## 4 Case Study  
The case study system, illustrated in Figure 5, consists of three distinct control areas, each managed by different generation companies (Gencos). In each control area, three nonreheat turbine units are used for power generation. To simplify the modeling process, these units are represented as a single equivalent generator per control area, ensuring the essential dynamics of power generation and load balancing are maintained. The tie  
Figure 4: Diagram of the proposed MCS-PSO approach. line synchronizing coefficients, which regulate power exchange between the control areas in the three-area LFC system, are listed in Table 2. Additionally, Table 3 presents the system parameters, describing the operational characteristics of the generators and control mechanisms within the interconnected system. Given the inherent nonlinearity and the degraded, time-varying characteristics of the 3-area LFC system under cyber-attack conditions, the model is implemented in MATLAB/SIMULINK instead of using a state-space representation. Traditional state-space representations, while useful for linear systems, often fail to capture the intricate dynamics and rapid changes induced by cyber threats, thus limiting their effectiveness in such applications. The system parameters are all expressed in per unit values, with a base \begin{table}
\begin{tabular}{c c} \hline \hline
**Area** & \(T_{ij}\) **(p.u./rad)** \\ \hline
1 & \(T_{12}=0.04,\ T_{13}=0.024\) \\ \hline
2 & \(T_{21}=0.04,\ T_{23}=0.05\) \\ \hline
3 & \(T_{31}=0.05,\ T_{32}=0.024\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tie-line synchronizing coefficients. Figure 5: A schematic framework of the 3-area LFC system. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameters** & \multicolumn{8}{c}{**Gencos**} \\ \hline \(MVA_{base}\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline Rate (MW) & 1100 & 900 & 1200 & 1000 & 800 & 1000 & 850 & 1000 & 1020 \\ \hline D (p.u./Hz) & 0.014 & 0.014 & 0.015 & 0.015 & 0.015 & 0.016 & 0.016 & 0.016 \\ \hline \(T_{T}\) (s) & 0.4 & 0.36 & 0.42 & 0.3 & 0.4 & 0.41 & 0.3 & 0.4 & 0.41 \\ \hline \(T_{H}\) (s) & 0.08 & 0.06 & 0.07 & 0.08 & 0.06 & 0.07 & 0.08 & 0.06 & 0.07 \\ \hline R (p.u./Hz) & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 \\ \hline B (p.u./Hz) & 1.014 & 1.014 & 1.014 & 1.015 & 1.015 & 1.015 & 1.016 & 1.016 & 1.016 \\ \hline \(T_{P}\) (p.u. s) & 0.15 & 0.15 & 0.15 & 0.166 & 0.166 & 0.166 & 0.1667 & 0.1667 \\ \hline \(\alpha\) & 0.6 & 0 & 0.4 & 0.4 & 0.4 & 0.2 & 0 & 0.5 & 0.5 \\ \hline Ramp rate (MW) & 12 & 0 & 8 & 8 & 8 & 4 & 0 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Corresponding system parameters. of 1000 MW MVA. Each control area utilizes a PI controller. The system's nominal frequency is 60 Hz, and the speed governor's dead band is configured to a maximum value of 0.06%. In this scenario, significant step increases in demand are introduced at t = 0 s: \(\delta P_{D_{1}}=-0.05\) p.u. (\(-50\) MW), \(\delta P_{D_{2}}=0.1\) p.u. (\(100\) MW), and \(\delta P_{D_{3}}=0.06\) p.u. (\(60\) MW). The regulation requirement for each area is established at 100 MW [60]. The degradation path of the control system, accounting for FDIA, is described as follows, with each time unit \(T_{0}\) representing a period of two years. The degradation path for the three turbines (\(n=3\)) is outlined as follows:  
\[d_{a_{n}}(T_{o})=d_{a_{n}}(0)+\lambda T_{o}+\sigma B(T_{o}) \tag{18}\]  
where \(\sigma B(T_{o})\sim N\big{(}0,0.005^{2}\cdot T_{o}\big{)}\) and \(\lambda\sim N(0.01,0.002)\), which are general for turbines [63], specifically \(d_{a_{n}}(0)=0\). The degradation path of the frequency sensor subject to FDIA is  
\[d_{s}(T_{o})=d_{s}(0)+\varsigma T_{o}+\sigma B(T_{o}) \tag{19}\]  
where \(\varsigma=0.005\) and \(\sigma B(T_{o})\sim N\big{(}0,0.002^{2}\cdot T_{o}\big{)}\), specifically \(d_{s}(0)=0\); the FDIA implemented on the frequency sensor is \(\omega\sim N(0,0.01^{2}\cdot T_{o})\). The degradation model of the above 3-area LFC system considering the effects of FDIA is therefore established.