## 3 First Generation: Virtual Inertia Topologies
### Other Topologies  
Some other topologies that have been proposed in the literature are--virtual synchronous machine, referred to as \"VISMA\" in the literature, Institute of Electrical Power Engineering (IEPE's) topology, Kawasaki Heavy Industries (KHI) lab's topology, synchronous power controllers (SPC), virtual oscillators, inducverters, etc. The basic concept of inertia emulation remains the same in all these techniques. The VISMA topology as proposed in [67] uses \(d\)-\(q\) (synchronous reference frame) based  
Figure 10: Schematic for frequency droop control. mathematical model of a SG. This model when implemented in the digital controller of a power inverter replicates the dynamics of a SG. Instantaneous measurements of the grid voltage are used to compute the stator currents of the virtual machine and these currents are injected through a hysteresis current control approach using a power inverter. However, concerns with numerical instability have been reported with the VISMA model [68]. To improve robustness, a three-phase model has been proposed in [69] over a \(d\)-\(q\) based model. This is especially effective under unsymmetrical load conditions or rapid disturbances in the grid. A comparison between the VISMA algorithm implemented as a current source versus a voltage source has also been performed in [70]. The VISMA model implemented as a voltage source is referred to as IEPE's topology in the literature [28]. Instead of using voltage as input as with the VISMA topology, IEPE's topology uses the DG output current as input and generates reference voltages for the virtual machine. The IEPE topology is better suited for islanded operation, but transient currents particularly during the synchronization processes when operated in grid-connected mode can be difficult to deal with. In the KHI topology, instead of using detailed dynamic model of SG, an equivalent governor and automatic voltage regulator (AVR) model is implemented in a digital controller to generate voltage amplitude and phase reference for the virtual machine [71]. The reference is then used to generate current references based on algebraic-phasor representation of the SGs. Another popular topology for virtual inertia implementation is the SPC as proposed in [72; 73; 74]. The general structure of the control algorithm is similar to the structure proposed in the Ise lab's topology, but instead of operating the converter as a voltage controlled system or a current controlled system, it implements a cascaded control system, with an outer voltage loop and an inner current control loop through the use of a virtual admittance. In general, such a cascaded control structure provides inherent over-current protection during severe transient operating conditions. This is lacking in other open-loop approaches such as synchronverters or the Ise lab's topology [75] described previously. SPC also avoids the discontinuities encountered in solving the mathematical models, thus making the system more robust against numerical instabilities. The nested loop structure however does entail complexity in tuning the control system parameters. Furthermore, at its core, instead of using the swing equation for inertia emulation, a second order model with an over-damped response is proposed. This helps to reduce the oscillations in the system [55]. Improved forms of this second-order model was presented in [55; 76]. Inducverters [58] are one of the recent topologies that has been proposed which tries to mimic the behavior of induction generators instead of SGs. This method has the advantage of auto-synchronization without a PLL [77]. A virtual-inertia based static synchronous compensator (STATCOM) controller was proposed in [65] which behaves as synchronous condenser. The virtual inertia controller was used to exploit the fact that no PLL is required, hence providing improved voltage regulation compared to traditional STATCOMs with PLL units.