## 4 Case Study
### Reliability Evaluation and Discussion  
The 3-area LFC system is engineered to keep frequency deviations within \(\pm 0.3\) Hz. To meet operational standards, the system must also satisfy the following quality criteria:  
* \(PT_{max}\): the output of the LFC process reaches its peak or trough within a maximum time of 1 s, 2 s, and 1 s.
* \(PO_{max}\): In these three control areas, the maximum permissible frequency deviations are limited to 0.413%, 0.395%, and 0.365% of the nominal 60 Hz frequency.
* \(ST_{max}\): the maximum time in these three required for the frequency to stabilize within 0.025% of the nominal 60 Hz frequency is 10.02 s, 8.67 s, and 10.35 s.  
We have set the required number of Monte Carlo simulations, denoted as \(M\), at 1000 to achieve a precision margin of 0.005 and a confidence level of 95%. The TA precision interval of 0.05 is used, ensuring that 95% of the simulations fall within this range. Each MCS run is performed in the MATLAB/SIMULINK environment, where each replication takes about 0.01 s. As a result, the MCS-PSO process takes approximately 7 h to compute the optimal PI controller settings for each total runtime \(T_{0}\).  
Table 4 presents the parameters settings of the PI controllers, specifying their parameters for each of the three distinct control areas across various operational periods The quality parameters, assuming no component degradation, are evaluated based on the frequency-response curve shown in Figure 6.  
Table 5 provides the quality parameters for each control area under the conditions where there is no component degradation or FDIA cyber-attacks. The results demonstrate that the quality parameters of the three-area LFC system meet all operational requirements when degradation is absent. Following this, the reliability of the three-area LFC system is determined through event-based MCS. These simulations account for the total operational duration (\(T_{0}\)) and include the influences of both component degradation and FDIA  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Area** & \(K_{P}\) & \(K_{I}\) \\ \hline
1 & \(-6.96\times 10^{-4}\) & \(-0.3435\) \\ \hline
2 & \(-3.27\times 10^{-4}\) & \(-0.3340\) \\ \hline
3 & \(-1.60\times 10^{-4}\) & \(-0.3398\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The PI controller parameters of each area.  
cyber-attacks on system performance. Here, \(T_{0}=0\) signifies that the system is free from degradation. Generally, the lifetime of the components of the power system can extend up to 25 years for most commercial products [30]. Consequently, in this case study, the time step for \(T_{0}\) is set to 1 year, as degradation levels are unlikely to change significantly over a shorter period, such as a week, compared to the entire lifespan. Even with a one-week interval, the 7 h MCS-PSO process would still ensure the model's applicability. The results are summarized in Table 6. A reliability value of 0.7336 indicates that there is a probability of 73. 36% that the LFC system will meet all operational requirements. It is noted that the reliability of the LFC system shows minimal variation with increasing total runtime \(T_{0}\) during the first 6 years. This stability is due to the remaining performance of the system being sufficient to meet operational requirements with appropriate adjustments to the PI controller. However, reliability declines sharply as the system approaches the average lifespan of power systems, as the designed PI controller can no longer adequately compensate for the degradation effects through increased power output from the generators. This leads to a significant deterioration in the LFC process performance. Traditional control design methods struggle significantly with effectively managing component degradation.