Control of Chaos in a Single Machine Infinite Bus Power System Using the Discrete Sliding Mode Control Technique

Under certain conditions, power systems may exhibit chaotic behaviors which are harmful and undesirable. In this paper, the discrete time slidingmode control technique is used to control a chaotic power system.Theobjective of the control is to eliminate the chaotic oscillations and to bring order to the power system. Two discrete time sliding mode control (DSMC) schemes are proposed for a fourth order discrete time chaotic power system. The first DSMC control scheme is based on the well-known exponential reaching law.The second DSMC control scheme is based on the recently developed double power reaching law. It is shown that the states of the controlled system converge to their desired values. Simulation results are presented for different values of the gains of the controllers as well as for different initial conditions. These results indicate that both control schemes work well. However, the simulation results show that the second control scheme gave better results since it was able to greatly reduce the chattering problem.


Introduction
Chaos in power systems was observed by [1,2] over a range of loading conditions.Chaotic oscillations in power systems are harmful and undesirable.The increasing demand for electric power forces the power system to operate nearly close to its stability boundary.In this operating environment, a sudden disturbance can lead to a chaotic behavior [3][4][5][6].Chaos is related to many power system instability phenomena such as voltage collapse which occurs when the power system is heavily loaded.Voltage collapse is characterized by a slow change in the operating point of the system caused by an increase in loads which results in a gradual decrease in voltage magnitudes until a sharp accelerated drop in voltage occurs.Voltage collapse can result in catastrophic blackouts [7][8][9][10][11][12].Therefore, it is imperative to properly control the power system so that chaos is suppressed and chaotic oscillations are eliminated.
Chaos suppression in power systems has received the attention of many researchers.In recent years, many control methods were applied to suppress chaos and stabilize the voltage of the power systems [13][14][15][16][17][18][19].Adaptive control was proposed in [13].Fuzzy and neural control methods were discussed in [14][15][16].Linear and nonlinear state feedback controllers are developed in [18] for the control of the bifurcation phenomenon in a power system A passivity-based adaptive controller was used in [19] to suppress chaotic oscillations in a power system.
Due to its simple implementation, good transient response, and robustness to parameters uncertainties and to disturbances, the sliding mode control technique has been applied to many nonlinear systems such as soft landing control, trajectory tracking, motor control, and power system control [20][21][22][23][24][25].Power system chaos suppression using the sliding mode technique was proposed in the literature [26][27][28].In [26], an approach that combines time scale separation design and sliding mode control was proposed.High order sliding mode controller was reported in [27]; the controller design was based on backstepping method.
As a result of the extensive use of computers in the implementation of controllers, discrete time sliding mode control has generated a sizable amount or research interest.There is subtle difference between continuous time sliding mode control and discrete time sliding mode control.For the DSMC, the control signal is determined once in every sampling interval and it is held constant during the sampling period [29][30][31][32].Hence, we have the so-called quasi-sliding mode in DSMC.As a consequence, the design of discrete time sliding mode controllers needs to be carefully investigated.This paper presents two discrete time sliding mode control schemes to eliminate chaos in a power system; the proposed DSMC control schemes use different reaching laws.The developed controllers are applied to a discretized 4th order three-bus power system model.Simulation studies clearly show that the proposed controllers work well.
The rest of the paper is organized as follows.The mathematical model of the power system is presented in Section 2; simulation results are also presented to show that the discretized model of the system displays chaotic behavior under certain loading conditions.The control problem is formulated in Section 3. The design of two discrete sliding mode control schemes for suppression of chaos is developed in Sections 4 and 5; simulation results for the controlled system are also presented and discussed in these sections.Finally, the conclusion is given in Section 6.

