1 Introduction Parallel double-circuit lines erected on the same tower have advantages such as narrow outgoing corridors, less land occupation, and fast construction speed. Double-circuit lines account for a certain proportion in China's 220kV systems, and about one-third in the Kunming power grid of Yunnan Province. Phase selection and distance measurement for double-circuit lines have their own unique characteristics, and many scholars have made significant contributions to this research [1-4]. References [3, 4] report a single-sided power frequency method for locating cross-line faults in double-circuit lines, and the principles and methods for phase selection and distance measurement of double-circuit lines have been basically established. Theory and practice show that the fault location algorithm using single-sided power frequency measurements to locate short-circuit points on long transmission lines with dual-end power sources cannot guarantee accuracy when the fault is located more than half the line length. The main reasons are: ① transition resistance; ② line distributed capacitance; ③ changes in the operating impedance of the opposite system, etc. With the development of power communication technology, ranging algorithms utilizing information from both ends have been proposed [5-7]. There are two main types of these ranging algorithms: one uses the near-end voltage and current and the opposite-end current power frequency quantities [5], and the other uses the power frequency quantities of both voltage and current [6, 7]. Among them, the ranging algorithm that does not require sampling synchronization or sampling synchronization processing of the data on both ends [7] will have a more promising application prospect. This paper studies the method of using power frequency quantities on both ends for accurate fault location of double-circuit lines on the same tower. [b]2. Parameters and Phase Sequence Transformation of Double-Circuit Lines[/b] The following transformation matrices are used for symmetrically coupled double-circuit lines [img=154,44]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/g1501.gif[/img] (1a) [img=172,46]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/g1502.gif[/img] (1b) [img=288,68]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/g1503.gif[/img] （1c） Wherein, equation (1b) can decouple the symmetrical double loop, and equation (1a) can decouple the positive and negative sequences of the I and II loops of the double loop while still coupling the zero sequence. The phase sequence transformation of electrical quantities using equations (1a) and (1b) is denoted as col［E（p］, p=a，b，c，a′，b′，c′］ col［E（s］, s=0，1，2，0′，1′，2′］ (2a) and col［E（p］, p=a，b，c，a′，b′，c′］ col［E（s］, s=T0，T1，T2，F0，F1，F2］ (2b). Here, abc and a′b′c′ represent the phases of circuit I and circuit II, respectively. 012 and 0′1′2′ represent the zero, positive, and negative sequences corresponding to circuit I and circuit II, respectively. In terms of line parameters, the zero sequences of circuit I and circuit II are still coupled, while the positive and negative sequences are independent of each other. T012 and F012 represent the zero, positive, and negative sequences of the same-sequence and opposite-sequence quantities, respectively. The same-sequence and opposite-sequence quantities of the symmetrical six phases are independent of each other. Let the self-impedance, mutual impedance, and admittance of the two symmetrical six-phase single circuits after the lightning protection wire is removed be zs,m and ys,m respectively, and the mutual impedance and admittance between the two single circuits be z′m and y′s,m respectively. Then the propagation constant and characteristic impedance of each sequence quantity are γs=［（zs-zm）（ys-ym）］1/2 s=1,2,1′,2′，T1，T2，F1，F2 （3a） γT0=［（zs+2zm+3z′m）（ys+2ym+3y′m）］1/2 （3b） γF0=［（zs+2zm-3z′m）（ys+2ym-3y′m）］1/2 （3c） and Zcs=［（zs-zm）/（ys-ym）］1/2 s=1,2,1′,2′，T1，T2，F1，F2 (4a) ZCT0 = [(zs+2zm+3z′m)(ys+2ym+3y′m)]1/2 (4b) ZCF0 = [(zs+2zm-3z′m)(ys+2ym-3y′m)]1/2 (4c) 3. Fault location with non-zero reverse sequence current: First, the transformation between positive sequence (T012) and reverse sequence (F012) is introduced using equation (1b) to determine the fault type. This paper divides the 120 possible short circuits on the double-circuit line into two categories: faults with zero and non-zero reverse