Methods for Increasing 

Region 2 Power Capture on 

a Variable Speed HAWT

Preprint 

November 2003 • NREL/CP-500-34979 

K.E. Johnson and L.J. Fingersh 

National Renewable Energy Laboratory

M.J. Balas and L.Y. Pao 

University of Colorado 

To be presented at the 23rd ASME Wind Energy 

Symposium 

Reno, Nevada 

January 5–8, 2004 

National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, Colorado 80401-3393 

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1

METHODS FOR INCREASING REGION 2 POWER CAPTURE ON A VARIABLE SPEED HAWT 

K. E. Johnson and L. J. Fingersh 

National Renewable Energy Laboratory, MS 3811, 1617 Cole Boulevard, Golden, Colorado 80401-3393 

kathryn_johnson@nrel.gov, lee_fingersh@nrel.gov

M. J. Balas, Aerospace Engineering, and L. Y. Pao, Electrical Engineering 

University of Colorado, Boulder, Colorado 80309 

mark.balas@colorado.edu, pao@colorado.edu

ABSTRACT

The standard region 2 control scheme for a variable 

speed wind turbine, τc = kω2

, has several shortcomings 

that can result in significant power loss. The first of 

these is that there is no accurate way to determine the 

gain k; modeling programs are not accurate enough to 

represent all of the complex aerodynamics, and these 

aerodynamics change over time. Furthermore, it is not 

certain whether the value of k used in the standard 

control even provides for the maximum energy capture 

under real-world turbulent conditions. New control 

ideas are introduced to address these issues. First, it is 

shown in simulation that using smaller values of k than 

the standard can result in increased power capture. 

Second, an optimally tracking rotor control scheme 

improves upon the standard scheme by assisting the 

rotor speed in tracking wind speed fluctuations more 

rapidly. Finally, an adpative control scheme is 

proposed that allows for maximum power capture 

despite parameter uncertainty. 

NOMENCLATURE

ω rotor angular speed (rad/s) 

v wind speed (m/s) 

ρ air density (kg/m3) 

A rotor swept area (m2

) 

R rotor radius (m) 

J rotor inertia (kgm2

) 

Cq rotor torque coefficient 

Cp rotor power coefficient 

Cpmax maximum rotor power coefficient 

λ* tip speed ratio (TSR) corresponding to Cpmax

P rotor power (kW) 

Pcap captured power (kW) 

Pwind power available in the wind (kW) 

Pwy power available in the wind, with approximate 

yaw error factor included (kW) 

Pfrac turbine captured power divided by wind power 

(kW) 

Pfavg average captured power divided by average 

wind power over a given time period (kW)

τaero aerodynamic torque (Nm) 

τc generator (control) torque (Nm) 

β blade pitch (degrees) 

ψ yaw error (degrees) 

M adaptive gain (m5

/rad3) 

n number of steps in adaptation period 

K∆M positive gain on adaptation law (rad3

/m5) 

INTRODUCTION

 Variable speed wind turbines have three main 

regions of operation. The first, region 1, includes the 

time when the turbine is starting up. Region 2 is an 

operational mode during which it is desirable that the 

turbine capture as much power as possible from the 

wind. Region 3 is encountered when the wind speeds 

are high enough that the turbine must limit the fraction 

of the wind power captured so that safe electrical and 

mechanical loads are not exceeded. This paper focuses 

on region 2 operation, which accounts for more than 

50% of yearly energy capture for a typical modern 

turbine. 

 The standard region 2 control scheme for a variable 

speed wind turbine is given in Eq. (1)-(2). 2 τ c = kω (1) 

3

*

3 max

2

1

λ ρ

Cp k = AR (2) 

where λ* is the optimal tip speed ratio, and tip speed 

ratio is defined as 

v

ωR λ = . (3) 

The control law in Eq. (1) is intended to keep the 

turbine operating at the peak of its Cp-TSR-Pitch 

surface. Figure 1 is an example of this surface, created 

using the modeling software PROP [1] for the Controls 

Advanced Research Turbine (CART). PROP conducts 

its simulations using blade-element-momentum theory 

(see, e.g., [2]). It should be noted that Cp can be 

negative, as shown in Figure 1; negative Cp corresponds 

to motoring operation when the turbine behaves like a 

fan, putting energy back into the wind. The CART is a 

600 kW, two-bladed, upwind turbine at the National 

Renewable Energy Laboratory’s National Wind 

Technology Center (NWTC) and is the turbine test bed 

used in this research and for testing other advanced 
2

control algorithms. It is equipped with many more 

sensors than would be found on a commercial turbine, 

including a meteorological (met) tower about two rotor 

diameters away. This met tower is equipped with many 

weather-related sensors, including anemometers at hub 

height and at the heights of the blade tips when the 

rotor is oriented vertically. Figure 2 shows a photo of 

the CART, with the met tower on the left. 

-5

-2

1

4

7

10

13

16

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

-0.5000

-0.4000

-0.3000

-0.2000

-0.1000

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

Cp

TSR Pitch

Figure 1: Cp vs. TSR and Pitch for Controls 

Advanced Research Turbine (CART) 

Figure 2: CART at the NWTC 

The power coefficient is defined as the ratio of the 

turbine power to the power available in the wind: 

wind

p P

P C = , (4) 

where

3

2

1 P Av wind = ρ . (5)

From Eq. (4), it can be seen that rotor power is directly 

proportional to Cp; thus, operation at Cpmax is clearly 

desirable. The rotor power P can be defined in 

different ways, which can result in slightly different 

interpretations of Cp; however, the most common 

definition of P is 

P = τ aeroω (6) 

 There are two significant problems with this 

standard control. The first is that there is no accurate 

way to determine k, especially since blade 

aerodynamics can change significantly over time. 

