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.
The motion of the rotor of the generator is given by the swing equation such aswhere is the phase angle of the generator (radians), is the momentum of inertia of the generator, is the damping coefficient, is the mechanical power, and is the electrical power.
The generator rotor speed deviation is related to such thatThe generated electrical power can be written aswhere and are the magnitude and phase angle of the admittance between the second generator and the load busbar.
Using (1), (3), and (2), we getThe model of the load is as follows [16]:where and are the constant real and reactive powers of the induction motor, is the time constant of the motor, and and are the real and reactive components of the load. The constants , , , , and are empirical constants that characterize the load.
The real power supplied to the load bus by the network is expressed aswhere , , and 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 asUsing the above equations, we can writeCollecting (2), (4), (8), and (9), the model of the power system can be represented as follows:We will use the values of the parameters to be the same as in [16]. The parameters of the generator are such that , , , , , and . The parameters of the network are such that , , , , , , and The parameters of the load are such that , , , , , , , , and . 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:where the parameters – in the power system model (11) have the following values: , , , , , , , , , , , , , , , , , , and .
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 is a system parameter representing the reactive power in the system.
For ease of presentation, we define the state variables – such that . Hence, model of the system is expressed as follows:The fourth-order model of the power system (12) is simulated when and with . The simulation results are shown in Figures 2–7. The machine power angle versus time and the deviation in the frequency versus time are shown in Figures 2-3. The load power angle versus time and the load voltage versus time are shown in Figures 4-5. The machine power angle versus the deviation in the frequency and the machine power angle versus the deviation in the frequency and the load voltage are presented in Figures 6-7. It is evident from these figures that the power system is chaotic for this value of .
The objective of this paper is to design controllers for the power system so that chaos is suppressed.
Let , , , and represent the desired values of , , , and , 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:To facilitate the design of the controllers, we propose to use the following transformation:Note that , , and represent the errors between the actual and the desired states for the first three states of the system.
Remark 2. The choice of the states , , , and in (14) guarantees that when , , , and converge to zero as tends to infinity, then the states , , , and 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:where , , , and are such thatIn 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 , , , and to converge to zero as tends to infinity. Hence, the states , , , and will converge to their desired values, respectively.
3. Design of a Feedback Linearization Controller to Suppress Chaos
3.1. Design of the Controller
Let , , , and be positive scalars such that the polynomial is a Hurwitz polynomial.
Theorem 3. The feedback linearization controllerwhen applied to the transformed fourth-order power system given by (15) guarantees the asymptotic convergence of the states to zero as 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:whereIt can be easily checked that since , , , and are positive scalars such that is a Hurwitz polynomial, then the matrix is a stable matrix (i.e., its eigenvalues are located in the left half of the complex plane).
The solution of (18) is , where is the initial state at . Since is a stable matrix, then will converge to zero as .
Therefore, since converges to zero as , then the states of the power system will converge to their desired values as 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 and given by (17) are transformed back into the original coordinates such that
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 that . The controllers and given by (17) and (20) are applied to the power system (12). The desired states of the power system are such that . The simulation results are depicted in Figures 8–11. The machine power angle versus time and the deviation in the frequency versus time are shown in Figures 8-9, respectively. The load power angle versus time and the load voltage 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 sec. Hence, the proposed feedback linearization controller works well in suppressing chaos in the power system.
It is well known that chaotic systems are very sensitive to initial conditions. Therefore, we simulated the system with the same initial conditions except for 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 rad/sec.
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.
Let , , , , , and be positive scalars. Also, let the sliding surface be such thatAnd let the sliding surface be such that
Theorem 5. The sliding mode controller,when applied to the transformed power system model given by (15), ensures that the states , , , and converge to zero as tends to infinity.
Proof. Differentiating (21) with respect to time and using (15), it follows thatSubstituting by its value from (23), it follows thatAlso, differentiating (22) with respect to time, using the model of the transformed system in (15) and substituting by its value from (24), it follows thatThe dynamic equations (26)-(27) satisfy the following equation:Therefore, the dynamic equations (26)-(27) guarantee that for () when . 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 , are positive and the gains and are chosen to be large enough strictly positive. Hence, the dynamic equations (26)-(27) are guaranteed to reach the surfaces and in finite time.
Since is driven to zero in finite time, the state variables , , and are governed after such finite amount of time by the equation . Using the transformed model in (15), this equation is equivalent to the second-order differential equation: . The solution of this equation, , will asymptotically converge to as because the design parameters and are chosen to be positive scalars. In addition, since will asymptotically converge to , it can be concluded from the transformed system in (15) that will asymptotically converge to as . Similarly, will also converge to zero as . In addition, since is driven to zero in finite time, the variable 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 , , , and to zero as .
Thus, the controller (23)-(24) guarantees the asymptotic convergence of the states of the power system , , , and to their desired values as .
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 with 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 controllers and in (20) when and 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 versus time and the deviation in the frequency versus time are shown in Figures 16-17, respectively. The load power angle versus time and the load voltage 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 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.
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 where and are positive scalars. Also, let , , , and be positive scalars.
Theorem 7. The second-order sliding mode controllerwithwhen applied to the transformed model of the power system given by (15), ensures that the states , , , and converge to zero as tends to infinity.
Proof. Differentiating the sliding surface given by (30) with respect to time and using the transformed model of the power system given by (15), and substituting by its value from (32), it follows thatAlso, differentiating the sliding surface given by (31) with respect to time, using the transformed model of the power system given by (15), and substituting by its value from (33), it follows thatHence, the dynamics that govern the sliding surfaces and are as follows:The dynamics in (37) correspond to the super-twisting algorithm [54, 55]. The finite time convergence to zero of for was proven in [54, 55] by using the Lyapunov function:where is a positive definite matrix and the vector is such as .
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 , , , and are positive.
Since and 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 , , , and converge to zero as .
Thus, the controller (32)–(34) guarantees the asymptotic convergence of , , , and to their desired values as .
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 Section 3. The controllers and 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 versus time and the deviation in the frequency versus time are shown in Figures 24-25, respectively. The load power angle versus time and the load voltage 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 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.
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 parameter . 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by Kuwait University under Research Grant no. EE03/13.