control of variable-speed wind turbines

8/3/2019 Control of Variable-speed Wind Turbines

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70 IEEE CONTROL SYSTEMS MAGAZINE JUNE 2006 1066-033X/06/$20.002006IEEE

Wind energy is the fastest-growing energy

source in the world, with worldwide

wind-generation capacity tripling in the

five 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, could

reduce the cost of wind energy. In contrast to constant-

speed turbines (see Wind Turbine Development and

Types of Turbines), variable-speed wind turbines are

designed to follow wind-speed variations in lowwinds to maximize aerodynamic efficiency. Standard

control laws [2] require that complex aerodynamic

properties be well known so that the variable-speed

turbine can maximize energy capture; in practice,

uncertainties limit the efficient energy capture of a

variable-speed turbine. The turbine used as a model

for this articles research is the Controls Advanced

Research Turbine (CART) pictured in Figure 1. CART

is located in Golden, Colorado, at the U.S. National

Renewable Energy Laboratorys National Wind Tech-

nology Center (see The National Renewable Energy

Laboratory and National

Wind Technology Center).

A modern utility-scale

wind turbine, as shown in

Figure 2, has several levels of control systems. On the

uppermost level, a supervisory controller monitors

the turbine and wind resource to determine when

the wind speed is sufficient to start up the turbine

and when, due to high winds, the turbine must be

shut 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. Generator

torque control, performed using the power electron-

ics, determines how much torque is extracted from

the turbine, specifically, the high-speed shaft. The

extracted torque opposes the aerodynamic torque

provided by the wind and, thus, indirectly regulates

the turbine speed. Depending on the pitch actuators

and type of generator and power electronics, blade

pitch 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


2/12

Yaw control, which rotates the nacelle to point into the

wind, is slower than generator torque control and blade

pitch control. Due to its slowness, yaw control is of less

interest to control engineers than generator torque control

and blade pitch angle control.

On the lowest control level are the internal generator,

power electronics, and pitch actuator controllers, which

operate at higher rates than the turbine-level control. These

low-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 generator

and 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 generator

and power electronics, but the turbine-level control does

not. For example, CART has a squirrel-cage induction gen-

erator and full-processing pulse-width modulation power

electronics. If the generator torque controller controls the

high-speed shaft torque, then the stability analysis of the

turbine-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 Turbines true optimal torque control gain (possiblyunknown) (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 (m10)

b Damping coefficient (kg-m2/s)

fs Sampling frequency (Hz)

k Adaptive controllers discrete-time index

n Number of steps in adaptation period

v Wind speed (m/s)

Blade pitch angle (deg)M Positive gain in gain adaptation law (m

5)

Tip-speed ratio (TSR) (dimensionless)

TSR corresponding to Cpmax (dimensionless)

Air density (kg/m3)

aero Aerodynamic torque (N-m)

c Generator (control) torque (N-m)

Yaw error (deg)

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 themodern 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 machines fixed generator frequencyforces the turbines mechanical components to absorb much of

the increased energy of a wind gust until the turbines 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


3/12

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 of

operation. A stopped turbine or a turbine that is just starting

up is considered to be operating in region 1. Region 2 is an

operational mode with the objective of maximizing wind

energy capture. In region 3, which encompasses high wind

speeds, the turbine must limit the captured wind power so

that safe electrical and mechanical loads are not exceeded.

For each region, the solid curve in Figure 3 illustrates the

desired 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 the

ratio of the aerodynamic rotor power P to the power Pwindavailable from the wind, that is,

Cp =P

Pwind. (1)

The available power Pwind is given by

Pwind =12 Av

3, (2)

where is the air density,A is the rotor swept area, and v is

the wind speed. The aerodynamic rotor power is given by

P= aero, (3)

where aero is the aerodynamic

torque applied to the rotor by the

wind and is the rotor angular

speed. In Figure 3, the dotted wind

power curve represents the power

in the unimpeded wind passingthrough the rotor swept area,

whereas the solid curve represents

the power extracted by a typical

variable-speed turbine. Because

the wind can change speed more

quickly than the turbine, there

does not exist a static relationship

between wind speed and turbine

power in dynamic conditions.

However, under steady-state con-

ditions, a static relationship exists;

the turbine power curve plotted in

Figure 3 represents the power ver-sus wind speed relationship for a

turbine withCp = 0.4.

