## 6 Conclusions  
This study presents a method for enhancing dynamic performance in power systems using virtual synchronous generators (VSGs). A multi-objective optimization approach is proposed that minimizes the rate of change of frequency (RoCoF) and maximizes various controllability Gramian-based metrics. It is worth highlighting that the values of multi-objective optimization are based on the Pareto front, as it provides a diverse set of solutions that balance multiple criteria. In this study, the point closest to the origin and the knee point solutions are chosen for comparison. However, any point along the Pareto front could be chosen depending on the importance or weight of the criterion, offering flexibility in decision-making based on specific application needs. Simulation results show that without damping ratio constraints, the determinant gives the best controllability, while the maximum eigenvalue is best for reducing RoCoF. With damping ratio constraints, solutions are more balanced but come with higher objective values. Among the tested metrics with damping ratio constraints, the maximum eigenvalue achieves the lowest RoCoF. The trace metric offers a strong compromise between minimizing RoCoF and maximizing controllability. Additional analyses, including damping ratios, time constants, damped frequencies, and time-domain responses (frequency deviations and rotor angles), confirm the comparative findings. By considering both stability and operational flexibility, this research contributes to designing resilient and adaptive power systems that can maintain stability under varying conditions and high renewable energy penetration. Future research will focus on advancing multi-criteria optimization strategies to optimize network performance. The goal will be to optimize key parameters such as system inertia, damping, and the operational modes of converter-interfaced power sources, whether grid-following (GFL) or grid-forming (GFM). The proposed method will be tested \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Generator Type**} & \multirow{2}{*}{**Controllability Metric**} & \multicolumn{2}{c}{\(J_{1}\) **(Controllability)**} & \multicolumn{2}{c}{\(J_{2}\) **(RoCoF)**} \\ \cline{3-6}  & & \(J_{1}\)**origin** & \(J_{1}\)**knee** & \(J_{2}\)**origin** & \(J_{2}\)**knee** \\ \hline \multirow{6}{*}{\(\text{VSG}_{1}\), VSG\({}_{2}\), VSG\({}_{3}\)} & EV\({}_{\text{min}}\) & 0.9129 & 0.9182 & 0.9672 & 0.9631 \\  & EV\({}_{\text{2 min}}\) & 0.9467 & 0.9467 & 0.9384 & 0.9384 \\  & EV\({}_{\text{max}}\) & 0.8748 & 0.8930 & 0.7267 & 0.7220 \\  & TR & 0.6705 & 0.7714 & 0.9870 & 0.9379 \\  & DET & 0.8397 & 0.8465 & 0.9761 & 0.9710 \\  & EV\({}_{\text{min}}\) & 0.9421 & 0.9565 & 0.9880 & 0.9770 \\  & EV\({}_{\text{2 min}}\) & 0.9376 & 0.9383 & 0.9683 & 0.9678 \\  & EV\({}_{\text{max}}\) & 0.8859 & 0.9554 & 0.9015 & 0.8851 \\  & TR & 0.7470 & 0.8402 & 0.9996 & 0.9555 \\  & DET & 0.8909 & 0.9208 & 0.9796 & 0.9659 \\  & EV\({}_{\text{min}}\) & 0.9999 & 0.9800 & 0.9737 & 0.9964 \\  & EV\({}_{\text{2 min}}\) & 0.9594 & 0.9594 & 0.9553 & 0.9553 \\ SG\({}_{1}\), SG\({}_{2}\), VSG\({}_{3}\) & EV\({}_{\text{max}}\) & 0.9954 & 0.9978 & 0.9992 & 0.9984 \\  & TR & 0.7364 & 0.8305 & 0.9995 & 0.9552 \\  & DET & 0.9525 & 0.9525 & 0.9552 & 0.9552 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of objective function values for VSGs and mixed-type generators (with damping ratio constraints). for full-order VSG models on larger systems to find relevant relationships between the controllability, RoCoF, and damping of the response, and to find a balance between the optimization cost and the model order needed to obtain feasible results. Moreover, we also aim to investigate the extent to which the inclusion of nonlinear dynamics can improve the accuracy and generalizability of the proposed method. Conceptualization, A.S. and B.P. ; methodology, A.S. and B.P. ; formal analysis, A.S., J.R. and P.K. ; writing--original draft preparation, A.S.; writing--review and editing, A.S., B.P., J.R. and P.K.