## 4 Simulation Results and Discussion
### Scenario 1: A Two-Area Power System (TAPS) Integrating Hydropower Plants (Figure 2)
In the graph illustrating the results for area 1, the PID-ZN method shows that at 1 s, when a load disturbance of 0.05 pu \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Area** & **Method** & \(\mathbf{K_{p}}\) & \(\mathbf{K_{i}}\) & \(\mathbf{K_{d}}\) \\ \hline \multirow{2}{*}{1} & PID-ZN & \(-2.7273\) & \(-1.9841\) & \(-0.9372\) \\  & PID-PSO & \(-6.5052\) & \(-8.5875\) & \(0.7574\) \\ \hline \multirow{2}{*}{2} & PID-ZN & \(-2.7273\) & \(-1.9841\) & \(-0.9372\) \\  & PID-PSO & \(0.2876\) & \(-2.1088\) & \(-0.0644\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PID parameters for scenario 1. Figure 2: Model of frequency control system for two hydropower plants integration. frequency drops sharply to approximately \(-0.23\) Hz at 3 s. It then suddenly rises to around 0.05 Hz and, in the subsequent period, the oscillation amplitude gradually decreases. By 18 s, the system frequency is adjusted to a balanced state. Meanwhile, the results of the PID-PSO method indicate that at 1 s, with the same load disturbance of 0.05 pu, the system frequency oscillates with a larger amplitude compared to the PID-ZN method by about 0.04 Hz. However, in terms of settling time, this method enables the system frequency to stabilize faster, achieving a steady state at 14 s. Similarly, for the frequency control results of area 2, the PID-PSO method once again outperforms the PID-ZN method. Although the initial oscillation amplitude is larger by approximately 0.1 Hz, the settling time is shorter, with 13 s compared to 18 s for the PID-ZN method. In both areas, the PID-PSO method demonstrates significantly smaller steady-state errors compared to the PID-ZN method. These results highlight the superiority of the PSO algorithm in optimizing PID parameters, which significantly improves frequency control performance, reduces settling time, and enhances accuracy in both areas (Table 3). The convergence plot (Figure 4) shows the PSO algorithm's effectiveness, with the objective function value dropping quickly. In the first five iterations, the value decreases from about \(10^{8}\) to below \(10^{2}\), which indicates a substantial improvement in solutions early on. After that, the decrease slows down, and by the 24 th iteration, the value stays nearly the same, showing that the algorithm has converged. This result suggests that the algorithm can reach a good solution in a short time while keeping the computation efficient. Remark 1: The TAPS considered in this case has a more general structure compared to those presented in [4; 19], which makes it more suitable for the application of LFC in real power systems. Figure 3: Frequency deviation simulation result of scenario 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Area 1**} & \multicolumn{2}{c}{**Area 2**} \\ \cline{2-5}
**Method** & **Steady-State** & **Settling Time** & **Steady-State** & **Settling Time** \\  & **Error (Hz)** & **(s)** & **Error (Hz)** & **(s)** \\ \hline PID-ZN & \(24.32\times 10^{-5}\) & 18 & \(24.32\times 10^{-5}\) & 18 \\ PID-PSO & \(6.973\times 10^{-5}\) & 14 & \(6.969\times 10^{-5}\) & 13 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of simulation results for scenario 1. Scenario 2: A Three-Area Power System Combining Hydropover, Reheat Thermal, and Non-Reheat Thermal Power Plants, with Continuous Load Variations (Figure 5)  
The comparison between the PSO-optimized PID technique and the traditional PID method, following the frequency regulation simulation for a two-area power system, clearly demonstrates the superior performance of the PSO algorithm. The PID-PSO controller not only balances the frequency more effectively but also achieves much faster stabilization. Building on these results, the study takes a step further by applying continuous load variations into a more complex system with the following three interconnected areas: a hydropower plant, a reheat thermal power plant, and a non-reheat thermal power plant.