Maximizing Wind Turbine Energy Capture using Multivariable Extremum Seeking Control

Maximizing Wind Turbine Energy Capture usingMultivariable Extremum Seeking Control

Justin Creaby1, Yaoyu Li2,* and John E. Seem3

1Vestas Wind Systems, 66 Church St., Leatherhead, KT22 8DP, UK2Department of Mechanical Engineering, University of Wisconsin, 3200 N. Cramer St., Milwaukee,WI 53211, USA3Controls Group, Johnson Controls, Inc., 507 E. Michigan St., PO Box 423, Milwaukee,WI 53201-0423, USAE-mail: [email protected], [email protected], [email protected]

WIND ENGINEERING VOLUME 33, NO. 4, 2009 PP 361–388 361

ABSTRACTMaximizing energy capture has become an important issue as more turbines are installed in

low wind areas. This paper investigates the application of extremum seeking control (ESC)

to maximizing the energy capture of variable speed wind turbines. The optimal control

torque and pitch angle are searched via ESC based on the measurement of the rotor power.

The advantage of this method is the independency from accurate turbine modelling and

wind measurement. Simulation was conducted on FAST for a wind turbine dynamic model,

under smooth, turbulent and field recorded wind profiles. The simulation results

demonstrated significant improvement in energy capture compared to the standard control

with fixed reference. An anti-windup ESC was applied to overcome the integral windup due

to actuator saturation which would otherwise disable the ESC process. Finally, the

integrator and high-pass filter resetting schemes were applied to improve the transient

under the abrupt changes of wind.

NOMENCLATURECp Power Coefficient

d1(t) Demodulation Signal

d2(t) Dither Signal

FHP(s) High Pass Filter

FI(s) Input Dynamics

FLP(s) Low Pass Filter

FO(s) Sensor Dynamics

K(s) Compensator in ESC

kr Torque Gain

l(t, u) Measurement of Cost Function

M Adaptive Gain

n(t) Noise

Pfavg Fractional Average Power

Q Hessian matrix of the cost function

GREEK SYMBOLSαi Phase Angle between Dither and Demodulating Signals for the i-th input

γ Adaptation Gain

θi Combination of Phase Angle αi and the Phase Response of the i-th Channel of Input

Dynamics

ρ Air Density (kg/m3)

τc Control Torque (Nm)

ωi Dither Frequencies

ωr Rotor Speed (rad/s)

1. INTRODUCTIONRenewable energy has become a major focus for energy and environment sustainability.

Among all the existing renewable energy technologies, wind is considered the most

appropriate energy source for utility level power generation. The US targets 20% wind based

electricity generation, i.e. about 300 GW, by 2030 [1]. The European Technology Platform for

Wind Energy (TPWind) envisions the coverage of electricity generation up to 12–14% by 2020

and 25% of by 2030 [2]. With such scale of wind power penetration, more turbines will be

installed in the medium to low wind areas. In order to reduce the cost-of-energy (COE), it is

critical to increase the efficiency of wind power generation. In addition to improving the

aerodynamic design and site optimization, development of cost-effective control strategies is

an important aspect. Johnson et al. [3] estimated that every percent of energy loss during

turbine operation would cost $100 million per year, based on 100GW of wind based power

generation. Improving the control algorithms on a commercial turbine is a cost-effective

measure of enhancing the wind energy capture. Improved control algorithms can also reduce

hardware and maintenance costs. The objective of this study is to develop a cost-effective control

strategy for maximizing the energy capture when the wind speed is below the rated speed.

The power generation for wind turbines is heavily affected by the aerodynamic

characteristics of wind-turbine interaction. The power coefficient Cp, i.e. the ratio of the

aerodynamic rotor power to the wind power, is mainly affected by the blade pitch angle and

the rotor speed. The rotor speed is often replaced by the tip-speed ratio (TSR), which is

defined as the ratio between the rotational speed of blade tip and the wind speed. The

relationship between the power coefficient and the pitch and TSR is illustrated by a simulation

result as shown in Fig. 1 [4].

On the pitch-speed-Cp surface in Figure 1, there is an optimal pitch and TSR for achieving

the maximum power output. The energy capture can be maximized by controlling the pitch

and rotor speed [4]. With the advancement in power conversion electronics, the rectifier and

converter can convert the variable speed power into the fixed frequency power for grid

integration. The variable-pitch technology has been limited to small turbines, while the

variable-pitch-variable-speed turbines have become the mainstream for utility power

generation.

Previously, most control strategies for maximizing energy capture have relied on some

kind of power map similar to Fig. 1 [5–8], and which have limitations in practice. First, the

results from computational fluid dynamics (CFD) based simulation are questionable in

accuracy. The aerodynamic interactions between wind and the turbine blades are quite

complicated. The wind field around a wind turbine has significant stochastic components due

to turbulence. An experimental study conducted by the US National Renewable Energy

Laboratory (NREL) reported large discrepancies between computational modeling results

and the actual measurement [9]. Second, wind measurement is often inaccurate. The wind

362 MAXIMIZING WIND TURBINE ENERGY CAPTURE USING MULTIVARIABLE EXTREMUM SEEKING CONTROL

field within the turbine rotation sweep has remarkable variations as the turbine size is quite

large for most utility level turbines [10] [11]. Thirdly, model wind turbine characteristics

change with time due to surface wear, dirt, bug or ice buildup, and in consequence, the

aerodynamics and the associated power map will change. Therefore, control strategies that

can adapt to the actual operation are more appropriate for enhancing energy capture.

Adaptive control based on power output has gained attention for maximizing the energy

capture against the inaccurate modeling and unreliable wind measurement. Recently,

adaptive torque control was studied by Johnson et al. [12] [3]. Torque has corresponding

relationship with speed, but it is easier to control than the shaft speed. In their study, the

control torque was chosen to be proportional to the square of the rotor speed. The adaptation

law was developed based on the power signal averaged every 3 hours. According to [3], such

long time period was chosen in part because of the difficulty of obtaining a high correlation

between measurements of wind speed over the entire swept area of the rotor and at

the anemometer which can be located either on the turbine’s nacelle or on a separate

meteorological tower. Another reason for the long adaptation period is that, since the turbine

changes speed at a much slower rate than the wind, the slow responses must be averaged

over time. The test converged to the expected torque gain eventually. However, due to the

use of very long adaptation period (3 hours), the convergence time was about 40 hours. It is

desirable to develop an adaptive control strategy with faster convergence behavior.

Furthermore, the adaptation was based on the fractional power which is defined as the ratio

of the captured power to the available wind power. The latter term was determined by the

cubic law of wind power that relates available wind power to the third power of wind

velocity. Such dependency makes the method sensitive to wind measurement. For field

operation, sensor error and faults in wind measurement could bring forth difficulty.

Therefore, it is also ideal to use adaptive control scheme that is not sensitive to wind

measurement sensor errors.

WIND ENGINEERING VOLUME 33, NO. 4, 2009 363

Figure 1: The relationship between rotor power coefficient, TSR and pitch angle for a typical turbine.

