## 3 First Generation: Virtual Inertia Topologies
### Virtual Synchronous Generators: A Frequency-Power Response Based Topology  
The main idea behind virtual synchronous generators (VSG) is to emulate the inertial response characteristics of a SG in a DG system, specifically the ability to respond to frequency changes [25; 44]. This emulates the release/absorption of kinetic energy similar to that of a SG, thus presenting the DG units as a dispatchable source [45; 46]. Compared to traditional droop controllers which provide only frequency regulation, the VSG approach is able to provide dynamic frequency control [21]. This dynamic control is based on the derivative of the frequency measurement and behaves similarly to inertial power release/absorption by a SG during a power imbalance. Thus, the VSG is a dispatchable current source that regulates its output based on system frequency changes. This is one of the simplest approaches to implement virtual inertia in DG systems as it does not incorporate all the detailed equations involved in a SG. However, operating multiple DG units as current sources is known to result in instability [47]. The output power of the VSG converter is controlled using Equation (12):  
\[P_{VSG}=K_{D}\Delta\omega+K_{I}\frac{d\Delta\omega}{dt} \tag{12}\]  
where, \(\Delta\omega\) and \(\frac{d\Delta\omega}{dt}\) represent the change in angular frequency and the corresponding rate-of-change. \(K_{D}\) and \(K_{I}\) represent the damping and the inertial constant, respectively. The damping constant is similar to the frequency droop and helps return the frequency to a steady-state value and reduce the frequency nadir. The inertial constant arrests the ROCOF by providing fast dynamic frequency response based on the frequency derivative. This feature is especially important in an isolated grid where the initial ROCOF can be very high, leading to unnecessary triggering of protection relays. The VSG topology is illustrated in Figure 9. A PLL is used to measure the change in system frequency and ROCOF [45]. Then, using Equation (12), the active power reference for the inverter is computed. The current references are then generated for the current controller based on this reference power. The topology illustrated here assumes a direct-quadrature (_d-q_) based current control approach, but any other current control techniques (as described in [48; 49]) may be used. For _d-q_ control, _d_-axis current reference can be calculated as [22]:  
\[I_{d}^{*}=\frac{2}{3}(\frac{V_{d}P_{VSG}-V_{q}Q}{V_{d}^{2}+V_{q}^{2}}) \tag{13}\]  
Figure 8: Ise Lab’s topology: (**a**) overall schematic showing operating principle; (**b**) the governor model to compute input power. where, \(V_{d}\) and \(V_{q}\) are the \(d-axis\) and \(q-axis\) components of the measured grid voltage \(v\). The \(q\)-axis current reference \(I_{q}^{*}\) and the reactive power \(Q\) is set to zero as it is assumed that only the active power is being controlled. The current controller based on the grid current feedback generates the gate signals to drive the inverter. Thus, the inverter behaves as a current-controlled voltage source inverter [13; 48]. This topology is used by the European VSYNC research group [45; 50] and has demonstrated the effectiveness of inertia emulation using VSG topology through real-time simulations [51] and several field tests [52]. In [22], an experimental verification of the topology is presented for remote microgrid applications. The VSG topology has also been widely employed for virtual inertia emulation from wind systems as reported in [6; 53; 54]. The main drawback of this topology is that it cannot be implemented in islanded modes where the virtual inertia unit has to operate as a grid forming unit. Moreover, the system emulates inertia during frequency variations, but not in input power variations [55]. Accurate measurement of the frequency derivative through PLLs can be challenging for this kind of implementation [56; 57]. The performance of PLLs can degrade and compete against each other, especially in weak grids [58; 59].