sequence current. Fault analysis and calculation show that faults with zero reverse-sequence current are the following same-phase cross-line faults: ① Type AA′ faults not only have zero reverse-sequence current, but also have the same load condition, making them electrically indistinguishable; ② Types AA′-G, BCB′C′, ABCA′B′C′, and ABCA′B′C′-G faults, when the transition impedance on a single same-phase line is equal, have zero reverse-sequence current. Faults with non-zero reverse-sequence current are: ① Arbitrary short circuit on a single circuit; ② Cross-line faults of non-same-phase lines, such as AA′B′-G; ③ Cross-line faults of same-phase lines but with asymmetrical transition impedances on the same-phase lines. Using formula (1b) to calculate the six-sequence currents of T012 and F012, if max［｜Is｜，s=F0,F1,F2］≤Iε, then it is a same-phase cross-line fault with zero reverse-sequence current and symmetrical transition impedance, where Iε is a floating threshold (large power supply side) or a fixed threshold (feeder). The following introduces three algorithms for fault location with non-zero reverse-sequence current. The double-circuit line and its fault same-sequence and reverse-sequence distribution parameter network are shown in Figure 1. As can be seen from the fault analysis, any fault with non-zero reverse-sequence current has an F1 sequence component. The following three algorithms all utilize the F1 sequence component. [img=194,140]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1501.1.gif[/img][img=362,132]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1501.2.gif[/img] Fig. 1 Coupled double-circuit line and superimposed T012 and F012 sequence circuit Algorithm 1 From Figure 1c, the fault location function and location equation for non-zero reverse current can be constructed as MF(x) = |I(s)Mshγsx| - |I(s)Nshγs(lx)| s = F1 (5a) and MF(x) = 0 (5b), where |*| is the modulus operator. When using MF(x) = 0 to locate the AB′ fault, its location function curve is shown in Figure 2. In order to place all the curves on the same coordinate, the values ​​on the vertical axis have been linearized. The curves of other non-zero reverse current faults are similar to those in Figure 2. Observing equation (5), it can be seen that the nonlinearity of the MF(x) function with respect to x is relatively weak. It can be proved that there is no pseudo-root problem for MF(x) = 0 in actual long transmission lines. Because when constructing the distance measurement equation, the voltage on both sides of the fault point is moduloed, and the current on both sides of the input MF(x) = 0 equation does not need to be sampled synchronously or sampled synchronously. [img=291,245]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1502.1.gif[/img][img=298,259]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1502.2.gif[/img] Fig. 2 Curve of fault location MF(x) Algorithm 2 Substituting the approximate relationships shγsx≈γsx and shγs(lx)≈γs(lx) into equation (5), since the order of Z(F012)M,N is always zero, this approximation is equivalent to not considering the distributed capacitance of the line. At this time, the equation MF(x)=0 is simplified to the distance measurement formula x=l/(1+｜IM(s)｜/｜IN(s)｜) s=F1 (6) When calculating the fault AB′ using the fault location approximation formula (6), the fault point xf moves from x=0 to x=l. The absolute error curve of the location within the entire line length range is shown in Figure 3 (Δx=x＊f-xf, x＊f is the location result). Because in equation (6), IM and N(F1) can be expressed as I(F1)M=I(F1)ff1(x,p) and I(F1)N=I(F1)ff2(x,p) respectively, where F1 is the set of line parameters, p={γF1,ZCF1,l}, so the current ratio |IM(F1)|/|IN(F1)| is only a function of the fault distance x and the line parameter p, and is not related to the fault boundary current. Analysis and calculation show that the shape of the approximate difference curve of equation (1b) is shown in Figure 3. Under the premise that the line is determined, the error of the approximate ranging formula (6) in locating any non-zero reverse current fault is determined at each point along the line, and the longer the line, the larger the positioning error amplitude of the approximate formula (6) (Figure 3). Algorithm 2 can be summarized as: ① Approximately calculate the xf*′ value by equation (6); ② Compensation, that is, use (xf*′-Δx) as the positioning conclusion. For faults near the sides of the line or around 0.5l, as well as for medium-length lines, no compensation is required. [img=290,256]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1503.1.gif[/img][img=295,273]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1503.2.gif[/img] Fig.3 Error setting in shunt capacitance current of transmission line Algorithm 3 Define the ratio of the amplitude of F1 sequence current on both sides of the line as ki(x)=｜IM(F1)｜/｜IN(F1)=｜shγF1(lx)｜/ ｜shγF1x｜ （7） As mentioned earlier, for a given line, the ratio of the F1 sequence current components on both sides, ki(x), is only a function of the fault location x and is independent of the fault boundary current. When xf varies on [0, l], ki(x) is monotonic. Figure 4 only shows the variation of the current ratio ki(x) in the intervals [0.05l, 0.1l] and [0.2l, l]. Because ki(x) is independent of the system impedance on both sides and the fault boundary current, once the line is determined, the variation of ki(x) is uniquely determined. Furthermore, ki(x) is monotonic on [0, l]. Therefore, the specific value of the ratio of the F1 sequence current amplitudes on both sides of the line after the fault, ki, can be used to find the corresponding fault location on the ki(x) curve. For actual lines, the line is very likely not strictly symmetrical. By using several short-circuit fault recordings, a curve similar to that shown in Figure 4 can be fitted, achieving accurate fault location. [img=296,472]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1504.gif[/img] Fig. 4 Curve of ki(x) [b]4 Digital Simulation of Fault Location[/b] Ignoring the system and line parameters on both sides, when an ABA′B′-G cross-line fault of {Rf} = (1, 5, 2, 4, 10) Ω occurs at xf = 0.2l = 80.0 km, the voltage waveforms of phases a and b at terminal M are shown in Figure 5. Data sampling between the two sides does not need to be synchronized; the data is taken from the second cycle after the fault, with a sampling frequency of 600 Hz. The digital filtering algorithm is a comprehensive filtering algorithm combining first-order differential and full-wave Fourier algorithms. To examine the adaptability of the method to cross-line faults with asymmetric resistance, only some results of locating cross-line faults with asymmetric filter resistance using Algorithm 1 (denoted as AⅠ) and Algorithm 2 (denoted as AⅡ) are listed in Table 1, and their transition resistances are shown in Table 2. [img=313,403]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1505.gif[/img][font=宋体]　Figure 5. A and B phase voltage waveform at end M during a same-phase inter-circuit ABA′B′-G fault. Table 1. Results of the same-phase inter-circuit fault location tested using transient data. [table][tr][td=2,3][font=SimSun] Assuming Fault [/font][/td][td=3,1][font=SimSun] x[sub][size=2]f [/sub]=80km[/size][/font][/td][td=3,1 ][font=SimSun] x[sub][size=2]f [/sub]=200km[/size] [/font][/td][td=3,1][font=SimSun] x[sub][size=2]f [/sub]=320km[/size][/font] [tr][tr][td=3,1][font=SimSun] {R[sub][size=2]f [/sub]}[/size][/font] [/td][td=3,1][font=SimSun] {R[sub][size=2]f [/sub]}[/size][/font] [/td][td=3,1][font=SimSun] {R[sub][size=2]f [/sub]}[/size][/font] [/td][tr][td][font=SimSun] 1# [/font] [/td][td][font=SimSun] 2# [/font] [/td][td][font=SimSun] 3# [/font] [/td][td][font=SimSun] 1# [/font] [/td][td][font=SimSun] 2# [/font] 3# 1# 2 # 3 # AA′-G x f* /km AⅠ AⅡ 80.06 81.29 79.96 81.16 [/td][td][font=宋体][size=3]79.96 81.16[/size][/font] [/td][td][font=宋体][size=3]199.96 200.00[/size][/font] [/td][td][font=宋体][size=3]200.01 200.00[/size][/font] [/td][td][font=宋体][size=3]199.96 200.01[/size][/font] [/td][td][font=宋体][size=3]319.91 318.71[/size][/font] [/td][td][font=宋体][size=3]320.00 318.82[/size][/font] [/td][td][font=宋体][size=3]320.36 318.84[/size][/font] [/td][/tr][tr][td][font=宋体][size=3]ABA′B′-G x [sub]f[/sub][sup]＊[/sup] /km[/size][/font] [/td][td][font=宋体][size=3]AⅠ AⅡ[/size][/font] [/td][td][font=宋体][size=3]79.96 80.98[/size][/font] [/td][td][font=宋体][size=3]80.16 81.32[/size][/font] [/td][td][font=宋体][size=3]80.06 81.23[/size][/font] [/td][td][font=宋体][size=3]197.96 200.01[/size][/font] [/td][td][font=宋体][size=3]199.96 200.00[/size][/font] [/td][td][font=宋体][size=3]199.96 200.00[/size][/font] [/td][td][font=宋体][size=3]320.36 318.28[/size][/font] [/td][td][font=宋体][size=3]319.71 