## 4 Simulation Results and Discussion
### Scenario (1)  
In this scenario, the power system being analyzed (Figure 2) experiences a disruption in its load with an amplitude of \(\Delta P_{L}\) = 0.1 pu at \(t\) = 1 s, as depicted in Figure 6. The fundamental parameters pertaining to the DOSA and FOPD-FOPID cascaded controller are presented in Table 2. Figure 7 illustrates the optimization process of FOPD-FOPID controller parameters using different algorithms, namely DOSA, MSA, PSO, ABC, and GTO. The objective function utilized is _ISE_. As demonstrated by Figure 7, it can be observed that the DOSA algorithm yields a swifter convergence rate than the others mentioned. In terms of _ISE_ values obtained from the respective algorithms (DOSA: 6.8 \(\times\) 10\({}^{-6}\); GTO: 9 \(\times\) 10\({}^{-6}\); MSA: 9.1 \(\times\) 10\({}^{-6}\); ABC: 9.9 \(\times\) 10\({}^{-6}\); and PSO: 10 \(\times\) 10\({}^{-6}\)), they are displayed in numeric format for easier comprehension. In Table 3, the values of optimized parameters of  
Figure 5: Utilization of the DOSA to optimize the parameters of the FOPD–FOPID controller.  
the FOPD-FOPID controller considering the _ISE_ objective function for DOSA, GTO, MSA, ABC, and PSO algorithms are shown. According to Figure 7 and Table 3, the DOSA can be used to optimize the FOPD-FOPID controller parameters. In this scenario, load disruptions and wind turbine disruptions have been applied to the power system according to Figure 8. In Figure 9a-e, the FR of the power system using different control methods to load and wind turbine disruptions is shown. According to Figure 9a, the MFD and ST based on LFC and SMES based on the proposed controller (optimized FOPD-FOPID controller using the DOSA) is equal to 0.0009 Hz and 4.2 s, respectively. The MFD and ST based on controller 1 are equal to 0.0018 Hz and 5 s, respectively; the MFD and ST based on controller 2 are equal to 0.0173 Hz and 19 s, respectively; the MFD and ST based on controller 3 are equal to 0.021 Hz and 38 s, respectively; and the MFD and ST based on controller 4 are equal to 0.0476 Hz and 90 s, respectively (Figure 9b-e). Based on the outcomes of this particular scenario, it is evident that the proposed controller surpasses its counterparts in effectively mitigating power system deviations, and it has also reduced the ST of frequency deviations caused by disruptions on the power system.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & **Value** & \multicolumn{2}{c}{**Parameter**} & \multicolumn{2}{c}{**Value**} \\ \hline Population of owls & 100 & \(K_{P1,\min},K_{d1,\min},K_{P2,\min},K_{l,\min},K_{l2,\min}\) & 0 \\ \hline Forest range for capacity & [0,1000] & \(K_{P1,\max},K_{d1,\max},K_{P2,\max},K_{l,\max},K_{l,\max},K_{l2,\max}\) & 100 \\ \hline \(\alpha\) & 1 & \(\lambda_{\min},\mu_{1,2,\min}\) & 0 \\ \hline Iterations & \multirow{2}{*}{100} & \multirow{2}{*}{\(\lambda_{\max},\mu_{1,2,\max}\)} & \multirow{2}{*}{1} \\ (stop criteria) & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fundamental parameters pertaining to the DOSA and FOPD–FOPID cascaded controller.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Controller** & \(K_{P1}\) & \(\mu_{1}\) & \(K_{d1}\) & \(K_{P2}\) & \(K_{I}\) & \(K_{d2}\) & \(\lambda\) & \(\mu_{2}\) & _ISE_ \\ \hline DOSA–FOPD–FOPID & 91.55 & 0.65 & 88.91 & 98.22 & 91.44 & 86.35 & 0.56 & 0.74 & \(6.8\times 10^{-6}\) \\ \hline GTO–FOPD–FOPID & 89.13 & 0.58 & 83.66 & 89.55 & 83.87 & 84.18 & 0.40 & 0.42 & \(9\times 10^{-6}\) \\ \hline MSA–FOPD–FOPID & 85.82 & 0.62 & 87.44 & 92.34 & 90.56 & 75.79 & 0.46 & 0.40 & \(9.1\times 10^{-6}\) \\ \hline ABC–FOPD–FOPID & 68.23 & 0.49 & 91.23 & 86.25 & 78.45 & 76.39 & 0.43 & 0.38 & \(9.9\times 10^{-6}\) \\ \hline PSO–FOPD–FO & 70.65 & 0.47 & 81.77 & 75.21 & 79.92 & 71.36 & 0.39 & 0.48 & \(10\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Values of parameters of the FOPD–FOPID controller using different algorithms.  
Figure 6: Load disruptions applied to the power system.