## 4 Simulation Results
### Solution Results and Comparison  
To verify the effectiveness of the proposed approach, the performances of conventional OPF, VSCOPF, and P-VSCOPF with respect to CM were compared for the modified IEEE-24 bus system. In our study, all three methods were utilized to consider the optimal installation of the wind farm in bus 3, which is obtained through a CDF based on VSI. However, both conventional OPF and VSCOPF, which take into account only the mean values of loads and the wind farm's output, do not consider the probabilistic effects between the loads and the wind farm's variability. Meanwhile, P-VSCOPF consists of two parts: an MCS for selecting both the wind output and load input variables and MOPSO for solving the multi-objective optimization problem. It was implemented based on 1000 MCS trials. Table 3 presents the optimal solution results for P-VSCOPF. Note that these results represent the mean values of 1000 MCS trials based on MOPSO. The power generation (\(P_{G}\)) in bus 3, which was connected to the wind farm, was 295.1 MW. This value was determined by the performance of 1000 MCSs with probabilistic variables taken into account. Figure 8 reveals the estimates of the Pareto optimal front, which relates \(Pay_{ISO}\) and the loading parameter \(\lambda_{c}\). Note that the Pareto optimal front of the test system exhibits highly nonlinear relationships between the social welfare and the voltage stability margin. A suitable range for the weighting factors is approximated to be 0.4-0.8. In fact, for values smaller than 0.4, system security can almost be ignored, and for values higher than 0.8, market power will happen. On the other hand, with respect to the weighting factors within this range, an importance of between 0.4 and 0.8 was assigned to the voltage security. Therefore, we assume that the results of our work apply to test systems with \(w=0.6\). \begin{table}
\begin{tabular}{c c c} \hline
**Rank** & **Between Buses** & **VSI** \\ \hline
1 & 3–24 & 1121.57 \\
2 & 23–20 & 735.85 \\
3 & 17–16 & 698.50 \\
4 & 16–14 & 473.11 \\
5 & 19–20 & 475.72 \\
6 & 19–16 & 350.73 \\
7 & 18–17 & 335.22 \\
8 & 21–15 & 304.80 \\
9 & 10–6 & 274.51 \\
10 & 1–2 & 223.31 \\ \hline \end{tabular}
\end{table} 
Table 1: Ranking results and corresponding VSI values. \begin{table}
\begin{tabular}{c c c c} \hline
**Bus** & **CDF** & **Bus** & **CDF** \\ \hline
1 & 0.4330 & 13 & \(-\)0.0768 \\
2 & 0.4853 & 14 & \(-\)0.3205 \\
3 & 1.3113 & 15 & \(-\)0.7625 \\
4 & 0.3358 & 16 & \(-\)0.634 \\
5 & 0.0359 & 17 & \(-\)0.5492 \\
6 & \(-\)0.4014 & 18 & \(-\)0.5419 \\
7 & \(-\)0.0498 & 19 & \(-\)0.5507 \\
8 & 0.1511 & 20 & \(-\)0.4097 \\
9 & 0.3992 & 21 & \(-\)0.6040 \\
10 & 0.1580 & 22 & \(-\)0.5824 \\
11 & \(-\)0.0463 & 23 & \(-\)0.3308 \\
12 & 0.0232 & 24 & \(-\)1.8239 \\ \hline \end{tabular}
\end{table} 
Table 2: CDFs for lines between buses 3 and 24. Figure 9 compares the LMPs and the NCPs. As shown in Figure 9a, for CM, the LMP differences were less with P-VSCOPF than they were with either the conventional OPF or VSCOPF. It is clear that the congestion problems were resolved by minimizing the LMP differences and then maintaining them at the lowest possible levels. As shown in Figure 9b, the range of NCP values was \(-0.6954\) to \(0.5962\) $/MWh. Here, NCP values at bus 13 (the slack bus) were zero. Lower NCP values indicate lower congestion in the test system. The congestion mitigation of P-VSCOPF was the most effective among those of all of the methods, since the NCP of P-VSCOPF was the lowest. A comparison of the results obtained in each case reveals that the best solution for CM was obtained with P-VSCOPF. Consequently, appropriate realizations of both the voltage stability constraints and the contingency analyses not only result in a better distribution of electricity prices but also reduce the influence of system congestion. \begin{table}
\begin{tabular}{c c c c c c} \  
Table 4 shows a comparison of the optimal solutions for the three methods.