Dynamics Modeling and Periodic Control of ...

Dynamics Modeling and Periodic Control of Horizontal-Axis Wind Turbines Karl A. Stol B.E. (Hons), University of Canterbury, N.Z., 1996 M.S., University of Colorado, 1998

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 Aerospace Engineering Sciences 2001

ii

This thesis entitled: Dynamics Modeling and Periodic Control of Horizontal-Axis Wind Turbines written by Karl A. Stol has been approved for the Department of Aerospace Engineering Sciences

_________________________________ Mark Balas

_________________________________ Gunjit Bir

__________________ 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.

iii Stol, Karl A. (Ph.D., Aerospace Engineering) Dynamics Modeling and Periodic Control of Horizontal-Axis Wind Turbines Thesis directed by Professor Mark Balas

ABSTRACT

The development of large multi-megawatt wind turbines has increased the need for active feedback control to meet multiple performance objectives. Power regulation is still of prime concern but there is an increasing interest in mitigating loads for these very large, dynamically soft and highly integrated power systems. This work explores the opportunities for utilizing state-space modeling, modal analysis, and multi-objective controllers in advanced horizontal-axis wind turbines. A linear state-space representation of a generic, multiple degree-of-freedom wind turbine is developed to test various control methods and paradigms. The structural model, SymDyn, provides for limited flexibility in the tower, drive train and blades assuming a rigid component architecture with joint springs and dampers. Equations of motion are derived symbolically, verified by numerical simulation, and implemented in the Matlab with Simulink computational environment. AeroDyn, an industry-standard aerodynamics package for wind turbines, provides the aerodynamic load data through interfaced subroutines. Linearization of the structural model produces state equations with periodic coefficients due to the interaction of rotating and non-rotating components. Floquet theory is used to extract the necessary modal properties and several parametric studies identify the damping levels and dominant dynamic coupling influences. Two separate issues of control design are investigated: full-state feedback and state estimation. Periodic gains are developed using time-varying LQR techniques and many different time-invariant control designs are constructed, including a classical PID controller.

iv Disturbance accommodating control (DAC) allows the estimation of wind speed for minimization of the disturbance effects on the system. Controllers are tested in simulation for multiple objectives using measurement of rotor position and rotor speed only and actuation of independent blade pitch. It is found that periodic control is capable of reducing cyclic blade bending moments while regulating speed but that optimal performance requires additional sensor information. Periodic control is also the only design found that could successfully control the yaw alignment although blade loads are increased as a consequence. When speed regulation is the only performance objective then a time-invariant state-space design or PID is appropriate.

v

To my ever supportive parents, Alex and Linda

vi ACKNOWLEDGMENTS First, I would like to thank my advisor and thesis committee chair, Professor Mark Balas. His outstanding teaching abilities in control theory have made my journey so much more enjoyable. Through his patience and many hours of consultation my research work has finally come to fruition. I have worked closely with many at the National Wind Technology Center and to these people I am sincerely indebted. Gunjit Bir has coached me through my dual-life as a wind turbine dynamicist and has provided a great deal of inspiration. Maureen Hand, Alan Wright, Marshall Buhl, and Lee Fingersh have all provided technical counsel when I’ve needed it most. Thank you also to Michael Robinson for his suggestions in the final hours of thesis preparation. I am grateful for the financial support provided by a number of sources.

The

Templin Traveling Scholarship and the William Georgetti Scholarship from New Zealand first opened the door to my graduate study in Colorado. Following this I have been fortunate to have the support of the National Renewable Energy Laboratory through continuous research contracts. Finally, I would like to express my love and appreciation for my fiancée Maria. Her understanding and encouragement have made all my endeavors worthwhile.

vii CONTENTS

CHAPTER 1.

INTRODUCTION .............................................................................................. 1 1.1

Wind Turbine Architecture ........................................................................2

1.2

System Dynamics and Cyclic Load Variations..........................................3

1.3

Dynamics Modeling ...................................................................................5

1.4

Control .......................................................................................................6 1.4.1 Control Objectives and Actuation Methods ...................................6 1.4.2 Control Strategies ...........................................................................8

1.5 2.

Scope and Outline .................................................................................... 10

MODEL DEVELOPMENT............................................................................. 13 2.1

Introduction.............................................................................................. 13

2.2

Structural Model Geometry and Properties.............................................. 13

2.3

Kinematics ............................................................................................... 18

2.4

Equations of Motion Derivation............................................................... 23

2.5

Verification .............................................................................................. 28

2.6

Aerodynamics .......................................................................................... 32 2.6.1 Implementation in Matlab ............................................................ 34 2.6.2 Example Results ........................................................................... 36

2.7

Nonlinear Aeroelastic Model ................................................................... 39 2.7.1 Example Results ........................................................................... 39

2.8

Linearization ............................................................................................ 41

2.9

Summary .................................................................................................. 43

viii 3.

MODAL ANALYSIS STUDY ......................................................................... 45 3.1

Introduction.............................................................................................. 45

3.2

Background .............................................................................................. 45

3.3

Floquet Approach..................................................................................... 46

3.4

Structure-Only Analysis........................................................................... 49 3.4.1 One-DOF Model: Yaw only......................................................... 52 3.4.2 Three-DOF Model: Teeter and Flap only..................................... 54 3.4.3 Four-dof Model: Yaw, Teeter, and Flap only............................... 56 3.4.4 Seven-DOF Model ....................................................................... 62 3.4.5 Summary ...................................................................................... 63

3.5 4.

Aeroelastic Analysis ................................................................................ 64

FULL-STATE FEEDBACK CONTROL....................................................... 67 4.1

Introduction.............................................................................................. 67

4.2

Periodic Gain Design ............................................................................... 68 4.2.1 Solving the Periodic Riccati Equation.......................................... 71

4.3

Constant Gain Design .............................................................................. 71

4.4

Disturbance Rejection .............................................................................. 75

4.5

Controller Implementation ....................................................................... 76

4.6

Results...................................................................................................... 77 4.6.1 Speed Regulation.......................................................................... 80 4.6.1.1 Linear Plant.................................................................... 81 4.6.1.2 Nonlinear Plant .............................................................. 82 4.6.2 Blade Load Mitigation ................................................................. 87 4.6.3 Yaw Directional Control .............................................................. 91

4.7

Summary .................................................................................................. 96

ix 5.

STATE AND DISTURBANCE ESTIMATION ............................................ 98 5.1

Introduction.............................................................................................. 98

5.2

Periodic Estimator Design ....................................................................... 98

5.3

Time-Invariant Estimator Design........................................................... 101

5.4

Estimator Implementation...................................................................... 103

5.5

Results.................................................................................................... 104 5.5.1 Linear Plant ................................................................................ 105 5.5.2 Nonlinear Plant........................................................................... 108

5.6 6.

Summary ................................................................................................ 113

CONCLUSIONS AND RECOMMENDATIONS ...................................... 115

REFERENCES ....................................................................................................... 119 APPENDIX A. SAMPLE NONLINEAR SYMDYN EQUATIONS OF MOTION ............ 122 B.

ALTERNATIVE DERIVATION OF THE SYMDYN EQUATIONS OF MOTION............................................................................................... 126

C. TURBINE PROPERTY DATA..................................................................... 129 D. AERODYN INTERFACE SOURCE CODE ............................................... 130 E.

ADDITIONAL CONTROL STUDIES......................................................... 139 E.1.

Example for Constant Gain Optimization.............................................. 139

E.2.

Frozen Model Parametric Studies .......................................................... 142 E.2.1 Speed Regulation........................................................................ 142 E.2.2 Blade Load Mitigation ............................................................... 143 E.2.3 Yaw Directional Control ............................................................ 143

x LIST OF COMMON SYMBOLS

t

Time

j

−1

I

Identity matrix

w

Horizontal hub-height wind speed

ρ

Air density

R

Radius of the rotor

A

Area of the rotor

λ

Tip speed ratio

Cp

Power coefficient

Cq

Torque coefficient

Ω

Rotor speed

Ω0

Nominal rotor speed

N

Number of degrees-of-freedom

Nb

Number of blades

q

Vector of degrees-of-freedom (angular positions)

q&

Vector of angular velocities

&q&

Vector of angular accelerations

τ1

Tower fore-aft angle

τ2

Tower lateral angle

τ3

Tower twist angle

γ

Nacelle yaw angle

η

Nacelle tilt angle

xi ψ

Rotor azimuth angle (0° when blade #1 is in the 12 o’clock position)

ψg

Shaft compliance angle

φ

Hub teeter angle

βi

Blade #i flap angle

Tg

Generator torque

Ta

Aerodynamic torque about the shaft

M(q)

Mass matrix in the nonlinear equations of motion

M(t), G(t), K(t)

Mass, gyroscopic, and stiffness matrices in the linear equations of motion

q op , q& op , &q& op

Angular position, velocity, and acceleration vectors at the operating point

∆q, ∆q& , ∆&q&

Perturbed angular position, velocity, and acceleration vectors

wop

Wind speed at the operating point

∆w

Perturbed wind speed

θc

Collective full-span blade pitch angle

θ

Vector of full-span blade pitch angles

θop

Vector of blade pitch angles at the operating point

∆θ

Vector of perturbed blade pitch angles

A(t)

State matrix

B(t)

Control input matrix

Bd(t)

Wind/disturbance input matrix

x, x(t)

State vector

u

Control input (vector of perturbed blade pitch angles)

ud

Wind/disturbance input

n

Number of states (dimension of the state-space system)

s

Laplace operator

Φ(t,0)

State transition matrix

xii T

Time period of the system dynamics

A

Time-invariant matrix in Floquet Theory

P(t)

Periodic matrix in Floquet Theory (in Chapter 3) Solution of the periodic Riccati equation (in Chapter 4)

λi

Characteristic exponent

σi

Characteristic multiplier

Λ

Diagonal matrix of characteristic exponents

Σ

Diagonal matrix of characteristic multipliers

ξi

Modal damping coefficient (real part of characteristic exponent)

ωi

Modal frequency coefficient (imaginary part of characteristic exponent)

Re(z)

Real part of the complex number z

p

Per-rotor-revolution

J

Quadratic cost function

J*

Optimal quadratic cost

∆J

Quadratic cost variation from J*

Jsim

Approximation to the quadratic cost found by simulation

Q

Matrix weighting on the states (LQR)

R

Matrix weighting on the inputs (LQR)

QE

Matrix weighting on the states (dual LQR)

RE

Matrix weighting on the outputs (dual LQR)

ℜ

Field of real numbers

C

Field of complex numbers

G(t), G*(t)

Optimal periodic state gain

Gd(t)

Periodic disturbance gain

G

Arbitrary constant state gain

xiii G, G d

Constant state and disturbance gains calculated by the mean of G(t)

Gf, Gd f

Constant state and disturbance gains calculated by freezing the state matrices

Ga, Ga f

Constant state and disturbance calculated by averaging the state matrices

Af, Bf, Bd f

Frozen representation of A(t), B(t), Bd(t)

Aa, Ba, Bd a

Averaged representations of A(t), B(t), Bd(t)

tf

Time within the period that the state matrices are frozen

ψf

Freezing azimuth angle

KP, KI, KD

Constant scalar gains in PID control

u*

Ideal control input (when state and disturbance information is known)

y

Vector of plant outputs

C(t)

Periodic output matrix

xˆ

State estimate

uˆ d

Wind/disturbance input estimate

yˆ

Plant output estimate

zd

Disturbance generator state vector

F, Θ

Disturbance generator state matrices

zˆ d

Disturbance generator state estimate

) ) ) A( t ), B( t ), C( t )

Augmented state matrices

) ) K ( t ) , K( t ) *

Optimal periodic estimator gain

K

) Constant estimator gain calculated by the mean of K( t )

) Kf

Constant estimator gain calculated by freezing the state matrices

) Ka

Constant estimator gain calculated by averaging the state matrices

xiv LIST OF TABLES

Table 2-1: Denavit-Hartenburg parameters for the SymDyn model ...................................... 20 Table 2-2: Verification test cases ........................................................................................... 29 Table 2-3: Hub-referenced wind field parameters in AeroDyn.............................................. 33 Table 3-1: Turbine models analyzed ...................................................................................... 50 Table 3-2: Characteristic exponents for the four degree-of-freedom model, normalized by the nominal rotor speed, Ω0 = 57.5 rpm. ................................... 64 Table 4-1: Quadratic cost comparison between the controllers ............................................. 82 Table 4-2: Speed regulation performance results ................................................................... 86 Table 4-3: Blade load mitigation performance results with the 4 d.o.f. nonlinear plant ........ 90 Table 4-4: Blade load mitigation performance results with the 5 d.o.f. nonlinear plant ........ 90 Table 4-5: Yaw control performance results .......................................................................... 94 Table 5-1: Time-invariant DAC performance results with the 4 d.o.f. nonlinear plant. RMS speed error in rpm, ADC in deg/s, and FRR in deg/s.............................. 109 Table 5-2: DAC performance results with the 4 d.o.f. nonlinear plant ................................ 110 Table 5-3: DAC performance results with the 5 d.o.f. nonlinear plant (teetered rotor) ....... 112 Table C-1: SymDyn property values: * verification model, ** all other models.................... 129

xv LIST OF FIGURES

Figure 1-1: Components of a large modern horizontal-axis wind turbine................................ 3 Figure 1-2: Spatial components of a wind field ....................................................................... 4 Figure 1-3: Relationship between the dynamic subsystems of a wind turbine......................... 5 Figure 1-4: Regions of turbine operation and ideal power output............................................ 7 Figure 2-1: SymDyn degrees-of-freedom and geometry. (a) Tower components. (b) Nacelle components. (c) Hub with δ3 angle in the front view. ..................... 15 Figure 2-2: Illustration of the choice of hinge-offset for the tower and blade. Solid line shows the modeshape and the dashed line depicts the least-squares fit. ............ 16 Figure 2-3: SymDyn center-of-mass and applied load locations for a two-bladed rotor ....... 17 Figure 2-4: SymDyn reference frame locations ..................................................................... 21 Figure 2-5: Simulink block diagram for the nonlinear structural dynamics subsystem ......... 27 Figure 2-6: Verification study results (9 DOF); tower fore-aft angle response ..................... 30 Figure 2-7: Verification study results (9 DOF); tower lateral angle response ....................... 30 Figure 2-8: Verification study results (9 DOF); tower twist angle response.......................... 30 Figure 2-9: Verification study results (9 DOF); yaw angle response..................................... 31 Figure 2-10: Verification study results (9 DOF); shaft twist angle response ......................... 31 Figure 2-11: Verification study results (9 DOF); hub teeter angle response.......................... 31 Figure 2-12: Verification study results (9 DOF); blade #1 flap angle response..................... 32 Figure 2-13: Vertical wind shear (power law) in AeroDyn.................................................... 33 Figure 2-14: Summary of the Matlab-SymDyn-Aerodyn interface ....................................... 35 Figure 2-15: Simulink block diagram for the aerodynamics subsystem ................................ 36 Figure 2-16: Torque coefficient contour plot ......................................................................... 37 Figure 2-17: Power coefficient contour plot .......................................................................... 37

xvi Figure 2-18: Desired blade pitch variation versus wind speed for constant aerodynamic & = 57.5 rpm.................................................................. 38 torque (Tg = 14.0 Nm, ψ Figure 2-19: Simulink block diagram for the nonlinear aeroelastic subsystem ..................... 39 Figure 2-20: Tower fore-aft angle variation in the periodic steady-state solution ................. 40 Figure 2-21: Blade #1 flap variation in the periodic steady-state solution............................. 41 Figure 2-22: Blade #2 flap variation in the periodic steady-state solution............................. 41 Figure 2-23: Rotor speed variation in the periodic steady-state solution ............................... 41 Figure 3-1: Yaw modal damping variation from conventional modal analysis ..................... 53 Figure 3-2: Yaw response due to unit velocity initial condition ............................................ 54 Figure 3-3: Fan-plot for the three-DOF model (Ω0 = 57.5 rpm) ............................................ 56 Figure 3-4: Effect of teeter damping on teeter and differential-flap mode stability............... 58 Figure 3-5: Effect of precone and teeter stiffness on teeter mode stability ............................ 59 Figure 3-6: Effect of yaw stiffness on modal damping .......................................................... 60 Figure 3-7: Effect of yaw stiffness on modal frequencies...................................................... 60 Figure 3-8: Effect of yaw damping on modal damping.......................................................... 61 Figure 3-9: Effect of yaw damping on modal frequencies ..................................................... 61 Figure 3-10: Open-loop characteristic exponents for the aeroelastic model, normalized by the nominal rotor speed, Ω0 = 57.5 rpm. .................................... 65 Figure 4-1: Simulink block diagram for periodic gain control............................................... 77 Figure 4-2: Simulink block diagram for constant gain control .............................................. 77 Figure 4-3: Simulink block diagram for the closed-loop system (periodic control with the nonlinear plant) ......................................................... 79 Figure 4-4: Wind disturbance input........................................................................................ 80 Figure 4-5: Periodic state gain, G(t), and disturbance gain, Gd(t), for speed regulation, Q0 = 1. Solid line: gains for θ1, dashed line: gains for θ2. .................................. 83

xvii Figure 4-6: Closed-loop characteristic exponents for speed regulation (Q0 = 1), normalized by the nominal rotor speed, Ω0 = 57.5 rpm. .................................... 84 Figure 4-7: Rotor speed response for speed regulation .......................................................... 86 Figure 4-8: Blade #1 pitch usage for speed regulation........................................................... 87 Figure 4-9: Periodic state gain, G(t), and disturbance gain, Gd(t), for blade load mitigation. Solid line: gains for θ1, dashed line: gains for θ2................................................ 89 Figure 4-10: Blade #1 bending moment for the blade load mitigation study (4 d.o.f. plant) . 90 Figure 4-11: Blade #1 bending moment for the blade load mitigation study (5 d.o.f. plant) . 91 Figure 4-12: Periodic state gain, G(t), and disturbance gain, Gd(t), for yaw control. Solid line: gains for θ1, dashed line: gains for θ2................................................ 93 Figure 4-13: Yaw response for the yaw control study ........................................................... 94 Figure 4-14: Blade #1 bending moment for the yaw control study........................................ 95 Figure 5-1: Simulink block diagram for the periodic DAC controller in closed-loop ......... 104 )

Figure 5-2: Components of the periodic estimator gain, K(t ) , for each output channel....... 106 Figure 5-3: Wind speed estimator performance with the linear plant .................................. 107 Figure 5-4: Tower fore-aft rate estimator performance with the linear plant....................... 108 Figure 5-5: Blade #1 flap rate estimator performance with the linear plant......................... 108 Figure 5-6: Blade #1 bending moment with the 4 d.o.f. nonlinear plant.............................. 110 Figure 5-7: Wind speed estimator performance of the periodic DAC controller with the 4 d.o.f. nonlinear plant...................................................................................... 111 Figure 5-8: Blade #1 flap rate estimator performance of the periodic DAC controller with the 4 d.o.f. nonlinear plant........................................................................ 111 Figure 5-9: Blade #1 bending moment with the 5 d.o.f. nonlinear plant (teeter active)....... 112 Figure E-1: Periodic gain for the example system ............................................................... 139 Figure E-2: Variation of suboptimality, ∆J, with constant gain values ................................ 141

xviii Figure E-3: Quadratic cost variation with freezing azimuth position (speed regulation case) ..................................................................................... 143 Figure E-4: Quadratic cost variation with freezing azimuth position (blade load mitigation case).............................................................................. 143 Figure E-5: Quadratic cost variation with freezing azimuth position (yaw control case) .... 144

1 1. CHAPTER 1 INTRODUCTION

The basic premise of a modern wind turbine is to capture as much of the available kinetic energy from a given wind field for the lowest possible cost. The cost of energy (COE) metric most often used is simply expressed as the sum of all machine costs including manufacture, installation and maintenance divided by the total energy capture over the machine operational life. Modern machines can produce energy from 4 to 5 ¢/kWh and are designed to operate in excess of 20 years. The designer’s challenge is to produce the lowest COE machine without sacrificing structural life or reliability. Machine cost, as with any mass produced product, is primarily a function of weight. Cost reduction is most readily achieved by eliminating material. The adverse consequence is a reduction in strength, stiffness and an increase in machine dynamics. Ultimately, the turbine design and cost is driven by aerodynamic and gravity loads experienced over the design life and the material required to adequately handle the resulting cyclic and stochastic shear stresses. With very large turbines, the problem is exacerbated. The machine weight increases with the length cubed while energy capture increases with the length squared.

Final

“optimized” designs must minimize COE by balancing numerous technical and market requirements. Even greater pressure to achieve weight savings is placed on engineers and designers for these multi-megawatt machines. Competitive designs require reduced safety factor margins from a more thorough understanding and prediction of loads as well as active control to mitigate loads. This work examines both the applicability of advanced control methods to these large wind turbine architectures and explores the potential for effective power and load mitigation control.

2 1.1 Wind Turbine Architecture Shown in Figure 1-1 is the typical configuration of a large horizontal-axis wind turbine. Rotor diameters are in excess of 70 m and are projected to exceed 100 m in the next 10 years. Tower heights for machines located in the Midwest of the U.S. are already at 80100 m and will approach 150 m in the next 5 years. The machines are designed to capture the maximum amount of energy taking advantage of increasing wind speed with increasing tower height. When compared with civil engineering structures of comparable size, these multimegawatt machines are both light and flexible and possess extremely energetic structural dynamics. The nacelle houses the drive train connecting the rotor hub to the generator. This coupling may be direct through a single shaft (direct drive) or through a gearbox which increases the rotation rate of the low speed shaft in order to drive the generator at a higher RPM. A yaw bearing on top of the tower allows the nacelle to rotate and follow changes in wind direction. Downwind turbines, where the wind flows through the tower to the rotor hub, like that pictured, are dynamically stable. The nacelle is free to yaw and will track changes in wind direction. In contrast, upwind machines require active yaw control to remain upwind with changes in direction. Although any number of blades can be used, two- and three-bladed rotors are the most common for large machines due to increased aerodynamic efficiency and solidity (blade length to width ratio). The blades have optimized airfoil cross-sections and develop lift and drag aerodynamic forces not unlike the wing of an aircraft. These blade designs are usually optimized to produce maximum power at a single blade tip speed ratio (blade tip speed/wind velocity - λ), thus maximum power extraction for any given wind speed is achieved for a single hub rotation rate. This requires the machine to be operated variable speed in order to extract maximum kinetic energy.

Most large turbines have active blade pitch control

(rotation about the longitudinal blade axis) as a means to control aerodynamic loads.

3

Wind direction Drive train

Nacelle

Blade

Generator

Hub

Tower

> 80 m

Rotor

Figure 1-1: Components of a large modern horizontal-axis wind turbine

1.2 System Dynamics and Cyclic Load Variations The stochastic nature of the wind resource provides the largest variation in system loading. The wind varies in both magnitude and direction, each effect produces a change in the loads transferred through the blades, low-speed shaft, generator and tower. For modeling purposes it is convenient to characterize the wind as having both a deterministic and stochastic component. The stochastic component consists of a time-varying, zero-mean random variation of wind speeds over the rotor area. The time-varying loads produced by stochastic component are intuitively clear. Periodic loading arising from the deterministic component warrants further discussion. The deterministic component is time-invariant, increasing monotonically in height in accordance with the standard power law planetary boundary layer characterization. This increase gives rise to the vertical velocity shear shown in Figure 1-2 and introduces a strong

4 cyclic variation in load with blade rotation as the blade rotates from the top (maximum load) to the bottom (minimum load) of the machine.

+

Stochastic variation

Deterministic variation

Figure 1-2: Spatial components of a wind field

A similar effect occurs with yaw error. Turbines perfectly aligned with the wind (zero yaw error) would experience no difference in blade loading from side to side with blade rotation. As the machine orientation changes relative to the wind direction, the rotating blade experiences a velocity change proportional to the yaw angle. This velocity difference again produces a cyclic one-per-revolution load variation from side to side. Downwind configurations introduce another periodic load. The momentum deficit produced by the tower produces a wake that can be represented using a simple Prandtl mixing length model. A cyclic load variation occurs as the blade passes into and out of the wake deficit. Rather than the more gradual load effect from the velocity gradients described above, this periodic effect is impulsive and occurs over a limited portion of the rotation cycle. The wind field is just one element in the dynamics of the entire system. Figure 1-3 illustrates the relationship between all the elements. As described, wind induces loads on the turbine blades, causing the structure to flex and consequently change the blades’ orientation in the wind. This aeroelastic coupling can be both beneficial, producing useful output power, or undesirable when internal loads lead to structural failure. The final component is the

5 automatic control system running in real-time on the turbine. It uses sensors to determine the continuous condition of the structure and/or wind and commands changes in the structure through actuators. The controller objective is then to maximize output power while reducing adverse loads. Control concepts will be described in more depth shortly. Suffice to say that the interaction of the controller with the rest of the system is very important to the health of the turbine. Therefore, the controller should be designed with as much information about the aeroelastic dynamics as practical. Electrical Power Blade Loads

Wind Field

Internal Loads Structural Dynamics

Aerodynamics Motion Actuation

Sensing Control

Figure 1-3: Relationship between the dynamic subsystems of a wind turbine

1.3 Dynamics Modeling Many modeling tools are capable of simulating the dynamic response of wind turbines with various degrees of component flexibility. YawDyn, developed at the University of Utah, models the blades as rigid components connected by springs and dampers at the hub [1]. FAST, from Oregon State University, models structure flexibility by allowing a limited number of predetermined vibration mode shapes [2]. ADAMS is a commercial code from Mechanical Dynamics, Inc. capable of modeling a much larger number of degrees-offreedom including fully flexible components [3]. All of these structural dynamics models utilize AeroDyn, an aerodynamics model based on blade element momentum theory to obtain blade load data [1].

