## 4 Transient Current Characteristics of Wind Turbines after Short-Circuit Fault  
The control strategy based on switching the virtual resistor can suppress the overcurrent of the GSC when the short-circuit fault occurs. For the tuning of virtual resistor parameters, it is essential to study the transient current characteristics of the VS-controlled wind turbine during the grid short-circuit fault. Figure 4 shows an equivalent circuit diagram of the grid-connected wind turbine system during the three-phase short-circuit fault, where the wind turbine and the grid are all represented by voltage sources, the grid voltage is \(\mathrm{e_{g}}\), the AC-side voltage of the wind turbine is ug, and the ground-connection resistance at the fault point is R.  
Before the three-phase short-circuit fault occurs, the switch \(\mathrm{S_{G}}\) in Figure 3 is opened. Correspondingly, the switch \(\mathrm{S_{1}}\) is closed and the switch \(\mathrm{S_{2}}\) is opened in Figure 4, which means the virtual resistor, \(\mathrm{R_{v}}\), is not introduced to the main circuit and the fault point is not grounded. When the three-phase short-circuit fault occurs, the switch \(\mathrm{S_{2}}\) is closed, which means the fault point is grounded through the resistor, R. In order to suppress the overcurrent of the wind turbine, the switch \(\mathrm{S_{G}}\) in Figure 3 is closed. Correspondingly, the switch \(\mathrm{S_{1}}\) in Figure 4 is opened and the virtual resistor, \(\mathrm{R_{v}}\), is put into the main circuit. According to Figure 4, taking the phase A as an example, the equivalent circuit diagram of the grid-connected wind turbine system in the s domain after the three-phase short-circuit fault occurs is shown in Figure 5, where the grid voltage is \(\mathrm{E_{g}}\)(s), the equiva  
Figure 4: Equivalent circuit diagram of the grid-connected wind turbine system during the three-phase short-circuit fault. lent AC-side voltage of the wind turbine is U\({}_{\rm gA}\)(s), the output current magnitude of the wind turbine before the short-circuit fault is I\({}_{\rm gm}\), the output current of the wind turbine after the short-circuit fault is I\({}_{\rm 1A}\)(s), and the grid-side current is I\({}_{\rm 2A}\)(s). The time-domain grid voltage, e\({}_{\rm gA}\), of the phase A can be expressed as  
\[e_{gA}=E_{gm}\cos(\omega t+\varphi) \tag{1}\]  
where E\({}_{\rm gm}\) is the amplitude of the grid voltage and \(\varphi\) is the phase of the grid voltage. The AC-side equivalent phase A voltage, u\({}_{\rm gA}\), of the wind turbine in the time domain can be expressed as  
\[u_{gA}=U_{gm}\cos(\omega t+\theta_{u}) \tag{2}\]  
where U\({}_{\rm gm}\) is the amplitude of the wind turbine's AC-side equivalent voltage, \(\theta_{\rm u}\) is the phase and \(\theta_{\rm u}=\varphi+\delta_{u}\), and \(\delta_{\rm u}\) is the phase difference that the voltage, u\({}_{\rm gA}\), leads ahead of the grid voltage, e\({}_{\rm gA}\). The phase A output current of the wind turbine in the time domain before the short-circuit fault is  
\[i_{gA}=I_{gm}\cos(\omega t+\theta_{i}) \tag{3}\]  
where I\({}_{\rm gm}\) is the amplitude of the wind turbine's output current before the short-circuit fault, \(\theta_{\rm i}\) is the phase and \(\theta_{\rm i}=\varphi+\delta_{\rm i}\), and \(\delta_{\rm i}\) is the phase difference that the output current, i\({}_{\rm gA}\), leads ahead of the grid voltage, e\({}_{\rm gA}\). Assuming that the short-circuit fault occurs at the time 0, changing the grid voltage phase \(\varphi\) in the range of 0 to \(2\pi\) corresponds to the occurrence of three-phase short-circuit fault at any time. Converting the voltages e\({}_{\rm gA}\) and u\({}_{\rm gA}\) in (1) and (2) to the s domain, the expressions of E\({}_{\rm gA}\)(s) and U\({}_{\rm gA}\)(s) in Figure 5 are  
\[E_{gA}(s)=E_{gm}\frac{s\cos\varphi-\omega_{g}\sin\varphi}{s^{2}+\omega_{g}^{2 }} \tag{4}\]  
\[U_{gA}(s)=U_{gm}\frac{s\cos\theta_{u}-\omega_{g}\sin\theta_{u}}{s^{2}+\omega_{ g}^{2}} \tag{5}\]  
where \(\omega_{g}\) is the angular frequency of the grid, i.e., \(100\pi\) rad/s.