# Neural Adaptive Decentralized Coordinated Control with Fault-Tolerant Capability for DFIGs under Stochastic Disturbances

Authors: Xiao-ming Li; Xiu-yu Zhang; Hong Cao; Zhong-wei Lin; Yu-guang Niu; Jian-guo Wang
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

At present, most methodologies proposed to control over double fed induction generators (DFIGs) are based on single machine model, where the interactions from network have been neglected. Considering this, this paper proposes a decentralized coordinated control of DFIG based on the neural interaction measurement observer. An artificial neural network is employed to approximate the nonlinear model of DFIG, and the approximation error due to neural approximation has been considered. A robust stabilization technique is also proposed to override the effect of approximation error. AH2 controller and a H∞ controller are employed to achieve specified engineering purposes, respectively. Then, the controller design is formulated as a mixed H2/H∞ optimization with constrains of regional pole placement and proportional plus integral (PI) structure, which can be solved easily by using linear matrix inequality (LMI) technology. The results of simulations are presented and discussed, which show the capabilities of DFIG with the proposed control strategy to fault-tolerant control of the maximum power point tracking (MPPT) under slight sensor faults, low voltage ride-through (LVRT), and its contribution to power system transient stability support.


---

## Body

## 1. Introduction

During the last decade, wind power has shown world’s fastest growing rate compared to any other electric power generations, which causes the share of wind power to reach a considerable level [1]. DFIG is becoming the dominant type used in wind farms (WFs) for its maximizing wind energy conversion and flexible control to network support [2]. For ensuring that DFIG is integrated into the power network reliably and efficiently, it is necessary to provide DFIGs with suitable control strategies.Power system is a geographically extensive large-scale system, and its controller design is commonly based on decentralized approach which only depends on local signals [3–5]. However, this simple approach reduces the controller capability and even leads to stability problems [6]. Considering this, a few of decentralized coordinated control strategies of power system have been proposed [7–11]. A hierarchical decentralized coordinated control strategy is proposed to control the excitation system of synchronous generator (SG), where the interaction terms are considered as bounded disturbances which are suppressed by a H∞ controller [8]. A direct feedback linearization based decentralized coordinated control of excitation and steam valve is proposed, where the upper bound of interaction terms is estimated [9]. A multiagent system based strategy is also used to control a multimachine power system [10, 11]. According methodologies used, decentralized coordinated control strategies of power system can be divided into two types, with and without communication system support. For the first type, the interaction terms are commonly considered as bounded disturbances which are completely suppressed, where the involved coordinated information has been neglected, while the second method needs communication system support, which may bring new stability problems caused by communication time delay and communication system fault.It is generally recognized that mode decomposition based decentralized coordinated control strategy is more suitable for control over power systems, where the interaction term is modelled as a coordinated signal [12, 13]. This allows the system-wide state feedback control strategy to be replaced by using local state feedback control, which is a desired performance for power system controller design. This paper proposes a neural observer based decentralized coordinated control of DFIG, where a neural controller is used to compute the weightings. The mode decomposition technology is used to modelling power system and a mixed H2/H∞ suboptimal control with regional pole placement and PI structure is employed to control a DFIG-based wind turbine. More concretely, the main contribution consists of the following aspects:(i)
The mode decomposition is used to modelling power system, and the interaction measurement model of DFIG is introduced (where interaction measurement term has been considered as a coordinated signal). An ANN-based weighting controller is proposed to approximate the nonlinear model of DFIG, which achieves a closed-loop nonlinear adaptive approximation.(ii)
The neural observer is proposed to approximate the nonlinear model of DFIG, where the approximation error due to the proposed neural approximation has been considered. A robust stabilization technique is proposed to override the effect of approximation error.(iii)
For improving the fault-tolerant capability, aH∞ controller is employed to cope with the slight faults represented by bounded stochastic disturbances, and a H2 controller with PI structure is also employed to achieve specified engineering purposes. Then, the controller design is formulated as a mixed H2/H∞ suboptimal problem with regional pole placement which is used to further improve damping performance.(iv)
The proposed control strategy combines the merits of conventional PI control, robust stabilization control, and mixedH2/H∞ optimization. Simulation results show that the proposed controller not only improves the MPPT control with fault-tolerant capability bus also enhances system damping and LVRT capability, which greatly improves power system transient stability.The rest part of this paper is arranged as follows. The neural interaction measurement observer of DFIG is proposed in Section2. In Section 3, the mixed H2/H∞ control with regional pole placement based on the obtained interaction measurement model is proposed. In Section 4, simulation results are presented and discussed, which demonstrate the capabilities of the proposed control strategy to enhance MPPT performance under external disturbances and its contribution on power system transient stability support. Finally, the conclusions are drawn in the Section 5.


## 2. Neural Adaptive Interaction Measurement Observer of DFIG

The proposed control strategy shown in Figure1 is comprised of two parts, the neural interaction measurement observer of DFIG and the mixed H2/H∞ controller. The neural interaction measurement observer is established at chosen operating conditions by considering the interactions from network, and a neural weighting controller is proposed to compute the weightings according the approximation error. Based on the obtained observer, the H∞ controller and H2 controller are designed separately for specified engineering purposes. Then, the controller design is formulated as a mixed H2/H∞ suboptimal problem with the constrains of PI controller structure and regional pole placement, and it can be solved easily by using LMI technology.Figure 1
Neural PI control scheme.


### 2.1. DFIG Model with Stochastic Disturbances

For obtaining a good balance between the accuracy and simplification, theith DFIG nonlinear model is chosen as a third-order model [15], where the stator dynamic has been neglected.Dynamic equations:(1)dωridt=12HtotiTmi-TeidEqi′dt=-sωsiEdi′+ωsiLmiLrrivdri-1T0i′Eqi′-Xsi-Xsi′idsidEdi′dt=sωsiEqi′-ωsiLmiLrrivqri-1T0i′Edi′+Xsi-Xsi′iqsi.Output equations:(2)Psi=-Edi′idsi-Eqi′iqsiQsi=Edi′iqsi-Eqi′idsi,where(3)Edi′=-Rsiidsi+Xsi′iqsi+udsiEqi′=-Rsiiqsi-Xsi′idsi+uqsiidsi=-Ei′Bii+∑j=1j≠iNEj′Yijcos⁡δij-αijiqsi=Ei′Gii+∑j=1j≠iNEj′Yijsin⁡δij-αij.Xsi′=ωsLssi-Lmi2/Lrri, δij=δi-δj, αij=90-φij, Ei′ is the internal voltage of the ith generator, δi is the angle between the Ei′ and the x-axis of the synchronous coordinates, φij is the angle of impedance Zij(Zij=(Yij)-1), and N is the number of generators of a multimachine power system.According the above equations, theith DFIG nonlinear model with unmeasurable stochastic disturbances wi and vi can be written as the following compact form:(4)x˙i=fixi+gixiui+dxiEj′,cosδij-αij+wiyi=hixi+dyiEj′,sin⁡δij-αij+vi,where xi=ωriEqi′Edi′T is the state variable, yi=PsiQsiT is the output variable, ui=uqriudriT is the input signal (control vector), and dxi(|Ej′|,cos⁡(δij-αij)) and dyi(|Ej′|,sin⁡(δij-αij)) are the interaction terms from network.


