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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 [1]. 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 [2], 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 ii Z are the zero sequence self-impedances of the lines, and ij Z (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 12 Z 11 Z 22 Z nn Z 1n Z 2n Z 1 I 2 I n I 1 U 2 U n U n U 2 U 1 U Figure 1. The model of transmission lines with mutual in- ductances. Copyright © 2013 SciRes. EPE J. J. SU ET AL. 1436 sequence current increment i I 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 121111 21 222222 12 33 12 in in i iiiin n nninnnn ZZZ Z I U ZZZZ I U ZZZ Z I U ZZZ Z I U (1) Simplify Equation (1) as: Z IU (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 i u 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, L jj ,1 ,,ij ni iu i u 12 12 RjL 11 11 R jL 22 22 R jL nn nn R jL 11nn RjL 22nn RjL 1 u 2 u n u n u 2 u 1 u 1 i 2 i n i Figure 2. The model of transmission lines with mutual in- ductances by differential method. line’s zero sequence voltage difference, which iii uu u . Equation set of the differential method is described in Equation (3). 12 1 111121212 11 1 12 1 121222222 22 2 12 1112 22 n nn n n nn n nnn n n nnn nnn di di iR LiRL dt dt di iR Lu dt di di iR LiRL dt dt di iR Lu dt di di iR LiRL dt dt di iR Lu dt (3) where (1 i ik ) , , and , , i () i ik (1) i ik(1) i uk() i uk (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 i didt by [(ik 1) ( ii ik 1)]2T s . 1112111 121 1 12222212222 12 12 11 1 22 () () () [( 1)( 1)]2( [( 1)(1)]2 [( 1)( 1)]2 nn nn n nnnnn nnn nn RR RLL L ik RRRikLLL ik RR RLL L ikikTu ik ikT ik ikT 2 ) () () n k uk uk (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. 1437 derivative terms in Equation (3) by integral terms. 1 1 1 1 1 11111111 111 12112111 221 111111 ()() () () ()() () () ()() k k k k k k k k k k t kk t t nnnnk nk t t kk t t nnnnknk t t nnkk t nn n RidtLitit RidtLititu RidtLitit RidtLitit u RidtLitit Ri 1 1 () () k k t nnnknkn t dtLi ti tu 1k k t t 1 1 2 k k t t 1k k t t (5) Trapezoidal rule is used to calculate the integral value approximately. Therefore, Equation (5) is transformed into Equation (6). Where s T 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. 11 11 121 22 12 222 12 11 12111 12 22222 12 (1)()2 (1) ()2 (1) ()2 (1)() (1) () (1) ( s n s n nn nnnns n n nn nn nn ik ikT RR R ik ikTRR R RRRik ikT LLL ik ik LLLikik ik ik LL L 11 22 ) (1) ()2 (1)()2 (1)() 2 s s nns uk ukT uk ukT uk 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 Tline 1 Tl i ne 1 Ib11 Ic1 1 Ua11 Ub11 Uc11 Ia2 1 Ib 2 1 Ic2 1 Ua21 Ub21 Uc21 Ib 1 2 Ic1 2 Ua12 Ub12 Uc12 Ia22 Ib22 Ic22 L2 L1 V A V A V A P+ j Q V A TLine T R=0 1.0 [ ohm] BRK Figure 3. PSCAD simulation model. Manual Entry of Y,Z 0 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% cr r ZZ error Z , where c Z is the calculated value and r Z 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 [8]. 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)]2 ii s 4.2. 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 i didt . 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 -1 0 1 2 3 sampling frequency （kHz） error （ % ） algebraic method without harmonic algebraic method with harmonic differential method without harmonic differential method with harmonic integral method without harmonic integral 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. Owing to the filter characteristic of Fourier algorithm, th e algebr aic method has the highest accuracy; the algebraic method is preferred in actual engineering application. REFERENCES [1] Y. P. Chen, C. X. Zhang and Z. J. Hu, “Zero-seqence Parameter Measurement under Operation of Transmission Line with Mutual Inductance Based on GPS Technol- ogy,” Automation of Electric Power Systerms, Vol. 22, No. 6, 1998, pp. 43-45. [2] Z. J. Hu, Y. P. Chen and C. X. Zhang, “A New Method of Parameters Live Line Measurement of Transmission Lines Based on GPS Technology,” Power System Tech- nology, Vol. 23, No. 9, 1999, pp. 4 -7. [3] C. N. Wang, Z. R. Liang and Z. Y. Xue, “Live Line Measurement of Zero - sequence Parameters for Trans- mission Lines with Mutual Inductance,” Electric Power Automation Equipment, Vol. 28, No. 4, 2008, pp. 87-90. [4] J. M. Wang, “Research and Realization of Live Line Measurement of Zero-sequence Parameters for Transmis- sion Lines with Mutual Inductance,” Wuhan: Wuhan university, 2010 [5] C. N. Wang, “Study and Experiment of Zero-sequence Mutual Inductance Parameter Measurement of Transmis- sion Line,” Beijing: North China Electric Power Univer- sity, 2006. [6] Z. J. Hu, Y. P. Chen and W. Xu, “Principles and Realiza- tion of Live Line Measurement of Parameters of Trans- mission Lines with Mutual Inductance Based on Differ- ential Equations,” Proceedings of the CSEE, Vol. 25, No. 2, 2005, pp. 28-33 [7] N. Hu, Z. J. Hu and G. Zheng, “Live Line Parameters Measurement of Transmission Lines with Mutual Induc- tance Based on GPS Technology and Integral Equations,” Power System Technology, Vol. 29, No. 16, 2005, pp. 76-79. [8] C. Huang, “Study of Zero-sequence Mutual Inductance Parameter Measurement of Overhead Line,” Nanjing: Nanjing University of Science and Technology, 2010. [9] J. Q. Guo, Z. J. Hu and J. X. Guo, “Study on Ze- ro-sequence Distributed Parameter Live Line Measure- ment of Transmission Lines with Mutual Inductance,” High Voltage Apparatus, Vol. 41, No. 6, 2005, pp. 425-427. [10] P. Mao, C. X. Zhang and Y. Y. Wang, “A Capacitive Current Compensation Algorithm on Zero-sequence Pa- rameters Live Line Measurement of Transmission Lines,” Automation of Electric Power Systems, Vol. 34, No. 17, 2010, pp. 55-58. [11] Z. J. Hu, “New Method of Live Line Measuring the Pa- rameters of T-Connection Transmission Lines with Mu- tual Inductance,” 2009 IEEE Power and Energy Society General Meeting, PES '09, 2009 IEEE Power and Energy Society General Meeting, PES '09,2009. Copyright © 2013 SciRes. EPE |