## 5 Simulation and Analysis
### Case Study 1  
The model of the two-area LFC system with an ESS and PV generation is shown in Figure 2, and the parameters of the two-area power system are listed in Table 2.  
In this subsection, the model of the two-area LFC system with an ESS and PV generation is taken as an example to examine the impact of large frequency perturbations and continuous random disturbances on the performance of the LFC system. First, the large frequency perturbation of area 1 and area 2 is set to 0.016 per unit (p.u) at \(t=5\) s. Furthermore, amplitude-limited continuous random disturbances of area 1 and area 2, with the upper limit set to 0.001 p.u, are added, occurring at \(t=5\) s \(\sim 15\) s. The simulation runs for 80 s, providing sufficient time to analyze the frequency response. To demonstrate the effectiveness of the proposed method under disturbances of different magnitudes, the large frequency disturbance of area 1 and area 2 is set to 0.02 p.u at \(t=5\) s. Moreover, continuous random disturbances of area 1 and area 2, with the upper limit set to 0.002 p.u, are introduced, occurring at \(t=5\) s \(\sim 15\) s. The simulation results are depicted in Figure 3a-d. Key performance metrics derived from simulations are compared in Tables 3 and 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Area1** & **Area2** & **Parameters** & **Area1** & **Area2** \\ \hline \(T_{gi}\) & 0.08 & 0.09 & \(k_{1}\) & 0.3 & 0.4 \\ \(T_{chi}\) & 0.35 & 0.4 & \(k_{2}\) & 0.6 & 0.5 \\ \(D_{i}\) & 0.02 & 0.01 & \(k_{3}\) & 0.1 & 0.1 \\ \(M_{i}\) & 0.3 & 0.3 & \(\lambda_{pi}\) & 120 & 120 \\ \(\beta_{i}\) & 0.4 & 0.4 & \(a\) & 99.5 & 98 \\ \(R_{i}\) & 5 & 5 & \(b\) & \(-50\) & \(-50\) \\ \(T_{ij}\) & 0.2 & 0.22 & \(c\) & 0.5 & 0.5 \\ \(T_{bi}\) & 1 & 1 & \(\lambda_{bi}\) & 10 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the two-area power system model. Figure 2: Two-area power system model diagram. As can be seen from Figure 3, the maximum frequency deviations of the system based on PID and SMC are about \(-0.323\) Hz and \(-0.276\) Hz, respectively, and the response times are about 47.75 s and 42.48 s, respectively. The response time of the system based on GFTSMC is roughly 22.81 s and is relatively short compared with the preceding two control \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Area 1**} & \multicolumn{2}{c}{**Area 2**} \\ \cline{2-5}  & **Steady-State** & **Response Time** & **Steady-State** & **Response Time** \\  & **Error (Hz)** & **(s)** & **Error (Hz)** & **(s)** \\ \hline PID & \(9.12\times 10^{-4}\) & 48.36 & 8.79 \(\times 10^{-4}\) & 47.18 \\ SMC & \(5.92\times 10^{-4}\) & 42.71 & \(5.72\times 10^{-4}\) & 40.07 \\ GFTSMC & \(3.58\times 10^{-4}\) & 23.93 & \(3.21\times 10^{-4}\) & 23.41 \\ AGFTSMC & \(1.82\times 10^{-4}\) & 17.50 & \(1.63\times 10^{-4}\) & 14.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of simulation results when the large frequency disturbance is set to 0.02 p.u. Figure 3: Results of case 1. (**a**) Frequency deviations of area \(1\) when the large frequency disturbance is set to 0.016 p.u. (**b**) Frequency deviations of area 2 when the large frequency disturbance is set to 0.016 p.u. (**c**) Frequency deviations of area 1 when the large frequency disturbance is set to 0.02 p.u. (**d**) Frequency deviations of area 2 when the large frequency disturbance is set to 0.02 p.u.