−0.2

−0.4

−0.6

−0.8

−1

−1.20

2 4 6

Tip-speed ratio

Pitch8 10 12

14 15

10

5

0

−5

0

0.2

0.4

Power map

Cp

Another class of control schemes with some adaptation capability is the linear parameter

varying (LPV) control. Recently, Bianchi et al. have developed a gain-scheduling control

strategy for variable-speed turbine operation using a quasi-LPV model [13] [14]. The essence of

this method is to design an H∞ robust controller with the gains adaptive to the parameter

change within a polytope (convex set). For the case of variable-speed wind turbine, the wind

speed and the rotor speed were chosen as the gain-scheduling parameters. Due to such

adaptation, a higher gain can be applied for specific operating point than a constant-gain

controller, thus a better performance could be achieved. The underlying control design

problem was set as a reference tracking problem, i.e. the reference needs to be obtained from

the power map. Therefore this approach may have aforementioned limitation in maximizing

energy capture for actual operation.

More recently, Frost et al. [15] [16] applied the direct model reference adaptive control

(DMRAC) approach to speed regulation problem for Region 3 operation based on collective

pitching. The objective of the adaptive pitch controller was to regulate generator speed and to

reject step disturbances. Model reference adaptive control is a good fit for the speed regulation

problem for Region 3 operation, as the suitable reference is known. However, for Region 2

operation, such reference model is hard to obtain in accurate fashion, and thus self-optimizing

control strategies that can search for unknown and even time-varying optimal reference points

would be more suitable. In this study, we have considered the Extremum Seeking Control.

The ESC [17–19], as a self-optimizing control strategy, has recently drawn attention for wind

turbine control by some researchers [20] [21]. The ESC, as described in the next section, can

search for the unknown and/or time-varying input parameters for optimizing a

performance index. ESC can be considered as a dynamic realization of the gradient search

through the usage of dithering signals. This method has demonstrated its unique benefit

dealing with the systems that are hard to model due to the complex physical and/or chemical

processes involved. ESC has faster searching transient than the static optimization via the

steepest descent, and thus it has been applied to a number of control applications with

nontrivial system dynamics. Komatsu et al. [20] disclosed a brief study of ESC based pitch

control for wind turbine output maximization. A simple case of quadratic map was presented,

and it appeared that the optimal pitch angle under different wind speeds remained the same,

which did not reflect the fact that the optimal pitch varies with wind speed. In addition, this

pitch-only control can be applied only to constant speed wind turbines, which is limited to the

small turbines [21]. Since the variable-speed turbines have dominated the overall wind power

capacity, the impact of ESC would not be as significant if not covering this class. For variable-

speed turbines, simultaneously searching for both the optimal pitch angle and the optimal

rotor speed is necessary for maximizing the energy capture. Furthermore, the simulation

platform in [20] [21] was too simple to reflect the realistic wind input and turbine dynamics.

Under abrupt change of operating condition such as realistic wind, conventional ESC may

experience a long transient and thus some remedy is in need. In addition, as the ESC contains

an integral component by design, the conventional ESC may be disabled by integral windup

due to actuation saturation.

This paper presents a new multi-variable extremum seeking control scheme that

maximizes the energy capture of variable-pitch-variable-speed wind turbines. This improves

the ESC turbine operation in fluctuating wind and with actuator saturation. In addition to the

standard design of ESC, ESC with anti-windup and input resetting were also applied to deal

with the integral windup and to improve the transient performance under abrupt change of

wind speed and direction. The proposed control schemes were validated through simulation

on NREL’s FAST (Fatigue, Aerodynamics, Structures and Turbulence) software [22].



The remainder of this paper is organized as follows. Section 2 describes the principle

and design guidelines for multi-variable extremum seeking control, the anti-windup ESC

to overcome the integral windup inherent to the general ESC loop, and two resetting

schemes for improving the transient performance of ESC under abrupt change of

performance map. Section 3 presents the simulation results for single-input and multi-

input ESC based turbine control, the anti-windup ESC and resetting ESC. Section 4

concludes the current work.

2. MULTI VARIABLE EXTREME SEEKING CONTROLThis paper describes a novel multi-variable extremum seeking control scheme for

maximizing the energy capture of variable-speed-variable-pitch wind turbines. The input

parameters are blade pitch angle and generator torque. The objective of the control strategy

is is to maximize the mechanical power output by searching for the optimal pitch angle and

rotor torque for either a constant or time-varying wind speed. The basic idea of ESC will be

described first, followed by the ESC scheme for the wind turbine control.

2.1. Overview of Extremum Seeking ControlExtremum seeking control deals with the on-line optimization problem of finding an

optimizing input uopt (t) for the generally unknown time-varying cost function l (t, u), where

u(t) ∈ Rm is the input parameter vector, i.e.

(1)

Figure 2 shows the block diagram for a typical ESC system [19]. The measurement of the

cost function l(t, u), denoted by y (t), is corrupted by noise n (t). The transfer function FI (s)

denotes the linear dynamics of the mechanism that command the control or optimization

parameter vector u(t). FO (s) denotes the transfer function of the sensor dynamics that

measure the cost function, which is often a low-pass filter for removing noise from

the measurement. The dithering and demodulating signals are denoted by

and , respectively, where

ωi are the dithering frequencies for each input parameter channel, and αi are the phase angles

introduced intentionally between the dithering and demodulating signals. The signal vector

d2(t) contains the perturbation or dither signals used to extract the gradient of the cost

function l (t, u). These signals work in conjunction with the high-pass filter FHP (s), the

demodulating signal and the low-pass filter FLP (s), to produce ad t t tTm1 1( ) [ sin( )...sin( )]= ω ω

d t t tTm1 1( ) [ sin( )...sin( )]= ω ωd t a t a tT

m m m2 1 1 1( ) [ sin( )... sin( )]= + +ω α ω α

u t l t uoptu m

( ) arg min ( , )=∈ℜ

FHP(s)

d1 d2

FLP(s) K(s)

FI (s)l(t, u)

− ∫

Fo(s)y(t) u(t )

n(t )

+

++

+

û

Figure 2: Block diagram of the ESC algorithm.


vector-valued signal proportional to the gradient, , of the cost function at the input of

the multivariable integrator, where û is the control input based on the gradient estimation. By

integrating the gradient signal, asymptotic stability of the closed loop system will make the

gradient vanish, i.e., achieving the optimality. Adding compensator K (s) may enhance the

transient performance by compensating the input/output dynamics. Detailed explanation of

ESC can be found in [17–19].

The earliest version of ESC can be dated back to Leblanc’s work in 1922 [23]. There was

great interest in this subject in 1950s and 1960’s [24–26]. The research conducted by Krstic and

his coworkers in the past decade ignited a resurgence of extremum seeking control [17, 18].

Krstic and Wang first provided the stability proof for general SISO nonlinear plants based on

averaging and singular perturbation methods [17]. More design issues were addressed in

another paper by Krstic [18]. Later, the stability proof was extended to discrete-time

situation [27]. The proposed ESC framework has been applied to various applications, such as

maximizing biomass production rate [28], maximizing pressure rise in axial flow compressor [29],

minimizing acoustic pressure oscillation to enhance combustion stability [30], minimizing the

power demand in formation flight [31], and minimizing limit cycling [32], among others. The

extremum seeking control was also studied along different paths. Özgüner and his coworkers

combined ESC with sliding mode control [33–35] to explore the application in the vehicle ABS

control. Based on the assumption of quadratic functional form with a finite number of

parameters, Banavar et al. developed an ESC scheme with an adaptation procedure of on-line

identifying the parameters in the assumed function [36–38].

