## 3 Impedance Model of Converters with Improved Virtual Impedance Control Strategy
### DC-Side Impedance Modeling of DC-Voltage-Controlled Converter
\right]}^{G^{d}_{PLL}}\Delta u^{s}_{sd}\left] \tag{18}\]  
where  
\[tf_{PLL}\ =\ k_{pPLL}+\frac{k_{pPLL}}{s} \tag{19}\]  
\[G_{PLL}\ =\ \frac{tf_{pLL}}{s+u_{sd0}f_{pLL}} \tag{20}\]  
Here, \(tf_{PLL}\) is the PLL transfer function and \(k_{pPLL}\) and \(k_{ipLL}\) are the proportional and integral gains of the PLL compensator, respectively. The linearized equation of the power balance between the AC and DC sides of the converter is expressed as  
\[\Delta i_{dc}\ =\ 1.5(\left[ \begin{array}{cc}m_{d0}&m_{q10}\end{array} \right] \left[ \begin{array}{c}\Delta i^{s}_{sd}\\ \Delta i^{s}_{sq}\end{array} \right]+\left[ \begin{array}{cc}i_{sd0}&i_{sq0} \end{array} \right]\left[ \begin{array}{c}\Delta m^{s}_{d}\\ \Delta m^{s}_{q}\end{array} \right]) \tag{21}\]  
The small signal expression of the improved virtual impedance compensator is expressed as  
\[\Delta i_{dcn}\ =\ -\frac{k_{i}}{s}\Delta v_{dc} \tag{22}\]  
\[\Delta v_{dcref}\ =\ -(R_{sq}+sL_{eq})(\frac{k_{i}}{s}\Delta v_{dc}+\Delta i_{dc}). \tag{23}\]  
Inserting (22) and (23) into (10), the relation between the AC current reference values \(\Delta i_{sd,ref}\), \(\Delta i_{sq,ref}\), DC voltage \(\Delta v_{dc}\), and DC current \(\Delta i_{dc}\) is expressed as  
\[\left[ \begin{array}{c}\Delta i_{sd,ref}\\ \Delta i_{sq,ref}\end{array} \right]=\left[ \begin{array}{cc}H_{vdc}&H_{idc} \\ 0&0\end{array} \right]\left[ \begin{array}{c}\Delta v_{dc}\\ \Delta i_{dc}\end{array} \right] \tag{24}\]  
where  
\[H_{idc}\ =\ -(k_{pvdc}+\frac{k_{idc}}{s})(\frac{k_{i}R_{eq}}{s}+k_{i}L_{ eq}+1) \tag{25}\]  
\[H_{idc}\ =\ -(k_{pvdc}+\frac{k_{idc}}{s})(R_{sq}+sL_{eq}) \tag{26}\]  
Formula (5) can be written as  
\[\left[ \begin{array}{c}\Delta i^{s}_{sd}\\ \Delta i^{s}_{sq}\end{array} \right]\ =\ -Y_{g}(s)\left[ \begin{array}{c}\Delta i^{s}_{ sd}\\ \Delta i^{s}_{sq}\end{array} \right]. \tag{27}\]  
Inserting (27) into (6), the relation between the AC voltage and AC current can be given as  
\[\left[ \begin{array}{c}\Delta u^{s}_{sd}\\ \Delta u^{s}_{sq}\end{array} \right]\ =\ \overbrace{\left(Y_{cf}(s)-Y_{g}(s)\right)^{-1}}^{Z_{s}} \Delta^{s}_{sd}\  
Considering PLL, inserting (16)-(18) and (28) into (11), the modulation index can be written as  
\[\left[ \begin{array}{c}\Delta m_{d}^{s}\\ \Delta m_{q}^{s}\end{array} \right]v_{dc0}\ =\ -G_{pwnm}G_{cc}\left[ \begin{array}{cc}H_{ vdc}&H_{ikc}\\ 0&0\end{array} \right]\left[ \begin{array}{c}\Delta v_{dc}\\ \Delta i_{dc}\end{array} \right]+G_{z}^{vdc}(s)\left[ \begin{array}{c}\Delta _{sd}^{s}\\ \Delta_{sq}^{s}\end{array} \right] \tag{29}\]  
where  
\[G_{z}^{vdc}(s)\ =\ G_{pwnm}(G_{cc}+Z_{del})+Z_{s}(s)G_{c}^{i}. \tag{30}\]  
\[G_{c}^{i}\ =\ (G_{pwnm}G_{PLL}^{v}+G_{pwnm}(G_{cc}+Z_{del})G_{PLL}^{i}+v_{d \varepsilon 0}G_{PLL}^{d}) \tag{31}\]  
The relation between the DC voltage, DC current, and AC currents can be obtained from (4) by inserting (28) and (29):  
\[Y_{AC}^{Vdc}(s)\left[ \begin{array}{c}\Delta v_{dc}\\ \Delta i_{dc}\end{array} \right]\ =\ \left[ \begin{array}{c}\Delta i_{sd}^{s}\\ \Delta i_{sq}^{s}\end{array} \right] \tag{32}\]  
where  
\[Y_{AC}^{Vdc}(s)\ =\ \left(Z_{s}(s)-Z_{0}(s)-G_{z}^{vdc}(s)\right)^{-1}\left( \begin{array}{cc}m_{d0}&0\\ m_{q0}&0\end{array} \right]+G_{pwn}G_{cc}\left[ \begin{array}{cc}H_{vdc}&H_{ idc}\\ 0&0\end{array} \right]) \tag{33}\]  
and \(Y_{AC}^{Vdc}\) can be expressed as a \(2^{\bullet}2\) order matrix:  
\[Y_{AC}^{Vdc}\ =\ \left[ \begin{array}{cc}Y_{1}&Y_{2}\\ Y_{3}&Y_{4}\end{array} \right] \tag{34}\]  
Inserting (32) into (4), yields  
\[\left[ \begin{array}{c}\Delta m_{d}^{s}\\ \Delta m_{q}^{s}\end{array} \right]\ =\ \frac{1}{v_{dc0}}((Z_{s}(s)-Z_{0}(s))Y_{AC}^{Vdc}(s)- \left[ \begin{array}{cc}m_{d0}&0\\ m_{q0}&0\end{array} \right])\left[ \begin{array}{c}\Delta v_{dc}\\ \Delta i_{dc}\end{array} \right]. \tag{35}\]  
In the power balance relation, by inserting (34) and (35) into (21), (21) can be replaced by  
\[