**1. Introduction**

220 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

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When you capture and plot a signal, you get only a graph of amplitude versus time. Sometimes, you need frequency and phase information, too. However, you need to know whenever in a waveform the certain characteristics occur. Signal processing could help, but you need to know which type of processing to apply to solve your data-analysis problem.

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 Geophisicist, and the team at the Marseille Theoretical Physics Center working under A. Grossmann, a theoretical phisicist, in France. They provided a way of thinking for wavelets based on physical intuition. They also proved that with nearly any wave shape they could recover the signal exactly from its transform (Graps, 1995). In other words, the transform of a signal does not change the information content presented in the signal.

The wavelet functions are created from a single charasteristic shape, known as the mother wavelet function, by dialating and shifting the window. Wavelets are oscillating transforms of short duration amplitude decaying to zero at both ends. Like the sine wave in Fourier transform (FT), the mother wavelet *ψ*(*t*) is the basic block to represent a signal in WT. However, unlike the FT whose applications are fixed as either sine or cosine functions, the mother wavelet, *ψ*(*t*), has many possible functions. Fig. 1 shows some of the popular wavelets including Daubechies, Harr, Coiflet, and Symlet. Dilation involves the stretching and compressing the mother wavelet in time. The wavelet can be expanded to a coarse scale to analyze low frequency, long duration features in the signal. On the other hand, it can be shrunk to a fine scale to analyze high frequency, short duration features of a signal. It is this ability of wavelets to change the scale of observation to study different scale features is its hallmark.

The WT of a signal is generated by finding linear combinations of wavelet functions to represent a signal. The weights of these linear combinations are termed as wavelet coefficients. Reconstruction of a signal from these wavelet coefficients arises from a much older theory known as Calderon's reproducing activity (Grossmann & Morlet, 1984).

Application of Wavelet Analysis in Power Systems 223

 \* , , 0 *CWT a b x t t dt a a b*

where *x*(*t*) is the signal to be analyzed, *ψa,b*(*t*) is the mother wavelet shifted by a factor (*b*), scaled by a factor (*a*), large and low scales are respectively correspondence with low and

*t b <sup>t</sup> a and b*

CWT generates a huge amount of data in the form of wavelet coefficients with respect to change in scale and position. This leads to large computational burden. To overcome this limitation, DWT is used. In other words, in practice, application of the WT is achieved in digital computers by applying DWT on discretized samples. The DWT uses scale and position values based on powers of two, called dyadic dilations and translations. To do this, the scaling and translation parameters are discreted as *a*=*a*0*<sup>m</sup>* and *b*=*nb*0*a*0*<sup>m</sup>*, where *a*0>1, *b*0>0,

> \* , *DWT m n x t t dt* ,

based only on subsamples of the CWT, makes the analysis much more efficient, easy to implement and has fast computation time, at the same time, with the DWT, the original signal can be recovered fully from its DWT with no loss of data. Note a continuous-time signal can be represented in a discrete form as long as the sampling frequency is chosen properly. This is done by using the sampling theorem, termed the Nyquist theorem: the sampling frequency used to turn the continuous signal into a discrete signal must be twice

To implement the DWT, (Mallat, 1989) developed an approach called the Mallat algorithm or Mallat's Multi-Resolution Analysis (MRA). In this approach the signal to be analyzed (decomposed) is passed through finite impulse response (FIR) high-pass filters (HPF) and low-pass filters (LPF) with different cutoff frequencies at different levels. In wavelet analysis the low frequency content is called the approximation (A) and the high frequency content is called the details (D). This procedure can be repeated to decompose the approximation

The WPT is a generalization of wavelet decomposition that offers a richer range of possibilities for signal analysis. In WPT, the details as well as the approximation can be split

as large as the highest frequency present in the signal (Oppenheim & Schafer, 1989).

obtained at each level until the desired level is reached as shown in Fig. 2.

*m n*

*t a t na b a* is the discretized mother wavelet. The DWT,

(1)

(3)

(2)

, <sup>1</sup> <sup>0</sup> *a b*

*a a*

 

**2.1 Continuous wavelet transform (CWT)** 

**2.2 Discrete wavelet transform (DWT)** 

and *m*, *n* are integers, then the DWT is defined as:

where /2 , 0 00 0 / *m mm*

**2.3 Wavelet Packet Transform (WPT)** 

 

*m n* 

as shown in Fig. 3.

high frequencies, and \* stands for complex conjugation.