Multi-parameter tuning is of more practical importance and also presents more

complexity. Rotea [19] and Walsh [39] first studied multi-parameter ESC with constant

parameters. Rotea [19] considered the system setting with corrupting noise for the output

measurement. Simple averaging analysis was conducted in stability proof. The design

problem turned out to be one for the linear time-invariant (LTI) plant after averaging, and a

set of design guidelines was summarized. Later, Ariyur and Krstic [40, 41] presented their

multi-parameter ESC for general time-varying parameters. A systematic design algorithm

based on standard LTI control was developed. Uncertainty of the second order derivative was

included in their design. The scheme is applicable for proper output dynamics and thus faster

perturbation is possible. An analytical procedure was proposed to quantify the level of

design difficulty in terms of the number of parameters and the shape of the unknown

equilibrium. In addition, it was noted that the idea of driving the gradient of a nonlinear map

to zero in extremum seeking was generalized to driving the gradient to any given reference.

A later paper [42] showed the application of this idea to compressor instability control.

In the past several years, more ESC applications have been reported. Li et al. described a

multi-parameter ESC application for maximizing the cooling power of a tunable

thermoacoustic cooler [43]. For a solar power system, the maximum power point tracking was

obtained using the extremum seeking with the panel voltage/current [44]. In [45], the user’s

power output was maximized for an exercise machine by optimizing the velocity set point via

ESC. ESC was also applied to search on-line for parameter settings that minimize fuel

consumption of a dual independent variable cam timing engine running at fixed speed and

torque in a dynamometer test-cell [46]. For a biogas production application described in [47],

the performance of a laboratory-scale up-flow anaerobic fixed-bed reactor was greatly

improved via ESC. For fluid system control, the drag force exerted on a bluff-body was

reduced by using extremum-seeking control [48]. An interesting extension of ESC is in the PID

auto-tuning area [49].

∂∂

l

u( )û


Previous investigations on ESC based wind turbine control for enhancing energy capture

[20, 21], as mentioned in Section I, have been relatively simple and appeared only for small

wind turbines. Its application for the large turbines with variable speed capability, currently

the main stream type, has not been reported. In addition, none of the reported research on

adaptive wind turbine control has dealt with the important issues for the practical operation,

such as the possible windup problem due to the actuator saturation and the slow convergence

rate for rapidly changing wind speed. In this research, a multivariable ESC scheme is proposed

for maximizing the energy capture of wind turbines, including the solutions to the above

practical issues.

Design of multivariable ESC system requires the determination of several parameters,

including the dither frequencies, the dither magnitudes, the phase angles between the dither

and demodulation signals, the high-pass filter, the low-pass filters and the compensator. In this

study, the design guidelines for multivariable ESC provided by [19] are followed, which are

summarized as follows.

(1) The dither frequencies should be selected well within the bandwidth of the input

dynamics, and the frequencies should be distinct for individual input channels.

(2) The high-pass filter should be designed to pass the dither frequency, while the low-

pass filter should attenuate the dithering frequency and more significantly for its

second harmonics. The high-pass filter should pass both frequencies and the low

pass filter must stop both frequencies.

(3) For diagonal input dynamics FI (s), the phase angle αi between the dither and

demodulation signals is selected to satisfy , for i = 1, …,

m, where FI , i (s) denotes the ith diagonal entry of FI (s), and m is the number of inputs.

(4) The ESC stability can be determined from the characteristic equation of the

averaged system:

. (2)

where Q is the Hessian of the cost function near the optimum, G(s) is the

concatenation of the input dynamics, the dynamic compensator, the integrator and

the low pass filter:

, (3)

and R is given by

, (4)

where .

(5) The dither amplitude should be selected large enough such that the dithered output

has distinguishable amplitude from the noisy measurement.

2.2. Extremum Seeking Control for Maximizing Energy Capture of Wind TurbinesFigure 3 shows the block diagram for the ESC based wind turbine control for maximizing the

energy capture. The proposed ESC scheme aims to tune both blade pitch and rotor

torque. The input parameters of the ESC loop are the pitch reference and the rotor torque

θ α ωi i Ii iF j= + ∠ ( )

R

a j

a j

I

m I m

= 1

2

1 1 1 1

1

| ( ) | cos( )

| ( ) | cos(

,

,

F

F

ω θ

ωO

θθm )

G s s K s s sI LP( ) ( ) ( ) ( ) /= −F F

det ( )I G s RQ− = 0

− < ∠ + <( / ) ( ) ( / ),π ω α π2 2FI i i ij


reference. The output variable is the rotor power. Notice that the signal flow after the high pass

filter FHP (s) is a two-dimensional vector, highlighted with bold lines. For the actual turbine, the

pitch angle can be measured by the encoder installed at the pitching shaft, and the inner loop

controller GCP (s) is used for pitch servo. The rotor torque can be determined from current, and

regulated by the inner loop controller GCS (s). The idea of torque control in [3] proved to be

effective in speeding up the tuning performance and thus adopted in this study. The control

torque is selected to be proportional to the square of the rotor speed, i.e. τ =kτωr2, where is ωr is

the rotor speed. The torque gain kτ is the tuning input, instead of the torque directly.

Although ESC does not require knowledge of the optimal reference input, knowledge of

the input dynamics and shape profile (second-order derivative) information is important for

successful design. Also, as ESC can deal with local optimum only, validation of the existence of

global optimum within the operating region is important. All this information needs to be

investigated via quality dynamic modeling of the turbine system. In this study, the wind

turbine dynamics is modeled by the NREL’s FAST software [20]. FAST provides a

streamlined prediction of wind-turbine loads and responses, and is suitable for modeling

both two- and three-bladed, horizontal-axis wind turbines.

2.3. Integral Windup and Anti-windup ESCSaturation occurs in any control system because the dynamic range of all real actuators is

limited. The averaging analysis of ESC in [17] showed that, at a larger time scale, the ESC can

be deemed as a linear system regulating the gradient signal with a proportional-integral

(PI) controller. When saturation occurs in the ESC loop, integrator wind-up is inevitable, and

in consequence, leads to the windup phenomena. For wind turbine operation, both pitch

angle and rotor speed can be limited, due to the design and operational constraints. There

are always times that pitch angle and/or rotor speed have to be saturated for quite some

time, and thus integral windup would occur if ESC were applied. The ESC could thus be

disabled after the rotor speed or pitch angle saturation is removed. An anti-windup ESC

using the back-calculation method [48] is proposed as shown in Figure 4.

+

+

+

++

+

−

+

+

n

uT

uP

d1T d2T

FLPT(s) −

−

FHP (s)

FLPP (s)

�� (� )

�� (� )

�� (� )

�� (� )

d1Pd 2P

Pitch

RotorPower

��

��

PitchReference

TorqueReference

RotorTorque

∫

∫

Figure 3: Block diagram of ESC based wind turbine control for maximizing energy capture.


