STANDARD AND ADAPTIVE TECHNIQUES
FOR MAXIMIZING ENERGY CAPTURE
KATHRYN E. JOHNSON, LUCY Y. PAO, MARK J. BALAS, and LEE J. FINGERSH
W
ind 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 present a myriad of control problems that, if solved, could
reduce the cost of wind energy. In contrast to constantspeed turbines (see “Wind Turbine Development and
Types of Turbines”), variable-speed wind turbines are
designed to follow wind-speed variations in low
winds 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 article’s 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 Laboratory’s National Wind Technology 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 discrete if-then variety. On the middle level is turbine
control, which includes generator torque control,
blade pitch control, and yaw control. Generator
torque control, performed using the power electronics, 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 operate quickly relative to the rotor-speed time constant.
NATIONAL RENEWABLE ENERGY LABORATORY
70 IEEE CONTROL SYSTEMS MAGAZINE
»
JUNE 2006
1066-033X/06/$20.00©2006IEEE
Wind Turbine Development and Types of Turbines
W
ind-powered machines have been used by humans for centuries. Most familiar are the historical many-bladed windmills
used for milling grain, the earliest versions of which appeared
during the 12th century [21]. Water-pumping wind machines
appeared in the United States in the mid-19th century, while the
modern era of wind turbine generators began in the 1970s [21].
These modern horizontal-axis wind turbines typically have two or
three blades and can be either upwind (with the rotor spinning on
the upwind side of the tower) or downwind. Horizontal-axis wind
turbines range in size from small home-based turbines of a few
hundred watts to utility-scale turbines up to several megawatts.
Most modern utility-scale turbines operate in variable-speed
mode with the turbine speed changing continuously in response
to wind gusts and lulls. Although costly power electronics are
required to convert the variable-frequency power to the fixed utility grid frequency, variable-speed turbines can spend more time
operating at maximum aerodynamic efficiency than constant-
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 perspective of the turbine-level control. For example, the gener-
speed turbines. In addition, variable-speed turbines often endure
smaller power fluctuations and operating loads than constantspeed turbines. Constant-speed turbines are connected directly
to the utility grid, which eliminates the requirement for power electronics. A constant-speed machine’s fixed generator frequency
forces the turbine’s mechanical components to absorb much of
the increased energy of a wind gust until the turbine’s power regulation system can respond. On a variable-speed machine, however, 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 provide 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 turbines to minimize the cost of wind energy is a complex task, and
much research remains to be done to improve the controllers.
ator and power electronics controllers regulate the generator
and power electronics variables to achieve the desired generator 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 generator 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
Nomenclature
A
Cp
Cpmax
Cq
J
K
M
M+
M∗
P
P0
Pcap
Pfavg
Pwind
Rotor swept area (m2 )
Rotor power coefficient (dimensionless)
Maximum rotor power coefficient (dimensionless)
Rotor torque coefficient (dimensionless)
Rotor inertia (kg-m2 )
Standard torque control gain (kg-m2 )
Adaptive torque control gain (m5 )
Simulation-derived prediction of optimal torque
control gain (m5 )
Turbine’s true optimal torque control gain (possibly
unknown) (m5 )
Turbine (rotor) power (kW)
Symmetric quadratic curve coefficient (dimensionless)
Captured power (kW)
Average captured power divided by average wind
power over a given time period (dimensionless)
Power available in the wind (kW)
Pwy
R
a
b
fs
Power available in the wind, with approximate yaw
error factor included (kW)
Rotor radius (m)
Symmetric quadratic curve coefficient (m−10 )
Damping coefficient (kg-m2 /s)
k
n
v
β
γM
Sampling frequency (Hz)
Adaptive controller’s discrete-time index
Number of steps in adaptation period
Wind speed (m/s)
Blade pitch angle (deg)
Positive gain in gain adaptation law (m−5 )
λ
λ∗
ρ
Tip-speed ratio (TSR) (dimensionless)
TSR corresponding to Cpmax (dimensionless)
Air density (kg/m3 )
τaero
Aerodynamic torque (N-m)
Generator (control) torque (N-m)
Yaw error (deg)
Rotor angular speed (rad/s)
τc
ψ
ω
JUNE 2006
«
IEEE CONTROL SYSTEMS MAGAZINE 71
this project, we ignore the particulars of the high- and lowlevel 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 variable-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 Pwind
available from the wind, that is,
Cp =
P
.
Pwind
(1)
The available power Pwind is given by
Pwind = 12 ρAv3 ,
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 turbine’s cost
by reducing the stress on its components.
where ρ is the air density, A is the rotor swept area, and v is
the wind speed. The aerodynamic rotor power is given by
P = τaero ω,
Pitch
Low-Speed
Shaft
Rotor
Gear Box
Generator
Wind
Direction
Anemometer
Controller
Brake
Yaw Drive
Wind Vane
Yaw Motor
High-Speed
Shaft
Blades Tower
Nacelle
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 turbine’s high torques and extensive operating hours. The power electronics for a variable-speed turbine are usually located at the base of the tower. (Drawing courtesy of
the U.S. Department of Energy.)
