# Robust H2-Optimal TS Fuzzy Controller Design for a Wind Energy Conversion System

Authors: Maher Kharrat; Sahbi Abderrahim; Moez Allouche
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

In this paper, robust optimal control of an uncertain wind energy conversion system (WECS), described by a Takagi–Sugeno fuzzy model, is proposed to guarantee the maximum power trajectory tracking (MPTT). The design of the fuzzy optimal control is based on a quadratic criterion related to the maximum power tracking error. The proposed fuzzy controller allows to regulate the rectified direct current (DC) voltage of the variable-speed wind turbine by adjusting the duty cycle of a boost converter. Linear matrix inequality (LMI) conditions, ensuring the optimalH2 tracking error, are proposed for the controller gain design. Simulation as well as experimental tests are performed for efficiency evaluation of the established MPPT fuzzy control scheme under variable wind speed conditions.


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## Body

## 1. Introduction

Over the past few decades, global energy consumption has increased dramatically resulting in limited reserve capacity availability for the foreseeable future and a notable increase of the environment pollution. The field of renewable energy, as an alternative solution, has gained a lot of interest since its inexhaustible resources which creates no pollution, waste, or contamination risks to air and water. Among available renewable resources, the wind energy is considered as one of the most promising renewable resources with an average annual growth rate of 20% [1]. A wind turbine converts wind energy into electricity using the aerodynamic force from the rotor blades. The kinetic energy generated by the wind is then converted into electricity through the electric generator located at the top of the hub [2, 3]. Electrical energy can be produced from a wind farm connected to the grid or from a small wind electrical system that could meet some domestic uses including water pumping on farms and ranches. In this paper, the designed WECS is composed of a wind turbine coupled to a permanent magnet synchronous generator (PMSG) that feeds a resistive load using a three-diode bridge rectifier connected to a boost converter. The latter is considered as a maximum power tracker, allowing the adjustment of the load impedance to the wind generator source. A maximum power point tracking (MPPT) algorithm is used in the WECS to realize the optimal matching. Moreover, a fuzzy reference model is used to generate the optimum DC output current of the rectifier Idcopt leading to the maximum power operation [4].Variable-speed wind turbines are able to operate at an optimal rotation speed as a function of the wind speed. In the literature, new formulations of the wind turbine optimum operation have been proposed, which express the rectified DC voltage as a linear function of the generator rotor speed [5]. Hence, the structure of the control scheme could be then simplified resulting in system cost reduction since it will be based on the use of a static DC/DC converter. As a result, the duty cycle of the converter can be then adjusted to regulate the DC voltage, thus guaranteeing optimal operation of the WECS.Generally, the MPPT algorithms can be classified into two major methods: indirect and direct. The indirect methods require the knowledge of the characteristics of the wind turbine, from which one could deduce the optimum speed of the generator for each value of wind speed [6, 7]. Among them, one could cite power signal feedback (PSF) control [8], TSR control, sliding mode control [9], fuzzy logic control [10, 11], and so on. Moreover, the indirect methods are considered as simple and intuitive, as they are highly based on the accuracy of the wind speed measurement which constitutes a challenge for such methods.Direct methods, like PO and Hill Climbing Techniques, are characterized by a fast control algorithm to find the maximum power point. However, this algorithm seems to be too sensitive to all abrupt wind speed changes and presents energy loss near the maximum power point [12, 13]. Applied successfully in photovoltaic systems, these methods are based on the search of extreme points which depend on the step-size setting. However, the uncertain and random nature of the wind profiles and the fast dynamic behaviours of the generator make the application of these methods difficult in wind energy conversion systems and require significant modification.Recently, the Takagi–Sugeno (TS) fuzzy approach, for the control of complex systems, has attracted increasing interest from the scientific community. It represents the nonlinear system by a convex combination of local linear model, interpolated by nonlinear weighting functions [14, 15]. In this context, many studies have been conducted regarding the maximum power point tracking problem when applied to WECS [16–18].This paper proposes a novel TS fuzzy MPPT controller using the DC-link voltage under quickly changing wind speed conditions. By controlling the rectified DC voltage, the TS fuzzy MPPT controller generates the optimal duty cycle forcing the permanent magnet synchronous generator (PMSG) to work at the maximum power trajectory. The robustness is tested taking into account system uncertainties as well as the rapid change in wind speed.The remainder of this paper is structured as follows. In Section2, the structure of the wind power conversion system is presented. Section 3 is reserved to the TS fuzzy MPPT controller development to achieve the wind energy MPPT control. Sections 4 and 5 provide simulation and experimental results to illustrate the efficiency the proposed method. Finally, a general conclusion is given in Section 6.


