# Optimal Fault Tolerant Control of Large-Scale Wind Turbines in the Case of the Pitch Actuator Partial Faults

Authors: Younes Ait El Maati; Lhoussain El Bahir
Journal: Complexity
 (2020)

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## Abstract

In this paper, an adaptive fault tolerant control strategy is proposed to deal with the three pitch actuator faults in the large-scale wind turbines. Firstly, a simultaneous state and fault estimation was performed through a suitable LMI (linear matrix inequality) based optimal strategy. Hereafter, the new control law is designed using the previously estimated fault information. The actuator efficiency estimator uses as design parameters, respectively, the performance indexγ against the wind and the learning rate Ξ. of the fault estimation algorithm. The study shows that the choice of the previous two parameters impacts the response time of the fault estimation and the correlation of the tracking error with the wind. The aim is to choose a small fault estimation response time while keeping a weak correlation between the tracking error and the wind turbulence noise. Finally, a tuning strategy is elaborated to choose the suitable γ and Ξ to match the reconfiguration objective.


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## Body

## 1. Introduction

The increasing energy demand leads to the production of large wind turbines. Thus, more and more vibrational behaviors could be expected in the different degrees of freedom (DOFs). Figures1 and 2 show the most important DOF. The excitation of the DOFs leads to increase in loads and forces on the structure. The aim is to reduce the loads in order to protect the structure against fatigue and damage.Figure 1
Blade pitch degree of freedom.Figure 2
Blade flap and tower fore-aft degrees of freedom.The wind turbine operates in two different regions, low and high winds. In the first region, so called partial load region, the aim is to maximize the extracted power [1, 2]. This was achieved by controlling the generator torque to maintain an optimum ratio between the tip speed of the blades and the operating wind speed [3]. In the second region, so called full load region, the wind turbine was controlled to reduce loads and keep the power at its rated value. Practically, the constant power was maintained through a fixed speed and a fixed generator torque. This has been obtained by pitching the blades from or toward wind to adjust the speed of the rotor, while applying a constant generator torque [4]. For example, the authors in [5] proposed an anticipative control scheme of a wind turbine working in the maximum power region. The idea is to use the wind speed as a generator of the set point of the MPP (maximal power point) control.


### 1.1. The Pitch Actuator Control of Wind Turbines in the Full Load Region

In the full load region, the high turbulence occurs and produces big deflection movements on the blades. The pitch actuators were used to attenuate the loads in the full load region [6–8].Many papers have studied the loads exerted by the wind on the different parts of the wind turbine structure. In [9], the authors discussed and identified which wind turbine components can benefit from advanced control algorithms and also presented results from a preliminary loads case analysis using a baseline controller.From the discussed papers above, it can be noticed that the wind turbines in the megawatt size are expensive, and hence their reliability is expected to be high to generate as much energy as possible. This type of wind turbines is expected to have reduced downtimes while keeping energy production. Usually, the turbines are turned off even during simple faults to wait for service.


### 1.2. Fault Detection and Tolerant Control in Wind Turbines

The statistics in [10] show that gearbox faults are the most critical ones in wind turbines, essentially due to its relatively high stopping time and severity on the whole structure. Another type of wind turbine faults is of electrical root. For example, the wind turbine can face short circuit faults in the wires or inverter over voltage faults. It can also experience hard over fault in the generator or some sensor faults like drift or bias.The last reason creates the need to introduce fault detection, isolation, and accommodation systems within the existing classical control systems in order to reduce stop time even with limited energy production [11]. Early detection has been the spot of the control industry studies especially using residual generation and fault analysis. The paper [12] presents an overview of the fault diagnosis method categories, with an application to the wind turbine pitch fault control. For more details, see also [13, 14].The model-based methods are one of the most suitable methods for the study of the fault tolerant control of the wind turbines. In fact, the availability of the large-scale wind turbine model is no longer a problem especially with the existence of highly trusted wind environment software and emulators such as FAST (Fatigue, Aerodynamics, Structures, and Turbulence).In this paper, the FAST code is used to test the proposed strategy. FAST is an aeroelastic simulator capable of predicting the fatigue loads of two- and three-bladed flexible with 24 degrees of freedom [15, 16]. The code uses a modal approach in combination with Kane dynamics to develop the equations of motion. FAST helps considerably researchers and engineers in testing wind load reduction algorithms [17]. This software provides a linearized model parameter about a chosen operating point with more than 40 measures of outputs (rotor speed, generator speed, and tower and blade displacement).The pitch servomotor is the most critical actuator in the wind turbine [18, 19]. As discussed earlier, it is used to regulate the power and to attenuate the vibrations in the high wind region. If this actuator loses its efficiency, it will no longer provide the sufficient control efforts to attenuate the vibrations and regulate the power. Based on the last reason, a new control law is to be constructed to deal with the fault. The new control efforts could be produced by online tuning of the controller parameters or by changing the baseline control law using an estimation of the fault [20–22].In the wind turbines’ control field, in the pitch actuator fault, the authors inreference [23] proposed a fault tolerant control strategy to deal with the drop of the hydraulic liquid in the servo pitches of the rotor. In the same field, another type of fault is the imbalance in the rotor due to icing or destruction of one or more blades. This fault not only reduces the aerodynamic efficiency of the turbine and its power output but can also lead to large increases of loads on the drivetrain, blades, and tower. For this, the authors in [24] used the FAST software to model aerodynamic imbalance in a sample 5 MW offshore wind turbine. It has been concluded that the combination of blade and nacelle measurements, obtained from instrumentation already placed on the structure, can be formulated into an algorithm used to detect and locate the imbalance (see also [25]).Another essential component is the anemometer. The anemometers are used in the feedforward control of the wind turbines. In [26], the authors proposed a new method to estimate the wind speed that could be substituted for anemometers in control loops. Moreover, the authors in [27] proposed a robust observer to generate residuals for the fault turbine moment sensors which provides a measurement of the loads applied on the structure. In [28], the authors have proposed a method to estimate the gearbox efficiency in the wind turbines. The idea is to detect the drop in the efficiency and generate a suitable residual to indicate the fault occurrence (see also [29, 30]).Given the very disturbed wind nature, several optimization methods could be used to minimize the effect of the disturbance on the outputs. In [31], a multiagent optimal method based on reinforcement learning was proposed to control a grid connected energy system.The fault isolation and tolerant control applied to the wind turbines has been illustrated in many papers. For example, in [32, 33], a codesign of the observers and decision mechanisms for satisfying certain trade-off between different isolation performance indices has been established. The performance indices were the false isolation rates, the isolation times, and the minimum size of the isolable faults. The considered fault was additive on the pitch actuator. The paper [34] used jump observers to detect and isolate the faults. The false alarm rate is upper bounded by means of Markov’s inequality. The trade-offs between the minimum detectable faults, the false alarm rate, and the response time to faults of the fault diagnoser have been explored. In the previous cited works, the assumption of slow dynamics faults was considered which limits the usage of the strategies to slow faults dynamics.In this paper, the following contributions have been developed:(i)
An integrator-based simultaneous states and fault estimator algorithm is designed.(ii)
The stability and the robustness of the algorithm is proven for the partial fault of the pitch actuators using Lyapunov theory.(iii)
A fault tolerant control including a baseline multiobjective state feedback regulator is designed. The performance of the method depends on two parameters to be tuned,γ and Ξ.(iv)
To ensure a best performance of the proposed control solution, two mappings are proposed to tune the design parametersγ and Ξ.(v)
The evaluation of the proposed control strategy is performed on the 1.5 MW wind turbine benchmark model.This paper is organized as follows. In Section2, the wind turbine model used in the benchmark is presented. In Section 3, the wind turbine fault and state estimation is performed and the nominal multiobjective regulator is designed. The wind turbine adaptive fault tolerant control in the faulty production is studied and the tuning strategy is proposed in Section 4. Some concluding remarks are presented in Section 5.


## 1.1. The Pitch Actuator Control of Wind Turbines in the Full Load Region

In the full load region, the high turbulence occurs and produces big deflection movements on the blades. The pitch actuators were used to attenuate the loads in the full load region [6–8].Many papers have studied the loads exerted by the wind on the different parts of the wind turbine structure. In [9], the authors discussed and identified which wind turbine components can benefit from advanced control algorithms and also presented results from a preliminary loads case analysis using a baseline controller.From the discussed papers above, it can be noticed that the wind turbines in the megawatt size are expensive, and hence their reliability is expected to be high to generate as much energy as possible. This type of wind turbines is expected to have reduced downtimes while keeping energy production. Usually, the turbines are turned off even during simple faults to wait for service.


## 1.2. Fault Detection and Tolerant Control in Wind Turbines

The statistics in [10] show that gearbox faults are the most critical ones in wind turbines, essentially due to its relatively high stopping time and severity on the whole structure. Another type of wind turbine faults is of electrical root. For example, the wind turbine can face short circuit faults in the wires or inverter over voltage faults. It can also experience hard over fault in the generator or some sensor faults like drift or bias.The last reason creates the need to introduce fault detection, isolation, and accommodation systems within the existing classical control systems in order to reduce stop time even with limited energy production [11]. Early detection has been the spot of the control industry studies especially using residual generation and fault analysis. The paper [12] presents an overview of the fault diagnosis method categories, with an application to the wind turbine pitch fault control. For more details, see also [13, 14].The model-based methods are one of the most suitable methods for the study of the fault tolerant control of the wind turbines. In fact, the availability of the large-scale wind turbine model is no longer a problem especially with the existence of highly trusted wind environment software and emulators such as FAST (Fatigue, Aerodynamics, Structures, and Turbulence).In this paper, the FAST code is used to test the proposed strategy. FAST is an aeroelastic simulator capable of predicting the fatigue loads of two- and three-bladed flexible with 24 degrees of freedom [15, 16]. The code uses a modal approach in combination with Kane dynamics to develop the equations of motion. FAST helps considerably researchers and engineers in testing wind load reduction algorithms [17]. This software provides a linearized model parameter about a chosen operating point with more than 40 measures of outputs (rotor speed, generator speed, and tower and blade displacement).The pitch servomotor is the most critical actuator in the wind turbine [18, 19]. As discussed earlier, it is used to regulate the power and to attenuate the vibrations in the high wind region. If this actuator loses its efficiency, it will no longer provide the sufficient control efforts to attenuate the vibrations and regulate the power. Based on the last reason, a new control law is to be constructed to deal with the fault. The new control efforts could be produced by online tuning of the controller parameters or by changing the baseline control law using an estimation of the fault [20–22].In the wind turbines’ control field, in the pitch actuator fault, the authors inreference [23] proposed a fault tolerant control strategy to deal with the drop of the hydraulic liquid in the servo pitches of the rotor. In the same field, another type of fault is the imbalance in the rotor due to icing or destruction of one or more blades. This fault not only reduces the aerodynamic efficiency of the turbine and its power output but can also lead to large increases of loads on the drivetrain, blades, and tower. For this, the authors in [24] used the FAST software to model aerodynamic imbalance in a sample 5 MW offshore wind turbine. It has been concluded that the combination of blade and nacelle measurements, obtained from instrumentation already placed on the structure, can be formulated into an algorithm used to detect and locate the imbalance (see also [25]).Another essential component is the anemometer. The anemometers are used in the feedforward control of the wind turbines. In [26], the authors proposed a new method to estimate the wind speed that could be substituted for anemometers in control loops. Moreover, the authors in [27] proposed a robust observer to generate residuals for the fault turbine moment sensors which provides a measurement of the loads applied on the structure. In [28], the authors have proposed a method to estimate the gearbox efficiency in the wind turbines. The idea is to detect the drop in the efficiency and generate a suitable residual to indicate the fault occurrence (see also [29, 30]).Given the very disturbed wind nature, several optimization methods could be used to minimize the effect of the disturbance on the outputs. In [31], a multiagent optimal method based on reinforcement learning was proposed to control a grid connected energy system.The fault isolation and tolerant control applied to the wind turbines has been illustrated in many papers. For example, in [32, 33], a codesign of the observers and decision mechanisms for satisfying certain trade-off between different isolation performance indices has been established. The performance indices were the false isolation rates, the isolation times, and the minimum size of the isolable faults. The considered fault was additive on the pitch actuator. The paper [34] used jump observers to detect and isolate the faults. The false alarm rate is upper bounded by means of Markov’s inequality. The trade-offs between the minimum detectable faults, the false alarm rate, and the response time to faults of the fault diagnoser have been explored. In the previous cited works, the assumption of slow dynamics faults was considered which limits the usage of the strategies to slow faults dynamics.In this paper, the following contributions have been developed:(i)
An integrator-based simultaneous states and fault estimator algorithm is designed.(ii)
The stability and the robustness of the algorithm is proven for the partial fault of the pitch actuators using Lyapunov theory.(iii)
A fault tolerant control including a baseline multiobjective state feedback regulator is designed. The performance of the method depends on two parameters to be tuned,γ and Ξ.(iv)
To ensure a best performance of the proposed control solution, two mappings are proposed to tune the design parametersγ and Ξ.(v)
The evaluation of the proposed control strategy is performed on the 1.5 MW wind turbine benchmark model.This paper is organized as follows. In Section2, the wind turbine model used in the benchmark is presented. In Section 3, the wind turbine fault and state estimation is performed and the nominal multiobjective regulator is designed. The wind turbine adaptive fault tolerant control in the faulty production is studied and the tuning strategy is proposed in Section 4. Some concluding remarks are presented in Section 5.


