#### Abstract

This paper focuses on the swing oscillation process of the synchronous generator rotors in a three-machine power system. With the help of bifurcation diagram, time history, phase portrait, Poincaré section, and frequency spectrum, the complex dynamical behaviors and their evolution process are detected clearly in this power system with varying perturbation related parameters and different system parameters. Furthermore, combining the qualitative and quantitative characteristics of the chaotic motion, different paths leading to chaos coexisting in this system have been found. The Wolf method has been introduced to calculate the corresponding largest Lyapunov exponent, which is used to verify the occurrence of chaotic motion. These results obtained in this paper will contribute to a better understanding of nonlinear dynamic behaviors of synchronous generator rotors in a three-machine power system.

#### 1. Introduction

When power system suffers disturbances, it can cause parameters to change. This will result in the system exhibiting abundant nonlinear dynamic behaviors including chaotic behavior. Chaotic oscillation in power system may cause voltage collapse and even catastrophic blackouts. Consequently, chaotic oscillation in the power system often makes a great threat to the stability of the power grid [1, 2].

Discovery of chaos can enrich our understanding of complex and unpredictable nonlinear behaviors arising in power system; therefore, it is an important part of power system stability research. In the last decades, the nonlinear dynamic behaviors including chaotic oscillation in power system have attracted much attention of the researchers. For single-machine infinite bus (SMIB) power system, Wei et al. [3, 4] examined how a Gaussian white noise affected the dynamic behaviors of power system by means of a random Melnikov method. Zhu and Mohler [5] made a Hopf bifurcation analysis with subsynchronous resonance (SSR). Nayfeh et al. [6, 7] investigated the period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism) on a single-machine quasi-infinite bus system. Duan et al. [8] studied the bifurcations associated with subsynchronous resonance of a SMIB power system with series of capacitor compensation. Chen et al. [9] studied the chaotic control and identification problem of a SMIB power system, where the power of the machine was assumed to be a simple harmonic quantity. Moreover, in order to discuss bifurcations of periodic orbits and homo-(hetero-) clinic orbits of dynamical systems, Melnikov’s method has been effectively applied in the classical SMIB power system models [10–14].

For a two-machine power system, under some particular conditions, Yuan and Sun [15] studied the occurrence of chaotic phenomenon by using Melnikov’s method. Ueda et al. [16] investigated a two-generator electric power system model by considering that the infinite bus maintains a voltage of fixed amplitude with a small periodic fluctuation in the phase angle.

For a three-machine power system, Majidabad et al. [17] designed two novel nonlinear fractional-order sliding mode controllers and applied them in a three-machine power system with two types of faults. Jiang et al. [18] proposed a Weierstrass-based numerical method for computing damping torque of a three-machine power system during transient period to determine appropriate and accurate damping terms for power system dynamic simulation. The bifurcation phenomena in a power system with three machines and four buses were investigated by applying bifurcation theory and harmonic balance method in [19]. In addition, transient stability analysis of a three-machine nine bus power system was carried out by considering a three-phase fault at busbars 7 and 4 with the effect of various fault-clearing times [20].

Although the chaotic oscillations in simple power system (especially for SMIB) have been studied with many mathematical models and theories, to the best of our knowledge, the chaotic oscillation research of three-machine power system with power disturbance is not systemic and thorough enough. In order to deal with the issue mentioned above, this paper provides an efficient analysis method of assessing the dynamic impacts of critical system parameters on dynamic characteristics of the power system. The dynamic characteristics of the power system model with varying different system parameters and perturbation related parameters are investigated for the first time.

The synchronous generators are the main source of energy for the power system and they are also the core of the entire grid. The motivation of this paper is to detect the effect of several critical system parameters on the nonlinear dynamic characteristics of synchronous generators in three-machine power system. Firstly, the swing equations describing the motions of the synchronous generator rotors are established. Based on the swing equations, a more in-depth research on the dynamical properties of the system has been carried out. In addition, the Wolf method is introduced to calculate the largest Lyapunov exponent, which enables us to identify the bifurcation points and to analyze the dynamic characteristics of the system influenced by the power disturbance. With the aid of bifurcation diagram, time history, phase portrait, Poincaré section, and frequency spectrum, the detailed numerical simulations are also carried out.

The remainder of this paper is organized as follows. In Section 2, the swing equations describing the motions of the synchronous generator rotors are formulated. In Section 3, in order to investigate the effects of the system parameters and perturbation related parameters on the dynamic characteristics of the system, some numerical simulations are carried out. In Section 4, some discussions have been made. Finally, concluding remarks of this paper are presented in Section 5.

