Discussing the dynamical properties of various power system models is of significant importance in order to understand its complete behavior. Even though there are many literatures discussing about the chaotic behavior shown by phase converter circuits, none of them have reported the hazardous phenomenon of multistability. In this paper, we derive the fractional-order model of a phase converter circuit and investigate its dynamics. Bifurcation of the system with the parameters and fractional order are investigated. A forward and backward continuation scheme is adopted to display various coexisting attractors; the property of multistability is also discussed. Using forward and backward continuation, various coexisting attractors and the property of multistability are discussed. Two different sliding mode controllers for controlling chaotic oscillations with model disturbances and parameter uncertainties are derived, and the effectiveness of the controllers is discussed with numerical simulations.

The nonlinear dynamical systems are described using nonlinear differential equations and represented in state (phase) space. The system state in steady state can be defined by equilibrium point and by limit cycles in state space; when the system is subjected to aperiodic oscillations, the system can be in quasiperiodic or in chaotic state. For different parameter values of a nonlinear system, the location and the number of equilibrium points can change. According to Lyapunov, regular attractors (steady states, limit cycles) are fully stable or fully unstable, but this is not true for chaotic attractors. Deterministic chaos is a phenomenon when small discrepancies in the initial conditions lead to an unpredictable behavior. The average rate of expansion along the principle axes are known as Lyapunov exponents. The presence of at least one positive Lyapunov exponent confirms chaos in the system. The chaotic behavior in electrical drive systems are noted as undesirable.

Most of the power systems are composed of ordinary differential equations; the system state changes (drastic changes in current or voltage decreasing or increasing rate) lead to the nonlinear dynamical behavior and makes the circuit modeling complex [

Considerable works are identified on stabilizing such systems. Notable stabilization techniques that have been tested on electrical systems are time delayed approach [

Fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary (noninteger) order. Exponential laws are a classical approach to study dynamical changes in systems, but there are many systems where dynamics undergo faster or slower changes than exponential laws. Fractional-order systems play a significant role to describe the irregular changes in dynamics. Naturally, dynamical systems are composed in the form of fractional order, for example, the voltage-current relation of a semi-infinite lossy transmission line [

Chaos suppression research findings on the permanent magnet synchronous generator (PMSG) in wind turbine [

In this paper, we consider the phase converter circuit with saturable inductors as described in [

The equivalent circuit of the single- to three-phase converter circuit.

The magneto motive force

We also assume that the variables are restricted to

For simplicity, let us assume (symmetrical case)

Using the relations discussed in (

The PCC system described in (

This paper is aimed at analyzing the PCC system in its fractional-order form and also proposing sliding mode controllers to suppress chaotic oscillations with external disturbances and parameter uncertainties. To derive the fractional-order PCC (FOPCC), we have three commonly used definition of the fractional-order differential operator: Grünwald-Letnikov (GL), Riemann-Liouville, and Caputo [

As can be seen in the literatures, the Caputo method has been widely used for numerical solutions of the fractional-order systems. But the GL method has benefits over the other methods of solving fractional orders due to the smoothness of the resultant approximations [

For numerical calculations, the above equation is modified as

Let the general form of the 3D fractional-order system be defined as

In order to simulate system (

Let us define the FOPCC oscillator as

Using (

For the parameter values

2D phase plots of the FOPCC system for (a)

We discuss the bifurcation of the FOPCC with respect to parameters and fractional order. The autonomous FOPCC system shown in (

(a, b) The bifurcation of the FOPCC system with

Figure

Multistability and coexisting attractors in physical systems play a significant role as such oscillations may create hazardous effects in system behavior. There are few discussions on the multistable properties of mechanical systems such as the controllability of multistability in a quasiperiodically driven system [

To derive the multistability plots of the FOPCC, the bifurcation diagram is obtained by plotting local maxima of the coordinate

(a) Bifurcation plots of the FOPCC system with parameter

Coexisting attractors exhibited by the FOPCC system. (a) Coexisting attractor and period-1 limit cycle for

Recent literatures have shown that fractional-order controls are effective in suppressing chaos compared to their integer-order counterparts [

We discuss the chaos suppression problem of FOPCC with two different scenarios. In the first one, we assume that the FOPCC system has model uncertainties and external disturbances and design a fractional-order no-chattering sliding mode controller to suppress chaotic oscillations. In the second scenario, we assume that the FOPCC system has parameter uncertainties and design a fractional-order adaptive sliding mode controller to suppress chaos. In both cases, we use the Lyapunov approach to show that the designed controllers are effective in suppressing chaos in finite time

In this section, we derive the chaos control for the FOPCC system considered with external disturbances. We use the no-chattering sliding mode control defined in [

Let the nonlinear generalized equation of the FOPCC system (

The error dynamics of the system defined in (

The proportional integral sliding surface can be defined as

In the sliding surface, we know the condition that

Using (

The controller satisfying the sliding condition can be defined as

We use the Lyapunov approach to derive the stability of the controller and to show that the controller can make the FOPCC system reach the sliding condition. Let us define the Lyapunov function as

The Lyapunov first derivative can be defined as

The direct numerical solution of (

Using (

Using the designed sliding mode controller (

Further approximating (

As per the boundedness of the disturbance and the parameters,

If we assume

If the positive constant

The Lyapunov derivative is negative semidefinite, confirming that there exists a finite time

For the FOPCC system,

For numerical simulations, the initial conditions of the nonautonomous FOPCC system are taken as

For simplicity, the desired signal is taken as

The time history of the states with the controller in effect when

In this section, we assume that the FOPCC system has uncertain parameters and hence we derive an adaptive sliding mode controller which could estimate the parameters and suppress chaotic oscillations. Let us define the FOPCC system with adaptive sliding mode controllers as

The uncertainties in the parameters are estimated with the parameter estimates

The dynamics of the parameter estimates are derived as

Let us define the proportional integral sliding surface as

The fractional-order sliding dynamics can be derived from (

Let us define the adaptive controllers which can make the FOPCC system with parameter uncertainties reach the sliding condition as

The stability of the controllers is established using the Lyapunov stability approach. The Lyapunov candidate function can be defined as

The dynamics of the Lyapunov candidate function is

From the definition of fractional calculus,

The direct numerical simulation of (

Using the controller (

Let us define the parameter update laws as

Using the Parameter update laws in (

The Lyapunov first derivative in (

For numerical simulations, we use the initial conditions of the state variables as

The time history of the parameter estimates of the FOPCC system.

The time history of the controlled states of the FOPCC system.

A fractional-order phase converter circuit is derived from its integer-order mathematical model, and various dynamical properties are discussed. Bifurcation analysis of the fractional-order system with forward and backward continuation shows the existence of multistability. Such coexisting attractors in a power system model were not explored earlier, and multistability existence is hazardous in any physical nonlinear system and needs to be further investigated. Coexisting attractors shown by the fractional-order system are shown but not limited to only those discussed as there may be other initial conditions and parameters that can contribute to coexistence. To suppress chaotic oscillations, two different scenarios are discussed. In the first scenario, we use a chattering free sliding mode controller to suppress chaotic oscillations with disturbances, and in the second scenario, we use an adaptive sliding mode controller to control chaos with parameter uncertainties. Numerical simulations are conducted to validate the effectiveness of the controllers.

All the numerical simulation parameters are mentioned in the respective text part, and there are no additional data requirements for the simulation results.

The authors declare that there is no conflict of interest in publishing the paper.