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

Authors: Muthana T. Alrifai; Mohamed Zribi
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 s represent the sliding surface. One continuous approximation of the sign function of the sliding surface sign⁡(s) is defined as follows: 2/πarctan⁡(μs), where μ is a positive constant and the approximation error can be decreased by increasing μ [39]. Other approximation functions of the sign function which were used to alleviate chattering include s/|s|+ϵ and tanh⁡(s/ϵ), with ϵ 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 Section2. 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 Figure1. In this system, one generator bus is an infinite busbar, while the other one has a constant voltage magnitude Em. The Thevenin equivalent model representing the infinite busbar is denoted by Eo, Yo, and θo. The Thevenin equivalent model representing the second generator is denoted by Em, Ym, and θm. 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 VL and δ, respectively.Figure 1
A three-bus power system.The motion of the rotor of the generator is given by the swing equation such as(1)Mδ¨m+dmδ˙m=Pm-Pe,where δm is the phase angle of the generator (radians), M is the momentum of inertia of the generator, dm is the damping coefficient, Pm is the mechanical power, and Pe is the electrical power.The generator rotor speed deviationωm is related to δm such that(2)δ˙m=ωm.The generated electrical power Pe can be written as(3)Pe=-Em2Ymsin⁡θm-EmVLYmsin⁡δ-δm+θm,where Ym and θm are the magnitude and phase angle of the admittance between the second generator and the load busbar.Using (1), (3), and (2), we get(4)Mω˙m=-dmωm+Pm+EmVLYmsin⁡δ-δm-θm+Em2Ymsin⁡θm.The model of the load is as follows [16]:(5)P=P0+P1+kpwδ˙+kpvVL+TV˙L,Q=Q0+Q1+kqwδ˙+kqvVL+kqv2VL2,where P0 and Q0 are the constant real and reactive powers of the induction motor, T is the time constant of the motor, and P1 and Q1 are the real and reactive components of the PQ load. The constants kpw, kpv, kqw, kqv, and kqv2 are empirical constants that characterize the load.The real power supplied to the load bus by the network is expressed as(6)P=-E0′Y0′VLsin⁡δ+θ0′-EmYmVLsin⁡δ-δm+θm+Y0′sin⁡θ0+Ymsin⁡θmVL2,where E0′, Y0′, and θ0′ are the adjusted values of the Thevenin equivalent model representing the infinite bus in consideration of the capacitor.The reactive power supplied to the load bus by the network is expressed as(7)Q=E0′Y0′VLcos⁡δ+θ0′+EmYmVLcos⁡δ-δm+θm-Y0′cos⁡θ0+Ymcos⁡θmVL2.Using the above equations, we can write(8)kqwδ˙=E0′Y0′VLcos⁡δ+θ0′+EmYmVLcos⁡δ-δm+θm-kqvVL-Q0-Q1-kqv2+Y0′cos⁡θ0′+Ymcos⁡θmVL2,(9)TkqwkpvV˙L=kpwkqv2VL2+kpwkqv-kqwkpvVL-kqwE0′Y0′VLsin⁡δ+θ0′+EmYmVLsin⁡δ-δm+θm+kqwY0′sin⁡θ0′+Ymsin⁡θmVL2-P0-P1-kpwE0′Y0′VLcos⁡δ+θ0′+EmYmVLcos⁡δ-δm+θm+kpwY0′cos⁡θ0′+Ymcos⁡θmVL2+Q0+Q1.Collecting (2), (4), (8), and (9), the model of the power system can be represented as follows:(10)δ˙m=ωm,Mω˙m=-dmωm+Pm+EmVLYmsin⁡δ-δm-θm+Em2Ymsin⁡θm,kqwδ˙=E0′Y0′VLcos⁡δ+θ0′+EmYmVLcos⁡δ-δm+θm-kqvVL-Q0-Q1-kqv2+Y0′cos⁡θ0′+Ymcos⁡θmVL2,TkqwkpvV˙L=kpwkqv2VL2+kpwkqv-kqwkpvVL-kqwE0′Y0′VLsin⁡δ+θ0′+EmYmVLsin⁡δ-δm+θm+kqwY0′sin⁡θ0′+Ymsin⁡θmVL2-P0-P1-kpwE0′Y0′VLcos⁡δ+θ0′+EmYmVLcos⁡δ-δm+θm+kpwY0′cos⁡θ0′+Ymcos⁡θmVL2+Q0+Q1.We will use the values of the parameters to be the same as in [16]. The parameters of the generator are such that Ym=5, θm=-5, Em=1, Pm=1, dm=0.05, and M=0.3. The parameters of the network are such that Y0=20, θ0=-5, E0=1, C=12, Y0′=8, θ0′=-12, and E0′=2.5 The parameters of the load are such that kpw=0.4, kpv=0.3, kqw=-0.03, kqv=-2.8, kqv2=2.1, T=8.5, P0=0.6, Q0=1.3, and P1=0.0. All the parameters values are in per unit except the angles which are in radians.Therefore, the model of the power system can be written as follows:(11)δ˙m=ωm,ω˙m=p1VLsin⁡δ-δm+p2+p3ωm+p4,δ˙=p5VLcos⁡δ-δm-p2+p6VLcos⁡δ-p7+p8VL2+p9VL+p10Q1+p11+u1,V˙L=p12VLcos⁡δ-δm-p13+p14VLcos⁡δ-p15+p16VL2+p17VL+p18Q1+p19+u2,where