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The second-order chaotic oscillation system model 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.

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 [

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 [

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 [

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.

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:

The origin of system (

The origin of system (

If there exists a continuous function

for some

any solution

For any nonnegative real numbers, that is,_{1},_{2}, …,_{N}≥0, the following inequality holds:

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},_{2} and_{1},_{2} are equivalent generators and main transformers of system, respectively.

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:_{s} and_{m} are electromagnetic power and mechanical power of generator, respectively;

Then the fast terminal sliding mode surface can be designed as_{0}, _{0} >0, and_{0},_{0} are positive odd numbers. The derivative of the system (

For system (_{0} = (_{0}, another global fast terminal sliding mode is taken as the following form:

System (

Firstly, substitute (

Consider the following Lyapunov candidate function as

By using the designed controller

Here, owing to the fact that

According to Lemma

In this section, numerical simulations are performed to demonstrate the effectiveness and the superiority of the proposed control method. System (_{0}_{0} =2,_{0} =5,_{0} =9, _{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_{1},_{2},_{3}) =(0.0174, 0, −0.0374). There is one positive Lyapunov exponent, which validates the existence of chaotic attractor. Figure

Phase portrait of chaotic power system.

Time responses of state variables in chaotic power system.

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 _{1}=64.59s. Figure _{1}. Figure

Time responses of state variables in chaotic power system with proposed controller.

Phase portrait of chaotic power system with proposed controller.

Tuning parameter of terminal attractor

Time response of the proposed controller

To explore the influence of different initial conditions on the control effect, the simulation has been done. Figure _{1}.

The time responses of state variables with different S.

Figures

The time responses of state variables with different values of

Figures

Record of the numerical values of the settling time and the maximum deviation.

System state variable | Settling time | Maximum deviation | |
---|---|---|---|

| Proposed control | 12.06s | -0.0236 |

FTSM control | 12.76s | -0.0480 | |

Finite-time control | 12.31s | -0.0304 | |

| |||

| Proposed control | 11.47s | 0.0787 |

FTSM control | 11.97s | 0.1662 | |

Finite-time control | 11.81s | 0.1353 |

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

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 high-order system with noise perturbation.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (Grant no. 51607179).