Energies 2014, 7(3), 1750-1769; doi:10.3390/en7031750

Article
H Fault Tolerant Control of WECS Based on the PWA Model
Yun-Tao Shi *, Qi Kou , De-Hui Sun , Zheng-Xi Li , Shu-Juan Qiao and Yan-Jiao Hou
Key Lab of Field Bus and Automation of Beijing, North China University of Technology, Beijing 100144, China; E-Mails: kouqi0627@gmail.com (Q.K.); sundehui@ncut.edu.cn (D.-H.S.); lzx@ncut.edu.cn (Z.-X.L.); qiaoshujuan001@163.com (S.-J.Q.); houyanjiao0730@163.com (Y.-J.H.)
Author Contributions Yuntao Shi designed the simulations and the study, Shujuan Qiao contributed to the derivation of Theorem 1 for actuator fault tolerant control of WECS, and Yanjiao Hou contributed to the derivation of Theorem 2 for sensor fault tolerant control of WECS. All authors performed the simulations, carried out data analysis, discussed the results and contributed to writing the paper.
*
Author to whom correspondence should be addressed; E-Mail: shiyuntao@ncut.edu.cn; Tel.: +86-10-8880-2520; Fax: +86-10-8880-3372.
Received: 26 January 2014; in revised form: 13 March 2014 / Accepted: 19 March 2014 /
Published: 24 March 2014

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.
Keywords:
H fault tolerant control; piecewise affine; wind energy conversion system; H robust control

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 [69]. 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 βref(t) and generator torque reference Tgref(t), respectively. The outputs of the system are generator power Pg(t) and high-speed shaft speed ωg(t).

2.2. Wind Model

The equation of wind ν(t) is:

v ( t ) = v s ( t ) + v m ( t )
where νm(t) is the low-frequency component (describing long term, low-frequency variations), i.e., average wind speed; and νs(t) is the turbulence component (corresponding to fast, high-frequency variations).

2.3. Aerodynamics Model

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

P w = 1 2 ρ A ν 3
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 Pa is:

P a = P w C p ( λ , β )
where Cp(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:

λ = R ω r ν
where ωr 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:

{ ω ˙ r = T r J r ω r D s J r + ω g D s J r N g δ K s J r ω ˙ g = ω r D s J g N g ω g D s J g N g 2 + δ K s J g T g J g
where Jr and Jg are the moments of inertia of the low-speed shaft and the high-speed shaft; Ks is the torsion stiffness of the drive train; Ds is the torsion damping coefficient of the drive train; Ng is the gear ratio; δ is the twist of the flexible drive train with δ ˙ = ω r ω g N g.

2.5. Pitch System Model

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

{ β ˙ = β ˙ β ¨ = ω n 2 β 2 ξ ω n β ˙ + ω n 2 β r e f

2.6. Generator and Converter Model

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

T ˙ g = 1 τ g T g + 1 τ g T g r e f
where τg is the time constant.

The real-time power is described by:

P g = η g ω g T g

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

[ ω ˙ r ω ˙ g β ˙ β ¨ T ˙ g ] = [ T r J r ω r D s J r + ω g D s J r N g δ K s J r ω r D s J g N g ω g D s J g N g 2 + δ K s J g T g J g β ˙ ω n 2 β 2 ξ ω n β ˙ 1 τ g T g ] + [ 0 0 0 0 0 0 ω n 2 0 0 1 τ g ] [ β r e f T g r e f ]

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 < νcutin, the system stops.

If ν ∈ [νcutin, νrated], the system needs to maximize the wind harvested power. If ν > νrated, 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 νcutin, 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 Pg to the rated value Pgnom and limit the high-speed shaft speed ωg remaining between [ωgnom ωgmax] 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 νcutout, 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 ωgopt and the optimal generator power Pgopt. 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 Cp(λ, β) is unknown under a certain wind speed. It can calculate the optimal generator speed ωgopt and the optimal generator power Pgopt. 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:

x k + 1 = A i x k + B i u k + B i w w k + a i z k = C i x k + D i u k + D i w w k , [ x k u k ] χ i , x k χ ¯ j
where xk ∈ ℝn is the state; uk ∈ ℝm is the control input; wk ∈ ℝr is a disturbance signal and zk ∈ ℝm is a performance output. The set Energies 07 01750i1 ⊆ ℝ n+m of every possible vector [ x k T u k T ] T is ℝn+m or a polyhedron containing the origin; { χ i } i = 1 s is a polyhedral partition of Energies 07 01750i1 and ai ∈ ℝn are constant vectors. Each χi 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:
χ i = { [ x T u T ] T such that F i x x f i x and F i u u f i u }
and:
χ ¯ i = { x such that F i x x f i x }
S j = { i that x , u with x χ ¯ j , [ x T u T ] T χ i }
where Sj is the set of all indices i such that χi is a cell containing a vector [xT uT]T with x χ ¯ j is satisfied. Denote I = {1, …, s}, which is the set of indices of the cells χi; Denote Energies 07 01750i2 = {1, …, t}, which is the set of indices of the cells χ̅j. It is important to see that:
j = 1 t S j = I