72 IEEE CONTROL SYSTEMS MAGAZINE
»
JUNE 2006
(2)
(3)
where τaero is the aerodynamic
torque applied to the rotor by the
wind and ω is the rotor angular
speed. In Figure 3, the dotted wind
power curve represents the power
in the unimpeded wind passing
through 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 conditions, a static relationship exists;
the turbine power curve plotted in
Figure 3 represents the power versus wind speed relationship for a
turbine with Cp = 0.4.
Classical techniques such as proportional, 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
STANDARD VARIABLE-SPEED CONTROL LAW
For variable-speed wind turbines operating in region 2, the
control objective is to maximize energy capture by operating the turbine at the peak of the Cp-TSR-pitch surface of
the rotor, shown in Figure 4. The power coefficient Cp(λ, β)
is a function of the tip-speed ratio (TSR) λ and the blade
pitch β. The TSR λ is defined as
λ=
ωR
.
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 Cp
surface downward, reducing energy capture. In this section, 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 the
600-kW two-bladed, upwind CART. Unfortunately, modeling tools such as PROP are of questionable accuracy; in
fact, an NREL study [15] comparing wind turbine modeling codes reports large discrepancies and an unknown
level of uncertainty. Therefore, computer models are unreliable 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 τc (that is, the generator torque) be given by
The National Renewable Energy Laboratory
and National Wind Technology Center
T
he National Renewable Energy Laboratory (NREL) is a
part of the U.S. Department of Energy (DOE) Office of
Energy Efficiency and Renewable Energy. Located in Golden, Colorado, the laboratory began operating in 1977 as
the Solar Energy Research Institute (SERI) and attained
the national laboratory classification in 1991 when SERI
was renamed NREL. NREL’s mission statement summarizes the laboratory’s research: “NREL develops renewable
energy and energy efficiency technologies and practices,
advances related science and engineering, and transfers
knowledge and innovations to address the nation’s energy
and environmental goals.”
The National Wind Technology Center (NWTC) supports
the U.S. wind industry by performing applied research and
testing in conjunction with its industry partners. These industry partners range from large commercial turbine manufacturers to small distributed wind system developers, all of whom
share the goal of reducing the cost of wind energy. The
NWTC’s facilities include numerous turbine test pads, which
currently test turbines ranging from 300 W to 600 kW; a
dynamometer facility for testing advanced drive trains; an
industrial user facility for testing new blade designs; a hybrid
test facility, which allows testing of energy systems consisting of wind combined with solar, diesel, or other electricity
sources; and two advanced research turbines. Together with
NWTC’s wind industry partners, researchers at the NWTC
have helped to bring the cost of large-scale wind energy
down from about US$0.80/kW-h in 1980 (today’s dollars) to
US$0.04–US$0.06/kW-h today.
2,000
1,800
1,600
Wind Power
Cp = 1
1,400
Power (kW)
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 turbine manufacturers begun to incorporate more modern and
advanced control methods in commercial turbines. In part, the
gap between the research and commercial turbine communities is a result of the fact that few theoretically advanced controllers have been successfully tested on real turbines.
In this article, we analyze the stability of a control system that has been tested on CART, focusing on adaptive
generator torque control with constant blade pitch to maximize 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 torque
controller. We begin with a review of nonadaptive controllers, continue with a discussion of the adaptive controller of [2], and then proceed to the stability analysis.
1,200
High
Wind
Cutout
1,000
Turbine Power
800
Region 3
600
Region 2
400
Cp = 0.4
200 Region 1
0
0
5
10
15
Wind Speed (m/s)
20
25
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 electrical and mechanical loads are not exceeded.
JUNE 2006
«
IEEE CONTROL SYSTEMS MAGAZINE 73
τc = Kω2 ,
(5)
where the gain K is given by
K=
Cpmax
1
ρAR3 3 ,
2
λ∗
R is the rotor radius, and λ∗ is the tip-speed ratio at which
the maximum power coefficient Cpmax occurs.
Next, assuming that the rotor is rigid, the angular acceleration ω̇ is given by
(6)
0.0
−0.1
−5
−1
7
11
15
Tip-Speed Ratio λ
13
11
9
7
3
5
3
1
−0.2
−0.3
−0.4
−0.5
Power Coefficient Cp
0.4
0.1
Pitch β (deg)
0.4
Power Coefficient Cp
τaero =
1
ρARCq(λ, β)v2 ,
2
(8)
where
Cq(λ, β) =
Cp(λ, β)
λ
(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
ω̇ =
FIGURE 4 Cp versus tip-speed ratio and pitch for CART. Since turbine 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 turbine’s 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.
1
ρAR3 ω2
2J
Cp(λ, β)
λ3
−
Cpmax
λ3∗
.
