# Chaotic Threshold for a Class of Power System Model

Authors: Xiaodong Wang; Zhenyong Lu; Caiqin Song
Journal: Shock and Vibration
 (2019)

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## Abstract

This paper deals with the bifurcation and chaotic dynamic characteristic of a single-machine infinite-bus (SMIB) power system under two kinds of harmonic excitation disturbance, which are induced by the external periodic load and the outer mechanical disturbance. By applying Melnikov’s method, the threshold value for the occurrence of chaotic motion is provided. In addition, the chaotic boundary surface is given. The efficiency of the criteria for chaotic motion obtained in this paper is verified by bifurcation diagram, phase portraits, Poincaré section, and frequency spectrum. The results obtained in this paper will provide a better understanding of the nonlinear dynamic behaviors for this class of SMIB power system subjected to two kinds of harmonic excitation components.


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## Body

## 1. Introduction

With the rapid development of economy and the increasing power consumption, electric power systems have become more huge and complicated. Stability problems have become more complex as interconnections become more extensive [1]. Recently, the modern power system is forced to operate close to its stability limit. Power system is exposed to disturbances of varying intensity and some of the disturbances or cascading propagation of initial disturbance. Such disturbance can lead the power system to lose its partial or total stability. Therefore, many researchers focus on the stability of electric power systems in scientific and engineering studies [2–5], especially on an equivalent single-machine infinite-bus (SMIB) power system. The model of the swing equation has attracted great interest in the last decades due to its wide application in the research of stability of such power system.By using a random Melnikov method and numerical simulation, Wei et al. [6, 7] examined how a Gaussian white noise affects the dynamic behaviors of the power system. Zhu et al. [8] made a Hopf bifurcation analysis for a SMIB power system with subsynchronous resonance (SSR) by applying the Hopf bifurcation theorem. Nayfeh et al. [9, 10] investigated a single-machine quasi-infinite bus bar system, and they used numerical simulations to exhibit some of the complicated responses of the generator, including the period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism). Duan et al. [11] investigated the bifurcations associated with subsynchronous resonance of a SMIB power system with a series of capacitor compensation. Chen et al. [12] studied the chaotic control and identification problem of an SMIB power system, where the power of the machine was assumed to be a simple harmonic quantity. Moreover, Melnikov’s method works effectively for discussing bifurcations of periodic orbits and homo- (hetero-) clinic orbits for dynamical systems. For instance, this method has been applied in a classical SMIB power system model [13, 14]. Zhou and Chen [15] investigated the mechanism and parametric conditions for chaotic motions of a SMIB power system, and the critical curves separating the chaotic and nonchaotic regions have been given. Chaotic and subharmonic oscillations of the SMIB power system were discussed in [16] by computing Melnikov functions with the residue of a complex function and elliptic integrals. Wang et al. [17] investigated a SMIB power system under a periodic load disturbance and obtained the threshold for the onset of chaos by using Melnikov’s method.With the development of electric power industry, a lot of new equipment has been used in the power system. For example, a valve on a steam turbine was used, which can take quick action as a device for regulating the power of a prime mover. When the valve is under periodic perturbation, meanwhile, the system is subjected by the periodic load power, and there exists two kinds of periodic excitation. To the best of our knowledge, there are few literature considering dynamic characteristic of a single-machine infinite-bus power system under two kinds of periodic excitation disturbance. In addition, when the well-known Melnikov’s method is employed to obtain the threshold value for the occurrence of chaotic motion, the analytic expressions of the homoclinic orbits need to be given. It is worth pointing out that the analytical expression of the homoclinic orbits in the mathematical model of the SMIB power system is difficult to obtain. Because of this, the mechanical power input to the synchronous machine is assumed to be a very small quantity or zero in some existing literature, which is different from the real SMIB power system in engineering practice. In this case, the original homoclinic orbits have been changed into heteroclinic obits, so the intrinsic characteristic of the original real system has been changed. Although we have investigated the nonlinear dynamic analysis of a SMIB system subjected to a periodic load disturbance before in [17], in this paper, we extend a SMIB system subjected to two kinds of disturbance. So the purpose of this paper is to investigate the bifurcation and chaotic characteristic of it. The work in this paper differs from the existing literature in two aspects: one is the two kinds of excitation components, and the other one is the way of obtaining the threshold value for the occurrence of chaotic motion by applying the Melnikov method.