2.4. Resetting ESC for Transient Performance ImprovementSudden change of wind speed and direction often occurs in a gusty wind field. When an abrupt

change occurs, it can cause the corresponding change for the performance (power) map, and

thus the optimum shifts away from the previous optimum. Under such circumstances, ESC may

take a long time to converge to the new optimum. In order to improve the transient

performance, two resetting schemes were proposed. Figure 5 shows the first scheme, the

integrator resetting. Large magnitude change of gradient caused by sudden changes in wind

speed is detected and the output of the integrator is then reset to zero for some time duration. The

y

n

u

− µ∫

FHP(s)

Fo(s) l (t, u) FI(s)

FLP(s)

d1(t ) d2(t )

++

++

GradientDetector

0

n

y u

++

++

++

+

−

Actuator Saturation

FLPT (s)

I(t, u)

Gain

FI (s)

− µ∫FHP (s)

d1(t ) d 2(t )

FO (s)

Figure 5: Block diagram of ESC with integrator resetting.

d1(t ) d2(t )

+

+

+

+n

y u

ResettingLogic

Step-ChangeDetector

FHP (s) FLP(s )

Fo (s) l (t, u ) FI(s)

− µ∫

Figure 6: Block diagram of ESC with high-pass filter resetting.

Figure 4: Block diagram of anti-windup ESC.


other scheme is to reset the input to the high-pass filter in the ESC loop, as shown in Figure 6. This

scheme also detects a step change in the output signal and resets the input to the high pass filter

to some predefined value.

3. SIMULATION STUDYIn order to validate the proposed ESC based wind turbine control for maximizing energy

capture, simulation has been conducted on the NREL’s FAST software [20] and Matlab®

Simulink. FAST is a medium complexity horizontal axis wind turbine design code which can

calculate aerodynamic and mechanical properties. FAST has the ability to interface with

Simulink which allows easy development of turbine controls. Aerodyn [49], also developed by

NREL, is used to simulate aerodynamic forces on the turbine blades. TurbSim [50], also

developed by NREL, was used to generate turbulent wind data. Figure 7 shows a block

diagram describing the interaction among these codes and the relevant data flow. TurbSim

creates turbulent wind profiles for simulations. Aerodyn takes the wind data output from

Turbsim to calculate the aerodynamic loads on the turbine. FAST (Fatigue, Aerodynamics,

Structures, and Turbulence) takes this loading information and applies it to the nonlinear wind

turbine model to solve for the relevant equations of motion. Through the interface with

Matlab SIMULINK, the Controller block provides the feedback control function to regulate the

behavior of the wind turbine. The FAST output of the controlled turbine will then return to the

input end of AeroDyn.

The turbine model for this study is that of the CART (Controls Advanced Research Turbine)

facility located at NREL in Golden, Colorado. The CART facility is a two-blade 600 kW variable-

speed-variable-pitch turbine well instrumented for control and data acquisition purpose.

3.1. Pitch-Torque ESC Design for CART Turbine ModelThe input dynamics for the turbine power output were determined from the step responses of

the generator torque and pitch angle inputs, respectively. Figure 8 shows the step responses

for the wind speed of 10 m/s.

(a) Torque Input Dynamics

(b) Pitching Input Dynamics

The step responses indicated first-order system behavior for both input channels, which

can be modeled as

(5a)F ssIT

T

( ) =+

1

1τ

TurbineInformation

(e.g. Rotor Speed,Pitch Angle ...)

TurbSim AeroDyn Fast

Controller

WindData

BladeLoads

Pitch &ControlTorque

RotorPower

Figure 7: Illustration of FAST based simulation for wind turbine control.


. (5b)

For control torque dynamics, the blade pitch was fixed at −1°, and the value of kτ was

stepped up and down a number of times of different step sizes. The time constant τk was

calculated for each step change, which ranged from 6.3 to 7.7 seconds. Similarly, the time

constant for the pitching input dynamics was determined ranging from 5.2 to 5.7 seconds. As

an average of measurement, τT and τP were set to be 5.5 seconds and 6.63 seconds,

respectively.

F ssIP

P

( ) =+

1

1τ

180160140120100

50

Torq

ue g

ain

100 150 200 250 300 350 400 450 500 550

400350

300

250

20050 100 150 200 250 300

Time (sec)

(a) Torque input dynamics

350 400 450 500 550

40

35

3050

Rot

or s

peed

Rot

or p

ower

100 150 200 250 300 350 400 450 500 550

Simulated dataFitted data

Time (sec)

(b) Pitching input dynamics

43210

−1−2−3

50 100 150 200 250 300 350 400 450 500 550

Pitc

h

290280

270

260

250

240

23050 100 150 200 250 300 350 400 450 500 550

Rot

or p

ower

Simulated dataFitted data

Figure 8: Rotor power with various step changes to the pitch angle and torque gain.


The fundamental period of the dither signals was set at 10 times the time constant of its

input dynamics: and . The following high-pass filter FHP (s) was selected to pass frequencies

higher than the lowest dithering frequency, i.e. ωk :

(6)

The following low pass filter FLP (s) was chosen to attenuate the dither frequencies:

(7)

The low pass filter has magnitude attenuation of 10.7 dB at ωT and 23 dB at 2ωk. Figure 9 shows

the Bode plot of the input dynamics, high-pass and low-pass filter, with the dithering frequencies

labeled. The dithering amplitude for the multivariable pitch ranges 0.2° to 2° as the turbulence

intensity varies from 0% to 20%. The dithering amplitude for the multivariable torque gain ranges

2 to 20 as the turbulence intensity varies from 0% to 20%. The phase angles were chosen as:

, leading to , and .

3.2. Simulation Results of Single Input ESCThis sub-section presents the simulation results for single-input ESC, i.e. when only pitch or

torque control is applied. The single-input ESC is applicable for simpler turbine operations

such as fixed-speed or variable-speed with almost fixed optimum pitch angle. The design

process is similar to, although simpler than, the two-input ESC design presented in Section 3.1.

The details of the design process are omitted due to length limit.

Following the design procedure similar to Section 3.1, the dither frequency was selected as

. The high pass filter FHPP (s) was chosen to be

, (8)F ss

s sHPP ( )

. . .=

+ × × +

2

2 22 0 58 0 064 0 064

ωP rad s= 0 0806. /

θ α ωP P IP PF j= +∠ ≈ −( ) .0 05°θ α ωT T IT TF j= +∠ ≈( ) .0 05°α α πT P rad= = 0 178. ( )

F ss s

LP ( ).

. . .=

+ × × +0 05

2 0 6 0 05 0 05

2

2 2

Fs

s sHP =

+ × × +

2

2 22 0 58 0 045 0 045. . .

Figure 9: Input dynamics, high-pass and low-pass filter for pitch-torque ESC.

0

−20

−40

−60

−80

20

Frequency (rad/sec)

Pha

se (

deg)

Mag

nitu

de (

dB)

Pitch dither frequency Torque dither frequency

1359045

0−45−90

−135−180

10−3 10−2 10−1 100 101

180

Input dynamicsHigh pass filterLow pass filter


5000

11

10.5

10

Win

d sp

eed

(m/s

)

9.5

91000 1500 2000 2500 3000 3500

5000

2

1

Pitc

h (d

eg)

0

−1

1000 1500 2000 2500 3000 3500

5000

300

220

280

Rot

or p

ower

(kW

)

260

240

1000 1500Time (sec)

2000 2500 3000 3500

Turbine with standard controlTurbine with ESC control

Figure 10: Pitch ESC under a smooth 10 m/s wind.

with nearly unit gain at ωP . The low pass filter FLPP (s) was designed as

. (9)

with magnitude attenuation in excess of 8 dB at ωP and 20 dB at 2ωP . To improve tracking, the

phase angle between the dither and demodulating signal is chosen to be: ,

which results in .

