## 6 Computational Validation
### Results for the 33- and 69-Node Test Feeders  
Table 5 lists the computational efficiency of the proposed MBF method as compared to the classical approaches reported in the literature. These results were reached after performing 1000 consecutive evaluations, which confirms the following:  
* The proposed approach is the most efficient in terms of processing times as compared with the classical approaches as it only takes 1.32 ms for completion in the case of the 33-node test feeder and 5.37 ms for completion in the case of the 69-node test feeder, which implies that it  
The NR and LM methods belong to the same class as both comprise the use of Jacobian matrices. Therefore, their performance with respect to the number of iterations and the processing times is very similar. Even if the proposed approach requires double the number of iterations, in terms of processing times, we can confirm that the MBF is the best numerical method for distribution power-flow analysis as compared with the NR and LM approaches.  
In terms of the power losses calculation, it is important to mention that all the numerical methods listed in Table 1 arrive at the same solution with negligible errors of estimation (lower than \(1\times 10^{-10}\)). This implies that each of them is suitable for implementation; nevertheless, our proposed method, i.e., the MBF reformulation, is the most desirable approach due to its speediest performance.  
Figure 7 presents the voltage profiles for the 33- and 69-node test feeders for the proposed MBF and NR power-flow methods. We can observe that these plots confirm that, numerically speaking, both these methods demonstrate the same performance in terms of voltage determination. However, the main advantage of the MBF when compared to the NR approach (see Table 1) is its faster calculation times as it guarantees convergence on well-defined voltage conditions.  
To demonstrate that the proposed MBF approach guarantees convergence in the 33- and 69-node test feeders, Figure 8 presents the behavior of the \(\theta\)-coefficient. From the results in Figure 8, it is possible to confirm that hypothesis (19) is fulfilled, and the proposed BFM guarantees the convergence of the power-flow problem. In the case of the 33-node test feeder, the maximum coefficient occurs at node 30 with a value of 0.027, while in the case of the 69-node test feeder, this occurs at node 61 with a value of 0.062. These values facilitate the verification that \(\theta\) is comprehended between 0 and 1 in the case of both the test feeders.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multicolumn{4}{c}{**33-node Test Feeder**} \\ \hline
**Method** & **Proc. time [ms]** & **Iterations** & **Losses [p.u]** \\ \hline
**GS** & 441.974 & 2313 & 2.110 \\
**AG**\((\alpha=1.82)\) & 38.555 & 227 & 2.110 \\
**NR** & 10.751 & 5 & 2.110 \\
**LM** & 10.882 & 5 & 2.110 \\ \hline
**MBF** & 1.323 & 10 & 2.110 \\ \hline \multicolumn{4}{c}{**69-node Test Feeder**} \\ \hline
**GS** & 31107.756 & 49031 & 2.422 \\
**AG**\((\alpha=1.92)\) & 1662.691 & 2455 & 2.422 \\
**NR** & 38.303 & 5 & 2.422 \\
**LM** & 42.719 & 5 & 2.422 \\ \hline
**MBF** & 5.369 & 10 & 2.422 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Numerical results for the 33- and 69-node test systems.  
Figure 8: Behavior of the \(\theta\)-coefficient: (**a**) 33-node test system, and (**b**) 69-node test feeder.  
Figure 7: Voltage profiles: (**a**) 33-node test system, and (**b**) 69-node test feeder.