## 4 Use Cases
### Kumamoto System  
The Kumamoto system consists of 15 buses, including 1 root bus and 14 PQ buses. The network operates at a base voltage of 6.6 kV and a base power of 10 MVA. Detailed data on the buses and branches comprising this system are provided in [53]. For this study, additional branches were considered to enable the reconfiguration of the network, as shown in Figure 4. A list of these additional branches is provided in Table 1.  
To maintain a systematic approach, and given that this is a simulated scenario, a fixed percentage has been applied to the initial value of the distribution system to determine the maximum and minimum power values for all buses, which is set at 50%. However, it should be noted that these values could be different or randomly assigned for each bus, as the optimizer can work with any specified power range. The cost associated with power losses is 0.4 EUR/kW, and the cost associated with flexible power is 0.04 EUR/kVA. The algorithm code is executed in Matlab on a PC with an AMD Ryzen 5 2500U processor, Radeon Vega Mobile Gfx at 2.00 GHz, and 16.0 GB of RAM. Once the algorithm has been applied to the Kumamoto system in Figure 4 and has optimized all feasible radial systems, the optimal system is presented, and the overall results are evaluated. The runtime was 52.8784 s.  
A total of 432 feasible systems were found, and all of them were optimized. The system labeled as number 175 had the lowest total costs and is therefore considered the optimal configuration. In Figure 4(a),b, the topology and voltage profile of this optimal system are shown, respectively. Compared to the initial system, lines 4-7 and 8-11 are disconnected, while lines 2-10 and 3-13 are connected. The initial system exhibits 0.006484695 p.u. (64.85 kW) of power loss, with a cost of EUR 25.9387, while the optimal system shows 0.0028515 p.u. (28.5 kW) with a cost of EUR 11.406210. For the initial system, the total cost is equal to the cost of losses since no flexible power is being used. In contrast, in the optimized system, there is a cost of EUR 0.000013, indicating minimal use of flexible power. In the optimal system, power losses are reduced by 43.9% compared to the initial system, resulting in significant cost savings. The reason for the minimal use of flexible power in this system is due to the low profitability it would represent for the total calculation of losses, i.e., using flexible power would not have a sufficiently profitable impact to justify its use. In Figure 5(a), the total power losses for each optimized system are shown, compared to the initial system (indicated by a dashed black line). It can be observed that a significant number of optimized configurations exhibit lower losses compared to the initial system, demonstrating the effectiveness of the optimization algorithm in improving system efficiency in several cases. However, 309 systems out of a total of 432 show higher losses after optimization. In the graph, system number 175, highlighted in red, stands out as the optimal system, achieving the lowest losses. On the other hand, Figure 5(b) represents the total costs associated with each optimized system. Comparing both graphs, it is noted that the relationship between power losses and costs is not directly proportional. This is because, in many cases, the algorithm has made considerable use of flexible power to  
Figure 5: Reconfigured system number 175 (optimal system). (**a**) Topology, (**b**) voltage profile. reduce costs, which explains why, although some systems present higher losses, they are compensated with a reduction in total costs. In Figure 7, the surface of voltage profiles for each bus in the different optimized systems is illustrated. This three-dimensional graph allows visualization of how voltages vary depending on the system configuration and the buses considered. It is evident that not all configurations meet the voltage constraints, as voltage peaks outside the permitted range (\(0.95\leq V\leq 1.05\)) can be observed.