Classical techniques such as pro-

portional, integral, and derivative

(PID) control of blade pitch [3] are

typically used to limit power and

speed on both the low-speed shaft

and high-speed shaft for turbines

operating 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 the

tower. 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 capable

of withstanding a wind turbines 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 of

the 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 is

a 600-kW turbine with a 43.3-m rotor diameter used in advanced

control experiments. The aim of these control experiments is to

reduce the cost of wind energy, either by increasing the amount of

energy extracted from the wind or by decreasing the turbines cost

by reducing the stress on its components.


4/12

generator torque control [4] is usually used in region 2. In [5],

disturbance accommodating control is used to limit power and

speed in region 3. The reduction of mechanical loads on the

tower and blades is another area of turbine control research

[6][8]. Finally, [9][12] use adaptive control to compensate for

unknown and time-varying parameters in regions 2 and 3.

Although specific techniques for controlling modern turbines

are usually proprietary, we believe that only recently have tur-

bine manufacturers begun to incorporate more modern and

advanced control methods in commercial turbines. In part, the

gap 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 adaptive

generator torque control with constant blade pitch to maxi-

mize energy capture of a variable-speed wind turbine

operating in region 2. In [2], an adaptive strategy is shown

to improve wind turbine performance. The focus of this

article 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 LAW

For variable-speed wind turbines operating in region 2, the

control objective is to maximize energy capture by operat-

ing the turbine at the peak of the Cp-TSR-pitch surface of

the rotor, shown in Figure 4. The power coefficientCp(, )

is a function of the tip-speed ratio (TSR) and the blade

pitch . The TSR is defined as

=R

v . (4)

Since, by (1), rotor power P increases with Cp, operation at

the maximum power coefficient Cpmax is desirable. We note

that Cp can be negative, which corresponds to operating

the generator in reverse as a motor while drawing power

from the utility grid. Also, the Cp surface changes when

the condition of the blade surface changes. For example,

icing or residue buildup on the blade typically shifts the Cpsurface 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]. The

PROP simulation was performed to estimate Cp for the600-kW two-bladed, upwind CART. Unfortunately, mod-

eling tools such as PROP are of questionable accuracy; in

fact, an NREL study [15] comparing wind turbine model-

ing codes reports large discrepancies and an unknown

level 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 the

control torque c (that is, the generator torque) be given by


The National Renewable Energy Laboratoryand National Wind Technology Center

The 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. NRELs mission statement summa-

rizes the laboratorys research: NREL develops renewable

energy and energy efficiency technologies and practices,

advances related science and engineering, and transfers

knowledge and innovations to address the nations 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 whomshare the goal of reducing the cost of wind energy. The

NWTCs 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

NWTCs 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 (todays dollars) to

US$0.04US$0.06/kW-h today.

FIGURE 3 Illustrative steady-state power curves. A variable-speed

turbine attempts to maximize energy capture while operating in

region 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)

Power(kW)

Wind PowerCp= 1

15 20 25

Region 1

Region 2

Region 3

Turbine Power

HighWind

Cutout

Cp= 0.4


5/12

c = K2, (5)

where the gain K is given by

K =1

2AR3

Cpmax

3, (6)

R is the rotor radius, and is the tip-speed ratio at which

the maximum power coefficientCpmax occurs.

Next, assuming that the rotor is rigid, the angular accel-

eration is given by

=1

J(aero c), (7)

where J is the combined rotational inertia of the rotor,

gearbox, generator, and shafts and the aerodynamic torque

aero , derived from (1)(4), is given by

aero =1

2ARCq(,)v

2, (8)

where

Cq(,) =Cp(,)

(9)

is the rotor torque coefficient. Since CART has a fairly rigid

rotor, the rigid body model (7) is a valid approximation for

the rotor dynamics.Now, substituting (8) and (5) into (7) and using (9) and

(4) yields

=1

2JAR32

Cp(,)

3

Cpmax

3

. (10)