The pitch ESC was first simulated for a constant wind of 10 m/s without turbulence. The

dithering amplitude aP was selected to be 0.2° for steady wind. The static map obtained from

FAST simulation showed that the optimum pitch angle occurs at −1.1°. To test the proposed

controller, the pitch angle was initially set at a non-optimal value of +2°. As shown in Figure 10,

the pitch angle converges to the optimum −1.1° and the power is thus maximized. The setting

time was about 420 seconds and the steady state errors for pitch and power were 0.1° and 0.5

kW, respectively. For the simulated period, the energy capture improvement was 5.7%.

The simulation was then repeated with turbulent wind of 2% turbulence intensity, with the

initial pitch angle set at +2° again. The dithering amplitude had to be increased to 0.5° for such

turbulence intensity. Figure 11 shows the result. The pitch converges on the optimum and the

power is maximized. Convergence occurs within 1500 seconds. The pitch angle converges to

within 0.15°. The power capture improvement is 4.7%.

For the torque ESC controller, the dithering frequency was chosen to be .

The high pass filter, FHPT (s) was designed as

, (10)

with nearly unit gain at ωT . The low pass filter is designed to attenuate the dither frequency.

The following low pass filter was chosen:

, (11)F ss s

LPT ( ).

* . * . * .=

+ +0 055

2 0 6 0 055 0 055

2

2 2

F ss

s sHPT ( )

. . .=

+ × × × +

2

2 22 0 58 0 071 0 071

ωT rad s= 0 0887. /

θ α ωP P IP PF j= +∠ ≈( ) 0°α πP rad= 0 179. ( )

F ss s

LPP ( ).

. . .=

+ × × +0 055

2 0 6 0 055 0 055

2

2 2

with magnitude attenuation in excess of 8 dB at ωk and 20 dB at 2ωk . The phase angle between

dithering and demodulating signals was chosen as , resulting in

. The dithering amplitude was selected as 5 for steady winds, but has to

increase up to 20 as turbulence intensity reaches 20%.

The nominal value of kτ can be calculated from the following equation from [3],

. (14)

For the CART turbine, with ρ = 1 kg/m3, R = 21.65m, Cpmax = 0.4, λ* = 7.5 and the gearbox ratio

of 43.2, we obtained kτ to be 164 m5/rad3. This value was used as the initial value of kτ for the

τ ρλ

ω ωτc

pAR

Ck= =1

23

3

2 2max

*

θ α ωk k Ik kF j= +∠ ≈( ) 0°α πk rad= 0 179. ( )


5000

10.5

10

Win

d sp

eed

(m/s

)

9.5

1000 1500 2000 2500 3000 3500

5000

2

1

Pitc

h (d

eg)

0

−1

1000 1500 2000 2500 3000 3500

5000

280

300

Rot

or p

ower

(kW

)

260

240

1000 1500Time (sec)

Turbine with ESC control

Turbine with standard control

2000 2500 3000 3500

Figure 11: Pitch ESC under 10 m/s wind with 2% turbulence.

5000

10.5

11

10

Win

d sp

eed

(m/s

)

9.5

91000 1500 2000 2500 3000 3500

5000

160

140

Torq

ue g

ain

k

120

1001000 1500 2000 2500 3000 3500

5000

280

300

Rot

or p

ower

(kW

)

260

2401000 1500

Time (sec)

Turbine with ESC control Turbine with standard control

2000 2500 3000 3500

Figure 12: Torque ESC with a smooth 10 m/s wind.

simulation of the torque ESC. The numerical simulation on FAST showed that the optimal kτ

based on the computational models was 110. This value is far from the estimated value of k.

Figure 12 shows the simulation results under a constant 10 m/s wind, with the torque gain kτ

converged to the optimum within 200 seconds, and thus the rotor power was maximized. The

steady state errors of the torque gain and rotor power were about 5 and 0.1 kW, respectively.

The improvement in energy capture compared to the case with the torque gain fixed at the

nominal optimum of 164, was about 3.6% for the simulated period. Figure 13 shows the

simulation results of a turbulent wind with 2% turbulence intensity. The initial value of kτ was

set again at 164. The value of kτ converged within about the ±10 band of the optimum in about

500 seconds. The improvement in power capture was 3.1%.


5000

10.5

10

Win

d sp

eed

(m/s

)

9.5

1000 1500 2000 2500 3000 3500

5000

160140

Torq

ue g

ain

k

120100

80

1000 1500 2000 2500 3000 3500

5000

300

320

Rot

or p

ower

(kW

)

280

260

1000 1500Time (sec)



2000 2500 3000 3500

Figure 13: Torque ESC under 2% turbulent wind.

10Win

d sp

eed

(m/s

)

72 3 4 5

89

10

10

Torq

ue g

ain

k

1002 3 4 5

120140160

10

Pitc

h an

gle

(¡)

2 3 4 5

−1

01

10

Rot

or p

ower

(kW

)

2 3Time (Hrs)

4 5

150100

200250



Figure 14: Pitch-torque ESC results under smooth wind with step changes.

3.3. Simulation Results of Pitch-torque ESCThe two-input (pitch-torque) ESC described in Section 3.1 was tested on the CART model. The

initial pitch angle and torque gain were set to +1° and 164, respectively. Figure 14 shows the

simulation results under a staircase smooth wind input of 6 hours long. The wind file used is

smooth with a step change in speed approximately every 30 minutes. The pitch converged to

within 0.1° of the optimum and the value of k converged to between 100 and 120. The obtained

power is within 99% of the maximum. In this case the energy captured was increased by 8.8%.

The settling time was approximately 20 minutes. The same simulation was conducted for

constant 10 m/s wind with 2% turbulence, as shown in Figure 15. Both parameters converged to

the optimum in approximately 30 minutes. The improvement in power capture was about 7%.


0.50

Win

d sp

eed

(m/s

)

9.51 1.5 2 2.5

10

10.5

0.50

Torq

ue g

ain

k

501 1.5 2 2.5

100

150

200

0.50

Pitc

h an

gle

(¡)

1 1.5 2 2.5

−2

0

2

0.50

Rot

or p

ower

(kW

)

1 1.5Time (Hrs)

2 2.5

280260240

300320


Figure 15: Pitch-torque ESC results under constant wind with 2% turbulence.

16

Win

d sp

eed

(m/s

) 141210

8642

0 0.5 1 1.5Time (hr)

2 2.5 3

16

Win

d sp

eed

(m/s

) 141210

8642

0 0.5 1 1.5 2 2.5 3

TurbSim wind Real wind 1

TurbSim wind Real wind 2

Figure 16: Wind profiles of TURBSIM wind and real wind from CART site.

Increasing the dither amplitude was critical for improving the transient performance with

turbulent wind.

