## 2 Modeling and Analytical Benchmarking of Equivalent Virtual Inertia for DFIGs  
### Dynamic Model of DFIG with Inertia Control Methods  
In the synchronous reference frame, the DFIG model with stator flux orientation is represented by a simplified fifth-order mathematical model, including flux, electromagnetic torque, and rotor motion equations.  
\[\left\{\begin{array}{l}\psi_{d,s}=L_{s}i_{d,s}+L_{m}i_{d,r}=\psi_{s}\\ \psi_{d,s}=L_{s}i_{d,s}+L_{m}i_{d,r}=0\end{array}\right. \tag{1}\]  
\[P_{e}=\frac{3 pL_{m}}{4\omega_{s}L_{s}}\psi_{d,s}i_{d,r} \tag{2}\]  
\[2H_{wind}\frac{d\omega_{r}}{dt}=P_{m}-P_{e} \tag{3}\]  
where \(\psi_{s}\) is the stator flux linkage; \(\psi_{d,s}\) is the \(d\)-axis component of \(\psi_{s}\); \(\psi_{d,s}\) is the \(q\)-axis component of \(\psi_{s}\); \(i_{d,s}\), \(i_{d,r}\), \(i_{q,s}\) and \(i_{q,r}\) are the \(d\)-axis and \(q\)-axis components of the currents in the stator and rotor windings, respectively; \(L_{m}\) is the mutual inductance between the stator and rotor; \(L_{s}\) is the stator self-inductance; \(p\) is the number of pole pairs of the wind turbine; \(\omega_{s}\) is the grid angular frequency; \(\omega_{r}\) is the rotor speed; \(H_{wind}\) is the inherent inertia time constant of the WT; \(P_{m}\) is the mechanical power of the WT; and \(P_{e}\) is the electromagnetic power of the WT.  
To enhance the frequency stability of power systems, DFIGs are often equipped with supplementary inertia control strategies. Among them, virtual inertia control [36; 37] and integrated inertia control [38] are widely implemented in practical wind power applications. Consequently, they are selected in this work as representative cases to evaluate the effectiveness and applicability of the proposed identification methodology under limited model transparency.  
The virtual inertia control involves the incorporation of a frequency differential signal into the RSC, which facilitates dynamic adjustments to the rotor speed and active power output, thereby emulating the inertia response characteristic of synchronous machines; this can be expressed as follows:  
\[\Delta P_{1}^{*}=-k_{d_{1}}\Bigg{(}\frac{\Delta\omega_{s}\cdot s}{1+T_{f} \cdot s}\Bigg{)} \tag{4}\]  
where \(\Delta\omega_{s}\) is the change in angular frequency; \(k_{d1}\) is the virtual inertia control gain; and \(T_{f}\) is the time constant of the low-pass filter.  
The integrated inertia control combines virtual inertia control and droop control. It adjusts the WT power output based on frequency deviation and the rate of change. When the grid frequency exceeds nominal, the power command becomes negative, reducing output and accelerating the rotor. During a frequency drop, the rotor slows, converting kinetic energy into active power for short-term grid support; this can be derived as follows:  
\[\Delta P_{2}^{*}=-k_{p}\Delta\omega_{s}-k_{d2}\Bigg{(}\frac{\Delta\omega_{s} \cdot s}{1+T_{f}\cdot s}\Bigg{)} \tag{5}\]  
where \(k_{p}\) and \(k_{d2}\) represent the proportional and derivative coefficients of the combined inertia control, respectively.  
Meanwhile, the transfer function expressions for the reference electromagnetic power under MPPT module are given as follows [39]:  
\[\Delta P_{3}^{*}=\Delta\omega_{r}\bigg{(}k_{p1}+\frac{k_{l1}}{s}\bigg{)} \tag{6}\]  
where \(\Delta P_{3}^{*}\) is the reference electromagnetic power under the MPPT module; \(\Delta\omega_{r}\) is the change in the wind turbine rotor speed; \(k_{l1}\) and \(k_{p1}\) are the integral and proportional gains of the speed controller, respectively.  
Figure 1 shows the simplified DFIG model under different inertia control strategies. Considering that the current inner loop responds much faster than the electromechanical transient process of the generator, the converter's inner loop is simplified as a first-order inertia element in this study: \(G_{s}(s)=1/(\tau s+1)\), \(\tau\) is the converter time constant.  
Figure 1: Inertial control model of DFIG.