## 5 Result and Discussion
### Performance Test  
The multi-objective performance of the MNSGA-II was evaluated by comparing the Pareto solutions obtained using the NSGA-II and MNSGA-II for 30 simulations performed on both the IEEE test system and actual power system. The convergence metric, spread/diversity metric, inverted generational distance, and minimum spacing metric were calculated for the non-dominated solutions obtained using the NSGA-II and MNSGA-II [38]. A total of 30 independent trials were conducted to select the appropriate values for these parameters and the optimal parameters selected are presented in Table 10. In this case, the initial weight factor, \(w\), was set to 0.5. Figure 12 displays the optimization results of the Pareto solution obtained in Case 2 among the particle swarm optimization (PSO) [39], adaptive modified PSO (AMPSO) [40], strength Pareto evolutionary algorithm (SPEA) [41], genetic algorithm (GA) [42], NSGA-II [43], and MNSGA-II on the test system and actual system. Note that the third point converged as the Pareto solution in the case of MNSGA-II for both systems. Table 11 indicates the comparison results for each system. According to the comparison results, it is clear that the DSO operating cost and MGO profit were lower in the proposed MNSGA-II than in the other algorithms for both systems. These results may be small in terms of the overall operation, but a value is not small in terms of the cost regarding the margin of the tie-line between the DN and MG for sharing the exchange information. As shown in Table 11, the run time of the proposed approach was significantly reduced compared to the others, particularly for large power \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**IEEE Test System**} & \multicolumn{2}{c}{**Actual Power System**} \\ \cline{2-5}  & **NSGA-II** & **MNSGA-II** & **NSGA-II** & **MNSGA-II** \\ \hline Population size & 100 & 100 & 200 & 200 \\ Max. no. of generations & 30 & 30 & 30 & 30 \\ Crossover probability & 0.8 & 0.8 & 0.9 & 0.9 \\ Mutation probability & 1/12 & 1/12 & 1/75 & 1/75 \\ Crossover index & 1 & 1 & 2 & 2 \\ Mutation index & 10 & 10 & 20 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Parameters used for non-dominated sorting genetic algorithm II (NSGA-II) and modified version of the NSGA-II (MNSGA-II). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multicolumn{2}{c}{**Operation Cost (\$/h)**} & \multirow{2}{*}{**Note**} \\ \cline{2-3} \cline{5-5}  & **DSO** & **MGO** & \\ \hline Scenario A & 2,527,983 & 1,387,562 & No bi-level model with Case 2 \\ Scenario B & 2,320,075 & 1,450,047 & Proposed bi-level model with Case 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of operational costs for each Scenario in Case 2. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Operational Cost**} & \multirow{2}{*}{**Profit (MGO)**} \\  & **(DSO)** & \\ \hline Case 1 & 2,548,231 & 988,274 \\ Case 2 & 2,320,075 & 980,610 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison optimal results for each case. systems, owing to the parallel processing based on the bi-level model. These results demonstrate that the proposed approach is suitable for the requirements of realistic power system operation.