STANDARD AND ADAPTIVE TECHNIQUES FOR ...ece.colorado.edu/.../JohnsonPaoBalasFingersh_IEEE_CSM.pdf70 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2006 1066-033X/06/$20.00©2006IEEE W ind energy

70 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2006 1066-033X/06/$20.00©2006IEEE

Wind energy is the fastest-growing energysource in the world, with worldwidewind-generation capacity tripling in thefive years leading up to 2004 [1].Because wind turbines are large, flexible

structures operating in noisy environments, they pre-sent a myriad of control problems that, if solved, couldreduce the cost of wind energy. In contrast to constant-speed turbines (see “Wind Turbine Development andTypes of Turbines”), variable-speed wind turbines aredesigned to follow wind-speed variations in lowwinds to maximize aerodynamic efficiency. Standardcontrol laws [2] require that complex aerodynamicproperties be well known so that the variable-speedturbine can maximize energy capture; in practice,uncertainties limit the efficient energy capture of avariable-speed turbine. The turbine used as a modelfor this article’s research is the Controls AdvancedResearch Turbine (CART) pictured in Figure 1. CARTis located in Golden, Colorado, at the U.S. NationalRenewable Energy Laboratory’s National Wind Tech-nology Center (see “The National Renewable Energy

Laboratory and NationalWind Technology Center”).

A modern utility-scalewind turbine, as shown in

Figure 2, has several levels of control systems. On theuppermost level, a supervisory controller monitorsthe turbine and wind resource to determine whenthe wind speed is sufficient to start up the turbineand when, due to high winds, the turbine must beshut down for safety. This type of control is the dis-crete if-then variety. On the middle level is turbinecontrol, which includes generator torque control,blade pitch control, and yaw control. Generatortorque control, performed using the power electron-ics, determines how much torque is extracted fromthe turbine, specifically, the high-speed shaft. Theextracted torque opposes the aerodynamic torqueprovided by the wind and, thus, indirectly regulatesthe turbine speed. Depending on the pitch actuatorsand type of generator and power electronics, bladepitch control and generator torque control can oper-ate quickly relative to the rotor-speed time constant.

STANDARD AND ADAPTIVE TECHNIQUES FOR MAXIMIZING ENERGY CAPTURE

KATHRYN E. JOHNSON, LUCY Y. PAO, MARK J. BALAS, and LEE J. FINGERSH

NATIONAL RENEWABLE ENERGY LABORATORY

Yaw control, which rotates the nacelle to point into thewind, is slower than generator torque control and bladepitch control. Due to its slowness, yaw control is of lessinterest to control engineers than generator torque controland blade pitch angle control.

On the lowest control level are the internal generator,power electronics, and pitch actuator controllers, whichoperate at higher rates than the turbine-level control. Theselow-level controllers operate as black boxes from the per-spective of the turbine-level control. For example, the gener-

ator and power electronics controllers regulate the generatorand power electronics variables to achieve the desired gen-erator torque, as determined by the turbine-level control.The low-level controllers depend on the types of generatorand power electronics, but the turbine-level control doesnot. For example, CART has a squirrel-cage induction gen-erator and full-processing pulse-width modulation powerelectronics. If the generator torque controller controls thehigh-speed shaft torque, then the stability analysis of theturbine-level control does not depend on these details. In

A Rotor swept area (m2)

Cp Rotor power coefficient (dimensionless)

Cpmax Maximum rotor power coefficient (dimensionless)

Cq Rotor torque coefficient (dimensionless)

J Rotor inertia (kg-m2)

K Standard torque control gain (kg-m2)

M Adaptive torque control gain (m5)

M+ Simulation-derived prediction of optimal torque

control gain (m5)

M∗ Turbine’s true optimal torque control gain (possibly

unknown) (m5)

P Turbine (rotor) power (kW)

P0 Symmetric quadratic curve coefficient (dimensionless)

Pcap Captured power (kW)

Pfavg Average captured power divided by average wind

power over a given time period (dimensionless)

Pwind Power available in the wind (kW)

Pwy Power available in the wind, with approximate yaw

error factor included (kW)

R Rotor radius (m)

a Symmetric quadratic curve coefficient (m−10)

b Damping coefficient (kg-m2/s)

fs Sampling frequency (Hz)

k Adaptive controller’s discrete-time index

n Number of steps in adaptation period

v Wind speed (m/s)

β Blade pitch angle (deg)

γ�M Positive gain in gain adaptation law (m−5)

λ Tip-speed ratio (TSR) (dimensionless)

λ∗ TSR corresponding to Cpmax (dimensionless)

ρ Air density (kg/m3)

τaero Aerodynamic torque (N-m)

τc Generator (control) torque (N-m)

ψ Yaw error (deg)

ω Rotor angular speed (rad/s)

JUNE 2006 « IEEE CONTROL SYSTEMS MAGAZINE 71

Wind-powered machines have been used by humans for cen-

turies. Most familiar are the historical many-bladed windmills

used for milling grain, the earliest versions of which appeared

during the 12th century [21]. Water-pumping wind machines

appeared in the United States in the mid-19th century, while the

modern era of wind turbine generators began in the 1970s [21].

These modern horizontal-axis wind turbines typically have two or

three blades and can be either upwind (with the rotor spinning on

the upwind side of the tower) or downwind. Horizontal-axis wind

turbines range in size from small home-based turbines of a few

hundred watts to utility-scale turbines up to several megawatts.

Most modern utility-scale turbines operate in variable-speed

mode with the turbine speed changing continuously in response

to wind gusts and lulls. Although costly power electronics are

required to convert the variable-frequency power to the fixed utili-

ty grid frequency, variable-speed turbines can spend more time

operating at maximum aerodynamic efficiency than constant-

speed turbines. In addition, variable-speed turbines often endure

smaller power fluctuations and operating loads than constant-

speed turbines. Constant-speed turbines are connected directly

to the utility grid, which eliminates the requirement for power elec-

tronics. A constant-speed machine’s fixed generator frequency

forces the turbine’s mechanical components to absorb much of

the increased energy of a wind gust until the turbine’s power reg-

ulation system can respond. On a variable-speed machine, how-

ever, the rotor speed can increase, absorbing a great deal of

energy due to the large rotational inertia of the rotor.

For modern turbines and power electronics systems, the

increased efficiency and lower loads of variable-speed turbines pro-

vide enough benefit to make the power electronics cost effective.

The wind industry trend is thus to design and build variable-speed

turbines for utility-scale installations. Controlling these modern tur-

bines to minimize the cost of wind energy is a complex task, and

much research remains to be done to improve the controllers.

