Controlling Chaos through Period-Doubling Bifurcations in Attitude Dynamics for Power Systems

)is paper addresses the complex nonlinear dynamics involved in controlling chaos in power systems using bifurcation diagrams, time responses, phase portraits, Poincaré maps, and frequency spectra. Our results revealed that nonlinearities in power systems produce period-doubling bifurcations, which can lead to chaotic motion. Analysis based on the Lyapunov exponent and Lyapunov dimension was used to identify the onset of chaotic behavior. We also developed a continuous feedback control method based on synchronization characteristics for suppressing of chaotic oscillations. )e results of our simulation support the feasibility of using the proposed method. )e robustness of parametric perturbations on a power system with synchronization control was analyzed using bifurcation diagrams and Lyapunov stability theory.


Introduction
e characteristics of power systems are inherently nonlinear; this is due to the nonlinearity of synchronous generators. Power systems are usually described using a nonlinear dynamical system of equations and system parameters, where any change in one of the parameters can destabilize the entire system, resulting in chaotic motion and eventual voltage collapse as well as catastrophic blackout. Various researchers have studied voltage collapse in electric power systems [1][2][3][4][5].
Chaotic behavior must be accepted in some situations; however, it is normally undesirable, as it degrades performance and restricts the operating range of electric and mechanical devices. Beginning with Ott et al. [6], researchers [7,8] have sought to develop control methods for converting chaotic motions into periodic orbits or steady states. Researchers [9,10] have made particular progress in single-machine infinite bus systems (SMIB) by applying the classical generator model. In this work, we sought to convert chaotic behavior into periodic motion in dynamic power systems based on multiple machines. e synchronization of chaos has received considerable attention [11][12][13][14][15][16]. Usually, a periodic system is referred to as the master (drive) system and a chaotic system as the slave (response) system. e principle underlying synchronization is the control of a slave system using the output of a master system, such that the output of the slave system follows the output of the master system asymptotically. Continuous feedback control methods based on the properties of synchronization, such as the scheme developed by Pyragas [17] and Kapitaniak [18], can be used to convert chaotic motion into stable motion by using feedback in conjunction with a periodic external force.
is approach has been used in numerous chaotic systems [19,20]. Chang and Hu [19] used synchronization to lessen chaotic behavior in an automotive suspension system. Chang and Lue [20] used synchronization to achieve stable equilibrium in a magnetic levitation system.
Modern nonlinear theories of bifurcation and chaos have been developed to study nonlinear systems, and numerous studies on nonlinear dynamics in power systems have been published [21][22][23][24][25][26][27]. In this work, we employed bifurcation diagrams, phase portraits, Poincaré maps, and frequency spectra to examine the rich nonlinear dynamics of power systems. We also adopted advanced algorithms to compute Lyapunov exponents of smooth dynamical systems [28] to identify instances of chaos. en, we developed a continuous feedback control method based on synchronization characteristics to suppress chaotic oscillations in power systems. We then performed simulations to assess the feasibility and efficiency of the scheme. Finally, the design of the feedback controller was validated through the application of optimal control and Lyapunov stability theories, guaranteeing the global stability of nonlinear error systems [29][30][31].
is paper is organized as follows. Section 2 describes a power system model involving three synchronous generators with a resistive load configuration. Section 3 describes the complex dynamic behavior of a power system, which was elucidated using numerical analysis methods, such as bifurcation diagrams, phase portraits, Poincaré maps, and frequency spectra. Section 4 presents the Lyapunov exponent used to determine whether the system exhibits chaotic motion. A synchronization control technique for controlling chaos in power systems is described in Section 5. Section 6 reveals the effects of parameter errors on the performance of the proposed synchronization control system. Conclusions are outlined in Section 7.

Mathematical Modeling of Swing Equations for Three Machines
Synchronous generators are the most critical energy sources in power systems; however, they are also the source of most dynamic fluctuations. A power system model involving three synchronous generators with a resistive load configuration is considered, as illustrated in Figure 1. Following [32,33], the dynamics of this nonlinear power system can be expressed as follows: Now, a special case is considered in which swing equations are applied for dynamics analysis of the three machines. We assume that Machine 1 has large inertia, where M 1 � (M 1 /ε), ε ≪ 1. e transmission line joining Machines 2 and 3 is shorter than the other lines. Similarly, the external power P 1 is proportionally larger: P 1 � P 1 /ε. Under these assumptions, the conservative swing equations for three machines can be written as follows: In accordance with [32,33], δ 1 can be expressed as follows: where μ 2 � (M 2 /M 1 ) and μ 3 � (M 3 /M 1 ). Substituting equation (3) into equations (2a)-(2f ), we can construct an autonomous differential equation for δ 2 , δ 3 , ω 2 , and ω 3 , thereby eliminating δ 1 and ω 1 as follows:

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In accordance with [32,33], α k can be expressed as follows: where P k indicates the constant real power, K f � L k /M k , and L k is a load-frequency coefficient. Substituting equation (6) into equations (5a)-(5d), we obtain Assuming y 1 � δ 2 , y 2 � ω 2 , y 3 � δ 3 , and y 4 � ω 3 are the state variables, such that the state-space model of the swing equation can be written as follows: e numerical values of all parameters in equations (8a)-(8d) are listed in Table 1 [33].

