# Stability, Chaos Detection, and Quenching Chaos in the Swing Equation System

Authors: Shun-Chang Chang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The main objective of this study is to explore the complex nonlinear dynamics and chaos control in power systems. The rich dynamics of power systems were observed over a range of parameter values in the bifurcation diagram. Also, a variety of periodic solutions and nonlinear phenomena could be expressed using various numerical skills, such as time responses, phase portraits, Poincaré maps, and frequency spectra. They have also shown that power systems can undergo a cascade of period-doubling bifurcations prior to the onset of chaos. In this study, the Lyapunov exponent and Lyapunov dimension were employed to identify the onset of chaotic motion. Also, state feedback control and dither signal control were applied to quench the chaotic behavior of power systems. Some simulation results were shown to demonstrate the effectiveness of these proposed control approaches.


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## Body

## 1. Introduction

The characteristics of power systems are well known to be inherently nonlinear owing to the nonlinearity of synchronous generators. The most important issue in power system operation is the prevention of voltage collapse. Various works have already studied the voltage collapse in electric power systems [1–4]. A power system is typically described by a nonlinear dynamical system of equations, including system parameters. Altering one of these parameters changes the power system dynamics that exhibit chaos motion, leading to voltage collapse. However, the chaotic motion in a power system may destabilize it, leading to voltage collapses and even catastrophic blackouts. Modern nonlinear theories of bifurcation and chaos are widely adopted in the studies of nonlinear systems, and chaos dynamics in power systems have been widely studied [5–14]. This work presents several numerical schemes, including bifurcation diagrams, phase portraits, Poincaré maps, and frequency spectra, to clearly explain the rich nonlinear dynamics in power systems. Additionally, the Lyapunov exponents of smooth dynamical systems were calculated using highly developed algorithms [15–17] with the objective of determining whether a system exhibits chaos or not.Chaotic behaviors in power systems are considered undesirable due to the restrictions they impose on the operating ranges of electrical and mechanical devices. The dynamics of a power system become unstable when they exhibit chaotic motions. If instability is not properly controlled, it causes voltage collapse, eventually leading to blackouts [12]. Hence, in many engineering applications, control approaches are developed to convert chaotic motions into periodic orbits or steady states. Since the pioneering work of Ott et al. [18] in controlling chaos, many modified methods and other approaches have been successively proposed [19–26]. Various control algorithms have also been presented to control the chaos of power systems [8, 27–30]. This work proposed converting chaotic behaviors into periodic motions to improve the performance of system dynamics with multiple machine power system chaotic behaviors. Also, chaotic motions in a power system were inhibited using state feedback control [21, 31] and dither signal control [32], and the simulation results were presented to confirm the feasibility and efficacy of the proposed control approaches.


