## 2 Analysis of the DFIG at Fundamental Frequency  
The general configuration of a DFIG-based wind turbine connected to an electrical grid is shown in Figure 1. The electrical single-equivalent phase circuit of the DFIG is illustrated in Figure 2.  
Figure 1: Configuration of a DFIG connected to the electrical grid.  
Figure 2: Equivalent circuit of the induction generator in steady-state seen from stator.  
The circuit displays a conventional steady-state equivalent model, with all parameters referring to the stator. The parameters of Figure 2 were previously described. The equation describing the system is given by Equation (1),  
\[\begin{array}{c}\mathbf{V_{s}}\ =\ -(r_{s}+j\omega_{es}L_{ls})\mathbf{I_{s}}+j \omega_{es}L_{M}\mathbf{I_{r}}\\ \mathbf{V^{{}^{\prime}}_{r}}\ =\ {r^{\prime}}_{r}/s+j\omega_{es}(L^{{}^{\prime}}_{lr}+L _{M})\mathbf{I^{{}^{\prime}}_{r}}-j\omega_{es}L_{M}\mathbf{I_{s}}\end{array} \tag{1}\]  
where \(s\) is defined as the relative velocity between the magnetic field produced by the currents injected in the stator and the mechanical velocity of the rotor using Equation (2),  
\[s\ =\ \frac{\omega_{es}-\omega_{r}}{\omega_{es}} \tag{2}\]  
From Equation (1), the stator and rotor current phasors in the time-domain are obtained by Equation (3):  
\[\begin{array}{c}\mathbf{i_{s}}=|l_{s}|\cos(\omega_{es}t+\varphi_{s})\\ \mathbf{i_{r}}=|l^{\prime}r|\cos\left(s\omega_{es}t+{\varphi^{\prime}}_{r}- \theta_{eff}\right)\end{array} \tag{3}\]  
The importance of the slip is analyzed as follows: if the machine is stopped, the shaft's mechanical velocity is zero (\(\omega_{r}\ =\ 0\)), resulting in a slip value of 1 (\(s\ =\ 1\)) obtained from Equation (2). In this condition the machine operates as a blocked rotor. On the contrary, if the rotor rotates at the magnetic field velocity, then the slip is zero (\(s\) = 0). In this case, the magnetic field velocity is termed the synchronous velocity of the machine. If the rotor rotates higher than the synchronous velocity, the slip becomes negative, and the machine operates as a generator. Conversely, if the velocity of the magnetic field of the stator exceeds that of the rotor, the machine operates as a motor. Therefore, the slip defines the operation of the machine. Therefore, when all parameters of the induction generator and the electrical grid are known, the slip determines the point of operation.  
To ensure a complete analysis of the DFIG, the effects of the generator's rotor must be considered. The steady-state equivalent circuit model is depicted in Figure 3, with all parameters referring to the rotor [3].  
Equation (4) describes the system given by,  
\[\begin{array}{c}0=r^{\prime}s/s_{r}+j\omega_{er}(L^{\prime}{}_{ls}+L_{M}) \mathbf{I^{{}^{\prime}}}_{s}-j\omega_{er}L_{M}\mathbf{I_{r}}\\ \mathbf{V_{r}}=r_{r}+j\omega_{er}(L_{lr}+L_{M})\mathbf{I_{r}}-j\omega_{es}L_{ M}\mathbf{I_{s}}\end{array} \tag{4}\]  
Since the rotor is moving, the slip seen from it is given by Equation (5),  
\[s_{r}=\frac{\omega_{er}+\omega_{r}}{\omega_{er}} \tag{5}\]  
Considering Equation (4), the stator and rotor currents phasors in the time domain are obtained by Equation (6),  
\[\begin{array}{c}\mathbf{i_{r}}=|l_{r}|\cos(\omega_{er}t+\varphi_{r})\\ \mathbf{i_{s}}=|l^{\prime}s|\cos\left(s\varphi_{r}\omega_{er}t+{\varphi^{ \prime}}_{s}+\theta_{eff}\right)\end{array} \tag{6}\]  
Figure 3: Steady-state equivalent circuit of the generator seen from the rotor.  
Thus, the complete solution for a DFIG at fundamental frequency considering the stator and rotor effects is given by Equation (7),  
\[\begin{split}&\mathbf{i}_{\mathrm{s}}=|I_{\mathrm{s}}|\cos(\omega_{ \mathrm{es}}t+\varphi_{\mathrm{s}})+|I^{\prime}_{\mathrm{s}}|\cos\Bigl{(}s_{r} \omega_{\mathrm{ref}}t+\varphi^{\prime}_{\mathrm{s}}+\theta_{eff}\Bigr{)}\\ &\mathbf{i}_{\mathrm{r}}=|I_{r}|\cos(\omega_{\mathrm{ref}}t+ \varphi_{r})+|I^{\prime}_{r}|\cos\Bigl{(}s\omega_{\mathrm{es}}t+\varphi^{\prime }_{\mathrm{r}}-\theta_{eff}\Bigr{)}\end{split} \tag{7}\]