2. Wavelet transform (WT)

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 1980s, 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].

### 2.1 Continuous wavelet transform (CWT)

The short-time Fourier transform (STFT) of the continuous signal x(t) can be seen as the Fourier Transform (FT) of the signal with windowed x(t).g(t τ) or also as a signal decomposition <sup>x</sup>(t) into basis functions g(t <sup>τ</sup>).ejwt. The functions

## Wavelet Transform Analysis to Applications in Electric Power Systems DOI: http://dx.doi.org/10.5772/intechopen.85274

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].

In the FT case, the base functions are complex sinusoid e �jwt with a windows 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 variable displacement in time τ, is represented by b.

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:

$$
\psi\_{a,b}(t) = \frac{1}{\sqrt{a}} \cdot \psi\left(\frac{t-b}{a}\right) \tag{1}
$$

In Eq. (1), the constant 1= ffiffiffi a p is used to normalize the energy and ensure that the 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 satisfy the admissibility criterion given by [7]:

$$C\_{\Psi} = \int\_{-\infty}^{+\infty} \frac{\left| \widehat{\boldsymbol{\mu}}^{\cdot} \left( \boldsymbol{\mathcal{y}} \right) \right|^{2}}{\left| \boldsymbol{\mathcal{y}} \right|} d\boldsymbol{\mathfrak{y}} \leq \infty \tag{2}$$

where ψ \_ ð Þ<sup>y</sup> is the FT of the <sup>ψ</sup> (t). This means that if <sup>ψ</sup> \_ is a continuous function, then C<sup>ψ</sup> is finite only ifψ (0) = 0, i.e., [7]:

$$\int\_{-\infty}^{+\infty} \varphi(t) \, dt = 0\tag{3}$$

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:

$$(\text{CWT})(a,b) = \int\_{-\infty}^{+\infty} x(t) \cdot \mathbb{1}\_{a,b}(t) \, dt = \frac{1}{\sqrt{a}} \int\_{-\infty}^{+\infty} x(t) \cdot \mathbb{1} \left(\frac{t-b}{a}\right) dt\tag{4}$$

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 wavelet [9].

## 2.2 Discrete wavelet transform (DWT)

### 2.2.1 DWT definition

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

community in the last years because they are better suited for the analysis of certain

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

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

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 1980s, 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 short-time Fourier transform (STFT) of the continuous signal x(t) can be seen as the Fourier Transform (FT) of the signal with windowed x(t).g(t τ) or also as a signal decomposition <sup>x</sup>(t) into basis functions g(t <sup>τ</sup>).ejwt. The functions

2.1 Continuous wavelet transform (CWT)

78

types of transient waveforms than the other transform approaches.

transient or the state of the postdisturbance system.

2. Wavelet transform (WT)

signal [1].

Wavelet Transform and Complexity

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 1970s [10]. The DWT of the continuous signal x(t) is given by:

$$(DWT)(m, p) = \int\_{-\infty}^{+\infty} \varkappa(t) \cdot \boldsymbol{\nu}\_{m, p} \, dt \tag{5}$$

where ψm,p form bases of wavelet functions, created from a translated and dilated of the mother wavelet using the dilation m and translation p parameters, respectively.

Thus, ψm,p is defined as:

$$
\psi\_{m,p} = \frac{1}{\sqrt{a\_0^m}} \varphi \left( \frac{t - pb\_0 a\_0^m}{a\_0^m} \right) \tag{6}
$$

The DWT of a discrete signal x[n] is derived from CWT and defined as [9]:

$$(DWT)(m,k) = \frac{1}{\sqrt{a}} \sum\_{n} \mathbf{x}[n] \cdot \mathbf{g}\left(\frac{k - nb\_0 a\_0^m}{a\_0^m}\right) \tag{7}$$

where g(\*) is the mother wavelets and x[n] is the discretized signal function.