#### 2. System Configuration and Modelling

Synchronous motor as the significant equipment is the key to studying the dynamic characteristic of the power system. Consider a general three-machine infinite bus power system as shown in Figure 1; the swing equations describing the generator rotors are written as Here is the rotor angle of the generator, is the synchronous speed, is the deviation between the synchronous speed and the rotor angular velocity, is the generator rotor inertia, is the mechanical input power to the generator, is the electromagnetic power, and is the damping of the generator. is the magnitude of voltage behind the transient reactance of the generator. are the self-conductance and self-admittance of the node. are the mutual conductance and mutual admittance of the node.

Taking into account the practical engineering application, an assumption is made about the impedance angle; the electromagnetic power is denoted in the format as

Therefore, it can be seen that the expression of every electromagnetic output power is a multivariable function related to the relative angle between its own rotor angle and the other two generator angles.

Considering that the generators are connected with the infinite bus node, combining (1) and (3), let ; the following equation can be obtained: is the voltage of the busbar, is the phase angle of the busbar, and is the admittance of the machine to the infinite busbar. Due to the fluctuation of reactive and active power, the fluctuation of infinite bus voltage is caused in the following form:where is the magnitude of the voltage of the busbar, is the disturbance magnitude of the voltage of the busbar, and is the disturbance frequency.

#### 3. Numerical Simulations

In order to get the effects of the critical system parameters on the dynamic characteristics of this power system, the fourth-order Runge-Kutta method is employed in the following computations. According to the qualitative and quantitative characteristics of chaotic oscillations, we try to find out the road to chaos and to understand the oscillation response process of the synchronous generator rotor. In the following, with the aid of bifurcation diagram, time history, phase portrait, Poincaré map, and frequency spectrum, the characteristics of swing oscillation of each generator rotor are illustrated. In addition, the Wolf method is introduced to calculate the largest Lyapunov exponent, which enables us to identify the bifurcation points and different oscillation states [21].

##### 3.1. Influence of Disturbance Amplitude of Infinite Bus Voltage

The calculation parameters of the considered system are* H*_{1} = 1.37,* H*_{2} = 2,* H*_{3} = 1.37,* D*_{1} = 0.008,* D*_{2} = 0.008,* D*_{3} = 0.008, = 120* π*,

*P*

_{1}= 1,

*P*

_{2}= 0.5,

*P*

_{3}= 1,

*E*

_{1}= 1.27,

*E*

_{2}= 1.27,

*E*

_{3}= 1.27, = 1, Ω = 2,

*B*

_{12}= 0.1,

*B*

_{13}= 0.6,

*B*

_{23}= 1,

*B*

_{14}= 2,

*B*

_{24}= 1.55, and

*B*

_{34}= 2. In this subsection, the influences of the infinite bus voltage

*V*

_{B1 }on the dynamic behaviors of the three-machine power system are presented. The bifurcation diagram is plotted in Figure 2(a) for different values of to show the responses of the system under different infinite bus voltage. It can be seen that the oscillation response of the system leads to chaotic motion through the period-doubling bifurcation, and the chaotic parameter region is very narrow. Especially after the parameter crosses the chaotic parameter region, the system exhibits a rotating motion orbit, and then the generator loses synchronization, and in this case the system is unstable. According to the Wolf method, the Lyapunov exponent is calculated, and the corresponding Lyapunov exponent spectrum is shown in Figure 2(b) with varying values of . In the case of , the largest Lyapunov component is always negative, which is corresponding to periodic motion window in the bifurcation diagram. This proves that system swing oscillation starts with periodic motion. When , , and , the largest Lyapunov component will change from negative to zero and finally to positive. It indicates that the period-doubling bifurcation behavior occurs. If , the largest Lyapunov component is positive. This proves that in the system chaotic motion occurs. Comparing Figures 2(a) and 2(b), it can be found that has the same value at every bifurcation point.

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For the purpose of clearly displaying the evolving process of swing oscillation as the infinite bus voltage changes, Figures 3–9 show the time history, phase portrait, Poincaré map, and Frequency spectrum. In the case of , the dynamic behavior of the system is periodic 1 motion. Form = 0.5 to = 0.63, it is worthy to point out that the dynamic behaviors are also periodic 1 motion, but there exist many super-harmonic components in the frequency spectrum. As shown in Figures 6–8, in the case of , , and , the system presents periodic 2 oscillation, periodic 4 oscillation, and periodic 8 oscillation, respectively. In addition, the response of the system is chaotic motion at (shown in Figure 9). In this case, the corresponding largest Lyapunov exponent is 0.0082; it is further confirmed that chaotic oscillation has occurred.