Wind Turbine Development and Types of Turbines

Nomenclature

this project, we ignore the particulars of the high- and low-level controls and focus on the turbine-level control.

Variable-speed wind turbines have three main regions ofoperation. A stopped turbine or a turbine that is just startingup is considered to be operating in region 1. Region 2 is anoperational mode with the objective of maximizing windenergy capture. In region 3, which encompasses high windspeeds, the turbine must limit the captured wind power sothat safe electrical and mechanical loads are not exceeded.For each region, the solid curve in Figure 3 illustrates thedesired power-versus-wind-speed relationship for a vari-able-speed wind turbine with a 43.3-m rotor diameter.

In Figure 3, the power coefficient Cp is defined as theratio of the aerodynamic rotor power P to the power Pwindavailable from the wind, that is,

Cp = PPwind

. (1)

The available power Pwind is given by

Pwind = 12ρAv3, (2)

where ρ is the air density, A is the rotor swept area, and v isthe wind speed. The aerodynamic rotor power is given by

P = τaeroω, (3)

where τaero is the aerodynamictorque applied to the rotor by thewind and ω is the rotor angularspeed. In Figure 3, the dotted windpower curve represents the powerin the unimpeded wind passingthrough the rotor swept area,whereas the solid curve representsthe power extracted by a typicalvariable-speed turbine. Becausethe wind can change speed morequickly than the turbine, theredoes not exist a static relationshipbetween wind speed and turbinepower in dynamic conditions.However, under steady-state con-ditions, a static relationship exists;the turbine power curve plotted inFigure 3 represents the power ver-sus wind speed relationship for aturbine with Cp = 0.4.

Classical techniques such as pro-portional, integral, and derivative(PID) control of blade pitch [3] aretypically used to limit power andspeed on both the low-speed shaftand high-speed shaft for turbinesoperating in region 3, while

72 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2006

FIGURE 2 Major components of an upwind turbine, in which the wind hits the rotor before thetower. Unlike CART, this turbine rotor has three blades. Most turbines have a fixed-ratio gearbox,as shown, rather than a transmission, since it is not economical to build a transmission capableof withstanding a wind turbine’s high torques and extensive operating hours. The power electron-ics for a variable-speed turbine are usually located at the base of the tower. (Drawing courtesy ofthe U.S. Department of Energy.)

Pitch

WindDirection

Low-SpeedShaft

Rotor

Brake

Gear Box

Yaw Drive

Yaw Motor

Blades Tower

High-SpeedShaft

Nacelle

Wind Vane

Controller

GeneratorAnemometer

FIGURE 1 CART at the National Wind Technology Center. CART isa 600-kW turbine with a 43.3-m rotor diameter used in advancedcontrol experiments. The aim of these control experiments is toreduce the cost of wind energy, either by increasing the amount ofenergy extracted from the wind or by decreasing the turbine’s costby reducing the stress on its components.

generator torque control [4] is usually used in region 2. In [5],disturbance accommodating control is used to limit power andspeed in region 3. The reduction of mechanical loads on thetower and blades is another area of turbine control research[6]–[8]. Finally, [9]–[12] use adaptive control to compensate forunknown and time-varying parameters in regions 2 and 3.Although specific techniques for controlling modern turbinesare usually proprietary, we believe that only recently have tur-bine manufacturers begun to incorporate more modern andadvanced control methods in commercial turbines. In part, thegap between the research and commercial turbine communi-ties is a result of the fact that few theoretically advanced con-trollers have been successfully tested on real turbines.

In this article, we analyze the stability of a control sys-tem that has been tested on CART, focusing on adaptivegenerator torque control with constant blade pitch to maxi-mize energy capture of a variable-speed wind turbineoperating in region 2. In [2], an adaptive strategy is shownto improve wind turbine performance. The focus of thisarticle is stability analysis of the adaptive generator torquecontroller. We begin with a review of nonadaptive con-trollers, continue with a discussion of the adaptive con-troller of [2], and then proceed to the stability analysis.

STANDARD VARIABLE-SPEED CONTROL LAWFor variable-speed wind turbines operating in region 2, thecontrol objective is to maximize energy capture by operat-ing the turbine at the peak of the Cp-TSR-pitch surface ofthe rotor, shown in Figure 4. The power coefficient Cp(λ, β)

is a function of the tip-speed ratio (TSR) λ and the bladepitch β . The TSR λ is defined as

λ = ωRv

. (4)

Since, by (1), rotor power P increases with Cp, operation atthe maximum power coefficient Cpmax is desirable. We notethat Cp can be negative, which corresponds to operatingthe generator in reverse as a motor while drawing powerfrom the utility grid. Also, the Cp surface changes whenthe condition of the blade surface changes. For example,icing or residue buildup on the blade typically shifts the Cp

surface downward, reducing energy capture. In this sec-tion, we assume the blades are clean.

Figure 4 is based on the modeling software PROP [13],which uses blade-element momentum theory [14]. ThePROP simulation was performed to estimate Cp for the600-kW two-bladed, upwind CART. Unfortunately, mod-eling tools such as PROP are of questionable accuracy; infact, an NREL study [15] comparing wind turbine model-ing codes reports large discrepancies and an unknownlevel of uncertainty. Therefore, computer models are unre-liable for fixed-gain controller synthesis.

A control law, which we refer to as the standard control,for region 2 operation of variable-speed turbines is to let thecontrol torque τc (that is, the generator torque) be given by


The National Renewable Energy Laboratory and National Wind Technology Center

T he National Renewable Energy Laboratory (NREL) is a

part of the U.S. Department of Energy (DOE) Office of

Energy Efficiency and Renewable Energy. Located in Gold-

en, Colorado, the laboratory began operating in 1977 as

the Solar Energy Research Institute (SERI) and attained

the national laboratory classification in 1991 when SERI

was renamed NREL. NREL’s mission statement summa-

rizes the laboratory’s research: “NREL develops renewable

energy and energy efficiency technologies and practices,

advances related science and engineering, and transfers

knowledge and innovations to address the nation’s energy

and environmental goals.”

The National Wind Technology Center (NWTC) supports

the U.S. wind industry by performing applied research and

testing in conjunction with its industry partners. These indus-

try partners range from large commercial turbine manufactur-

ers to small distributed wind system developers, all of whom

share the goal of reducing the cost of wind energy. The

NWTC’s facilities include numerous turbine test pads, which

currently test turbines ranging from 300 W to 600 kW; a

dynamometer facility for testing advanced drive trains; an

industrial user facility for testing new blade designs; a hybrid

test facility, which allows testing of energy systems consist-

ing of wind combined with solar, diesel, or other electricity

sources; and two advanced research turbines. Together with

NWTC’s wind industry partners, researchers at the NWTC

have helped to bring the cost of large-scale wind energy

down from about US$0.80/kW-h in 1980 (today’s dollars) to

US$0.04–US$0.06/kW-h today.