Since the rotor inertia J, the air density , the rotor swept

area A, the rotor radius R, and the squared rotor speed 2

are nonnegative, the sign of the angular acceleration

depends on the sign of the difference in (10). When the tip-

speed ratio > , it follows from (10) and the fact that

Cp Cpmax that is negative and the rotor decelerates

toward = . On the other hand, when < and

Cp >Cpmax

33, (11)

it follows that is positive. The curve

F() =Cpmax

33

is plotted as the dotted line in Figure 5, and CARTs PROP-

derived Cp curve for a fixed pitch of 1 is the solid

line. A pitch angle of 0 means that the blade chord line

is approximately parallel to the rotor plane, although theexact angle depends on the amount of twist of the blade

and the distance between the blade root and the chord line

where the pitch angle is measured. The solid line in Figure 5

is a two-dimensional slice of Figure 4. The inequality (11)

is satisfied for tip-speed ratios ranging from about 3.3 to

7.5. Thus, as long as CART has a tip-speed ratio of at least

3.3, the standard control law (5) causes the speed of a well-

characterized turbine to approach the optimal tip-speed

ratio. 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 is

ideally operated at the peak of the surface. Blade pitch angle is a

control variable, whereas tip-speed ratio is controlled indirectly using

generator torque control. A turbines Cp surface can change due to

icing, blade erosion, and residue buildup. Negative Cp corresponds

to motoring operation during which the turbine draws energy from

the utility grid.

0.5

0.4

0.3

0.2

PowerCoefficientCp

0.1

1

13

15 1

17

3 1

535

79

11

0.1

0.2

0.3

0.4

0.5

0.0

Tip-Speed Ratio Pitch (deg)

FIGURE 5 CARTs power coefficient Cp versus tip-speed ratio and

cubic function F. The intersection of the solid and dotted lines at

the tip-speed ratio = 7.5 indicates the optimal operating point in

terms of energy capture. The cubic function F is derived from the

standard 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 = 7.5 is locally

asymptotically stable.

2 4

0.4

0.3

0.2

0.1

0.0 6 8

PowerCoefficientCp

10 12 14

F()

Tip-Speed Ratio


6/12

conditions, this behavior occurs in an averaged sense

under time-varying wind conditions. We refer to the gain

K corresponding to optimum tip-speed ratio operation as

the optimal K.

When the tip-speed ratio < 3.3, the inequality (11) is

no longer satisfied, and the angular acceleration is nega-

tive. In this case, the rotor speed slows toward zero.

However, most turbines have separate control mechanisms

to ensure that a low tip-speed ratio < 3.3 does not drive

the rotor speed to zero when the wind speed is adequate

for energy production. This article is concerned only with

the torque control and, hence, does not consider these sep-

arate control mechanisms. While the critical tip-speed

ratios and control mechanisms are different for different

turbines, the dynamics presented here approximate all

variable-speed turbines using the standard control law (5).

The above discussion assumes that the turbines prop-

erties used to calculate the gain K in (6) are accurate, which

is rarely the case. Also, over time, debris buildup and

blade erosion change the Cp surface and thus Cpmax , withthe same effect as a suboptimally chosen K. The sensitivity

of energy loss to errors in and the maximum power

coefficient Cpmax is considered in [4], which concludes that

a 5% error in the optimal tip-speed ratio can cause a sig-

nificant energy loss of 13% in region 2. If the United

States meets the American Wind Energy Associations goal

of 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 development

and investigation of an adaptive control approach that can

improve energy capture.

ADAPTIVE CONTROLFor region 2 operation, we now consider the adaptive con-

troller [2] given by

c =

0, < 0,

M2, 0,(12)

where the adaptive gain M replaces A, R, Cpmax , and in

(6). The air density is kept separate because air density is

time varying and measurable.

The control law (12) is defined separately for positive

and negative regions of the rotor speed because it is

undesirable to apply torque control when the turbine is

spinning in reverse. Reverse operation can cause excessive

wear 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) = 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 controllers discrete time

step. The fractional average power Pfavg, given by

Pfavg(k) =

1n

ni=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 wind

power. Pfavg is computed at each adaptive control time step

k, 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 =1

2Av3 (cos )3 , (17)

where is the yaw error, that is, the error between the

wind direction and the yaw position of the turbine. Pcap is

the captured power, given by

Pcap = c +J, (18)

which is also computed at 100 Hz. The yaw error factor

(cos )3 in (17) shows that yaw errors reduce the power

available to the turbine. The term c in the captured power

Pcap is the generator power while J 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 periods

of operation in region 2. Testing on CART indicates that

the 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 period

is required in part because of the difficulty of obtaining ahigh 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 turbines

nacelle or on a separate meteorological tower [16]. Another

reason 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 of

the adaptive gain M to its optimal value M, the gain that

results in maximum energy capture. As M moves toward

the peak of the curve in Figure 6, a given adaptation step

M results in a smaller |Pfavg|because (dPfavg)/(d M) 0

as M 0. Thus, |M| decreases as the optimal gain isapproached. The exponent 1/2 is chosen based on simula-

tion, and selection of M > 0 is discussed below.