The mother wavelets may be dilated and translated discretely by selecting the scaling and translation parameters a = a<sup>0</sup> <sup>m</sup> and b = nb0a0 <sup>m</sup> respectively (with fixed constants a0>1, b0>1, m and n belonging the set of positive integers).

### 2.2.2 Multi-resolution analysis (MRA)

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. The temporal resolutions, Δt, and frequency, Δf, indicate the precision time and frequency in the analysis of the signal. Both parameters vary in terms of time and frequency, respectively, in signal analysis using WT. 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. Thus, tax is that Δf is proportional to f, i.e.,

$$\frac{\Delta f}{f} = c \tag{8}$$

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. Thus, the DWT can be implemented by multistage filter bank named MRA [11], as illustrated on Figure 2. The Mallat algorithm consists of series of high-pass and the low-pass filters that decompose the original

Comparison of (a) the STFT uniform frequency coverage to (b) the logarithmic coverage of the DWT.

Wavelet Transform Analysis to Applications in Electric Power Systems

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

signal x[n] into approximation a(n) and detail d(n) coefficient, each one

sampling rate of a signal by adding new samples to the signal.

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

corresponding to a frequency bandwidth.

Figure 1.

Figure 2.

81

DWT filter bank framework.

where c is constant.

The main difference between DWT and STFT is the time-scaling parameter. The result is geometric scaling, i.e., 1, 1/a, 1/a2 , …; and translation by 0, n, 2n, and so on. This scaling gives the DWT logarithmic frequency coverage in contrast to the uniform frequency coverage of the STFT, as compared in Figure 1.

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]. 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,

Wavelet Transform Analysis to Applications in Electric Power Systems DOI: http://dx.doi.org/10.5772/intechopen.85274

Figure 2. DWT filter bank framework.

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. Thus, the DWT can be implemented by multistage filter bank named MRA [11], as illustrated on Figure 2. 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.

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 1970s [10]. The DWT of the continuous signal x(t)

> þ ð∞

�∞

<sup>t</sup> � pb0am 0 am 0 � �

> <sup>k</sup> � nb0a<sup>m</sup> 0

am 0 � �

<sup>f</sup> <sup>¼</sup> <sup>c</sup> (8)

, …; and translation by 0, n, 2n, and so on.

x n½ �� g

<sup>m</sup> and b = nb0a0

where ψm,p form bases of wavelet functions, created from a translated and dilated of the mother wavelet using the dilation m and translation p parameters,

> ffiffiffiffiffiffi am 0 <sup>p</sup> <sup>ψ</sup>

> > ffiffiffi <sup>a</sup> <sup>p</sup> <sup>∑</sup> n

The DWT of a discrete signal x[n] is derived from CWT and defined as [9]:

where g(\*) is the mother wavelets and x[n] is the discretized signal function. The mother wavelets may be dilated and translated discretely by selecting the

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. The temporal resolutions, Δt, and frequency, Δf, indicate the precision time and frequency in the analysis of the signal. Both parameters vary in terms of time and frequency, respectively, in signal analysis using WT. 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

Δf

This scaling gives the DWT logarithmic frequency coverage in contrast to the

uniform frequency coverage of the STFT, as compared in Figure 1.

The main difference between DWT and STFT is the time-scaling parameter. The

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]. 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,

x tðÞ� ψ m,p dt (5)

(6)

(7)

<sup>m</sup> respectively (with fixed

ð Þ DWT ð Þ¼ m; p

<sup>ψ</sup> m,p <sup>¼</sup> <sup>1</sup>

constants a0>1, b0>1, m and n belonging the set of positive integers).

ð Þ DWT ð Þ¼ <sup>m</sup>; <sup>k</sup> <sup>1</sup>

the filters bank. Thus, tax is that Δf is proportional to f, i.e.,

is given by:

respectively.

Thus, ψm,p is defined as:

Wavelet Transform and Complexity

scaling and translation parameters a = a<sup>0</sup>

2.2.2 Multi-resolution analysis (MRA)

where c is constant.