the parameters p1–p19 in the power system model (11) have the following values: p1=16.66667, p2=0.087266, p3=-0.16667, p4=1.88074, p5=-166.6667, p6=-666.66667, p7=0.20944, p8=496.87181, p9=-93.3333, p10=33.33333, p11=43.33333, p12=26.21722, p13=0.01241, p14=104.868887, p15=0.13458, p16=-78.7638, p17=14.52288, p18=-5.22876, and p19=-7.03268.Note that the control inputs for the system in (11) are added to the third and fourth differential equations of the system. Also, recall that Q1 is a system parameter representing the reactive power in the system.For ease of presentation, we define the state variablesx1–x4 such that x1x2x3x4T=δmωmδVLT. Hence, model of the system is expressed as follows:(12)x˙1=x2,x˙2=p1x4sin⁡x3-x1+p2+p3x2+p4,x˙3=p5x4cos⁡x3-x1-p2+p6x4cos⁡x3-p7+p8x42+p9x4+p10Q1+p11+u1,x˙4=p12x4cos⁡x3-x1-p13+p14x4cos⁡x3-p15+p16x42+p17x4+p18Q1+p19+u2.The fourth-order model of the power system (12) is simulated when u1=u2=0 and with Q1=11.37. The simulation results are shown in Figures 2–7. The machine power angle δm versus time and the deviation in the frequency ωm versus time are shown in Figures 2-3. The load power angle δ versus time and the load voltage VL versus time are shown in Figures 4-5. The machine power angle δm versus the deviation in the frequency ωm and the machine power angle δm versus the deviation in the frequency ωm and the load voltage VL are presented in Figures 6-7. It is evident from these figures that the power system is chaotic for this value of Q1.Figure 2
The machine power angleδm versus time (no control).Figure 3
The deviation in the frequencyωm versus time (no control).Figure 4
The load power angleδ versus time (no control).Figure 5
The load voltageVL versus time (no control).Figure 6
The machine power angleδm versus the deviation in the frequency ωm (no control).Figure 7
The machine power angleδm versus the deviation in the frequency ωm and the load voltage VL (no control).The objective of this paper is to design controllers for the power system so that chaos is suppressed.Letx1d, x2d, x3d, and x4d represent the desired values of x1, x2, x3, and x4, respectively.Remark 1.
It is clear from the dynamic model of the system given in (12) that the desired (equilibrium) values of the states of the power system satisfy the following two equations:(13)x2d=0,p1x4dsin⁡x3d-x1d+p2+p4=0.To facilitate the design of the controllers, we propose to use the following transformation:(14)z1=x1-x1d,z2=x2,z3=p1x4sin⁡x3-x1+p2-p1x4dsin⁡x3d-x1d+p2+p3x2,z4=x3-x3d.Note that z1(t), z2(t), and z4(t) represent the errors between the actual and the desired states for the first three states of the system.Remark 2.
The choice of the statesz1(t), z2(t), z3(t), and z4(t) in (14) guarantees that when z1(t), z2(t), z3(t), and z4(t) converge to zero as t tends to infinity, then the states x1(t), x2(t), x3(t), and x4(t) converge to their desired values, respectively.Using the transformation given in (14), the model of the power system given by the equations in (12) can be written as follows:(15)z˙1=z2,z˙2=z3,z˙3=f1+u¯1,z˙4=f2+u¯2,where f1, f2, u¯1, and u¯2 are such that(16)f1=p1p12x4cos⁡x3-x1-p13sin⁡x3-x1+p2+p1p14x4cos⁡x3-p15+p16x42+p17x4+p18Q1+p19sin⁡x3-x1+p2+p1x4cos⁡x3-x1+p2p5x4cos⁡x3-x1-p2+p6x4cos⁡x3-p7+p8x42+p1x4cos⁡x3-x1+p2p9x4+p10Q1+p11-x2+p3p1x4sin⁡x3-x1+p2+p32x2+p3p4,f2=p5x4cos⁡x3-x1-p2+p6x4cos⁡x3-p7+p8x42+p9x4+p10Q1+p11,u¯1=p1u2sin⁡x3-x1+p2+p1u1x4cos⁡x3-x1+p2,u¯2=u1.In the following three sections, we will propose three control schemes to suppress chaos in the power system. We will use the transformed model (15) to design the controllers. The controllers will be designed to force the states z1(t), z2(t), z3(t), and z4(t) to converge to zero as t tends to infinity. Hence, the states x1(t), x2(t), x3(t), and x4(t) will converge to their desired values, respectively.