Second, even when it is assumed that k can be 

accurately determined via simulation or experiments, 

wind speed fluctuations force the turbine to operate off 

the peak of its Cp-λ curve much of the time, resulting in 

less energy capture. This paper addresses these issues 

in the following three ways: 

 First, the authors perform simulations to address 

the question of whether the k used in the standard 

control actually results in optimum power capture for 

all wind conditions. These simulations indicate that, for 

a steady wind, this k does indeed result in the optimum 

power capture. However, each increase in wind 

turbulence results in a corresponding decrease in the 

optimal operating point of the standard control law gain 

k, especially for turbines with large rotational inertias. 

 Next, the optimally tracking rotor (OTR) 

controller, which was originally proposed by Fingersh 

and Carlin [3], is examined in simulation. This control 

scheme reduces the amount of time the turbine requires 

to regain its optimal tip-speed-ratio by using generator 

torque to assist in acceleration or deceleration in 

response to wind gusts. Simulations are used to verify 

the effectiveness of OTR control; real data has been 

collected, as well, and is discussed in [4]. Optimally 

tracking rotor control assumes that the k used in Eq. (1) 

is accurate and based on the peak of the Cp-λ curve. 

 Finally, an adaptive controller is proposed that 

reduces the negative effects of both the uncertainty 

regarding k and the change in optimal operating point 

due to turbulence. The adaptive controller will seek the 

gain that maximizes power capture, on average, 

regardless of whether this gain corresponds to the 

maximum of the Cp-λ curve for the turbine. Both 

simulations and real data are used to verify the 

effectiveness of the adaptive controller. 

 Before the new control ideas are presented, 

however, an examination of the shortcomings of the 

existing torque control law is in order. 
3

STANDARD CONTROL

 An understanding of how the standard torque 

control law given by Eq. (1) causes operation to occur, 

on average, at the optimum tip speed ratio (assuming 

the Cp-λ curve is known exactly) will be useful in 

understanding the results in this paper. To study the 

dynamics of the system, consider a simple rigid body 

model of a wind turbine. The equation relating net 

torque and angular acceleration is 

( ) aero c Jω = τ −τ

1 & (7) 

where 

( ) 2 , 2

1 ARC v τ aero = ρ q λ β (8) 

and 

( ) ( )

λ

λ β λ β , , p

q

C

C = . (9)

While very simple, this rigid body model is a fair 

approximation for the CART, which has a very rigid 

rotor. Now, substitute Eq. (8) and Eq. (1) into Eq. (7): 

( ) 













 = − 2

3

*

2 3 max

2

1 , 21 1

ω

λ

ω ρ λ β ρ p

q

C

ARC v AR

J & . (10) 

Then, simplify Eq. (10) further with Eq. (9) and Eq. (3): 

( ) 













 = − 3

*

max

3

3 2 ,

2

1

λ λ

λ β

ω ρ ω

Cp Cp AR

J & (11) 

Because J, ρ, A, R, and ω2

 are all nonnegative, the sign 

of ω& depends on the sign of the difference within the 

parentheses. By definition, Cp ≤ Cpmax. Thus, when λ > 

λ*, ω& will be negative, and the rotor will decelerate 

towards λ = λ* (since wind is constant in this example). 

On the other hand, if λ < λ*, ω& will be positive when 

3

3

*

max λ

λ

p

p

C

C ≥ . (12) 

The curve 

3 3

3

*

max λ λ

λ

F

C

C p

p = = (13) 

is plotted as the dotted line in Figure 3, and the CART’s 

Cp-λ curve for a fixed pitch of -1° is the solid line. 

Note that the solid line is simply a two-dimensional 

slice of Figure 1. Thus, the inequality in Eq. (12) is 

satisfied for tip speed ratios ranging from about 3.3 to 

7.5. As long as the CART has a tip speed ratio of 3.3 or 

greater, its standard control law (Eq. (1)) will cause its 

speed to approach the optimum tip speed ratio. 

When λ < 3.3, the requirement given by Eq. (12) is 

no longer satisfied, and ω& < 0. In this case, the rotor 

speed would actually slow towards zero, and, by Eq. 

(11), the turbine would remain stopped once ω = 0. 

However, most turbines have other control mechanisms 

in place to prevent this from occurring (otherwise, the 

turbines would never start in the first place). The 

CART, for example, uses blade pitch to change the 

aerodynamic torque at low tip speed ratios. While the 

specific tip speed ratio numbers and control 

mechanisms are different for different turbines, the 

general idea of the dynamics presented here hold for all 

variable speed turbines. 

2 4 6 8 10 12 14 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Tip Speed Ratio

Cp

CART Cp vs. TSR

Cp = F*TSR3

Figure 3: CART Cp vs. TSR and Cubic TSR 

Unfortunately, the modeling tools used to 

determine the Cp-TSR-Pitch surface for most wind 

turbines are not perfectly accurate, and fixed controllers 

designed based on these modeling tools are generally 

still sub-optimal. Figure 4 presents a graphical 

depiction of how sub-optimal performance results in 

energy loss. 

2 4 6 8 10 12 14 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Actual 

Operating

Point 

TSR

Cp

Best 

Possible 

Operating

Point 

Assumed Operating Point 

Assumed 

Cp-TSR 

Curve 

Actual Cp-TSR Curve 

Figure 4: Sub-Optimal Operation Due to Inaccurate 

Cp-TSR Curve 

Assume, for example, that the dotted curve in 

Figure 4 is the actual Cp surface and the solid curve is 

the best guess made by the control engineer based on 

the modeling tools available. In this case, the control 

torque τc will be designed to drive steady-state 

operation to the asterisk at the peak of the solid curve. 

Because τc is directly proportional to Cpmax, as given in 

Eq. (1)-(2), the net result is that the control torque will 
4

be higher than warranted by the dotted (real) curve. 

Then, considering Eq. (7) and for a given aerodynamic 

torque τaero, a higher control torque τc will result in a 

smaller acceleration. All else being equal, the end 

result of the discrepancy between the two curves will be 

that the turbine will operate at a slightly lower speed 

(and thus tip-speed-ratio) than if the solid curve were 

correct. For this example, steady-state operation will 

occur at the black circle on the dotted curve, which is 

lower than the real system’s best possible operating 

point represented by the asterisk on the dotted curve. It 

should be emphasized that true steady state operation, 

as described above, occurs only for constant wind, 

which is never the case. 

