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Applicationof Wavelet Analysis in Power System Disturbance ModelingJ. Liu P. PillayDepartmentof Electrical & Computer EngineeringClarkson University

Adjunct Professor, University of Cape Town, S.A.Potsdam, NY 13699-5720

Abstract: his paper shows that wavelets can beused to model a variety of power system transientsusing a fraction of the total number of coefficients.This modeling technique paves the way for faultclassification and improved protection.1. Introduction

Power quality disturbances include problems causedby voltage sags, voltage swells , capacitor switch ingtransients, voltage notches, and supply discontinuitiescaused by autoreclose protection. All these faults aregenerally transitory, i.e., of a short-term duration.Traditionally, the identification of the transients hasbeen based on the visual inspectio n of the disturbancein the time domain. This is time-consuming and also hasits limitation in practical applications, especially when alarge amount of transient data is to be analyzed.Additional complications arise when several kinds oftransients occur at the same time. A reliable way ofclassifying different faults is becoming a major concernin this field, particularly with deregulation and the useof power quality based rates.It is a well-known fact in the signal processing areathat the Fourier Transform is a powerful tool for theanalysis of periodic information. The drawback is thatits coefficients do not have inherent time inform ation.

Wavelets have been shown to have advantages overthe Fourier Transform where time information isneeded. It translates the time-domain function into arepresentation localized not only in frequency but alsoin time. The use of Wavelet transform to analyze non-stationary harmonic distortions has been proposed [13.This approach is a more powerful technique and canbetter category and analyze, in a more effective way,m p y types of non-stationary voltage distortions orpower quality deviations[2]. Several works haveaddressed its application in the power area. A dyadic-orthonormal wavelet transform analysis is used to detectand localize various types of power qualitydisturbances, including harmonic distortion [3]. A key

idea is to decompose a given disturbance into othersignals which represent a smoothed version and adetailed version of the original signal 141. In [5], thewavelet transform is used to detect and quantifynonsinusoidal power transients, especially voltage. In[6], a new w avelet transform based p rocedure for powerquality analysis is presented, which is based on themultiresolution signal d ecomposition and reconstructionby means of the Time D iscrete Wavelet Transform. Theoptimized decomposition of signals in frequencysubbands allows the most relevant disturbances inelectrical power systems to be not only detected,localized, and classified but also estimated. In [7], theability of wavelets to reconstruct transients wasdemonstrated. [8] shows how wavelets can be used toaccurately reconstruct non-stationary power systemdisturbances and the contribution of individualcoefficients to the reconstruction process. In addition,it should be possible for the wavelet coefficients to beused to categorize different types of disturbances.With the combinations of wavelet and signalprocessing tools, w avelets has been extensively used ina variety of pow er fieldsp-131.Section 2 presents the basic w avelet theory. Section 3shows that several typical power transients can bereconstructed by a limited number of waveletcoefficients. Section 4 h as the conclusion.

2. Wavelet TransformThe ability of the Wavelet Transform to localize bothtime and frequency makes it possible to simultaneouslydetermine sharp transitions of the signals and thelocalization of their occurrence. In the WaveletTransform, a set of wavelet functions is chosen as thebasis, similar to the sinusoidal function for the FourierTransform. The set of basis functions is obtained bydilating and shifting a prototype, called a mother

wavelet, on different levels. Wavelet coefficients arethe projection of the original function onto the basis.Fig.2.1 shows a typical example from the D aubechies

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wavelet family, which is frequently used in powertransients analysis because of its property of compactsupport.

0 0.5 1 1.5 2 2.54.dFigure 2.1. Daubechies 4 mother wavelet

The w avelet transform [141 can be though t of as theinner product of the original signal with the basisfunction w(t):

Where

In discrete cases, the selection of a=2 , b=k2 (k,m Zresults in the dyadic wavelet transform.The definition of the Wavelet Transform shows thatthe wavelet analysis is a measure of similarity betweenthe basis functions (wavelets) and the original function.The calculated coefficients indicate how close thefunction is to the daughter wavelet at the particularscale. If the function has a major contribution at aparticular scale, the coefficient computed at this point inthe time-scale plane will be a relatively large number.An invertible transform can be obtained by a simpledyadic translation and dilation scheme, which is basedon functions y~~,~(t)=2j%+r(2Jt-k),here j,k E Z. That is,there exist functions w such that ( ~ ~ k ,,k 2 orms acomplete orthonormal basis of L2(R), which is the spaceof the square integrable functions. For a function fEL2(R), there is a wavelet expansion.

