## 5 Simulation Results and Discussion
### IEEE Nine-Bus Case Study  
The IEEE nine-bus system comprises three generators and nine lines, as detailed in reference [38]. The simulation is based on a time step of 0.0001 s, with the rotor angle of generator 1 serving as the reference angle for the study. Additionally, a controllable UPFC device with an optimized coefficient controller is installed between buses 6 and 9 to manage the voltage and power flow between these buses, as shown in Figure 9.  
The three-dimensional Pareto distribution chart of the final objective functions includes the total production cost (axis X), voltage deviations (axis Y), and total frequency deviations (axis Z) for the optimal and coordinated operation of PSSs and UPFCs, as shown in Figure 10. As observed in this chart, the most suitable operating point with the lowest production cost is equal to 1107.3, at which point there are the highest voltage and frequency deviations. On the other hand, the most suitable operating point with the lowest voltage deviation is equal to 0.145, at which point there are the highest production costs and frequency deviations. Additionally, the most suitable operating point with the lowest frequency deviation corresponds to a frequency deviation of 0.0029, where the objective functions of operating costs and voltage deviations are far from optimal conditions. Therefore, it can be said that the objectives in this network are conflicting with each other. Therefore, since in the present network, all three objective functions have the same importance, and the coefficients of the controllers are selected for conditions where all three objective functions are in optimal agreement. Therefore, it can be said that the proposed method provides an effective framework for system recovery with the desired objectives in the event of a cyber physical-attack or disruption in the power system. Figures 11 and 12 are centered on the following two key aspects: technical factors (total speed deviations and voltage deviation) and the total generation cost of the generators within the system. These aspects are evaluated when the coefficients of the coordinated controller for the UPFC and PSSs are optimized using both PM and various previous state-of-the-art research approaches. These figures reveal that the system and generators are in an optimally stable condition within the first 5 s of real-time simulation. However, in this case study, a cyber-physical attack leads to the sudden disconnection of the line between buses 4 and 5 at the 35 s mark. Figure 9: Schematic of the simulated file of the IEEE 9\(-\)bus case. Figure 11: Total cost and total speed deviation of IEEE 9\(-\)bus network for scenario of line outage between buses 4 and 5. Figure 10: Three-dimensional Pareto diagram of optimal objective functions in the IEEE 9\(-\)bus network. As depicted in Figures 11 and 12, PM exhibits the quickest response in restoring the power system to a stable state, with minimal generation costs compared to other previous state-of-the-art (SOTA) techniques. Additionally, Figure 12 indicates that the best performance in controlling the bus voltage is related to PM, with minimum voltage deviation throughout the operation. Regarding this aspect, Figure 13 illustrates how the optimal generation point and rotor angle of the generators change after the mentioned line outage for the PM method. PM promptly identifies the new stable rotor angle in real time compared to prior research, demonstrating the high accuracy and capability of the proposed approach in estimating the optimal secure operation point of the system in the face of cyber-physical attacks. Figure 12: Voltage deviation of IEEE 9\(-\)bus network for scenario of line outage between buses 4 and 5. Figure 13: Output power and rotor-angle generators in IEEE 9\(-\)bus network (for PM condition) for scenario of line outage between buses 4 and 5.