 Even if it could be assumed that the initially chosen 

gain k was optimal, wind turbine blades will change 

over time due to issues like bug build-up and blade 

erosion, causing the same net result as a sub-optimally 

chosen initial k. A study by Fingersh and Carlin [3] 

shows how sensitive energy loss is to errors in the 

optimal tip-speed-ratio λ* and the maximum power 

coefficient Cpmax. The study showed that a very 

common 5% error in the optimal tip-speed-ratio λ*

alone can cause an energy loss of around 1% - 3% in 

region 2, which is a significant loss in this industry. 

 The next three sections describe three new methods 

that can be used to increase energy capture. The first is 

as simple as reducing the gain k given by Eq. (2) by a 

small amount, depending on the typical wind turbulence 

at a given wind turbine. 

EFFECTS OF TURBULENCE ON 

OPTIMAL OPERATING POINT

 The question has arisen as to whether the 

maximum power capture is actually achieved when the 

standard control law given by Eq. (1)–(2) is used. In 

fact, the following simulation results show that using a 

gain k that is 5% - 20% smaller than that given by Eq. 

(2) actually results in improved power capture, 

depending on the turbulence in the wind. The reason 

for this result is the fact that a turbine with high rotor 

inertia spends much of its region 2 operational time 

trying to regain the optimum tip speed ratio lost due to 

wind gusts and lulls; it simply cannot track wind speed 

perfectly. Because the power in the wind is 

proportional to the cube of the wind speed, it is more 

important to track wind gusts than lulls. This concept is 

explained further throughout this section. 

As is shown in Figure 5, a turbine with a very low 

rotor inertia (J = 1000 kg•m2

) has a much different 

optimal operating point than one like the CART, with J

= 388,500 kg•m2

. In Figure 5, “Nominal Power 

Capture” is the maximum power captured by the 

turbine with very low rotor inertia at k/M*

 = 1.0, all 

other physical parameters used in simulation being the 

same. In this plot, three different 100 Hz sampled wind 

data sets were used as inputs in simulations. The 

curves were each created by simulating the CART’s 

behavior 24 times, each simulation lasting one hour and 

having a different (fixed) value of the torque control 

gain, k. The nominal value, M*

, is given by 

3

*

* 3 max

2

1

λ

Cp M = AR , (14) 

which is simply Eq. (2) with the time varying air 

density ρ set equal to one for simplicity. For the low 

rotor inertia (J = 1000) case, the power capture is 

virtually identical no matter which wind is used as 

input—the lines are indistinguishable on the scale of 

Figure 5. However, there are noticeable differences in 

the three plots showing the normal inertia (J = 388,500) 

case. In these simulations, optimum power capture 

(indicated by the stars) is achieved at anywhere from 

89% to 93% of the nominal optimal control torque gain. 

According to these three curves, it is clear that captured 

power could be increased by 0.5% simply by reducing 

the gain k by an average of 10% below the nominal 

value. While 0.5% may not sound like an impressive 

number, the control modification required is so simple, 

costing nothing, that there is no question that this 

change is worthwhile. 

0.8 0.85 0.9 0.95 1 1.05 96.5

97

97.5

98

98.5

99

99.5

100

100.5

k/M*

Percent of Nominal Power Capture

J=1000 

J=388500, Wind Data Set 1

J=388500, Wind Data Set 2

J=388500, Wind Data Set 3

Figure 5: Average Power Capture for Normal and 

Low Inertia 

Why does the optimum torque gain differ for 

different wind inputs? Since the three curves from the 

low inertia simulations lie almost on top of each other 

and only the high inertia curves differ by wind input, it 

seems likely that the rotor inertia is the cause of the 

change in optimum gain from one wind speed to 

another. A higher inertia will result in slower tracking 

of the optimal tip speed ratio λ* in response to wind 

gusts and lulls. When characterizing the wind, 

turbulence intensity is one measure used to describe the 

amount of “gustiness” in the wind. It is usually defined 

as the standard deviation of the wind speed divided by 
5

the mean wind speed over 10 minutes. The following 

simulations, presented in Figure 6, show that the 

optimum operating point is closely related to a slightly 

modified measure of the wind’s turbulence intensity. 

This modified one-hour turbulence intensity uses 60 

minutes of data rather than 10 in the mean and standard 

deviation calculations. 

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Turbulence Intensity

k/M*

Wind Data Set 1

Wind Data Set 2

Figure 6: Optimum Torque Gain vs. Turbulence 

Intensity 

Figure 6 is created in a similar manner to Figure 5. 

Two of the wind data sets used to create Figure 5 were 

slightly modified to have the same mean (7.533 m/s), 

and then further scaled to create 15 different standard 

deviations per wind input, resulting in 15 different onehour turbulence intensities. The simulations were then 

run for each of these 15 one-hour turbulence intensities 

per wind data set, holding the mean wind speed 

constant throughout the experiment. Each simulation 

was one hour in length and used the normal inertia and 

other CART parameters. As shown by the “Wind Data 

Set 2” curve, for high one-hour turbulence intensities, 

the torque gain resulting in the greatest power capture 

can be nearly 20% lower than the nominal value given 

by Eq. (14). 

Similar simulations performed while varying the 

mean wind speed and holding the one-hour turbulence 

intensity constant showed that the mean wind speed 

does not significantly affect the optimum torque control 

gain. Thus, it can be concluded that it is the standard 

deviation of the wind speed that forms the correlation 

between turbulence intensity and optimal torque control 

gain. The fact that the two curves in Figure 6 do not lie 

on top of each other indicates that other wind 

characteristics also play a role. For example, the 

Richardson number and other factors may also affect 

the optimum gain, but these are not investigated in this 

paper. 