FIGURE 3 Illustrative steady-state power curves. A variable-speedturbine attempts to maximize energy capture while operating inregion 2. In region 3, the power is limited to ensure that safe electri-cal and mechanical loads are not exceeded.

2,000

1,800

1,600

1,400

1,200

1,000

800

600

400

200

00 5 10

Wind Speed (m/s)

Pow

er (

kW)

Wind PowerCp = 1

15 20 25

Region 1

Region 2

Region 3

Turbine Power

HighWind

Cutout

Cp = 0.4

τc = Kω2, (5)

where the gain K is given by

K = 12ρAR3 Cpmax

λ3∗, (6)

R is the rotor radius, and λ∗ is the tip-speed ratio at whichthe maximum power coefficient Cpmax occurs.

Next, assuming that the rotor is rigid, the angular accel-eration ω̇ is given by

ω̇ = 1J(τaero − τc), (7)

where J is the combined rotational inertia of the rotor,gearbox, generator, and shafts and the aerodynamic torqueτaero, derived from (1)–(4), is given by

τaero = 12ρARCq(λ, β)v2, (8)

where

Cq(λ, β) = Cp(λ, β)

λ(9)

is the rotor torque coefficient. Since CART has a fairly rigidrotor, the rigid body model (7) is a valid approximation forthe rotor dynamics.

Now, substituting (8) and (5) into (7) and using (9) and(4) yields

ω̇ = 12J

ρAR3ω2(

Cp(λ, β)

λ3 − Cpmax

λ3∗

). (10)

Since the rotor inertia J, the air density ρ, the rotor sweptarea A, the rotor radius R, and the squared rotor speed ω2

are nonnegative, the sign of the angular acceleration ω̇depends on the sign of the difference in (10). When the tip-speed ratio λ > λ∗ , it follows from (10) and the fact thatCp ≤ Cpmax that ω̇ is negative and the rotor deceleratestoward λ = λ∗. On the other hand, when λ < λ∗ and

Cp >Cpmax

λ3∗λ3, (11)

it follows that ω̇ is positive. The curve

F(λ) = Cpmax

λ3∗λ3

is plotted as the dotted line in Figure 5, and CART’s PROP-derived Cp − λ curve for a fixed pitch of −1◦ is the solidline. A pitch angle β of 0◦ means that the blade chord lineis approximately parallel to the rotor plane, although theexact angle depends on the amount of twist of the bladeand the distance between the blade root and the chord linewhere the pitch angle is measured. The solid line in Figure 5is a two-dimensional slice of Figure 4. The inequality (11)is satisfied for tip-speed ratios λ ranging from about 3.3 to7.5. Thus, as long as CART has a tip-speed ratio of at least3.3, the standard control law (5) causes the speed of a well-characterized turbine to approach the optimal tip-speedratio. Although easier to understand under constant wind


FIGURE 4 Cp versus tip-speed ratio and pitch for CART. Since tur-bine power is proportional to the power coefficient Cp, the turbine isideally operated at the peak of the surface. Blade pitch angle is acontrol variable, whereas tip-speed ratio is controlled indirectly usinggenerator torque control. A turbine’s Cp surface can change due toicing, blade erosion, and residue buildup. Negative Cp correspondsto motoring operation during which the turbine draws energy fromthe utility grid.

0.5

0.4

0.3

0.2

Pow

er C

oeffi

cien

t Cp

0.1

1

13 15

11

7

3 −1

−53 5

7

9

11

−0.1

−0.2−0.3−0.4

−0.5

0.0

Tip-Speed Ratio λ Pitch (deg)β

FIGURE 5 CART’s power coefficient Cp versus tip-speed ratio andcubic function F. The intersection of the solid and dotted lines atthe tip-speed ratio λ = 7.5 indicates the optimal operating point interms of energy capture. The cubic function F is derived from thestandard control law, and the intersection points of the cubic func-tion and Cp curve are equilibrium points of the turbine operation.Theorem 2 shows that the equilibrium point λ = 7.5 is locallyasymptotically stable.

2 4

0.4

0.3

0.2

0.1

0.06 8

Pow

er C

oeffi

cien

t Cp

10 12 14

F(λ)

Tip-Speed Ratio λ

conditions, this behavior occurs in an averaged senseunder time-varying wind conditions. We refer to the gainK corresponding to optimum tip-speed ratio operation asthe optimal K.

When the tip-speed ratio λ < 3.3, the inequality (11) isno longer satisfied, and the angular acceleration ω̇ is nega-tive. In this case, the rotor speed ω slows toward zero.However, most turbines have separate control mechanismsto ensure that a low tip-speed ratio λ < 3.3 does not drivethe rotor speed ω to zero when the wind speed is adequatefor energy production. This article is concerned only withthe torque control and, hence, does not consider these sep-arate control mechanisms. While the critical tip-speedratios and control mechanisms are different for differentturbines, the dynamics presented here approximate allvariable-speed turbines using the standard control law (5).

The above discussion assumes that the turbine’s prop-erties used to calculate the gain K in (6) are accurate, whichis rarely the case. Also, over time, debris buildup andblade erosion change the Cp surface and thus Cpmax , withthe same effect as a suboptimally chosen K. The sensitivityof energy loss to errors in λ∗ and the maximum powercoefficient Cpmax is considered in [4], which concludes thata 5% error in the optimal tip-speed ratio λ∗ can cause a sig-nificant energy loss of 1–3% in region 2. If the UnitedStates meets the American Wind Energy Association’s goalof 100,000 MW of installed wind capacity by 2020, a 3%loss in total energy would equal US$300 million per year.The potential for cost savings motivates the developmentand investigation of an adaptive control approach that canimprove energy capture.

ADAPTIVE CONTROLFor region 2 operation, we now consider the adaptive con-troller [2] given by

τc ={

0, ω < 0,

ρMω2, ω ≥ 0,(12)

where the adaptive gain M replaces A, R, Cpmax , and λ∗ in(6). The air density ρ is kept separate because air density istime varying and measurable.

The control law (12) is defined separately for positiveand negative regions of the rotor speed ω because it isundesirable to apply torque control when the turbine isspinning in reverse. Reverse operation can cause excessivewear on components that are designed for operation in onedirection.

