## 6 Computational Validation
### Inclusion of Voltage-Controlled Nodes  
One of the main advantages of the proposed MBF power-flow reformulation is the possibility of working with voltage-controlled nodes, i.e., nodes that control active power injection and the magnitude of the voltage profile. These nodes are common in power-system analyses [14], wherein multiple synchronous machines are interconnected via the transmission network. To realize the VC inclusion in the proposed model, we can apply the classical GS method, where the reactive power generation at the VC source nodes is calculated as follows:  
\[\mathbb{S}_{g_{k}}^{\text{VC},t}=P_{g_{k}}^{\text{VC}}+jQ_{g_{k}}^{\text{VC},t}, \tag{20a}\] \[Q_{g_{k}}^{\text{VC},t}=-\text{Imag}\left\{\mathbb{S}_{d_{k}}^{*,t}-\mathbb{V}_ {d,k}^{*,t}\sum_{j=1}^{n}\mathbb{Y}_{kj}\mathbb{V}_{d,j}^{*,t}\right\} \tag{20b}\]  
where \(\mathbb{S}_{g_{k}}^{\text{VC},t}\) is the apparent power injected at node \(k\) by the VC source, which can be split into its real and imaginary powers, i.e., \(P_{g_{k}}^{\text{VC}}\) and \(Q_{g_{k}}^{\text{VC},t}\), which are the active power constant and reactive power variable for maintaining a constant magnitude of the voltage output; \(\mathbb{Y}_{kj}\) is the admittance that relates nodes \(k\) and \(j\), respectively. The voltage-controlled nodes can be easily included in the proposed MBF method by modifying (12) as follows. \[\mathbb{V}_{d}^{t+1}=\mathbb{Z}_{dd}\mathbf{diag}^{-1}\left(\mathbb{V}_{d}^{ t,*}\right)\left(\mathbb{S}_{g}^{\text{VC},t}-\mathbb{S}_{d}^{*}\right)- \mathbb{b}. \tag{21}\]  
**Remark 2**: _Once all the voltages are calculated, in the case of the VC source, the magnitude of the voltage is fixed as a constant, and from (21), only the angle at each iteration is updated, i.e.,_  
\[\mathbb{V}_{g_{k}}^{t+1,VC}=\left|\mathbb{V}_{g_{k}}^{VC}\right|\angle\mathbb{ V}_{g_{k}}^{t+1,VC}.\]  
_being \(\mathbb{V}_{g_{k}}^{VC}\) the specified voltage in the VC source._  
To demonstrate the applicability of the proposed MBF power-flow method, we implement a small four-node test feeder with a slack node at bus 1 and a VC source at node 4. This system was initially presented in [14] to introduce the GS and NR methods. The comparison in this section is performed using the widely used power-system software called DigSILENT. Figure 9 presents the configuration of this test feeder and the DigSILENT results obtained after evaluating the AC power flow via the NR method. The information of the branches and loads for this four-node test feeder are obtained from [14] and are listed in Tables 6 and 7. Once the power-flow problem is solved using the proposed MBF method, the results reported in Table 8 are obtained. It should be noted that these are compared to the DigSILENT solution presented in Figure 9.  
The results presented in Table 8 confirm the possibility of using our proposal approach to solve power-flow problems in power systems by using the MBF reformulation with the main advantage that it is comparable with sophisticate software such as DigSILENT. In Appendix A has been illustrated the codification of the proposed MBF method for power systems that comprise voltage-controlled nodes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node** & **DigSILENT** & **MBF** & **Node** & **DigSILENT** & **MBF** \\ \hline
1 & 1.000\(\angle\)0.00\({}^{\circ}\) & 1.000\(\angle\)0.00\({}^{\circ}\) & 3 & 0.969\(\angle\)\(-\)1.87\({}^{\circ}\) & 0.969\(\angle\)\(-\)1.87\({}^{\circ}\) \\
2 & 0.982\(\angle\)\(-\)0.98\({}^{\circ}\) & 0.982\(\angle\)\(-\)0.98\({}^{\circ}\) & 4 & 1.020\(\angle\)\(-\)1.52\({}^{\circ}\) & 1.020\(\angle\)\(-\)1.52\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Voltages in the four-node test feeder.