## 3 Case Study
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Appliances** & **[W]** & **Appliances** & **[W]** & **Appliances** & **[W]** & **Appliances** & **[W]** \\ \hline Fridge & 350 & TV set (x3) & 38 (x3) & Stereo & 100 & Microwave oven & 1250 \\ CD player & 15 & DVD player & 34 & Iron & 1000 & Coffee maker & 800 \\ Clock & 2 & Digital TV receiver & 20 & Personal computer & 60 & Washer & 405 \\ Wireless phone & 2 & Oven & 2125 & Printer & 40 & Lighting & 15–100 \\ \hline \end{tabular}
\end{table} 
Table 2: List of CREST Demand Model appliances. Figure 4: Load profiles for one week of the year for different groups of: (**a**) 1 residential household and (**b**) 100 residential households. Since the planning of the microgrids will consider the size of assets (PV generator, storage systems and inverters), it is necessary to include their investment costs. These costs are included from annuities in Equation (42). \[A=VP\cdot\frac{r\cdot(r+1)^{n}}{(r+1)^{n}-1} \tag{42}\]  
where \(A\) is the annuity of the investment, \(VP\) is the present value of the investment, \(r\) is the discount rate and \(n\) is the number of years or period of analysis. Table 3 shows the annualized investment costs of the assets for the planned microgrids, considering a discount rate of 3%. The costs of on-grid, hybrid, inverter, charger and battery inverters and panels [45,46] are included. The energy purchase and sale prices have also been considered [47]. The lifetime corresponds to 20 years for the PV system with an on-grid inverter or hybrid inverter, 10 years for the batteries and 15 years for the inverter-charger. From the proposed models, hundreds of planning exercises were carried out to study the behavior of the optimization variables. For this, the following assumptions were taken into account:  
* A full year is analyzed, with a time interval of 1 h and considering a total project duration of 20 years. * A total of 100 different microgrids per group of 1, 10, 20 and 100 residential households were optimally determined with the objective of evaluating the variation in the results for different groups of clients. * For each proposed microgrid model, different reliability requirements are taken into account, with various simulated faults in the power grid. Thus, for the deterministic planning approach, faults of 4, 8 and 12 continuous hours of duration are considered, which are distributed throughout the year of analysis, according to the monthly historical probability of fault for a distribution circuit of the study zone. * For the stochastic planning of microgrids, explicit faults are considered according to the scenarios identified with [PERSON]'s agglomeration method (three different start times  
Figure 5: Average monthly solar generation profiles for the city of Santiago de Chile. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Cost** \\ \hline Solar PV system with on-grid inverter & USD 101.4/kW \\ Solar PV system with hybrid inverter & USD 103.5/kW \\ Lead acid battery & USD 13.8/kWh \\ Inverter charger & USD 11.3/kW \\ Energy purchase price & USD 0.124/kWh \\ Energy sale price & USD 0.068/kWh \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Base investment costs, annualized with a 3% discount rate. Source: [45, 46, 47]. for each group of users). For this planning approach, only faults with a duration of 8 h are considered. After the optimization process, sensitivity analyses are carried out for the investment costs of the microgrid assets. For this purpose, the costs for EoS in Table 4 are considered. The concept of EoS refers to reductions in investment costs per unit as the size of a facility increase [48]. Furthermore, a case will be established without reliability requirements, which will serve as a base case to compare the results of the proposed models. This case considers the installation of assets, assuming that there are no faults in the power grid (without active reliability constraints: Equations (22), (23), (38) and (39)).