The equations for the gain adaptation law are

M (k) = M (k − 1) + �M (k) , (13)

�M(k) = γ�M sgn [�M(k − 1)] sgn[�Pfavg(k)]

× |�Pfavg(k)|1/2, (14)

�Pfavg(k) = Pfavg(k) − Pfavg(k − 1), (15)

where k denotes the adaptive controller’s discrete timestep. The fractional average power Pfavg, given by

Pfavg(k) =1n

n∑i=1

Pcap((k − 1)n + i )

1n

n∑i=1

Pwy((k − 1)n + i ), (16)

is the ratio of the mean power captured to the mean windpower. Pfavg is computed at each adaptive control time stepk, where k is incremented once every n steps of region 2 oper-ation at the discrete-time torque control rate fs =100 Hz. Pwy, computed at 100 Hz, is the wind power given by

Pwy = 12ρAv3 (cos ψ)3 , (17)

where ψ is the yaw error, that is, the error between thewind direction and the yaw position of the turbine. Pcap isthe captured power, given by

Pcap = τcω + Jωω̇, (18)

which is also computed at 100 Hz. The yaw error factor(cos ψ)3 in (17) shows that yaw errors reduce the poweravailable to the turbine. The term τcω in the captured powerPcap is the generator power while Jωω̇ is the kinetic power(that is, the time derivative of the kinetic energy) of the rotor.

In (13), M is adapted after n time steps of 10-ms periodsof operation in region 2. Testing on CART indicates thatthe adaptation period must be on the order of hours; con-sequently, n = 1,080,000 steps, which corresponds to 3 h,is used in many CART experiments. This long time periodis required in part because of the difficulty of obtaining ahigh correlation between measurements of wind speedover the entire swept area of the rotor and at theanemometer, which can be located either on the turbine’snacelle or on a separate meteorological tower [16]. Anotherreason for the long adaptation period is that, since the tur-bine changes speed at a much slower rate than the wind,the slow responses must be averaged over time.

In (14), the factor |�Pfavg(k)|1/2 indicates the closeness ofthe adaptive gain M to its optimal value M∗, the gain thatresults in maximum energy capture. As M moves towardthe peak of the curve in Figure 6, a given adaptation step�M results in a smaller |�Pfavg| because (dPf avg)/(dM̃) → 0as M̃ → 0. Thus, |�M| decreases as the optimal gain isapproached. The exponent 1/2 is chosen based on simula-tion, and selection of γ�M > 0 is discussed below.

In (16), Pcap is used rather than the rotor aerodynamicpower P given by (3) because the sensor requirements forPcap are more consistent with the instrumentation normal-ly available on industrial turbines. The two definitions ofturbine power are closely related, differing only by themechanical losses in the turbine’s gearbox; these lossesmake Pcap < P by a small amount.


Figure 6 portrays the output of constant-wind-speedsimulations using the rigid body model (7) and the controltorque (12). The model and controller are simulated with26 different values of the gain M, where each simulationlasts 200 s with constant M for the duration of the simula-tion. The turbine’s power output for each of the 26 gainvalues is averaged over each 200-s simulation to producethe solid Pfavg curve in Figure 6. In Figure 6, M∗ = 174.5 isthe optimal gain based on the standard torque controlcoefficient K in (6) as well as the simulated power-coefficient Cp surface in Figure 4. Since these data are

obtained from simulations, the optimal gain M∗ is known.The error M̃ in M is given by

M̃ = M∗ − M.

The adaptive controller attempts to have the turbine powertrack the wind power, assuming that the maximum powercoefficient Cpmax and the optimal tip-speed ratio λ∗ areunknown. In contrast, adaptive controllers such as those in[10]–[11] focus on different uncertainties and assume someknowledge of the Cp surface, particularly λ∗ and Cpmax . In addi-tion, the averaging period used in this article is long comparedto the time periods used by the adaptive controller in [9].

Figure 7 shows data collected in the first year of adaptiveCART operation. Only region 2 data is plotted, and thechange in the adaptation period length from 10 min to 180min is apparent. The adaptation behavior with the longeradaptation period is significantly better than the behaviorwith the shorter adaptation period. The three discontinuitiesin the data reflect occasions where the adaptive controllerwas restarted due to a change in the method for calculatingPfavg and problems with sensors on CART. The last dozenadaptations oscillate about a value that is just less than 50%of the predicted optimal value M+ = 174.5 computed fromthe PROP model of CART. In comparison, the CART study[17], obtained with the turbine running in constant speedmode, gives a true optimal gain M∗ around 47% of the pre-dicted optimal value M+. The experimental results shown inFigure 7 indicate that modeling tools such as PROP [13] canlead to large errors in predicting the optimal value of thegain M. We now proceed with the stability analysis.

STABILITYWe now consider the stability of the closed-loop systemwith the adaptive torque gain control law. Some of theresults in this section appear in [18]. Although control ofCART’s torque is a discrete-time problem, we simplify thestability analyses of the torque control law (12) by assum-ing that the torque control is continuous time. This simpli-fication is valid because the control time step of 0.01 s ismuch smaller than the tip-speed ratio’s time constant,which depends on wind speed [19] and is about 4–8 s forCART operating in region 2 wind speeds of 6–12 m/s.Also, we assume that the adaptive control gain M > 0 isconstant in the torque control law (12) analysis; thisassumption is valid because the gain adaptation takesplace discretely and on a time scale several orders of mag-nitude slower than changes in the wind speed and rotorspeed (hours versus seconds). Thus, each result that isbased on a constant M assumption holds for the durationof each 3-h adaptation period. Furthermore, M is con-strained to be positive since the control torque (12) cannotbe negative. In all of these proofs, the air density ρ isassumed to be a positive constant. In reality, changes in airdensity are small, typically not much greater than 5%. A


FIGURE 7 Adaptive gain M normalized by the predicted optimal gainM+ during region 2 operation of CART. Discontinuities indicaterestarts of the gain adaptation law due to changes in the law andturbine sensor errors. In the second half of the data, M oscillatesaround the value 0.47 M+, which is approximately equal to the trueoptimal torque gain M∗.

1.5

1

0.5

Nor

mal

ized

M (

M/M

+ )

00 20 40

Time (h)60 80

FIGURE 6 Pfavg versus M̃ for the CART model. Pfavg is the ratio ofthe mean captured power to the mean wind power, while M̃ is theerror between the torque control gain M and its optimal value M∗.The shape of this curve is based on the shape of CART’s Cp − λ

curve. In the adaptive controller, the gain adaptation law convergesin part due to the shape of the Pfavg curve.

−100 −50 0 50 1000.30

0.32

0.34

0.36

0.38

0.40

0.42Pfavg

Gain Error M = M* − M

Fra

ctio

nal A

vera

ge P

ower

Pfa

vg

∼

simplified block diagram for these continuous-time sys-tems is given in Figure 8(a), where the linear plant is givenby (7) and the nonlinear controller is given by (12).