The CWT is defined as:

The attention of the signal processing community was caught when Mallat (Mallat, 1989) and Daubechies (Daubechies, 1988) established WT connections to discrete signal processing. To date various theories have been developed on various aspects of wavelets and it has been successfully applied in the areas of signal processing, medical imaging, data compression, image compression, sub-band coding, computer vision, and sound synthesis.

There is a plan in this chapter to study the WT applications in power systems. The content of the chapter is organized as follows:

The first part explains a brief definition of wavelet analysis, benefits and difficulties. The second part discusses wavelet applications in power systems. This section is consisted of the modeling guidelines of each application whose the goal is to introduce how to implement the wavelet analysis for different applications in power systems. Also there will be a literature review and one example for each application, separately. In the last part, the detailed analysis for two important applications of wavelet analysis, i.e. detection of the islanding state and fault location, will be illustrated by the authors.

Although, there have been a great effort in references to prove that one wavelet is more suitable than another, there have not been a comprehensive analysis involving a number of wavelets to prove the point of view suggested. Also, the method of comparison among them is not unified, such that a general conclusion is reached. In this chapter, algorithms are also presented to choose a suitable mother wavelet for power system studies.

In general, the properties of orthogonality, compactness support, and number of vanishing moments are required when analyzing electric power system waveforms for computing the power components. All these properties are well described in (Ibrahim, 2009).

Fig. 1. Some of the popular wavelets used for analysis

## **2. Wavelet transform**

There are several types of WTs and depending on the application, one method is preferred over the others. For a continuous input signal, the time and scale parameters are usually continuous, and hence the obvious choice is continuous wavelet transform (CWT). On the other hand, the discrete WT can be defined for discrete-time signals, leading to discrete wavelet transform (DWT).

#### **2.1 Continuous wavelet transform (CWT)**

The CWT is defined as:

222 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

The attention of the signal processing community was caught when Mallat (Mallat, 1989) and Daubechies (Daubechies, 1988) established WT connections to discrete signal processing. To date various theories have been developed on various aspects of wavelets and it has been successfully applied in the areas of signal processing, medical imaging, data compression, image compression, sub-band coding, computer vision, and sound synthesis. There is a plan in this chapter to study the WT applications in power systems. The content of

The first part explains a brief definition of wavelet analysis, benefits and difficulties. The second part discusses wavelet applications in power systems. This section is consisted of the modeling guidelines of each application whose the goal is to introduce how to implement the wavelet analysis for different applications in power systems. Also there will be a literature review and one example for each application, separately. In the last part, the detailed analysis for two important applications of wavelet analysis, i.e. detection of the

Although, there have been a great effort in references to prove that one wavelet is more suitable than another, there have not been a comprehensive analysis involving a number of wavelets to prove the point of view suggested. Also, the method of comparison among them is not unified, such that a general conclusion is reached. In this chapter, algorithms are also

In general, the properties of orthogonality, compactness support, and number of vanishing moments are required when analyzing electric power system waveforms for computing the

There are several types of WTs and depending on the application, one method is preferred over the others. For a continuous input signal, the time and scale parameters are usually continuous, and hence the obvious choice is continuous wavelet transform (CWT). On the other hand, the discrete WT can be defined for discrete-time signals, leading to discrete

islanding state and fault location, will be illustrated by the authors.

presented to choose a suitable mother wavelet for power system studies.

Fig. 1. Some of the popular wavelets used for analysis

**2. Wavelet transform** 

wavelet transform (DWT).

power components. All these properties are well described in (Ibrahim, 2009).

the chapter is organized as follows:

$$\text{CWT}\left(a,b\right) = \int\_{-\infty}^{+\infty} \mathbf{x}\left(t\right) \boldsymbol{\nu}^\*\big|\_{a,b}\left(t\right) dt \qquad a>0\tag{1}$$

where *x*(*t*) is the signal to be analyzed, *ψa,b*(*t*) is the mother wavelet shifted by a factor (*b*), scaled by a factor (*a*), large and low scales are respectively correspondence with low and high frequencies, and \* stands for complex conjugation.

$$
\psi\_{a,b}(t) = \frac{1}{\sqrt{a}} \psi\left(\frac{t-b}{a}\right) \qquad a > 0 \qquad \text{and} \qquad -\infty < b < +\infty \tag{2}
$$

