# Adaptive Dynamic Surface Integral Sliding Mode Fault-Tolerant Control for Multimachine Excitation Systems with SVC

Authors: Guoqiang Zhu; Shuang Ji; Zhiwei Li; Yilong Zhang
Journal: Complexity
 (2020)

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## Abstract

In this paper, an adaptive dynamic surface integral sliding mode fault-tolerant controller is designed for the multimachine power system with static var compensator (SVC) to overcome the problem of actuator failure. The main features of the proposed method are as follows: (1) By combining the dynamic surface control (DSC) method with integral sliding mode (ISM), the tracking errors of the system converge to the neighborhood of zero within a finite time, and the convergence speed, tracking accuracy, and anti-interference ability of the system are also significantly improved. (2) By introducing the failure factors, an adaptive fault-tolerant controller is designed to ensure the stability of the entire system after partial failure of the actuator. (3) By estimating the norm of the ideal weight vector of the radial basis function neural networks (RBFNNs), the computational burden of the controller is reduced. Finally, the simulation results show the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

The modern power system has the characteristics of large power grid, large unit, ultrahigh voltage, long distance power transmission, and high automatic control, which brings about great challenges to the stability operation of the power system [1–4]. The important role of generator excitation control in improving the stable operation of the power system has attracted widespread attention from researchers worldwide [5–8]. The traditional power system stabilizer (PSS) [9, 10] and linear optimal excitation control [11] are based on the approximate linearization model. Although the small disturbance stability problem can be improved, the effective suppression of large disturbances cannot be achieved. Moreover, due to parameter uncertainty, external interference, multidimensional and strong coupling, and strong nonlinear characteristics, designing a control system is a very challenging task. Therefore, advanced control strategies such as backstepping [12–15], sliding mode variable structure [16–18], and feedback linearization [19] have been proposed to solve the problem of power system excitation control.The adaptive backstepping technique can effectively handle the case where the parameters are unknown [20–22]. In [20], the infinite bus voltage and transmission line parameters of the single-machine infinite-bus (SMIB) power system are regarded as unknown, and an adaptive backstepping controller is proposed. A robust nonlinear adaptive backstepping excitation controller is designed in [21] by considering the influence of external disturbance based on the parameter uncertainty. However, the problem of “explosion of complexity” exists in the backstepping method. In order to solve this problem, the dynamic surface control method is used [23–27]. In [24], for a class of uncertain nonlinear systems with asymmetric dead zone nonlinearity, an antidisturbance dynamic surface control scheme is proposed, which effectively solves the problems of nonmatching external disturbance and unknown asymmetric dead zone nonlinearity. A first-order filter is introduced on the traditional backstepping method for the power system, and the controller based on the dynamic surface method is designed in [26]. The device effectively improves the stability of the power system. For large-scale multimachine systems with static var compensator (SVC), the decentralized dynamic surface quantization control scheme is proposed in [4] and the fuzzy dynamic surface sliding mode controller is designed in [7], which improves the robustness of the system. The partial feedback linearization (PFBL) technique is widely used in the design of excitation control systems by overcoming the limitations of direct feedback linearization (DFBL) and exact feedback linearization (EFBL) on rotor angle measurement problem. In [28], the nonlinear system model is transformed into a reduced-order linear part and a nonlinear dynamic autonomous part. The optimal control theory is used to design a linear state feedback stability controller for the reduced-order linear part, which enhances the transient stability limit of the power system.However, the feedback linearized excitation controller has a parameter sensitivity problem. Sliding mode control (SMC) overcomes the parameter sensitivity problem due to being less sensitive to changes in parameters and external disturbances [29–34]. In [35], for the multimachine power system with external disturbance, the adaptive method and the sliding mode variable structure method are combined based on the feedback linearization to design a decentralized coordinated adaptive sliding mode stabilizer. The adaptive gain of the super-twist algorithm is constructed by using the equivalent control theory, and the finite time stability of the closed-loop power system is realized in [33]. In [36], by combining adaptive fuzzy control with sliding mode control, the chattering problem caused by system uncertainty and dynamics is overcome, and the tracking error is converged to zero better and faster. In addition, intelligent controllers [37–43] are also widely used. The fuzzy logic rules [40] and the neural networks [44, 45] are utilized to design adaptive controller for nonlinear system with parameters uncertainty and external disturbances; the robustness and anti-interference ability of control system are improved.The static var compensator (SVC) control is also one of the effective and economical means of improving the stability of power systems. Generally, the generator excitation controller and the SVC controller are designed independently of each other, but the uncoordinated control of the SVC and the excitation will have a negative effect and even lead to system instability. The research on the excitation control problem of SMIB power system with SVC has achieved remarkable results. In [46], the sliding mode dynamic surface method and disturbance attenuation technique are adopted. Considering the external disturbance and parameter uncertainty, a nonlinear adaptive robust coordination controller for SVC and generator excitation is designed. The energy-based coordinated stability controller is constructed by using the Hamiltonian function method in [47, 48], which effectively improves the transient stability and voltage regulation performance of the power systems. Immersion and invariant (I&I) control method is widely used in the coordinated control of SVC and excitation. An I&I adaptive control method is adopted to ensure that all single-machine infinite-bus power system states are globally bounded and converge to a new stable equilibrium [3, 49]. However, in the actual operation of the power grid and power generation system, the system is more complicated and cannot be simply equivalent to a SMIB. Therefore, the excitation control of multimachine infinite-bus power systems with SVC has become the research emphasis of this paper.Inspired by the above research works, the objective of this paper is to design a new adaptive controller for generators with SVC to ensure the stability of the system and the performance of the transient power system when the system has parameter uncertainties and actuator failures. The controller design process includes two partial steps. First, the dynamic surface is combined with the integral sliding mode to design the fault-tolerant controller of the generator excitation controller. Second, the SVC controller is designed. The main contributions of the proposed method are summarized as follows: (1) By introducing integral sliding mode surface in the dynamic surface control process, the problem of “differential explosion” in the traditional backstepping method is solved, and the structure of the controller is simplified. The introduction of the integral sliding mode surface makes the tracking error of the system converge to zero within a finite time, thereby ensuring the convergence speed and tracking accuracy of the closed-loop system, and the anti-interference ability of the system is also improved. (2) By introducing the actuator failure factor, an adaptive fault-tolerant controller is designed to solve the problem of actuator failure and ensure that the system can still be stable after partial actuator failure. (3) By adopting the minimum parameter learning method, the norm of ideal weight vector of radial basis function neural networks (RBFNNs) is estimated online, the “dimension disaster” problem caused by the traditional neural network due to the estimated weight vector is avoided, and the calculation burden of the controller is reduced.The rest of this paper is organized as follows. In Section2, the mathematical model of the multimachine infinity-bus power systems with SVC is described. Coordination controller for generator excitation and SVC is designed in Section 3. The stability analysis is presented in Section 4. The effectiveness of the proposed scheme is illustrated by simulation in Section 5. The conclusion is drawn in Section 6.


