# Fast Frequency Response of a Doubly-Fed Induction Generator for System Inertia Support
## 3 Doubly-Fed Induction Generators Model and Control  
The mechanical input power (\(P_{m}\)) extracted from the wind is defined as:  
\[P_{m}=0.5\rho Av_{w}^{3}c_{P}(\lambda,\beta), \tag{3}\]  
where \(\rho\) means the air density, \(A\) means the swept area by the turbine, \(v_{w}\) indicates the wind speed, \(c_{p}\) means the power coefficient, \(\lambda\) indicates the tip-speed ratio, and \(\beta\) represents the pitch angle.  
\(c_{p}\) in Equation (3) is given by Equation (4), as in [16].  
\[c_{P}(\lambda,\beta)=0.645[0.00912\lambda+\frac{-5-0.4(2.5+\beta)+116\lambda_ {i}}{e^{21\lambda_{i}}}], \tag{4}\]  
where  
\[\lambda_{i}=\frac{1}{\lambda+0.08(2.5+\beta)}-\frac{0.035}{1+\left(2.5+\beta \right)^{3}}, \tag{5}\]  
and \(\lambda\) is given as:  
\[\lambda=\frac{\omega_{r}R}{v_{w}}, \tag{6}\]  
For indicating the dynamics between wind turbines and induction generators, a two mass model is employed [17].  
The dynamics between the low-speed shaft torque, \(T_{ls}\), and the mechanical torque, \(T_{m}\), can be represented as:  
\[2H_{t}\frac{d\omega_{t}}{dt}=T_{m}-T_{ls}, \tag{7}\]  
where \(H_{t}\) means the turbine inertia constant, and \(\omega_{t}\) indicates turbine rotor speed.  
The dynamics between electromagnetic torque of a generator, \(T_{em}\), and the high-speed shaft torque, \(T_{hs}\), is expressed:  
\[2H_{g}\frac{d\omega_{r}}{dt}=T_{hs}-T_{em}, \tag{8}\]  
where \(H_{g}\) indicates is the inertia constant of the induction generator, and \(\omega_{r}\) represents the generator rotor speed.  
\(T_{ls}\) is derived by:  
\[T_{ls}=K(\theta_{t}-\theta_{ls})+B(\omega_{t}-\omega_{ls}), \tag{9}\]  
where \(K\), \(\theta_{t}\), and \(\theta_{ls}\), respectively, are the spring constant, angular displacement of the wind turbine and angular displacement of the low speed shaft. \(B\) and \(\omega_{ls}\), respectively, are the damping constant and the low speed shaft rotor speed.  
The gear ratio, \(N\), can be obtained by:  
\[N=\frac{\omega_{r}}{\omega_{ls}}=\frac{T_{ls}}{T_{hs}}. \tag{10}\]  
As in Figure 1, the DFIG control system includes a pitch angle controller, a rotor-side controller, and a grid-side controller. The pitch angle controller focuses on preventing \(\omega_{r}\) from the maximum \(\omega_{r}\) and enables the de-loading operation according to the order from the high level controller. The rotor-side controller, which realizes the decoupled control of reactive and active powers, is used for keeping the voltage of stator at the reference value and to regulate the active power fed to the power grid. The gird-side controller regulates the DC-link voltage in this study.  
As represented in [18], the MPT operation power reference (\(P_{MPT}\)) is provided by:  
\[P_{MPT}=\frac{1}{2}\rho\pi R^{2}(\frac{\omega_{r}R}{\lambda_{opt}})^{3}c_{P, \max}=k_{g}\omega_{r}^{3}, \tag{11}\]  
where \(c_{P,\max}\) indicates the maximum value of \(c_{P}\) and is set to 0.5; \(\lambda_{opt}\) is the optimal value of \(\lambda\) and is set to 9.95; \(k_{g}\) is set to 0.512.  
As shown in Figure 2, the red dotted line indicates \(P_{MPT}\). The greed solid lines represent different mechanical power curves for various wind speeds.