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Finally, in Fig. 9 the differences between FEM and numerical integration in the results for the magnitude of the self impedance are shown. The horizontal SC cable arrangement in Fig. 2 was examined for the earth models IV 50% 40% case Ι case ΙΙΙ case ΙΙ case ΙV CONSTANTS/PARAMETERS support routine, due to the assumptions implemented in the latter calculation procedure. Application of FEM in cases of underground cables in a two-layered earth showed differences 15% - 40% from those obtained with the assumption of homogeneous earth. These differences cannot be disregarded. They justify the need for the development of a mathematical model for multi-layered earth in the case of underground conductors as well as methods for its numerical evaluation. case V case VI difference(%) 30% 20% 10% 0% 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 frequency(Hz) Fig. 8: Two-layered earth case. Differences in the magnitude of the mutual impedance. and VI. In these models the earth resistivities of the first and second layer have their maximum values. FEM results are compared to those corresponding for homogeneous earth having the resistivity of either the first or the second layer. difference (%) 50% 40% 30% 20% 10% caseIV, ρ=ρ1 case VI,ρ=ρ1 case IV, ρ=ρ2 case VI,ρ=ρ2 0% 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 frequency (Hz) Fig. 9: Two-layered earth case. Differences in the magnitude of the self impedances. VI. CONCLUSIONS The problem of the calculation of the earth return impedances of underground SC power cables was addressed in this contribution. The available approaches for the numerical evaluation of the infinite integrals and the corresponding assumptions and limitations are reported. A novel numerical integration scheme, based on proper combinations of integration methods, is proposed. The scheme proved to be numerically stable and efficient in all examined cases and lead to results whose validity was justified using a proper FEM formulation. These results showed differences 10%-25% against those computed from EMTP CABLE VII. REFERENCES [1] J. R. Carson, ‘Wave propagation in overhead wires with ground return’, Bell Syst. Tech. J., Nr. 5, pp.539-554, 1926. [2] F. Pollaczek, ‘Ueber das Feld einer unendlich langen wechselstromflossenen Einfachleitung’, Elektrische Nachrichtentechnik vol.3, pp. 339-359, 1926. [3] L. M. Wedepohl, D. J. Wilcox, ‘ Transient analysis of underground power-transmission systems. System model and wave propagation characteristics’, Proc. IEE, vol. 120, No. 2, pp. 253-260, 1973. [4] A. Ametani, ‘ A general formulation of impedance and admittance of cables’, IEEE Trans. on P.A.S , vol. PAS-99, No 3, pp. 902-910, 1980. [5] H. W. Dommel, ‘Electromagnetic Transients Program Reference Manual’, Boneville Power Administration, Portlant OR, 1986. [6] O. Saad, G. Gaba, M. Giroux, ‘ A closed-form approximation for ground return impedance of underground cables’, IEEE Trans. on Power Delivery, vol. PWRD-11, No 3, pp. 1536-1545, 1996. [7] T. T. Nguen, ‘ Earth return path impedances of underground cables – Pt.1 Numerical integration of infinite integrals, Pt. 2 Evaluations using neural networks’, Proc. IEE – Generation, Transm., Distr., vol. 145, No 6, pp. 621-633, 1998. [8] E. D. Sunde, ‘ Earth Conduction effects in transmission systems 2nd Edition’, Dover Publications, 1968, p. 113. [9] M. Nakagawa, A. Ametani, K. Iwamoto, ‘Further studies on wave propagation in overhead transmission lines with earth return: Impedance of stratified earth’, Proc. IEE, vol. 120, No 12, pp.1521-1528, 1973 [10] M. Nakagawa, K. Iwamoto, ‘ Earth return impedance for the multi-layer case’, IEEE Trans. on P.A.S , vol. PAS-95, No 2, pp. 671 - 676, 1976 [11] S. Cristina, M. Feliziani, ‘ A finite element technique for multiconductor cable parameters calculation’, IEEE Trans. on Magnetics, vol. 25, No. 4, pp. 2986-2988, 1989. [12] Y. Yin, H. W. Dommel, ‘ Calculation of frequency dependent impedances of underground power cables with finite element method’, IEEE Trans. on Magnetics, vol. 25, No. 