### 2.2. Approximation Error Considered Interaction Measurement Observer

The published literatures [16, 17] extend the classical SG based interaction measurement modelling to DFIG field. The interaction measurement model of DFIG with a certain weighting method can be written as the following form, where the model bank is established at chosen operating points (Table 3) [16]:(5)x˙i=∑k=1nμkAiikxi+Bikui+Twik+Imxikyi=∑k=1nμkCiikxi+Imyik,where Imxik=A2iik∑j=1,j≠inMijxj=A2iikΔIsi-Miixi and Imyik=C2iik∑j=1,j≠inMijxj=C2iikΔIsi-Miixi, Mij denotes the interaction matrix from the ith node to the jth node, μk(k=1,…,n) is the weighting for the kth model in the model bank, and n is the number of model.The termsImxik and Imyik are interaction measurement vectors, which represents interactions from network and can be regarded as coordinated signals. It is seen that Imxik and Imyik only depend on local signals, which allows system-wide state feedback control to be replaced by using local state feedback method.By combining (4)-(5), the approximation error considered interaction measurement model of DFIG can be rewritten as(6)x˙i=∑k=1nμkAiikxi+Bikui+Twik+Imxik+fixi-∑k=1nμkAiikxi+gixi-∑k=1nμkBikui+dxiEj′,δij-Imxik+Twik+wi=∑k=1nμkAiikxi+Bikui+Twik+Imxik+Δfi+Δgi+Δdxi+wiyi=∑k=1nμkCiikxi+Imyik+hixi-∑k=1nhkCiikxi+dyiEj′,δij-Imyik+vi=∑k=1nμkCiikxi+Imyik+Δhi+Δdyi+vi,where(7)Δfi=fixi-∑k=1nμkAiikxi(8)Δgi=gixi-∑k=1nμkBik(9)Δhi=hixi-∑k=1nμkCiikxi(10)Δdxi=dxiEj′,cos⁡δij-αij-∑k=1nμkImxik+Twik(11)Δdyi=dyiEj′,sin⁡δij-αij-∑k=1nμkImyik.In order to cope with the nonlinearity of DFIG, a neural observer is introduced to estimate the state variables of DFIG, where a neural controller is used to compute the weightings according the tracking error.According (5) and (6), the observer can be written as(12)x^˙i=∑k=1nμkAiikx^i+Bikui+Twik+Imxik+Likyi-y^i=∑k=1nμk∑g=1nμgAiikx^i+Bikui+Twik+Imxik+LikCiikei+LikΔhi+LikΔdyi+Likvi,where μk is the output of the ANN, y^i=∑k=1nμkCiikx^ik-Imyik is the observer output, ei=xi-x^i is the state estimation error, and(13)e˙i=x˙i-x^˙i=∑k=1nμk∑g=1nμgAiik-LikCiige+Δfi+Δgi+Δdxi-LikΔhi-LikΔdyi-Likvi+wi.


### 2.3. Neural Adaptive Weighting Controller

The Elman ANN can be described as the following equations [18]:(14)vm=WmUuin+WmCXmCXmH=f1vmXmC=αXm-1Hyout=f2WmOXmH,where WmU, WmC, and WmO are weight matrixes of input layer, context unit, and output layer, respectively, uin and yout are the input and output vectors, respectively, vm and XmH are the input and output vectors of hidden layer, respectively, XmC is the output vector of context unit, f1(•) and f2(•) are activation functions of hidden layer and output layer, and α is the self-feedback gain of context unit.This paper employs an ANN controller shown in Figure1 to approximate the nonlinear model of DFIG according the tracking error xei(m)=yi(m)-y^im, where m denotes the mth interval. The objective of the ANN controller is defined as(15)JNim=12xeiTmQNxeim+12UiTmRNUim,where QNT=QN>0 and RNT=RN>0 are weighting matrixes and Uim=μi,1∗m⋯μi,n∗m is the output vector of the ANN (which is also the weightings represented by vector form).The gradient descent method is employed to minimize the objective shown in (15). Then, the output layer weighting matrix of the ANN controller WO(m) can be updated as follows:(16)WOm=WOm-1-η∇WOmJNm,where η is the learning rate and ∇WOmJNi(m) is the gradient of JNi(m) with respect to WO(m).(17)∂JNim∂WOm=∂JNim∂Uim∂Uim∂WOm=-QNxemCTmx^i.m+I~myim∂Uim∂WOm,where y^im=∑k=1nμi,kmCiikmx^i.m+Imyikm=UimCTmx^i.m+I~myim is the observer output and CTm=Cii1m⋯CiinmT.The term∂Ui(m)/∂WO(m) can be computed by the backpropagation method and no difficulty is involved in it. With a similar approach, the weighting matrixes of input layer and context unit can be updated. Then, the weighting vector Ui (which is also the output of the Elman ANN) can be updated adaptively according the mathematic model of the Elman ANN shown in (14). It is noted that, for obtaining the reasonable weightings, the activation function of the output layer is a sigmoid function fx=1/1+e-x, so that 0<μk∗<1. By normalizing μk∗, the reasonable weightings can be obtained as μk=μk∗/∑g=1nμg∗ and ∑k=1nμk=1.It can be seen that the weightingμk is regulated adaptively according the tracking error via a closed-loop approach. Considering the nonlinearity of ANN, the proposed weighting controller can be regarded as an adaptive nonlinear controller, which provides a desired approximation performance.


## 2.1. DFIG Model with Stochastic Disturbances

For obtaining a good balance between the accuracy and simplification, theith DFIG nonlinear model is chosen as a third-order model [15], where the stator dynamic has been neglected.Dynamic equations:(1)dωridt=12HtotiTmi-TeidEqi′dt=-sωsiEdi′+ωsiLmiLrrivdri-1T0i′Eqi′-Xsi-Xsi′idsidEdi′dt=sωsiEqi′-ωsiLmiLrrivqri-1T0i′Edi′+Xsi-Xsi′iqsi.Output equations:(2)Psi=-Edi′idsi-Eqi′iqsiQsi=Edi′iqsi-Eqi′idsi,where(3)Edi′=-Rsiidsi+Xsi′iqsi+udsiEqi′=-Rsiiqsi-Xsi′idsi+uqsiidsi=-Ei′Bii+∑j=1j≠iNEj′Yijcos⁡δij-αijiqsi=Ei′Gii+∑j=1j≠iNEj′Yijsin⁡δij-αij.Xsi′=ωsLssi-Lmi2/Lrri, δij=δi-δj, αij=90-φij, Ei′ is the internal voltage of the ith generator, δi is the angle between the Ei′ and the x-axis of the synchronous coordinates, φij is the angle of impedance Zij(Zij=(Yij)-1), and N is the number of generators of a multimachine power system.According the above equations, theith DFIG nonlinear model with unmeasurable stochastic disturbances wi and vi can be written as the following compact form:(4)x˙i=fixi+gixiui+dxiEj′,cosδij-αij+wiyi=hixi+dyiEj′,sin⁡δij-αij+vi,where xi=ωriEqi′Edi′T is the state variable, yi=PsiQsiT is the output variable, ui=uqriudriT is the input signal (control vector), and dxi(|Ej′|,cos⁡(δij-αij)) and dyi(|Ej′|,sin⁡(δij-αij)) are the interaction terms from network.