Power System Characteristics
Numerical simulations based on equations (8a)-(8d) were used to elucidate the characteristics of the power system. e commercial package DIVPRK of IMSL was used to write a series of FORTRAN subroutines to solve the ordinary differential equation (ODE) [34]. e resulting bifurcation diagram displayed in Figure 2 shows that the first perioddoubling bifurcation occurred at approximately K f � 0.078, with chaotic motion appearing at approximately K f � 0.0109. More details on the various responses exhibited by the system are shown in Figures 3-6. Each response type is characterized using a phase portrait, Poincaré map, and frequency spectrum. When the parameter K f > 0.078, the Further decreasing K f resulted in a cascade of period-doubling bifurcations, leading to chaos. e resulting chatter vibration was sufficient to cause a voltage collapse. We used Poincaré maps and frequency spectra to characterize the chaotic behavior. Poincaré maps present chaotic motion as an infinite set of points (a strange attractor). Strange attractors and continuous Fourier spectra are strong indicators of chaos. Figures 6(a)-6(c) illustrate the chaotic behavior in detail.

Analysis of Chaotic Phenomena in Power Systems
In this section, Lyapunov exponents are used to overcome the limitations of the schemes used in Section 3 to identify the occurrence of chaotic motion. Every dynamic system involves a spectrum of Lyapunov exponents (λ) [28], which indicate changes in the length, area, and volume of a phase space. Determining whether a system exhibits characteristics Table 1: Physical parameters of a power system. of chaos only requires calculation of the largest Lyapunov exponent to determine whether, on average, the nearby trajectories diverge (λ > 0) or converge (λ < 0). Chaotic motion exhibits at least one positive Lyapunov exponent within a bounded system, whereas periodic motion exhibits no positive Lyapunov exponents.
In the current study, we adopted the algorithm proposed by Wolf et al. [28] for the calculation of Lyapunov exponents. Figure 7 illustrates the evolution of the largest Lyapunov exponent, in which the onset of chaotic motion occurs at approximately K f � 0.0109. At point P 3 , the gradual reduction in K f causes the sign of the largest Lyapunov exponent to change from negative to positive. At points P 1 and P 2 , the largest Lyapunov exponent approached zero, beyond which the system is susceptible to bifurcation. However, the Lyapunov exponent at these points provides no information pertaining to the class of bifurcation. We must therefore consult the bifurcation diagram presented in Figure 2. A    comparison of Figures 7 and 2 revealed that period-2 and period-4 bifurcations occur at P 1 and P 2 , respectively. When K f � 0.09, the Lyapunov exponents obtained using equations (8a)-(8d) are λ 1 � −0.000084, λ 2 � −0.0157341, λ 3 � −0.5120906, and λ 4 � −0.5977934. e negative value λ 1 + λ 2 + λ 3 + λ 4 � −1.1257021 indicates that the power system exhibits stable periodic motion in its current state. Denoting λ 1 ≥ · · · ≥ λ n as the Lyapunov exponents of a dynamical system, Kaplan and Yorke [35] provided a means by which to estimate the Lyapunov dimension d L :

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where j i�1 Periodic orbit is indicated by an integer-valued Lyapunov dimension, whereas the noninteger value of the Lyapunov dimension indicates chaotic motion. When K f � 0.09, using the technique in equations (8a)-(8d), returned a Lyapunov dimension of d L � 1, which indicates a system undergoing periodic motion. When K f decreased beyond the bifurcation point P 3 to K f � 0.009, the Lyapunov exponents were λ 1 � 0.0340716, λ 2 � −0.0000068, λ 3 � −0.3985904, and λ 4 � −0.5270599, and the Lyapunov dimension was d L � 2.0854. In this situation, the largest Lyapunov exponent was positive and the Lyapunov dimension was a noninteger, indicating that the power system was undergoing chaotic motion.

