## 2 Doubly-Fed Induction Generator  
The configuration of the DFIG is shown in Figure 3A. During the fault condition, a very high electromotive force (EMF) is induced in the rotor due to the electromagnetic interaction with the stator and the rotor of the DFIG, which leads to over-voltage and over-current in the rotor side. The over-voltage and the over-current could damage the RSC components if the semiconductor switches are continuously trigged in the fault condition. In this situation, the rotor high-transient over-current passes through the switches. Meanwhile, the excess active power, which is not able to be delivered to the power system in the fault condition, charges the DC-link capacitor and rapidly increases the DC-link voltage. So, with regard to Figure 3A, the DC braking chopper typically operates to dissipate the excess energy and keep the DC-link voltage (\(V_{DC}\)) in a constant value. It should be noted that, because of the operation of the DC braking chopper and the time-varying characteristics of the electromagnetic interaction with the stator and the rotor, the response of the DFIG is very non-linear. However, during the fault, the DC-link voltage is almost kept constant when the DC braking chopper operates. In a static stator-oriented reference frame, the rotor and the stator fluxes, \(\overset{\rightarrow}{\psi}_{r}\), \(\overset{\rightarrow}{\psi}_{s}\), and the rotor and the stator voltages, \(\overset{\rightarrow}{v}_{r}\), \(\overset{\rightarrow}{v}_{s}\), are expressed as follows with respect to the Park model of the DFIG [13]:  
\[\overset{\rightarrow}{v}_{s}=R_{s}\overset{\rightarrow}{i}_{s}+\frac{d \overset{\rightarrow}{\psi}_{s}}{dt} \tag{1}\]  
\[\overset{\rightarrow}{v}_{r}=R_{r}\overset{\rightarrow}{i}_{r}+\frac{d \overset{\rightarrow}{\psi}_{r}}{dt}-j\omega_{r}\overset{\rightarrow}{\psi }_{r} \tag{2}\]  
\[\overset{\rightarrow}{\psi}_{s}=L_{s}\overset{\rightarrow}{i}_{s}+L_{m} \overset{\rightarrow}{i}_{r} \tag{3}\]  
Figure 2: Diagram of the fault ride-through (FRT) capability enhancement by fault current limiting devices in the doubly-fed induction generator. FCL: Fault current limiter. \[\overrightarrow{\psi}_{r}=L_{m}\overset{\rightarrow}{i}_{s}+L_{r}\overset{ \rightarrow}{i}_{r} \tag{4}\]  
whereby \(\overset{\rightarrow}{i}\), \(\omega\), \(R\), and \(L\) represent current, angular frequency, resistance, and inductance, respectively. Additionally, stator, mutual, and rotor parameters are mentioned by subscripts of \(s\), \(m\), and \(r\), respectively. By means of (3) and (4), \(\overrightarrow{\psi}_{r}\) is computed with respect to \(\overset{\rightarrow}{i}_{r}\) and \(\overrightarrow{\psi}_{s}\). (5) expresses \(\overrightarrow{\psi}_{r}\) as follows:  
\[\left\{ \begin{array}{c}\overrightarrow{\psi}_{r}=\frac{L_{m}\overrightarrow {\psi}_{s}}{L_{s}}+\sigma L_{r}\overset{\rightarrow}{i}_{r}\\ \sigma=1-\frac{L_{m}^{2}}{L_{s}L_{r}}\end{array} \right. \tag{5}\]  
whereby the leakage coefficient is represented by \(\sigma\). To calculate the rotor voltage with respect to (2) and (5), the following expression is deduced:  
\[\overrightarrow{v}_{r}=\overbrace{L_{m}}\left(\frac{d}{dt}-j\omega_{r} \right)\overrightarrow{\psi}_{s}+\left(R_{r}+\sigma L_{r}(\frac{d}{dt}-j\omega _{r})\right)\overset{\rightarrow}{i}_{r} \tag{6}\]  
whereby \(\overset{\rightarrow}{v}_{r}\) denotes the rotor voltage when the rotor is in an open circuit condition and the rotor current is zero. If the reference frame is changed from a static stator-oriented reference frame to a rotor-oriented reference frame, the rotor voltage is expressed in (7):  
\[\overrightarrow{v}_{r}^{r}=\overrightarrow{v}_{ro}^{r}+\left(R_{r}+\sigma L_ {r}\frac{d}{dt}\right)\overset{\rightarrow}{i}_{r}^{r} \tag{7}\]  
whereby \(\overset{\rightarrow}{v}_{r}^{r}\), \(\overset{\rightarrow}{v}_{ro}^{r}\), and \(\overset{\rightarrow}{i}_{r}^{r}\) are the rotor voltage, the open circuit rotor voltage, and the rotor current in the rotor-oriented reference frame, respectively.