## 5 Applications
### Applications to Distribution Systems  
The resulting uncertainty case reduction strategies have been applied to the RESs penetrated test systems for calculating the total energy loss throughout the year, the lowest voltage and highest voltage values throughout the year, and the voltage out limits coefficient. Additionally, the detailed data of the year have been applied to the same problem. Moreover, the errors between each uncertainty case reduction strategy and the detailed data of the year have been calculated as shown in Tables 4 and 5. From the results, it is obtained that the lowest voltage values error does not exceed 4% in all reduction strategies. The highest voltage values error for all reduction strategies is very small and does not exceed 1%. Additionally, the energy losses over the whole year error does not  
Figure 14: Resulting uncertainty cases of the 64 uncertainty cases reduction strategy. Figure 15: One-year data through the sorting stages in the 81 uncertainty cases reduction strategy. exceed 5%. However, the error in voltage out limits coefficients is considered relatively large as it reached 17.8% in the 27 cases reduction strategy for 69 bus system. This is logical since the calculation of this coefficient depends on the summation of the system bus voltages over the whole year, and therefore if there is a percentage error in the voltage of each bus, the errors of all buses will be accumulated. Thus, the total error becomes large. This also explains the increase in this error by increasing the number of system buses as shown in the 64 and 81 cases reduction strategies. The 27 cases reduction strategy has error values larger than the 64 and 81 cases reduction strategies, but this strategy is characterized by the lowest uncertainty cases number. Thus, it is recommended in optimal planning studies as it is fast in testing an operation case over the whole year. The 64 and 81 cases reduction strategies compete for the lowest errors, as each one has its advantages. The 81 cases reduction strategy is more accurate in calculating the output of PV plants than the 64 cases reduction strategy. Moreover, the 81 cases reduction strategy is more representative than the 64 cases reduction strategy in the uncertainty parameter values throughout the year, as it divided the parameter values range into four sub-groups. Thus, the 81 cases reduction strategy is the best for the 69-bus system. However, in the 118-bus, the number of PV and WT units increases, where the summation of WT-rated power is larger than the summation of PV-rated power. Thus, in the 118-bus system, the 64 cases reduction strategy is the best. From all the previous results, it is obtained that the proposed cases reduction strategies in RESs uncertainty modeling in distribution systems are very effective and have an acceptable level of accuracy. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**System**} & **Modeling Method** & \multirow{2}{*}{\(\upsilon_{lowest}\)} & \multirow{2}{*}{\(\upsilon_{highest}\)} & \multirow{2}{*}{\(E_{loss}\)} & \multirow{2}{*}{\(\mathit{coff_{Vout}}\)} \\ \hline \multirow{4}{*}{69-bus} & All year data & 0.8784 & 1.0446 & 2.5543 \(\times 10^{8}\) & 108.1319 \\ \cline{2-6}  & 81 cases & 0.9063 & 1.0445 & 2.4597 \(\times 10^{8}\) & 99.7397 \\ \cline{2-6}  & 27 cases & 0.9121 & 1.0368 & 2.6527 \(\times 10^{8}\) & 88.7837 \\ \cline{2-6}  & 64 cases & 0.8995 & 1.0468 & 2.6776 \(\times 10^{8}\) & 97.3506 \\ \hline \multirow{4}{*}{118-bus} & All year data & 0.8760 & 1.0286 & 1.2142 \(\times 10^{9}\) & 217.2986 \\ \cline{2-6}  & 81 cases & 0.9058 & 1.0277 & 1.1743 \(\times 10^{9}\) & 193.3500 \\ \cline{2-6}  & 27 cases & 0.9091 & 1.0228 & 1.1663 \(\times 10^{9}\) & 180.8850 \\ \hline \multirow{2}{*}{2-6} & 64 cases & 0.8981 & 1.0283 & 1.1776 \(\times 10^{9}\) & 192.2738 \\ \cline{2-6}  & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The proposed reduction strategy results in distribution systems operation.