 Energy and Power Engineering, 2013, 5, 1435-1439 doi:10.4236/epe.2013.54B272 Published Online July 2013 (http://www.scirp.org/journal/epe) Simulation on Calculation Accuracy of Three Methods for Live Line Measuring the Parameters of Transmission Lines with Mutual Inductance Jianjun Su1, Ronghua Zhang1, Demin Cui1, Yongqiang Chai1, Xiaobo Li1, Chengxue Zhang2, Peiyan Li2, Zhijian Hu2, Yingying Hu2 1Dezhou Power Supply Company, Shandong Electric Power Group Co., Dezhou, China 2School of Electrical Engineering, Wuhan University, Wuhan, China Email: cxzhang@whu.edu.cn Received April, 2013 ABSTRACT Live line measurement methods can reduce the loss of power outages and eliminate interference. There are three live line measurement methods including integral method, differential method and algebraic method. A simulation model of two coupled parallel transmission lines spanning on the same towers is built in PSCAD and the calculation errors of these three methods are compared with different sampling frequencies by using of Matlab. The effect of harmonic on calculation is also involved. The simulation results indicate that harmonic has the least effect on the algebraic method which provides stable result and small error. Keywords: Lines with Mutual Inductance; Zero Sequence Parameters; Live Line Measurement; Algebraic Method; Differential Method; Integral Method 1. Introduction With the development of power system and limitation of transmission line corridor, the number of lines with mu-tual inductance increases. Zero sequence parameters of the lines, which include zero sequence self-impedance and mutual impedance, are important basis of relay set-ting so that the parameters’ precision has a significant effect on power system’s safe operation . These pa-rameters are mainly influenced by earthling resistance rate. Chinese relay rules specify that zero sequence pa-rameters of lines belong to 110kV and higher voltage levels must be measured [2, 3]. In the methods of live line measurement , there are two approaches to avoid the disadvantage that all the lines to be measured should be shut down? First, shut down one of the lines and add an external power source. Second, generate big zero se-quenc e current in th e way that open one phase bre aker of an operating line (about 0.5 seconds) by the protective relay, re-close the phase breaker automatically to restore normal operation. An over determined equation set used for calculating the parameters are obtained under differ-ent measurement modes. The set is solved by using least square method. There are three live line measurement methods, including integral method, differential method and algebraic method [4-8]. Data that algebraic method needs is sampled in a period, while several successive sampling points are needed by integral method and dif-ferential method [9-11]. This paper simulates all these three methods in different sampling frequencies, with and without harmonic, and analyses the measurement errors. The conclusion can help to choose a proper meas-urement method. 2. The Three Measurement Methods 2.1. Algebraic Method The model of n transmission lines with mutual induc-tance is shown in Figure 1. Where iiZ are the zero sequence self-impedances of the lines, and ijZ (ij) are the mutual impedances. While the zero sequence current increment is generated on a line, the other lines coupled with it will induct zero 12Z11Z22ZnnZ1nZ2nZ1I2InI1U2UnUnU2U1U Figure 1. The model of transmission lines with mutual in-ductances. Copyright © 2013 SciRes. EPE J. J. SU ET AL. 1436 sequence current increment iI and zero sequence volt-age increment i. The voltage-current characteristic of the lines with mutual inductance is described in Equa-tion (1). U11 12111121 22222212 3312inini iiiinn nninnnnZZZ ZIUZZZZIUZZZ ZIUZZZ ZIU (1) Simplify Equation (1) as: ZIU (2) where Z is the zero sequence impedance matrix, I and are the increment vector of zero sequence currents and voltages of all lines. UThe increments can be produced by adding large enough current on a shutdown line while the other lines are on operation. Different equations produced by dif-ferent measurement modes form the over determined equation set. The set is solved through least square method. The algebraic method excludes the influence of zero sequence voltage and current existed in the lines by using increment of voltage and current. The algebraic method needs at least half period sampling points. The algebraic method’s accuracy increases by eliminating harmonic through the Fourier method. 