## 2 Materials and Methods
The determination of the ampacities results from the thermal balance, or more precisely, from the Joule's heat portion supplied to the working conductor of the power cable so as to achieve the maximum permissible temperature. The critical temperature threshold for PVC material was determined to be 70 \({}^{\circ}\)C. By incrementally augmenting the heat flux applied to the internal surface of the insulation, the system's maximum temperature was elevated to meet this limit, which represents the maximum permissible temperature for cable insulation. To enhance calculation accuracy, a second-order discretization scheme was implemented for energy residue. In general, for the second order spatial discretization, quantities at cell faces are computed using a multidimensional linear reconstruction approach, whereas for the first order discretization, a cell is represented by its cell-center value. Only the energy equation was solved within the solver framework, with the flow and turbulence equations disabled. Convergence criteria were set to \"none\", and iterative computations were monitored via the energy equation residual chart until the residual exhibited negligible further changes, signalling computational stability. Table 2 shows the results of numerical simulations and a comparison of their results with data from the standard [19]. Examples of temperature distributions/fields around the cables can be found in Figure 4.  
Figure 3: View of the computational grid for the numerical model. Figure 4: Temperature fields (\({}^{\circ}\)C) around the analyzed three power cables (240 mm\({}^{2}\)) for the following load currents: (**a**) \(I_{\rm A}\) = 627.1 A, \(\rho_{\rm g}\) = 0.5 (K-m)/W; (**b**) \(I_{\rm A}\) = 357.5 A, \(\rho_{\rm g}\) = 2.0 (K-m)/W, (**c**) \(I_{\rm A}\) = 482.4 A, \(\rho_{\rm g}\) = 1.0 (K-m)/W. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cross-Section of** & **Soil** & **Numerical** & **Standard IEC** & **Difference between** \\
**All Three Cables,** & **Thermal Resistivity,** & **Simulations,** & **60364-5-52 [19],** & **Simulation Results and** \\
**mm\({}^{2}\)** & \(\rho_{\rm gg}\) (K-m)/W** & \(I_{\rm A\,A}\) & \(I_{\rm A\,A}\) & **the Standard [19] Data, \%** \\ \hline
35 & 0.5 & 219.4 & 207.0 & 6.0 \\ based on [3] & 1.0 & 169.8 & 165.0 & 2.9 \\  & 2.0 & 127.4 & 124.0 & 2.7 \\ \hline
120 & 0.5 & 431.3 & 414.0 & 4.2 \\  & 1.0 & 330.0 & 330.0 & 0.0 \\  & 2.0 & 245.2 & 246.4 & 0.5 \\ \hline
240 & 0.5 & 627.1 & 602.0 & 4.2 \\  & 1.0 & 482.4 & 480.0 & 0.5 \\  & 2.0 & 357.5 & 358.4 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the results of numerical model verification (\(I_{\rm A}\)—ampacity). Summarizing the results obtained in Table 2, it is possible to prove the high convergence of the numerical model with the values given in the standard [19]. Differences in ampacity \(I_{\mathrm{A}}\) amount to a maximum of 6% (for a thermal resistivity of soil equal to \(\rho_{\mathrm{g}}=0.5\) (K\(\cdot\)m)/W). It should be mentioned here that the authors of the standard [19] assume that the overall accuracy of the values included in the standards is around 5%. Based on the verification of the numerical model, it can be concluded that the presented model is correct and accurate enough. These results justify the development of a model of a system containing stabilized backfill.