# Overload Mitigation of Inertial Grid-Forming Inverters Under Frequency Excursions
## 4 Performance Evaluation of Overload Mitigation Strategies
### Controller Tuning Criteria  
The parameters of the PSL are selected to provide a frequency regulation of 5% (\(D=20\)) with an inertia of 5 s during normal operation. The damping term is set to achieve a damped response, with a \(\xi=1/\sqrt{2}\). The operational limits of the active power are set to \([-1,1]\) pu and are assumed either constant or slowly varying. Table 2 summarizes the parameters for each PSL configuration and overload mitigation strategy. The active power controllers are tuned based on the small-signal models developed along Section 3. For the three controllers, the equivalent inertia during overload mitigation is set to the minimum value defined in Section 2.2.2 in order to prevent adverse interactions. This value is 0.64 s for the defined grid conditions (\(K_{t}=4\)). In parallel PI, this is achieved by directly tuning its integral gain according to (14). In AL, inertia is indirectly shaped by the integral gain of the PLL dynamics and the filter inductance (22). Finally, in EFS, the integral gain of the internal PI is adjusted to ensure the minimum inertia (26), while the remaining 4.36 s of inertia are supplied by the external controller (28). \begin{table}
\begin{tabular}{l l l l l} \hline \hline Base values & & & & \\ \(S_{b}\) & 625 W & \(v_{b}\) & 200 V & \(\omega_{b}\) & 2\(\pi\)50 Hz \\ \hline Filter & & & & \\ \(L_{c}\) & 0.05 pu & \(L_{g}\) & 0.2 pu & \(C_{f}\) & 0.05 pu \\ \(R_{c}\) & 0.005 pu & \(R_{g}\) & 0.02 pu & \(R_{f}\) & 0.008 pu \\ \hline Grid & & & & \\ \(V_{g}\) & 1 pu & \(\omega_{g}\) & 1 pu & & \\ \hline Inverter & & & & \\ \(V_{dc}\) & 1.6 pu & \(f_{sw}\) & 10 kHz & \(T_{s}\) & 100 μs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters. \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Parallel PI** & **Angle Limiter** & **External Frequency Support** \\ \hline \(H\) (s) & 5 & 5 & 4.36 \\ \(D\) & 20 & 20 & 20 \\ \(k_{d}\) & 0.126 & 0.126 & 0.049 \\ \(k_{p}\) & 0.02 & 0.32 & 0.035 \\ \(k_{i}\) & 0.785 & 15.3 & 0.785 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overloading mitigation strategy tuning. The proportional gains of the controllers are adjusted according to the damping needs of the system. In parallel PI, it is tuned to keep the damping level of normal operation (15). In AL, the proportional gain of the PLL is tuned to yield a damping ratio \(\xi=1/\sqrt{2}\) in the closed-loop poles of the PLL dynamics, described in (19). In these conditions, the PLL bandwidth is close to 23 Hz. The power response during AL overload mitigation results in lower damping (0.32) due to the high \(L_{g}/L_{c}\) ratio, determined by (23). To further damp the active power response, the proportional gain should be doubled, increasing the PLL bandwidth to 35 Hz. The bandwidth of the PLL should be kept as low as possible to ensure stability in weak grids and reduce its sensitivity to voltage noise. Additionally, the AL requires a phase compensation of \(n=1.5\) samples. This compensation is related to the discretization of the controller and the effect of the PWM [42]. In the EFS strategy, the damping of both inner and outer controllers is set to \(\xi=1/\sqrt{2}\), adjusting both \(k_{d}\) in the outer controller and \(k_{p}\) in the inner controller. Although the EFS inertia is significantly higher than that of the inner power controller, its cascaded control structure introduces potential coupling between the two loops. While each controller is tuned independently based on the small-signal models from Section 3.3, this interaction causes deviations from the expected closed-loop dynamics. Figure 11(a) compares the \(\Delta P/\Delta\omega_{g}\) bode plot of the PSL without overloading mitigation and EFS strategy (see Appendix A for the transfer function). The damping of the dominant poles during normal operation is reduced, as illustrated in Figure 11(b). Additionally, the natural frequency shifts to approximately 1.2 Hz. Figure 12: \(\Delta P/\Delta\omega_{g}\) of a GFM without overload mitigation and EFS in normal operation.