## 3 First Generation: Virtual Inertia Topologies
### Synchrotroners: A Synchronous Generator Model Based Topology  
Synchronverters operate the inverter-based DG units as SGs representing the same dynamics from the point-of-view of the grid [29]. This is based on the notion that such a strategy allows traditional operation of the power system to be continued without major changes in the operation structure. The topology is well developed in the literature by [PERSON] [30]. A frequency drooping mechanism is used to regulate the power output from the inverter similar to how the SG regulates its power output [31]. The following basic equations are used to capture the dynamics of the SG:  
\[T_{e}=M_{f}i_{f}<i,\widetilde{sin}\theta> \tag{6}\]  
\[e=\dot{\theta}M_{f}i_{f}\widetilde{sin}\theta \tag{7}\]  
\[Q=-\dot{\theta}M_{f}i_{f}<i,\widetilde{cos}\theta> \tag{8}\]  
where, \(T_{e}\) is the electromagnetic torque of the synchronverter, \(M_{f}\) is the magnitude of the mutual inductance between the field coil and the stator coil, \(i_{f}\) is the field excitation current, \(\theta\) is the angle between the rotor axis and one of the phases of the stator winding, \(e\) is the no load voltage generated, and \(Q\) is the generated reactive power. In Equations (6) and (8), \(\langle\cdot,\cdot\rangle\) represents the standard inner product of two vectors in \(\mathbb{R}^{3}\). The three-phase stator current, \(i\), \(\widetilde{sin}\theta\), and \(\widetilde{cos}\theta\) are vectors defined as follows:  
\[i= \begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} ;\widetilde{sin}\theta= \begin{bmatrix}sin\theta\\ sin(\theta-\frac{2\pi}{3})\\ sin(\theta-\frac{4\pi}{3})\end{bmatrix} ;\widetilde{cos}\theta= \begin{bmatrix} cos\theta\\ cos(\theta-\frac{2\pi}{3})\\ cos(\theta-\frac{4\pi}{3})\end{bmatrix}  \tag{9}\]  
Equations (6)-(8) are first discretized and then solved in each control cycle in a digital controller to generate the gating signals for the DG unit under consideration. Figure 7a shows the basic schematic of the synchronverter. The dashed box represents the control part of the synchronverter, the details of which are illustrated in Figure 7b. The inverter output current \(i\) and grid voltage \(v\) are the feedback signals utilized to solve the differential equations within the controller. Additionally, the desired moment of inertia \(J\) and damping factor \(D_{p}\) can be set as desired. The selection of these parameters is crucial from the point-of-view of the stability of the system as shown in [32]. The frequency and voltage loops, as indicated in Figure 7b, are used to generate the control inputs--the mechanical torque, \(T_{m}\) and \(M_{f}I_{f}\). In the frequency loop, \(T_{m}\) is generated from the reference active power \(P^{*}\) based on the nominal angular frequency of the grid \(w_{n}\). The virtual angular frequency of the synchronverter \(w\) is thus generated by this loop which is integrated to calculate the phase command \(\theta\) and is used for the pulse width modulation (PWM). Similarly, in the voltage loop, the difference between the reference voltage \(v^{*}\) and and the amplitude of the grid voltage \(v\) is multiplied by a voltage drooping constant \(D_{q}\). This is added to the error between the reference reactive power \(Q^{*}\) and the reactive power \(Q\) calculated using (8). The resulting signal is then passed through an integrator with gain \(\frac{1}{K_{v}}\) to generate \(M_{f}I_{f}\). The outputs of the controller are \(e\) and \(\theta\) which are used for PWM generation. The underlying equations of a synchronverter topology form an enhanced phase locked loop (PLL) or a sinusoid-locked loop, making it inherently capable of maintaining synchronism with the terminal voltage [33]. Single phase variants of the synchronverter have also been designed in [34]. The basic version of synchronverter requires a PLL to initially synchronize with the grid, however the use of PLLs in weak grids is known to be prone to instabilities [35; 36; 37]. To counteract this, self-synchronized synchronverters are introduced in [38].