## 2. System Dynamic Models and Problem Statement

### 2.1. Mathematical Models

The dynamic model of multimachine infinite-bus power systems with SVC is given in this section, in which the dynamics of theith generator are described by a third-order differential equation, and the SVC is described by a first-order inertia equation. Then, the mathematical models of power system can be described as follows [3, 5]:(1)δ˙it=ωi,ω˙it=−Di2Hiωi+ωi02HiPmi−Pei,E˙qi′t=1Td0i′Efit−Eqit,with a SVC model(2)B˙Li=1TCi−BLi+BCi+uBi,and the electrical equations are as follows:(3)Eqit=Eqi′t+xdi−xdi′Idit,(4)Efit=keiufit,(5)Peit=∑j=1nEqi′tEqj′tBijsinδi−δj,(6)Qeit=−∑j=1nEqi′tEqj′tBijcosδi−δj,(7)Idit=−∑j=1nEqj′tBijcosδi−δj,(8)Iqit=∑j=1nEqj′tBijsinδi−δj,where the explanation of symbols used in the dynamic model (1)–(8) is given in Table 1.Table 1
The notation for multimachine power systems with SVC.
SymbolNomenclatureδiThe power angle of theith generator, in radωiThe relative speed of theithgenerator, in rad/sωi0The synchronous machine speed, in rad/sDiThe per unit damping constantHiThe inertia constant, in sPmiThe mechanical input power, in p.u.PeiThe electrical power, in p.u.Eqi′The transient EMF in the quadrature axis, in p.u.EfiThe equivalent EMF in the excitation coil, in p.uEqiThe EMF in the quadrature axis, in p.u.xdiThe direct axis reactance, in p.u.xdi′The direct axis transient reactance, in p.u.IdiThe direct axis current, in p.ukeiThe gain of the excitation amplifier, in p.u.ufiThe input of the SCR amplifier, in p.u.QeiThe reactive power, in p.u.IqiThe quadrature axis current, in p.u.BLiThe adjustable equivalent susceptance in SVCBCiThe initial value of adjustable susceptanceuBiThe input of SVC, in p.u.TCiThe time constant of adjusting system and SVCTd0i′The direct axis transient short-circuit time constant, in sBijTheith row and jth column element of nodal susceptance matrix at the internal nodes after eliminating all physical buses, in p.u.Remark 1.
Generally, in order to accurately describe a large-scale multimachine power system, a high-order complex nonlinear dynamic equation is required [28], which makes it very difficult to analyze the stability of the power system and design the excitation controller [3, 5]. In order to overcome such difficulties, a third-order power system model simplified by the integral manifold method is proposed in [5, 50]. This model makes the control system design and stability analysis feasible; and it has been widely used in the design of excitation controllers for multimachine power system [7, 26, 51–53]. The third-order power system mode and the first-order static reactive power compensator model adopted in this paper are derived from [3, 5].
In order to reduce the complexity of the controller design process in Section3, Eqi′t in the generator electrical dynamics is eliminated by differentiating Peit in (5):(9)P˙eit=1Td0i′Efit−xdi−xdi′IditIdit−Peit−Qeitωi+γiδ,ω.
LetΔPei=Pei−Pmi; we have(10)ΔP˙eit=P˙eit=−1Td0i′ΔPeit+1Td0i′ui+γiδ,ω.
The multimachine power systems model can be transformed into the following mathematical model [1, 5]:(11)δ˙it=ωi,ω˙it=−Di2Hiωi+ωi02HiPmi−Pei+di1,ΔP˙eit=−1Td0i′ΔPeit+1Td0i′ui+γiδ,ω+di2,where di1 and di2 are bounded uncertainties, including modeling errors, measurement errors, and external disturbances. ui is the control signal of the generator with(12)ui=EfitIqit−xdi−xdi′IditIqit−Pmi−Td0i′Qeitωi.
