Asymptotic Tracking Control for Wind Turbines in Variable Speed Mode

Ⅰ. INTRODUCTION

Renewable energy, especially wind energy, receives increasing attention over the past decade because of environmental concerns and pressures. Wind turbines, the electrical power generation plants from wind power, are structurally large and operating usually in harsh environments. How to efficiently control wind turbines is a challenging problem but of great significance for reducing the cost of wind energy.

Nowadays, most of the utility-scale turbines operate in variable speed mode. The speeds of wind turbines keep changing continuously in response to wind speed variation. The real-time adjustment of the turbine speed allows the wind turbine to operate longer at maximum aerodynamic efficiency, and hence capable of capturing more wind power than the constant speed operation. Additionally, variable speed turbines can filter out the generated power fluctuations and operating load changes that are caused by wind speed variation. Although variable speed wind turbines require complex and costly power electronics systems, the increasing efficiency provides enough benefit to compensate the cost and makes the required power electronics cost effective. As a result, the wind industry is inclined to design and build variable speed turbines [1]. On the other hand, the complex nature of the variable speed wind turbine system brings tremendous challenges, in particular, on how to effectively control variable speed turbines while reducing the cost of wind energy.

Fig. 1 shows a variable speed wind turbine structure and its major components. The wind comes across the rotor and causes it to spin. The low-speed shaft transfers the captured energy to the high speed shaft through the gearbox. The high speed shaft spins the generator and hence generates electricity. The yaw system is used to turn the nacelle so that the rotor faces directly into the wind.

Figure  1.  Wind turbine structure and major components (Figure courtesy of the U.S. Department of Energy [2]).

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A variable speed wind turbine has several levels of control systems. On the top level, a supervisory controller monitors the wind speed and direction to determine when the wind speed is sufficient to start up the turbine and when to shut down the wind turbine for safety due to too high wind speed. Typically, the operating modes of wind turbine are divided into three regions associated with wind speed, maximum allowable rotor speed and rated power. Fig. 2 shows a power curve versus wind speed for a wind turbine with three regions divided. $V_{\rm cut-in}$ is the {cut-in} wind speed at which the wind turbine starts to spin and $V_{\rm cut-off}$ is the wind speed at which the maximum allowable rotor speed is reached.

Figure  2.  Output power of a typical wind turbine operating in different wind speed regions, denoted by Region Ⅰ (from $V_{\rm cut-in}$ to $V_{\rm 12}$), Region Ⅱ (from $V_{\rm 12}$ to $V_N$), and Region Ⅲ (from $V_N$ to $V_{\rm cut-off}$), respectively.

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A wind turbine that is just starting up is considered to be operating in Region Ⅰ, i.e., the area from $V_{\rm cut-in}$ to $V_{\rm 12}$ shown in Fig. 2. In Region Ⅱ (the area from $V_{\rm 12}$ to $V_N$ of Fig. 2), the wind turbine is operating to maximize wind energy capture, which is realized by variable speed control with the pitch angle being usually kept constant. In Region Ⅲ ---the region ranges from $V_N$ to $V_{\rm cut-off}$ of Fig. 2, the turbine reaches and maintains its rated power $P_{N}$. The captured power cannot be further increased even with higher speed wind, due to electrical and mechanical safety concerns of wind turbine systems.

The lower level of wind turbine control systems is turbine speed control, which includes generator torque control and blade pitch control. Generator torque control is to determine how much torque needs to be extracted from the turbine given the present wind speed. The extracted torque offsets the aerodynamic torque caused by the wind, and thus indirectly regulates the turbine speed. On the other hand, pitch control, manipulated by the pitch actuators, adjusts the pitch angle to the wind inflow and consequently regulates the turbine speed. In this paper, we focus on turbine control in Region Ⅱ, that is, the region from $V_{\rm 12}$ to $V_N$ of Fig. 2, where pitch angle is usually kept constant and only torque control takes effect.

