# 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
As for the steady state after a fault in a synchronous rotating reference frame, the positive sequence voltage equations and flux linkage equations are shown in Equations (6) and (7). The negative sequence voltage equations and flux linkage equations are shown in Equations (8) and (9), respectively. The equivalent circuits of DFIG in a synchronous rotating reference frame are shown in Figure 3[17;33]. \[\left\{ \begin{array}{c}\overset{\rightarrow}{v}_{s1}=-R_{s}\overset{ \rightarrow}{i}_{sf1}+D\overset{\rightarrow}{\psi}_{sf1}+j\omega_{s} \overset{\rightarrow}{\psi}_{sf1}\\ \overset{\rightarrow}{v}_{r1}=R_{r}\overset{\rightarrow}{i}_{rf1}+D \overset{\rightarrow}{\psi}_{rf1}+j(\omega_{s}-\omega_{r})\overset{ \rightarrow}{\psi}_{rf1}\end{array} \right., \tag{6}\]  
\[\left\{ \begin{array}{c}\overset{\rightarrow}{\psi}_{sf1}=-L_{s}\overset{ \rightarrow}{i}_{sf1}+L_{m}\overset{\rightarrow}{i}_{rf1}\\ \overset{\rightarrow}{\psi}_{rf1}=-L_{m}\overset{\rightarrow}{i}_{sf1}+L_{r} \overset{\rightarrow}{i}_{rf1}\end{array} \right., \tag{7}\]Then, the post-fault rotor positive and negative sequence flux linkages are obtained in Equation (10), respectively. The detailed derivation is presented in Appendix A (ii). \[\left\{ \begin{array}{l}\stackrel{{\rightarrow}}{{\psi}}_{r1}= \stackrel{{\rightarrow}}{{\psi}}_{m1}+\stackrel{{ \rightarrow}}{{\psi}}_{rf1}=[\stackrel{{\rightarrow}}{{\psi}}_{ r0}-f_{1}(s)\frac{(1-A)V_{pr2}}{j\omega_{s}}e^{j\theta}]e^{-t/T_{r}{}^{\prime}+j\omega_{r}t}+f_{1}(s) \frac{(1-A)V_{pr2}}{j\omega_{s}}e^{j(\omega_{s}t+\theta)}\\ \stackrel{{\rightarrow}}{{\psi}}_{r2}=\stackrel{{ \rightarrow}}{{\psi}}_{m2}+\stackrel{{\rightarrow}}{{\psi}}_{rf2}= f_{2}(s)\frac{V_{2}}{j\omega_{s}}e^{j\theta^{\prime}}e^{-t/T_{r}{}^{\prime}+j\omega_{r}t}-f_{2}(s) \frac{V_{2}}{j\omega_{s}}e^{j(-\omega_{s}t+\theta^{\prime})}\end{array} \right., \tag{10}\]  
where \(T_{r}{}^{\prime}=L_{r}{}^{\prime}/R_{r}\) is the rotor decaying time constant; \(\stackrel{{\rightarrow}}{{\psi}}_{m1}\) and \(\stackrel{{\rightarrow}}{{\psi}}_{rf1}\) are the natural component and forced component of \(\stackrel{{\rightarrow}}{{\psi}}_{r1}\), respectively; and \(\stackrel{{\rightarrow}}{{\psi}}_{m2}\) and \(\stackrel{{\rightarrow}}{{\psi}}_{rf2}\) are the natural component and forced component of \(\stackrel{{\rightarrow}}{{\psi}}_{r2}\), respectively; \(f_{1}(s)=\frac{R_{r}L_{m}/L_{s}}{R_{r}+j(\omega_{s}-\omega_{r})L_{r}}\), \(f_{2}(s)=\frac{R_{r}L_{m}/L_{s}}{R_{r}-j(2-s)\omega_{s}L_{r}}\), and \(L_{r}{}^{\prime}=L_{r}-L_{m}{}^{2}/L_{s}\); \(\stackrel{{\rightarrow}}{{\psi}}_{r0}\) is the pre-fault rotor flux linkage. Substituting Equations (5) and (10) into Equation (3), the stator positive and negative sequence short circuit currents can be obtained:  
\[\left\{\begin{array}{l}\stackrel{{\rightarrow}}{{i}}_{s1}=- \frac{A\psi_{pr2}}{j\omega_{s}L_{r}}e^{-t/T_{r}{}^{\prime}+j\varphi}+\frac{L_{m} }{L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{ r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{r}L_{short circuit current on the transmission line of the wind farm side still exists under a ground fault. Figure 4 shows a real fault case where a single-phase ground fault occurred on the transmission line connected to a DFIG-based wind farm without LVRT capability. The wind power system details are presented in Appendix B. In Figure 4, \(\dot{I}_{W}\) is the wind farm side short circuit current on the transmission line and \(\dot{I}_{WA}\), \(\dot{I}_{WB}\), \(\dot{I}_{WC}\), and \(\dot{I}_{W0}\) are phase A, phase B, phase C, and zero sequence current of \(\dot{I}_{W}\), respectively; \(Z_{W0}\) is zero sequence impedance of the wind farm side, and it consists of the zero sequence impedance of the transformer grounded and transmission line; \(Z_{S0}\), \(Z_{S1}\), \(Z_{S2}\) are the zero sequence, positive sequence, and negative sequence impedance of the grid side, respectively; and \(\dot{E}_{S(0)}\) is the grid voltage before the fault.