Controlling Chaos through Synchronization
Although there are some benefits to identifying chaotic systems, the ultimate objective is to assume control over such systems. Avoiding chaos in dynamic systems requires transformation of chaotic motion into periodic motion. Pyragas [17] and Kapitaniak [18] proposed a simple yet effective time-continuous control method for the conversion of chaotic motion into periodic motion using time-continuous perturbation with a feedback mechanism. Figure 8 plots the control system proposed in this study, which comprises a feedback-controlled loop with external periodic perturbation. is method is explained briefly in the following paragraph.
Consider the following n-dimensional dynamic system: where x(t), y(t) ∈ R n is the state vector and F(t) is the input signal. Equation (11) without an input signal (F(t) � 0) is assumed to have a strange attractor, whereas periodic motion is observed in equation (12). rough the strategy presented in Figure 8, the periodic system (i.e., drive system) is synchronized with the chaotic system (i.e., response system). e difference between signals y(t) and x(t) yields the following control signal: where K indicates the feedback gain. When K f � 0.09 is selected as the drive system, equations (14a)-(14d) reveal period-1 motion.

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Synchronizing equations (14a)-(14d) and (15a)-(15d) involves introducing the control signal in equation (13) into equations (15a)-(15d) as feedback control, which yields the following coupled system: Equations (16a)-(16d) describe chaotic motion when K � 0 and K f � 0.008. e feedback gain K is adjusted between 0.0 and −1.0 to convert the dynamics of the system from chaotic motion into periodic motion. Figure 9 presents the resulting bifurcation diagram, which comprehensively explains the dynamic behavior of the system over a range of feedback gains. Stable periodic motion appears when K decreases below 0.005. Period-doubling bifurcations appear when K decreases to between approximately −0.135 and −0.375. A further decrease in K beyond −0.375 results in period-1 motion. e efficacy of the proposed system in controlling chaos was demonstrated by the application of a synchronization control signal (K � −0.8) after a period of 100 s, as displayed in Figure 10.

Effects of Parametric Perturbations on Power Systems Using Synchronization Control
We sought to understand the effects of parameter errors on the performance of the proposed synchronization control system by adding a sinusoidal perturbation directly to parameters β 21 , β 23 , β 31 , β 32 , D 2 , D 3 , P 2 , and P 3 in the periodic excitation (drive system). Assume that equations (14a)-(14d) are representative of the drive system; then, the corresponding controlled response system is given by where ε is the amplitude of the perturbation and ω is the angular frequency. Subtracting equations (14a)-(14d) from equations (17a)-(17d), we obtain the following error equation: _ e 2 � P 2 ε sin(ωt) − D 2 ε sin(ωt) + K f e 2 − β 21 ε sin(ωt)sin e 1 − β 23 ε sin(ωt)sin e 1 − e 3 + u 2 , where e 1 � z 1 − y 1 , e 2 � z 2 − y 2 , e 3 � z 3 − y 3 , and e 4 � z 4 − y 4 . Considering a Lyapunov function for equations (18a)-(18d), We obtain the first derivative of V(e) as follows:  Mathematical Problems in Engineering erefore, if we select is a negatively defined function, the error states are lim t⟶∞ e(t) � 0.
us, the states of the controlled response system and drive system are synchronized globally and asymptotically [36]. e correctness of these theoretical results was evaluated using simulations with the following perturbation parameters: ε � 0.05 and ω �125.6 rad/s. e results are, respectively, presented in Figures 11(a)-11(d), as follows: e 1 � z 1 − y 1 , e 2 � z 2 − y 2 , e 3 � z 3 − y 3 , and e 4 � z 4 − y 4 . e synchronization error eventually converges to zero, resulting in stabilization of the error system. is is a clear demonstration that despite parametric perturbations, the controller is able to synchronize the states of the drive and response systems. In other words, the proposed control method is robust to parameter mismatch in the power system. e bifurcation diagram in Figure 12 was also used to illustrate the effects of parametric perturbation. Again, the proposed control method was shown to suppress chaotic behavior under the perturbed parameters (e.g., β 21 , β 23 , β 31 , β 32 , D 2 , D 3 , P 2 , and P 3 ) in the periodic excitation using equations (14a)-(14d).

Conclusions
Phase portraits, Poincaré maps, and frequency spectra were used to explore the rich nonlinear dynamics in power systems as well as methods used to control chaos. Bifurcation diagrams revealed several nonlinear behaviors associated with the emergence of chaotic motion at lower values of K f . We also observed a cascade of period-doubling bifurcations prior to the onset of chaos. e most powerful approach to predicting chaotic motion involves computing the Lyapunov exponent and Lyapunov dimension. We developed a continuous feedback-control method based on synchronization for suppressing chaotic motion in power systems.
e robustness of parametric perturbation on a power system with synchronization control was analyzed using bifurcation diagrams and Lyapunov stability theory. We believe that efforts to control chaos based on nonlinear dynamics could help to prevent voltage collapse in power systems.

Data Availability
All data generated or analyzed during this study are included within this article.
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