## 2. The Wind Turbine Model

The wind turbine model used for the control contains seven wind turbine states as follows:(i)
The rotor speed(ii)
The bladei flap wise deflection(iii)
The bladei flap wise deflection velocitywhere i = 1, 2, 3.The state space model of the rotor about the operating point is given by(1)δx˙=Aδx+Bδu+Ddωδy=Cδx.The operatorδ means a variation about the operating point. A is the dynamics matrix, B is the input matrix, Dd is the disturbance matrix, and C is the measured output matrix. The numerical values of the matrices A, B, C, and Dd are given in the Appendix.The state vector and the pitch actions are, respectively, represented by the vectorsδx and δu:(2)δx=δΩrδx_def1δx˙_def1δx_def2δx˙_def2δx_def3δx˙_def3T,δu=δβ1δβ2δβ3T,where δx_defi and δx˙_defix˙_defi are, respectively, the blade deflection and deflection velocity associated to the ith blade. δy represents the measured rotor speed and deflection velocity. The vector ω is the wind disturbance on the considered states and is given by(3)ω=Vt−V0,where Vt is the instantaneous wind speed and V0 is the operating wind speed which is equal to 18 m/s in the high wind region. The vector is the rotor speed about the operating point and is given by(4)δΩr=Ωrt−Ωr0,where Ωrt is the measured rotor speed and the operating rotor speed is Ωr0=40rpm. The input pitches about operating point are equal to δβ1=δβ2=δβ3=βt−β0. βt is the measured pitch angle and the operating pitch angle is β0=9°.The numerical values of variables in the international system of measurement units are given.It is assumed that the measured outputs are the rotor speed and the blade flap deflection velocities of the three blades. In actual application like wind turbine blade analysis, the deflection velocities could be obtained using the derivative of the deflection displacement measure. This information could be directly obtained using a MEMS gyroscope sensor as in [35], or estimated using ultra-wideband signals as proposed in [36].The considered turbine is a Wind PACT 1.5 MW large scale. More details about the parameters of this wind turbine could be found in [37, 38]. The turbine parameters summary is given in Table 1.Table 1
Wind turbine parameters.
Characteristics
Value
Rated power (MW)
1.5
M
b
:Blade mass (kg)
4,230
D
b
:Blade stiffness (N·m/rad)
20760
Total rotor mass (kg)
32,016
J
r: rotor inertia (kg·m2)
2.9624 ∗ 106
J
g: generator inertia (kg·m2)
53.036 ∗ Ng2
N
g: gearbox ratio
87.965
R: rotor radius (m)
35


## 3. Wind Turbine Control Strategy in the Nominal Production

The nominal controller is designed for two objectives. On the one hand, the rotor speed should be regulated despite the wind changes. On the other hand, the blade’s deflections should be attenuated to prevent the structure from being damaged. The control scheme is summarized in Figure3. The blades’ deflection velocity and the rotor speed are both measured and fed to the robust state observer. Finally, the state feedback controller uses the estimated states to generate the necessary pitch angle variations responsible for regulating rotor speed and attenuating blade deflections. The adjective nominal means the baseline or the fault free situation. The system is linearized about the nominal operating point, and then the nominal controller is derived based on the pole placement method to achieve the nominal production. The opposite to nominal is faulty, which means the situation in which the fault occurs.Figure 3
The wind turbine control objective scheme.


### 3.1. The Global Wind Turbine State Space Model

The global state space representation in the equation (5) takes into account the pitch actuator faults. The actuator effectiveness fault is modeled as a multiplication of the input matrix B by an unknown fault matrix F, and the faulty system is then given by(5)x˙t=Axt+BFtut+D1ωt,yt=Cxt+D2ωt,where xt∈ℝn is the state vector; ut∈ℝm is the input vector; yt∈ℝpis the measured output with p≥m; and C and B have full rank (rank(C) = p and rank(B) = m). Ft∈ℝm×m, and Ft=diagf1t,f2t,…,fmt is the vector of faults on the actuators.


### 3.2. The Observer and the Baseline Controller Design

The three-blade flap deflection amplitude reduction and the rotor speed regulation are performed through a pole placement state feedback [39] as in equation (6). In fact, the nominal control law in equation (6) is based on the fault free (nominal) situation, in which the fault matrix F(t) equals the identity matrix which is constant.(6)u=Δβi_c=Kx^.The variablex^ in equation (6) is the state vector to estimate. The gains of controller K are design parameters to achieve the multiobjective wind turbine control. The variable x^ should be estimated robustly in the next paragraph.In Figure3, Tg0 is the generator torque input at the operating point and is kept constant in the region of high winds, β0 is the pitch angle input at the operating point, Δβi_c is the pitch angle input variations from the operating point (i = 1, 2, 3), K is the control gain, Δx˙def_i are the deflection velocity variations from the operating point (i = 1, 2, 3), ΔΩr is the rotor speed variations from the operating point, Δβi is the measured pitch angles and are input to the state observer, and Ft is the fault matrix.


#### 3.2.1. The Robust State Observer

The structure of the state observer is given by system (7) and illustrated by Figure 4:(7)x^˙t=Ax^t+BFtut−LCx^t−yt,where yt corresponds to the vector containing the measured deflection velocities Δx˙_defi and rotor speed ΔΩr. The vector ut corresponds to the measured three blade pitch angles. Ft is the matrix of the faults. x^t is the output of the observer.Figure 4
The state observer structure.x
^
t
∈
ℝ
n are the observer states, y^t∈ℝn are the observed outputs, F^t∈ℝm×m is the estimate of Ft, and Lt∈ℝn×p is the estimator gain to be designed.


#### 3.2.2. The Fault Matrix Estimator

The fault matrixFt has to be estimated. This estimate is F^t∈ℝm×m and has the following diagonal form: F^t=diagf^1,f^2,f^3, where f^i is the estimation of the pitch fault on the actuator of index i. The following integrator-based algorithm is proposed:(8)f^˙i=−ξirieytuit,where i = 1, 2, 3; Ξ=diagξ1,ξ2,…,ξm; Ξ∈ℝm×m is called the learning rate; and ri∈ℝ1×p. The parameter p is the number of measurements. uit is the control input in the direction to the pitch actuator number i.The adaptability of the estimation consists in the update of the fault estimate dynamics with time by acting on the learning rateΞ or the matrix R=r1r2r3T. The matrix R is an additional degree of freedom that can be used to refine the approximation BTP=RC in equation (15) in Section 3.2.3. An application of this matrix could be to refine equation (15) only in the direction of a selected measurement from matrix C for a fixed η. For example, to refine equation (15) in the direction of the blade deflections, R could be chosen equal to 000−470.41580.00100−470.4158000−470.4158 and the resulting product RC will then be equal to 000000−470.4158000000−470.4158000000−470.4158. In the resulting matrix RC, the second row vector became null. Indeed, this row represents the contribution of the rotor speed in the product matrix RC. The remaining vector rows 3, 5, and 6 represent the blade deflection velocities contribution in the product matrix RC. Another alternative could be to compute an initial guess of R using YALMIP and refine it according to the need. In the present paper, R was extracted using YALMIP.The learning rateΞ is selected and handled by the user to ensure the fault estimation performance requirements. It allows to achieve a compromise between a quick fault accommodation and the disturbance amplification. Indeed, for high values of Ξ, high frequency information, namely, disturbances, is integrated in the learning process and vice versa. The variable ey(t) is the output tracking error and is equal to the difference y^t−yt.


#### 3.2.3. Stability and Robustness of the Observer

Consider the states tracking errorext=x^t−xt, the output tracking error eyt=y^t−yt, and the fault tracking error eft=F^t−Ft.The states’ tracking error dynamics are given by the state space representation in the following equation:(9)e˙xt=A−LCext+Beftut+LD2−D1ωt,eyt=Cext−D2ωt.The first objective is to haveext⟶0 and eft⟶0 when t⟶+∞. The tracking errors’ convergence and stability could be ensured using the Lyapunov function Vet:(10)Vet=exTtPext+trefTtΞ−1eft.The condition of validity ofVet is that Ξ is chosen invertible.P is a symmetric positive definite matrix. The idea of the Lyapunov function choice is as the learning rate increases, the quadratic fault tracking error eft decreases and the fault estimate tends to the actual fault. Finally, as the P matrix decreases, the quadratic state tracking error eyt decreases and tends to the null vector.The second objective is to minimize the effect of the wind turbulence on the output tracking erroreyt, using a performance index γ>0 as(11)eyt2≤γωt2.The minimization of the wind turbulenceωt effect on the output error could be performed using the following objective function:(12)J=∫tf∞1γeyTteyt−γωtTωtdt.The considered fault is time invariant. This means thatF˙t=0 and then e˙ft=F^˙t. The derivative of Vet with respect to time is given by(13)V˙et=exTtPA−LC+A−LCTPext+2exTtPBeftut+2exTtPLD2−D1ωt+2treftΞ−1F^˙t.Writing equation (8) as a matrix form gives the following:(14)Ξ−1F^˙=ReytuTt=RCextuTt.To eliminate the term containing bothexTt and ut, equation (15) could be adopted. Indeed, by adopting (15) the second and fourth term in the development of the derivative of the Lyapunov cancel each other.(15)BTP=RC.In order to solve the condition (15), its LMI version is used as in the following lemma.Lemma 1 (see [40]).
The conditionBTP=RC has an LMI equivalent writing:(16)minimizeηsubject toηIrBTP−RC∗ηIn>0,where η is a dimensionless constant used to approximate BTP=RC as an LMI. The smaller η is, the accurate BTP=RC is in LMI writing.In the sequel, the symbol ‘∗’ means the transposition of the term in the same diagonal as “∗.” For example, in equation (16), “∗” is the transposition of the term BTP−RC.The derivative of the Lyapunov function becomes(17)V˙et=exTtPA−LC+A−LCTPext+2exTPLD2−D1ωt.By replacing the Lyapunov function derivatives in the cost function expression, the following inequality is obtained:(18)J≤∫tf∞1γeyTteyt−γωtTωtdt+V˙et=∫tf∞1γeyTteyt−γωtTωtdt+exTtPA−LC+A−LCTPext+2exTPLD2−D1ωt.There is no direct action of the wind on the outputs; then,D2=0. The objective function inequality:(19)J≤1γeyTteyt−γωTtωtdt−2exTtPD1ωt+exTtPA−LC+A−LCTPext=∫tf∞ζtTϒζtdt,where(20)ζt=extωt,ϒ=PA−LC+A−LCTP+1γCTC−PD1∗−γId.Finally, to ensure the stability of the states’ tracking error dynamics and the minimization of the wind turbulence on the output tracking error, the following inequality should hold:(21)ϒ<0.Using the Schur complement, the same previous inequality becomes(22)PA+ATP−YC−CTYT−PD1CT∗−γId0∗∗−γIp<0,where Y=PL, and then the error dynamic (9) satisfies the H∞ performance.Remark 1.
The adaptive algorithm (8) is proposed with the assumption that the input vector uit satisfies the persistent excitation condition, which means that there exist positive scalars μ1, μ2, and t0 such that for all t, the following inequality holds:(23)μ1≤∫tt+t0uisuisds≤μ2.
The poles of the designed observer could be placed through the linear matrix inequality in the following lemma.Lemma 2 (see [41]).
The eigenvalues of closed loop observed system dynamic matrixA−LC∈ℝn×nA∈Rn×n belong to the circular region Dα,τwith center α+j0 and radius τ if and only if there exists a symmetric positive definite matrix P∈ℝn×nP∈Rn×n such that the following condition holds:(24)−PPA−LC−αIn∗−τ2P<0.The final integrated problem to be solved for the symmetric positive definite matrixP is as the following lemma.Lemma 3.
AssumeD2 is null matrix and let a prescribed H∞ performance level γ be given. If there exists a symmetric positive definite matrix P∈ℝn×n and matrices Y∈ℝn×p and R∈ℝm×p such that the following condition holds:(25)PA+ATP−YC−CTYT−PD1CT∗−γId0∗∗−γIp<0−PPA−LC−αIn∗−τ2P<0,minimizeηsubject  toηIrBTP−RC∗ηIn>0,where Y=PL, then the error dynamic (9) satisfies the H∞ performance and the poles of the robust state observer are within the region Dα,τ.