(10)
Since the rotor inertia J, the air density ρ, the rotor swept
area A, the rotor radius R, and the squared rotor speed ω2
are nonnegative, the sign of the angular acceleration ω̇
depends on the sign of the difference in (10). When the tipspeed ratio λ > λ∗ , it follows from (10) and the fact that
Cp ≤ Cpmax that ω̇ is negative and the rotor decelerates
toward λ = λ∗ . On the other hand, when λ < λ∗ and
Cp >
Cpmax
λ3∗
λ3 ,
(11)
it follows that ω̇ is positive. The curve
0.3
F(λ)
0.2
F(λ) =
0.1
0.0
(7)
where J is the combined rotational inertia of the rotor,
gearbox, generator, and shafts and the aerodynamic torque
τaero , derived from (1)–(4), is given by
0.5
0.3
0.2
1
(τaero − τc ),
J
ω̇ =
2
4
6
8
10
Tip-Speed Ratio λ
12
14
FIGURE 5 CART’s power coefficient Cp versus tip-speed ratio and
cubic function F. The intersection of the solid and dotted lines at
the tip-speed ratio λ = 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 function and Cp curve are equilibrium points of the turbine operation.
Theorem 2 shows that the equilibrium point λ = 7.5 is locally
asymptotically stable.
74 IEEE CONTROL SYSTEMS MAGAZINE
»
JUNE 2006
Cpmax
λ3∗
λ3
is plotted as the dotted line in Figure 5, and CART’s PROPderived Cp − λ curve for a fixed pitch of −1◦ is the solid
line. A pitch angle β of 0◦ means that the blade chord line
is approximately parallel to the rotor plane, although the
exact 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 λ 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 wellcharacterized turbine to approach the optimal tip-speed
ratio. Although easier to understand under constant wind
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 λ < 3.3, the inequality (11) is
no longer satisfied, and the angular acceleration ω̇ is negative. In this case, the rotor speed ω slows toward zero.
However, most turbines have separate control mechanisms
to ensure that a low tip-speed ratio λ < 3.3 does not drive
the rotor speed ω 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 separate 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 turbine’s properties 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 , with
the same effect as a suboptimally chosen K. The sensitivity
of energy loss to errors in λ∗ and the maximum power
coefficient Cpmax is considered in [4], which concludes that
a 5% error in the optimal tip-speed ratio λ∗ can cause a significant energy loss of 1–3% in region 2. If the United
States meets the American Wind Energy Association’s 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 CONTROL
For region 2 operation, we now consider the adaptive controller [2] given by
0,
ω < 0,
τc =
(12)
ρMω2 , ω ≥ 0,
where the adaptive gain M replaces A, R, Cpmax , and λ∗ in
(6). The air density ρ 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 ω 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 one
direction.
The equations for the gain adaptation law are
M (k) = M (k − 1) + M (k) ,
(13)
M(k) = γM sgn [M(k − 1)] sgn[Pfavg (k)]
× |Pfavg (k)|1/2 ,
(14)
Pfavg (k) = Pfavg (k) − Pfavg (k − 1),
(15)
where k denotes the adaptive controller’s discrete time
step. The fractional average power Pfavg , given by
Pfavg (k) =
1
n
1
n
n
Pcap ((k − 1)n + i )
i=1
n
,
(16)
Pwy((k − 1)n + i )
i=1
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 operation at the discrete-time torque control rate fs =
100 Hz. Pwy, computed at 100 Hz, is the wind power given by
Pwy =
1
ρAv3 (cos ψ)3 ,
2
(17)
where ψ 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 = τc ω + Jωω̇,
(18)
which is also computed at 100 Hz. The yaw error factor
(cos ψ)3 in (17) shows that yaw errors reduce the power
available to the turbine. The term τc ω in the captured power
Pcap is the generator power while Jωω̇ is the kinetic power
(that is, the time derivative of the kinetic energy) of the rotor.
In (13), M is adapted after n time steps of 10-ms periods
of operation in region 2. Testing on CART indicates that
the adaptation period must be on the order of hours; consequently, 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 a
high correlation between measurements of wind speed
over the entire swept area of the rotor and at the
anemometer, which can be located either on the turbine’s
nacelle or on a separate meteorological tower [16]. Another
reason for the long adaptation period is that, since the turbine changes speed at a much slower rate than the wind,
the slow responses must be averaged over time.
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 (dP f avg)/(dM̃) → 0
as M̃ → 0. Thus, |M| decreases as the optimal gain is
approached. The exponent 1/2 is chosen based on simulation, and selection of γM > 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 normally available on industrial turbines. The two definitions of
turbine power are closely related, differing only by the
mechanical losses in the turbine’s gearbox; these losses
make Pcap < P by a small amount.