## 2. Dynamic Model of the Wind Energy Conversion System

The structure of the wind energy conversion system is illustrated in Figure1. It consists of a wind turbine connected to a PMSG that supplies a resistive load using a three-diode rectifier bridge associated to a boost converter.Figure 1
Wind energy conversion system structure.The recovered mechanical power, at the wind turbine shaft, is related to the wind speedVw as follows [2]:(1)Pt=12Cpλ,βρπR2Vw3,where ρ is the air density, R is the rotor radius, and Cpλ,β is the power coefficient characterizing the turbine conversion efficiency of the wind kinetic energy into mechanical energy. The function Cpλ,β, specific for each wind turbine, allows to classify the different types of wind turbines according to their maximal Cp value. This parameter depends on the tip-speed ratio λ and the pitch angle β as follows:(2)Cpλ,β=0.11151λi−0.58β−0.002β2.14−13.2exp−18.41λi,λi=1λ−0.02β−0.003β3+1−1,and the tip-speed ratio is expressed as [2](3)λ=ωmRVw.The generator power curves as a function of the rectified voltage, presented in Figure2, illustrate the presence of extreme points, for different values of wind speed. Hence, using the optimum tip-speed ratio λopt = 6.9, corresponding to each given wind speed, the optimal rotor speed is calculated as follows:(4)ωmopt=λoptVwR,leading thereafter to the determination of the optimum rectified DC voltage value. By forcing the wind turbine to operate under the optimal conditions, i.e., at λ = λopt, the torque generated by the wind turbine, leading to the maximum power transfer, could be expressed by [17, 19](5)Ct=Koptωm2,where(6)Kopt=0.5πρCR5λopt3pmax.Figure 2
Generated power curve in function of the DC rectified voltage.In order to convert the recovered mechanical energy into electrical energy, a PMSG, characterized by the following (d-q) dynamic model, is used:(7)ddtisd=−RsLdisd+npωmisq+1Ldvsd,ddtisq=−RsLdisq−npωmisd−npLdωmψf+1Ldvsq,ddtωm=1JCt−Cem,where Rs and Ld represent, respectively, the stator resistance and inductance in the direct axis; J is the total moment inertia; ψfis the fixed flux linked by the stator windings; np is the number of poles; and Cem is the electromagnetic torque expressed by(8)Cem=32npisqψf.The generator power, obtained at the rectifier output, is given by(9)Pg=IdcVdc,where Vdc is the DC side voltage, after 3-phase rectification, given by(10)Vdc=33πVsd2+Vsq2,and Idc is the rectified DC current expressed by(11)Idc=π23isd2+isq2.The power loss of the three-diode full bridge rectifier circuit has been neglected in these expressions.The topology of the power electronic circuit given in Figure1 is composed of an uncontrolled rectifier bridge coupled to a boost chopper. The function of the chopper is to facilitate the operation at a wide range of variable speed. The dynamic model of the converter may be expressed using two functional modes depending on the switch power state. For this reason, we propose the averaged bilinear model of the boost converter which leads thereafter to the TS model that serves to the control design step. Hence, using the rectified DC voltage Vdc(t), the inductor current IL(t), and the load voltage Vch(t), as state variables, the first operating mode could be determined upon the “ON” period, when the switch is closed according to the following differential equations:(12)dVdctdt=−1C1Vdct+1C1Idc,dILtdt=1LVdct−RLLILt,dVchtdt=−1R0C2Vcht.The second operating mode could also be determined during the “OFF” time when the control switch is open, according to the following differential equations:(13)dVdctdt=−1C1Vdct+1C1Idct,dILtdt=1LVdct−RLLILt−1LVcht,dVchtdt=1C2ILt−1R0C2Vcht.The reformulation, under state equation, could lead to the following state model:(14)x˙t=Akxt+Ewtk=1,2,A1=0−1C101L−RLL000−1R0C2A2=0−1C101L−RLL−1L01C2−1R0C2,E=1C100,wt=Idct,xt=VdctILtVcht.By referring to the bilinear models (13), the average model of the converter is obtained:(15)x˙t=A2xt+Bxtut+Ewt,ut∈0,1,where Bxt=0VC2t/LILt/C2 and ut is the duty ratio.