## 2.2. Approximation Error Considered Interaction Measurement Observer

The published literatures [16, 17] extend the classical SG based interaction measurement modelling to DFIG field. The interaction measurement model of DFIG with a certain weighting method can be written as the following form, where the model bank is established at chosen operating points (Table 3) [16]:(5)x˙i=∑k=1nμkAiikxi+Bikui+Twik+Imxikyi=∑k=1nμkCiikxi+Imyik,where Imxik=A2iik∑j=1,j≠inMijxj=A2iikΔIsi-Miixi and Imyik=C2iik∑j=1,j≠inMijxj=C2iikΔIsi-Miixi, Mij denotes the interaction matrix from the ith node to the jth node, μk(k=1,…,n) is the weighting for the kth model in the model bank, and n is the number of model.The termsImxik and Imyik are interaction measurement vectors, which represents interactions from network and can be regarded as coordinated signals. It is seen that Imxik and Imyik only depend on local signals, which allows system-wide state feedback control to be replaced by using local state feedback method.By combining (4)-(5), the approximation error considered interaction measurement model of DFIG can be rewritten as(6)x˙i=∑k=1nμkAiikxi+Bikui+Twik+Imxik+fixi-∑k=1nμkAiikxi+gixi-∑k=1nμkBikui+dxiEj′,δij-Imxik+Twik+wi=∑k=1nμkAiikxi+Bikui+Twik+Imxik+Δfi+Δgi+Δdxi+wiyi=∑k=1nμkCiikxi+Imyik+hixi-∑k=1nhkCiikxi+dyiEj′,δij-Imyik+vi=∑k=1nμkCiikxi+Imyik+Δhi+Δdyi+vi,where(7)Δfi=fixi-∑k=1nμkAiikxi(8)Δgi=gixi-∑k=1nμkBik(9)Δhi=hixi-∑k=1nμkCiikxi(10)Δdxi=dxiEj′,cos⁡δij-αij-∑k=1nμkImxik+Twik(11)Δdyi=dyiEj′,sin⁡δij-αij-∑k=1nμkImyik.In order to cope with the nonlinearity of DFIG, a neural observer is introduced to estimate the state variables of DFIG, where a neural controller is used to compute the weightings according the tracking error.According (5) and (6), the observer can be written as(12)x^˙i=∑k=1nμkAiikx^i+Bikui+Twik+Imxik+Likyi-y^i=∑k=1nμk∑g=1nμgAiikx^i+Bikui+Twik+Imxik+LikCiikei+LikΔhi+LikΔdyi+Likvi,where μk is the output of the ANN, y^i=∑k=1nμkCiikx^ik-Imyik is the observer output, ei=xi-x^i is the state estimation error, and(13)e˙i=x˙i-x^˙i=∑k=1nμk∑g=1nμgAiik-LikCiige+Δfi+Δgi+Δdxi-LikΔhi-LikΔdyi-Likvi+wi.


## 2.3. Neural Adaptive Weighting Controller

The Elman ANN can be described as the following equations [18]:(14)vm=WmUuin+WmCXmCXmH=f1vmXmC=αXm-1Hyout=f2WmOXmH,where WmU, WmC, and WmO are weight matrixes of input layer, context unit, and output layer, respectively, uin and yout are the input and output vectors, respectively, vm and XmH are the input and output vectors of hidden layer, respectively, XmC is the output vector of context unit, f1(•) and f2(•) are activation functions of hidden layer and output layer, and α is the self-feedback gain of context unit.This paper employs an ANN controller shown in Figure1 to approximate the nonlinear model of DFIG according the tracking error xei(m)=yi(m)-y^im, where m denotes the mth interval. The objective of the ANN controller is defined as(15)JNim=12xeiTmQNxeim+12UiTmRNUim,where QNT=QN>0 and RNT=RN>0 are weighting matrixes and Uim=μi,1∗m⋯μi,n∗m is the output vector of the ANN (which is also the weightings represented by vector form).The gradient descent method is employed to minimize the objective shown in (15). Then, the output layer weighting matrix of the ANN controller WO(m) can be updated as follows:(16)WOm=WOm-1-η∇WOmJNm,where η is the learning rate and ∇WOmJNi(m) is the gradient of JNi(m) with respect to WO(m).(17)∂JNim∂WOm=∂JNim∂Uim∂Uim∂WOm=-QNxemCTmx^i.m+I~myim∂Uim∂WOm,where y^im=∑k=1nμi,kmCiikmx^i.m+Imyikm=UimCTmx^i.m+I~myim is the observer output and CTm=Cii1m⋯CiinmT.The term∂Ui(m)/∂WO(m) can be computed by the backpropagation method and no difficulty is involved in it. With a similar approach, the weighting matrixes of input layer and context unit can be updated. Then, the weighting vector Ui (which is also the output of the Elman ANN) can be updated adaptively according the mathematic model of the Elman ANN shown in (14). It is noted that, for obtaining the reasonable weightings, the activation function of the output layer is a sigmoid function fx=1/1+e-x, so that 0<μk∗<1. By normalizing μk∗, the reasonable weightings can be obtained as μk=μk∗/∑g=1nμg∗ and ∑k=1nμk=1.It can be seen that the weightingμk is regulated adaptively according the tracking error via a closed-loop approach. Considering the nonlinearity of ANN, the proposed weighting controller can be regarded as an adaptive nonlinear controller, which provides a desired approximation performance.


## 3. Controller Design

In this paper, the controller of rotor-side converter is chosen as the same structure as the conventional PI controller for taking its natural advantages of tracking control.(18)ui=∑g=1nμgkpigx^i+kiig∫t0tfri-y^idt=∑g=1nμgkpigx^i+kiigxri,where kpi and kii are the respective proportion coefficient and integration coefficient, ri is the set point vector for the ith DFIG, xri=∫t0tfri-yidt is the integral of tracking error, and(19)x˙ri=ri-yi=ri-∑k=1nμikCiikx^i+ei+Imyik-Δhi-Δdyi-vi.By combining (12)–(19), the closed-loop system model can be written as(20)x^˙ie˙ix˙ri=∑k=1nμk∑g=1nμgAiik+BiikkpigLikCiigBikkiig0Aiik-jLikCiig0CiigCiig0x^ieixri+0LikI-Lik0-Iwivi+0Δfi0+0Δgi0+∑k=1nμkLikΔhi-∑k=1nμkLikΔhi-Δhi+00ri+∑k=1nμkImxik0-∑k=1nμkImyik+0Δdxi0+∑k=1nμkLikΔdyi-∑k=1nμkLikΔdyi-Δdyi+∑k=1nμkTwik00.By defining augment state vectorx-i=x^ieixriT, the compact form of (20) is(21)x-˙i=∑k=1nμk∑g=1nμgA-ikgx-i+E-ikw-i+Δf-i+Δg-i+Δh-ii+I-mi+Δd-xi+Δd-yi+T-wi+r-i,where (22)A-ikg=Aiik+BiikkpigLikCiigBikkiig0Aiik-jLikCiig0CiigCiig0,E-ik=0LikI-Lik0-IΔf-i=0Δfi0Δg-i=0Δgi0,r-=00riiΔd-xi=0Δdxi0,Δh-ii=∑k=1nμkLikΔhi-∑k=1nμkLikΔhi-Δhi,I-mi=∑k=1nμkImxik0-∑k=1nμkImyik,Δd-yi=∑k=1nμkLikΔdyi-∑k=1nμkLikΔdyi-Δdyi,T-wi=∑k=1nμkTwik00.Assumption 1.
There exist bounding matrixesΔAi, ΔBi, ΔCi, ΔDi, ΔEi, ΔFxi, and ΔFyi such that(23)Δfi2≤ΔAixi2(24)Δgi2≤∑k=1nμkΔBikpikx^i+kiikxri2(25)Δhi2≤ΔCixi2(26)Δdxi2≤ΔDixi2,(27)Δdyi2≤ΔEixi2,(28)∑k=1nμkImxik2≤∑k=1nμkΔFxikxi2(29)∑k=1nμkImyik2≤∑k=1nμkΔFyikxi2.
Since the parametersT-wi and r-i are limited by the capacity of a DFIG, their upper bounds can be easily determined as(30)∑k=1nμkTwik2≤ΔHixi2ri2≤ΔGixi2,where the details of ΔHi and ΔGi can be found in Appendix A.1.According (23)–(30), we have(31)Δf-iTΔf-i=ΔfiTΔfi≤ΔAixiTΔAixi=ΔAiΔAi0x-iTΔAiΔAi0x-i=φfix-iTφfix-i(32)r-iTr-i≤ΔGixiTΔGixi=ΔGiΔGi0x-iTΔGiΔGi0x-i=φrix-iTφrix-i(33)Δg-iTΔg-i=ΔgiTΔgi≤∑k=1nμkΔBikpikx^i+ΔBikkiikxriT∑k=1nμkΔBikpikx^i+ΔBikkiikxri=∑k=1nμkΔBikpik0ΔBikiikx-iT∑k=1nμkΔBikpik0ΔBikiikx-i≤∑k=1nμkφkgix-iTφkgix-i(34)Δh-iTΔh-i=2∑k=1nμkLikΔhiT∑k=1nμkLikΔhi+ΔhiTΔhi≤2∑k=1nμkΔCixiT∑k=1nμkΔCixi+ΔCixiTΔCixi=2∑k=1nμkLikΔCiLikΔCi0xiT∑k=1nμkLikΔCiLikΔCi0xi+ΔCiΔCi0x-iTΔCiΔCi0x-i≤2∑k=1nμkφh1ikx-iiTφh1ikx-ii+φh2ix-iTφh2ix-i(35)Δd-xiTΔd-xi≤ΔDixiTΔDixi=ΔDiΔDi0x-iTΔDiΔDi0x-i=φdxiX-iTφdxiX-i(36)Δd-yiTΔd-yi=2∑k=1nμkLikΔdyiT∑k=1nμkLikΔdyi+ΔdyiTΔdyi≤2∑k=1nμkΔEixiT∑k=1nμkΔEixi+ΔEixiTΔEixi=2∑k=1nμkLjiΔEiLjiΔEi0x-iT∑k=1nμkLjiΔEiLjiΔEi0x-i+ΔEiΔEi0x-iTΔEiΔEi0x-i≤2∑k=1nμkφdy1ikx-iTφdy1ikx-i+φdy2ix-iTφdy2ix-i(37)T-wiTT-wi=∑k=1nμkTwikT∑k=1nμkTwik≤ΔHixiTΔHixi=ΔHiΔHi0x-iTΔHiΔHi0x-i≤φdix-iTφdix-i(38)ΔI-miTΔI-mi=∑k=1nμkImxikT∑k=1nμkImxik+∑k=1nμkImyikT∑k=1nμkImyik≤∑k=1nμkΔFxikxiT∑k=1nμkΔFxikxi+∑k=1nμkΔFyikxiT∑k=1nμkΔFyikxi=∑k=1nμkΔFxikΔFxik0x-iT∑k=1nμkΔFxikΔFxik0x-i+∑k=1nμkΔFyikΔFyik0x-iT∑k=1nμkΔFyikΔFyik0x-i≤∑k=1nμkφjmxix-iTφjmxix-i+φjmyix-iTφjmyix-i,where φfi=ΔAiΔAi0, φgik=ΔBikpik0ΔBikiik, φh1ik=LikΔCiLikΔCi0, φh2i=ΔCiΔCi0, φri=ΔGiΔGi0, φdxi=ΔDiΔDi0, φdy1ik=LikΔEiLikΔEi0, φdy2i=ΔEiΔEi0, φmxik=ΔFxikΔFxik0, φmyik=ΔFyikΔFyik0, and φdi=ΔHiΔHi0.


