## 4 Optimization Method
### Case Study Descriptions and Simulation Results
#### 5.1.1 Case A: Deterministic Multi-Objective RPP without Considering the Loadability Factor  
In order to validate the efficiency of the proposed method for multi-objective RPP, the \(\varepsilon\)-constraint method is applied to a deterministic multi-objective RPP problem. The obtained results are also compared with the approach presented in [3]. The deterministic multi-objective RPP aims to minimize the total VAR cost and voltage stability index. Thus, it is expected to achieve a reduction in active power losses and an improvement in the voltage stability index. Table 4 shows the obtained results from deterministic multi-objective RPP without considering the loadability factor in IEEE 30-bus test system with the initial settings. The duration of the load for deterministic multi-objective RPP is assumed to be 8760 h for full-load condition and without changes in the load level. According to Table 4, 15 Pareto optimal solutions are obtained by the \(\varepsilon\)-constraint method. After that, the min-max approach chooses the fifth solution (highlighted row) as the BCS. The active power losses for the BCS are 4.9813 MW. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\chi\) & \(f_{1}\)(\$) & \(f_{2}\) & \(\overset{\wedge}{F}_{1}\) & \(\overset{\wedge}{F}_{2}\) & \(\mathbf{min}\overset{\wedge}{(F_{1},F_{2})}\) \\ \hline
1 & \(3.0334\times 10^{6}\) & 0.1241 & 0.0000 & 1.0000 & 0.0000 \\
2 & \(2.9064\times 10^{6}\) & 0.1241 & 0.3033 & 0.9286 & 0.3033 \\
3 & \(2.8182\times 10^{6}\) & 0.1246 & 0.5139 & 0.8571 & 0.5139 \\
4 & \(2.7532\times 10^{6}\) & 0.1249 & 0.6692 & 0.7857 & 0.6692 \\
5 & \(2.7084\times 10^{6}\) & 0.1252 & 0.7762 & 0.7143 & 0.7143 \\
6 & \(2.6802\times 10^{6}\) & 0.1255 & 0.8437 & 0.6429 & 0.6429 \\
7 & \(2.6611\times 10^{6}\) & 0.1257 & 0.8892 & 0.5714 & 0.5714 \\
8 & \(2.6496\times 10^{6}\) & 0.1260 & 0.9166 & 0.5000 & 0.5000 \\
9 & \(2.6415\times 10^{6}\) & 0.1263 & 0.9360 & 0.4286 & 0.4286 \\
10 & \(2.6350\times 10^{6}\) & 0.1266 & 0.9516 & 0.3571 & 0.3571 \\
11 & \(2.6294\times 10^{6}\) & 0.1269 & 0.9649 & 0.2857 & 0.2857 \\
12 & \(2.6247\times 10^{6}\) & 0.1271 & 0.9762 & 0.2143 & 0.2143 \\
13 & \(2.6207\times 10^{6}\) & 0.1274 & 0.9858 & 0.1429 & 0.1429 \\
14 & \(2.6174\times 10^{6}\) & 0.1277 & 0.9936 & 0.0714 & 0.0714 \\
15 & \(2.6147\times 10^{6}\) & 0.1280 & 1.0000 & 0.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Obtained Pareto optimal solutions for the deterministic multi-objective RPP without considering the loadability factor. \begin{table}
\begin{tabular}{c c} \hline \hline
**Control Variable** & **Value** \\ \hline \(V_{gi}^{min}\) (p.u.) & 0.900 \\ \(V_{gi}^{max}\) (p.u.) & 1.100 \\ \(t_{k}^{min}\) (p.u.) & 0.900 \\ \(t_{k}^{max}\) (p.u.) & 1.100 \\ \(Q_{Ci}^{min}\) (MVAR) & 0.000 \\ \(Q_{Ci}^{max}\) (MVAR) & 35.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Control variables limits. Considering the same operating condition, the \(\varepsilon\)-constraint method is compared with the Multi-Objective Differential Evolution (MODE) algorithm, which is recommended to solve the deterministic multi-objective RPP [3]. For the BCS, the results of the comparison are presented in Table 5. As it can be observed from Table 5, the \(\varepsilon\)-constraint method shows better performance compared with the MODE algorithm in minimizing total VAR cost and active power losses. The superiority of the \(\varepsilon\)-constraint method is confirmed by a 6.2578 % reduction in total VAR cost and a 9.3815 % decrease in the active power losses over the Base Case. However, as it can be observed from Table 5, the voltage stability index of the conventional method is better than the proposed approach. The optimal values of the control variables for Case A are represented in Table 6. As it can be observed, only one VAR source has a value of zero. Note that the fixed installation VAR cost is also considered for all VAR sources with the value of zero during the planning studies.