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

3.1.2. Stochastic PWA Model for WECS

The linearized drive train dynamics function is described as follows:

{ ω ˙ r = 1 3 J r T a β β + B d t N g J r ω g + ( B d t + B r J r + 1 J r T a ω r ) ω r + 1 3 J r T a ( t ) v r ν ( t ) ω ˙ g = 1 J g T g ( η d t B d t J g N g 2 + B g J g ) ω g + B d t N g J g ω r
where T a β, T a ω r and T a ( t ) ν r are the linearized parameters in different working points; Bg 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:

[ T ˙ g β ˙ β ¨ ω ˙ g ω ˙ r ] = [ A 11 0 0 0 0 0 0 I 0 0 0 A 54 A 55 0 0 a 71 0 0 a 77 B d t N g J g 0 a 84 0 B d t N g J r a 88 ] [ T g β β ˙ ω g ω r ] + [ 0 0 0 0 e 81 ] v ( t ) + [ B 11 0 0 0 0 B 42 0 0 0 0 ] [ T g r e f β r e f ]
where A 11 = 1 τ g, A 54 = ω n 2, A 55 = 2 ξ ω n, a 71 = 1 J g, a 77 = ( η d t B d t J g N g 2 + B g J g ), a 84 = 1 3 J r T a β, a 88 = B d t + B r J r + 1 J r T a ω r, B 11 = 1 τ g, B 42 = ω n 2, e 81 = 1 3 J r T a ( t ) ν 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 νm, which could be either constant or time varying as a result of a sophisticated forecasting method, and a stochastic component νs. From [14,15], the stochastic part νs 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 Sνs of νs can be written as Sνs (f, νm) = Sp (f) Sf (f, νm), where Sp (f) is the spectrum of the point wind and Sf (f, νm) 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:

{ ω ˙ 1 = ω 2 ω ˙ 2 = a 1 ω 1 + a 2 ω 2 + a 3 e
where ω1 = νs, e Energies 07 01750i3 (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:

[ T ˙ g β ˙ β ¨ ω ˙ g ω ˙ r ω ˙ 1 ω ˙ 2 ] = [ 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 B d t N g J g 0 0 0 a 84 0 B d t N g J r a 88 0 0 0 0 0 0 0 0 1 0 0 0 0 0 a 1 a 2 ] [ T g β β ˙ ω g ω r ω 1 ω 2 ] + [ B 11 0 0 0 0 0 0 0 0 B 42 0 0 0 0 ] [ T g * β * ] + [ 0 0 0 0 e 81 0 0 ] υ m + [ 0 0 0 0 e 81 0 a 3 ] e

Choose 3 working points for the parameters of WECS described as Figure 4. The values of T a β, T a ω r and T a ( t ) ν 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 [1720]. The parameters are shown in Tables 1 and 2.

3.1.3. WECS Actuator Fault and Sensor Fault Model

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

u j F = ( 1 α j ) u j , 0 α j α M j
where αj is the percentage of failure in the j'th actuator or sensor, αMj is the maximum loss in the j'th actuator or sensor. αj = 0 represents the case without faults in the j'th actuator or sensor, 0 < αj < 1 corresponds to the partial loss of the j'th actuator or sensor fault, and αj = 1 corresponds to the all loss of the j'th actuator or sensor fault. Define α with α = diag{ α1, α2, …, αm}. Then uF = Γu, where Γ = (Iα). The faults considered are shown in Table 3:

where Γa and Γs are both diagonal matrix of two elements.