### 3.1.H∞ Controller Design

TheH∞ control is the common solution for external disturbance rejection, of which objective can be defined as(39)J∞i=∫t0tfx-iTQ1ix-dt≤x-iTt0P1ix-t0+ρ2∫t0tfw-iTtw-itdt,where ρ is a prescribed attenuation level and weighting matrixes P1iT=P1i>0 and Q1iT=Q1i>0.A Lyapunov function for system of (21) is chosen as following form:(40)Vx-=x-TPx-.By differentiating (40), we obtain(41)V˙=x-˙iTPi1x-i+x-iTPi1x-˙=∑k=1nμk∑g=1nμgA-ikgx-i+E-ikw-i+ΔfghTP1ix-i+x-iTP1i∑k=1nμk∑g=1nμgA-ikgx-i+E-ikw-i+Δfgh=∑k=1nμk∑g=1nμgA-ikgx-iTP1ix-i+x-iTP1i∑k=1nμk∑g=1nμgA-ikgx-i+∑k=1nμkE-ikw-iTP1ix-i+x-iTP1i∑k=1nμkE-ikw-i+ΔfghTP1ix-i+x-iTP1iΔfgh,where Δfgh=Δf-i+Δg-i+Δh-i+r-i+I-mi+Δd-xi+Δd-yi+T-wi.Lemma 2.
Give two vectorsx∈Rn and y∈Rn; the following inequality is identical:(42)xTy+yTx-ξ2xTx-ξ-2yTy=-ξxT-ξ-1yTξxT-ξ-1y≤0ξ≠0.According Lemma2, the following inequalities can be obtained:(43)∑k=1nμkE-ikw-iTP1ix-i+x-iTP1i∑k=1nμkE-ikw-i≤ρ-2x-iTP1i∑k=1nμkE-ik∑k=1nμkE-ikTP1ix-i+ρ2w-iTw-i(44)ΔfghTP1ix-i+x-iTP1iΔfgh=Δf-i+Δg-i+Δh-i+r-i+I-mi+Δd-xi+Δd-yi+T-wiTP1ix-i+x-iTP1iΔf-i+Δg-i+Δh-i+r-i+I-mi+Δd-xi+Δd-yi+T-wi≤8x-iTP1iP1ix-i+Δf-iTΔf-i+Δg-iTΔg-i+Δh-iTΔh-i+r-iTr-i+I-miTI-mi+Δd-xiTΔd-xi+Δd-yiTΔd-yi+T-wiTT-wi.By using (31)–(37), (44) can be rewritten as(45)ΔfghTP1ix-i+x-iTP1iΔfgh≤φfix-iTφfix-i+∑k=1nμkφgikx-iTφgikx-i+2∑k=1nμkφh1ikx-iiTφh1ikx-ii+φh2ix-iTφh2ix-i+φrix-iTφrix-i+φdxix-iTφdxix-i+2∑k=1nμkφdy1ikx-iTφdy1ikx-i+φdy2ix-iTφdy2ix-i+φdix-iTφdix-i+∑k=1nμkφmxikx-iTφmxikx-i+φmyikx-iTφmyikx-i+8x-iTP1iP1ix-i.According (43) and (45), (41) can be rewritten as(46)V˙=x-˙iTPi1x-i+x-iTPi1x-˙≤∑k=1nμk∑g=1nμgx-iTA-ikgTP1i+P1iA-ikg+ρ-2P1iE-ikE-ikT+8IP1i+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyikx-i+ρ2w-iTw-i.Then, the following result can be obtained.Theorem 3.
In the nonlinear augmented system (21), if P1iT=P1i>0 is the common solution for the matrix inequality(47)A-ikgTP1i+P1iA-ikg+ρ-2P1iE-ikE-ikT+8IP1i+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik+Q1i<0for k,g=1,2,…,n, then the performance of the proposed H∞ controller shown in (39) is guaranteed for a prescribed ρ2.Proof.
From (47),(48)A-ikgTP1i+P1iA-ikg+ρ-2P1iE-ikE-ikT+8IP1i+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik<-Q1i.
From (46) and (48), we get(49)V˙=x-˙iTPi1x-i+x-iTPi1x-˙≤∑k=1nμk∑g=1nμgx-iT-Q1ix-i+ρ2w-iTw-i≤x-iT-Q1ix-i+ρ2w-iTw-i.
By integrating (49) from t=t0 to t=tf, we have (50)Vtf-Vt0≤-∫t0tfx-iTQ1ix-dt+ρ2∫t0tfw-iTw-idt.
Then,(51)∫t0tfx-iTQ1ix-dt≤Vt0-Vtf+ρ2∫t0tfw-iTw-idt≤x-iTP1ix-i+ρ2∫t0tfw-iTw-idt.
From (51), it is seen that, under the constrain of (47), the H∞ control performance is achieved with a prescribed ρ2.


