The effect of soil and cable backfill thermal conductivity on the temperature distribution in underground cable system

The paper presents a mathematical model of heat transfer in the underground cable system. The computations were performed for flat formation of power cables buried in the ground at a depth of 2 meters. The model allows determining the two-dimensional temperature distribution in the soil, thermal backfill and power cables. The simulations studied the effect of soil thermal conductivity on the maximum temperature of the cable conductor. Furthermore, the effect of thermal backfill soil conductivity on the cable conductor temperature was studied. Numerical analyses were performed based on a program written in MATLAB.


Introduction
When designing the underground electricity network, the thermal phenomena shall be considered. Due to the cable insulation meltdown occurrence, the cable conductor temperature shall not exceed 90 ºC. Since the data on soil thermal resistance is mostly uncertain, the cable engineers design the underground power system in such a way that the cable core temperature does not exceed the optimum temperature of cable operation (65 ºC). Therefore, the cable engineers in many cases use thermal backfill materials for locating the power cables. The cable backfill material exhibits higher thermal conductivity than the mother soil. Therefore, the heat dissipation from the buried cables to surroundings is enhanced, when compared to the cables situated in the native soil.
Thermal analysis of the underground power cables operation was performed in [1][2][3][4][5] [6][7][8][9][10]. The optimization of thermal backfill dimensions was presented in [6,7], and the computational studies of multilayered soil effect [8] and various cable placement type [9,10] were performed . This study presents a numerical model of underground power cable system. Three power cables arranged in the flat (in-line) formation and buried at a depth of 2 m are considered. The influence of soil thermal conductivity and thermal backfill conductivity on the maximum cable core temperature is studied. The numerical code developed in MATLAB software allows calculating the two-dimensional temperature distribution within the underground power cable system including cable core, thermal backfill, and soil. The presented mathematical model is efficient and straightforward in programming implementation.

Mathematical model of underground power cable system
The scheme of the computational domain is shown in Fig.  1:   The following dimensions are shown in Fig. 2 l -spacing between two consecutive cables (assumed as 0.4 m in this study), s -distance from the right edge of the backfill layer to the side cable axis (assumed as 0.4 m in this study), b − distance from the conductor's axis to the top of bedding layer (assumed as 0.4 m in this study), p − distance between the conductor's axis and the bottom of bedding layer (assumed as 0.4 m in this study), The cable conductor cross section Ac is selected from the XLPE HV cable design series provided by the cable producer, and equal to Ac = 1400 mm 2 . The computational scheme is given in Fig. 3a while the grid used in the analysis is shown in Fig. 3b The temperature at certain point with (x,y) coordinates is calculated from the heat conduction equation [8]: where: Tc -cable conductor temperature, ºC qv -volumetric heat source (valid only for cable conductor domain), W/m 3 . k -thermal conductivity, W/(m K), The thermal conductivity values for power cable layers are assumed according to Table 1. The soil thermal conductivity is varied from 0.5 W/(m K) to 1 W/(m K) and depends on the computational case. The thermal backfill conductivity is ranged from 1 W/(m K) to 3 W/(m K)). The volumetric heat source is given as: Where the heat losses ∆Q are calculated based on the current rating I = 1000 A and the alternating current (AC) electric resistance Re,AC: The altering current electrical resistance is a function of direct current electrical resistance Re,DC and skin and proximity coefficients denoted as ys and yp, respectively: with where, Re,ref and αe,ref are the reference cable conductor's electric resistance, and the temperature coefficient for the conductor material, both given at the reference temperature Tref = 20°C. The reference electric resistance of the cable conductor is given by: where, ρ20 -specific electrical resistance of copper conductor in 20°C, Ω ·m; for the following case study ρ20 = 1.  Where, ks and kp are skin and proximity effect correction factors equal to 0.435 and 0.37, respectively, for the case of segmented conductor type [8]. An alternating current frequency f = 50 Hz and l is the distance between adjacent conductor axes.