#### **2.2 Discrete wavelet transform (DWT)**

CWT generates a huge amount of data in the form of wavelet coefficients with respect to change in scale and position. This leads to large computational burden. To overcome this limitation, DWT is used. In other words, in practice, application of the WT is achieved in digital computers by applying DWT on discretized samples. The DWT uses scale and position values based on powers of two, called dyadic dilations and translations. To do this, the scaling and translation parameters are discreted as *a*=*a*0*<sup>m</sup>* and *b*=*nb*0*a*0*<sup>m</sup>*, where *a*0>1, *b*0>0, and *m*, *n* are integers, then the DWT is defined as:

$$DWT\left(m,n\right) = \int\_{-\infty}^{+\infty} \mathbf{x}\left(t\right) \boldsymbol{\nu}^\*\_{\ \boldsymbol{m},n}\left(t\right) dt\tag{3}$$

where /2 , 0 00 0 / *m mm m n t a t na b a* is the discretized mother wavelet. The DWT,

based only on subsamples of the CWT, makes the analysis much more efficient, easy to implement and has fast computation time, at the same time, with the DWT, the original signal can be recovered fully from its DWT with no loss of data. Note a continuous-time signal can be represented in a discrete form as long as the sampling frequency is chosen properly. This is done by using the sampling theorem, termed the Nyquist theorem: the sampling frequency used to turn the continuous signal into a discrete signal must be twice as large as the highest frequency present in the signal (Oppenheim & Schafer, 1989).

To implement the DWT, (Mallat, 1989) developed an approach called the Mallat algorithm or Mallat's Multi-Resolution Analysis (MRA). In this approach the signal to be analyzed (decomposed) is passed through finite impulse response (FIR) high-pass filters (HPF) and low-pass filters (LPF) with different cutoff frequencies at different levels. In wavelet analysis the low frequency content is called the approximation (A) and the high frequency content is called the details (D). This procedure can be repeated to decompose the approximation obtained at each level until the desired level is reached as shown in Fig. 2.

#### **2.3 Wavelet Packet Transform (WPT)**

The WPT is a generalization of wavelet decomposition that offers a richer range of possibilities for signal analysis. In WPT, the details as well as the approximation can be split as shown in Fig. 3.

Application of Wavelet Analysis in Power Systems 225

Fig. 4 shows the percentage of 196 IEEE papers based in each area (Source: search on IEEE Explore). One can conclude that most research are carried in the field of power quality and

Next sections present a general description of wavelet applications in the selected areas of

In the area of PQ, several studies have been carried out to detect and locate disturbances using the WT as a useful tool to analyze sag, swell, interruption, etc. of non-stationary signals. These disturbances are "slow changing" disturbances. Therefore, it contains only the spectral contents in the low frequency range. Therefore, examining WT coefficients (WTCs) in very high decomposition levels would help to determine the occurrence of the disturbance events as well as their occurring time. Counting on this, the DWT techniques

Theoretically, all scales of the WTCs may include all the features of the original signal. However, if all levels of the WTCs were taken as features, it would be difficult to classify diverse PQ events accurately within reasonable time, since it has the drawbacks of taking a longer time and too much memory for the recognition system to reach a proper recognition rate. Moreover, if only the first level of the WTCs were used, some significant features in the other levels of the WTCs may be ignored. Beside, with the advancement of PQ monitoring equipment, the amount of data over the past decade gathered by such monitoring systems has become huge in size. The large amount of data imposes practical problems in storage and communication from local monitors to the central processing computers. Data compression has hence become an essential and important issue in PQ area. A compression technique involves a transform to extract the feature contained in the data and a logic for removal of redundancy present in extracted features. For example, in (Liao, 2010) to effectively reduce the number of features representing PQ events, spectrum energies of the

Fig. 4. Percentage of wavelet articles in different areas of power system

have been widely used to analyze the disturbance events in power systems.

Power system transients

power system protection.

power systems.

**3.1 Power quality (PQ)** 

Fig. 2. Decomposition tree for DWT

Fig. 3. Decomposition tree for WPT

In view of the fact that WPT generates large number of nodes it increases the computational burden. In DWT only approximations are further decomposed thus reducing the level of decomposition and thereby computational attempts.