Asymptotic Stability of Zero Rotor SpeedFirst, we consider the asymptotic stability of the rotor-speed equilibrium ω = 0 in the absence of wind and inconstant wind. To minimize energy loss in wind turbines,friction and drag due to mechanical bearings, gear mesh,generator core losses, and air resistance are designed to beas small as possible. However, in the analysis of asymptot-ic stability of the equilibrium point ω = 0, we revise (7) sothat the angular acceleration ω̇ includes a damping termbω, where the damping coefficient b > 0, which yields

ω̇ = 1J (τaero − τc − bω). (19)

Using (8) and (12), (19) can be expanded to

ω̇ ={ 1

2J ρARCqv2 − bJ ω, ω < 0,

12J ρARCqv2 − ρ

J Mω2 − bJ ω, ω ≥ 0.

(20)

Theorem 1Suppose that the wind speed v = 0 and M > 0 are con-stant. Then the equilibrium ω = 0 of the closed-loop sys-tem (20) is asymptotically stable.

ProofFor the initial condition ω(0) = ω0 , the solution to (20)when v = 0 is

ω(t) ={

ω0e−bJ t, ω < 0,

ω0 b

(b+ρMω0)ebJ t−ρMω0

, ω ≥ 0.

Hence, ω → 0 as t → ∞. �We also note that when the damping coefficient b = 0

and the wind speed v = 0, (20) becomes

ω̇ ={

0, ω < 0,

− ρJ Mω2, ω ≥ 0,

which has the solution

ω(t) ={

ω0, ω < 0,J

ρMt+ Jω0

, ω ≥ 0.

In this case, ω → 0 holds only when the rotor is spinning in thepositive direction, which is normal operation for the turbine.

Asymptotic Stability of Rotor Speed with Constant, Positive Wind InputThe next stability result concerns the convergence of therotor speed ω to an equilibrium value under an idealized

constant, positive wind speed. This analysis is similar tothe one describing Figure 5 and given in (5)–(11). Onceagain, the plant is given by (19) and the nonlinear con-troller is given by (12). The adaptive controller (12) doesnot assume knowledge of the aerodynamic parametersCpmax and λ∗. Setting the ω ≥ 0 portion of (20) equal to zeroand solving for Cp in terms of λ using (4) and (9) yields

Cp = ρMλ3v + λ2bR12ρAR3v

≡ G(λ, M, b, v). (21)

The equilibrium points ω̇ = 0 of turbine operation are thusgiven by the intersection of with the turbine’s Cp − λ

curve. Figure 9 shows CART’s Cp − λ curve and two illus-trative G(λ, M, b, v) curves plotted using representativevalues of ρ, v, and b.

In Figure 9, the cubic functions G(λ, M, b, v) do not inter-sect the Cp curve at the peak of the curve when the adaptive


FIGURE 8 Control loops for (a) the aerodynamic torque τaero androtor speed ω and (b) the gain adaptation law. (a) Stability of thecontinuous-time control loop is analyzed by Theorems 1–3, whileTheorem 4 considers (b) the discrete-time adaptive loop.

−+

NonlinearController

LinearPlant

τc

τaero ω

−+

NonlinearController

PfavgNonlinearPlant

M

M*

(a) (b)

FIGURE 9 CART’s power coefficient Cp curve and cubic functionsfor two values of the adaptive gain M. When M is not equal to itsoptimum value M∗, the intersection of the Cp and G(λ, M) curvesdoes not occur at the peak of the Cp curve, which leads to subopti-mal energy capture. Similar to Figure 5, the intersection of eachcubic curve with the Cp curve is an equilibrium point of the systemfor the indicated adaptive gain M.

2 4

0.4

0.3

0.5

0.2

0.1

06 8

Tip-Speed Ratio

Pow

er C

oeffi

cien

t Cp

10 12 14

G( ,M)M = 1.3M*

G( ,M)M = 0.7M*

Cart Cp Versusλ λ

λ

λ

gain M �= M∗ ; thus, the equilibrium point of the system issuboptimal in terms of energy capture. Let λ2 be the highestvalue of λ for which the curve G(λ, M) intersects Cp(λ).Mathematically, λ2 is the tip-speed ratio for whichG(λ, M) > Cp(λ) for all λ > λ2. Let λ1 denote the next highestintersection point, that is, the value of λ for which0 < λ1 < λ2 and G(λ, M) < Cp(λ) for all λ1 < λ < λ2 andG(λ, M) > Cp(λ) for all λ < λ1 within a neighborhood of λ1.For the dashed curve M = 0.7 M∗ in Figure 9, these valuescorrespond to λ1 = 3.1 and λ2 = 8.4. The following resultshows that, for a constant wind input, the tip-speed ratio λ con-verges to λ2 as long as the initial value of λ is greater than λ1.

Theorem 2Suppose that the wind speed v and the adaptive gain M arepositive constants and λ1 > 0. Then the equilibrium pointλ = λ2 of the closed-loop system consisting of the plant

λ̇ = RJv

(τaero − τc − λ

bvR

)(22)

and the nonlinear controller (12) is locally asymptoticallystable with domain of attraction λ ∈ (λ1,∞).

ProofFirst note that ω > 0 for all 0 < λ1 < λ since ω = λv/R from(4). Define λ̃ = λ2 − λ and the Lyapunov candidateV = (1/2)λ̃2. For ω > 0,

V̇ = (λ − λ2)

(12J

ρAR2Cqv − 1JR

ρMλ2v − bJλ

)= (λ − λ2)h(Cq, v, λ).

(23)

Substituting Cp/λ for Cq in (23) and applying (21) yieldsh(Cq, v, λ) > 0 for all λ such that λ1 < λ < λ2 , that is,G(λ, M) < Cp(λ). Moreover, λ > λ2 gives G(λ, M) > Cp(λ) bydefinition of λ2, and therefore h(Cq, v, λ) < 0 by definition (21)of G. Thus, V̇ < 0 for all λ ∈ (λ1,∞) except λ = λ2, for whichV̇ = 0. Hence, the equilibrium point λ = λ2 of (22) is locallyasymptotically stable. Finally, it is easy to show that thedomain of attraction is (λ1,∞). Note that V̇ is bounded awayfrom zero on every connected, compact subinterval of (λ1,∞)

that does not contain λ2. Thus, the time required for λ to reachthe edge of the subinterval closest to λ2 is finite. Now, λ movesmonotonically toward λ2. If λ does not converge to λ2, then thetime it takes λ to reach the edge closest to λ2 of a subintervalnot containing λ2 must be infinite, which contradicts the earlierresult. Thus, the domain of attraction is (λ1,∞). �

The convergence of the tip-speed ratio λ to λ2 is equiva-lent to the convergence of the rotor speed ω to λ2v/R for aspecific wind speed v. Furthermore, when M = M∗ , thecurves G(λ, M) and Cp(λ) intersect at (λ∗, Cpmax) as shownfor the standard torque control in Figure 5; therefore, optimalenergy capture is achieved for the constant wind input case.

