H_∞ Fault Tolerant Control of WECS Based on the PWA Model

Abstract

The main contribution of this paper is the development of H_∞ fault tolerant control for a wind energy conversion system (WECS) based on the stochastic piecewise affine (PWA) model. In this paper the normal and fault stochastic PWA models for WECS including multiple working points at different wind speeds are established. A reliable piecewise linear quadratic regulator state feedback is designed for the fault tolerant actuator and sensor. A sufficient condition for the existence of the passive fault tolerant controller is derived based on some linear matrix inequalities (LMIs). It is shown that the H_∞ fault tolerant controller of WECS can control the wind turbine exposed to multiple simultaneous sensor faults or actuator faults; that is, the reliability of wind turbines can be improved.

1. Introduction

A wind energy conversion system is a mechanical electronic hydraulic integrated system which consists of rotor, drive train, gear box, generator and other mechanical equipment. WECS driven by stochastic wind power signal indicate nonlinear switching system properties. Control systems play a vital role in satisfying harvested power and load alleviation objectives in wind turbines. The performance of the designed controller can be easily interrupted by possible faults and failures in different parts of the system. Therefore, designing a fault-tolerant controller is very beneficial in wind turbine operations.

So far, the design of fault-tolerant control for wind turbine systems is still lacking in studies. In [1], Sloth et al. presented active and passive fault tolerant controllers for wind turbines. The linear parameter varying control design method is applied which leads to LMI-based optimization in case of active fault tolerant and BLMI in case of passive fault-tolerant problems. In [2], the authors gave the model-based fault detection and control loop reconfiguration for doubly fed induction generators. In [3], generator-converter fault-tolerant control for direct driven wind turbines was investigated. In [4], a fault-tolerant switched reluctance motor was designed for the blade pitch control system.

The above mentioned methods are all restricted to the nonlinear characteristics of wind turbines. In order to overcome the nonlinear characteristics of wind turbines, H_∞ fault tolerant strategy for WECS is proposed in this paper based on a stochastic PWA model framework.

Piecewise linear systems provide a powerful tool of analysis and design for nonlinear control systems. The piecewise linear system framework can be used to analyze smooth nonlinear systems with arbitrary accuracy. Many other classes of nonlinear systems can also be approximated by piecewise linear systems [5].

A number of results have been obtained in controller design of such piecewise continuous time linear systems during the last few years [6–9]. In the case of discrete time, the authors of [10] presented an approach for stabilization of piecewise linear systems based on a global quadratic Lyapunov function. In [5,11], the authors gave a number of results on stability analysis, controller design, H_∞ analysis, and H_∞ controller design for the piecewise linear systems based on a piecewise Lyapunov function. In [5], for H_∞ control synthesis, the affine term was treated as a disturbance. In a recent paper [12], a new method was presented to synthesize the H_∞ controller for the piecewise discrete time linear systems. However, studies solving the fault tolerant control problem of the WECS within the stochastic PWA framework are lacking. In this paper, the stochastic PWA normal, sensor fault and actuator fault models for WECS including multiple work regions are established. A reliable piecewise linear quadratic regulator state feedback is designed such that it can make the actuator and sensor faults tolerant. A sufficient condition for the existence of the passive fault tolerant controller is derived based on some LMIs. The paper is organized as follows: in Section 2, a dynamic model of WECS and the control strategy are briefly described. In Section 3, the normal and fault stochastic PWA models for WECS including multiple working points at different wind speeds are established. H_∞ fault tolerant control for WECS with actuator and sensor faults is also presented. In Section 4, some simulation results are presented. Section 5 gives our conclusions.

2. Literature Review

2.1. Model Structure of WECS

The structure of a WECS is described as Figure 1, in which the WECS inputs are wind speed ν(t), pitch angle reference β_ref(t) and generator torque reference T_gref(t), respectively. The outputs of the system are generator power P_g(t) and high-speed shaft speed ω_g(t).

2.2. Wind Model

The equation of wind ν(t) is:

$$v\left(t\right)=v_s\left(t\right)+v_m(t)$$

(1)

where ν_m(t) is the low-frequency component (describing long term, low-frequency variations), i.e., average wind speed; and ν_s(t) is the turbulence component (corresponding to fast, high-frequency variations).

2.3. Aerodynamics Model

The available power in the wind P_w can be expressed as:

$$P_w=\frac{1}{2}\rho A{\nu }^3$$

(2)

where A is the rotor swept area; ν is the rotor effective wind speed; ρ is the air density, which is assumed to be constant.

From the available power in the wind, the power captured by the rotor P_a is:

$$P_a=P_w{C}_p(\lambda ,\beta )$$

(3)

where C_p(λ, β) is the power coefficient, which depends on the tip-speed ratio λ and the pitch angle β.

The tip-speed ratio λ is defined as the ratio between the tip speed of the blades and the rotor effective wind speed:

$$\lambda =\frac{R{\omega }_r}{\nu }$$

(4)

where ω_r is the low-speed shaft speed; and R is the blade length.

2.4. Drive Train Model

The drive train model includes a low-speed shaft, a high-speed shaft, a gear box and flexible device. The drive train dynamics function is given:

$$\{\begin{array}{l}{\dot{\omega }}_r=\frac{T_r}{J_r}-\frac{{\omega }_r{D}_s}{J_r}+\frac{{\omega }_g{D}_s}{J_r{N}_g}-\frac{\delta K_s}{J_r}\hfill \\ \hfill \\ {\dot{\omega }}_g=\frac{{\omega }_r{D}_s}{J_g{N}_g}-\frac{{\omega }_g{D}_s}{J_g{N_g}^2}+\frac{\delta K_s}{J_g}-\frac{T_g}{J_g}\hfill \end{array}$$

(5)

where J_r and J_g are the moments of inertia of the low-speed shaft and the high-speed shaft; K_s is the torsion stiffness of the drive train; D_s is the torsion damping coefficient of the drive train; N_g is the gear ratio; δ is the twist of the flexible drive train with $\dot{\delta }={\omega }_r-\frac{{\omega }_g}{N_g}$.

