Development of a Spectral Theory for Analysis of Non-Stationary Pulse Stochastic Electromagnetic Processes in Devices of Electric Transport Systems

A spectral analysis of traction voltages and currents is a basis for estimation of electromagnetic compatibility level and quality of consumed power in electric transport systems. However, such an analysis is usually performed for steady-state modes and only under the condition that the time realization of voltages and currents, being deterministic for continuous quantities, have infinite length. De facto, the electric transport devices operate in non-stationary dynamic modes (starting up, coasting, acceleration, regenerative braking, stopping, wheel spin, voltage surges, etc.). As a result, the voltage across the traction motors and the current flowing through them are noncontinuous pulsed stochastic processes. It is necessary to add that in emergency modes the voltage and current are short-term single pulses. The paper presents the spectral analysis of such random sequences of pulses as well as their fronts and decays, the concepts of actual and instantaneous spectra. The analytical expressions for amplitudes and the initial phases of k-th harmonics are obtained using the discrete Fourier transformation. The numerical calculations of the spectral composition of stochastic pulse processes of voltage and current were performed for the DE1 and VL8 electric locomotives (Ukraine) as well as for trams operating on the routes of the city of Dnipro. The actual and instantaneous spectra, as well as the spectra of the full correlation functions and their “tails”, were determined for the electric traction voltages and currents.

I. Introduction

Currently, the spectral composition of traction voltages and currents , the quality indices of electric power and the level of electromagnetic compatibility of devices in the electric transport systems are defined; first, in steady-state modes and, second, with the condition that and realizations have infinite duration [1]–[5]. Also, the spectra of deterministic non-sinusoidal voltages and currents are mainly investigated. In particular, the noncanonical harmonic composition of non-sinusoidal periodic voltages and currents in the external traction power supply lines have been established by [6]–[8] using the classical Fourier analysis. The discrete spectral composition of the output voltage of traction substations equipped with 6- and 12-pulse diode rectifiers were investigated with the use of the same method in [9]. The harmonic composition of low-frequency pulsations of the output voltage of 6-pulse controlled (thyristor) rectifiers were thoroughly analysed in [10]. Harmonic spectra of traction current of locomotives with a thyristor converter and locomotive with resistor control was compared in [11], [12]; the authors also found the harmonics created by traction substation in the currents' variable component of the locomotive, rail circuits and return wire. Researches [13], [14] constitute exceptions in which the voltage and the traction-recuperation current are considered as continuous, endless, stationary random processes, so their spectra were defined by the formula of the direct Fourier transform as the spectrum of one realization with infinite duration of a random process of voltage or current:

$$\begin{equation*} S(j \omega)=\int\limits_{-\infty}^{\infty}f(t)e^{-j\omega t}dt. \tag{1} \end{equation*}$$