Nonlinear Analysis Of A Simple Amplitude-Phase Motion Equation Model

With the large-scale application of a large number of new energy sources,such as wind turbines and photovoltaics,and various power electronic devices,the power electronic system is becoming gradually a more power-electronics-based system.The dynamic equipment modeling in power systems mainly consider mathematical models to describe details of different equipments.These models do not reflect the dynamic physical mechanism inside the equipments and the simple relation between equipments and network,nor reveal the nonlinear characteristics of the equipments.The theory of amplitude-phase motion equation provides a new methodology for power system dynamic modeling,which describes the dynamic characteristics of equipments using the relationship between unbalanced power(including active power and reactive power)and the potential phase and amplitude.This theory has a clear physical meaning and universal applicability,and will play a crucial role in understanding the role of power equipments in power-electronics-based power dynamic system problems.This article will take a simple amplitude-phase motion equation as an example,consider nonlinear factors,and study the problems of bifurcation,parameter space and dynamic stability of the model,etc.The specific research content is as follows:(1)A third-order nonlinear mathematical model based on the theory of amplitude-phase motion equation is established,which is described as a simplest amplitude-phase motion equation model.Through theoretical analysis and calculation,that under different parameters,the system will have different types of bifurcation.Among them,under the condition of positive damping,saddle node bifurcation and homoclinic bifurcation occur,under the condition of negative damping,Hopf bifurcation occurs.(2)Based on the simple amplitude-phase motion equation model,the parameter space and the stability domain with positive damping are studied by numerical simulation.The parameter different regions(including the stable fixed point,the stable limit cycle,and their coexistence)are analyzed within the bistable(coexistent)region,the basin stability analysis of the fixed point is performed.Meanwhile,the dynamic behavior under the condition of negative damping is studied.We find that Hopf bifurcation occurs with the change of damping,and it will lead to an irregular oscillation and cause the system to collapse further with the decrease of damping.In addition,the influence of the change of inertia on the system is discussed,as well as some differences in the field of nonlinear dynamics and power systems.