6 These dynamics models are limited, providing only time-response information. For advanced linear control designs an explicit linear system description is required. This enables a characterization of the turbine flexibility so that the controller may positively interact with the structure. Another important use for a linearized system model is in the calculation of operating modes. Operating modes quantify damping levels in the turbine, from which one can determine stability. A modal analysis can also identify the dominant coupling between individual turbine states. While ADAMS can produce the modes of a wind turbine, the rotor must be in a parked position. The result is neglect of the significant gyroscopic coupling effects in a rotating turbine. When modeled correctly, the linear model of a two-bladed turbine has time-varying coefficients due to the asymmetric inertia properties of the rotor and spatial variations in the wind field. In fact the coefficients are periodic, with frequency equal to the rotor speed. Even a three-bladed turbine model is periodic if non-uniform wind conditions exist, such as wind shear.

As such, an operating modal analysis cannot use standard time-invariant

methods. For these cases, one must employ Floquet theory [4].

1.4 Control 1.4.1

Control Objectives and Actuation Methods Traditionally, the primary goal of a wind turbine control system is to regulate power

extraction from the wind. Presented next is an overview of the important principles of turbine operation to achieve this objective. The maximum kinetic power from the wind that is available to a turbine increases with the cube of the wind speed, w: Pwind = 12 ρAw 3

(1.1)

7 where ρ is the air density and A is the rotor swept area. Megawatt scale turbines have very different control strategies based on the region of operation shown in Figure 1-4.

Pwind Power

Rated Power

Wind Speed, w wcut-in Region I

Region II

wrated

wcut-out Region III

Figure 1-4: Regions of turbine operation and ideal power output

In Region I, below wcut-in, there is insufficient energy in the wind to overcome friction and other system losses to produce a net positive energy flow. Operationally, the wind speed must exceed the minimum wind speed for several minutes before a control system will allow the machine to come on-line. In Region II, output power increases with the cube of wind speed. At best a turbine can extract a maximum of 59.3% of Pwind. This upper bound, commonly referred to as the Betz limit [5], can be derived from basic fluid dynamics. In Region III, maximum rated power output is achieved. This limit is usually set by the generator, drive train, and power electronics designs. Excess energy above rated power cannot be used. The cut out wind speed is usually set as the upper limit for the thrust load on the machine and reflects both a blade and tower moment upper bound.

8 The control actuators are used to regulate power over the range of operating wind speeds. In a constant speed turbine, for which the generator is constrained to run at a fixed speed, there is no active control in Region II but blade pitch is often used to maintain constant aerodynamic torque in Region III. In variable speed turbines the generator can run at any speed. Here, control in Region II is usually by regulation of generator torque alone to achieve maximum aerodynamic efficiency, while either blade pitch or generator torque is used in Region III to maintain constant power. If blade pitch alone is used in Region III, the generator torque is often kept constant by power electronics. Then the objective becomes speed regulation, a term used often. Blade fatigue is responsible for many premature turbine failures. To prevent this from happening, active control can play a key role, due to the interaction of controller dynamics with the structure. In particular, fatigue failure of the blades is common and can be slowed or prevented by use of control for the mitigation of cyclic blade loads. Satisfying this control objective in addition to adequate power regulation is achievable with use of blade pitch. Examples to support this claim are provided in the next section. Another form of control actuation is the use of a yaw servo. Generally this is used for an upwind turbine to keep the rotor swept area perpendicular to the prevailing wind direction. Usually a mean wind direction is determined after some time period and the yaw motor is only engaged once a certain yaw direction error is observed. Use of the yaw servo for power regulation or load mitigation is very rare.

1.4.2

Control Strategies For the description of past and present control strategies, we begin by considering

what has become the industry standard, Proportional-integral-derivative (PID) control. PID has had widespread use because generally little knowledge of the plant dynamics is necessary prior to implementation. The design is single-input-single-output (SISO), meaning one plant

9 output is measured and only one actuator is commanded. There is no analytical basis for the choice of gains (of which there are three), although Hand has developed a systematic gain search procedure for wind turbines in [6]. Most instances of PID control has been for speed regulation in Region III. However, in [7] Rock, et. al., use PI control (PID with two gains) to actively suppress blade bending moments. In this study they use a predicted bending moment based on a look-up table and control collective blade pitch in Region II and III. They show results that indicate fatigue reduction is possible but did not test the controller on a more comprehensive simulation code. The biggest drawback of PID is that it cannot easily be extended to control multiinput-multi-output (MIMO) systems.

An attempt to do so is likely to lead to poor

performance or instability because the individual controllers may compete with one another, as with any form of decentralized control.

Modern control methods use state-space

representations of the system for design of gains and are applicable to MIMO systems with multiple objectives. The application of this approach to wind turbines is described next. Time-invariant Linear Quadratic Gaussian (LQG) techniques have been reported by Bossanyi [8] to be effective in power regulation of a constant speed machine. Similarly, Ekelund [9] has successfully implemented LQG for a variable speed machine with fixed pitch.

A different approach is taken by Kendall [10] who examines Disturbance

Accommodating Control (DAC) for variable speed, variable-pitch speed regulation in Region III. In this study the wind speed is estimated by the controller and used to minimize the wind disturbance effect on the system. Instead of LQG for the design of gains, a simple poleplacement approach is taken. Periodic control is essentially the use of time-varying feedback gains with a fixed time period.

In a wind turbine the period would correspond to the time of one rotor

revolution. While periodic control techniques have been applied to various fields, including helicopters [11] and spacecraft orbit optimization [12], wind turbines have received little

10 attention. This is despite the fundamental periodicity in the model dynamics, as mentioned earlier. One application to wind turbines by Ekelund [9] studies the use of continuous yaw control to improve tower damping and minimize hub teeter motion. The author of this paper showed that periodic control, designed using the Linear Quadratic Regulation (LQR) method, performed well compared to a PID controller. However, simulations were only performed with a linear plant and knowledge of all plant states was assumed. Also wind input was ignored. In another study by Liebst [13], LQR is used to design periodic gains for mitigation of blade bending moments through actuation of a partial span pitch system.

Again,

performance was measured on a linear plant only and estimation of the plant states was ignored.

1.5 Scope and Outline The research in this thesis was motivated by the following succession of issues. ƒ

Large, flexible wind turbines require active control to meet multiple performance objectives. It is clear that improvements in the design of wind turbine controllers are necessary to reduce the likelihood of fatigue failure while continuing to maximize energy capture. This is a multi-objective problem that can be dealt with in a systematic way using modern state-space control concepts. At the heart of this approach is a linear state-space model of the turbine.

ƒ

Current turbine dynamics models do not provide linear state-space representations. Modeling codes in use by the U.S. wind industry are designed as accurate simulations of real turbines and best serve as platforms upon which to test existing control designs.

Therefore, the need exists for a model that can produce state-space

representations. When any form of asymmetric wind or structural property exists, the state-space models are not time-invariant – they have periodic coefficients.

11 ƒ

The periodic nature of wind turbines is largely ignored in modal analyses and control designs. A correct modal analysis is important for determining stability, damping levels, and the dominant motion couplings. Most turbine control designs are based on timeinvariant concepts and are not optimized for use with periodic plants.

Based on these issues, the scope of my original contribution is defined by the following objectives.

1. Develop an approach to build state-space representations of wind turbine dynamics. These models must be capable of incorporating the degrees-of-freedom that need to be controlled, including tower and blade flexibility. It would be convenient if degrees-offreedom could be easily added or suppressed from the models to facilitate many different control studies. 2. Perform modal analysis studies using Floquet theory to correctly determine damping and dynamic couplings. 3. Apply periodic gain control techniques with wind estimation and assess what performance objectives would require them. 4. Develop tools for time-invariant control with wind estimation, optimized for use on periodic plants.

Chapter 1 is devoted to the development of the dynamics model from which statespace representations of wind turbines can be constructed. The structural dynamics and aerodynamics components are interfaced to form a nonlinear model. Linearization follows in a logical fashion. As the chapter progresses a two-bladed turbine model develops, capable of independent blade pitch actuation. This model is used in most of the subsequent studies. In

12 Chapter 2 a series of modal analysis studies are performed, primarily on the linear model without aerodynamics. Analysis of the two-bladed aeroelastic turbine model provides the open-loop properties for control studies. Chapter 3 deals with full-state feedback issues only. Results compare periodic and constant gain controller designs for use in speed regulation, blade load mitigation, and yaw directional control. Chapter 4 adds state and wind estimation fundamentals to round out with realizable designs. A single results case compares periodic and time-invariant controller performance with the objective of simultaneous speed regulation and blade load mitigation.

13 2. CHAPTER 2 MODEL DEVELOPMENT

2.1 Introduction This chapter describes the aeroelastic wind turbine model from which linear statespace representations are constructed. The largest contribution details the development of the structural dynamics component called SymDyn (for Symbolic Dynamics). It is so called because the equations of motion are derived explicitly in symbolic form. This allows much physical insight to be gained (for the smaller models) and provides for analytically exact linearization. Aerodynamics from an existing code is interfaced with SymDyn to complete the aeroelastic model. Transformation of the system into linear state-space form comprises the last section. Example numerical results are scattered throughout this chapter. They serve to illustrate the capabilities of the modeling approach and provide a progressively developed turbine model for use in subsequent chapters.

2.2 Structural Model Geometry and Properties A rigid-body approach is adopted to model the flexible components (tower, shaft, and blades) of a horizontal-axis wind turbine. This allows the model to have few degrees-offreedom, which in turn simplifies state-space descriptions and aids the subsequent control system design. One alternative approach, to more accurately model the flexible components, is to use a finite element or multi-body method, discretizing the structure with a large number of elements. Such an approach is taken when modeling with ADAMS [3] or with the code under development by Hodges [14]. Here the dynamics models have a large number of degrees-of-freedom, requiring a form of model reduction for subsequent control design. Yet another approach is to assume certain modeshapes for each flexible component – as per the

14 FAST code [2]. These modeshapes must be determined a priori and therefore are based on a certain operating condition (such as rotor speed, wind speed, etc.). The rigid-body approach can directly provide a suitable low-order model. It also allows simple derivation techniques to be used and aids physical interpretation. A schematic diagram of the SymDyn model is shown in Figure 2-1. The structure consists of rigid bodies connected by revolute joints, allowing single-axis rotation of each component relative to the next. While the tower may flex about three axes (fore-aft, lateral, and twist motions), the blade may only flap normal to the rotor plane, i.e. no in-plane (lag) or twist motion is considered. Note that the tower hinges are co-located, which assumes that the tower has an orthotropic cross-section. The nacelle tilt axis is assumed to intersect the yaw axis, as shown in Figure 2-1(b). The rotor precone, β0, is the angle that each blade is fixed relative to the vertical. Generally, this parameter is designed to alleviate blade root bending loads by considering the balance between centrifugal and aerodynamic loading under operating conditions. Precone also provides additional clearance between the rotor and the tower to prevent tower-strike. For a two-bladed rotor, the teeter axis is often skewed, as shown in Figure 2-1(c). This skew angle is commonly referred to as δ3. Positive δ3 improves teeter stability by pitching the blades as the rotor teeters. The total number of degrees-of-freedom allowable is N = 8+Nb, where Nb is the number of blades. This set of parameters adequately models the most significant dynamic behavior of a generic wind turbine. The notation is as follows.

q1 = τ 1

Tower fore-aft angle

q2 = τ 2

Tower lateral angle

q3 = τ 3

Tower twist angle

q4 = γ

Nacelle yaw angle

15 q5 = η

Nacelle tilt angle

q6 = ψ

Rotor azimuth angle (0° when blade #1 is in the 12 o’clock position)

q7 = ψg

Shaft compliance angle (relative to the rotor end of the shaft)

q8 = φ

Hub teeter angle

q9 = β1

Blade #1 flap angle

… qN = βNb Blade #Nb flap angle

β0

Blade β1

Tower

High speed shaft

Generator dt2

Hub dh2

η

τ3

dn1

ψ

φ

dh1 φ

ψg

τ2

dn2

δ3

γ

τ1

Bedframe

dt1

(a)

Low speed shaft

βΝb

(b)

(c)

Figure 2-1: SymDyn degrees-of-freedom and geometry. (a) Tower components. (b) Nacelle components. (c) Hub with δ3 angle in the front view.

Only the fundamental vibration modes of the flexible components (namely the tower, rotor shaft, and blades) are modeled.

This is accomplished by modeling each flexible

component by two rigid parts, separated by a hinge with linear stiffness and damping

16 properties. The hinge position, spring stiffness, and mass properties for the tower and blades are chosen using the following criteria. 1.

A least-squares fit of the single-hinge model to the modeshape of the original flexible component determines the hinge-offset parameters (dt1, dh1, and dh2) as illustrated in Figure 2-2. The modeshapes are calculated using finite-element representations of the actual components. Note that the hinge location for tower twist is arbitrary.

Lumped mass representing mass and inertia of the nacelle and rotor

Centrifugal loading due to rotation, Ω

Blade Hinge-offset

Tower

Ω

Figure 2-2: Illustration of the choice of hinge-offset for the tower and blade. Solid line shows the modeshape and the dashed line depicts the least-squares fit.

2.

Hinge spring stiffness is calculated to give the same tip deflection (lateral displacement or rotation) when a unit load is applied.

3.

The mass of the two portions of each component is consistent with the original, given the hinge-offset distance as the separating point. The lower portion of the tower remains stationary and therefore its mass is ineffectual. The inner portion of each blade remains fixed to the hub where the mass is lumped.

4.

The center-of-mass location of the moving portion of each component (the tip section) is consistent with the original component. The other portion has a center-of-mass location such that the center-of-mass of the entire component is conserved.

5.

The moment-of-inertia of the moving portion about the hinge is calculated to give the correct natural frequency (as per a finite-element analysis without centrifugal stiffening

17 effects or tip mass). The moment-of-inertia of the other portion makes up the difference to conserve the moment-of-inertia of the entire component. To model the compliance of the rotor shaft a single torsional spring is used. The entire shaft mass is lumped at the generator end with the correct center-of-mass location. Lateral moments-of-inertia, which are assumed equal, are also lumped at the generator end portion. This reduces the number of terms in the equations of motion without loss of accuracy. However, both portions of the shaft have polar moments-of-inertia defined. For the case when a gearbox is used these polar moments-of-inertia can be adjusted (by the square of the gear ratio) to capture the dynamics correctly. Center-of-mass locations for each of the SymDyn elements are illustrated in Figure 2-3. It is assumed that the bedframe c.o.m. (depicted by cp) lies along the yaw axis, while the nacelle c.o.m. (depicted by cn) lies along the shaft axis. Lumped with the nacelle component is the mass of the stationary parts of the generator. The mass and inertia of the rotating parts of the generator are lumped with the shaft.

Fb1x, Mb1x

cb Fb1z, Mb1z

Tg

ct

Fb1y, Mb1y

ch

cn

cs

cp

Fb2y, Mb2y Fb2z, Mb2z

Fb2x, Mb2x

Figure 2-3: SymDyn center-of-mass and applied load locations for a two-bladed rotor

18 The SymDyn model supports blade-fixed forces and moments as Cartesian components applied to each blade at the flap hinge, as shown in Figure 2-3. This allows inclusion of aerodynamic loads, as will be introduced in Section 2.6. The internal generator torque (Tg) is applied to the shaft, with a reaction torque on the nacelle. A dynamics model for the generator is not considered because the natural frequencies of this component are generally of high order and hence do not interact with the low frequencies of the structure. However, by treating the generator torque as an unknown input to the system we may incorporate a generator model in simulation studies, if so desired. For the studies presented in this thesis we prescribe the generator torque at all times.

2.3 Kinematics The model development makes use of the Denavit-Hartenberg (DH) convention for assignment of body-fixed coordinate systems. This convention is used in the robotics field to standardize kinematics of open linkage systems [15] and has also been applied to wind turbines at a preliminary level [16]. The application to wind turbines is fairly straightforward with the addition of branching and dummy reference frames, as will be evident shortly. The DH procedure assigns a coordinate system to each rigid-body (or link) beginning with the inertial frame. These are Cartesian reference frames with orthonormal unit vectors we denote by { xˆ i , yˆ i , zˆ i } for frame {i}. The zˆ i -axis is aligned with the rotation axis of the body. The direction of rotation is determined by the right-hand rule. Axis xˆ i is defined from zˆ i to zˆ i +1 and perpendicular to both. The origin location is then the intersection of the xˆ i and zˆ i axes. The yˆ i -axis completes the right-hand triad. This convention provides us with a

general form for the transformation matrix and position vector of each frame relative to the previous frame.

19

i −1

i −1

cos(γ i ) 0 − sin( γ i )    R i = cos(α i ) sin( γ i ) cos(α i ) cos(γ i ) − sin(α i )   sin(α i ) sin( γ i ) sin(α i ) cos(γ i ) cos(α i ) 

(2.1)

p i = [b i

(2.2)

− d i sin(α i ) d i cos(α i )]

T

where i-1Ri is the transformation from frame {i-1} (the left superscript) to frame {i} (the right subscript),

i −1

p i is the position vector from the origin of frame {i-1} to the origin of frame

{i}, and {αi, bi, γi, di} are referred to as the DH parameters for link i. These parameters are defined as follows.

αi is the rotation about xˆ i −1 to align zˆ i −1 to zˆ i , bi is the distance along xˆ i −1 from zˆ i −1 to zˆ i , γi is the rotation about zˆ i −1 to align xˆ i −1 to xˆ i , and di is the distance along zˆ i −1 from xˆ i −1 to xˆ i .

Given ci, the center-of-mass position vector of link i, referenced from its own frame, the position vector of this location from the origin is given by 0

ri =

∑( i

0

R j−1

j−1

j=1

)

p j + 0 R i ci

(2.3)

where 0Ri = 0R1 1R2. … i-1Ri, the transformation matrix from the inertial frame {0}, to frame {i}. The angular velocity vector of link i, referenced in its own frame, is defined by T ωi = [0 0 q& i ]

(2.4)

where q& i is a scalar unknown for the rate of rotation. When expressed and measured in the inertial frame, this vector becomes

20

0

ωi =

i

∑

0

R jωj .

(2.5)

j=1

Finally, the linear velocity of the center-of-mass of link i, referenced from the inertial frame, can be calculated from 0

vi =

∑( ω i −1

j=1

0

j

j

)

0

× 0 R j p j+1 + ωi × c i .

(2.6)

The reference frame positions for the SymDyn wind turbine model are illustrated in Figure 2-4 with associated DH parameters listed in Table 2-1. Notice that only one blade is shown; the kinematics expressions for the other (Nb-1) blades can be derived via the following transformation. Consider blade #j (with j>1). Its velocity can be derived from the velocity of blade #1 as follows:

q& 9  q 9 0 0 v 8+ j = v 9 ∋  q 6 δ  3

→ q& 8+ j → q 8+ j

(2.7)

→ q 6 + 2π( j − 1) / N b → δ 3 − 2π( j − 1) / N b

where the notation xØy means the variable x is replaced by the variable y in the original expression. Link i 1 2 3 4 5 6 7 6’ 8 8’ 9

Description Tower fore-aft Tower lateral Tower twist Nacelle yaw Nacelle tilt Shaft rotation Shaft compliance Dummy frame Hub teeter Dummy frame Blade #1 flap

αi -π/2 -π/2 -π/2 0 π/2 -π/2 0 π/2 δ3 δ3 0

bi 0 0 0 0 0 dn1 0 0 0 dh1 dh2

γi -π/2+q1 -π/2+q2 π/2+q3 π+q4 -π/2+q5 π+q6 q7 π/2 π+q8 π/2 q9

Table 2-1: Denavit-Hartenburg parameters for the SymDyn model

dI 0 0 0 dt2 0 dn2 -dn2 0 0 0 0

21

β0

{4} Nacelle yaw

x^9

z^9

z^4 x^4 z^5 x^7

z^7

dh2

x^8’ {8’} Dummy frame dn2

δ3 z^8’

dh1

x^6

z^6’ ^ x8 z^6

{6} Shaft rotation

{3} Tower twist x^2

z^2

z^1

{2} Tower lateral

{1} Tower fore-aft

dt1 z^0 x^0

{0} Inertial frame

Figure 2-4: SymDyn reference frame locations

z^8

x^6’ {6’} Dummy frame

indicates co-located coordinate axes

z^3

x^1

dn1

{7} Shaft compliance

dt2

x^3

{5} Nacelle tilt

x^5

{9} Blade #1 flap

{8} Hub teeter

δ3

22

Essentially this transformation rotates blade #1 through the appropriate azimuth angle (q6), where blade #j is located, while maintaining the correct teeter axis location by modification of the δ3 angle. The same method can be applied to derive the center-of-mass position and angular velocity vectors. A deviation from the standard DH convention is made in three instances. First, it is desirable to locate the inertial frame at ground level for correct wind velocity referencing and also to have the shown orientation to be consistent with model geometry presented in turbine literature. Therefore, it is necessary to set the first position vector as 0 p 1 = [0 0 dt1]T instead of using the result from expression (2.2). Second, it is essential to branch the topology of the structure at the shaft rotation frame {6} to allow both the shaft compliance angle (from frame {7}) and hub teeter angle (frame {8}) to be measured relative to the shaft. Third, as a consequence of branching, dummy reference frames are defined between frames {6} and {8}, denoted by frame {6’}, and between frames {8} and {9}, denoted by {8’}. From Figure 2-3, the center-of-mass position vectors are c1 = c2 = [0 0 0]T, c3 = [0 0 ct]T, c4 = [0 0 -cp]T, c5 = [dn1 -cn 0]T, c6 = [0 0 0]T, c7 = [0 0 -cs]T, c8 = [ch 0 0]T, and c9 = [cb 0 0]T.

The kinematics expressions, (2.3) through (2.6), can be very large and complicated, particularly for the links furthest from the inertial reference frame. Therefore, all symbolic

23 manipulations are performed using Mathematica, computer software developed by Wolfram Research Inc. Mathematica also provides a convenient and orderly way to document steps in the derivation process.

2.4 Equations of Motion Derivation The system equations of motion are derived using Lagrange’s energy method, which takes advantage of the rigid-body nature of the structure [18].

Kane's method is another

popular technique for such a system [14,17]. In this method, generalized speeds are the degrees-of-freedom, which are nonlinear functions of the physical coordinates and their time derivatives. The advantage here is that equations of motion are relatively compact and hence numerical integration is faster. When analyzing a system with few degrees-of-freedom, however, the saving in simulation time is not an overwhelming advantage. In addition, with Kane's method, much physical insight is lost because equations are not expressed in terms of physical quantities. It is often beneficial to be able to identify terms in the equations of motion to understand dynamic phenomena and for validation purposes. Lagrange's method is therefore the method of choice. The Lagrangian is defined as L = Kt + Kr – V

(2.8)

where Kt is the total kinetic energy due to translational motion, Kr is the total kinetic energy due to rotational motion, and V is the potential energy due to conservative forces (i.e. from gravity and joint springs). Non-conservative forces are dealt with separately. Given the kinematic expressions of the previous section, the energy functions can be expressed as

24

Kt =

∑( N

i =1

0

0

mi vi . vi

∑(

)

1 0 ωi 2

. 0 I i ωi

)

∑ (m g . (

0 0 0 ri − ri

)+

Kr =

N

i =1

V=

1 2

N

i

0

i =1

(2.9) 2 1 K iqi 2

)

where mi is the mass of link i, 0

Ii is the matrix containing the principle moments of inertia at the center-of-mass,

referenced from the inertial frame, i.e. 0Ii = 0Ri Ii 0Ri with I ix I i =  0  0

0 I iy 0

0 0  , I iz 

g = [0 0 g]T, the gravitational acceleration vector in the inertial frame, 0 0 ri

is the position vector of link i in its ‘resting’ state ( q = 0 ), and

K i is the linear spring constant for the joint of link i.

