# Fault Studies and Distance Protection of Transmission Lines Connected to DFIG-Based Wind Farms
## 2 Fault Analysis of DFIG-Based Wind Farm and Its Verification
### DFIG Short Circuit Current  
The transformer connecting the DFIG to the grid is generally a \(\Delta/\Upsilon\) connection; therefore, only the positive and negative sequence components of a DFIG during faults need to be analyzed. A space vector description was used to present the generalized induction machine equations [7; 16; 17]. Figure 1: Topology of the doubly fed induction generator (DFIG)-based wind farm. Figure 2: A typical DFIG with a rotor crowbar protection circuit. Using a generator convention for the stator windings and a motor convention for the rotor windings, the stator and rotor voltage equations and flux linkage equations in a stationary reference frame are expressed as:  
\[\left\{ \begin{array}{c}\overrightarrow{v}_{s}=-R_{s}\overset{\rightarrow}{i}_ {s}+D\overrightarrow{\psi}_{s}\\ \overrightarrow{v}_{r}=R_{r}\overset{\rightarrow}{i}_{r}+D\overrightarrow{ \psi}_{r}-j\omega_{r}\overset{\rightarrow}{\psi}_{r},\end{array} \right., \tag{1}\]  
\[\left\{ \begin{array}{c}\overrightarrow{\psi}_{s}=-L_{s}\overset{ \rightarrow}{i}_{s}+L_{m}\overset{\rightarrow}{i}_{r}\\ \overrightarrow{\psi}_{r}=-L_{m}\overset{\rightarrow}{i}_{s}+L_{r}\overset{ \rightarrow}{i}_{r}\end{array} \right., \tag{2}\]  
where \(\overrightarrow{v}\), \(\overset{\rightarrow}{i}\), and \(\overset{\rightarrow}{\psi}\) represent the voltage space vector, current space vector, and flux linkage space vector, respectively; \(\omega\) is the electrical angular velocity; \(R\) is the per-phase resistance; subscripts \(s\) and \(r\) denote the stator and rotor quantities, respectively; \(L_{s}\) and \(L_{r}\) are the per-phase stator and rotor self-inductances, respectively; and \(L_{m}\) is the per-phase mutual inductance; and \(D\) is the time-differential operator. By solving Equation (2), the current can be written as equation of the flux linkages as:  
\[\overset{\rightarrow}{i}_{s}=-\frac{1}{L_{s}}\overset{\rightarrow}{\psi}_{ s}+\frac{L_{m}}{L_{r}}\frac{1}{L_{s}}\overset{\rightarrow}{\psi}_{r}, \tag{3}\]  
where \(L_{s}{}^{\prime}=L_{s}-L_{m}{}^{2}/L_{r}\). Assuming that, at time \(t=0\), an asymmetrical short circuit fault occurs at the DFIG. The post-fault stator voltage is:  
\[\overset{\rightarrow}{v}_{s}=\overset{\rightarrow}{v}_{s1}+\overset{ \rightarrow}{v}_{s2}=(1-A)V_{pre}e^{j(\omega_{s}t+\theta)}+V_{2}e^{j(-\omega_ {s}t+\theta^{\prime})}, \tag{4}\]  
where \(\overset{\rightarrow}{v}_{s1}\) and \(\overset{\rightarrow}{v}_{s2}\) are the positive and negative sequence voltages, respectively; \(A\) is the voltage sag ratio; \(V_{pre}\) is the amplitude of the pre-fault stator voltage; \(V_{2}\) is the amplitude of the post-fault stator negative sequence voltage; and \(\theta\) and \(\theta^{\prime}\) are the phase angles of \(\overset{\rightarrow}{v}_{s1}\) and \(\overset{\rightarrow}{v}_{s2}\) at time \(t=0\), respectively. The post-fault stator positive and negative sequence flux linkage are obtained in Equation (5). The detailed derivation is presented in Appendix A (i). \[\left\{ \begin{array}{c}\overset{\rightarrow}{\psi}_{s1}=\overset{ \rightarrow}{\psi}_{s1}+\overset{\rightarrow}{\psi}_{sf1}=\frac{AV_{pre}}{ \int\omega_{s}}e^{-t/T_{s}{}^{\prime}+\dot{\theta}}+\frac{(1-A)V_{pre}}{\int \omega_{s}}e^{j(\omega_{s}t+\theta)}\\ \overset{\rightarrow}{\psi}_{s2}=\overset{\rightarrow}{\psi}_{s2}+\overset{ \rightarrow}{\psi}_{sf2}=\frac{V_{2}}{\int\omega_{s}}e^{-t/T_{s}{}^{\prime}+ \dot{\theta}^{\prime}}-\frac{V_{2}}{\int\omega_{s}}e^{j(-\omega_{s}t+\theta^{ \prime})}\end{array} \right., \tag{5}\]  
where \(T_{s}{}^{\prime}=L_{s}{}^{\prime}/R_{s}\) is the stator decaying time constant; \(\overset{\rightarrow}{\psi}_{sm1}\) and \(\overset{\rightarrow}{\psi}_{sf1}\) are the natural component and forced component of \(\overset{\rightarrow}{\psi}_{s1}\), respectively; and \(\overset{\rightarrow}{\psi}_{s2}\) and \(\overset{\rightarrow}{\psi}_{sf2}\) are the natural component and forced component of \(\overset{\rightarrow}{\psi}_{s2}\), respectively.