## 4 Results and Discussion  
This section presents OPF results obtained by ESDO algorithm and compared to PSO, GOA, HHO, MVO and HO techniques on the IEEE 30-bus power systems. The IEEE 30-bus evaluation system has 41 transmission lines, 6 generators at buses 1, 2, 5, 8, 11, and 13, 4 transformers with off-nominal tap ratio at lines 6-9, 6-10, 4-12, and 28-27, and 9 shunt Var compensators at buses 10, 12, 15, 17, 20, 21, 23, 24, and 29. At a base of 100 MVA, the overall load demands are Pload = 2.834 p.u. and Qload = 1.262 p.u. There are 24 control variables, including 5 generator active power outputs, 6 generator voltage magnitudes, 4 transformer tap settings, and 9 shunt Var compensator power injections. Figure 3 illustrates a single-line network design including the control variables' upper and lower limits. The system data was gathered from [56; 57; 58; 59]. Table 1 lists active and reactive power limitations in addition to generator cost coefficients, whereas Tables 2 and 3 offer system load and line data.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Bus No.**} & \multicolumn{2}{c}{**Active Power Limits**} & \multicolumn{2}{c}{**Reactive Power Limits**} & \multicolumn{2}{c}{**Cost Factors**} \\ \cline{2-9}  & \(\mathbf{P_{Gmin}}\) **(p.u.)** & \(\mathbf{P_{Gmax}}\) **(p.u.)** & \(\mathbf{Q_{Gmin}}\) **(p.u.)** & \(\mathbf{Q_{Gmax}}\) **(p.u.)** & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) \\ \hline
1 & 0.5 & 2.5 & \(-\)0.2 & 2 & 0 & 200 & 37.5 \\
2 & 0.2 & 0.8 & \(-\)0.2 & 1 & 0 & 175 & 175 \\
5 & 0.15 & 0.5 & \(-\)0.15 & 0.8 & 0 & 100 & 625 \\
8 & 0.1 & 0.35 & \(-\)0.15 & 0.6 & 0 & 325 & 83.4 \\
11 & 0.1 & 0.3 & \(-\)0.1 & 0.5 & 0 & 300 & 250 \\
13 & 0.12 & 0.4 & \(-\)0.15 & 0.6 & 0 & 300 & 250 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Generator data for IEEE 30-bus test system.  
Figure 3: IEEE 30-bus single-line network.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Bus Tag** & \
\begin{tabular}{c} **Active Power** \\ **P (p.u.)** \\ \end{tabular} &
\begin{tabular}{c} **Reactive Power** \\ **Q (p.u.)** \\ \end{tabular} \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 2: Load data for IEEE 30-bus test system.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Line Tag** & **From Bus** & **To Bus** & **R (p.u.)** & **X (p.u.)** & **B (p.u.)** & **Tap Settings** \\ \hline
1 & 1 & 2 & 0.0192 & 0.0575 & 0.0264 \\
2 & 1 & 3 & 0.0452 & 0.1852 & 0.0204 \\
3 & 2 & 4 & 0.0570 & 0.173  
In this part, four case studies will be provided to evaluate performance across various operating modes. In the first scenario, the OPF's aim was to reduce costs, but the second case demonstrates the performance of the proposed algorithm when the intention was to minimize transmission losses. Case 3 illustrates the behavior of the suggested algorithm when seeking to reduce voltage drop on transmission lines. The last scenario presents the behavior of the suggested algorithm when through multi-objective function that has three objectives: minimize cost, power losses, and voltage drops. The proposed algorithm's performance is compared to the PSO algorithm, which is a well-known technique for solving the OPF problem, MVO, GOA, HHO, and HO algorithms. All simulations are carried out using MATLAB \"2024a\".