### 3.2.H2 Controller Design

TheH2 controller (power regulator and automatic voltage regulator (AVR)) is developed, of which objective can be written as(52)min⁡J2iui=∫t0tfri-y^iTQ~2iri-y^i+uiTRiuidt.Since that the external disturbancesw-i have been efficiently eliminated by the proposed H∞ controller, the H2 controller should be designed without considering w-i. For the approximation errors have been considered, it is hard to obtain the optimal solution of (52). Thus, a suboptimal method is employed to minimize its upper bound.By substituting (18) into (52), we have (53)J2iui=∫t0tfri-y^iTQ~2iri-y^i+∑g=1nμgkpigx^i+kiigxriTRi∑g=1nμgkpigx^i+kiigxridt≤∫t0tfxriTQ~2ixri+∑g=1nμgK-igx-iTRi∑g=1nμgK-igx-i+dtdx-iTP2ix-idt+x-iTt0P2ix-it0-x-iTtfP2ix-itf≤∫t0tfxriTQ~2ixri+∑g=1nμgK-igx-iTRi∑g=1nμgK-igx-i+x-˙iTP2ix-i+x-iTP2ix-˙idt+x-iTt0P2ix-it0,where K-ig=kpik0kiik.From (41) and (45), (53) can be rewritten as(54)J2iui≤∫t0tfx-iTQ2ix-i+∑g=1nμgK-igx-iT·Ri∑g=1nμgK-igx-i+∑k=1nμk∑g=1nμgA-ikgx-iTP1ix-i+x-iTP1i∑k=1nμk∑g=1nμgA-ikgx-i+ΔfghTP1ix-i+x-iTP1iΔfgh∑g=1nμgK-igx-idt+x-iTt0P2ix-it0≤∫t0tf∑k=1nμk∑g=1nμgx-iTQ2i+K-igTRiK-ig+A-ikgTP1i+P1iA-ikg+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik·x-idt+x-iTt0P2ix-it0,if(55)Q2i+K-igTRiK-ig+A-ikgTP1i+P1iA-ikg+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik<0.From (55), the upper bound of the H2 objective is obtained as(56)J2iui≤∫t0tfx-iT-Q2ix-idt+x-iTt0P2ix-it0≤x-iTt0P2ix-it0Therefore, the suboptimalH2 control can be formulated as following minimization problem:(57)minP2ix-iTt0P2ix-it0Subject  toP2i>0and55.


### 3.3. MixedH2/H∞ Control with Regional Pole Placement

Since theH∞ and H2 controllers have been developed separately, the mixed H2/H∞ control is developed to satisfy both suboptimal H2 performance in (56) and H∞ performance in (39). The proposed mixed H2/H∞ controller can be formulated as the following suboptimization problem:(58)minPix-iTt0Pix-it0(59)Subject  toPi=P1i=P2i>0,47and55.In order to further improve DFIG damping performance, the poles of closed-loop system of (21) are placed within the region S(α,r,θ) shown in Figure 2, of which characteristic LMI can be written as following forms [19]:(60)A-ikgPi+PiA-ikgjkT+2αPi<0-rPiA-ikgPiA-ikgPi-rPi<0sin⁡θA-ikgPi+PiA-ikgTcos⁡θA-ikgPi-PiA-ikgTcos⁡θPiA-ikgT-A-ikgPisin⁡θA-ikgPi+PiA-ikgT<0.Figure 2
RegionS(α,r,θ).After solving the mixedH2/H∞ problem shown in (58)–(60), the attenuation level ρ2 can be minimized so that the performance degradation due to w-i is minimized; that is, (61)minPiρ2Subject  to(58)–(60).It should be pointed out that (59) and (60) are not convex, which can not be directly solved by using LMI technology. Fortunately, by using the Schur complement, those inequalities can be transferred into three eigenvalue problems with constrain of LMIs [20], which is convex and can be solved easily by using Matlab LMI toolbox.It is noted here that the stability of the closed-loop system in (21) can be guaranteed by (47) at the equilibrium x-it=0 without considering the disturbance w-i, of which proof is given in Appendix A.2.Then, the mixedH2/H∞ problem shown in (58)–(60) can be solved by using LMI technology. However, it is difficult to give the appropriate values of those bounding matrixes shown in (23)–(29). This paper adopts an iteration processer to obtain the suitable values of bounding matrixes ΔAi, ΔBi, ΔCi, ΔDi, ΔEi, ΔFxi, and ΔFyi [20].Assumption Correction Processer
(a)
Give an initial attenuation levelρ2 and the bounding matrixes, select weighting matrixes Q1i, Q2i, and Ri, and solve the problem in (58) to obtain the observer gain Lik and the controller parameters kpik and kiik.
(b)
Check assumption (23)–(29). If they are not satisfied, expand the bounds for all elements in ΔAi, ΔBi, ΔCi, ΔDi, ΔEi, ΔFxi, and ΔFyi, and then repeat (a)-(b).
(c)
Check positive definiteness ofPi11-1>0, Pi22>0, and Pi33-1>0 (where Pi=blkdiag⁡{Pi11,Pi22,Pi33}). If it is not satisfied, increase ρ and then repeat (b)-(c).
(d)
SubstitutePi and ρ into (59) and (60) to confirm the stability and verify those inequalities.When the appropriate values of the bounding matrixes have been obtained, the neural adaptive observer in (12) and mixed H2/H∞ neural PI controller in (18) can be constructed. It is noted that in the second step of the above iteration method, (26)-(27) involves global variables Ej′ and δij. Before constructing the observer and the controller, the correction of ΔDi and ΔEi should be solved in a decentralized approach.From (10) and (11), we get(62)Δdxi≤dxiEj′max,1-∑k=1nμkImxik+Twik≤ΔDixiΔdyi≤dyiEj′max,1-∑k=1nμkImyik≤ΔEixi.E
j
′
m
a
x is the prescribed value according the normal capacity of the jth generator, and Twi, Imxi, and Imyi can be computed by only using local signals. Thus, the correction of ΔDi and ΔEi of (26) and (27) can be replaced by (62), where the decentralized control is achieved.From the above derivations, it is seen that the neural weighting controller is proposed to cope with the nonlinearity of DFIG, and approximation error caused by neural approximation and parameter uncertainty has been considered and stabilized by a proposed robust controller. Based on the characteristics of power system, several advanced technologies have been integrated smoothly into the proposed neural PI controller, which leads to a multiobjective optimization in comparison with the conventional PI controller which is a signal-objective control.It is seen that the proposed neural PI controller has a similar structure as the conventional PI controller, which takes the natural advantage of conventional PI controller in tracking control. However, the proposed neural PI controller considers the interactions from network which is represented by only using local variables. This means system-wide feedback control can be replaced by only using local variables. Thus, the proposed controller can be regarded as a decentralized coordinated control, which is a desired result for the controller design of large-scale geographically extensive systems.


