## 4 Optimization Method
### Case Study Descriptions and Simulation Results
#### 5.1.3 Case C: Probabilistic Multi-Objective RPP Considering the Load Demand Uncertainty  
In Cases A and B, the multi-objective RPP in power systems is solved using the deterministic approach. However, with the increasing level of uncertainty, probabilistic multi-objective is required for the RPP problem. In Case C, the probabilistic multi-objective RPP considering three different scenarios for the load level is performed. Each scenario for the load level consists of two main parts: (1) probability of the load level and (2) duration of the load. The overall duration of the load is assumed to be 8760 h, which is the expected time horizon for the RPP. The specifications of the system loading are described in Table 9.  
The simulation results obtained from the probabilistic multi-objective RPP considering the load demand uncertainty are given in Table 10. As it can be observed from this table, 15 Pareto optimal solutions are generated using the \(\varepsilon\)-constraint method. Using the min-max approach, the eighth solution (highlighted row) is chosen as the BSC. It is worth mentioning that the expected active power losses are 9.5049 MW for the BCS. From Table 10, it is clear that the expected total VAR cost is reduced compared with the Base Case. The expected voltage stability index and the expected loadability factor also show improvement towards the initial conditions. However, with more considerations, it is revealed that the expected active power losses are increased. This fact stems from the evident increase in the loadability factor. As a common incidence in power systems, following the escalation of the loadability factor, the active power losses increase and the system becomes voltage unstable. Generally, from the power systems operators' perspective, monitoring of the voltage magnitude at the load buses as a way of preventing voltage collapse is in high priority. Therefore, the voltage profile of the load buses for each loading scenario is plotted for the BCS, as shown in Figure 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(X\) & \(f_{1}\)(\$) & \(f_{2}\) & \(f_{3}\) & \(\overset{\wedge}{F_{1}}\) & \(\overset{\wedge}{F_{2}}\) & \(\overset{\wedge}{F_{3}}\) & \(\min(\overset{\wedge}{F_{1}}\overset{\wedge}{F_{2}}\overset{\wedge}{F_{3}})\) \\ \hline
1 & \(1.3129\times 10^{6}\) & 0.1294 & 0.0000 & 1.0000 & 1.0000 & 0.0000 & 0.0000 \\
2 & \(1.4223\times 10^{6}\) & 0.1339 & 0.0237 & 0.9443 & 0.9197 & 0.0722 & 0.0722 \\
3 & \(1.5418\times 10^{6}\) & 0.1383 & 0.0473 & 0.8834 & 0.8395 & 0.1443 & 0.1443 \\
4 & \(1.6647\times 10^{6}\) & 0.1428 & 0.0710 & 0.8207 & 0.7592 & 0.2165 & 0.2165 \\
5 & \(1.7954\times 10^{6}\) & 0.1472 & 0.0946 & 0.7542 & 0.6790 & 0.2887 & 0.2887 \\
6 & \(1.9328\times 10^{6}\) & 0.1517 & 0.1183 & 0.6841 & 0.5987 & 0.3609 & 0.3609 \\
7 & \(2.0774\times 10^{6}\) & 0.1561 & 0.1420 & 0.6105 & 0.5185 & 0.4330 & 0.4330 \\
8 & \(2.2510\times 10^{6}\) & 0.1606 & 0.1688 & 0.5220 & 0.4382 & 0.5148 & 0.4382 \\
9 & \(2.3848\times 10^{6}\) & 0.1637 & 0.1893 & 0.4538 & 0.3818 & 0.5774 & 0.3818 \\
10 & \(2.5365\times 10^{6}\) & 0.1673 & 0.2131 & 0.3765 & 0.3172 & 0.6501 & 0.3172 \\
11 & \(2.7000\times 10^{6}\) & 0.1713 & 0.2379 & 0.2932 & 0.2447 & 0.7256 & 0.2447 \\
12 & \(2.9942\times 10^{6}\) & 0.1783 & 0.2852 & 0.1433 & 0.1187 & 0.8700 & 0.1187 \\
13 & \(3.1909\times 10^{6}\) & 0.1828 & 0.3143 & 0.0431 & 0.0376 & 0.9586 & 0.0376 \\
14 & \(3.2627\times 10^{6}\) & 0.1844 & 0.3252 & 0.0065 & 0.0072 & 0.9919 & 0.0065 \\
15 & \(3.2754\times 10^{6}\) & 0.1848 & 0.3278 & 0.0000 & 0.0000 & 1.000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Obtained Pareto optimal solutions for the probabilistic multi-objective RPP considering the load demand uncertainty. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{Scenario}\) & \(\mathbf{Level}\) of the Load & \(\mathbf{Probability}\) & \(\mathbf{Duration}\) of the Load (h) \\ \hline \(S_{1}\) & 0.95 & 0.1 & 2920 \\ \(S_{2}\) & 1.00 & 0.8 & 4380 \\ \(S_{3}\) & 1.05 & 0.1 & 1460 \\ \hline \hline \end{tabular}
\end{table}