## 3 First Generation: Virtual Inertia Topologies  
### Concept and Classification of Virtual Inertia Topologies  
The frequency variation in a power system after a frequency event/disturbance can be approximated by the swing equation [25]:  
\[P_{gen}-P_{load}=\frac{d(E_{K,E.})}{dt}=\frac{d(\frac{1}{2}J\omega_{g}^{2})}{dt} \tag{1}\]  
\[P_{gen}-P_{load}=J\omega_{g}\frac{d\omega_{g}}{dt} \tag{2}\]  
where, \(P_{gen}\) is the generated power, \(P_{load}\) is the power demand including losses, \(J\) is the total system inertia, and \(\omega_{g}\) is the system frequency. The inertia constant of the power system \(H\) is the kinetic energy normalized to apparent power \(S_{g}\) of the connected generators in the system:  
\[H=\frac{J\omega_{g}^{2}}{2S_{g}} \tag{3}\]  
Equation (2) can then be written as:  
\[\frac{2H}{\omega_{g}}\frac{d\omega_{g}}{dt}=\frac{P_{gen}-P_{load}}{S_{g}} \tag{4}\]  
Equation (4) can also be represented in terms of frequency (Hz) instead of angular frequency (rad/s) as follows:  
\[\frac{2H}{f}\frac{df}{dt}=\frac{P_{gen}-P_{load}}{S_{g}} \tag{5}\]  
where, \(\frac{df}{dt}\) is the ROCOF of the system. With reduced inertia, the ROCOF of the system increases which causes larger changes in frequency of the system in the same time-frame. Thus, the system requires additional inertia as more RESs are integrated into the power system. The concept of virtual inertia implementation using power electronic converters was first developed by [PERSON] and [PERSON] [26]. Many other topologies and approaches have been developed in the literature since.  
Virtual inertia is a combination of control algorithms, RESs, ESSs, and power electronics that emulates the inertia of a conventional power system [13]. The concept of virtual inertia is summarized in Figure 5. The core of the system is the virtual inertia algorithm that presents the various energy sources interfaced to the grid through power electronics converters as SGs. Most modern wind turbines are operated as variable speed wind turbines and interfaced through back-to-back converters, completely decoupling the inertia from the grid. Similarly, PV systems and ESSs have a DC-DC converter and an inverter in the front-end, and do not contribute to the inertial response [4; 27]. Virtual inertia systems based on current/voltage feedback from the inverter output generate appropriate gating signals to present these resources as SGs from the point-of-view of the grid [28]. Although the basic underlying concepts are similar among the various topologies in the literature, the implementation is quite varied based on the application and desired level of model sophistication. Some topologies try to mimic the exact behavior of the SGs through a detailed mathematical model that represent their dynamics. Other approaches try to simplify this by using just the swing equation to approximate the behavior of SGs, while others employ an approach which makes the DG units responsive to frequency changes in the power system. This section discusses the various topologies that have been proposed in literature. Figure 6 shows a general classification of various topologies that are available in the literature for virtual inertia implementation. Among the listed topologies, the synchronverter, the Ise lab's topology, the virtual synchronous generator (most popular in literature from each classification), and the droopcontrol were selected for a detailed description. A brief description of the remaining topologies is also presented under Section 3.6.  
Figure 5: Concept of virtual inertia.  
Figure 6: Classification of different topologies used for virtual inertia implementation.