Further simulation was performed using two sets of 3-hour actual wind data recorded at

the CART facility site. Figure 16 shows both sets of real wind data compared to wind data from

TurbSim with 16% turbulence. A wind with 16% turbulence was chosen because it is very


2

64

8101214

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3


Win

d sp

eed

(m/s

)

50

100

150

200

Torq

ue g

ain

(k)

−5

0

5

Pitc

h an

gle

(°)

200

400

600

Rot

or p

ower

(kW

)

Figure 17: Pitch-torque ESC results under real wind data #1 from CART site.

15

10

200150100

50

5

0 0.5 1 1.5 2 2.5 3

0 0.5Turbine with standard control Turbine with ESC control

Rot

or p

ower

(kW

)P

itch

angl

e(¡

)To

rque

gai

n(k

)W

ind

spee

d(m

/s)

1 1.5 2 2.5 3

5

0

−50 0.5 1 1.5 2 2.5 3

1000

600800

200400

0 0.5 1 1.5

Time (Hrs)

2 2.5 3

Figure 18: Pitch-torque ESC results under real wind data #2 from CART site.

similar in wind speed variation to that of the real wind. It appears that both wind files have

similar characteristics to that of the wind file created by TurbSim.

The pitch-torque ESC was simulated using the above actual wind data, respectively. For

the first wind data (Fig. 17), the initial pitch and torque gain were set at +5° and 180 kg.m3/rad3,

respectively. Such values were selected to clearly show convergence towards the optimum.

The convergence was achieved with longer transient, about 1.3 hours due to the increased

turbulence. For the 3 hour operation, the total increase in energy capture was 26%. The same

ESC setting was repeated for the second wind data as in Fig. 18. The energy capture improved

by 28%, with about 1.2 hours of convergence time. Compared to the previous cases, the

significant larger improvement in energy capture was mainly due to larger deviation of the

initial setting from the optimum. Nevertheless, the convergence time seemed quite improved

than that reported in [3].

As fatigue loading is critical for the reliability of wind turbines, it is important to inspect the

effect of any control strategy on the structural loading for the turbine. The effect of ESC on

structural loading was briefly investigated. For the pitch-torque simulation with approximately

15% turbulence as in Figure 18, four variables of interest were recorded: tower fore-aft, tower

side-to-side, blade flap and blade edge bending moment. Figure 19 shows the comparison of


200

180

160

140

120

100

80

60

40

20

00 0.5 1 1.5 2 2.5

Frequency (Hz)

(a) Tower side-to-side bending

3 3.5 4 4.5

Tow

er s

ide-

side

ben

ding

(dB

/Hz)

5

With ESCWithout ESC

200

180

160

140

120

100

80

60

40

20

00 0.5 1 1.5 2 2.5

Frequency (Hz)

(b) Tower fore-aft bending

3 3.5 4 4.5

Tow

er fo

re-a

ft be

ndin

g (d

B/H

z)

5

With ESCWithout ESC

Figure 19: (Continued)

the spectra of these variables before and after applying the ESC. Figure 20 shows the

difference spectra for the respective cases. No significant difference was found. Therefore, the

ESC did not bring forth sensible impact on structural loading.

3.4. Integral Windup and Anti-windup ESCA case study was simulated to illustrate the integral windup phenomenon due to actuator

saturation and remedial effect of the anti-windup ESC proposed in Section 2.3. For NREL’s

600kW CART model, an upper bound of 2300 N-m was assumed for the generator torque.

Figure 21 shows the simulated operation under smooth wind with step change every

35 minutes. For some change of wind speed, the demanded torque was limited by the

saturation bound, and thus the ESC windup occurred, as shown in the dashed line. The ESC

was disabled due to windup phenomenon. In comparison, the solid line in Figure 21

demonstrated the capability of the proposed anti-windup ESC in recovering the searching

process. Figure 22 shows the simulation with similar settings except that the wind input had 2%

turbulence. A similar situation was observed.


200

180

160

140

120

100

80

60

40

20

00 0.5 1 1.5 2 2.5

Frequency (Hz)

(c) Blade flap bending

3 3.5 4 4.5

Bla

de fl

ap b

endi

ng (

dB/H

z)

5

With ESCWithout ESC

200

180

160

140

120

100

80

60

40

20

00 0.5 1 1.5 2 2.5

Frequency (Hz)

(d) Blade edge bending

3 3.5 4 4.5

Bla

de e

dge

bend

ing

5

With ESCWithout ESC

Figure 19: Frequency response of blade and tower loading using pitch-torque ESC.

3.5. ESC with Integrator and High-pass Filter ResettingThe two resetting schemes proposed in Section 2.4 were simulated for the ESC operation on

the CART model. For the integrator resetting scheme, once the magnitude of the gradient

value exceeds ±10, the zero-reset is activated and held for 120 seconds. This effectively ignores

the signal which would have caused the ESC to deviate far from the optimal search. Two cases

of simulation were carried out. The first case used a smooth wind file with a sudden change in

wind speed at 2000 seconds, as shown in Figure 23. With the conventional ESC, the turbine

demonstrated a large deviation and took a long time to return to the optimal. The turbine with

integrator resetting could effectively ignore the sudden change and quickly reached the

region of the optimum. The second case is similar to the first except that 2% turbulence was

added. Figure 24 shows the simulation results of the second case, with the resetting time

extended to 130 seconds.

The high-pass filter resetting scheme was then simulated with both smooth and turbulent

wind inputs. The input to the high-pass filter was reset to 650 for 45 seconds immediately after

the step change in wind speed was detected. Figure 25 shows the simulation results for smooth


100

80

60

40

20

Tow

er s

ide-

side

ben

ding

diff

eren

ce (

dB/H

z)

0

−20

−40

−60

−80

−1000 0.5 1 1.5 2 2.5

Frequency (Hz)3 3.5 4 4.5 5

(a) Tower side-to-side bending

100

80

60

40

20

Tow

er fo

re-a

ft be

ndin

g di

ffere

nce

(dB

/Hz)

0

−20

−40

−60

−80

−1000 0.5 1 1.5 2 2.5

Frequency (Hz)

3 3.5 4 4.5 5

(b) Tower fore-aft bending

Figure 20: (Continued)

wind. The wind speed increased from 10 m/s to 13 m/s at 1000 second. The ESC with high-pass

filter resetting demonstrated transient approximately tens times faster than the conventional

ESC for the generator torque response. Figure 26 shows the simulation results with the similar

setting but 2% turbulence added. The generator torque was able to catch up about twice as

fast. However, the power capture enhancement was not as obvious.

4. CONCLUSIONS & DISCUSSIONThis paper presents a multi-variable ESC scheme for maximizing wind turbine energy

capture, which aims at the self-optimizing pitch and torque control of variable-pitch-variable-

speed turbines under time-varying wind. Both single-input and multi-input ESC schemes have

been developed, designed and simulated for enhancing the energy capture in smooth,

turbulent and field recorded wind. The simulation with field recorded wind data demon-strated

faster convergence compared to previously reported work. Significant improvement in

energy capture was observed compared to the standard control techniques without self


100

80

60

40

20

Bla

de fl

ap b

endi

ng d

iffer

ence

(dB

/Hz)

0

−20

−40

−60

−80

−1000 0.5 1 1.5 2 2.5


(c) Blade flap bending

100

80

60

40

20

Bla

de e

dge

bend

ing

diffe

renc

e (d

B/H

z)

0

−20

−40

−60

−80

−1000 0.5 1 1.5 2 2.5


(d) Blade edge bending

Figure 20: Difference spectra of blade and tower loading with and without pitch-torque ESC.


