Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 2703684, 13 pages

https://doi.org/10.1155/2018/2703684

## Sliding Mode Control of Chaos in a Single Machine Connected to an Infinite Bus Power System

Electrical Engineering Department, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait

Correspondence should be addressed to Mohamed Zribi; moc.liamg@01ibirz.demahom

Received 29 March 2017; Revised 18 October 2017; Accepted 7 March 2018; Published 17 April 2018

Academic Editor: Alberto Borboni

Copyright © 2018 Muthana T. Alrifai and Mohamed Zribi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the control of chaos in a power system. A fourth-order model is adopted for the power system. Three controllers are proposed to suppress the chaos and avoid voltage collapse. The controllers are a feedback linearization controller, a conventional sliding mode controller, and a second-order super-twisting sliding mode controller. It is shown that the proposed controllers guarantee the convergence of the states of the system to their desired values. Simulations studies are presented to show the effectiveness of the proposed control schemes.

#### 1. Introduction

Electric power systems are generally comprised of three-phase AC systems operating essentially at constant voltages. Voltage stability of a power system pertains to the ability of the system to maintain steady acceptable voltages at all buses of the power system under normal operating conditions and also after being subjected to some disturbances. Disturbances can be due to faults on the system, increase in the loads demand, or any other changes affecting the system conditions.

A power system enters a state of voltage instability when a disturbance acting on the system causes a progressive and uncontrollable change in the voltages of the buses. The sequence of events accompanying voltage instability may lead to a low unacceptable voltage profile in a significant part of the power system which in turn may lead to a voltage collapse or a voltage avalanche [1]. It should be mentioned that losses of loads and/or tripping of the transmission lines and the complete shutdown of the affected areas may follow a voltage collapse. In recent years, many parts of countries, such as the USA, Japan, the UK, and France, experienced major blackouts incidents which were associated with voltage collapse.

Moreover, chaos is a nonlinear phenomenon which may affect the stability of some systems. Chaotic oscillations are very sensitive to the parameters and to the initial conditions of the system. They are related to random, continuous, and bounded oscillations. Chaos has been widely investigated in many systems in different areas. Moreover, chaos is categorized as one of the top discoveries of the 20th century, and it is expected that chaos will draw more attention in future studies [2, 3].

Several researchers studied the chaotic phenomena in power systems. Early studies were mainly focused on interpreting the behavior of chaotic oscillations of power systems [4–15]. Routes to chaotic oscillations in power systems and relationship between chaos and power system instability were studied in [16–19]. It was reported in [19] that chaos can lead a power system to voltage instability and a voltage collapse when stability conditions are broken. Moreover, it was shown that chaos may possibly exist as an intermediate stage in the instability incident after a large disturbance affecting the power system. In [4], it was shown that voltage collapse phenomenon is linked to static and/or dynamic bifurcation. In addition, it was found that the nominal operating point undergoes dynamic bifurcations prior to the static bifurcation to which voltage collapse was attributed. Some studies considered the interaction of chaotic motion and the system dynamic components and the relation between the power system stability region and chaos. Clearly, chaos oscillations in power systems are harmful and should be suppressed or eliminated by using effective control measures. Several control strategies were developed for this purpose.

Since power systems are highly nonlinear, different nonlinear control schemes were used to curb or eliminate the chaotic oscillations in power systems; for example, see [20–37]. Global state feedback linearization was applied in [21] to control the chaotic behavior of the power system. Also, adaptive control was used to control chaos of power systems in [22]. Because of their robustness, sliding mode control (SMC) schemes were widely used to control different types of systems [35]. Several types of SMC control schemes were designed to control the chaos and to avoid voltage collapse in power systems; for example, see [27, 28, 36, 37]. However, SMC schemes suffer from the problem of chattering which is undesirable in practice.

Researchers have used many different ways to reduce or eliminate chattering. One solution which is used to reduce chattering is the use of a boundary layer [38]. Inside the boundary layer, the controller is chosen to be a continuous approximation of the switching controller. This solution involves the use of a saturation function to approximate the sign function in the controller. Let represent the sliding surface. One continuous approximation of the sign function of the sliding surface is defined as follows: , where is a positive constant and the approximation error can be decreased by increasing [39]. Other approximation functions of the sign function which were used to alleviate chattering include and , with being a properly chosen scalar. The introduction of a boundary layer eliminates the high-frequency chattering at the price of losing some degree of robustness. Another approach for chattering reduction is through the use of dynamic sliding mode controllers (DSMC) [40]. This is done through placing an integrator (or a strictly proper low pass filter) in front of the system to be controlled [41]. The advantage of this technique is that it does not sacrifice the control accuracy. In addition, high-order sliding mode (HOSM) controllers can also be used to reduce or eliminate the chattering phenomena. For example, researchers such as Bartolini et al. [42, 43], Levant [44], and Shtessel et al. [45] proposed controllers using second-order sliding mode techniques. Many other works dealing with the design of higher order sliding mode controllers and their applications were also presented; for example, see [46–51].

This paper proposes three nonlinear control schemes to suppress chaos and avoid voltage collapse in a power system. The proposed controllers are a feedback linearization controller, a conventional sliding mode controller, and a second-order super twisting sliding mode controller. These control schemes guarantee the convergence of the states of the system to their desired values. Simulations results indicate that the proposed controllers work well in eliminating the chaotic oscillations of the power system and hence preventing voltage collapse.

The rest of the paper is organized as follows. The model of a multimachine power system is derived in Section 2. A feedback linearization controller is presented in Section 3. Two sliding mode control schemes are proposed in Sections 4 and 5. Finally, the conclusion is given in Section 6.

#### 2. Dynamic Model of the Power System

The power system under consideration consists of a three-bus system as shown in Figure 1. In this system, one generator bus is an infinite busbar, while the other one has a constant voltage magnitude . The Thevenin equivalent model representing the infinite busbar is denoted by , , and . The Thevenin equivalent model representing the second generator is denoted by , , and . The load bus consists of an induction motor in parallel with a PQ (active, reactive) load. A fixed capacitor is also included with the load to increase the voltage up to near one per unit. The magnitude and phase angle of the load busbar voltage are denoted by and , respectively.