## 2 Modeling and Analytical Benchmarking of Equivalent Virtual Inertia for DFIGs
### Analytical Derivation of Equivalent Inertia Time Constant  
The virtual inertia time constant of a wind turbine is defined by the ratio of kinetic energy stored at the rated rotor speed to the generator's rated capacity and can be formulated as follows [32; 40]:  
\[H\ =\ \frac{E_{k}}{S_{N}}=\frac{1}{2}\frac{\omega_{s0}^{2}J_{eq}}{S_{N}}= \frac{\omega_{s0}\omega_{r0}}{\omega_{N}^{2}}\frac{\Delta\omega_{r}}{\Delta \omega_{s}}H_{wind} \tag{7}\]  
where \(E_{k}\) represents the kinetic energy stored in the generator rotor at the rated speed of a single wind turbine; \(S_{N}\) is the rated capacity of the wind turbine; \(\omega_{N}\) is the rated angular frequency of the wind turbine; \(\omega_{s0}\) and \(\omega_{r0}\) are the initial grid angular frequency and initial generator speed, respectively; and \(J_{eq}\) is the virtual equivalent moment of inertia of the wind turbine.  
Subsequently, the equivalent virtual inertia time constants corresponding to the two control strategies can be derived individually [41].  
\[H_{1}(s)=\frac{H_{wind}\omega_{s0}\omega_{r0}k_{d1}s^{2}}{\omega_{N}^{2}\Big{(} 1+T_{f}\cdot s\Big{)}\big{(}2H_{wind}s^{2}+k_{p1}s+k_{i1}\big{)}} \tag{8}\]  
\[H_{2}(s)=\frac{H_{wind}\omega_{s0}\omega_{r0}\big{(}k_{d2}s+k_{p}\big{)}s}{ \omega_{N}^{2}\Big{(}1+T_{f}\cdot s\Big{)}\big{(}2H_{wind}s^{2}+k_{p1}s+k_{i1 }\big{)}} \tag{9}\]  
where \(H_{1}\) denotes the equivalent virtual inertia time constant under virtual inertia control, and \(H_{2}\) represents the equivalent virtual inertia time constant under integrated inertia control.  
The above equation is the method for calculating the inertia time constant of a single wind turbine. For a wind farm consisting of \(n\) DFIG units, the wind farm can be equivalently represented as a single aggregated unit, with the equivalent virtual inertia given as follows [32]:  
\[H_{eq}=\frac{\sum\limits_{j=1}^{n}E_{kj}}{\sum\limits_{j=1}^{n}S_{Nj}}=\frac{ \frac{J_{eqq1}\omega_{s0}^{2}}{2}+\frac{J_{eqq2}\omega_{s0}^{2}}{2}+\ldots+ \frac{J_{eqm}\omega_{s0}^{2}}{2}}{S_{N1}+S_{N2}+\ldots+S_{Nn}}=\]  
\[\frac{H_{eqq1}S_{N1}+H_{eq2}S_{N2}+\ldots+H_{eqm}S_{Nn}}{S_{N1}+S_{N2}+\ldots+ S_{Nn}}= \tag{10}\]  
\[\frac{\sum\limits_{j=1}^{n}H_{equj}S_{Nj}}{\sum\limits_{j=1}^{n}S_{Nj}}=\frac{ \sum\limits_{j=1}^{n}H_{equj}}{n}\]  
where \(H_{eq}\) is the equivalent virtual inertia of the wind farm; \(H_{equj}\) is the virtual inertia of a single WT.  
By substituting Equation (10) into Equation (9), the frequency expression for the equivalent virtual inertia of the wind farm with integrated inertia control can be derived as follows:  
\[H_{eq}(s)=\frac{H_{wind}\omega_{s0}\omega_{r0}\big{(}k_{d2}s+k_{p}\big{)}s}{ \omega_{N}^{2}\Big{(}1+T_{f}\cdot s\Big{)}\big{(}2H_{wind}s^{2}+k_{p1}s+k_{i1} \big{)}}\cdot\frac{\sum\limits_{j=1}^{n}\omega_{r0j}}{n} \tag{11}\]  
where \(\omega_{r0j}\) is the initial angular velocity of the jth WT.  
Based on Equation (11), the equivalent virtual inertia of wind farms is time-varying, depending on the control parameters, inherent inertia time constant, and operational conditions. Typically, \(\omega_{N}\) can be obtained from the generator's technical manual; \(\omega_{s0}\) can be measured in real-time. Meanwhile, \(\omega_{r0}\) equals the reference rotor speed \(\omega_{ref}\) under steady-state conditions, which is derived from the power-speed curve. However, due to commercial confidentiality, in practice, the relevant inertia controller structure and parameters of some wind turbines are usually unknown. Therefore, the accurate evaluation of the DFIG wind farm's equivalent virtual inertia requires the inertia control structure and parameters of the WT to be identified.