## 4 Numerical Application  
The procedure described above was applied to a LV distribution network which is a Cigre benchmark system whose scheme is shown in Figure 2 and whose details are reported in [45]. The grid'snominal voltage is 400 V, it includes 41 busses and is connected to a 20 kV MV grid by means of three transformers of 500, 150 and 300-kVA. The three transformers supply a residential, a commercial and an industrial feeder, respectively. The price of energy, which is based on an existing tariff, is reported in Table 5[46]. DG units are supposed to be connected to the grid. They are four single-phase 5 kW PV units connected to the buses no.15 (phase no.2), no.19 (phase no.1), no.37 (phase no.3), no.40 (phase no.3) and one three phase 20 kW PV unit located at bus no.21. It is supposed that both single-phase and three-phase DESSs can be allocated whose base size is 4 kWh having a nominal discharging time of 5 h. DESSs are based on Li-Ion batteries technology having an efficiency of 90% in charging mode and 92% in discharging mode. The batteries are assumed having a maximum DoD of 80%. To maximize their lifetime, the batteries can have only one charging/discharging cycle per day. More specifically, they can charge during the off-peak hours indicated in Table 5 and discharge during the rest of the day. The battery's installation cost is supposed 600$/kWh whereas the replacement cost is assumed 250$/kWh. Two typical days were considered for the cost evaluation, one winter day and one summer day. Both effective rate of change and discount rate are assumed equal to 3%. An annual 2% growth is  
Figure 2: Cigrè low voltage (LV) test grid \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Summer Tariff**} & \multicolumn{2}{c}{**Winter Tariff**} \\ \cline{2-5}  & **Period** & **Price (\$/MWh)** & **Period** & **Price (\$/MWh)** \\ \hline
**On peak** & 12:00–18:00 & 542.04 & & \\ \hline
**Part Peak** & 8:30–12:00 & 252.90 & 8:30–21:30 & 161.96 \\
18:00–21:30 & 252.90 & & & \\ \hline
**Off Peak** & 21:30–8:30 & 142.54 & 21:30–8:30 & 132.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Time of Use (ToU) Tariff. assumed for the load demand. As an example, Figure 3 shows the per unit value profiles of three loads of the residential, industrial and commercial feeder, respectively. The locations of the loads and their peak power are reported in Table 6. These values were derived from [45] and properly modified to obtain critical conditions in terms of voltages and currents. Constant power factors of 0.95 (residential), 0.85 (industrial) and 0.9 (commercial) have been assumed during the simulations. In what follows three case studies have been reported referring to:  
1. a reference case without DESS;
2. DESSs are allocated by means of only a GA;
3. DESSs are allocated by means of the proposed procedure (GA applied for ABS). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bus** & **Phase** & **Active Power [kW]** & **Bus** & **Phase** & **Active Power [kW]** \\ \hline \multicolumn{5}{c}{**Residential Feeder**} & \multicolumn{5}{c}{**Commercial Feeder**} \\ \hline \multirow{3}{*}{12} & 1 & 2.4 & & 1 & 0.6 \\  & 2 & 1.9 & 33 & 2 & 0.6 \\  & 3 & 5.2 & & 3 & 3.6 \\ \hline \multirow{3}{*}{16} & 1 & 17.4 & & 1 & 1.2 \\  & 2 & 14 & 34 & 2 & 1.2 \\  & 3 & 38 & & 3 & 7.2 \\ \hline \multirow{3}{*}{17} & 1 & 8.7 & & 1 & 1.5 \\  & 2 & 7 & 35 & 2 & 1.5 \\  & 3 & 19.2 & & 3 & 9.0 \\ \hline \multirow{3}{*}{18} & 1 & 5.5 & & 1 & 0.72 \\  & 2 & 4.4 & 38 & 2 & 0.72 \\  & 3 & 12.2 & & 3 & 4.3 \\ \hline \multirow{3}{*}{19} & 1 & 7.4 & & 1 & 1.5 \\  & 2 & 6 & 39 & 2 & 1.5 \\  & 3 & 16.4 & & 3 & 9.0 \\ \hline \multicolumn{5}{c}{**Industrial feeder**} & \multicolumn{5}{c}{1} & 10.48 \\ \hline \multirow{3}{*}{21} & 1 & 14.2 & 40 & 2 & 0.48 \\  & 2 & 11.3 & & 3 & 2.9 \\ \cline{1-1}  & 3 & 31.2 & & 1 & 0.48 \\ \cline{1-1}  & & & 41 & 2 & 0.48 \\ \cline{1-1}  & & & & 3 & 2.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Peak Active Power and location of loads.