Adaptive Fixed-Time Fast Terminal Sliding Mode Control for Chaotic Oscillation in Power System

The second-order chaotic oscillation systemmodel is used to analyze the dynamic behavior of chaotic oscillations in power system. To suppress chaos and stabilize voltage within bounded time independent of initial condition, an adaptive fixed-time fast terminal sliding mode chaos control strategy is proposed. Compared with the conventional fast terminal sliding mode control strategy and finite-time control strategy, the proposed scheme has advantages in terms of convergence time and maximum deviation. Finally, simulation results are given to demonstrate the effectiveness of the proposed control scheme and the superior performance.


Introduction
As a typical multivariable and strongly coupled nonlinear system, power system exhibits a lot of nonlinear dynamic behaviors during operation, such as low-frequency oscillation, bifurcation, and chaos [1].If the amplitude of the disturbance satisfies certain conditions, the power system will appear with continuous and random chaotic oscillations which may lead to instability of the system, voltage collapse, and even catastrophic blackouts [2,3].Therefore, it is imperative to study control schemes for chaos suppression in power system.In 1993, Chiang et al. [4] observed the chaotic behavior under different load conditions in the power system firstly.Since then, chaos control had become a hot issue of academia.The control methods of chaotic oscillation can be divided into two aspects: guiding the development of the chaotic oscillation to the expected orbit and suppressing the occurrence of system chaos [5,6].At present, the commonly used control methods include parameter perturbation method (OGY) [7][8][9][10], feedback control [11][12][13][14], adaptive control [15][16][17], fuzzy control [18][19][20], and backstepping control [21].These approaches have important theoretical and practical significance to guarantee the stability and secure operation of power system while some defects did exist at the same time.For example, the backstepping control was proposed to systematize and structure the design process of the system's Lyapunov functions and controllers through the reverse design.But its structure is very complex, and the complexity of regression matrix would become stronger especially when the nonlinear damping existed for the system parameters uncertainty [22,23].Fuzzy control is difficult to adapt to the requirements of large-scale adjustment.It needs to constantly adjust the control rules and parameters [24][25][26][27].Moreover, all the above control methods can only reach the asymptotic stability; that is, the convergence time cannot be allocated in advance.From the point of view of the operation of the power system, the oscillation is acceptable if it is damped within a limited time.
Fast terminal sliding mode (FTSM) control can achieve system stability in finite time.It has the advantages of strong robustness to external disturbances and parameter disturbances [28,29].The fast terminal sliding mode (FTSM) control can converge the system to the equilibrium point on the sliding mode surface in finite time while the traditional sliding mode control can only ensure that it reaches the sliding mode surface during the same period [30].Finite-time control can also keep the system stable within bounded time [31].Because the finite-time control has stronger robustness, better anti-interface, and faster convergence, it has become a research hotspot of academia in recent years.Cai et al. [32] applied the finite-time theory for the first time to deal with the generalized synchronization problem of different order chaotic systems.Khaki et al. [33] proposed a novel fuzzy finite-time variable structure controller for double integrator power systems to damp the complicated chaotic oscillations of an interconnected power system, when such oscillations can be made by load perturbation of a power system working on its stability edges.The above two control methods can make the system stable in finite time but also subject to the limitless of finite-time stability.The system stability is affected by the initial condition while it is difficult to obtain accurate parameters in practical project.
Fixed-time stability is an extension of finite-time stability.Compared with the above control approaches, fixed-time control can not only maintain stronger robustness and better anti-interference and ensure a clear upper bound of settling time, but also make the system globally uniformly ultimately bounded stable one with any initial conditions [34].Polyakov firstly proposed the concept of fixed-time in 2012 [35].At present, due to this attractive property, fixed-time control has begun to be widely used in hydraulic turbine governing systems [36], PMSM [37], multiagent systems [38], aircraft systems [39], chaotic systems [40], and so on.Ni et al. [40] firstly applied the STATCOM to suppress chaos of power system and authors designed a fixed-time dynamic surface high-order sliding mode control approach to achieve semiglobally fixed-timely uniformly ultimately bounded stabilization.In view of the discussion of the above control schemes, this paper presents an adaptive fixed-time fast terminal sliding mode control scheme which can accelerate the convergence, suppress chaos in the power system, and avoid voltage collapse.The main advantages of proposed controller are that it can guarantee the system stability in finite time independent of initial state and the settling time can be directly calculated.
The structure of this paper is as follows: the second section gives some theorems and lemmas to facilitate the derivation of the system control method in the later text.In the third section, an adaptive fixed-time fast terminal sliding mode control scheme is proposed to suppress chaos in the power system.The fourth section provides the simulation results in this paper to demonstrate the effectiveness of the proposed control scheme.The fifth section has made some summary conclusion based on the above work.