Based on the SymDyn model assumptions described in Section 2.2, simplifications are made to the mass and inertia matrices. m1 = 0, I1 = 0 ,

m3 = m t ,

 I t lat  I3 =  0  0 

I n x  m5 = m n , I5 =  0  0 

m 2 = 0, 0 I t lat 0

0 In y 0

I2 = 0 ,

0   0 , I p z 

0   0 , I t long 

I p x  m4 = mp , I4 =  0  0 

0   0 , I n z 

0 0 0    m 6 = 0, I 6 = 0 0 0 , 0 0 I s long   

0 Ip y 0

25 I s lat  m7 = ms , I7 =  0  0 

0 I s lat 0

0   0 , I g long 

m 9 = m10 = ... = m N = m b , I 9 = I10

I h x  m8 = m h , I8 =  0  0 

I b long  = ... = I N =  0  0 

0 Ib − mb cb 0

0 Ih y 0

0   0 , I h z 

  0 , 2 Ib − mb cb  0

2

where Ib is the moment of inertia of a blade about its flap hinge, not its center-of-mass. In general, the subscripts {t, p, n, s, h, b} refer to the tower, bedframe, nacelle, shaft, hub, and blade respectively. The Lagrange’s equations of motion are formed from d  ∂L  d t  ∂ q& i

 ∂L  − = Q i , i = 1, 2, ..., N  ∂ qi

(2.10)

where Qi are the generalized forces due to nonconservative loads. The contributions are due to internal joint damping moments and applied loads. Qi = Qdamp i + Qload i

(2.11)

The joint damping component is given by Q damp i = C i q& i

(2.12)

where Ci is the linear damping coefficient. The generalized forces due to applied loads at the center of mass of link j, are given by 0

Q load i , j =

( R F ) . ∂∂ q&v + ( R 0

j

j

0

j

i

0

) ∂∂ q&ω

j Mj .

j

(2.13)

i

where Fj and Mj are the 3×1 force and moment vectors, aligned with the reference frame of link j. Summing over all bodies gives

26

Q load i =

N

∑Q

load i , j

.

(2.14)

j=1

With reference to the desired applied load locations in Figure 2-3 we have the following force and moment vectors. F1 = F2 = F3 = F4 = 0,

M1 = M2 = M3 = M4 = 0,

F5 = 0,

M5 = [0

F6 = 0,

M6 = 0,

F7 = 0,

M7 = [0

F8 = 0,

M8 = 0,

F9 = [Fb1x Fb1y Fb1z]T,

M9 = [Mb1x Mb1y Mb1z]T,

…

…

FN = [Fb’Nb’x Fb’Nb’y Fb’Nb’z]T,

MN = [Mb’Nb’x Mb’Nb’y Mb’Nb’z]T.

Tg

0]T,

0 -Tg]T, (2.15)

Since the blade loads are not be applied at the blade center of mass locations, an adjustment must be made to the linear velocity terms used in (2.13) to correct for this. The nonlinear equations of motion for the entire structure are very large and it would be overwhelming to present them all here. Instead, a sample set of equations is provided in Appendix A. At this stage it is convenient to represent the structural dynamics by the following vector equation

M(q ) &q& + f (q, q& , Tg , L) = 0 , where q = [q 1 q& =

dq dt

q2 , &q& =

... q N ] , T

d2 q dt 2

,

M is the familiar N×N mass matrix in terms of moments of inertia,

(2.16)

27 Tg is the generator torque, and L = [Fb1x … Mb1z … Fb’Nb’x … Mb’Nb’z]T, the vector of 6×Nb blade loads from (2.15).

Equation (2.16) lends itself conveniently to numerical integration once turbine properties are chosen. The computer application, Matlab with Simulink, by Mathworks Inc., is ideal to perform simulation studies, due to its graphical user interface and modular architecture. Implemented in Simulink, the subsystem block diagram for nonlinear dynamics (2.16) is illustrated in Figure 2-5. The ‘MATLAB Function’ block contains a reference to a script file where the equations of motion are defined. The initial conditions for simulation are set in the ‘Integrator’ blocks.

Nonlinear Structural Dynamics Nonlinear Structural Gen. Torque 2

1 qdot

Tg L

1 Aero. Loads

MATLAB Function

qdotdot

qdotdot = -Minv*f(q, qdot, Tg, L) q

1 s Integrator

qdot

1

q

s Integrator

2 q

qdot

Figure 2-5: Simulink block diagram for the nonlinear structural dynamics subsystem

An alternative approach for deriving the SymDyn equations of motion has been developed and is described in Appendix B. Instead of deriving the equations of motion symbolically (which can be extremely large for many degrees-of-freedom), we form the necessary expressions numerically from first principles. The advantage here is that the required storage for expressions is less and far fewer calculations are made for reduced order models. To realize the runtime savings, the expressions would need to be coded in a lowerlevel programming language, such as FORTRAN. To allow access to the equations and

28 perform the numerical integration in Matlab, one would require an interface – known as a MEX file in Matlab.

A description of this type of configuration is provided in the

Aerodynamics section. However, all simulations in this thesis are performed using the original implementation method.

2.5 Verification To verify the nonlinear SymDyn equations of motion we make a comparison of simulation results with ADAMS, version 9.1 [3].

The commercial ADAMS code, by

Mechanical Dynamics Inc., has been well verified to industry standards and is well capable of modeling the rigid-body turbine configuration of SymDyn. Structural properties resemble a generic teetered-rotor, two-bladed turbine, although the accuracy of the values to match a real turbine are immaterial for this study. A listing of all properties can be found in the first data column of Table C-1 (Appendix C). Numerical integration of the SymDyn equations of motion is performed in Matlab with Simulink using the default Runge-Kutta scheme with variable time steps. The ADAMS model time response is calculated using the default Gear integration method, also with variable time steps. A run time of 2.0 seconds was chosen to capture a sufficient sample of dynamic behavior. Reference [19] describes the verification study for cases 1-5 listed in Table 2-2. Results show almost exact agreement in each case, with minor differences later attributed to modeling errors in ADAMS.

We now augment the study with an additional case that

includes the effect of applied loads (case 6). All possible degrees-of-freedom except tilt are active in SymDyn, resulting in 9 degrees-of-freedom. Rotor speed has an initial condition of 57.5 rpm but is otherwise unconstrained, while all other initial conditions are zero. The timevarying forces on the blades are defined as follows.

29 Fb1x = Fb1y = Fb1z = 1000 + 200 cos(ω F t )

Fb 2 x = Fb 2 y = Fb 2 z = 1000 + 200 sin (ω F t )

M b1x = M b1y = M b1z = 10000 + 2000 cos(ω F t )

where ωF = 2 Hz (120 rpm).

M b 2 x = M b 2 y = M b 2 z = 10000 + 2000 sin (ω F t )

Test Case 1 2 3 4 5 6

Degrees-of-freedom 2 (β1, β2) 3 (φ, β1, β2) 4 (γ, φ, β1, β2) 5 (τ1, γ, φ, β1, β2) 7 (τ1, τ2, τ3, γ, φ, β1, β2) 9 (τ1, τ2, τ3, γ, ψ, ψg, φ, β1, β2)

Applied Loads No No No No No Yes

Initial Conditions 0 0 0 0 0 & ψ(0) = 57.5 rpm

Table 2-2: Verification test cases

The simulation results, comparing the ADAMS and SymDyn model responses, are shown in Figures 2-6 through 2-12. While the results agree very well, there is still a small but noticeable difference in many of the plots, particularly in the shaft twist response (Figure 2-10). In a separate trial case, using the same model but without applied loads, the ADAMS and SymDyn models match perfectly. The cause of the error in the present test case must then be due to the application of loads, either in ADAMS or SymDyn, or a numerical integration issue. Upon examination of the symbolic load expressions for SymDyn one discovers that they are analytically exact. This leads to the conclusion that it is the modeling in ADAMS that is likely at fault – possibly user error. Besides this unresolved discrepancy, all validation test cases show well enough agreement to conclude that confidence can be placed in SymDyn structural dynamics results.

30

Tower fore-aft angle [deg]

0.025 ADAMS SymDyn

0.02 0.015 0.01 0.005 0

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

Figure 2-6: Verification study results (9 DOF); tower fore-aft angle response x 10

Tower lateral angle [deg]

1.5

-3

ADAMS SymDyn

1 0.5 0 -0.5 -1 -1.5

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

Figure 2-7: Verification study results (9 DOF); tower lateral angle response

Tower twist angle [deg]

5

x 10

-5

ADAMS SymDyn

0

-5

-10

-15

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

Figure 2-8: Verification study results (9 DOF); tower twist angle response

2

31

0.2 ADAMS SymDyn

Yaw angle [deg]

0

-0.2

-0.4

-0.6

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

Figure 2-9: Verification study results (9 DOF); yaw angle response

Shaft twist angle [deg]

0 ADAMS SymDyn

-5

-10

-15

-20

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

Figure 2-10: Verification study results (9 DOF); shaft twist angle response

Teeter angle [deg]

1 ADAMS SymDyn

0.5

0

-0.5

-1

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

Figure 2-11: Verification study results (9 DOF); hub teeter angle response

2

32

Blade #1 flap angle [deg]

0 ADAMS SymDyn

-0.5 -1 -1.5 -2 -2.5

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

Figure 2-12: Verification study results (9 DOF); blade #1 flap angle response

2.6 Aerodynamics It would be ideal to have an analytic model of aerodynamics, as has been developed for the turbine structure, to facilitate linearization. However, the complexity of wind flow over modern turbine blades makes it difficult to characterize the distributed aerodynamic loads accurately. Instead, we make use of AeroDyn, an existing simulation code developed at the University of Utah [1]. Due to its popularity in the U.S. wind industry, AeroDyn has been coupled with many structural dynamics models, including YawDyn, FAST, and ADAMS. The FORTRAN source code for AeroDyn is in the public domain, which promotes feedback to the developers as well as frequent revisions. Essentially, the existing AeroDyn code uses blade element momentum theory to calculate lift and drag forces at prescribed elements along each blade length. Adjustments are made to take into account the vortex drag at the blade tips and induction factors are calculated in both the axial and tangential directions. Dynamic inflow and dynamic stall options are available but are not used in the present implementation with SymDyn. In doing so, loads are calculated in a quasi-steady fashion, which assumes that local wind velocity and blade loading are in equilibrium at each instant of time.

33 For the purposes of specifying the dominant wind characteristics, a simple hubreferenced wind field is used, defined by the parameters given in Table 2-3. This consists of a uniform wind field over the rotor swept area, modified by horizontal and vertical shear factors, as shown in Figure 2-13. A further description can be found in AeroDyn literature [1]. AeroDyn Symbol V δ VZ HSHR VSHR VlinSHR VG

Description Horizontal hub-height wind speed Wind direction Vertical wind speed Horizontal shear factor (linear) Vertical shear factor (power law) Vertical shear factor (linear) Gust speed

Table 2-3: Hub-referenced wind field parameters in AeroDyn

V

zhh

V(z)

V(z) = V.(z z hh ) VSHR

z

Figure 2-13: Vertical wind shear (power law) in AeroDyn

For simplification it is assumed that all wind parameters are constant in time except the horizontal wind speed. The aerodynamic loads are now a function of rotor position, rotor velocity, blade pitch and wind speed only. This is summarized by the following load vector function.

(

L = L q, q& , w, θ

)

(2.17)

34 where w is the horizontal wind speed (V in AeroDyn), and θ = [θ1 … θNb]T is the vector of full-span blade pitch angles. Positive pitch is defined as the direction that rotates the leading edge of the blade into the wind, toward feather.

Therefore, increasing pitch decreases the

angle of attack. For a twisted blade, the angle between the plane of rotation and the chord of the blade element is the sum of the element twist angle and the pitch angle. In some literature, this total angle is referred to as the pitch angle. Reference [1] provides further details.

2.6.1

Implementation in Matlab To interface the aerodynamics with SymDyn, a number of FORTRAN subroutines

are written to access the main AeroDyn code, aerosubs.f. The new subroutines are in fact modifications of two YawDyn source files; aero.f and bldfm.f. These files are listed in Appendix D. The aero.f program essentially loops over the number of blades, calculates the aerodynamic loads on each blade with calls to bldfm.f, and assigns values to the load vector (L in Eqn (2.17)). The bldfm.f program loops over the elements in each blade to calculate the lift and drag forces. Essential to this calculation are three kinematic expressions; the position vector of the element relative to the tower top, the velocity of the element expressed in the blade reference frame, and the velocity of the wind, also expressed in the blade reference frame. These expressions can be easily deduced from SymDyn as follows. First consider blade #1 and element i, located a distance xi from the flap hinge. The relative position vector of this element is given by c b → x i 0 0 ( r9 − r4 ) ∋   cp → 0

(2.18)

where the notation cbØxi means that the variable cb is replaced by xi in the expression. The velocity of the element in the blade frame is given by

35

−0 R 9

T0

v9 ∋ cb → x i .

(2.19)

The negative sign appears because it is the velocity of the air relative to the blade that is required. Finally, the velocity of the wind in the blade frame is given by 0

R 9 [VX VY VZ] T

T

(2.20)

where VX, VY, and VZ are the x,y,z components of the wind in the inertial frame. Expressions for the other blades can be derived using the transformation described in the kinematics section (see Eqn (2.7)). To allow FORTRAN subroutines to be called from within Matlab they must first be compiled into an executable file. Since Matlab is based on the C++ programming language, all workspace variables that are called in or out must have memory dynamically allocated for them.

This is achieved with the Matlab specific gateway program, aerog.f, listed in

Appendix D (Listing 3). The complete executable file is known as a MEX file in Matlab. Figure 2-14 summarizes the Matlab-SymDyn-AeroDyn interface. In addition to the input provided by Matlab, AeroDyn requires airfoil data (a lift/drag coefficient table) and blade element data. These properties are provided by the input file, yawdyn.ipt. An example of this file is provided in Appendix D (Listing 4).

yawdyn.ipt

Blade element geometry and AeroDyn options

q, q& , w , θ aerog.f

MATLAB

aero.f

bldfm.f

aerosubs.f

L Matlab Interface (Gateway)

SymDyn Interface

Compiled as executable (MEX file) Figure 2-14: Summary of the Matlab-SymDyn-Aerodyn interface

AeroDyn Subroutines

36 The Simulink block diagram for the aerodynamics subsystem is illustrated in Figure 2-15. The ‘MATLAB Function’ block contains a call to the compiled MEX file and the ‘wparams’ input contains all the wind parameters from Table 2-3, except horizontal wind speed. In this form, aerodynamic loads can be calculated within simulations and results are directly accessible from the Matlab workspace. Aerodynamics 4 q

3 qdot

MATLAB Function

2 pitch

1

1 L

AeroDyn MEX

w wparams

Figure 2-15: Simulink block diagram for the aerodynamics subsystem

2.6.2

Example Results Using the aerodynamic subsystem in Matlab, a number of common aerodynamic

characteristics can be derived. We use the two-bladed turbine properties provided in the input file of Listing 4 (Appendix D). Each blade is made up of 15 elements from 3 different airfoil data tables with twist included. For the calculation of aerodynamic loads we assume the structure is undeformed and the rotor is spinning at a constant speed, ψ& = Ω . We also assume that the wind field is characterized by hub-referenced parameters as described earlier. In particular, the vertical shear factor (VSHR) is constant at 0.2, the horizontal wind speed is a variable, w, and all other quantities are zero. The torque coefficient is defined as Cq =

Ta 1 2

ρπR 3 w 2

(2.21)

37 where Ta is the aerodynamic torque on the rotor shaft, ρ is the air density (1.0251 kg/m3), and R is the rotor radius (13.0 m). The power coefficient is defined as Cp =

Ta Ω 1 2

(= λ C q ) .

ρπR 2 w 3

(2.22)

Both coefficients are often presented as functions of tip speed ratio, defined as λ = (ΩR)/w, and collective blade pitch, θc. For the given properties, the contour plots for the torque and power coefficients are shown in Figures 2-16 and 2-17 respectively. The torque coefficient plot illustrates the regions of stable and unstable operation, demarcated by the dashed line in the figure. At low λ and low θc the blades are stalled. In this region, an increase in rotor speed (with constant wind speed) increases the aerodynamic torque, which

15

15

10

10

Collective pitch, θc [deg]

Collective pitch, θc [deg]

further accelerates the rotor, hence local instability. In the stable region, the reverse is true.

Stable

5

Unstable 0

0.04

0.03 0.02

0.01

0

-5

-10

0

5

10

15

20

Tip Speed Ratio, λ

Figure 2-16: Torque coefficient contour plot

5 0

0.3

0.2

0.1 0

-5

0

5

10

15

20

Tip Speed Ratio, λ

Figure 2-17: Power coefficient contour plot

The power coefficient plot provides the most important information to turbine designers, as it pertains to potential energy production. For a given wind speed, the peak aerodynamic power occurs at λ ≈ 8.7 and θc ≈ 0°, capturing 34% of the available wind kinetic

38 energy over the rotor swept area. The turbine is most efficient when operating at the power coefficient peak, which corresponds to the main control objective for Region II on a variable speed machine. Note that at the peak, pitch control is ineffective, which leaves generator torque as the desirable control method. Beyond the wind speed at which maximum allowable power is produced (Region III), the turbine must be operated off the peak. By keeping generator torque constant in this region, blade pitch can be utilized to maintain constant rotor speed and therefore constant output power. For the modeled turbine, the desired operating rotor speed is Ω0 = 57.5 rpm, resulting in a rated wind speed of 9 m/s. Under ideal conditions the blade pitch variation with wind speed is shown in Figure 2-18. Below rated wind speed, the pitch can be kept constant at 0° for peak power efficiency. Feasibly, one could schedule pitch angles based on known wind speed and use generator torque to adjust for variations in turbine speed due to aeroelastic effects. Such an approach is taken by [7]. A problem encountered with this method is the large fluctuations in power and therefore an increase in undesirable structural loads. In the present study, rotor speed is regulated by active pitch control only.

25

Collective pitch [deg]

20 15 10 5 0

8

10

12

14

16

18

20

22

24

Wind speed [m/s]

Figure 2-18: Desired blade pitch variation versus wind speed for constant aerodynamic torque & = 57.5 rpm). Chosen operating point marked with an ‘X”. (Tg = 14.0 kNm, ψ

39 2.7 Nonlinear Aeroelastic Model We now combine SymDyn and AeroDyn to form the complete aeroelastic turbine model. From Eqns (2.16) and (2.17), the vector representation of the equations of motion is

(

(

))

M(q) &q& + f q, q& , Tg , L q, q& , w , θ = 0 .

(2.23)

Implemented in Matlab, the block diagram for the complete nonlinear turbine model is shown in Figure 2-19.

Nonlinear Aeroelastic Dynamics

1 Wind Speed

w theta

2 Blade Pitch

Aero Loads, L

qdot

qdot q Aerodynamics

Tg Generator Torque

q

2 qdot

Nonlinear Structural Dynamics

1 q

Figure 2-19: Simulink block diagram for the nonlinear aeroelastic subsystem

2.7.1

Example Results Consider the two-bladed turbine model referred to in Section 2.6, with structural

properties as listed in the second column of Table C-1 (Appendix C), and the following four active degrees-of-freedom. q = [τ1

ψ β1 β 2 ]

T

(tower fore-aft angle, azimuth angle, blade #1 flap, blade #2 flap)

Based on the ideal blade pitch profile in Figure 2-18 and a cut-out wind speed of 22.5 m/s, the variation of pitch with wind speed is fairly linear above 10 m/s. A suitable operating

40 point for control design is shown by an ‘X’ in the figure, defined by wop = 16 m/s and θop = [15° 15°]T. Recall that the assumed rotor speed is Ω0 = 57.5 rpm. A time response for the four degrees-of freedom nonlinear model is calculated with the given operating wind and pitch inputs until a steady-state solution is found. From (2.23) we find the solution of

(

)

(

M(q op ) &q& op + f q op , q& op , Tg , L q op , q& op , w op , θ op = 0 .

(2.24)

Due to deflection of the tower and blades and the vertical wind shear, the steady-state solution is periodic. The periodic, T, is equal to the time taken for one rotor revolution,

& op , is also 2π/Ω0 = 1.04 seconds. The actual rotor speed component of the solution, ψ periodic, but we are able to ensure that the mean of the variation is equal to our desired speed, Ω0, by selection of generator torque.

For a rotor speed of 57.5 rpm, the required generator

torque is Tg = 13.4 kNm (not the ideal torque of 14.0 kNm, as used for Figure 2-18 when the structure was undeformed). Selected components of the steady-state solution (q op , q& op , &q& op )

Tower fore-aft angle, τ1 [deg]

are plotted in Figures 2-20 through 2-23. 0.066 0.065 0.064 0.063 0.062

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [sec]

Figure 2-20: Tower fore-aft angle variation in the periodic steady-state solution

Blade #1 flap angle, β1 [deg]

41

-2

-2.5

-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [sec]

Blade #2 flap angle, β2 [deg]

Figure 2-21: Blade #1 flap variation in the periodic steady-state solution

-2

-2.5

-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [sec]

Figure 2-22: Blade #2 flap variation in the periodic steady-state solution

Rotor speed [rpm]

57.51 57.505 57.5 57.495 57.49

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [sec]

Figure 2-23: Rotor speed variation in the periodic steady-state solution

2.8 Linearization A linear representation is obtained by first perturbing each of the system variables

(q, q& , &q&, w , θ) about a desired operating point (q op , q& op , &q& op , w op , θ op ) .

42

q = q op + ∆q, q& = q& op + ∆q& , &q& = &q& op + ∆&q&, w = w op + ∆w, θ = θ op + ∆θ

Since generator torque, Tg, is assumed to be a known constant, a perturbation of this variable is not required. Substituting the above expressions into (2.23) and expanding as a Taylor series approximation, the linear system that results is M ( t ) ∆ &q& + G ( t ) ∆ q& + K ( t ) ∆ q = E ( t ) ∆w + H ( t ) ∆ θ

(2.25)

where ∆q is the vector of perturbed angular displacements, ∆w is the perturbed wind speed, ∆θ is the vector of perturbed blade pitch angles, M ( t ) = M (q op ), the mass matrix,  ∂f ∂f ∂ L   , the damping/gyroscopic matrix, + G(t) =   ∂ q& ∂ L ∂ q&    op  ∂M (q ) ∂ f ∂ f ∂ L  &q& + + K(t) =  , the stiffness matrix,  ∂q ∂ q ∂ L ∂ q   op E( t ) = − H( t ) = −

∂f ∂ L ∂ L ∂w

op

∂f ∂L ∂ L ∂θ

op

, the wind transmission vector , and , the blade pitch transmission matrix.

Finding the Jacobian matrices for the structural dynamics (∂f/∂q, …, ∂f/∂L) is straightforward because SymDyn provides a symbolic representation of terms and therefore derivatives are exact. Without a symbolic form for aerodynamics, calculating the remaining Jacobians (∂L/∂q, …, ∂L/∂θ) requires numerical differentiation. Cubic spline interpolation with periodic end conditions is used to minimize numerical errors resulting from this calculation.

43 The linear system parameters (M, G, K, E, H) are functions of the chosen operating point. Due to the inherent periodicity of a wind turbine, the linear set of equations are periodic in time, with period, T. That is, M(t) = M(t+T), G(t) = G(t+T), etc. This does not imply that the solution of the system, ∆q , is periodic. For control system design and modal analysis, it is convenient now to transform the linear equations into first-order state-space form. x& = A ( t ) x + B( t ) u + B d ( t ) u d

(2.26)

where

I N× N   0 A( t ) =  N×−N1 −1  , the state matrix, − M K − M G  0  B( t ) =  N−×1N b , the control input matrix, M H   0 B d ( t ) =  N−1×1 , the wind input matrix, M E  T x (t) = ∆q ∆ q& , the state vector of perturbed coordinates,

[

]

u = ∆θ, the control input (vector of blade pitch angles), and ud = ∆w, the wind disturbance input.

The dimension of this system is n = 2N, where N is the number of degrees-offreedom. Reducing the order of the system is achieved by removing the rows and columns from the state matrices, corresponding to the degrees-of-freedom (and velocities) that one wishes to restrain. This is identical to the technique employed in the finite element method to constrain boundary degrees-of-freedom [20].

2.9 Summary The SymDyn model is built in a systematic fashion. Kinematics are derived using the Denavit-Hartenberg convention adopted from the robotics field. By using this method it

44 is a straightforward process to restructure the model, such as if a new degree-of-freedom was required.