4, pp. 3025-3027, 1989. [13] G. K. Papagiannis, D. G. Triantafyllidis, D. P. Labridis, ‘ A One-Step Finite Element Formulation for the Modeling of Single and Double Circuit Transmission Lines’, IEEE Trans. On Power Systems, vol 15, Nr. 1, pp.33-38, 2000. [14] P. Davies, P. Rabinowitch, ‘ Methods of Numerical Integration – 2nd Edition’, Academic Press, 1984, pp. 104, 223 [15] D. Labridis, P. Dokopoulos: ‘Finite Element Computation of Field, Losses and Forces in a Three-Phase Gas Cable with Non-Symmetrical Conductor Arrangement’, IEEE Trans. on Power Delivery, Vol. PWRD- 3, No.4, pp. 1326-1333, 1988. [16] J. L. del Alamo: ‘A Comparison among different techniques to achieve an optimum estimation of electrical grounding parameters in two-layered earth’, IEEE Trans. on Power Delivery, vol. PWRD-8, No. 4, pp. 1890- 1899, 1993. VIII. BIOGRAPHIES Grigoris Papagiannis (S’79-M’88) was born in Thessaloniki, Greece, on September 23, 1956. He received his Dipl.-Eng. Degree and the Ph.D. degree from the Department of Electrical Engineering at the Aristotle University of Thessaloniki, in 1979 and 1998 respectively. Since 1981 he has been working as research assistant and since 1998 as Lecturer at the Power Systems Laboratory of the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Greece. His
special interests are power systems modeling and computation of electromagnetic transients. Dimitrios Tsiamitros was born in Kozani, Greece, on March 19, 1979. He received his Dipl.-Eng. Degree from the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, in 2001. Since 2001 he has been a postgraduate student at the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki. His special interests are power system modeling and computation of electromagnetic transients. Georgios Andreou (S' 98-A' 02) was born in Thessaloniki, Greece, on August 16, 1976. He received his Dipl.-Eng. Degree from the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, in 2000. Since 2001 he has been a postgraduate student at the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki. His special interests are power system analysis and powerline communications. Dimitris Labridis (S' 88-M' 90-SM' 00) was born in Thessaloniki, Greece, on July 26, 1958. He received the Dipl.-Eng. Degree and the Ph.D. degree from the Department of Electrical Engineering at the Aristotle University of Thessaloniki, in 1981 and 1989 respectively. During 1982-2001 he has been working at first as a research assistant and later as Lecturer and as Assistant Professor at the Department of Electrical Engineering at the Aristotle University of Thessaloniki, Greece. Since 2001 he has been Associate Professor at the same Department. His special interests are power system analysis with special emphasis on the simulation of transmission and distribution systems, electromagnetic and thermal field analysis, numerical methods in engineering, artificial intelligence applications in power systems and powerline communications. Petros S. Dokopoulos (M’77) was born in Athens, Greece, in September 1939.He received the Dipl. Eng. degree from the Technical University of Athens in 1962 and the Ph.D. degree from the University of Brunswick, Germany, in 1967. During 1962–1967 he was with the Laboratory for High Voltage and Transmission at the University of Brunswick, Germany, during 1967–1974 with the Nuclear Research Center at Julich, Germany, and during 1974–1978 with the Joint European Torus. Since 1978 he has been Full Professor at the Department of Electrical Engineering at the Aristotle University of Thessaloniki, Greece. He has worked as consultant to Brown Boveri and Cie, Mannheim, Germany, to Siemens, Erlagen, Germany, to Public Power Corporation, Greece and to National Telecommunication Organization and construction companies in Greece. His scientific fields of interest are dielectrics, power switches, power generation (conventional and fusion), transmission, distribution and control in power systems. Prof. Dokopoulos has 81 publications and 7 patents on these subjects.
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