In the existing literature, there are some works that investigated modeling and control of variable speed wind turbine, such as [1], [3]-[10], just to name a few. Adaptive control design for wind turbines was studied in [3]. Based on the same model, a robust backstepping approach has been applied for the control system design [4]. Nonlinear power regulation for wind turbine was considered by [7]. Sliding mode and disturbance rejection controls were proposed in [6] and [10], respectively. The paper [5] developed a nonlinear control approach without wind speed measurement, yielding a robust nonlinear controller. More recently, the work [11] has developed a new maximum power point tracking (MPPT) algorithm for variable-speed wind turbine systems, by taking advantage of the rotor inertia power, while the paper [12] presented a MPPT algorithm that eradicates the problems that exist in the conventional hill-climbing MPPT technique. The proposed MPPT is a sensorless adaptive algorithm with higher control efficiency and faster tracking under rapid wind change. without requiring prior knowledge of the system's characteristics.

Wind turbines have been modeled as a one-mass drive-train system in several papers; see, for instance, [1], [5], and [6]. Although the one-mass drive-train model has a simple structure and is easy for the control design, it cannot capture the dynamics of torsional effects, which has a significant influence on the power fluctuations and the interaction of the wind turbine with the grid. Moreover, oversimplifying the models of wind turbine drive train could introduce significant error in dynamic behavior and stability [13]. Incorporating torsional effects, a linear system based on a two-mass drive-train model has been proposed in [9] for stability analysis. However, taking into account the aerodynamics, the wind turbine system is actually a nonlinear plant that is not in the so-called normal form [14]. This certainly raises a challenging issue on the design of nonlinear controllers for wind turbine systems described by a two-mass drive-train model, without linearizing the model.

In this paper, we present a geometric nonlinear control method to study the global tracking problem for variable speed wind turbines. The wind turbine is modeled as a two-mass drive-train system with generator torque control, which can be reduced to a one-mass drive-train system by assuming that the low speed shaft is perfectly rigid. Using the two-mass drive-train model, variable speed control schemes are investigated for Region Ⅱ. We design nonlinear tracking controllers to achieve global asymptotic tracking for a prescribed rotor speed reference signal so that maximum wind power capture can be yielded. Due to the difficulty of torsional angle measurement, we further develop an observer-based control scheme by using only the information of rotor speed to fulfill global asymptotic output tracking. Simulation results are given to illustrate the effectiveness of the proposed nonlinear control schemes.

Ⅱ. WIND TURBINE MODELING

The power extraction of wind turbine is a function of three main factors: the wind power available, the power curve of the machine and the ability of the machine to respond to wind fluctuation. As explained in [15] and [16], the available wind power is given by

and the power captured by the wind turbine is

where $\rho$ is the air density, $R_w$ is the blade tip radius, $V_w$ is the wind speed and $C_p$ is the power coefficient, which is a function of the tip-speed ratio $\lambda$ and the pitch angle $\beta$. Note that the tip-speed ratio is defined as

where $\omega$ is the rotor speed.

From the governing equation (2) for the generated power out of the wind turbine, it is clear that if the rotor speed is kept as a constant, a change in the wind speed will change the tip-speed ratio, leading to the changes of power coefficient $C_p$ as well as the generated power out of the wind turbine. However, if the rotor speed is adjusted according to the wind speed variation, then the tip-speed ratio can be maintained at an optimal point, which could yield maximum power output from the system [3]. Consequently, we have the following relationship between captured wind power, rotor speed and rotor torque $T_a$,

where

As shown in Fig. 3, the dynamics of wind turbine drive-train system can be modeled as a two-mass drive-train system. Driven by the aerodynamic torque $T_a$, the rotor of the wind turbine runs at the speed $\omega$. The aeroturbine converts wind energy into mechanical energy, the gearbox serves to increase the speed $\omega_g$ and decrease the torque from low speed shaft to the high speed shaft. The generator is to convert mechanical energy into electrical energy. $J_r$, $J_g$, $T_{ls}$, $T_{hs}$ and $T_e$ denote rotor inertia, generator inertia, low speed shaft torque, high speed shaft torque and generator torque, respectively, $b_r$, $b_g$, $C_s$ and $K_s$ are the damping and torsional coefficients.