Also, there is little human intervention during the derivation process, as all

symbolic manipulations are performed in Mathematica. From the many simulation results comparing SymDyn and ADAMS, confidence can be placed in the equations. One of the many conveniences of Mathematica is that the derived equations of motion (linear or nonlinear) can be exported in Matlab format allowing simulation in the more numerically efficient environment. Since the equations are in symbolic form, the process of reducing the order of the system is as simple as setting the unwanted degrees-offreedom to zero. Of course, by importing a large order system in Matlab, the equations can be very long, which slows down the simulations immensely. Therefore, to simulate large SymDyn models, use of the alternate described derivation method is recommended. Implementing the aerodynamics as a separate system in Matlab utilizes the modular architecture of the simulation environment, Simulink. As newer versions of the AeroDyn subroutines become available, the new FORTRAN code simply needs to be recompiled with the interface files. If extra degrees-of-freedom are added to the SymDyn structure, then these will need to be included in the position and velocity expressions coded in AeroDyn. This may prove to be somewhat labor intensive for those not familiar with FORTRAN and Mathematica.

45 3. CHAPTER 3 MODAL ANALYSIS STUDY

3.1 Introduction This chapter introduces the modal analysis techniques for linear periodic systems, based on Floquet-Lyapunov theory.

Later we present modal analysis results for the

developed wind turbine model. Section 3.2 outlines methods used in the past to derive the operating modes of a wind turbine.

Section 3.3 describes the Floquet approach.

The

following two sections (3.4 and 3.5) provide numerical results, first a detailed study without aerodynamic effects and then a short example using the complete aeroelastic turbine model.

3.2 Background The modes of a wind turbine are often determined with its rotor in a parked position. While such modes help validate a turbine’s structural properties, they do not capture the dominant centrifugal and gyroscopic effects associated with an operating turbine. Operating turbine modes identify mechanisms and couplings that underlie critical loading conditions. If correctly identified, these modes help in designing efficient controls that mitigate loads and improve stability. Because of the complexity of the dynamics of wind turbines, characterized by rotating components coupled with stationary components, only a limited number of researchers [21-28] have attempted computation of operating turbine modes. These attempts use time-intensive simulations, followed by post-processing of time response data to extract the modal frequencies, and implicitly approximate the turbine as a time-invariant system. As described in the previous chapter, a wind turbine is inherently time varying. If the turbine has three or more blades and aerodynamics is ignored, the structural symmetry of its rotor ensures that a multi-blade coordinate transformation [29] is sufficient to transform the system time-periodic equations into a set of time-invariant equations.

46 Therefore, a conventional modal identification technique may be used to compute the modes. An earlier paper [28] used this approach to compute the operating modes of a three-bladed wind turbine. However, due to lack of a turbine analytical model in an explicit form, modal computations required a time-intensive post-processing of data from several simulations. The development of the SymDyn structural model, which provides the linear equations of motion in an explicit form, allows a direct modal analysis. While SymDyn does not capture highorder vibration modes in the tower and blades, it does retain the dominant physics of a rotating wind turbine. Use of SymDyn for modal analysis studies has been previously presented in [30] and [31].

3.3 Floquet Approach The Floquet-Lyapunov theory for periodic systems is briefly outlined in this section. For further details, one could refer to [4], [29] or [30]. Consider the homogenous form of the state-space SymDyn equations of motion (2.26). x& = A( t ) x .

(3.1)

[

Recall that x = ∆q ∆q&

]

T

is the state vector of perturbed coordinates, and A(t) is the

periodic state matrix with period T, i.e. A(t) = A(t+T). The solution of the linear time-varying system, (3.1), can be found in terms of the state transition matrix, Φ ( t ,0) :

x ( t ) = Φ (t ,0) x (0) .

(3.2)

47 The essence of Floquet-Lyapunov theory for periodic systems is that the state transition matrix can be decomposed into a periodic matrix and the matrix exponential of a constant matrix: Φ(t ,0 ) = P(t ) e

At

(3.3)

where P( t + T ) = P( t ), P(0) = I, and A=

1 ln (Φ (T,0 )) T

(3.4)

Since P( t ) is bounded, the stability of x ( t ) is governed by the eigenvalues of A , referred to as the characteristic exponents of A( t ) . The eigenvalue problem is

A = VΛV −1

(3.5)

where Λ is a diagonal matrix containing the characteristic exponents, λ i . Now consider the eigenanalysis of Φ (T,0) , sharing the same modal matrix, V : Φ(T,0) = V Σ V −1

.

(3.6)

Here, Σ is the diagonal matrix of eigenvalues, termed the characteristic multipliers, σ i , of the system. From (3.4) the relationship between (3.5) and (3.6) is Λ=

1 ln Σ T

(3.7)

which suggests a convenient method to calculate the characteristic exponents:

 Im(σ i ) 1 1 + 2πk  ln σ i + j  tan −1 Re(σ i ) T T  = ζ i + jω i .

λi =

(3.8)

where j = − 1 . The unknown integer, k , appears because there are infinite branches in the logarithm of a complex number. Note that when σ i is complex, then λ i is complex with a

48 corresponding complex conjugate (since Φ is real). However, when σ i is real, then λ i is, in general, complex and there need not be a conjugate to form a complex pair. This behavior of periodic systems is detailed further in [29] and will become apparent in later results. In view of the foregoing development, the Floquet approach for modal analysis comprises the following four steps. 1. Compute the transition matrix after one period, Φ (T,0) . From (3.1) and (3.2) the transition matrix can be computed by integrating & (t ,0 ) = A (t ) Φ (t ,0 ) Φ

(3.9)

with linearly independent initial conditions; Φ (0,0) = I , the identity matrix. 2. Compute the characteristic multipliers, σ i , by eigenanalysis of Φ (T,0) . From (3.6), the eigenanalysis of Φ (T,0) computes both the modal matrix, V , and matrix of characteristic multipliers, Σ . 3. Compute damping and frequency for each mode. From (3.8), the modal damping coefficient is given by ςi =

1 ln σ i , T

(3.10)

and the modal frequency is given by ωi =

 Im(σ i ) 1  tan −1 + 2πk  . T Re( σ i ) 

(3.11)

The unknown, k, can be determined by time response of the system and subsequent frequency identification. The system is stable when ζ i < 0 for all modes. 4. Identify the modes. The modal matrix, V , computed in step 2 contains the eigenvectors for all modes. Comparing the entries in each modeshape helps to identify the dominant motion. Note

49 that the actual modeshapes are computed from P( t )V and therefore are periodic in time. Since P(0) = I , the modal matrix provides the system modeshapes at t = 0 .

A key step in the application of the Floquet analysis is the computation of the state transition matrix over one period. This poses no problem for a system with a small number of degrees-of-freedom.

However, if the number of degrees-of-freedom become large,

computational cost can become overwhelming. A few researchers [33-36] have proposed innovative extensions of the Floquet theory to deal with large-order systems and improve accuracy. The models we are analyzing in this thesis have, at the most, seven degrees-offreedom. Therefore, it is convenient to apply the classical Floquet approach directly.

3.4 Structure-Only Analysis In this section, modal properties for free vibration of the SymDyn model are investigated. All gravity and aerodynamic loads are set to zero. As a consequence, we assume zero values for the operating point for linearization ( q op ( t ) = q& op ( t ) = &q& op ( t ) = 0 in the derivation of Section 2.8). The exceptions are the operating rotor speed, which is assumed to & op ( t ) = Ω , and the operating azimuth position, ψ op = Ω t . be constant ψ

(To simplify

notation in this section, we drop the ‘op’ subscript for the azimuth position and use the symbol ψ.) The governing equations of motion then have a period of T = 2π Ω . A two-bladed turbine design is analyzed with the geometric, mass, stiffness, and damping properties listed in second column of Table C-1 (Appendix C). This is consistent with the other numerical results presented up to this point.

Four different sized SymDyn

models are considered with increasing complexity, as listed in Table 3-1. The first turbine model has only the yaw degree-of-freedom active. This is the simplest model that exhibits

50 periodicity. The three-DOF model contains hub teeter and blade flap degrees-of-freedom. The four-DOF model adds yaw and the seven-DOF model also allows motion of the tower.

Model one-DOF three-DOF four-DOF seven-DOF

Degrees-of-freedom (γ) (φ, β1, β2) (γ, φ, β1, β2) (τ1, τ2, τ3, γ, φ, β1, β2)

Table 3-1: Turbine models analyzed

As an example of the terms that appear in the linear turbine model, the equations of motion for the four-DOF model is presented next.  m γγ m  γφ  m γβ  − m γβ

m γφ m φφ m φβ − m φβ

m γβ m φβ m ββ 0

− m γβ   g γγ g − m φβ  φγ ∆&q& + Ω  g βγ 0    m ββ  − g βγ

where ∆q = [∆γ ∆φ ∆β1

∆β 2 ]

T

g γφ g φφ 0 0

g γβ 0 g ββ 0

− g γβ  k γγ  0 0  ∆q& + Ω 2   0 0    g ββ   0

k γφ k φφ k φβ − k φβ

k γβ k φβ k ββ 0

− k γβ  − k φβ  ∆q = 0 0   k ββ  (3.12)

The following terms use the simplified notation

.

s β = sin β 0 and c β = cos β 0 . Other symbol definitions can be found in Appendix C.

The entries in the mass matrix are:

(

)

mγγ = I nx + I s _ lat + m n c n + m s c s + m h (d n 2 − c h ) + cos 2 ψ I hy + 2 s β I b + 2c β I blong + 2

2

2

(

2

(

2

)

) ))))

sin 2 ψ (I hz + 2 I b ) + 2 m b (d n 2 − d h1 ) + sin 2 ψ d h 2 d h 2 + 2c β c b + 2 s β c b (d n 2 − d h1 ) 2

(

(

(

mγφ = sinψ I hz + 2 I b + mh c h (c h − d n 2 ) + 2mb d h1 + d h 2 − d h1 d n 2 + cb 2c β d h 2 + s β (d n 2 − 2d h1

(

2

2

2

2

(

2

mφφ = I hz + mh c h + 2I b + 2mb d h1 + d h 2 + 2cb c β d h 2 − s β d h1

(

(

))

))

mγβ = sin ψ I b + mb cb c β d h 2 + s β (d n 2 − d h1 )

(

mφβ = I b + mb cb c β d h 2 − s β d h1

)

m ββ = I b

The entries in the gyroscopic/damping matrix are:

(

2

(

)

(

))

g γγ = sin 2ψ I hz − I hy + 2c β I b − I blong + 2mb d h 2 d h 2 + 2c β c b +

Cγ Ω

51

( )) ]

)

g γφ = cosψ [ I hz + I hy − I hx + 2m h c h (c h − d n 2 ) + 2 s β 2 I b − I blong + 2c β I blong + 2

(

4mb d h1 − d h1 d n 2 + s β c b (d n 2 − 2d h1

( ( = cosψ (I − I

2

)

2

)

g γβ = cosψ s β 2 I b − I blong + c β I blong + 2 s β mb cb (d n 2 − d h1 ) 2

g φγ

g φφ =

hz

2

2

hy

(

)

))

Cφ Ω

( (

)

2

2

g βγ = cosψ c β 2 I b − I blong + s β I blong + 4c β mb cb d h 2

g ββ =

(

2

+ I hx + 2c β 2 I b − I blong + 2 s β I blong + 4mb d h 2 d h 2 + 2c β cb

)

Cβ Ω

The entries in the stiffness matrix are: k γγ =

Kγ Ω2

(

)(

2

k γφ = sin ψ [ I hx − I hy + 2 I b − I blong c β − s β

(

)

2

) + m c (d h h

− c h ) + 2mb ( d h 2 − d h1 + d h1 d n 2 + 2

n2

2

c b s β (2d h1 − d n 2 ) + 2c β d h 2 )]

((

)(

2

k γβ = sin ψ I b − I blong c β − s β

2

) + m c (s b b

(

2

)(

β

2

(d h1 − d n 2 ) + c β d h 2 ))

kφφ = I hx − I hy − m h c h + 2 I b − I blong c β − s β

(

)(

2

(

)(

2

kφβ = I b − I blong c β − s β

k ββ = I b − I blong c β − s β

2

) + m c (s b b

2

)+ c

β

β

2

) + 2m (d

d h1 + c β d h 2

mb c b d h 2 +

b

2 h2

2

(

))

− d h1 + 2cb s β d h1 + c β d h 2 +

Kφ Ω2

)

Kβ Ω2

Note that the inertia, gyroscopic, and stiffness matrices depend on the rotor azimuth position, ψ , and on the turbine geometric, mass, stiffness, and damping properties. Also note that the gyroscopic matrix is multiplied by Ω , the rotor speed, and the stiffness matrix is multiplied by Ω 2 . The physical interpretation of all terms appearing above is outside the intent of this thesis. However, a few observations deserve mention here and in the analysis of selected results later. All the gyroscopic terms, with one exception, cross couple the degrees-of-

52 freedom. This implies that, owing to rotor rotation, a motion in one degree-of-freedom would induce a gyroscopic motion in another degree-of-freedom.

The exception is the direct

gyroscopic term, g γγ . A yaw motion of the nacelle obviously causes the spinning two-bladed rotor also to yaw, and this induces a gyroscopic moment about the tower yaw axis. The important point to note is that this self-induced gyroscopic moment is a multiple of sin 2ψ and therefore shows 2p variation, where p denotes per rotor revolution. Rotors with three of more blades do not experience this variation due to symmetry. All the other terms in the matrix represent gyroscopic cross coupling with a 1p variation. We also note that the flap and teeter motions are gyroscopically decoupled. This is because both of these motions occur about parallel axes and are referenced to the same hub-fixed rotating frame.

3.4.1

One-DOF Model: Yaw only Assuming zero yaw stiffness and damping, and locking all degrees-of-freedom

except yaw, expressions in the previous section provide the single governing equation:

m γγ (ψ) ∆&γ& + g γγ (ψ) ∆γ& = 0 .

(3.13)

The yaw inertia, m γγ , and the gyroscopic damping, g γγ , show a 2ψ periodicity. Because ψ = Ω t , the azimuth represents non-dimensionalized time. Researchers unfamiliar with periodic systems may attempt a conventional eigenanalysis with the intent to compute timevarying eigenvalues. Such an analysis provides two eigenvalues. One has zero real and imaginary parts, implying an undamped rigid-body yaw mode. The other eigenvalue has a zero imaginary part, implying zero frequency, and an azimuth-dependent real part, implying a time-varying damping. Figure 3-1 shows the azimuth-dependent real part from this analysis, normalized with respect to the rotor speed. (Henceforth, all the eigenvalues will be shown normalized with respect to the rotor speed.) The yaw damping shows a 2p variation. In this

53 form we cannot deduce information about the stability of the system and therefore the conventional eigenanalysis is not appropriate.

The actual mechanism involved is

conservation of yaw angular momentum, while the total system energy is not conserved. Consider the yaw response shown in Figure 3-2 due to a unit velocity initial condition. The yaw rate exhibits a 2p variation to conserve the angular momentum about the yaw axis given the 2p variation in inertia, m γγ . The instantaneous total energy is not conserved due to a periodic transfer of energy between the generator and the nacelle. This is made possible by the constraint that rotor speed is constant.

Real Realpart partof ofeigenvalue eigenvalue //Ω Ω 00

1.5 1 0.5 0 -0.5

0

45

90

135

180

225

270

315

360

-1 -1.5 Rotorazimuth azimuth angle, angle, ψ [deg] [deg] Rotor

Figure 3-1: Yaw modal damping variation from conventional modal analysis

The Floquet approach provides the correct modal analysis of the system. This approach yields two zero characteristic exponents implying two rigid-body modes, both undamped. The first mode is a constant yaw-position rigid-body mode and the other is a constant yaw-velocity mode modulated by a time-periodic modal amplitude. The modal amplitude for this second mode is contained in the periodic modeshape. The Floquet analysis result is consistent with the yaw response in Figure 3-2, which shows zero damping behavior.

54 1.2

Yaw Response

1 0.8

Yaw rate

0.6 0.4 Yaw angle

0.2 0 0

45

90

135

180

225

270

315

360

Rotorazimuth azimuthangle, angle,ψψ[deg] [deg] Rotor

Figure 3-2: Yaw response due to unit velocity initial condition

3.4.2

Three-DOF Model: Teeter and Flap only For this model, all joint damping values are assumed to be zero and there is no teeter

spring. The sixth-order linear equations of motion for this system are time-invariant (not periodic) because all degrees-of-freedom are in the rotating frame. Therefore, both a standard eigenanalysis and Floquet analysis produce the same results. At the nominal rotor speed,

Ω 0 = 57.5 rpm , the three pairs of complex conjugate characteristic exponents are: {0 ± 3.227j, 0 ± 6.364j, 0 ± j0.993}.

As in the rest of this chapter, these modes are normalized with respect to the rotor speed. The zero real parts of the exponents imply that all modes are undamped. The following vectors and their conjugates represent the corresponding modeshapes. The velocity components are removed.

Teeter Blade #1 flap Blade #2 flap

 0∠0 0   0.26∠0 0   1.0∠0 0   0 0  0  1.0∠0 , 1.0∠180 , 0.002∠180 . 1.0∠0 0   1.0∠0 0   0.002∠0 0       

The entries in each vector represent the amplitude and phase contributions from each degreeof-freedom. The first mode is the collective-flap mode, wherein the blades flap in-phase and with equal amplitude. The second mode is the differential-flap mode and here there is a

55 significant coupling of the rotor differential motion with the teeter motion. The third mode is predominantly a teeter mode with very little contribution from the rotor differential motion. The fan-plot in Figure 3-3 shows that all modal frequencies increase with the rotor speed. The collective mode frequency is predominantly dictated by the centrifugal stiffening effect associated with the k ββ term in the stiffness matrix (see (3.12)). Due to complete decoupling of flap deflection in this mode we can in fact calculate the normalized modal frequency by the square root of ( k ββ / m ββ ). The frequency asymptotically approaches 1.70p as the flap spring stiffness disappears from k ββ :

k ββ lim  ωβ  lim   = Ω → ∞  Ω  Ω → ∞ m ββ =

((

)(

)

1 2 2 I b − I blong c β − s β + c β m b c b d h 2 Ib

)

(3.14)

= 1.7048 using the properties in Table C-1.

The differential mode frequency is determined by the hub oscillation resisted by the centrifugal pulling of the two blades. One need not be inordinately concerned with the differential mode. Its high frequency implies that it would require substantial energy for sustenance, and even a small amount of flap damping would easily suppress it (as will be shown in the next test case). The teeter mode frequency follows the 0.993p trend for all speeds. In fact, it would also be 0.993p if the flap spring were made rigid, owing to the weak blade-flap coupling. This property allows us to predict the normalized teeter frequency very accurately using the square root of ( k φφ / m φφ ) from (3.12).

Normalized modal frequency, ω /Ω0 Normalized modal frequency, ω/Ω 0

56

9 8 7 6 5 4 3 2 1 0

Differential Flap

Collective Flap

1.70p 1p

Teeter

0

0.5

1

1.5

2

2.5

Normalized rotor Rotorspeed, Speed, Normalized /Ω0 0 Ω Ω/Ω

Figure 3-3: Fan-plot for the three-DOF model (Ω0 = 57.5 rpm)

3.4.3

Four-dof Model: Yaw, Teeter, and Flap only This model represents an eighth-order periodic system. At the nominal rotor speed

the Floquet analysis yields the following characteristic exponents: {0, 0, 0±3.227j, 0±6.371j, -0.0956-1j, 0.0956+1j} The first two exponents correspond to the same yaw modes that were calculated in the first test case; two rigid-body modes. Neither teeter nor flap motions participate in these modes. The second and third pair of exponents represent the rotor collective and differential-flap modes and are hardly affected by the yaw degree-of-freedom. The last two eigenvalues represent a split teeter mode, with one being stable and the other unstable respectively. The stable mode has a decay rate of 0.0956p and the unstable mode has a growth rate of equal magnitude. However, both modes have exactly the same frequency of 1p. The stable mode pumps energy into the unstable mode. Because there is no external source of excitation, this is a self-excited instability. The corresponding modeshapes for these two modes are:  1.0∠1800   1.0∠0 0   0   0   0.076∠0 ,  0.076∠0 . Blade #1 flap  0.0003∠0 0   0.0003∠0 0      Blade #2 flap 0.0003∠1800  0.0003∠1800  Yaw Teeter

57 The vectors are identical except that in the first, corresponding to the stable teeter mode, the yaw motion is out of phase with teeter motion. In the unstable mode the yaw motion is in-phase with teeter motion. Recall that the actual system modeshapes are periodic and that those calculated here correspond to a single rotor orientation, ψ = 0° , when the teeter axis is perpendicular to the yaw axis. At this instant the gyroscopic moment transfer from the rotor to the nacelle is at a maximum. This explains why, although we are analyzing a teeter mode, that the yaw amplitude is about 13 times the teeter amplitude. The large angular momentum associated with the spinning rotor induces large gyroscopic moments about the yaw axis with even a small change in the rotor plane. The mechanism underlying gyroscopic moment transfer is actually somewhat more complicated because the teeter-hinge direction alters periodically as the rotor spins. We have shown that by adding the yaw degree-of-freedom the system is made unstable.

In practice, free-yaw teetered-rotor wind turbines are stable, due to large

aerodynamic forces that oppose structure deflection and velocity. For now we investigate the effect of structural stiffness and damping properties on stability. Adding blade damping at the flap hinges has the expected result of increased damping factors in both the collective and differential flap modes; particularly the latter, which has a higher natural frequency.

For example, with a flap damping of Cβ = 10

kNms/rad the collective and differential flap characteristic exponents become {-0.565 ± 3.177j, -2.373 ± 5.913j} respectively. The yaw and teeter modes are not noticeably affected at all, due to weak blade-flap coupling.

58 Including teeter damping alone has no effect on the yaw and collective flap modes but does stabilize the teeter and differential flap modes, as shown in Figure 3-4. Recall that a positive real part of a characteristic exponent indicates instability. Here we see that the differential mode and one of the teeter modes are stable for all values of teeter damping, while the second teeter mode is only stable for damping values of 31 kNms/rad or greater. Meanwhile, the teeter mode frequencies remain locked at 1p, while the differential mode frequency decreases slightly from 6.371p to 6.348p over the teeter damping range (not

Real partpart of teeter mode, Real of mode, ζ/Ωζ0 /Ω0

plotted). 0.1 Teeter (2) 0 0

10

20

30

40

50

-0.1 Differential Flap -0.2 Teeter (1) -0.3 -0.4 Teeter Teeterdamping, Damping,CφC[kNms/rad] φ [kNms/rad]

Figure 3-4: Effect of teeter damping on teeter and differential-flap mode stability

Earlier results were obtained with the blades set at the nominal precone of β 0 = 7° and zero teeter stiffness, K φ = 0. The next study investigates what happens to teeter stability when the precone angle is changed between 0° and 10°, and the teeter stiffness is changed between 0 kNm/rad and 20 kNm/rad. All the damping values and the yaw stiffness are still assumed zero. The variation of teeter mode stability is illustrated in Figure 3-5. In fact, two teeter modes exist but we only plot the unstable one – the stable mode has a real part of equal magnitude but opposite sign. For the case when the teeter stiffness is zero we see that the teeter mode is unstable for all precone angles examined. There is, however, a point at which the exponential growth rate is minimal. This occurs at a precone angle of 1.59°. With this

59 knowledge, wind turbine designers could maximize natural stability of the structure. As the teeter spring stiffness is increased beyond approximately 4 kNm/rad an increasing range of precone angles exist at which the teeter mode is critically stable (real part of characteristic exponent is zero). Increasing teeter stiffness to at least 17 kNm/rad provides critical stability

ζ /Ωζ0/Ω Real part mode, Real part of of teeter mode, 0

at the nominal precone angle of 7°.

0.14 0.12 0.1 0.08

K φ = 0 kNm/rad Kteet

0.06 0.04

5

0.02

5

10

15

2 20

0 0

2

4 6 Preconeangle, angle,ββ00[deg] [deg] Precone

8

10

Figure 3-5: Effect of precone and teeter stiffness on teeter mode stability

Next, we study the effect of yaw stiffness on the turbine modal behavior (with the precone angle kept fixed at 7°). As expected, the rotor collective and differential modes remain relatively unaffected. We therefore focus our attention on the yaw and teeter modes. Figure 3-6 shows the modal damping variation with yaw stiffness and Figure 3-7 shows the modal frequency variation (note the two vertical axes in the latter plot). Here we show only the unstable modes - the stable modes simply trace reflections in the horizontal axes. At zero yaw stiffness we notice the unstable teeter mode with frequency 1p and the rigid-body yaw mode with zero damping and frequency, as identified earlier. As the yaw stiffness increases within the range 0< K γ 0

∆G (t )

2

(4.13)

While this may not be a very tight bound on ∆J it suggests that the cost function will be reduced when sup t>0 max

∆G (t )

t ∈ [0, T ]

2

∆G (t )

or equivalently (4.14) 2

is minimized.

This problem is obviously easier than solving the original minimization problem, (4.10). One technicality is that K and σ in (4.13) are functions of the choice of constant gain, G, and therefore the factor K / 2σ may increase during minimization of (4.14). It is also true that we are not guaranteed of finding the global minimum to the original optimization problem via this second problem.