JUNE 2006
«
IEEE CONTROL SYSTEMS MAGAZINE 75
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 simulation. The turbine’s 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 powercoefficient Cp surface in Figure 4. Since these data are
Fractional Average Power Pfavg
0.42
Pfavg
0.40
0.38
0.36
0.34
0.32
0.30
−100
−50
0
∼
50
100
Gain Error M = M* − M
FIGURE 6 Pfavg versus M̃ for the CART model. Pfavg is the ratio of
the mean captured power to the mean wind power, while M̃ is the
error between the torque control gain M and its optimal value M ∗ .
The shape of this curve is based on the shape of CART’s Cp − λ
curve. In the adaptive controller, the gain adaptation law converges
in part due to the shape of the Pfavg curve.
Normalized M (M/M+)
1.5
1
0.5
0
0
20
40
Time (h)
60
80
FIGURE 7 Adaptive gain M normalized 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, M oscillates
around the value 0.47 M + , which is approximately equal to the true
optimal torque gain M ∗ .
76 IEEE CONTROL SYSTEMS MAGAZINE
»
JUNE 2006
obtained from simulations, the optimal gain M∗ is known.
The error M̃ in M is given by
M̃ = M∗ − M.
The adaptive controller attempts to have the turbine power
track the wind power, assuming that the maximum power
coefficient Cpmax and the optimal tip-speed ratio λ∗ are
unknown. In contrast, adaptive controllers such as those in
[10]–[11] focus on different uncertainties and assume some
knowledge of the Cp surface, particularly λ∗ and Cpmax . In addition, 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 controller
was 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 predicted optimal value M+ . 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.
STABILITY
We 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
CART’s torque is a discrete-time problem, we simplify the
stability analyses of the torque control law (12) by assuming that the torque control is continuous time. This simplification is valid because the control time step of 0.01 s is
much smaller than the tip-speed ratio’s time constant,
which depends on wind speed [19] and is about 4–8 s for
CART operating in region 2 wind speeds of 6–12 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 takes
place discretely and on a time scale several orders of magnitude 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 constrained to be positive since the control torque (12) cannot
be negative. In all of these proofs, the air density ρ is
assumed to be a positive constant. In reality, changes in air
density are small, typically not much greater than 5%. A
simplified block diagram for these continuous-time systems 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 Speed
First, we consider the asymptotic stability of the rotorspeed equilibrium ω = 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 asymptotic stability of the equilibrium point ω = 0, we revise (7) so
that the angular acceleration ω̇ includes a damping term
bω , where the damping coefficient b > 0, which yields
ω̇ = 1J (τaero − τc − bω).
(19)
Using (8) and (12), (19) can be expanded to
ω̇ =
1
2J
1
2J
b
J ω,
Mω2 − bJ
ρARCq v2 −
ρARCq
v2
ρ
J
−
ω < 0,
ω,
ω ≥ 0.
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 controller is given by (12). The adaptive controller (12) does
not assume knowledge of the aerodynamic parameters
Cpmax and λ∗ . Setting the ω ≥ 0 portion of (20) equal to zero
and solving for Cp in terms of λ using (4) and (9) yields
Cp =
ρMλ3 v + λ2 bR
1
3
2 ρAR v
≡ G(λ, M, b, v).
The equilibrium points ω̇ = 0 of turbine operation are thus
given by the intersection of with the turbine’s Cp − λ
curve. Figure 9 shows CART’s Cp − λ curve and two illustrative G(λ, M, b, v) curves plotted using representative
values of ρ, v, and b.
In Figure 9, the cubic functions G(λ, M, b, v) do not intersect the Cp curve at the peak of the curve when the adaptive
(20)
τaero
+
Theorem 1
ω
Linear
Plant
−
Suppose that the wind speed v = 0 and M > 0 are constant. Then the equilibrium ω = 0 of the closed-loop system (20) is asymptotically stable.
τc Nonlinear
Controller
M*
+
−
M
(a)
Proof
For the initial condition ω(0) = ω0 , the solution to (20)
when v = 0 is
ω(t) =
− bJ t
ω0 e
ω0 b
,
bt
(b+ρMω0 )e J −ρMω0
,
(21)
Nonlinear
Plant
Pfavg
Nonlinear
Controller
(b)
FIGURE 8 Control loops for (a) the aerodynamic torque τaero and
rotor speed ω and (b) the gain adaptation law. (a) Stability of the
continuous-time control loop is analyzed by Theorems 1–3, while
Theorem 4 considers (b) the discrete-time adaptive loop.
ω < 0,
ω ≥ 0.
Hence, ω → 0 as t → ∞.
b
=
0
We also note that when the damping coefficient
v
=
0,
and the wind speed
(20) becomes
ω̇ =
0,
ω < 0,
− ρJ Mω2 , ω ≥ 0,
which has the solution
ω(t) =
ω0 ,
J
ρMt +
J
ω0
,
ω < 0,
ω ≥ 0.