We acknowledge that zero and constant wind speedsnever occur in the field. However, wind speeds near zero dooccur during turbine operation, causing a shutdown whenthe wind speed is close to zero for a sufficiently long time.These results are useful for developing an understanding ofthe torque control law, although the cases are idealized.

Input-Output StabilityNext, we show that a bounded input (squared wind speedv2) to the system produces a bounded output (rotor speedω). All wind turbines have a maximum safe operatingspeed, and often pitch control is used to prevent the tur-bine from operating at speeds above this maximum. Nev-ertheless, an enhanced understanding of the wind turbinecontrol system can be achieved by examining whether thetorque control (12) bounds the turbine speed. The follow-ing result considers a time-varying wind speed v.

For T > 0, we use the standard the definition of the L2norm of v(t) given by

‖v‖L2[0,T] =√∫ T

0‖v(t)‖2dt.

Theorem 3Suppose the rotor torque coefficient Cq ≤ 1, the adaptivegain M > 0 is constant, and consider the closed-loop tur-bine system (12), (19) with input given by the squared windspeed v2 and output given by the rotor speed ω. Then, forall finite T > 0, the system (12), (19) is L2 stable on [0, T].

ProofConsider the kinetic energy EK = (1/2)Jω2 of the rotorand define

V = 1ρAR

Jω2, (24)

where ρ > 0 is a constant. The time derivative of (24) is

V̇ ={

Cqv2ω − b12 ρAR

ω2, ω < 0,

Cqv2ω − b12 ρAR

ω2 − M12 AR

ω3, ω ≥ 0.(25)

Let δ = b/((1/2)ρAR) . Then, since Cq ≤ 1 and(M/((1/2)AR))ω3 ≥ 0 for ω ≥ 0, it follows that

V̇ ≤ v2ω − δω2. (26)

Thus, Lemma 6.5 in [20] implies that the wind turbine sys-tem, from the squared wind speed v2 to the rotor speed ω,is finite-gain L2 stable over [0, T]. �

The restriction that T be finite is necessary due to thenature of the wind speed v(t). Since wind speed v(t) > 0can hold at all times, it is possible that v(t) /∈ L2[0,∞].Thus, T must be finite to guarantee that proof of L2 stabilityin [0, T] makes sense.


The condition Cq ≤ 1 is usually satisfied for modernturbines in normal region 2 operation. The Betz limit [14],which is the theoretical maximum power coefficient Cp forany real turbine, has a value of Cp = 16/27. SinceCq = Cp/λ [see (9)], it follows that Cq ≤ 1 for λ ≥ 16/27.When λ ≤ 16/27, it follows from the definition of tip-speedratio λ in (4) that ω = λv/R ≤ (16/27)v/R.

For finite η > 0 and λ ∈ [0, T],L∞ , that is, boundedinput, bounded output stability of ω with respect to theinput v is given by ω = λv/R.

Theorem 3 shows that a wind turbine is not a perpetu-al motion machine. Since the assumption that M is con-stant holds only for the duration of an adaptation period, Theorem 3 shows that the energy produced by a turbineis less than that contained in the wind over each adapta-tion period.

Convergence of the Gain Adaptation AlgorithmThe final stability analysis examines convergence of theadaptive gain M → M∗ using the gain adaptation law(13)–(15). Figure 8(b) shows a simplified block diagramfor this system, where the nonlinear plant is the fractionalaverage power Pfavg versus torque gain error M̃ relation-ship shown in Figure 6 and the nonlinear controller isgiven by (13)–(15).

We make two assumptions before studying the stabilityproperties of the gain adaptation law.

Assumption 1The optimum torque control gain M∗ is constant.

The turbine’s aerodynamic parameters, and thus M∗ ,change with time due to blade erosion, residue buildup,and related events. However, we can assume that M∗ isconstant because the turbine’s physical changes are typical-ly noticeable only over months or years, whereas the gainadaptation law has an adaptation period of less than a day.

Assumption 2The Pfavg versus M̃ curve has a maximum at M̃ = 0, is con-tinuously differentiable, and is strictly monotonicallyincreasing on M̃ < 0 and strictly monotonically decreasingon M̃ > 0. Experimental data [17] support this assumption.

For the initial conditions M0, Pfavg0, �M0 , and �M1 ,k > 2 is the time frame of interest in the convergence analy-sis. Theorem 4 covers only the time k > 2 because the firsttwo steps are more influenced by the initial guesses than bythe turbine’s aerodynamic properties.

We begin the convergence analysis by considering howthe adaptive gain can diverge, that is, |M̃| → ∞ as k → ∞.One possibility is |M̃k| > |M̃k −1| with either sgn(M̃k) = 1or sgn(M̃k) = −1 for all k > 2. However, it is easy to showthat this scenario cannot occur with the gain adaptationlaw (13)–(15). Indeed, the adaptive torque gain error M̃cannot take two consecutive steps in the wrong (incorrect)direction for all k > 2, as shown by the following result.

Theorem 4Let k > 2. Under Assumptions 1 and 2 and the gain adap-tation law (13)–(15), |M̃k +1| > |M̃k| > |M̃k −1| never occurswhen sgn(M̃k +1) = sgn(M̃k) = sgn(M̃k −1).

ProofSuppose M̃k +1 > M̃k > M̃k −1 and sgn(M̃k +1) = sgn(M̃k) =sgn(M̃k −1) = 1 for some k > 2. Note that M̃k > M̃k −1 gives

M̃k − M̃k −1 = −�Mk > 0, (27)

which implies that �Mk < 0. Furthermore, M̃k +1 > M̃k gives

M̃k +1 − M̃k = −�Mk +1 > 0, (28)

which implies that �Mk +1 < 0. By (16)–(18), Pfavg k +1 is cal-culated at the end of the adaptation interval during whichM = Mk ; thus, Pfavg k +1 is calculated from data collectedwhile the torque gain error was M̃k . Since M̃k > M̃k −1 ,Assumption 2 implies Pfavg k +1 < Pfavg k . Therefore, by (15),

�Pfavg k +1 < 0. (29)

In (27) and (29), sgn(�Mk) = sgn(Pfavg k +1) = −1. Thus, by(14), sgn(�Mk +1) = 1, contradicting (28). Thus, it is impossible for both M̃k +1 > M̃k > M̃k −1 and sgn(M̃k +1) = sgn(M̃k) = sgn(M̃k −1) = 1 to be true. A similarargument can be used for negative values of M̃. �

Since the sign of the adaptation step �M cannot beincorrect for two consecutive steps, the gain γ�M, whichaffects the magnitude of �M, is the critical factor in deter-mining whether the adaptive gain diverges. Figure 10shows an example in which the gain γ�M is large enoughto cause the adaptive gain M to diverge. In this example,|M̃k +1| > |M̃k −1| for all k > 2, although both |M̃k +1| > |M̃k|and |M̃k +1| < |M̃k| occur when k > 2.


