## 4 Results
### Time-Domain Results  
The aim of this subsection is to use time-domain simulations to demonstrate improved system performance with optimized parameters, i.e., to confirm higher controllability, lower RoCoF, and more damped oscillations.  
Figure 6 presents the time evolution of the mean RoCoF for the initial system and the optimized knee point solution on the Pareto front for the maximum eigenvalue metric with damping ratio constraints. Among the various metrics, optimization based on the maximum eigenvalue of the controllability Gramian gives the most favorable results, achieving the lowest RoCoF, which is presented here. Note that the RoCoF time responses were calculated from the frequencies obtained from the initial condition test (\(\textbf{x}_{0}=[1,1,1,0,0]^{\text{T}}\)). The optimized response shows improved RoCoF characteristics, reflecting enhanced system performance under disturbances.  
Figure 4: Comparison of initial parameters, closest-to-origin solutions, and knee points across Pareto fronts (with and without \(\zeta\) constraints) for: (**a**) \(H_{1}\), \(H_{2}\), \(H_{3}\), and (**b**) \(D_{1}\), \(D_{2}\), \(D_{3}\).  
Figure 5: (**a**) Damping ratios (\(\zeta\)); (**b**) time constants (\(\tau\)); and (**c**) damped frequencies (\(f_{d}\)) across Pareto fronts with and without damping ratio constraints.  
Figure 7 shows the time-domain responses of the system states (\(\Delta f_{1},\ \Delta f_{2},\ \Delta f_{3},\ \Delta\delta_{21},\ \Delta\delta_{3 1}\)) for the step input change of all inputs (\(\Delta P_{m1},\ \Delta P_{m2},\ \Delta P_{m3}\)) from 0 to 0.01 p.u., while \(\mathbf{x}_{0}=[0,0,0,0,0]^{\mathrm{T}}\). Each subplot represents the response of a single state variable over a 50 s simulation period. Results are presented for the initial system parameters and the knee point parameters without constraints on damping ratios for the determinant controllability metric. The optimized results confirm improved controllability compared to the initial system as larger final values of frequencies are reached. However, optimizing without considering damping ratio constraints can lead to instability, even if the system is more easily controllable, which is reflected in less damped and prolonged oscillations in rotor angles when compared to the initial case.  
Figure 8 shows the time-domain responses of the system states (\(\Delta f_{1},\ \Delta f_{2},\ \Delta f_{3},\ \Delta\delta_{21},\ \Delta\delta_{3 1}\)) for the step input change of all inputs (\(\Delta P_{m1},\ \Delta P_{m2},\ \Delta P_{m3}\)) from 0 to 0.01 p.u., \(\mathbf{x}_{0}=[0,0,0,0,0]^{\mathrm{T}}\). Results are presented for the initial system parameters and the knee point parameters with constraints on damping ratios for the three controllability metrics. EV\({}_{\mathrm{max}}\) results in the lowest frequency deviations overall. TR shows a good balance between maximizing controllability, minimizing RoCoF and the damping of oscillations. DET with  
Figure 6: Time evolution of mean RoCoF for initial and optimal VSG parameters using controllability metric EV\({}_{\mathrm{max}}\) (with damping ratio constraints).  
the knee point parameters has the highest frequency deviations, but it results in the best controllability overall.