2.5. Pitch System Model

The pitch system can be modeled by a second-order transfer function [13]:

$$\{\begin{array}{l}\dot{\beta }=\dot{\beta }\hfill \\ \ddot{\beta }=-{\omega }_n^2\beta -2\xi {\omega }_n\dot{\beta }+{\omega }_n^2{\beta }_{ref}\hfill \end{array}$$

(6)

2.6. Generator and Converter Model

The generator and converter dynamics can be modeled by a first-order transfer function:

$${\dot{T}}_g=-\frac{1}{{\tau }_g}{T}_g+\frac{1}{{\tau }_g}{T}_{gref}$$

(7)

where τ_g is the time constant.

The real-time power is described by:

$$P_g={\eta }_g{\omega }_g{T}_g$$

(8)

Then, the dynamics of the WECS can be obtained by combining Equations (5)–(7):

$$\left[\begin{array}{c}{\dot{\omega }}_r\\ {\dot{\omega }}_g\\ \dot{\beta }\\ \ddot{\beta }\\ {\dot{T}}_g\end{array}\right]=\left[\begin{array}{c}\frac{T_r}{J_r}-\frac{{\omega }_r{D}_s}{J_r}+\frac{{\omega }_g{D}_s}{J_r{N}_g}-\frac{\delta K_s}{J_r}\\ \frac{{\omega }_r{D}_s}{J_g{N}_g}-\frac{{\omega }_g{D}_s}{J_g{N_g}^2}+\frac{\delta K_s}{J_g}-\frac{T_g}{J_g}\\ \dot{\beta }\\ -{\omega }_n^2\beta -2\xi {\omega }_n\dot{\beta }\\ -\frac{1}{{\tau }_g}{T}_g\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ {\omega }_n^2& 0\\ 0& \frac{1}{{\tau }_g}\end{array}\right]\left[\begin{array}{c}{\beta }_{ref}\\ T_{gref}\end{array}\right]$$

(9)

2.7. Control Strategy

The basic control strategy is described as Figure 2. The control requirements for the power and the speed are different in different wind regions. If ν < ν_cut–in, the system stops.

If ν ∈ [ν_cut–in, ν_rated], the system needs to maximize the wind harvested power. If ν > ν_rated, the system needs to limit power to the rated and maintain the stability of the system.

Figure 3 shows the tendency between the control variables and the controlled variables of the WECS in different workspaces, and the following gives the explanations of each workspace:

A-B

For wind speed under ν_cut–in, the system stops.

B-C

During the partial load region, the generator control is the only active control and aims at maximizing the energy captured from the wind and/or at limiting the rotational speed at rated. This is possible by continuously accelerating or decelerating the generator speed in such a way that the optimum tip speed ratio is tracked.

C-D

During the full load region, the control objective is to keep the real-time power P_g to the rated value P_gnom and limit the high-speed shaft speed ω_g remaining between [ω_gnom ω_gmax] at the same time. The main purpose of pitch control is to limit wind capture that is to limit power and avoid the mechanical load increasing.

D-E

When the wind speed is close to ν_cut–out, changing the value of the pitch angle to minimize the wind energy. The system needs to be disconnected from the grid and brake to stop.

In this paper, we use Maximum Power Point Tracking (MPPT) algorithm to obtain the optimal generator speed ω_gopt and the optimal generator power P_gopt. The so-called MPPT is a very reliable and robust control method, which covers an entire class of extremum search algorithms to maintain the optimal operating point when the curve C_p(λ, β) is unknown under a certain wind speed. It can calculate the optimal generator speed ω_gopt and the optimal generator power P_gopt. The control loop we presented should track the optimal set points of MPPT's outputs.

3. The Proposed Method

3.1. Fault Modelling of WECS within Stochastic PWA Model Framework

In summary, WECS is mainly based on four regions for modeling and control. According to the different wind speeds, models and control strategies need to switch. This section introduces the basic principles of PWA, and then gives the idea of WECS modeling.

3.1.1. Stochastic PWA Model Form

A linear stochastic discrete-time PWA system is defined by the state-space equation:

$$\begin{array}{ll}\begin{array}{l}{x}_{k+1}=A_i{x}_k+B_i{u}_k+B_i^w{w}_k+a_i\hfill \\ z_k=C_i{x}_k+D_i{u}_k+D_i^w{w}_k\hfill \end{array},\hfill & \left[\begin{array}{c}{x}_k\\ u_k\end{array}\right]\in {\chi }_i,x_k\in {\overline{\chi }}_j\hfill \end{array}$$

(10)

where x_k ∈ ℝⁿ is the state; u_k ∈ ℝ^m is the control input; w_k ∈ ℝ^r is a disturbance signal and z_k ∈ ℝ^m is a performance output. The set ⊆ ℝ ^n+m of every possible vector ${\left[\begin{array}{cc}{x}_k^T& u_k^T\end{array}\right]}^T$ is ℝ^n+m or a polyhedron containing the origin; ${\left\{{\chi }_i\right\}}_{i=1}^s$ is a polyhedral partition of and a_i ∈ ℝⁿ are constant vectors. Each χ_i is regarded as a cell. For simplicity, assume that the cells are polyhedral defined by matrices $F_i^x$, $F_i^u$, $f_i^x$ and $f_i^u$ as follows:

$${\chi }_i=\{{[\begin{array}{cc}{x}^T& u^T\end{array}]}^T\text{such that}{F}_i^{x}x\ge f_i^x\text{and}{F}_i^{u}u\ge f_i^u\}$$

(11)

and:

$${\overline{\chi }}_i=\{x\text{such that}{F}_i^{x}x\ge f_i^x\}$$

(12)

$$S_j=\{i\text{that}\exists x,u\text{with x}\in {\overline{\chi }}_j,{[x^T{u}^T]}^T\in {\chi }_i\}$$

(13)

where S_j is the set of all indices i such that χ_i is a cell containing a vector [x^T u^T]^T with $x\in {\overline{\chi }}_j$ is satisfied. Denote I = {1, …, s}, which is the set of indices of the cells χ_i; Denote = {1, …, t}, which is the set of indices of the cells χ̅_j. It is important to see that:

$$\underset{j=1}{\overset{t}{\cup }}{S}_j=I$$

(14)

Furthermore, if cells χ_i have the structure pointed out in Equation (11), then the sets S_j are disjoint. If cells χ_i have a more complicate structure (for instance when mixed state-input constraints are used to define each cell χ_i), then the set S_j could be overlapping.