FIGURE 10 Adaptive gain steps in an unstable case. The numbers1–9 indicate the discrete-time steps. In this case, the gain γ�M inthe gain adaptation algorithm (13)–(15) is too large, and thus thegain adaptation law diverges.

−20 −10−15 −5 0 105

9

5

1

7

36 2 48

20150.25

0.30

0.35

0.40

0.45

0.50


Fra

ctio

nal A

vera

ge P

ower

Pfa

vg

∼

Since M diverges if |�Mk| > |M̃k−1| for all k > 2, weconsider

|�Mk| = |M̃k −1|, M̃k −1 �= 0 (30)

to be the critical case, or the marginal stability case. Define yk by

yk ≡ aM̃2k−1 + P0, (31)

where yk is a curve satisfying Assumptions 1–2 whose formis better known than Pfavgk

. In (31), a < 0 and P0 is a realnumber; (30) can be solved for the critical gain γ�M. Forconsistency with the discrete-time indices in the equation(16) for Pfavgk

, yk is a function of M̃k −1 rather than of M̃k.

In the critical gain scenario of this example, the systemalternates among the three points plotted in Figure 11. If�Mk = M̃k−1, then the error M̃k = 0 by (13). Substituting ykfor Pfavgk

in (15) and considering (14), the gain γ�M is suchthat �Mk+1 = �Mk , resulting in M̃k+1 = −M̃k−1. Followingthe equations through one more step shows that M̃k+2 = 0,and the adaptive gain alternates among these three points.Thus, an upper bound on the gain γ�M for stability can befound by equating

�Mk = M̃k−1 = −M̃k+1

and solving for γ�M in terms of a with M̃k = 0, which yields

γ�M =√

1|a| . (32)

Thus, if 0 < γ�M < |a|−1/2, then the gain adaptation law(13)–(15) does not cause divergence of the adaptivetorque control gain error M̃ on the curve (31). In fact,s ince γ�M = |a|−1/2 is the marginal stabil i ty case,0 < γ�M < |a|−1/2 yields M̃ → 0. Since this bound on γ�M

depends on the magnitude |a|, every gain γ�M chosenfor a given value of a in (31) also guarantees conver-gence of the adaptive gain M on a curve with a smallervalue of a.

We can state a similar result for a curve that is not even[as in (31)], that is, one for which Pfavg(M̃) = Pfavg(−M̃)

does not hold. If the gain γ�M is chosen to guarantee con-vergence based on the slope of the steeper side of thecurve, then γ�M guarantees convergence over the entirecurve. Thus, for an arbitrary Pfavg versus M̃ curve, thereexists γ�M > 0 that guarantees convergence of the adap-tive gain M, and this gain γ�M depends on the steepness ofthe Pfavg versus M̃ curve.

Since there are no turbines for which the Pfavg versus M̃curve is well known, an approximation of the curve is nec-essary to control each turbine. The more conservative thechoice of γ�M, the more likely it is that M converges to M∗since the gain adaptation law (13)–(15) is more robust toerrors in the approximated Pfavg versus M̃ curve for small-er γ�M. However, a smaller γ�M also results in smallerstep sizes and thus might cause the convergence to occurmore slowly.

An example of the choice of γ�M is provided in Figure12. The coefficients a and P0 of (31) are chosen so that (31)fits snugly inside the Pfavg curve, being coincident atM̃ = 0 and satisfying y < Pfavg for M̃ �= 0. In this case,a = −0.00001 m−10. Thus, the maximum allowable gainγ�M for stability is 316 m−5. The gain used in testing onCART before this stability analysis was performed wasγ�M = 100 m−5, which was determined empirically fromsimulations and early hardware testing. Although actualturbine results indicate stable performance of the adaptivecontrol law, this stability analysis provides further reassur-ance and guidelines in choosing γ�M.


FIGURE 12 CART Pfavg versus M̃ curve and symmetric inset curve.The curve labeled Pfavg is identical to the curve shown in Figure 6,while the quadratic curve labeled y is added to illustrate the methodfor selecting the adaptive gain γ�M . When the quadratic curve ischosen such that y(M̃) ≤ Pfavg(M̃) and y(M̃) = Pfavg(M̃) if and onlyif M̃ = 0, the upper limit on γ�M for stability of the gain adaptationlaw is a function of the coefficient of the squared term in (31).

−100 −50 0 50 1000.30

0.32

0.34

0.36

0.38

0.40

0.42Pfavg

y


Fra

ctio

nal A

vera

ge P

ower

Pfa

vg

∼

FIGURE 11 Finding the critical gain γ�M . Marginal stability of thegain adaptation law, defined as oscillation among three points onthe y curve (31), occurs when the step size �Mk has the samemagnitude as the error M̃k−1 for a symmetric curve.

−50 0 50

0.40

0.41

0.42

0.43


(Mk, yk+1)

−∆Mk−∆Mk+1

Fra

ctio

nal A

vera

ge P

ower

Pfa

vg

∼

(Mk+1, yk+2)

∼

∼

(Mk−1, yk)

∼

CONCLUSIONSThis article considers an adaptive control scheme previouslydeveloped for region 2 control of a variable-speed wind tur-bine. In this article, we addressed the question of theoreticalstability of the torque controller, showing that the rotor speedis asymptotically stable under the torque control law (12) in theconstant wind speed input case and L2 stable with respect totime-varying wind input. Further, we derived a method forselecting γ�M in the gain adaptation law (13)–(15) to guaranteeconvergence of the adaptive gain M to its optimal value M∗.

ACKNOWLEDGMENTSThis work was supported in part by the U.S. Department ofEnergy through the National Renewable Energy Laboratoryunder contract DE-AC36-99G010337, the University of Col-orado at Boulder, and the American Society for EngineeringEducation. We would also like to acknowledge Prof. DaleLawrence and Dr. Vishwesh Kulkarni for their suggestionson improving our article.