Where vj.r>JAt)v'jr(t)dt. That is at each scale J,there is a projection o f f on some linear space. Thisspace can be alternatively spanned by translates of anaccordingly dilated and shifted version of a so-calledscaling function I . Let &k(t)=2mI$ (2't-k). Now thewavelet expansion of the functionfis

3. Reconstruction of Power TransientsTypical power transients include momentaryinterruptions, such as autoreclosure, voltage sags andswells, which are the results of power system faults,equipment failure and control malfunctions. Oscillatorytransients occur when there is a sudden, non-powerfrequency change in the steady state condition ofvoltage, current, or both, that include both positive and

negative polarity values. A typical example of theoscillatory transient is caused by capacitor switching.Figure 3.1 shows an autoreclosure signal for a periodof 0.2 seconds with 2 cycles of power outage. After theWavelet Transform is carried out, the coefficients areused to reconstruct the original signal. In this case, only2% of the total number of coefficients are used and agood approxim ation is obtained.

I

im50g odo

1Wo am aw o m 0.m 0.1 0.12 0.14 ai 8 0.18 0.2Th k r o n d

Figure3.1 A utoreclosure and its5% reconstruction withDB4 mother wavelet.One measure of the accuracy of the reconstruction,commonly used in the signal processing is

error=norm x- z)lnorm(x)*100 (3.1)that is, the norm of the difference between the originalsignal x and reconstructed signal z, divided by the norm

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of the original signal x. The effective reconstructionrequires the error as small as possible.In the autoreclosure case, the relative error ofreconstruction is 4.47%. The reconstructed signalretains approximately 95% of the energy in the originalsignal with 5% of the coefficients.

The result above is obtained via Wavelet Transformusing the DB4 mother wavelet. If a higher order of theDaubechies mother wavelet is used, the reconstructionwill be more effective. Figure 3.2 gives thereconstruction of an autoreclosure using the DBlOmother wavelet. The relative error according to equation(3.1) s 1.21 . That means that only 5 of thecoefficients retain 98.8% of the energy in the originalsignal. The effectiveness comes from the increase ofthe order of Daubechies mother wavelet. The selectionof the optimal mothe r wavelet is a major concern in thepractical implementations.

100

505 03

1mo om O.M 0.00 0.06 0.1 0.12 0.14 0.18 o.ia 0.2

nm in a m:igure3.2Autoreclosure and its 5 reconstruction H

DB 10 mother wavelet.

1m.p1m

100o.I00

igure 3.3 Capacitor switching and 5 reconstructwith DB4 mother wavelet

Figures 3.3-3.5 show the reconstruction of acapacitor switching, voltage sag and voltage swell with5 of the wavelet coefficients. The related error withdifferent Daubechies mother wavelet is given in Table1. It is clear that the use of high order mother waveletreduces the reconstruction error.I 100

50I

Odo

U. U . v . U v Yo 0.02 aw 0.w o.00 ai 0.12 014 o.ie 0.18 0.2.rm .

imso

1 0do U U v U .U

o 0.02 0.w 0.w 0.w ai 0.12 0.14 a i e a i 8 0.2100 'nme n iaOn IFigure 3.4 Voltage sag and its 5% reconstruction withDB4 mother wavelet

. u . v . . J0 0.02 0.M 0.06 0.M 0.1 0 .12 0.14 0.18 0.18 0.2

5 0f1mI V V . I . I . I J

0 0 m 0.04 0.06 0.M 0.1 0.12 0.14 0.18 0.18 02Nn n1s

P'igure 3.5 Voltage swell and its 5 reconstruction W IDB4 mother wavelet hTable 1 Reconstruction error of four types of powertransients with different Daubechies mother w avelet and5% of the total coefficients ( )