In (16), Pcap is used rather than the rotor aerodynamic

power P given by (3) because the sensor requirements for

Pcap are more consistent with the instrumentation normal-

ly available on industrial turbines. The two definitions of

turbine power are closely related, differing only by the

mechanical losses in the turbines gearbox; these losses

make Pcap < P by a small amount.



7/12

Figure 6 portrays the output of constant-wind-speed

simulations using the rigid body model (7) and the control

torque (12). The model and controller are simulated with

26 different values of the gain M, where each simulation

lasts 200 s with constant M for the duration of the simula-

tion. The turbines power output for each of the 26 gain

values is averaged over each 200-s simulation to produce

the solid Pfavg curve in Figure 6. In Figure 6, M = 174.5 is

the optimal gain based on the standard torque control

coefficient 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 M inM is given by

M = M M.

The adaptive controller attempts to have the turbine power

track the wind power, assuming that the maximum power

coefficient Cpmax and the optimal tip-speed ratio are

unknown. In contrast, adaptive controllers such as those in

[10][11] focus on different uncertainties and assume some

knowledge of theCp surface, particularly andCpmax . In addi-

tion, the averaging period used in this article is long compared

to the time periods used by the adaptive controller in [9].

Figure 7 shows data collected in the first year of adaptive

CART operation. Only region 2 data is plotted, and the

change in the adaptation period length from 10 min to 180

min is apparent. The adaptation behavior with the longer

adaptation period is significantly better than the behavior

with the shorter adaptation period. The three discontinuities

in the data reflect occasions where the adaptive controllerwas restarted due to a change in the method for calculating

Pfavg and problems with sensors on CART. The last dozen

adaptations oscillate about a value that is just less than 50%

of the predicted optimal value M+ = 174.5 computed from

the PROP model of CART. In comparison, the CART study

[17], obtained with the turbine running in constant speed

mode, gives a true optimal gain M around 47% of the pre-

dicted optimal valueM+. The experimental results shown in

Figure 7 indicate that modeling tools such as PROP [13] can

lead to large errors in predicting the optimal value of the

gain M. We now proceed with the stability analysis.

STABILITYWe now consider the stability of the closed-loop system

with the adaptive torque gain control law. Some of the

results in this section appear in [18]. Although control of

CARTs torque is a discrete-time problem, we simplify the

stability 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 is

much smaller than the tip-speed ratios time constant,

which depends on wind speed [19] and is about 48 s for

CART operating in region 2 wind speeds of 612 m/s.

Also, we assume that the adaptive control gain M> 0 is

constant in the torque control law (12) analysis; this

assumption 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 rotor

speed (hours versus seconds). Thus, each result that is

based on a constant M assumption holds for the duration

of each 3-h adaptation period. Furthermore, M is con-

strained to be positive since the control torque (12) cannot

be negative. In all of these proofs, the air density is

assumed to be a positive constant. In reality, changes in air

density are small, typically not much greater than 5%. A


FIGURE 7 Adaptive gain Mnormalized by the predicted optimal gain

M+ during region 2 operation of CART. Discontinuities indicate

restarts of the gain adaptation law due to changes in the law and

turbine sensor errors. In the second half of the data, Moscillates

around the value 0.47 M+, which is approximately equal to the true

optimal torque gain M.

1.5

1

0.5

NormalizedM(M/M+)

00 20 40

Time (h)

60 80

FIGURE 6 Pfavg versus M for the CART model. Pfavg is the ratio of

the mean captured power to the mean wind power, while M is the

error between the torque control gain Mand its optimal value M.