AUTHOR INFORMATIONKathryn E. Johnson ([email protected]) received the B.S.degree in electrical engineering from Clarkson University in2000 and the M.S. and Ph.D. degrees in electrical engineeringfrom the University of Colorado in 2002 and 2004, respective-ly. In 2005, she completed a postdoctoral research assignmentstudying adaptive control of variable-speed wind turbines atthe National Renewable Energy Laboratory’s National WindTechnology Center. That fall, she was appointed Clare BootheLuce Assistant Professor at the Colorado School of Mines inthe Division of Engineering. Her research interests are in con-trol systems and control applications. She can be contacted atColorado School of Mines, Division of Engineering, 1610 Illi-nois St., Golden, CO 80401 USA.

Lucy Y. Pao received the B.S., M.S., and Ph.D. degreesin electrical engineering from Stanford University. She iscurrently a professor of electrical and computer engineer-ing at the University of Colorado at Boulder. She has pub-lished over 120 journal and conference papers in the areaof control systems. Her awards include the Best Commer-cial Potential Award at the 2004 International Symposiumon Haptic Interfaces for Virtual Environments and Teleop-erator Systems as well as the Best Paper Award at the 2005World Haptics Conference. She was the program chair forthe 2004 American Control Conference, and she is current-ly an elected member on the IEEE Control Systems SocietyBoard of Governors.

Mark J. Balas has made theoretical contributions inlinear and nonlinear systems, especially in the control ofdistributed and large-scale systems, aerospace structurecontrol, and variable-speed, horizontal-axis wind turbinecontrol for electric power generation. He is a Fellow ofthe IEEE and the AIAA. He is currently head of the Elec-trical and Computer Engineering Department at the Uni-versity of Wyoming.

Lee J. Fingersh received the B.S. and M.S. degrees inelectrical engineering from the University of Colorado in1993 and 1995, respectively. He has been employed atNREL since 1993, working in the fields of aerodynamicstesting, power electronics, electric machines, energy stor-age, and controls. Most recently, he has been responsiblefor a large controls field testing project and its associatedtest machine, the Controls Advanced Research Turbine.

REFERENCES[1] “Global wind energy installations climb steadily,” American Wind Ener-gy Association’s Wind Power Outlook 2005 [Online], Mar. 2004, p. 6. Avail-able: http://www.awea.org/pubs/documents/Outlook%202005.pdf[2] K. Johnson, L. Fingersh, M. Balas, and L. Pao, “Methods for increasingregion 2 power capture on a variable speed wind turbine,” J. Solar EnergyEng., vol. 126, no. 4, pp. 1092–1100, 2004.[3] J. Svensson and E. Ulen, “The control system of WTS-3 instrumentationand testing,” in Proc. 4th Int. Symp. Wind Energy Systems, Stockholm,Sweden, 1982, pp. 195–215.[4] L. Fingersh and P. Carlin, “Results from the NREL variable-speed testbed,” in Proc. 17th ASME Wind Energy Symp.}, Reno, NV, 1998, pp. 233–237.[5] K. Stol and M. Balas, “Periodic disturbance accommodating control forspeed regulation of wind turbines,” in Proc. 21st ASME Wind Energy Symp.,Reno, NV, 2002, pp. 310–320.[6] A. Eggers, H. Ashley, K. Chaney, S. Rock, and R. Digumarthi, “Effects of cou-pled rotor-tower motions on aerodynamic control of fluctuating loads on light-weight HAWTs,” in Proc. 17th ASME Wind Energy Symp., Reno, NV, 1998, pp.113–122.[7] M. Hand, “Load mitigation control design for a wind turbine operatingin the path of vortices,” in Proc. Science of Making Torque from Wind 2004 Spe-cial Topic Conf., Delft, The Netherlands, 2004 [CD-ROM]. [8] A. Wright and M. Balas, “Design of controls to attenuate loads in theControls Advanced Research Turbine,” J. Solar Energy Eng., vol. 126, no. 4,pp. 1083–1091, 2004.[9] S. Bhowmik, R. Spée, and J. Enslin, “Performance optimization for dou-bly-fed wind power generation systems,” IEEE Trans. Ind. Applicat., vol. 35,no. 4, pp. 949–958, 1999.[10] J. Freeman and M. Balas, “An investigation of variable speed horizontal-axis wind turbines using direct model-reference adaptive control,” in Proc.18th ASME Wind Energy Symp., Reno, NV, 1999, pp. 66–76.[11] Y. Song, B. Dhinakaran, and X. Bao, “Variable speed control of wind tur-bines using nonlinear and adaptive algorithms,” J. Wind Eng. Ind. Aerodyn.,vol. 85, no. 3, pp. 293–308, 2000.[12] M. Simoes, B. Bose, and R. Spiegel, “Fuzzy logic based intelligent con-trol of a variable speed cage machine wind generation system,” IEEE Trans.Power Electron., vol. 12, no. 1, pp. 87–95, 1997.[13] S. Walker and R. Wilson, Performance Analysis Program for Propeller TypeWind Turbines. Corvallis, OR: Oregon State Univ., 1976.[14] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind EnergyHandbook. New York: Wiley, 2001.[15] D. Simms, S. Schreck, M. Hand, and L. Fingersh, “NREL unsteady aero-dynamics experiment in the NASA-Ames wind tunnel: A comparison ofpredictions to measurements,” NREL Rep. TP-500-29494, 2001 [Online].Available: http://www.nrel.gov/docs/fy01osti/29494.pdf.[16] K. Johnson, L. Fingersh, L. Pao, and M. Balas, “Adaptive torque controlof variable speed wind turbines for increased region 2 energy capture,” inProc. 2005 ASME Wind Energy Symp., Reno, NV, 2005, pp. 66–76.[17] L. Fingersh and K. Johnson, “Baseline results and future plans for theNREL controls advanced research turbine,” in Proc. 23rd ASME Wind EnergySymp., Reno, NV, 2004, pp. 87–93.[18] K. Johnson, L. Pao, M. Balas, V. Kulkarni, and L. Fingersh, “Stabilityanalysis of an adaptive torque controller for variable speed wind turbines,”in Proc. IEEE Conf. on Decision and Control, Atlantis, Paradise Island,Bahamas, 2004, vol. 4, pp. 4016–4021.[19] K. Pierce, “Control method for improved energy capture below ratedpower,” Proc. ASME/JSME Joint Fluids Engineering Conf., San Francisco, CA,1999, pp. 1041–1048.[20] H. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002, p. 242.[21] P. Gipe, Wind Energy Comes of Age. New York: Wiley, 1995.


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