Fractional-Order Hyperbolic Tangent Sliding Mode Control for Chaotic Oscillation in Power System

Chaotic oscillation will occur in power system when there exist periodic load disturbances. In order to analyze the chaotic oscillation characteristics and suppression method, this paper establishes the simplified mathematical model of the interconnected two-machine power system and analyzes the nonlinear dynamic behaviors, such as phase diagram, dissipation, bifurcation map, power spectrum, and Lyapunov exponents. Based on fractional calculus and sliding mode control theory, the fractional-order hyperbolic tangent sliding mode control is proposed to realize the chaotic oscillation control of the power system. Numerical simulation results show that the proposed method can not only suppresses the chaotic oscillation but also reduce the convergence time and suppress the chattering phenomenon and has strong robustness.


Introduction
With the increasing scale of power system, the nonlinear characteristics are more complex in power systems. Chaotic oscillation and even blackouts will occur in power system due to a number of reasons, such as periodic load disturbances, parameter variation, voltage collapse, frequency collapse, and transient instability [1][2][3][4][5]. How to effectively control chaos in the power system has become a common problem to be considered both theoretically and practically. Chaotic oscillation is an important phenomenon that has attracted considerable attention in recent years. e mechanism causing chaotic oscillation is described and the difference between chaotic oscillation and loss of stability is distinguished [6]. e nonlinear dynamic characteristics of a simple three-bus power system mathematic model are studied, including phase diagrams and a bifurcation map [7]. Reference [8] proposed a conventional power network system and demonstrated the chaotic behavior under some special operating conditions. e reasons for chaos in the power system have been investigated employing the theory of nonlinear dynamic systems based on a simple power system model. e critical conditions and parameter region in which chaos occurs have been worked out using the Melnikov function calculation method [9]. Based on the fractional calculus, [10] reported the dynamic analysis of a fractional-order power system and established its numerical simulations which are provided to demonstrate the feasibility and efficacy of the analysis.
Based on the above analysis of the chaotic oscillation mechanism in power system, there are two aspects to suppress the chaotic oscillation: one is controlling the system's behavior to the expected orbit and the other is suppressing the occurrence of the chaotic oscillations in power system. In recent years, various nonlinear control strategies have been applied to chaos control in power systems, such as LS-SVM method [11], the finite-time stability theory method [12], ANFIS method [13], adaptive control method [14][15][16][17], and sliding mode control method [18][19][20][21][22][23][24][25][26]. In a practical application, the sliding mode control (SMC) has the advantages of a fast dynamic response, good stability, and strong antidisturbance ability, and it is widely used in many control systems. e main problem hindering the engineering application of the conventional sliding mode control is the well-known chattering phenomenon. In order to solve the chattering problem of SMC, [22] proposed an equivalent fuzzy fast terminal sliding mode control method, and this method can not only reduce the chattering but also accelerate the convergence time. In order to reduce chattering phenomenon of SMC, [23] designed a sliding mode observer to suppress the chaotic oscillation in power system by using the hyperbolic tangent function instead of the symbolic function. Reference [24] presented a fast fixed-time nonsingular terminal sliding mode control method to suppress chaotic oscillation in power systems, and the proposed control scheme achieved system stabilization within bounded time independent of initial condition and has an advantage in convergence rate over existing result of fixedtime stable control method. Based on fixed-time stability theory, [25] designed a fixed-time integral sliding mode controller to ensure the precise convergence of the state variables of controlled system and overcame the drawback of convergence time growing unboundedly as the initial value increases in finite time controller. Reference [26] proposed a new adaptive fuzzy sliding mode control design strategy for the control of a special class of three-dimensional fractionalorder chaotic systems with uncertainties and external disturbance.
e design methodology is developed in two stages: first, an adaptive sliding mode control law is proposed for the class of fractional-order chaotic systems without uncertainties, and then a fuzzy logic system is used to estimate the control compensation effort to be added in the case of uncertainties on the system's model.
All of the above references are designed by using integerorder sliding mode controllers, but the fractional-order sliding mode control increases two control freedom degrees and makes the controller design more flexibility. In recent years, fractional-order sliding mode control has been applied to other fields and has better control result. Reference [27] proposed a novel optimized fractional-order controller for control of chaos in power system and discussed the steps to optimize the order of fractional controller. e proposed controller perturbed the dynamics of the nonlinear power system and pushed it to nonchaotic and bounded stable state. Reference [28] proposed a novel discrete-time fractional-order sliding mode control scheme which guarantees the desired tracking performance of a linear motor control system; and a better performance is achieved due to the memory effect of the fractional calculus. Reference [29] proposed a fractional-order sliding mode control strategy for grid-connected doubly fed induction generator. e simulation results fully exhibited the effectiveness of the proposed method and indicated that the fractional-order sliding mode control strategy has superior performance over that of the conventional sliding mode control scheme. As the contributions to the fractional-order sliding mode control investigation efforts, [30] proposed a new adaptation law for fractional-order sliding mode control addressing the synchronization problem for a class of nonlinear fractionalorder systems with chaotic behavior. e main innovation in the proposed control design concerns the choice of a sliding surface with two adjustable parameters, leading easily to an efficient adaptation law for the sliding mode control controller. erefore, the design of sliding mode controller using fractional calculus theory can suppress the chaotic oscillation in power system effectively. e rest of this paper is organized as follows. System description and the basic chaotic oscillation characteristics are analyzed in Section 2 and Section 3, respectively. Section 4 gives some mathematical preliminaries. In Section 5, the fractional-order hyperbolic tangent sliding mode controller is proposed to suppress the chaotic oscillation in power system, and the theoretical analysis and proof are given. Section 6 gives the comparison of the proposed controller with the traditional sliding mode controllers. Conclusions end the paper in Section 7.