3.1.2. Stochastic PWA Model for WECS

The linearized drive train dynamics function is described as follows:

$$\{\begin{array}{ll}{\dot{\omega }}_r\hfill & =\frac{1}{3J_r}\frac{\partial T_a}{\partial \beta }\beta +\frac{B_{dt}}{N_g{J}_r}{\omega }_g+\left(-\frac{B_{dt}+B_r}{J_r}+\frac{1}{J_r}\frac{\partial T_a}{\partial {\omega }_r}\right){\omega }_r+\frac{1}{3J_r}\frac{\partial T_a\left(t\right)}{\partial v_r}\nu \left(t\right)\hfill \\ {\dot{\omega }}_g\hfill & =-\frac{1}{J_g}{T}_g-\left(\frac{{\eta }_{dt}{B}_{dt}}{J_g{N}_g^2}+\frac{B_g}{J_g}\right){\omega }_g+\frac{B_{dt}}{N_g{J}_g}{\omega }_r\hfill \end{array}$$

(15)

where $\frac{\partial T_a}{\partial \beta }$, $\frac{\partial T_a}{\partial {\omega }_r}$ and $\frac{\partial T_a\left(t\right)}{\partial {\nu }_r}$ are the linearized parameters in different working points; B_g is the viscous friction of the high speed shaft.

Then a linearized overall state space model describing the dynamics of the WECS can be given:

$$\left[\begin{array}{c}{\dot{T}}_g\\ \dot{\beta }\\ \ddot{\beta }\\ {\dot{\omega }}_g\\ {\dot{\omega }}_r\end{array}\right]=\left[\begin{array}{ccccc}{A}_{11}& 0& 0& 0& 0\\ 0& 0& I& 0& 0\\ 0& A_{54}& A_{55}& 0& 0\\ a_{71}& 0& 0& a_{77}& \frac{B_{dt}}{N_g{J}_g}\\ 0& a_{84}& 0& \frac{B_{dt}}{N_g{J}_r}& a_{88}\end{array}\right]\left[\begin{array}{c}{T}_g\\ \beta \\ \dot{\beta }\\ {\omega }_g\\ {\omega }_r\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ e_{81}\end{array}\right]v(t)+\left[\begin{array}{cc}{B}_{11}& 0\\ 0& 0\\ 0& B_{42}\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{T}_{gref}\\ {\beta }_{ref}\end{array}\right]$$

(16)

where $A_{11}=-\frac{1}{{\tau }_g}$, $A_{54}=-{\omega }_n^2$, $A_{55}=-2\xi {\omega }_n$, $a_{71}=-\frac{1}{J_g}$, $a_{77}=-\left(\frac{{\eta }_{dt}{B}_{dt}}{J_g{N}_g^2}+\frac{B_g}{J_g}\right)$, $a_{84}=\frac{1}{3J_r}\frac{\partial T_a}{\partial \beta }$, $a_{88}=-\frac{B_{dt}+B_r}{J_r}+\frac{1}{J_r}\frac{\partial T_a}{\partial {\omega }_r}$, $B_{11}=\frac{1}{{\tau }_g}$, $B_{42}={\omega }_n^2$, $e_{81}=\frac{1}{3J_r}\frac{\partial T_a\left(t\right)}{\partial {\nu }_r}$.

Although wind can provide the energy that drives the wind turbine, due to its intermittent nature, it also acts as a disturbance. Hence, the effective wind (Equation (1)) can also be considered as a superposition of the mean wind speed ν_m, which could be either constant or time varying as a result of a sophisticated forecasting method, and a stochastic component ν_s. From [14,15], the stochastic part ν_s can be considered as the point wind after a second-order filter, which models the effect of the disc-shaped area swept by the rotor blades. In the frequency domain the power density spectrum S_νs of ν_s can be written as S_νs (f, ν_m) = S_p (f) S_f (f, ν_m), where S_p (f) is the spectrum of the point wind and S_f (f, ν_m) denotes the filter, which depends on the mean wind speed. This nonlinear expression can be then approximated by a linear second-order transfer function driven by a white noise process:

$$\{\begin{array}{ll}{\dot{\omega }}_1\hfill & ={\omega }_2\hfill \\ {\dot{\omega }}_2\hfill & =-a_1{\omega }_1+a_2{\omega }_2+a_{3}e\hfill \end{array}$$

(17)

where ω₁ = ν_s, e ∈ (0,1); and a₁, a₂, a₃ are parameters depending on the mean wind speed.

Combining with Equation (17), model (16) can be translated into the following form:

$$\left[\begin{array}{c}{\dot{T}}_g\\ \dot{\beta }\\ \ddot{\beta }\\ {\dot{\omega }}_g\\ {\dot{\omega }}_r\\ {\dot{\omega }}_1\\ {\dot{\omega }}_2\end{array}\right]=\left[\begin{array}{ccccccc}{A}_{11}& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0\\ 0& A_{54}& A_{55}& 0& 0& 0& 0\\ a_{71}& 0& 0& a_{77}& \frac{B_{dt}}{N_g{J}_g}& 0& 0\\ 0& a_{84}& 0& \frac{B_{dt}}{N_g{J}_r}& a_{88}& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& -a_1& -a_2\end{array}\right]\left[\begin{array}{c}{T}_g\\ \beta \\ \dot{\beta }\\ {\omega }_g\\ {\omega }_r\\ {\omega }_1\\ {\omega }_2\end{array}\right]+\left[\begin{array}{cc}{B}_{11}& 0\\ \begin{array}{c}0\\ \begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}{B}_{42}\\ 0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}\end{array}\right]\left[\begin{array}{c}{T}_g*\\ \beta *\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ e_{81}\\ 0\\ 0\end{array}\right]{\upsilon }_m+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ e_{81}\\ 0\\ a_3\end{array}\right]e$$

(18)

Choose 3 working points for the parameters of WECS described as Figure 4. The values of $\frac{\partial T_a}{\partial \beta }$, $\frac{\partial T_a}{\partial {\omega }_r}$ and $\frac{\partial T_a\left(t\right)}{\partial {\nu }_r}$ in different working points can be obtained according to different wind speeds by an effective wind estimator [16], where the realization method is described in [17–20]. The parameters are shown in

Table 1. Model parameters.

