## 5 Discussion
### Impact of Load Variation  
In addition, the influence of load variations in the amount of \(\pm 10\%\) and \(\pm 20\%\) was evaluated. Only the baseline case was considered, i.e., with initial values of the VSG parameters, for which the objective values are \(J_{1}=J_{2}=1\). When comparing the objective values to the results under changes in the operating point, it is observed that the changes \begin{table}
\begin{tabular}{l c c c c c} \hline
**Performance Aspect** & \(\text{EV}_{\text{min}}\) & \(\text{EV}_{\text{2 min}}\) & \(\text{EV}_{\text{max}}\) & **TR** & **DET** \\ \hline Controllability (\(J_{1}\)) & \(\pm\) & \(\pm\) & + & + & + \\ RoCoF (\(J_{2}\)) & \(-\) & \(\pm\) & + & \(\pm\) & \(-\) \\ Damping (\(\zeta\)) & \(+\) & \(\pm\) & \(-\) & + & + \\ Response time (\(\tau\)) & \(\pm\) & \(\pm\) & \(-\) & + & \(\pm\) \\ Damped frequency (\(f_{d}\)) & \(\pm\) & \(\pm\) & \(\pm\) & + & + \\ Overall & \(\pm\) & \(\pm\) & \(\pm\) & + & \(\pm\) \\ performance & \(\pm\) & \(\pm\) & \(\pm\) & + & \(\pm\) \\ \hline \multicolumn{6}{l}{\(+=\text{good performance, }\pm\) = moderate/acceptable, \(-=\text{weaker}/\text{less desirable performance}\).} \\ \end{tabular}
\end{table} 
Table 2: Controllability metric performance comparison (optimization with damping ratio constraints). Figure 9: Participation factors (absolute values) obtained for initial parameters (ini) and optimal parameters (knee point) for different controllability metrics. in the objective functions \(J_{1}\) (controllability) and \(J_{2}\) (RoCoF) are generally minimal for the EV\({}_{\rm min}\) and EV\({}_{\rm 2 min}\) metrics. In these scenarios, \(J_{1}\) values remain high, typically above 0.90, and \(J_{2}\) values are very close to 1.0, indicating that both system controllability and frequency stability are well preserved despite load changes. For metric EV\({}_{\rm max}\), although the \(J_{\rm 2 max}\) value is lower (around 0.72), load variations cause only minor additional changes. In contrast, the DET and TR metrics show a bit more sensitivity to load variation. Specifically, the DET metric shows a notable drop in \(J_{\rm 1 min}\) (from a baseline minimum of around 0.82 down to around 0.71 at +20% load variation), although \(J_{2}\) remains robust. The TR matric shows changes in \(J_{\rm 1 min}\) (0.78 at \(-\)10% load variation compared to 0.66 at baseline). Except for the mentioned objective function values in the metrics DET and TR, the values of both objectives did not show significant changes as the load varied, and the data points for different load conditions were closely grouped together. This indicates that, in this scenario, the proposed method is relatively insensitive to changes in load. Moreover, to expand the sensitivity analysis, the larger variations in load conditions (beyond \(\pm\)20%) were included. The EV\({}_{\rm min}\) metric remains relatively stable across most load levels but exhibits increased deviation in \(J_{\rm 1 min}\), particularly at \(-\)40% (0.95) and +30% (0.94) compared to the nominal value of 0.91. A notable drop in \(J_{\rm 1 max}\) also occurs at +50% (0.90) relative to the baseline of 0.97. In the case of the EV\({}_{\rm 2 min}\) metric, the worst deviations from nominal appear at \(-\)50% load, where \(J_{\rm 1 min}\) increases to 0.94 (from 0.90 nominal) and \(J_{\rm 2 min}\) drops to 0.95 (from 0.97 nominal). The EV\({}_{\rm max}\) metric shows its poorest performance at +40% and +50%, where both \(J_{\rm 1 max}\) and \(J_{\rm 2 max}\) increase significantly, most notably, \(J_{\rm 2 max}\) reaches 0.997, compared to its nominal value of 0.73, suggesting a substantial deterioration under high loading. For the TR metric, the most significant deviations from the nominal values occur at \(-\)30%, where \(J_{\rm 1 min}\) increases to 0.75 (from 0.66) and \(J_{\rm 2 max}\) decreases to 0.96 (from 0.99). Additionally, \(J_{\rm 1 max}\) shows its greatest deviation at \(-\)50%, increasing by +0.08.