74 One hypothesis is that the best constant gain is always the mean of the optimal periodic gain. G=

1 T G * (t ) dt T 0

∫

(4.15)

The simple example provided in Appendix E.1 proves that this is not the case. Here it is shown that the minimum of ∆J occurs at a constant gain that is distinct from both the mean of the periodic gain and from the solution of the reduced optimization problem (4.11). Since a numerical search for optimal constant gains of a wind turbine is difficult and case specific, gain averaging is the most convenient method so far and will be employed in later studies. An entirely different approach for constant gain calculation is based on finding a suitable time-invariant representation of the periodic system and using LQR techniques. In this thesis we explore the following two options.

1. Averaged Model

x& a = A a x a + Ba u + Bd a u d

(4.16)

where 1 T A(t ) dt, T 0 1 T Ba = B(t ) dt, and T 0 1 T Bd a = B d (t ) dt . T 0 Aa =

∫

∫

∫

Then the state feedback gain, Ga, is calculated from time-invariant LQR with (Aa, Ba) and the weighting matrices (Q, R).

2. Frozen Model x& f = A f x f + Bf u + B d f u d

where

(4.17)

75 Af = A(tf), Bf = B(tf), Bd f = Bd(tf), which produces the gain, Gf, following time-invariant LQR. The freezing time, tf ∈ [0,T], could be selected to correspond to some rotor azimuth position, ψf, by tf = ψf/Ω0, where Ω0 is the mean operating rotor speed. Still, there is no intuitive freezing azimuth angle that would produce the most representative time-invariant model for control. Instead, it is wise to perform parametric control studies and compare performance at various freezing angles. Such a study will be described in the results section.

4.4 Disturbance Rejection Up until this point we have ignored the effect of wind disturbance on the system in the controller designs. Recall that the perturbed wind speed, ud, enters the linear periodic system through the wind input matrix, Bd. x& = A ( t ) x + B( t ) u + B d ( t ) u d

(4.18)

To cancel this disturbance we may assume that the wind speed is known and augment the optimal control input.

u = G * (t) x + G d (t ) u d

(4.19)

The closed-loop system is then

x& = (A(t ) + B(t )G * ( t ) ) x + (B( t )G d ( t ) + B d ( t ) ) u d .

(4.20)

If the disturbance entered the system through the same channel as the control input, i.e. Bd(t) = B(t), then the cancellation would be exact and we could set Gd(t) = -I, the negative

76 of the identity matrix. Alternatively, if there were as many independent control inputs as states, i.e. B(t) was square and nonsingular, then exact cancellation would be accomplished by Gd(t) = -B(t)-1Bd(t). Since we have few and different input channels, the disturbance effect can only be minimized. In a least-squares sense, this is achieved by

G d ( t ) = −B( t ) + B d ( t )

(4.21)

where B(t)+ = B(t)T (B(t) B(t)T)-1, the Moore-Penrose pseudoinverse of B(t). For the constant gain designs there are corresponding disturbance gain calculations.

Mean gain method: Gd =

1 T G d (t ) dt T 0

∫

(4.22)

Averaged model method: +

G d a = −B a B d a

(4.23)

where Ba+ = BaT (Ba Ba T)-1.

Frozen model method: +

G d f = −B f B d f

(4.24)

where Bf+ = BfT (Bf Bf T)-1.

4.5 Controller Implementation For simulation studies, the various control designs are implemented in Matlab with Simulink. Figure 4-1 illustrates the block diagram logic for periodic control. Note that the periodic gains, G(t) and Gd(t) are stored as tables and scheduling is achieved using the actual azimuth position, ψ, instead of time. Since the system is periodic solely because of the

77 changing rotor position, the use of ψ in this way is a logical approach. The alternative would be to use clock time, but this fails to produce the appropriate gains when rotor speed is allowed to vary. Implementation of constant gains is much simpler, as illustrated in Figure 4-2. The gain pair (G, Gd) is set to (G, G d ) when using the mean gain calculation approach, (Ga, Gd a) for the averaged model method, and (Gf, Gd f) for the frozen model method.

Periodic Gain Control Wind Perturbation 1 ud

Azimuth

3 psi

Gd*ud

MATLAB Function Gd( t )

States

2 x

Control Input

Gx*x

1 u

G( t ) * x

Figure 4-1: Simulink block diagram for periodic gain control Constant Gain Control Wind Perturbation

1 ud

Gd Gd

States 2 x

G G

Control Input

1 u

Figure 4-2: Simulink block diagram for constant gain control

4.6 Results A number of different simulation studies are presented in this section using the SymDyn with AeroDyn aeroelastic model and the control designs just described. Three different control objectives are investigated in order to evaluate the effectiveness of periodic

78 control versus constant gain methods. In all cases, speed regulation in Region III is the prime concern. Consistent with the series of numerical examples presented in the earlier chapters, the analyzed turbine is a downwind, two-bladed, teetered hub machine, with properties from Appendix C. The different control objectives call for models of various dimensions, drawn from the following six degrees-of-freedom. Note that the drive train is always assumed rigid, so the shaft compliance angle, ψg, is omitted.

τ1

Tower fore-aft angle

γ

Nacelle yaw angle

ψ

Rotor azimuth angle

φ

Hub teeter angle

β1

Blade #1 flap angle

β2

Blade #2 flap angle

The chosen operating point for Region III operation is also consistent with the earlier example in Section 2.7. In particular, the operating wind speed is wop = 16 m/s and the operating pitch angles are θop = [15° 15°]T. The desired rotor speed is Ω0 = 57.5 rpm, which is achieved on average with a constant generator torque of Tg = 13.4 kNm. The subsequent period of rotor rotation is T = 1.04 seconds. For regulation goals, the following angle and velocity vectors define the set point. They are based on the mean of the steady-state responses (q op , q& op ) for the earlier four degree-of-freedom model (τ1, ψ, β1, β2). These values (with t = 0) also serve as initial conditions for simulation in the nonlinear plant.

79 q set = [τ1set

γ set

ψ set ( t ) φ set

β1set

β 2set ]

T

= [0.06° 0° Ω 0 t 0° − 2.4° − 2.4°]

T

[

q& set = τ& 1set

γ& set

= [0 0 Ω 0

ψ& set

φ& set

β& 1set

β& 2set

]

T

(4.25)

0 0 0]

T

The Simulink block diagram for the periodic controller in closed-loop with the nonlinear plant is shown in Figure 4-3.

The constant gain controller in closed-loop is

implemented in a similar fashion, except that azimuth position is not required to synchronize gains. The way that SymDyn is implemented in Matlab allows the nonlinear plant to allow any or all of the degrees-of-freedom listed above. An omitted degree-of-freedom represents a locked hinge in the model. Note that actuator dynamics have been neglected in the system. It is believed that modern electro-mechanical pitch servos have fast enough response times that their influence would be almost negligible in the simulations. A possible limitation imposed by such actuators is a maximum pitch rate, in the order of 10-20 deg/s. The maximum pitch rate is tested for in the upcoming results.

w

w ind Wind Input w _op

Operating Wind Speed

theta_op ud u

x

Operating Pitch

w

q

Pitch

qdot

psi Nonlinear Plant

Periodic Gain Controller psi

K

Extract Azimuth Position q states, x

qdot Subtract Set Point

Figure 4-3: Simulink block diagram for the closed-loop system (periodic control with the nonlinear plant)

80 The wind disturbance input used in most of the simulation cases is based on actual data taken from the National Wind Technology Center in Colorado [6]. Figure 4-4 illustrates the 100-second turbulence history defining the horizontal wind speed parameter, w. The data is sampled at 1 Hz, which essentially describes a series of ramp disturbances to the system. A constant vertical wind shear factor of 0.2 provides spatial variation of wind speed across the rotor, as per the aerodynamics description (Section 2.6). 26 24

Wind Windspeed, speed,ww[m/s] [m/s]

22 20 18 16 14 12 10 8

0

10

20

30

40

50 Time [s]

60

70

80

90

100

Figure 4-4: Wind disturbance input

4.6.1

Speed Regulation The first control objective is speed regulation only. With the LQR approach, stability

of the linear system is guaranteed, which amounts to implicit regulation of the other states also, but with much less emphasis. The ( q = [τ1

control

model

is

chosen

to

include

four

degrees-of-freedom

ψ β1 β 2 ] ), resulting in eight states. A single state model is really all that is T

necessary to build a speed-regulating controller. However, when applied to a plant with tower flexibility there is risk of destabilizing the tower fore-aft coupled mode, owing to its light damping. Results to this effect have been presented in an earlier paper [5]. Therefore, it is wise to include the tower and blade flap degrees-of-freedom in the control model.

81 The weightings for LQR design are chosen to have the following form: Q = Q0.diag(0,1,0,0, 0,1,0,0) R = diag(1, 1) where Q0 is a constant and diag(.) is shorthand for a diagonal matrix with the provided components.

Here only the azimuth and rotor speed perturbation states are weighted.

Regulating the azimuth perturbation improves the periodic controller performance, since the actual azimuth position is used for gain synchronization.

4.6.1.1 Linear Plant First, we consider a linear periodic plant with the same active degrees-of-freedom as the control model. Wind input is assumed constant at the operating value, 16 m/s. The closed-loop dynamics should then correspond to the theoretical representations upon which the gains are designed. As the only control performance measure we use an approximation of the quadratic cost function, given by J sim =

t ∫0 (x sim

T

T

)

Q x + u R u dt

(4.26)

where tsim is a large enough time to allow for convergence of the stable integrand. Each state has an initial condition of 1.0 to allow excitation of all the variables. This choice is arbitrary for a completely linear system and obviously the time responses will bear little resemblance to the output of a nonlinear wind turbine. A comparison of the various control designs at two values of Q0 is presented in Table 4-1. The costs associated with the periodic gain are consistent with the exact result, J* = x(0)TP(0)x(0), validating the periodic gain calculation and the simulation models. Also consistent with theory is that the constant gains produce costs that are greater than J*. Results at both values of Q0 show the same trend, although at the higher Q0 (associated with higher

82 gains), the spread is greater. Both the averaged gain, G , and the gain from the frozen model method, Gf, perform almost identically.

The gain from the averaged model approach, Ga,

performs somewhat less well but still comparably in the low Q0 case. Gain G(t), periodic G Ga Gf (ψf = 105°)

Cost, Jsim Q0 = 1 Q0 = 1000 12.37 1.077×103 12.44 1.092×103 12.71 12.45

3.199×103 1.094×103

Table 4-1: Quadratic cost comparison between the controllers

The frozen model approach deserves special mention. Selection of the freezing azimuth angle is not intuitive but instead is based on the results of a parametric study, over all angles. As shown in Appendix E.2, the optimum freezing angle depends on both the model parameters and the LQR weightings. In fact, it is possible to destabilize the closed-loop system if the wrong freezing angle is chosen, particularly when the periodic gains are centered about zero, as with the yaw control case in the Appendix. Since use of Ga provides the least favorable speed regulation results and because Gf is expensive to calculate case-by-case, averaging the periodic gains to calculate G is chosen as the representative constant gain approach. All subsequent studies use G for comparison with periodic control.

4.6.1.2 Nonlinear Plant We now compare controller performance on the nonlinear plant containing the degrees-of-freedom of the control model with the addition of hub teeter. This extra motion is included to more closely resemble the response of common two-bladed turbines. Teeter is not added as a degree-of-freedom in the control model because, if the controller were attempting to regulate this motion, it would be working against the natural behavior of the teeter hinge to reduce asymmetric loads.

83 The same periodic gains from the previous section for Q0 = 1 are used. A plot of these gains versus azimuth position is shown in Figure 4-5. With two independent pitch

1.1

G(∆τ1)

1.05 1 0.72

G(∆ψ)

0.71 0.7 0.06

G(∆β2)

0.04 0.02 0.06

G(∆β1)

0.04 0.02

. G(∆τ1)

-0.05 -0.1 -0.15

. G(∆ψ)

0.78 0.76

. G(∆β1)

0 -1 -2

. G(∆β2)

-3

x 10

0

-3

x 10

-1 -2 0.015

Gd

0.01 0.005

0

90

180

270

360

ψ [deg]

Figure 4-5: Periodic state gain, G(t), and disturbance gain, Gd(t), for speed regulation, Q0 = 1. Solid line: gains for θ1, dashed line: gains for θ2.

84 controls there are a total of sixteen periodic state gain components plus two disturbance gain components. High mean gains are associated with the rotor speed and azimuth perturbation states as expected. However, the highest gains correspond to the tower fore-aft angle, which is a result of indirect coupling between the tower motion and rotor speed. Since tower motion is coupled with blade flap, which in turn is coupled to rotor speed by centrifugal stiffening, the gains reflect the use of tower position information to regulate speed. Such coupling is present in the Floquet modeshapes. Also, the closed-loop characteristic exponents in Figure 4-6 suggest that the damping in the tower fore-aft mode is being reduced by the control design as a sacrifice for improved damping in the rotor speed error. To improve tower damping, one would have to weight the appropriate entry in the Q matrix. The other periodic gain components in Figure 4-5 have low mean values and contribute little to speed regulation. The variations in the disturbance gain components reflect the variations in wind speed over a rotor revolution due to shear. 4 C losed-loop Open-loop

3

Differential flap

Collective flap

2

Tower fore-aft

Imag

Imag

1

Rotor speed perturbation

0

Azimuth perturbation

-1 -2 -3 -4 -1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Real Real

Figure 4-6: Closed-loop characteristic exponents for speed regulation (Q0 = 1), normalized by the nominal rotor speed, Ω0 = 57.5 rpm.

85 A time response comparison is made between the periodic and constant gain controllers with the nonlinear plant and turbulent wind input. The initial conditions for the plant are the same as the set point. As performance measures we use root-mean-square (RMS) of speed error and actuator duty cycle (ADC). ADC is defined as the total number of degrees pitched by the blades divided by the simulation time. It is closely related to the mean of the absolute pitch rate. Since there are two blades, it is the maximum value of the two that is used. Generally, the better the speed regulation (lower RMS speed error) the higher the control usage (higher ADC). This is the typical design trade-off. As a baseline for comparison, a simple PID controller is also designed.

This

controller is limited to single-input-single-output problems and therefore we use rotor speed output from the plant and actuate collective blade pitch.

A constrained minimization

algorithm calculates the proportional, integral, and derivative gains. The objective is to minimize ADC while producing the same RMS as for the periodic gain controller. The results for this case are the gains KP = 0.76, KI = 1.55, and KD = 0.08. Performance results for each of the controllers are listed in Table 4-2. Since speed error RMS is the same in each case, regulation performance can be judged by differences in ADC. While the periodic controller is guaranteed to be optimal with a linear periodic plant without wind disturbance, the same cannot be said here. A consolation is that we are using different performance measures now – a quadratic control usage term is much different than ADC. Regardless, the results show that constant gain control and PID perform similarly but noticeably better than periodic control. The time traces for rotor speed and blade #1 pitch are shown in Figure 4-7 and Figure 4-8, over a short time scale. The rotor speed from PID appears to have larger variations but over the full 100 second interval has the same RMS as the other controllers. All the rotor speed traces exhibit a 2p (where p means per-rotorrevolution) variation, which is a direct result of the vertical wind shear present. The blade pitch traces illustrate the additional actuator commands used by the periodic controller,

86 causing the high ADC. Were blade #2 pitch response plotted on the same figure, one would see the same trend but with the variations 180° out-of-phase for the periodic controller and no difference for the other controllers, since they are acting in a collective fashion. As a check, the maximum pitch rates for the periodic, constant gain, and PID controllers are 10 deg/s, 5 deg/s, and 6 deg/s respectively. These are expected to be well in the range of a modern pitch servo. Controller Periodic, G(t) Constant, G PID

Speed RMS (rpm) 0.154 0.154 0.154

ADC (deg/s) 1.88 1.34 1.40

Table 4-2: Speed regulation performance results

The periodic controller performs unsatisfactorily when speed regulation is the only objective. The high inertia of the rotor essentially filters out the larger pitch commands, resulting in the same speed RMS as the time-invariant controllers.

58 Periodic Constant PID

Rotor Speed [rpm]

Rotor speed [rpm]

57.9 57.8 57.7 57.6 57.5 57.4 57.3 55

55.5

56

56.5

57

57.5 Time [sec]

58

58.5

59

Figure 4-7: Rotor speed response for speed regulation

59.5

60

87 26 Periodic Constant PID

Blade pitch, θ [deg]

Blade #1 pitch, θ1 1 [deg]

25 24 23 22 21 20 19 55

55.5

56

56.5

57

57.5 Time [sec]

58

58.5

59

59.5

60

Figure 4-8: Blade #1 pitch usage for speed regulation

4.6.2

Blade Load Mitigation This section investigates the potential for periodic and constant gain full-state

feedback controllers to minimize blade loading. Of particular interest is the magnitude and frequency of cyclic loads as these contribute to premature failure of the blades by fatigue. Using the SymDyn model, the out-of-plane bending moment at the flap hinge of blade #i is given simply by the product Kβi βi(t), where Kβi is the flap spring stiffness and βi(t) is the flap angle. This is not the bending moment at the blade root, which will be somewhat greater, but the variations will be identical. Assuming that the contribution to blade loading from higher order blade modes is negligible, then SymDyn provides a reasonable estimate of trends in root loading of a fully flexible turbine. Since it is the cyclic loads that are of interest a suitable LQR quadratic cost function should include a weighting on the blade flap rate, not flap angle. Also, with flap motion being tightly coupled with tower fore-aft motion, a weighting on the tower fore-aft rate state would also be appropriate. The positive side effect of this is the increase in tower damping, improving the stability margin. The weighting matrices have the following form:

88 Q = diag(0,1,0,0, 10,1,1,1) R = diag(1, 1)

Speed regulation is still a control objective; hence the weightings on azimuth and rotor speed perturbation states are retained. A plot of the periodic gains is shown in Figure 4-9. The larger gains are associated with the tower fore-aft and blade flap states as expected. While most gains are either strictly positive or negative, the ∆τ1 gain briefly changes sign over the period. This suggests that a constant gain controller would be ineffective at certain azimuth positions during rotation. A comparison of the periodic and constant gain controllers is made for two different nonlinear plants. The first uses the same four active degrees-of-freedom as the control model ( q = [τ1

ψ β1 β 2 ] ),

( q = [τ1

ψ φ β1 β 2 ] ) as in the previous section.

T

while

the

T

second

also

include

the

teeter

motion

A new performance measure is

necessary to quantify the cyclic blade loads from the time-response. We use the flap rate RMS (FRR), defined as the root-mean-square of the flap rate from both blades. Results using the first nonlinear plant are listed in Table 4-3. The PID controller (with KP = 0.42, KI = 0.71, and KD = 0.08) is designed using the same method as described earlier, without regard to the blade loads. This provides a basis from which to compare the state-space controllers. Speed regulation performance for all control methods is very similar but there are definite differences in load mitigation performance. The periodic controller is able to reduce blade flap rates by 13% (in FRR) over the constant gain control and by 61% over PID. Plots of the blade bending moment variations over the entire simulation are given in Figure 4-10, supporting the FRR results. When the teeter degree-of-freedom is added to the plant, there are noticeable differences (Table 4-4). Hub teeter is able to passively reduce blade loads, alleviating the

89 demand on active feedback control. Hence, the actuator usage and flap rates for both statespace controllers is less than with the previous plant. Another consequence of hub teetering is that periodic gains are no longer as favorable for load mitigation. Use of constant gains gives the same FRR but with 20% less actuator use.

10

G(∆τ1)

0 -10

G(∆ψ)

0.75 0.7 0.65

G(∆β1)

3 2.5 2

G(∆β2)

3 2.5 2

. G(∆τ1)

2 1 0

. G(∆ψ)

1

0.95

. G(∆β1)

1 0.5 0

. G(∆β2)

1 0.5 0 0.015

Gd

0.01 0.005

0

90

180

270

360

ψ [deg]

Figure 4-9: Periodic state gain, G(t), and disturbance gain, Gd(t), for blade load mitigation. Solid line: gains for θ1, dashed line: gains for θ2.

90

Controller

Speed RMS (rpm) 0.317 0.317 0.317

Periodic, G(t) Constant, G PID

ADC (deg/s) 4.67 4.60 1.17

FRR (deg/s) 0.86 0.99 2.22

Table 4-3: Blade load mitigation performance results with the 4 d.o.f. nonlinear plant

Blade #1 bending moment [kNm]

30

Periodic gain Constant gain PID

28 26 24 22 20 18 16 55

55.5

56

56.5

57

57.5 Time [sec]

58

58.5

59

59.5

60

Figure 4-10: Blade #1 bending moment for the blade load mitigation study (4 d.o.f. plant)

Controller Periodic, G(t) Constant, G PID

Speed RMS (rpm) 0.317 0.317 0.317

ADC (deg/s) 2.16 1.70 1.28

FRR (deg/s) 0.36 0.36 0.52

Table 4-4: Blade load mitigation performance results with the 5 d.o.f. nonlinear plant

Performance from the same PID controller (with KP = 0.42, KI = 0.71, and KD = 0.08) is also provided in Table 4-4.

The results show that, at the expense of additional actuator

usage, the full-state feedback controllers still reduce cyclic blade loading. The time-response plots in Figure 4-11 support this conclusion. Notice that, while mean variations in bending moment are similar for each controller, the PID design has higher frequency content resulting in a 44% higher FRR. The comparison is not completely fair because the model-based controllers have access to the wind speed, while the PID controller uses only rotor speed

91 information. The design and performance of an observer to estimate the states and wind is the focus of the next chapter. The intention here is to illustrate the potential of model-based control only.

Blade #1 bending moment [kNm]

27 Peiodic gain Constant gain PID

26 25 24 23 22 21 55

55.5

56

56.5

57

57.5 Time [sec]

58

58.5

59

59.5

60

Figure 4-11: Blade #1 bending moment for the blade load mitigation study (5 d.o.f. plant)

4.6.3

Yaw Directional Control With the ability to pitch the blades independently we are able to apply asymmetric

aerodynamic loads on the rotor. This means that a controller has the potential to yaw the nacelle to increase the capture of energy during changes in the wind direction. Downwind turbines are generally stable in yaw and are often operated without yaw control to naturally follow the wind direction changes. However, due to wind shear, downwind turbines rarely have their rotor plane perpendicular to the prevailing wind direction. Also, large machines are very slow to respond due to the large rotor inertia and gyroscopic stability. Therefore, an improvement to energy capture is possible if responsive yaw control was employed. This section is a study of the feasibility of yaw control via blade pitch. It does not consider the effect of different wind conditions or the impact on energy capture, both of which are very important if the application of this type of controller were to be attempted.

92 For the control model, four degrees-of-freedom are active: yaw, azimuth position, and flap motion of each blade, q = [γ ψ β1 β 2 ] . As with the previous control model, T

linearization is about a periodic operating point, calculated from the steady-state solution of the nonlinear equations. Regulation of both yaw position and rotor speed is achieved through appropriate weightings in the Q matrix: Q = diag(10,1,0,0, 0,1,0,0) R = diag(1, 1)

In this case, the ∆γ state is heavily weighted to generate an aggressive yaw controller. The resulting periodic gains are shown in Figure 4-12. What is most noticeable, compared to the previous designs, is that there are gain components that are centered around zero. These correspond to the yaw position and velocity perturbation gains, enabling blade pitch to generate a moment about the yaw axis. For example, consider a positive yaw error. Over the first 180° of rotation the positive gain causes a positive blade pitch (or reduced angle-ofattack).

This reduces the loads normal to the blade. Over the same range of azimuth, the

second blade has an increased angle of attach and higher normal loads. The net effect is a negative moment about the yaw axis to reduce the positive yaw error. Over the second half of the azimuth range, the blades change position and the gain variations repeat with opposite sign. The gains would be maximum at an azimuth position of 90°, when the yaw moment arm is greatest, if it were not for the rotor moment of inertia about the yaw axis changing with azimuth. Yaw acceleration is greatest when the blades are vertical (0° azimuth) and therefore there is a compromise reflected in the gain magnitudes.

93

G(∆γ)

5 0 -5

G(∆ψ)

0.8 0.7

G(∆β1)

2 0 -2

G(∆β2)

2 0 -2

. G(∆γ)

2 0 -2

. G(∆ψ)

0.9 0.8 0.7

. 0.2 G(∆β1) 0 -0.2 0.2 . G(∆β2) 0 -0.2

Gd

0.015 0.01 0.005

0

90

180

270

360

ψ [deg]

Figure 4-12: Periodic state gain, G(t), and disturbance gain, Gd(t), for yaw control. Solid line: gains for θ1, dashed line: gains for θ2.

94 Performance of the periodic and constant gain methods is compared on a nonlinear plant containing six degrees-of-freedom, q = [τ1

γ ψ φ β1 β 2 ] . The standard wind T

disturbance is used with constant direction, corresponding to γ = 0°.

