| DC-DC converters are the core parts of many power electronic devices,the reliability of which has always been an important issue.Discussing theories of more efficient analytical models and exploring nonlinear dynamical characteristics can provide guidance for the design and analysis of DC-DC converters,which also has important practical significance.In recent years,with the exploration of researchers,people found that using mathematical models based on fractional calculus can describe some natural phenomena more accurately,and the potential fractional characteristics of some circuit elements are also found one after another.Accordingly,a systematic study is presented in this work,in which a modeling and analyzing method based on fractional calculus for power electronic systems is proposed,the applicability of different definitions for fractional derivation is discussed,and nonlinear dynamical analysis of DC-DC converters is provided along with the fractional-order characteristic research of circuit elements.The main contributions of this dissertation lay in:(1)A modeling and analysis method based on the extended harmonic balance principle is brought out for fractional-order systems,and the solving process of an ideal fractional-order DC–DC Boost converter is presented based on the extended equivalent small parameter method,which can be used as a guide to a detailed implementation process of the proposed method.By utilizing the method,the approximate analytical solution can be obtained without considering complex definitions of fractional calculus,and a reference for the steady-state and nonlinear dynamical analysis of fractional-order DC–DC converters can be supplied.Based on the proposed modeling and analysis method,the applicability of two commonly used definitions of fractional derivative is discussed.By combining the derivation processes of definitions and operating principles of power electronic systems,Riemann-Liouville derivative is proved to be more reasonable for steady-state analysis of fractional-order DC-DC converters.Theoretical results are supported by approximate analytical solutions,simulating results of time-domain numerical algorithm,frequency-domain approximate solutions and experimental outcomes.(2)Nonlinear dynamical characteristics of arbitrary order DC-DC converters are studied.One key factor affecting such characteristics,the orders of energe-storage components,is explored.Based on the FFT analyzing result of arbitrary order DC-DC converters in different dynamic states and ergodicity of chaotic systems,a view point is put forward,that is,the state variables of UPO-1 is made up of the DC component and harmonics with integer times of switching frequency,juust like those of stable period-1 orbit.Thus,according to the idea,an unreported approach is proposed for extracting the information of the unstable period-1 orbit(UPO-1)of DC-DC converters,which is mainly based on a harmonic-content related hypothesis of the converters in chaotic state,and by which detailed guidance information with clear physical meaning can be provided for the design of chaos controllers of DC-DC converters.(3)By exploring the principle of electrochemical reactions of Aluminum electrolytic capacitors,the fractional-order characteristics of capacitance and equivalent series resistance are revealed,which are important factors affecting the performance of converters.Also,the parameter estimating and correcting method is offered for the fractional equivalent circuit model of the capacitors.The effectiveness and superiority of the fractional-oder model for estimating state of the converters are certified by comparative works with traditional integer-order models.(4)Nonlinear dynamical characteristics of DC-DC Boost converter are analyzed based on fractional piecewise-smooth modeling method,in which the fractional equivalent circuit model of Aluminum electrolytic capacitors is utilized.Moreover,the theoretical basis for nonlinear dynamic phenomenon analysis,such as stability criterion of fractional system,the largest Lyapunov exponent of fractional piece-wise smooth dynamical systems,and the mechanism of boundary collision bifurcation are discussed in the work,by which the design-oriented principles of component selection and control function design can be derived.Simulation results and comparison show that some nonlinear dynamical phenomena can be predicted accurately by the proposed fractional piecewise-smooth modeling and analyzing method. |
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