## 2 VSC System Modeling and Quantitative Evaluation Method of System Stability  
### VSC Impedance Modeling  
Impedance stability theory requires obtaining the grid-connected equipment and grid impedance under small signals as the basis. This section first derives the impedance model of the VSC. Figure 1 shows the topology and control block diagram of the VSC. Since the dynamic response of the current and PLL of VSC is faster than the DC voltage loop, and this article mainly considers the oscillations caused by the current loop and the PLL, the DC voltage control is replaced by a DC voltage source \(V_{dc}\). In Figure 1, \(v_{ox}\) and \(i_{ox}\) (\(x\) = \(a\),\(b\),\(c\), the latter variables are expressed in the same way) are the VSC output voltage and current, respectively. \(u_{pc}\) is the point of common coupling (PCC) voltage. \(U_{gx}\) is the grid voltage. \(L_{f}\) is the filter inductance. Grid impedance is equivalent to an inductance \(L_{g}\). \(i_{ad}\) and \(i_{ox}\) are the values of VSC output current in \(dq\) coordinate system. \(i_{drf}\) and \(i_{grf}\) are the reference values of the grid-connected currents. \(G_{i}\)(s)= \(k_{\text{P-c}}\) + \(k_{\text{i-c}}\)/s is the current proportional-integral (PI) control link, \(k_{\text{P-c}}\) is the proportional coefficient, and \(k_{\text{i-c}}\) is the integral coefficient. PLL is employed to detect the voltage and frequency of the power grid and ensure the synchronization between the VSC and power grid, \(\theta_{pll}\) is the output angle of PLL, \(G_{pll}\) = \(k_{\text{P-PLL}}\) + \(k_{\text{i-PLL}}\)/s is the PI controller, where \(k_{\text{P-PLL}}\) is the proportional coefficient and \(k_{\text{i-PLL}}\) is the integral coefficient. \(v_{md}\) and \(v_{mq}\) are the output modulation signals of the regulator in the \(dq\) coordinate system. It should be pointed out that the frequency range considered in this article is within 500 Hz which is much smaller than the switching frequency of VSC. Therefore, the switching process of modulation can be simplified and regarded as linearized when modeling [26]. Thus, the expressions of \(v_{md}\) and \(v_{mq}\) can be written as Equation (1). \[\left\{ \begin{array}{l}v_{md}=\frac{u_{pcd}-\omega_{1}L_{feq}}{K_{PWM}}\\ v_{mq}=\frac{u_{pcd}+\omega_{1}^{*}L_{fdd}}{K_{PWM}}\end{array} \right. \tag{1}\]  
where \(u_{pcd}\) is the d-axis voltage component of \(u_{pcc}\), \(u_{pccq}\) is the q-axis voltage component of \(u_{pcc}\), and \(\omega_{1}\) is the fundamental wave angular frequency. \(K_{PWM}\) is the equivalent gain of the VSC, and its expression is shown in Equation (2), where \(U_{tri}\) is the carrier amplitude. \[K_{PWM}=\frac{V_{dc}}{U_{tri}} \tag{2}\]  
The impedance modeling of VSC starts from the modeling of PLL. For the PLL, if it is ideal, the PLL detects the magnitude and phase of grid voltage without errors, and the output of PLL is identical to the phase of grid voltage, which can be noted as \(\theta_{pll}^{*}\)  
Figure 1: VSC structure and control system. Thus, the ideal \(dq\) coordinate system can be obtained. When considering the small signal perturbations, there is a \(\Delta\theta\) difference between the output of PLL and the phase of grid voltage, the output of PLL \(\theta_{pll}^{c}\) at this time can be written as Equation (3),  
\[\theta_{pll}^{c}=\theta_{pll}^{s}+\Delta\theta \tag{3}\]  
Therefore, when considering the small signal perturbations, the \(dq\) transformation achieved by the VSC control is nonideal, the intuitive descriptions for the ideal \(dq\) coordinate system, and nonideal \(dq\) coordinate system are presented in Figure 2.  
For any variable \(x\), when they are transferred into the \(dq\) coordinate system by the control of VSC, their expressions can be expressed as Equation (4).