In this case, ω → 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 Input
The next stability result concerns the convergence of the
rotor speed ω to an equilibrium value under an idealized
Power Coefficient Cp
0.5
G(λ ,M)
M = 1.3M*
0.4
Cart Cp Versus λ
0.3
G(λ ,M)
M = 0.7M*
0.2
0.1
0
2
4
6
8
10
Tip-Speed Ratio λ
12
14
FIGURE 9 CART’s 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(λ, M) curves
does not occur at the peak of the Cp curve, which leads to suboptimal 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.
JUNE 2006
«
IEEE CONTROL SYSTEMS MAGAZINE 77
gain M = M∗ ; thus, the equilibrium point of the system is
suboptimal in terms of energy capture. Let λ2 be the highest
value of λ for which the curve G(λ, M) intersects Cp(λ).
Mathematically, λ2 is the tip-speed ratio for which
G(λ, M) > Cp(λ) for all λ > λ2 . Let λ1 denote the next highest
intersection point, that is, the value of λ for which
0 < λ1 < λ2 and G(λ, M) < Cp(λ) for all λ1 < λ < λ2 and
G(λ, M) > Cp(λ) for all λ < λ1 within a neighborhood of λ1 .
For the dashed curve M = 0.7 M∗ in Figure 9, these values
correspond to λ1 = 3.1 and λ2 = 8.4. The following result
shows that, for a constant wind input, the tip-speed ratio λ converges to λ2 as long as the initial value of λ is greater than λ1 .
Theorem 2
Suppose that the wind speed v and the adaptive gain M are
positive constants and λ1 > 0. Then the equilibrium point
λ = λ2 of the closed-loop system consisting of the plant
λ̇ =
R
bv
τaero − τc − λ
Jv
R
(22)
and the nonlinear controller (12) is locally asymptotically
stable with domain of attraction λ ∈ (λ1 , ∞).
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 Stability
Next, we show that a bounded input (squared wind speed
v2 ) to the system produces a bounded output (rotor speed
ω). All wind turbines have a maximum safe operating
speed, and often pitch control is used to prevent the turbine from operating at speeds above this maximum. Nevertheless, an enhanced understanding of the wind turbine
control system can be achieved by examining whether the
torque control (12) bounds the turbine speed. The following result considers a time-varying wind speed v.
For T > 0, we use the standard the definition of the L2
norm of v(t) given by
T
vL2 [0,T] =
v(t)2 dt.
0
Theorem 3
Proof
First note that ω > 0 for all 0 < λ1 < λ since ω = λv/R from
(4). Define λ̃ = λ2 − λ and the Lyapunov candidate
V = (1/2)λ̃2 . For ω > 0,
V̇ = (λ − λ2 )
1
1
b
ρAR2 Cq v −
ρMλ2 v − λ
2J
JR
J
(23)
= (λ − λ2 )h(Cq, v, λ).
Substituting Cp/λ for Cq in (23) and applying (21) yields
h(Cq, v, λ) > 0 for all λ such that λ1 < λ < λ2 , that is,
G(λ, M) < Cp(λ). Moreover, λ > λ2 gives G(λ, M) > Cp(λ) by
definition of λ2 , and therefore h(Cq, v, λ) < 0 by definition (21)
of G. Thus, V̇ < 0 for all λ ∈ (λ1 , ∞) except λ = λ2 , for which
V̇ = 0. Hence, the equilibrium point λ = λ2 of (22) is locally
asymptotically stable. Finally, it is easy to show that the
domain of attraction is (λ1 , ∞). Note that V̇ is bounded away
from zero on every connected, compact subinterval of (λ1 , ∞)
that does not contain λ2 . Thus, the time required for λ to reach
the edge of the subinterval closest to λ2 is finite. Now, λ moves
monotonically toward λ2 . If λ does not converge to λ2 , then the
time it takes λ to reach the edge closest to λ2 of a subinterval
not containing λ2 must be infinite, which contradicts the earlier
result. Thus, the domain of attraction is (λ1 , ∞).
λ
λ
The convergence of the tip-speed ratio to 2 is equivalent to the convergence of the rotor speed ω to λ2 v/R for a
specific wind speed v. Furthermore, when M = M∗ , the
curves G(λ, M) and Cp(λ) intersect at (λ∗ , Cpmax ) as shown
for the standard torque control in Figure 5; therefore, optimal
energy capture is achieved for the constant wind input case.
78 IEEE CONTROL SYSTEMS MAGAZINE
»
JUNE 2006
Suppose the rotor torque coefficient Cq ≤ 1, the adaptive
gain M > 0 is constant, and consider the closed-loop turbine system (12), (19) with input given by the squared wind
speed v2 and output given by the rotor speed ω. Then, for
all finite T > 0, the system (12), (19) is L2 stable on [0, T].
Proof
Consider the kinetic energy EK = (1/2)Jω2 of the rotor
and define
V=
1
Jω2 ,
ρAR
(24)
where ρ > 0 is a constant. The time derivative of (24) is
V̇ =
Cq v2 ω −
Cq
v2 ω
−
b
1
2 ρAR
b
ω2
1
2 ρAR
ω2 ,
−
M
ω3 ,
1
2 AR
ω < 0,
ω ≥ 0.