System Description
Reference [9] presented a dual-unit power system that consists of generator, transformer, transmission line, and power load. By simplifying the presented exact model, the second-order mathematical model of the interconnected two-machine power system is obtained, which is shown in Figure 1.
In Figure 1, 1 and 2 are the equivalent generators; 3 and 4 are the equivalent transformers of systems S1 and S2, respectively. 5 is power load, 6 is the circuit breaker, and 7 is the transmission line. According to the analysis of the above interconnected two-machine power system, we can get the second-order mathematical model, which is shown as follows: where δ is the electrical angle between generators 1 and 2, ω is the corresponding relative angle speed, H is the equivalent moment inertia of S1, P S is electromagnetic power of generator 1, P m is mechanical power of the generator 2, D is the damping coefficient, and P e cos t is disturbance of S1. Here, Equation (1) can be simplified as follows: When a � 1, b � 0.02, e � 0.2, F � 0.2593, chaotic oscillation will occur in the power system, and the corresponding system parameters of system (1) are H � 100 kg m 2 , P s � 100W, D � 2 N m s/rad, P m � 20 W, and P e � 25.93 W. e time-domain waveforms and phase diagram of system (1) are shown in Figures 2 and 3, respectively.
As can be seen from Figures 2 and 3, it is known that the dynamic behavior of the power system is complex. It appears that chaotic oscillation is accompanied by voltage instability and frequency oscillation under the above parameter conditions. e existence of chaos and chaos control of the interconnected two-machine power system will be investigated to suppress chaotic oscillation.

Basic Characteristics
3.1. Dissipative Analysis. By calculation, the divergence of system (1) can be obtained: and, from equation (3), we can get that system (1) is dissipative, and the trajectory is bounded and converged in the exponential form (dV/dt) � e − (D/H)t . In other words, the initial volume element V 0 converges to V 0 e − (D/H)t at t. When t ⟶ ∞, the trajectories of each volume element will converge to zero according to the exponential rate − (D/H).
at is, all trajectories would be restricted to the zero volume set and fixed on an attractor. us, chaotic attractor exists in system (1).

Lyapunov Exponent.
e Lyapunov exponents of a chaotic trajectory have at least one positive value, which corresponds to the sensitive dependence feature. is feature distinguishes a strange attractor from the other types of steady-state behaviors. By calculation, the computation Lyapunov exponents are LE1 � 0.0174, LE2 � 0, and LE3 � − 0.0374 at P m � 20W. System (1) is a chaotic system with the positive Lyapunov exponent [31,32]. e mechanical power P m has a great influence on stable operation in power system with excitation limitation, which determines the speed of the generator. To better understand the dynamics of power system (1), the Lyapunov exponents with the parameter P m are shown in Figure 4. As can be seen, when the mechanical power P m is located in (17.38, 55.1), (56.25, 7.15), (69.9, 76), (80.34, 80.68), and (81.32, 100), the largest Lyapunov exponent of system (1) is larger than zero, and the system is chaotic.