Figure  3.  The two-mass drive-train model consists of turbine aerodynamics, low speed shaft, gearbox, high speed shaft and generator.

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The dynamics of the rotor is characterized by the first order differential equation

The low shaft speed results from the torsion and friction effects due to the difference between $\omega$ and $\omega_{ls}$

The generator is driven by the high speed shaft torque $T_{hs}$ and braked by the generator electromagnetic torque $T_e$.

Through the gearbox, the low shaft speed $\omega_{ls}$ is increased by the gearbox ratio $n_g$ to obtain the generator speed $\omega_g$ while the low speed shaft $T_{ls}$ torque is augmented.

Let $\theta_k=\theta-\theta_{ls}$. In view of the equations (4)-(8), the dynamics of the two-mass drive-train model can be represented by

If a perfectly rigid low speed shaft is assumed, then $\omega=\omega_{ls}$. In this case, the two-mass drive-train model reduces to a one-mass drive-train model shown in Fig. 4. The governing dynamic equation is given by

Figure  4.  The one-mass drive-train model.

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where $J_{t} =J_{r}+n_{g}^{2}J_{g},$ $b_{t} =b_{r}+n_{g}^{2}b_{g},$ $T_{g} =n_{g}T_{e}$.

Using (4), the dynamics of the one-mass drive-train model can be expressed as

Ⅲ. CONTROL DESIGN

It is known that if the rotor speed is adjusted according to the wind speed variation, then the tip-speed ratio can be maintained at an optimal point, which could yield maximum power output from the system. A wind turbine control block diagram is shown in Fig. 5. The rotor speed of the wind turbine is controlled by the load torque. According to the feedback information of the current rotor speed, the control input $T_e$ is adjusted such that the rotor speed $\omega$ can track the desired rotor speed $\omega^*(t)$. Therefore, the tracking control problem in Region Ⅱ of Fig. 2 can be stated as follows. Design, if possible, a generator control torque $T_g$ or $T_e$ such that the rotor speed $\omega$ of the wind turbine globally asymptotically tracks the desired speed $\omega^{*}(t)$. Throughout this work, it is assumed that $\omega^*(t)$, $\dot{\omega}^*(t)$ and $\ddot{\omega}^*(t)$ are bounded.

Figure  5.  A block diagram for wind turbine feedback control system.

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A State Feedback Design for the Two-mass Drive-train Model

For the two-mass drive-train model (9)-(11) of wind turbine system, the control objective is to force the output signal $\omega$ to track a prescribed reference signal $\omega^{*}(t)$. For the convenience of application of the geometric control method, the nonlinear system (9)-(11) can be transformed into a normal form [14] by introducing the change of coordinates

Obviously, the transform above is a globally diffeomorphism. As a result, the nonlinear system in the new coordinates $(z, \xi_1, \xi_2)$ is expressed as

where

the system output is $\xi_1=\omega$, $\phi_{1}(\xi _{1})$ is a quadratic function of $\xi_1$, and $\phi_{2}(z, \xi_{1}, \xi_2)$ is linear with respect to $z,$ $\xi_1$ and $\xi_2$.

Clearly, the nonlinear system (17)-(20) is minimum-phase because the zero dynamics $\dot z= -({K_s}/{C_s})z$ is asymptotically stable. However, both the system output $\xi_1$ and the state $\xi_2$ drive the zero-dynamic system (17). This makes the global asymptotic tracking problem nontrivial. On the other hand, because the zero-dynamic system (17) is linear, it might be possible to utilize the idea of partial state feedback design [17] -[19], together with the tracking control strategy exploited in [20], to achieve global output tracking with stability. It turns out this idea works, as illustrated in the rest of this section.

