# The Equivalence between Successive Approximations and
## 6 Numerical Validation  
The numerical validations are implemented using a personal computer with an INTEL(R) Core(TM) \(i7-7700\) processor at 3.60 GHz, 8 GB RAM, running a 64-bits Windows 10 operative system. The simulation software used is MATLAB 2020\(b\). In addition, the processing time presented in this research are the average values after 10,000 repetitions for the studied and comparative methods. Besides, to determine the effectiveness of each of the studied and comparative methods, we evaluate the amount of active power losses on the distribution network (see Equation (24)). Note that we consider \(\epsilon=1\times 10^{-10}\) as the convergence error between two consecutive iterations of voltage variables:  
\[p_{\text{loss}}=\text{real}\Big{\{}\mathbf{J}|^{T}\mathbb{Z}_{p}|\mathbf{J}| \Big{\}}=\text{real}\Big{\{}\mathbf{V}^{\intercal}\mathbb{V}^{\intercal} \mathbf{V}^{\intercal}\mathbf{V}^{\intercal}\Big{\}}. \tag{24}\]  
Table 1 reports the numerical results of the studied load flow methods, i.e., the SA and the MBF load flow approaches as well as the results of the comparative methods. Figure 1: Electrical configuration of the test feeders: (**a**) 33-node test system and (**b**) 69-node test system. The results in Table 1 allows for the following remarks:  
* The conventional GS approach exhibits the worst performance in terms of processing times and the number of iterations required to solve the load flow problem in both test feeders. In addition, the accelerated version of the GS approach allows for improvement in its performance (i.e., the GS behavior) by reducing processing times by about 91.28 % and 94.66% for the 33- and 69-bus distribution grids, while the number of iterations is reduced by about 90.19% and 94.99% respectively;
* The classical NR and LM load flow approaches have the same numerical performance regarding the total iterations required to solve the load flow problem in both test feeders with 5 iterations in both cases. The low number of iterations in these methods is attributed to the fact that both work with the Jacobian matrix, which contains information about the direction of the maximum change of load flow equations, a situation that does not occur for the remainder of the load flow methods, as these do not use information of the derivatives of the load flow equations. With regard to processing times, the NR and LM methods are very similar, as in the case of the system with 33 buses, the difference is lower than 0.15 ms, and in the case of the system composed of 69 buses, this difference is about 4 41 ms;
* The numerical results for the 33- and 69-node test feeders obtained for the SA and the MBF load flow methods, i.e., the equivalent load flow approaches, is the same regarding the number of iterations, as these take 10 iterations to solve the load flow problem in both test feeders however, regarding the processing times, we can observe that the SA is the faster approach as demonstrated in [12]. Even if both methods are equivalent, the SA approach calculates the admittance matrix, i.e., \(\mathbb{V}\), using the direct method by adding the inverse of the conductance at this line, while the MBF method uses matricial calculations for the products between the incidence matrix and the primitive matrices, which consumes additional processing times. It is also important to mention that regarding the estimation of the total grid losses, all the methods find an equal numerical solution with an error lower than \(1\times 10^{-6}\), which can be considered negligible for any practical purpose, which entails that all the approaches reported in Table 1 are suitable for solving the load flow problem in electrical distribution networks. On the other hand, Table 2 presents the behavior of the convergence errors of the SA and the MBF load flow methods for the largest studied test feeder.