Why is the optimal torque control gain lower for 

higher turbulence intensities? Consider this example: 

imagine that the constant wind has a speed of 10 m/s, 

and the CART is operating in steady state with ω = 

3.46 rad/s (for a tip speed ratio of λ* = 7.5). At some 

time, there is a step change in wind speed to 11 m/s, 

which causes the instantaneous tip speed ratio to drop 

to λ = 6.82. The torque control law will cause the rotor 

to accelerate towards λ = 7.5, but in the meantime, the 

power loss is proportional to the cube of the wind 

speed, or 1331 (m/s)3, by Eq. (5). Similarly, if there is 

a step change in wind speed down to 9 m/s, the power 

loss is proportional to 93

 = 729. Because of this cubic 

law relating wind speed and power, it is more important 

to quickly regain the optimal tip speed ratio following 

wind gusts than lulls. Since it is safe to assume that, in 

general, a given wind input will have a similar number 

of gusts and lulls, using a torque control gain that is 

slightly smaller than the one given by Eq. (2) is more 

likely to make Eq. (7) positive, which means the rotor 

will accelerate more easily in response to a gust. 

 Turbulence intensity is related to the geological 

features in a given area, but there are many other 

atmospheric factors that influence the turbulence at any 

given time. Thus, for many wind sites, it may not be 

practical to choose a constant gain reduction factor for 

the torque controller. The next section describes an 

alternative approach to the standard control law given 

by Eq. (1). This approach uses generator torque to 

assist in the acceleration and deceleration in response to 

wind gusts. 

OPTIMALLY TRACKING ROTOR CONTROL

 Because OTR control was described in depth by 

Fingersh and Carlin [3], only a brief discussion of its 

motivation will be provided here before simulation 

results are presented. The idea for OTR control came 

from the observation that significant power loss 

occurred in region 2 as a result of the turbine’s inability 

to accelerate and decelerate quickly in response to wind 

speed changes. Fingersh and Carlin [3] observed a gap 

of as much as 10% between the optimal Cp in region 2 

and the Cp actually achieved by the turbine. In order to 

achieve a higher Cp, it is possible to use the generator 

torque to assist in acceleration or deceleration, thereby 

causing the turbine to spend more time closer to the 

optimum tip speed ratio. The authors proposed 

replacing the standard region 2 control law (Eq. (1)) 

with the following: 

( ) 2 2 τ c = kω − G τ aero − kω (15) 

where τaero can be determined by rearranging Eq. (7). 

The gain G can be selected to provide a tradeoff 

between acceleration or deceleration rate and other 

considerations, such as preventing the turbine from 

actually drawing power from the grid (motoring). 

 Figure 7 was created by running 24 one hour 

simulations for each of three wind inputs for each of the 

six values of G plotted (i.e., 432 total simulations). The 
6

results of the three wind inputs, the same as those used 

to create Figure 5, were averaged before plotting to 

reduce clutter in Figure 7. The nominal case is G = 0, 

which reduces to the standard control law, and k/M*

 = 

1.0. Each curve’s maximum is marked with a star. 

Because of the effects of turbulence intensity, the 

maximum on the G = 0 curve is located at k/M*

 = 0.91 

rather than k/M*

 = 1.0. Notice that the maxima for the 

OTR control cases occur at slightly higher k/M*

 than 

the maximum of the nominal standard control law 

curve; this is because the generator assistance in 

acceleration and deceleration helps to alleviate the 

negative effects of the wind gusts and lulls discussed in 

the previous section. 

0.8 0.85 0.9 0.95 1 1.05 99.5

100

100.5

101

101.5

k/M*

Percent of Nominal (G=0) Power Capture

G = 0.2

G = 0.4

G = 0.6

G = 0.8

G = 1.0

Nominal Case

(J = 388500,

G = 0.0) 

Figure 7: Improved Power Capture with OTR 

Control 

 Even assuming nothing is known about the 

optimum gain k for a given situation (i.e., the 

turbulence intensity is unknown, so k/M*

 = 1), the OTR 

control algorithm captures more power than the 

standard region 2 controller. For an aggressive G = 1, 

the average increase in power capture over nominal is 

about 1.2%, as shown by the G = 1 curve in Figure 7. 

More information on OTR control, including results 

from experiments on the CART, can be found in [4]. 

 The focus of the next section is the implementation 

of an adaptive controller that moves the steady state 

operating point to the asterisk on the dotted curve in 

Figure 4. 

ADAPTIVE CONTROL

A small number of papers have been published 

regarding adaptive control of wind turbines (including 

[5]-[11]), but most involve region 3 control, and very 

few attempts have been made to test these advanced 

controls on real turbines. One researcher, Bossanyi [8], 

[9], has performed such tests, studying his adaptive 

control scheme on a real turbine, but his focus was on 

pitch control in region 3. Enslin and Van Wyk [10] 

have tested an adaptive direct speed controller on a 

small wind turbine, but their direct speed control is very 

dissimilar to the standard region 2 torque control given 

by Eq. (1). Bhowmik, Spée, and Enslin [11] have also 

conducted research that involves direct speed control of 

a variable speed turbine in order to maintain the optimal 

tip-speed-ratio, but they used a DC machine set up to 

simulate a wind turbine rather than an actual turbine in 

their experiments. In any case, commanding speed 

explicitly at a high control rate (as in [10], [11]) would 

cause excessive torques on the drive train of a turbine 

with a large rotational inertia like the CART. 

 One solution to the problem of parameter 

uncertainty in region 2 control is to use a form of 

Model Reference Adaptive Control (MRAC) [12]. If a 

suitable model could be found, the gain k in the control 

torque equation (Eq. (1)) could be adapted in order to 

drive the turbine to follow the model. In theory, a 

designer could make an educated initial guess regarding 

the values of the uncertain parameters and the controller 

could adapt k towards the point of maximum power 

capture. Assuming the adaptation process continued 

throughout the lifetime of the turbine, this controller 

would be able to track changes in the blade parameters 

over time. 

