1. Introduction

Electromagnetics transients in electric power systems (EPS) are generally caused by lightning discharges and/or certain operating conditions, such as faults in equipment and transmission lines, switching of electric power system devices, voltage sags, capacitor switching, and transmission line energization and de-energization. Faulted EPS signals are associated with fast electromagnetic transients and are typically nonperiodic with high-frequency oscillations. These characteristics present a problem for traditional Fourier analysis because it assumes a periodic signal and a wide-band signal that require denser sampling and longer time periods to maintain good resolution in low frequencies. Wavelet transform (WT), on the other hand, is a powerful tool in the analysis of transient phenomena in power systems. 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. This ability to tailor the frequency resolution can greatly facilitate the detection of signal features that may be useful in characterizing the transient cause or the state of the postdisturbance electrical system.

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.

based term refers to a complete set of functions that, when combined on the sum

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 wave-

> 1 ffiffiffi <sup>a</sup> <sup>p</sup> � <sup>ψ</sup>

energy of ψ a,b(t) is independent of the dilation level [6]. The wavelet is derived from operations such as dilating and translating the mother wavelet, ψ, which must

> ψ \_ ð Þ<sup>y</sup> � � � � 2

t � b a

a p is used to normalize the energy and ensure that the

�jwt with a windows

� � (1)

j j <sup>y</sup> dy ≤ ∞ (2)

ψð Þt dt ¼ 0 (3)

\_ is a continuous function,

with specific weight can be used to then construct a certain sign [5]. In the FT case, the base functions are complex sinusoid e

Wavelet Transform Analysis to Applications in Electric Power Systems

ψa,bðÞ¼ t

C<sup>ψ</sup> ¼

\_ ð Þ<sup>y</sup> is the FT of the <sup>ψ</sup> (t). This means that if <sup>ψ</sup>

þ ð∞

�∞

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

x tðÞ� <sup>ψ</sup><sup>a</sup>:<sup>b</sup>ð Þ<sup>t</sup> dt <sup>¼</sup> <sup>1</sup>

where the scale factor a and the translation factor b are continuous variables. 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

The redundancy of information and the enormous computational effort to calculate all possible translations and scales of CWT restricts its use. An alternative to this analysis is the discretization of the scale and translation factors, leading to the

ffiffiffi a p þ ð∞

�∞

x tðÞ� ψ

t � b a

� �dt (4)

þ ð∞

�∞

variable displacement in time τ, is represented by b.

DOI: http://dx.doi.org/10.5772/intechopen.85274

lets are represented by:

where ψ

defined as:

wavelet [9].

79

2.2.1 DWT definition

In Eq. (1), the constant 1= ffiffiffi

satisfy the admissibility criterion given by [7]:

then C<sup>ψ</sup> is finite only ifψ (0) = 0, i.e., [7]:

ð Þ CWT ð Þ¼ a; b

2.2 Discrete wavelet transform (DWT)

þ ð∞

�∞

centered on the τ time. The WT is described in terms of its basic functions, called wavelet or mother wavelet, and variable frequency w is replaced by an ever-escalating variable factor a (which represents the swelling) and, generally, to

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 first recorded mention of what we now call a "wavelet" seems to be in 1909, in a thesis by A. Haar. The concept of wavelets in its present theoretical form was first proposed by J. Morlet, a Geophysicist, and the team at the Marseille Theoretical Physics Center working under A. 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 loworder 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.