## 3. TS Fuzzy Reference Model-Based MPPT Control Design

Figure3 provides the fuzzy control structure used in the present study. It consists of a TS reference model, which generates the reference state xr leading to the maximum power trajectory. It includes an MPP searching module, delivering the optimal DC rectified current.Figure 3
Fuzzy control strategy for MPPT.In order to fulfil the LMI conditions, one could exploit Lemma1 demonstrating the following usual matrix property.Lemma 1.
Consider two matrices A and B of appropriate dimensions; there exists a positive scalar that verifies the following inequality:(16)ATB+BTA≤εATA+ε−1BTB.


### 3.1. MPP Searching Algorithm

The MPP searching algorithm is applied for the generation of the optimal rectified currentIdcopt ensuring the optimal operations. Once the steady state is established, the torque, developed by the wind turbine, is similar to the generator electromagnetic torque, which could lead to determine the expression of the q-axis current as follows:(17)isq=2Koptωm23npψf,orIdc=π23isd2+isq2.Hence, by adopting the vector control strategy of the synchronous machine, which consists of imposingisd = 0, the DC side current of the rectifier becomes(18)Idc=πKopt33npψfωm2.Therefore, by driving the WECS at the maximum power trajectory, the optimal DC side current of the rectifier could be obtained online via (18), by equating ωm=ωmopt:(19)Idcopt=πKopt33npψfωmopt2.Therefore, considering the optimal DC side current of the rectifier as a control input, the TS fuzzy reference model can provide the reference states allowing to the optimal operation condition. Here, the main objective of the established fuzzy controller is to guarantee the convergence of the tracking errorxt−xrt towards zero for all wind speed variations. To achieve the wind energy MPPT control, the TS fuzzy MPPT controller is proposed in the following section.


### 3.2. Uncertain TS Model

The multi-model, inspired from the Takagi–Sugeno approach, is used to describe the nonlinear systems as the form of interpolation of several linear local models. Each local model is considered as a dynamic LTI (linear invariant time) system valid around an operating point [14, 15]. In this context, nonlinear model (15) contains two premise variables zkt=xkt∈zkmin,zkmax deduced from the input matrix Bxt with k = 1, 2:(20)z1t=Vcht,z2t=ILt.Define now the membership functionsFij(z (t)) of each premise variable zk(t) in the fuzzy subsets Fij as follows:(21)Fk,minzt=zkt−minzktmaxzkt−minzkt,Fk,maxzt=maxzkt−zktmaxzkt−minzkt.The uncertain fuzzy model could be formulated by the following “If-Then” fuzzy rules.RuleRi: if z1tisF1,i and z2tisF2,i, then(22)x˙t=A2xt+Biut+Ewt,i=1,2,3,4,where the vectors Bii=1,2,3,4 are given by(23)B1=0VC2mintLILmintC2,B2=0VC2maxtLILmintC2,B3=0VC2mintLILmaxtC2,B4=0VC2maxtLILmaxtC2.Thus, the inference of the four fuzzy rules leads to obtaining an uncertain TS multi-model given by the following nonlinear state representation:(24)x˙t=∑i=14hiztA2+ΔA2xt+Bi+ΔBiut+Ewt,where ΔA2 and ΔBi are matrices representing the parametric uncertainties, such as ΔA2=DAFitEA and ΔBi=DBiFitEBi, DA,Bi and EA,Bi are a constant matrices with well-determined dimension, and Fit is an unknown function satisfying for all variable t, FitFiTt<I.


