## 3 Small-Signal Admittance Model in dq-Domain  
Figure 3 shows the model of the current controller, the feedforward terms, the PWM modulator and the filter choke of the grid side converter in the dq frame. The measurement delay is very small compared to the PWM delay and can be neglected. An outer dc-link voltage control loop is implemented in the GSC and is responsible to keep the dc-link voltage constant. The bandwidth of this outer control loop is very low compared to the fundamental frequency. This paper focuses on the oscillations around and above the fundamental frequency. Therefore, such low-bandwidth outer loops can be neglected [13]. As it is assumed that the dc-link voltage is constant, the dynamics of the dc-link controller have been neglected. Because of the dynamics of the grid synchronization loop, the dq domain model comprises a fixed global frame and a local controller frame. In the model, the measurement delay is neglected, but the PWM delay (\(G_{d}\)) is included as defined in [30]. The grid voltage is added to the output current controller (\(G_{cc}\)) through a high-pass/low-pass filter (\(G_{f}\)). Figure 3: Dq small signal model of the current controller, PLL and the filter inductance (L\({}_{\ell}\)). Figure 2: Grid side converter and its control, as typically used in wind turbines. According to Figure 3, the small signal output currents and dq output voltage can be obtained by the following matrices:  
\[\left[ \begin{array}{cc}sL_{f}&-\omega_{0}L_{f}\\ +\omega_{0}L_{f}&sL_{f}\end{array} \right]\left[ \begin{array}{cc}\Delta i^{g} _{q}\\ \Delta i^{g}_{q}\end{array} \right]=\left[ \begin{array}{cc}G_{d}&0\\ 0&G_{d}\end{array} \right]\left[ \begin{array}{cc}\Delta v^{inv-g}_{q}\\ \Delta v^{inv-g}_{q}\end{array} \right]+\left[ \begin{array}{cc}-1&0\\ 0&-1\end{array} \right]\left[ \begin{array}{cc}\Delta v^{g}_{q}\\ \Delta v^{g}_{q}\end{array} \right] \tag{1}\]  
\[\left[ \begin{array}{c}\Delta v^{inv-c}_{q}\\ \Delta v^{inv-c}_{q}\end{array} \right]=\left[ \begin{array}{cc}-G_{cc}&0\\ 0&-G_{cc}\end{array} \right]\ \left[ \begin{array}{cc}\Delta i^{c}_{d}\\ \Delta i^{c}_{q}\end{array} \right]+\left[ \begin{array}{cc}G_{f}&0\\ 0&G_{f}\end{array} \right]\left[ \begin{array}{c}\Delta v^{c}_{d}\\ \Delta v^{c}_{q}\end{array} \right] \tag{2}\]  
By neglecting the dynamics of the grid synchronization loop, e.g., PLL, the dq frame of the plant and the dq frame of the controller would be the same, i.e.,  
\[\left[ \begin{array}{c}\Delta i^{c}_{d}\\ \Delta i^{g}_{q}\end{array} \right]=\left[ \begin{array}{c}\Delta i^{g}_{d}\\ \Delta i^{g}_{q}\end{array} \right]=\left[ \begin{array}{c}\Delta i^{c}_{d}\\ \Delta i^{g}_{q}\end{array} \right]=\left[ \begin{array}{c}\Delta v^{c}_{d}\\ \Delta v^{c}_{q}\end{array} \right]=\left[ \begin{array}{c}\Delta v^{g}_{d}\\ \Delta v^{g}_{q}\end{array} \right]=\left[ \begin{array}{c}\Delta v^{inv-g}_{q }\\ \Delta v^{inv-c}_{q}\end{array} \right]=\left[ \begin{array}{c}\Delta v^{inv}_{d} \\ \Delta v^{inv}_{q}\end{array} \right] \tag{3}\]  
By substituting (2) and (3) in (1), the admittance matrix can be obtained as  
\[-\left[ \begin{array}{c}\Delta i^{c}_{d}\\ \Delta i^{g}_{q}\end{array} \right]=Y_{dq}\left[ \begin{array}{c}\Delta v_{d} \\ \Delta v_{q}\end{array} \right]\Rightarrow Y_{dq}=-\left[ \begin{array}{cc}sL_{ f}+G_{d}G_{cc}&-\omega_{0}L_{f}\\ +\omega_{0}L_{f}&sL_{f}+G_{d}G_{cc}\end{array} \right]^{-1}\left[ \begin{array}{ cc}G_{d}G_{f}-1&0\\ 0&G_{d}G_{f}-1\end{array} \right] \tag{4}\]  
where the dynamics of the grid synchronization loop are ignored. For the grid synchronization loop, the synchronous reference frame phase-locked loop (SRF-PLL) can be used and the small signal model is shown in Figure 4.  
When small-signal perturbations are added to the \(V_{g}\) voltage, the estimated angle will have a small error (\(\Delta\theta\)) between the local (controller) and global (plant) frames because of the PLL dynamics. In Figure 4b,\(f_{dq}{}^{c}\) corresponds to the estimated angle and \(f_{dq}{}^{g}\) corresponds to the actual angle.