Boundary condition
The symmetry pattern of the heat conduction equation solution is expected. Therefore only the half of the computational domain is considered. Hence, the square domain with a height of 10.0 m and width of 5.0 m is used in the computations (Fig. 3). It is assumed that the right, left and bottom edges of the boundary region are perfectly insulated. At the top edge (the ground level), the temperature Tg is set to 20°C. This temperature value is specified by the standards [11,12]. The mathematical description of the applied boundary conditions is given by Eq. (9). The Finite Element Method is applied to solve the heat conduction equation Eq. (1). The linear triangular elements are used to discretize the computational domain. Since the volumetric heat source depends on the temperature, the system of nonlinear equation, comprising from FEM discretization, needs to be solved. Therefore, the Jacobi iteration method is used. The computational domain mesh was created using a PDE toolbox of MATLAB software. The FEM solver was developed by the authors, and implement to the discretized model. The computations performed in MATLAB allow to determine the temperature distribution within entire underground power cable system.

Soil thermal conductivity effect on temperature distribution in underground power cable system
The variation in the mother ground thermal conductivity changes the intensity of the heat transfer from power cables. The larger the conductivity, the faster the soil receives the heat, and thus also lowers the temperature of the cable conductor. The soil thermal conductivity changes due to the decrease/increase of the water content (e.g. caused by rains or droughts). The lower the moisture content, the lower the soil thermal conductivity. Therefore, it may happen that the thermal calculations performed for moist soil that meets the security requirements do not satisfy them when the soil is dry. Figure 4 presents the variation of cable core temperature with the increase in the mother ground thermal conductivity from 0.5 W/(m K) to 1 W/(m K). It is assumed that the cables are placed in thermal backfill (Fluidized Thermal Backfill) with a thermal conductivity of 1.54 W/(m K).  Figure 4 shows the effect of soil thermal conductivity on cable core temperature (central cable conductor). It can be seen that for the analyzed computational case, the two times decrease in the soil thermal conductivity results in the 15ºC increase of cable core temperature.
The temperature distribution within the cable system (Fig. 5) illustrates well the major impact of soil thermal conductivity on heat transfer. It can be observed that a properly designed cable system for the specified location (with high thermal conductivity), may not perform well in the region where

Cable backfill thermal conductivity effect on temperature distribution in underground power cable system
Variation in the thermal conductivity of cable backfill has similar effects (as the soil thermal conductivity changes) on the cable core temperature. Cable backfill has a higher thermal conductivity than the mother ground, and therefore enhances the heat dissipation from buried cables. The study was performed for the soil thermal conductivity of 0.5 W/(m K) and cable backfill thermal conductivity varied in a range of 1 W/(m K) to 3 W/(m K). The first value (1 W/(m K)) refers to the conventional thermal backfill i.e. sand-cement mixture in proportions of 12:1. The last value (3 W/(m K)) relates to the POWERCRETE TM thermal backfill produced by Heidelberg Cement Group.
It should be noted that the higher the backfill thermal conductivity, the increased costs of the cable backfill. Therefore, during the design of underground power cable system, the economic conditions shall be considered. Figure 6 shows that the increase in the cable backfill thermal conductivity from 1 W/(m K) to 3 W/(m K) causes the decrease in cable conductor temperature of 7ºC. Fig. 6. Variation of cable core temperature with the increase in cable backfill thermal conductivity (the soil thermal conductivity is equal to 0.5 W/(m K). Figure 7 presents the effect of cable backfill thermal conductivity on temperature distribution within the underground power cable system. The value of soil thermal conductivity is assumed as 0.5 W/(m K). It can be seen, that for the analyzed system, when the thermal backfill with high thermal conductivity is applied, the temperature gradients within the soil and cable backfill are lower. Hence, the heat is dissipated in a more efficient way from power cables to the surroundings.  This example shows that if the cable backfill is not used, but only the mother ground, the cable conductor would reach the temperature of 77,53ºC. It is much higher than the cable design temperature of 65ºC. This example shows how important is to use the thermal backfill layer. When the cable backfill is used instead of mother ground, the cable core temperature is 14ºC lower.

Conclusions
This paper presents the thermal analysis of underground power cable system. The case when the cables are arranged in flat formation (in-line) and buried at a depth of 2 m underground was studied. The Finite Element model that allowed determining the temperature distribution within the system was developed. The Mother ground from 0.5 W/(m K) to 1 W/(m K) Based on the performed analysis it can be concluded that: -soil and thermal backfill conductivity play an important role in cable line design -the higher the soil thermal conductivity, the lower the cable conductor temperature -the replacement of mother ground with the thermal backfill material may reduce the cable core temperature up to 14ºC.