### 3.3. MPPT Reference Model

In order to force the WECS to pursue the maximum power trajectory, we consider a reference model in which the optimal rectified currentIdcopt is considered as a control input. The reference model providing the desired trajectory is given by(25)x˙rt=Arxrt+EIpvopt,or equivalently(26)x˙rt=Arxrt+rt,Ar=0−1C101L−RLL−1L1−uopt01C21−uopt−1RC2,rt=IdcoptC100,uopt=VdcoptRIdcopt.Model (24) is nonlinear and can be represented by a convex combination of two linear submodels which originates from the premise variable zr=1−uopt. It can be formulated using the following two rules:(i)
Rule 1: ifzrtisNmin, then x˙rt=Ar1xt+rt.(ii)
Rule 2: ifzrtisNmax, then x˙rt=Ar2xt+rt.The membership functions are expressed as follows:(27)Nminzrt=zrt−zr,minzr,max−zr,min,Nmaxzrt=1−Nminzrt,and the reference state matrices are given by(28)Ar1=0−1C101L−RLL−1Lzr,min01C2zr,min−1R0C2,Ar2=0−1C101L−RLL−1Lzr,max01C2zr,max−1R0C2.Finally, nonlinear model (26) can be described by a TS reference model as follows:(29)x˙rt=∑k=12hkzrtArkxrt+rt.


