# Stabilization for Damping Multimachine Power System with Time-Varying Delays and Sector Saturating Actuator

Authors: Linlin Ma; Yanping Liang; Jian Chen
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper studies the stabilization problem for damping multimachine power system with time-varying delays and sector saturating actuator. The multivariable proportional plus derivative (PD) type stabilizer is designed by transforming the problem of PD controller design to that of state feedback stabilizer design for a system in descriptor form. A new sufficient condition of closed-loop multimachine power system asymptomatic stability is derived based on the Lyapunov theory. Computer simulation of a two-machine power system has verified the effectiveness and efficiency of the proposed approach.


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## Body

## 1. Introduction

To cope with the increasing demand for quality electric power, excitation control, power system stabilizer (PSS), and other power system controllers are playing important roles in power system stability and maintaining dynamic performance. Conventional PSS is mainly designed based on a linear model and considered one operating point. Recently, to interconnect large energy pools connecting neighboring electric grids together and transmit bulk energy during peak times of load demand can satisfy the growing demand for energy [1]. But it introduces some modes of electromechanical oscillations and frequency deviations within the range of 0.2–2 Hz in the power system which will make power system more complicated [2, 3]. A conventional PSS cannot guarantee to have the best performance. Hence, a variety of control strategies have been used to obtain PSS, such as lead-lag controller [4], variable structure controller [5–7], robust controller [8], PID controller [9–11], and fuzzy logic controller [10]. Most of the controllers are nonlinear. Some researchers have designed the PSS by using searching algorithms such as genetic algorithms [10, 12], particle swarm optimization [13, 14], and chaotic optimization algorithm [15, 16]. But these algorithms are hard to program and are not sure to find the optimum solutions.It is well known that the amplitude of the controller is always bounded in the real world [17]. So it is very necessary that the actuator saturation is taken into consideration. Time-delay is very common in power systems which can be a source of instability of performance degradation [18]. Multimachine power system with time-varying delay and sector saturating actuator [19] is a complex interconnected large-scale system that is composed of many electric devices and mechanical components with a better description of real world. The state feedback control problem for such a system is addressed by [19] based on the LMI methods. However, the conditions in [19] are conservative because of the amplifying technique to deal with the nonlinear terms in the conditions. Moreover, we usually cannot find the state feedback controller to satisfy demand when system becomes more complex.The purpose of this paper is to design a PD controller for damping multimachine power systems with time-varying delay and sector saturating actuator. Under a descriptor transformation, the problem of PD type controller design is transformed into the state feedback controller design for a descriptor system. Then, a new sufficient condition is derived for the admissible of the descriptor system based on the Lyapunov theory. Compared with the existing LMI methods in [19], our method introduces more relax matrix variables. Therefore, it is less conservative. Compared with some of the nondeterministic methods, our method has the advantages of low complexity.This paper is organized as follows. Section2 is the problem formulation and preliminaries. Section 3 gives the main results. Section 4 provides an example to show the merits and effectiveness of the results and Section 5 concludes this paper.Notation. Rn denotes n-dimensional Euclidean space; the superscripts -1 and T denote the matrix inverse and transpose, respectively; X>0(X≥0) means that X is positive definite (positive semidefinite); the star ∗ denotes the symmetric term in a matrix.


