## 4 Computational Validation
### Results in the 21-Bus Grid  
Considering that the solution reached by the hyperbolic approximation took fewer iterations and ensured the same numerical results as those of the successive approximation power flow method with fewer iterations, here, we only present the results reached by our proposal. Table 6 presents the voltage profile performance for the 21-bus grid in the solidly grounded and non-grounded simulation scenarios. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Node** & \(+\)**Pole (V)** & **0 Pole (V)** & \(-\)**Pole (V)** \\ \hline \multicolumn{4}{c}{Grounded neutral} \\ \hline  
\hline \multicolumn{4}{c}{Power losses} & 91.2701 kW \\ \hline \hline \multicolumn{4}{c}{Number of iterations} & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Voltages for both simulation cases in the 21-bus system. The numerical results in Table 6 show that: (i) The numbers of iterations with the hyperbolic approximation were four for the solidly grounded case and 13 for the non-grounded scenario, while the successive approximation power flow method took 10 iterations in the solidly grounded case and 13 in the non-grounded scenario. (ii) The effect of the neutral wire grounded in all of the nodes of the network was very positive in terms of the amount power loss, since for this scenario, this was 91.2701 kW, while in the non-grounded case, the power loss increased until 95.4237 kW, i.e., 4.2136 kW of additional losses. (iii) The voltage regulation when the neutral wire was solidly grounded was defined by the positive pole at node 17, since it had a low voltage profile with a magnitude of 890.1027 pu, which defined a regulation of 10.9897% for the 21-bus system in this simulation case. In addition, when the neutral wire was non-grounded, the minimum voltage was maintained at node 17 with a magnitude of 888.2594 V for the positive pole, which produced a regulation in the whole system of 11.1741%, which was worse than that in the solidly grounded case; however, this situation was explained by the additional voltage droops in the neutral wire due to the absence of the grounded system at each node. (iv) In the non-grounded neutral wire scenario, the voltage at this conductor reached about 24.3408 V at node 17; even though this is a low voltage value, it can cause the misoperation of electronic devices in areas near this node, since power electronic devices can be sensitive to variations in the voltage magnitudes of their local references. To demonstrate the ability of the proposed hyperbolic approximation to deal with the power flow problem in meshed DC bipolar configurations, here, we add two lines to the 21-bus system that allow the radial configuration to become a meshed one. The added distribution lines connect node 7 with node 19 and node 11 with node 16, with resistances per pole of 0.082 and 0.037 \(\Omega\), respectively. The numerical simulations of the meshed configuration for the 21-bus system showed that: (i) The total power losses in this system with a solidly grounded neutral wire were \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Node** & **+Pole (V)** & **0 Pole (V)** & **\(-\)Pole (V)** \\ \hline \multicolumn{4}{c}{Non-grounded neutral} \\ \hline
1 & 1000 & 0 & \(-\)1000 \\
2 & 996.2821 & \(-\)1.6193 & \(-\)994.6628 \\  
\hline \multicolumn{4}{c}{Power losses} & \multicolumn{3}{c}{95.4237 kW} \\ \hline \multicolumn{4}{c}{Number of iterations} & \multicolumn{3}{c}{4} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._75.1832 kW, while for the non-grounded scenario, these were 78.7372 kW, i.e., as expected, there was an increment in the power losses of 3.5540 kW due to the currents that flowed through the neutral conductor. (ii) The performances of the voltage profiles in both simulation cases were very similar, as can be seen in Figure 5; however, there were some differences when comparing the voltage magnitudes between the positive and negative poles.