### 3.4. MPPT Controller Design

To guarantee the MPPT of the rectifier output power, one should verify that the boost converter statex(t) follows the reference state xr(t) despite the wind speed and system parameters variations. Consequently, the tracking problem could be then transformed into a tracking error feedback control defined as follows [12, 13]:(30)x˙rt=∑k=12hkzrtArkxrt+rt.Substituting control law (30) in (24) and considering reference model (29), the dynamics of the tracking error can be obtained as follows:(31)e˙rt=∑i=18∑j=18hizthjztA2+ΔA2+Bi+ΔBiKjert+A2−Ari+ΔA2xrt+Ewt−rt.Finally, considering the augmented state vectorx¯t=ertxrtT, one could obtain the augmented system describing the closed-loop dynamics given by the following equation:(32)x¯˙t=∑i=18∑j=18hizthjztG¯ijx¯t+E¯W¯t,where(33)G¯ij=A¯ij+ΔA¯ij=A2+BiKjA2−Ari0Ari+ΔA2+ΔBiKjΔA200,E¯=E−I0I,W¯t=wtrt.In this work, the H2-optimal control problem is proposed. It consists of finding a multi-controller Ki which stabilizes closed-loop system (32) and minimizing the cost function J related to the tracking error ert. Based on fuzzy augmented system (32), the H2 tracking performance without considering the effect of general disturbance W¯t can be represented by the following equation:(34)J=∫0∞ertTQert+utTRut.The tracking performance criteria (34) could be then reformulated as follows:(35)J=∫0∞x¯tTQ¯x¯t+utTRut≤α,where Q¯=Q000, α is a positive scalar, and Q is a positive matrix.The following theorem summarizes the LMI conditions that ensure the stability of augmented system (35) with the MPP tracking performance under constraint (34).Theorem 1.
The closed-loop wind energy conversion system (32) is asymptotically stable under the MPP tracking control law (30), satisfying the quadratic criterion (34), if there are symmetric definite positive matrices X1=P1−1,P2, matrices Yi, and scalars α,εA1,εA2,εBi1 verifying the following optimization problem:(36)minαP1,P2,subject to the following constrained inequalities:(37)xr0TP2xr0−αIerT0er0−X1<0,(38)Π11X1EATYjEBiTAi−Ari−YiX1∗−εA1−10000∗∗−εBi1−1000∗∗∗Π4400∗∗∗∗−R−10∗∗∗∗∗−Q−1<0,where(39)Π11=X1AiT+AiX1+YjTBiT+BiYj+εA1−1+εA2−1DADAT+εBi1−1DBiDBiT,Π44=AriTP2+P2Ari+εA1EAEAT,Ki=YiX1−1.Proof 1.
Consider the following quadratic Lyapunov function:(40)Vx¯t=x¯TtP¯x¯t,where P¯=diagP1,P2. In order to obtain an optimal solution, with regard to the quadratic criterion (35), one requires checking the following inequality:(41)V˙x¯t+x¯TtQ¯x¯t+uTtRut<0.
By integrating (41), one could obtain(42)Vx¯t0∞+∫0∞x¯TtQ¯x¯t+uTtRutdt<0,which can be rewritten in the following form:(43)∫0∞x¯TtQ¯x¯t+uTtRutdt<x¯T0P¯x¯0.
Note that if inequality (43) is satisfied, this leads to suboptimal solution, bound by the initial conditions of the Lyapunov candidate function. In order to reach the LMI conditions (37), we introduce a scalar α>0 to be minimized such that(44)J≤x¯T0P¯x¯0≤α.
Considering closed-loop system (32) and neglecting the effect of the general disturbance W¯t, the time derivative of function (40) leads to following inequality:(45)∑i=18∑j=18hizthjztx¯TtG¯ijTP¯+P¯G¯ij+Q¯+I0TKjTRKjI0x¯t<0.
Inequality (45) could be rewritten in a developed form with the notation defined in (32) such that(46)ϒ¯ij+Δϒ¯ij<0,where(47)ϒ¯ij=P1A2+BiKj+A2+BiKjTP1+QP1A2−AriKj∗AriTP2+P2Ari0∗∗−R−1,Δϒ¯ij=P1ΔAi+ΔBiKj+ΔAi+ΔBiKjTP1P1ΔAi0∗00∗∗0.
By considering the variable changeYj=KjX1, pre and postmultiplying BMI (46) by diagP1−1,I,I, and using well-known Lemma 1, Δϒ¯ij can be bound as follows:(48)Δϒ¯ij≤εA1−1+εA2−1DADAT+εBi1−1DBiDBiT+εA1X1EATEAX1+εBi1YjTEBiEBiTYj000εA2EATEA0000.
By applying Schur complement in (46) and using condition (48), we get LMI (37).


## 3.1. MPP Searching Algorithm

The MPP searching algorithm is applied for the generation of the optimal rectified currentIdcopt ensuring the optimal operations. Once the steady state is established, the torque, developed by the wind turbine, is similar to the generator electromagnetic torque, which could lead to determine the expression of the q-axis current as follows:(17)isq=2Koptωm23npψf,orIdc=π23isd2+isq2.Hence, by adopting the vector control strategy of the synchronous machine, which consists of imposingisd = 0, the DC side current of the rectifier becomes(18)Idc=πKopt33npψfωm2.Therefore, by driving the WECS at the maximum power trajectory, the optimal DC side current of the rectifier could be obtained online via (18), by equating ωm=ωmopt:(19)Idcopt=πKopt33npψfωmopt2.Therefore, considering the optimal DC side current of the rectifier as a control input, the TS fuzzy reference model can provide the reference states allowing to the optimal operation condition. Here, the main objective of the established fuzzy controller is to guarantee the convergence of the tracking errorxt−xrt towards zero for all wind speed variations. To achieve the wind energy MPPT control, the TS fuzzy MPPT controller is proposed in the following section.