## 3. Design of a Feedback Linearization Controller to Suppress Chaos

### 3.1. Design of the Controller

Letα1, α2, α3, and α4 be positive scalars such that the polynomial P(s)=s3+α3s2+α2s+α1 is a Hurwitz polynomial.Theorem 3.
The feedback linearization controller(17)u¯1=-f1-α1z1-α2z2-α3z3,u¯2=-f2-α4z4when applied to the transformed fourth-order power system given by (15) guarantees the asymptotic convergence of the states z(t)=z1tz2tz3tz4tT to zero as t tends to infinity.Proof.
The closed loop system when the control scheme given by (17) is applied to the transformed power system given by (15) can be written as follows:(18)z˙=Atz,where(19)At=01000010-α1-α2-α30000-α4.It can be easily checked that since α1, α2, α3, and α4 are positive scalars such that P(s)=s3+α3s2+α2s+α1 is a Hurwitz polynomial, then the matrix At is a stable matrix (i.e., its eigenvalues are located in the left half of the complex plane).
The solution of (18) is z(t)=exp⁡(Att)z(0), where z(0) is the initial state at t=0. Since At is a stable matrix, then z(t) will converge to zero as t→∞.Therefore, sincez(t) converges to zero as t→∞, then the states of the power system will converge to their desired values as t→∞ because of the choice of the transformation given by (14).Remark 4.
The design of the controller scheme is accomplished using the transformed dynamic model given by (15). Then the controllers u¯1 and u¯2 given by (17) are transformed back into the original coordinates such that(20)u1=u¯2,u2=u¯1-p1u¯2x4cos⁡x3-x1+p2p1∗sin⁡x3-x1+p2.


### 3.2. Simulation Results

The performance of the closed loop system when using the proposed feedback linearization controller was simulated using the MATLAB software. The initial states of the system are such thatx10x20x30x40T=0.3rad0rad/s0.2rad0.97p.u.T. The controllers u1 and u2 given by (17) and (20) are applied to the power system (12). The desired states of the power system are such that xd=x1dx2dx3dx4dT=0.34rad0rad/s0.1365rad0.9728p.u.T. The simulation results are depicted in Figures 8–11. The machine power angle δm versus time and the deviation in the frequency ωm versus time are shown in Figures 8-9, respectively. The load power angle δ versus time and the load voltage VL versus time are shown in Figures 10-11, respectively. It is clear from these figures that the states of the power system reach their desired values in less than 3 sec. Hence, the proposed feedback linearization controller works well in suppressing chaos in the power system.Figure 8
The machine power angleδm versus time using the feedback linearization controller when ωm(0)=0.Figure 9
The deviation in the frequencyωm versus time using the feedback linearization controller when ωm(0)=0.Figure 10
The load power angleδ versus time using the feedback linearization controller when ωm(0)=0.Figure 11
The load voltageVL versus time using the feedback linearization controller when ωm(0)=0.It is well known that chaotic systems are very sensitive to initial conditions. Therefore, we simulated the system with the same initial conditions except forx2(0)=ωm(0)=0.2 rad/sec. We checked that this initial condition still results in a chaotic system. The simulation results are shown in Figures 12–15. These figures show that the proposed controller still works but the transient responses are not as good as when x2(0)=ωm(0)=0 rad/sec.Figure 12
The machine power angleδm versus time using the feedback linearization controller when ωm(0)=0.2.Figure 13
The deviation in the frequencyωm versus time using the feedback linearization controller when ωm(0)=0.2.Figure 14
The load power angleδ versus time using the feedback linearization controller when ωm(0)=0.2.Figure 15
The load voltageVL versus time using the feedback linearization controller when ωm(0)=0.2.Feedback linearization controllers are generally not very robust to unmodeled dynamics and external robustness. Therefore, we will use sliding mode controllers in the next two sections to suppress chaos in power systems. Sliding mode controllers will be used because they are well known for their robustness features.