## 2. The Formulation of a Class of SMIB Power System

In this paper, we investigate a class of single-machine infinite-bus (SMIB) power system. Figure1 shows its configuration. In Figure 1, by using a transmission line, the synchronous machine S is transferring power to the infinite bus. We denote the equivalent main transformer of system 1 and system 2 as 3 and 4, respectively. We denote load, circuit breaker, and systematic tie line as 5, 6, and 7, respectively. In [1], the angle dynamics of the synchronous generator is described by the swing equation. So, it can be expressed as(1)dδdt=Δω=ω−ωR,2HωRdΔωdt=PM−PE,where δ is the rotor angle, ωR is the synchronous angular velocity, ω is the angular speed, and Δω is the angular speed deviation. PE,PM, and H represent the electrical power, mechanical power, and the inertia constant of the machine, respectively.Figure 1
SMIB power system with two kinds of disturbance.As we all know, suppose the generator is a nonsalient pole generator, and the electrical powerPE can be written as(2)PE=UEXsinδ=PSsinδ,in which the reference phasor of the infinite bus bar is represented as U∠0°, the voltage of the machine is denoted by E, and X stands for the reactance of the system.Assume system (1) is under two kinds of disturbance: a periodic load with the amplitude Pd and frequency Ω1 and an external periodic mechanical excitation disturbance with the amplitude Pf and frequency Ω2. D is the damping of the synchronous generator. The following equation can be obtained:(3)dδdt=ω−ωR,Hdωdt=Pm−Pmaxsinδ−Dω−ωR+PdcosΩ1t+PfcosΩ2t.For the convenience of analysis, the following transforms are introduced:(4)τ=tPmaxH,xτ=δt,yτ=HPmaxωt−ωR,c=DωrHPmax,0<ρ=PmPmax≤1,f1=PdPmax,ω1=Ω1HPmax,f2=PfPmax,ω2=Ω2HPmax.Then, system (3) changes into(5)dxdτ=y,dydτ=−sinx−cy+ρ+f1cosω1τ+f2cosω2τ.In order to obtain the basic behavior of the presented SMIB system in this paper, first we study a case with no damping and no external excitation. Now letc=0,f1=0,andf2=0; therefore, Equation (6) can be changed into the following form:(6)dxdτ=y,dydτ=−sinx+ρ.Thus, we can obtain the equilibriums of system (6):(7)E:x1,0=arcsinρ,0,S:x2,0=π−arcsinρ,0.Hence, it is easily concluded that the equilibriumE:x1,0=arcsinρ,0 is a center, and the equilibrium S:x2,0=π−arcsinρ,0 is a saddle point.It is observed from Figure2 that the phase portraits of the studied system (6) for different initial conditions are shown. The phase trajectory marked as circle “1” is oscillating orbits, whose angular displacement is usually smaller and the angular velocity changes its sign twice per period. At the same time, we denote the trajectory “3” as a typical rotational motion, whose angular velocity does not change its sign and its angular displacement grows with time, and we denote the trajectory “2” as the homoclinic orbit.Figure 2
Phase trajectories for the unperturbed system (5).


## 3. Melnikov Function of the SMIB Power System

In this section, we will study the condition for the emergence of chaotic motion in Equation (5) by applying Melnikov’s method, which has been successfully applied to the analysis of chaos in many nonlinear systems [18–21].In order to give the transversality condition by using the Melnikov’s method, the SMIB system (5) is rewritten as(8)dxdτ=y,dydτ=−sinx+ρ+ε−cy+f1cosω1τ+f2cosω2τ.Now we assumeε=0, and then Hamiltonian function of Equation (8) can be given as(9)Hx,y=12y2−cosx−ρx+H0,in which H0 stands for an integral constant.From Hamiltonian function (9), we can plot the phase portrait as shown in Figure 2. If we let HxE,yE=0, it yields to(10)H0=arcsinρ+cosarcsinρ.Therefore, the homoclinic orbitSCABS in Figure 2 is suitable for the following equation:(11)12y2−cosx−ρx=cosarcsinρ−ρπ+ρarcsinρ=^Hρ.We use the parametric functionx0τ,y0τ to stand for the homoclinic orbit, that is,(12)limτ⟶±∞x0τ,y0τ=S,yτ=±2Hρ+2cosx0τ+2ρx0τ.(1)
We denotex00,y00 as the coordinate of point A in Figure 2, in which y00=0 and x00 satisfy the equation ρx00+cosx00−ρπ+ρarcsinρ+cosarcsinρ=0.(2)
Ifτ∈−∞,0, we use x0τ,y0τ representing the lower part of the homoclinic orbit SCA. Contrarily, if τ∈0,∞, x0τ,y0τ denotes the upper part of the homoclinic orbit ABS, in which x0τ and y0τ are the even function and odd function, respectively.Now, if we introduce the following notation in system (8), Equation (8) has been changed into(13)X˙=fX+εgX,y=x˙,X=xy,fX=y−sinx+ρ,gX=0−cy+f1cosω1τ+f2cosω2τ.With regard to the homoclinic orbit, we obtain(14)trDf=01−cosx0=0,fXτ∧gXτ,τ+τ0=−cy2τ+yτf1cosω1τ0+τ+f2cosω2τ0+τ,in which the operator ∧ is defined as(15)a∧b=a1b2−a2b1,for any a=a1,a2T and b=b1,b2T.Therefore, the Melnikov function of Equation (8) is expressed as(16)Mτ0=∫−∞∞fXτ∧gXτ,τ+τ0dτ=∫−∞∞−cy2τ+yτf1cosω1τ0+τ+f2cosω2τ0+τdτ=−c∫−∞+∞y2τdτ+f1∫−∞+∞yτcosω1τ0+τdτ+f2∫−∞+∞yτcosω2τ0+τdτ=−cK+f1sinω1τ0Ihom1ω1+f2sinω2τ0Ihom2ω2,in which(17)K=∫−∞+∞y02τdτ=2∫0+∞y02τdτ=2∫xAxSy0τdx=2∫xAxS2Hρ+2cosx+2ρxdx,(18)Ihom1ω1=−2∫0+∞y0τsinω1τdτ=−2∫xAxSsinω1±∫xAxdθ2Hρ+2cosθ+2ρθdx,(19)Ihom2ω2=−2∫0+∞y0τsinω2τdτ=−2∫xAxSsinω2±∫xAxdθ2Hρ+2cosθ+2ρθdx.In Equations (12)–(14), xA stands for the horizontal ordinate of the points A and xS denotes the horizontal ordinates of the point S, in which the homoclinic orbit in Figure 2 intersects the x axis.Thus, we can see that(20)Mτ0=−cK+f1sinω1τ0Ihom1ω1+f2sinω2τ0Ihom2ω2=0,has a simple zero root for τ¯0 if and only if the following inequality holds(21)c<f1Ihom1ω1+f2Ihom2ω2K=Rf1,f2,ω1,ω2.The Melnikovian-detected chaotic boundary surface of equation (8) is obtained, as shown in Figure 3. When the system parameters traverse this boundary surface, the chaotic motions will be generated, i.e., in the area of c≤Rf1,f2,ω1,ω2.Figure 3
The boundary surface of chaotic motion.