## 3.1.H∞ Controller Design

TheH∞ control is the common solution for external disturbance rejection, of which objective can be defined as(39)J∞i=∫t0tfx-iTQ1ix-dt≤x-iTt0P1ix-t0+ρ2∫t0tfw-iTtw-itdt,where ρ is a prescribed attenuation level and weighting matrixes P1iT=P1i>0 and Q1iT=Q1i>0.A Lyapunov function for system of (21) is chosen as following form:(40)Vx-=x-TPx-.By differentiating (40), we obtain(41)V˙=x-˙iTPi1x-i+x-iTPi1x-˙=∑k=1nμk∑g=1nμgA-ikgx-i+E-ikw-i+ΔfghTP1ix-i+x-iTP1i∑k=1nμk∑g=1nμgA-ikgx-i+E-ikw-i+Δfgh=∑k=1nμk∑g=1nμgA-ikgx-iTP1ix-i+x-iTP1i∑k=1nμk∑g=1nμgA-ikgx-i+∑k=1nμkE-ikw-iTP1ix-i+x-iTP1i∑k=1nμkE-ikw-i+ΔfghTP1ix-i+x-iTP1iΔfgh,where Δfgh=Δf-i+Δg-i+Δh-i+r-i+I-mi+Δd-xi+Δd-yi+T-wi.Lemma 2.
Give two vectorsx∈Rn and y∈Rn; the following inequality is identical:(42)xTy+yTx-ξ2xTx-ξ-2yTy=-ξxT-ξ-1yTξxT-ξ-1y≤0ξ≠0.According Lemma2, the following inequalities can be obtained:(43)∑k=1nμkE-ikw-iTP1ix-i+x-iTP1i∑k=1nμkE-ikw-i≤ρ-2x-iTP1i∑k=1nμkE-ik∑k=1nμkE-ikTP1ix-i+ρ2w-iTw-i(44)ΔfghTP1ix-i+x-iTP1iΔfgh=Δf-i+Δg-i+Δh-i+r-i+I-mi+Δd-xi+Δd-yi+T-wiTP1ix-i+x-iTP1iΔf-i+Δg-i+Δh-i+r-i+I-mi+Δd-xi+Δd-yi+T-wi≤8x-iTP1iP1ix-i+Δf-iTΔf-i+Δg-iTΔg-i+Δh-iTΔh-i+r-iTr-i+I-miTI-mi+Δd-xiTΔd-xi+Δd-yiTΔd-yi+T-wiTT-wi.By using (31)–(37), (44) can be rewritten as(45)ΔfghTP1ix-i+x-iTP1iΔfgh≤φfix-iTφfix-i+∑k=1nμkφgikx-iTφgikx-i+2∑k=1nμkφh1ikx-iiTφh1ikx-ii+φh2ix-iTφh2ix-i+φrix-iTφrix-i+φdxix-iTφdxix-i+2∑k=1nμkφdy1ikx-iTφdy1ikx-i+φdy2ix-iTφdy2ix-i+φdix-iTφdix-i+∑k=1nμkφmxikx-iTφmxikx-i+φmyikx-iTφmyikx-i+8x-iTP1iP1ix-i.According (43) and (45), (41) can be rewritten as(46)V˙=x-˙iTPi1x-i+x-iTPi1x-˙≤∑k=1nμk∑g=1nμgx-iTA-ikgTP1i+P1iA-ikg+ρ-2P1iE-ikE-ikT+8IP1i+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyikx-i+ρ2w-iTw-i.Then, the following result can be obtained.Theorem 3.
In the nonlinear augmented system (21), if P1iT=P1i>0 is the common solution for the matrix inequality(47)A-ikgTP1i+P1iA-ikg+ρ-2P1iE-ikE-ikT+8IP1i+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik+Q1i<0for k,g=1,2,…,n, then the performance of the proposed H∞ controller shown in (39) is guaranteed for a prescribed ρ2.Proof.
From (47),(48)A-ikgTP1i+P1iA-ikg+ρ-2P1iE-ikE-ikT+8IP1i+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik<-Q1i.
From (46) and (48), we get(49)V˙=x-˙iTPi1x-i+x-iTPi1x-˙≤∑k=1nμk∑g=1nμgx-iT-Q1ix-i+ρ2w-iTw-i≤x-iT-Q1ix-i+ρ2w-iTw-i.
By integrating (49) from t=t0 to t=tf, we have (50)Vtf-Vt0≤-∫t0tfx-iTQ1ix-dt+ρ2∫t0tfw-iTw-idt.
Then,(51)∫t0tfx-iTQ1ix-dt≤Vt0-Vtf+ρ2∫t0tfw-iTw-idt≤x-iTP1ix-i+ρ2∫t0tfw-iTw-idt.
From (51), it is seen that, under the constrain of (47), the H∞ control performance is achieved with a prescribed ρ2.


## 3.2.H2 Controller Design

TheH2 controller (power regulator and automatic voltage regulator (AVR)) is developed, of which objective can be written as(52)min⁡J2iui=∫t0tfri-y^iTQ~2iri-y^i+uiTRiuidt.Since that the external disturbancesw-i have been efficiently eliminated by the proposed H∞ controller, the H2 controller should be designed without considering w-i. For the approximation errors have been considered, it is hard to obtain the optimal solution of (52). Thus, a suboptimal method is employed to minimize its upper bound.By substituting (18) into (52), we have (53)J2iui=∫t0tfri-y^iTQ~2iri-y^i+∑g=1nμgkpigx^i+kiigxriTRi∑g=1nμgkpigx^i+kiigxridt≤∫t0tfxriTQ~2ixri+∑g=1nμgK-igx-iTRi∑g=1nμgK-igx-i+dtdx-iTP2ix-idt+x-iTt0P2ix-it0-x-iTtfP2ix-itf≤∫t0tfxriTQ~2ixri+∑g=1nμgK-igx-iTRi∑g=1nμgK-igx-i+x-˙iTP2ix-i+x-iTP2ix-˙idt+x-iTt0P2ix-it0,where K-ig=kpik0kiik.From (41) and (45), (53) can be rewritten as(54)J2iui≤∫t0tfx-iTQ2ix-i+∑g=1nμgK-igx-iT·Ri∑g=1nμgK-igx-i+∑k=1nμk∑g=1nμgA-ikgx-iTP1ix-i+x-iTP1i∑k=1nμk∑g=1nμgA-ikgx-i+ΔfghTP1ix-i+x-iTP1iΔfgh∑g=1nμgK-igx-idt+x-iTt0P2ix-it0≤∫t0tf∑k=1nμk∑g=1nμgx-iTQ2i+K-igTRiK-ig+A-ikgTP1i+P1iA-ikg+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik·x-idt+x-iTt0P2ix-it0,if(55)Q2i+K-igTRiK-ig+A-ikgTP1i+P1iA-ikg+φfiTφfi+φgikTφgik+2φh1ikTφh1ik+φh2iTφh2i+φriTφri+φdxiTφdxi+2φdy1ikTφdy1ik+φdy2iTφdy2i+φdiTφdi+φmxikTφmxik+φmyikTφmyik<0.From (55), the upper bound of the H2 objective is obtained as(56)J2iui≤∫t0tfx-iT-Q2ix-idt+x-iTt0P2ix-it0≤x-iTt0P2ix-it0Therefore, the suboptimalH2 control can be formulated as following minimization problem:(57)minP2ix-iTt0P2ix-it0Subject  toP2i>0and55.