By considering the failure fault of the actuator at timetf, the generator excitation voltage Efit can be defined as(13)Efit=kei1−βiufi,where βi is the failure factor. So(14)ui′=EfitIqit−xdi−xdi′IditIqit−Pmi−Td0i′Qeitωi−βiEfitIqit.
So the multimachine power systems model by considering the failure fault can be described as follows:(15)δ˙it=ωi,ω˙it=−Di2Hiωi+ωi02HiPmi−Pei+di1,ΔP˙eit=−1Td0i′ΔPeit+1Td0i′ui+γiδ,ω+di2+Fi,where(16)Fi=−βi1Td0i′EfiIqi,γiδ,ω=∑j=1nEqi′tE˙qj′tBijsinδi−δj−∑j=1nEqi′tEqj′tBijcosδi−δjωj.
Here, the bound of the interconnection termγiδ,ω satisfies(17)γiδ,ω≤∑j=1nγi1jsinδj+γi2ωj≤∑j=1nγi1jδj+γi2ωj,where(18)γi1j≜4p1ijTd0j′minPeimax,whenj≠i,∑j=1,j≠in4p1ijTd0j′minPeimax,whenj=i,γi2≜p2ijQeimax,with p1ij and p2ij being the constants with values of either 1 or 0. Peit and Qeit are readily measurable variables. From (5)–(8), we can obtain that(19)Pei=Eqi′Iqi,Qei=−Eqi′Idi.
Define the following state variables for coordinate transformation:(20)xi1=δi−δi0,xi2=ωi−ωi0,xi3=ΔPei=Pei−Pmi,xi4=Vmi−Vrefi,where Pmi=Pmi0 is a constant and(21)Vmi=X2iEqi′2+X1i2+2X1iX2iEqi′cosxi1Xd∑i′,where Vmi is the accessing point voltage and Vrefi is the corresponding reference voltage of the SVC:(22)X1i=xdi′+XTi,Xd∑i′=X1i+X2i+X1iX2iBLi−BCi,where X2i and XTi are the transmission line reactance and the transformer reactance, respectively. According to (2) and (15) and (20), we can obtain the following mathematical model of multimachine power systems with SVC:(23)x˙i1=xi2,x˙i2=fi2x¯i2−gi2xi3+di1,x˙i3=fi3x¯i3+gi3ui+γiδ,ω+di2+Fi,yi1=xi1,(24)x˙i4=gi4uBi′+fi4x¯i4,yi2=xi4,where yi1 is the output of multimachine power systems, yi2 is the output of the SVC, and(25)fi2x¯i2=−Di2Hixi2,fi3x¯i3=−1Td0i′xi3,fi4x¯i4=−sinxi1X1iX2ixi4+VrefiXd∑i′2xi2Eqi′−X1iX2ixi4+VrefiXd∑i′TCi−BLi+BCi+X2i2Eqi′+X1iX2icosxi1xi4+VrefiXd∑i′2−Xd∑iTd0iXd∑i′Eqi′+1Td0iXdi−Xdi′Xd∑i′cosxi1+X2i2xi3+X1iX2iVsicosδiTd0iVmiXd∑i′2ui,gi2=ωi02Hi,gi3=1Td0i′,gi4=X1iX2iTCiXd∑i′,uBi′=−xi4uBi,with(26)Xd∑i=X3i+X2i+X3iX2iBLi−BCi,X3i=xdi+XTi.
Since the damping coefficientDi is difficult to measure accurately, Di is treated as an uncertainty parameter, and the nonlinear system with uncertain parameters (1) is transformed by equation (20) into a feedback form as shown in equation (23), so that the dynamic surface integral sliding mode method can be used to design the controller of the system equation (23). We can make the following two assumptions in this paper.Assumption 1.
The signs ofgij,i=1,2,j=2,3,4, are unknown bounded positive parameters, and there are known positive constants gmax and gmin such that gmax≥gij≥gmin; fi2x¯i2, fi3x¯i3, and fi4x¯i4 are unknown continuous functions.Assumption 2.
The reference signalxi1d is bounded and smooth function, its first and second derivatives both exist, and there is a positive real number Bi0 that satisfies xi1d2+x˙i1d2+x¨i1d2≤Bi0.Remark 2.
Sincegij,i=1,2,j=2,3,4, are unknown bounded positive parameters and the reference signal xi1d is a bounded smoothing function, Assumptions 1 and 2 are reasonable and common assumptions in [24, 26].


