## 4 Transient Current Characteristics of Wind Turbines after Short-Circuit Fault
In Figure 5, the voltage equation of the current path 1 is  
\[U_{gA}(s)=R_{v}I_{1A}(s)+sL_{f}I_{1A}(s)+L_{f}I_{gm}\cos\theta_{i}+R[I_{1A}(s )-I_{2A}(s)] \tag{6}\]  
In the same way, the voltage equation of the current path 2 in Figure 5 is  
\[E_{gA}(s)=-R_{line}I_{2A}(s)+L_{line}I_{gm}\cos\theta_{i}-sL_{line}I_{2A}(s)+ R[I_{1A}(s)-I_{2A}(s)] \tag{7}\]  
Figure 5: Equivalent circuit diagram of the grid-connected wind turbine system in the s domain after the three-phase short-circuit fault occurs. Combining Equations (6) and (7), eliminating I\({}_{\text{ZA}}\)(s), the output current, I\({}_{\text{IA}}\)(s), of the wind turbine after the short-circuit fault can be derived as  
\[I_{1A}(s)=\frac{\frac{(R+R_{\text{Rine}}+R_{\text{Line}})}{R}\left[L_{gA}(s)-L_ {f}I_{gm}\cos\theta_{i}\right]+L_{line}I_{gm}\cos\theta_{i}-E_{gA}(s)}{\frac{(R +R_{\text{Line}}+S+L_{line})(R+R_{\text{v}}+S+L_{line})}{R}-R} \tag{8}\]  
It can be seen from Equation (8) that the virtual resistance, R\({}_{\text{v}}\), is located in the denominator of the equation, which means that increasing the virtual resistance, R\({}_{\text{v}}\), can suppress the overcurrent of the wind turbine after the fault occurs. Carrying out the inverse Laplace transform of Equation (8), the phase A output current, i\({}_{\text{IA}}\)(t), of the wind turbine in the time domain after the short-circuit fault can be obtained. Since the expression of i\({}_{\text{IA}}\)(t) is extremely complicated, an actual grid-connected wind turbine system is taken as an example, where the grid-side filter inductance, L\({}_{\text{f}}\), is 0.038 mH, the ground-connected resistance, R, at the fault point is 0.1 \(\Omega\), the virtual resistance, R\({}_{\text{v}}\), is 0.3 \(\Omega\), the wind turbine operates in the state of unit power factor and the rated active power, P\({}_{\text{g}}\), is 2 MW, the grid voltage phase, \(\varphi\), is 0, and the short-circuit ratio (SCR) of the grid is 2.5, i.e., L\({}_{\text{line}}\) is 0.304 mH and R\({}_{\text{line}}\) is 0.045 \(\Omega\). Substituting these above parameters into Equation (8), the output current, i\({}_{\text{IA}}\)(t), of the wind turbine after the short-circuit fault occurs is derived as  
\[e_{gA}=E_{gm}\cos(\omega t+\varphi) \tag{1}\]0  
It can be seen from Equation (9) that the output current, i\({}_{\text{IA}}\)(t), after the fault can be derived into two parts, where part I is the steady-state component of the short-circuit current, and part II is the transient component of the short-circuit current. Due to the exponential decay component contained in part II, the transient component of the short-circuit current quickly decays to zero. The above analysis shows that the output current of the wind turbine quickly enters a steady state after the fault occurs, and the transition process is short. In order to study how the timing of the fault occurrence impacts the maximum value, I1 max, of the wind turbine's output current, Figure 6 shows the curve of the maximum value, I1 max, with different grid voltage phases, \(\varphi\). It can be seen from Figure 6 that as the grid voltage phase, \(\varphi\), varies from 0 to 2\(\pi\), the value of I1 max varies in the range of 1462 A to 2184 A, which indicates the maximum value of the wind turbine's output current during the short-circuit fault is between the steady-state current before the fault and the steady-state current after the fault. Therefore, in order to limit the output overcurrent of the wind turbine after the short-circuit fault, the value of the virtual resistance, R\({}_{\text{v}}\), can be designed with the goal to ensure that the post-fault output steady-state current does not exceed its pre-fault value. \[e_{gA}=E_{gm}\cos(\omega t+\varphi) \tag{1}\]1 Maximum value of the wind turbine’s output current after the fault with different grid voltage phases, \(\varphi\).