## 2. The Problem Description and Modeling of Special Swing Equations for Three Machines

A power system model of three synchronous generators with a resistive load configuration was considered, as shown in Figure1. Synchronous generators are the most important energy source for power systems. The study of interconnected power systems with synchronous motors as significant equipment is the key to studying the dynamic characteristics of power systems. The governing differential equations of the power system in Figure 1 can be expressed as follows [33, 34]:(1a)δ˙1=ω˙1,(1b)M1ω˙1=−D1ω1+P1−F12sinδ1−δ2−F13sinδ1−δ3,(1c)δ˙2=ω˙2,(1d)M2ω˙2=−D2ω2+P2−F21sinδ2−δ1−F23sinδ2−δ3,(1e)δ˙3=ω˙3,(1f)M3ω˙3=−D3ω3+P3−F31sinδ3−δ1−F32sinδ3−δ2.Figure 1
Schematic of a power system.A special case of the swing equations was considered for three machines. It was assumed that machine 1 has a large inertia, i.e.,M1=M¯1/ε and ε≪1. The transmission line joining machines 2 and 3 is shorter than that of the other lines. Similarly, the external power P1 is proportionally larger: P1=P¯1/ε. With these assumptions, the conservative swing equations for the three machines are(2a)δ˙1=ω˙1,(2b)M¯1ω˙1=−εD1ω1+P¯1−εF12sinδ1−δ2−εF13sinδ1−δ3,(2c)δ˙2=ω˙2,(2d)M2ω˙2=−D2ω2+P2−F21sinδ2−δ1−F23sinδ2−δ3,(2e)δ˙3=ω˙3,(2f)M3ω˙3=−D3ω3+P3−F31sinδ3−δ1−F32δ3−δ2.From [33, 34], δ1 can be expressed as follows:(3)δ1=−εμ2δ2−εμ3δ3,where μ2=M2/M¯1 and μ3=M3/M¯1.By substituting equation (3) in equations (2a)–(2f), write an autonomous differential equation for δ2, δ3, ω2, and ω3 and eliminate δ1 and ω1 as follows:(4a)δ˙2=ω˙2,(4b)ω˙2=−D2ω2+α¯2−β21sinδ21+εμ2+εμ3δ3−β23sinδ2−δ3,(4c)δ˙3=ω˙3,(4d)ω˙3=−D3ω3+α¯3−β31sinδ21+εμ2+εμ3δ3−β32sinδ3−δ2,where α¯2=P2/M2, β21=F21/M2, β23=F23/M2, α¯3=P3/M3, β31=F31/M3, and β32=F32/M3.For convenience, letε = 0, and simplify equations (4a)–(4d) into(5a)δ˙2=ω˙2,(5b)ω˙2=−D2ω2+α¯2−β21sinδ2−β23sinδ2−δ3,(5c)δ˙3=ω˙3,(5d)ω˙3=−D3ω3+α¯3−β31sinδ3−β32sinδ3−δ2.From [33, 34], express α¯k as follows:(6)α¯k=P¯k−Kfωk,k=2,3,where Pk is a constant real power and Kf = Lk/Mk. Lk is a load-frequency coefficient. By substituting equation (6) in (5a)–(5d),(7a)δ˙2=ω˙2,(7b)ω˙2=−D2ω2+P¯2−Kfω2−β21sinδ2−β23sinδ2−δ3,(7c)δ˙3=ω˙3,(7d)ω˙3=−D3ω3+P¯3−Kfω3−β31sinδ3−β32sinδ3−δ2.Withy1=δ2, y2=ω2, y3=δ3, and y4=ω3 as the state variables, the state equations of the swing equation can be written as follows:(8a)y˙1=y2,(8b)y˙2=P¯2−D2+Kfy2−β21siny1−β23siny1−y3,(8c)y˙3=y4,(8d)y˙4=P¯3−D3+Kfy4−β31siny3−β32siny3−y1.Table1 [31] lists the numerical values for all the parameters in equations (8a)–(8d).Table 1
Physical parameters of a power system.
ParameterValueβ21−2β23−1β31−1β32−1D20.2D30.5P¯20.3P¯30.5


## 3. Overall Characteristics of the Power System: Simulation Results and Discussion

Numerical simulations were executed based on equations (8a)–(8d) to clearly understand the overall characteristics of the power system. The commercial package DIVPRK of IMSL in the FORTRAN subroutines was utilized for the mathematic applications to solve the ordinary differential equation problems [35]. Figure 2 presents the resulting bifurcation diagram, which clearly reveals that the first period-doubling bifurcation occurred at about Kf = 0.078 and that a chaotic motion appeared approximately below Kf = 0.0109. Figures 3–6 display various responses exhibited by this system in detail, where each type of response was characterized in detail using a phase portrait, a Poincaré map, and a frequency spectrum. The equilibrium point of equations (8a)–(8d) was stable with Kf > 0.078, revealing that no chatter vibration occurred. Figures 3(a)–3(d) show the period-1 motion. Also, Figures 4(a)–4(d) show a cascade of period-doubling bifurcations with new frequency components at Ω/2, 3Ω/2, 5Ω/2…, causing a series of subharmonic components. Figures 5(a)–5(d) depict the first period-four bifurcation, which occurred when Kf fell below Kf ≈ 0.0275. A cascade of chaos-inducing period-doubling bifurcations then appeared as Kf continued to fall in Figure 2, resulting in a chatter vibration that could cause a voltage collapse, thereby significantly reducing the power system performance and possibly causing catastrophic blackouts. Two descriptors, the Poincaré map and frequency spectrum, could be utilized to characterize the essence of the chaotic behavior. The Poincaré map shows an infinite set of points called a strange attractor. Simultaneously, the frequency spectrum of the chaotic motion is a continuous broad spectrum. These two main features, strange attractors and continuous-type Fourier spectrum, are strong indicators of chaos. Figures 6(a)–6(d) clearly reveal the chaotic behavior in detail.Figure 2
Bifurcation diagram of the rotor angular velocity againstKf.Figure 3
Period-1 orbit of the numerical simulation results forKf = 0.09: (a) time responses; (b) phase portrait; (c) Poincaré map; (d) frequency spectrum.
(a)(b)(c)(d)Figure 4
Period-2 orbit of the numerical simulation results forKf = 0.06: (a) time responses; (b) phase portrait; (c) Poincaré map; (d) frequency spectrum.
(a)(b)(c)(d)Figure 5
Period-4 orbit of the numerical simulation resultsKf = 0.02: (a) time responses; (b) phase portrait; (c) Poincaré map; (d) frequency spectrum.
(a)(b)(c)(d)Figure 6
Chaotic motion of the numerical simulation results forKf = 0.009: (a) time responses; (b) phase portrait; (c) Poincaré map; (d) frequency spectrum.
(a)(b)(c)(d)