### 2.2. Radial Basis Function Neural Networks (RBFNNs)

The unknown nonlinear continuous functions are approximated by the RBFNNs in this study. The general form of RBFNNs can be expressed as(27)yξ=WTεξ,where ξ∈Rn is the input of RBFNNs, yξ∈R is the output of RBFNNs, W∈RN is the adjustable weight vector, and εξ∈RN is the nonlinear vector function.In general, a continuous nonlinear functionF:Ωξ⟶R with Ωξ∈Rn being a compact set and an approximation error σm>0 are given such that Fξ−W∗Tεξ≤σm,∀ξ∈Ωξ. Therefore, Fξ can be described as Fξ=W∗Tεξ+σ∗,∀ξ∈Ωξ∈Rn, where σ∗ is the approximation error and satisfies σ∗≤σm. The optimal weight vector W∗ for analytical purposes can be defined as(28)W∗=argminW∈RNsupξ∈ΩξFξ−WTεξ≤σm.


## 2.1. Mathematical Models

The dynamic model of multimachine infinite-bus power systems with SVC is given in this section, in which the dynamics of theith generator are described by a third-order differential equation, and the SVC is described by a first-order inertia equation. Then, the mathematical models of power system can be described as follows [3, 5]:(1)δ˙it=ωi,ω˙it=−Di2Hiωi+ωi02HiPmi−Pei,E˙qi′t=1Td0i′Efit−Eqit,with a SVC model(2)B˙Li=1TCi−BLi+BCi+uBi,and the electrical equations are as follows:(3)Eqit=Eqi′t+xdi−xdi′Idit,(4)Efit=keiufit,(5)Peit=∑j=1nEqi′tEqj′tBijsinδi−δj,(6)Qeit=−∑j=1nEqi′tEqj′tBijcosδi−δj,(7)Idit=−∑j=1nEqj′tBijcosδi−δj,(8)Iqit=∑j=1nEqj′tBijsinδi−δj,where the explanation of symbols used in the dynamic model (1)–(8) is given in Table 1.Table 1
The notation for multimachine power systems with SVC.
SymbolNomenclatureδiThe power angle of theith generator, in radωiThe relative speed of theithgenerator, in rad/sωi0The synchronous machine speed, in rad/sDiThe per unit damping constantHiThe inertia constant, in sPmiThe mechanical input power, in p.u.PeiThe electrical power, in p.u.Eqi′The transient EMF in the quadrature axis, in p.u.EfiThe equivalent EMF in the excitation coil, in p.uEqiThe EMF in the quadrature axis, in p.u.xdiThe direct axis reactance, in p.u.xdi′The direct axis transient reactance, in p.u.IdiThe direct axis current, in p.ukeiThe gain of the excitation amplifier, in p.u.ufiThe input of the SCR amplifier, in p.u.QeiThe reactive power, in p.u.IqiThe quadrature axis current, in p.u.BLiThe adjustable equivalent susceptance in SVCBCiThe initial value of adjustable susceptanceuBiThe input of SVC, in p.u.TCiThe time constant of adjusting system and SVCTd0i′The direct axis transient short-circuit time constant, in sBijTheith row and jth column element of nodal susceptance matrix at the internal nodes after eliminating all physical buses, in p.u.Remark 1.
Generally, in order to accurately describe a large-scale multimachine power system, a high-order complex nonlinear dynamic equation is required [28], which makes it very difficult to analyze the stability of the power system and design the excitation controller [3, 5]. In order to overcome such difficulties, a third-order power system model simplified by the integral manifold method is proposed in [5, 50]. This model makes the control system design and stability analysis feasible; and it has been widely used in the design of excitation controllers for multimachine power system [7, 26, 51–53]. The third-order power system mode and the first-order static reactive power compensator model adopted in this paper are derived from [3, 5].
In order to reduce the complexity of the controller design process in Section3, Eqi′t in the generator electrical dynamics is eliminated by differentiating Peit in (5):(9)P˙eit=1Td0i′Efit−xdi−xdi′IditIdit−Peit−Qeitωi+γiδ,ω.