Yaw control

performance is then based on the RMS of yaw position, given that the set point is also at 0°. Results for the different controllers over the 100-second simulation are listed in Table 4-5 with yaw responses plotted in Figure 4-13. The PID controller (with KP = 0.95, KI = 1.31, and KD = 0.04) is designed only for speed regulation and represents the case when active yaw control is neglected. The yaw misalignment due to wind shear is apparent in the time response.

Controller Periodic, G(t) Constant, G PID

Speed RMS (rpm) 0.154 0.154 0.154

ADC (deg/s) 3.46 1.37 1.39

Yaw RMS (deg) 0.3 2.1 1.3

Table 4-5: Yaw control performance results 1 Periodic gain Constant gain PID

0.5

Yaw angle [deg]

0 -0.5 -1 -1.5 -2 -2.5 -3

0

10

20

30

40

50 Time [sec]

60

70

80

90

100

Figure 4-13: Yaw response for the yaw control study

The results show that time-varying gains improve directional control performance. The constant gain controller, while pitching the blades independently, cannot do so to

95 regulate yaw position and velocity. In fact, the gains for these states are close to zero, since they are calculated as averaged periodic gains. Yaw error is actually worse as a result of constant gain control than if PID is used. The obvious drawback to using pitch to control yaw is the large increase in actuator usage, reflected by the ADC value for the periodic controller. The maximum pitch rate is much higher than with PID (25 deg/s compared to 5 deg/s) which in this case would likely exceed the limits on a real pitch servo. In practical implementation one may choose to design a less aggressive controller, which would solve this problem. Another undesirable side effect observed is the increase in blade flap bending moments as illustrated in Figure 4-14. Over the 100-second simulation time, the flap rate RMS for the periodic controller is 0.90 deg/s, compared to 0.42 deg/s for PID. The bending moments would be higher still if the teeter degree-of-freedom were not included in the plant. However, the teeter degree-of-freedom also reduces the yaw controller’s effectiveness, since direct loads from the rotor are decoupled from the non-rotating components. Therefore, there is a trade-off between blade loads and yaw regulation that must be considered.

Blade #1 bending moment [kNm]

27 Periodic gain Constant gain PID

26 25 24 23 22 21 55

55.5

56

56.5

57

57.5 Time [sec]

58

58.5

59

Figure 4-14: Blade #1 bending moment for the yaw control study

59.5

60

96 4.7 Summary This chapter has presented the designs for full-state feedback control with known wind.

The periodic controller is designed using the linear quadratic regulator (LQR)

approach and compared to constant gain control under various conditions. In the design of constant gains an initial search was undertaken to find an analytic result that would optimize quadratic cost performance when the plant is periodic. A closedform solution to this problem could not be found. However, by examination of a simple numerical example it was proved that the constant gain solution is not simply the mean of the periodic gains, G . To search for an optimal solution for a larger state-space model would be very computationally expensive and dependent on both the model parameters and control objective. Therefore, we discarded this approach from further consideration in controller comparisons. We explored two other constant gain calculation methods by finding time-invariant versions of the periodic state-space equations. The first used the average of the periodic state matrices and the other involved ‘freezing’ in time, corresponding to a chosen azimuth position. The second method requires a parametric study to select the best freezing azimuth position, due to the wide range of results produced and the possibility of choosing a destabilizing gain. A comparison of the various methods on a linear plant in speed regulation showed that they all performed similarly. G was consistently the best and the gains produced by averaging the model were the worst, particularly at high gains. Using the mean periodic gain, G , was used for comparison with periodic control for the remaining studies. Three performance objectives were investigated; speed regulation alone, speed regulation with blade load mitigation, and speed regulation with yaw direction control. Performance compared three controllers; periodic gain, constant gain ( G ), and PID. A

97 fluctuating wind speed disturbance was input to a nonlinear plant. All controllers except PID commanded independent blade pitch. The results are summarized below.

Speed regulation alone: Constant gain control and PID were similar and both outperformed periodic control. Blade load mitigation:

The goal here was to reduce cyclic blade bending moments,

quantified by flap rate RMS.

The periodic controller outperformed constant gain

control when teeter was locked but both controllers were similar when teeter was free. Both outperformed PID. Yaw directional control: The objective was to keep the turbine pointed in the direction of the wind, counteracting the effect of vertical shear. Only the periodic controller could produce asymmetric loads on the rotor and therefore could achieve some degree of regulation. This was at the expense of blade loads, which were much higher compared to the baseline PID response. The constant gain controller made yaw misalignment worse.

98 CHAPTER 5 5. STATE AND DISTURBANCE ESTIMATION

5.1 Introduction So far we have considered only full-state feedback control, where the state vector in the control model is known for all time. In the practical implementation of a control system not all states would be measured and so would require an observer component. The wind disturbance input is another element. In the previous chapter we assumed that the wind speed information was available in order that the controller could minimize fluctuations. In this chapter, the observer design is augmented to allow estimation of the wind. Theory is based on the disturbance accommodating control (DAC) approach, developed by Johnson [40] and explored in wind turbines by Balas, et. al. [41]. Both periodic and time-invariant estimators are developed. Results at the end of the chapter compare the model-based controllers to PID for a multi-objective problem.

5.2 Periodic Estimator Design From (2.26), the linear state-space description of the wind turbine plant model is x& = A ( t ) x + B( t ) u + B d ( t ) u d y = C( t ) x

(5.1)

where y ∈ ℜp is the vector of p plant outputs and C(t) ∈ ℜp×n is the periodic output matrix, with the same period, T, as A(t), B(t), and Bd(t). From (4.19), the ideal control input takes the form

u * = G * (t) x + G d (t) u d

(5.2)

where G*(t) is the optimal periodic gain matrix, designed by full-state linear quadratic regulation (LQR) and Gd(t) is the disturbance canceling gain matrix from (4.21).

99 Given that the state vector, x, and disturbance input, ud, are not all measurable, the realizable control law is

u = G * (t ) xˆ + G d ( t ) uˆ d

(5.3)

where xˆ and uˆ d are the state and disturbance estimates respectively. A new linear system is constructed as a model for the disturbance; known as a disturbance generator. z& d = F z d u d = Θ zd

(5.4)

where F and Θ are chosen so that (5.4) produces the desired disturbance waveform. To generate rapidly changing wind speeds for wind turbine applications a step waveform is appropriate – requiring F = 0 and Θ = 1 (so zd = ud). If it were known that the wind existed with a narrow range of frequency content, then one could choose a series of sinusoidal functions and design the disturbance generator appropriately. Combining (5.1) with (5.4) and assuming a step disturbance,  x&  )  x  )  u&  = A ( t )  u  + B( t ) u  d  d ) x y = C( t )   u d 

(5.5)

) ) A( t ) B d ( t )  )  B( t )  where A ( t ) =  , B( t ) =  , and C( t ) = [C( t ) 0] .   0   0  0 

It follows that a periodic estimator system can be designed for (5.5) as a logical extension to the time-invariant case [39]: )  x&ˆ  )  xˆ  )  &  = A ( t )   + B( t ) u + K ( t )( y − yˆ)  uˆ d   uˆ d  )  xˆ  yˆ = C( t )    uˆ d 

) T where K( t ) = [K( t ) K d (t )] , the composition of state and disturbance estimator gains.

(5.6)

100 The error vector, defined as e = [xˆ uˆ d ]T − [x u d ]T , is governed by

(

)

) ) ) e& = A( t ) − K( t ) C(t ) e .

(5.7)

From the time-varying version of the separation principle, the dynamics of (5.7) are independent of the closed-loop dynamics with full-state feedback. For the calculation of the

) estimator gain, K( t ) , it is convenient to use the duality property and apply LQR techniques. With time-varying systems, the dual of the state matrices (A(t), B(t), C(t)) is (AT(-t), CT(-t), BT(-t)). It follows that the dual Riccati equation to solve is ) ) ) −1 ) P& E ( t ) + A ( t ) PE ( t ) + PE ( t ) A ( t ) T − PE ( t )C( t ) T R E C( t ) PE ( t ) + Q E = 0 PE ( t final ) = PE

(5.8)

final

and that the optimal periodic estimator gain is formed from ) ) −1 K ( t ) = PE ( t )C( t ) T R E . *

(5.9)

Bittanti’s theorem from the previous chapter holds and we are assured of stabilizing

) ) ) ) the estimator system (5.8) when (A(t ), C( t )) is detectable and no (A(t ), L) -unreachable )) ) characteristic exponents of A(t) have zero real parts. Here LLT = Q E and L ∈ ℜn×m is full column rank. If system noise information is known then the weightings QE and RE can be chosen to equal terms in the plant output noise covariance matrix. This is a common technique for time-invariant systems and the design is referred to as LQG (for Linear Quadratic Gaussian). For the designs in this thesis, QE and RE, are assumed to be diagonal, with entries chosen for acceptable estimator performance.

101 5.3 Time-Invariant Estimator Design The complete control system comprises three components; linear full-state feedback gains, linear state estimator gains, and a linear plant model within the observer. In the previous section these components were all periodic in time and the design of each followed the standard state-space approach, albeit time-varying. For a time-invariant controller, the constant full-state feedback and estimator gains can be designed using the methods described in the previous chapter. However, the choice of an appropriate time-invariant plant model appears less trivial. This is because it is not clear what the optimal choice should be in terms of performance in estimating a periodic plant. In this thesis we explore the averaged model and frozen model approaches that had been developed earlier for the construction of full-state feedback gains. They are reiterated below for completeness with the addition of the output parameters.

1. Averaged Model

x& a = A a x a + Ba u + Bd a u d y a = Ca x a where T

1

Aa =

∫ A(t) dt , T

Ba =

1

0

∫ B(t) dt,

T 0

Bd a = Ca =

T

1

T

1

∫B T 0

T

d ( t ) dt,

∫ C(t) dt.

T 0

2. Frozen Model

and

(5.10)

102 x& f = A f x f + Bf u + Bd f u d y f = Cf x f

(5.11)

where Af = A(tf), Bf = B(tf), Bd f = Bd(tf), Cf = C(tf), and tf = ψf/Ω0.

Once again, the choice of freezing azimuth angle, ψf, to produce the most representative time-invariant model is not intuitive. A parametric control study to compare performance at various freezing angles is appropriate. Obviously the design would be very specific to the turbine model and chosen control objectives, not unlike PID control. To simplify the design, the freezing azimuth angle is chosen from full-state feedback performance results (as in Appendix E.2) instead of performance of the model as an observer. To calculate constant state and disturbance estimator gains, we investigate two approaches.

1.

Dual LQR of the time-invariant plant model. The same QE and RE matrix weightings from the periodic control design are used to

) ) calculate the constant estimator gains: K a using the averaged model and K f using the frozen model.

2.

Mean of periodic estimator gains.

103

K=

1 T) K(t ) dt T 0

∫

(5.12)

The same approaches are available to calculate the constant full-state feedback gains. Therefore, for each plant model used (averaged or frozen) there are four combinations for calculating gains that are explored. In closed-loop with the linear periodic plant, the system dynamics are governed by B( t ) G B( t ) G d   x   B d ( t )   x&   A ( t )  xˆ&  =  KC ( t ) A + BG − KC BG + B   xˆ  +  0  u , d d  d        uˆ d   0   uˆ& d   K d C( t ) 0 − K dC

(5.13)

where the matrices (A, B, Bd, C, G, Gd, K, Kd) represent time-invariant parameters, E.g. (Aa, Ba, Bd a, Ca, Ga, Gd a, Ka, Kd a). Even after manipulation and a change of variables in (5.13), it is not possible to decouple the plant from the estimator model. Therefore, the separation principle does not apply as with the fully periodic case. From (5.13) the characteristic exponents can be calculated, providing information on the stability of the closed-loop system.

5.4 Estimator Implementation From (5.6), the periodic estimator system to be implemented is  xˆ&   A ( t ) − K ( t ) C( t ) B d ( t )   xˆ   B( t ) K ( t )   u  + & =  . 0   uˆ d   0 K d ( t )   y   uˆ d   − K d ( t ) C( t )

(5.14)

Initial conditions for both the state and disturbance estimates are set to zero. Figure 5-1 illustrates the estimator in closed-loop with the nonlinear turbine plant. Note that the azimuth signal, measured directly from the plant, is now used to synchronize three essential components of the system; the full-state feedback gains (G(t) and Gd(t)), the estimator gains (K(t) and Kd(t)), as well as the linear periodic plant model within the estimator subsystem. Implementation of the time-invariant estimator follows in a similar fashion.

104

w

w ind Wind Input

theta_op

ud^ u

x^

Operating Pitch

w

q

Pitch

qdot

psi Nonlinear Dynamics

Periodic Gain Controller psi

Extract Azimuth Position

psi

x^

K

u ud^

q

y

x

C

State & Disturbance Estimator

C

qdot

Subtract Set Point

Figure 5-1: Simulink block diagram for the periodic DAC controller in closed-loop

5.5 Results The simulation studies in this section serve as an application of the periodic and timeinvariant estimator designs. To exercise the ability of the state-space approach, a multiobjective control strategy is examined, consisting of simultaneous speed regulation and blade load mitigation. Results follow the full-state feedback case from the last chapter (Section 4.6.2). We begin by summarizing the configuration.

Control model: Four active degrees-of-freedom; tower fore-aft, azimuth position, and flap of each blade ( q = [τ1

ψ β1 β 2 ] ). T

Full-state feedback LQR weightings:

Q = diag(0,1,0,0, 10,1,1,1) and R = diag(1, 1).

Wind disturbance: 100-second horizontal wind speed input as used earlier.

The

measured

plant

states

are

azimuth )

and )

rotor

speed

perturbation

T ( y = [∆ψ ∆ψ& ] ), which ensures observability of (A(t ), C(t )) for Bittanti’s theorem. The

105

) ) reachability condition on (A(t ), L(t )) is met when weighting is placed on the disturbance generator state in QE. )

To design the periodic estimator gain, K(t ) , the following LQR matrix weightings are used. QE = diag(0,1,0,0, 0,105,0,0, 108) RE = diag(1, 1)

The nonzero diagonal entries in QE correspond to the azimuth perturbation, rotor speed perturbation, and wind disturbance generator states. This design is a result of many simulation runs with the nonlinear turbine model to achieve adequate state and wind estimation. Plots of the resulting periodic gains are shown in Figure 5-2, grouped by output

) channel (or columns of K( t ) ). The gains with the highest mean values correspond to the weighted states in QE.

5.5.1

Linear Plant We first investigate the performance of the periodic estimator on the linear periodic

plant model. The wind input provides a stochastic disturbance while the initial conditions are x(0) = 0. Response of the wind estimator over the entire simulation period in Figure 5-3 indicates very good performance. If viewed on a smaller time scale, one would observe small deviations from the actual wind, in the order of 0.7 m/s maximum speed difference or a time lag of 0.2 seconds. The azimuth position and rotor speed estimates are also excellent, which is not surprising since these are measured outputs of the plant and are merely filtered by the estimator.

106

. ∆ψ Output

∆ψ Output -3

2.2

K(∆τ1)

x 10

0.7

2

0.65

1.8

K(∆ψ)

K(∆β1)

K(∆β2)

. K(∆τ1)

. K(∆ψ)

1.4142

0.9997

1.4142

0.9997

1.4142 0.04

15

0.9997

0.02

10

0

5

0.04

15

0.02

10

0 x 10-3 2

5 1.8

0

1.7

-2

1.6

0.9997

105.5

0.9997

105

0.9997

. K(∆β1)

. K(∆β2)

Kd

104.5

0.2

60

0

40

-0.2 0.2

20 60

0

40

-0.2

20

7.8

3200

7.6

3150

7.4

0

90

180

270

360

3100

0

90

ψ [deg]

180

270

ψ [deg]

)

Figure 5-2: Components of the periodic estimator gain, K(t ) , for each output channel

360

107 26 24

Actual Estimated

Wind speed [m/s]

22 20 18 16 14 12 10 8

0

10

20

30

40

50 Time [s]

60

70

80

90

100

Figure 5-3: Wind speed estimator performance with the linear plant

Estimates for the other states are less than perfect, as illustrated in Figure 5-4 and Figure 5-5 for the tower fore-aft and blade #1 flap rates. While the estimator is capable of following the mean variations of the actual states, higher frequency excursions are not tracked. Adding weighting to these states in QE does not improve performance – in fact larger overshoots cause further degradation. This shows that, despite the fact that the system is completely observable, the estimator properties can not be improved by essentially

(

)

) ) ) increasing the stability margin of A(t ) − K(t ) C(t ) .

The likely reason for the slow

estimator is the existence of disturbance leakage into the system and the fact that the disturbance is not a single step input as modeled. Recall that with only two actuators and eight plant states, exact cancellation of the disturbance input is not possible when the control and disturbance channels are not identical, i.e. Bd(t) ≠ B(t) (from Section 4.4). Therefore, much of the wind turbulence is able to excite the plant states without its effect to be estimated or attenuated. As will be shown shortly, nonlinear plant dynamics corrode performance further.

108

Tower fore-aft rate [deg/s]

0.15 Actual Estimated

0.1

0.05

0

-0.05

-0.1 50

51

52

53

54

55 Time [s]

56

57

58

59

60

Figure 5-4: Tower fore-aft rate estimator performance with the linear plant

2 Actual Estimated

Blade #1 flap rate [deg/s]

1.5 1 0.5 0 -0.5 -1 -1.5 -2 50

51

52

53

54

55 Time [s]

56

57

58

59

60

Figure 5-5: Blade #1 flap rate estimator performance with the linear plant

5.5.2

Nonlinear Plant This section compares the periodic and time-invariant estimators on the nonlinear

plant. We begin with a plant of the same dimension as the controller, four active degrees-offreedom, then allow teeter motion later. In the design of a time-invariant estimator we examine both of the model construction methods (averaged and frozen models) and each of the four gain calculation methods, as described earlier in this chapter. The same measures of performance from the full-state feedback analysis are applied, with the flap rate RMS (FRR) used to quantify cyclic blade bending moments from the simulations.

109 Results for the various time-invariant estimator designs, using the four degree-offreedom plant, are listed in Table 5-1. Choice of freezing azimuth angle, ψf = 95°, to generate the frozen model is based solely on the parametric study in Appendix E.2. The obvious finding from the table is that all estimator designs perform very much alike. Whether the averaged model or frozen model is used, there does not appear to be a difference. However, this result is dependent on the choice of control model and objectives. For example, I have shown in [42] that inclusion of the yaw degree-of-freedom can render the averaged model unusable as an observer. It is also conceivable that the frozen model would generate a poor observer in similar cases.

Frozen Model

Estimator Gain

) Kf K

Averaged Model

Estimator Gain

) Ka K

Full-State Feedback Gain ) Gf G RMS: 0.432 RMS: 0.437 ADC: 1.28 ADC: 1.29 FRR: 2.26 FRR: 2.27 RMS: 0.430 RMS: 0.435 ADC: 1.27 ADC: 1.28 FRR: 2.27 FRR: 2.27 Full-State Feedback Gain ) G Ga RMS: 0.432 ADC: 1.26 FRR: 2.27 RMS: 0.431 * ADC: 1.26 FRR: 2.27

RMS: 0.437 ADC: 1.27 FRR: 2.27 RMS: 0.436 ADC: 1.27 FRR: 2.27

Table 5-1: Time-invariant DAC performance results with the 4 d.o.f. nonlinear plant. RMS speed error in rpm, ADC in deg/s, and FRR in deg/s

For comparison to the periodic observer, the time-invariant design marked with an asterisk in Table 5-1 is used. Results from both DAC designs and an optimized PID design is presented in Table 5-2. The collective pitch PID controller (with KP = 0.62, KI = 0.26, and KD = 0.0) is designed for similar RMS speed error as the periodic controller but with minimum actuator usage. The trend in performance measures suggests that the periodic DAC

110 design reduces cyclic blade loads (by 13% in FRR) over either time-invariant DAC or PID. The trade-off is that additional pitch authority is required – a maximum pitch rate of 11 deg/s. Time response data for blade bending moment, shown in Figure 5-6, supports the periodic controller’s performance.

Controller Periodic DAC Time-invariant DAC PID

Speed RMS (rpm) 0.434 0.436 0.435

ADC (deg/s) 2.00 1.27 1.14

FRR (deg/s) 1.96 2.27 2.24

Table 5-2: DAC performance results with the 4 d.o.f. nonlinear plant

Blade #1 bending moment [kNm]

30

Periodic DAC Time-invariant DAC PID

28 26 24 22 20 18 16 55

55.5

56

56.5

57

57.5 Time [sec]

58

58.5

59

59.5

60

Figure 5-6: Blade #1 bending moment with the 4 d.o.f. nonlinear plant

The influence of nonlinearities in the plant is evident in the wind speed estimation plot, Figure 5-7.

Aerodynamic loads increase with the square of the wind speed and

therefore, when the turbine operates far away from the operating speed (16 m/s in this model), the observer model cannot estimate the disturbance as accurately. In the practical implementation of any wind turbine controller it is often beneficial to design gains about several operating points and use gain scheduling to switch between them as necessary. Implementing scheduled periodic state-space gains would not be so different, particularly when azimuth angle is used to synchronize the gains within each rotation.

111

30 Actual Estimated

Wind speed [m/s]

25 20 15 10 5 0

0

10

20

30

40

50 Time [s]

60

70

80

90

100

Figure 5-7: Wind speed estimator performance of the periodic DAC controller with the 4 d.o.f. nonlinear plant

Nonlinear plant dynamics also affects the ability to estimate blade flap rates, Figure 5-8, as compared to the linear plant response discussed earlier. It appears that only the mean variations in flap rate are estimated adequately and little information can be detected about higher frequency rates. For improved estimation and subsequent load mitigation, additional plant output measurements would be necessary. Options for obtaining such measurements will be discussed shortly.

Blade #1 flap rate [deg/s]

3

Actual Estimated

2 1 0 -1 -2 -3 55

55.5

56

56.5

57

57.5 Time [s]

58

58.5

59

59.5

60

Figure 5-8: Blade #1 flap rate estimator performance of the periodic DAC controller with the 4 d.o.f. nonlinear plant

112 Next we investigate DAC performance when the rotor is free to teeter – the common configuration for a two-bladed downwind turbine. The nonlinear plant now has five active degrees-of-freedom and the same observer and PID designs are compared. Results are given in Table 5-3 and the blade bending moment response is shown in Figure 5-9. Clearly the flap rate RMS is reduced for all controllers, but now it appears that the model-based designs cannot improve upon simply using PID speed control. Essentially, the teeter hinge decouples the blades from the rotor shaft, preventing any blade motion information from showing up in the azimuth and rotor speed signals. This lack of observability can only be corrected by measurement of additional plant outputs. A few of the practical options are discussed next.

Controller Periodic DAC Time-invariant DAC PID

Speed RMS (rpm) 0.434 0.436 0.434

ADC (deg/s) 2.00 1.28 1.23

FRR (deg/s) 0.75 0.73 0.52

Table 5-3: DAC performance results with the 5 d.o.f. nonlinear plant (teetered rotor)

Blade #1 bending moment [kNm]

28 Periodic DAC Time-invariant DAC PID

27 26 25 24 23 22 21 20 55

55.5

56

56.5

57

57.5 Time [sec]

58

58.5

59

59.5

60

Figure 5-9: Blade #1 bending moment with the 5 d.o.f. nonlinear plant (teeter active)

Blade root strain gauges: Strain gauges attached at the roots of each blade provide a measurement of the blade bending moments. This can be related to flap deflection and, with less confidence and more noise, flap rate. The biggest disadvantage with this type

113 of sensor is that they are often unreliable and subject to fatigue failure with cyclic loading. There is also the issue of transferring the data from the rotor to the nacelle, requiring slip rings or other such additional hardware. Of course, this is not an issue if electro-mechanical pitch actuators are employed, as slip rings would already be in place. Blade tip accelerometers: With accelerometers at the blade tips one can measure absolute acceleration there. However, centrifugal loads also contribute to this signal as a function of tip displacement. Without an additional estimate of displacement it is difficult to reconstruct the acceleration due to flap motion only. Also, turbines with a teeter hinge, fast yaw rates, or flexible tower, would require an additional sensor (such as an accelerometer at the hub) in order to calculate relative motion of the blades. As with strain gauges, hardware to transfer information from the rotating frame may also be needed. Tower top accelerometer: An accelerometer in the nacelle can provide a measure of tower motion. Due to dynamic coupling with the blades, active damping of the tower can potentially reduce blade loads. However, differential or asymmetric blade motion will not be captured and this may limit the effectiveness of a tower sensor.