#### 3.2.4. The State Observer and Fault Simulation Results

The natural frequency of the blade isfn=1/2πDb/Mb=1.511Hz. Mb is the blade mass and Db is the stiffness of the blade. It gives a natural pulsation of ωn=9.49rad/s. All the frequencies above the natural frequency should be filtered by the observer. However, the observer should amplify the gain corresponding to ωn for the estimation of the blades’ deflections. The Bode diagrams of the designed observer in the system (7) are illustrated in Figures 5(a) and 5(b). The considered channel links the pitch actuator’s input (u1t=Δβ1 in Figure 4) with the blade deflection observed state (y3t=Δxdef_1 in Figure 4).Figure 5
(a) Bode diagrams of the designed observer for different performance levels. (b) Zoom on the Bode diagrams of the designed observer for different performance levels. (c) The state observer pole placement for LMI regionα,τ=−20,20 in the complex axis.
(a)
(b)
(c)Figures5(a) and 5(b) illustrate the designed observer Bode diagram function of the performance level γ corresponding to the system in equation (7). The Bode diagrams are divided into two main regions. The first region is about the natural pulsation of the blades ωn=9.49rad/s. In this region, the magnitude presents maximal values, which means that the blades’ deflections are perfectly observed. The second region is where the pulsations are less than the natural pulsation of the blades. As γ becomes smaller (closer to values near 0), the gain of the observer is reduced and the low frequencies of the wind speed are not amplified. The advantage is to prevent penetrating the amplified low frequencies contained in the rotor speed to the fault estimation algorithm through the residual eyt. The effect of the amplified low frequencies is reducing the convergence speed of the estimation algorithm in equation (8). As a conclusion, the convergence speed of the fault estimation could be tuned by the performance index γ in addition to the tuning by the learning rate Ξ as illustrated in Figure 6(a), where the learning rate is fixed at 0.1 and γ takes the values 3, 1, and 0.3.Figure 6
(a) Estimation of the pitch 1 actuator effectiveness for different performance levels. (b) Estimation of the pitch 1 actuator effectiveness for different learning rates. (c) Fault estimate with dynamics faster than system dynamics. (d) Zoom on fault estimate with dynamics faster than system dynamics.
(a)
(b)
(c)
(d)Figures7 and 8 illustrate the tracking errors in the faulty and the fault tolerant case for the rotor speed and the blade 1 deflection velocity as samples. The observation quality is defined by the states’ tracking error. It could be noticed that a sufficient level of observation is achieved for γ=0.3 and ξi=0.1 (i = 1, 2, 3).Figure 7
The rotor speed tracking error.Figure 8
The estimated blade 1 deflection velocity tracking error.For the rotor speed, in the fault tolerant case, the tracking error has a mean value of−3.96×10−4 rad/s and a standard deviation of 6.841×10−4rad/s. For the faulty case, the mean value increases to −0.002 rad/s with a standard deviation of 9.8893×10−4rad/s. For the blade deflection velocity, in the fault tolerant case, the tracking error has a mean of −0.058 m/s and a standard deviation of 0.4973 m/s. For the faulty case, the mean value increases to −0.0823 m/s and a standard deviation of 0.6595 m/s.Figure5(c) illustrates the result of pole placement of the state observer. The chosen region is of center 20 and radius 20. The resulting closed loop eigenvalues are −33.4089;−24.7069±8.6379i;−26.6204±18.7259i;−26.6204±18.7259iT.In Figure6(a), the effectiveness of actuator number 1 means 1−f1. f1 represents the decrease in the output of the actuator.In Figures7 and 8, the notations ey1 correspond to the rotor speed component and ey2 corresponds to the blade deflection velocity component in the tracking error vector ey.Figures6(a) and 8 correspond to the first faulty scenario of Section 4 and Figure 7 corresponds to the third faulty scenario.Figure6(b) illustrates the fault estimation of the blade 1 actuator for several values of learning rate ξ1. As the learning rate increases, the estimate of the fault converges rapidly to the actual fault (magenta).Remark: There is a coupling effect between the fault estimation and the state observer due to the fault tolerant control (FTC) correction. Such an effect can be neglected as long as the state observer is slower than the fault estimation. This can be achieved by choosing properly the learning rate.


#### 3.2.5. Fault Scenario with Dynamics Faster than the System Dynamics

In this section, a sinusoidal loss of efficiency is considered with frequency higher than 1.5 Hz of blades. Figures6(c) and 6(d) show the result of fault estimation in that case. In the proposed strategy, by only tuning the learning rate to suitable higher values, the faults with dynamics faster than the system dynamics could be reconfigured without need to redesign the fault estimate algorithm as recommended in [33]. The amplitude of the fault is 0.1 and the frequency is 12 rad/s higher than 9.49 rad/s of the blades.In Figure7, the most important issue is the change of the mean rotor speed error. This information is the only one needed by the algorithm about the fault on the rotor speed. In Figure 8, the change is obvious on the amplitude of the blades at the natural frequency 1.5 Hz. This last information could be easily deduced by computing the value 1/period. Indeed, in Figure 8, the period of the signals is equal to 0.66 seconds giving the blades a natural frequency of 1.5 Hz. In the faulty case, this increase in the mean and the standard deviation of the tracking error is integrated by the algorithm in equation (8) to deduce the fault amplitude.


#### 3.2.6. The Baseline Controller Design

Initially, the eigenvalues of the dynamics matrixA of the system are equal to the vector λ0 as follows:(26)λ0=−0.1043−5.2649±8.1929i−5.1247±7.9957i−5.2649±8.1929iT.The eigenvalue−0.1043 corresponds to the rotor speed dynamic. The last three eigenvalues represent the three blades’ dynamics. These eigenvalues are double conjugates because the blades represent an oscillatory system. The objective of the control is to reduce the oscillations of the blades by choosing new eigenvalues bigger in absolute value than the initial eigenvalues and with negative real parts to ensure stability of the closed loop. The design control parameter K is computed so that the closed loop eigenvalues λA+BK match the desired eigenvalues λd:(27)λd=λA+BK.In the present paper, the desired eigenvalues of the nominal closed loop are(28)λd=−0.2−24.7−24.7−26−26−27.3−27.3T.The design steps of the nominal control are as follows:(i)
The controllability matrix of the pair (A, B) is of full rank 7. So, the system is controllable.(ii)
The control gain is defined asK=K11…K17K21…K27K31…K37. The gain K is then substituted in the desired characteristic polynomial equation:(29)detsIn−A−BK=∏j=17s−λd,j,where In is the identity matrix and s is the Laplace operator. The integer n is the number of states; it is equal to 7 in this paper. λd,j is the jth element of the desired closed loop eigenvalue vector λd.(i)
Equation (29) is solved by equating the identical powers on both sides. This allows extracting the value of the control matrix K.Finally, the obtained control gainK is applied to the system according to the scheme of Figure 3. The results of the regulation are depicted in Figures 9–12. Figures 9 and 10 illustrate that the speed regulation objective is achieved. In fact, as the rotor speed increases (decreases), the blade pitch angle increases (decreases) in order to prevent wind from driving the wind turbine (in order to provide more wind to drive the wind turbine). The expected achieved performance is of 90% of attenuation. In the open loop, a difference from the operating point of 10 rpm at the instant 150 s is denoted. After applying the control strategy, the difference becomes 1 rpm, which corresponds to 90% of attenuation.Figure 9
Rotor speed (rpm).Figure 10
Zoom on the rotor speed (rpm) and the blade pitch angles (degrees) in the region [200, 300] sec.
(a)
(b)Figure 11
Blade deflection (meters).Figure 12
Blade 1 deflection (meters) and blade 1 pitch action.Figures11 and 12 show that when the deflection increases (the red signal), the pitch action (green) increases to attenuate the deflection as presented in the same figure (blue signal). It can be concluded that the blade pitch action is composed of two signals: a low frequency control action responsible for rotor speed regulation and a high frequency control action responsible for mitigating the blade deflections.Figures9–12 correspond to the nominal case (healthy case). In this case, no fault is occurring.


## 3.1. The Global Wind Turbine State Space Model

The global state space representation in the equation (5) takes into account the pitch actuator faults. The actuator effectiveness fault is modeled as a multiplication of the input matrix B by an unknown fault matrix F, and the faulty system is then given by(5)x˙t=Axt+BFtut+D1ωt,yt=Cxt+D2ωt,where xt∈ℝn is the state vector; ut∈ℝm is the input vector; yt∈ℝpis the measured output with p≥m; and C and B have full rank (rank(C) = p and rank(B) = m). Ft∈ℝm×m, and Ft=diagf1t,f2t,…,fmt is the vector of faults on the actuators.


## 3.2. The Observer and the Baseline Controller Design

The three-blade flap deflection amplitude reduction and the rotor speed regulation are performed through a pole placement state feedback [39] as in equation (6). In fact, the nominal control law in equation (6) is based on the fault free (nominal) situation, in which the fault matrix F(t) equals the identity matrix which is constant.(6)u=Δβi_c=Kx^.The variablex^ in equation (6) is the state vector to estimate. The gains of controller K are design parameters to achieve the multiobjective wind turbine control. The variable x^ should be estimated robustly in the next paragraph.In Figure3, Tg0 is the generator torque input at the operating point and is kept constant in the region of high winds, β0 is the pitch angle input at the operating point, Δβi_c is the pitch angle input variations from the operating point (i = 1, 2, 3), K is the control gain, Δx˙def_i are the deflection velocity variations from the operating point (i = 1, 2, 3), ΔΩr is the rotor speed variations from the operating point, Δβi is the measured pitch angles and are input to the state observer, and Ft is the fault matrix.


### 3.2.1. The Robust State Observer

The structure of the state observer is given by system (7) and illustrated by Figure 4:(7)x^˙t=Ax^t+BFtut−LCx^t−yt,where yt corresponds to the vector containing the measured deflection velocities Δx˙_defi and rotor speed ΔΩr. The vector ut corresponds to the measured three blade pitch angles. Ft is the matrix of the faults. x^t is the output of the observer.Figure 4
The state observer structure.x
^
t
∈
ℝ
n are the observer states, y^t∈ℝn are the observed outputs, F^t∈ℝm×m is the estimate of Ft, and Lt∈ℝn×p is the estimator gain to be designed.