LetΔPei=Pei−Pmi; we have(10)ΔP˙eit=P˙eit=−1Td0i′ΔPeit+1Td0i′ui+γiδ,ω.
The multimachine power systems model can be transformed into the following mathematical model [1, 5]:(11)δ˙it=ωi,ω˙it=−Di2Hiωi+ωi02HiPmi−Pei+di1,ΔP˙eit=−1Td0i′ΔPeit+1Td0i′ui+γiδ,ω+di2,where di1 and di2 are bounded uncertainties, including modeling errors, measurement errors, and external disturbances. ui is the control signal of the generator with(12)ui=EfitIqit−xdi−xdi′IditIqit−Pmi−Td0i′Qeitωi.
By considering the failure fault of the actuator at timetf, the generator excitation voltage Efit can be defined as(13)Efit=kei1−βiufi,where βi is the failure factor. So(14)ui′=EfitIqit−xdi−xdi′IditIqit−Pmi−Td0i′Qeitωi−βiEfitIqit.
So the multimachine power systems model by considering the failure fault can be described as follows:(15)δ˙it=ωi,ω˙it=−Di2Hiωi+ωi02HiPmi−Pei+di1,ΔP˙eit=−1Td0i′ΔPeit+1Td0i′ui+γiδ,ω+di2+Fi,where(16)Fi=−βi1Td0i′EfiIqi,γiδ,ω=∑j=1nEqi′tE˙qj′tBijsinδi−δj−∑j=1nEqi′tEqj′tBijcosδi−δjωj.
Here, the bound of the interconnection termγiδ,ω satisfies(17)γiδ,ω≤∑j=1nγi1jsinδj+γi2ωj≤∑j=1nγi1jδj+γi2ωj,where(18)γi1j≜4p1ijTd0j′minPeimax,whenj≠i,∑j=1,j≠in4p1ijTd0j′minPeimax,whenj=i,γi2≜p2ijQeimax,with p1ij and p2ij being the constants with values of either 1 or 0. Peit and Qeit are readily measurable variables. From (5)–(8), we can obtain that(19)Pei=Eqi′Iqi,Qei=−Eqi′Idi.
Define the following state variables for coordinate transformation:(20)xi1=δi−δi0,xi2=ωi−ωi0,xi3=ΔPei=Pei−Pmi,xi4=Vmi−Vrefi,where Pmi=Pmi0 is a constant and(21)Vmi=X2iEqi′2+X1i2+2X1iX2iEqi′cosxi1Xd∑i′,where Vmi is the accessing point voltage and Vrefi is the corresponding reference voltage of the SVC:(22)X1i=xdi′+XTi,Xd∑i′=X1i+X2i+X1iX2iBLi−BCi,where X2i and XTi are the transmission line reactance and the transformer reactance, respectively. According to (2) and (15) and (20), we can obtain the following mathematical model of multimachine power systems with SVC:(23)x˙i1=xi2,x˙i2=fi2x¯i2−gi2xi3+di1,x˙i3=fi3x¯i3+gi3ui+γiδ,ω+di2+Fi,yi1=xi1,(24)x˙i4=gi4uBi′+fi4x¯i4,yi2=xi4,where yi1 is the output of multimachine power systems, yi2 is the output of the SVC, and(25)fi2x¯i2=−Di2Hixi2,fi3x¯i3=−1Td0i′xi3,fi4x¯i4=−sinxi1X1iX2ixi4+VrefiXd∑i′2xi2Eqi′−X1iX2ixi4+VrefiXd∑i′TCi−BLi+BCi+X2i2Eqi′+X1iX2icosxi1xi4+VrefiXd∑i′2−Xd∑iTd0iXd∑i′Eqi′+1Td0iXdi−Xdi′Xd∑i′cosxi1+X2i2xi3+X1iX2iVsicosδiTd0iVmiXd∑i′2ui,gi2=ωi02Hi,gi3=1Td0i′,gi4=X1iX2iTCiXd∑i′,uBi′=−xi4uBi,with(26)Xd∑i=X3i+X2i+X3iX2iBLi−BCi,X3i=xdi+XTi.
Since the damping coefficientDi is difficult to measure accurately, Di is treated as an uncertainty parameter, and the nonlinear system with uncertain parameters (1) is transformed by equation (20) into a feedback form as shown in equation (23), so that the dynamic surface integral sliding mode method can be used to design the controller of the system equation (23). We can make the following two assumptions in this paper.Assumption 1.
The signs ofgij,i=1,2,j=2,3,4, are unknown bounded positive parameters, and there are known positive constants gmax and gmin such that gmax≥gij≥gmin; fi2x¯i2, fi3x¯i3, and fi4x¯i4 are unknown continuous functions.Assumption 2.
The reference signalxi1d is bounded and smooth function, its first and second derivatives both exist, and there is a positive real number Bi0 that satisfies xi1d2+x˙i1d2+x¨i1d2≤Bi0.Remark 2.
Sincegij,i=1,2,j=2,3,4, are unknown bounded positive parameters and the reference signal xi1d is a bounded smoothing function, Assumptions 1 and 2 are reasonable and common assumptions in [24, 26].