To sum up, the wind turbine maximum power capture problem boils down to a global output tracking problem for the minimum-phase nonlinear system (17)-(20). The following condition characterizes what type of the reference signals can be tracked for variable speed wind turbines.

Assumption 1: A family of reference signals to be tracked, namely, $y_r(t)=\omega^*(t)$, is assumed to be a bounded continuous function of time $t$ with bounded first and second derivatives.

The first result of this paper is devoted to the global asymptotic tracking problem, illustrating how the problem can be solved by state feedback for the minimum-phase nonlinear system (17)-(20).

Theorem 1: Consider the equivalent wind turbine system (17)-(20). Given a prescribed reference speed signal $\omega^*(t)$ satisfying Assumption 1, there exists a state feedback controller $v=\beta (z, \xi _{1}, \xi_{2}, y_{r}, \dot{y}_{r}, \ddot{y}_{r})$, with $\beta (0, 0, 0, y_{r}, \dot{y}_{r}, \ddot{y}_{r})=0$, such that the rotor speed $\xi_1=\omega$ asymptotically tracks the desired speed $y_r(t)=\omega^*(t)$, while maintaining global stability of the closed-loop system.

Proof: We begin with our proof by introducing the change of coordinates as done in [20]:

where $\xi _{2}^{d} =\dot{y}_{r}-\phi _{1}(y_{r}),$ $v^{d} =\dot{\xi}_{2}^{d}-\phi _{2}(z, y_{r}, \xi _{2}^{d})$.

Then, the nonlinear system (17)-(19) can be represented as

where

with $\hat{\phi }_{1}(0, y_{r})=0$ and $\hat{\phi }_{2}(0, 0, y_{r}, \xi _{2}^{d})=0$.

The asymptotic tracking control problem is to design a state feedback controller (using only the information of $e_1$ and $e_2$) for the nonlinear system (21)-(23) such that $e_1\rightarrow0$ as $t$ tends to $+\infty$, while maintaining global stability of the closed-loop system. Such a partial state feedback controller can be designed step by step by using the Lyapunov stability theory.

In the first step, consider the Lyapunov function

Taking a time derivative yields

where the virtual controller

Next, consider the Lyapunov function

whose time derivative along the system (22) and (23) satisfies

Obviously, the partial state feedback controller

with $\beta_{2}(0, 0, y_{r}, \dot{y}_{r})=0$ renders $\dot{V}_{2}$ negative definite. In fact,

which implies global asymptotic stability of $(e_1, e_2)$-subsystem of the nonlinear system (21)-(23).

Note that the zero dynamics $\dot z= -({K_s}/{C_s})z$ of the nonlinear system (21)-(23) is asymptotically stable. In addition, the $z$-system (21) also satisfies the BIBO (bounded-input bounded output) property when viewing $e_1,$ $e_2$ and $\delta _{0}(y_{r}, \dot{y}_{r})$ as the system inputs. Consequently, the state $z(t)$ is uniformly bounded $\forall t$, simply because $(e_1(t), e_2(t))\rightarrow 0$ as $t$ tends to $+\infty$ and $\delta _{0}(y_{r}, \dot{y}_{r})$ is bounded. Thus, the nonlinear system (21)-(23) is globally stabilizable by the partial state feedback controller (25), with an asymptotic convergent property of the components $(e_1, e_2)$. In particular,

In conclusion, the rotor speed of the wind turbine described by the two-mass drive-train model (17)-(19) can asymptotically track a prescribed speed $y_r(t)=\omega^*(t)$ by the smooth state feedback controller

with $\beta (0, 0, 0, y_{r}, \dot{y}_{r}, \ddot{y}_{r})=0$.

B Observer Based Control for the Two-mass Drive-train Model

The state feedback tracking controller (26) requires the information of all the system states, namely, $\theta_k$, $\omega$ and $\omega_g$. However, the torsional angle $\theta_k$ is usually difficult to be measured in practice. For this reason, we propose an observer based control design, using $\omega$ and $\omega_g$ information only, to achieve asymptotic tracking for a given reference speed signal $y_r(t)$ $=$ $\omega^*(t)$.