Modeling of the Power System
The power system studied in this paper is depicted in Figure 1; it consists of two generator buses and one load bus.One of the generator buses is an infinite busbar while the other generator bus has a constant voltage magnitude   .The Thevenin equivalent model representing the infinite busbar (external power system) is denoted by   ,   , and   .The Thevenin equivalent model for the second generator (internal power system) is denoted by   ,   ,   , and   .The load bus consists of an induction motor in parallel with a PQ load.Also, a fixed capacitor is added in parallel with the load to increase the voltage up to near one per unit.The magnitude of the load busbar voltage is denoted by  and phase angle of the load busbar voltage is denoted by .
The differential equation that describes the motion of the rotor of the synchronous generator can be written as follows: where  is the momentum of inertia of the generator,   is a modified damping coefficient,   is the accelerating power,   is the input mechanical power, and   is the output electrical power.
The electric power   is given as where  is the load busbar voltage and  * is the complex conjugate of the current of the load busbar.
Referring to Figure 1, the equation of the electric power is as follows: where   and   are the magnitude and phase angle of the admittance between the second generator and the load busbar.
The induction motor is as proposed in [2].Hence, the load model is described as follows: where the constant real and reactive powers of the induction motor are  0 and  0 ; the real and reactive components of the  load are  1 and  1 .The time constant of the motor is .The constants parameters   ,  V ,   ,  V , and  V2 denote empirical constants that characterize the model of the load.
The real power supplied to the load bus by the network is as follows: where   0 ,   0 , and   0 are the adjusted values of the Thevenin equivalent model of the infinite bus taking into account the added capacitor.
The reactive power supplied to the load bus by the network can be written as Hence, the dynamic model of the power system can be described as follows: Also,   is related to the generator rotor speed deviation  such that Using ( 1), (3), and (8), we get Using ( 7), (8), and ( 9), the model of the fourth order power system can be written as follows: Define the state variables  1 ,  2 ,  3 , and  4 such that  1 = ,  2 = ,  3 =   , and  4 = .Also, recall that  1 is a system parameter representing the reactive power in the system.The parameters of the power system are obtained from [2].The parameters of the generator are such that  = 0.3,   = 0.05,   = 1,   = 1,   = 5, and   = −5.The parameters of the network are such that  0 = 1,  0 = 20,  0 = −5,   0 = 2.5,   0 = 8, and   0 = −12.The empirical constants of the load are such that   = 0.4,  V = 0.3,   = −0.03, V = −2.8, V2 = 2.1,  = 8.5,  0 = 0.6,  0 = 1.3,  1 = 0.0, and  = 12.All the parameters values are in per unit except the angles which are in degrees.
Then, the equations of the power system given by ( 10) can be written in compact form as follows: This paper deals with the design of discrete time sliding mode control schemes to suppress chaos present in the power system.Therefore, the continuous time model of the system given by ( 11) is discretized using the Euler method.The discretized dynamic model of the power system is as follows: where  For  1 = 11.379, the discretized model of the power system given by ( 12) has Lyapunov exponents:  1 = 0.2793,    2 = 0.001 ≈ 0,  3 = −4.24,and  4 = −100.74.Since there is one positive Lyapunov exponent, two negative Lyapunov exponents, and one Lyapunov exponent which is approximately zero, then it can be concluded that the discretized model of the power system ( 12) is chaotic when  1 = 11.379.
In addition, the discretized model of the power system (12) is simulated using the MATLAB software.The simulation results are presented in Figures 2-9.Figures 2-5 show the trajectories of the load phase angle versus time, the load voltage versus time, the generator phase angle versus time, and the generator rotor speed deviation versus time, respectively.Also, 3D plots of the states of the system are shown in Figures 6-9.Hence, the simulation results clearly show that the discretized model of the power system given by ( 12) is indeed chaotic when  1 = 11.379.

Problem Formulation
The objective of this paper is to drive the states of the system  1 = ,  2 = ,  3 =   , and  4 =  to their desired constant values  1 ,  2 ,  3 , and  4 , respectively.Note that the desired value of  4 =  should be zero; that is,  4 = 0. We will denote by   the vector of desired values such that To be able to control the discretized model of the chaotic power system in (12), control inputs need to be added to the system.Adding control inputs to the first, the second, and the fourth difference equations of the system, the discretized dynamic model of the power system with control inputs is such that Define the regulation error vector Using ( 13) and ( 14), the error dynamic model of the power system can be written as follows: Two DSMC schemes will be designed to drive the states of the system , ,   , and , to converge to their desired constant values  1 ,  2 ,  3 , and  4 , respectively.This is done by forcing the error vector () to converge to (0, 0, 0, 0).The design of the two control schemes will be discussed in detail in the next two sections.