However, finding such a suitable model for region 

2 control is not a simple task. One obvious choice is to 

measure the wind speed and use it to calculate the wind 

power available to the rotor. Unfortunately, it is 

impossible to measure the wind at all points on the rotor 

plane, and the wind passing through a circle with a 

radius of 20 meters or more can be very non-uniform. 

Nevertheless, advances are being made in wind 

estimation (for example, [13]), and it is possible that the 

available measurements and estimates may still be 

accurate enough to allow a MRA controller of this type 

to operate satisfactorily (i.e., maximize the fraction of 

power in the wind captured by the turbine). 

Gain Adaptation Algorithm

 The research presented here uses a simple, highly 

intuitive gain adaptation algorithm. In region 2, the 

adaptive control is very similar to the non-adaptive case 

presented in Eq. (1)-(2): 2 τ c = ρMω . (16) 

The adaptive gain M incorporates all of the terms in the 

non-adaptive torque control gain k except the air 

density ρ. ρ is kept separate because it is time varying 

(with changes in the weather) and completely 

uncontrollable. M is adapted after a certain number n

of time steps of operation in region 2; n is selected to be 

large enough to average out high frequency wind 

variations and the slowness of the turbine’s response to 

wind gusts and lulls. Tests on the CART thus far 
7

indicate that the averaging period defined by n will 

need to be on the order of tens of minutes to hours. 

A simulation was conducted using a rigid body 

model relating net torque and angular acceleration as in 

Eq. (7) and using Eq. (16) for the control torque. This 

simulation was run for 200 seconds with each of 26 

different values of the gain M, and the turbine’s 

behavior for each of the 26 gain values was averaged to 

produce Figure 8. The resulting curve led to the 

development of the gain adaptation law described next. 

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

Normalized M (M/M*)

Mean P frac

Figure 8: Mean Fractional Power vs. Normalized M

 The instantaneous fractional power captured is 

given by 

wy

cap

frac P

P

P = (17) 

where 

P τ ω Jωω& cap = c + (18) 

ρ ( ) ψ 3 3 cos

2

1 P Av wy = . (19) 

The first term in Pcap is the generator power and the 

second is the kinetic power (i.e., the time derivative of 

the kinetic energy) of the rotor. The yaw error factor 

cos3

(ψ) is a necessary component of the available 

power calculation because a yaw error can significantly 

reduce the amount of power that the turbine is 

physically able to capture. While active control of the 

turbine’s yaw position is used to minimize the steadystate yaw error, that yaw control is not part of this 

research. The reason for incorporating the cos3

 factor 

into Eq. (19) is to prevent slowness in the yaw control 

from compromising the adaptive torque control, which 

would be the case if the gain adaptation law used an 

inaccurate measure of available wind power. Though 

cos3

 is only an approximation of the complex 

aerodynamic effects of yaw error, it is very close 

(within a few percent up to a 20° yaw error) to the 

actual power reduction caused by yaw error, as shown 

in Figure 9. Figure 9 plots the fraction of available 

power for a given yaw error for two cases: the cos3

approximation and real data collected from NREL’s 

Unsteady Aerodynamics experiment [14]. It should be 

noted that the Unsteady Aerodynamics experimental 

data, collected in a wind tunnel, gives virtually identical 

results to an aerodynamic simulation run with the 

CART’s parameters using the modeling tool SymDyn 

[15]; thus, the SymDyn curve has been omitted from 

the plot. 

0 10 20 30 40 50 60 70 80 90 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

Yaw Error (degrees)

Fractional Available Power

cos3 Approximation

Unsteady Aerodynamics Data

Figure 9: Decrease in Available Power vs. Yaw 

Error 

Given that a properly operating turbine will have 

zero mean yaw error and that fluctuations in 

instantaneous wind direction (and thus yaw error) will 

be very similar from one adaptation period to the next, 

these small percentages will cause no more than a very 

small fraction of a percent difference in the calculation 

of Pwy for any two adaptation periods. Thus, using the 

cos3

 approximation to the yaw error factor is not 

expected to cause problems for the gain adaptation law. 

 The only difference between Pwind, defined in Eq. 

(5), and Pwy is the cos3

(ψ) factor. Whenever ψ is zero, 

as in the simulation producing Figure 8, the two 

definitions become identical. The reason that Pcap is 

used in the calculation of Pfrac rather than the turbine 

power P given in Eq. (6) is that the sensor requirements 

are more reasonable, given the instrumentation 

normally available on an industrial turbine. The two 

definitions of the turbine’s captured power are closely 

related, differing only by the inevitable mechanical 

losses in the turbine’s gearbox that make Pcap < P by a 

small amount. Given that fact and the fact that Pwind ≥

Pwy, it is impossible to state definitively whether Pfrac < 

Cp (Eq. (4)) or vice versa at any given instant. 

However, it can be stated that they are closely related. 

The adaptive controller begins by changing M by 

some value ∆M. At the end of the adaptation period, 

the controller evaluates the turbine’s performance, 

averaging the captured power and the wind power and 

computing the ratio of the averages. If the fraction of 

the averaged powers Pfrac is greater than the mean 
8

fractional power in the preceding adaptation period, the 

controller selects a new ∆M of the same sign as the 

previous one. This process continues in the same 

manner until the fractional power is less than that of the 

preceding adaptation period. At that point, the new ∆M

is calculated to have the opposite sign of the previous 

∆M. Eventually, M should converge toward M*, the 

optimal gain. 

 The mathematical equations implementing this 

gain adaptation are 

M () ( ) () k = M k − n + ∆M k , (20)

( ) [ ] ( ) [ ] ( ) ( )

M

favg favg

K

sign M k n sign P k P k

M k

∆

∆ − ∆ ∆

∆ =

1/ 2

, (21)

P () () ( ) k P k P k n ∆ favg = favg − favg − , (22)

and 

( )

( )

∑ ( )

∑

=

=

− +

− +

= n

i

wy

n

i

cap

favg

P k n i n

P k n i n P k

1

1

1

1

. (23)

In Eq. (21), the |∆Pfavg(k)|

1⁄2 factor is an indicator of the 

closeness of M to the optimal operating point. When M

is such that operation is near the peak of the curve 

shown in Figure 8, a given ∆M will cause a smaller 

|∆Pfavg| due to the flatter nature of the curve near its 

peak. Thus, ∆M will get smaller and smaller as the 

optimal point is approached. The exponent 1⁄2 was 

chosen based on empirical results in simulation. 