### 3.2.2. The Fault Matrix Estimator

The fault matrixFt has to be estimated. This estimate is F^t∈ℝm×m and has the following diagonal form: F^t=diagf^1,f^2,f^3, where f^i is the estimation of the pitch fault on the actuator of index i. The following integrator-based algorithm is proposed:(8)f^˙i=−ξirieytuit,where i = 1, 2, 3; Ξ=diagξ1,ξ2,…,ξm; Ξ∈ℝm×m is called the learning rate; and ri∈ℝ1×p. The parameter p is the number of measurements. uit is the control input in the direction to the pitch actuator number i.The adaptability of the estimation consists in the update of the fault estimate dynamics with time by acting on the learning rateΞ or the matrix R=r1r2r3T. The matrix R is an additional degree of freedom that can be used to refine the approximation BTP=RC in equation (15) in Section 3.2.3. An application of this matrix could be to refine equation (15) only in the direction of a selected measurement from matrix C for a fixed η. For example, to refine equation (15) in the direction of the blade deflections, R could be chosen equal to 000−470.41580.00100−470.4158000−470.4158 and the resulting product RC will then be equal to 000000−470.4158000000−470.4158000000−470.4158. In the resulting matrix RC, the second row vector became null. Indeed, this row represents the contribution of the rotor speed in the product matrix RC. The remaining vector rows 3, 5, and 6 represent the blade deflection velocities contribution in the product matrix RC. Another alternative could be to compute an initial guess of R using YALMIP and refine it according to the need. In the present paper, R was extracted using YALMIP.The learning rateΞ is selected and handled by the user to ensure the fault estimation performance requirements. It allows to achieve a compromise between a quick fault accommodation and the disturbance amplification. Indeed, for high values of Ξ, high frequency information, namely, disturbances, is integrated in the learning process and vice versa. The variable ey(t) is the output tracking error and is equal to the difference y^t−yt.


### 3.2.3. Stability and Robustness of the Observer

Consider the states tracking errorext=x^t−xt, the output tracking error eyt=y^t−yt, and the fault tracking error eft=F^t−Ft.The states’ tracking error dynamics are given by the state space representation in the following equation:(9)e˙xt=A−LCext+Beftut+LD2−D1ωt,eyt=Cext−D2ωt.The first objective is to haveext⟶0 and eft⟶0 when t⟶+∞. The tracking errors’ convergence and stability could be ensured using the Lyapunov function Vet:(10)Vet=exTtPext+trefTtΞ−1eft.The condition of validity ofVet is that Ξ is chosen invertible.P is a symmetric positive definite matrix. The idea of the Lyapunov function choice is as the learning rate increases, the quadratic fault tracking error eft decreases and the fault estimate tends to the actual fault. Finally, as the P matrix decreases, the quadratic state tracking error eyt decreases and tends to the null vector.The second objective is to minimize the effect of the wind turbulence on the output tracking erroreyt, using a performance index γ>0 as(11)eyt2≤γωt2.The minimization of the wind turbulenceωt effect on the output error could be performed using the following objective function:(12)J=∫tf∞1γeyTteyt−γωtTωtdt.The considered fault is time invariant. This means thatF˙t=0 and then e˙ft=F^˙t. The derivative of Vet with respect to time is given by(13)V˙et=exTtPA−LC+A−LCTPext+2exTtPBeftut+2exTtPLD2−D1ωt+2treftΞ−1F^˙t.Writing equation (8) as a matrix form gives the following:(14)Ξ−1F^˙=ReytuTt=RCextuTt.To eliminate the term containing bothexTt and ut, equation (15) could be adopted. Indeed, by adopting (15) the second and fourth term in the development of the derivative of the Lyapunov cancel each other.(15)BTP=RC.In order to solve the condition (15), its LMI version is used as in the following lemma.Lemma 1 (see [40]).
The conditionBTP=RC has an LMI equivalent writing:(16)minimizeηsubject toηIrBTP−RC∗ηIn>0,where η is a dimensionless constant used to approximate BTP=RC as an LMI. The smaller η is, the accurate BTP=RC is in LMI writing.In the sequel, the symbol ‘∗’ means the transposition of the term in the same diagonal as “∗.” For example, in equation (16), “∗” is the transposition of the term BTP−RC.The derivative of the Lyapunov function becomes(17)V˙et=exTtPA−LC+A−LCTPext+2exTPLD2−D1ωt.By replacing the Lyapunov function derivatives in the cost function expression, the following inequality is obtained:(18)J≤∫tf∞1γeyTteyt−γωtTωtdt+V˙et=∫tf∞1γeyTteyt−γωtTωtdt+exTtPA−LC+A−LCTPext+2exTPLD2−D1ωt.There is no direct action of the wind on the outputs; then,D2=0. The objective function inequality:(19)J≤1γeyTteyt−γωTtωtdt−2exTtPD1ωt+exTtPA−LC+A−LCTPext=∫tf∞ζtTϒζtdt,where(20)ζt=extωt,ϒ=PA−LC+A−LCTP+1γCTC−PD1∗−γId.Finally, to ensure the stability of the states’ tracking error dynamics and the minimization of the wind turbulence on the output tracking error, the following inequality should hold:(21)ϒ<0.Using the Schur complement, the same previous inequality becomes(22)PA+ATP−YC−CTYT−PD1CT∗−γId0∗∗−γIp<0,where Y=PL, and then the error dynamic (9) satisfies the H∞ performance.Remark 1.
The adaptive algorithm (8) is proposed with the assumption that the input vector uit satisfies the persistent excitation condition, which means that there exist positive scalars μ1, μ2, and t0 such that for all t, the following inequality holds:(23)μ1≤∫tt+t0uisuisds≤μ2.
The poles of the designed observer could be placed through the linear matrix inequality in the following lemma.Lemma 2 (see [41]).
The eigenvalues of closed loop observed system dynamic matrixA−LC∈ℝn×nA∈Rn×n belong to the circular region Dα,τwith center α+j0 and radius τ if and only if there exists a symmetric positive definite matrix P∈ℝn×nP∈Rn×n such that the following condition holds:(24)−PPA−LC−αIn∗−τ2P<0.The final integrated problem to be solved for the symmetric positive definite matrixP is as the following lemma.Lemma 3.
AssumeD2 is null matrix and let a prescribed H∞ performance level γ be given. If there exists a symmetric positive definite matrix P∈ℝn×n and matrices Y∈ℝn×p and R∈ℝm×p such that the following condition holds:(25)PA+ATP−YC−CTYT−PD1CT∗−γId0∗∗−γIp<0−PPA−LC−αIn∗−τ2P<0,minimizeηsubject  toηIrBTP−RC∗ηIn>0,where Y=PL, then the error dynamic (9) satisfies the H∞ performance and the poles of the robust state observer are within the region Dα,τ.


### 3.2.4. The State Observer and Fault Simulation Results

The natural frequency of the blade isfn=1/2πDb/Mb=1.511Hz. Mb is the blade mass and Db is the stiffness of the blade. It gives a natural pulsation of ωn=9.49rad/s. All the frequencies above the natural frequency should be filtered by the observer. However, the observer should amplify the gain corresponding to ωn for the estimation of the blades’ deflections. The Bode diagrams of the designed observer in the system (7) are illustrated in Figures 5(a) and 5(b). The considered channel links the pitch actuator’s input (u1t=Δβ1 in Figure 4) with the blade deflection observed state (y3t=Δxdef_1 in Figure 4).Figure 5
(a) Bode diagrams of the designed observer for different performance levels. (b) Zoom on the Bode diagrams of the designed observer for different performance levels. (c) The state observer pole placement for LMI regionα,τ=−20,20 in the complex axis.
(a)
(b)
(c)Figures5(a) and 5(b) illustrate the designed observer Bode diagram function of the performance level γ corresponding to the system in equation (7). The Bode diagrams are divided into two main regions. The first region is about the natural pulsation of the blades ωn=9.49rad/s. In this region, the magnitude presents maximal values, which means that the blades’ deflections are perfectly observed. The second region is where the pulsations are less than the natural pulsation of the blades. As γ becomes smaller (closer to values near 0), the gain of the observer is reduced and the low frequencies of the wind speed are not amplified. The advantage is to prevent penetrating the amplified low frequencies contained in the rotor speed to the fault estimation algorithm through the residual eyt. The effect of the amplified low frequencies is reducing the convergence speed of the estimation algorithm in equation (8). As a conclusion, the convergence speed of the fault estimation could be tuned by the performance index γ in addition to the tuning by the learning rate Ξ as illustrated in Figure 6(a), where the learning rate is fixed at 0.1 and γ takes the values 3, 1, and 0.3.Figure 6
(a) Estimation of the pitch 1 actuator effectiveness for different performance levels. (b) Estimation of the pitch 1 actuator effectiveness for different learning rates. (c) Fault estimate with dynamics faster than system dynamics. (d) Zoom on fault estimate with dynamics faster than system dynamics.
(a)
(b)
(c)
(d)Figures7 and 8 illustrate the tracking errors in the faulty and the fault tolerant case for the rotor speed and the blade 1 deflection velocity as samples. The observation quality is defined by the states’ tracking error. It could be noticed that a sufficient level of observation is achieved for γ=0.3 and ξi=0.1 (i = 1, 2, 3).Figure 7
The rotor speed tracking error.Figure 8
The estimated blade 1 deflection velocity tracking error.For the rotor speed, in the fault tolerant case, the tracking error has a mean value of−3.96×10−4 rad/s and a standard deviation of 6.841×10−4rad/s. For the faulty case, the mean value increases to −0.002 rad/s with a standard deviation of 9.8893×10−4rad/s. For the blade deflection velocity, in the fault tolerant case, the tracking error has a mean of −0.058 m/s and a standard deviation of 0.4973 m/s. For the faulty case, the mean value increases to −0.0823 m/s and a standard deviation of 0.6595 m/s.Figure5(c) illustrates the result of pole placement of the state observer. The chosen region is of center 20 and radius 20. The resulting closed loop eigenvalues are −33.4089;−24.7069±8.6379i;−26.6204±18.7259i;−26.6204±18.7259iT.In Figure6(a), the effectiveness of actuator number 1 means 1−f1. f1 represents the decrease in the output of the actuator.In Figures7 and 8, the notations ey1 correspond to the rotor speed component and ey2 corresponds to the blade deflection velocity component in the tracking error vector ey.Figures6(a) and 8 correspond to the first faulty scenario of Section 4 and Figure 7 corresponds to the third faulty scenario.Figure6(b) illustrates the fault estimation of the blade 1 actuator for several values of learning rate ξ1. As the learning rate increases, the estimate of the fault converges rapidly to the actual fault (magenta).Remark: There is a coupling effect between the fault estimation and the state observer due to the fault tolerant control (FTC) correction. Such an effect can be neglected as long as the state observer is slower than the fault estimation. This can be achieved by choosing properly the learning rate.


### 3.2.5. Fault Scenario with Dynamics Faster than the System Dynamics

In this section, a sinusoidal loss of efficiency is considered with frequency higher than 1.5 Hz of blades. Figures6(c) and 6(d) show the result of fault estimation in that case. In the proposed strategy, by only tuning the learning rate to suitable higher values, the faults with dynamics faster than the system dynamics could be reconfigured without need to redesign the fault estimate algorithm as recommended in [33]. The amplitude of the fault is 0.1 and the frequency is 12 rad/s higher than 9.49 rad/s of the blades.In Figure7, the most important issue is the change of the mean rotor speed error. This information is the only one needed by the algorithm about the fault on the rotor speed. In Figure 8, the change is obvious on the amplitude of the blades at the natural frequency 1.5 Hz. This last information could be easily deduced by computing the value 1/period. Indeed, in Figure 8, the period of the signals is equal to 0.66 seconds giving the blades a natural frequency of 1.5 Hz. In the faulty case, this increase in the mean and the standard deviation of the tracking error is integrated by the algorithm in equation (8) to deduce the fault amplitude.