Bifurcation Diagram.
Power system is a complex nonlinear dynamic system; bifurcation phenomenon will occur with the changes of parameters. Bifurcation is the main route to chaos from the stable state, and it will cause the system to lose stability. e bifurcation map is used to analyze the dynamic characteristics of the nonlinear system when the system parameter varies [33]. It is known that the system parameter disturbance power amplitude P e must change all the time and the system conditions can change when the system is disturbed by inevitable disturbances.
us, the bifurcation map is shown in Figure 5 with the disturbance power amplitude P e varying. From the  Mathematical Problems in Engineering bifurcation map in Figure 5, we can see that chaotic oscillation will occur in the system when the disturbance power amplitude P e is under some certain condition.

Power Spectrum.
Chaotic oscillation is the nonperiodic motion; its power spectrum is continuous and there will appear a new crossover or multiplier peak frequency in bifurcation processing. e power spectrum of system (1) could be calculated and analyzed by the following indirect method. First, the autocorrelation function of the sample data should be calculated as follows: en, the power spectrum is obtained by the Fourier transform for autocorrelation function, and it is shown in Figure 6. From the simulation results, the power spectrum of system (1) is a continuous spectral line without obvious peak value. us, chaotic oscillation occurs in system (1).

Mathematical Preliminaries
At present, there are several definitions of the fractionalorder differential systems. Two commonly used definitions are Grünwald-Letnikov (GL) definition and Riemann-Liouville (RL) definition.
e best-known RL definition of fractional-order system can be expressed as [34] where n is an integer value satisfying n − 1 ≤ q < n, and Γ(·) is the Γ-function.
Consider a general fractional-order nonlinear dynamical system as follows: where X ∈ R n (∈ N), A ∈ R n×n , 0 < q ≤ 1. [35], we know the following:

Lemma 1. For a given autonomous linear system of fractional-order system (6) with
(1) e system is asymptotically stable if and only if |arg(λ i (A))| > (απ/2), i � 1, 2, . . . , n, where arg(λ i (A)) denotes the argument of the eigenvalues λ i of A (2) e system is stable if and only if either it is asymptotically stable or those critical eigenvalues satisfying |arg(λ i (A))| � (απ/2) have geometric multiplicity of one Lemma 2 (see [36,37]). For the nonlinear fractional-order system (6) with the order of 0 < q ≤ 1, if there exists a real symmetric positive definite matrix P satisfying . , x n (t)), then system (6) is asymptotically locally stable.
Lemma 3 (see [38]). When ε > 0, for any value χ ∈ R and a constant ε > 0, there is an inequality that is established in the following: Lemma 4 (see [30]). Let x � 0 be a point of equilibrium for the fractional-order system D α Lipschitz with a Lipschitz constant l > 0 and α ∈ (0, 1).

Suppose that there exists a Lyapunov function V(t, x(t)) such that
where a 1 , a 2 , a 3 , and b are positive constants. en the equilibrium point of the fractional-order system is Mittag-Leffler (asymptotically) stable.
Lemma 5 (see [39]). For a continuous system, If there is a continuous positive definite function V: R n ⟶ R and there exists a neighborhood U 0 ⊆ R n of the origin point satisfying (10), system (9) is globally finite time stable.

Controller Design
To eliminate chaotic oscillation in power system (1), a sliding mode controller is proposed to keep the power system operating on a stable motion. e simplified controlled system is described as follows: where f(x(t)) � − a sin x 1 (t) − bx 2 (t) + e, d(t) � F cos t, and u(t) is the designed sliding mode controller. Assume that the uncertain perturbation term is bounded that |d(t)| ≤ F, where F is a positive constant.