 It may seem that this simple adaptation scheme is 

rather slow. However, in the decades-long life of a 

wind turbine, several hours, days, or even weeks spent 

identifying the initial optimal operating point are not 

significant in terms of power lost or time wasted. 

Additionally, the aerodynamic changes that occur to all 

turbine blades typically take place on an even slower 

time scale than that of the adaptive controller. Thus, an 

adaptation period on the order of hours should not be a 

significant problem for a commercial turbine designer. 

 Although this gain adaptation scheme has not been 

theoretically proven to work (i.e., to maximize the 

fractional power captured by the turbine), simulations 

have demonstrated the desired behavior, and real 

turbine experiments have demonstrated an increase in 

power capture over the standard control scheme. While 

stability of the gain adaptation law has also not been 

proven theoretically, both simulations and experimental 

data have indicated that the system is stable. Further, 

there are many safety mechanisms protecting the 

turbine independent of the control technique described 

in this paper, and any system that changes over the long 

time intervals typical of this adaptation scheme is 

unlikely to cause catastrophic damage due to instability. 

At this point, the controller’s intuitive nature and ease 

of implementation far outweigh its drawbacks. 

 One main difference between the adaptive control 

described in this research and some of the adaptive 

controllers developed in previous research is which 

parameters are assumed to be unknown. In this 

research, the controller attempts to have the turbine 

power track the wind power but assumes Cpmax and λ*

are unknown. In contrast, previous adaptive controllers 

such as those presented in [5]-[6] force the turbine to 

track a desired rotor speed in region 2. However, in 

order to develop a model for desired rotor speed, it is 

necessary to have some knowledge of the Cp surface, 

particularly Cpmax and λ*. As discussed previously in 

this research, the fact that those two parameters are not 

well known is a major source of energy loss in region 2. 

An additional difference among the various adaptive 

controllers is the lengthy averaging period used in this 

research, compared to the very short time periods used 

in previous adaptive controllers. These short time 

periods may cause the adaptive controllers to have 

trouble distinguishing between performance changes 

caused by changes in the controller and those caused by 

wind gusts and lulls. 

Simulations

 Numerous simulations have been run to 

demonstrate that the controller results in the desired 

turbine behavior. Like the simulation used in 

producing Figure 8, these were performed using a 

simple, single-integrator model (Eq. (7)) of a horizontal 

axis wind turbine (HAWT) with the same 

characteristics as the CART. The simulations were 

done using Matlab’s® modeling software, Simulink®. 

A simplified Simulink model is shown in Figure 10. 

Net Torque w, wdot

Turbine Model

Control Torque

Wind

w, wdot

Pfavg

Pfavg Calculation

Wind

Measured

Wind Speed &

Wind Direction

w

M

Control Torque

Generator

Torque Control

Pfavg M

Gain Adaptation

Wind Aero. Torque

Aerodynamic Torque

Calculation

Figure 10: Simplified Simulink Model of CART and 

Adaptive Controller 

Equations (20)-(22) are implemented in the Gain 

Adaptation block. Similarly, Eq. (16) is computed in 

the Generator Torque Control block. The Aerodynamic 

Torque Calculation block implements Eq. (8), using a 

surface similar to the one in Figure 1 to determine Cq, 

given the relationship between Cp and Cq as described 

in Eq. (9). The Turbine Model block integrates Eq. (7) 
9

to obtain the angular velocity ω. Finally, the Pfavg 

Calculation block performs Eq. (18), (19), and (23). 

Some simulations have used measured wind data, 

while others have used simulated wind data. Where 

possible, the Simulink model uses the same control as 

would be implemented on the CART. One of the 

simulation outputs, which used a constant wind input, is 

plotted in Figure 11, with simulation parameters listed 

in Table 1. 

0 1 2 3 4 5 6

x 104

0.98

0.985

0.99

0.995

1

1.005

time (sec)

M/M*

0 1 2 3 4 5 6

x 104

-5

0

5

10

15

20 x 10-3

deltaM/M*

time (sec)

Figure 11: Simulation Results with Constant Wind 

Input 

Table 1: Simulation Parameters 

Parameter Value 

J 388,500 kg•m2

R 21.65 m 

M* 174.48† m5/rad3

K∆M 10-3 rad3

/m5

ρ 1.0 kg/m3

fs 100 Hz 

Figure 11 uses an adaptation period of 20 minutes, 

meaning that n, the number of steps between 

adaptations, is equal to 120,000. Clearly, the bulk of 

the changes in this particular case occur in the first 

dozen adaptation periods. After that, the value of M

oscillates near the expected value, M*

. Because the 

adaptation scheme uses only the most recent past value 

in addition to the current value of the adaptation 

parameters, these oscillations are not unexpected. 

Future work will investigate incorporating additional 

past values to reduce the oscillations. In Figure 11, 

both M and ∆M are normalized by M*

, which, 

according to Figure 1 is nominally the ideal value. 

Because the wind is constant (turbulence intensity = 0), 

note that optimal behavior occurs at M = M*

. However, 

that may not be true for the real turbine, for which M*

 is 

 

†

 This value is referenced to the High Speed Shaft. On the rotor side 

of the gearbox, M*

 = 7530.41 m5/rad3. 

not precisely known, and is not true for simulations 

using real (turbulent) wind data as an input, as shown in 

Figure 12. 

