Adaptive Notch Filter-Based Time-frequency Analysis Of Signals In Power System

The voltage and the current in a power system is commonly regarded as an almost periodic signal of which the time-frequency analysis mainly concerns the different sinusoidal components that are tracked and estimated for exact amplitudes, initial phases as well as the frequencies, and thus forms part of the basis of the evaluation and protection of power system. In this dissertation, a new method for time-frequency analysis that is in nature a nonlinear differential dynamic system in contrary to the traditional integral methods is put forward and may be called as frequency adaptive comb filter.A linear sinusoid tracer that is a two-dimensional linear ordinary differential equation deduced by using least square error principle, gradient descent method and rotation transformation can yield the estimates of amplitude and initial phase. The transient and steady state responses are studied with descriptions of the physical significances of the parameters such as frequency and bandwidth as well as their effects on the transient and steady state performance. The comparison of the simulation results validated the superior performance of this algorithm to DFT and nonlinear sinusoid tracers.The concept of an adjustable-bandwidth multi-frequency linear comb filter composed of a number of two-dimensional linear sinusoid tracers connected in parallel is then put forward and with its asymptotic stability proved. The expression of the frequency response and the effect of the bandwidth on transient and steady state performance are provided. It shows that the instantaneous value as well as the amplitude and the phase angle of each sinusoidal component can be accurately estimated while the frequencies of all the components whether harmonic or inter-harmonic of input signal are located on the division of the filter. The better performance than FFT and nonlinear comb filter is validated by simulation results comparison.The method of voltage flicker measurement recommended by IEC brings theoretical error while the voltage signal consists of harmonics and inter-harmonics, so a two-stage linear comb filter is tried to detect the fundamental voltage flicker. The first stage is used to decompose the input voltage to obtain the fluctuating fundamental signal with the deletion of the harmonics and inter-harmonics. The second stage is used to analyze such signal to get the fundamental amplitude and its fluctuation. Then the instantaneous flicker level can be obtained by weighted filtering, squaring and smoothing. Simulation verified that the new method can effectively restrain the flicker error caused by inter-harmonics and provide good ability of anti-jamming.Based on two-dimensional linear sinusoid tracer, two algorithms for frequency estimation, i.e. the normalized and the non-normalized, are presented. Being a three-dimensional adaptive notch filter, these two estimators form slow integral manifold of which the existence and asymptotic stability are proved. The influence of the bandwidth and the frequency adaptive gain on transient and steady state performance are discussed. The frequency, amplitude and phase angle of an unknown sinusoid signal can be exactly estimated. The decreased amplitude results the slow transient response of the non-normalized estimator while the speed of the normalized algorithm is hardly influenced by input amplitude due to the better robustness.A linear comb filter based non-normalized fundamental frequency estimator is also proposed. The method forms slow adaptive integral manifold of which the existence and stability are proved by Lyapunov stability theorem and averaging method. With the frequency parameters of the filter adjusted to the same orders of the harmonics and inter-harmonics in the input signal, then the integral manifold is uniformly asymptotically stable and the fundamental frequency as well as the amplitudes of all the harmonics and inter-harmonics can be precisely traced in exponential convergence. The validity of the proposed algorithm is verified by simulation and it is pointed out that better noise characteristic can be achieved by decreasing bandwidth and adaptive gain.It is put forward at last an algorithm of frequency adaptive comb filter. The algorithm includes two coupled nonlinear differential equations respectively for state estimate and frequency updating, and the decoupling is carried out by slow integral manifolds to reach an almost periodic nonlinear dynamic system for multi frequencies estimation, then the nonlinear autonomous equation for the frequency estimation deduced by averaging. With the frequency estimate space given, the equilibrium of the average system can be an isolated equilibrium, a continuum of equilibriums, or non-strict equilibrium. The exponential stability of the isolated equilibrium, the semi-stability of the continuum of equilibriums, and the robustness of system with no equilibrium are investigated respectively. The convergence speed of the frequencies and the amplitudes is governed by the adaptive gain and the bandwidth parameter respectively. The frequency axis is divided into a number of zones corresponding to the high and low limits of the frequency estimates setup beforehand. The frequency, amplitude and phase angle of each sinusoidal component can be exactly estimated if it is in one of the zones. The algorithm is validated by simulations.Finally, the implementation of the proposed algorithms without simulation tools or in embedded systems is discussed. The new algorithm can be implemented by forth order Runge-Kutta method for discrete time signal with the convergence and stability verified.