Doubly Fed Induction Generator
The doubly fed induction generator modeling equations are shown in \eqref{eq:volt_eq_array_dfig}~\cite{roucoDynamicPatternsModel2006}. The equations are referred to a synchronous rotating frame. The rotor and stator voltage equations are shown in \eqref{eq:volt_eq_array_dfig}
\begin{equation}\left\{
\begin{array}{ccl}
v_s & = & R_s \cdot i_s + \dfrac{1}{\omega_b}\dfrac{\mathrm{d}\psi_s}{\mathrm{d}t} + j\cdot\omega_s \cdot \psi_s \\
v_r & = & R_r \cdot i_r + \dfrac{1}{\omega_b}\dfrac{\mathrm{d}\psi_r}{\mathrm{d}t} + j\cdot s \cdot \omega_s \cdot \psi_r \\
\end{array} \right. \label{eq:volt_eq_array_dfig}
\end{equation}
In which $s$ is the slip frequency, i.e. the difference between the stator and rotor frequencies,
\begin{equation}
s = \dfrac{\omega_s-\omega_r}{\omega_s}
\end{equation}
Further, the flux-current relationships are shown in \eqref{eq:flux_current_dfig}.
\begin{equation}\left\{
\begin{array}{ccl}
\psi_s = L_{ss}\cdot i_s + L_m \cdot i_r \\
\psi_r = L_m \cdot i_s + L_{rr} \cdot i_r
\end{array} \right. \label{eq:flux_current_dfig}
\end{equation}
The following equations govern the rotor dynamics and electromagnetic torque,
\begin{equation}
\begin{array}{rcl}
\dfrac{\mathrm{d}s}{\mathrm{d}t} & = & \dfrac{t_m - t_e}{ 2\cdot H \cdot \omega_s}\\
t_e & = & \mathrm{Im}\{i_s\cdot \psi_s ^* \} = \mathrm{Im}\{-i_r \cdot \psi_r ^*\}
\end{array} \label{eq:dfig_swing_torque}
\end{equation}
As can be seen in Fig.~\ref{fig:basecase}, the output of the grid-side converter is connected to an L filter. The voltage equation of the inductance is,
\begin{equation}
v_s = R_a\cdot i_a +\dfrac{1}{\omega_b}\dfrac{\mathrm{d}\psi_a}{\mathrm{d}t}+j\cdot \omega_s\cdot \psi_a + v_a
\end{equation}
being
\begin{equation}
\psi_a = L_a\cdot i_a \label{eq:flux_current_La}
\end{equation}
$v_a$ the voltage across the inductance, and \eqref{eq:flux_current_La} relating the flux and the current across the inductance.
Finally, the DC-link capacitor voltage equation utilized to connect the grid-side converter and the rotor-side converter is shown in \eqref{eq:dc-link_cap},
\begin{equation}
\frac{1}{2}C \dfrac{\mathrm{d}(v_c^2)}{\mathrm{d}t} = p_r - p_a = \mathrm{Re}\{v_r\cdot i_r^*\}-\mathrm{Re}\{v_a\cdot i_a^*\}
\end{equation}
in which $C$ is the capacitor capacitance.
The control techniques utilized in this work for the doubly fed induction generator are taken from~\cite{roucoDynamicPatternsModel2006}. Further, a Phase-Locked-Loop taken from~\cite{avila-martinezImpactPLLControl2020}. The grid-side converter control is responsible for controlling the DC-link capacitor voltage with the direct-axis current component $i_{ad}^*$. The reactive power injected with the grid-side converter is controlled with the quadrature-axis current component. In this work, all the reactive power will be injected through the rotor, and therefore $i_{aq}* = 0$. The rotor-side converter is utilized to control the reactive power injection to the grid with the quadrature-axis current component $i_{rq}^*$, and the direct-axis current component $i_{rd}^*$ is utilized to control the electromagnetic torque, i.e. the active power, injected to the grid.