Brush Fault Analysis for Indian DC Traction Locomotive Using DWT-Based Multi-resolution Analysis

Abstract

This paper deals with the brush fault analysis in an Indian DC traction locomotive, monitoring the rectifier input current to the traction motor. The current input to the rectifier has been analysed using multi-resolution analysis of discrete wavelet transform-based statistical parameter analysis to identify the best fit solution for fault diagnosis in the system. Based on the optimized results, an algorithm has been built. Cross-validation of the assessment technique has been done using real field data analysis in professional software. The algorithm has been seen to work well for the practical case studies with less error in time of analysis. Also the operation time of the algorithm has been seen to be within 1.5 s from data acquisition to fault analysis, which is satisfactorily less and can be used for effective fault diagnosis in traction locomotives.

Appendix: Mathematical Description of the Work

The signals have been analysed using MRA of DWT considering Daubechies 20 (Db 20) as the mother wavelet and decomposing the signal into 9 decomposition levels.

Each level of decomposition has been analysed using continuous wavelet transform (CWT), and parameter estimation has been done comparing the discrete RMS value of each level coefficients. The WT of a continuous signal $x(t)$ is defined as [20, 21]:

$${\text{WT}}\,(a,b) = \frac{1}{\sqrt a }\int\limits_{ - \infty }^{\infty } {f(t)\,\,g\left( {\frac{t - b}{a}} \right)} \,{\text{d}}(t)$$

(2)

Let ‘$a$’ and ‘$b$’ be the dilation and translation parameters. The time extent of the wavelet $g\left( {\frac{t - b}{a}} \right)$ is expanded or contracted in time depending on whether $a > 1$ or $a < 1$. A value of $a > 1$ or $a < 1$ expands or contracts $g\left( t \right)$ in time and decreases or increases the frequency of the oscillations in $g\left( {\frac{t - b}{a}} \right)$. Hence, as ‘$a$‘ is ranged over some interval, usually beginning with unity and increasing, the input is analysed by an increasingly dilated function that is becoming less and less focused in time. In DWT, the scale and translation variables are discrete and in reconstruction process, the independent variable breaks down into small segments for the ease of the computer implementations. DWT gives a number of wavelet coefficients depending upon the integer number of the discrete step in scale and translation, denoted by $m$ and $n$. If $a_{0}$ and $b_{0}$ are the segmentation step sizes for the scale and translation, respectively, the scale and translation of RMS of these parameters will be $a = a_{0}^{m}$ and $b = n\,b_{0} \,a_{0}^{m}$. The discrete wavelet coefficients are henceforth given by

$${\text{DWT}}(m,n) = \int_{ - \infty }^{\infty } {\frac{1}{{\sqrt {a_{0}^{m} } }}} f(t)\,g(a_{0}^{ - m} t - nb_{0} )\,{\text{d(}}t)$$

(3)

where $g(a_{0}^{ - m} t - nb_{0} )\,$ denotes the discrete wavelet with scale and translation. Usually, the MRA technique implements the decomposition of a signal into its high- and low-frequency components, which are collectively known as high-pass and low-pass filters of MRA, respectively.

The recorded current has been assessed using DWT, and the approximate and detailed coefficients have been assessed computing the mean, mode and median values. Mean is the average of any data set. Median of a data set is dependent on whether the number of elements in the data set is odd or even. If the number of elements is even, then the median is the average of two middle terms. Mode for a data set is the element that occurs the most often in a data series. Any data set can also have two mode values since two or more elements in a data series can have equal frequency which is a logical occurrence in real time. From the best fit level of the statistical value computation, a suitable feature has been extracted to be used in the algorithm for effective brush fault assessment in the DC traction locomotive.