# Solution Algorithms for the Capacitated Location Tree Problem with Interconnections
## 6 Case Study  
The CLTPI was inspired by the need to redesign an electrical network for an electric public transportation system. The goal was to obtain a robust network capable of partial functioning in case of substantial energy disruptions due to different causes. The network requires connecting traction substations (clients) with the high-voltage electrical substations (plants). The electricity transportation system serves more than 1,057,461,875 passengers per year. The service is available from 5:00 a.m. to 11:59 p.m. during work days. On Saturday, the service starts at 7:00 a.m. and ends at 11:59 p.m. Finally, on Sundays and holidays, the service is available from 7:00 a.m. to 11:59 p.m. Failure to provide the service would strand thousands of people per hour, resulting in work delays and a decline in the local economy. The network extends over 226.49 km, distributed around the city. Within these kilometers, there are 174 traction substations. A traction substation converts electrical energy from the general network into the appropriate voltage, current, and frequency so that it can be used by the vehicles on the network. The power required by the traction substations is provided by high-voltage electrical substations. In the context of the CLTPI, the traction substations represent the clients, while the high-voltage electrical substations represent the plants. The problem of energy shortages in the transportation system lies in the growth of the population using this kind of transportation to travel around the city. Their growth  
Figure 8: Deviation to the best LB for instances with 100 vertices or more. Figure 7: Deviation to the best LB for instances with at most 50 vertices. means that track substations require more energy consumption. Energy consumption is increasing, causing the high-voltage electrical substations to be unable at some point to supply track substations' demand. This consumption by track substations and the capacity of the high-voltage electrical substations being unable to provide what is requested leads to a redesign of connections according to the capacities of each high-voltage electrical substation and track substation's demand. Track substation demand considers the peak hours, allowing the network operator to know the maximum demand that produces the highest energy consumption. The excess consumption produces energy shortages and, in some cases, the shutdown of the transportation system, causing travel delays. After a shutdown, the system needs to be restarted, leading to an excess demand for current, which could heat the connections and cause power shortages again. Nine electrical substations were considered for the case study. Three are already functioning (plants 0, 1, and 2), so their opening cost is zero. The other six have a specific opening cost and a given capacity. As for the track substations, 174 were considered. As previously mentioned, the idea was to redesign the network to obtain a robust one capable of partial functioning in case of critical energy disruptions. The strategy consisted of selecting high-voltage electrical substations that provide enough energy to provide the required power to all the track substations; it was not necessary to open all the high-voltage electrical substations, but all the track substations needed to receive the required energy from the selected high-voltage electrical substations. It was essential to maintain the system's stability while operating the electrical network, trying not to saturate the high-voltage electrical substations to prevent power outages or shortages. Table 6 presents the details for the high-voltage electrical substations. As previously mentioned, plants 0, 1, and 2 are operational and have zero opening costs. The capacities are expressed in MegaVolts-Ampers (MVA), and the opening costs are expressed in money units. The opening cost for each potential facility considers building and grid connection costs and other associated costs. For every track substation, a demand of 4.4 MVA was considered.