Parameter	Value	Unit	Parameter	Value	Unit
Jr	90,000	kg · m2	Pg,nom	225	kW
Jg	10	kg · m2	ωr,nom	4.29	rad/s
Ks	8 × 106	Nm/rad	ωg,nom	105.534	rad/s
Ds	8 × 104	kg · m2/(rad · s)	ωr,min	3.5	rad/s
Ng	24.6	-	ωg,min	86.1	rad/s
	14.5	m	θmin	0	deg
τθ	0.15	s	θmax	25	deg
τΓ	0.1	s	|θ˙|max	10	deg/s

and

Table 2. Parameters of linearized model in different working points.

Wind Speed/ms−1	Parameters
	a84	a88	e81	a1	a2	a3
V = 7.5	0.409	0.50	1.90	0.3125	2.92	0.9375
V = 10	0.479	0.53	2.31	0.33	3.65	2.3
V = 16	0.833	0.53	2.50	0.625	5	5

.

3.1.3. WECS Actuator Fault and Sensor Fault Model

In this section, we consider both actuator and sensor faults. Let u_j denote the j'th actuator or sensor, and u_j^F denote the failed j'th actuator or sensor. We model a loss of gain in an actuator or sensor as:

$${u_j}^F=(1-{\alpha }_j)u_j,0\le {\alpha }_j\le {\alpha }_{Mj}$$

(19)

where α_j is the percentage of failure in the j'th actuator or sensor, α_Mj is the maximum loss in the j'th actuator or sensor. α_j = 0 represents the case without faults in the j'th actuator or sensor, 0 < α_j < 1 corresponds to the partial loss of the j'th actuator or sensor fault, and α_j = 1 corresponds to the all loss of the j'th actuator or sensor fault. Define α with α = diag{ α₁, α₂, …, α_m}. Then u^F = Γu, where Γ = (I − α). The faults considered are shown in

Table 3. The faults considered.

Fault	Type	Fault Description
actuator	gain Factor	Γa
sensor	gain Factor	Γs

:

where Γ_a and Γ_s are both diagonal matrix of two elements.

The PWA model of the system with the loss of gain Γ_a in actuators can be described by:

$$\begin{array}{ll}\begin{array}{l}{x}_{k+1}=A_i{x}_k+B_i{\Gamma }_a{u}_k+B_i^w{w}_k+a_i\hfill \\ z_k=C_i{x}_k+D_i{u}_k+D_i^w{w}_k\hfill \end{array},\hfill & \left[\begin{array}{c}{x}_k\\ u_k\end{array}\right]\in {\chi }_i,x_k\in {\overline{\chi }}_j\hfill \end{array}$$

(20)

and the PWA model of the system with the loss of gain Γ_s in sensors can be described by:

$$\begin{array}{ll}\begin{array}{l}{x}_{k+1}=A_i{x}_k+B_i{u}_k+B_i^w{w}_k+a_i\hfill \\ z_k=C_i{\Gamma }_s{x}_k+D_i{u}_k+D_i^w{w}_k\hfill \end{array},\hfill & \left[\begin{array}{c}{x}_k\\ u_k\end{array}\right]\in {\chi }_i,x_k\in {\overline{\chi }}_j\hfill \end{array}$$

(21)

3.2. H_∞ Fault Tolerant Control for WECS

3.2.1. Fault Tolerant Control for Actuator Fault

Consider the PWA actuator faults system (20), define the input signal w_k as follows:

$${\tilde{w}}_k=\left[\begin{array}{c}{w}_k\\ a_i\end{array}\right]$$

(22)

Thus, system (20) can be rewritten as:

$$\begin{array}{ll}\begin{array}{l}{x}_{k+1}=A_i{x}_k+B_i{\Gamma }_a{u}_k+{\tilde{B}}_i^w{\tilde{w}}_k\hfill \\ z_k=C_i{x}_k+D_i{u}_k+{\tilde{D}}_i^w{\tilde{w}}_k\hfill \end{array},\hfill & \left[\begin{array}{c}{x}_k\\ u_k\end{array}\right]\in {\chi }_i,x_k\in {\overline{\chi }}_j\hfill \end{array}$$

(23)

where:

$${\tilde{B}}_i^w=\left[\begin{array}{cc}{B}_i^w& I\end{array}\right],{\tilde{D}}_i^w=\left[\begin{array}{cc}{D}_i^w& I\end{array}\right]$$

(24)

The H_∞ framework considered here is based on a finite horizon definition of the l₂ gain and, consequently, the proposed extension of the disturbance input is sensible [21–24].

Clearly, it is possible to apply the control approach proposed in [5] directly to the extended system (Equation (23)). This can be conservative because a_i is not an unknown disturbance but a known term. Unfortunately, in general, a_i is known only when the control signal u_k has already been calculated. Under the standard assumption:

$$a_i={\overline{a}}_j,\forall i\in S_j,\forall j\in \Im$$

(25)

an alternative control strategy can be proposed. More precisely, the control is assumed to have the following structure:

$$u_k=\left[\begin{array}{cc}{K}_j^1& K_j^2\end{array}\right]\left[\begin{array}{c}{x}_k\\ {\overline{a}}_j\end{array}\right],x_k\in {\overline{\chi }}_j$$

(26)

In this way, the controller can also consider the displacement term a_i = Dw̃_k, where D = [0 I].

By applying the control law (26) to the PWA system (23), we can obtain the closed loop PWA actuator fault system:

$$\begin{array}{ll}\begin{array}{l}{x}_{k+1}={\mathcal{A}}_{ij}{x}_k+{\tilde{B}}_{ij}^w{\tilde{w}}_k\hfill \\ z_k={\mathcal{C}}_{ij}{x}_k+{\tilde{D}}_{ij}^w{\tilde{w}}_k\hfill \end{array},\hfill & \left[\begin{array}{c}{x}_k\\ u_k\end{array}\right]\in {\chi }_i,x_k\in {\overline{\chi }}_j\hfill \end{array}$$

(27)

where:

$$\begin{array}{ll}\begin{array}{l}{\mathcal{A}}_{ij}=A_i+B_i{\Gamma }_a{K}_j^1,\hfill \\ {\mathcal{C}}_{ij}=C_i+D_i{K}_j^1,\hfill \end{array}\hfill & \begin{array}{l}{\tilde{B}}_{ij}^w={\tilde{B}}_i^w+B_i{K}_j^{2}D\hfill \\ {\tilde{D}}_{ij}^w={\tilde{D}}_i^w+D_i{K}_j^{2}D\hfill \end{array}\hfill \end{array}$$

