On the other hand, the waveforms associated with fast electromagnetic transients are typically nonperiodic and contain both high frequency oscillations and localized superimposed impulses on power frequency and its harmonics. These characteristics present problems for traditional Fourier analysis because the latter assumes a periodic signal that needs longer time periods to maintain good resolution in the low frequency.

In this sense, WT has received great attention in power community in the last years because they are better suited for the analysis of certain types of transient waveforms than the other transform approaches. Many books and papers have been written that explain WT of signals and can be read for further understanding of the basics of wavelet theory. The concept of wavelets in its present theoretical form was first proposed by J. Grossmann, a theoretical physicist, in France. They provided a way of thinking for wavelets based on physical intuition.

In other words, the transform of a signal does not change the information content presented in the signal [ 1 ]. Thus, in the first part, this chapter presents an overview of the main characteristic of wavelet transform for the transient signal analysis and the application on electric power system. The property of multiresolution in time and frequency provided by wavelets allows accurate time location of transient components while simultaneously retaining information about the fundamental frequency and its low-order harmonics. This property of the wavelet transform facilitates the detection of physically relevant features in transient signal to characterize the source of the transient or the state of the postdisturbance system.

Initially, we will discuss the performance, advantages, and limitations of the WT in electric power system application, where the basic wavelet theory is presented. Additionally, the main publications carried out in this field will be analyzed and classified by the next areas: power system protection, power quality disturbances, power system transient, partial discharge, load forecasting, faults detection, and power system measurement. Finally, a comprehensive analysis related to the advantages and disadvantages of the WT in relation to other tools is performed.

The wavelet transform theory is based on analysis of signal using varying scales in the time domain and frequency. Formalization was carried out in the s, based on the generalization of familiar concepts. The wavelet term was introduced by French geophysicist Jean Morlet. The seismic data analyzed by Morlet exhibit frequency component that changed rapidly over time, for which the Fourier Transform FT is not appropriate as an analysis tool. Thus, with the help of theoretical physicist Croatian Alex Grossmann, Morlet introduced a new transform which allows the location of high-frequency events with a better temporal resolution [ 2 ].

Faulted EPS signals are associated with fast electromagnetic transients and are typically nonperiodic and with high-frequency oscillations. This characteristic presents a problem for traditional Fourier analysis because it assumes a periodic signal and a wide-band signal that require more dense sampling and longer time periods to maintain good resolution in the low frequencies [ 3 ].

The WT is a powerful tool in the analysis of transient phenomena in power system. It has the ability to extract information from the transient signals simultaneously in both time and frequency domains and has replaced the Fourier analysis in many applications [ 4 ]. The functions based term refers to a complete set of functions that, when combined on the sum with specific weight can be used to then construct a certain sign [ 5 ].

## Automation of Electric Power Systems

The main characteristic of the WT is that it uses a variable window to scan the frequency spectrum, increasing the temporal resolution of the analysis. The wavelets are represented by:. In Eq. Thus, it is evident that WT has a zero rating property that increases the degrees of freedom, allowing the introduction of the dilation parameter of the window [ 8 ].

The continuous wavelet transform CWT of the continuous signal x t is defined as:. The WT coefficient is an expansion and a particular shift represents how well the original signal x t corresponds to the translated and dilated mother wavelet. Thus, the coefficient group of CWT a,b associated with a particular signal is the wavelet representation of the original signal x t in relation to the mother wavelet [ 9 ]. The redundancy of information and the enormous computational effort to calculate all possible translations and scales of CWT restricts its use.

## Applying blockchain technology to electric power systems

An alternative to this analysis is the discretization of the scale and translation factors, leading to the DWT. There are several ways to introduce the concept of DWT, the main are the decomposition bands and the decomposition pyramid or Multi-Resolution Analysis -MRA , developed in the late s [ 10 ].

The DWT of the continuous signal x t is given by:. The problems of temporal resolution and frequency found in the analysis of signals with the STFT best resolution in time at the expense of a lower resolution in frequency and vice-versa can be reduced through a multi-resolution analysis MRA provided by WT. Both parameters vary in terms of time and frequency, respectively, in signal analysis using WT.