80

result is geometric scaling, i.e., 1, 1/a, 1/a2

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]:

$$\varkappa[n] \* h[n] = \sum\_{k = -\infty}^{\infty} \varkappa[k] \cdot h[n - k] \tag{9}$$

simultaneous localization in time and frequency domain. The second main advantage of wavelets is that, using fast wavelet transform, it is computationally very fast. Wavelets have the great advantage of being able to separate the fine details in a signal. Very small wavelets can be used to isolate very fine details in a signal, while very large wavelets can identify coarse details. A wavelet transform can be used to decompose a signal into component wavelets. In wavelet theory, it is often possible

The Fourier transform shows up in a remarkable number of areas outside classic signal processing. Even taking this into account, we think that it is safe to say that the mathematics of wavelets is much larger than that of the Fourier transform. In fact, the mathematics of wavelets encompasses the Fourier transform. The size of wavelet theory is matched by the size of the application area. Initial wavelet applications involved signal processing and filtering. However, wavelets have been applied in many other areas including nonlinear regression and compression. An offshoot of wavelet compression allows the amount of determinism in a time series

to obtain a good approximation of the given function f by using only a few coefficients, which is a great achievement when compared to Fourier transform. Wavelet theory is capable of revealing aspects of data that other signal analysis techniques miss like trends, breakdown points, and discontinuities in higher derivatives and self-similarity. It can often compress or de-noise a signal without

Wavelet Transform Analysis to Applications in Electric Power Systems

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

4. Wavelets transform application in electric power system

Refs. [1, 13, 14] conducted studies related to this chapter. These authors also present a literature review on the application of WT in power electrical systems. By means of the bibliographic review, it is possible to highlight certain topics

Figure 3 shows the percentage of publications in each area. The areas in which more works have been developed are the power quality and protection field. The next section presents a general description of wavelet application in the selected areas of power systems. There are more works in these areas; however, no details will be entered due to space issues and that the approach to the topic used

Partial discharges are difficult to detect because of their short duration, high frequency, and low amplitude. However, the use of WT can not only detect them

appreciable degradation [12].

to be estimated [12].

of interest for researchers:

• Partial discharges

• Load forecasting

WT is similar.

83

4.1 Partial discharges

• Transient in electrical systems

• Power system protection

• Power system measurement

• Power quality

A half band low-pass filter removes all frequencies that are above half of the highest frequency in the signal. For example, if a signal has a maximum of 1000 Hz component, then half band low-pass filtering removes all the frequencies above 500 Hz. However, it should always be remembered that the unit of frequency for discrete time signals is radians.

After passing the signal through a half band low-pass filter, half of the samples can be eliminated according to the Nyquist's rule. 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]:

$$\mathcal{Y}[n] = \sum\_{k=-\infty}^{\infty} h[k] \cdot \varkappa[n-k] \tag{10}$$

The decomposition of the signal into different frequency bands is simply obtained by successive high-pass and low-pass filtering of the time domain signal. The original signal x[n] is first passed through a half band high-pass filter g[n] and a low-pass filter h[n]. After the filtering, half of the samples can be eliminated according to the Nyquist's rule, since the signal now has a highest frequency of p/2 radians instead of p. The signal can therefore be sub-sampled by 2, simply by discarding every other sample. This constitutes one level of decomposition and can mathematically be expressed as follows [2]:

$$\mathcal{Y}\_{high}[k] = \sum\_{n} \mathbf{x}[n] \cdot \mathbf{g}[2k - n] \tag{11}$$

$$\mathcal{Y}\_{low}[k] = \sum\_{n} \mathfrak{x}[n] \cdot h[2k - n] \tag{12}$$

where yhigh[k] and ylow[k] are the outputs of the high-pass and low-pass filters, respectively, after sub-sampling by 2.

## 3. Wavelets theory advantage

In [12], an application of WT and its advantages compared to Fourier transform is presented. One of the main advantages of wavelets is that they offer a
