## 4 Numerical Application
### Case Study 2  
In this case the support of the distributed resources is considered and DESSs are allocated by means of a GA applied over the set constituted by all network's buses. A maximum value of 440 kWh was imposed to the capacity that could be allocated in the whole network and a maximum value of 12 kWh was imposed to the capacity that could be placed at the single bus. Both single-phase and three-phase DESSs were allocated whose location and capacity are reported in Table 7. Six three-phase DESSs were allocated, two of them in the residential feeder and four in the commercial feeder. Many single-phase devices were also allocated in both these feeders whereas no DESS were allocated in the industrial feeder. The overall DESS's capacity resulted equal to 332 kWh that allowed a reduction of the total cost of more than 5% compared to case 1. Moreover, thanks to the support of the resources, the operation of the grid improved in terms of unbalances, line currents and bus voltages. This can be observed in Figures 7-9 which show, still referring to the summer typical day of the last year, the hourly unbalance factor at all buses, the line currents and the voltage magnitude (both referring to the residential feeder at hour 21.00), respectively. candidate buses with reference to voltage deviation and unbalance factor and assuming a threshold limit of 10%. Regarding the line currents, the loading constraints criterion was adopted to identify the candidate buses where the critical events were identified by considering the overload events of line current greater than the line ampacity. The resulting ABSs are shown in Table 8, with reference to each feeder. In Table 9 the results of the proposed approach in terms of location and capacity of the allocated DESSs are reported. Eight three-phase DESSs were allocated, five of them in the residential feeder, two in the commercial feeder and one in the industrial feeder. Six single-phase devices were also allocated in both residential and commercial feeders whereas no single-phase DESSs were allocated in the industrial feeder. The overall DESS's capacity resulted equal to 344 kWh that allowed a reduction of the total cost of more than 6% compared to case 1. By comparing Tables 8 and 9, it is interesting to note that DESSs are placed in almost all of the candidate buses. Moreover, Table 9, compared to Table 7, shows that the size of the DESSs allocated in case 3 is generally larger than that of the DESSs allocated in case 2. This allowed a significant reduction of the objective function compared to case 2 still preserving the network constraints, as shown in Figures 10-12. These figures show the hourly unbalance factor at all buses, the line currents and the bus voltages, respectively. As an example of further results obtained, Figures 13-15 show the active and reactive power of the controllable distributed resources. More specifically, still referring to the summer typical day of the last year, the phase active power daily profiles of the DESS placed at bus no.37 are reported in Figure 13. The corresponding stored energy profile is show in Figure 14. Figure 15 reports the reactive power support provided by the DESS placed at bus no.37 (only phase 1 is shown) and the reactive power supplied by the DG unit located at the bus no.15 (phase 2). As clearly shown in these figures, the constraints imposed on the resources are satisfied. Furthermore, even if the figure is not reported here for the sake of conciseness, the constraints imposed by the rated powers of the interfacing converters of both DESSs and DG units are satisfied. Finally, aimed at highlighting the advantages that can be drawn by the use of the proposed hybrid ABS and GA solution method, in Figure 16 the best values of the fitness function for all of the populations created and analyzed by the GA (case 2) and the ABS/GA (case 3) are reported versus the number of individuals investigated (particularly, the values are reported in per unit with reference to the value assumed in case 1).