwidth for providing accurate measurements: feedforward controller B, when receiving inputs from lidar configuration B, works best in terms of simulation performance when lowpass filtered with a cutoff frequency of 0.06 rad/m multiplied by wind speed in m/s. But when

107 92 Figure 5.3: Turbine loads using individual pitch baseline feedback control alone [8] (method A ), and with added feedforward controller A (method AAA), in individual pitch and collective pitch versions. Percents displayed are the average of the eight bars. Feedforward controller A reduces overall loads by 4.9% in the individual pitch version. The results are averages from FAST simulations across 155 wind files representing AR1 through AR vertical [m] horizontal [m] upwind [m] Figure 5.4: Lidar configuration B pulsed lidar scanning pattern. Figure courtesy of D. Schlipf.

108 93 receiving inputs from lidar configuration A, the spatial cutoff frequency is reduced to 0.04 rad/m. Lidar configuration B likely provides a more accurate wind speed preview because measurements are taken at five different radii instead of just one. This was originally done with pulsed lidar, with all 5 measurements taken at the same time. A CW lidar could produce equivalent measurements by refocusing between the five points, one after the other, if the CW lidar s sample rate was at least 5 times that of the pulsed lidar. Lidar configuration A detects the blade effective wind speed needed by feedforward controller A. On the contrary, lidar configuration B is trying to capture the rotor-effective wind characteristics required by feedforward controller B. The correlation of one single measurement ahead of each blade with each blade s effective wind speed should be greater than the correlation of those three point measurements with the effective wind characteristics of the whole rotor disk. Therefore feedforward controller A combined with lidar configuration A can compensate loads above the the once per revolution (1P) frequency, if a good correlation with the blade effective wind speed can be obtained above the 1P frequency. To capture comparable correlation with the rotor-effective wind characteristics for feedforward controller B, more measurement points are necessary. Lidar configuration B was designed to improve the correlation with the rotor-effective wind characteristics of wind speed and horizontal and vertical shears. Nonetheless, using lidar configuration A also shows a good correlation with the rotor-effective wind characteristics. Lidar configuration A has the advantage that it retains more information on how the wind speed varies with azimuth. Blade loads can be caused by both wind speed differences over azimuth and wind speed differences in the x-direction (up/downwind). For changes in the x-direction, we estimate that our bandwidth for good measurements is around 0.1 or 0.2 Hz (at 13 m/s). For changes over azimuth however, our bandwidth depends on the rate at which our lidar samples and spins: the number of lidar measurements per lidar revolution. This could easily translate to higher than 0.1 or 0.2 Hz as the blade sees it. For example, a low-level jet can cause a high wind speed at hub height and lower wind speeds at both the top and bottom of the rotor plane. As the blades spin through this, they see a 2P, or 0.4 Hz load. We will show below that when using

109 lidar configuration A, we do use 2P measurements to reduce these 2P loads. Lidar configuration B takes enough measurements to capture this low level jet, but the information is lost when these 94 measurements are simplified into the 1P d 0HV components [70] (average wind speed and vertical and horizontal linear shears, as in (5.1)). Using only the 1P d 0HV components of wind measurements as in lidar configuration B can be called cyclic feedforward. The advantage of cyclic feedforward is that it can be used with different lidar types and scan patterns, and is more independent of the preview time/scan distance. It should work for all wind speeds and can easily be adapted in real applications. To combine the advantages of configurations A and B, cyclic feedforward could be modified to reduce 2P loads in addition to 1P loads. This would involve transforming the ring of measurements into 2P components in addition to 1P components by additionally using a 2P MBC transformation. After capturing the d 0HV wind measurements, feedforward controller B first delays them so that the remaining preview time is just enough to compensate for the phase delay of the actuator, lowpass filter, and the turbine. The measurements are then sent to the feedforward controller, which is simply a lowpass filter followed by a set of static gains. The filter cutoff frequency is determined based on the correlation between the lidar measurements and the turbine effective wind speed. Then for v 0, there is a lookup table from steady-state wind speed to steady-state pitch to get u 0, the average feedforward signal. The horizontal and vertical wind shears, δ H and δ V, are each multiplied by a scalar that is optimized to yield the best u H and u V blade pitch control components that cancel the wind disturbance. The blade pitch components u H and u V are added to the feedback controller in the multi-blade coordinate (MBC) domain. This simpler design has the benefit of being easily tunable, which is very important when dealing with modeling uncertainty. Originally, simulation for method BBB was done with a stochastic full-field wind (23 23 grid, t = 0.25s) with a mean velocity of 16 m/s and a turbulence intensity of 18%. The results presented in Figure 5.5 show greatly reduced tower and blade loads. When originally studied separately, with different wind fields, different performance metrics, and slightly different baseline feedback controllers, method BBB vs method B appeared to have significantly greater load