Using the same change of coordinates (14)-(16) for the nonlinear system (9)-(11), we arrive at (17)-(20) in $(z,$ $\xi_1,$ $\xi_2)$ coordinates. Note that $\xi_2$ is not available for feedback design because it involves $\theta_k$ and the only measurable state is $\xi_1$. Keeping this in mind, the problem boils down to the design of an output feedback tracking controller for the nonlinear system (17)-(20) by using $\xi_1$ information only.

The following theorem is the main result of this work, showing how asymptotic tracking can be achieved by output feedback for the minimum-phase nonlinear system (17)-(20).

Theorem 2: Consider an equivalent wind turbine system (17)$-$(20). Given a prescribed reference speed signal $\omega^*(t)$ satisfying Assumption 1, there exists a smooth output feedback controller of the form

with $\Gamma (0, 0)=0$ and $\gamma (0, 0, y_{r}, \dot{y}_{r}, \ddot{y_{r}})=0$, such that the rotor speed $y=\xi_1=\omega$ globally asymptotically follows the desired speed $y_{r}(t)=\omega^*(t)$, while maintaining global stability of the closed-loop system.

Proof: This theorem can be proved by designing an output feedback controller with a reduced-order observer. Following the idea in [20], [21], we first construct a reduced-order observer to estimate the states $z$ and $\xi_2$. Due to the presence of the zero dynamics that were not considered in [20], the design of tracking controller here also involves the estimation of the state $z$ of the zero-dynamics. Thanks to the linearity of the zero-dynamic equation (17) and the minimum-phase property, an output feedback tracking controller for the wind turbine system (17)-(20) can be constructed. For the convenience of the reader, we break up the design into two parts.

Part Ⅰ: Reduced-order Observer Design.

Consider the error dynamic system (21)-(23) in $(z, e_1, e_2)$ coordinates. Introduce the rescaling transformation [20], [21]

where $M>0$ is a gain constant to be determined later on.

In the $(\eta _{1}, \eta _{2})$ coordinate, (21)-(23) can be rewritten as

Define

where $L_1,$ $L_2>0$ are constants to be determined later.

Naturally, the estimations of $z$ and $\eta_2$ can be given by

where $\hat{\chi}_1$ and $\hat{\chi}_2$ are the estimations of $\chi _{1}$ and $\chi _{2}$, respectively, generated by the reduced-order observer

Introduce the estimate errors

Obviously, the errors satisfy the dynamic equation

Recall that the function $\hat{\phi}_2(\cdot)$ is linear with respect to $\eta_1$, $\eta_2$, $y_r$ and $\dot{y_r}$. Hence, it is easy to check that

where $c>0$ is a known constant.

Consider the Lyapunov function

whose time derivative satisfies

Choosing

results in

which implies asymptotic stability of the error dynamics (34).

Part Ⅱ: Output Feedback Design.

To facilitate the design of an output tracking controller, we observe from (30)-(32) and (34) that the closed-loop system can be expressed as

with $\eta_1=e_1$ and $\hat{\eta}_2={\hat{e}_2}/{M}$ (in view of the rescaling transformation (29)).

Note that $(\eta_1, \hat \eta_2)$-subsystem of (37) can be represented in $(e_1, \hat{e}_2)$-coordinate as

For the system (38), consider the Lyapunov function

with $W_{0}(0, 0, y_r)=0$, where ${e}_{2}^{*}$ is given by (24).

A straightforward calculation shows that along the trajectories of (38), the time derivative of $W_{0}$ satisfies

Clearly, the implementable controller

is such that

Now, consider the Lyapunov function

for the closed-loop system (37).

In view of (36) and (40), it is easy to see that

which, in turn, implies asymptotic stability of $\varepsilon$-dynamics of (37) and $(e_1, \hat e_2)$-subsystem, and hence asymptotic stability of $(e_1, e_2)$-subsystem as well.

