# Overload Mitigation of Inertial Grid-Forming Inverters Under Frequency Excursions
## 2 Grid-Forming Inverter Modelling and Overload Mechanisms
### Overload Mechanisms in GFM Inverters
#### 2.2.2 Inertial Power  
Inertia counteracts changes in grid frequency by injecting an active power that opposes RoCoF. A common approach to evaluate frequency dynamics and inertial response involves modelling a GFM connected to an inductive grid. In these conditions, the active power and the phase shift between the inverter and the grid, \(\delta\), are related in (4), and voltage amplitude and reactive power coupling can be neglected. The quasi-stationary small-signal model of the system in per unit notation is given in Figure 3, where (4) is linearized to the synchronization constant \(K_{t}\). This constant is inversely proportional to the total inductance when phase shifts are small and frequency and voltage values are close to its rated values (5). \[P=\frac{EV_{g}}{\omega_{r}(L_{c}+L_{g})}\sin\delta \tag{4}\]  
\[\Delta P=\left(\frac{E_{0}V_{g0}}{\omega_{r0}(L_{c}+L_{g})}\cos\delta_{0} \right)\Delta\delta\approx\underbrace{\frac{1}{L_{c}+L_{g}}}_{K_{t}}\Delta\delta \tag{5}\]  
The active power response under grid frequency perturbations, denoted as \(\omega_{g}\), is given in (6). From this transfer function, two main sources of overload are identified:  
Figure 3: Small-signal model for the frequency dynamics of the proposed GFM inverter. Figure 2: Frequency regulation limitation. (**a**) P-\(\omega_{r}\) droop characteristic curve. (**b**) Virtual power strategy. \[\frac{\Delta P}{\Delta\omega_{g}}=-\frac{K_{t}\omega_{b}(2 Hs+D)}{2 Hs^{2}+(K_{t} \omega_{b}k_{d}+D)s+K_{t}\omega_{b}} \tag{6}\]
* High inertia: The inertial power \(\Delta P_{H}\) is obtained using the final value theorem of \(\Delta P/\Delta\omega_{g}\) under a ramp input when the frequency regulation is disabled (\(D=0\)). This power is proportional to the inertia constant \(H\) and the \(RoCoF\) (slope of the ramp input): \[\Delta P_{H}=\lim_{s\to 0}s\frac{\Delta P}{\Delta\omega_{g}}\frac{RoCoF}{s^{2}}=-2H \cdot RoCoF\] (7)
* Limited damping: A poorly damped system can result in increased overload due to the power overshoot that occurs during frequency deviations. If the frequency regulation is disabled, the response of the GFM inverter to grid frequency ramps follows the second-order system in (8). \[\frac{\Delta P}{s\Delta\omega_{g}}=-\frac{K_{t}\omega_{b}(2H)}{2 Hs^{2}+K_{t} \omega_{b}k_{d}s+K_{t}\omega_{b}}\] (8) In this second-order system, the damping \(\xi\) mainly depends on \(k_{d}\), as shown in (9). The maximum power peak, \(\Delta P_{Hmax}\), can be estimated using the overshoot ratio, \(M_{p}\), according to (10). When the system is critically damped (\(\xi=1\)), the maximum power peak is equal to the inertial power \(\Delta P_{Hmax}=\Delta P_{H}\). \[\xi=\frac{\sqrt{2 HK_{t}\omega_{b}}}{4H}k_{d}\] (9)  
\[\Delta P_{Hmax}=(1+M_{p})\Delta P_{H}=\left(1+e^{-\frac{\pi\xi}{\sqrt{1-\xi^{ 2}}}}\right)\Delta P_{H} \tag{10}\]  
Figure 4a relates the grid frequency RoCoF, inertia constant, and the inertial power according to (7). However, in the figure, inertial power is interpreted as overload. This is because, when the inverter operates at its operational limits, the inertial power can be regarded as the active power that surpasses the imposed limits, hence determining the active power overload. As expected from the previous equation, the overloading power increases linearly with \(H\) and the RoCoF of the perturbation. Since the overloading can be exacerbated due to the poor damping, the overload obtained from Figure 4a needs to be scaled to account for the overshoot, according to (10). The scaling factor, \(1+M_{p}\), for different \(\xi\) values is plotted in Figure 4b. A poorly damped system can increase the active power overloading up to 50%. For systems with damping above 0.7, the overshoot can be neglected. Figure 4 serves as a tool to size the overloading capability of a GFM inverter, considering different inertia constants and allowed grid frequency RoCoFs. A GFM inverter with high inertia should be oversized to handle a peak active power when operating near its operational limits.