Preliminaries
In this paper, for the convenience of analysis, we firstly introduce a necessary definition and some lemmas which play an important role in design process.
Consider the following differential equation system: where  ∈   is the system state variable; f :  + ×   →   is a smooth nonlinear function.Assume the origin is an equilibrium point of (1).
Definition 2 (see [35,43]).The origin of system ( 1) is said to be fixed-time stable equilibrium point if it is globally finite-time stable with bounded convergence time ( 0 ); that is, there exists a bounded positive constant  max such that ( 0 ) <  max satisfies.

Main Results
Ignoring the dynamic process of excitation loop and damping winding, it is assumed that the mechanical power of generator is always the same in transient process, and the transient salient effect of generator is not considered.The external factors of system, such as disturbance's influence on the system, are mainly considered.The equivalent circuit of power grid second-order chaotic oscillation system is presented in Figure 1, where G 1 , G 2 and T 1 , T 2 are equivalent generators and main transformers of system, respectively.
Figure 1: Equivalent circuit of power grid two-order chaotic oscillation system.
The power system is composed of two generator buses and one load bus.Here, the second-order nonlinear mathematical model of the synchronous generators used in this paper is as follows: where  and  are state variables that denote relative angle and angular frequency between equivalent generators, respectively; H and D refer to generator inertia and damping coefficient, respectively; P  and P  are electromagnetic power and mechanical power of generator, respectively;   and  represent amplitude and frequency of load disturbance.To simplify the model, we make the following transformation: where  =   /,  = /,  =   /,  =   /, [ 1 ,  2 ] = [, ], () = - sin  1 - 2 + , and u is control input.Then the fast terminal sliding mode surface can be designed as where parameters satisfy  0 ,  0 >0, and q 0 , p 0 are positive odd numbers.The derivative of the system (7) gives For system (6), we can see that X 0 = (z, 0) is a point of equilibrium of the system, where z is a constant.To realize the global stability of system (6) based on the fixed-time stability theory in the case of the equilibrium X 0 , another global fast terminal sliding mode is taken as the following form: where parameters satisfy ,  >0, 0<  <1,  >1, and q< p. p and q are positive odd numbers.According to the above analysis, we propose our main results in the following theorem.
Theorem 5. System ( 6) can be stabilized in fixed-time under the following adaptive controller: where F is the estimated value of the uncertain parameter and k is the tuning parameter of the terminal attractor satisfying the following adaptive laws: where  is the arbitrary constant.
Proof.Firstly, substitute ( 10) into (8) to get the following formula which will be used directly in the process of proof: Consider the following Lyapunov candidate function as By using the designed controller u and the appropriate tuning parameters, the time derivative of V can be obtained as 4