## 2.2. Radial Basis Function Neural Networks (RBFNNs)

The unknown nonlinear continuous functions are approximated by the RBFNNs in this study. The general form of RBFNNs can be expressed as(27)yξ=WTεξ,where ξ∈Rn is the input of RBFNNs, yξ∈R is the output of RBFNNs, W∈RN is the adjustable weight vector, and εξ∈RN is the nonlinear vector function.In general, a continuous nonlinear functionF:Ωξ⟶R with Ωξ∈Rn being a compact set and an approximation error σm>0 are given such that Fξ−W∗Tεξ≤σm,∀ξ∈Ωξ. Therefore, Fξ can be described as Fξ=W∗Tεξ+σ∗,∀ξ∈Ωξ∈Rn, where σ∗ is the approximation error and satisfies σ∗≤σm. The optimal weight vector W∗ for analytical purposes can be defined as(28)W∗=argminW∈RNsupξ∈ΩξFξ−WTεξ≤σm.


## 3. Process of the Controller Design

The design process of the controller is divided into two parts: the design of control law for multimachine excitation systems and the design of control law for SVC. The detailed design procedure is described in the following steps. Step 1 and 2 apply the dynamic surface method, and Step 3 uses integral sliding mode control. According to (24), we know that the model of SVC is of the first order. Therefore, the dynamic surface control method is used to design the controller of SVC in Step 4. The design scheme of the adaptive dynamic surface integral sliding mode fault-tolerant controller is shown in Figure 1, and the design steps are displayed in Table 2.Figure 1
Schematic diagram of the proposed control scheme.Table 2
Adaptive dynamic surface integral sliding mode fault-tolerant control.
Alternative variablesStep 1zi1=xi1−xi1d(T2.1)x¯i2=−ci1zi1+x˙i1d(T2.2)τi2x˙i2d+xi2d=x¯i2xi2d0=x¯i20(T2.3)Step 2zi2=xi2−xi2d(T2.4)x¯i3=ci2zi2+1/2zi2εi2Tεi2v^i2(T2.5)v^˙i2=ri21/2zi22εi2Tεi2−λi2v^i2(T2.6)τi3x˙i3d+xi3d=x¯i3xi3d0=x¯i30(T2.7)Step 3zi3=xi3−xi3d(T2.8)Si3=zi3+ki3∫0tzi3τdτ(T2.9)ui=−ci3Si3−1/2v^i3εi3Tεi3Si3−θ^gi3ki3⋅zi3−ηsgnSi3(T2.10)v^˙i3=ri31/2Si32εi3Tεi3−λi3v^i3(T2.11)θ^˙gi3=ρgi3Si3ki3⋅zi3−δi3θ^gi3(T2.12)Step 4zi4=xi4−Vrefi(T2.13)uBi′=−ci4zi4−1/2v^i4εi4Tεi4zi4(T2.14)v^˙i4=ri41/2zi42εi4Tεi4−λi4v^i4(T2.15)In Table2, zijj=1,2,3,4 are the jth surface errors with xi1d being the desired power angle of the ith generator. Si3 represents the designed integral sliding mode surface. x¯i2 and x¯i3 are the virtual control laws in Step 1 and Step 2. (T2.3) and (T2.7) are first-order low-pass filters, which are used to overcome the differential explosion problem in backstepping method. τi2 and τi3 are the time constants of (T2.3) and (T2.7). v^i2, v^i3, v^i4, and θ^gi3 are adaptive laws. ui and uBi′ are the actual control laws of generator excitation and SVC, respectively. cijj=1,2,3,4, ril, λill=2,3,4ρgi3, δi3, and ki3 are positive design parameters. v˜il=v^il−vil∗l=2,3,4 and v^ill=2,3,4 are the estimation of vil∗=Wil∗2 with Wil∗ being the optimal weight vector of the RBF neural networks in (A.9), (A.17), and (A.23). θ˜gi3=θ^gi3−θgi3∗ with θ^gi3 being the estimation of θgi3∗=1/gi3.Remark 3.
A neural network based adaptive robust controller is developed by using the traditional backstepping design method in [2, 54]. Although the semiglobal ultimate uniform boundedness of all signals in the control system is achieved, there is a problem of “explosion of complexity.” In this paper, the introduction of a first-order low-pass filter overcomes the problem.Remark 4.
The values of time constantsτi2 and τi3 should be chosen to be as small as possible. According to the actual condition, their values are usually between 0.001 and 0.1.Remark 5.
In order to achieve better transient performance while maintaining the tracking accuracy of the control, the integral sliding mode surfaceSi3 is adopted in Step 3, which greatly improves the robustness of the system.Remark 6.
The saturation functionsat⋅ is used instead of the symbol function sgn⋅ to attenuate the chattering phenomenon of sliding mode control. The expressions of the sat⋅ function can be written as satSi3=signSi3ifSi3>φ,satSi3=Si3/φifSi3≤φ, where φ>0.Remark 7.
The adaptive fault-tolerant controller proposed in this paper is based on the dynamic surface control (DSC) technique [25]. Subsystems (23) and (24) are third-order system and first-order system, respectively. Therefore, the controller design process consists of 4 steps, and the actual control input signals (T2.10) and (T2.14) are obtained in the 3rd and 4th steps. In addition, the detailed information of the DSC method is described in some literatures, such as [13, 15, 24]. In order to make the article more concise, the specific design process of DSC is omitted, and only the necessary intermediate variables are given in Table 2.


## 4. Stability Analysis

The stability analysis of the multimachine power systems with SVC will be given in this section. First of all, define the filter errorsyi2e and yi3e:(29)yi2e=xi2d−x¯i2=xi2d−−ci1zi1+x˙i1d,(30)yi3e=xi3d−x¯i3=xi3d−ci2zi2+zi2v^i2εi2Tεi22.From (T2.3) and (29), we have(31)x˙i2d=x¯i2−xi2dτi2=−yi2eτi2.Similarly, from (T2.7) and (30), we have(32)x˙i3d=x¯i3−xi3dτi3=−yi3eτi3.There are nonnegative continuous functionsBi2 and Bi3, and(33)y˙i2e+yi2eτi2≤Bi2zi1,zi2,yi2e,xi1d,x˙i1d,x¨i1d,(34)y˙i3e+yi3eτi3≤Bi3zi1,zi2,zi3,yi2e,yi3e,v^i2,xi1d,x˙i1d,x¨i1d,where(35)Bi2=ci1z˙i1−x¨i1d=ci1zi2+yi2e−ci1zi1−x¨i1dBi3=−ci2z˙i2+z˙i2v^i2εi2Tεi22+zi2v^˙i2εi2Tεi22+v^i2εi2Tεi2∑l=12∂εi2∂xilx˙il+∂εi2∂zi2z˙i2.Consider the following Lyapunov function condition:(36)V=∑k=14Vik+∑i=1n12yi2e2+∑i=1n12yi3e2.Theorem 1.
Consider the closed-loop control system, including multimachine systems model with actuator failure (23), SVC system model (24), first-order low-pass filters (T2.3) and (T2.7), actual control laws (T2.10) and (T2.14), and adaptive update laws (T2.6), (T2.11), (T2.12), and (T2.15). If Assumptions 1 and 2 are satisfied and the initial condition satisfies V0≤p, where p is arbitrary positive constant, then all closed-loop signals such as zi1,zi2,zi3,zi4,vi2,vi3,vi4,θgi3 are semiglobally uniformly bounded and the tracking error can converge to an arbitrarily small range by adjusting parameters ci1,ci2,ci3,ci4,λi2,λi3,λi4,δi3,ki3,τi2,τi3,i=1,2. In addition, the integral sliding mode surface shown in (T2.9) can eliminate the steady-state error of the system tracking and ensure the robustness of the system.Proof.
The specific proof can be seen in the Appendix.