(28)

From the above, we can have the following main results:

Lemma 1

Consider system (27) with zero initial condition x₀ = 0, if there exists a function V(x, u) = x^T P_ix for [x^T u^T]^T ∈ χ_i with $P_i=P_i^T>0$, satisfying the dissipativity inequality:

$$\forall w\in {ℝ}^r,\forall k\ge 0,V\left(x_{k+1},u_{k+1}\right)-V\left(x_k,u_k\right)<W\left(z_k,w_k\right)$$

(29)

with supply rate:

$$W_{\infty }\left(z,w\right)=\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right),\gamma >0$$

(30)

that is:

$$\forall k,V\left(x_{k+1},u_{k+1}\right)-V\left(x_k,u_k\right)<\left({\gamma }^2{\Vert w_k\Vert }^2-{\Vert z_k\Vert }^2\right),$$

(31)

then, the H_∞ performance condition:

$$\sum _{\text{k}=0}^{\text{N}}{\Vert z_k\Vert }^2<{\gamma }^2\sum _{\text{k}=0}^{\text{N}}{\Vert w_k\Vert }^2$$

(32)

is satisfied.

Furthermore, if the following matrix inequalities:

$$\forall j\in \mathcal{J},,\forall i\in S_j,\forall l\mathit{\text{with}}\left(l,i\right)\in S,M_{l,ij}<0$$

(33)

are satisfied, where:

$$M_{l,ij}=\left[\begin{array}{cc}{A}_{ij}^T{P}_i{A}_{ij}-P_j+C_{ij}^T{C}_{ij}& C_{ij}^T{D}_j^w+A_{ij}^T{P}_i{B}_j^w\\ D_j^{wT}{C}_{ij}+B_j^{wT}{P}_i{A}_{ij}& B_j^{wT}{P}_i{B}_j^w+D_j^{wT}{D}_j^w-{\gamma }^{2}I\end{array}\right]$$

(34)

then, condition (31) is fulfilled. Thus, system (27) is PWQ stable.

Proof

By recalling that x₀ = 0, from Equation (31) it follows that, ∀N ≥ 0:

$$V\left(x_{N+1},u_{N+1}\right)<\sum _{k=0}^N\left({\gamma }^2{\Vert w_k\Vert }^2-{\Vert z_k\Vert }^2\right)$$

(35)

Since V(x_N+1, u_N+1) > 0, then condition:

$$\sum _{k=0}^N{\Vert z_k\Vert }^2<{\gamma }^2\sum _{k=0}^N{\Vert w_k\Vert }^2$$

(36)

is met.

Moreover, if ${\left[\begin{array}{cc}{x}_k^T& u_k^T\end{array}\right]}^T\in {\chi }_i$ and ${\left[\begin{array}{cc}{x}_{k+1}^T& u_{k+1}^T\end{array}\right]}^T\in {\chi }_l$, then:

$$\forall w_k,\left[\begin{array}{cc}{x}_k^T& w_k^T\end{array}\right]M_{l,ij}{\left[\begin{array}{cc}{x}_k^T& w_k^T\end{array}\right]}^T<0$$

(37)

Obviously, if condition (33) holds, inequality (37) is satisfied and ${\mathcal{C}}_{ij}^T{\mathcal{C}}_{ij}>0$. For $A_{ij}^T{P}_l{A}_{ij}-P_i+C_{ij}^T{C}_{ij}$, we can state that:

$$\forall j\in \mathcal{J},\forall i\in S_j,\forall l\mathit{\text{with}}\left(l,i\right)\in S,{\mathcal{A}}_{ij}^T{P}_i{\mathcal{A}}_{ij}-P_i<0$$

(38)

This implies that system (27) is PWQ stable.

Now we focus on the possibility of finding a state-feedback control law of the type (26) for system (23). The main result is summarized in the following theorem.

Theorem 1

For PWA system (20), there exists a state feedback control law of type (26), which can guarantee PWQ Lyapunov stability and fulfil the dissipativity constraint:

$$\forall w\in {ℝ}^r,\forall k\ge 0,V\left(x_{k+1}\right)-V\left(x_k,u_k\right)<W\left(z_k,w_k\right)$$

(39)

with supply rate:

$${\tilde{W}}_{\infty }\left(z_k,w_k\right)=({\gamma }^2{\Vert {\left[\begin{array}{cc}{w}_k^T& a_i^T\end{array}\right]}^T\Vert }^2-{\Vert z_k\Vert }^2)=({\gamma }^2{\Vert (w_k\Vert }^2+{\Vert a_i\Vert }^2)-{\Vert z_k\Vert }^2),\gamma >0,\left[\begin{array}{c}{x}_k\\ u_k\end{array}\right]\in {\chi }_i$$

(40)

If there exist matrices $Q_i=Q_i^T>0$ with i ∈ I and matrices G_j, Y_j, $K_j^2$ with j ∈ , such that ∀j ∈ , ∀i ∈ S_j, ∀l with (l, i) ∈ S_all:

$$\left[\begin{array}{cccc}{Q}_i-G_j-G_j^T& {\left(A_i{G}_j+B_i{\mathrm{\Gamma }}_a{Y}_j\right)}^T& {\left(C_i{G}_j+D_i{Y}_j\right)}^T& 0\\ A_i{G}_j+B_i{\mathrm{\Gamma }}_a{Y}_j& -Q_l& 0& {\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D\\ C_i{G}_j+D_i{Y}_j& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {({\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D)}^T& {({\tilde{D}}_i^w+D_i{K}_j^{2}D)}^T& -{\gamma }^{2}I\end{array}\right]<0$$

(41)

holds, then the feedback gains matrices $K_j^1$ with j ∈ are given by:

$$K_j^1=Y_j{G}_j^{-1}.$$

(42)

Proof

Using Schur's lemma, then Equation (33) can be rewritten as follows:

$$\left[\begin{array}{cccc}-P_i& {\left(A_i+B_i{\mathrm{\Gamma }}_a{K}_j^1\right)}^T& {\left(C_i+D_i{K}_j^1\right)}^T& 0\\ A_i+B_i{\mathrm{\Gamma }}_a{K}_j^1& P_l^{-1}& 0& {\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D\\ C_i+D_i{K}_j^1& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {({\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D)}^T& {({\tilde{D}}_i^w+D_i{K}_j^{2}D)}^T& -{\gamma }^{2}I\end{array}\right]<0$$

(43)

where $A_{ij}=A_i+B_i{\mathrm{\Gamma }}_a{K}_j^1$, $C_{ij}=C_i+D_i{K}_j^1$.