## 3.2. Uncertain TS Model

The multi-model, inspired from the Takagi–Sugeno approach, is used to describe the nonlinear systems as the form of interpolation of several linear local models. Each local model is considered as a dynamic LTI (linear invariant time) system valid around an operating point [14, 15]. In this context, nonlinear model (15) contains two premise variables zkt=xkt∈zkmin,zkmax deduced from the input matrix Bxt with k = 1, 2:(20)z1t=Vcht,z2t=ILt.Define now the membership functionsFij(z (t)) of each premise variable zk(t) in the fuzzy subsets Fij as follows:(21)Fk,minzt=zkt−minzktmaxzkt−minzkt,Fk,maxzt=maxzkt−zktmaxzkt−minzkt.The uncertain fuzzy model could be formulated by the following “If-Then” fuzzy rules.RuleRi: if z1tisF1,i and z2tisF2,i, then(22)x˙t=A2xt+Biut+Ewt,i=1,2,3,4,where the vectors Bii=1,2,3,4 are given by(23)B1=0VC2mintLILmintC2,B2=0VC2maxtLILmintC2,B3=0VC2mintLILmaxtC2,B4=0VC2maxtLILmaxtC2.Thus, the inference of the four fuzzy rules leads to obtaining an uncertain TS multi-model given by the following nonlinear state representation:(24)x˙t=∑i=14hiztA2+ΔA2xt+Bi+ΔBiut+Ewt,where ΔA2 and ΔBi are matrices representing the parametric uncertainties, such as ΔA2=DAFitEA and ΔBi=DBiFitEBi, DA,Bi and EA,Bi are a constant matrices with well-determined dimension, and Fit is an unknown function satisfying for all variable t, FitFiTt<I.


## 3.3. MPPT Reference Model

In order to force the WECS to pursue the maximum power trajectory, we consider a reference model in which the optimal rectified currentIdcopt is considered as a control input. The reference model providing the desired trajectory is given by(25)x˙rt=Arxrt+EIpvopt,or equivalently(26)x˙rt=Arxrt+rt,Ar=0−1C101L−RLL−1L1−uopt01C21−uopt−1RC2,rt=IdcoptC100,uopt=VdcoptRIdcopt.Model (24) is nonlinear and can be represented by a convex combination of two linear submodels which originates from the premise variable zr=1−uopt. It can be formulated using the following two rules:(i)
Rule 1: ifzrtisNmin, then x˙rt=Ar1xt+rt.(ii)
Rule 2: ifzrtisNmax, then x˙rt=Ar2xt+rt.The membership functions are expressed as follows:(27)Nminzrt=zrt−zr,minzr,max−zr,min,Nmaxzrt=1−Nminzrt,and the reference state matrices are given by(28)Ar1=0−1C101L−RLL−1Lzr,min01C2zr,min−1R0C2,Ar2=0−1C101L−RLL−1Lzr,max01C2zr,max−1R0C2.Finally, nonlinear model (26) can be described by a TS reference model as follows:(29)x˙rt=∑k=12hkzrtArkxrt+rt.


