IEEE Transactions on Power Delivery, Vol. 8, No. 3, July 1993 761 THERMAL ANALYSIS OF POWER CABLES IN MULTI-LAYERED SOIL Part 1: Theoretical Model M.A. Hanna and A.Y. Chikhani Department of Electrical and Computer Engineering Royal Military College of Canada Kingston, Ontario, Canada ABSTRACT The heat dissipation and temperature distribution in multi- layered soil surrounding a buried cable is calculated using the finite difference method and the energy conservation principle. The unrealistic simplifying assumptions, used in the published models for calculating the heat dissipation from the underground cable system, are not needed in the proposed model. The numerical technique proposed in this paper is very suitable to model any real cable installation configuration in multi- layered soil. The development of the model, and the effect of the parameters that influence the conversion and the stability of the numerical solution of the heat dissipation from the underground cable system, is studied in this paper. The application of the model is given in the second part of this work submitted at the same time for publication. 1. INTRODUCTION The calculation of the heat dissipation from buried cables has been the subject of many papers [l-61. The basic limitation on the power-carrying capacity of the underground cable is that the ampacity of the cable which is determined by temperature limitations and, hence, heat dissipation in the soil surrounding the buried cable. The accurate prediction of thermal conditions in the underground cable system is essential for economical cable system design. Because of the high capital costs of underground transmission systems, small differences in predicting cable ratings- can represent a considerable saving in investment. As a result, over the last decade, a considerable effort has been devoted to developing an accurate method for calculating the thermal distribution of underground cables. The pioneer work by Neher [7] solved the energy equation in its differential form for special cases of buried cable in different layers of soil in order to obtain the heat dissipation and the temperature distribution in the soil surrounding the cable. An analytical solution for the energy equation was found after using various simplifying assumptions. The solution for the energy equation is known as the Kennelly’s Equation [7]. The concept of imaging was later used to improve the validity of Kennelly’s Equation [8]. However, due to the assumptionsused in developing the solution to the energy equation, this method was restricted to ideal cableltrench conditions, where the soil is homogeneous with single-valued thermal conductivity and the ground surface is an isothermal plane. A modified analytical equation was later proposed which used Kennelly’shypothesis in conjunction with the principle of superposition to improve the accuracy of the method, 92 SM 563-7 PWRD by the IEEE Insulated Conductors Committee of the IEEE Power Engineering Society for presentation at the IEEE/ PES 1992 Summer Meeting, Seattle, WA, July 12-16, 1992. Manuscript submitted January 30, 1992; made available for printing April 16, 1992. A paper recommended and approved M.M.A. Salama Department of Electrical Engineering University of Waterloo Waterloo, Ontario, Canada especially when the thermal network has more than one source of heat [8]. However, this method also failed to predict the temperature distribution of practical underground cable for different seasons. Due to the complexity of the problem, analytical methods generally have limitations in accurately predicting the temperature distribution of the underpund cable system. This complexity arises from the many parameters involved and the interaction between their effects on the problem. Analytical methods were used in combination with empirical formulas to construct a variety of design tables, which are now widely used in cable installation design. The purpose of this paper is to develop a more accurate method for calculating the thermal fields surrounding underground cables, and ultimately to construct design tables or charts for an extended range of cable configurations that can be used readily by the power engineers. The basic limitations existed in the application of the available design tables are the limited range of validity; (113 to 3) of the heightCwidth ratio of the trench dimension and the assumption of isothermal boundaries of the trench [lo]. Recently, more reliable methods, based on numerical techniques, have dealt with the calculation of the thermal fields of underground cables [5,6,9,101. These methods attempt to simulate the underground cable system heat model using a mesh or a grid followed by numerical techniques to obtain the heat dissipation. EL-Kady et.al. [lo] used the finite element method (FEM) to solve the problem of heat dissipation from the underpund cable for ranges of heighvwidth of the trench greater than 3 and he avoided using the assumption of isothermal trench boundaries. His method resulted in extended values of the geometric factor G,, which can be used for calculating the external thermal resistance between cables in duct banks. The Gb-factorcan be directly used in the Neher-McGrath analysis [lo]. Due to the complexity of the method and the computationaleffort involved, EL-Kady, et.al. [lo] were unable to construct a generalized G,,-factor, and the data available from the analysis was only valid for the cases studied in their paper [lo]. The effect of the multi-soil trench system, the moisture migration, and environmental conditions were not considered in their analysis. In an attempt to reduce the computational effort, Gela, eta1 [6] used the boundary element method (BEM) for the solution of the heat dissipation from u n d e r m d cable. The BEM has the advantage of consideringthe region boundaries rather than its interior, which reduces the dimension of the problem from three-dimensional to two- dimensional. The BEM can handle the boundaries at infinity directly truncating the region. However, when this method is used to calculate the thermal fields from the underground cables, unjustified constraints have to be implemented to facilitate the numerical solution, such as neglecting the convective boundary at the ground surface. The computational effort becomes extremely large when the BEM attempts to handle a practical trench design with multi-soil system (trench filling,backfill and mother soil). The dissimilar properties of the soil necessitates that the boundaries between them be included in the calculation as well, which increases the number of the unknowns in the BEM. This will increase the size of the problem and, hence, the computational effort since the matrices involved in the BEM are a function of the unknowns. These matrices are full and need full-size inversion sub-routines [ 111. 0885-8977/93$03.00 0 1992 IEEE