Now, let $Q_l=P_l^{-1}$ and $Y_j=K_j^1{G}_j$, we can obtain inequality (41) by multiplying (43) from the left by diag $\left(G_j^{T}III\right)$ and the right by diag (G_j I I I):

Let:

$$H=\left[\begin{array}{cccc}{G}_j^T& 0& 0& 0\\ 0& I& 0& 0\\ 0& 0& I& 0\\ 0& 0& 0& I\end{array}\right]$$

(44)

then:

$$\begin{array}{l}H*\left[\begin{array}{cccc}-P_i& {(A_i+B_i{\mathrm{\Gamma }}_a{K}_j^1)}^T& {(C_i+D_i{K}_j^1)}^T& 0\\ A_i+B_i{\mathrm{\Gamma }}_a{K}_j^1& -P_l^{-1}& 0& {\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D\\ C_i+D_i{K}_j^1& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {({\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D)}^T& {({\tilde{D}}_i^w+D_i{K}_j^{2}D)}^T& -{\gamma }^{2}I\end{array}\right]*H^T\hfill \\ =\left[\begin{array}{cccc}-G_j^T{P}_i{G}_j& G_j^T{(A_i+B_i{\mathrm{\Gamma }}_a{K}_j^1)}^T& G_j^T{(C_i+D_i{K}_j^1)}^T& 0\\ (A_i+B_i{\mathrm{\Gamma }}_a{K}_j^1)G_j& -P_l^{-1}& 0& {\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D\\ (C_i+D_i{K}_j^1)G_j& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {({\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D)}^T& {({\tilde{D}}_i^w+D_i{K}_j^{2}D)}^T& -{\gamma }^{2}I\end{array}\right]\hfill \\ =\left[\begin{array}{cccc}-G_j^T{P}_i{G}_j& {(A_i{G}_j+B_i{\mathrm{\Gamma }}_a{Y}_j)}^T& {(C_i{G}_j+D_i{Y}_j)}^T& 0\\ A_i{G}_j+B_i{\mathrm{\Gamma }}_a{Y}_j& -Q_l& 0& {\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D\\ C_i{G}_j+D_i{Y}_j& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {({\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D)}^T& {({\tilde{D}}_i^w+D_i{K}_j^{2}D)}^T& -{\gamma }^{2}I\end{array}\right]<0\hfill \end{array}$$

(45)

Since $0<G_j+G_j^T-Q_i\le G_j^T{Q}_i^{-1}{G}_j$, Equation (45) implies:

$$\left[\begin{array}{cccc}{Q}_i-G_j-G_j^T& {(A_i{G}_j+B_i{\mathrm{\Gamma }}_a{Y}_j)}^T& {(C_i{G}_j+D_i{Y}_j)}^T& 0\\ A_i{G}_j+B_i{\mathrm{\Gamma }}_a{Y}_j& -Q_l& 0& {\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D\\ C_i{G}_j+D_i{Y}_j& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {({\tilde{B}}_i^w+B_i{\mathrm{\Gamma }}_a{K}_j^{2}D)}^T& {({\tilde{D}}_i^w+D_i{K}_j^{2}D)}^T& -{\gamma }^{2}I\end{array}\right]<0$$

(46)

It is obvious to get Q_i > 0, ∀i ∈ I, then the control matrixes $K_j^1$ can be reconstructed as $K_j^1=Y_j{G}_j^{-1}$.

3.2.2. Fault Tolerant Control for Sensor Fault

Theorem 2

For PWA system (21), there exists a state feedback control law of type (26), which can guarantee PWQ Lyapunov stability and fulfil the dissipativity constraint:

$$\forall w\in {ℝ}^r,\forall k\ge 0,V\left(x_{k+1}\right)-V\left(x_k,u_k\right)<W\left(z_k,w_k\right)$$

(47)

with supply rate:

$${\tilde{W}}_{\infty }\left(z_k,w_k\right)=({\gamma }^2{\Vert {[w_k^T{a}_i^T]}^T\Vert }^2-{\Vert z_k\Vert }^2)=({\gamma }^2{\Vert (w_k\Vert }^2+{\Vert a_i\Vert }^2)-{\Vert z_k\Vert }^2),\gamma >0,\left[\begin{array}{c}{x}_k\\ u_k\end{array}\right]\in {\chi }_i$$

(48)

If there exist matrices $Q_i=Q_i^T>0$ with i ∈ I and matrices G_j, Y_j, $K_j^2$ with j ∈ , such that ∀j ∈ , ∀i ∈ S_j, ∀l with (l, i) ∈ S_all:

$$\left[\begin{array}{cccc}{Q}_i-G_j-G_j^T& {(A_i{G}_j+B_i{Y}_j)}^T& {(C_i{\mathrm{\Gamma }}_s{G}_j+D_i{Y}_j)}^T& 0\\ A_i{G}_j+B_i{Y}_j& -Q_l& 0& {\tilde{B}}_i^w+B_i{K}_j^{2}D\\ C_i{\mathrm{\Gamma }}_s{G}_j+D_i{Y}_j& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {({\tilde{B}}_i^w+B_i{K}_j^{2}D)}^T& {({\tilde{D}}_i^w+D_i{K}_j^{2}D)}^T& -{\gamma }^{2}I\end{array}\right]<0$$

(49)

holds, then the feedback gains matrices $K_j^1$ with j ∈ are given by:

$$K_j^1=Y_j{G}_j^{-1}$$

(50)

Proof

Similar to the proof of Theorem 1, Equation (33) can be rewritten as follows:

$$\left[\begin{array}{cccc}-P_i& {(A_i+B_i{K}_j^1)}^T& {(C_i{\mathrm{\Gamma }}_s+D_i{K}_j^1)}^T& 0\\ A_i+B_i{K}_j^1& -P_l^{-1}& 0& {\tilde{B}}_i^w+B_i{K}_j^{2}D\\ C_i{\mathrm{\Gamma }}_s+D_i{K}_j^1& 0& -I& {\tilde{D}}_i^w+D_i{K}_j^{2}D\\ 0& {\mathit{(}{\tilde{B}}_i^w+B_i{K}_j^{2}D\mathit{)}}^T& {\mathit{(}{\tilde{D}}_i^w+D_i{K}_j^{2}D\mathit{)}}^T& -{\gamma }^{2}I\end{array}\right]<0$$

(51)

where $A_{ij}=A_i+B_i{K}_j^1$, $C_i=C_i{\Gamma }_s+D_i{K}_j^1$.