## 3.4. MPPT Controller Design

To guarantee the MPPT of the rectifier output power, one should verify that the boost converter statex(t) follows the reference state xr(t) despite the wind speed and system parameters variations. Consequently, the tracking problem could be then transformed into a tracking error feedback control defined as follows [12, 13]:(30)x˙rt=∑k=12hkzrtArkxrt+rt.Substituting control law (30) in (24) and considering reference model (29), the dynamics of the tracking error can be obtained as follows:(31)e˙rt=∑i=18∑j=18hizthjztA2+ΔA2+Bi+ΔBiKjert+A2−Ari+ΔA2xrt+Ewt−rt.Finally, considering the augmented state vectorx¯t=ertxrtT, one could obtain the augmented system describing the closed-loop dynamics given by the following equation:(32)x¯˙t=∑i=18∑j=18hizthjztG¯ijx¯t+E¯W¯t,where(33)G¯ij=A¯ij+ΔA¯ij=A2+BiKjA2−Ari0Ari+ΔA2+ΔBiKjΔA200,E¯=E−I0I,W¯t=wtrt.In this work, the H2-optimal control problem is proposed. It consists of finding a multi-controller Ki which stabilizes closed-loop system (32) and minimizing the cost function J related to the tracking error ert. Based on fuzzy augmented system (32), the H2 tracking performance without considering the effect of general disturbance W¯t can be represented by the following equation:(34)J=∫0∞ertTQert+utTRut.The tracking performance criteria (34) could be then reformulated as follows:(35)J=∫0∞x¯tTQ¯x¯t+utTRut≤α,where Q¯=Q000, α is a positive scalar, and Q is a positive matrix.The following theorem summarizes the LMI conditions that ensure the stability of augmented system (35) with the MPP tracking performance under constraint (34).Theorem 1.
The closed-loop wind energy conversion system (32) is asymptotically stable under the MPP tracking control law (30), satisfying the quadratic criterion (34), if there are symmetric definite positive matrices X1=P1−1,P2, matrices Yi, and scalars α,εA1,εA2,εBi1 verifying the following optimization problem:(36)minαP1,P2,subject to the following constrained inequalities:(37)xr0TP2xr0−αIerT0er0−X1<0,(38)Π11X1EATYjEBiTAi−Ari−YiX1∗−εA1−10000∗∗−εBi1−1000∗∗∗Π4400∗∗∗∗−R−10∗∗∗∗∗−Q−1<0,where(39)Π11=X1AiT+AiX1+YjTBiT+BiYj+εA1−1+εA2−1DADAT+εBi1−1DBiDBiT,Π44=AriTP2+P2Ari+εA1EAEAT,Ki=YiX1−1.Proof 1.
Consider the following quadratic Lyapunov function:(40)Vx¯t=x¯TtP¯x¯t,where P¯=diagP1,P2. In order to obtain an optimal solution, with regard to the quadratic criterion (35), one requires checking the following inequality:(41)V˙x¯t+x¯TtQ¯x¯t+uTtRut<0.
By integrating (41), one could obtain(42)Vx¯t0∞+∫0∞x¯TtQ¯x¯t+uTtRutdt<0,which can be rewritten in the following form:(43)∫0∞x¯TtQ¯x¯t+uTtRutdt<x¯T0P¯x¯0.
Note that if inequality (43) is satisfied, this leads to suboptimal solution, bound by the initial conditions of the Lyapunov candidate function. In order to reach the LMI conditions (37), we introduce a scalar α>0 to be minimized such that(44)J≤x¯T0P¯x¯0≤α.
Considering closed-loop system (32) and neglecting the effect of the general disturbance W¯t, the time derivative of function (40) leads to following inequality:(45)∑i=18∑j=18hizthjztx¯TtG¯ijTP¯+P¯G¯ij+Q¯+I0TKjTRKjI0x¯t<0.
Inequality (45) could be rewritten in a developed form with the notation defined in (32) such that(46)ϒ¯ij+Δϒ¯ij<0,where(47)ϒ¯ij=P1A2+BiKj+A2+BiKjTP1+QP1A2−AriKj∗AriTP2+P2Ari0∗∗−R−1,Δϒ¯ij=P1ΔAi+ΔBiKj+ΔAi+ΔBiKjTP1P1ΔAi0∗00∗∗0.
By considering the variable changeYj=KjX1, pre and postmultiplying BMI (46) by diagP1−1,I,I, and using well-known Lemma 1, Δϒ¯ij can be bound as follows:(48)Δϒ¯ij≤εA1−1+εA2−1DADAT+εBi1−1DBiDBiT+εA1X1EATEAX1+εBi1YjTEBiEBiTYj000εA2EATEA0000.
By applying Schur complement in (46) and using condition (48), we get LMI (37).