Let δ = b/((1/2)ρAR) . Then, since Cq ≤ 1
(M/((1/2)AR))ω3 ≥ 0 for ω ≥ 0, it follows that
V̇ ≤ v2 ω − δω2 .
(25)
and
(26)
Thus, Lemma 6.5 in [20] implies that the wind turbine system, from the squared wind speed v2 to the rotor speed ω,
is finite-gain L2 stable over [0, T].
T
The restriction that be finite is necessary due to the
nature of the wind speed v(t). Since wind speed v(t) > 0
/ L2 [0, ∞].
can hold at all times, it is possible that v(t) ∈
Thus, T must be finite to guarantee that proof of L2 stability
in [0, T] makes sense.
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/λ [see (9)], it follows that Cq ≤ 1 for λ ≥ 16/27.
When λ ≤ 16/27, it follows from the definition of tip-speed
ratio λ in (4) that ω = λv/R ≤ (16/27)v/R.
For finite η > 0 and λ ∈ [0, T], L∞ , that is, bounded
input, bounded output stability of ω with respect to the
input v is given by ω = λv/R.
Theorem 3 shows that a wind turbine is not a perpetual motion machine. Since the assumption that M is constant 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 adaptation period.
Convergence of the Gain Adaptation Algorithm
The 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 M̃ relationship 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 turbine’s aerodynamic parameters, and thus M∗ ,
change with time due to blade erosion, residue buildup,
and related events. However, we can assume that M∗ is
constant because the turbine’s physical changes are typically noticeable only over months or years, whereas the gain
adaptation law has an adaptation period of less than a day.
Theorem 4
Let k > 2. Under Assumptions 1 and 2 and the gain adaptation law (13)–(15), |M̃k +1 | > |M̃k | > |M̃k −1 | never occurs
when sgn(M̃k +1 ) = sgn(M̃k ) = sgn(M̃k −1 ).
Proof
Suppose M̃k +1 > M̃k > M̃k −1 and sgn(M̃k +1 ) = sgn(M̃k ) =
sgn(M̃k −1 ) = 1 for some k > 2. Note that M̃k > M̃k −1 gives
M̃k − M̃k −1 = −Mk > 0,
(27)
which implies that Mk < 0. Furthermore, M̃k +1 > M̃k gives
M̃k +1 − M̃k = −Mk +1 > 0,
(28)
which implies that Mk +1 < 0. By (16)–(18), Pfavg k +1 is calculated 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 M̃k . Since M̃k > M̃k −1 ,
Assumption 2 implies Pfavg k +1 < Pfavg k . Therefore, by (15),
Pfavg k +1 < 0.
(29)
In (27) and (29), sgn(Mk ) = sgn(Pfavg k +1 ) = −1. Thus, by
(14), sgn(Mk +1 ) = 1 , contradicting (28). Thus, it is
impossible for both M̃k +1 > M̃k > M̃k −1 and sgn
(M̃k +1 ) = sgn(M̃k ) = sgn(M̃k −1 ) = 1 to be true. A similar
argument can be used for negative values of M̃.
M
Since the sign of the adaptation step
cannot be
incorrect for two consecutive steps, the gain γM , which
affects the magnitude of M, is the critical factor in determining whether the adaptive gain diverges. Figure 10
shows an example in which the gain γM is large enough
to cause the adaptive gain M to diverge. In this example,
|M̃k +1 | > |M̃k −1 | for all k > 2, although both |M̃k +1 | > |M̃k |
and |M̃k +1 | < |M̃k | occur when k > 2.
The Pfavg versus M̃ curve has a maximum at M̃ = 0, is continuously differentiable, and is strictly monotonically
increasing on M̃ < 0 and strictly monotonically decreasing
on M̃ > 0. Experimental data [17] support this assumption.
For the initial conditions M0 , Pfavg0 , M0 , and M1 ,
k > 2 is the time frame of interest in the convergence analysis. Theorem 4 covers only the time k > 2 because the first
two steps are more influenced by the initial guesses than by
the turbine’s aerodynamic properties.
We begin the convergence analysis by considering how
the adaptive gain can diverge, that is, |M̃| → ∞ as k → ∞.
One possibility is |M̃k | > |M̃k −1 | with either sgn(M̃k ) = 1
or sgn(M̃k ) = −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 M̃
cannot take two consecutive steps in the wrong (incorrect)
direction for all k > 2, as shown by the following result.
Fractional Average Power Pfavg
Assumption 2
0.50
6 2 48 1
3
0.45
5
0.40
7
0.35
0.30
9
0.25
−20 −15 −10
−5
0
5
10
∼
Gain Error M = M* − M
15
20
FIGURE 10 Adaptive gain steps in an unstable case. The numbers
1–9 indicate the discrete-time steps. In this case, the gain γM in
the gain adaptation algorithm (13)–(15) is too large, and thus the
gain adaptation law diverges.
