## 1 Introduction
### Construction of Parameter Stability Region  
According to the results of the participation factor analysis, \(\rm{h_{p1}}\), \(\rm{h_{p1}}\), \(\rm{h_{p3}}\), \(\rm{h_{3}}\), \(\rm{b_{p1}}\), \(\rm{b_{i1}}\) are all dominant variables, and the PSR is constructed according to the above six variables, on principle. For the sake of visual clarity, the q-axis proportional coefficient \(\rm{h_{p3}}\) of GSC of D-PMSG and the d-axis outer-loop proportional coefficient \(\rm{b_{p1}}\) of SEC of VSC-HVDC, which are the controller parameters that represent the coupling in the renewable power base (source side) and HVDC transmission system (grid side), respectively, and affect the studied SSO/SupSO with relatively high sensitivity according to the sensitivity calculation, are taken as dominant variables for PSR construction to demonstrate the method proposed in this paper, so as to visually display the two-dimensional PSR on the plane, which is \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **SSO-1** & **SSO-2** & **SupSO** \\ \hline Frequency (Hz) & 13.24 & 20.20 & 59.38 \\ Damping ratio & 0.8764 & 0.0966 & 0.8008 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Frequency and damping ratio corresponding to oscillation modes after order reduction. more intuitive. Besides this, the PSR can be built via other strongly correlated dominant variables. The method does not construct the PSR for each SSO mode individually, but for multiple SSO modes together. The functional relationships of each matrix element with respect to h\({}_{\text{p3}}\) and b\({}_{\text{p1}}\) are approximated, and are shown in Appendix A. It can be seen that the controller parameters h\({}_{\text{p3}}\) and b\({}_{\text{p1}}\) mainly affect the elements of columns 2-8 in rows 2, 4 and 8, and all of them are functions of h\({}_{\text{p3}}\) and b\({}_{\text{p1}}\). The set of parameters h\({}_{\text{p3}}\) = 1.1, b\({}_{\text{p1}}\) = 0.2 is substituted into the state matrix \(\mathbf{A}\), shown in Tables A1 and A2, and then the 8-order numerical state matrix is obtained, then the corresponding right eigenvector matrix \(\mathbf{P}\) is obtained by the initial eigenvalue calculation. A transition matrix \(\mathbf{B}\) (\(\mathbf{P}^{-1}\mathbf{A}\mathbf{P}\)) is formed via Equation (10). According to Equations (2) and (3), the stability constraints to be satisfied for the PSR are obtained as follows. Constraint of the center of disk  
\[\text{Re}(b_{ii})\leq 0,\quad i=1\,\sim n \tag{12}\]  
where \(b_{ii}\) is the diagonal element of the matrix \(\mathbf{B}\), and \(n\) is the matrix dimension. Constraints for the distance from the center of a disk to the imaginary axis and radius  
\[R_{i}-|\text{Re}(b_{ii})|=\left|\sum_{j=1,j\
eq i}^{n}b_{ij}\right|-|\text{Re} (b_{ii})|\leq 0,\quad i=1\,\sim n \tag{13}\]  
where \(R_{i}\) is the radius of the \(i\)-th disk and \(b_{ij}\) is the non-diagonal element of the matrix \(\mathbf{B}\). The intersection of h\({}_{\text{p3}}\)-b\({}_{\text{p1}}\) parameter sets which satisfy the above stability constraints forms the PSR, and its boundary can be obtained by taking the equal sign in Equations (12) and (13). The PSR of h\({}_{\text{p3}}\)-b\({}_{\text{p1}}\) obtained via the above constraints is shown as the shadowed area in Figure 3.  
The PSR obtained by point-wise calculation of eigenvalues is also given here for comparison (where h\({}_{\text{p3}}\) is 0-5, b\({}_{\text{p1}}\) is 0-5, and the interval is 0.01). The compared results are shown in Figure 4.  
In Figure 4, the PSR obtained by the point-wise calculation of eigenvalues is surrounded by the blue line and the coordinate axis, and the black shadowed area is the PSR obtained by the proposed method. It can be seen that the PSR obtained by the proposed method is located inside the point-wise PSR, which illustrates  
Figure 3: The parameter stability region (PSR) of h\({}_{\text{p3}}\)-b\({}_{\text{p1}}\) (parameter equilibrium point: h\({}_{\text{p3}}\) = 1.1, b\({}_{\text{p1}}\) = 0.2). proposed method.