## 2. Problem Formulation and Preliminaries

ConsiderN-machine power system with time-varying delays and input constraints which is described by the interconnection of N subsystems as follows:(1)x˙it=Aixit+Biusit+∑j=1NpijGijgijxit,xjt-τijt,usit=satuit,xit=ϕit,t∈-τ,0,where xi(t)=ΔδitΔWitΔEqi′tΔEfdi(t)  ΔVDCi(t)T, ui(t) = [Δmi], ui(t) is the control input vector to the actuator, and usi(t) is the control input vector to the plant.g i j ( x i ( t ) , x j ( t - τ i j ( t ) ) is the nonlinear function vector characterizing the interconnection between ith generator and jth generator with(2)gijxit,xjt-τijt=-cos⁡βiδΔδit-Δδjt-τijt=-cos⁡βiδWxit-xjt-τijt,where τij(t) is the time-varying delay and satisfied 0⩽τij(t)⩽τ⩽∞, τ˙ij(t)⩽τ˙∗<1, and βi(δ)=(δi(t)-δj(t-τij(t))+δi0-δj0)/2, W=[10000].The nominal system matrices are represented as follows:(3)
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.In this modeling, the single-machine infinite bus is modeled by Heffron-Phillips model which is shown in Figure 1.Figure 1
Heffron-Phillips model for single-machine power system connected to infinite bus along with SSSC series in the transmission line.The nonlinear saturation functionusi(t) is considered to be inside sector (ai,1) and is shown in Figure 1, where 0⩽ai⩽1.From Figure2, it is obvious that usi(t)-0.5(1+ai)ui(t)=Δtiui(t)⇒usi(t)=ψiui(t), where ψi=0.5(1+ai)+Δti, Δti is a real number which varying between -0.5(1-a¯i) and 0.5(1+a¯i), ai<a¯i<1.Figure 2
Sector saturation function.The control law for a PD controller is(4)uit=Kpixit+KDix˙it.Substituting (4) into (1), we have(5)I-ψiBiKDix˙it=Ai+ψiBiKpixit+∑j=1NpijGijgijxit,xjt-τijt,xit=ϕit,t∈-τ,0.Taking the inverse of the left-hand side of (5), we obtain(6)x˙it=I-ψiBiKDi-1Ai+ψiBiKpixit+∑j=1NpijGijgijxit,xjt-τijt,xit=ϕit,t∈-τ,0.Properly selecting the controller gainsKpi and KDi, so that the closed-loop systems are stable, then we have the PD controller design. It is obvious from (6) that the PD controller is nonlinear. Some researchers have designed such a controller with the aid of searching algorithms [10]. A huge amount of computation burden is foreseeable. In the following, we introduce a new state variable x¯i(t)=xiTtx˙iTtT, then system (1) with controller (4) is transformed into the following PD control system:(7)Ex-.it=A¯ix¯it+B¯iu¯sit+∑j=1NpijG¯ijg¯ijx¯it,x¯jt-τijt,u¯sit=satu¯it,u¯it=K¯ix¯it,where(8)A¯i=0IAi-I,B¯i=0Bi,G¯ij=0Gij,E=In000,g¯ijx¯it,x¯jt-τijt=gijxit,xjt-τijt=-cos⁡βiδW¯x¯it-x¯jt-τijt,K¯i=[KpiKDi], and W¯=[W0].System (7) is a descriptor system as rank(E)=n<dim(E),u¯i(t)=Kpixi(t)+KDix˙i(t)=K¯ix¯i(t); then, we have(9)Ex-.it=A¯i+ψiB¯iK¯ix¯it+∑j=1NpijG¯ijg¯ijx¯it,x¯jt-τijt=A¯i+ψiB¯iK¯ix¯it-cos⁡βiδ∑j=1NpijG¯ijW¯x¯it-x¯jt-τijt=A¯ci-∑j=1NAdijx¯it+∑j=1NAdijx¯jt-τijt,where Adij=cos⁡βi(δ)pijG¯ijW¯ and A¯ci=A¯i+ψiB¯iK¯i.The following definition and lemmas will be useful in this paper.Definition 1 (see [20]).
(i) Descriptor system(10)Ex˙t=Axt+∑j=1NAdjxt-djis said to be regular and impulse-free, if pair (E,A) is regular and impulse-free.
(ii) System (10) is said to be stable if for any scalar ε>0, there exists a scalar δ(ε)>0 such that, for any compatible initial conditions ϕ(k) satisfying sup-d≤t≤0ϕ(t)≤δ(ε), solution x(t) of system (10) satisfies x(t)≤ε for any t≥0; moreover, limt→∞x(t)=0.
(iii) System (10) is said to be admissible if it is regular, impulse-free, and stable.Lemma 2 (see [18]).
For any constant matrixM>0, scalar γ>0, and vector function W:[0,γ]→Rn, such that integrations concerned are well defined, the following inequality holds:(11)∫0γWsdsTM∫0γWsds⩽γ∫0γWTsMWsds.Lemma 3.
Given any matricesD and E with appropriate dimensions, the inequality(12)DE+ETDT⩽DTDT+ETT-1Eholds for any matrix T>0.