## 4. Lyapunov Exponent and Lyapunov Dimension for Analyzing the Chaos in the Power System

As described in Section3, the chaotic motion in power systems is difficult to identify using traditional approaches. This section describes the use of Lyapunov exponents to verify the occurrence of chaos in power systems. Every dynamic system involves a spectrum of Lyapunov exponents (λ) [15], indicating changes in the length, area, and volume within a phase space. To determine whether a system exhibits the characteristics of chaos, only the largest Lyapunov exponent needs to be calculated to determine whether nearby trajectories diverge (λ > 0) or converge (λ < 0) on average. Any bounded motion in a system with at least one positive Lyapunov exponent is defined as chaotic, whereas periodic motions exhibit no positive Lyapunov exponents.Figure7 plots the evolutions of the largest Lyapunov exponent in the power system, as computed using the algorithm proposed by Wolf et al. [15]. This figure reveals that the onset of the chaotic motion occurred at approximately Kf = 0.0109. At point P3, the sign of the largest Lyapunov exponent changed from negative to positive as the parameter Kf slowly decreased. At points P1 and P2, the largest Lyapunov exponents approached zero, which is beyond the point at which the system can undergo bifurcation. Nonetheless, the Lyapunov exponent at that point provided no indication of the involved bifurcation type, thereby necessitating the application of a bifurcation diagram, as shown in Figure 2. A conducted comparison of Figures 7 and 2 indicated the occurrence of period-2 bifurcation at P1 and period-4 bifurcation at P2. When Kf = 0.09, the Lyapunov exponents obtained using equations (8a)–(8d) were λ1 = −0.000084, λ2 = −0.0157341, λ3 = −0.5120906, and λ4 = −0.5977934, and their sum was λ1 + λ2 + λ3 + λ4 = −1.1257021, which is negative, indicating that the power system had stable periodic motion. By denoting λ1≥⋯≥λn as the Lyapunov exponents of a dynamical system, Kaplan and Yorke [36] expressed the estimate of the Lyapunov dimension dL as(9)dL=j+1λj+1∑i=1jλi,where∑i=1jλi>0and∑i=1j+1λi<0.Figure 7
Evolutions of the largest Lyapunov exponent versusKf.Thus, the Lyapunov dimension is an integer for periodic orbits and a noninteger for chaotic motion. Using equations (8a)–(8d) with Kf = 0.09, this calculation yielded a Lyapunov dimension of dL = 1. Therefore, this system exhibited periodic motion, as the Lyapunov dimension was an integer. When the parameter Kf continuously decreased across bifurcation point P3, for example, Kf = 0.009, the Lyapunov exponents were λ1 = 0.0340716, λ2 = −0.0000068, λ3 = −0.3985904, and λ4 = −0.5270599, and the Lyapunov dimension was dL = 2.0854. Obviously, the power system could be demonstrated to exhibit chaotic motion, as the largest Lyapunov exponent was positive and the Lyapunov dimension was a noninteger.


## 5. Quenching Chaos in the Power System

Analyzing and predicting the behaviors of chaotic systems are beneficial, but a system needs to be controlled to maximize its benefits. Improving the performance of a dynamic system and avoiding chaotic motion both require periodic motion, which is more important when working under specific conditions. This section presents two control methods, state feedback control [21, 31] and dither control [32], to suppress the chaos in the used power system in this study.