110 95 reduction than method AAA vs method A. To specifically compare the two feedforward controllers, we simulated method AAA and method BAA in the same wind fields, using the same performance metrics. Results are shown in Figure 5.6 for a less turbulent 13 m/s wind field, and in Figure 5.7 for a more turbulent 14 m/s wind field. Both feedforward controllers perform similarly, with feedforward controller A showing slightly more load reduction in the 14 m/s wind, and feedforward controller B showing slightly more load reduction in the 13 m/s wind. The more turbulent 14 m/s wind allows for more load reduction, with both controllers averaging about 9% overall, versus 5% in the 13 m/s wind. Blade root and tower base load reductions were the original performance metrics of feedforward controller B, and looked at individually, these two measures consistently have greater load reduction than the average of all 8 bars. Because performance for feedforward controller A was originally measured using this 8-bar average, this accounts for some of the originally perceived differences in the performance of the two controllers. Both controllers improve rotor speed regulation compared to baseline as shown by the lower peak rotor speeds, and, in the more turbulent 14 m/s wind, they also very slightly improve power capture. Feedforward controller B has a lower RMS pitch rate than A but does not regulate rotor speed quite as well as A. Feedforward controller A reduces tower top DEL more than feedforward controller B. As will be shown in Figure 5.10 below, much of feedforward controller A s tower top DEL reduction is at 0.6 Hz, which translates to 0.4 Hz in the rotating (blade) coordinate system [70]. Therefore the better tower top load reduction by feedforward controller A may be due to the increase in its magnitude at 0.4 Hz, which is shown in Figure 5.8 and discussed further below. This same pitch-rate activity vs. rotor-speed error tradeoff that is appearing for feedforward control also appears when comparing feedback controllers A and B. Feedback controller B was designed to use reduced feedback gains when combined with a feedforward controller. This greatly reduces RMS pitch rate and somewhat increases peak rotor speed. In some cases the reduced feedback gains also lead to reduced loads. Overall this feedback gain reduction appears beneficial because it reduces pitching action and allows the feedforward controller more control authority.

111 96 [knm 2 /Hz] 10 8 out of plane blade root bending moment Method _B_ Method BBB 10 5 tower base pitching moment 10 9 [knm 2 /Hz] pitch actuator speed [deg 2 /s 2 /Hz] frequency [Hz] Figure 5.5: Power spectral densities for method BBB under its original simulation conditions. Figure courtesy of D. Schlipf.

112 Figure 5.6: Turbine loads using individual pitch baseline feedback control alone ( A ) [8], with added individual pitch feedforward controller A, and with added individual pitch feedforward controller B. The rated rotor speed is 12.1 rpm, and the rated power is kw. x-accel, y-accel, and z-accel are respectively fore-aft, side-to-side, and up-down nacelle accelerations. The wind field used is AR2 s13, from the Great Plains set, at 13 m/s average wind speed. 97

113 Figure 5.7: Turbine loads using individual pitch baseline feedback control alone ( A ) [8], with added individual pitch feedforward controller A, and with added individual pitch feedforward controller B. The rated rotor speed is 12.1 rpm, and the rated power is kw. x-accel, y-accel, and z-accel are respectively fore-aft, side-to-side, and up-down nacelle accelerations. The wind field used is the IEC Kaimal NTM spectral model with a Class A turbulence level, at 14 m/s average wind speed. 98