5.6 Summary This chapter has presented the design of a periodic state estimator and wind disturbance estimator via disturbance accommodating control (DAC) and linear quadratic regulation (LQR) techniques. Comparisons with various time-invariant estimator designs have been made. Constant gains for the estimator may be designed via the methods introduced in the full-state feedback control chapter. However, to be completely time-invariant one must also choose a constant parameter turbine model in order to estimate states.

Two logical

approaches were taken; averaging the periodic state matrices and freezing the matrices at an

114 azimuth position.

The choice of freezing azimuth angle was made on the basis of a

parametric study involving only full-state feedback. Simulation results were presented for a multi-objective problem; simultaneous speed regulation and blade load mitigation. Periodic control of a linear plant identifies good wind estimation but poor estimation of tower and blade flap motion. This was attributed to leakage of the disturbance into the plant since exact cancellation is not possible. With a nonlinear plant, all the time-invariant estimators performed almost identically. A comparison was then made to periodic control and PID. The periodic controller performed the best in terms of reduction of cyclic blade loads but must use far more actuator authority to do so. When the teeter degree-of-freedom is added to the plant, periodic control loses its advantage and performs the same as time-invariant control. In this case however, both of the state-space based controllers increase the amount of cyclic blade loading when PID is the baseline. Use of additional plant sensors is recommended to achieve the level of load reduction evident in the full-state feedback case.

115 CHAPTER 6 6. CONCLUSIONS AND RECOMMENDATIONS

The SymDyn structural dynamics model is built on the assumption that the dominant physics of a wind turbine can be captured with only the first bending mode in the tower and blades.

By doing so relatively few degrees-of-freedom carry through the derivation,

facilitating the subsequent modal analyses and control designs.

In the process, much

information can be extracted about the interaction of the dynamic subsystems that make up a wind turbine. Both SymDyn and AeroDyn subroutines are implemented in a modular style through Simulink subsystems in Matlab.

This allows study of either system independently and

facilitates potential upgrades. There is also the capability of suppressing any degree-offreedom as necessary to reduce the model dimensions. Chapter 3 introduced the Floquet theory to calculate the characteristic exponents of a periodic system. The parametric studies presented in the structure-only analysis section illustrate the capability of SymDyn to efficiently calculate modal properties. Knowledge of the many forms of dynamic coupling that exist in wind turbines is important if resonant frequencies are to be considered in the overall design. The aeroelastic system analysis at the end of the chapter allowed identification of the damping that results solely from aerodynamic effects. We noted that, while the blade flap modes are highly damped, the tower fore-aft motion is not. One of the important goals of this research was to develop advanced control concepts for wind turbines. This has been achieved through application of optimal periodic control theory and the design of many different methods for the construction of time-invariant control systems. Both periodic and time-invariant designs were compared to PID for three different performance objectives. There is no clear winner for all cases and therefore, for practical

116 implementation it would be advantageous to explore all options. Studies on independent pitch control in Region III led to the following conclusions.

ƒ

If speed regulation is the only control objective then periodic control is not the most appropriate method. Either PID or time-invariant state-space control can regulate speed to the same extent but with lower actuator usage.

ƒ

When fatigue of the blades is an issue then the control objective should include the mitigation of cyclic blade loads. In the full-state feedback case, periodic control gives the best performance, followed by time-invariant state-space control. When the controller must estimate states from only azimuth position and rotor speed then controller performance is degraded. A teetered-hub, which naturally reduces the cyclic blade loads, reduces controller performance by preventing estimation of blade motion. In this case, the use of additional sensors would be necessary to produce the level of load attenuation witnessed with full-state feedback. Three-bladed turbines use a rigid hub and it is here that load mitigation may be most effective.

ƒ

Yaw directional control by blade pitch was also assessed.

Here, only the periodic

controller was capable of producing asymmetric loads about the yaw axis, which is required to align the rotor axis with the wind direction. However, cyclic blade loads increased with this method. For a downwind turbine, yaw control could be used to improve the yaw response rate to large changes in wind direction, increasing energy capture. For an upwind turbine, yaw control by blade pitch could reduce the demands on the yaw servo. It is clear that the yaw servo could not be dispensed with entirely because, at low or zero wind speeds, blade pitching would have no control authority.

Although the results do suggest applications for periodic control, there is the issue of implementation. Periodic control requires knowledge of the rotor azimuth position at all

117 times to synchronize feedback gains and the periodic estimator model. Therefore, additional hardware may be required on the turbine. For a variable speed machine that measures rotor speed optically it is possible to estimate azimuth fairly accurately.

The application of

periodic control to any turbine that doesn’t already measure rotor speed may not be economically feasible. Also, it is through independent blade pitch that periodic control is most effective. The rotor would require electro-mechanical pitch actuators installed for each blade. This further limits the application of periodic control to only newer turbine designs. For the cases when time-variant state-space control is the most suitable approach, a number of design methods have been introduced in this thesis. One of the most simple ways to calculate constant state and estimator gains is to average the periodic gains. This method appears to perform well in most situations. For the construction of a time-invariant estimator model, the methods of averaging and freezing the periodic state matrices have been proposed. Both have produced good results, meaning that the states of a periodic plant can be estimated well with a time-invariant model. In the estimator studies it was assumed that azimuth position was measured, even in the time-invariant case. This has been done only so that a fair comparison can be made to periodic control. In the actual implementation of a time-invariant controller this would not be necessary. An additional technicality is the possibility that the plant states and disturbance would no longer be observable. In this case the azimuth position state could be removed from the model. The use of disturbance accommodating control has been invaluable for the estimation of wind speed disturbance. With this information the state-space controllers have been able to partially cancel the effect of wind fluctuations on the system. It has been shown that trends in the wind speed have been estimated well, but for extreme wind speeds the errors are larger. For practical implementation, control gains could be designed for several operating points based on the expected range of wind speed. Then gain scheduling could be employed using

118 estimated wind speed to determine when gains should be switched. Both regulation and estimation properties would be improved with this approach. A direction for future research is the testing of the control designs on a more comprehensive simulation model and on an actual wind turbine. The important issues to be dealt with will include higher order vibration modes in the structure, operation in Region II, and the influence of sensor noise. For control in Region II, generator torque could take over the task of speed regulation while load mitigation or other objectives could be accomplished with pitch actuation.

119 REFERENCES [1] Hansen, A.C., 1998, User’s Guide to the Wind Turbine Dynamics Computer Programs YawDyn and AeroDyn for ADAMS, Version 11.0, Mechanical Engineering Department, University of Utah, Salt Lake City, UT. [2] Wilson, R.E., 1999, “Technical and User’s Manual for the FAST_AD Advanced Dynamics Code,” OSU/NREL Report 99-01, Oregon State University, Corvallis, OR. [3] Mechanical Dynamics, Inc., 1998, Using ADAMS/Solver (v9.1), Mechanical Dynamics, Inc. Ann Arbor, MI. [4] Wiberg, D.M., 1971, State Space and Linear Systems, McGraw-Hill, New York, NY. [5] Hau, E., 2000, Windturbines, Springer, Berlin, Germany. [6] Hand, M. and Balas, M., 2000, “Systematic controller design methodology for variablespeed wind turbines,” Wind Engineering, 24, pp. 169-187. [7] Rock, S.M., Eggers, A.J., Moriarty, P.J., and Chaney, K., 2000, “Tradeoffs in Active Control of Aerodynamic Power and Loads on a HAWT Rotor,” Proc. 19th ASME Wind Energy Symposium, Reno, NV, pp. 75-83. [8] Bassanyi, E.A., 2000, “Developments in Closed Loop Controller Design for Wind Turbines,” Proc. 19th ASME Wind Energy Symposium, Reno, NV, pp. 64-74. [9] Ekelund, T., 2000, “Yaw control for reduction of structural dynamic loads in wind turbines,” J. Wind Eng. and Industrial Aerodynamics, 85, Elsevier Science Publishers B.V., Amsterdam, Netherlands, p 241-262. [10] Kendall, L., Balas, M., Lee, Y.J., and Fingersh, L.J., 1997, “Application of Proportional-Integral and Disturbance Accommodating Control of Variable Speed Variable Pitch Horizontal Axis Wind Turbines,” Wind Engineering, 21, pp. 21-38. [11] McKillip, R., 1984, “Periodic Control of the Individual-Blade-Control Helicopter Rotor,” Ph.D. thesis, MIT, Cambridge, MA. [12] Jensen, K. E., Fahroo, F., and Ross, I. M., 1998, “Application of optimal periodic control theory to the orbit reboost problem,” Proc. AAS/AIAA Space Flight Mechanics Meeting, Univelt, Inc., San Diego, CA, pp. 935-945. [13] Liebst, B.S., 1983, “Pitch Control for Large-Scale Wind Turbines,” J. Energy, 7, pp. 182-192. [14] Hodges, D. H. and Patil, M., 2000, “Multi Flexible Body Analysis for Application to Wind Turbine Control Design,” Proc. 19thASME Wind Energy Symposium, Reno, NV, pp. 110-120. [15] Craig, J.J., 1955, Introduction to Robotics, Addison-Wesley Publishing Company, New York, NY.

120 [16] Wu, K.C., 1998, “An Approach to the Development and Analysis of Wind Turbine Control Algorithms,” Sandia National Laboratories Contractor Report SAND98-0668, Albuquerque, NM. [17] Kane, T.R. and Levinson, D.A., 1985, Dynamics, theory and applications, McGrawHill, New York, NY. [18] Meirovitch, L., 1997, Principles and Techniques of Vibrations, Prentice Hall, NJ. [19] Stol, K. and Bir, G., 2000, “Validation of a Symbolic Wind Turbine Structural Dynamics Model,” Proc. 19th ASME Wind Energy Symposium, Reno, NV, pp. 41-48. [20] Astley, R.J., 1992, Finite Elements in Solids and Structures, Chapman and Hall, London. [21] James, G.H., 1994, “Extraction of Modal Parameters from an Operating HAWT using the Natural Excitation Technique (NExT),” Proc. 13th ASME Wind Energy Symposium, pp. 227-232. [22] Malcolm, D.J. and James, G.H., 1996, “Identification of Natural Operating Modes of HAWTs from Modeling Data,” Proc. 15th ASME Wind Energy Symposium, pp. 24-31. [23] Lobitz, D.W. and Sullivan, W.N., 1980, “VAWTDYN: A Numerical Package for the Dynamic Analysis of Vertical Axis Wind Turbine,” Technical Report SAND-80-0085, Sandia National Laboratories, Albuquerque NM. [24] Carne, T.G., Lobitz, D.W., Nord, A.R., and Watson, R.A., 1982, “Finite Element Analysis and Modal Testing of a Rotating Wind Turbine,” Presented at the AIAA/ASME/ASCE/AHS Structures - Structural; Dynamics and Materials Conference, New Orleans LA. [25] Carne, T.G., Martinez, D.R., and Ibrahim, S.R., 1983, “Modal Identification of a Rotating Blade System,” Technical Report SAND82–2115, UC–32, Sandia National Laboratories, Albuquerque NM. [26] James, G.H., Carne, T.G., and Lauffer J.P., 1993, “The Natural Excitation Technique (NExT) for Modal Parameter Extraction from Operating Wind Turbines,” Technical Report SAND92–1666, UC–261, Sandia National Laboratories, Albuquerque NM. [27] Yamane, T. and Thresher, R.W., 1987, “Coupled Rotor/Tower Stability Analysis of a 6Meter Experimental Wind Turbine,” Proc. 6th ASME Wind Energy Symposium, Dallas TX. [28] Bir, G.S. and Butterfield, C.P., 1997, ”Modal Dynamics of a Next-Generation FlexibleRotor Soft-Tower Wind Turbine,” Proc. 15th International Modal Analysis Conference, Orlando FL. [29] Johnson, W. J., 1980, Helicopter Theory, Princeton University Press, Princeton, NJ, pp. 369-377.

121 [30] Bir, G. and Stol, K., 2000, “Modal Analysis of a Teetered-Rotor Wind Turbine using the Floquet Approach,” Proc. 19th ASME Wind Energy Symposium, Reno, NV, pp. 2333. [31] Stol, K. and Bir, G., (2002), “Floquet Modal Analysis of a Teetered-Rotor Wind Turbine,” Accepted for J. Sol. Energy Eng., 124. [32] Nayfeh, A.H. and Mook, D.T, 1979, Nonlinear Oscillations, John Wiley & Sons, New York NY. [33] Sinha, S.C. and Wu D.H., 1991, “An Efficient Computational Scheme for the Analysis of Periodic Systems,” J. Sound and Vibration, 151, pp. 91-117. [34] Bauchau, O.A. and Nikishkov, Y.G., 1998, “An Implicit Floquet Analysis for Rotorcraft Stability Evaluation,” Proc. 54th AHS Annual Forum, Washington DC. [35] Holley, W.E. and Bahrami, M., 1981, “Periodic Linear Systems Forced by White Noise with Wind Turbine Applications,” Proc. 15th Asilomar Conference on Circuits, Systems, and Computers, Pacific Grove CA. [36] Peter, D. A., 1994, “Fast Floquet Theory and Trim for Multi-Blade Rotorcraft,” J. American Helicopter Society, 39-4, pp. 82-89. [37] Brunovsky, P., 1969, “Controllability and Linear Closed-loop Controls in Linear Periodic Systems,” J. Differential Equations, 6, pp. 296-313. [38] Bittanti, S., Laub, A.J., and Willems, J.C. (eds.), 1991, The Riccati Equation, Springer Verlag, Berlin, pp. 127-162. [39] Kwakernaak, H. and Sivan, R., 1972, Linear Optimal Control Systems, Wiley Interscience, New York, NY. [40] Johnson, C.D., 1976, “Theory of Disturbance Accommodating Controllers,” Advances in Control and Dynamic Systems, 12, pp. 387-489. [41] Balas, M., and Lee, Y., 1996, “Stable Disturbance Accommodating Control of LargeScale Systems Using Singular Perturbations with Application to Variable Speed Wind Turbines,” Proc. 30th Conf. on Information Sciences and Systems, Princeton, NJ. [42] Stol, K. and Balas, M., 2002, “Periodic Disturbance Accommodating Control for Speed Regulation of Wind Turbines,” Proc. 21st ASME Wind Energy Symposium, Reno, NV.

122 APPENDIX A A. SAMPLE NONLINEAR SYMDYN EQUATIONS OF MOTION

Degrees-of-freedom: q = [τ1

ψ β1 β 2 ]

T

(tower fore-aft, azimuth, blade #1 flap, blade #2 flap)

A definition of the symbols in the following Mathematica expressions can be found in Appendix C. Note that some of the symbols cannot be represented exactly as defined.

Tower fore-aft (τ1) equation:

123

124 Azimuth (ψ) equation:

125 Blade #1 flap (β1) equation:

Blade #2 flap (β2) equation:

126 APPENDIX B B. ALTERNATIVE DERIVATION OF THE SYMDYN EQUATIONS OF MOTION

We define V ∈ ℜ3×N and Ω ∈ ℜ3×N as coefficients of the velocity vector (2.6) and angular velocity vector (2.5) respectively.

⇒

0

v i = V i q&

0

v& i = V&i q& + Vi &q&

⇒

0

ωi = Ωi q&

0

& i = Ω& i q& + Ωi &q& ω

(B.1)

From Lagrange’s equations of motion (2.10) and the supporting expressions, (2.8) and (2.9), the equation of motion for link j is given by d  ∂ ∂ Kti + Kri − K t i + K r i − Vi    ∂q j  dt  ∂ q& j i =1  N

∑

(

)

(



) = Q .

(B.2)

j



We now calculate each term in (B.2).  d ∂ d  ∂ K ti  =   dt  ∂ q& j dt  ∂ q& j   = mi

1 2

 0 m i v i .0 v i   

0 d  ∂ v i 0  . vi  dt  ∂ q& j 

(B.3)

(

= m i V&i (:, j) .0 v i + V i(:, j) .0 v& i

)

where V i(:, j) is the j th column of V i

T T T = m i  V&i (:, j) V i + V i (:, j) V&i  q& + m iV i(:, j) V i &q&  

 d ∂ d  ∂ Kri  =    dt  ∂ q& j dt  ∂ q& j  

1 0 ωi .I i 0 ωi 2

   

where I i ≡ 0 I i to simplify notation

0 d  ∂ ωi 0  = .I i ωi  dt  ∂ q& j  d = where Ωi(:, j) is the j th column of Ωi Ωi(:, j) .I i 0 ωi dt = Ω& i(:, j) . I i Ωi q& + Ωi(:, j) .&I i Ωi q& + Ωi(:, j) .I i Ω& i q& + Ωi &q&

(

)

(

=  Ω& i(:, j) I i Ωi + Ωi(:, j)  T

T

(&I Ω i

i

)

)

T + I i Ω& i  q& + Ωi(:, j) I i Ωi &q& 

(B.4)

127

( )

∂ ∂ Kti = ∂q j ∂q j = mi

(

0

m i v i .0 v i

1 2

)

∂ 0 vi 0 . vi ∂q j

( )

T

= m i V i ' q& V i q&

( )

(B.5)

∂ ∂ K ri = ∂q j ∂q j

(

where V i ' ≡

0 1 0 ωi .I i ωi 2

∂I 0 1 0 ωi . i ωi 2 ∂q j

((

(Ω q& ) I ')Ω q&

= Ωi ' q&

)

T

Ii +

1 2

1, k = j q& k =   0, otherwise

T

i

i

i

where Ωi ' ≡

I i ' ≡ &I i ∂ (Vi ) = − ∂ m i g . 0 r i ∂q j ∂q j = −m i g . 0 v i T

= −m i g Vi Q load j =

N

∑ i =1

=

0

T

Fi V j

N

∂V i

∑ ∂q j=1

j

.q& j

∂ Ωi = Ω& i ∂q j

(B.6) 1, k = j q& k =   0, otherwise

,

1, k = j q& k =   0, otherwise

ignoring the joint stiffness term in (2.9)

(B.7)

1, k = j q& k =   0, otherwise

(:, j)

 0 ∂ 0vi 0 ∂ 0 ωi  + Mi.  Fi .  ∂ q& j ∂ q& j  

∑[ N

since V&i =

)

∂ 0 ωi 0 .I i ωi + ∂q j

=

∂V i & =V i ∂q j

(:, j)

T

+ 0Mi Ωj

(:, j)

]

ignoring the joint damping term (B.8)

i =1

Combining (B.2) through (B.8) with joint stiffness and damping terms, the equation of motion for link j is given by M j &q& + f j = 0 where

c.f. (2.16)

(B.9)

128

Mj =

N

∑ m V

i i

(:, j) T

i =1

T V i + Ωi(:, j) I i Ωi  

(

(B.10)

)

T   & (:, j) (:, j) T &    Vi  +    m i  Vi − Vi ' q& Vi + Vi   q& − N  fj =  Ω& (:, j) − Ω ' q& T I − 1 q& T Ω T I ' + Ω (:, j) T &I  Ω + Ω (:, j) T I Ω&   + C j q& j + K j q j  i i i i i i i i i 2  i  i i =1      T (:, j) T (:, j) T (:, j) 0 0 m i g Vi − F j V j − M j Ω j  

∑

(

)

Forming V i , V&i , Ωi , Ω& i , I i , and &I i : Let Ri ≡

i-1

Ri ∈ ℜ3×N, the transformation matrix for the ith link relative to the (i-1)th

link – a function of qi only. Define Si ∈ ℜ3×N and Ti ∈ ℜ3×N from R& i ≡ S i q& i , and S& i ≡ Ti q& i . Then V i , V&i , Ωi , Ω& i , I i , and &I i can be formed as compact symbolic functions of Ri, Si, and Ti (as well as constant geometric and mass properties).

Vi (R 1 , K , R i , S1 , K , Si ) Ωi (R 1 , K , R i )

I i (R 1 , K , R i )

V&i (R 1 , K , R i , S1 , K , Si , T1 , K , Ti ), Ω& (R , K , R , S , K , S ), i

1

i

1

i

&I (R , K , R , S , K , S ) i 1 i 1 i

129 APPENDIX C C. TURBINE PROPERTY DATA Parameter Description Units Value* Value** Nominal rotor speed rpm 57.5 57.5 Ω0 g Acceleration due to gravity m/s2 9.81 9.81 Fixed tilt angle deg 0 0 η0 Teeter hinge angular offset deg 0 0 δ3 Blade precone angle deg 7 7 β0 dt1 Fore-aft tower hinge offset m 3 9.332 dt2 Tower fore-aft hinge to tilt axis distance m 22 15.468 dn1 Tilt axis to shaft axis distance m 0 0 dn2 Yaw axis to teeter axis longitudinal distance m 2.388 2.388 dh1 Teeter axis to flap hinge longitudinal dist. m 0.2 -0.375 dh2 Shaft axis to flap hinge distance m 1.8 5.783 ct C.O.M. distance of tower m 7 9.931 cp C.O.M. distance of bedplate m 0 0 cn C.O.M. distance of nacelle m 0.763 0.789 cs C.O.M. distance of shafts m -0.386 -0.386 ch C.O.M. distance of hub m 0 0.139 cb C.O.M. distance of blade m 4.16 2.864 mt Mass of tower kg 10000 3022 mp Mass of bedplate kg 0 0 mn Mass of nacelle, including generator housing kg 6900 6900 ms Mass of both shafts kg 800 800 mh Mass of hub kg 1240 1804 mb Mass of each blade kg 455 173 It_lat M.O.I. of tower lateral axis kgm2 1041 1×107 It_long M.O.I. of tower longitudinal axis kgm2 41.8 1×107 2 Ipx, Ipy, Ipz M.O.I. of bedplate kgm 0 0 Inx M.O.I. of nacelle about vertical axis kgm2 7721 7738 Iny M.O.I. of nacelle about longitudinal axis kgm2 110 110 Inz M.O.I. of nacelle about lateral axis kgm2 7721 7738 Is_lat M.O.I. of both shafts about lateral axis kgm2 1793 1793 Is_long M.O.I. of low speed shaft about long. axis kgm2 50 50 Ig_long M.O.I. of high speed shaft about long. axis kgm2 30 30 Ihx M.O.I. of hub about shaft axis kgm2 5 7536 Ihy M.O.I. of hub about lateral axis kgm2 50 30 Ihz M.O.I. of hub about teeter hinge kgm2 50 7561 Ib M.O.I. of blade about flap hinge kgm2 12588 1469 Ib_long M.O.I. of blade about pitch axis kgm2 4 0 Tower fore-aft and lateral hinge stiffness Nm/rad 2.72×108 1.14×108 Kτ1, Kτ2 Tower twist hinge stiffness Nm/rad 1.78×108 4.35×107 Kτ3 4 Nacelle yaw hinge stiffness Nm/rad 0 3.28×10 Kγ Shaft compliance hinge stiffness Nm/rad 100 N/A Kψg Hub teeter hinge stiffness Nm/rad 0 3.28×104 Kφ Blade flap hinge stiffness Nm/rad 2.675×106 3.998×105 Kβ1, Kβ2 Nms/rad 0 0 Cτ1, Cτ2, Cτ3 Tower hinge damping 3 Nacelle yaw hinge damping Nms/rad 0 3.28×10 Cγ Shaft compliance hinge damping Nms/rad 0 N/A Cψg Hub teeter hinge damping Nms/rad 0 3.28×103 Cφ Blade flap hinge damping Nms/rad 0 0 Cβ1, Cβ2 * ** Table C-1: SymDyn property values: verification model, all other models

130 APPENDIX D D. AERODYN INTERFACE SOURCE CODE

Listing 1: aero.f file

C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C

**************************************************************** AERO (modified from AERO subroutine in ‘yawdyn.f’) Calculates all aerodynamic blade loads, LOADS, given vectors containing: SymDyn angles, qv, velocities, qvdot, blade pitch angles, pitchin, and hub-height wind field, WIN Assumes 4+Nb DOFs: tower fore-aft, yaw, azimuth, teeter, flap of each blade (Nb blades) 5/4/99: K. Stol Created from AERO subroutine v11.03 (11/4/98) 11/16/00 Updated code to YawDyn/AeroDyn v11.36 (7/17/00) 11/19/00 Changed load variable names, to be consistent with SymDyn (Fbx ... Mbz) Now Fbx is non-zero, with contribution from non-flapping blade section LOADS is declared for Nb=3. If Nb=2 then elements 13 to 16 are zero. LOADS(1) = Fb1x LOADS(2) = Fb1y LOADS(3) = Fb1z LOADS(4) = Fb2x LOADS(5) = Fb2y LOADS(6) = Fb2z LOADS(7) = Fb3x LOADS(8) = Fb3y LOADS(9) = Fb3z LOADS(10) = Mb1x LOADS(11) = Mb1y LOADS(12) = Mb1z LOADS(13) = Mb2x LOADS(14) = Mb2y LOADS(15) = Mb2z LOADS(16) = Mb3x LOADS(17) = Mb3y LOADS(18) = Mb3z 7/12/01 Corrected for teetered-rotor case. NOTE: Teeter must only be used with Nb=2 or results will be incorrect! ***************************************************************** SUBROUTINE AERO( qv, qdot, pitchin, WIN, LOADS )

