A Method for the Calculation of Frequency-Dependent Transmission-Line Smooth Transformation Matrix

Document Type : Original Article

Authors

Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran

Abstract

Modeling of transmission lines for the analysis of electromagnetic transient behavior requires the calculation of a modal transformation matrix in a wide range of frequencies. In this paper, a comprehensive framework for smooth calculating the transformed matrix is ​​presented. Accordingly, at each frequency, a general set for eigen vectors corresponding to each eigen value is computed. Then, on the basis of a smoothing criterion and an optimization algorithm, at each frequency the set of eigen vectors are chosen arbitrarily, avoiding sudden mutations and smoothing the fitted function for different elements The conversion matrix is​​guaranteed. The proposed algorithm is implemented on an underground cable 3 phases system and overhead line in order to obtain a transformation matrix. The simulation results indicate the effectiveness of the algorithm used to calculate the transformation matrix in a very smooth and precise manner. Also, this algorithm will continue to show significant accuracy in a situation where the frequency sampling rate is reduced, so that it exhibits acceptable tracking capability at low sampling rates.

Keywords


[1]      T. Noda, N. Nagaoka and A. Ametani, “Phase domain modeling of frequency-dependent transmission lines by means of an ARMA model,”IEEE Trans. Power Delivery, vol. 11, pp. 401–411, Jan. 1996.
[2]      H. V. Nguyen, H. W. Dommel and J. R. Marti, “Direct phase-domain modeling of frequency-dependent overhead transmission lines,” IEEE Trans. Power Delivery, vol. 12, pp. 1335–1342, July 1997
[3]      A. Morched, B. Gustavsen and M. Tartibi, “A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables,” IEEE Trans. Power Del., vol. 14, no. 3, pp.1032–1038, Jul. 1999.
[4]      L. Marti, “Simulation of transients in underground cables with frequency-dependent modal transformation matrices,” IEEE Trans.Power Del., vol. 3, no. 3, pp. 1099–1110, Jul. 1988.
[5]      B. Gustavsen and A. Semlyen, “Simulation of transmission line transients using vector fitting and modal decomposition,” IEEE Trans.Power Del., vol. 13, no. 2, pp. 605–614, Apr. 1998.
[6]      T. Kauffmann, I. Kocar and J. Mahseredjian, “New investigations on the method of characteristics for the evaluation of line transients,” Electric Power Systems Research, vol. 160, pp. 243–250, 2018.
[7]      L. M. Wedepohl and S. E. T. Mohamed, “Multiconductor transmission lines. Theory of natural modes and fourier integral applied to transient analysis,”  Electrical Engineers, Proceedings of the Institution of, vol. 116, no. 9, pp. 1553-1563, September 1969.
[8]      احسان دشتیان، مجید اخوت، حمید آرزومند، «تخمین کانال MIMO با استفاده از QRD و الگوریتم وفقی LMS»، مجله مهندسی برق دانشگاه تبریز، مقالات آماده انتشار، 1397.
[9]      T. Noda, “Application of Frequency-Partitioning Fitting to the Phase-Domain Frequency-Dependent Modeling of Underground Cables,” IEEE Transactions on Power Delivery, vol. 31, no. 4, pp. 1776-1777, Aug. 2016.
[10]      H. Ye and K. Strunz, “Multi-Scale and Frequency-Dependent Modeling of Electric Power Transmission Lines,” IEEE Transactions on Power Delivery, vol. 33, no. 1, pp. 32–41, 2018.
[11]      M. Y. Tomasevich and A. C. Lima, “Investigation on the limitation of closed-form expressions for wideband modeling of overhead transmission lines,” Electric Power Systems Research, vol. 130, pp. 113–123, 2016.
[12]      L. M. Wedepohl, H. V. Nguyen and G. D. Irwin, “Frequency-dependent transformation matrices for untransposed transmission lines using Newton-Raphson method,” IEEE Trans. Power Del., vol. 11, no. 3, pp.1538–1546, Jul. 1996.
[13]      T. T. Nguyen and H. Y. Chan, “Evaluation of modal transformation matrices for overhead transmission lines and underground cables by optimization method,” IEEE Trans. Power Del., vol. 17, no. 1, pp. 200–209, Jan. 2002.
[14]      A. I. Chrysochos, T. A. Papadopoulos and G. K. Papagiannis, "Robust Calculation of Frequency-Dependent Transmission-Line Transformation Matrices Using the Levenberg–Marquardt Method," IEEE Trans. Power Del., vol. 29, no. 4, pp. 1621-1629, Aug. 2014.
[15]      T. Noda, “Numerical technique for accurate evaluation of overhead line and underground cable constants,” Inst. Elect. Eng. Jpn. Trans. Elect. Electron. Eng., vol. 3, no. 5, pp. 549–559, 2008.
[16]      S. Fan, Y. Li, X. Li and L. Bi, “A method for the calculation of frequency-dependent transmission line transformation matrices,” IEEE Trans. Power Syst., vol. 24, no. 2, pp. 552–560, May 2009.
[17]      L. M. Wedepohl, “Application of matrix methods to the solution of travelling-wave phenomena in polyphase systems,” Proc. Inst. Elect. Eng., vol. 110, pp. 2200–2212, Dec. 1963.
[18]      A. Hoshmeh and U. Schmidt, “A Full Frequency-Dependent Cable Model for the Calculation of Fast Transients,” Energies, vol. 10, no. 8, p. 1158–1176, Jul. 2017.
[19]      A. Ametani, T. Yoneda, Y. Baba and N. Nagaoka, “An Investigation of Earth-Return Impedance between Overhead and Underground Conductors and Its Approximation,” IEEE Trans. Electromagn. Compat, vol. 51, no. 3, pp. 860-867, Aug. 2009.
[20]      T. Theodoulidis, “Exact solution of pollaczek’s integral for evaluation of earth-return impedance for underground conductors,” IEEE Trans. Electromagn. Compat, vol. 54, no. 4, pp. 806–814, Aug. 2012.
[21]      A .Ametani, T .Ohno and N   .Nagaoka, Cable System Transients:Theory, Modeling and Simulation , Wiley-IEEE Press, pp.550–829, 2015.
[22]      J. P. Bickford, N. Mullineux and J. R. Reed, Computation of Power System Transients. London, U.K.: Peregrinus, 1976.
[23]      T. Noda, “Application of Frequency-Partitioning Fitting to the Phase-Domain Frequency-Dependent Modeling of Overhead Transmission Lines, ”  in IEEE Transactions on Power Delivery, vol. 30, no. 1, pp. 174-183, Feb. 2015.
[24]      عباس کارگر، فهیمه صیاد شهرکی، جعفر سلطانی، «خازن­گذاری بهینه در شبکه توزیع دارای اغتشاش هارمونیکی برای تنظیم ولتاژ و کاهش تلفات با استفاده از PSO»، مجله مهندسی برق دانشگاه تبریز، دوره 41، شماره 1، صفحات 33-43، بهار 1390.
[25]      G. W. Stewart and J. G. Sun, Matrix Perturbation Theory. New York: Academic, 1990.
[26]      K. Singh, “General Vector Spaces, ” in Linear Algebra: Step by Step, Oxford, Oxford University Press, pp. 191-274, 2015.
[27]      J. A. Brandao Faria and J. F. Borges da Silva, “Wave Propagation in Polyphase Transmission Lines a General Solution to Include Cases Where Ordinary Modal Theory Fails, ”  IEEE Power Engineering Review, Vols. PER-6, no. 4, pp. 45-46, 1986.