## 4 Simulation Results and Discussion
### Scenario 1: A Two-Area Power System (TAPS) Integrating Hydropower Plants (Figure 2)
From 40 to 60 s, all three graphs experience pronounced swings around the 40 s point, then gradually recover toward the 0 pu level. While the red and blue graphs show minor ripples before stabilizing toward the end, the green graph maintains a lower amplitude of oscillation throughout. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Area** & \(\mathbf{K_{p}}\) & \(\mathbf{K_{i}}\) & \(\mathbf{K_{d}}\) \\ \hline
1 & \(-2.8563\) & \(-2.7593\) & \(-2.9135\) \\
2 & \(-1.7872\) & \(-2.9356\) & \(-0.2980\) \\
3 & \(-2.9861\) & \(-0.6005\) & \(-2.9815\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: PID parameters for scenario 2. Figure 6: Continuous load variation scenario. Figure 7: Frequency deviation simulation result of scenario 2. The convergence plot (Figure 9) demonstrates the performance of the PSO algorithm, where the objective function value drops rapidly within the initial iterations. As iterations progress, the rate of decrease slows down, and by approximately the eighth iteration, the value stabilizes, which means the algorithm has effectively converged. This behavior reflects the algorithm's ability to find an optimal solution for the given optimization problem. Remark 2: In contrast with the other approaches, the three-area power system combining the hydropower, reheat thermal, and non-reheat thermal power plants considered in this method has a more wide-ranging structure than the PS construction assumed in [12; 17; 18]. Furthermore, the proposed regulator built on PSO achieves better transient conditions in terms of zero frequency deviation error, small overshot, and settling time, which is useful to guarantee stability and reliability for the real PS. Scenario 3: A Three-Area Power System Combining Hydropower, Reheat Thermal, and Non-Reheat Thermal Power Plants, with the Integration of Wind Power (Figure 10)  
In the third simulation scenario, the system still consists of three interconnected areas--a hydropower plant, a reheat thermal power plant, and a non-reheat thermal power plant. However, the key difference in this simulation is the integration of a renewable energy source, specifically wind power. Inspired by the study in [34], this scenario will examine  
Figure 8: Tie-line power simulation result of scenario 2. Figure 9: Convergence of PSO for PID frequency control (scenario 2). the impact of wind speed variations on the output power of the wind energy system. This factor directly affects the frequency regulation and stability of the system. The incorporation of renewable energy takes the research a step further, providing a more realistic perspective on the system's ability to maintain frequency stability as more renewable energy sources are integrated. Through this simulation, the PSO-optimized PID controller will be evaluated more comprehensively. This assessment is particularly important under real-world operating conditions where wind energy fluctuations can significantly impact power balance and system frequency stability. The parameters in the simulation model and the conditions for the PSO algorithm remain unchanged from Scenario 2. However, this simulation also incorporates wind power, requiring a wind fluctuation scenario. In this setup, wind speeds range from 3 m/s to 6 m/s, varying over time with random increases and decreases. These fluctuations help represent real-world wind conditions. The graph below shows how wind speed changes throughout the simulation (Figure 11). With a total simulation time of 60 s, the load is set to fluctuate at 0.05 pu at the 1 s mark. The simulation is conducted, and the results are obtained as follows (Table 6):  
Figure 10: Model of frequency control system for hydropower, reheat, and non-reheat thermal power plant integration with wind power. During the first 0-5 s, the frequency deviation of all three areas fluctuates significantly, with the maximum deviation reaching approximately \(-0.13\) Hz, where area 2 experiences the largest deviation.