## 2 Inertia of the Electric Power System  
Each rotating machine (motor or generator) that is directly connected to the grid provides a certain level of inertia to the system. In this regard, industrial consumers with their induction motors also contribute some inertia to the system. However, the majority comes from synchronous generators within conventional power plants. The inertia of a synchronous generator involves the rotating masses of the drive turbine, shafts, and the generator rotor, where kinetic energy is stored. The parameter used to quantify inertia is called the inertia constant. It is expressed in seconds and represents the time required for the unit to accelerate from rest to a synchronous speed when driven by a constant torque equivalent to the ratio of the rated apparent power and synchronous speed. To assess the inertial response, the inertia constant can also be defined as the time during which the machine can provide active power corresponding to the rated apparent power solely through the stored kinetic energy. As RESs replace conventional generators, the system's inertia response diminishes, leading to a faster RoCoF and lower frequency nadirs within a very short timeframe. Lower-inertia systems show significantly lower-frequency nadirs and a higher RoCoF. Such situations can trigger frequency relays and, in the worst-case scenario, cascade failures. The solution for these scenarios involves incorporating virtual inertia into the system [26]. Virtual inertia combines control algorithms, renewable energy sources, energy storage systems (ESS) like batteries, and power electronics to mimic the inertia of traditional synchronous generators. This approach aims to compensate for the decreasing natural inertia in EPS caused by the integration of RESs. This shift is due to the increasing dominance of inverter technology in EPS, replacing conventional synchronous generators. Figure 1 presents the evolution of modern power systems with dominant inverter technology. While CIGs typically lack the traditional frequency response processes found in SGs, such as inertia response and speed governor control, implementing virtual inertia through power converter control allows the acquisition of non-rotating inertia. This approach, utilized in CIG units with virtual inertia control, opens the possibility of defining inertia considering CIGs in power systems. \[H_{\text{sys}}=\frac{\sum_{i=1}^{N_{\text{SG}}}H_{\text{SG},i}\cdot S_{B,i}+ \sum_{j=1}^{N_{\text{CIG}}}H_{\text{CIG},j}\cdot S_{B,j}}{\sum_{i=1}^{N_{\text{ SG}}}S_{B,j}+\sum_{j=1}^{N_{\text{CIG}}}S_{B,j}} \tag{1}\]  
where  
* \(H_{\text{sys}}\) is the equivalent apparent inertia constant of the system [s];
* \(S_{B,i}\) is the rated apparent power of \(i\)th SG [MVA];
* \(S_{B,j}\) is the rated apparent power of \(j\)th CIG [MVA];
* \(H_{\text{SG},i}\) is the inertia constant of \(i\)th SG based on its own rated apparent power [s];
* \(H_{\text{CIG},i}\) is the virtual inertia of \(j\)th CIG based on its own rated apparent power [s];
* \(N_{\text{SG}}\) is the number of SGs in the power system;
* \(N_{\text{CIG}}\) is the number of CIG units in the power system. The typical values of inertia constant vary based on the types of SGs: around 1.75 to 4.75 s for hydroelectric power plants, 2 to 10 s for thermal power plants, and approximately 4 s for nuclear power plants [6]. The advent of WAMS enabled the derivation of the CoI frequency, offering a comprehensive view of the system's frequency response to estimate disturbances resulting from production loss using the generator rotor swing equation [27]. To apply the generator rotor swing equation to a multi-generator system, and the following specific mathematical  
Figure 1: Progression towards an electric energy system with dominant inverter technology. manipulations as outlined in references [28; 29], the following equation for the generator rotor swing can be formulated:  
\[2H_{Col}\frac{df_{Col}}{dt}+D_{Col}\Delta f_{Col}=\Delta p_{Col} \tag{2}\]  
where \(D\) represents the damping of the system.