318.52[/size][/font] [/td][td][font=宋体][size=3]319.96 318.79[/size][/font] [/td][/tr][tr][td][font=宋体][size=3]ABCA′B′C′-G x [sub]f[/sub][sup]＊[/sup] /km[/size][/font] [/td][td][font=宋体][size=3]AⅠ AⅡ[/size][/font] [/td][td][font=宋体][size=3]77.96 79.08[/size][/font] [/td][td][font=宋体][size=3]77.16 77.65[/size][/font] [/td][td][font=宋体][size=3]80.36 81.52[/size][/font] [/td][td][font=宋体][size=3]199.96 200.00[/size][/font] [/td][td][font=宋体][size=3]200.36 200.00[/size][/font] [/td][td][font=宋体][size=3]200.36 200.01[/size][/font] [td][font=SimSun][size=3]321.26 320.04[/size][/font] [td][td][font=SimSun][size=3]320.36 319.27[/size][/font] [td][td][font=SimSun][size=3]319.96 318.76[/size][/font][/td][/tr][/table] [table][tr][td=1,2][font=SimSun] Fault Type [/font] [/td][td=3,1][font=SimSun] {R[sub][size=2]f [/sub]}/Ω[/size][/font] [/td][/tr][tr][td][font=SimSun] 1# [/font] [/td][td][font=SimSun] 2# [/font] 3# AA′-G 1, 5, 10 20, 40, 50 50 , 60, 100 ABA′B′ -G 1 , 5, 2, 4, 10 10, 5, 5, 10, 20 50 , 60, 60 , 50, 100 [/td][/tr][tr][td][font=SimSun] ABCA′B′C′-G [/font][/td][td][font=SimSun] 1, 5, 10, 10 , 5, 1, 10 [/font][/td][td][font=SimSun] 10, 20, 30, 30, 20, 10, 50 [/font][/td][td][font=SimSun] 5. Fault Location with Zero Reverse Sequence Current[/b] As previously discussed, the reverse sequence current is zero only when the transition impedances connected to the same phases are the same. For faults with zero reverse sequence current, the transformation relationship (1a) is used, that is, the distance measurement algorithm is constructed on the positive sequence distributed parameter line corresponding to the single-circuit line of the double circuit. The positive sequence network corresponding to this type of fault is shown in Figure 6. The main positioning function and the combined positioning equation can be constructed as MP(x) = |VMchγx-ZCIMIshγx| - |VNchγ(lx)-ZCINIshγ(lx)| (8a) and MP(x) = 0 (8b). The labels of the positive sequence quantities have been hidden. Similarly, due to the introduction of modulo operation, the data on both sides do not need to be sampled synchronously or processed for sampling synchronization. The positioning function MP(x) curve is shown in Figure 7. It can be proved that for actual long lines, MP(x) = 0 has at most two roots on [0, l]. Starting from x = 0 and x = l, the NR iteration can converge to the two roots, the true and the false, so that the two roots can be easily obtained (if there are two roots). Analysis and calculation show that if MP(x) = 0 has two roots, the difference between the true and the false roots is large. If there are two roots, the false roots are eliminated by the approximate positioning formula introduced below. [img=205,114]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1506.gif[/img][align=left] Fig.6 Positive sequence network[/align][img=278,234]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1507.gif[/img] Fig.7 Curve of MP(x) The positive sequence labels are hidden, and it is assumed that M and N are in the same measurement coordinate, and the positive sequence components are (VM,IM)exp(jδ) and (VN,IN) respectively. Where δ is the phase angle between the respective measurement reference frames at both ends, and neglecting the distributed capacitance of the line, the fault distance can be deduced as [img=349,36]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/g1504.gif[/img] （9a） or [img=342,36]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/g1505.gif[/img] （9b） Where k1=RLRe（IM）-XLIm（IM） k2=RLIm（IM）+XLRe（IM） k3=lRLRe（IN）-lXLIm（IN） k4=lRLIm（IN）+lXLRe（IN） RL=Re（ZC） XL=Im（ZC） [img=210,35]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/g1506.gif[/img][align=left] a=k4Re（VM）-k3Im（VM）-lk2Re（VN）+ lk1Im（VN）+l（k2k3-k1k4） b=-k3Re（VM）-k4Im（VM）-lk1Re（VN）- lk2Im（VN）+l（k1k3-k2k4） c=lk1Im（VN）-lk2Re（VM）+k4Re（VN）- k3Im（VN） Using the above formula to calculate the AA′-G fault, the absolute error curve of the calculation result when xf changes from 0 to l is shown in Figure 8. [/align][img=310,563]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dgjsxb/dgjs99/dgjs9906/image6/1508.gif[/img] Fig.8 Error curve of AA′-G fault location Analysis and calculation show that: ① Under the premise that the impedance of the system on both sides is determined, the longer the line, the larger the amplitude of the absolute error Δx curve of the location calculation