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##### 3.2. Influence of Damping

In this section, the damping on the dynamic characteristics of the rotor system is discussed. For the sake of comparison, herein, choose the damping coefficient as the bifurcation parameter; the other parameter values are the same as the ones in the previous section. The bifurcation diagram and the corresponding largest Lyapunov exponent diagram are illustrated in Figures 10(a) and 10(b), respectively.

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From Figure 10, it can be seen that, as the damping coefficient increases, the dynamic responses of the system undergo an inversed bifurcation process, namely, chaotic motion→periodic 8 motion→periodic 4 motion→periodic 2 motion→periodic 1 motion. Therefore, increasing the damping coefficient can prevent the occurrence of chaotic motion. Figures 11–14 have given the time history, phase portrait, Poincaré map, and frequency spectrum under different values of damping. These figures have clearly shown the evolution process of the dynamic response of the system as the damping changes.

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##### 3.3. Influence of the Basic Voltage

Due to the active and reactive power of the load change, it may cause a change in the basic value of the infinite bus voltage. Thus, it also affects the dynamic behavior of the system. Choose as the bifurcation parameter; Figure 15 shows the bifurcation diagram and the corresponding largest Lyapunov exponent of the dynamic response, as changes from small to large. From Figure 15, it can be seen that the dynamic response of the system also presents an inversed bifurcation process.

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##### 3.4. Influence of the Perturbation Frequency

It is interesting to note that, at , a new route to chaos via quasi-periodic torus rupture in this system has been observed. As changes from small to large, Figures 16 and 17 have given bifurcation diagram and corresponding largest Lyapunov exponent diagram, respectively. Moreover, the results show that the swing oscillation process is periodic, quasi-periodic, chaotic, and out-of-step. In order to show the evolution process more clearly, the time history, phase diagram, Poincaré section, and frequency spectrum are plotted (Figures 18–21). More concretely, in the case of , the dynamic response is period 1 motion (shown in Figure 18). When , the dynamic response is in almost periodic motion; the corresponding Poincaré section is a circle (shown in Figure 20). In the case of , there exists chaotic motion (Figure 21). After the parameter passes through chaotic region, the synchronous generator loses synchronization.

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#### 4. Discussions

The chaotic dynamics may appear in power system, which make a great threat to the stability of the power system. Therefore, many researchers pay much attention to the dynamic behavior and chaotic mechanism, especially for SMIB power system [3–9, 22]. From the point of nonlinear dynamic analysis, a few works have addressed the dynamic characteristic about three-machine power system subjected to load disturbance. In this paper, a relatively deep and systematic study of dynamic response in three-machine power system subjected to load disturbance has been carried out. This paper aims to demonstrate the complete transition process of different dynamic behaviors by combining qualitative and quantitative analysis. Comparing the results about simple power system (e.g., SMIB power system) [9–15], a new route to chaos via quasi-periodic torus rupture has been found. Moreover, it is worthy to point out that, in our considered system, there exist many super-harmonic components in the frequency spectrum of the swing oscillation response. Attention should be paid to these new phenomena in engineering practice. In addition, for three-machine power system, our future work will investigate the mechanism of chaotic oscillation occurrence by improving Melnikov analysis, such that effective measures can be taken in time to avoid system collapse.

#### 5. Conclusions

This paper has investigated the effects of critical system parameters on dynamic characteristics of synchronous generator rotors in a three-machine power system subjected to load disturbance. The swing equations describing the motions of the synchronous generator rotors are established. Based on these swing equations, with the help of bifurcation diagrams, largest Lyapunov exponent spectrums, phase portraits, Poincaré map, and frequency spectrum, the influence of system parameters on dynamic behaviors is shown clearly. The Wolf method is introduced to calculate the largest Lyapunov exponent, which is used to verify the occurrence of chaotic motion. Moreover, different paths leading to chaos coexisting in this system have been found. They are period-doubling cascading bifurcations to chaos induced by changing the infinite bus voltage magnitude and quasi-periodic torus rupture to chaos induced by changing the disturbance frequency of the infinite bus voltage. These results will contribute to a better understanding of the nonlinear dynamic behaviors of synchronous generator rotors in three-machine power system.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

#### Acknowledgments

This research is supported by Higher Educational Science and Technology Program of Shandong Province, China (Grant no. J18KA235), Shandong Provincial Natural Science Foundation, China (Grants nos. ZR2016AP06, ZR2018QA005, ZR2018BA018, and ZR2018BA021), and National Natural Science Foundation of China (Grants nos. 11501246 and 61703251).