0 1 2 3 4 5 6 7 8 9 10

x 105

0.5

0.75

1

1.25

1.5

1.75

2

Noramlized M (M/M*)

0 1 2 3 4 5 6 7 8 9 10

x 105

-0.4

-0.2

0

0.2

0.4

deltaM/M*

time (sec)

Figure 12: Simulation Results with Real Wind Data 

as Input 

The turbulence at the NWTC during the winter wind 

season tends to be higher than the average at a typical 

wind site; thus, it is expected that using measured wind 

speeds will result in M converging to some value 

approximately 20% lower than M*

. This turbulence 

effect can be seen in the top plot of Figure 12. Further, 

it is clear that convergence under real wind input does 

not occur as quickly as it does with constant wind, 

though it seems that the system is stable in some sense, 

since the normalized ∆M is oscillating around zero. It 

is also important to note that Figure 12 required the use 

of a much larger n than Figure 11 due to the timevarying wind. In the case of Figure 12, n = 4,680,000, 

which translates to 13 hours. 

Real Turbine Experiments

The CART is an invaluable tool for determining 

the benefit of this new adaptive controller to the turbine 

industry. Issues that may not arise in simulation must 

be solved before any new controller can be trusted by 

the industry. For example, in the first significant 

experiment on the CART, the nacelle anemometer was 

used in the calculation of Pwy (Eq. (19)). This 

anemometer would be available on a commercial wind 

turbine. Unfortunately, it seems that simply using the 

measured wind speed on the nacelle results in improper 

adaptation. Although the aerodynamics behind the 

rotor and around the nacelle are very complicated while 

the turbine is operating, it is believed that the major 

error in the nacelle anemometer measurements is due to 

the well known axial induction factor. The effect of the 

axial induction factor is that, for a given upwind wind 

speed, a faster spinning rotor (i.e., higher tip speed 

ratio) results in slower wind downwind of the rotor. 

Thus, the measured Pwy actually appears lower than the 
10

true available wind power, and the value of Pfrac given 

by Eq. (17) is misleadingly larger. 

Figure 13 shows data from nearly all of the 

adaptations that occurred during the four months 

between the initial implementation and the end of the 

wind season. 

0 5 10 15

x 104

0

50

100

150

200

250

Time in Region 2 (seconds)

M

Best Data

Estimated CART M*

Figure 13: Adaptations of M on the CART 

The adaptation period used to collect the data 

plotted in Figure 13 is 10 minutes for the first 11 

adaptations, 20 minutes for the next nine adaptations, 

30 minutes for the following 51 adaptations, and 60 

minutes (n = 360,000) for the remaining nine. During 

the time covered by Figure 13, there were numerous 

changes made to the adaptive controller as minor errors 

were discovered; only to the right of the vertical dotted 

line is the algorithm implemented exactly as described 

in Eq. (20)-(23). In addition, there were several 

problems with sensors on the CART discovered. For 

example, the drop in M between about 550 and 900 

minutes is partly a result of an erroneous wind direction 

sensor. As a result of changes in the control algorithm 

and sensor failures, the value of M was re-set three 

times during the four months; these times are marked 

by the stars along the plot. Finally, it is evident that 

even a 30-minute adaptation period is too short to result 

in convergence of M, which is consistent with the 13-

hour adaptation period used to create Figure 12. 

Unfortunately, the wind resources did not last long 

enough to extend the data collection beyond the nine 

adaptations with 60-minute adaptation periods. Finally, 

in the interest of clarity, the time scale in Figure 13 

reflects only the time spent operating in region 2. 

The data collected on the CART is, unfortunately, 

insufficient for showing that the proposed adaptive 

controller does or does not work in terms of whether or 

not M converges to an optimum value. However, it is 

still possible to compare this controller with the 

standard non-adaptive controller (Eq. (1)) in terms of 

power capture. In fact, based on data analysis, it has 

been determined that the gain k used in the standard 

variable speed region 2 CART controller may have 

been close to 100% higher than the true optimum value 

for the turbine, resulting in significant power loss. 

Figure 14 provides a plot comparing the mean low 

speed shaft power vs. mean wind speed for the standard 

controller and the adaptive controller. The standard 

controller data is actually a random, representative 

sampling of the data collected using the standard 

controller and presented in [4], since using all of that 

data makes the plot very crowded and difficult to read. 

The adaptive controller data includes those data sets for 

which M < 99.5. For mean wind speeds below about 13 

m/s, the power curve follows the expected cubic 

relationship between wind speed and power. Above 

that wind speed, the turbine frequently operates in 

region 3, and the power limiting control algorithm (not 

discussed in this research) takes over, which causes the 

flattening of the curve. 

2 4 6 8 10 12 14 16 18 0

100

200

300

400

500

600

700

Mean Wind Speed (m/s)

Mean LSS Power (kW)

Standard Controller

Adaptive Controller

Figure 14: 10-Minute Averages of Low Speed Shaft 

Power vs. Wind Speed for Standard and Adaptive 

Control 

As is clearly evident in Figure 14, there is some overlap 

between the controllers. However, there is a distinct 

curve of adaptive control data lying above the standard 

control data. This curve shows that, for the values of M

that are less than 43% of the CART’s predicted M*

, 

more power is captured. In fact, analysis of the 

CART’s constant speed controller data in [4] has shown 

that the optimum tip speed ratio λ* for the CART is 

probably closer to 9.5 than the predicted 7.5, and the 

Cpmax is about 0.45 rather than 0.425. Given the 

equation for M*

 (Eq. (14)), these errors result in a 

torque control gain about 92% higher than the optimum 

gain for maximum power capture.

CONCLUDING REMARKS

 An analysis of the standard region 2 control 

scheme (τc = kω2

) for variable speed wind turbines 

reveals some of its shortcomings, specifically those 
11

caused by the turbulence in the wind and the difficulty 

in determining the gain k. Three new control ideas 

were proposed to address these shortcomings. First, a 

very simple scheme reduced k by 5% - 20% in order to 

better capture the power available in wind gusts. This 

change can result in a 0.5% increase in power capture 

with virtually no cost, and can be implemented on any 

existing turbine that uses the standard torque controller 

in region 2. Next, an optimally tracking rotor (OTR) 

control algorithm was tested in simulation. The OTR 

controller uses generator torque to assist in acceleration 

and deceleration of the rotor in response to wind gusts 

and lulls. It can result in a 1% or more increase in 

power capture, depending on various parameters. 