## 3. Main Results

In this section, we will give the following condition for system (9).Theorem 4.
The delay descriptor system (9) is admissible with K¯i=YiXi-T if there exist matrices Xi=X1iX2i0X3i, Qij>0, T>0, Yi,Uij>0, (i,j=1,2,…,N), such that the following inequalities hold:(13)EXiT=XiET⩾0,(14)symA¯iXiT+ψiB¯iYi∗⋯∗∗+τ∑j=1NpjiETQjiE-1-τ˙∗τ∑j=1NpjiETQjiE+∑j=1NpjiUji+∑j=1NpijG¯ijW¯TW¯TG¯ijT1-τ˙∗τETQ1iE-α1-τ˙∗U1i-1-τ˙∗τETQ1iE⋮⋱1-τ˙∗τETQNiE-α1-τ˙∗UNi-1-τ˙∗τETQNiEXiT-TN-1<0,(15)-1-α1-τ˙∗UijXiXiT-T<0,where α is a fixed scalar which satisfies 0<α<1.Proof.
Firstly, we prove that system (9) with PD gain matrices K¯i is regular and impulse-free. System (9) can be rewritten as(16)E¯X˙t=A~+ψB~K~-∑i=1N∑j=1NA~dijXt+∑i=1N∑j=1NA~dijXt-τijt,where X(t)=x¯1Tt,x¯2Tt,…,x¯NTtT, matrices E¯=diag{E,E,…,E}, A~=diag{A¯1,A¯2,…,A¯N}, ψ=diag{ψ1,ψ2,…,ψN}, B~=diag{B¯1,B¯2,…,B¯N}, K~=diag{K¯1,K¯2,…,K¯N}, and(17)A~dij=0⋯0⋯00⋯cos⁡βiδpijG¯ijW¯⋯00⋯0⋯0ij.
From (14), it is easy to see that(18)symA¯iXiT+ψiBiYi-1-τ˙∗τ∑j=1NpjiETQjiE=symA¯i+ψiB¯iKiXiT-1-τ˙∗τ∑j=1NpjiETQjiE<0.That is(19)symA~+ψB~K~XT-1-τ˙∗τETdiag∑j=1NpjiQjiE¯<0,where X=diag{X1,…,XN}. Since E¯ is descriptor, there exist nonsingular matrices G and H such that(20)GE¯H=InN000.Suppose(21)GA~+ψB~K~H=A11A12A21A22,GXH=X11X120X22.Equation (19) yields sym{A22X22T}<0, which implies that the pair (E¯,A~+ψB~K~) is regular and impulse-free. By Definition 1, system (16) is regular and impulse-free. It also shows that system (9) is regular and impulse-free.
Next, we will show that system (9) is stable. Consider a Lyapunov-Krasovskii functional as(22)Vt=∑i=1NxiTtP11ixit+∑j=1Npij∫t-τijttx-jTξUijx¯jξdξ+∫-τijt0∫t+αtx-.jTξETQijEx-.jTξdξdα.Taking the time derivative of V(t) along the solution of system (9) yields(23)V˙t=∑i=1Nx˙iTtP11ixit+xiTtP11ix˙it+∑j=1Npijx¯jTtUijx¯jt-1-τ˙ijtx¯jTtt-τijtUijx¯jt-τijt+τijtx¯jTtETQijEx¯jt-1-τ˙ijt∫t-τijttx-.jαETQijEx¯jαdα.By Lemma 2, we obtain(24)-1-τ˙ijt∫t-τijttx-.jαETQijEx-.jαdα⩽-1-τ˙ijtτijtx¯jt-xjt-τijtTETQijEx¯jt-x¯jt-τijt.Then,(25)V˙t⩽∑i=1Nx-.iTtETPix¯it+x¯iTtPiTEx¯it+∑j=1Npijx¯jTtUijx¯jt-1-τ˙∗x¯jTt-τijtUijx¯jt-τijt+τx¯jTtETQijEx¯jt-1-τ˙∗τx¯jt-x¯jt-τijtTETQijEx¯jt-x¯jt-τijt=∑i=1Nx¯iTtA¯i+ψiB¯iK¯iTPi+PiTA¯i+ψiB¯iK¯ix¯it+symx¯iTtPiT-cos⁡βiδ∑j=1NpijG¯ijW¯x¯it-x¯jt-τijt+∑j=1Npijx¯jTtUijx¯jt-∑j=1N1-τ˙∗pijx¯jTt-τijtUijx¯jt-τijt+∑j=1Npijτx¯jTtETQijEx¯jt-∑j=1Npij1-τ˙∗τx¯jt-x¯jt-τijtTETQijEx¯jt-x¯jt-τijt,where(26)Pi=P11iP12i0P22i.It is obvious that V˙(t)<0 can be obtained by the following equation:(27)V-.=diagV˙1t,…,V˙Nt=XTtA~+ψB~K~TP+PTA~+ψB~K~Xt+sym-XTtPT∑i=1N∑j=1NA~dijXt+XTtPT∑i=1N∑j=1NA~dijXt-τijt+XTtdiag∑j=1Npj1Uj1,…,∑j=1NpjNUjNXt-∑i=1N∑j=1N1-τ˙∗XTt-τijtUijXt-τijt+τXTtdiag∑j=1Npj1ETQj1E,…,∑j=1NpjNETQjNEXt-∑i=1N∑j=1Npij1-τ˙∗τXt-Xt-τijtTQijXt-Xt-τijt,where(28)P=diagP1,P2,…,PN,Uij=0⋯0⋯00⋯pjiUji⋯00⋯0⋯0ij,Qij=0⋯0⋯00⋯ETQjiE⋯00⋯0⋯0ij.