The shape of this curve is based on the shape of CARTs Cp

curve. In the adaptive controller, the gain adaptation law converges

in 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

0.42

Pfavg

Gain Error M= M* M

FractionalAve

ragePowerPfavg


8/12

simplified block diagram for these continuous-time sys-

tems is given in Figure 8(a), where the linear plant is given

by (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 = 0 in the absence of wind and in

constant 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 be

as small as possible. However, in the analysis of asymptot-

ic stability of the equilibrium point = 0, we revise (7) so

that the angular acceleration includes a damping term

b, where the damping coefficient b > 0, which yields

= 1J(aero c b). (19)

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

=

12JARCqv2 bJ , < 0,

12JARCqv

2 J M

2 bJ , 0.(20)

Theorem 1

Suppose that the wind speed v = 0 and M > 0 are con-

stant. Then the equilibrium = 0 of the closed-loop sys-

tem (20) is asymptotically stable.

Proof

For the initial condition (0) = 0 , the solution to (20)

when v = 0 is

( t) =

0e b

J t, < 0,0b

(b+M0)ebJtM0

, 0.

Hence, 0 as t .

We also note that when the damping coefficient b = 0

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

=

0, < 0,

JM

2, 0,

which has the solution

( t) =

0, < 0,JMt+ J

0

, 0.

In this case, 0 holds only when the rotor is spinning in the

positive 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 the

rotor speed to an equilibrium value under an idealized

constant, positive wind speed. This analysis is similar to

the one describing Figure 5 and given in (5)(11). Once

again, the plant is given by (19) and the nonlinear con-

troller is given by (12). The adaptive controller (12) does

not assume knowledge of the aerodynamic parameters

Cpmax and . Setting the 0 portion of (20) equal to zero

and solving for Cp in terms of using (4) and (9) yields

Cp =M3v + 2bR

12 AR

3v G(,M, b, v). (21)

The equilibrium points = 0 of turbine operation are thus

given by the intersection of with the turbines Cp

curve. Figure 9 shows CARTsCp curve and two illus-

trative G(,M, b, v) curves plotted using representative

values of , v, and b.

In Figure 9, the cubic functions G(,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 aero and

rotor speed and (b) the gain adaptation law. (a) Stability of the

continuous-time control loop is analyzed by Theorems 13, while

Theorem 4 considers (b) the discrete-time adaptive loop.

+

NonlinearController

LinearPlant

c

aero

+

NonlinearController

PfavgNonlinearPlant

M

M*

(a) (b)

FIGURE 9 CARTs power coefficient Cp curve and cubic functions

for two values of the adaptive gain M. When M is not equal to its

optimum value M, the intersection of the Cp and G(, M) curves

does not occur at the peak of the Cp curve, which leads to subopti-

mal energy capture. Similar to Figure 5, the intersection of each

cubic curve with the Cp curve is an equilibrium point of the system

for the indicated adaptive gain M.

2 4

0.4

0.3

0.5

0.2

0.1

06 8

Tip-Speed Ratio

PowerCoefficientCp

10 12 14

G( ,M)M= 1.3M*

G( ,M)M= 0.7M*

Cart CpVersus


9/12

gain M = M ; thus, the equilibrium point of the system is

suboptimal in terms of energy capture. Let 2 be the highest

value of for which the curve G(,M) intersects Cp().

Mathematically, 2 is the tip-speed ratio for which

G(,M) > Cp() for all > 2. Let 1 denote the next highest

intersection point, that is, the value of for which

0 < 1 < 2 and G(,M) < Cp() for all 1 < < 2 and

G(,M) > Cp() for all < 1 within a neighborhood of 1.

For the dashed curve M = 0.7M in Figure 9, these values

correspond to 1 = 3.1 and 2 = 8.4. The following result

shows that, for a constant wind input, the tip-speed ratio con-

verges to 2 as long as the initial value of is greater than 1.

Theorem 2

Suppose that the wind speed v and the adaptive gain M are

positive constants and 1 > 0. Then the equilibrium point

= 2 of the closed-loop system consisting of the plant

=R

Jvaero c

bv

R (22)

and the nonlinear controller (12) is locally asymptotically

stable with domain of attraction (1,).

Proof

First note that > 0 for all 0 < 1 < since = v/R from

(4). Define = 2 and the Lyapunov candidate

V= (1/2)2 . For > 0,

V= ( 2)

1

2JAR2Cqv

1

JRM2v

b

J

= ( 2)h(Cq, v, ).