C

Initialization from Yawdyn.f USE USE USE

FF_Wind Outputs Identify

INCLUDE 'aerodyn.inc' INCLUDE 'bedoes.inc' DIMENSION CHARACTER*80 CHARACTER*80

QP ( 2*MAXBLD+2 ) FOILNM ( MAXELEM ) TITLE

131 DIMENSION FN(0:NRING) DIMENSION RM(NRING) C

! from YAWDYN main program (in Yawdyn.f)

Declaration of SymDyn related variables REAL*8 variables used for MATLAB REAL*8 qv(10), qdot(10), pitchin(3), WIN(7), LOADS(18) REAL PSIIN, qv2(10), qdot2(10) REAL Fbx, Fby, Fbz, Mbx, Mby, Mbz

C

Copy REAL*8 variables to REAL variables to be consistent with AeroDyn DO 13 I = 1, 10 qv2(I) = qv(I) qdot2(I) = qdot(I) 13 CONTINUE

C

Initialization from original AERO subroutine (in Yawdyn.f) YAWRAT = qdot2(2)

C

Initialization from YAWDYN main program (in Yawdyn.f) TAB PI PIBY2 TWOPI DEG TIME AVGINFL AVGINFL1 ADAMSRUN IDADAMS

C

! Yaw rate [rad/s] used in VELD subroutine

= = = = = = = = = =

CHAR(9) 4. * ATAN( 1. ) PI / 2. 2.0 * PI 180. / PI 0.0D0 0.0 0.0 .TRUE. ! Emulate ADAMS interface (for input file reading only) 1

Initialize output vector DO 11 I = 1, 18 LOADS(I) = 0.0 11 CONTINUE

C

Open file to write to - all write(*,*) changed to write(92,*) for MATLAB OPEN ( UNIT=92, FILE='yawdyn.opt', IOSTAT=IERR )

C

Read input file 'yawdyn.ipt' CALL YAWIN( SECTOR, N, TOLER, QP, NUMFOIL, FOILNM, TITLE )

C

C C

Add missing variables and calculations because part of YAWIN not executed RH = 0.0 BLENGTH = RELM( NELM ) + DR( NELM )/2.0 R = RH + BLENGTH * COSPC Calculate tip loss constants for each blade element Use mid-element value if located at tip or axis of rotation DO 46 I = 1, NELM RL = RH + RELM(I) * COSPC IF( RL .EQ. 0.0 ) RL = DR(I) / 2. TLCNST(I) = 0.5 * B * ( R - RL ) / RL IF( TLCNST(I) .LT. 0.001 ) & TLCNST(I) = 0.5 * B * DR(I) / 2.0 * COSPC / RL 46 CONTINUE PSI = PsiInit PSIIN = qv2(3)

C

Set up the initial values (from YawDyn v11.15) FY4 = 0.0 NSECT = SECTOR ISCREEN = SECTOR / 8

132 C

Overwrite ‘yawdyn.wnd’ data V DELTA VZ HSHR VSHR VLinShr VGUST

= = = = = = =

WIN(1) WIN(2)/DEG WIN(3) WIN(4) WIN(5) WIN(6) WIN(7)

DO 12 I = 5, 10 qv2(I) = qv2(I) + PC 12 CONTINUE C C C

! Add precone to blade flap since not included in ! MATHEMATICA expressions

Calculate blade forces DO 10 IBLADE = NB, 1, -1 qv2(3) = PSIIN + FLOAT(IBLADE-1) * TWOPI/NB qv2(4) = -qv2(4) ! Quick way to ensure blade#2 has -ve teet angle qdot2(4) = -qdot2(4) ! and velocity (blade#1 is unchanged) PITCH(IBLADE) = pitchin(IBLADE) &

CALL BLDFM(qv2, qdot2, Fbx, Fby, Fbz, Mbx, Mby, Mbz, RM, AVEL) AVGINFL1 = AVGINFL1 LOADS(IBLADE*3 - 2) LOADS(IBLADE*3 - 1) LOADS(IBLADE*3) LOADS(IBLADE*3 + 7) LOADS(IBLADE*3 + 8) LOADS(IBLADE*3 + 9)

+ = = = = = =

AVEL LOADS(IBLADE*3 LOADS(IBLADE*3 LOADS(IBLADE*3) LOADS(IBLADE*3 + LOADS(IBLADE*3 + LOADS(IBLADE*3 +

2) + Fbx 1) + Fby + Fbz 7) + Mbx 8) + Mby 9) + Mbz

10 CONTINUE AVGINFL = AVGINFL1 / B CLOSE(92) RETURN END

! Calculated but not stored by MATLAB ! Used in skewed wake calculation (SKEW)

133 Listing 2: bldfm.f file

C C C C C C C C C C C C C C C C C C C C C C

********************************************** BLDFM (modified from BLDFM subroutine in ‘yawdyn.f’) Calculates the forces and moments for a particular blade Assumes 4+Nb DOFs: tower fore-aft, yaw, azimuth, teeter, flap of each blade (Nb blades) Assumes the following geometric properties from SymDyn: dn1 = 0, dn2 = 2.388m, dh1 = -0.375m, dh2 = 5.783m 6/15/00 K. Stol Created from BLDFM subroutine v11.03 (11/4/98) 11/18/00 K. Stol Updated to YawDyn/AeroDyn v11.36 (7/17/00) Changed load variable names to be consistent with SymDyn Fbx (no YawDyn equivalent), Fby (FN(0)), Fbz (-FT), Mbx (PMOM), Mby (EMOM), Mbz (FM) ********************************************** &

SUBROUTINE BLDFM(qv, qdot, Fbx, Fby, Fbz, Mbx, Mby, Mbz, RM, AVEL) USE

FF_Wind

INCLUDE 'aerodyn.inc' DIMENSION FN(0:NRING) DIMENSION RM(NRING) REAL qv(10), qdot(10), sq(10), cq(10) REAL beta, sb, cb, bdot, sb2, cb2 REAL Fbx, Fby, Fbz, Mbx, Mby, Mbz Fbx Fby Fbz Mbx Mby Mbz AVEL

= = = = = = =

0. 0. 0. 0. 0. 0. 0.

DO 20 IRING = 1, NRING FN(IRING) = 0. RM(IRING) = 0. 20 CONTINUE C C

AVEL is the induced velocity in the normal direction from the momentum equation.

C C

Get ambient wind velocities relative to ground reference frame First get the spatial mean velocities (name ends in BAR) IF( FFWindFlag VXGBAR = VYGBAR = VZGBAR = ELSE VXGBAR = VYGBAR = VZGBAR = ENDIF DO 11 I = 1,10

) THEN MeanFFWS 0.0 0.0 V * COS( DELTA ) -V * SIN( DELTA ) VZ

134 sq(I) = SIN(qv(I)) cq(I) = COS(qv(I)) 11 CONTINUE SQ3 = -sq(2) CQ3 = cq(2)

! Q3 = -yaw angle (reversed sign). Kept because of ! reference in VEL and VELD, ! although not used if WAKE = FALSE in VEL

TILT = 0 CTILT = COS(TILT) STILT = SIN(TILT)

! ensures tilt is always zero regardless of ! value in 'yawdyn.ipt'

CALL VELD c

If Dynamic Inflow is used, no skewed wake correction is made. IF ( .NOT. DYNINFL ) CALL GETSKEW DO 10 J = 1, NELM

C

PITNOW is the local blade element pitch (including twist) PITNOW = PITCH( IBLADE ) + TWIST( J )

C C

RLOCAL is the radius of the blade element measured from the axis of rotation RLOCAL = RH + RELM(J) * cq(IBLADE+4)

C C C C

8/8/00 New variable XELM measures blade element distance from flap hinge position, not blade root as assumed in YawDyn - this allows for non-flapping blade section XELM = RELM(J) - (5.783 - RH)/COS(PC) IF (XELM .LT. 0.0) THEN beta = PC bdot = 0 ELSE beta = qv(IBLADE+4) bdot = qdot(IBLADE+4) ENDIF sb = SIN(beta) cb = COS(beta)

C C C

! dh2 = 5.783

Get the Cartesian coordinates of the blade element in the ground (inertial) reference frame Imported SymDyn expressions from MATHEMATICA & & & & & & & & & & & & & & &

C

! not used if WAKE = FALSE

XGRND = 2.388*cq(1)*cq(2) + 5.783*(cq(3)*cq(4)*sq(1) + cq(1)* cq(4)*sq(2)*sq(3) + cq(1)*cq(2)*sq(4)) - 0.375*((-1)* cq(1)*cq(2)*cq(4) + cq(3)*sq(1)*sq(4) + cq(1)*sq(2)*sq( 3)*sq(4)) + (cq(1)*cq(2)*cq(4)*sb + cb*cq(3)*cq(4)*sq( 1) + cb*cq(1)*cq(4)*sq(2)*sq(3) + cb*cq(1)*cq(2)*sq(4) + (-1)*cq(3)*sb*sq(1)*sq(4) + (-1)*cq(1)*sb*sq(2)*sq(3) *sq(4))*XELM YGRND = 2.388*sq(2) + 5.783*((-1)*cq(2)*cq(4)*sq(3) + sq(2)*sq( 4)) - 0.375*((-1)*cq(4)*sq(2) + (-1)*cq(2)*sq(3)*sq(4) ) + (cq(4)*sb*sq(2) + (-1)*cb*cq(2)*cq(4)*sq(3) + cb* sq(2)*sq(4) + cq(2)*sb*sq(3)*sq(4))*XELM ZGRND = -2.388*cq(2)*sq(1) + 5.783*(cq(1)*cq(3)*cq(4) + (-1)* cq(4)*sq(1)*sq(2)*sq(3) + (-1)*cq(2)*sq(1)*sq(4)) + -0.375*(cq(2)*cq(4)*sq(1) + cq(1)*cq(3)*sq(4) + (-1)* sq(1)*sq(2)*sq(3)*sq(4)) + (cb*cq(1)*cq(3)*cq(4) + (-1) *cq(2)*cq(4)*sb*sq(1) + (-1)*cb*cq(4)*sq(1)*sq(2)*sq(3) + (-1)*cq(1)*cq(3)*sb*sq(4) + (-1)*cb*cq(2)*sq(1)*sq( 4) + sb*sq(1)*sq(2)*sq(3)*sq(4))*XELM Get wind and rotational velocities CALL VEL( VNROTOR2, .FALSE. )

135 ************************************************************************ c akihiro 01/24/00 revived for v11.31b c akihiro 11/05/99 change for v11.23 ************************************************************************ c calculate the spatial average of the turbulent wind speeds IF ( FFWindFlag ) THEN nVFFbar = nVFFbar + 1 VFFbar( 1 ) = ( VFFbar( 1 ) * ( nVFFbar -1 ) + VX ) / nVFFbar VFFbar( 2 ) = ( VFFbar( 2 ) * ( nVFFbar -1 ) + VY ) / nVFFbar VFFbar( 3 ) = ( VFFbar( 3 ) * ( nVFFbar -1 ) + VZ ) / nVFFbar END IF ************************************************************************ C

Get velocities relative to blade element

C C

Get normal and tangential velocity components treat wind and blade motions separately

C C C C C C

VTW VNW VTB VNB

Tangential component of wind Normal component of wind Tangential component of blade element abs. vel. Normal component of blade element abs. vel.

Imported expressions from MATHEMATICA notebook, aerodyn_expr.nb & & & & & & & & & & & & & & & & & & & & & & &

VTW = ((-1)*cq(1)*cq(3)*sq(2) + sq(1)*sq(3))*VX + cq(2)*cq(3) *VY + (cq(3)*sq(1)*sq(2) + cq(1)*sq(3))*VZ VNW = (cb*cq(1)*cq(2)*cq(4) + (-1)*cq(3)*cq(4)*sb*sq(1) + ( -1)*cq(1)*cq(4)*sb*sq(2)*sq(3) + (-1)*cq(1)*cq(2)*sb* sq(4) + (-1)*cb*cq(3)*sq(1)*sq(4) + (-1)*cb*cq(1)*sq(2) *sq(3)*sq(4))*VX + (cb*cq(4)*sq(2) + cq(2)*cq(4)*sb*sq( 3) + (-1)*sb*sq(2)*sq(4) + cb*cq(2)*sq(3)*sq(4))*VY + ( (-1)*cq(1)*cq(3)*cq(4)*sb + (-1)*cb*cq(2)*cq(4)*sq(1) + cq(4)*sb*sq(1)*sq(2)*sq(3) + (-1)*cb*cq(1)*cq(3)*sq(4) + cq(2)*sb*sq(1)*sq(4) + cb*sq(1)*sq(2)*sq(3)*sq(4))* VZ VTB = qdot(2)*(-2.388*cq(3) - 0.375*cq(3)*cq(4) - 5.783*cq( 3)*sq(4) + ((-1)*cq(3)*cq(4)*sb + (-1)*cb*cq(3)*sq(4))* XELM) + qdot(3)*(5.783*cq(4) - 0.375*sq(4) + (cb*cq(4) + (-1)*sb*sq(4))*XELM) + qdot(1)*(15.468*cq(3)*sq(2) + 2.388*cq(2)*sq(3) - 0.375*((-1)*cq(2)*cq(4)*sq(3) + sq(2)*sq(4)) + 5.783*(cq(4)*sq(2) + cq(2)*sq(3)*sq(4)) + (cb*cq(4)*sq(2) + cq(2)*cq(4)*sb*sq(3) + (-1)*sb*sq( 2)*sq(4) + cb*cq(2)*sq(3)*sq(4))*XELM) VNB = qdot(4)*(-5.783*cb - 0.375*sb + (-1)*XELM) + (-1)* bdot*XELM + qdot(1)*(-5.783*cb*cq(2)*cq(3) - 0.375*cq( 2)*cq(3)*sb - 2.388*cq(2)*(cq(3)*cq(4)*sb + cb*cq(3)* sq(4)) + 15.468*((-1)*cb*cq(2)*cq(4) + cq(4)*sb*sq(2)* sq(3) + cq(2)*sb*sq(4) + cb*sq(2)*sq(3)*sq(4)) + (-1)* cq(2)*cq(3)*XELM) + qdot(2)*(-5.783*cb*sq(3) - 0.375* sb*sq(3) + 2.388*((-1)*cq(4)*sb*sq(3) + (-1)*cb*sq(3)* sq(4)) + (-1)*sq(3)*XELM) VT

C C C

= VTW + VTB

Get blade element forces and induced velocity qv(3) is the azimuth position azimuth and VNROTOR2 not used if Induction Factor Model is not WAKE &

CALL ELEMFRC( qv(3), RLOCAL, J, VNROTOR2, VT, VNW, VNB, DFN, DFT, PMA, VINDUCD ) AVEL

c c c

= = = =

= AVEL + VINDUCD * RLOCAL * DR(J)

Accumulate blade element forces cb2 = COS(qv(IBLADE + 4) - PC) sb2 = SIN(qv(IBLADE + 4) - PC) IF (XELM .LT. 0.0) THEN

! blade flap angle used (since PC added ! in qv(IBLADE + 4) earlier)

! Add contributions from non-flapping blade section

136 Fbx = Fbx - DFN*sb2 Fby = Fby + DFN*cb2 Mbx = Mbx + PMA*cb2 - DFT*XELM*sb2 Mby = Mby + DFT*XELM*cb2 - PMA*sb2 ELSE ! Add contributions from flapping blade section Fby = Fby + DFN Mbx = Mbx + PMA Mby = Mby + DFT*XELM ENDIF Fbz = Fbz - DFT ! Aero blade edge load Mbz = Mbz + DFN*XELM ! Aero blade edge moment C

Store the element IF ( NRING .EQ. 1 FN(1) = FN(1) RM(1) = RM(1) ELSE FN(J) = DFN RM(J) = DFN * ENDIF 10 CONTINUE AVEL = AVEL * 2.0/R/R RETURN END

normal forces and the flap bending moment. ) THEN + DFN + DFN * RLOCAL RLOCAL

137 Listing 3: aerog.f gateway file

C C C C C C C C C C C C C C

**************************************************** Gateway program for interface with MATLAB Calls aero.f to calculate blade forces 4/25/99 K. Stol Modified from similar code written by M. Hand 8/30/99 K. Stol Added vector for all blade loads (loads) Assumes 3 blades 8/31/99 K. Stol Added wind speed as input (wind) 10/1/99 K. Stol Allows position vector(q) and velocity vector(qdot) as input **************************************************** SUBROUTINE mexFunction(nlhs,plhs,nrhs,prhs) INTEGER plhs(*),prhs(*) INTEGER nlhs,nrhs INTEGER mxCreateFull, mxGetPr

C

Declare variables for this subroutine INTEGER qp, qdotp, pitchp, windp, loadsp

C

Check for proper number of arguments IF (nrhs.ne.4) then CALL mexErrMsgTxt('4 inputs needed: q, qdot, pitch, wind') ELSEIF (nlhs.ne.1) then CALL mexErrMsgTxt('1 output needed: loads(18)') ENDIF

C

Create matrix for output plhs(1) = mxCreateFull(18,1,0)

C

Dereference arguments to get array pointers. qp = mxGetPr(prhs(1)) qdotp = mxGetPr(prhs(2)) pitchp = mxGetPr(prhs(3)) windp = mxGetPr(prhs(4)) loadsp = mxGetPr(plhs(1))

C

Call the computation routine (aero.f) CALL aero(%val(qp),%val(qdotp),%val(pitchp),%val(windp), + %val(loadsp)) RETURN END

138 Listing 4: yawdyn.ipt file Notes Input file INTERACT STEADY NO_CM SWIRL 0.005 EQUIL HH yawdyn.wnd SI 0.0 1.52 2 1.0251 24.8 2.388 0 7.0 0.004 3 new35.dat new65.dat new85.dat 15 0.6553 1.9004 3.08 4.194 5.177 6.0945 7.0119 7.9294 8.7813 9.5021 10.1575 10.8128 11.4681 12.1234 12.7787 END

Simulation mode: INTERACTive, BATCH Dynamic stall model: BEDDOES, STEADY Aerodynamic pitching moment included?: USE_CM or NO_CM Induction Factor Model: NONE, WAKE, SWIRL ATOLER, Tolerance for induction factor convergence Dynamic inflow model: DYNIN, EQUILibrium Wind data file type: HH (Hub height) or FF (Full Field) SIUNIT, select units: SI or ENGLISH Tower shadow deficit fraction Tower shadow width Number of blades Air density Hub height above ground Distance from yaw axis to hub (not used if WAKE = FALSE) Shaft tilt angle (deg) Rotor precone angle (deg) Aerodynamics time step (sec) Number of airfoil data files you wish to use

Number of blade elements per blade 1.3106 6.043 0.791 1 1.1796 5.27 0.981 1 1.1796 4.946 1.12 1 1.0485 4.31 1.149 1 0.9174 3.53 1.114 2 0.9174 2.63 1.056 2 0.9174 1.915 1.004 2 0.9174 1.299 0.971 2 0.7864 0.725 0.928 2 0.6553 0.375 0.845 2 0.6553 0.177 0.76 3 0.6553 0.1 0.687 3 0.6553 0.061 0.622 3 0.6553 0.043 0.537 3 0.6553 0.018 0.432 3

NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT NOPRINT

Not used . . . . . . Not used . . Not used = Nb . = dt1+dt2-dn1 = dn2 = η0 = β0 Not used . Airfoil Data files RELM, DR, Twist, Chord, Airfoil#

139 APPENDIX E E. ADDITIONAL CONTROL STUDIES

E.1. Example for Constant Gain Optimization Consider the single state periodic system x& = A( t ) x + B( t )u with

(E.1)

A(t) = (1+sin(2πt)) and B(t) = 1.

The period of the system is T = 1 and the open-loop characteristic exponent is at λ = 1. Using the periodic LQR techniques in Chapter 4, and the weightings Q = 1, R = 1, the resulting optimal periodic gain is shown in Figure E-1.

This produces a closed-loop

characteristic exponent of λ = -0.23. -1.5

G (t)

-2

-2.5

*

-3

-3.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

Figure E-1: Periodic gain for the example system

Using a constant full-state feedback gain, G, results in the suboptimal cost ∆J =

∞ T T x ∆G R ∆G x dt 0

∫

where ∆G = G – G*(t).

(E.2)

140 ∆J can be calculated by numerical integration of (E.2) from zero until some large enough time to allow for convergence. The variation of ∆J over a range of constant gains, G ∈ [-2, -4], is plotted in Figure E-2. We now explore the results from solving the two optimization problems in Section 4.2 as well as from using the mean gain.

1.

The global optimization problem (4.10) is to find the constant gain that minimizes ∆J. From Figure E-2 this occurs at a gain of Gopt1 = –2.75, giving ∆J1 = 0.108.

2.

The reduced optimization problem (4.14) is to find the constant gain that minimizes max ∆G (t ) t ∈ [0, T ]

2

(E.3)

For a single state system, (E.3) is solved by Gopt2 =

min  1  max  G * (t ) + G * (t ) t ∈ [0, T ] 2  t ∈ [0, T ] 

= -2.48 for the given periodic gain. Then ∆J2 = 0.126. 3.

Finally, the mean of the periodic gain is G = -2.43, giving ∆J3 = 0.134.

Results from the three gain calculation methods are shown on Figure E-2. The main conclusion from this example is that the three methods produce similar but distinct gains. In particular, simply averaging the optimal periodic gain does not provide the minimum cost.

141

0.35 0.3

∆J∆J

0.25 0.2 0.15

∆J1 0.1 -4

-3.8

-3.6

-3.4

-3.2

-3

-2.8

∆J2 -2.6

∆J3 -2.4

-2.2

G

Figure E-2: Variation of suboptimality, ∆J, with constant gain values

-2

142 E.2. Frozen Model Parametric Studies From Section 4.3 the frozen model is defined as x& f = A f x f + Bf u + B d f u d

(E-4)

where Af = A(tf), Bf = B(tf), Bd f = Bd(tf), and tf = ψf/Ω0. ψf is called the freezing azimuth angle. The constant full-state feedback gain is then calculated from time-invariant LQR. The best freezing angle provides the lowest quadratic cost, Jsim, given in equation (4.26). State and input time response information is determined by numerical integration of the closed-loop system x& = A(t )x + Bu u = Gf x

.

(E-5)

E.2.1. Speed Regulation Consider the speed regulation case with q = [τ1

ψ β1 β1 ] , Q = diag(0,1,0,0, T

0,1,0,0), R = diag(1,1), and initial conditions x(0) = [1 … 1]T. The profile of Jsim over the azimuth range of 0° to 180° is plotted in Figure E-3. The profile is periodic and repeats every 180°. By Inspection, the minimum Jsim occurs at ψf = 105°, corresponding to a cost of 12.45 (6% higher than the optimal cost using periodic gains, 12.37).

143

14 13.8 13.6

JJsim sim

13.4 13.2 13 12.8 12.6 12.4

0

20

40

60

80

100

ψf

120

140

160

180

ψf

Figure E-3: Quadratic cost variation with freezing azimuth position (speed regulation case)

E.2.2. Blade Load Mitigation For blade load mitigation we use q = [τ1

ψ β1 β1 ] , Q = diag(0,1,0,0, 10,1,1,1), T

R = diag(1,1), and the same unit initial conditions as above. The plot of Jsim is shown in Figure E-4. The minimum cost is 719, occurring at ψf = 95°. This is 4% higher than the optimal cost using periodic gains, J* = 693. 750 745 740

JJsim sim

735 730 725 720 715

0

20

40

60

80

100

120

140

160

180

ψf [deg]ψf

Figure E-4: Quadratic cost variation with freezing azimuth position (blade load mitigation case)

E.2.3. Yaw Directional Control Now consider the yaw control case with q = [γ ψ β1 β1 ] , Q = diag(10,1,0,0, T

0,1,0,0), R = diag(1,1), and identical initial conditions. The corresponding Jsim profile is

144 shown in Figure E-5. For the freezing azimuth positions that produce an unstable closed-loop system, the cost is arbitrary set to 300. In fact there are only small ranges of freezing angles for with the system is stable.

It is clear that the minimum Jsim occurs at an angle of

approximately ψf = 5°, with a corresponding cost of 14.5 (101% greater than the optimal periodic gain cost of 7.2). Examination of the periodic gains for this case, plotted in Figure 4-12, reveals that the primary components are centered on zero. As explained in Section 4.6.3, a constant gain cannot mimic the sign changes over one rotation and therefore cannot be as effective. In the current situation, freezing the periodic model produces a sign change in the subsequent gains, resulting in instability. 350 300 250

Jsim J

sim

200 150 100 50 0

0

20

40

60

80

100

ψf

120

140

160

180

ψf

Figure E-5: Quadratic cost variation with freezing azimuth position (yaw control case)