It is obtained from Lemma3 that(29)sym-XTtPT∑i=1N∑j=1NA~dijXt+XTtPT∑i=1N∑j=1NA~dijXt-τijt⩽XTtdiagP1T∑j=1Np1jG¯1jW¯TW¯TG1jTP1,…,PNT∑j=1NpNjG¯NjW¯TW¯TGNjTPNXt+XTtdiag∑j=1Np1jT-1,…,∑j=1NpNjT-1Xt+∑i=1N∑j=1NpijXTt-τijtTXt-τijt,where T=diag{0,…,0,T-1,0,…,0}.
Substituting (29) into (27), we obtain(30)V-.⩽XTtΦXt+∑i=1N∑j=1NpijXTt-τijtT-1-τ˙∗Uij-1-τ˙∗τQijXt-τijt+∑i=1N∑j=1Npij1-τ˙∗τsymXTtQijXt-τijt=ξ¯TtΞξ¯t,where ξ¯t=XTtXTt-τiji,j=1,…,N,i≠jT,(31)Φ=A~+ψB~K~TP+PTA~+ψB~K~+diagP1T∑j=1Np1jG¯1jW¯TW¯TG1jTP1,…,PNT∑j=1NpNjG¯NjW¯TW¯TGNjTPN+diag∑j=1Np1jT-1,…,∑j=1NpNjT-1+diag∑j=1Npj1Uj1,…,∑j=1NpjNUjN-1-τ˙∗τdiag∑j=1Npj1ETQj1E,…,∑j=1NpjNETQjNE+τdiag∑j=1Npj1ETQj1E,…,∑j=1NpjNETQjNE,(32)Ξ¯=Φ∗⋯∗∑i=1N1-τ˙∗τQ1i∑i=1NT-1-τ˙∗U1i-1-τ˙∗τQ1i⋮⋱∑i=1N1-τ˙∗τQNi∑i=1NT-1-τ˙∗UNi-1-τ˙∗τQNi<0.From (30), we obtain that(33)V˙t⩽∑i=1NξiTtΞiξit,(34)T-1-1-α1-τ˙∗Uji<0,i≠j,where ξit=x¯iTt,x¯iTt-τ1it,…,x¯iTt-τi-1it,x¯iTt-τi+1it,…,x¯iTt-τNitT and(35)Ξ=symA¯i+ψiB¯iK¯iTPi∗⋯∗+PiT∑j=1NpijG¯ijW¯TW¯TG¯ijTPi+∑j=1NpijT-1+∑j=1NpjiUji+τ∑j=1NpjiETQjiE-1-τ˙∗τ∑j=1NpjiETQjiE1-τ˙∗τ∑j=1NETQ1iE-α1-τ˙∗U1i-1-τ˙∗τ∑j=1NETQ1iE⋮⋱1-τ˙∗τ∑j=1NETQNiE-α1-τ˙∗UNi-1-τ˙∗τ∑j=1NETQNiE<0.Pre- and postmultiply (45) by diag{Xi,Xi,…,Xi} and its transpose, respectively, where Xi=Pi-T, and define Yi=K¯iXiT, Qji=XiQjiXiT, and Uji=XiUjiXiT; we arrive at(36)symA¯iXiT+ψiB¯iYi+∑j=1NpjiUji∗⋯∗+∑j=1NτpjiETQjiE-∑j=1N1-τ˙∗τpjiETQjiE+∑j=1NpijG¯ijW¯TW¯TGijT+Xi∑j=1NpjiT-1XiT1-τ˙∗τETQ1iE-α1-τ˙∗U1i-1-τ˙∗τETQ1iE⋮⋱1-τ˙∗τETQNiE-α1-τ˙∗UNi-1-τ˙∗τETQNiE<0.
By Schur complement, (44) is equivalent to (14).
Pre-and postmultiplying (34) by Xi and its transpose, respectively, and by Schur complement, (34) is equivalent to (15).
SinceA22 is nonsingular, there exist two nonsingular matrices G¯ and H¯ such that(37)G¯E¯H¯=InN000,G¯A+ψBKH¯=A100InN,H¯TE¯TPE¯H¯=P¯1000,G¯A~dijH¯=Adij11Adij12Adij21Adij22,H¯TPH¯=P¯11P¯120P¯22,H¯TUjiH¯=Uji11Uji12Uji21Uji22.Denote Vt=H¯-1Xt=[v1(t)v2(t)], where Q1(t)∈RnN and v2(t)∈RnN. We obtain(38)-Xt2⩽-H¯-1-2Vt2⩽-H¯-1-2v1t2.
By (30), we obtain that there exists scalar ε>0 such that(39)V˙t<-εXt2,which implies(40)λmin⁡P¯1v1t2-V0⩽∫0tV˙sds⩽-εH¯-1-2∫0tv1s2ds.
Then,∑i=1Nv1(t)2⩽V(0)/λmin⁡(P¯1), and ∫0t‍v1(s)2ds⩽1/εV(0)H¯-1-2.
Thus,(41)limt→∞v1t=0.Next, we will prove that limt→∞v2(t)=0, which is equivalent to the stability of system (5).