### 3.2.6. The Baseline Controller Design

Initially, the eigenvalues of the dynamics matrixA of the system are equal to the vector λ0 as follows:(26)λ0=−0.1043−5.2649±8.1929i−5.1247±7.9957i−5.2649±8.1929iT.The eigenvalue−0.1043 corresponds to the rotor speed dynamic. The last three eigenvalues represent the three blades’ dynamics. These eigenvalues are double conjugates because the blades represent an oscillatory system. The objective of the control is to reduce the oscillations of the blades by choosing new eigenvalues bigger in absolute value than the initial eigenvalues and with negative real parts to ensure stability of the closed loop. The design control parameter K is computed so that the closed loop eigenvalues λA+BK match the desired eigenvalues λd:(27)λd=λA+BK.In the present paper, the desired eigenvalues of the nominal closed loop are(28)λd=−0.2−24.7−24.7−26−26−27.3−27.3T.The design steps of the nominal control are as follows:(i)
The controllability matrix of the pair (A, B) is of full rank 7. So, the system is controllable.(ii)
The control gain is defined asK=K11…K17K21…K27K31…K37. The gain K is then substituted in the desired characteristic polynomial equation:(29)detsIn−A−BK=∏j=17s−λd,j,where In is the identity matrix and s is the Laplace operator. The integer n is the number of states; it is equal to 7 in this paper. λd,j is the jth element of the desired closed loop eigenvalue vector λd.(i)
Equation (29) is solved by equating the identical powers on both sides. This allows extracting the value of the control matrix K.Finally, the obtained control gainK is applied to the system according to the scheme of Figure 3. The results of the regulation are depicted in Figures 9–12. Figures 9 and 10 illustrate that the speed regulation objective is achieved. In fact, as the rotor speed increases (decreases), the blade pitch angle increases (decreases) in order to prevent wind from driving the wind turbine (in order to provide more wind to drive the wind turbine). The expected achieved performance is of 90% of attenuation. In the open loop, a difference from the operating point of 10 rpm at the instant 150 s is denoted. After applying the control strategy, the difference becomes 1 rpm, which corresponds to 90% of attenuation.Figure 9
Rotor speed (rpm).Figure 10
Zoom on the rotor speed (rpm) and the blade pitch angles (degrees) in the region [200, 300] sec.
(a)
(b)Figure 11
Blade deflection (meters).Figure 12
Blade 1 deflection (meters) and blade 1 pitch action.Figures11 and 12 show that when the deflection increases (the red signal), the pitch action (green) increases to attenuate the deflection as presented in the same figure (blue signal). It can be concluded that the blade pitch action is composed of two signals: a low frequency control action responsible for rotor speed regulation and a high frequency control action responsible for mitigating the blade deflections.Figures9–12 correspond to the nominal case (healthy case). In this case, no fault is occurring.


## 3.2.1. The Robust State Observer

The structure of the state observer is given by system (7) and illustrated by Figure 4:(7)x^˙t=Ax^t+BFtut−LCx^t−yt,where yt corresponds to the vector containing the measured deflection velocities Δx˙_defi and rotor speed ΔΩr. The vector ut corresponds to the measured three blade pitch angles. Ft is the matrix of the faults. x^t is the output of the observer.Figure 4
The state observer structure.x
^
t
∈
ℝ
n are the observer states, y^t∈ℝn are the observed outputs, F^t∈ℝm×m is the estimate of Ft, and Lt∈ℝn×p is the estimator gain to be designed.


## 3.2.2. The Fault Matrix Estimator

The fault matrixFt has to be estimated. This estimate is F^t∈ℝm×m and has the following diagonal form: F^t=diagf^1,f^2,f^3, where f^i is the estimation of the pitch fault on the actuator of index i. The following integrator-based algorithm is proposed:(8)f^˙i=−ξirieytuit,where i = 1, 2, 3; Ξ=diagξ1,ξ2,…,ξm; Ξ∈ℝm×m is called the learning rate; and ri∈ℝ1×p. The parameter p is the number of measurements. uit is the control input in the direction to the pitch actuator number i.The adaptability of the estimation consists in the update of the fault estimate dynamics with time by acting on the learning rateΞ or the matrix R=r1r2r3T. The matrix R is an additional degree of freedom that can be used to refine the approximation BTP=RC in equation (15) in Section 3.2.3. An application of this matrix could be to refine equation (15) only in the direction of a selected measurement from matrix C for a fixed η. For example, to refine equation (15) in the direction of the blade deflections, R could be chosen equal to 000−470.41580.00100−470.4158000−470.4158 and the resulting product RC will then be equal to 000000−470.4158000000−470.4158000000−470.4158. In the resulting matrix RC, the second row vector became null. Indeed, this row represents the contribution of the rotor speed in the product matrix RC. The remaining vector rows 3, 5, and 6 represent the blade deflection velocities contribution in the product matrix RC. Another alternative could be to compute an initial guess of R using YALMIP and refine it according to the need. In the present paper, R was extracted using YALMIP.The learning rateΞ is selected and handled by the user to ensure the fault estimation performance requirements. It allows to achieve a compromise between a quick fault accommodation and the disturbance amplification. Indeed, for high values of Ξ, high frequency information, namely, disturbances, is integrated in the learning process and vice versa. The variable ey(t) is the output tracking error and is equal to the difference y^t−yt.


## 3.2.3. Stability and Robustness of the Observer

Consider the states tracking errorext=x^t−xt, the output tracking error eyt=y^t−yt, and the fault tracking error eft=F^t−Ft.The states’ tracking error dynamics are given by the state space representation in the following equation:(9)e˙xt=A−LCext+Beftut+LD2−D1ωt,eyt=Cext−D2ωt.The first objective is to haveext⟶0 and eft⟶0 when t⟶+∞. The tracking errors’ convergence and stability could be ensured using the Lyapunov function Vet:(10)Vet=exTtPext+trefTtΞ−1eft.The condition of validity ofVet is that Ξ is chosen invertible.P is a symmetric positive definite matrix. The idea of the Lyapunov function choice is as the learning rate increases, the quadratic fault tracking error eft decreases and the fault estimate tends to the actual fault. Finally, as the P matrix decreases, the quadratic state tracking error eyt decreases and tends to the null vector.The second objective is to minimize the effect of the wind turbulence on the output tracking erroreyt, using a performance index γ>0 as(11)eyt2≤γωt2.The minimization of the wind turbulenceωt effect on the output error could be performed using the following objective function:(12)J=∫tf∞1γeyTteyt−γωtTωtdt.The considered fault is time invariant. This means thatF˙t=0 and then e˙ft=F^˙t. The derivative of Vet with respect to time is given by(13)V˙et=exTtPA−LC+A−LCTPext+2exTtPBeftut+2exTtPLD2−D1ωt+2treftΞ−1F^˙t.Writing equation (8) as a matrix form gives the following:(14)Ξ−1F^˙=ReytuTt=RCextuTt.To eliminate the term containing bothexTt and ut, equation (15) could be adopted. Indeed, by adopting (15) the second and fourth term in the development of the derivative of the Lyapunov cancel each other.(15)BTP=RC.In order to solve the condition (15), its LMI version is used as in the following lemma.Lemma 1 (see [40]).
The conditionBTP=RC has an LMI equivalent writing:(16)minimizeηsubject toηIrBTP−RC∗ηIn>0,where η is a dimensionless constant used to approximate BTP=RC as an LMI. The smaller η is, the accurate BTP=RC is in LMI writing.In the sequel, the symbol ‘∗’ means the transposition of the term in the same diagonal as “∗.” For example, in equation (16), “∗” is the transposition of the term BTP−RC.The derivative of the Lyapunov function becomes(17)V˙et=exTtPA−LC+A−LCTPext+2exTPLD2−D1ωt.By replacing the Lyapunov function derivatives in the cost function expression, the following inequality is obtained:(18)J≤∫tf∞1γeyTteyt−γωtTωtdt+V˙et=∫tf∞1γeyTteyt−γωtTωtdt+exTtPA−LC+A−LCTPext+2exTPLD2−D1ωt.There is no direct action of the wind on the outputs; then,D2=0. The objective function inequality:(19)J≤1γeyTteyt−γωTtωtdt−2exTtPD1ωt+exTtPA−LC+A−LCTPext=∫tf∞ζtTϒζtdt,where(20)ζt=extωt,ϒ=PA−LC+A−LCTP+1γCTC−PD1∗−γId.Finally, to ensure the stability of the states’ tracking error dynamics and the minimization of the wind turbulence on the output tracking error, the following inequality should hold:(21)ϒ<0.Using the Schur complement, the same previous inequality becomes(22)PA+ATP−YC−CTYT−PD1CT∗−γId0∗∗−γIp<0,where Y=PL, and then the error dynamic (9) satisfies the H∞ performance.Remark 1.
The adaptive algorithm (8) is proposed with the assumption that the input vector uit satisfies the persistent excitation condition, which means that there exist positive scalars μ1, μ2, and t0 such that for all t, the following inequality holds:(23)μ1≤∫tt+t0uisuisds≤μ2.
The poles of the designed observer could be placed through the linear matrix inequality in the following lemma.Lemma 2 (see [41]).
The eigenvalues of closed loop observed system dynamic matrixA−LC∈ℝn×nA∈Rn×n belong to the circular region Dα,τwith center α+j0 and radius τ if and only if there exists a symmetric positive definite matrix P∈ℝn×nP∈Rn×n such that the following condition holds:(24)−PPA−LC−αIn∗−τ2P<0.The final integrated problem to be solved for the symmetric positive definite matrixP is as the following lemma.Lemma 3.
AssumeD2 is null matrix and let a prescribed H∞ performance level γ be given. If there exists a symmetric positive definite matrix P∈ℝn×n and matrices Y∈ℝn×p and R∈ℝm×p such that the following condition holds:(25)PA+ATP−YC−CTYT−PD1CT∗−γId0∗∗−γIp<0−PPA−LC−αIn∗−τ2P<0,minimizeηsubject  toηIrBTP−RC∗ηIn>0,where Y=PL, then the error dynamic (9) satisfies the H∞ performance and the poles of the robust state observer are within the region Dα,τ.