## 3.1. Design of the Controller

Letα1, α2, α3, and α4 be positive scalars such that the polynomial P(s)=s3+α3s2+α2s+α1 is a Hurwitz polynomial.Theorem 3.
The feedback linearization controller(17)u¯1=-f1-α1z1-α2z2-α3z3,u¯2=-f2-α4z4when applied to the transformed fourth-order power system given by (15) guarantees the asymptotic convergence of the states z(t)=z1tz2tz3tz4tT to zero as t tends to infinity.Proof.
The closed loop system when the control scheme given by (17) is applied to the transformed power system given by (15) can be written as follows:(18)z˙=Atz,where(19)At=01000010-α1-α2-α30000-α4.It can be easily checked that since α1, α2, α3, and α4 are positive scalars such that P(s)=s3+α3s2+α2s+α1 is a Hurwitz polynomial, then the matrix At is a stable matrix (i.e., its eigenvalues are located in the left half of the complex plane).
The solution of (18) is z(t)=exp⁡(Att)z(0), where z(0) is the initial state at t=0. Since At is a stable matrix, then z(t) will converge to zero as t→∞.Therefore, sincez(t) converges to zero as t→∞, then the states of the power system will converge to their desired values as t→∞ because of the choice of the transformation given by (14).Remark 4.
The design of the controller scheme is accomplished using the transformed dynamic model given by (15). Then the controllers u¯1 and u¯2 given by (17) are transformed back into the original coordinates such that(20)u1=u¯2,u2=u¯1-p1u¯2x4cos⁡x3-x1+p2p1∗sin⁡x3-x1+p2.


## 3.2. Simulation Results

The performance of the closed loop system when using the proposed feedback linearization controller was simulated using the MATLAB software. The initial states of the system are such thatx10x20x30x40T=0.3rad0rad/s0.2rad0.97p.u.T. The controllers u1 and u2 given by (17) and (20) are applied to the power system (12). The desired states of the power system are such that xd=x1dx2dx3dx4dT=0.34rad0rad/s0.1365rad0.9728p.u.T. The simulation results are depicted in Figures 8–11. The machine power angle δm versus time and the deviation in the frequency ωm versus time are shown in Figures 8-9, respectively. The load power angle δ versus time and the load voltage VL versus time are shown in Figures 10-11, respectively. It is clear from these figures that the states of the power system reach their desired values in less than 3 sec. Hence, the proposed feedback linearization controller works well in suppressing chaos in the power system.Figure 8
The machine power angleδm versus time using the feedback linearization controller when ωm(0)=0.Figure 9
The deviation in the frequencyωm versus time using the feedback linearization controller when ωm(0)=0.Figure 10
The load power angleδ versus time using the feedback linearization controller when ωm(0)=0.Figure 11
The load voltageVL versus time using the feedback linearization controller when ωm(0)=0.It is well known that chaotic systems are very sensitive to initial conditions. Therefore, we simulated the system with the same initial conditions except forx2(0)=ωm(0)=0.2 rad/sec. We checked that this initial condition still results in a chaotic system. The simulation results are shown in Figures 12–15. These figures show that the proposed controller still works but the transient responses are not as good as when x2(0)=ωm(0)=0 rad/sec.Figure 12
The machine power angleδm versus time using the feedback linearization controller when ωm(0)=0.2.Figure 13
The deviation in the frequencyωm versus time using the feedback linearization controller when ωm(0)=0.2.Figure 14
The load power angleδ versus time using the feedback linearization controller when ωm(0)=0.2.Figure 15
The load voltageVL versus time using the feedback linearization controller when ωm(0)=0.2.Feedback linearization controllers are generally not very robust to unmodeled dynamics and external robustness. Therefore, we will use sliding mode controllers in the next two sections to suppress chaos in power systems. Sliding mode controllers will be used because they are well known for their robustness features.