Pre-and postmultiplying (32) by diag{H¯T,H¯T,…,H¯T} and its transpose, respectively, we obtain(42)P¯22+P¯22T+∑i=1N∑j=1NUji22∗⋯∗∑i=1NP¯22TAdi122-∑i=1N1-τ∗U1i22⋮⋱∑i=1NP¯22TAdiN22-∑i=1N1-τ∗UNi22<0.Pre- and postmultiplying (42) by -∑i=1N‍∑j=1N‍Adij22TI⋯I and its transpose, respectively, we obtain ∑i=1N‍∑j=1N‍Adij22T∑i=1N‍∑j=1N‍Uji22∑i=1N‍∑j=1N‍Adij22-∑i=1N‍∑j=1N‍(1-τ∗)Uji22<0, which implies that(43)ρ∑i=1N∑j=1NAdij22<1.
Using the expression in (37), the singular delay system (16) can be decomposed as(44)v˙1t=A1v1t+∑i=1N∑j=1NAdij11v1t-τijt+∑i=1N∑j=1NAdij12v2t-τijt,(45)0=v2t+∑i=1N∑j=1NAdij21v1t-τijt+∑i=1N∑j=1NAdij22v2t-τijt.Noting that, for any t>0, there exists positive integer k.
Such thatkτ-τ⩽t⩽kτ, and considering (44), we have(46)v2t=-∑i=1N∑j=1NAdij22kv2t-kτ-∑l=1k∑i=1N∑j=1NAdij22l-1∑i=1N∑j=1NAdij21v1t-kτ.This, together with (43) and (41), we obtain limt→∞v2(t)=0, and this completes the proof.Remark 5.
Theorem4 provides a PD control method for system (1). An LMI based criterion is obtained by transforming a regular system into the state feedback stabilizer design for a descriptor system. It is worth noting that if A¯i, B¯i, G¯ij, W¯, and E are replaced with Ai, Bi, Gij, W, and In, the state feedback controller can be solved by the following corollary. It is obvious that Theorem 4 has wider range of application.Corollary 6.
The delay system (5) with KDi=0 is stable with Kpi=YiXi-T if there exist matrices Xi>0, Qij>0, T>0, Yi,Uij>0, (i,j=1,2,…,N), such that the following inequalities hold:(47)symAiXiT+ψiBiYi∗⋯∗∗+τ∑j=1NpjiQji-1-τ˙∗τ∑j=1NpjiQji+∑j=1NpjiUji+∑j=1NpijGijWTWTGijT1-τ˙∗τQ1i-α1-τ˙∗U1i-1-τ˙∗τQ1i⋮⋱1-τ˙∗τQNi-α1-τ˙∗UNi-1-τ˙∗τQNiXiT-TN-1<0,(48)-1-α1-τ˙∗UijXiXiT-T<0,where α is a fixed scalar which satisfies 0<α<1.Remark 7.
Both Theorem4 and Corollary 6 are LMIs. The solutions of Xi, Yi are obtained, and the corresponding controller gain matrices are derived as K¯i=YiXi-T. Our method is a deterministic method which can be solved easier than some of the nondeterministic methods, such as the genetic algorithm [10] and particle swarm optimization [13].Remark 8.
Compared with Theorem2 in [19], Corollary 6 is less conservative in two aspects. Firstly, By using constraint (48), Uij, and Ti in Corollary 6 can be variables matrices, while relatives uij, vi in Theorem 2 in [19] are fixed scalars. Secondly, the integral term -(τ˙ij(t))∫t-τij(t)t‍x-.j(α)ETQijEx-.j(α)dα in the proof is enlarged by using Lemma 2 instead of being removed in the proof in [19].