13

12

11

5

0

22002400

−5

Win

d sp

eed

(m/s

) In

tegr

ator

inpu

t

Gen

.to

rque

(N

m)

Rot

orpo

wer

(kW

)

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

200018001600

600

500

400

300

Standard ESC Anti-windup ESC

Figure 21: Illustration of windup and anti-windup ESC in control torque under smooth wind.

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

13

12

11

50

22002400

−5

Win

d sp

eed

(m/s

) In

tegr

ator

inpu

t

Gen

.to

rque

(N

m)

Rot

orpo

wer

(kW

)

2000180016001400

600500400300

Time (Hrs)

Anti-windup ESC Standard ESC

Figure 22: Illustration of windup and anti-windup ESC in control torque under 2% turbulent wind.

optimization of reference. To deal with the integral windup inherent for ESC operation, an

anti-windup ESC scheme was proposed based on the back-calculation method, and then

validated with simulation. In order to improve the transient performance under abrupt

change of wind speed and wind direction, two resetting schemes were proposed: integrator

and high-pass filter resetting. Simulation results showed that both methods can improve the

transient performance, while the integrator resetting seems to work better. The presented

ESC approach has good fit for the objective of maximizing energy capture for Region 2

operation. For Region 3 operation where speed regulation and disturbance rejection are the

primary goals, model reference adaptive control [15] [16] can be a better solution with faster

transient performance. The two adaptive control strategies could be used in combination

for actual operation.


13121110

10050

0−50

100

300200100

600

400

200

500

−50

Win

d sp

eed

(m/s

) G

radi

ent

Inte

grat

orin

put

Tor

que

gain

Rot

or p

ower

(kW

)

3500300025002000Time (sec)

150010005000

3500300025002000150010005000

3500300025002000150010005000

3500300025002000150010005000

3500300025002000150010005000

No integrator reset

Integrator reset

Figure 23: Simulation result for ESC with integrator resetting under smooth wind.

13

Win

d sp

eed

(m/s

) 121110

3500300025002000150010005000

100

Gra

dien

t

500

−503500300025002000150010005000

100

150100

50

600500400300

Inte

grat

orin

put

Tor

que

gain

Rot

or p

ower

(kW

)

500

−503500300025002000150010005000

3500300025002000150010005000

3500300025002000Time (sec)

150010005000

No integrator reset

Integrator reset

Figure 24: Simulation result for ESC with integrator resetting under 2% turbulent wind.


0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000Time (sec)

2500 3000 3500

10111213

Win

d sp

eed

(m/s

)

300400500600700

HP

F in

put

2000

2500

3000

Gen

. tor

que

(Nm

)

300400500600700

Rot

or p

ower

(kW

)

No HPF

HPF Input reset

Input reset

Figure 25: ESC with high-pass filter resetting under smooth wind.

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000Time (sec)

2500 3000 3500

10111213

Win

d sp

eed

(m/s

)

300400500600700

HP

F in

put

2000

2500

3000

Gen

. tor

que

(Nm

)

300400500600

Rot

or p

ower

(kW

)

No

HPF Input reset

HPF Input reset

Figure 26: ESC with high-pass filter resetting under 2% turbulent wind.

ACKNOWLEDGEMENTSThis research has been partly supported by We Energies and Johnson Controls, Inc. The

authors are grateful for this support. Special thanks go to NREL researchers Lee Jay Fingersh,

Jason Jonkman and Alan Wright for their great help in the acquisition of the CART wind data,

the CART models and use of FAST.

REFERENCES1. US Department of Energy. 20% Wind Energy by 2030: Increasing Wind Energy’s

Contribution to U.S. Electricity Supply. DOE/GO-102008-2567. July 2008.

2. European Wind Energy Technology Platform. Strategic Research Agenda: Market

Deployment Strategy from 2008 to 2030. July 2008.

3. Johnson, K.E., Pao, L.Y., Balas, M.J. and Fingersh, L.J. Control of Variable-Speed Wind

Turbines: Standard and Adaptive Techniques for Maximizing Energy Capture. IEEE

Control Systems Magazine, 2006, 2266(3), 70–81.

4. Johnson, K.E., Fingersh, L.J., Balas, M.J. and Pao, L.Y. Methods for Increasing Region 2

Power Capture on a Variable-Speed Wind Turbine. Journal of Solar Energy

Engineering, 2004, 112266(4): 1092–1100.

5. Leithead, W.E. and Conner, B. Control of Variable Speed Wind Turbines: Dynamic

Models. International Journal of Control, 2000, 7733(13), 1173–1188.

6. Leithead, W.E. and Conner, B. Control of Variable Speed Wind Turbines: Design Task.

International Journal of Control, 2000, 7733(10), 1189–1212.

7. Holley, W., Rock, S., and Chaney, K., Control of Variable Speed Wind Turbines below

Rated Wind Speed. Proceedings of ASME/JSME Joint Fluids Engineering Conference,

San Francisco, 1999.

8. Senjyu, T., Sakamoto, R., Urasaki, N., Funabashi, T., Fujita, H. and Sekine, H., Output

Power Leveling of Wind Turbine Generator for All Operating Regions by Pitch Angle

Control. IEEE Transaction on Energy Conversion, 2006, 2211(2), 467–475.

9. Simms, D., Schreck, S., Hand, M. and Fingersh, L.J. NREL Unsteady Aerodynamics

Experiment in the NASA-Ames Wind Tunnel: A Comparison of Predictions to

Measurements. NREL/TP-500-29494, 2001.

10. Hansen, M.O.L., Aerodynamics of Wind Turbines, 2nd edition 2007.

11. Burton, T., Sharpe, D., Jenkins, N. and Bossanyi, E. Wind Energy Handbook, 1st Edition,

2001, Wiley.

12. Johnson, K.E., Adaptive Torque Control of Variable Speed Wind Turbines. Ph.D. Thesis,

University of Colorado at Boulder, 2004.

13. Bianchi, F.D., Mantz, R.J. and Christiansen, C.F. Gain Scheduling Control of Variable-

Speed Wind Energy Conversion Systems Using Quasi-LPV Models. Control Engineering

Practice, 2005, 1133(2), 247–255.

14. Bianchi, F.D., Battista, H.D. and Mantz, R.J. Wind Turbine Control Systems: Principles,

Modeling and Gain Scheduling Design, 2006, Springer.

15. Frost, S.A., Balas, M.J. and Wright, A.D. Direct Adaptive Control of Utility-Scale Wind

Turbines for Enhanced Energy Capture and Aerodynamic Load Mitigation. International

Journal of Robust and Nonlinear Control, Vol. 19, No. 1, pp. 59–71, January 2009.

16. Frost, S.A., Balas, M.J. and Wright, A.D. Adaptive Control of a Utility-Scale Wind Turbine

Operating in Region 3. AIAA 2009-480, Proceedings of ASME Wind Energy

Symposium, Orlando, FL, February 2009.

