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
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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
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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.)
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(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.
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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 ω˙ 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 ω˙ =
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 ω˙ 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 ω˙ 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 ω˙ 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
ω˙ =
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.
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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 ω˙ 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 + Jωω,
(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 Jωω˙ 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 ˜ →0 M results in a smaller |Pfavg | because (dP f avg)/(dM) ˜ M → 0. |M| Thus, decreases as the optimal gain is as 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.
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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 M˜ for the CART model. Pfavg is the ratio of the mean captured power to the mean wind power, while M˜ is the error between the torque control gain 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 ∗ .
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obtained from simulations, the optimal gain M∗ is known. ˜ in M is given by The error M ˜ = M∗ − M. M The adaptive controller attempts to have the turbine power track the wind power, assuming that the maximum power coefficient Cpmax and the optimal tip-speed ratio 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 ω˙ includes a damping term bω , where the damping coefficient b > 0, which yields ω˙ = 1J (τaero − τc − bω).
(19)
Using (8) and (12), (19) can be expanded to ω˙ =
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 ω˙ = 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 ω˙ =
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.
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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 R bv τaero − τc − λ 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 λ˜ = λ2 − λ and the Lyapunov candidate V = (1/2)λ˜ 2 . For ω > 0, V˙ = (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, V˙ < 0 for all λ ∈ (λ1 , ∞) except λ = λ2 , for which V˙ = 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 V˙ 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.
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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 V˙ =
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 V˙ ≤ 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 ˜ relationaverage power Pfavg versus torque gain error M ship shown in Figure 6 and the nonlinear controller is given by (13)–(15). We make two assumptions before studying the stability properties of the gain adaptation law.
Assumption 1
The optimum torque control gain M∗ is constant. The 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 adap˜ k | > |M ˜ k −1 | never occurs ˜ k +1 | > |M tation law (13)–(15), |M ˜ k +1 ) = sgn(M ˜ k ) = sgn(M ˜ k −1 ). when sgn(M
Proof
˜ k +1 > M ˜k >M ˜ k −1 and sgn(M ˜ k) = ˜ k +1 ) = sgn(M Suppose M ˜k >M ˜ k −1 ) = 1 for some k > 2. Note that M ˜ k −1 gives sgn(M ˜k −M ˜ k −1 = −Mk > 0, M
(27)
˜ k +1 > M ˜ k gives which implies that Mk < 0. Furthermore, M ˜ k +1 − M ˜ k = −Mk +1 > 0, M
(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 ˜ k . Since M ˜k >M ˜ k −1 , while the torque gain error was M 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 ˜ k +1 > M ˜k >M ˜ k −1 and sgn impossible for both M ˜ k ) = sgn(M ˜ k −1 ) = 1 to be true. A similar ˜ k +1 ) = sgn(M (M ˜ argument can be used for negative values of M.
Since the sign of the adaptation step M cannot be incorrect for two consecutive steps, the gain 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, ˜ k +1 | > |M ˜ k −1 | for all k > 2, although both |M ˜ k +1 | > |M ˜ k| |M ˜ ˜ | < | M | k > 2. | M occur when and k +1 k
˜ = 0, is con˜ curve has a maximum at M The Pfavg versus M tinuously differentiable, and is strictly monotonically ˜ < 0 and strictly monotonically decreasing increasing on M ˜ > 0. Experimental data [17] support this assumption. on M 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 ˜ → ∞ as k → ∞. the adaptive gain can diverge, that is, |M| ˜ k | > |M ˜ k −1 | with either sgn(M ˜ k) = 1 One possibility is |M ˜ ) = −1 k > 2. M for all However, it is easy to show or sgn( k that this scenario cannot occur with the gain adaptation ˜ law (13)–(15). Indeed, the adaptive torque gain error M cannot take two consecutive steps in the wrong (incorrect) direction for all k > 2, as shown by the following result.
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.
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In the critical gain scenario of this example, the system alternates among the three points plotted in Figure 11. If ˜ k−1 , then the error M ˜ k = 0 by (13). Substituting yk Mk = M P for favgk in (15) and considering (14), the gain γM is such ˜ k+1 = −M ˜ k−1 . Following that Mk+1 = Mk , resulting in M ˜ k+2 = 0, the equations through one more step shows that M 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
˜ k−1 = −M ˜ k+1 Mk = M
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 M˜ k−1 for a symmetric curve.
˜ k−1 | for all k > 2, we Since M diverges if |Mk | > |M consider ˜ k −1 |, |Mk | = |M
˜ k −1 = 0 M
(30)
to be the critical case, or the marginal stability case. Define yk by ˜ 2 + P0 , yk ≡ aM k−1
(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 ˜ k −1 rather than of M ˜ k. (16) for Pfavgk , yk is a function of M
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 M˜ 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 ˜ and y(M) ˜ = Pfavg (M) ˜ if and only ˜ ≤ Pfavg (M) chosen such that y(M) if M˜ = 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).
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˜ k = 0, which yields and solving for γM in terms of a with M 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 ˜ on the curve (31). In fact, torque control gain error M 1 since γM = |a|− /2 is the marginal stability case, 1 ˜ → 0. Since this bound on γM 0 < γM < |a|− /2 yields M 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 ˜ = Pfavg (−M) ˜ [as in (31)], that is, one for which Pfavg (M)
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, there curve. Thus, for an arbitrary Pfavg versus M exists γM > 0 that guarantees convergence of the adaptive gain M, and this gain γM depends on the steepness of ˜ curve. the Pfavg versus M ˜ Since there are no turbines for which the Pfavg versus M 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 ˜ curve for smallerrors in the approximated Pfavg versus M er 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 ˜ = 0. In this case, ˜ = 0 and satisfying y < Pfavg for M M 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 (
[email protected]) received the B.S. degree in electrical engineering from Clarkson University in 2000 and the M.S. and Ph.D. degrees in electrical engineering from the University of Colorado in 2002 and 2004, 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.
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