## 4. Design of a Conventional Sliding Mode Controller to Suppress Chaos

### 4.1. Design of the Controller

The first step in the design of sliding mode controllers is the choice of the sliding surfaces. As the power system has two inputs; then two sliding surfaces need to be designed.Letβ1, β2, η1, η2, K1, and K2 be positive scalars. Also, let the sliding surface S1 be such that(21)S1=z3+β1z1+β2z2.And let the sliding surface S2 be such that(22)S2=z4.Theorem 5.
The sliding mode controller,(23)u¯1=-f1-β1z2-β2z3-η1z3+β1z1+β2z2-K1sign⁡z3+β1z1+β2z2,(24)u¯2=-f2-η2z4-K2sign⁡z4when applied to the transformed power system model given by (15), ensures that the states z1(t), z2(t), z3(t), and z4(t) converge to zero as t tends to infinity.Proof.
Differentiating (21) with respect to time and using (15), it follows that(25)S˙1=z˙3+β1z˙1+β2z˙2=f1+u¯1+β1z2+β2z3.Substituting u¯1 by its value from (23), it follows that(26)S˙1=f1+u¯1+β1z2+β2z3=f1+β1z2+β2z3-f1-β1z2-β2z3-η1z3+β1z1+β2z2-K1sign⁡z3+β1z1+β2z2=-η1z3+β1z1+β2z2-K1sign⁡z3+β1z1+β2z2=-η1S1-K1sign⁡S1.Also, differentiating (22) with respect to time, using the model of the transformed system in (15) and substituting u¯2 by its value from (24), it follows that(27)S˙2=z˙4=f2+u¯2=-η2z4-K2sign⁡z4=-η2S2-K2sign⁡S2.The dynamic equations (26)-(27) satisfy the following equation:(28)SiS˙i=-ηiSi2-KiSisign⁡Si=-ηiSi2-KiSifori=1,2.Therefore, the dynamic equations (26)-(27) guarantee that SiS˙i<0 for (i=1,2) when Si≠0. Also, it can be checked that the trajectories associated with the discontinuous dynamics (26) and (27) exhibit a finite time reachability to zero from any given initial condition provided that the scalars η1, η2 are positive and the gains K1 and K2 are chosen to be large enough strictly positive. Hence, the dynamic equations (26)-(27) are guaranteed to reach the surfaces S1=0 and S2=0 in finite time.
SinceS1 is driven to zero in finite time, the state variables z1(t), z2(t), and z3(t) are governed after such finite amount of time by the equation z3+β1z1+β2z2=0. Using the transformed model in (15), this equation is equivalent to the second-order differential equation: z¨1=-β2z˙1-β1z1. The solution of this equation, z1(t), will asymptotically converge to 0 as t→∞ because the design parameters β1 and β2 are chosen to be positive scalars. In addition, since z1(t) will asymptotically converge to 0, it can be concluded from the transformed system in (15) that z2(t) will asymptotically converge to 0 as t→∞. Similarly, z3(t) will also converge to zero as t→∞. In addition, since S2 is driven to zero in finite time, the variable z4(t) will converge to zero in finite time.
Therefore, it can be concluded that the sliding mode controller (23)-(24) guarantees the convergence of the state variables z1(t), z2(t), z3(t), and z4(t) to zero as t→∞.Thus, the controller (23)-(24) guarantees the asymptotic convergence of the states of the power system x1(t), x2(t), x3(t), and x4(t) to their desired values as t→∞.Remark 6.
To reduce the chattering associated with the proposed sliding mode controller, the switching function sign can be replaced by the saturation function, such that(29)sat⁡Si=sign⁡SiifSi>ϵSiϵifSi≤ϵ with ϵ being a small positive number.


### 4.2. Simulation Results

The performance of the closed loop system when using the conventional sliding mode controller was simulated using the MATLAB software. The initial and the desired states of the power system are the same as in the previous section. The controllersu1 and u2 in (20) when u¯1 and u¯2 are given by (23)-(24) are applied to the power system (12). The simulation results are depicted in Figures 16–19. The machine power angle δm versus time and the deviation in the frequency ωm versus time are shown in Figures 16-17, respectively. The load power angle δ versus time and the load voltage VL versus time are depicted in Figures 18-19, respectively. These figures indicate that the states of the power system converge to their desired values in less than 3 sec. Therefore, it can be concluded that the proposed conventional sliding mode controller works well. Moreover, we simulated the responses of system with the same initial conditions except for x2(0)=ωm(0)=0.2 rad/sec. This initial condition still results in a chaotic system. The simulation results are shown in Figures 20–23. Again, these figures show that the proposed controller still work but the transient responses suffer a little bit.Figure 16
The machine power angleδm versus time using the conventional sliding mode controller when ωm(0)=0.Figure 17
The deviation in the frequencyωm versus time using the conventional sliding mode controller when ωm(0)=0.Figure 18
The load power angleδ versus time using the conventional sliding mode controller when ωm(0)=0.Figure 19
The load voltageVL versus time using the conventional sliding mode controller when ωm(0)=0.Figure 20
The machine power angleδm versus time using the conventional sliding mode controller when ωm(0)=0.2.Figure 21
The deviation in the frequencyωm versus time using the conventional sliding mode controller when ωm(0)=0.2.Figure 22
The load power angleδ versus time using the conventional sliding mode controller when ωm(0)=0.2.Figure 23
The load voltageVL versus time using the conventional sliding mode controller when ωm(0)=0.2.The responses of the states of the system suffer from the well-known chattering problem. Therefore, in the next section, we will use a second-order sliding mode controller [52] to reduce the chattering problem.