Now, let $Q_l=P_l^{-1}$ and $Y_j=K_j^1{G}_j$, we can also obtain inequality (49) by multiplying (51) from the left by diag $\left(G_j^{T}III\right)$ and the right by diag (G_j I I I).The rest proof process is similar to Theorem 1, omitted here.

4. Simulation Results

According to the modeling method described in part 3, we can obtain the following PWA model of WECS:

If 0 < E₁x_k ≤ 8 (Region 1), then:

$$\left[\begin{array}{c}{\dot{T}}_g\\ \dot{\beta }\\ \ddot{\beta }\\ {\dot{\omega }}_g\\ {\dot{\omega }}_r\\ {\dot{w}}_1\\ {\dot{w}}_2\end{array}\right]=\left[\begin{array}{ccccccc}-10& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& -123.4& -13.332& 0& 0& 0& 0\\ -0.1& 0& 0& -13.2& 325.2& 0& 0\\ 0& -0.409& 0& 0.036& -0.5& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& -0.3125& -2.92\end{array}\right]\left[\begin{array}{l}{T}_g\\ \beta \\ \dot{\beta }\\ {\omega }_g\\ {\omega }_r\\ {\omega }_1\\ {\omega }_2\end{array}\right]+\left[\begin{array}{cc}10& 0\\ 0& 0\\ 0& 123.4\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{l}{T}_{gref}\\ {\beta }_{ref}\end{array}\right]+\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 1.9\\ 0\\ 0\end{array}\right]\times 7.5+\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 1.9\\ 0\\ 0.9325\end{array}\right]e$$

(52

If 8 < E₁x_k ≤ 12 (Region 2), then:

$$\left[\begin{array}{c}{\dot{T}}_g\\ \dot{\beta }\\ \ddot{\beta }\\ {\dot{\omega }}_g\\ {\dot{\omega }}_r\\ {\dot{w}}_1\\ {\dot{w}}_2\end{array}\right]=\left[\begin{array}{ccccccc}-10& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& -123.4& -13.332& 0& 0& 0& 0\\ -0.1& 0& 0& -13.2& 325.2& 0& 0\\ 0& -0.479& 0& 0.036& -0.53& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& -0.33& -3.65\end{array}\right]\left[\begin{array}{l}{T}_g\\ \beta \\ \dot{\beta }\\ {\omega }_g\\ {\omega }_r\\ {\omega }_1\\ {\omega }_2\end{array}\right]+\left[\begin{array}{cc}10& 0\\ 0& 0\\ 0& 123.4\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{l}{T}_{gref}\\ {\beta }_{ref}\end{array}\right]+\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 2.31\\ 0\\ 0\end{array}\right]\times 10+\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 2.31\\ 0\\ 2.3\end{array}\right]e$$

(53)

If 12 < E₁x_k ≤ 18 (Region 3), then:

$$\left[\begin{array}{c}{\dot{T}}_g\\ \dot{\beta }\\ \ddot{\beta }\\ {\dot{\omega }}_g\\ {\dot{\omega }}_r\\ {\dot{w}}_1\\ {\dot{w}}_2\end{array}\right]=\left[\begin{array}{ccccccc}-10& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& -123.4& -13.332& 0& 0& 0& 0\\ -0.1& 0& 0& -13.2& 325.2& 0& 0\\ 0& -0.833& 0& 0.036& -0.53& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& -0.625& -5\end{array}\right]\left[\begin{array}{l}{T}_g\\ \beta \\ \dot{\beta }\\ {\omega }_g\\ {\omega }_r\\ {\omega }_1\\ {\omega }_2\end{array}\right]+\left[\begin{array}{cc}10& 0\\ 0& 0\\ 0& 123.4\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{l}{T}_{g\mathit{\text{ref}}}\\ {\beta }_{\mathit{\text{ref}}}\end{array}\right]+\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 2.5\\ 0\\ 0\end{array}\right]\times 16+\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 2.5\\ 0\\ 5\end{array}\right]e$$

(54)

where E₁ = [0 0 0 0 0 1 0].

According to Theorem 1 and Theorem 2, we can successfully obtain the piecewise linear state feedback matrixes K₁, K₂, K₃ of WECS system in normal, sensor failure and actuator failure cases. The feedback matrixes calculated in the three cases are shown as follows:

The normal WECS:

$$K_1=\left[\begin{array}{ccccccccc}-0.0859& -0.0149& -0.0009& 0.0018& 0.2721& 0.0000& 0.0000& -0.0066& -0.0247\\ -0.0334& -0.1671& -0.0121& 0.0080& 2.4130& 0.0000& 0.0000& -0.0130& 0.0000\end{array}\right]$$

(55)

$$K_2=\left[\begin{array}{ccccccccc}7.6202& -0.7650& -0.0482& 0.0031& 0.1483& 0.0000& 0.0000& -0.4561& -0.8690\\ 81.4923& -24.3547& -1.6292& 1.4002& 396.2525& 0.0000& 0.0000& -17.8043& -0.0099\end{array}\right]$$

(56)

$$K_3=\left[\begin{array}{ccccccccc}0.3367& -0.0142& -0.0009& -0.0024& 0.2854& 0.0000& 0.0000& -0.0055& -0.0142\\ 3.9585& -0.1744& -0.0186& -0.0379& 3.7296& 0.0000& 0.0000& -0.0441& 0.0000\end{array}\right]$$

(57)

Actuator within fault tolerance:

$$K_1=\left[\begin{array}{ccccccccc}-0.1431& -0.0248& -0.0015& 0.0030& 0.4536& 0.0000& 0.0000& -0.0110& -0.0411\\ -0.0556& -0.2784& -0.0201& 0.0134& 4.0216& 0.0000& 0.0000& -0.0217& 0.0000\end{array}\right]$$