17. Krstic, M. and Wang, H.-H. Stability of Extremum Seeking Feedback for General

Nonlinear Dynamic Systems. Automatica, 2000, 3366(4), 595–601.

18. Krstic, M. Performance Improvement and Limitations in Extremum Seeking Control.

Systems and Control Letters, 2000, 3399(5), 313–326.


19. Rotea, M.A. Analysis of Multivariable Extremum Seeking Algorithms. Proceedings of

2000 American Control Conference, 2000, Chicago, IL, 433–437.

20. Komatsu, M., Miyamoto, H., Ohmori, H. and Sano, A. Output Maximization Control of

Wind Turbine Based on Extremum Control Strategy. Proceedings of the 2001 American

Control Conference, 1739–1740.

21. Ishii, C., Hashimoto, H. and Ohmori, H. Modeling of Variable Pitch Micro Wind Turbine

and Its Output Optimization Control with Adaptive Extremum Control Scheme.

Transactions of the Japan Society of Mechanical Engineers, Part C, 2003, 6699(11),

3034–3040.

22. Jonkman, J.M. and Buhl, M. FAST User’s Guide. Technical Report NREL/EL-500-38230,

2005.

23. Leblanc, M. Sur l’electrification des Chemins de fer au Moyen de Courants Alternatifs de

Frequence Elevee. Revue Generale de l’Electricite, 1922.

24. Sternby, J. Extremum Control Systems: An Area for Adaptive Control. Preprints of the

Joint American Control Conference. 1980: San Francisco, CA.

25. Åström, K.J. and Wittenmark, B. Adaptive Control, 2nd ed., 1995, Addison-Wesley,

Reading, MA.

26. Tsien, H.S., Engineering Cybernetics, 1954, McGraw-Hill, New York.

27. Choi, J.-Y., Krstic, M., Ariyur, K.B. and Lee, J.S. Extremum Seeking Control for Discrete-

time Systems. IEEE Transactions on Automatic Control, 2002, 4477(2), 318–323.

28. Wang, H.-H., Krstic, M. and Bastin, G. Optimizing Bioreactors by Extremum Seeking.

International Journal of Adaptive Control and Signal Processing, 1999, 1133(8), 651–669.

29. Wang, H.-H., Yeung, S. and Krstic, M. Experimental Application of Extremum Seeking on

an Axial-flow Compressor. IEEE Transactions on Control Systems Technology, 2000,

88(2), 300–309.

30. Banaszuk, A., Zhang, Y. and Jacobson, C.A. Adaptive Control of Combustion Instability

Using Extremum-Seeking. Proceedings of the 2000 American Control Conference,

Chicago, IL, 2000, 416–422.

31. Binetti, P., Ariyur, K.B., Krstic, M. and Bernelli, F. Control of Formation Flight via

Extremum Seeking. Proceedings of the 2002 American Control Conference, 2002,

2848–2853.

32. Wang, H.-H. and Krstic, M. Extremum Seeking for Limit Cycle Minimization. IEEE

Transactions on Automatic Control, 2000, 4455(12), 2432–2437.

33. Drakunov, S., Özgüner, Ü., Dix, P. and Ashrafi, B. ABS Control Using Optimum Search via

Sliding Modes. IEEE Transactions on Control Systems Technology, 1995, 33(1), 79–85.

34. Yu, H. and Özgüner, Ü. Extremum-seeking Control Strategy for ABS System with Time

Delay. Proceedings of the 2002 American Control Conference, 2002, 3753–3758.

35. Yu, H. and Özgüner, Ü. Extremum-seeking Control via Sliding Mode with Periodic

Search Signals. Proceedings of the 41st IEEE Conference on Decision and Control, 2002,

323–328.

36. Speyer, J.L., Banavar, R.N., Chichka, D.F. and Rhee, I. Extremum Seeking Loops with

Assumed Functions. Proceedings of the 39th IEEE Conference on Decision and Control,

2000, 142–147.

37. Banavar, R.N., Chichka, D.F. and Speyer, J.L. Functional Feedback in an Extremum

Seeking Loop, Proceedings of the 40th IEEE Conference on Decision and Control, 2001,

1361–1321.


38. Banavar, R.N. Extremum Seeking Loops with Assumed Functions: Estimation and

Control. Proceedings of the 2002 American Control Conference, Anchorage, AK, 2002,

3159–3164.

39. Walsh, G.C., On the Application of Multi-parameter Extremum Seeking Control.

Proceedings of the 2000 American Control Conference, Chicago, IL, 2000, 411–415.

40. Ariyur, K.B. and Krstic, M. Analysis and Design of Multivariable Extremum Seeking.

Proceedings of the 2002 American Control Conference, Anchorage, AK, 2002,

2903–2908.

41. Ariyur, K.B. and Krstic, M. Real-time Optimization by Extremum-Seeking Control, 2003,

Wiley Interscience.

42. Ariyur, K.B. and Krstic, M. Slope Seeking and Application to Compressor Instability

Control. Proceedings of the 41st IEEE Conference on Decision and Control, 2002,

3690–3697.

43. Li, Y., Rotea, M.A., Chiu, G., Mongeau, L.G., Paek, I.-S. Extremum Seeking Control of a

Tunable Thermoacoustic Cooler. IEEE Transactions on Control Systems Technology,

2005, 1133(4), 527–536.

44. Leyva, R., Alonso, C., Queinnec, I., Cid-Pastor, A., Lagrange, D. and Marinez-Salamero, L.

MPPT of Photovoltaic Systems Using Extremum-Seeking Control. IEEE Transactions on

Aerospace and Electronic Systems, 2006, 4422(1), 249–258.

45. Zhang, X.T., Dawson, D.M., Dixon, W.E. and Xian, B., Extremum-seeking Nonlinear

Controllers for a Human Exercise Machine. IEEE/ASME Transactions on Mechatronics,

2006, 1111(2), 233–240.

46. Popovic, D., Jankovic, M., Magner, S. and Teel, A.R. Extremum Seeking Methods for

Optimization of Variable Cam Timing Engine Operation. IEEE Transactions on Control

Systems Technology, 2006, 1144(3), 398–407.

47. Liu, J., Olsson, G. and Mattiasson, B. Extremum-seeking with Variable Gain Control for

Intensifying Biogas Production in Anaerobic Fermentation. Water Science and

Technology, 2006, 5533(4-5), 35–44.

48. Beaudoin, J.F., Cadot, O., Aider, J.L. and Wesfreid, J.E. Bluff-body Drag Reduction by

Extremum-Seeking Control. Journal of Fluids and Structures, 2006, 2222(6–7), 973–978.

49. Killingsworth, N.J. and Krstic, M. PID Tuning Using Extremum Seeking: Online, Model

Free Performance Optimization. IEEE Control Systems Magazine, 2006, 2266(1), 70–79.

50. Åström, K.J. and Rundqwist, L. Integrator Windup and How to Avoid It. Proceedings of

1989 American Control Conference, 1989, 1693–1698.

51. Laino, D. and Hansen, C. AeroDyn User’s Guide, Version 12.50, 2002, NREL.

52. Jonkman, B.J. and Buhl, M.L. TurbSim User’s Guide, Version 1.21, 2007, NREL.


Maximizing Wind Turbine Energy Capture using Multivariable Extremum Seeking Control

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