2.2. Differential Method The model of n transmission lines with mutual induc-tance is shown in Figure 2. Where ii and ii are the zero sequence self-resis- tance and self-inductance of the -th line, ij and ij are the zero sequence mutual resistance and inductance between the i-th and the -th line (), i is the instantaneous value of the i-th line’s zero se-quence current, i and iu are the instantaneous val-ues of zero sequence voltage of the i-th line’s head and end separately, is The instantaneous value of the i-th R Li R,2,Ljj,1 ,,ij niiuiu 12 12RjL11 11RjL22 22RjLnn nnRjL11nnRjL22nnRjL1u2ununu2u1u1i2ini Figure 2. The model of transmission lines with mutual in-ductances by differential method. line’s zero sequence voltage difference, which iiiuu u. Equation set of the differential method is described in Equation (3). 121 11112121211 1121 12122222222 2121112 22nnn nnnn nnnn nnnnn nnndi diiR LiRLdt dtdiiR Ludtdi diiR LiRLdt dtdiiR Ludtdi diiR LiRLdt dtdiiR Ludt    (3) where (1iik ), , and , , i()iik (1)iik(1)iuk()iuk(uk 1) are separately the zero sequence current and voltage of three successive sampling points. Equation (4) is the matrix form of Equation (3), and is discretized in the way of replacing the derivative terms ididt by [(ik 1) (iiik 1)]2Ts . 1112111 12111222221222212 1211 122()()()[( 1)( 1)]2([( 1)(1)]2[( 1)( 1)]2nnnnnnnnnn nnnnnRR RLL LikRRRikLLLikRR RLL LikikTuik ikTik ikT         2)()()nkukuk (4) An equation can be achieved with any three successive sampling points. Parameters of the lines can be solved from the over determined equation set obtained through different measurement modes. For only three sampling points needed in the differen-tial method, much more equations can be obtained by sampling a series successive points. Different equation sets can be obtained by sampling different series of points. The accuracy of differential method can be en-hanced by averaging the results solved from these sets. 2.3. Integral Method Equation (5) is the integral equation set of the live line measurement. It is formed in the way of replacing the Copyright © 2013 SciRes. EPE J. J. SU ET AL. 1437derivative terms in Equation (3) by integral terms. 111111111111111112112111221111111()()() ()()()() ()()()kkkkkkkkkktkkttnnnnk nkttkkttnnnnknkttnnkktnn nRidtLititRidtLitituRidtLititRidtLitit uRidtLititRi  11() ()kktnnnknkntdtLi ti tu1kktt112kktt1kktt (5) Trapezoidal rule is used to calculate the integral value approximately. Therefore, Equation (5) is transformed into Equation (6). Where sT is the sampling period? Only two sampling points are needed. Much more equations can be obtained by sampling a series of successive points. Therefore, the accuracy of integral method will be enhanced. 1111 1212212 2221211 1211112 2222212(1)()2(1) ()2(1) ()2(1)()(1) ()(1) (snsnnn nnnnsnnnnnn nnik ikTRR Rik ikTRR RRRRik ikTLLL ik ikLLLikikik ikLL L 1122)(1) ()2(1)()2(1)() 2ssnnsuk ukTuk ukTuk ukT (6) The influence of distributed capacitance is ignored in all these three methods. 3. PSCAD Simulation Model A simulation model built in PSCAD is shown in Figure 3. In the model, there are two coupled parallel transmis-sion lines spanning on the same towers. The lengths of the lines are both 50 km. All the lin es are shut down and connected with an external zero sequence power sources in turn where L1 is the line that operates normally. The head end is connected with a 110 kV three-phase power source. The tail end is connected with 50 MW active load and 10 Mvar reactive load. Tail end of L2 is three-phase connected and grounding. L2’s head end is three-phase connected and an external voltage source is applied with. PSCAD describes the line’s characteristic in RLC mode. Transmission line is represented by the Bergeron model which separates the line into several distributed  type modules. This model assumes that the line’s self-impedance and mutual impedance per unit length is constant and frequency-independent. The parameters per unit length in RLC mode are shown in Figure 4. The reference values of zero sequence impedances are obtained according to the input parameters. The self- impedances of L1 and L2 are 8.479+ j66.385, and their module values are 66.920 . The mutual imped-ance between L1 and L2 is 6.750+ j34.500, and its module value is 35.154. 4. Calculation Result and Error Analysis In the PSCAD model, an external zero sequence power source is connected to the shutdown line. There are two types of the source. Type 1 only outputs fundamental voltage while type 2 outputs both fundamental and har-monic voltage. This section illustrates the influence and analyzes the errors of the both types. 4.1. Type 1 The output voltage of type 1 is 1 kV. Simulation lasts 0.5 s. Data sampling begins at 0.4 s. Data of a whole period is used by algebraic method. Several successive sampling points are used by differential method and integral method separately. The results of differen tial method and integral method are achieved in the way of averaging the measurement results. Table 1 shows the lines’ self-im- pedance and mutual impedance calculated through three Tline1Tl i ne1Ib11Ic1 1Ua11Ub11Uc11 Ia2 1Ib 2 1Ic2 1Ua21Ub21Uc21Ib 1 2Ic1 2Ua12Ub12Uc12Ia22Ib22Ic22L2L1VAVAVAP+jQVATLineTR=01.0 [ ohm]BRK Figure 3. PSCAD simulation model. Manual Entry of Y,Z0 Se quen c e R:0 Se quen c e Mutua l R:+ve S equence R:0.36294e-4 [ ohm/m]0 Se quen c e XL:0 Se quen c e Mutua l XL:+ve S equence XC :0 Se quen c e XC:0 Se quen c e Mutua l XC:0.5031e-3 [ohm/m]302. 