## 6 Numerical Validation
### Numerical Results  
The numerical experiments were performed on a modified version of the CIGRE low-voltage test system [29]. The convex model was solved using CVX, a package for specifying and solving convex programs [30]. Table 3 shows the configuration of phase-balancing achieved by the proposed model in generation and demand nodes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Load/Generation** & **Probability** \\ \hline
1 & low/low & 0.2210 \\
2 & low/medium & 0.0443 \\
3 & low/high & 0.0676 \\
4 & medium/low & 0.2767 \\
5 & medium/medium & 0.0554 \\
6 & medium/high & 0.0845 \\
7 & high/low & 0.0845 \\
8 & high/medium & 0.0332 \\
9 & high/high & 0.0507 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Scenarios of generation and load.  
Figure 4: Modified CIGRE low voltage benchmark test system.  
Figure 5 shows the initial power losses for each scenario and their corresponding power loss reduction with the configuration achieved by the MI-QP model.  
From Figure 5, it can be observed that the MI-QP model reduced system power losses by, on average, 41.74 %; this indicates that, in any scenario, the system power losses always will reduce concerning the base case. The MI-QP model's solution guarantees the global optimum for all scenarios with a robust configuration.  
To demonstrate that the proposed approach reaches the global optimum of the problem, we evaluate all the possible phase-swapping configurations using an exhaustive search method based on nested loops, which takes about 46 h to evaluate 34 million combinations. However, with the proposed MI-QP, the global optimum finding takes about 104.17 s, demonstrating the efficiency of the proposed approach regarding the required processing times to solve this complex optimization problem.  
Figure 6 depicts the box-plot of thee-phase voltages, before and after the phase-balancing. This type of plot, provides a visual representation of the results under different scenarios; it also shows the minimum, maximum, median and first quartile. After the phase-balancing, the voltage profiles are enhanced by reducing the average unbalanced index from 0.3354 to 0.2921. In addition, the voltages moved to acceptable ranges between 0.9 and 1.1 pu. For example, the average of the voltages in Nodes 8, 9 and 10 was below 0.9, in the original system, and changed to values between 0.9 to 1.0 pu in the balanced system. There are, some scenarios with voltages below 0.9, even after balancing; however, this behavior occurs only in few critical scenarios.  
It is worth mentioning that the proposed MI-QP model can deal with planning and operating problems associated with phase balancing. However, the proposed model is most suitable for the planning stage, where a working group will configure all the phases before setting up the grid under operation. In the grid operation, all the grid nodes must have a three-phase device composed of switches to change the phase configuration. This option is possible in practice, but it is expensive for low-voltage applications and may introduce unwanted transient behavior.  
Figure 5: Power loss behavior for each one of the scenarios evaluated.