^{1}

^{*}

^{2}

In this paper, we aim to control an instable chaotic oscillation in power system that is considered to be small system by using a linear state feedback controller. First we will analyze the stability of the mentioned power system by means of modern nonlinear theory (Bifurcation and Chaos). Our model is based on a three bus power system that consists of multi generators containing both dynamic and static loads. They are considered to be in the form of an induction motor in parallel with a capacitor, as well as a combination of constant power along with load impedance, PQ. We consider the load reactive power as the control parameter. At this stage, after changing the control parameter, the study showed that the system is experiencing a subcritical Hopf bifurcation point. This leads to a chaos within the system period doubling path. We then discuss the system controllability and present that the all chaotic oscillations fade away through the linear controller that we impose on the system.

In the past so many researchers have used the voltage collapse case study within a power system. Here, the power system performs as a significant part of the sequential event that is accompanied by an unstable voltage profile [

Throughout the past most of the work that was conducted would have linked the voltage collapse to a static bifurcation done by Kwatny et al. [

Combining both control theories along with the applications of bifurcation theory is a challenge and great interest to many researchers in the design of controllers for nonlinear systems that exhibit bifurcations models. Both Abed and Fu [

Recent studies are directed at combining control of bifurcation along with chaos in dynamical system simultaneously. Thus, the emphasis will be weighted on design techniques that will result into a prescribed nonlinear dynamic system for the controlled processes. Moreover, bifurcation control stimulates its usage by a control input modifying its bifurcation characteristics for the parameterized system. The control model will be considered for a static or a dynamic feedback and with an open loop system.

Our objective is to control, stabilize, and delay a given bifurcation. We also aim for reducing the bifurcation amplitude, getting a bifurcation solution, and finally optimizing the performance index located near the bifurcation.

In the end, we reshape the bifurcation diagram of the system [

The small power system model is known to be highly nonlinear dynamic system. It is represented by an algebraic along with a set of ordinary differential equations. In this paper, we followed the work done by Nayfeh et al. [_{m}. After Nayfeh et al. [

Now, we add the linear control systems, thus the state-space equations are coupled to the previous four equations resulting into:

where x_{1} and x_{3} are the power angles of the machine and the load, respectively. Furthermore, x_{2} is the load voltage, and x_{4} is the radian frequency of the load, while x_{5}, x_{6}, x_{7}, and x_{8} are the exciter, amplifier, proportional controller, and the sensor, respectively._{ }The above eight equations provide an equilibrium solution along with a dynamic solution to our model.

The equilibrium solutions are found by setting of the system of equations (1)-(8) to be equal to zeros. The stability of these solutions depends on the eigenvalues of the Jacobian matrix of set of equations (2)-(8) evaluated each at the equilibrium point.

Parameter | Value |
---|---|

k_{pw} | 0.4 |

k_{pv} | 0.3 |

k_{qw} | −0.03 |

k_{qv} | −2.8 |

k_{qv2} | 2.1 |

T | 8.5 |

P_{o} | 0.6 |

Q_{o} | 1.3 |

P_{1} | 0.0 |

Y_{o} | 20.0 |

θ_{o} | −5.0^{0} |

E_{o} | 1.0 |

C | 12.0 |

Y_{o}^{`} | 8.0 |

θ_{o}^{`} | −12.0^{0 } |

E_{o}^{`} | 2.5 |

Y_{m} | 5.0 |

θ_{m} | −5.0^{0} |

E_{m} | 1.0 |

P_{m} | 1.0 |

d_{m} | 0.05 |

T_{g} | 1.4 |

K_{g} | 0.8 |

T_{e} | 0.4 |

K_{e} | 1 |

T_{a} | 0.1 |

K_{a} | 9 |

K_{p} | 0.7 |

T_{r} | 0.05 |

K_{r} | 1 |

M | 0.3 |

Nayfeh et al. [_{1} and H_{2} at the control parameters Q_{1} = 6.9929 and 7.2229, respectively, and a saddle-node bifurcation SN at Q_{1} = 7.2238. They have also found that the two Hopf bifurcation points H_{1} and H_{2} are subcritical and supercritical points respectively, as shown in the bifurcation diagram in _{1} ≤ 6.9449957, the system response may be constant or dynamic (limit-cycle or chaos), depending on the initial conditions. Because the chaos is very sensitive to initial condition.

In order to control and eliminate the chaotic oscillations, a linear control via proportional controller must be added to the system. This linear controller will fade away the eighth order system of equations (1)-(8). The subcritical Hopf bifurcation, H_{1} and hence control chaos and voltage collapse there, a nonlinear feedback controller is designed by Nayfeh et al. [^{3} and it is added to equation (2). They show that by increasing the value of K, we will reduce the amplitude of the limit-cycle born at H_{1}. Furthermore, by choosing a gain K larger than a critical value of 58.56, one can suppress the period-doubling bifurcations and hence the chaos and its associated crisis bifurcation and voltage collapse as shown in

The modern nonlinear theories of bifurcation and chaos are applied to a small three bus power system. The results showed that the system, without any controller, has a dangerous subcritical Hopf bifurcation point. That means when 6.9105 ≤ Q_{1} ≤ 6.9449957, the system response may be constant or dynamic (limit-cycle or chaos), depending on the initial conditions. A comparison between the uncontrolled and controlled system was discussed and investigated. The study has revealed that the linear controller stabilizes the system by eliminating all chaotic oscillations. Hence, the system became completely stable over the whole range of the control parameter point Q_{1}.

We would like to acknowledge the funding of this project (08-ENE416-5) by the Science and Technology Unit at Taibah University through the National Science, Technology, and Innovation Plan for Saudi Arabia.