## 3 Calculation of System Inertia and Permissible Active Power Imbalance  
Inertia can be defined as the resistance of a body having a certain mass to a change in its velocity and the direction of its velocity. The inertia of an EPS consists of all the rotating parts of the electrical equipment synchronously connected to it, such as synchronous generators and their turbines. The higher the mass of generators and turbines, the higher the inertia of the EPS. Due to the inertia of the EPS, when there is a shortage of power generation in the system, the frequency of the EPS changes not suddenly but gradually over time [21; 22; 23; 24; 25]. This is because kinetic energy accumulated in the rotating mass is fed to the EPS in the form of electricity as it slows down. Kinetic energy of a rotating mass is described by the following equation  
\[E_{\mathrm{k}}=\frac{\int\cdot\omega_{\mathrm{m}}^{2}}{2}, \tag{1}\]  
where \(\mathrm{\textit{J}}\) is moment of inertia of generator and turbine, \(\omega_{\mathrm{\textit{m}}}\) is mechanical rotation speed of generator. The additional electrical energy in the transmission system reduces the active power imbalance, which in turn slows down the frequency variation in the EPS. The higher inertia of the EPS, the slower frequency variation, which gives more additional time for the frequency regulators to start operating. The relationship between the power balance of the system and the generator rotation speed is described as follows [26; 27]  
\[\mathrm{\textit{J}}\cdot\omega_{\mathrm{m}}\frac{d\omega_{\mathrm{m}}}{dt}=P_{ \mathrm{m}}-P_{\mathrm{e}}, \tag{2}\]  
where \(P_{\mathrm{m}}\) is mechanical power of the generator, and \(P_{\mathrm{e}}\) is load power. The left-hand side of the equation is the derivative of the kinetic energy (\(E_{k}\)) of the turbine and generator. This energy is usually expressed in proportion to the generator's rated power and is called the inertia constant \(H_{\mathrm{G}}\). The inertia constant indicates how long the generator can supply its rated power using only its available kinetic energy. The inertia constant is calculated as follows  
\[H_{\mathrm{G}}=\frac{\frac{\mathrm{\textit{J}}\cdot\omega_{\mathrm{m}}^{2}}{ 2}}{S_{\mathrm{g}}}=\frac{E_{\mathrm{k}}}{S_{\mathrm{g}}}, \tag{3}\]  
where \(S_{\mathrm{g}}\) is apparent nominal power of the generator. Expressing Equation (2) in relative units (\({}^{-}\)) and applying Equation (3) gives the following result:  
\[2\cdot H_{\mathrm{G}}\cdot\overline{\omega}_{\mathrm{m}}\frac{d\overline{\omega}_ {\mathrm{m}}}{dt}=\overline{P}_{\mathrm{m}}-\overline{P}_{\mathrm{e}}. \tag{4}\]  
Since the frequency is the same throughout the EPS, all individual generators can be described as a single generator (\(\overline{P}_{\mathrm{G}}\)) with a shared load (\(\overline{P}_{\mathrm{L}}\)):  
\[2\cdot H_{\mathrm{system}}\cdot\overline{\omega}_{\mathrm{m}}\frac{d\overline {\omega}_{\mathrm{m}}}{dt}=\overline{P}_{\mathrm{m}}-\overline{P}_{\mathrm{e}}, \tag{5}\]  
where \(H_{\mathrm{system}}\) is inertia constant of the total EPS:  
\[H_{\mathrm{system}}=\frac{\sum H_{\mathrm{G}}\cdot S_{\mathrm{G}}}{\sum S_{ \mathrm{G}}}=\frac{\sum E_{\mathrm{k}}}{S_{\mathrm{system}}}, \tag{6}\]  
where \(S_{\mathrm{system}}\) is total installed apparent power of the generators in the EPS. To explore the effect of inertia on the frequency variation in the EPS, a digital simulation study was carried out where three generators with a generating capacity of 100 MW each were successively connected to the EPS. One generator has the FCR function switched on and the other two generators have the FCR function switched off. The EPS has an additional load of 10 MW connected. The simulation results are shown in Figures 2 and 3:  
1. Only one generator (G1) is switched on with the FCR function enabled;
2. One generator (G1) with the FCR function switched on and two generators (G2 and G3) with the FCR function switched off. The arrows in Figure 3 show the trajectory of the EPS frequency variation when the FCR is not activated.