result described by formula (9); ② Δx can be expressed as the function Δx=ψ(x,NP), where NP is the set of positive sequence parameters of the system and the line, that is, NP={γ1,ZC1,l,ZM1,ZN1}. Δx is independent of the short-circuit type, the magnitude and nature of the transition impedance. Thus, for long lines, the main location equation MP(x) = 0 is solved using NR iterations starting at x = 0 and x = l. If there are two roots, x(δ) is used as an auxiliary calculation, and the true roots of MP(x) = 0 are removed by the value of x(δ). MP(x) = 0 is denoted as Algorithm 1 (A1), and x(δ) is denoted as Algorithm 2 (A2). The transient simulation results for fault location on a line with l = 400 km are listed in Table 3. Table 3. Results of AA′-G fault location tested using transient data. [table][tr][td=1,3][font=SimSun] Assumed Fault [/font][/td][td=3,1][font=SimSun] x[sub][size=2]f [/sub]=80km[/size][/font][/td][td=3,1][font=SimSun] x[sub][size=2]f [/sub]=200km[/size][/font][/td][td=3,1][font=SimSun] x[sub][size=2]f [/sub]=320km[/size][/font][/tr][tr][td=3,1][font=SimSun] {R[sub][size=2]f [/sub]}[/size][/font][/td][td=3,1][font=SimSun] {R[sub][size=2]f [/sub]}[/size][/font] [/td][td=3,1][font=SimSun] {R[sub][size=2]f [/sub]}[/size][/font] [/td][/tr][tr][td][font=SimSun] 1# [/font] [/td][td][font=SimSun] 2# [/font] [/td][td][font=SimSun] 3# [/font] [/td][td][font=SimSun] 1# [/font] [/td][td][font=SimSun] 2# [/font] [/td][td][font=SimSun] 3# [/font] [/td][td][font=SimSun] 1# [/font] [/td][td][font=SimSun] 2# [/font] 3# A1 xf* /km 78.91 80.37 80.72 199.68 200.24 200.25 319.86 319.86 319.44 A2 xf* /km 82.24 83.30 83.35 199.27 199.55 199.51 [/td][td][font=SimSun] 316.89 [/font] [/td][td][font=SimSun] 316.86 [/font] [/td][td][font=SimSun] 316.67 [/font] [/td][/tr][/table] It is worth pointing out that the positioning algorithm proposed in reference [8] using the "difference between two currents" is essentially using the reverse sequence current. Obviously, for cross-line faults with zero reverse sequence current, the ranging algorithm in reference [8] will fail. [b]6 Conclusion[/b] (1) Possible short circuits on double-circuit lines are divided into two types of faults: zero and non-zero reverse sequence current. The line wave parameters used are all spatial mode parameters, which are different from ground mode parameters, laying the foundation for accurate fault location. The influence of line distributed capacitance on fault ranging accuracy is considered and completely overcome. There is no need for sampling synchronization or sampling synchronization processing between the two data ends, and no real-time communication is required. (2) For fault location with non-zero reverse current, algorithm 1 or 3 can be selected for long lines, and algorithm 2 or 3 can be selected for medium and short lines. (3) For fault location with zero reverse current, the fault location can be directly calculated using the formula x (δ) for medium and short lines. For long lines, the main location equation MP (x) = 0 can be selected, and NR can be iterated 2 to 5 times from both sides. If two roots appear, the pseudo-roots are eliminated by approximate estimation of x (δ). [b]References[/b] 1 Sonam, Ge Yaozhong. A new method for analyzing open-circuit faults in double-circuit lines on the same pole using the six-sequence component composite sequence network method. Automation of Electric Power Systems, 1992, 16(3): 15-21 2 Sonam, Ge Yaozhong, et al. Six-sequence phase selection principle for double-circuit lines on the same pole. Proceedings of the CSEE, 1991, 11(6): 1-9 3 Lu Jiping, Ye Yilin. Accurate distance measurement algorithm for cross-line faults of parallel double-circuit lines erected on the same tower. Proceedings of the CSEE, 1992, 12(6): 18-24 4 Sonam, Ge Yaozhong. Accurate fault location method for double-circuit lines on the same pole. Proceedings of the CSEE, 1992, 12(3): 1-9 5 Dong Xinzhou, Ge Yaozhong. A fault location algorithm for high-voltage transmission lines using electrical quantities at both ends. Automation of Electric Power Systems, 1995, 19(8): 47-53 6 Johns AT, Jumali S. Accurate fault location technique for power transmission lines. IEE Proc., 1990, 137-cc6 7 Novosel D, Hart DG, et al. Unsynchronized two-terminal fault location estimation. IEEE Trans., 1996, PWRD-11: 130～137 8 Nagaswa T, et al. Trans.,1992,PWRD-7(3):1516～1532