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

x k + 1 = A i x k + B i Γ a u k + B i w w k + a i z k = C i x k + D i u k + D i w w k , [ x k u k ] χ i , x k χ ¯ j
and the PWA model of the system with the loss of gain Γs in sensors can be described by:
x k + 1 = A i x k + B i u k + B i w w k + a i z k = C i Γ s x k + D i u k + D i w w k , [ x k u k ] χ i , x k χ ¯ j

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 wk as follows:

w ˜ k = [ w k a i ]
Thus, system (20) can be rewritten as:
x k + 1 = A i x k + B i Γ a u k + B ˜ i w w ˜ k z k = C i x k + D i u k + D ˜ i w w ˜ k , [ x k u k ] χ i , x k χ ¯ j
where:
B ˜ i w = [ B i w I ] , D ˜ i w = [ D i w I ]

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 [2124].

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

a i = a ¯ j , i S j , j
an alternative control strategy can be proposed. More precisely, the control is assumed to have the following structure:
u k = [ K j 1 K j 2 ] [ x k a ¯ j ] , x k χ ¯ j

In this way, the controller can also consider the displacement term ai = Dw̃k, 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:

x k + 1 = A i j x k + B ˜ i j w w ˜ k z k = C i j x k + D ˜ i j w w ˜ k , [ x k u k ] χ i , x k χ ¯ j
where:
A i j = A i + B i Γ a K j 1 , C i j = C i + D i K j 1 , B ˜ i j w = B ˜ i w + B i K j 2 D D ˜ i j w = D ˜ i w + D i K j 2 D

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) = xT Pix for [xT uT]Tχi with P i = P i T > 0, satisfying the dissipativity inequality:

w r , k 0 , V ( x k + 1 , u k + 1 ) V ( x k , u k ) < W ( z k , w k )
with supply rate:
W ( z , w ) = ( γ 2 w 2 z 2 ) , γ > 0
that is:
k , V ( x k + 1 , u k + 1 ) V ( x k , u k ) < ( γ 2 w k 2 z k 2 ) ,
then, the H performance condition:
k = 0 N z k 2 < γ 2 k = 0 N w k 2
is satisfied.

Furthermore, if the following matrix inequalities:

j J , , i S j , l with ( l , i ) S , M l , i j < 0
are satisfied, where:
M l , i j = [ A i j T P i A i j P j + C i j T C i j C i j T D j w + A i j T P i B j w D j w T C i j + B j w T P i A i j B j w T P i B j w + D j w T D j w γ 2 I ]
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 ( x N + 1 , u N + 1 ) < k = 0 N ( γ 2 w k 2 z k 2 )

Since V(xN+1, uN+1) > 0, then condition:

k = 0 N z k 2 < γ 2 k = 0 N w k 2
is met.

Moreover, if [ x k T u k T ] T χ i and [ x k + 1 T u k + 1 T ] T χ l, then:

w k , [ x k T w k T ] M l , i j [ x k T w k T ] T < 0

Obviously, if condition (33) holds, inequality (37) is satisfied and C i j T C i j > 0. For A i j T P l A i j P i + C i j T C i j, we can state that:

j J , i S j , l with ( l , i ) S , A i j T P i A i j P i < 0

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:

w r , k 0 , V ( x k + 1 ) V ( x k , u k ) < W ( z k , w k )
with supply rate:
W ˜ ( z k , w k ) = ( γ 2 [ w k T a i T ] T 2 z k 2 ) = ( γ 2 ( w k 2 + a i 2 ) z k 2 ) , γ > 0 , [ x k u k ] χ i

If there exist matrices Q i = Q i T > 0 with iI and matrices Gj, Yj, K j 2 with j Energies 07 01750i2, such that ∀j Energies 07 01750i2, ∀iSj, ∀l with (l, i) ∈ Sall:

[ Q i G j G j T ( A i G j + B i Γ a Y j ) T ( C i G j + D i Y j ) T 0 A i G j + B i Γ a Y j Q l 0 B ˜ i w + B i Γ a K j 2 D C i G j + D i Y j 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i Γ a K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] < 0
holds, then the feedback gains matrices K j 1 with j Energies 07 01750i2 are given by:
K j 1 = Y j G j 1 .

Proof

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

[ P i ( A i + B i Γ a K j 1 ) T ( C i + D i K j 1 ) T 0 A i + B i Γ a K j 1 P l 1 0 B ˜ i w + B i Γ a K j 2 D C i + D i K j 1 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i Γ a K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] < 0
where A i j = A i + B i Γ a K j 1, C i j = 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 ( G j T I I I ) and the right by diag (Gj I I I):

Let:

H = [ G j T 0 0 0 0 I 0 0 0 0 I 0 0 0 0 I ]
then:
H * [ P i ( A i + B i Γ a K j 1 ) T ( C i + D i K j 1 ) T 0 A i + B i Γ a K j 1 P l 1 0 B ˜ i w + B i Γ a K j 2 D C i + D i K j 1 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i Γ a K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] * H T = [ G j T P i G j G j T ( A i + B i Γ a K j 1 ) T G j T ( C i + D i K j 1 ) T 0 ( A i + B i Γ a K j 1 ) G j P l 1 0 B ˜ i w + B i Γ a K j 2 D ( C i + D i K j 1 ) G j 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i Γ a K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] = [ G j T P i G j ( A i G j + B i Γ a Y j ) T ( C i G j + D i Y j ) T 0 A i G j + B i Γ a Y j Q l 0 B ˜ i w + B i Γ a K j 2 D C i G j + D i Y j 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i Γ a K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] < 0

Since 0 < G j + G j T Q i G j T Q i 1 G j, Equation (45) implies:

[ Q i G j G j T ( A i G j + B i Γ a Y j ) T ( C i G j + D i Y j ) T 0 A i G j + B i Γ a Y j Q l 0 B ˜ i w + B i Γ a K j 2 D C i G j + D i Y j 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i Γ a K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] < 0
It is obvious to get Qi > 0, ∀iI, 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:

w r , k 0 , V ( x k + 1 ) V ( x k , u k ) < W ( z k , w k )
with supply rate:
W ˜ ( z k , w k ) = ( γ 2 [ w k T a i T ] T 2 z k 2 ) = ( γ 2 ( w k 2 + a i 2 ) z k 2 ) , γ > 0 , [ x k u k ] χ i

If there exist matrices Q i = Q i T > 0 with iI and matrices Gj, Yj, K j 2 with j Energies 07 01750i2, such that ∀j Energies 07 01750i2, ∀iSj, ∀l with (l, i) ∈ Sall:

[ Q i G j G j T ( A i G j + B i Y j ) T ( C i Γ s G j + D i Y j ) T 0 A i G j + B i Y j Q l 0 B ˜ i w + B i K j 2 D C i Γ s G j + D i Y j 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] < 0
holds, then the feedback gains matrices K j 1 with j Energies 07 01750i2 are given by:
K j 1 = Y j G j 1

Proof

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

[ P i ( A i + B i K j 1 ) T ( C i Γ s + D i K j 1 ) T 0 A i + B i K j 1 P l 1 0 B ˜ i w + B i K j 2 D C i Γ s + D i K j 1 0 I D ˜ i w + D i K j 2 D 0 ( B ˜ i w + B i K j 2 D ) T ( D ˜ i w + D i K j 2 D ) T γ 2 I ] < 0
where A i j = A i + B i K j 1, C i = C i Γ 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 ( G j T I I I ) and the right by diag (Gj 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 < E1xk ≤ 8 (Region 1), then:

[ T ˙ g β ˙ β ¨ ω ˙ g ω ˙ r w ˙ 1 w ˙ 2 ] = [ 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 ] [ T g β β ˙ ω g ω r ω 1 ω 2 ] + [ 10 0 0 0 0 123.4 0 0 0 0 0 0 0 0 ] [ T g r e f β r e f ] + [ 0 0 0 0 1.9 0 0 ] × 7.5 + [ 0 0 0 0 1.9 0 0.9325 ] e

If 8 < E1xk ≤ 12 (Region 2), then:

[ T ˙ g β ˙ β ¨ ω ˙ g ω ˙ r w ˙ 1 w ˙ 2 ] = [ 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 ] [ T g β β ˙ ω g ω r ω 1 ω 2 ] + [ 10 0 0 0 0 123.4 0 0 0 0 0 0 0 0 ] [ T g r e f β r e f ] + [ 0 0 0 0 2.31 0 0 ] × 10 + [ 0 0 0 0 2.31 0 2.3 ] e

If 12 < E1xk ≤ 18 (Region 3), then:

[ T ˙ g β ˙ β ¨ ω ˙ g ω ˙ r w ˙ 1 w ˙ 2 ] = [ 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 ] [ T g β β ˙ ω g ω r ω 1 ω 2 ] + [ 10 0 0 0 0 123.4 0 0 0 0 0 0 0 0 ] [ T g ref β ref ] + [ 0 0 0 0 2.5 0 0 ] × 16 + [ 0 0 0 0 2.5 0 5 ] e
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 = [ 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 ]
K 2 = [ 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 ]
K 3 = [ 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 ]
Actuator within fault tolerance:
K 1 = [ 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 ]
K 2 = [ 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 ]
K 3 = [ 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 ]
Sensor within fault tolerance:
K 1 = [ 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 ]
K 2 = [ 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 ]
K 3 = [ 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 ]

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 νm and wind speed turbulent νs ( ν = νm + νs ). νm = 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 Tg. 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 βf and the generator gain fault Tgf (let gain loss factor Γ a = [ 0.6 0 0 0.6 ]), 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 Pgf and generator speed ωgf 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 Γa. 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 Γ s = [ 1.2 0 0 1.2 ]) including generator speed sensor fault wgf and generator power sensor fault pgf. 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 pgf and generator speed wgf 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.