JUNE 2006
«
IEEE CONTROL SYSTEMS MAGAZINE 79
In the critical gain scenario of this example, the system
alternates among the three points plotted in Figure 11. If
Mk = M̃k−1 , then the error M̃k = 0 by (13). Substituting yk
for Pfavgk in (15) and considering (14), the gain γM is such
that Mk+1 = Mk , resulting in M̃k+1 = −M̃k−1 . Following
the equations through one more step shows that M̃k+2 = 0,
and the adaptive gain alternates among these three points.
Thus, an upper bound on the gain γM for stability can be
found by equating
Fractional Average Power Pfavg
0.43
∼
(Mk, yk+1)
0.42
∼
(Mk+1, yk+2)
0.41
∼
(Mk−1, yk)
−∆Mk+1
−∆Mk
Mk = M̃k−1 = −M̃k+1
0.40
−50
0
∼
Gain Error M = M* − M
50
FIGURE 11 Finding the critical gain γM . Marginal stability of the
gain adaptation law, defined as oscillation among three points on
the y curve (31), occurs when the step size Mk has the same
magnitude as the error M̃k−1 for a symmetric curve.
Since M diverges if |Mk | > |M̃k−1 | for all k > 2, we
consider
|Mk | = |M̃k −1 |,
M̃k −1 = 0
(30)
to be the critical case, or the marginal stability case. Define yk by
yk ≡ aM̃2k−1 + P0 ,
(31)
Fractional Average Power Pfavg
where yk is a curve satisfying Assumptions 1–2 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 γM . For
consistency with the discrete-time indices in the equation
(16) for Pfavgk , yk is a function of M̃k −1 rather than of M̃k .
0.42
Pfavg
0.40
0.38
y
0.36
0.34
0.32
0.30
−100
−50
0
50
∼
Gain Error M = M* − M
100
FIGURE 12 CART Pfavg versus M̃ curve and symmetric inset curve.
The curve labeled Pfavg is identical to the curve shown in Figure 6,
while the quadratic curve labeled y is added to illustrate the method
for selecting the adaptive gain γM . When the quadratic curve is
chosen such that y(M̃) ≤ Pfavg (M̃) and y(M̃) = Pfavg (M̃) if and only
if M̃ = 0, the upper limit on γM for stability of the gain adaptation
law is a function of the coefficient of the squared term in (31).
80 IEEE CONTROL SYSTEMS MAGAZINE
»
JUNE 2006
and solving for γM in terms of a with M̃k = 0, which yields
1
γM = |a|
.
(32)
1
Thus, if 0 < γM < |a|− /2 , then the gain adaptation law
(13)–(15) does not cause divergence of the adaptive
torque control gain error M̃ on the curve (31). In fact,
1
since γM = |a|− /2 is the marginal stability case,
1
0 < γM < |a|− /2 yields M̃ → 0. Since this bound on γM
depends on the magnitude |a|, every gain γM chosen
for a given value of a in (31) also guarantees convergence 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 (M̃) = Pfavg (−M̃)
does not hold. If the gain γM is chosen to guarantee convergence based on the slope of the steeper side of the
curve, then γM guarantees convergence over the entire
curve. Thus, for an arbitrary Pfavg versus M̃ curve, there
exists γM > 0 that guarantees convergence of the adaptive gain M, and this gain γM depends on the steepness of
the Pfavg versus M̃ curve.
Since there are no turbines for which the Pfavg versus M̃
curve is well known, an approximation of the curve is necessary to control each turbine. The more conservative the
choice of γM , 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 M̃ curve for smaller γM . However, a smaller γM also results in smaller
step sizes and thus might cause the convergence to occur
more slowly.
An example of the choice of γM 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 at
M̃ = 0 and satisfying y < Pfavg for M̃ = 0. In this case,
a = −0.00001 m−10 . Thus, the maximum allowable gain
γM for stability is 316 m−5 . The gain used in testing on
CART before this stability analysis was performed was
γM = 100 m−5 , 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 reassurance and guidelines in choosing γM .
CONCLUSIONS
This article considers an adaptive control scheme previously
developed for region 2 control of a variable-speed wind turbine. 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
selecting γM in the gain adaptation law (13)–(15) to guarantee
convergence of the adaptive gain M to its optimal value M∗ .
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 Colorado 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 (kjohnson@mines.edu) 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, respectively. In 2005, she completed a postdoctoral research assignment
studying adaptive control of variable-speed wind turbines at
the National Renewable Energy Laboratory’s 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 control systems and control applications. She can be contacted at
Colorado School of Mines, Division of Engineering, 1610 Illinois 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 engineering at the University of Colorado at Boulder. She has published over 120 journal and conference papers in the area
of control systems. Her awards include the Best Commercial Potential Award at the 2004 International Symposium
on Haptic Interfaces for Virtual Environments and Teleoperator 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 currently an elected member on the IEEE Control Systems Society
Board 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 Electrical and Computer Engineering Department at the University 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 storage, 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.