(58)

$$K_2=\left[\begin{array}{ccccccccc}12.7003& -1.2750& -0.0804& 0.0052& 24.4886& 0.0000& 0.0000& -0.7601& -1.4483\\ 135.8205& -40.5912& -2.7154& 2.3337& 660.4208& 0.0000& 0.0000& -29.6738& -0.0166\end{array}\right]$$

(59)

$$K_3=\left[\begin{array}{ccccccccc}0.1833& -0.0077& -0.0005& -0.0013& 0.1554& 0.0000& 0.0000& -0.0030& -0.0078\\ 2.1305& -0.0967& -0.0102& -0.0202& 2.0492& 0.0000& 0.0000& 0.0222& 0.0000\end{array}\right]$$

(60)

Sensor within fault tolerance:

$$K_1=\left[\begin{array}{ccccccccc}-0.1110& -0.0138& -0.0009& 0.0019& 0.2541& 0.0000& 0.0000& -0.0053& -0.0247\\ 0.0000& -0.1721& -0.0124& 0.0079& 2.4826& 0.0000& 0.0000& -0.0184& 0.0000\end{array}\right]$$

(61)

$$K_2=\left[\begin{array}{ccccccccc}-0.1652& -0.0208& -0.0013& 0.0030& 0.3894& 0.0000& 0.0000& -0.0086& -0.0363\\ 0.0000& -0.2548& -0.0183& 0.0119& 3.7666& 0.0000& 0.0000& -0.0269& 0.0000\end{array}\right]$$

(62)

$$K_3=\left[\begin{array}{ccccccccc}-0.1682& -0.0210& -0.0013& 0.0030& 0.3930& 0.0000& 0.0000& -0.0085& -0.0377\\ -0.0008& -0.2175& -0.0159& 0.0062& 2.9796& 0.0000& 0.0000& 0.0228& 0.0001\end{array}\right]$$

(63)

In these three cases, feedback gains K₁, K₂, K₃ correspond to Regions 1, 2, 3, respectively. The feedback gains K₁, K₂, K₃ are designed according to the stochastic PWA model of WECS. When the model switches, corresponding the feedback gain switches. Then the piecewise linear state feedbacks can guarantee the WECS systems meeting PWQ Lyapunov stability.

4.1. Validation of H_∞ Control for the Normal WECS

Figure 5 demonstrates the simulation results of WECS fault free dynamic response which is regulated by the robust H_∞ controller designed based on Theorem 1. Figure 5a is the test wind speed signal which is consisted of mean wind speed ν_m and wind speed turbulent ν_s ( ν = ν_m + ν_s ). ν_m = 7.5, 10, 16 (m/s), which is switched at time of 100 s, 200 s, 300 s, respectively. The tip speed ratio and the optimal generator power set point calculated by MPPT algorithm are given by Figure 5b,c, respectively, which are under the control of reliable H_∞ controller designed based on the stochastic PWA model of WECS.

The generator speed and generator power responses are shown in Figure 5d,e, from which we can conclude the real generator power tracks the optimal set point quite well and the generator speed is controlled with good performance. Figure 5f,g give the H_∞ controller output of pitch angle β and generator torque T_g. Figure 5 show the H_∞ controller of WECS designed based on stochastic PWA model combining with MPPT module works quite well.

4.2. Validation of H_∞ Control for WECS with Actuators Fault

Figure 6a–c show that with the pitch system actuator gain fault β_f and the generator gain fault T_gf (let gain loss factor ${\mathrm{\Gamma }}_a=\left[\begin{array}{cc}0.6& 0\\ 0& 0.6\end{array}\right]$), the MPPT module works quite well. Comparing with Figure 5a–c, there are little changes.

However, comparing with Figure 5d,e, the performance of generator power P_gf and generator speed ω_gf have deteriorated, and begin to show the trend of losing stability. The control variables display wide range oscillations in Figure 6f,g. It is seen that the normal H_∞ controller for WECS cannot deal with the actuators fault with gain factor loss Γ_a. Comparing Figure 6d,e, Figure 7d,e demonstrate the response of generator speed and generator power under the control of H_∞ fault tolerant controller designed based on Theorem1 based on the stochastic PWA actuator fault model of WECS, from which we can see that the performance with fault tolerant design significantly is better than the normal controller.

4.3. Validation of H_∞ Control for WECS with Sensors Fault

Figures 8 and 9 demonstrate the simulation results of WECS with sensor gain factor fault (let gain loss factor ${\mathrm{\Gamma }}_s=\left[\begin{array}{cc}1.2& 0\\ 0& 1.2\end{array}\right]$) including generator speed sensor fault w_gf and generator power sensor fault p_gf. In Figure 8, the dynamic behaviors of WECS under the normal controller without fault tolerant design for two sensor fault are demonstrated. In Figure 9 the H_∞ fault tolerant controller is synthesized based on the stochastic PWA model of WECS with sensor fault using the Theorem 2. In Figure 8d,e, the performances of generator power p_gf and generator speed w_gf have deteriorated and totally lose stability. The control variables which are shown in Figure 8e,f display wide range oscillations. While in Figure 9e,f, with the help of fault tolerant controller, the outputs of WECS with sensor faults can obtain better performance than the normal controller. But the generator power and generator speed cannot track the optimal set point of MPPT module's output. This problem should be further investigated.

5. Conclusions

From the discussion in Section 3, it is shown that the stochastic PWA model offers an ideal framework for capturing the stochastic property of wind speed and the nonlinear dynamics of WECS. The pictures that emerge from the proposed method and simulation results in Sections 3 and 4, suggest that the presented H_∞ fault tolerant control method offers an effective tool to deal with synthesis problem of WECS with sensors and actuator faults. However, the control strategies presented in this paper are mainly investigated for the overall model of WECS which overlooks the nonlinear and high-order dynamic properties of generator and converter. Future research will focus on developing an inner loop vector controller for generator torque control to deal with the nonlinear and high-order generator model. So far, we only finished the simulation work for this paper. We have developed a laboratory setup of a 10 KW WECS simulator which would provide a benchmark for further experimental investigation of the WECS control algorithm we presented in the paper.