## 3 Impedance Model of Converters with Improved Virtual Impedance Control Strategy  
### DC-Side Impedance Modeling of DC-Voltage-Controlled Converter  
The linearized dynamic equations of the DC voltage-controlled converter are expressed as  
\[\left[ \begin{array}{c}m_{d0}\\ m_{q0}\end{array} \right]\Delta v_{dc}+v_{dc0}\left[ \begin{array}{c}\Delta m_{ d}^{s}\\ \Delta m_{q}^{s}\end{array} \right]+Z_{0}(s)\left[ \begin{array}{c}\Delta_{sd}^{s }\\ \Delta_{sq}^{s}\end{array} \right]=\left[ \begin{array}{c}\Delta u_{sd}^{s}\\ \Delta u_{sq}^{s}\end{array} \right] \tag{4}\]  
\[\left[ \begin{array}{c}\Delta u_{sd}^{s}\\ \Delta u_{sq}^{s}\end{array} \right]=-Z_{g}(s)\left[ \begin{array}{c}\Delta_{ gd}^{s}\\ \Delta_{sq}^{s}\end{array} \right] \tag{5}\]  
\[\left[ \begin{array}{c}\Delta_{gd}^{s}\\ \Delta_{gq}^{s}\end{array} \right]=\left[ \begin{array}{c}\Delta_{sd}^{s}\\ \Delta_{sq}^{s}\end{array} \right]+Y_{cf}(s)\left[ \begin{array}{c}\Delta n_{ sd}^{s}\\ \Delta n_{sq}^{s}\end{array} \right] \tag{6}\]  
where  
\[Z_{0}=\left[ \begin{array}{cc}R_{c}+sL_{c}&-\omega_{0}L_{c}\\ \omega_{0}L_{c}&R_{c}+sL_{c}\end{array} \right] \tag{7}\]  
\[Z_{g}=\left[ \begin{array}{cc}R_{n}+sL_{n}&-\omega_{0}L_{n}\\ \omega_{0}L_{n}&R_{n}+sL_{n}\end{array} \right] \tag{8}\]  
\[Y_{cf}(s)=\left[ \begin{array}{cc}sC_{f}&-\omega_{0}C_{f}\\ \omega_{0}\tilde{C}_{f}&sC_{f}\end{array} \right] \tag{9}\]  
Its controller and modulator linearized equations are expressed as  
\[\left[ \begin{array}{c}\Delta i_{sd,ref}\\ \Delta i_{sq,ref}\end{array} \right]=\left[ \begin{array}{c}k_{pndc}+\frac{k_ {ink}}{s}\\ 0\end{array} \right](\Delta v_{dcf}-\Delta v_{dc}) \tag{10}\]  
\[v_{dc0}\left[ \begin{array}{c}\Delta m_{d}^{c}\\ \Delta m_{q}^{s}\end{array} \right]=-G_{psum}G_{cc}\left[ \begin{array}{c} \Delta i_{sd,ref}\\ \Delta i_{sq,ref}\end{array} \right]+G_{psum}\left[ \begin{array}{c}\Delta u_{ sd}^{c}\\ \Delta u_{sq}^{s}\end{array} \right]+G_{psum}(G_{cc}+Z_{dd})\left[ \begin{array}{c} \Delta i_{sd}^{c}\\ \Delta i_{sq}^{s}\end{array} \right] \tag{11}\]  
where \(k_{pndc}\) and \(k_{inlc}\) are the proportional and integral gains of the DC voltage controller, respectively, \(G_{psum}\) is the PWM delay, and \(G_{cc}\) is the current compensator transfer function, where \(k_{pc}\) and \(k_{ic}\) are the proportional and integral gains of the current compensator, respectively. \[G_{psum}=\left[ \begin{array}{cc}H_{psum}&0\\ 0&H_{psum}\end{array} \right] \tag{12}\]  
\[H_{psum}=e^{-sT_{s}}\frac{1-e^{-sT_{s}}}{sT_{s}} \tag{13}\]  
\[G_{cc}=\left[ \begin{array}{cc}k_{pc}+\frac{k_{c}}{s}&0\\ 0&k_{pc}+\frac{k_{c}}{s}\end{array} \right] \tag{14}\]  
\[Z_{dd}=\left[ \begin{array}{cc}0&\omega_{pLL}L_{c}\\ -\omega_{pLL}L_{c}&0\end{array} \right] \tag{15}\]  
The variables of the controller are based on the output of the PLL reference, whereas the variables of the circuit are based on the system reference. The relationship of physical quantity between the PLL and the system references is expressed as [13]  
\[\left[ \begin{array}{c}\Delta i_{sd}^{c}\\ \Delta i_{sq}^{s}\end{array} \right]=\left[ \begin{array}{c}\Delta i_{sd}^{s} \\ \Delta i_{sq}^{s}\end{array} \right]+\widehat{\left[ \begin{array}{cc}0&G_{PLL} (s)i_{sq0}\\ 0&-G_{PLL}(s)i_{sd0}\end{array} \right]}\left[ \begin{array}{c}\Delta i_{sd}^{s} \\ \Delta n_{sq}^{s}\end{array} \right] \tag{16}\]\[\left[ \begin{array}{c}\Delta u^{c}_{sd}\\ \Delta u^{c}_{sq}\end{array} \right]=\overbrace{\left[ \begin{array}{cc}1&G_{PLL}(s)u_{sq0}\\ 0&1-G_{PLL}(s)u_{sq0}\end{array} \right]}^{G^{p}_{PLL}}\left[ \begin{array}{c} \Delta u^{s}_{sd}\\ \Delta u^{s}_{sq}\end{array} \right] \tag{17}\]  
\[\left[ \begin{array}{c}\Delta m^{c}_{d}\\ \Delta m^{c}_{q}\end{array} \right]=\left[ \begin{array}{c}\Delta m^{s}_{d}\\ \Delta m^{s}_{q}\end{array} \right]-\overbrace{\left[ \begin{array}{cc}0&-G_{PLL}(s)m_{q0}\\ 0&G_{PLL}(s)m_{d0}\end{array}