## 4.1. Design of the Controller

The first step in the design of sliding mode controllers is the choice of the sliding surfaces. As the power system has two inputs; then two sliding surfaces need to be designed.Letβ1, β2, η1, η2, K1, and K2 be positive scalars. Also, let the sliding surface S1 be such that(21)S1=z3+β1z1+β2z2.And let the sliding surface S2 be such that(22)S2=z4.Theorem 5.
The sliding mode controller,(23)u¯1=-f1-β1z2-β2z3-η1z3+β1z1+β2z2-K1sign⁡z3+β1z1+β2z2,(24)u¯2=-f2-η2z4-K2sign⁡z4when applied to the transformed power system model given by (15), ensures that the states z1(t), z2(t), z3(t), and z4(t) converge to zero as t tends to infinity.Proof.
Differentiating (21) with respect to time and using (15), it follows that(25)S˙1=z˙3+β1z˙1+β2z˙2=f1+u¯1+β1z2+β2z3.Substituting u¯1 by its value from (23), it follows that(26)S˙1=f1+u¯1+β1z2+β2z3=f1+β1z2+β2z3-f1-β1z2-β2z3-η1z3+β1z1+β2z2-K1sign⁡z3+β1z1+β2z2=-η1z3+β1z1+β2z2-K1sign⁡z3+β1z1+β2z2=-η1S1-K1sign⁡S1.Also, differentiating (22) with respect to time, using the model of the transformed system in (15) and substituting u¯2 by its value from (24), it follows that(27)S˙2=z˙4=f2+u¯2=-η2z4-K2sign⁡z4=-η2S2-K2sign⁡S2.The dynamic equations (26)-(27) satisfy the following equation:(28)SiS˙i=-ηiSi2-KiSisign⁡Si=-ηiSi2-KiSifori=1,2.Therefore, the dynamic equations (26)-(27) guarantee that SiS˙i<0 for (i=1,2) when Si≠0. Also, it can be checked that the trajectories associated with the discontinuous dynamics (26) and (27) exhibit a finite time reachability to zero from any given initial condition provided that the scalars η1, η2 are positive and the gains K1 and K2 are chosen to be large enough strictly positive. Hence, the dynamic equations (26)-(27) are guaranteed to reach the surfaces S1=0 and S2=0 in finite time.
SinceS1 is driven to zero in finite time, the state variables z1(t), z2(t), and z3(t) are governed after such finite amount of time by the equation z3+β1z1+β2z2=0. Using the transformed model in (15), this equation is equivalent to the second-order differential equation: z¨1=-β2z˙1-β1z1. The solution of this equation, z1(t), will asymptotically converge to 0 as t→∞ because the design parameters β1 and β2 are chosen to be positive scalars. In addition, since z1(t) will asymptotically converge to 0, it can be concluded from the transformed system in (15) that z2(t) will asymptotically converge to 0 as t→∞. Similarly, z3(t) will also converge to zero as t→∞. In addition, since S2 is driven to zero in finite time, the variable z4(t) will converge to zero in finite time.
Therefore, it can be concluded that the sliding mode controller (23)-(24) guarantees the convergence of the state variables z1(t), z2(t), z3(t), and z4(t) to zero as t→∞.Thus, the controller (23)-(24) guarantees the asymptotic convergence of the states of the power system x1(t), x2(t), x3(t), and x4(t) to their desired values as t→∞.Remark 6.
To reduce the chattering associated with the proposed sliding mode controller, the switching function sign can be replaced by the saturation function, such that(29)sat⁡Si=sign⁡SiifSi>ϵSiϵifSi≤ϵ with ϵ being a small positive number.


## 4.2. Simulation Results

The performance of the closed loop system when using the conventional sliding mode controller was simulated using the MATLAB software. The initial and the desired states of the power system are the same as in the previous section. The controllersu1 and u2 in (20) when u¯1 and u¯2 are given by (23)-(24) are applied to the power system (12). The simulation results are depicted in Figures 16–19. The machine power angle δm versus time and the deviation in the frequency ωm versus time are shown in Figures 16-17, respectively. The load power angle δ versus time and the load voltage VL versus time are depicted in Figures 18-19, respectively. These figures indicate that the states of the power system converge to their desired values in less than 3 sec. Therefore, it can be concluded that the proposed conventional sliding mode controller works well. Moreover, we simulated the responses of system with the same initial conditions except for x2(0)=ωm(0)=0.2 rad/sec. This initial condition still results in a chaotic system. The simulation results are shown in Figures 20–23. Again, these figures show that the proposed controller still work but the transient responses suffer a little bit.Figure 16
The machine power angleδm versus time using the conventional sliding mode controller when ωm(0)=0.Figure 17
The deviation in the frequencyωm versus time using the conventional sliding mode controller when ωm(0)=0.Figure 18
The load power angleδ versus time using the conventional sliding mode controller when ωm(0)=0.Figure 19
The load voltageVL versus time using the conventional sliding mode controller when ωm(0)=0.Figure 20
The machine power angleδm versus time using the conventional sliding mode controller when ωm(0)=0.2.Figure 21
The deviation in the frequencyωm versus time using the conventional sliding mode controller when ωm(0)=0.2.Figure 22
The load power angleδ versus time using the conventional sliding mode controller when ωm(0)=0.2.Figure 23
The load voltageVL versus time using the conventional sliding mode controller when ωm(0)=0.2.The responses of the states of the system suffer from the well-known chattering problem. Therefore, in the next section, we will use a second-order sliding mode controller [52] to reduce the chattering problem.