### 5.1. State Feedback Control

Cai et al. [21, 31] proposed a simple and effective method for converting chaos into periodic motion at a steady state using the linear-state feedback of an available system variable. For an n-dimensional dynamic system, this method can be briefly summarized as follows:(10)x˙=fx,t,where x(t) ∈ Rn is the state vector and f = (f1,…, fi,…, fn), where fi is a linear or nonlinear function and f includes at least one nonlinear function. If fk(x, t) is the key nonlinear function that leads to chaotic motion in equation (10), only one term of the state feedback of an available system variable xm was added to the equation that includes fk(x, t) as follows:(11)x˙k=fkx,t+Kxm,k,m∈1,2,…,n,where K is the feedback gain, and the other functions keep their original forms.Equations (8a)–(8d) can be rewritten as follows with the state feedback control:(12a)y˙1=y2,(12b)y˙2=P¯2−D2+Kfy2−β21siny1−β23siny1−y3+Ky2,(12c)y˙3=y4,(12d)y˙4=P¯3−D3+Kfy4−β31siny3−β32siny3−y1+Ky4.Without the state feedback control, equations (8a)–(8d) exhibited chaotic behavior under the parameter Kf = 0.009. Considering that the effect of the state feedback control was added to the right-hand side of equations (8a)–(8d), by decreasing the feedback gain K from 0 to −0.1, the chaotic behavior disappeared at certain feedback gains. Figure 8 presents the resulting bifurcation diagram, which comprehensively explains the dynamic behavior of the controlled power system over a range of feedback gains. Chaotic motion appeared when K < −0.0008, and stable periodic motion appeared when K decreased beyond −0.0008. Period-doubling bifurcations appeared when K decreased to between about −0.0699 and −0.0008. A further decrease in K beyond −0.0699 resulted in a period-1 motion. The efficacy of the proposed system in controlling chaos was demonstrated by applying a control signal after 300 seconds, as shown in Figure 9. Therefore, to suppress the occurrence of chaos, the simple state feedback of an available system variable can be sufficient to disturb the balance of dynamic behaviors in a chaotic system.Figure 8
Bifurcation diagram of the rotor angular velocity againstK for the power system with the state feedback control, where K indicates the feedback gain.Figure 9
Transforming chaotic motion into a period-1 orbit forK = −0.08 and Kf = 0.009: (a) phase portrait of the controlled system; (b) time responses of the controlled system. A state feedback control signal was introduced after 300 s.
(a)(b)


### 5.2. Dither Control

This section describes how to control motion in a chaotic system by injecting another external input dither signal to only modify nonlinear terms. A dither signal averages nonlinearity due to its high frequency and periodic nature. Researchers have developed dither smoothing methods [32, 37] to stabilize chaotic systems, and popular dither signals were proposed in [38]. The simplest dither signal is a square-wave dither signal in which the frequency and amplitude are 2000 rad/s and W, respectively, in front of the nonlinearity f(.). Therefore, the effective value of μ and the output of the nonlinear element can be expressed as(13)μ=12fy+W+fy−W.Consequently, the system equation can be expressed as(14)y˙=μ.Considering the influence of the dither signal control added to systems (8a)–(8d) under the parameter Kf = 0.009, by raising the amplitude of the square-wave dither signal from W = 0 to W = 0.5, the dynamics changed from chaotic behaviors to periodic motions. Figure 10 displays the evolution of the bifurcation diagram. Figure 11(a) depicts the time response of the displacement with the amplitude of the square-wave dither signal W = 0.2. The chaotic behavior system was transformed into a period-1 orbit. Figure 11(b) illustrates the phase portrait of the controlled system. Notably, the system exhibited a chaotic behavior before dither was introduced but exhibited a periodic motion afterward.Figure 10
Bifurcation diagram of the rotor angular velocity againstW for the power system with square-wave dither, where W represents the dither amplitude.Figure 11
A square-wave dither signal was injected to control the chaotic motion of the power system forW = 0.2 and Kf = 0.009: (a) time responses of the controlled system; (b) phase portrait of the controlled system.
(a)(b)


## 5.1. State Feedback Control

Cai et al. [21, 31] proposed a simple and effective method for converting chaos into periodic motion at a steady state using the linear-state feedback of an available system variable. For an n-dimensional dynamic system, this method can be briefly summarized as follows:(10)x˙=fx,t,where x(t) ∈ Rn is the state vector and f = (f1,…, fi,…, fn), where fi is a linear or nonlinear function and f includes at least one nonlinear function. If fk(x, t) is the key nonlinear function that leads to chaotic motion in equation (10), only one term of the state feedback of an available system variable xm was added to the equation that includes fk(x, t) as follows:(11)x˙k=fkx,t+Kxm,k,m∈1,2,…,n,where K is the feedback gain, and the other functions keep their original forms.Equations (8a)–(8d) can be rewritten as follows with the state feedback control:(12a)y˙1=y2,(12b)y˙2=P¯2−D2+Kfy2−β21siny1−β23siny1−y3+Ky2,(12c)y˙3=y4,(12d)y˙4=P¯3−D3+Kfy4−β31siny3−β32siny3−y1+Ky4.Without the state feedback control, equations (8a)–(8d) exhibited chaotic behavior under the parameter Kf = 0.009. Considering that the effect of the state feedback control was added to the right-hand side of equations (8a)–(8d), by decreasing the feedback gain K from 0 to −0.1, the chaotic behavior disappeared at certain feedback gains. Figure 8 presents the resulting bifurcation diagram, which comprehensively explains the dynamic behavior of the controlled power system over a range of feedback gains. Chaotic motion appeared when K < −0.0008, and stable periodic motion appeared when K decreased beyond −0.0008. Period-doubling bifurcations appeared when K decreased to between about −0.0699 and −0.0008. A further decrease in K beyond −0.0699 resulted in a period-1 motion. The efficacy of the proposed system in controlling chaos was demonstrated by applying a control signal after 300 seconds, as shown in Figure 9. Therefore, to suppress the occurrence of chaos, the simple state feedback of an available system variable can be sufficient to disturb the balance of dynamic behaviors in a chaotic system.Figure 8
Bifurcation diagram of the rotor angular velocity againstK for the power system with the state feedback control, where K indicates the feedback gain.Figure 9
Transforming chaotic motion into a period-1 orbit forK = −0.08 and Kf = 0.009: (a) phase portrait of the controlled system; (b) time responses of the controlled system. A state feedback control signal was introduced after 300 s.
(a)(b)