This work was simultaneously supported by the National Natural Science Foundation of China under Grant No. 61304049, No.61174116, and the Beijing Natural Science Foundation Program 4132021.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Sloth, C.; Esbensen, T.; Stoustrup, J. Active and Passive Fault-Tolerant LPV Control of Wind Turbines. Proceedings of the 2010 American Control Conference, Baltimore, MD, USA, 30 June 2010–2 July 2010; pp. 4640–4646.
  2. Rothenhagen, K.; Fuchs, F.W. Doubly fed induction generator model-based sensor fault detection and control loop reconfiguration. IEEE Trans. Ind. Electron. 2009, 56, 4229–4238, doi:10.1109/TIE.2009.2013683.
  3. Parker, M.A.; Ran, L.; Finney, S.J. Distributed control of a fault-tolerant modular multilevel inverter for direct-drive wind turbine grid interfacing. IEEE Trans. Ind. Electron. 2013, 60, 509–522, doi:10.1109/TIE.2012.2186774.
  4. Ruba, M.; Szabo, L.; Jurca, F. Fault Tolerant Switched Reluctance Machine for Wind Turbine Blade Pitch Control. Proceedings of the 2009 International Conference on Clean Electrical Power, Capri, Italy, 9–11 June 2009; pp. 721–726.
  5. Cuzzola, F.A.; Morari, M. A Generalized Approach for Analysis and Control of Discrete-Time Piecewise Affine and Hybrid Systems. In Hybrid Systems: Computation and Control Lecture Notes in Computer Sciences; Springer: Berlin/Heidelberg, Germany, 2001; pp. 189–203.
  6. Johansson, M.; Rantzer, A. Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 1998, 43, 555–559, doi:10.1109/9.664157.
  7. Rantzer, A. Piecewise linear quadratic optimal control. IEEE Trans. Autom. Control 2000, 45, 629–637, doi:10.1109/9.847100.
  8. Hassibi, A.; Boyd, S. Quadratic Stabilization and Control of Piecewise-Linear Systems. Proceedings of the American Control Conference, Philadelphia, PA, USA, 21–26 June 1998; pp. 3659–3664.
  9. Pettersson, S.; Lennartson, B. LMI for Stability and Robustness of Hybrid Systems. Proceedings of the American Control Conference, Albuquerque, NM, USA, 4–6 June 1997; pp. 1714–1718.
  10. Slupphaug, O.; Foss, B.A. Constrained quadratic stabilization of discrete time uncertain nonlinear multi-model systems using piecewise affine state feedback. Int. J. Control 1999, 72, 686–701, doi:10.1080/002071799220867.
  11. Mignone, D.; Ferrari-Trecate, G.; Morari, M. Stability and Stabilization of Piecewise Affine and Hybrid Systems: An LMI Approach. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, 12–15 December 2000; pp. 504–510.
  12. Gang, F. Stability analysis of piecewise discrete-time linear systems. IEEE Trans. Autom. Control 2002, 47, 1108–1112, doi:10.1109/TAC.2002.800666.
  13. Esbensen, T.; Sloth, C. Fault Diagnosis and Fault-Tolerant Control of Wind Turbine; Aalborg University: Aalborg, Denmark, 2009.
  14. Højstrup, J. Velocity spectra in the unstable planetary boundary layer. J. Atmos. Sci. 1982, 39, 2239–2248, doi:10.1175/1520-0469(1982)039<2239:VSITUP>2.0.CO;2.
  15. Thomsen, S. Nonlinear Control of a Wind Turbine; Technical University of Denmark: Kgs. Lyngby Copenhagen, Denmark, 2006.
  16. Østergaard, K.Z.; Brath, P.; Stoustrup, J. Estimation of effective wind speed. J. Phys. 2007, 75, 1–9.
  17. Burkart, R.M.; Margellos, K.; Lygeros, J. Nonlinear Control of Wind Turbines: An Approach Based on Switched Linear Systems and Feedback Linearization. Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA, 12–15 December 2011; pp. 5485–5490.
  18. Odgaard, P.F.; Aalborg, U.; Stoustrup, J.; Kinnaert, M. Fault-tolerant control of wind turbines: A benchmark model. IEEE Trans. Control Syst. Technol. 2013, 21, 1168–1182, doi:10.1109/TCST.2013.2259235.
  19. Sloth, C.; Esbensen, T.; Stoustrup, J. Robust and fault-tolerant linear parameter-varying control of wind turbines. Mechatronics 2011, 21, 645–659, doi:10.1016/j.mechatronics.2011.02.001.
  20. Rotondo, D.; Nejjari, F.; Puig, V.; Blesa, J. Fault Tolerant Control of the Wind Turbine Benchmark Using Virtual Sensors/Actuators. Proceedings of the 8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, National Autonomous University of Mexico, Mexico, 29–31 August 2012; pp. 114–119.
  21. Zhang, H.B.; Dang, C.H.; Zhang, J. H control of piecewise-linear systems under unreliable communication links. Circuits Syst. Sign. Process. 2012, 31, 1297–1318, doi:10.1007/s00034-011-9377-7.
  22. Munteanu, I.; Cutululis, N.-A.; Bratcu, A.L.; CeangĂ, E. Optimal Control of Wind Energy Systems; Springer: London, UK, 2008.
  23. Zhang, H.; Shi, Y. Observer-based H feedback control for time-varying discrete-time systems with intermittent measurements and input constraints. ASME J. Dyn. Syst. Meas. Control. 2012, 134, doi:10.1115/1.4006070.
  24. Shi, Y.; Huang, J.; Yu, B. Robust tracking control of networked control systems: Application to a networked DC motor. IEEE Trans. Ind. Electron. 2013, 60, 5864–5874, doi:10.1109/TIE.2012.2233692.
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Figure 1. The structure of WECS.