Numerical Simulations
In this section, numerical simulations are performed to demonstrate the effectiveness and the superiority of the proposed control method.System (6) is selected for simulation.Without loss of generality, the system parameters are a =1, b =0.02, and c =0.2 and controller parameters are  0 =1,  0 =2, q 0 =5, p 0 =9, =20, =15, =0.5, =1.5, p=3, q=1, =0.3, and k(0)=0.2.The initial values of the state variable are S ( 0 ,  0 ) = (0.43,0.003).In the power system, the damping coefficient and the inertial coefficient of the generator are often constant, and the variation of the load disturbance can usually cause the chaotic oscillation of the power system.Further, when amplitude of periodic load disturbance F varies to 0.2593, the power system exhibits chaotic oscillations which are displayed in Figures 2 and 3.As the parameters are selected, Lyapunov exponents can be calculated as ( 1 , 2 , 3 ) =(0.0174, 0, −0.0374).There is one positive Lyapunov exponent, which validates the existence of chaotic attractor.Figure 2 is the phase portrait of chaotic power system and it shows chaotic behavior of the power system clearly.Figure 3 shows the time responses of state variables in chaotic power system.As can be seen from Figure 3, the time responses of state variables are in an irregular and aperiodic oscillatory state and their trajectories are unpredictable after a long period of time.
The chaotic oscillation state has great harm to the power system.Circuit voltage and current waveform distortion caused by the oscillation, particularly over voltage, will cause severe local instability with potentially serious impact and damage to the power system.Therefore, an immediate control action needs to be activated to suppress chaos.Simulations are conducted to examine the proposed controller performance in terms of suppressing chaos in power system and stabilizing the power system to its desired operating point.The time responses of state variables and the phase portrait of chaotic power system under proposed controller are presented in Figures 4 and 5, respectively.Figures 6  and 7 are the time response of tuning parameter of the terminal attractor and the time response of the proposed controller u.As Figure 7 shows, the controller was added at 50s.Further, the bound of convergence time T can be calculated by taking the controller parameters we selected into (17).After calculation, we get T=14.59s.Therefore, the theoretical limit of stabilization time in the numerical simulations is t 1 =64.59s.Figure 4 shows that the control objectives have been controlled to the stability value.The system state variables  and  reached stable states at 59.2s and 53.3s, respectively, and they are all smaller than t 1 .Figure 5 depicts that the control objective has been stabilized to its desired operating point and the chaotic oscillation has been suppressed completely, which verifies the effectiveness of proposed controller.
To explore the influence of different initial conditions on the control effect, the simulation has been done.Figure 8 shows the time responses of state variables when the initial states of system are S1=(0.43,0.003),S2=(0.45,0.005),and S3=(0.47,0.007).As can be seen from Figure 8, the oscillation amplitude and overshoot are different, but the difference between the settling times is very small.The system state  The simulated condition for the controllers is chosen to be the same.In Figures 9(a) and 9(b), where  = 1.5 and  = 0.6, 0.7, 0.8, 0.9, it is shown that, for both  and , the smaller the value of  is, the faster the convergence rate is.In Figures 9(c) and 9(d), where  = 0.6 and  = 1.5, 1.6, 1.7, 1.8, it is shown that the law of  is the same as that of .In other words, the convergence time is a decreasing function of the values of  and  for both  and , which is consistent with the theoretical analysis of the maximum stable time T of the system in the previous section.In addition, it is found from Figures 9(a) and 9(c) that the values of  and  affect the stability value of the system state variable .Therefore, we can get the stability value of the system state variable  we need by adjusting the values of the controller parameters  and .
Figures 10(a) and 10(b) are the comparisons of time responses of state variables in chaotic power system with the proposed controller, the fast terminal sliding mode (FTSM) controller, and finite-time controller.To make a fair comparison, the initial conditions, parameters, and the tuning parameters are identical.The simulation time is set to 20s, and the controllers are added at 10s.Figures 10(a) and 10(b) show that the settling time of proposed control scheme is smaller than that of the other two.Moreover, the time response curves of system state variables ,  under proposed control scheme have smaller maximum deviation than that under the other two, which demonstrates the proposed controller has better control effect.Table 1 records the exact numerical values of the settling time and maximum deviation in the simulation results, which verifies the conclusion more accurately.

Conclusion
In this paper, to investigate the problem of chaos suppression and voltage stabilization in chaotic power system, a new control scheme based on the fixed-time stability theory is proposed.An adaptive fixed-time fast terminal sliding mode chaos control strategy is presented to design controller.Simulation results illustrate the effectiveness and superiority of the proposed controller.Compared with the conventional fast terminal sliding mode control strategy and finite-time control strategy, the proposed controller has more advantages in the aspect of convergence time and maximum deviation.Moreover, the settling time is independent of the initial state and can be directly calculated.Note that the present study did not consider the effect of noise perturbation; future research will extend the proposed control strategy to highorder system with noise perturbation.

Figure 9 :
Figure 9: The time responses of state variables with different values of  and .

Figure 10 :
Figure 10: Time responses of state variables in chaotic power system with different controllers.

Table 1 :
Record of the numerical values of the settling time and the maximum deviation.) show the variations of the system state variables  and  with different values of  and .