The Bifurcation Analysis And Control Of Simple Electric System

The dynamic characteristics of electric system is researched in this paper. By choosing classical dual-machine three-node system as the model, making full use of theoretical methods and applying to Matcont and Matlab, the matching evaluation of different variables and parameters under the different bifurcation behaviors is analyzed, which will be beneficial to improving the system and designing the controller, meanwhile, can provide the practical reference for matching parameters. This paper is divided into six chapters.The first chapter aims to formulate the meaning, the background and the present statement of the chosen topic, which discusses the research meaning of the whole article. The present academic achievements in this field are collated.The second chapter is fundamental knowledge. It introduces the main concept, methods,and tools used by the article in detail including the equilibrium solution manifold method,direct method, center manifold theory and Lyapunov index method. Particularly, the equilibrium solution manifold method, center manifold theory and Lyapunov index method offer theoretical basis to the bifurcation with the single parameter, while direct method is suitable for the bifurcations with the single parameter and double parameters.The main content of the third chapter is divided into four parts:(1)In order to obtain the equilibrium solution manifold of the system, the third chapter uses Runge Kutta method, Newton iterative method and parabolic interpolation method;(2)Based on the equilibrium solution manifold method and Matcont, the article analyzes the bifurcation situations considering1 Q as the single parameter of the system, which includes static bifurcation points and Hopf bifurcation points;(3)The existence of SNB is demonstrated by introducing the relevant theoretic methods;(4)By using linear transformation method, center manifold theory and direct method, the existence and type of the Hopf bifurcation points are demonstrated and estimated.This chapter puts the qualitative analyses of the research questions on the first step, then by the simulation of pictures to explain the further questions and conduct practical researches.Based on the single parameter bifurcation, the article makes the analyses of the system by treating1 Q and1P as double parameters to research the bifurcation situations of the system comprehensively. Firstly, the existence of Hopf extreme point is analyzed by conducting mathematical calculation. Secondly, by means of Matcont and Matlab the dynamics of the system are analyzed. By explaining the pictures makes it demonstrated that1 Q and1P have some effect on the operation of the system.The control problem of the bifurcation with a single parameter is researched in the fifth chapter. Add a nonlinear controller to the system so as to transform sub-critical Hopfbifurcation into super-critical Hopf bifurcation, which can improve the stability of the system.And using the simulation is to prove the effectiveness of the controller.The whole article expands on the dynamic characteristics of the system under the different variables and parameters, and the running condition of every target under certain specific bifurcation behavior, which is the corresponding data. The data will be of value to the improvement and control of the system.