(23)

Substituting Cp/ for Cq in (23) and applying (21) yields

h(Cq, v,) > 0 for all such that 1 < < 2 , that is,

G(,M) < Cp(). Moreover, > 2 givesG(,M) > Cp()by

definition of 2, and therefore h(Cq, v,) < 0by definition (21)

of G. Thus, V< 0 for all (1,) except = 2, for which

V= 0. Hence, the equilibrium point = 2 of (22) is locally

asymptotically stable. Finally, it is easy to show that the

domain of attraction is (1,). Note that V is bounded away

from zero on every connected, compact subinterval of (1,)

that does not contain 2. Thus, the time required for to reach

the edge of the subinterval closest to 2 is finite. Now, moves

monotonically toward 2. If does not converge to 2, then thetime it takes to reach the edge closest to 2 of a subinterval

not containing 2 must be infinite, which contradicts the earlier

result. Thus, the domain of attraction is (1,).

The convergence of the tip-speed ratio to 2 is equiva-

lent to the convergence of the rotor speed to 2v/R for a

specific wind speed v. Furthermore, when M = M , the

curves G(,M) and Cp() intersect at (,Cpmax ) as shown

for the standard torque control in Figure 5; therefore, optimal

energy capture is achieved for the constant wind input case.

We acknowledge that zero and constant wind speeds

never occur in the field. However, wind speeds near zero do

occur during turbine operation, causing a shutdown when

the wind speed is close to zero for a sufficiently long time.

These results are useful for developing an understanding of

the torque control law, although the cases are idealized.

Input-Output StabilityNext, we show that a bounded input (squared wind speed

v2) to the system produces a bounded output (rotor speed

). All wind turbines have a maximum safe operating

speed, 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 turbine

control system can be achieved by examining whether the

torque 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

vL2[0,T] =

T0v(t)2dt.

Theorem 3

Suppose the rotor torque coefficient Cq 1, the adaptive

gain M > 0 is constant, and consider the closed-loop tur-

bine system (12), (19) with input given by the squared wind

speed v2 and output given by the rotor speed . Then, for

all finite T> 0, the system (12), (19) is L2 stable on [0,T].

Proof

Consider the kinetic energy EK = (1/2)J2 of the rotor

and define

V=1

ARJ2, (24)

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

V=

Cqv2

b12 AR

2, < 0,

Cqv2 b1

2AR

2 M12AR

3, 0.(25)

Le t = b/((1/2)AR) . Then, since Cq 1 an d

(M/((1/2)AR))3 0 for 0, it follows that

V v2 2. (26)

Thus, Lemma 6.5 in [20] implies that the wind turbine sys-

tem, from the squared wind speed v2 to the rotor speed ,

is finite-gain L2 stable over [0,T].

The restriction that T be finite is necessary due to the

nature of the wind speed v(t). Since wind speed v(t) > 0

can hold at all times, it is possible that v(t) / L2[0,].

Thus, Tmust be finite to guarantee that proof of L2 stability

in [0,T] makes sense.



10/12

The condition Cq 1 is usually satisfied for modern

turbines in normal region 2 operation. The Betz limit [14],

which is the theoretical maximum power coefficient Cp for

any real turbine, has a value of Cp = 16/27. Since

Cq = Cp/ [see (9)], it follows that Cq 1 for 16/27.

When 16/27, it follows from the definition of tip-speed

ratio in (4) that = v/R (16/27)v/R.

For finite > 0 and [0,T],L , that is, bounded

input, bounded output stability of with respect to the

input v is given by = v/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 turbine

is less than that contained in the wind over each adapta-

tion period.

Convergence of the Gain Adaptation AlgorithmThe final stability analysis examines convergence of the

adaptive gain M

M

using the gain adaptation law(13)(15). Figure 8(b) shows a simplified block diagram

for this system, where the nonlinear plant is the fractional

average power Pfavg versus torque gain error M relation-

ship shown in Figure 6 and the nonlinear controller is

given by (13)(15).

We make two assumptions before studying the stability

properties of the gain adaptation law.

Assumption 1

The optimum torque control gain M is constant.

The turbines 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 turbines physical changes are typical-

ly noticeable only over months or years, whereas the gain

adaptation law has an adaptation period of less than a day.

Assumption 2

The Pfavg versus M curve has a maximum at M = 0, is con-

tinuously differentiable, and is strictly monotonically

increasing on M < 0 and strictly monotonically decreasing

on M > 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 > 2because the first

two steps are more influenced by the initial guesses than bythe turbines aerodynamic properties.

