# 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 governor model refers to the active power synchronization loop used in GFM converters to emulate the swing equation of an SG. As shown in Figure 2, this block takes the actual power \(P_{G}\) and its reference \(P_{G}^{*}\) as inputs, along with the active current \(I_{aG}\) and its maximum limit \(I_{aG}^{max}\). The output of this block is the power angle \(\delta\), representing the angular position of the internal voltage vector \(Ee^{\delta s}\) with respect to the \(DQ\) stationary reference frame (see Figure 3a). The dynamic equation for \(\delta\) is obtained by integrating the rotational speed \(\omega\) with respect to the synchronously rotating reference \(\omega_{0}\) in rad/s [34], as given by Equation (19):  
\[\frac{1}{\omega_{0}}\frac{d\delta}{dt}=\Delta\omega \tag{19}\]  
where \(\Delta\omega\) is the difference between \(\omega\) and \(\omega_{0}\) in p.u. As can be seen in Figure 7, the frequency increment \(\Delta\omega\) is composed of the sum of three components. The first component, \(\omega_{1}\), is the output of the first-order block, representing the swing equation, defined by the inertia constant \(H\) and damping factor \(D\). The dynamic of \(\omega_{1}\) is given by Equation (20), which is derived from the classical power swing equation used to model frequency behavior in electrical power systems. This formulation has been extensively studied in the context of synchronous generators and grid-forming converters, particularly in low-inertia systems, where its influence on frequency stability is crucial [2; 34]. The equation captures the interplay between system inertia, damping, and power imbalances, making it fundamental for maintaining stable frequency control. \[2H\frac{d\omega_{1}}{dt}+D\omega_{1}=P_{G}^{*}-P_{G} \tag{20}\]  
Both parameters, \(H\) and \(D\), indicate the active power variation of the GFM during a frequency event to ensure its synchronization. In the case of the inertia constant (\(H\)), the GFM converter exchanges extra active power only when the frequency is changing, while the damping factor (\(D\)) comes into play when a frequency deviation occurs. One of the key advantages of GFM over real SGs is that \(H\) and \(D\) can be readily changed during operation, with the only restriction being having sufficient energy on the DC bus to provide the required inertia and frequency response. However, it is not always possible to choose any set of \(H\) and \(D\) parameters since, in some cases, the active power oscillation can  
Figure 6: Reactive current limiter (RCL) block. become unacceptable [17]. For example, when designing a GFM with only inertial response (\(D=0\)), the transfer function of the active power under a frequency change presents two complex conjugate poles on the imaginary axis, representing a critically stable system with an undamped oscillatory mode. As depicted in Figure 7, a new component, \(\Delta\omega_{2}\), is added to the frequency \(\omega_{1}\). This signal is derived from the PSS block, where the actual active power \(P_{G}\) is filtered through a washout filter with a time constant \(T_{w}\) and a gain constant \(K_{w}\). Consequently, the output signal \(\Delta\omega_{2}\) is only non-null during changes in \(P_{G}\) and is zero in steady state. This signal has a negative sign, indicating that the PSS counteracts active power changes. The dynamic equations of the PSS block are given by Equations (21) and (22). \[T_{w}\frac{d\Delta z}{dt}=P_{G}-\Delta z \tag{21}\] \[\Delta\omega_{2}=K_{w}(P_{G}-\Delta z) \tag{22}\]  
where \(\Delta z\) is an auxiliary variable. While conventional grid-forming control strategies already incorporate a high damping coefficient (\(D\)), the addition of a power system stabilizer (PSS) is justified in cases where synthetic inertia effects introduce low-frequency oscillation modes [2]. These oscillations are particularly relevant in weak grids, where sudden disturbances can lead to prolonged transient instabilities.