## 4. Numerical Simulations

In this section, numerical simulations have been conducted to verify the theoretical analysis and to better understand the changes of the dynamic characteristic for the considered SMIB power system as the periodic load and the external periodic mechanical excitation vary. We useDynamics [22] to investigate the complicated dynamic behaviors of the SMIB power system by considering the bifurcation diagram. With the help of the software Matlab, the time histories, Poincaré maps, and frequency spectrums are plotted by using the fourth-order Runge–Kutta method. As an illustrative example, consider a specific system with the system parameters, and they are Rated MVA = 160, Rated PF = 0.85, Rated KV = 15, ωR=120π, and xd′=0.35. The rest of the parameters of the SMIB power system are (converted to per unit) M=1, PM=0.5,D=0.25,U=1,xT=0.65,andX=xd′+xT=1. We choose parameter f2 as a bifurcation parameter, which is corresponding to the external periodic mechanical excitation, and the other system parameters are fixed. Bifurcation diagram (Figure 4) are plotted to investigate the effects of varying the selected system parameter f2 for typical ranges.Figure 4
Bifurcation diagram with different values of parameterf2.As the excitation amplitudef2 increases, the rotor angle oscillation changes in terms of the sequence of period-1 motion, period-2 motion, and chaotic motion. Figure 5 shows the phase portrait, Poincare map, and frequency spectrum of period-1 motion for f2=0. Figures 6–8 show the phase portrait, Poincaré map, and frequency spectrum of period-2 motion for f2=0.1, f2=0.5, and f2=0.8, respectively. The differences between them are the different motion forms. The reason for that lies in different excitation amplitudes. It is easily to see that the shape of the periodic trajectory subjected to two-frequency excitation becomes more complex and more irregular. Figure 9 shows the phase portrait, Poincare map, and frequency spectrum of period-4 motion for f2=1.0. Figure 10 shows the phase portrait, Poincaré map, and frequency spectrum of chaotic motion for f2=1.1. In the case of f2=1.1 and the other chosen parameters, it can be easily verified that the parameter values satisfy the condition (21).Figure 5
Dynamic response (f2=0): (a) phase portrait; (b) Poincaré map; (c) frequency spectrum.
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Dynamic response (f2=0.1): (a) phase portrait; (b) Poincaré map; (c) frequency spectrum.
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Dynamic response (f2=0.5): (a) phase portrait; (b) Poincaré map; (c) frequency spectrum.
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Dynamic response (f2=0.8): (a) phase portrait; (b) Poincaré map; (c) frequency spectrum.
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Dynamic response (f2=1.0): (a) phase portrait; (b) Poincaré map; (c) frequency spectrum.
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Dynamic response (f2=1.1): (a) phase portrait; (b) Poincaré map; (c) frequency spectrum.
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## 5. Conclusions

This paper has investigated the dynamical behaviors of a classical single-machine infinite-bus power system by applying theoretical analysis and numerical simulations. Our results extend the existing research by considering two kinds of excitation components, which is related to the external mechanical and electrical disturbance. With the help of phase portraits, Poincaré maps, and frequency spectrums, the transition process of different dynamic behaviors is shown. Melnikov’s method is employed to give the threshold value of chaotic motion occurrence. In the considered system, there exists the phenomenon of period-doubling cascading bifurcation to chaos induced by the external excitation. These obtained results can give an overall understanding of the complex dynamic behaviors for this system.



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