# RMS Modeling and Control of a Grid-Forming E-STATCOM for Power System Stability in Isolated Grids
## 3 GFM Converter Modeling and Control
### Governor Model
The PSS proposed in this work enhances power oscillation damping, ensuring faster frequency stabilization after large power imbalances. Similarly, as indicated in Figure 7, another component in the frequency increment, \(\Delta\omega_{3}\), is introduced in the active power synchronization loop to limit the active current. The control logic for this active current limiter (ACL) block is shown in Figure 8. The maximum current limit, \(I_{aG}^{max}\), is multiplied by \(L_{p}\) and \(L_{n}\), two inputs derived from the ESS control block. \(L_{p}\) is set to \(1\) when the DC voltage, \(V_{dc}\), is within limits, and \(0\) if \(V_{dc}\leq V_{dc,min}\), preventing further discharge of the ESS to avoid a reduction in \(V_{dc}\). Conversely, \(L_{n}\) is set to \(-1\) when \(V_{dc}\) is within limits, and \(0\) if \(V_{dc}\geq V_{dc,max}\). When this condition is met, the ESS charging stops, and \(V_{dc}\) is limited to its maximum value, \(V_{dc,max}\). Figure 7: GFM governor model. As shown in Figure 8, if \(-l_{aG}^{max}<I_{aG}<I_{aG}^{max}\), both upper and lower limiters saturate to zero, and therefore \(\Delta\omega_{3}=0\). However, when \(I_{aG}>I_{aG}^{max}\), the upper limiter saturates to \(-k\), and the lower limiter to 0, so that \(\Delta\omega_{3}=-k\), i.e., the power angle \(\delta\) is reduced, and therefore the active current is limited. On the contrary, when \(I_{aG}<-I_{aG}^{max}\), the upper limiter saturates to 0, and the lower limiter to \(k\), so that \(\Delta\omega_{3}=k\). In this case, the power angle is negative, and its absolute value is reduced. Finally, \(K_{p}\) is a gain that determines the time response to reach the saturation limits when \(I_{aG}\) exceeds \(I_{aG}^{max}\) or \(-I_{aG}^{max}\). The higher \(K_{p}\) is, the shorter the time response. One of the challenges of separate active and reactive current limitations is ensuring robustness under large phase shifts, where a simple proportional control \(K_{p}\) may not be sufficient. To address this issue, the active current limiter (ACL) in this work dynamically adjusts \(\Delta\omega_{3}\) based on the ESS state and the active current limits \(I_{aG}^{max}\). This prevents excessive phase shifts that could destabilize the synchronization process. The simulation results in Section 5 confirm that the proposed ACL effectively limits current overshoots while maintaining stable power exchange, even under significant disturbances. The ACL/RCL control logic operates by sequentially verifying the active and reactive current limits to adjust the power angle (\(\delta\)) and modulation index (\(m_{a}\)). The process begins by checking the active current (\(I_{aG}\)). If it exceeds the predefined limit (\(I_{aG}^{max}\)), the power angle \(\delta\) is adjusted accordingly to regulate the current. Otherwise, \(\delta\) remains unchanged. Next, the reactive current (\(I_{rG}\)) is evaluated. If it surpasses its threshold (\(I_{rG}^{max}\)), the modulation index (\(m_{a}\)) is modified to bring the current within acceptable levels. If the reactive current is within limits, \(m_{a}\) remains unchanged. Finally, the updated values of \(\delta\) and \(m_{a}\) are applied to the converter control to ensure proper operation.