## 4. Numerical Simulation

Simulation experiments are performed to assess the effectiveness of the proposed MPP fuzzy tracking controller. The tests are carried out using a TS reference model as well as a variable wind speed profile as illustrated in Figure4. The robustness study is performed by assuming a variation of 10% of the boost converter parameters compared to their nominal values. By setting the control parameters: Q = diag {0.01, 0.01, 0.01} and R = 0.00005, LMIs (37) and (38) of Theorem 1 are solved, leading to the following controller gains:(49)K1=−0.49561.7851−0.3718,K2=−0.50361.7682−0.3822,K3=−0.50371.7665−0.3902,K4=−0.48991.7831−0.3802.Figure 4
Wind speed profile.The optimal parameter related to the H2-optimal performance is α∗=minαP1,P2=0.987.In order to evaluate the performance of the developed fuzzy controller, we compared it to the PO method. The latter is considered as a direct method requiring only rectified DC currentIdc and voltage Vdc measurements. Figure 5 shows the evolution, as a function of the time, of the rectified DC voltage Vdc(t), the inductor current IL(t), and the load voltage Vch(t) for both tested methods. The variations of the power generator, rotor speed, and duty cycle, as a function of the time, are given in Figure 6.Figure 5
Time evolution of (a) rectified DC voltage, (b) inductance current, and (c) load voltage.
(a)(b)(c)Figure 6
Time evolution of (a) output power, (b) rotor speed, and (c) control signal.
(a)(b)(c)Compared to the PO method with a fixed step of 0.01 V, the TS fuzzy controller provides good dynamic performance. One could notice that it converges to the reference state more rapidly without oscillation around the MPP. This will increase the efficiency of energy production from the generator to the load. The dynamic performance of the PO algorithm can be further improved with a variable step size, but at the cost of increased computational complexity. The estimated average output powers are, respectively, 255.7 W and 247.2 W for the TS controller and the PO algorithm.From the simulation results, it could be noticed that the proposed fuzzy controller exhibits the following:(i)
A fast convergence speed of the power to the MPP.(ii)
A low tracking accuracy.(iii)
A smoother response in steady state without oscillation around the MPP resulting in a stable power generation.


## 5. Experimental Results

To validate the developed fuzzy control law, a wind emulator bench installed in the LIAS laboratory has been used to reproduce the behaviour of the wind turbine. The test bench structure, shown in Figure7, is composed of two permanent magnet synchronous machines. The first machine, operating as a motor, is controlled in torque mode using a Leroy-Somer industrial speed driver to emulate the behaviour of the turbine. It receives the reference couple considered as input control to provide the reference three-phase voltages. The second machine, operating as a generator, is coupled to three-phase diode bridge generating the DC rectified voltage. The latter is connected to the boost converter which is controlled by the dSPACE RTI1005 board. A LEM-PR30 current probe and a ST1000 differential voltage sensor are used to measure the rectified DC current and the rectified DC voltage, respectively.Figure 7
Experimental test bench schema of the wind turbine emulator.The experimental results related to the proposed fuzzy MPPT algorithm take the same wind speed curve shown in Figure4. The optimal duty cycle is supplied to the MOSFET switch of the boost converter which is a periodic switching pulse with a switching frequency of 10 kHz.The obtained curves, presented in Figures8(a) and 8(b), confirm the convergence of the real state variables, composed of rectified DC voltage Vdc, the inductor current IL, and the load voltage Vch, to the desired trajectory. The experimental results demonstrate that the maximum power, captured by the wind turbine is reached, and hence, the WECS system can operate under its optimum performance. Moreover, Figure 8(c) illustrates the fast convergence of the generator power towards the reference power trajectory Pdcref, despite the wind speed variation. Hence, the experimental results proved that the proposed MPPT fuzzy control law provides robustness and the highest MPPT efficiency.Figure 8
The response of (a) rectified voltage and inductance current, (b) load voltage and duty cycle, and (c) generator power.
(a)(b)(c)


## 6. Conclusion

In this paper, we propose LMI stabilization conditions for the tracking of the maximum power trajectory when applied to an uncertain wind energy conversion system described by TS models. The considered fuzzy control law structure, expressed by the tracking error, is based on the use of a TS reference model ensuring the maximum power tracking. By minimizing a quadratic cost function, related to the tracking error, optimal synthesis conditions leading to the controller gains were obtained. The efficiency of the proposed MPPT fuzzy controller is firstly tested by numerical simulation under Matlab environment and then validated experimentally using the wind emulator-based test bench.



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*Source: 5428109-2022-05-27.xml*