We begin the convergence analysis by considering how

the adaptive gain can diverge, that is, | M| as k .

One possibility is | Mk| > | Mk1| with either sgn( Mk) = 1

or sgn( Mk) = 1 for all k > 2. However, it is easy to show

that this scenario cannot occur with the gain adaptation

law (13)(15). Indeed, the adaptive torque gain error M

cannot take two consecutive steps in the wrong (incorrect)

direction for all k > 2, as shown by the following result.

Theorem 4

Let k > 2. Under Assumptions 1 and 2 and the gain adap-

tation law (13)(15), | Mk+1| > | Mk| > | Mk1| never occurs

when sgn( Mk+1) = sgn( Mk) = sgn( Mk1).

Proof

Suppose Mk+1 > Mk > Mk1 and sgn( Mk+1) = sgn( Mk) =

sgn( Mk1) = 1 for some k > 2. Note that Mk > Mk1 gives

Mk Mk1 = Mk > 0, (27)

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

Mk+1 Mk = 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 which

M = Mk ; thus, Pfavg k+1 is calculated from data collected

while the torque gain error was Mk . Since Mk > Mk1 ,

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 Mk+1 > Mk > Mk1 and sgn

( Mk+1) = sgn( Mk) = sgn( Mk1) = 1 to be true. A similar

argument can be used for negative values of M.

Since the sign of the adaptation step M cannot be

incorrect for two consecutive steps, the gain M, which

affects the magnitude of M, is the critical factor in deter-

mining whether the adaptive gain diverges. Figure 10

shows an example in which the gain M is large enoughto cause the adaptive gain M to diverge. In this example,

| Mk+1| > | Mk1| for all k > 2, although both | Mk+1| > | Mk|

and | Mk+1| < | Mk| occur when k > 2.


FIGURE 10 Adaptive gain steps in an unstable case. The numbers

19 indicate the discrete-time steps. In this case, the gain M in

the gain adaptation algorithm (13)(15) is too large, and thus the

gain adaptation law diverges.

20 1015 5 0 105

9

5

1

7

36 2 48

20150.25

0.30

0.35

0.40

0.45

0.50

Gain Error M= M* M

FractionalA

veragePowerPfavg


11/12

Since M diverges if |Mk| > | Mk1| for all k > 2, we

consider

|Mk| = | Mk1|, Mk1 = 0 (30)

to be the critical case, or the marginal stability case. Defineykby

yk a M2k1 + P0, (31)

whereyk is a curve satisfying Assumptions 12 whose form

is better known than Pfavgk . In (31), a < 0 and P0 is a real

number; (30) can be solved for the critical gain M. For

consistency with the discrete-time indices in the equation(16) for Pfavgk , yk is a function of

Mk1 rather than of Mk.

In the critical gain scenario of this example, the system

alternates among the three points plotted in Figure 11. If

Mk = Mk1 , then the error Mk = 0 by (13). Substituting ykfor Pfavgk in (15) and considering (14), the gain M is such

that Mk+1 = Mk, resulting in Mk+1 = Mk1. Following

the equations through one more step shows that Mk+2 = 0,

and the adaptive gain alternates among these three points.

Thus, an upper bound on the gain M for stability can be

found by equating

Mk = Mk1 = Mk+1

and solving for M in terms of a with Mk = 0, which yields

M =

1|a| . (32)

Thus, if 0 < M < |a|1/2 , then the gain adaptation law

(13)(15) does not cause divergence of the adaptive

torque control gain error M on the curve (31). In fact,

since M =|a|

1/2

is the marginal stabil i ty case,0 < M < |a|1/2 yields M 0. Since this bound on M

depends on the magnitude |a|, every gain M chosen

for a given value of a in (31) also guarantees conver-

gence of the adaptive gain M on a curve with a smaller

value of a.

We can state a similar result for a curve that is not even

[as in (31)], that is, one for which Pfavg( M) = Pfavg( M)

does not hold. If the gain M is chosen to guarantee con-

vergence based on the slope of the steeper side of the

curve, then M guarantees convergence over the entire

curve. Thus, for an arbitrary Pfavg versus M curve, there

exists M > 0 that guarantees convergence of the adap-

tive gainM, and this gain M depends on the steepness ofthe Pfavg versus M curve.

