# LVRT and Stability Enhancement of Grid-Tied Wind Farm Using DFIG-Based Wind Turbine
## 3 Design of the Wind Turbine and Drive Train Model  
The aerodynamic power output in the WT model is as follows [8; 33]:  
\[P_{\omega}=0.5\rho\pi R^{2}V_{\omega}{}^{3}C_{p}(\lambda,\beta) \tag{1}\]  
where \(P_{\omega}\), \(\rho\), \(R\), \(V_{\omega}\), and \(C_{p}\) denoted as the extracted wind power, air density, rotor blade radius, wind speed, and power coefficient, respectively.  
\(C_{p}\) can be calculated from the following equation [31]:  
\[C_{p}(\lambda,\beta)=c_{1}\left(\frac{c_{2}}{\lambda_{i}}-c_{3}\beta-c_{4} \right)e^{-\frac{\alpha}{\lambda_{i}}}+c_{6\lambda} \tag{2}\]  
\[\frac{1}{\lambda_{i}}=\frac{1}{\lambda-0.08\beta}-\frac{0.035}{\beta^{3}+1} \tag{3}\]  
\[\lambda=\frac{\omega_{r}R}{V_{\omega}} \tag{4}\]  
\[T_{\omega}=\frac{P_{\omega}}{\omega_{r}}, \tag{5}\]  
where \(\lambda\), \(T_{\omega}\), and \(\beta\) represent the tip speed ratio, WT torque, and pitch angle, respectively. Furthermore, \(c_{1}\)-\(c_{6}\) present the characteristic coefficients of the WT [34], and \(\omega_{r}\) presents the rotational speed of the WT.  
Figure 2 illustrates the \(C_{p}\) vs. \(\lambda\) characteristics that are achieved by solving Equation (2) with various \(\beta\) values. The optimum power coefficient \((C_{popt})\) and the optimum tip speed ratio \((\lambda_{opt})\) are equal to 0.48 and 8.1, respectively, when the \(\beta\) is equal to zero degrees.  
Figures 3 and 4 illustrate the designs of the FSWT and VSWT blade pitch control schemes, respectively [35]. The output power of SCIG was controlled by the pitch controller in FSWT so that it did not surpass the rated power and in VSWT, the pitch controller regulates the rotor speed of the DFIG so that it did not surpass the nominal condition. A first-order transfer function consisting of a pitch rate limiter characterized the pitch actuator control loop. Moreover, to control the tracking error, a PI controller was used.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Doubly Fed Induction Generator (DFIG)** & **Squirrel Cage Induction Generator (SCIG)** \\ \hline MVA & 15 & MVA & 35 \\ \(R_{s}\) (pu) & 0.007 & \(R_{1}\) (pu) & 0.01 \\ \(R_{r}\) (pu) & 0.005 & \(X_{1}\) (pu) & 0.1 \\ \(L_{st}\) (pu) & 0.171 & \(X_{m}\) (pu) & 3.5 \\ \(L_{rt}\) (pu) & 0.156 & \(R_{21}\) (pu) & 0.035 \\ \(L_{m}\) (pu) & 2.9 & \(R_{22}\) (pu) & 0.014 \\ - & - & \(X_{21}\) (pu) & 0.03 \\ - & - & \(X_{22}\) (pu) & 0.089 \\ - & - & H (s) & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of wind generators.  
Since the model of the two-mass shaft was adequate for the dynamic study of grid-tied WTs, therefore, it was used in this research as indicated in Figure 5[36]. The motion laws are defined as:  
\[\frac{d\theta_{S}}{dt}=\omega_{t}-\omega_{g} \tag{6}\]  
\[\frac{d\omega_{t}}{dt}=\frac{1}{2H_{t}}(T_{t}-K_{\rm s}\theta_{S}) \tag{7}\]  
\[\frac{d\omega_{g}}{dt}=\frac{1}{2H_{g}}(-T_{e}+K_{\rm s}\theta_{S}) \tag{8}\]  
Figure 4: Pitch control scheme for variable-speed wind turbines (VSWT).  
Figure 3: Pitch control scheme for fixed-speed wind turbines (FSWT).  
Figure 2: \(C_{p}\) vs. \(\lambda\) characteristics with various \(\beta\).  
Here, mechanical torque and equivalent turbine blade inertia transferred to the generator side is denoted as \(T_{t}\) and \(H_{t}\), respectively. Electromagnetic torque is denoted as \(T_{e}\), the rotational speed of turbine and generator are denoted as \(\omega_{t}\) and \(\omega_{g}\), sequentially. \(\theta_{s}\) represents the shaft angular displacement between the two ends. The stiffness of the shaft is denoted as \(K_{s}\), and \(H_{g}\) stands for the generator inertia.  
The curve of the maximum power point tracking (MPPT) for the VSWT-DFIG is illustrated in Figure 6[33].