Design of the Sliding Mode Controller.
First, x d (t) is the control input of the interconnected two-machine power system (1), the error is defined as e(t) � x 1 (t) − x d (t), and the sliding mode switching function is designed as follows: where c should satisfy Hurwitz condition, that is, c > 0; then e controller u(t) can be designed as u(t) � u eq (t) + u sw (t) based on sliding mode control theory; and the equivalent controller u eq (t) can be obtained by _ e switching controller is designed as en, the sliding mode controller u(t) becomes Lyapunov function is designed as V S (t) � 0.5s 2 (t); thus, Substitute (14) into (15), and we can obtain In other words, the Lyapunov function Mathematical Problems in Engineering satisfied that V s (t) is positive definite and _ V s (t) is seminegative definite. e system is stable.

Design of the Fractional-Order Hyperbolic Tangent Sliding
Mode Controller. Because switching function is discontinuous in traditional sliding mode control, it may cause the unexpected chattering phenomenon in actual controller design. Compared with the traditional switching function, the continuous smooth hyperbolic tangent function can soften the controller output characteristic and the fractionalorder controller can make the system have better performance. us, the fractional-order sliding surface and hyperbolic tangent sliding control law are designed to reduce the chattering in sliding mode control. e fractional-order sliding mode function is designed as follows: e controller u(t) can be designed as u(t) � u eq (t) + u sw (t). e fractional-order equivalent controller u eq (t) can be obtained by _ e hyperbolic tangent switching controller u sw (t) is designed: en, the controller u(t) becomes According to Lemma 3, it can be obtained that at is, It can be obtained that Based on d(t) ≤ K, (25) can be written in the following form: According to the above analysis and combining fractional stability in Lemmas 4 and 5, we can obtain that the system is globally finite time stable with fractional-order sliding mode controller and lim t⟶∞ e(t) � _ e(t) � 0. e stability of the controller is verified.

Numerical Simulation
In order to verify the control effect of the proposed control method, the fractional-order hyperbolic tangent sliding mode controller is established by MATLAB/Simulink. Selecting the system parameters as a � 1, b � 0.02, e � 0.2, and F � 0.2593, the tracking target is x d � sin t, the controller parameters are designed as c � 25, K � 2, η � 20, and ε � 0.02, and the fractional-order parameter is chosen as α � 0.8. Simulation time is set as 10 s, and the fractionalorder hyperbolic tangent sliding mode controller is added to system (11). Simulation results of the time-domain waveform are shown in Figure 7, and the corresponding error waveforms are shown in Figure 8. In order to compare with the control effect between the designed controllers, simulation results of the traditional sliding model control, the hyperbolic tangent sliding mode control, and the fractionalorder hyperbolic tangent sliding mode control are shown in Figures 9-11, respectively [40].
Compared with the above simulation results, we can get the following: (1) From Figures 7 and 8, it can be seen that the chaotic interconnected two-machine power system is restored to the expected state when the designed fractional-order hyperbolic tangent sliding mode controller is added to the chaotic system. Simulation results show that the power system stability is realized, and the chaotic oscillation is suppressed effectively by the proposed controller. (2) By comparing with the simulation results in Figures 9 and 10, the chattering phenomenon is reduced by the continuous smooth hyperbolic tangent functions, and the system control performance is further improved by the proposed fractional-order sliding surface. By calculation in Figure 10, the angle speed relative errors of the three different sliding mode control are 0.91%, 0.626%, and 0.21%. (3) Compared with the results of the controller output in Figure 11, it can be seen that the continuous smooth hyperbolic tangent function can soften the controller output performance; and the fractional-order sliding    Mathematical Problems in Engineering surface combined with the hyperbolic tangent function can further improve the system's stability.

Conclusions
e interconnected two-machine power system will have chaotic oscillation characteristics under some certain parameters. By analyzing the dynamic characteristics of the chaotic oscillation power system, the chaotic oscillation parameters are obtained. In order to suppress the chaotic oscillation, the fractional-order hyperbolic tangent sliding mode controller is designed to control the chaotic power system. Simulation results show that the continuous smooth hyperbolic tangent function reduced the chattering phenomenon and the fractional-order sliding surface improved the system control performance. e proposed fractionalorder hyperbolic tangent sliding mode controller can suppress the chaotic oscillation and shorten the convergence time of the chaotic power system. e designed controller has the advantages of fast response, good stability, and strong robustness. In the future, the proposed control method of this paper will be applied to other related control fields to improve the control performance.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest. fractional-order nonlinear systems with unknown control Mathematical Problems in Engineering 9