## 3.2.4. The State Observer and Fault Simulation Results

The natural frequency of the blade isfn=1/2πDb/Mb=1.511Hz. Mb is the blade mass and Db is the stiffness of the blade. It gives a natural pulsation of ωn=9.49rad/s. All the frequencies above the natural frequency should be filtered by the observer. However, the observer should amplify the gain corresponding to ωn for the estimation of the blades’ deflections. The Bode diagrams of the designed observer in the system (7) are illustrated in Figures 5(a) and 5(b). The considered channel links the pitch actuator’s input (u1t=Δβ1 in Figure 4) with the blade deflection observed state (y3t=Δxdef_1 in Figure 4).Figure 5
(a) Bode diagrams of the designed observer for different performance levels. (b) Zoom on the Bode diagrams of the designed observer for different performance levels. (c) The state observer pole placement for LMI regionα,τ=−20,20 in the complex axis.
(a)
(b)
(c)Figures5(a) and 5(b) illustrate the designed observer Bode diagram function of the performance level γ corresponding to the system in equation (7). The Bode diagrams are divided into two main regions. The first region is about the natural pulsation of the blades ωn=9.49rad/s. In this region, the magnitude presents maximal values, which means that the blades’ deflections are perfectly observed. The second region is where the pulsations are less than the natural pulsation of the blades. As γ becomes smaller (closer to values near 0), the gain of the observer is reduced and the low frequencies of the wind speed are not amplified. The advantage is to prevent penetrating the amplified low frequencies contained in the rotor speed to the fault estimation algorithm through the residual eyt. The effect of the amplified low frequencies is reducing the convergence speed of the estimation algorithm in equation (8). As a conclusion, the convergence speed of the fault estimation could be tuned by the performance index γ in addition to the tuning by the learning rate Ξ as illustrated in Figure 6(a), where the learning rate is fixed at 0.1 and γ takes the values 3, 1, and 0.3.Figure 6
(a) Estimation of the pitch 1 actuator effectiveness for different performance levels. (b) Estimation of the pitch 1 actuator effectiveness for different learning rates. (c) Fault estimate with dynamics faster than system dynamics. (d) Zoom on fault estimate with dynamics faster than system dynamics.
(a)
(b)
(c)
(d)Figures7 and 8 illustrate the tracking errors in the faulty and the fault tolerant case for the rotor speed and the blade 1 deflection velocity as samples. The observation quality is defined by the states’ tracking error. It could be noticed that a sufficient level of observation is achieved for γ=0.3 and ξi=0.1 (i = 1, 2, 3).Figure 7
The rotor speed tracking error.Figure 8
The estimated blade 1 deflection velocity tracking error.For the rotor speed, in the fault tolerant case, the tracking error has a mean value of−3.96×10−4 rad/s and a standard deviation of 6.841×10−4rad/s. For the faulty case, the mean value increases to −0.002 rad/s with a standard deviation of 9.8893×10−4rad/s. For the blade deflection velocity, in the fault tolerant case, the tracking error has a mean of −0.058 m/s and a standard deviation of 0.4973 m/s. For the faulty case, the mean value increases to −0.0823 m/s and a standard deviation of 0.6595 m/s.Figure5(c) illustrates the result of pole placement of the state observer. The chosen region is of center 20 and radius 20. The resulting closed loop eigenvalues are −33.4089;−24.7069±8.6379i;−26.6204±18.7259i;−26.6204±18.7259iT.In Figure6(a), the effectiveness of actuator number 1 means 1−f1. f1 represents the decrease in the output of the actuator.In Figures7 and 8, the notations ey1 correspond to the rotor speed component and ey2 corresponds to the blade deflection velocity component in the tracking error vector ey.Figures6(a) and 8 correspond to the first faulty scenario of Section 4 and Figure 7 corresponds to the third faulty scenario.Figure6(b) illustrates the fault estimation of the blade 1 actuator for several values of learning rate ξ1. As the learning rate increases, the estimate of the fault converges rapidly to the actual fault (magenta).Remark: There is a coupling effect between the fault estimation and the state observer due to the fault tolerant control (FTC) correction. Such an effect can be neglected as long as the state observer is slower than the fault estimation. This can be achieved by choosing properly the learning rate.


## 3.2.5. Fault Scenario with Dynamics Faster than the System Dynamics

In this section, a sinusoidal loss of efficiency is considered with frequency higher than 1.5 Hz of blades. Figures6(c) and 6(d) show the result of fault estimation in that case. In the proposed strategy, by only tuning the learning rate to suitable higher values, the faults with dynamics faster than the system dynamics could be reconfigured without need to redesign the fault estimate algorithm as recommended in [33]. The amplitude of the fault is 0.1 and the frequency is 12 rad/s higher than 9.49 rad/s of the blades.In Figure7, the most important issue is the change of the mean rotor speed error. This information is the only one needed by the algorithm about the fault on the rotor speed. In Figure 8, the change is obvious on the amplitude of the blades at the natural frequency 1.5 Hz. This last information could be easily deduced by computing the value 1/period. Indeed, in Figure 8, the period of the signals is equal to 0.66 seconds giving the blades a natural frequency of 1.5 Hz. In the faulty case, this increase in the mean and the standard deviation of the tracking error is integrated by the algorithm in equation (8) to deduce the fault amplitude.


## 3.2.6. The Baseline Controller Design

Initially, the eigenvalues of the dynamics matrixA of the system are equal to the vector λ0 as follows:(26)λ0=−0.1043−5.2649±8.1929i−5.1247±7.9957i−5.2649±8.1929iT.The eigenvalue−0.1043 corresponds to the rotor speed dynamic. The last three eigenvalues represent the three blades’ dynamics. These eigenvalues are double conjugates because the blades represent an oscillatory system. The objective of the control is to reduce the oscillations of the blades by choosing new eigenvalues bigger in absolute value than the initial eigenvalues and with negative real parts to ensure stability of the closed loop. The design control parameter K is computed so that the closed loop eigenvalues λA+BK match the desired eigenvalues λd:(27)λd=λA+BK.In the present paper, the desired eigenvalues of the nominal closed loop are(28)λd=−0.2−24.7−24.7−26−26−27.3−27.3T.The design steps of the nominal control are as follows:(i)
The controllability matrix of the pair (A, B) is of full rank 7. So, the system is controllable.(ii)
The control gain is defined asK=K11…K17K21…K27K31…K37. The gain K is then substituted in the desired characteristic polynomial equation:(29)detsIn−A−BK=∏j=17s−λd,j,where In is the identity matrix and s is the Laplace operator. The integer n is the number of states; it is equal to 7 in this paper. λd,j is the jth element of the desired closed loop eigenvalue vector λd.(i)
Equation (29) is solved by equating the identical powers on both sides. This allows extracting the value of the control matrix K.Finally, the obtained control gainK is applied to the system according to the scheme of Figure 3. The results of the regulation are depicted in Figures 9–12. Figures 9 and 10 illustrate that the speed regulation objective is achieved. In fact, as the rotor speed increases (decreases), the blade pitch angle increases (decreases) in order to prevent wind from driving the wind turbine (in order to provide more wind to drive the wind turbine). The expected achieved performance is of 90% of attenuation. In the open loop, a difference from the operating point of 10 rpm at the instant 150 s is denoted. After applying the control strategy, the difference becomes 1 rpm, which corresponds to 90% of attenuation.Figure 9
Rotor speed (rpm).Figure 10
Zoom on the rotor speed (rpm) and the blade pitch angles (degrees) in the region [200, 300] sec.
(a)
(b)Figure 11
Blade deflection (meters).Figure 12
Blade 1 deflection (meters) and blade 1 pitch action.Figures11 and 12 show that when the deflection increases (the red signal), the pitch action (green) increases to attenuate the deflection as presented in the same figure (blue signal). It can be concluded that the blade pitch action is composed of two signals: a low frequency control action responsible for rotor speed regulation and a high frequency control action responsible for mitigating the blade deflections.Figures9–12 correspond to the nominal case (healthy case). In this case, no fault is occurring.


## 4. Wind Turbine Optimal Adaptive Fault Tolerant Control in the Faulty Operation

The considered fault is the loss of efficiency in the three pitch actuators. The scenario created in the FAST software is a drop of 55% in the effectiveness of the pitch actuator 1 at the instant 90 s, 40% in the effectiveness of the pitch actuator 2 at the instant 150 s, and 60% in the effectiveness of the pitch actuator 3 at the instant 200 s.


### 4.1. The Fault Mechanism

The considered pitch actuator is hydraulic based. In Figure13(a), the prefix p denotes the piston side, the prefix r denotes the rod side, and the prefix v denotes the proportional valve side. Ap and Ar correspond, respectively, to the surfaces of the piston side and the rod side. The pump pumps oil from the accumulator with a pressure ps which passes across the proportional valve with a mass flow Qv. Finally, the air arrives to the piston side with the pressure pp and thrusts the rod linked to the blades. The trust force is Fcyl and is given by(30)Fcyl=ppAp−prAr−Ffric,where Ffric is the friction force of the piston on the cylinder and is linearly dependent to the displacement velocity of the piston.Figure 13
(a) The pitch actuator model. (b) The wind turbine control objective scheme.
(a)
(b)For the same rod pressurepr and friction Ffric, the considered pitch loss of efficiency fault occurs when the pressure pp drops due to the ageing of the pump station. This ageing appears for example because of excessive Joule effect (heat) losses in the wirings of the pump motor. Under these conditions, the thrust force Fcyl of the rod on the blade is no longer sufficient to actuate the blade. The thrust force Fcyl is represented by the action matrix B in equation (5). The fault is modeled as a loss of efficiency percentage (equal or less than 100%) multiplied by the action matrix B. As a result, the pitch efforts are no longer sufficient for damping the oscillations of the blades. The aim of the algorithm in equation (8) is to track that multiplicative percentage with a sufficient precision and response time.After that, equation (31) comes to amplify the supplied power (control) to the pump to compensate the lost energy due to Joule effect. The amplification is performed mathematically by dividing the old control signal by the estimated faulty efficiency.


### 4.2. The Fault Tolerant Controller

The fault tolerant control signal is obtained by dividing the initial baseline control signal by the estimated fault:(31)uft=F^t−1ut.To avoid the division by zero in the case of zero efficiency,F^t should be replaced by a small number (for example, 10−4). The scheme of the global fault tolerant control is given by Figure 13(b).Before being fed to the pitch actuator, the effectiveness is estimated by the adaptive algorithm. Then, the control signal is divided by the estimated efficiency to compensate the drop in the actual efficiency of the pitch actuator.Remark: The different LMIs of the algorithm are resolved through the YALMIP tool [42].Δ
β
i is the pitch angle input variations from the operating point for the ith actuator (i = 1, 2, 3), Δx˙_defi are the deflection velocity variations from the operating point for the ith blade, and ΔΩr is the rotor speed variations from the operating point.Δ
β
i
_
F is the faulty pitch angle input variations from the operating point for each actuator, f^i is the estimated efficiency for the ith actuator, and x^ is the estimated state variations of the wind turbine from the operating point.


### 4.3. The Three Pitch Actuators Loss of Effectiveness Faulty Scenario

Let us consider the blade without loss of generality. Figure14 illustrates the blade 1 deflection.Figure 14
The bade deflections in the faulty deflections, nominal deflections, and fault tolerant deflections.In the faulty situation, the deflections increase from 1.5 meters to 4 meters because the pitch actuator could not answer completely to the corrective action of the baseline regulator against the wind. According to [43], the maximal allowable industrial deflection for the considered wind turbine is of 3 m. In the faulty case, this limit is exceeded which increases the risk of damage of the blade.After applying the fault tolerant control strategy, the blade deflections are perfectly reconfigured.LetFs be the effect of the pitch angle i on the blade i deflection. The effect of the strategy on the blades deflections is interpreted as an additional zero at the origin in the transfer function Fs. In Figure 15, it can be noticed that when s tends to zero, sFs tends to zero. The reconfiguration time for each fault estimation of index i is dictated by the product ξi×ri.Figure 15
The effect of the reconfiguration on the blades.The rapidity of the algorithm depends on the learning rateΞ and the matrix R. As studied earlier in the paper, in addition to the learning rate Ξ, even the performance level γ could influence the response time of the fault estimation through the low frequencies of the wind. In fact, for a given γ, a matrix P depending on γ could be found through the LMI in equation (22). Then, by using the LMI in equation (15), the matrix R is obtained and used to compute the estimation F^t using equation (8). As a conclusion, the parameter γ influences the response time of the algorithm through the matrix R.The derivative form of the block betweenΔβi_c and Fs in Figure 15 has a risk of turbulence amplification. For this reason, a study to choose the learning rate Ξ and the performance level γ function of the correlation between the tracking error ey_2t (blade 1 deflection velocity as an example) and the wind turbulence is proposed in the next section.


### 4.4. The Choice ofγ and Ξ Based on the Correlation with Turbulence and the Estimation Response Time Criteria

In this section, empirical boundaries are proposed for choosing the performance levelγ and the learning rate Ξ based on two criteria. Namely, the correlation between the wind speed ω and the residual ey is considered. This variable is crucial to investigate because it could be amplified if decided to use the performance level γ to tune the estimation response time. The second criterion is the rapidity of convergence which is important to ensure because only short times of convergence are recommended for a fast recovery of the fault.


