## 4 Case Study
### Results for PI Controllers and Discussions  
The MCS is applied to four different cases. These systems are built with identical internal control frameworks and similar component degradation. However, their differenti  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} \(T_{0}\) \\ **Index** \\ \end{tabular} } & \multirow{2}{*}{**0**} & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline \multirow{3}{*}{Reliability} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.9998 & 0.9961 & 0.9458 & 0.7336 & 0.3502 \\ \cline{2-11}  & 1 & 1 & 1 & 1 & 1 & 1 & 0.9998 & 0.9961 & 0.9458 & 0.7336 & 0.3502 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Reliability evaluation of the 3-area power system.  
Figure 6: The quality parameters for each area.  
ation lies in the control strategies implemented--parameters \(K_{P_{1}},K_{I_{1}},K_{P_{2}},K_{I_{2}},K_{P_{3}}\), and \(K_{I_{3}}\) within their PI controllers. Details of the PI controller parameters for each scenario are listed in Table 7.  
Event-based MCS simulations are next applied to each of the four cases, evaluating the effects of component degradation and FDIA cyber-attacks on system reliability across various total operational periods (\(T_{0}\)). The results are summarized in Table 8.  
The reliability curves for each case are illustrated in Figure 7. Three key conclusions can be drawn from these results. Firstly, the event-based MCS method effectively estimates the reliability of the 3-area LFC system, utilizing the proposed time-varying degraded model. Secondly, systems with identical degraded components and structural configurations but different controller parameters exhibit varying reliability dynamics. Lastly, optimizing the controller parameters can significantly enhance the reliability of LFC systems in multi-area power systems.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(K_{\mathbf{P1}}\)} & \multirow{2}{*}{\(K_{\mathbf{I1}}\)} & \multirow{2}{*}{\(K_{\mathbf{P2}}\)} & \multirow{2}{*}{\(K_{\mathbf{I2}}\)} & \multirow{2}{*}{\(K_{\mathbf{P3}}\)} & \multirow{2}{*}{\(K_{\mathbf{I3}}\)} \\ \hline
1 & \(-5.69\times 10^{-4}\) & \(-0.3435\) & \(-8.86\times 10^{-4}\) & \(-0.3340\) & \(-9.60\times 10^{-4}\) & \(-0.3398\) \\ \hline
2 & \(-6.96\times 10^{-4}\) & \(-0.3435\) & \(-3.27\times 10^{-4}\) & \(-0.3334\) & \(-1.60\times 10^{-4}\) & \(-0.3398\) \\ \hline
3 & \(-9.26\times 10^{-3}\) & \(-0.3391\) & \(-7.36\times 10^{-5}\) & \(-0.3298\) & \(-3.84\times 10^{-5}\) & \(-0.3385\) \\ \hline
4 & \(-1.38\times 10^{-2}\) & \(-0.3672\) & \(-1.01\times 10^{-4}\) & \(-0.3534\) & \(-5.96\times 10^{-2}\) & \(-0.3746\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameters of PI controllers used in each case.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(T_{\mathbf{O}}\)} & \multicolumn{8}{c}{**Case**} \\ \cline{2-13}  & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.9960 & 0.9460 & 0.7760 & 0.4062 \\ \hline
2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.9998 & 0.9961 & 0.9458 & 0.7336 & 0.3502 \\ \hline
3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.9963 & 0.9109 & 0.5433 & 0.2625 \\ \hline
4 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.9980 & 0.9740 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Reliability evaluation of each case in Table 7 as a function of \(T_{\mathrm{O}}\).  
Figure 7: The comparison of system reliability with and without optimization.