A DSMC Scheme Based on the Exponential Reaching Law
4.1.Design of the Controller.The first step in the design of a sliding mode controller is the choice of the sliding surfaces.
Since the system has three control inputs, we need to choose three sliding surfaces.The sliding surfaces  1 (),  2 (), and  3 () are chosen such that where  is a positive design parameter such that || < 1.Also, define the sgn function such that sgn Let the positive parameters , , and   be such that 1 −   > 0. Proposition 1 gives the first discrete sliding mode control scheme.
Proposition 1.The following sliding mode control controller when applied to the discretized model of the power system (13) guarantees the convergence of the states of the system to their desired values.
Proof.Using the sliding surfaces given by ( 16), the dynamical model of the errors in (15), and the control scheme given by (18), we obtain Therefore, ( 19) can be written as follows: Sarpturk et al. [33] showed that a necessary and sufficient condition assuring both sliding motion and convergence onto the th hyperplane may be stated as The condition given by (21) guarantees that all the trajectories will enter and remain within a domain of decreasing or at the worst case nonincreasing radius.The inequality in (21) can be decomposed into the following two inequalities: (  ( + 1) +   ()) sgn (  ()) > 0.
Conditions ( 22) and ( 23) result in a lower bound and in an upper bound for the control action, respectively.While condition (22) implies that the closed-loop system should be moving in the direction of the sliding surface, condition (23) implies that the closed-loop system is not allowed to go too far in that direction.Moreover, it was shown by Gao et al. [34] that the discrete system given by ( 20) meets the existence and the reaching condition |  ( + 1)| < |  ()|.Also, it was specified in [34] that the system given by (20) has the following properties.
(1) Starting from any initial condition, the trajectories of system (20) will move toward the switching surface and cross it in finite time.(2) After crossing the switching surface, the trajectories of system (20) will cross the plane again in every successive sampling period, resulting in a zigzag motion about the switching surface.The size of each successive zigzag step is nonincreasing and the trajectories stay within a specified band called the quasi-sliding band.
It was stated in [34] that the quasi-sliding band (QSMB) is given by Also, the width of the QSMB is It is worth mentioning that a desirable reaching mode response can be achieved by adjusting the parameters  and .

Simulation Results.
The designed sliding mode controller in (18) is applied to the discretized model of the chaotic power system given by (13).The performance of the controlled system is simulated using the MATLAB/SIMULINK software.The parameters of the system are taken from [2].In the simulation, the initial states are taken to be (0) = 0.2, (0) = 0.97,   (0) = 0.3, and (0) = 0.The desired states are taken to be  1 = 0.1426,  2 = 0.8788,  3 = 0.3587, and  4 = 0.The sampling time   is taken to be   = 0.001.We simulated the performances of the controlled power system for 3 different cases.For the first case, the parameters of the controller are taken to be  = 1,  = 0.001, and  = 0.    show the trajectories of the load phase angle versus time, the load voltage versus time, the generator phase angle versus time, and the generator rotor speed deviation versus time.The errors versus time are shown in Figure 14.It is clear that the errors converge to zero in about 5 seconds.Therefore, it can be concluded that the proposed discrete sliding mode controller is able to suppress chaos in the power system.In addition, the controllers  1 ,  2 , and  3 versus time are presented in Figures 15-17.It is clear that some of the trajectories of the system as well as the controllers display a zigzag motion of the trajectories about the switching surfaces; the magnitude of the zigzag motion is higher for the generator rotor speed deviation and for the control law  3 .For the second case, the parameters of the controller are taken to be  = 5,  = 0.01, and  = 0.   less than one second.It is noted that the response of the system is faster than the previous case.However, some of the trajectories of the system display a zigzag motion of the trajectories about the switching surfaces; the magnitude of the zigzag motion is higher for the generator rotor speed deviation.For the third case, the parameters of the controller are taken to be  = 1,  = 0.001, and  = 0.1.However, the initial condition of () is taken to be (0) = 0.2.It is well known that chaotic systems are very sensitive to  initial conditions.We made sure that the discretized model of the power system is chaotic for this initial condition.The simulation results are shown in Figures 23-27.Figures 23-26 show the trajectories of the load phase angle versus time, the load voltage versus time, the generator phase angle versus time, and the generator rotor speed deviation versus time.The errors versus time are shown in Figure 27.It is clear that the errors converge to zero in about 5 seconds.We can conclude that the controller is robust with respect to changes in the initial conditions.Therefore, it can be concluded that the proposed discrete sliding mode controller is able to suppress chaos in the power system.The next section proposes a controller to lessen the chattering problem of this controller.

The DSMC Scheme Based on the Double Power Reaching Law
In this section, we will use the double power reaching law [35] to design a discrete sliding mode controller for the discretized model of the chaotic power system.The proposed controller greatly reduces the zigzag motion of the trajectories of the system about the switching surfaces.16) will be used in the design of the second DSMC control scheme.
Let the positive parameters  1 ,  2 , ,   , ,  be such that 1 −   > 0, 0 <  < 1, and  > 1.The following proposition gives the second DSMC control scheme.Proposition 2. The following sliding mode control scheme when applied to the discretized model of the power system (13) guarantees the convergence of the states of the system to their desired values.
Therefore, it is guaranteed that all the trajectories of system (31) will enter and remain within a domain of decreasing or at the worst case nonincreasing radius.Hence, the discrete system in (31) meets the existence and the reaching condition It should be mentioned that Zhao et al. [35] analyzed the reaching rate of the double power reaching law and showed that it possesses better features than those of the conventional exponential reaching law as it reduces the chattering of the trajectories of the controlled system.