#### 4.4.1. The Correlation between the Wind Turbulence and the Residualey

The expression of the correlation of the two signals is given by(32)Corr=Eey×ω−EeyEωSDey×SDω,where Ex denotes the expectation of the signal x and SDx denotes the standard deviation of the signal x. The correlation Corr belongs to the interval [−1, 1]. It obeys to the weak and strong correlation. If the coefficient Corr∈−0.50∪00.5, it is called a weak correlation. If the coefficient Corr∈−1−0.5∪0.51, it is called a strong correlation.


#### 4.4.2. The Estimation Response Time of the Algorithm

The response time of the fault estimation algorithm in equation (8) for each actuator of index i is given by(33)υi=ξi×ri.Many simulations have been made on the FAST software in order to derive empirical data onυ and Corr helping to choose the appropriate values of the two parameters γ and Ξ. Then, an interpolation between data points has been performed to define mappings linking the tracking error and the wind disturbance.Figures16 and 17 show the evolution of the correlation and fault estimation response time with the performance level and the learning rate for the considered wind turbine in Table 1. From Figure 18, one can conclude that for γ less than 1, this parameter influences the correlation more than the response time of the algorithm. However, when γ exceeds 2.5, the correlation becomes more influenced also by Ξ as γ increases. For γ less than 2.25, the response time of the algorithm is influenced by both Ξ and γ. But, as γ increases, only Ξ influences the response time of the algorithm. In order to guarantee a fast reconfiguration of the system performance, one should choose a small response time of the algorithm by choosing a suitable pair (γ, Ξ). However, one should guaranty that neither the resulting performance level nor the learning rate amplifies the correlation between the tracking error ey and the wind turbulence ω.Figure 16
The correlation between measuring error and the turbulence with respect to the learning rate and the performance level.Figure 17
The effectiveness estimation response time with respect to the learning rate and the performance level.Figure 18
Maximal correlation zones vs. learning rate and performance level.For the last reason, the two parameters should obey to the following empirical law. For example, ifγ is in [0.9, 1.4] interval, Ξ should not be in [2.21, 9.84] (zone 1 in Figure 18) to avoid high correlation, and with the same idea, if γ is chosen in [1.4, 2.02], Ξ should not be in the interval [8.317, 9.84] (zone 2 in Figure 18).An example of a choice of the pair (γ, Ξ) leading to a fast and less correlated response time is indicated in Figure 19 by (γ, Ξ) = (2.25, 1.25).Figure 19
The effectiveness estimation algorithm response time vs. learning rate and performance level.


#### 4.4.3. Discussion with respect to Some Existing Methods

The proposed strategy is not complicated and uses an integral structure with one Lyapunov function to prove stability and convergence of both state estimation error and fault estimation error. The proposed approach uses a reduced number of measurements. Indeed, only the rotor speed and the individual blades’ deflection velocity are needed. This reduces the information needs with respect to other techniques such as the one in [44] (requiring also the generator speed measurement). It does not either require the information about the presumed fault size as in [45].The residual-based techniques in [45–47] deal only with fault detection tasks. Contrariwise, the proposed approach deals with both fault estimation and fault isolation suitable for active fault tolerant control strategies. Indeed, the individual deflection velocity measurement of each blade allows a natural isolation of the fault blade without the need of additional isolation methods. This method avoids the intrinsic delays of online least-squares methods, which could have a negative impact on the fault estimation response time.The proposed strategy is equipped with a tuning method to avoid amplification of wind turbulence in the direction of the high learning rates. This issue is not sufficiently discussed in other works related to wind turbines.Contrariwise to other strategies, the proposed method could deal with faults having dynamics faster than the system dynamics only by tuning the learning rate suitably.


#### 4.4.4. Comparison of the Proposed Strategy with the Virtual Actuator with State Residual Integrator

A new scenario of pitch fault on blade 1 is considered for the comparison between the proposed strategy and a virtual actuator-based strategy (details of this method are not the objective of the paper, [48]). In this section, the considered fault happend at the instant 15 seconds. This fault represents pitch actuator loss of efficiency of 55%.Figure20 illustrates the structure of the strategy used for comparison. The principle of this strategy is to use a model reference and to compute the difference between the estimated faulty states and the model reference states. This difference is integrated and used as new control law. Equation (8) was used to derive the faulty matrix Bf. The difference with respect to the proposed strategy is in the integration of the state residual xΔ instead of the division in equation (30) and the use of reference model.Figure 20
The simplified virtual actuator strategy with state residualxΔ integration.Figure21 illustrates a comparison between the proposed strategy and the virtual actuator with integrating the state residual in terms of blade fault reconfiguration. In some regions where the wind speed makes high amplitudes of high frequencies, which means the effort needed for integration is higher, the performance of the virtual actuator becomes poor in the interval [16 s; 26 s]. This could be explained by the existence of two integrators, the first presented in equation (8) used to estimate the fault needed by Bf and the second to integrate the state residual. In conclusion, the virtual actuator with integration of state residual is not always suitable for integrator-based fault estimators.Figure 21
The comparison of the faulty blade 1 deflection between the proposed FTC and the residual integration FTC.


#### 4.4.5. Digital Implementation of the Strategy

The equations to be implemented are the baseline control law in equation (6), the observer in equation (7), and the fault estimation in equation (8). In order to digitalize the continuous equations, the first-order approximation in equation (34) is used. If z˙ is the derivative, it could be approximated by(34)z˙t=zt+T−ztT,where T is the sampling step. Hence, the strategy could be digitalized for each iteration of index k as follows.The digitalized baseline controller (from equation (6)) is given by(35)uk=Kx^k.The digitalized robust observer (from equation (7)) is given by(36)xk+1=Adx^k+BdFkuk+Ldy^k−yk,where Ad, Bd, and Ld are, respectively, the state matrix, the input matrix, and the observer matrix. The expressions of the previous matrix are given by Ad=In+TA; Bd=TB; and Ld=TL.The digitalized fault estimation (equation (8)) for each actuator of index i is given by(37)fk+1,i=fk,i+Tξk,irk,iey,k,iuk,i.


## 4.1. The Fault Mechanism

The considered pitch actuator is hydraulic based. In Figure13(a), the prefix p denotes the piston side, the prefix r denotes the rod side, and the prefix v denotes the proportional valve side. Ap and Ar correspond, respectively, to the surfaces of the piston side and the rod side. The pump pumps oil from the accumulator with a pressure ps which passes across the proportional valve with a mass flow Qv. Finally, the air arrives to the piston side with the pressure pp and thrusts the rod linked to the blades. The trust force is Fcyl and is given by(30)Fcyl=ppAp−prAr−Ffric,where Ffric is the friction force of the piston on the cylinder and is linearly dependent to the displacement velocity of the piston.Figure 13
(a) The pitch actuator model. (b) The wind turbine control objective scheme.
(a)
(b)For the same rod pressurepr and friction Ffric, the considered pitch loss of efficiency fault occurs when the pressure pp drops due to the ageing of the pump station. This ageing appears for example because of excessive Joule effect (heat) losses in the wirings of the pump motor. Under these conditions, the thrust force Fcyl of the rod on the blade is no longer sufficient to actuate the blade. The thrust force Fcyl is represented by the action matrix B in equation (5). The fault is modeled as a loss of efficiency percentage (equal or less than 100%) multiplied by the action matrix B. As a result, the pitch efforts are no longer sufficient for damping the oscillations of the blades. The aim of the algorithm in equation (8) is to track that multiplicative percentage with a sufficient precision and response time.After that, equation (31) comes to amplify the supplied power (control) to the pump to compensate the lost energy due to Joule effect. The amplification is performed mathematically by dividing the old control signal by the estimated faulty efficiency.


## 4.2. The Fault Tolerant Controller

The fault tolerant control signal is obtained by dividing the initial baseline control signal by the estimated fault:(31)uft=F^t−1ut.To avoid the division by zero in the case of zero efficiency,F^t should be replaced by a small number (for example, 10−4). The scheme of the global fault tolerant control is given by Figure 13(b).Before being fed to the pitch actuator, the effectiveness is estimated by the adaptive algorithm. Then, the control signal is divided by the estimated efficiency to compensate the drop in the actual efficiency of the pitch actuator.Remark: The different LMIs of the algorithm are resolved through the YALMIP tool [42].Δ
β
i is the pitch angle input variations from the operating point for the ith actuator (i = 1, 2, 3), Δx˙_defi are the deflection velocity variations from the operating point for the ith blade, and ΔΩr is the rotor speed variations from the operating point.Δ
β
i
_
F is the faulty pitch angle input variations from the operating point for each actuator, f^i is the estimated efficiency for the ith actuator, and x^ is the estimated state variations of the wind turbine from the operating point.


## 4.3. The Three Pitch Actuators Loss of Effectiveness Faulty Scenario

Let us consider the blade without loss of generality. Figure14 illustrates the blade 1 deflection.Figure 14
The bade deflections in the faulty deflections, nominal deflections, and fault tolerant deflections.In the faulty situation, the deflections increase from 1.5 meters to 4 meters because the pitch actuator could not answer completely to the corrective action of the baseline regulator against the wind. According to [43], the maximal allowable industrial deflection for the considered wind turbine is of 3 m. In the faulty case, this limit is exceeded which increases the risk of damage of the blade.After applying the fault tolerant control strategy, the blade deflections are perfectly reconfigured.LetFs be the effect of the pitch angle i on the blade i deflection. The effect of the strategy on the blades deflections is interpreted as an additional zero at the origin in the transfer function Fs. In Figure 15, it can be noticed that when s tends to zero, sFs tends to zero. The reconfiguration time for each fault estimation of index i is dictated by the product ξi×ri.Figure 15
The effect of the reconfiguration on the blades.The rapidity of the algorithm depends on the learning rateΞ and the matrix R. As studied earlier in the paper, in addition to the learning rate Ξ, even the performance level γ could influence the response time of the fault estimation through the low frequencies of the wind. In fact, for a given γ, a matrix P depending on γ could be found through the LMI in equation (22). Then, by using the LMI in equation (15), the matrix R is obtained and used to compute the estimation F^t using equation (8). As a conclusion, the parameter γ influences the response time of the algorithm through the matrix R.The derivative form of the block betweenΔβi_c and Fs in Figure 15 has a risk of turbulence amplification. For this reason, a study to choose the learning rate Ξ and the performance level γ function of the correlation between the tracking error ey_2t (blade 1 deflection velocity as an example) and the wind turbulence is proposed in the next section.


## 4.4. The Choice ofγ and Ξ Based on the Correlation with Turbulence and the Estimation Response Time Criteria

In this section, empirical boundaries are proposed for choosing the performance levelγ and the learning rate Ξ based on two criteria. Namely, the correlation between the wind speed ω and the residual ey is considered. This variable is crucial to investigate because it could be amplified if decided to use the performance level γ to tune the estimation response time. The second criterion is the rapidity of convergence which is important to ensure because only short times of convergence are recommended for a fast recovery of the fault.


### 4.4.1. The Correlation between the Wind Turbulence and the Residualey

The expression of the correlation of the two signals is given by(32)Corr=Eey×ω−EeyEωSDey×SDω,where Ex denotes the expectation of the signal x and SDx denotes the standard deviation of the signal x. The correlation Corr belongs to the interval [−1, 1]. It obeys to the weak and strong correlation. If the coefficient Corr∈−0.50∪00.5, it is called a weak correlation. If the coefficient Corr∈−1−0.5∪0.51, it is called a strong correlation.


