## 2 VSG-Based Frequency Control
### Emulation of IR and PFR  
The inertial response of the SGs is instantaneous, and without prerequisite measurements. For traditional SGs, the inertia is provided by the kinetic energy stored in the rotation equipment. The inertia constant \(H_{SG}\) indicates the inertia property, which can be expressed as  
\[H_{SG}=\frac{J_{SG}\omega_{r}^{2}}{2 VA_{base}} \tag{1}\]  
where \(J_{SG}\) is the moment of inertia, \(\omega_{r}\) is the angular speed of the rotor, \(VA_{base}\) is the rated power. Similar to \(H_{SG}\), the virtual inertia constant \(H_{vir}\) of PEIs can be expressed as the ratio of the provided energy to the rated power.  
\[H_{vir}=\frac{J_{vir}\omega_{m}^{2}}{2 VA_{base}} \tag{2}\]  
where \(J_{vir}\) is the virtual moment of inertia, \(\omega_{m}\) is the virtual angular speed.  
The inertial response follows [PERSON]'s law of motion, which can be expressed as the swing equation  
\[P_{m}-P_{e}=\frac{d\Delta\omega_{r}/dt}{2H_{SG}}+D\Delta\omega_{r} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & IR & PFR & SFR & TFR \\ \hline Time scale & Instantaneous & Seconds-30 s & 10–30 min & Hours-day ahead \\ \hline Activation & ROCOF & \
\begin{tabular}{c} Frequency deviation crosses \\ a preset deadband \\ \end{tabular} &
\begin{tabular}{c} Frequency deviation last \\ over a preset period \\ \end{tabular} & Economic dispatching \\ \hline Reference & Naturally provided & Frequency deviation & Frequency deviation & Load reference \\ \hline Control parameter & \(H\) & \(R_{SG}\) & \(K_{I}\) & \(P_{scheduled}\) \\ \hline Participants & All SGs & All SGs with headroom & Selected SG unit & All SGs \\ \hline Executor & Rotating equipment & Speed governor & AGC & Turbine \\ \hline Energy resource & Kinetic & Primary input & Primary input & Primary input \\ \hline Restoration & \multicolumn{3}{c}{The restoration of rotor speed is coupled with the frequency recovery of the system.} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Conventional multiple temporal frequency control.  
where \(D\) is the damping coefficient of frequency sensitive load, \(P_{m}\) and \(P_{e}\) are the virtual mechanical and electrical power, \(\omega_{r}\) is virtual angular speed. When the swing equation is represented by the inertia constant, all the parameters are per-unit values. Then the inertia support can be expressed as  
\[\Delta P_{IR}=-2H_{vir}\frac{d\Delta f}{df} \tag{4}\]  
The PFR of SGs is provided by the turbine-governor system. In per unit value, the droop property for an SG with a typical reheat steam turbine is  
\[\Delta Y=-\frac{1}{R}\times\frac{1}{1+sT_{G}}\times\frac{1+sF_{HP}T_{RH}}{(1+ sT_{CH})(1+sT_{RH})} \tag{5}\]  
where \(R\) is speed droop, \(Y\) is the valve position, \(F_{HP}\), \(T_{CH}\), and \(T_{RH}\) are typical parameters for a reheat steam turbine. The emulation for the properties of turbines of SGs is necessary when the VSG-based PEIs are operated in parallel with traditional power plants. In steady state, the supporting power by PFR is  
\[\Delta P_{PFR}=-\frac{1}{R}\Delta f \tag{6}\]  
The implementation of the emulation control strategies for IR and PFR is executed based on the mathematical model of SGs mentioned before. When the VSG algorithm is activated, the control block of the angular speed deviation is shown in Figure 1.