## 4 Results and Analysis
### Solution of the Deterministic Planning Problem
#### 4.1.2 Results for the Cases with Reliability Requirements
It can be seen that for the microgrids presented, the optimization algorithm decides to install panels with on-grid inverter technology, an inverter charger and batteries. In the same way as for planning without reliability requirements, a reduction in investment costs of groups of 10, 20 and 100 clients is observed, with respect to the investment cost of the one-client microgrid, due to the diversity of consumption (18.8%, 21.2% and 23.3% of costs reduction) and owing to EoS (12.0% on average of additional costs reduction). It should be noted that the results presented in Table 6 correspond to a single optimization exercise for each group of residential households (one of the 100 exercises performed for each group of \(n\) households), to show the alternative configuration that in each case the optimization algorithm decides to consider. Additionally, in order to show how the microgrid is operating, the one-client microgrid is taken as an example. The graphical results considering the specific 8 h fault, with a start time at 906 h, is shown in Figure 9.  
It is observed that the batteries provide energy in some instants of no fault. It is also highlighted that when the microgrid operates in connected mode, power was injected into the power grid, generating income for the household. The participation of the power grid is low compared to solar PV generation (the microgrid tries to use the main grid as little as possible and take advantage of the solar resource to supply all the demand). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Clients of Each Microgrid** & **1** & **10** & **20** & **100** \\ \hline Panel power [kW] & 2.98 & 24.1 & 46.2 & 225 \\ Panel power per household [kW] & 2.98 & 2.41 & 2.31 & 2.25 \\ Inverter technology & On-grid & On-grid & On-grid & On-grid \\ Battery capacity per household [kWh] & 2.37 & 1.92 & 2.11 & 2.05 \\ Inverter charger capacity per household [kW] & 0.35 & 0.34 & 0.29 & 0.28 \\ Investment cost per client (without EoS) [USD/year] & 338.4 & 274.7 & 266.6 & 259.6 \\ Investment cost per client (with EoS) [USD/year] & 338.4 & 234.6 & 227.9 & 221.3 \\ Cost function per household (with EoS) [USD/year] & 196.7 & 145.5 & 133.6 & 133.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results for an 8 h fault (start time: 906 h, Case 1). Figure 9: Operation graph of a household’s microgrid for an 8 h fault (start time: 906 h, Case 1). Results for Microgrids Planning with Reliability Requirements Only for Critical Loads (Case 2)  
The results of hundreds of microgrids for Case 2 (ensuring supply of a critical demand during a fault) are presented in Figure 10. The diversity of consumption, by grouping a different number of households and the EoS associated with investment costs of microgrid infrastructure, allows for a joint reduction in investment costs for the microgrids of 10, 20 and 100 clients, equivalent to 21.1%, 24.0% and 29.0%, compared to the average cost of one-client microgrids and taking the 90% protection level as a reference. The rate of increase in investment costs, as protection level and duration of faults increase, is lower than the rate at which the increase in costs occurs in Case 1. Precisely, in Figure 10b, for 100 residential households, the increase in the investment average cost, which is the result of preparing the microgrids for longer hours of operation in islanded mode, is 13.6% for faults from 4 to 12 h (12% less than in Case 1). By increasing the duration of the fault, the amount of critical demand to be satisfied does not increase significantly, i.e., since the critical demand is low and scattered during the day, additional hours of interruption do not significantly increase the energy requirements, and therefore  
Figure 10: Investment cost per household as a function of (**a**) protection level for faults of 8 h duration and (**b**) duration of faults with 90% protection level (Case 2).