## 2 VSC System Modeling and Quantitative Evaluation Method of System Stability
### VSC Impedance Modeling
\[x^{c}=e^{-\mathrm{i}\Delta\theta}x^{s}\approx(1-\mathrm{i}\Delta\theta)x^{s} \tag{4}\]  
where the superscript \(c\) is used to represent the variable in the nonideal \(dq\) coordinate system, and the superscript \(s\) is used to represent the variable in the ideal \(dq\) coordinate system. Thus, based on Equation (4), the PCC voltage in the \(dq\) coordinate system built by the control of VSC can be written as Equation (5). \[\left\{ \begin{array}{l}\Delta u_{pcc}^{c}=\Delta u_{pcc}^{s}+\Delta\theta u _{pcc}^{s}\\ \Delta u_{pcc}^{c}=-\Delta\theta u_{pcc}^{s}+\Delta u_{pcc}^{s}\end{array} \right. \tag{5}\]  
where \(\Delta\) denotes the small signal perturbation. Subscript \(pccd\) denotes the \(d\)-axis component of the system PCC and \(pccq\) denotes the \(q\)-axis component of the system PCC. The subscript \(0\) denotes the steady-state value. Meanwhile, based on the transfer function block of PLL in Figure 1, \(\Delta\theta\) can be obtained as Equation (6). \[\Delta\theta=\frac{G_{pll}(s)\Delta u_{pccq}^{c}}{s} \tag{6}\]  
Thus, based on Equations (5) and (6), the small-signal model of PLL can be obtained as Equation (7). \(T_{PLL}\) is the transfer function of PLL, in which the physical significance of \(T_{PLL}\) can be described as the transfer function between the perturbation of PCC voltage and the phase error of PLL in VSC control. \[\Delta\theta=\frac{G_{pll}(s)}{s+G_{pll}(s)u_{pccd0}^{s}}\Delta u_{pccq}^{s}=T _{PLL}\Delta u_{pccq}^{s} \tag{7}\]  
Similar to \(u_{pcc}\), the VSC output current and the modulation signal output by current control in the ideal and nonideal \(dq\) coordinate system can be described as Equations (8) and (9), where \(G_{ylm}\)(s) represents the transfer function between the output current and the small signal perturbation of the PCC voltage, \(G_{y2m}\)(s) represents the transfer functions between the output modulation signals of VSC, and the small signal perturbation of the PCC voltage. Figure 2: Relationship between ideal and nonideal \(dq\) coordinate system considering disturbance. \[\left[ \begin{array}{c}\Delta i_{od}^{s}\\ \Delta i_{eq}^{s}\end{array} \right]=\left[ \begin{array}{c}\Delta i_{od}^{c}\\ \Delta i_{eq}^{c}\end{array} \right]+\left[ \begin{array}{cc}0&-T_{PLL}i_{eq 0}^{s}\\ 0&T_{PLL}i_{eq0}^{s}\end{array} \right]\left[ \begin{array}{c}\Delta i_{od}^{ s}\\ \Delta i_{eq}^{s}\end{array} \right]=\left[ \begin{array}{c}\Delta i_{od}^{c}\\ \Delta i_{eq}^{c}\end{array} \right]+\mathbf{G}_{y1m}(s)\left[ \begin{array}{c} \Delta i_{pccd}^{s}\\ \Delta i_{pccq}^{s}\end{array} \right] \tag{8}\]  
\[\left[ \begin{array}{c}\Delta v_{md}^{s}\\ \Delta v_{mq}^{s}\end{array} \right]=\left[ \begin{array}{cc}\Delta v_{md}^{c} \\ \Delta v_{mq}^{s}\end{array} \right]+\left[ \begin{array}{cc}0&-T_{PLL}v_{mq 0}^{s}\\ 0&T_{PLL}v_{md0}^{s}\end{array} \right]\left[ \begin{array}{c}\Delta u_{pccd}^ {s}\\ \Delta u_{pccq}^{s}\end{array} \right]=\left[ \begin{array}{c}\Delta v_{md}^{c} \\ \Delta v_{mq}^{s}\end{array} \right]+\mathbf{G}_{y2m}(s)\left[ \begin{array}{c} \Delta u_{pccq}^{s}\\ \Delta u_{pccq}^{s}\end{array} \right] \tag{9}\]  
With the above derivations, the small-signal model of VSC considering PLL can be obtained as shown in Figure 3. The blue region represents the controlled object in reality, the orange region represents the control system of VSC in the artificially constructed synchronous coordinate system. However, according to the above analyses, the small signal perturbation of PCC voltage makes the artificially constructed synchronous coordinate system not synchronized with the controlled object, the variables in reality that need to enter into the control system (orange region) through PLL phase detection should be modified by Equations (8) and (9), which is the red region in Figure 3.  
In Figure 3, the mathematical model of VSC also considers the effects of time delay [27; 28] and introduces an equivalent model of time delay. \(\mathbf{G}_{im}\)(s), \(\mathbf{G}_{dem}\)(s), and \(\mathbf{Y}_{Lm}\) are shown in (10)-(12).