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   . Fractional calculus provides a good instrument to describe the memory, hereditary, non-locality and self-similarity properties of various materials and processes. Many chaotic systems, such as Lorenz system  , Chua’s system  , Duffing system  , Rössler system  , Chen system  and so on, still remain chaotic when their equations become fractional.
Chaotic phenomena have been observed in power systems during the past few decades  -  . Chaos causes electromechanical oscillations to behave randomly, which are harmful to the secure and stable operation of power systems, and even produce undesired negative consequences, such as angle divergence, voltage collapse and system splitting  . So far, almost all the studies of dynamics of power systems are concerned with the integer-order models, and there are little research results on fractional modeling and control design of power systems. Tan et al. studied the dynamics of a fractional-order interconnected power system and found that the system became chaotic when the fractional order is no less than 0.88  . Sun and Li investigated the chaotic and bifurcation phenomena in a fractional-order three-bus power system and the existence of chaos was demonstrated for different orders  .
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 is the Gamma function.
The Caputo fractional derivative satisfies the following properties:
where C, and are real constants.
Lemma 1.  -  Consider the fractional-order system
where and. The equilibrium point of system (3) is locally asymptotically stable if all the eigenvalues of the Jacobian matrix satisfy
Lemma 2.  Let be a continuous differentiable function. Then, at any instant the following inequality holds
A continuous function is referred as class-K if it is strictly increasing and  .
Lemma 3. (Fractional-order extension of Lyapunov direct method  ) Let be an equilibrium point of the nonautonomous fractional-order system
with initial condition. Assume that is a Lyapunov candidate and are class-K functions. Then is asymptotically stable if the following conditions hold
where and denotes an arbitrary norm.
3. The FOSMIB Power System
In  Chen et al. analyzed the angle dynamics of the classical single-machine infinite-bus (SMIB) power system, which is governed by the so-called swing equation
where M is the moment of inertia, D is the damping constant, is the maximum power of generator and is the power of the machine.
Let and, then Equation (9) can be rewritten as
where, and are positive constant parameters. When, , and, the SMIB power system is chaotic.
Here, we consider the fractional-order single-machine infinite-bus (FOSMIB) power system
where is the fractional order. When, system (11) is the original integer-order SMIB power system.
The autonomous system (11) (as) has two equilibrium points: and. For the equilibrium point O, the Jacobian matrix is
and its eigenvalues are
In both cases,. According to Lemma 1, O is asymptotically stable.
For the equilibrium point E, the Jacobian matrix is
and its eigenvalues are
It can be seen that and. In accordance with Lemma 1, E is unstable.
4. Dynamic Analysis of the FOSMIB Power System
In this section, we use the Adams-Bashforth-Moulton predictor-corrector algorithm proposed by Diethelm et al. in  -  to solve the FOSMIB power system (11). The dynamics are numerically analyzed by means of bifurcation diagrams, phase portraits and Lyapunov exponents. In the following simulations, parameter f is chosen as bifurcation parameter and the other parameters are fixed at, ,. The initial conditions are selected as,.
First, let, and vary f from 2.4 to 3.5. The corresponding bifurcation diagram is plotted in Figure 1(a), from which a period-doubling route to chaos can be found. To confirm chaos, the largest Lyapunov exponent (LLE) is calculated using Wolf
Figure 1. Bifurcation diagram and the LLE versus f for q = 0.95: (a) Bifurcation diagram; (b) The LLE.
algorithm  and plotted in Figure 1(b). The FOSMIB power system is chaotic over most of the range, where the LLEs are positive. The phase portraits for different values of f are plotted in Figure 2. With the increase of f from 2.4, period-1, period-2 and period-4 orbits are obtained at, and, respectively. After a cascade of period-doubling bifurcations, the system loses its stability and enters chaos at. As f increases further, the system becomes stable again via inverse period-doubling bifurcations.
Now, let, , , and vary q from 0.87 to 1. The resulting bifurcation diagram is plotted in Figure 3(a), which indicates period-doubling bifurcations and chaos. The fractional-order SMIB power system is chaotic over most of the range, where the LLEs are positive as shown in Figure 3(b). The phase portraits for different values of q are plotted in Figure 4. With the increase of q from 0.87, period-1, period-2 and period-4 orbits are obtained at, and, respectively. As q increases further, after a cascade of period-doubling bifurcations, a chaotic attractor is obtained at.
5. Adaptive Backstepping Control of Chaos
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.
5.1. Controller Design
Consider the controlled FOSMIB power system
Figure 2. Phase portraits for different values of f: (a) f = 2.5; (b) f = 2.55; (c) f = 2.61; (d) f = 2.65.
Figure 3. Bifurcation diagram and the LLE versus q for f = 2.8: (a) Bifurcation diagram; (b) The LLE.
Figure 4. Phase portraits for different values of q: (a) q = 0.88; (b) q = 0.893; (c) q = 0.913; (d) q = 0.92.
where and the parameter f is unknown. The backstepping design procedure consists of two steps.
Step 1. Define. Its derivative is given by
where, is the virtual control to be defined later.
Select the candidate Lyapunov function as
Now, applying Lemma 2, it can be found that
Define the virtual control as
where is a positive constant, which leads to. Substituting Equation (20) into Equation (17) and inequality (19), we have
Step 2. The derivative of is expressed as
where is the estimate of. Choose the candidate Lyapunov function as
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 is a positive constant. Substituting Equation (26) and Equation (27) into Equation (23) and inequality (25), we have
According to Lemma 3, the closed-loop error system is asymptotically stable at the origin. It means that, with the proposed controller and adaptive law, the FOSMIB power system is asymptotically stable at the equilibrium point.
5.2. Simulation Results
In the simulation, the fractional order q is equal to 0.95. The parameters of system (16) are taken as, , and. The parameters of the controller (26) and the adaptive law (27) are chosen as and. The initial conditions are taken as,. The initial parameter estimate is given by. The closed-loop system consisted of Equations ((16), (26) and (27)) is solved by using the predictor-corrector algorithm. The simulation results are shown in Figure 5.
The time-domain waveforms the states of the controlled system (16) are shown in Figure 5(a) and Figure 5(b). The FOSMIB power system has experienced chaotic behavior before the controller is put into effect. By activating the controller u at, the chaotic behavior is suppressed and the controlled system converges to the equilibrium point quickly. The parameter estimate is converged to as shown in Figure 5(c) and the controller u is bounded as shown in Figure 5(d). From Figure 5, it can be seen that the proposed controller is feasible for suppressing chaos in the FOSMIB power system.
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.
Figure 5. The time-domain waveforms of the controlled system (16).
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).