DR8 1.28 2 88 n.98 0.86I DBlO 1.21 1.98 0.61 0.64An other example of a power disturbance is thewaveform distortion, which is a steady state deviationfrom an ideal sine wave of power frequency principallycharacterized by the spectral content of the deviation. A

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example given in this paper is voltage notching, whichis a periodic voltage disturbance caused by the normaloperation of power electronics devices when current iscommutated from one phase to another.In figure 3.6,a voltage notch is given. S ince notching

occurs continuously, it can be characterized by itsharmonic spectrum. When wavelet reconstruction isused, more coefficients are needed to achieve the sameaccuracy than for the case of short term powertransients. For a 20% reconstruction, the error is 4.2%with the DB4 mother wavelet. For DB10, the error isreduced to 3.1%.N e c h

0 0.02 0 04 0.W 0 08 0.1 0.12 0.14 0.18 0.16 0.220 rrOrY1lUniDnnm. 1 .urn

SM

0t-5m

0 0.02 0.04 0 04 0.08 0.1 0.12 0.14 0.16 0.18 0.2hme in.U

Figure 3.6 Notching and its 20% reconstructionwithDB4 mother wavelet

4. ConclusionThis paper shows that wavelets can be used toreconstruct non-stationary power transients as well as

periodic disturbances. With the appropriate selection ofthe mother wavelet, the reconstruction accuracy can beimproved. The modeling can form the basis fordisturbace classification and protection.

5. Reference[ l] P.Riberio, R.Ceilo and M.J. Smotyi FutureAnalysis Tools for Power Quality, Proc. PQA93PECON IV Conf., USA, Nov16-19, 1993, pp.7-3.1[2] P.F. Ribeiro, Wavelet transform: an advanced toolfor analyzing non-stationary harmonic distortions inpower systems, Proceedings of the IEEE InternationalConference on Harmonics in Power systems, Bologna,Italy, September 19 94

to 7-3.10.

[3] Robertson, D.C., Wavelet and electromagneticpower system transients, IEEERES 1995 SummerMeeting, 95 SM 391-3 PWRD.[4] Santoso, S., Power quality assessment via wavelettransform analysis, IEEEYPES 1995 Summer Meeting,[5]Tunaboylu,N.S.,Collins,E.R.,he w avelet TransformApproach to Detect and Quantify Voltage Sags, Proc.ICHQ P, October 1996, pp619-624[6]Angrisani, L., Daponte, P., A New WaveletTransform Based Procedure For Electrical PowerQuality Analysis, Proc. ICHQP, October 1996, pp608-614[7] Pillay,P., Bhattacharjee, A., Application of Waveletsto Model Short-time Power System Disturbances, 96[8] P.Pillay,P.Ribeiro, Q.Pan, Power Quality ModelingUsing Wavelets, Proc. ICHQP, October, 1996[9] S Santoso et al., Power Quality DisturbanceIdentification using Wavelet Transforms and ArtificialNeural Networks, Proc. ICHQ P, October 1996,[ l o ] S Santoso et al, Power Qualtiy DisturbnaceWaveform Recognition Using Wavelet-based NeuralClassifier, PE-599-PWRD-0-01-1997, IEEE PESWinter Meeting 1997[111 L.Angrisani, P .Daponte, A Measurement MethodBased on the Wavelet Transform for Power QualityAnalysis, PE-056-PW RD-0-1-1998, IEEE PES WinterMeeting 1998.[12] F. H. Magnago, A. Abur, Fault Location UsingWavelets PE-303-PW RD-0-12-1997, IEEE PESWinter Meeting 1998.[131 S.Pandey,L.Satish, Multiresolution SignalDecomposition: A New Tool For Detection In PowerTransformers During Impulse Tests, PE - 128-PWRD-O-[14] Daubechies, I., Ten Lectures on Wavelets, KluwverAcademic Publishers, 1993

95 SM 371-5 PWRD

WM 284-0 PWRS, 1996

~ ~ 6 1 5 - 68.

11-1997

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