## 5 Outcomes and Valuation
### Sensitivity Assessment for a Higher Value of Disturbance  
An examination of sensitivity is performed to detect the robustness of the SHO-augmented TIDN-FOID subordinate controller attributes, focusing on the core characteristics and wide-ranging amendments of the scheme circumstances in T-T-Bd in arena-1, T-T-GPPt in arena-2, and T-H in arena-3 scheme, as well as with RFB and time delay. The inspected scheme was unveiled with 3% step load agitation in arena-1. The optimized attributes of the TIDN-FOID controller shown in Table 11 were obtained by engaging SHO. The results for the premium values similar to basic and wide-ranging outcomes are shown in Figure 10. The outcomes are somewhat comparable, which shows that there is no supplementary rearranging of unsurpassed values when circumstances are altered. \begin{table}
\begin{tabular}{c c} \hline \hline
**System Condition** & **Damping Ratio (z)** \\ \hline Hydrothermal system excluding biodiesel and & \\ GPPt, time delay, and RFB & \\ \hline Hydrothermal system including biodiesel and & \\ GPPt, time delay, and RFB & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Value of the damping ratio for different conditions of the system using the TIDN-FOID controller. \begin{table}
\begin{tabular}{c c} \hline \hline
**System Condition** & **Eigenvalue** \\ \hline \(-0.1000+0.0000i-3.3333+0.0000i-0.1000+0.0000i-3.3333+0.0000i-0.1000+0.0000i\) \\ \(-3.3333+0.0000i-0.1000+0.0000i-3.3333+0.0000i-0.1000+0.0000i\) \\ \(-12.5000+0.0000i-12.5000+0.0000i-12.5000+0.0000i-12.5000+0.0000i-12.5000+0.0000i\) \\ \(-59.9198+6.5359i-59.9198-6.53559i-56.2001+3.4843i-56.2001-3.4843i-55.8472+7.2876i\) \\ \(-55.8472-7.2876i-49.9874+0.0000i-49.9739+0.0000i-49.9747+0.0000i-36.4359+0.0000i\) \\ \(-35.1881+0.0000i-34.7088+0.0000i-33.4630+0.0000i-33.6403+0.0000i-33.7199+0.0000i\) \\ \(-50.0000+0.0000i-30.0215+0.0000i-28.7628+0.0000i-27.9137+0.0000i-50.0000i\) \\ \(-50.0000+0.0000i-16.8387+3.1713i-16.8387-3.1713i-17.2647+0.0000i-15.2569+0.0000i\) \\ \(-15.4193+0.0000i-15.4773+0.4694i-15.4773-0.4694i-13.8376+0.0000i-13.6473+0.0000i\) \\ \(-8.3891+6.2010i-8.3891-6.2010i-12.4523+0.0000i-13.1265+0.0000i-7.9307+0.0000i\) \\ \(-7.5106+0.0000i-7.1518+0.0000i-7.1170+0.0000i-6.8575+0.1307i-6.8575-0.1307i\) \\ \(-6.3711+0.0000i-5.9147+0.0000i-5.5548+0.0000i-19.044+2.1300i-19.9044-2.1300i\) \\ \(-0.1910+2.1081i-0.1910-2.1081i-3.5930+0.0000i-3.5387+0.0000i-3.2840+0.0000i\) \\ \(-3.2919+0.0000i-3.2353+0.1174i-3.2353-0.1174i-2.9165+0.0000i-2.7281+0.0111i\) \\ Hydrothermal system & \(-2.7281-0.0111i-2.1935+0.0000i-2.0714+0.1040i-2.0714-0.1040i-1.5607+0.5131i\) \\ including biodiesel and GPPt, & \(-1.5607-0.5131i-1.1884+0.1830i-1.1884-0.1830i-1.5489+0.0392i-15.489-0.0392i\) \\ time delay, and RFB & \(-1.4783+0.0575i-1.4773-0.0575i-1.4719+0.0000i-1.4191+0.0000i-0.8067+0.0561i\) \\ \(-0.8067-0.0561i-0.8258+0.0000i-0.6371+0.0291i-0.6371-0.0291i-0.6895+0.0092i\) \\ \(-0.6895-0.0092i-0.6800+0.0000i-0.6539+0.0000i-0.3959+0.0000i-0.3713+0.0117i\) \\ \(-0.3713-0.0117i-0.2945+0.0065i-0.2945-0.0065i-0.3172+0.0023i-0.3172-0.0023i\) \\ \(-0.3128+0.0125i-0. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name of Controller** & **Corresponding Gains and Parameters** & **Area-1** & **Area-2** & **Area-3** \\ \hline  & \(K_{pl}\) * & 0.7016 & 0.6892 & 0.6895 \\  & \(K_{li}\) * & 0.6985 & 0.7985 & 0.8789 \\  & \(K_{Di}\) * & 0.4016 & 0.5014 & 0.6014 \\  & \(N_{i}\) * & 62.25 & 61.28 & 62.36 \\ TIDN-FOID & \(n_{i}\) * & 2.1111 & 2.1875 & 2.8955 \\  & \(K_{Fil}\) * & 0.8111 & 0.7895 & 0.7884 \\  & \(\lambda_{i}\) * & 0.0216 & 0.0356 & 0.0389 \\  & \(K_{FD1}\) * & 0.7896 & 0.7989 & 0.7954 \\  & \(\mu_{i}\) * & 0.1589 & 0.1586 & 0.2478 \\ \hline \hline \end{tabular}  
* Signify the optimum values.  
\end{table} 
Table 11: Related attributes of TIDN-FOID subordinate controllers for structure-4 (i.e., including biodiesel, GPPt, time delay (\(\tau_{d}\)), and RFB) at 3% step load disturbance.