151 [Mohm*m]0. 169 58e- 3 [oh m/m]0. 132 77e- 2 [oh m/m]1590. 33 [Mohm*m]0.135e-3 [ohm/m]0.069e-2 [ohm/m]5056.0 [Mohm*m]+ve S equence XL : Figure 4. The reference values of the lines’ parameters. Copyright © 2013 SciRes. EPE J. J. SU ET AL. Copyright © 2013 SciRes. EPE 1438 methods and their errors under different sampling fre-quencies. Errors are calculated by algebraic method. Errors of the other two methods dete-riorate apparently compared with the ones without har-monic, and get bigger as the sampling frequency de-creases. Curves of errors changing with sampling fre-quency are shown in Figure 5. 100%crrZZerror Z, where cZ is the calculated value and rZ is the refer-ence value. 4.3. Error Analysis The simulation model contains two 50 km lines. All the three methods ignore the influence of distributed capaci-tance. Therefore, the errors caused by distributed capaci-tance are contained in the results . The effect of distribution capacitance is included in errors. Table 1 indicates that the errors of algebraic method are the smallest. Errors of the other two methods get bigger as the sampling frequency decreases. Curves of errors changing with sampling frequency are shown in Figure 5. The algebraic method utilizes the data of a period. Calculated after Fourier filtering, the algebraic method is not affected by harmonic. Therefore, it has the highest accuracy. And its error gets bigger as the sampling fre-quency decreases. In differential method, principle error exits due to using [( 1)( 1)]2ii s4.2. Type 2 The voltage output by Type 2 contains 8% 3rd, 5% 5th and 5% 7th harmonic. Table 2 shows the lines’ zero se-quence parameters calculated with the three methods and their errors under different sampling frequencies. ik ikT to approxi-mate ididt . In integral method, principle error exits due to using trapezoid area to approximate integral value. Errors of the two methods both increase as the sampling freque n c y d e c r e a s e s . Table 2 indicates that harmonic has no influence on Table 1. The calculated values with power source type 1. Sampling Fre quency 1 kHz 2 kHz 5 kHz 10 kHz Measurement Method Zero Sequence Impedance() |Z| Error (%)|Z| Error (%)|Z| Error (%) |Z| Error (%)Self-Impedance 66.972 0.071 66.972 0.071 66.970 0.068 66.970 0.068 Algebraic Method Mutual Impedance 35.179 0.072 35.180 0.072 35.180 0.072 35.180 0.072 Self-Impedance 68.171 1.725 67.237 0.468 67.016 0.138 66.985 0.090 Differential Method Mutual Impedance 35.734 1.650 35.333 0.510 35.212 0.168 35.185 0.075 Self-Impedance 66.427 -0.742 66.832 -0.137 66.975 0.076 66.974 0.074 Integral Method Mutual Impedance 34.900 -0.718 35.1175 -0.104 35.181 0.076 35.181 0.075 Table 2. The calculated values with power source type 2. Sampling Fre quency 1 kHz 2 kHz 5 kHz 10 kHz Measurement Method Zero Sequence Impedance() |Z| Error (%)|Z| Error (%)|Z| Error (%) |Z| Error (%)Self-Impedance 66.972 0.072 66.972 0.071 66.970 0.068 66.970 0.068 Algebraic Method Mutual Impedance 35.179 0.072 35.180 0.072 35.180 0.072 35.180 0.072 Self-Impedance 68.548 2.427 67.536 0.914 67.116 0.286 67.018 0.141 Differential Method Mutual Impedance 36.008 2.430 35.438 0.809 35.247 0.265 35.213 0.168 Self-Impedance 66.164 -1.136 66.733 -0.285 66.987 0.094 66.986 0.092 Integral Method Mutual Impedance 34.800 -1.006 35.083 -0.204 35.183 0.081 35.182 0.080 J. J. SU ET AL. 1439 12345678910-2-10123sampling frequency （kHz）error（%） algebraic method without harmonicalgebraic method with harmonicdifferential method without harmonicdifferential method with harmonicintegral method without harmonicintegral method with harmonic Figure 5. Curves of errors changing with sampling fre-quency. As the lines is short and the results of both differential and integral methods are achieved in the way of averag-ing three results of measurement, the errors of all the three methods are less than 0.5% in 5 kHz sampling fre-quency. The algebraic method is the most accurate one. 5. Conclusions A simulation model of two double-circuit lines spanning on the same towers is built in PSCAD. The zero se-quence self-impedance and mutual impedance of the lines are calculated through the algebraic method, dif-ferential method and integral method. Errors of these methods are analyzed in two conditions that the external power sour ce with or w ithout harmonics. Prin ciple errors exist due to the approximate calculation in the differen-tial and integral methods. Therefore, errors increase along with the decreasing of sampling frequency. More-over, errors rise when harmonic is involved. 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