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This paper has numerically studied the dynamical behaviors of a fractional-order single-machine infinite-bus (FOSMIB) power system. Periodic motions, period- doubling bifurcations and chaotic attractors are observed in the FOSMIB power system. The existence of chaotic behavior is affirmed by the positive largest Lyapunov exponent (LLE). Based on the fractional-order backstepping method, an adaptive controller is proposed to suppress chaos in the FOSMIB power system. Numerical simulation results demonstrate the validity of the proposed controller.

As a mathematical branch with a history of over 300 years, fractional calculus and its applications to physics and engineering have attracted increasing attentions in recent years [

Chaotic phenomena have been observed in power systems during the past few decades [

In this paper, we numerically investigate the chaotic dynamics of a fractional-order single-machine infinite-bus (FOSMIB) power system. Period-doubling bifurcation and chaos are observed in FOSMIB power system and the existence of chaos is confirmed by evaluating the largest Lyapunov exponent (LLE). Based on the fractional-order backstepping method, an adaptive controller is presented to suppress chaos in the FOSMIB power system, and the effectiveness of the proposed controller is proved by the numerical simulation results.

The rest of the paper is organized as follows. Some definitions and lemmas about fractional calculus are introduced in Section 2. The dynamics of the FOSMIB power system are analyzed in Section 3. An adaptive controller is designed using the fractional-order backstepping method to suppress chaos in the FOSMIB power system in Section 4. Finally, conclusions are addressed in Section 5.

There are several different definitions of fractional derivatives. The most appropriate one for practical problems is the Caputo definition. The Caputo fractional derivative is given by

where m is integer and

The Caputo fractional derivative satisfies the following properties:

where C,

Lemma 1. [

where

Lemma 2. [

A continuous function

Lemma 3. (Fractional-order extension of Lyapunov direct method [

with initial condition

where

In [

where M is the moment of inertia, D is the damping constant,

Let

where

Here, we consider the fractional-order single-machine infinite-bus (FOSMIB) power system

where

The autonomous system (11) (as

and its eigenvalues are

In both cases,

For the equilibrium point E, the Jacobian matrix is

and its eigenvalues are

It can be seen that

In this section, we use the Adams-Bashforth-Moulton predictor-corrector algorithm proposed by Diethelm et al. in [

First, let

algorithm [

Now, let

In this section, an active controller is designed using fractional-order backstepping method to suppress chaos in the FOSMIB power system and stabilize it to the unstable equilibrium point

Consider the controlled FOSMIB power system

where

Step 1. Define

where

Select the candidate Lyapunov function as

Now, applying Lemma 2, it can be found that

Define the virtual control

where

Step 2. The derivative of

where

where k is a positive constant, which can adjust the speed of the adaptive law. Using Lemma 2, it can be found that

Choose the control input and the adaptive law as

where

According to Lemma 3, the closed-loop error system is asymptotically stable at the origin

In the simulation, the fractional order q is equal to 0.95. The parameters of system (16) are taken as

The time-domain waveforms the states of the controlled system (16) are shown in

In this paper, we have numerically investigated the FOSMIB power system. The parameter f and the fractional order q are selected as bifurcation parameters respectively. Complex dynamical behaviors, such as periodic orbits, period-doubling bifurcations and chaotic attractors, are observed in the FOSMIB power system. The LLE is calculated using Wolf algorithm to confirm the existence of chaos. Furthermore, by exploiting the fractional-order backstepping method, we propose an adaptive controller to suppress chaos in the FOSMIB power system. The effectiveness of the presented controller is verified by numerical simulation results.

The work was supported by the Natural Science Foundation of Henan Province, China (Grant No. 14A120005) and Excellent Young Scientist Development Foundation of Zhengzhou University, China (Grant No. 1421319086).

Liang, Z.H. and Gao, J.F. (2016) Chaos in a Fractional-Or- der Single-Machine Infinite-Bus Power System and Its Adaptive Backstepping Control. International Journal of Modern Nonlinear Theory and Application, 5, 122-131. http://dx.doi.org/10.4236/ijmnta.2016.53013