REFERENCES
[1] “Global wind energy installations climb steadily,” American Wind Energy Association’s Wind Power Outlook 2005 [Online], Mar. 2004, p. 6. Available: http://www.awea.org/pubs/documents/Outlook%202005.pdf
[2] K. Johnson, L. Fingersh, M. Balas, and L. Pao, “Methods for increasing
region 2 power capture on a variable speed wind turbine,” J. Solar Energy
Eng., vol. 126, no. 4, pp. 1092–1100, 2004.
[3] J. Svensson and E. Ulen, “The control system of WTS-3 instrumentation
and testing,” in Proc. 4th Int. Symp. Wind Energy Systems, Stockholm,
Sweden, 1982, pp. 195–215.
[4] L. Fingersh and P. Carlin, “Results from the NREL variable-speed test
bed,” in Proc. 17th ASME Wind Energy Symp.}, Reno, NV, 1998, pp. 233–237.
[5] K. Stol and M. Balas, “Periodic disturbance accommodating control for
speed regulation of wind turbines,” in Proc. 21st ASME Wind Energy Symp.,
Reno, NV, 2002, pp. 310–320.
[6] A. Eggers, H. Ashley, K. Chaney, S. Rock, and R. Digumarthi, “Effects of coupled rotor-tower motions on aerodynamic control of fluctuating loads on lightweight HAWTs,” in Proc. 17th ASME Wind Energy Symp., Reno, NV, 1998, pp.
113–122.
[7] M. Hand, “Load mitigation control design for a wind turbine operating
in the path of vortices,” in Proc. Science of Making Torque from Wind 2004 Special Topic Conf., Delft, The Netherlands, 2004 [CD-ROM].
[8] A. Wright and M. Balas, “Design of controls to attenuate loads in the
Controls Advanced Research Turbine,” J. Solar Energy Eng., vol. 126, no. 4,
pp. 1083–1091, 2004.
[9] S. Bhowmik, R. Spée, and J. Enslin, “Performance optimization for doubly-fed wind power generation systems,” IEEE Trans. Ind. Applicat., vol. 35,
no. 4, pp. 949–958, 1999.
[10] J. Freeman and M. Balas, “An investigation of variable speed horizontalaxis wind turbines using direct model-reference adaptive control,” in Proc.
18th ASME Wind Energy Symp., Reno, NV, 1999, pp. 66–76.
[11] Y. Song, B. Dhinakaran, and X. Bao, “Variable speed control of wind turbines using nonlinear and adaptive algorithms,” J. Wind Eng. Ind. Aerodyn.,
vol. 85, no. 3, pp. 293–308, 2000.
[12] M. Simoes, B. Bose, and R. Spiegel, “Fuzzy logic based intelligent control of a variable speed cage machine wind generation system,” IEEE Trans.
Power Electron., vol. 12, no. 1, pp. 87–95, 1997.
[13] S. Walker and R. Wilson, Performance Analysis Program for Propeller Type
Wind Turbines. Corvallis, OR: Oregon State Univ., 1976.
[14] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind Energy
Handbook. New York: Wiley, 2001.
[15] D. Simms, S. Schreck, M. Hand, and L. Fingersh, “NREL unsteady aerodynamics experiment in the NASA-Ames wind tunnel: A comparison of
predictions to measurements,” NREL Rep. TP-500-29494, 2001 [Online].
Available: http://www.nrel.gov/docs/fy01osti/29494.pdf.
[16] K. Johnson, L. Fingersh, L. Pao, and M. Balas, “Adaptive torque control
of variable speed wind turbines for increased region 2 energy capture,” in
Proc. 2005 ASME Wind Energy Symp., Reno, NV, 2005, pp. 66–76.
[17] L. Fingersh and K. Johnson, “Baseline results and future plans for the
NREL controls advanced research turbine,” in Proc. 23rd ASME Wind Energy
Symp., Reno, NV, 2004, pp. 87–93.
[18] K. Johnson, L. Pao, M. Balas, V. Kulkarni, and L. Fingersh, “Stability
analysis of an adaptive torque controller for variable speed wind turbines,”
in Proc. IEEE Conf. on Decision and Control, Atlantis, Paradise Island,
Bahamas, 2004, vol. 4, pp. 4016–4021.
[19] K. Pierce, “Control method for improved energy capture below rated
power,” Proc. ASME/JSME Joint Fluids Engineering Conf., San Francisco, CA,
1999, pp. 1041–1048.
[20] H. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: PrenticeHall, 2002, p. 242.
[21] P. Gipe, Wind Energy Comes of Age. New York: Wiley, 1995.
JUNE 2006
«
IEEE CONTROL SYSTEMS MAGAZINE 81