## 4. Simulation

In this section, a two-machine infinite bus example system is chosen to show the effectiveness of the proposed method, which is shown in Figure3. The system parameters used in the simulation are as follows:(49)p11=p22=0,p12=p21=1,D1=5,D2=3,A1=0379.2000000-0.3169-0.8333-0.11230-0.0041-0.00990-0.22660.1983-0.004812.70000-951.7000-200.0000-24.4000-0.175900.030200.0257,A2=0379.2000000-0.4054-0.5000-0.04630-0.0041-0.04950-0.02830.1983-0.0048-12.70000-551.7000-200.0000-24.4000-0.27590000.0257,B1=B2=01.0000-3.00000.80004.0000T,G12=G13=00-2.700T,G21=G23=00-2.300T.Figure 3
Connecting diagram of two-machine system.If we seta1=a2=0.3 and a´1=a´2=0.5, so ψ1=ψ2 varies between 0.4 and 0.9. The upper bound of delay τ=5 and τ˙∗=0.5. The method in [19] fails to find a state feedback controller for this system. According to Theorem 4, the PD controller can be solved as(50)KD1=0.20570.0837-0.3127-0.14290.3615,Kp1=-645.2015-107.9008-156.5715-33.0835-94.0722,KD2=0.10300.0874-0.3113-0.20680.3635,Kp2=-559.78915.6352-95.1653-35.7473-74.1156.The close-loop state trajectories of generator 1-2 are shown in Figure4.Figure 4
The close-loop state trajectories.


## 5. Conclusion

In this paper, a decentralized PD control scheme has been proposed to deal with the time-delay multimachine power system with sector saturating actuator. A sufficient condition of closed-loop system asymptomatic stability is presented in terms of LMIs, which can be solved easily by LMI toolbox. Then, a sufficient condition of state feedback control is also obtained which is less conservative than that in [19]. A two-machine infinite bus system is considered as an example, and the simulation result shows the effectiveness of proposed method.



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