## 2 Mathematical Modeling of WECS Based on DFIG  
Doubly-fed induction generators (DFIG) are widely used in wind power systems due to their stability, robustness and relatively low cost. As a result, the doubly fed wind power generation system is taken as an example [24]--WECS based on DFIG--as shown in Figure 1. The wind power is converted to mechanical energy by a wind turbine, and the DFIG is driven by the drive system to generate electric energy and then transmitted to the power grid through an AC-DC converter.  
Assuming that the wind turbine is an ideal state, according to [PERSON] theory, the mechanical power of the wind turbine is as follows [25]:  
\[P_{wt}=0.5\pi R^{2}\rho v^{3}C_{p}(\lambda,\beta) \tag{1}\]  
The wind torque produced by the wind turbine is as follows [26]:  
\[\Gamma_{wt}=\frac{P_{wt}}{\Omega_{l}}=0.5\pi\rho v^{2}R^{3}C_{\Gamma}(\lambda,\beta) \tag{2}\]  
where \(\rho\) is the air density, \(v\) is the wind speed, \(R\) is the blade length, \(C_{p}(\lambda,\beta)\) is the conversion coefficient of wind energy. \(\lambda\) is the ratio of blade tip velocity to wind speed; that is, \(\lambda=\Omega_{l}(R/v)\), \(\Omega_{l}\) is the angular velocity of wind turbine rotor(low speed shaft), and \(C_{\Gamma}(\lambda,\beta)\) is the torque coefficient, satisfying \(C_{\Gamma}(\lambda,\beta)=C_{p}(\lambda,\beta)/\lambda\).  
In order to simplify the modeling, the rigid model is adopted in the connection between the high-speed axis and the low speed axis [27]; the equation of motion of the transmission system Equations (3) and (4) is as follows:  
\[J_{h}\frac{d\Omega_{h}}{d_{t}}=\frac{\eta}{i_{0}}\Gamma_{wt}-\Gamma_{G} \tag{3}\]  
Figure 1: Wind energy conversion system (WECS) based on doubly-fed induction generators (DFIG).  
\[J_{t}\frac{d\Omega_{t}}{d_{t}}=\Gamma_{wt}-\frac{i_{0}}{\eta}\Gamma_{G} \tag{4}\]  
where \(\Omega_{t}\) is the rotor speed (high speed shaft) of generator, that is, \(\Omega_{t}=i_{o}\times\Omega_{t}\), \(i_{o}\) is the gear transmission ratio, \(\Gamma_{G}\) is the electromagnetic torque of generator, \(\eta\) is the transmission efficiency, \(J_{t}\) is the high speed shaft inertia, \(J_{t}\) is the low speed shaft inertia.  
Some dynamic characteristics of the model are neglected and, considering that the electromagnetic time constant is much smaller than the mechanical time constant, the dynamic process of electromagnetic response of generator is ignored when modeling the system [28]. The state equation of WECS is modeled as follows:  
\[\left\{\begin{array}{l}\dot{x}_{p}(t)=A_{p}x_{p}(t)+B_{p}u(t)\\ y_{p}(t)=C_{p}x_{p}(t)\end{array}\right. \tag{5}\]  
where \(x_{p}(t)=\left[\begin{array}{cc}\Omega_{t}&\Gamma_{G}\end{array}\right]^{T}\), \(u(t)=\Gamma_{ref}^{*}\), \(A_{p}=\left[\begin{array}{cc}\frac{\Gamma_{wt}(i_{0}\Omega_{t},v)}{i_{0} \cdot I_{t}\cdot\Omega_{t}}&-\frac{1}{I_{t}}\\ 0&-\frac{1}{I_{G}}\end{array}\right]\), \(B_{p}=\left[\begin{array}{cc}0\\ \frac{1}{I_{G}}\end{array}\right]\), \(C_{p}=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right]\); \(x_{p}(t)\) is the state vector, \(u(t)\) is the input vector, \(y_{p}(t)\) is the output vector, \(\Gamma_{ref}^{*}\) is the reference value of the electromagnetic torque of the generator, and \(T_{G}\) is the electromagnetic time constant [29].