## 5. Simulation Result and Analysis

Simulation studies are meticulously discussed in this section. In order to verify the control performance of the dynamic surface integral sliding mode coordination controller designed in this paper, the dynamic simulation of the 2-machine power system with SVC shown in Figure2 is performed, where XT1 and XT2 represent the transformers and 2XL1 and 2XL2 represent the transmission lines. Some simulation parameters are selected as in [26].Figure 2
The structure of 2-machine power system with SVC.The design parameters of the control laws including the virtual control laws and the final control laws are selected asc11=8,c21=8,c12=3,c22=3,c13=60,c23=60,c14=20,c24=20. The updated laws are chosen as λ12=λ22=0.3, r12=r22=2, λ13=λ23=0.3, r13=r23=2, ρg13=ρg23=0.6, δ13=δ23=1, λ14=λ24=0.5, and r14=r24=1.The parameters of the integral sliding mode control are selected ask13=k23=0.5. The time parameters of the low-pass filter are chosen as τ12=τ22=0.005 and τ13=τ23=0.005. In addition, the interference signals are selected as di1=0.001cos2t and di2=0.001sintcos2t.In order to verify the effectiveness of the designed controller, the tracking performances of the following four controllers are compared in Figures3–5 under three conditions. (1) Two generators adopt the excitation controller with SVC designed by the traditional adaptive backstepping control (TABSC) method. (2) Two generators adopt the coordinated controller designed by the adaptive dynamic surface control (DSC) method. (3) Two generators adopt the coordinated controller designed by the adaptive sliding mode control (SMC) method. (4) The two generators adopt the coordination controller designed by dynamic surface integral sliding mode control (DSISMC) method in this paper.Figure 3
The tracking error of power anglesδi,i=1,2 by using four controllers for the two generators in Case 1.Figure 4
The tracking error of power anglesδi,i=1,2, by using four controllers for the two generators in Case 2.Figure 5
The tracking error of power anglesδi,i=1,2, by using four controllers for the two generators in Case 3.Case 1.
The operating points are selected as follows:(37)δ10=40.415∘,ω10=314.06rad/s,Pm10=1.02p.u.,Vref1=1.15p.u.,δ20=40.285∘,ω20=313.94rad/s,Pm20=0.98p.u.,Vref2=1.05p.u..
Figures3 and 6–11 show the simulation results in Case 1. Figure 3 shows the tracking error of power angle δi under four controllers; it can be seen that, compared to the controllers designed by the other three methods, the dynamic surface integral sliding mode method adopted in this paper has better dynamic performance in terms of overshoot and stable operation time. The time required to achieve stabilization is shortened by 1∼2 s, and the fluctuation range is also significantly improved. According to Figure 3, the steady state (9∼10) can be selected to obtain the maximum value of tracking error (MVTE) and the root mean square value of tracking error (RMSVTE) for the four control schemes as shown in Table 3. Figures 6–11 show the power angles δ1,δ2, the rotated speeds ω1,ω2, the electrical powers Pe1,Pe2, the control inputs u1,u2 of the excitation system, the accessing voltages Vm1,Vm2, and control inputs uB1,uB2 of SVC. It is proven by Figures 6–11 that the controller designed by combining dynamic surface and integral sliding mode has good control effect.Figure 6
The power anglesδi,i=1,2 of the two generators in Case 1.Figure 7
The rotated speedωi,i=1,2 of the two generators in Case 1.Figure 8
The electrical powerPei,i=1,2 of the two generators in Case 1.Figure 9
The control inputui,i=1,2 of the two generators in Case 1.Figure 10
The accessing voltageVmi,i=1,2 of the SVC for the two generators in Case 1.Figure 11
The control inputuBi,i=1,2 of the SVC in Case 1.Table 3
The MVTE and RMSVTE in Case1.
Types of errorProposed schemeTBSC schemeDSC schemeSMC schemeMVTE of G#13.3449e−067.9840e−062.3967e−056.3628e−06MVTE of G#23.3402e−061.5855e−052.8119e−056.6305e−06RMSVTE of G#12.8195e−066.0995e−061.7739e−055.3544e−06RMSVTE of G#22.8147e−061.1737e−052.0606e−055.5828e−06Case 2.
The operating points are selected as follows:(38)δ10=40.415∘,ω10=314.12rad/s,Pm10=1.01p.u.,Vref1=1.15p.u.,δ20=40.285∘,ω20=314.16rad/s,Pm20=1.018p.u.,Vref2=1.05p.u..