Since there are no turbines for which the Pfavg versus M

curve is well known, an approximation of the curve is nec-

essary to control each turbine. The more conservative the

choice of M, the more likely it is that M converges to M

since the gain adaptation law (13)(15) is more robust to

errors in the approximated Pfavg versus M curve for small-

er M. However, a smaller M also results in smaller

step sizes and thus might cause the convergence to occur

more slowly.

An example of the choice of M is provided in Figure

12. The coefficients a and P0 of (31) are chosen so that (31)

fits snugly inside the Pfavg curve, being coincident atM= 0 and satisfying y < Pfavg for M = 0. In this case,

a= 0.00001 m10 . Thus, the maximum allowable gain

M for stability is 316 m5. The gain used in testing on

CART before this stability analysis was performed was

M = 100 m5, which was determined empirically from

simulations and early hardware testing. Although actual

turbine results indicate stable performance of the adaptive

control law, this stability analysis provides further reassur-

ance and guidelines in choosing M.


FIGURE 12 CART Pfavg versus Mcurve and symmetric inset curve.

The curve labeled Pfavg is identical to the curve shown in Figure 6,

while the quadratic curve labeled yis added to illustrate the method

for selecting the adaptive gain M. When the quadratic curve is

chosen such that y(M) Pfavg(M) and y(M) = Pfavg(M) if and only

if M= 0, the upper limit on M for stability of the gain adaptation

law 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

Gain Error M= M* M

Fra

ctionalAveragePowerPfavg

FIGURE 11 Finding the critical gain M. Marginal stability of the

gain adaptation law, defined as oscillation among three points on

the ycurve (31), occurs when the step size Mk has the same

magnitude as the error Mk1 for a symmetric curve.

50 0 50

0.40

0.41

0.42

0.43

Gain Error M= M* M

(Mk,yk+1)

MkMk+1

FractionalAveragePowerPfavg

(Mk+1,yk+2)

(Mk1,yk)


12/12

CONCLUSIONS

This article considers an adaptive control scheme previously

developed for region 2 control of a variable-speed wind tur-

bine. In this article, we addressed the question of theoretical

stability of the torque controller, showing that the rotor speed

is asymptotically stable under the torque control law (12) in the

constant wind speed input case and L2 stable with respect to

time-varying wind input. Further, we derived a method for

selectingM in the gain adaptation law (13)(15) to guarantee

convergence of the adaptive gainM to its optimal valueM.

ACKNOWLEDGMENTS

This work was supported in part by the U.S. Department of

Energy through the National Renewable Energy Laboratory

under contract DE-AC36-99G010337, the University of Col-

orado at Boulder, and the American Society for Engineering

Education. We would also like to acknowledge Prof. Dale

Lawrence and Dr. Vishwesh Kulkarni for their suggestions

on improving our article.

AUTHOR INFORMATION

Kathryn E. Johnson ([email protected]) received the B.S.

degree in electrical engineering from Clarkson University in

2000 and the M.S. and Ph.D. degrees in electrical engineering

from the University of Colorado in 2002 and 2004, respective-

ly. In 2005, she completed a postdoctoral research assignment

studying adaptive control of variable-speed wind turbines at

the National Renewable Energy Laboratorys National Wind

Technology Center. That fall, she was appointed Clare Boothe

Luce Assistant Professor at the Colorado School of Mines in

the Division of Engineering. Her research interests are in con-

trol systems and control applications. She can be contacted at

Colorado 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. degrees

in electrical engineering from Stanford University. She is

currently 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 area

of control systems. Her awards include the Best Commer-

cial Potential Award at the 2004 International Symposium

on Haptic Interfaces for Virtual Environments and Teleop-

erator Systems as well as the Best Paper Award at the 2005

World Haptics Conference. She was the program chair for

the 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 in

linear and nonlinear systems, especially in the control of

distributed and large-scale systems, aerospace structure

control, and variable-speed, horizontal-axis wind turbine

control for electric power generation. He is a Fellow of

the 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 in

electrical engineering from the University of Colorado in

1993 and 1995, respectively. He has been employed at

NREL since 1993, working in the fields of aerodynamics

testing, power electronics, electric machines, energy stor-

age, and controls. Most recently, he has been responsible

for a large controls field testing project and its associated

test machine, the Controls Advanced Research Turbine.

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