# Cable Temperature Prediction Based on RF-GPR for Digital Twin Applications
## 2 Finite Element Calculation Model of Cable Temperature  
The cable laid in the cable trench is a cross-linked polyethylene three-core cable; the specific model is YJV22-8.7/10-3 \(\times\) 240 mm\({}^{2}\). According to the internal structure of the cable, the cable can be divided into the following parts: cable core, semiconductor layer, insulation layer, metal shielding layer, filling layer, armor and sheath layer. Since the axial length of the cable is much larger than the radial length of the cable, the two-dimensional plane model is used in the analysis of the cable model. The finite element model is shown in Figure 1, and the cable model inside the cable trench is shown in Figure 2. The finite element software used in this paper was Comsol Multiphysics 5.6. Free triangle mesh was used for the cable part, and boundary layer mesh was used for the inner wall and surface of the cable channel. The total number of meshes in the finite element calculation model was 92,856. Figure 1: Finite element calculation model of cable trench laying. Figure 2: Finite element calculation model of cable temperature. Two-dimensional steady-state electromagnetic and temperature field coupling calculation is carried out for 10 kV AC cables, in which the cable core loss, dielectric loss, metal shield loss and armor loss generated by the cables themselves are used as the heat source [22], and the partial differential equation of the temperature inside the cables can be obtained as shown below:  
\[-\
abla\cdot(k\
abla T)=Q_{h} \tag{1}\]  
where \(Q_{h}\) is the total cable loss, and \(k\) is the thermal conductivity. In the cable trench laying environment set up in this paper, because there is contact between the cable and the air inside the cable trench, it is necessary to consider the influence of the air flow in the cable trench on the heat dissipation. The conductor loss, insulation loss and armor loss of the cable are the main sources of heat, which affect the flow of air in the cable trench and then transfer the heat through conduction and convection. The thermal conductivity differential equation is shown in Equation (2), and the convection differential equation is shown in Equation (3). \[\lambda\left(\frac{\partial^{2}T_{s}}{\partial x^{2}}+\frac{\partial^{2}T_{s} }{\partial y^{2}}\right)+Q=0 \tag{2}\]  
where \(\lambda\) is the thermal conductivity of the medium, \(T_{s}\) is the medium temperature, and \(Q\) is the heating rate per unit volume of the medium. \[Q=h\Delta t \tag{3}\]  
where \(h\) is the convective heat transfer coefficient, and \(\Delta t\) is the temperature difference between a solid and a fluid. Heat transfer between the cable trench and soil is a solid heat transfer problem, so the lower boundary of the finite element model of the cable trench belongs to the first type of boundary condition, and the temperature is set as 20 \({}^{\circ}\)C for the deep soil. The formula is shown as follows:  
\[\left\{ \begin{aligned} & T|_{\Gamma}=T_{w}\\ & T|_{\Gamma}=f(x,y,z,t)\end{aligned} \right. \tag{4}\]  
where \(\Gamma\) is the object boundary, \(T_{w}\) is the boundary temperature, and \(f(x,y,z,t)\) is the boundary temperature function. The left and right boundaries of the finite element calculation model of the cable trench belong to the second type of boundary conditions. The heat flux value of the boundary method phase is set as 0, so the boundary is regarded as an adiabatic surface. The formula is shown as follows:  
\[\left\{ \begin{aligned} &\lambda\frac{\partial T}{\partial n}\Big{|}_{ \Gamma}+q_{w}=0\\ &\lambda\frac{\partial T}{\partial n}\Big{|}_{\Gamma}+g(x,y,z,t)= 0\end{aligned} \right. \tag{5}\]  
where \(q_{w}\) is the known boundary heat flux value, and \(g(x,y,z,t)\) is the boundary heat flux function. The upper boundary of the finite element model of the cable trench belongs to the third type of boundary conditions. The external air temperature is set as the fluid temperature, and the heat transfer coefficient is the convective heat transfer coefficient in the air.