Finally, a model reference adaptive control scheme was 

been developed and tested both in simulation and on the 

CART. This controller uses information on power 

available in the wind in an attempt to maximize power 

captured by the turbine. The main advantage of the 

adaptive controller is the potential for increased longterm energy capture over a fixed-gain controller by 

eliminating the problem of parameter uncertainty in k. 

The adaptive controller differs from the simple 

reduction in k described above because it does not 

depend on the initial determination of k that is near the 

true value. 

ACKNOWLEDGEMENTS

This work was funded by the U. S. Department of 

Energy (through the National Renewable Energy 

Laboratory under contract number DE-AC36-

99G010337), the University of Colorado at Boulder, 

and the American Society for Engineering Education. 

REFERENCES

[1] Walker, S. N. and Wilson, R. E., 1976, 

Performance Analysis Program for Propeller Type 

Wind Turbines, Oregon State University, Corvallis, 

Oregon. 

[2] Burton, T., Sharpe, D., Jenkins, N., and Bossanyi, 

E., 2001, Wind Energy Handbook, John Wiley & Sons, 

New York. 

[3] Fingersh, L. J. and Carlin, P. W., 1998, “Results 

from the NREL Variable-Speed Test Bed,” 

Proceedings of the 17th ASME Wind Energy 

Symposium, Reno, NV, pp. 233-237. 

[4] Fingersh, L. J. and Johnson, K. E., 2004, “Baseline 

Results and Future Plans for the NREL Controls 

Advanced Research Turbine.” Submitted for 

publication in the Proceedings of the 23rd ASME Wind 

Energy Symposium.

[5] Freeman, J. B. and Balas, M. J., 1999, “An 

Investigation of Variable Speed Horizontal-Axis Wind 

Turbines Using Direct Model-Reference Adaptive 

Control,” Proceedings of the 18th ASME Wind Energy 

Symposium, Reno, NV, pp. 66-76. 

[6] Song, Y. D., Dhinakaran, B., and Bao, X.Y., 2000, 

“Variable Speed Control of Wind Turbines Using 

Nonlinear and Adaptive Algorithms,” Journal of Wind 

Engineering and Industrial Aerodynamics, 85, pp. 293-

308. 

[7] Dambrosio, L. and Fortunato, B., 1999, “One-StepAhead Adaptive Control of a Wind-Driven, 

Synchronous Generator System,” Energy, 24, pp. 9-20. 

[8] Bossanyi, E. A., 1987, “Adaptive Pitch Control for 

a 250 kW Wind Turbine,” Proceedings of the British 

Wind Energy Conference, pp. 85-92. 

[9] Bossanyi, E. A., 1989, “Adaptive Control of the 

MS2 Wind Turbine. Practical Results,” Wind 

Engineering, 13, No. 5, pp. 259-273. 

[10] Enslin, J. H. R. and Van Wyk, J. D., 1992, “A 

Study of a Wind Power Converter with MicroComputer Based Maximal Power Control Utilising an 

Over-Synchronous Electronic Scherbius Cascade,” 

Renewable Energy, 2, pp. 551-562. 

[11] Bhowmik, S., Spée, R., and Enslin, J. H. R, 1999, 

“Performance Optimization for Doubly-Fed Wind 

Power Generation Systems,” IEEE Transactions on 

Industry Applications, 35, No. 4, pp. 949-958. 

[12] Narendra, K. and Annaswamy, A., 1989, Stable 

Adaptive Systems, Prentice-Hall, Englewood Cliffs, NJ. 

[13] Balas, M. J., Lee, Y. J., and Kendall, L., 1998, 

“Disturbance Tracking Control Theory with 

Application to Horizontal Axis Wind Turbines,” 

Proceedings of the 17th ASME Wind Energy 

Symposium, Reno, NV, pp. 95-99. 

[14] Fingersh, L. J., Simms, D., Hand, M., Jager, D., 

Cotrell, J., Robinson, M., Schreck, S., and Larwood, S., 

2001, “Wind Tunnel Testing of NREL’s Unsteady 

Aerodynamics Experiment,” Proceedings of the 20th

ASME Wind Energy Symposium, Reno, NV, pp. 194-

200. 

[15] Stol, K. A., and Bir, G. S. “SymDyn User’s 

Guide,” NREL/EL-500-33845. National Renewable 

Energy Laboratory, Golden, Colorado, 2003. 
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Conference Paper 

4. TITLE AND SUBTITLE

Methods for Increasing Region 2 Power Capture on a Variable Speed HAWT: Preprint 

6. AUTHOR(S) 

K.E. Johnson, L.J. Fingersh, M.J. Balas, and L.Y. Pao 

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13. ABSTRACT (Maximum 200 words)

The standard region 2 control scheme for a variable speed wind turbine, τc = kω2, has several shortcomings that can result in significant power 

loss. The first of these is that there is no accurate way to determine the gain k; modeling programs are not accurate enough to represent all of the 

complex aerodynamics, and these aerodynamics change over time. Furthermore, it is not certain whether the value of k used in the standard 

control even provides for the maximum energy capture under real-world turbulent conditions. New control ideas are introduced to address these 

issues. First, it is shown in simulation that using smaller values of k than the standard can result in increased power capture. Second, an 

optimally tracking rotor control scheme improves upon the standard scheme by assisting the rotor speed in tracking wind speed fluctuations 

more rapidly. Finally, an adpative control scheme is proposed that allows for maximum power capture despite parameter uncertainty. 

15. NUMBER OF PAGES 14. SUBJECT TERMS 

wind turbine; variable speed wind turbine; wind turbine control system; renewable 

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