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In the STFT, a higher temporal resolution could be achieved at the expense of frequency resolution. Intuitively, when the analysis is done from the point of view of filters series, the temporal resolution should increase increasing the center frequency of the filters bank. The result is geometric scaling, i. The CWT follows exactly these concepts and adds the simplification of the scale, where all the impulse responses of the filter bank are defined as dilated versions of a mother wavelet [ 10 ].

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The CWT is a correlation between a wavelet at different scales and the signal with the scale or the frequency being used as a measure of similarity. The CWT is computed by changing the scale of the analysis window, shifting the window in time, multiplying by the signal, and integrating over all times. In the discrete case, filters of different cut-off frequencies are used to analyze the signal at different scales. The signal is passed through a series of high-pass filters to analyze the high frequencies, and it is passed through a series of low-pass filters to analyze the low frequencies.

The Mallat algorithm consists of series of high-pass and the low-pass filters that decompose the original signal x [ n ] into approximation a n and detail d n coefficient, each one corresponding to a frequency bandwidth. The resolution of the signal, which is a measure of the amount of detail information in the signal, is changed by the filtering operations, and the scale is changed by up-sampling and down-sampling sub-sampling operations. Sub-sampling a signal corresponds to reducing the sampling rate or removing some of the samples of the signal.

For other hand, up-sampling a signal corresponds to increasing the sampling rate of a signal by adding new samples to the signal. The procedure starts with passing this signal x [ n ] through a half band digital low-pass filter with impulse response h [ n ]. Filtering a signal corresponds to the mathematical operation of convolution of the signal with the impulse response of the filter.

The convolution operation in discrete time is defined as follows [ 2 ]:. A half band low-pass filter removes all frequencies that are above half of the highest frequency in the signal. However, it should always be remembered that the unit of frequency for discrete time signals is radians. Simply discarding every other sample will subsample the signal by two, and the signal will then have half the number of points. The scale of the signal is now doubled. Note that the low-pass filtering removes the high frequency information but leaves the scale unchanged. Only the sub-sampling process changes the scale.

Resolution, on the other hand, is related to the amount of information in the signal, and therefore, it is affected by the filtering operations. Half band low-pass filtering removes half of the frequencies, which can be interpreted as losing half of the information. Therefore, the resolution is halved after the filtering operation.

Note, however, the sub-sampling operation after filtering does not affect the resolution, since removing half of the spectral components from the signal makes half the number of samples redundant anyway. Half of the samples can be discarded without any loss of information. This procedure can mathematically be expressed as [ 2 ]:.

Physical Science and Engineering Chevron Right. Electrical Engineering. Electric Power Systems. Offered By. University at Buffalo. The State University of New York. About this Course 21, recent views. Flexible deadlines. Flexible deadlines Reset deadlines in accordance to your schedule.

Beginner Level. Hours to complete. Available languages. English Subtitles: English. Chevron Left. Syllabus - What you will learn from this course. Show All. Video 6 videos. Basic Electricity: Concepts 6m. Basic Electrical Properties 9m. Simple Circuits 3m. Ohm's Law 5m. AC Current 2m. Reading 2 readings. Acknowledgements 10m. Quiz 6 practice exercises.

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Self-Check 4m. Evolution of the number of total citation per document and external citation per document i. International Collaboration accounts for the articles that have been produced by researchers from several countries. The chart shows the ratio of a journal's documents signed by researchers from more than one country; that is including more than one country address. Not every article in a journal is considered primary research and therefore "citable", this chart shows the ratio of a journal's articles including substantial research research articles, conference papers and reviews in three year windows vs.

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Ratio of a journal's items, grouped in three years windows, that have been cited at least once vs. The purpose is to have a forum in which general doubts about the processes of publication in the journal, experiences and other issues derived from the publication of papers are resolved.

## Electrical Power Systems

For topics on particular articles, maintain the dialogue through the usual channels with your editor. Year SJR 0. Citations per document.