### 4.4.2. The Estimation Response Time of the Algorithm

The response time of the fault estimation algorithm in equation (8) for each actuator of index i is given by(33)υi=ξi×ri.Many simulations have been made on the FAST software in order to derive empirical data onυ and Corr helping to choose the appropriate values of the two parameters γ and Ξ. Then, an interpolation between data points has been performed to define mappings linking the tracking error and the wind disturbance.Figures16 and 17 show the evolution of the correlation and fault estimation response time with the performance level and the learning rate for the considered wind turbine in Table 1. From Figure 18, one can conclude that for γ less than 1, this parameter influences the correlation more than the response time of the algorithm. However, when γ exceeds 2.5, the correlation becomes more influenced also by Ξ as γ increases. For γ less than 2.25, the response time of the algorithm is influenced by both Ξ and γ. But, as γ increases, only Ξ influences the response time of the algorithm. In order to guarantee a fast reconfiguration of the system performance, one should choose a small response time of the algorithm by choosing a suitable pair (γ, Ξ). However, one should guaranty that neither the resulting performance level nor the learning rate amplifies the correlation between the tracking error ey and the wind turbulence ω.Figure 16
The correlation between measuring error and the turbulence with respect to the learning rate and the performance level.Figure 17
The effectiveness estimation response time with respect to the learning rate and the performance level.Figure 18
Maximal correlation zones vs. learning rate and performance level.For the last reason, the two parameters should obey to the following empirical law. For example, ifγ is in [0.9, 1.4] interval, Ξ should not be in [2.21, 9.84] (zone 1 in Figure 18) to avoid high correlation, and with the same idea, if γ is chosen in [1.4, 2.02], Ξ should not be in the interval [8.317, 9.84] (zone 2 in Figure 18).An example of a choice of the pair (γ, Ξ) leading to a fast and less correlated response time is indicated in Figure 19 by (γ, Ξ) = (2.25, 1.25).Figure 19
The effectiveness estimation algorithm response time vs. learning rate and performance level.


### 4.4.3. Discussion with respect to Some Existing Methods

The proposed strategy is not complicated and uses an integral structure with one Lyapunov function to prove stability and convergence of both state estimation error and fault estimation error. The proposed approach uses a reduced number of measurements. Indeed, only the rotor speed and the individual blades’ deflection velocity are needed. This reduces the information needs with respect to other techniques such as the one in [44] (requiring also the generator speed measurement). It does not either require the information about the presumed fault size as in [45].The residual-based techniques in [45–47] deal only with fault detection tasks. Contrariwise, the proposed approach deals with both fault estimation and fault isolation suitable for active fault tolerant control strategies. Indeed, the individual deflection velocity measurement of each blade allows a natural isolation of the fault blade without the need of additional isolation methods. This method avoids the intrinsic delays of online least-squares methods, which could have a negative impact on the fault estimation response time.The proposed strategy is equipped with a tuning method to avoid amplification of wind turbulence in the direction of the high learning rates. This issue is not sufficiently discussed in other works related to wind turbines.Contrariwise to other strategies, the proposed method could deal with faults having dynamics faster than the system dynamics only by tuning the learning rate suitably.


### 4.4.4. Comparison of the Proposed Strategy with the Virtual Actuator with State Residual Integrator

A new scenario of pitch fault on blade 1 is considered for the comparison between the proposed strategy and a virtual actuator-based strategy (details of this method are not the objective of the paper, [48]). In this section, the considered fault happend at the instant 15 seconds. This fault represents pitch actuator loss of efficiency of 55%.Figure20 illustrates the structure of the strategy used for comparison. The principle of this strategy is to use a model reference and to compute the difference between the estimated faulty states and the model reference states. This difference is integrated and used as new control law. Equation (8) was used to derive the faulty matrix Bf. The difference with respect to the proposed strategy is in the integration of the state residual xΔ instead of the division in equation (30) and the use of reference model.Figure 20
The simplified virtual actuator strategy with state residualxΔ integration.Figure21 illustrates a comparison between the proposed strategy and the virtual actuator with integrating the state residual in terms of blade fault reconfiguration. In some regions where the wind speed makes high amplitudes of high frequencies, which means the effort needed for integration is higher, the performance of the virtual actuator becomes poor in the interval [16 s; 26 s]. This could be explained by the existence of two integrators, the first presented in equation (8) used to estimate the fault needed by Bf and the second to integrate the state residual. In conclusion, the virtual actuator with integration of state residual is not always suitable for integrator-based fault estimators.Figure 21
The comparison of the faulty blade 1 deflection between the proposed FTC and the residual integration FTC.


### 4.4.5. Digital Implementation of the Strategy

The equations to be implemented are the baseline control law in equation (6), the observer in equation (7), and the fault estimation in equation (8). In order to digitalize the continuous equations, the first-order approximation in equation (34) is used. If z˙ is the derivative, it could be approximated by(34)z˙t=zt+T−ztT,where T is the sampling step. Hence, the strategy could be digitalized for each iteration of index k as follows.The digitalized baseline controller (from equation (6)) is given by(35)uk=Kx^k.The digitalized robust observer (from equation (7)) is given by(36)xk+1=Adx^k+BdFkuk+Ldy^k−yk,where Ad, Bd, and Ld are, respectively, the state matrix, the input matrix, and the observer matrix. The expressions of the previous matrix are given by Ad=In+TA; Bd=TB; and Ld=TL.The digitalized fault estimation (equation (8)) for each actuator of index i is given by(37)fk+1,i=fk,i+Tξk,irk,iey,k,iuk,i.


## 4.4.1. The Correlation between the Wind Turbulence and the Residualey

The expression of the correlation of the two signals is given by(32)Corr=Eey×ω−EeyEωSDey×SDω,where Ex denotes the expectation of the signal x and SDx denotes the standard deviation of the signal x. The correlation Corr belongs to the interval [−1, 1]. It obeys to the weak and strong correlation. If the coefficient Corr∈−0.50∪00.5, it is called a weak correlation. If the coefficient Corr∈−1−0.5∪0.51, it is called a strong correlation.


## 4.4.2. The Estimation Response Time of the Algorithm

The response time of the fault estimation algorithm in equation (8) for each actuator of index i is given by(33)υi=ξi×ri.Many simulations have been made on the FAST software in order to derive empirical data onυ and Corr helping to choose the appropriate values of the two parameters γ and Ξ. Then, an interpolation between data points has been performed to define mappings linking the tracking error and the wind disturbance.Figures16 and 17 show the evolution of the correlation and fault estimation response time with the performance level and the learning rate for the considered wind turbine in Table 1. From Figure 18, one can conclude that for γ less than 1, this parameter influences the correlation more than the response time of the algorithm. However, when γ exceeds 2.5, the correlation becomes more influenced also by Ξ as γ increases. For γ less than 2.25, the response time of the algorithm is influenced by both Ξ and γ. But, as γ increases, only Ξ influences the response time of the algorithm. In order to guarantee a fast reconfiguration of the system performance, one should choose a small response time of the algorithm by choosing a suitable pair (γ, Ξ). However, one should guaranty that neither the resulting performance level nor the learning rate amplifies the correlation between the tracking error ey and the wind turbulence ω.Figure 16
The correlation between measuring error and the turbulence with respect to the learning rate and the performance level.Figure 17
The effectiveness estimation response time with respect to the learning rate and the performance level.Figure 18
Maximal correlation zones vs. learning rate and performance level.For the last reason, the two parameters should obey to the following empirical law. For example, ifγ is in [0.9, 1.4] interval, Ξ should not be in [2.21, 9.84] (zone 1 in Figure 18) to avoid high correlation, and with the same idea, if γ is chosen in [1.4, 2.02], Ξ should not be in the interval [8.317, 9.84] (zone 2 in Figure 18).An example of a choice of the pair (γ, Ξ) leading to a fast and less correlated response time is indicated in Figure 19 by (γ, Ξ) = (2.25, 1.25).Figure 19
The effectiveness estimation algorithm response time vs. learning rate and performance level.


## 4.4.3. Discussion with respect to Some Existing Methods

The proposed strategy is not complicated and uses an integral structure with one Lyapunov function to prove stability and convergence of both state estimation error and fault estimation error. The proposed approach uses a reduced number of measurements. Indeed, only the rotor speed and the individual blades’ deflection velocity are needed. This reduces the information needs with respect to other techniques such as the one in [44] (requiring also the generator speed measurement). It does not either require the information about the presumed fault size as in [45].The residual-based techniques in [45–47] deal only with fault detection tasks. Contrariwise, the proposed approach deals with both fault estimation and fault isolation suitable for active fault tolerant control strategies. Indeed, the individual deflection velocity measurement of each blade allows a natural isolation of the fault blade without the need of additional isolation methods. This method avoids the intrinsic delays of online least-squares methods, which could have a negative impact on the fault estimation response time.The proposed strategy is equipped with a tuning method to avoid amplification of wind turbulence in the direction of the high learning rates. This issue is not sufficiently discussed in other works related to wind turbines.Contrariwise to other strategies, the proposed method could deal with faults having dynamics faster than the system dynamics only by tuning the learning rate suitably.


## 4.4.4. Comparison of the Proposed Strategy with the Virtual Actuator with State Residual Integrator

A new scenario of pitch fault on blade 1 is considered for the comparison between the proposed strategy and a virtual actuator-based strategy (details of this method are not the objective of the paper, [48]). In this section, the considered fault happend at the instant 15 seconds. This fault represents pitch actuator loss of efficiency of 55%.Figure20 illustrates the structure of the strategy used for comparison. The principle of this strategy is to use a model reference and to compute the difference between the estimated faulty states and the model reference states. This difference is integrated and used as new control law. Equation (8) was used to derive the faulty matrix Bf. The difference with respect to the proposed strategy is in the integration of the state residual xΔ instead of the division in equation (30) and the use of reference model.Figure 20
The simplified virtual actuator strategy with state residualxΔ integration.Figure21 illustrates a comparison between the proposed strategy and the virtual actuator with integrating the state residual in terms of blade fault reconfiguration. In some regions where the wind speed makes high amplitudes of high frequencies, which means the effort needed for integration is higher, the performance of the virtual actuator becomes poor in the interval [16 s; 26 s]. This could be explained by the existence of two integrators, the first presented in equation (8) used to estimate the fault needed by Bf and the second to integrate the state residual. In conclusion, the virtual actuator with integration of state residual is not always suitable for integrator-based fault estimators.Figure 21
The comparison of the faulty blade 1 deflection between the proposed FTC and the residual integration FTC.


## 4.4.5. Digital Implementation of the Strategy

The equations to be implemented are the baseline control law in equation (6), the observer in equation (7), and the fault estimation in equation (8). In order to digitalize the continuous equations, the first-order approximation in equation (34) is used. If z˙ is the derivative, it could be approximated by(34)z˙t=zt+T−ztT,where T is the sampling step. Hence, the strategy could be digitalized for each iteration of index k as follows.The digitalized baseline controller (from equation (6)) is given by(35)uk=Kx^k.The digitalized robust observer (from equation (7)) is given by(36)xk+1=Adx^k+BdFkuk+Ldy^k−yk,where Ad, Bd, and Ld are, respectively, the state matrix, the input matrix, and the observer matrix. The expressions of the previous matrix are given by Ad=In+TA; Bd=TB; and Ld=TL.The digitalized fault estimation (equation (8)) for each actuator of index i is given by(37)fk+1,i=fk,i+Tξk,irk,iey,k,iuk,i.


## 5. Conclusion

In this paper, the partial failure of pitch actuators in wind turbine was considered. The LMI-based fault tolerant control was applied to the considered fault. The method presents the advantages that it estimates at the same time the states and the fault. The method convergence is proved through Lyapunov theory. Two key design parameters were identified in the integrated design, namely, the performance level and the learning rate. It has been shown that the performance levelγ of the observer and the learning rate Ξ impact indirectly the loads on the turbine. The parameters have an effect on the availability time of the fault estimation. In some zones of the pair (γ, Ξ), it has been found that as the learning rate increases, the fault becomes quickly available and the fault tolerant strategy accommodates rapidly the prefault load reduction. In other zones, the robustness γ enhances the loads reduction. Moreover, the performance level γ provides another degree of freedom to the adaptive algorithm in addition to the learning rate Ξ. An empirical design method was proposed to impose both a given correlation of the measuring error to the wind turbulence and a given response time of the effectiveness estimation algorithm. The proposed method helps choosing the desired short response time of the effectiveness estimation algorithm with sufficiently good correlation level to the wind turbulence.



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*Source: 6210407-2020-02-12.xml*