使用連續小波變換在配電系統中故障定位畢業論文(編輯修改稿)

　

【文章內容簡介】 。2. Fault location information provided by continuouswavelettransformThe CWT of a signal s(t) is the integral of the productbetween s(t) and the socalled daughterwavelets, whichare time translated and scale expanded/pressed versions of a function having ?nite energy w(t), calledmotherwavelet. This process, equivalent to a scalar product, produces wavelet coe?cients C(a, b), which can be seenas ‘‘similarity indexes’’ between the signal and the socalleddaughterwavelet located at position b (time shifting factor)and positive scale a:Z11Unlike DWT, CWT can operate at any scale, speci?callyfrom that of the original signal up to some maximum scale.CWT is also continuous in terms of shifting: during putation, the analyzing wavelet is shifted smoothly overthe full domain of the analyzed function.The CWTanalysis is performed in time domain on thevoltage transients recorded after the fault in a bus of thedistribution network.The analyzed part of the transient recorded signal s(t),which can correspond to a voltage or current faulttransient, has a limited duration (few milliseconds) corresponding to the product between the sampling time Tsand the number of samples N. The numerical implementation of the CWT to signal s(t) is a matrix C(a,b) de?ned asp wtb240。1222。C240。a。 b222。 188。1s240。t222。a???adtfollows:1where * denotes plex conjugation.Eq. (1) can be expressed also in frequency domain (.,C240。a。 iTs222。 188。 Tspj??????ajX1nw240。i222。Tsas240。nTs222。[5]):i 188。 0。 1。 . . . 。 N :n188。0240。5222。F 240。C240。a。 b222。222。 188。p???aW 240。a x222。S240。x222。240。2222。The sum of the squared values of all coe?cients corresponding to the same scale, which is henceforth calledwhere F(C(a,b)), S(x) and W(x) are the Fourier transformsof C(a, b), s(t) and w(t), respectively.Eq. (2) shows that if the motherwavelet is a bandpass?lter function in the frequencydomain, the use of CWT inthe frequencydomain allows for the identi?cation of thelocal features of the signal. According to the Fourier transform theory, if the center frequency of the motherwaveletW(x) is F0, then the one of W(ax) is F0/a. Therefore, di?erent scales allows the extraction of di?erent frequenciesfrom the original signal – larger scale values correspondingto lower frequencies – given by the ratio between center frequency and bandwidth. Opposite to the windowedFourieranalysis where the frequency resolution is constant anddepends on the width of the chosen window, in the waveletapproach the width of the window varies as a function of a,thus allowing a kind of timewindowed analysis, which isdependent to the values of scale a.As known, the use of CWT, allows the use of arbitrarymotherwavelets which must satisfy the ‘admissibilitycondition’:CWTsignal energy ECWT(a), identi?es a ‘scalogram’ whichprovides the weight of each frequency ponent:X1ECWT240。a222。 188。 240。C240。a。 nTs222。222。2: 240。6222。n188。0By inspecting the relative maximum peaks of the obtainedscalogram ECWT(a), the most signi?cant frequency ponents of the signal are detected. From now on, thesefrequency ponents are called ‘CWTidenti?ed frequencies’ of the transient. The CWTidenti?ed frequencies canbe correlated to the propagation phenomena of the faultoriginated waves, traveling along the lines, and to theirre?ections at discontinuity points. For each fault location,some theoretical frequency values are calculated as a function of the length of the path covered by the travelingwaves, of the propagation velocities along the lines andof the type of re?ections. The match between these valuesthe CWTidenti?ed frequencies can provide useful information for the fault location.610A. Borghetti et al. / Electrical Power and Energy Systems 28 (2006) 608–617It is worth noting that the propagation of travelingwaves in multiconductor transmission lines, involves thepresence of di?erent propagation speeds. In this respect,the CWTbased analysis has been carried out separatelyon the various modes present in the voltage transientrecorded at the observation point.Eqs. (7) and (8), written in the frequency domain, summarize the modal transformation, as a way to make diagonal thematrixes given by the products between the impedance andadmittance perunitlength matrixes, namely [Z0][Y0] and[Y0][Z0]. These matrixes are not equal, but have the sameeigenvalues that, squared, form diagonal matrix [c]2.h icase of nonsymmetrical faults. The fault transients areobtained making reference to the distribution system con?guration shown in Fig. 1, modeled by means of the electromagnetic transient program EMTPRV [12,13]. Somedetails and data of the model are given in Appendix.For the case of balanced lines, transformation matrixes[Te] and [Ti] de?ned in (8), are identical and the elementscan be real numbers and they correspond to the Clarke’s(0, a, b) transformation matrix [14]. For the case of unbalanced lines, real matrix can be still inferred by using theprocedure implemented in EMTPbased programs [15]. Inview of the vertical symmetry of the conductor con?guration of the considered overhead line (see Fig. 9 of thed2Vphdx2h i188。 189。Z0189。Y0189。Vph240。7222。Appendix), the simulated transients relevant to balancedand unbalanced lines do not di?er signi?cantly.d2Iphdx2188。 189。Y0189。Z0189。Iph189。Vph 188。 189。Te189。Vm189。Iph 188。 189。Ti189。Imh i240。8222。. Balanced faultsFig. 2 shows the simulated voltage transients at threed2Vmdx2h i188。 189。c2189。Vm240。9222。di?erent observation points of the network of Fig. 1,namely bus 2, bus 3 and bus 4, due to a zeroimpedanced2Imdx2188。 189。c2189。Imthreephase fault at bus 1, ., the terminal end of the mainCoe?cients ciare the propaga