# Performance Tuning for Power Electronic Interfaces Under VSG Control
## 2 VSG Emulation Control Strategies for VSIs with Frequency Control  
For the IBG, the external characteristics are largely determined by the supplementary controller in the control structure. In inchoate studies, among the VSG implementations in different orders, the simplest second-order model is with better stability in transients [34] and can be combined with virtually any VSI control strategies involving a cascade structure [35]. By the implementation of the VSG algorithm, the IBG is capable of performing IR and PFR in the same way as the SGs. The equations of motion, describing the effect of unbalance between the mechanical torque and the electromagnetic torque on rotation, could be found in many classical textbooks (such as [16]). The inertial response of the SGs is instantaneous 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 \(J\) 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 analogically expressed as the ratio of the provided energy to its 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,  
\[P_{m}-P_{e}=\frac{d\Delta\omega_{r}/dt}{2H_{SG}}+D\Delta\omega_{r} \tag{3}\]where \(D\) is the damping coefficient of frequency sensitive load, \(P_{m}\) and \(P_{e}\) are the virtual mechanical and electrical power (when the initial status is assumed as 0, and the second-order terms are neglected), \(\omega_{r}\) is virtual angular speed. When the swing equation is represented by inertia constant, all the parameters are per-unit values. The PFR of SGs is provided by the turbine-governor system [16]. In per unit value, for a typical SG with a reheat steam turbine, the droop property is  
\[\Delta Y=-\frac{1}{R}\times\frac{1}{1+sT_{G}}\times\frac{1+sF_{HP}T_{RH}}{(1+sT _{CH})(1+sT_{RH})}\Delta\omega_{r} \tag{4}\]  
where \(R\) is the speed droop, \(Y\) is the valve position, \(T_{G}\) is the time constant of speed governor, and \(F_{HP}\), \(T_{RH}\), \(T_{CH}\) are typical parameters for a reheat steam turbine. The emulation for the properties of turbine-speed governor systems of SGs is necessary when the VSG-based PEIs are operated in parallel with traditional power plants. By the exploitation of conventional theories of the SG-based power system, the VSG algorithm also enables the PEIs to perform SFR and TFR. The SFR kicks in when the frequency deviation lasts for a preset time period after the frequency is stabilized by the IR and PFR. The TFR is activated by the system operator (SO) to set the load reference for each generation unit. For traditional SGs, the SFR and TFR are executed by the regulation of load reference settings, which is shown in Figure 1a. The variables of load references also pass through the blocks of the turbine-governor system. However, because only specific units in the power system are selected to perform automatic generation control (AGC) regulation, when the emulation of AGC is implemented on VSG-PEIs, the load reference can be directly fed into the block of the virtual swing equation, as shown in Figure 1b. In Figure 1b, \(\tau\) is the time delay of SFR, \(K_{I}\) is the coefficients of the integral controller. The system frequency indicates the overall power balance between the generation and the demand at any instantaneous time. In the traditional SG-dominated power system, active power-frequency control is a series of multiple temporal processes that can be divided into four stages [16].