## 4 Analysis and Simulation Verification
### Optimization of One Parameter
Furthermore, according to the change degree of Nyquist curves, parameters are sorted as follows: \(k_{\mathrm{ip1}}>k_{\mathrm{ip3}}>k_{\mathrm{ip2}}>k_{\mathrm{pllp2}}>k_{ \mathrm{pllp3}}>k_{\mathrm{pllp1}}\). This validation confirms the correctness and effectiveness of the proposed parameter sensitivity analysis method. \begin{table}
\begin{tabular}{c  
To improve the stability of system A, parameter optimization is performed on the stability-weakest parameter based on the third criterion of the parameter optimization rules. From Table 1, the absolute sensitivity value of the \(k_{\text{lp1}}\) parameter is significantly larger than that of other parameters; thereby, \(k_{\text{ip1}}\) is the stability-weakest parameter. Combining Equation (26) and Figure 9b, the calculation yields a new value for \(k_{\text{ip1}}\) as 1.800 \(\times\) 10\({}^{-3}\). The return ratio matrix of the new system is re-obtained to construct the \(g(f)\) function, as shown in Figure 12. It can be seen that the function value of the lowest point for the \(g(f)\) function is \(-2.682\)\(\times\) 10\({}^{-1}\), which is larger than that of the previous \(g(f)\) function, but still less than 0. Therefore, parameter optimization needs to be continued. Figure 11: Nyquist curves of system A with different VSC control parameters: (**a**) integrated graph; (**b**) partial enlargement of (**a**); (**c**) partial enlargement of (**b**). Figure 10: Nyquist curves of system A with different \(k_{\text{ip1}}\) parameters: (**a**) integrated graph; (**b**) partial enlargement (**a**). Based on the flowchart shown in Figure 8, the parameter optimization value of \(k_{\text{ip1}}\) is finally calculated as 4.600 \(\times\)\(10^{-4}\). Figure 13 shows system A's function \(g(f)\) under final parameter optimization, and its function value at the lowest point is 4.700 \(\times\)\(10^{-3}\), which indicates the system is stable. To verify the theoretical analysis results mentioned above, the time-domain simulations based on MATLAB/Simulink are performed. Figure 14 shows the PCC voltage before and after modifying different control parameters of the VSCs while keeping other parameters constant. Figure 14 presents the waveform of PCC voltage during the optimization of \(k_{\text{ip1}}\). System A is oscillated for 2 s. At 2 s, \(k_{\text{ip1}}\) is optimized to 1.800 \(\times\)\(10^{-3}\), reducing system A oscillations but still being unstable. At 2.2 s, \(k_{\text{ip1}}\) is further optimized to 4.600 \(\times\)\(10^{-4}\), and system A regains stability. Parameter optimization of weak stability parameters can effectively improve the stability of the system. Furthermore, Figure 14b-f shows that as \(k_{\text{ip2}}\), \(k_{\text{ip3}}\), \(k_{\text{p1}\text{p1}}\), \(k_{\text{p1}\text{p2}}\), and \(k_{\text{p1}\text{p3}}\) decrease, the oscillation of the PCC voltage is significantly improved. This is consistent with the sign results of parameter sensitivity calculated in Table 1.  
Figure 14: _Cont._  
Figure 12: \(g(f)\) function after the first parameter optimization (\(k_{\text{ip1}}=1.800\)\(\times\)\(10^{-3}\)). Figure 13: \(g(f)\) function of system A under final parameter optimization (\(k_{\text{ip1}}=4.600\)\(\times\)\(10^{-4}\)). Figure 8:0 _Cont._  
The parameter change of \(k_{\text{ip2}}\) and \(k_{\text{ip3}}\) is the same, but the change in harmonic content for \(k_{\text{ip3}}\) (24.75%) is greater than that for \(k_{\text{ip2}}\) (20.59%). It indicates that \(k_{\text{ip3}}\) has a greater impact on system A stability compared to \(k_{\text{ip2}}\). The corresponding change in harmonic content for \(k_{\text{ip1}i=1,\,2,\,3}\) is smaller than that for \(k_{\text{ip1}i=1,\,2,\,3}\) when the variation of \(k_{\text{p1}\text{ip}}\) is larger than that for \(k_{\text{ip1}}\). Thus, \(k_{\text{p1}\text{ip}}\) has a smaller impact on system A stability compared to \(k_{\text{ipi}}\). The remaining parameters are analyzed using the same method.