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Fundamentals and Literature Review of Wavelet Transform in Power Quality Issues

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Fundamentals and Literature Review of Wavelet Transform in Power Quality Issues Vol. 5(1), pp. 1-8 May, 2013 Journal of Electrical and Electronics DOI 10.5897/JEEER2013.0435 ISSN 2141 – 2367 ©2013 Academic Journals Engineering Research http://www.academicjournals.org/JEEER Review Fundamentals and literature review of wavelet transform in power quality issues Lütfü SARIBULUT 1*, Ahmet TEKE 2, Mohammad BARGHI LATRAN 2 and Mehmet TÜMAY 2 1Department of Electrical-Electronics Engineering, Adana Science and Technology University, Adana/TURKEY. 2Department of Electrical-Electronics Engineering, Çukurova University, Adana/TURKEY. Accepted 12 May, 2013 The effects of power quality (PQ) disturbances on power utilities and power customers are increasing day by day with widespread use of power electronics devices. These disturbances should be detected rapid and accurate. Wavelet transform (WT) has become a popular method to detect, classify and analyze various types of PQ disturbances in power systems. In this study, the fundamentals of WT, its distinctions from Fourier transform and the results of an extensive literature review of published studies are presented. The results of the literature review reveal that the detection of PQ disturbances using WT is a very powerful method and has been used in PQ mitigation devices effectively (that is, Custom Power, Flexible AC Transmission System, Flexible , Reliable and Intelligent Energy Delivery System). Key words: Wavelet transform, power quality disturbances, literature review, feature extraction. INTRODUCTION The term wavelet means a small wave. In addition, a finance. The ability of wavelet analysis deals with both wave refers the condition that this function is an the stationary and non-stationary data, their localization oscillatory. The other term of Wavelet transform (WT) is in time and the decomposing and analyzing the the mother wavelet. It implies that the functions used in punctuation in variable signal. While a review of the the decomposition and reconstruction processes are possible future contributions of WT is provided to the derived from one main function or the wavelet. In other engineering discipline, there are explored two ways in words, the mother wavelet is a prototype for generating which wavelets might be used to enhance the empirical the other wavelet functions. The term translation (τ) is toolkit of our profession in engineering (Crowley, 2005). used in the same sense as it was used in short time These are summarized in the following: Fourier transform (STFT). It is related to the location of the window, as the window is shifted through the signal. It Time scale versus frequency: An examination of data is visualized in Figure 1 (Amara, 1995; Robi, 2012). sets to assess the presence and ebb and the flow of The application of WT has become a popular for some frequency components is potentially valuable (Crowley, time under the various names of multirate-sampling 2005). (Quadrature Mirror Filters) QMF in electrical engineering (Sarkar et al., 1998). WT possesses many desirable Time scale decomposition: Recognition components of properties that are useful in engineering, economics and the signal can be possibly found at disaggregate (scale) *Corresponding author. E-mail: [email protected]. 2 J. Electrical Electron. Eng. Res. Window Window Amplitude Amplitude time time Figure 1. Window is shifted through the signal. Signal Signal Wavelet Wavelet (a) (b) Figure 2. Wavelet function with it’s shifted and dilated versions. level rather than an aggregate level forecasting: CWT is given in the following (Wikipedia, 2012). disaggregate forecasting, establishing global versus local aspects of series (Crowley, 2005). 1 ∗ t −τ ψ(,) τs= ∫ xt () ψ dt (1) s s In this study, the basic principles of WT especially Continuous Wavelet Transform (CWT) and Discrete The scale (s) and τ parameters are the parameters of Wavelet Transform (DWT), the properties of most used CWT functions. The function ψ(τ,s ) named as mother Haar and Daubechies wavelets are presented. The wavelet is the transforming function. Let, x(t) be the similarity and contrast points of WT with the Fourier signal to be analyzed. The mother wavelet is chosen to Transform (FT) are also given in this study. The recent serve as a prototype for all windows in the process. All studies with the subject of WT applications in power the windows used for the dilated, compressed and shifted quality (PQ) issues are examined in details in the last part are the versions of the mother wavelet. There are many of the work. functions used for this purpose. Once the mother wavelet is