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1、On the use of continuous-wavelet transform for fault location in distribution power systems A. Borghetti a, S. Corsib, C.A. Nuccia,*, M. Paolonea, L. Perettoa, R. Tinarellia a Department of Electrical Engineering, University of Bologna, viale Risorgimento, 2, 40136 Bologna, Italy b CESI, Milan, Ital
2、y Received 31 March 2006; accepted 31 March 2006 Abstract The paper illustrates a procedure based on the continuous-wavelet transform (CWT) for the analysis of voltage transients due to line faults, and discusses its application to fault location in power distribution systems. The analysis carried o
3、ut shows that correlation exists between typical frequencies of the CWT-transformed signals and specifi c paths in the network covered by the traveling waves originated by the fault. The paper presents a procedure for determining fault location in MV distribution systems, which exploits the above-me
4、n- tioned correlation. The MV distribution system analysed in the paper is accurately represented by means of an EMTP model; various fault types and network characteristics are examined. The paper presents also the basic concepts of a measurement and fault location prototype system with distributed
5、architecture. ? 2006 Elsevier Ltd. All rights reserved. Keywords: Fault location; Power distribution systems; Continuous-wavelet transform; Electromagnetic transients; Distributed measurement systems 1. Introduction Fault location in MV distribution network is a research topic that is receiving incr
6、eased attention in recent years, due both to the most severe power quality requirements and to the availability of improved measurement and mon- itoring systems. In addition, the increasing installation of distributed generation resources in the network requires the overhaul of traditional procedure
7、s based on automatic switching systems. The most promising approach for the problem of inter- est appears to be the application of appropriate signal pro- cessingtechniquestothevoltage/currenttransients produced by short circuit events and recorded at one or more locations in the distribution system
8、. Recent contributions to the subject are based on the use of the wavelet transform (e.g., 14), usually adopting the discrete-wavelet transform (DWT), due to its straightfor- ward implementation and the reduced computational time it requires. In this paper, use is made of the continuous-wavelet tran
9、sform (CWT) algorithm. As known, compared to the DWT algorithm, the CWT one allows performing a more detailed and continuous analysis of the spectrum energy of the fault transient. Such a feature is used to detect indi- vidual frequencies that characterize the voltage transients generated by the fau
10、lt. These frequencies can be used for inferring the location of the fault, being the network topol- ogy, the wave propagation velocity along the lines and the fault type known. The proposed CWT-based fault location procedure is conceived to be combined with a measurement system aimed at acquiring bo
11、th the starting time of the transient and the relevant waveforms. Thepaperisstructuredasfollows.Section3 introduces the proposed correlation between the results of the CWT-analysis and specifi c paths along the net- work covered by the traveling waves originated by the fault. 0142-0615/$ - see front
12、 matter ? 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2006.03.001 * Corresponding author. Tel.: +39 51 2093479; fax: +39 51 2093470. E-mail address: carloalberto.nucciunibo.it (C.A. Nucci). Electrical Power and Energy Systems 28 (2006) 608617 Section 4 presents the application to a
13、distribution sys- tem for both the case of symmetrical faults and non-sym- metrical ones. It also presents the results obtained for diff erent neutral grounding characteristics and fault loca- tions. The CWT-based procedure is applied in such a sec- tion to computer simulation results obtained with
14、a detailed EMTP (electromagnetic transient program) model of the distribution system, whose characteristics and data are reported in Appendix. Section 5 describes the basic characteristics of the earlier mentioned measurement system with distributed architec- ture for the acquisition of voltage tran
15、sients. The conclusions summarize the results obtained with the proposed approach and identify the main aspects requiring additional research eff orts. 2. Fault location information provided by continuous-wavelet transform The CWT of a signal s(t) is the integral of the product between s(t) and the
