【正文】 , for di?erent types of unbalanced faults at various locations, also with nonzero fault velocity equal to the speed of light with those identi?ed from the peaks in Fig. 3. If the CWTanalysis is applied to the voltage transients recorded in a di?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 results of the CWTanalysis at bus 2 for the previous case of a zeroimpedance threephase fault at bus 1. For this observation point three paths are of interest: L3 + L4, with opposite sign re?ections at the fault location and at the bus 2, L1 + L2 + L4, with re?ection at the line terminations having the same sign, and L2 + L4 + L5 with re?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 therefore 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 re?ections at the fault location and at the main feeder sending end and L1 + L5, with re?ection coe?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 re?ections at the fault location (bus 2) and at the main feeder sending end (bus 4), (b) L1 + L2 + L3 and (c) L1 + L5, with re?ections at the line terminations. The CWTanalysis, performed by using the Morlet motherwavelet, is able to detect only the frequencies asso ciated with two paths, namely the ?rst and the third ones, while the frequency peak associated with the second path appears to be hidden by the ?rst peak due to the large ?lter amplitude related to the adopted motherwavelet. The in?uence of the presence of distributed generation has been also investigated. The analysis is repeated for bal anced faults at bus 1 and bus 5 in presence of a generator connected at bus 2 through a transformer. The CWTiden ti?ed frequency values are very similar to those of Tables 1–3, showing that the presence of the generator does not impedances. Fig. 7 illustrates the results of the CWTanalysis of the voltage transients due to a phasetoground fault at bus 1, both for the case of a grounded and ungrounded neutral. Also in this case, the considered paths are those illus trated in Fig. 1. The phase velocity of mode 0 is however signi?cantly lower than speed of light (as shown in Table 9 of Appendix). This velocity is used to evaluate the theo retical frequency values, which, in Table 5, are pared with those identi?ed by the CWTanalysis. For the same system in Fig. 1 and Table 6 shows the results for a phasetoground fault at bus 5. The simulations and the analysis are repeated also by taking into account a fault resistance equal to 10 X, and quite the same results as those shown in Tables 5 and 6 have been obtained. Also the presence of unbalanced loads does not appear to have evident impacts on the results. Table 7 shows the results for the case of a phaseto phase fault, and Table 8 shows the results obtained for a two phasetoground fault. For both cases, two fault loca tions are examined: bus 1 and bus 5. The neutral is consid ered ungrounded. Although some of the results show some limits of the adoption of the Morletwavelet, namely those relevant to fault at the laterals of the work in Fig. 1 (., balanced fault at bus 2), overall a reasonably good match between the theoretical values and the CWTidenti?ed frequencies has been achieved. Such a match encourages the develop ment of a fault location system exploiting this information. Section 4 is devoted to this subject. 4. Measurement system with distributed architecture The described CWTbased algorithm is conceived to be bined with a distributed measurement system. Each unit, located at some suitable busses of the distribution work, is equipped with a GPS synchronization device and is able to acquire both the starting instant of the transient and the relevant waveform. A measurement unit of the faultlocation distributed s。 10 L1 + L2 + L4 2 bi : 240。 di?erent observation points of the work of Fig. 1, namely bus 2, bus 3 and bus 4, due to a zeroimpedance d2Im dx2 188。 189。Ti189。Te189。Z0189。7222。 189。2: 240。C240。 The sum of the squared values of all coe?cients corre sponding to the same scale, which is henceforth called where F(C(a,b)), 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 motherwavelet is a bandpass ?lter function in the frequencydomain, the use of CWT in the frequencydomain allows for the identi?cation of the local features of the signal. According to the Fourier trans form theory, if the center frequency of the motherwavelet W(x) is F0, then the one of W(ax) is F0/a. Therefore, di?er ent scales allows the extraction of di?erent frequencies from the original signal – larger scale values corresponding to lower frequencies – given by the ratio between center fre quency and bandwidth. Opposite to the windowedFourier 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 timewindowed analysis, which is dependent to the values of scale a. As known, the use of CWT, allows the use of arbitrary motherwavelets which must satisfy the ‘a(chǎn)dmissibility condition’: 使用連續(xù)小波變換在配電系統(tǒng)中故障定