2594 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 4, OCTOBER 2005 A Systematic Approach to the Evaluation of the Inﬂuence of Multilayered Earth on Overhead Power Transmission Lines Grigoris K. Papagiannis, Member, IEEE, Dimitrios A. Tsiamitros, Dimitris P. Labridis, Senior Member, IEEE, and Petros S. Dokopoulos, Member, IEEE Abstract—The inﬂuence of earth stratiﬁcation on overhead power transmission line impedances is investigated in this paper. A systematic comparison of existing approaches is done, while results are also obtained using a ﬁnite-element method formula- tion. A novel numerical integration technique is proposed for the calculation of the inﬁnite integrals involved. Typical single- and double-circuit line conﬁgurations are examined for a combination of layer depths and earth resistivities over a wide frequency range. The inﬂuence of the layer depth is also investigated. Results show signiﬁcant differences from those, corresponding to the case of homogeneous earth. Using the multilayered earth return imped- ances in transient simulations, the transient responses show that differences occur mainly in cases of asymmetrical faults, justifying the need for a detailed earth model implementation. Index Terms—Electromagnetic transient analysis, ﬁnite element method, nonhomogeneous earth, power transmission lines. I. INTRODUCTION I N TRANSIENT simulations, detailed transmission line modeling is required. Transmission line modeling is strongly inﬂuenced by the presence of the resistive earth re- turn path, which is included in transmission line impedances through proper correction terms. The latter are calculated using the widely accepted Carson’s formulas [1] developed for the case of semi-inﬁnite, homogeneous earth. In practice, earth is composed of several layers of different electromagnetic properties. Wise and Sunde [9] developed new formulas for the two-layer earth case under certain simpliﬁca- tions. In 1966, Wedepohl and Wasley [2] calculated the line impedances for a two-layer earth model using a double com- plex Fourier transform. In 1973, Nakagawa [3] proposed a more rigorous and general solution for the multilayer earth case, al- lowing the relative permittivity and permeability of each layer to be different from unity. Nakagawa’s earth model is imple- mented in the electromagnetic transients program (EMTP) [11]. Finally, Moghram [4] extended in 1998 the solution of [2] for the case of a three-layer earth. The ﬁnite element method (FEM) is a numerical method widely used for the solution of the electromagnetic ﬁeld equa- tions in a region, regardless of the geometric complexity. In a recently proposed method by some of the authors [5], the Manuscript received May 5, 2004; revised November 11, 2004. Paper no. TPWRD-00214-2004. The authors are with the Power Systems Laboratory, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: grigoris@eng.auth.gr). Digital Object Identiﬁer 10.1109/TPWRD.2005.855448 Fig. 1. Two conductors over a three-layer earth. electromagnetic ﬁeld variables have been calculated using the FEM and properly linked with the transmission line equivalent circuit parameters. The new method is capable of handling cases of terrain surface irregularities and nonhomogeneous, stratiﬁed soil, where most classical methods usually fail. This paper presents a generalized methodology capable of handling all different approaches for the overhead transmission line impedance calculation, in cases where earth is considered to be multilayered. Instead of using inﬁnite series approxima- tions for the evaluation of the inﬁnite integrals, a direct nu- merical integration scheme is adopted. The new method is ap- plied to typical single- and double-circuit overhead transmis- sion line conﬁgurations for varying soil parameters and over a wide frequency range. The obtained results show signiﬁcant differences when checked against those obtained by Carson’s formulas under the assumption of homogeneous earth. The va- lidity of the results is justiﬁed using a suitable FEM formula- tion for the multilayered earth case. Finally the new impedances are used in typical transient simulations in order to evaluate the inﬂuence of the earth stratiﬁcation on the actual voltages and currents. The resulting differences reveal that earth stratiﬁca- tion must be taken into account in the transmission line models, when zero sequence components may occur. II. PROBLEM FORMULATION In the general case of a two conductors arrangement of Fig. 1, earth is assumed to consist of three layers. The distances of the two conductors from the earth surface are and respectively, while their horizontal distance is . The ﬁrst earth 0885-8977/$20.00 © 2005 IEEE