## 5. Design of a Second-Order Sliding Mode Controller to Suppress Chaos

### 5.1. Design of the Controller

The controllers given by (23) and (24) suffer from chattering. To remedy this problem, we propose to use the second-order super-twisting sliding mode controller [53]. This controller maintains the robustness features of sliding mode controllers while producing less chattering compared to the conventional sliding mode controller.We will use the same sliding surfaces as in the previous section, mainly(30)S1=z3+β1z1+β2z2,(31)S2=z4,where β1 and β2 are positive scalars. Also, let K¯1, K¯2, W1, and W2 be positive scalars.Theorem 7.
The second-order sliding mode controller(32)u¯1=-f1-β1z2-β2z3-K¯1S11/2sign⁡S1+ν1,(33)u¯2=-f2-K¯2S21/2sign⁡S2+ν2with(34)ν˙1=-W1sign⁡S1,ν˙2=-W2sign⁡S2when applied to the transformed model of the power system given by (15), ensures that the states z1(t), z2(t), z3(t), and z4(t) converge to zero as t tends to infinity.Proof.
Differentiating the sliding surfaceS1 given by (30) with respect to time and using the transformed model of the power system given by (15), and substituting u¯1 by its value from (32), it follows that(35)S˙1=z˙3+β1z˙1+β2z˙2=f1+u¯1+β1z2+β2z3=f1+β1z2+β2z3-f1-β1z2-β2z3-K¯1S11/2sign⁡S1+ν1=-K¯1S11/2sign⁡S1+ν1.Also, differentiating the sliding surface S2 given by (31) with respect to time, using the transformed model of the power system given by (15), and substituting u¯2 by its value from (33), it follows that(36)S˙2=z˙4=f2+u¯2=-K¯2S21/2sign⁡S2+ν2.Hence, the dynamics that govern the sliding surfaces S1 and S2 are as follows:(37)S˙i=-K¯iSi1/2sign⁡Si+νi,ν˙i=-Wisign⁡Si,fori=1,2.The dynamics in (37) correspond to the super-twisting algorithm [54, 55]. The finite time convergence to zero of Si,νiT for (i=1,2) was proven in [54, 55] by using the Lyapunov function:(38)Vi=ξiTPiξi,where Pi(i=1,2) is a positive definite matrix and the vector ξi is such as ξi=Si1/2sign⁡Si,νiT(i=1,2).
Therefore, the trajectories associated with the discontinuous dynamics (35) and (36) exhibit a finite time reachability to zero from any given initial condition provided that the constants K¯1, K¯2, W1, and W2 are positive.
SinceS1 and S2 are driven to zero in finite time and following the same arguments as the ones used to prove the previous sliding mode controller, it can be concluded that z1(t), z2(t), z3(t), and z4(t) converge to zero as t→∞.Thus, the controller (32)–(34) guarantees the asymptotic convergence of x1(t), x2(t), x3(t), and x4(t) to their desired values as t→∞.


### 5.2. Simulation Results

The performance of the closed loop system when using the second-order super-twisting sliding mode controller was simulated. The initial and desired states of the power system are the same as in Section3. The controllers u1 and u2 given by the relationship (20) and (32)–(34) are applied to the power system (12). The simulation results are depicted in Figures 24–27. The machine power angle δm versus time and the deviation in the frequency ωm versus time are shown in Figures 24-25, respectively. The load power angle δ versus time and the load voltage VL versus time are shown in Figures 26-27, respectively. These figures clearly indicate that the states of the system reached their desired values in less than 2 sec. Also, the depicted responses do not suffer from the chattering problem. Hence, it can be concluded that the proposed second-order sliding mode controller works well. Moreover, we simulated the responses of system with the same initial conditions except for x2(0)=ωm(0)=0.2 rad/sec. This initial condition still results in a chaotic system. The simulation results are shown in Figures 28–31. Again, these figures show that the proposed controller still work but the transient responses suffer a little bit.Figure 24
The machine power angleδm versus time using the second-order sliding mode controller when ωm(0)=0.Figure 25
The deviation in the frequencyωm versus time using the second-order sliding mode controller when ωm(0)=0.Figure 26
The load power angleδ versus time using the second-order sliding mode controller when ωm(0)=0.Figure 27
The load voltageVL versus time using the second-order sliding mode controller when ωm(0)=0.Figure 28
The machine power angleδm versus time using the second-order sliding mode controller when ωm(0)=0.2.Figure 29
The deviation in the frequencyωm versus time using the second-order sliding mode controller when ωm(0)=0.2.Figure 30
The load power angleδ versus time using the second-order sliding mode controller when ωm(0)=0.2.Figure 31
The load voltageVL versus time using the second-order sliding mode controller when ωm(0)=0.2.