Simulation Results.
The designed sliding mode control law in ( 29) is applied to the discretized model of the chaotic power system given by (13).The performance of the controlled system is simulated using the MATLAB/SIMULINK software.In the simulation, the initial states, the desired states, and the sampling time are taken to be the same as in the previous section.Again, we simulated the performances of the controlled power system for 3 different cases.For the first case, the parameters of the controller are taken to be  = 1,  1 = 0.001,  2 = 0.001,  = 0.5,  = 2, and  = 0.1.The simulation results are shown in Figures 28-35.Figures 28-31 show the trajectories of the load phase angle versus time, the load voltage versus time, the generator phase angle versus time, and the generator rotor speed deviation versus time.The errors versus time are shown in Figure 32.It is clear that the errors converge to zero in about 5 seconds.Therefore, it can be concluded that the proposed discrete sliding mode controller is able to suppress chaos in the power system.In addition, the controllers  1 ,  2 , and  3 versus time are presented in Figures 33-35.It is clear that chattering was greatly reduced.For the second case, the parameters of the controller are taken to be  = 5,  1 = 0.01,  2 = 0.01,  = 0.5,  = 2, and    It is clear that the errors converge to zero in less than one second.It is noted that the response of the system is faster than the previous case; also chattering was greatly reduced.For the third case, the parameters of the controller are taken to be  = 5,  1 = 0.001,  2 = 0.001,  = 0.5,  = 2, and  = 0.1.The initial condition of () is taken to be (0) = 0.2.The simulation results are shown in Figures 41-45.Figures 41-44 show the trajectories of the load phase angle versus time, the load voltage versus time, the generator phase angle versus time, and the generator rotor speed deviation versus time.The errors versus time are shown in Figure 45.It is clear that the errors converge to zero in about 1.5 seconds.We can conclude that the controller is robust with respect to changes in the initial conditions.Therefore, it can be concluded that the proposed second discrete sliding mode controller is able to suppress chaos in the power system.

Conclusion
This paper dealt with the suppression of chaos in a discretized model of a chaotic power system.The discrete sliding mode control technique was used.Two controllers were proposed.The first controller uses an exponential reaching law; the second controller uses a double power reaching law.Extensive simulation studies were presented to illustrate the effectiveness of the proposed controllers.The simulation studies show that the two controllers work well.Also, the simulation results indicate that some of the trajectories of the discretized model of the power system when controlled using the first control scheme display zigzag motion about the switching surfaces.However, the second proposed controller greatly reduces the zigzag motion of the trajectories of the system about the switching surfaces.Therefore, the second control scheme gave better results than the first controller.Moreover, simulation studies show that the proposed controllers are robust with respect to changes in the initial conditions.Future research will tackle the control of chaos in power systems using observer-based sliding mode control schemes.

Figure 1 :
Figure 1: Schematic of a power system.

Figure 2 :
Figure 2: The load phase angle versus time without control.

Figure 3 :
Figure 3: The load voltage versus time without control.

Figure 4 :
Figure 4: The generator phase angle versus time without control.

Figure 10 :
Figure 10: The load phase angle versus time using the first control law (case 1).

3 Figure 17 :
Figure 17: The controller  3 versus time using the first control law (case 1).

1 .Figure 18 :
Figure 18: The load phase angle versus time using the first control law (case 2).

Figure 19 :Figure 20 :
Figure 19: The load voltage versus time using the first control law (case 2).

16 Figure 21 :
Figure 21: The generator rotor speed deviation versus time using the first control law (case 2).

4 Figure 22 :
Figure 22: The errors versus time using the first control law (case 2).

Figure 23 :
Figure 23: The load phase angle versus time using the first control law (case 3).

Figure 24 :Figure 25 :
Figure 24: The load voltage versus time using the first control law (case 3).

Figure 26 :Figure 27 :. 10 Discrete
Figure 26: The generator rotor speed deviation versus time using the first control law (case 3).

Figure 28 :
Figure 28: The load angle versus time using the second control law (case 1).

Figure 29 :
Figure 29: The load voltage versus time using the second control law (case 1).

Figure 30 :Figure 31 :Figure 32 :
Figure 30: The generator phase angle versus time using the second control law (case 1).

Figure 33 :
Figure 33: The controller  1 versus time using the second control law (case 1).

Figure 38 :Figure 39 :
Figure 38: The generator phase angle versus time using the second control law (case 2).

4 Figure 40 :
Figure 40: The errors versus time using the second control law (case 2).

4 Figure 45 :
Figure 45: The errors versus time using the second control law (case 3).