In the second case, it is assumed that the short circuit fault occurred on the transmission line att=5s and disappeared after 0.2 seconds. Figures 4 and 12–17 show the simulation results in this case. Figure 4 shows the power angle tracking error zi1,i=1,2, of the four control schemes in the case of the short circuit; it can be seen that, after the short circuit fault disappears, the method adopted in this paper can make the tracking error quickly converge to the working point, so as to achieve the expected tracking performance of power angles. Similarly, according to Figure 4, the MVTE and RMSVTE in Case 2 can be given in Table 4. Figures 12–17 illustrate the curves of the power angles δ1,δ2, the rotated speeds ω1,ω2, the electrical powers Pe1,Pe2, the control inputs u1,u2 of the excitation system, the accessing voltages Vm1,Vm2, and the control inputs uB1,uB2 of SVC under the case that the unexpected short circuit fault occurred on the transmission line at t=5s and disappeared at t=5.2s.Figure 12
The power anglesδi,i=1,2, of the two generators in Case 2.Figure 13
The rotated speedsωi,i=1,2, of the two generators in Case 2.Figure 14
The electrical powersPei,i=1,2, of the two generators in Case 2.Figure 15
The control inputsui,i=1,2, of the two generators in Case 2.Figure 16
The accessing voltagesVmi,i=1,2, of the SVC for the two generators in Case 2.Figure 17
The control inputsuBi,i=1,2, of the SVC in Case 2.Table 4
The MVTE and RMSVTE in Case2.
Types of errorProposed schemeTBSC schemeDSC schemeSMC schemeMVTE of G#13.4536e−067.6832e−062.4130e−058.3619e−06MVTE of G#23.2387e−061.5550e−052.7959e−056.8147e−06RMSVTE of G#12.9299e−065.8961e−061.8058e−055.5876e−06RMSVTE of G#22.7118e−061.0816e−052.0317e−055.6189e−06Case 3.
The operating points are selected as follows:(39)δ10=40.4∘,ω10=314.10,Pm10=1.02p.u.,Vref1=1.15p.u.,δ20=40.2∘,ω20=314.06rad/s,Pm20=1.00p.u.,Vref2=1.05p.u..
In this case, consider that an actuator failure fault occurs at timetf=4.8s, so u¯fi can be assumed such that u¯fi=1−βiufi,t≥tf, u¯fi is the input of the SCR amplifier when failure occurs, and the multimachine excitation systems models with SVC in the case of failure fault are equations (23) and (24). The failure factors β1=0.2,β2=0.4. Figures 5 and 18–23 show the simulation results in Case 3. Figure 5 illustrates the power angle tracking error zi1,i=1,2, of the four control schemes in the case of the failure fault; it can be seen that when the failure fault occurs, the method designed in this paper has good tracking performance. Similarly, Table 5 shows the MVTE and RMVSTE in Case 3. Figures 18–21 show the curves of the power angles δ1,δ2, the rotated speeds ω1,ω2, the electrical powers Pe1,Pe2, and the control inputs u1,u2 of the excitation system under the case that the failure fault occurred at t=4.8s, respectively. Figures 22 and 23 illustrate the curves of the accessing voltages Vm1,Vm2 and the control inputs uB1,uB2 of SVC in the third case.Figure 18
The power anglesδi,i=1,2, of the two generators in Case 3.Figure 19
The rotated speedsωi,i=1,2, of the two generators in Case 3.Figure 20
The electrical powersPei,i=1,2, of the two generators in Case 3.Figure 21
The control inputsui,i=1,2, of the two generators in Case 3.Figure 22
The accessing voltagesVmi,i=1,2, of the SVC for the two generators in Case 3.Figure 23
The control inputsuBi,i=1,2, of the SVC in Case 3.Table 5
The MVTE and RMSVTE in Case3.
Types of errorProposed schemeTBSC schemeDSC schemeSMC schemeMVTE of G#15.7220e−064.8496e−042.2845e−042.8019e−04MVTE of G#28.2102e−062.5091e−035.0814e−045.6473e−04RMSVTE of G#15.2248e−064.8282e−042.2052e−042.7854e−04RMSVTE of G#27.6584e−062.5137e−034.9848e−045.6299e−04


## 6. Conclusion

In this paper, the method of combining adaptive dynamic surface with integral sliding mode fault-tolerant control is proposed to solve the excitation control problem of multimachine infinite-bus power systems with SVC by considering parameter uncertainties, external disturbances, and actuator failure. Applying the RBFNNs to approximate the unknown nonlinear function not only overcomes the uncertainty of the model but also reduces the computational burden. The designed controller solves the “explosion of complexity” problem of the traditional backstepping method and makes the tracking error converge to zero in a limited time, thereby improving the steady-state tracking accuracy of the system, while ensuring that the system still has good stability after the failure. Simulation results show the effectiveness of the proposed control algorithm. Future work will focus on the design of a disturbance observer based adaptive output feedback control scheme for multimachine power systems by introducing a fixed-time control method to improve the robustness of the system.



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