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Figure 1. The structure of WECS.
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Figure 2. The control strategy for wind turbine.

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Figure 2. The control strategy for wind turbine.
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Figure 3. The tendency of the β, Tg and Pg following the wind speed.

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Figure 3. The tendency of the β, Tg and Pg following the wind speed.
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Figure 4. Wind speed vs. the corresponding working point.

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Figure 4. Wind speed vs. the corresponding working point.
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Figure 5. H reliable control for normal WECS based on stochastic PWA model: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).

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Figure 5. H reliable control for normal WECS based on stochastic PWA model: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).
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Figure 6. Actuators fault of WECS without fault tolerant control: (a) wind speed (m/s); (b) the tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) the pitch angle (°); (g) the generator torque (Nm).

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Figure 6. Actuators fault of WECS without fault tolerant control: (a) wind speed (m/s); (b) the tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) the pitch angle (°); (g) the generator torque (Nm).
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Figure 7. H fault tolerant control for actuators fault of WECS: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).

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Figure 7. H fault tolerant control for actuators fault of WECS: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).
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Figure 8. Sensors fault of WECS without fault tolerant control: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).

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Figure 8. Sensors fault of WECS without fault tolerant control: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg setpiont (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).
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Figure 9. H fault tolerant control for sensors fault of WECS: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg set point (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).

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Figure 9. H fault tolerant control for sensors fault of WECS: (a) wind speed (m/s); (b) tip speed ratio; (c) optimal ωg set point (rad/s); (d) generator speed (rad/s); (e) generator power (KW); (f) pitch angle (°); (g) generator torque (Nm).
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Table Table 1. Model parameters.

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Table 1. Model parameters.
ParameterValueUnitParameterValueUnit
Jr90,000kg · m2Pg,nom225kW
Jg10kg · m2ωr,nom4.29rad/s
Ks8 × 106Nm/radωg,nom105.534rad/s
Ds8 × 104kg · m2/(rad · s)ωr,min3.5rad/s
Ng24.6-ωg,min86.1rad/s
Energies 07 01750i414.5mθmin0deg
τθ0.15sθmax25deg
τΓ0.1s|θ˙|max10deg/s
Table Table 2. Parameters of linearized model in different working points.

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Table 2. Parameters of linearized model in different working points.
Wind Speed/ms−1Parameters

a84a88e81a1a2a3
V = 7.50.4090.501.900.31252.920.9375
V = 100.4790.532.310.333.652.3
V = 160.8330.532.500.62555
Table Table 3. The faults considered.

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Table 3. The faults considered.
FaultTypeFault Description
actuatorgain FactorΓa
sensorgain FactorΓs
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