16、so-called daughter-wavelets, which are time translated and scale expanded/compressed ver- sions of a function having fi nite energy w(t), called mother-wavelet. This process, equivalent to a scalar prod- uct, produces wavelet coeffi cients C(a,b), which can be seen as similarity indexes between the
17、signal and the so-called daughter-wavelet located at position b (time shifting factor) and positive scale a: Ca;b Z 1 ?1 st 1 ffi ffiffi a p w? t ? b a ? dt1 where*denotes complex conjugation. Eq. (1) can be expressed also in frequency domain (e.g., 5): FCa;b ffi ffiffi a p W?a ? xSx2 where F(C(a,b)
18、, S(x) and W(x) are the Fourier transforms of C(a,b), s(t) and w(t), respectively. Eq. (2) shows that if the mother-wavelet is a band-pass fi lter function in the frequency-domain, the use of CWT in the frequency-domain allows for the identifi cation of the local features of the signal. According to
19、 the Fourier trans- form theory, if the center frequency of the mother-wavelet W(x) is F0, then the one of W(ax) is F0 /a. Therefore, diff er- ent scales allows the extraction of diff erent frequencies from the original signal larger scale values corresponding to lower frequencies given by the ratio
20、 between center fre- quency and bandwidth. Opposite to the windowed-Fourier analysis where the frequency resolution is constant and depends on the width of the chosen window, in the wavelet approach the width of the window varies as a function of a, thus allowing a kind of time-windowed analysis, wh
21、ich is dependent to the values of scale a. As known, the use of CWT, allows the use of arbitrary mother-wavelets which must satisfy the admissibility condition: Ch Z 1 ?1 jWxj x 2 dx 13 Eq. (3) is satisfi ed by the two following conditions: mean value of w(t) equal to zero; fast decrease to zero of
22、w(t) for t ! 1. Provided that the mother-wavelet satisfi es specifi c condi- tions, in particular the orthogonality one, the signal can also be reconstructed from the transform coeffi cients. Several mother-wavelet has been used in the literature (e.g., 611), in this paper, the so-called Morlet-wave
23、let is chosen as mother one w(t): wt e?t 2=2ej2pF0t: 4 Unlike DWT, CWT can operate at any scale, specifi cally from that of the original signal up to some maximum scale. CWT is also continuous in terms of shifting: during com- putation, the analyzing wavelet is shifted smoothly over the full domain
24、of the analyzed function. The CWT-analysis is performed in time domain on the voltage transients recorded after the fault in a bus of the distribution network. The analyzed part of the transient recorded signal s(t), which can correspond to a voltage or current fault- transient, has a limited durati
25、on (few milliseconds) corre- sponding to the product between the sampling time Ts and the number of samples N. The numerical implementa- tion of the CWT to signal s(t) is a matrix C(a,b) defi ned as follows: Ca;iTs Ts 1 ffi ffi ffi ffi ffiffi jaj p X N?1 n0 w? n ? iTs a ? snTs; i 0;1;.;N:5 The sum o
26、f the squared values of all coeffi cients corre- sponding to the same scale, which is henceforth called CWT-signal energy ECWT (a), identifi es a scalogram which provides the weight of each frequency component: ECWTa X N?1 n0 Ca;nTs2:6 By inspecting the relative maximum peaks of the obtained scalogr
27、am ECWT (a), the most signifi cant frequency compo- nents of the signal are detected. From now on, these frequency components are called CWT-identifi ed frequen- cies of the transient. The CWT-identifi ed frequencies can be correlated to the propagation phenomena of the fault- originated waves, trav
28、eling along the lines, and to their refl ections at discontinuity points. For each fault location, some theoretical frequency values are calculated as a func- tion of the length of the path covered by the traveling waves, of the propagation velocities along the lines and of the type of refl ections.
29、 The match between these values the CWT-identifi ed frequencies can provide useful infor- mation for the fault location. A. Borghetti et al. / Electrical Power and Energy Systems 28 (2006) 608617609 It is worth noting that the propagation of traveling waves in multiconductor transmission lines, invo
30、lves the presence of diff erent propagation speeds. In this respect, the CWT-based analysis has been carried out separately on the various modes present in the voltage transient recorded at the observation point. Eqs.(7)and(8),writteninthefrequencydomain,summa- rizethemodaltransformation,asawaytomak
31、ediagonalthe matrixes given by the products between the impedance and admittance per-unit-length matrixes, namely Z0Y0 and Y0Z0. These matrixes are not equal, but have the same eigenvalues that, squared, form diagonal matrix c2. d2Vph dx2 hi Z0?Y 0?Vph? d2Iph dx2 hi Y 0?Z0?Iph? 7 V ph? T e?V m? Iph?