## 5.2. Dither Control

This section describes how to control motion in a chaotic system by injecting another external input dither signal to only modify nonlinear terms. A dither signal averages nonlinearity due to its high frequency and periodic nature. Researchers have developed dither smoothing methods [32, 37] to stabilize chaotic systems, and popular dither signals were proposed in [38]. The simplest dither signal is a square-wave dither signal in which the frequency and amplitude are 2000 rad/s and W, respectively, in front of the nonlinearity f(.). Therefore, the effective value of μ and the output of the nonlinear element can be expressed as(13)μ=12fy+W+fy−W.Consequently, the system equation can be expressed as(14)y˙=μ.Considering the influence of the dither signal control added to systems (8a)–(8d) under the parameter Kf = 0.009, by raising the amplitude of the square-wave dither signal from W = 0 to W = 0.5, the dynamics changed from chaotic behaviors to periodic motions. Figure 10 displays the evolution of the bifurcation diagram. Figure 11(a) depicts the time response of the displacement with the amplitude of the square-wave dither signal W = 0.2. The chaotic behavior system was transformed into a period-1 orbit. Figure 11(b) illustrates the phase portrait of the controlled system. Notably, the system exhibited a chaotic behavior before dither was introduced but exhibited a periodic motion afterward.Figure 10
Bifurcation diagram of the rotor angular velocity againstW for the power system with square-wave dither, where W represents the dither amplitude.Figure 11
A square-wave dither signal was injected to control the chaotic motion of the power system forW = 0.2 and Kf = 0.009: (a) time responses of the controlled system; (b) phase portrait of the controlled system.
(a)(b)


## 6. Conclusions

This work addressed the rich nonlinear dynamics and chaos control in power systems. The resulting bifurcation diagram showed many nonlinear behaviors, indicating that the tested power system exhibited chaotic motion at lowerKf, meaning that the system could undergo a cascade of period-doubling bifurcations prior to the onset of chaos. Numerical approaches, including phase portraits, Poincaré maps, and frequency spectra, were employed to explore the dynamics of the used power system. The most powerful algorithm to determine whether a power system is in chaotic motion or not is to use the Lyapunov exponent and Lyapunov dimension. The presence of a chaotic behavior was generic for certain nonlinearities, parameter ranges, and external forces, and it may need to be avoided or controlled to improve the performance of the power system. The state feedback control scheme was simple and effective for chaos suppression, and it can be implemented by adding the feedback of suitable variables to the original system with sufficient control gain to quench chaos development. Additionally, the square wave of the dither signal can be applied to efficiently convert a chaotic motion into a periodic orbit by injecting a dither signal in front of the nonlinearity of the power system. We believe that studying the nonlinear dynamics and chaos control in power systems will help prevent voltage collapses and advance the development of smart power systems.Other numerous methods for chaos control have been devised, such as synchronization control, time-delayed feedback control, neurofuzzy control, adaptive control, and bang-bang control. In this study, two chaos control strategies, state feedback control and dither control, were implemented to control the chaotic behavior of a power system. The effectiveness of these proposed chaos control strategies was illustrated through numerical simulations. Also, a simple control method was presented to convert chaos into periodic motion using the linear-state feedback of an available system variable. Overall, it was found that compared with other chaos control methods, the state feedback control technique is simple and can be easily implemented in chaos suppression. Finally, the robustness of the parametric perturbation on a chaotic system with various chaos control methods will be studied in my future work.



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