## 2 DFIG-Based Wind Turbine System
### Fault Characteristics of DFIG
#### 2.3.2 Fault Current of the DFIG  
Assuming a fault voltage dip occurs in the DFIG at \(t\)= 0, the DFIG terminal voltage during the fault is  
\[\overset{\rightarrow}{u}_{1}=(1-A)U_{s}e^{i(\omega_{s}t+\theta)}, \tag{5}\]  
where \(A\) is the depth of the DFIG voltage dips; \(U_{s}\) is the amplitude of the DFIG terminal voltage before the fault; and \(\theta\) is the phase of the DFIG terminal voltage at 0 s.  
Considering that the stator flux cannot suddenly change when the fault occurs, and neglecting the stator resistance, the stator flux during fault periods can be obtained from Equation (1):  
\[\overset{\rightarrow}{\psi}_{s}=\frac{AU_{s}}{j\omega_{s}}e^{-t/T_{s}^{ \prime}+j\theta}+\frac{(1-A)U_{s}}{j\omega_{s}}e^{j(\omega_{s}t+\theta)}, \tag{6}\]  
where \(T_{s}^{\prime}=L_{s}^{\prime}/R_{s}\), \(L_{s}^{\prime}=L_{s}-L_{m}^{2}/L_{r}\); \(T_{s}^{\prime}\) is the decay time constant of the stator circuit; and \(L_{s}^{\prime}\) is the transient reactance of the DFIG stator. Assuming that the crowbar circuit is instantaneously activated when the fault occurs, and the equivalent circuit to the DFIG with the crowbar circuit in a d-q synchronous rotating reference frame is shown in Figure 2. Thus, after the fault, the voltage equation and flux equation can be obtained as follows:\[\left\{ \begin{array}{l}(1-A)\overset{\rightarrow}{u}_{s}=-R_{s}\overset{ \rightarrow}{i}_{sf}+D\overset{\rightarrow}{\psi}_{sf}+j\omega_{s}\overset{ \rightarrow}{\psi}_{sf}\\ 0=\ (R_{r}+R_{cb})\overset{\rightarrow}{i}_{rf}+D\overset{\rightarrow}{\psi}_{ rf}+j(\omega_{s}-\omega_{r})\overset{\rightarrow}{\psi}_{rf}\\ \overset{\rightarrow}{\psi}_{sf}=-L_{s}\overset{\rightarrow}{i}_{sf}+L_{m} \overset{\rightarrow}{i}_{rf}\\ \overset{\rightarrow}{\psi}_{rf}=-L_{m}\overset{\rightarrow}{i}_{sf}+L_{r} \overset{\rightarrow}{i}_{rf}\end{array} \right., \tag{7}\]  
where \(R_{cb}\) is the crowbar resistance; \(\overset{\rightarrow}{i}_{sf}\) and \(\overset{\rightarrow}{i}_{rf}\) are the steady-state currents of the DFIG stator and rotor after the fault; and \(\overset{\rightarrow}{\psi}_{sf}\) and \(\overset{\rightarrow}{\psi}_{rf}\) are the steady-state flux of the DFIG stator and rotor after the fault. In the d-q synchronous rotating reference frame, \(D\overset{\rightarrow}{\psi}_{rf}=0\). Additionally, the DFIG rotor's steady-state flux can be calculated using Equation (7):  
\[\overset{\rightarrow}{\psi}_{rf}=f(s,R_{cb})\overset{\rightarrow}{\psi}_{ sf}=f(s,R_{cb})\frac{(1-A)U_{s}}{j\omega_{s}}e^{j(\omega_{s}t+\theta)}, \tag{8}\]  
\[f(s,R_{cb})=\frac{(R_{r}+R_{cb})L_{m}}{[(R_{r}+R_{cb})+j(\omega_{s}-\omega_{r}) L_{r}{}^{\prime}]L_{s}}, \tag{9}\]  
where \(L_{r}{}^{\prime}=L_{r}-L_{m}{}^{2}/L_{s}\), which represents the transient reactance of the DFIG rotor. Considering that the rotor flux cannot suddenly change at the time of the fault, the flux of the DFIG rotor can be expressed as  
\[\overset{\rightarrow}{\psi}_{r}=[\overset{\rightarrow}{\psi}_{r0}e^{j( \alpha+\theta)}-f(s,R_{cb})\frac{(1-A)U_{s}}{j\omega_{s}}e^{i\theta}]e^{-t/T_{ r}+j\omega_{s}t}-+f(s,R_{cb})\frac{(1-A)U_{s}}{j\omega_{s}}e^{j(\omega_{s}t+ \theta)}, \tag{10}\]  
where \(T_{r}{}^{\prime}=L_{r}{}^{\prime}/(R_{r}+R_{cb})\), which represents the decay time constant of the rotor circuit. \(\overset{\rightarrow}{\psi}_{r0}\) is the rotor flux before the fault.