32、 Ti?Im? 8 d2Vm dx2 hi c?2V m? d2Im dx2 hi c?2Im? 9 Coeffi cients ciare the propagation constant of mode i. They are complex numbers ci= ai+ jbiwhere aiis the attenuation constant and biis the phase constant of mode i. The phase velocity of mode i is given by vi x bi :10 The columns of transformation
33、 matrixes Te and Ti, that make diagonal matrix Z0Y0 and matrix Y0Z0, respec- tively, are given by corresponding linear independent eigenvectors. 3. Application of the CWT-based fault location procedure The proposed method is fi rstly applied to the simple case of a balanced (three-phase) fault and t
34、hen extended to the case of non-symmetrical faults. The fault transients are obtained making reference to the distribution system con- fi guration shown in Fig. 1, modeled by means of the elec- tromagnetic transient program EMTP-RV 12,13. Some details and data of the model are given in Appendix. For
35、 the case of balanced lines, transformation matrixes Te and Ti defi ned in (8), are identical and the elements can be real numbers and they correspond to the Clarkes (0,a,b) transformation matrix 14. For the case of unbal- anced lines, real matrix can be still inferred by using the procedure impleme
36、nted in EMTP-based programs 15. In view of the vertical symmetry of the conductor confi gura- tion of the considered overhead line (see Fig. 9 of the Appendix), the simulated transients relevant to balanced and unbalanced lines do not diff er signifi cantly. 3.1. Balanced faults Fig. 2 shows the sim
37、ulated voltage transients at three diff erent observation points of the network of Fig. 1, namely bus 2, bus 3 and bus 4, due to a zero-impedance three-phase fault at bus 1, i.e., the terminal end of the main feeder. Fig. 1 also illustrates six paths covered by traveling waves originated by a fault
38、at bus 1. The traveling waves are refl ected at the line terminations and at the fault loca- tion. Paths with partial refl ections at the point where more lines converged are here disregarded. Only three paths (namely paths 1, 2 and 3) reach the observation point, assumed at bus 4, i.e., the sending
39、 end of the main feeder. It is possible to correlate each path to characteristic fre- quencies of the fault transient recorded at the observation point by means of the following remarks: path 1 is associ- ated to a period given by a traveling time equal to four times L1 + L2 + L3 divided by the prop
40、agation speed, as the traveling wave experience refl ections of opposite sign at the fault location (bus 1) and at the sending end of the main feeder (bus 4). For paths 2 and 3, the associated 150 kV L3_5km L4_2km 12 150/20 Tr_1_20 MVA BUS1 Load BUS2 L1_2kmL2_3km L5_1km BUS3 Path 1 Path 2 Path 3 Pat
41、h 4 Path 5 BUS4 Load Load BUS5 Fig. 1. Power distribution network confi guration and paths covered by traveling waves caused by a fault at bus 1. 610A. Borghetti et al. / Electrical Power and Energy Systems 28 (2006) 608617 periods are given by the traveling time relevant to the dou- ble path length
42、s (L1 + L2 + L4 and L1 + L5, respec- tively), as the traveling wave is refl ected at the line terminations. Fig. 3 presents the results of the CWT-analysis of the voltage transient of Fig. 2 at the observation point (bus 4). Table 1 compares the results inferred theoretically by assuming, in a fi rs
43、t approximation, the traveling wave -20 -15 -10 -5 0 5 10 15 20 0.0000.0050.0100.0150.0200.0250.030 Time (s) Phase voltage (kV) bus 4 bus 1 bus 2 bus 3 -20 -15 -10 -5 0 5 10 15 20 0.01490.01500.01510.01520.01530.0154 Time (s) Phase voltage (kV) bus 4 bus 1 bus 2 bus 3 (a) (b) Fig. 2. Voltage transie
44、nts on a phase due a three-phase fault at bus 1 as observed at three diff erent nodes (bus 2, bus 3 and bus 4), of the power distribution network shown in Fig. 1: (a) general behavior and (b) detailed view. 0.01 0.10 1.00 0102030405060 Frequency (kHz) CWT signal energy (p.u.) Fig. 3. Results of the
45、CWT-analysis of the voltage transient of Fig. 2 at bus 4. The values are in per unit with respect to the maximum (1.25 1012V2s). A. Borghetti et al. / Electrical Power and Energy Systems 28 (2006) 608617611 velocity equal to the speed of light with those identifi ed from the peaks in Fig. 3. If the
46、CWT-analysis is applied to the voltage transients recorded in a diff erent observation point, in other words we are considering a measurement system with distributed architecture (see Section 5), it is possible to increase the information relevant to the fault location. Fig. 4 and Table 2 show the r
47、esults of the CWT-analysis at bus 2 for the previous case of a zero-impedance three-phase fault at bus 1. For this observation point three paths are of interest: L3 + L4, with opposite sign refl ections at the fault location and at the bus 2, L1 + L2 + L4, with refl ection at the line terminations h
48、aving the same sign, and L2 + L4 + L5 with refl ection at the line terminations having the same sign. As it can be seen, by joining the information provided by this observation point with those of bus 4, two fault locations can be obtained and an increase of the reliability of the procedure therefor
49、e achieved. Fig. 5 and Table 3 show the results for the case of a bal- anced fault at bus 5. In this case only two paths are of interest: L1 + L2, with opposite sign refl ections at the fault location and at the main feeder sending end and L1 + L5, with refl ection coeffi cient of the same sign at the line terminations. Fig. 6 and Table 4 show the results for the case of a bal- anced fault at bus 2, which is the termination of a lateral. In this case three paths are of interest: (a) L1 + L2 + L4, with opposite sign refl ections at the fault location (bus 2) andatt