## 4 Results
If one or more eigenvalues are located nearer to the imaginary axis line, then the amplitude of the oscillation modes is very high. The Scenario A-1 eigenvalue sectors are displayed in Figure 17. Overall, all methods were successful in moving the eigenvalue location to the left side of the phase plan, indicating that all systems were stable. In this case, PIDF-EPSCA outperformed the other three strategies by allocating the majority of the eigenvalue locations to the far left. One of the eigenvalues from the other three mapping methods was placed closest to the origin. The coordinates, which were nearly at the same location from \(-0.6427\) p.u to \(-0.5339\) p.u on the real axis, were thought to be in a less stable condition than PIDF-EPSCA, which was at \(-0.6687\) p.u. The output of PIDF-EPSCA further demonstrated that the eigenvalue sector was moved to the real-imaginary plane's nearest point towards the real-axis. The fact that the PIDF-MFO and PIDF-EP systems both had a pair of eigenvalues that were closest to the imaginary axis and furthest from the real axis, respectively, supports the notion that these two approaches were the worst. Thus, PIDF-EPSCA was confirmed as the suitable option for improving SMIB system stability. Figure 16: Speed deviation response of the TCSC-PIDF controller using PIDF-EPSCA, PIDF-SCA, PIDF-MFO, and PIDF-EP techniques for Scenario A-1. Table 3 shows the N, \(K_{\rm P}\), \(K_{\rm D}\), and \(K_{\rm I}\) parameters' objective functions, number of iterations, and settling times for the four optimisation techniques for Scenario A-1. In comparison to existing optimisation techniques, the suggested hybrid EPSCA strategy offered improved damping of angle deviation and eigenvalue location. In terms of eigenvalue location, the suggested EPSCA method offered a greater shift towards the real axis of the s-plane to reduce the abrupt variation in the amplitude of angle deviation and boosts system stability by moving the eigenvalues farther to the left of an imaginary axis line. In contrast to other optimisation techniques, the PIDF-EPSCA system needed nine iterations to converge the solution (Table 3). Unlike other optimisation techniques, the PIDF-SCA method needed 25 iterations to converge the maximum objective function (Table 3). The MFO and EP systems converged on the results after 60 and 10 iterations, respectively. The minimum objective function value of the PIDF-EP system resulted in a higher convergence rate. The suggested method had greater computing efficiency than the PIDF-EP system and provided a superior objective function than the three other techniques. As a result, the suggested optimisation strategy, which uses MO as the objective function, was able to simultaneously improve both DF and DR by giving them equal weight in the development process for a system with improved damping capabilities. This  
Figure 17: Eigenvalue distribution of the TCSC-PIDF controller system using PIDF-EPSCA, PIDF-SCA, PIDF-PRO, and PIDF-EP techniques for Scenario A-1. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Type** & **PIFF-EPSCA** & **PIDF-SCA** & **PIDF-MFO** & **PIDF-EP** \\ \hline \multirow{4}{*}{Parameters optimised} & \(K_{\rm P}=0.024275\) & \(K_{\rm P}=0.023936\) & \(K_{\rm P}=0.038666\) & \(K_{\rm P}=0.039688\) \\  & \(K_{\rm D}=0.000531\) & \(K_{\rm D}=0.000318\) & \(K_{\rm D}=0.000202\) & \(K_{\rm D}=0.000169\) \\  & \(K_{\rm I}=0.111213\) & \(K_{\rm I}=0.018067\) & \(K_{\rm I}=0.084392\) & \(K_{\rm I}=0.084817\) \\  & \(N=122.1625\) & \(N=102.0015\) & \(N=184.5104\) & \(N=184.2895\) \\ \hline \multirow{3}{*}{Objective function value} & 0.1740 & 0.1338 & 0.1240 & 0.1253 \\  & DR \(=0.9041\) & DR \(=0.9070\) & DR \(=0.8179\) & DR \(=0.8258\) \\ \cline{1-1}  & DF \(=-0.6687\) & DF \(=-0.6427\) & DF \(=-0.5660\) & DF \(=-0.5339\) \\ \hline No. of iterations & 9 & 25 & 60 & 10 \\ \hline Settling time (s) & 1.35 & 2 & 2.35 & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of PIDF-EPSCA, PIDF-SCA, PIDF-MFO, and PIDF-EP for Scenario A-1.