chosen in CWT, the computation can be started any value of s and computed for all values of s (smaller CONTINUOUS WAVELET TRANSFORM and/or larger than 1). Such as, the procedure can be started from scale s=1 and it is continued with increasing Unlike FT, CWT possesses the ability to construct a time- the values of s. In this situation, the analyzing of the frequency representation of a signal that offers very good signal is started from the high frequencies and proceeded time and frequency localization. In other words, CWT is towards the low frequencies. This procedure is repeated used to divide a continuous-time function into the equal- for every value of s. Every computation for a given value time intervals and equal-frequency intervals. In CWT, the of s fills the corresponding single row of the time-scale signal is multiplied with the wavelets, same as STFT and plane (Robi, 2012). This process is summarized in Figure the transform is computed separately for different parts of 2. the time-domain signal. The mathematical definition of An accurate signal analysis is the proper analysis of Saribulut 3 y Haar Wavelet 1 There are two functions that play a primary role in the wavelet analysis. These are the scaling function ( ϕ) and wavelet ( ψ). These functions generate a family of func- tions that can be used to break up or reconstruct the signal. To emphasize the marriage involved in building φ(x ) this family, ϕ is sometimes called as father wavelet and ψ as mother wavelet (Boggess, 2001). The simplest wavelet analysis is based on the Haar scaling function 0 0 illustrated in Figure 3. x The Haar scaling function is defined as 0 1 1,if 0≤ x ≤ 1 φ(x ) = (2) Figure 3. Scaling function of Haar wavelet. 0,elsewhere . The function ϕ(x-k) is obtained by shifted versions of ϕ(x) to the right by k units (assuming k is positive). Let, V0 be 1 the space of functions in the form of ϕ(x-k). It is given in the following. φ(2x ) ∑ akφ( x− k ) (3) k∈ z where k can range over any finite set of positive or negative integers. The thinner blocks are required to analyze the high- frequency signals. The function ϕ(2x ), whose width is half 0 0 of ϕ(x), is illustrated in Figure 4. x The simplest ψ(x) of Haar wavelet is the function that 0 1 consists of two blocks and it can be written as (4). 2 ψ(x )= φ (2 x ) − φ (2( x − 1 2)) (4) Figure 4. Graph of φ(2x ) . =φ(2)x − φ (2 x − 1) Haar wavelets have one particularly desirable property that they are zero everywhere except on a small interval. each trace. FT is not a good tool for this purpose. It can If the function is closed everywhere outside of a bounded only provide frequency information of the oscillations that interval, it has compact support. If a function has compact comprise the signal. It gives no direct information about support, the smallest closed interval on which the when an oscillation is occurred. Another tool for signal function has non-zero values is called the support of the analysis is STFT. The full-time interval is divided into a function (Aboufadel et al., 1999). If k ≠ 0 and ψ(x) is number of small, equal-time intervals. These are orthogonal to ϕ(x), then the compact support of ψ(x) and individually analyzed by using FT. The result contains the compact support of ϕ(x-k) do not overlap in the time and frequency information. However, there is a Cartesian coordinate system. problem with this approach. The equal-time intervals are ∞ not adjustable. When the time intervals are very-short- So, ∫ ψ(x ) φ ( x− k ) dx = 0. Consequently, ψ(x) duration, high-frequency bursts are hard to detect −∞ (Sarkar, 1998; Morlet, 1982a, b). belongs to V1 and it is orthogonal to V0. Then, ψ(x) is CWT is a good tool for analysis the signal included the called Haar wavelet. It is given in Figure 5. harmonics both the time-domain and the frequency- In general, nth generation of daughters will have 2n domain. The signal, that its frequency is changed with wavelets and it is defined as (5). time, is detected easily when the time intervals are very- short-duration. The most common wavelets are () (2n ), 0 21n (5) summarized in the following. ψn, k t= ψ tk − ≤≤− k 4 J. Electrical Electron. Eng. Res. 1.5 the signal in the time-domain. The mathematical expression of DWT is given in the following. 1 1 t− nb a m ( ) 0 0 (7) 0.5 ψ m, n t = m m a a0 0 0 where the integers m and n control the wavelet dilation -0.5 and translation respectively, a0 is a specified fixed dilation step parameter which should be greater than one and b0 -1 is the location parameter which should be greater than zero (Addison, 2002). -1.5 The control parameters m and n contain the set of 0 0.5 1 positive and negative integers. WT of a continuous signal x(t) gives the decomposition wavelets by using the Figure 5.
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