## 5.1. Design of the Controller

The controllers given by (23) and (24) suffer from chattering. To remedy this problem, we propose to use the second-order super-twisting sliding mode controller [53]. This controller maintains the robustness features of sliding mode controllers while producing less chattering compared to the conventional sliding mode controller.We will use the same sliding surfaces as in the previous section, mainly(30)S1=z3+β1z1+β2z2,(31)S2=z4,where β1 and β2 are positive scalars. Also, let K¯1, K¯2, W1, and W2 be positive scalars.Theorem 7.
The second-order sliding mode controller(32)u¯1=-f1-β1z2-β2z3-K¯1S11/2sign⁡S1+ν1,(33)u¯2=-f2-K¯2S21/2sign⁡S2+ν2with(34)ν˙1=-W1sign⁡S1,ν˙2=-W2sign⁡S2when applied to the transformed model of the power system given by (15), ensures that the states z1(t), z2(t), z3(t), and z4(t) converge to zero as t tends to infinity.Proof.
Differentiating the sliding surfaceS1 given by (30) with respect to time and using the transformed model of the power system given by (15), and substituting u¯1 by its value from (32), it follows that(35)S˙1=z˙3+β1z˙1+β2z˙2=f1+u¯1+β1z2+β2z3=f1+β1z2+β2z3-f1-β1z2-β2z3-K¯1S11/2sign⁡S1+ν1=-K¯1S11/2sign⁡S1+ν1.Also, differentiating the sliding surface S2 given by (31) with respect to time, using the transformed model of the power system given by (15), and substituting u¯2 by its value from (33), it follows that(36)S˙2=z˙4=f2+u¯2=-K¯2S21/2sign⁡S2+ν2.Hence, the dynamics that govern the sliding surfaces S1 and S2 are as follows:(37)S˙i=-K¯iSi1/2sign⁡Si+νi,ν˙i=-Wisign⁡Si,fori=1,2.The dynamics in (37) correspond to the super-twisting algorithm [54, 55]. The finite time convergence to zero of Si,νiT for (i=1,2) was proven in [54, 55] by using the Lyapunov function:(38)Vi=ξiTPiξi,where Pi(i=1,2) is a positive definite matrix and the vector ξi is such as ξi=Si1/2sign⁡Si,νiT(i=1,2).
Therefore, the trajectories associated with the discontinuous dynamics (35) and (36) exhibit a finite time reachability to zero from any given initial condition provided that the constants K¯1, K¯2, W1, and W2 are positive.
SinceS1 and S2 are driven to zero in finite time and following the same arguments as the ones used to prove the previous sliding mode controller, it can be concluded that z1(t), z2(t), z3(t), and z4(t) converge to zero as t→∞.Thus, the controller (32)–(34) guarantees the asymptotic convergence of x1(t), x2(t), x3(t), and x4(t) to their desired values as t→∞.


## 5.2. Simulation Results

The performance of the closed loop system when using the second-order super-twisting sliding mode controller was simulated. The initial and desired states of the power system are the same as in Section3. The controllers u1 and u2 given by the relationship (20) and (32)–(34) are applied to the power system (12). The simulation results are depicted in Figures 24–27. The machine power angle δm versus time and the deviation in the frequency ωm versus time are shown in Figures 24-25, respectively. The load power angle δ versus time and the load voltage VL versus time are shown in Figures 26-27, respectively. These figures clearly indicate that the states of the system reached their desired values in less than 2 sec. Also, the depicted responses do not suffer from the chattering problem. Hence, it can be concluded that the proposed second-order sliding mode controller works well. Moreover, we simulated the responses of system with the same initial conditions except for x2(0)=ωm(0)=0.2 rad/sec. This initial condition still results in a chaotic system. The simulation results are shown in Figures 28–31. Again, these figures show that the proposed controller still work but the transient responses suffer a little bit.Figure 24
The machine power angleδm versus time using the second-order sliding mode controller when ωm(0)=0.Figure 25
The deviation in the frequencyωm versus time using the second-order sliding mode controller when ωm(0)=0.Figure 26
The load power angleδ versus time using the second-order sliding mode controller when ωm(0)=0.Figure 27
The load voltageVL versus time using the second-order sliding mode controller when ωm(0)=0.Figure 28
The machine power angleδm versus time using the second-order sliding mode controller when ωm(0)=0.2.Figure 29
The deviation in the frequencyωm versus time using the second-order sliding mode controller when ωm(0)=0.2.Figure 30
The load power angleδ versus time using the second-order sliding mode controller when ωm(0)=0.2.Figure 31
The load voltageVL versus time using the second-order sliding mode controller when ωm(0)=0.2.


## 6. Conclusion

The problem of controlling chaos in a power system is studied in this paper. The power system is modeled using four first-order differential equations. Simulation studies indicate that the system is chaotic for some values of the parameterQ1. Three controllers are designed to suppress chaos in the system. The first controller is a feedback linearization controller. The second and third are of the sliding mode type. The simulation results show that the proposed control schemes work well. However, the second-order super-twisting sliding mode controller is the best among the three controllers since it possesses robustness features while having the advantage of less chattering as compared to the conventional sliding mode controller.Future work will address the problem of suppressing chaos using other types of sliding mode control schemes as well as observer based controllers.



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*Source: 2703684-2018-04-17.xml*