## 2 VSC System Modeling and Quantitative Evaluation Method of System Stability
### Quantitative Evaluation Method of VSC System Stability  
To quantitatively evaluate the stability of the VSC system, this article proposes a stability quantitative evaluation method based on the relationship between Nyquist curve and the unit circle. According to the generalized Nyquist criterion, the stability of the VSC system can be determined by whether the eigenvalue of the ratio matrix \(\mathbf{L}\)(s) surrounds \((-1,\mathrm{j0})\)[10]. \(\mathbf{L}\)(s) is shown in Equation (15). \[\mathbf{L}(s)=\mathbf{Z}_{g}/\mathbf{Z}_{VSC} \tag{15}\]  
A 7 kW weak grid-connected VSC is analyzed as an example here. The parameters of the grid-connected VSC are shown in Table 1.  
The Nyquist curves of the eigenvalue of \(\mathbf{L}\)(s) are shown in Figure 5. As seen in Figure 5a, the Nyquist curve of \(\lambda_{2}\)(s) is closer to \((-1,\mathrm{j0})\) than \(\lambda_{1}\)(s), which means that the system stability is determined by \(\lambda_{2}\)(s). Figure 5b shows that \(\lambda_{2}\)(s) and the unit circle have two symmetric intersection points A and B. The distances between A (or B) and \((-1,\mathrm{j0})\) can reveal the system stability. The system stability is weaker if the A (or B) and \((-1,\mathrm{j0})\) are closer [29]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline DC Voltage & \(V_{dc}\) & 700 V \\ Filter Inductance & \(L_{f}\) & 3 mH \\ Grid Voltage(line-to-line) & \(U_{g}\) & 380 V \\ Grid Impedance & \(L_{g}\) & 12.8 mH \\ Grid Frequency & \(f_{g}\) & 50 Hz \\ Switching Frequency & \(f_{sw}\) & 15 kHz \\ Sampling Frequency & \(f_{s}\) & 15 kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main system parameters. Figure 4: Validation of the VSC impedance model. Figure 6 gives the Nyquist curve \(\lambda_{2}\)(s) under four sets of current controller parameters, the curves are denoted as \(\lambda_{2-x}\)(s), where \(x\) is the number of the curves (\(x\) = 1,2,3,4). Figure 6a shows the overall plot of \(\lambda_{2-x}\)(s) (\(x\) = 1,2,3,4), Figure 6b shows the enlarged plot of \(\lambda_{2-x}\)(s) (\(x\) = 1,2,3,4) around (\(-\)1, j0), and the current controller parameters of \(\lambda_{2-x}\)(s) are given in Figure 6b. As shown in Figure 6b, note the intersection of \(\lambda_{2-x}\)(s) without an envelope (\(-\)1, j0) and the unit circle as A\({}_{x}\)(\(x\) = 1,2,3). Then, connect point A\({}_{x}\) and the origin of the coordinates (O), note the angle between the line OA\({}_{x_{x}}\) and the negative real axis as \(\phi_{x}\)(\(x\) = 1,2,3). With this approach, the distance between the intersection and (\(-\)1, j0) can be intuitively reflected by \(\phi_{x}\). From the change of curve \(\lambda_{2-x}\)(s), the system stability decreases to instability, \(\phi\) will decrease to 0\({}^{\circ}\), and then \(\lambda_{2}\)(s) will surround (\(-\)1, j0) (as \(\lambda_{2-4}\)(s)). Thus, it can be concluded that the system is more stable if \(\phi\) is larger. If the angle \(\phi\) is negative for \(\lambda_{2-x}\)(s) which means the curve of \(\lambda_{2-x}\)(s) will surround (\(-\)1, j0), such as \(\lambda_{2-x}\)(s), the grid-connected VSC will be unstable at this time. To illustrate the unstable phenomenon caused by\(\lambda_{2-4}\)(s) in Figure 6. Figure 7 gives a Bode plot of \(\mathbf{Z}_{g}\) and \(\mathbf{Z}_{VSC}\) for the case of curve \(\lambda_{2-4}\)(s) in Figure 6. The stability of the grid-connected system can be determined by the phase difference at the magnitude intersection of \(\mathbf{Z}_{VSC}\) and \(\mathbf{Z}_{g}\). If the phase difference is less than 180\({}^{\circ}\), the system is stable, otherwise, the system is unstable, and the frequency of the magnitude intersection is the oscillation frequency [30]. Figure 7 shows that the phase difference at the magnitude intersection of \(\mathbf{Z}_{g}\) and \(Z_{qq}\) is greater than 180\({}^{\circ}\), the system is unstable. The oscillation frequency of the system under \(dq\) axis is 125 Hz, corresponding to 175 Hz and 75 Hz in the stationary coordinate system.