## 3.3. MixedH2/H∞ Control with Regional Pole Placement

Since theH∞ and H2 controllers have been developed separately, the mixed H2/H∞ control is developed to satisfy both suboptimal H2 performance in (56) and H∞ performance in (39). The proposed mixed H2/H∞ controller can be formulated as the following suboptimization problem:(58)minPix-iTt0Pix-it0(59)Subject  toPi=P1i=P2i>0,47and55.In order to further improve DFIG damping performance, the poles of closed-loop system of (21) are placed within the region S(α,r,θ) shown in Figure 2, of which characteristic LMI can be written as following forms [19]:(60)A-ikgPi+PiA-ikgjkT+2αPi<0-rPiA-ikgPiA-ikgPi-rPi<0sin⁡θA-ikgPi+PiA-ikgTcos⁡θA-ikgPi-PiA-ikgTcos⁡θPiA-ikgT-A-ikgPisin⁡θA-ikgPi+PiA-ikgT<0.Figure 2
RegionS(α,r,θ).After solving the mixedH2/H∞ problem shown in (58)–(60), the attenuation level ρ2 can be minimized so that the performance degradation due to w-i is minimized; that is, (61)minPiρ2Subject  to(58)–(60).It should be pointed out that (59) and (60) are not convex, which can not be directly solved by using LMI technology. Fortunately, by using the Schur complement, those inequalities can be transferred into three eigenvalue problems with constrain of LMIs [20], which is convex and can be solved easily by using Matlab LMI toolbox.It is noted here that the stability of the closed-loop system in (21) can be guaranteed by (47) at the equilibrium x-it=0 without considering the disturbance w-i, of which proof is given in Appendix A.2.Then, the mixedH2/H∞ problem shown in (58)–(60) can be solved by using LMI technology. However, it is difficult to give the appropriate values of those bounding matrixes shown in (23)–(29). This paper adopts an iteration processer to obtain the suitable values of bounding matrixes ΔAi, ΔBi, ΔCi, ΔDi, ΔEi, ΔFxi, and ΔFyi [20].Assumption Correction Processer
(a)
Give an initial attenuation levelρ2 and the bounding matrixes, select weighting matrixes Q1i, Q2i, and Ri, and solve the problem in (58) to obtain the observer gain Lik and the controller parameters kpik and kiik.
(b)
Check assumption (23)–(29). If they are not satisfied, expand the bounds for all elements in ΔAi, ΔBi, ΔCi, ΔDi, ΔEi, ΔFxi, and ΔFyi, and then repeat (a)-(b).
(c)
Check positive definiteness ofPi11-1>0, Pi22>0, and Pi33-1>0 (where Pi=blkdiag⁡{Pi11,Pi22,Pi33}). If it is not satisfied, increase ρ and then repeat (b)-(c).
(d)
SubstitutePi and ρ into (59) and (60) to confirm the stability and verify those inequalities.When the appropriate values of the bounding matrixes have been obtained, the neural adaptive observer in (12) and mixed H2/H∞ neural PI controller in (18) can be constructed. It is noted that in the second step of the above iteration method, (26)-(27) involves global variables Ej′ and δij. Before constructing the observer and the controller, the correction of ΔDi and ΔEi should be solved in a decentralized approach.From (10) and (11), we get(62)Δdxi≤dxiEj′max,1-∑k=1nμkImxik+Twik≤ΔDixiΔdyi≤dyiEj′max,1-∑k=1nμkImyik≤ΔEixi.E
j
′
m
a
x is the prescribed value according the normal capacity of the jth generator, and Twi, Imxi, and Imyi can be computed by only using local signals. Thus, the correction of ΔDi and ΔEi of (26) and (27) can be replaced by (62), where the decentralized control is achieved.From the above derivations, it is seen that the neural weighting controller is proposed to cope with the nonlinearity of DFIG, and approximation error caused by neural approximation and parameter uncertainty has been considered and stabilized by a proposed robust controller. Based on the characteristics of power system, several advanced technologies have been integrated smoothly into the proposed neural PI controller, which leads to a multiobjective optimization in comparison with the conventional PI controller which is a signal-objective control.It is seen that the proposed neural PI controller has a similar structure as the conventional PI controller, which takes the natural advantage of conventional PI controller in tracking control. However, the proposed neural PI controller considers the interactions from network which is represented by only using local variables. This means system-wide feedback control can be replaced by only using local variables. Thus, the proposed controller can be regarded as a decentralized coordinated control, which is a desired result for the controller design of large-scale geographically extensive systems.


## 4. Simulations

For assessing the performance of the proposed controller, a multimachine power system shown in Figure3 is modelled in Matlab/Simulink, and the parameters of DFIG are given in Tables 1 and 2. The power system model consists of two fields, the load center comprised by two SGs, and the remote terminal comprised of two DFIG-based WFs. Those two fields are connected by the transmission line L5 with a long distance of 200 km to investigate the proposed controller capabilities in a weak power system, which is difficult to guarantee the LVRT capability of DFIG, especially under sensor fault case. In order to restore the terminal voltage of WFs, a Var compensator (VC) is connected to the common coupling point (CCP) of WFs Bus B3-2.Table 1
Parameter of DFIG (per unit:Sbase=10 WM; Vbase=575 V).
Parameter
R
s
L
s
R
r
L
r
L
m
H
t
o
t
Converter capacity
Value in pu
0.007
0.171
0.005
0.156
2.9
5.04
50%Table 2
Parameter of the conventional PI controller [14].
Parameter
Power regulator
Voltage regulator
Rotor current regulators
Proportion coefficient
k
p
p
=
1
k
v
p
=
1.25
k
i
d
p
=
k
i
q
p
=
0.3
Integration coefficient
k
p
i
=
100
k
v
i
=
300
k
i
d
i
=
k
i
q
i
=
7Table 3
Operating points chosen for establishing the model bank (motor convention).
Operating point
u
d
s
u
q
s
i
d
s
i
q
s
v
d
r
v
q
r
P
e
Q
e
ω
r
=
0.80
−1.0
0.0
0.3
0.34
−0.32
−0.03
−0.2
0.36
ω
r
=
0.92
0.1
−1.0
−0.4
0.4
0.03
−0.1
−0.4
0.36
ω
r
=
0.97
−0.39
0.93
0.51
−0.3
−0.02
0.03
−0.46
0.36
ω
r
=
1.03
1.0
−0.14
−0.64
−0.28
−0.04
−0.003
−0.62
0.37
ω
r
=
1.12
−1.0
−0.19
0.57
0.48
0.09
0.04
−0.71
0.37
ω
r
=
1.20
−0.45
0.9
0.65
−0.49
0.13
−0.15
−0.89
0.36Figure 3
Multimachine power system model for assessing the performance of the proposed controller.In this section, the capabilities of the proposed control strategy are assessed under small disturbance and large disturbance, respectively. The small disturbance is identified as a slight sensor fault, which is mimicked by a bounded stochastic disturbance shown in Figure4. The large disturbance is the slight sensor fault plus three-phase ground faults. For comparison purpose, the responses with the conventional PI (CPI) controller is also presented and discussed. In order to simplify the introduction, the proposed control strategy is identified as the neural PI (NPI) controller.Figure 4
Stochastic disturbancew-i.


### 4.1. Responses to Small Disturbances

In this subsection, a slight sensor fault represented by a stochastic bounded disturbance shown in Figure4 is applied in the sensor of active power. The WF1 responses with the proposed neural PI controller and the conventional PI controller are shown in Figure 5.Figure 5
MPPT performances with sensor fault: (a) NPI controller; (b) CPI controller.
(a)
(b)It can be seen that the MPPT performance with the CPI controller is reduced drastically under the disturbancew-i. (Pe of Figure 5(b)), which causes the dc-link voltage vdc oscillated seriously (vdc of Figure 5(b)). However, the MPPT performance under the same sensor fault is still acceptable when the NPI controller is installed. It can be seen that the effect of the disturbance w-i has been efficiently suppressed by the H∞ controller, and the oscillation bound of Pe with the NPI controller is narrow (Pe of Figure 5(a)), which helps to smooth the dc-link voltage (vdc of Figure 5(a)). It can be seen that the oscillation of vdc is very small when the NPI controller is used.In order to show this difference directly, the integral of absolute error (IAE) defined asIAE=∫t0tfPeref-Pedt is used to evaluate the MPPT performances with different controllers. The IAE is 251 when the NPI controller is used. However, the value with the CPI controller is 526, which is two times of that with the NPI controller.It is concluded that the MPPT performance under the slight sensor fault has been considerably improved by the NPI controller, which is valuable for WFs installed in the remote regions without timely maintenance.


### 4.2. Responses to Large Disturbances

In this subsection, a three-phase ground fault is applied in the middle of line L1 att=0+, and it is cleared after 0.1 s. The responses of WF1 with the NPI and CPI controllers are shown in Figure 6(a). For illustrating the contribution to network supports, the response of Bus B1-2 which is the CCP of SGs is also presented in Figure 6(b).Figure 6
Fault responses ats1=0.2 pu: NPI controller (full line); CPI controller (dashed line).
(a)
(b)It is seen that both the NPI controller and CPI controller can provide acceptable damping performance; however the NPI controller is better (Pe of Figure 6(a)). Since the NPI controller has achieved an effective control of internal voltage vector by reducing its angle jump, the terminal voltage drop is smaller in comparison with the CPI controller used. The smaller terminal voltage drop makes DFIG output more active power in the faults. Thus, less active power is accumulated in the dc-link, which reduces the peak value of dc-link voltage (vdc of Figure 6(a)). Terminal under voltage and dc-link over voltage are regarded as two major reasons to limit LVRT capability of DFIG, which have been considerably improved by the NPI controller.Figure6(b) shows the contribution of the NPI controller to network supports, such as improved system damping (Pe of Figure 6(b)), better terminal voltage recovery capability (vs of Figure 6(b)), and system frequency support (Hz of Figure 6(b)).