Finally, examine $z$-subsystem of the system (21). Obviously, the state $z(t)$ remains uniformly bounded $\forall t$. This conclusion follows from the BIBO property of linear systems and the facts that $\delta _{0}(y_{r}, \dot{y}_{r})$ is bounded and $(e_1(t), e_2(t))\rightarrow 0$ as $t$ goes to $+\infty$. In summary, the closed-loop system formed by (21)-(23), (33) and (39) is globally stable. Moreover, $(e_1, e_2)$-subsystem is asymptotically stable at $(e_1, e_2)=(0, 0)$. Therefore, asymptotic output tracking for the minimum-phase nonlinear system (17)-(19) can be achieved by an output feedback controller that is composed of the reduced-order observer (33) and the realizable controller (39) or, equivalently,

with $\gamma (0, 0, 0, y_{r}, \dot{y}_{r}, \ddot{y_{r}})=0$.

By construction, the output feedback controller (33) and (41) is of the form (27) and (28).

Remark 1: For the closed-loop system represented by the $(z, e)$-coordinates and $\varepsilon$-dynamics, global asymptotic stability is only achieved for the $(e, \varepsilon)$-subsystem while global stability of the $z$-dynamics is obtained. In other words, the proposed output feedback control strategy ensures not only global asymptotic tracking for the output signal $\xi_1$ of the wind turbine system, but also global stability of the closed-loop system.

So far it has been proved that a nonlinear state feedback controller and an observer-based output feedback controller can be designed for the two-mass drive-train model, globally asymptotically tracking a prescribed rotor speed reference signal as long as the signal to be tracked is bounded with bounded two-time derivatives. The proposed control scheme is applicable in Region Ⅱ of wind turbine operation (see Fig. 2), achieving the maximum wind power capture. The effectiveness of the tracking control scheme presented in this paper will be validated by simulations in the next subsection.

C Simulation Study

A simulation study is conducted to illustrate the proposed tracking control schemes. The wind turbine under study is with 1.7 MW rated power and the diameter of its blades is 35 meters [22]. The physical parameters are given in Table Ⅰ.

Table  Ⅰ.  WIND TURBINE'S PARAMETERS

Symbol	Value	Symbol	Value
$J_r$	$4\times10^6$ kg$\cdot$m$^2$	$J_g$	20 kg$\cdot$m$^2$
$b_r$	980 N$\cdot$m/s	$b_g$	.2 N$\cdot$m/s
$n_g$	38.06	$k_w$	$1.14\times10^5$ kg$\cdot$m$^2$
$K_s$	$10\times6$ N$\cdot$m	$C_s$	500 N$\cdot$m/s

 | Show Table

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The rotor speed is required to track the desirable reference signal

The simulation results with the state and output feedback controllers are shown in Figs. 6 and 7, respectively. It clearly demonstrates that the proposed control schemes are capable of achieving asymptotic tracking, while maintaining stability of the closed-loop system. As indicated in the simulation results, the settling time of the closed-loop system in the case of either state or output feedback is pretty short and reasonable, which is what we expect.

Figure  6.  Rotor speed tracking with state feedback controller.

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Figure  7.  Rotor speed tracking with output feedback controller.

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Ⅳ. CONCLUSIONS

Variable speed control for wind turbine is necessary to increase the efficiency of wind power generation. In this paper, wind turbine system modeling and nonlinear tracking control have been investigated. A wind turbine is modeled as a two-mass drive-train system, which can be reduced to a one-mass drive-train model if the low speed shaft is assumed to be perfectly rigid. By introducing a suitable change of coordinates, the two-mass drive-train system can be transformed into a nonlinear system in a tractable form. By taking advantage of the minimum-phase property and the linearity of the zero-dynamic system of wind turbine, we have developed both state and output feedback tracking controllers that can achieve asymptotic tracking for the rotor speed with global stability. Analysis and simulations have indicated that the proposed methods are capable of achieving satisfactory rotor speed tracking, with global stability of the closed-loop system.