### 4.3. Contribution on Transient Stability

A three-phase ground fault is applied in the terminal of transformer T3 att=0+, and it is cleared after 0.1 s. The fault is closer to the WF1, which means that the disturbance is more serious. The system responses are shown in Figure 7.Figure 7
Fault responses ats1=-0.122 pu: NPI controller (full line); CPI controller (dashed line).
(a)
(b)
(c)It is seen that, under such a large disturbance, the WF1 with the CPI controller is tripped att=0.1084 s for terminal under voltage (|vs| of Figure 7(a)) and its output active power drops to zero at the same time (Pe of Figure 7(a)). The trip of WF1 leads to surplus reactive power, which rises the terminal voltage of WF2 and triggers the terminal over voltage protection to trip the WF2 at t=0.2481 (Pe of Figure 7(b)). The trip of WFs makes power system lose large-scale active power in a very short time. Since the large inertial of thermal power plant, the SGs are not capable of generating the corresponding active power immediately, which drops the rotor speed of SG1 ωr from 1 pu to 0.9954 pu (ωr of Figure 7(c)) very quickly. This may cause the SG1 to be tripped and leads to imbalance of active power, which leads to frequency collapse and further worsens power system transient stability.It is seen that, as opposed to the CPI controller, the NPI controller ensures that WF1 can be connected to the grid with acceptable rotor speed (ωr of Figure 7(a)), which provides DFIG with continuing network support capability to balance the active power and reactive power. Thus, the power system frequency can be operated in a permissible range. It is noticeable that the NPI controller provides the system with good terminal voltage recovery capability (vs of Figure 7).Figures6 and 7 show the capabilities of the NPI controller to improve system damping, MPPT, LVRT, and its contribution to network support at both subsynchronous and supper synchronous conditions.


## 4.1. Responses to Small Disturbances

In this subsection, a slight sensor fault represented by a stochastic bounded disturbance shown in Figure4 is applied in the sensor of active power. The WF1 responses with the proposed neural PI controller and the conventional PI controller are shown in Figure 5.Figure 5
MPPT performances with sensor fault: (a) NPI controller; (b) CPI controller.
(a)
(b)It can be seen that the MPPT performance with the CPI controller is reduced drastically under the disturbancew-i. (Pe of Figure 5(b)), which causes the dc-link voltage vdc oscillated seriously (vdc of Figure 5(b)). However, the MPPT performance under the same sensor fault is still acceptable when the NPI controller is installed. It can be seen that the effect of the disturbance w-i has been efficiently suppressed by the H∞ controller, and the oscillation bound of Pe with the NPI controller is narrow (Pe of Figure 5(a)), which helps to smooth the dc-link voltage (vdc of Figure 5(a)). It can be seen that the oscillation of vdc is very small when the NPI controller is used.In order to show this difference directly, the integral of absolute error (IAE) defined asIAE=∫t0tfPeref-Pedt is used to evaluate the MPPT performances with different controllers. The IAE is 251 when the NPI controller is used. However, the value with the CPI controller is 526, which is two times of that with the NPI controller.It is concluded that the MPPT performance under the slight sensor fault has been considerably improved by the NPI controller, which is valuable for WFs installed in the remote regions without timely maintenance.


## 4.2. Responses to Large Disturbances

In this subsection, a three-phase ground fault is applied in the middle of line L1 att=0+, and it is cleared after 0.1 s. The responses of WF1 with the NPI and CPI controllers are shown in Figure 6(a). For illustrating the contribution to network supports, the response of Bus B1-2 which is the CCP of SGs is also presented in Figure 6(b).Figure 6
Fault responses ats1=0.2 pu: NPI controller (full line); CPI controller (dashed line).
(a)
(b)It is seen that both the NPI controller and CPI controller can provide acceptable damping performance; however the NPI controller is better (Pe of Figure 6(a)). Since the NPI controller has achieved an effective control of internal voltage vector by reducing its angle jump, the terminal voltage drop is smaller in comparison with the CPI controller used. The smaller terminal voltage drop makes DFIG output more active power in the faults. Thus, less active power is accumulated in the dc-link, which reduces the peak value of dc-link voltage (vdc of Figure 6(a)). Terminal under voltage and dc-link over voltage are regarded as two major reasons to limit LVRT capability of DFIG, which have been considerably improved by the NPI controller.Figure6(b) shows the contribution of the NPI controller to network supports, such as improved system damping (Pe of Figure 6(b)), better terminal voltage recovery capability (vs of Figure 6(b)), and system frequency support (Hz of Figure 6(b)).


## 4.3. Contribution on Transient Stability

A three-phase ground fault is applied in the terminal of transformer T3 att=0+, and it is cleared after 0.1 s. The fault is closer to the WF1, which means that the disturbance is more serious. The system responses are shown in Figure 7.Figure 7
Fault responses ats1=-0.122 pu: NPI controller (full line); CPI controller (dashed line).
(a)
(b)
(c)It is seen that, under such a large disturbance, the WF1 with the CPI controller is tripped att=0.1084 s for terminal under voltage (|vs| of Figure 7(a)) and its output active power drops to zero at the same time (Pe of Figure 7(a)). The trip of WF1 leads to surplus reactive power, which rises the terminal voltage of WF2 and triggers the terminal over voltage protection to trip the WF2 at t=0.2481 (Pe of Figure 7(b)). The trip of WFs makes power system lose large-scale active power in a very short time. Since the large inertial of thermal power plant, the SGs are not capable of generating the corresponding active power immediately, which drops the rotor speed of SG1 ωr from 1 pu to 0.9954 pu (ωr of Figure 7(c)) very quickly. This may cause the SG1 to be tripped and leads to imbalance of active power, which leads to frequency collapse and further worsens power system transient stability.It is seen that, as opposed to the CPI controller, the NPI controller ensures that WF1 can be connected to the grid with acceptable rotor speed (ωr of Figure 7(a)), which provides DFIG with continuing network support capability to balance the active power and reactive power. Thus, the power system frequency can be operated in a permissible range. It is noticeable that the NPI controller provides the system with good terminal voltage recovery capability (vs of Figure 7).Figures6 and 7 show the capabilities of the NPI controller to improve system damping, MPPT, LVRT, and its contribution to network support at both subsynchronous and supper synchronous conditions.


## 5. Conclusions

This paper proposes an adaptive neural decentralized coordinated control of DFIG, where a neural interaction measurement observer is proposed to approximate the nonlinear model of DFIG. The approximation error due to neural approximation has been considered, and a robust stabilization technique is also proposed to override the effect of the approximation error. For considering the slight sensor fault represented by stochastic disturbance, theH∞ controller is employed to suppress the fault effect. The H2 controller is also employed to achieve specified engineering purposes. Then, the proposed controller can be formulated as a mixed H2/H∞ optimization problem with constrains of PI structure and regional pole placement, which can be solved by using LMI technique. Simulation results are presented and discussed, which demonstrate the capabilities of the proposed control strategy to system damping, voltage recovery and LVRT, and its contributions on power system transient stability, especially for frequency support.This paper demonstrates that, in comparison with the conventional PI controller, the proposed control strategy provides DFIG with the greater capabilities of fault-tolerant control of MPPT and continuing network support during power system fault conditions.



---

*Source: 6271341-2017-10-10.xml*