## 4 Analysis and Simulation Verification  
### Optimization of One Parameter  
To verify the effectiveness of the proposed method for analyzing and improving the stability of the multi-converter grid-tied system, a three-VSC system is built as shown in Figure 3. Differentiated control parameters for VSCs are employed to emphasize the distinctions between different VSCs. The system under the grid parameters and control parameters of VSCs presented in Table A1 (Appendix A) is defined as system A. Obtaining the return ratio matrix \(\mathbf{L}(s)\) of system A, the \(h(f)\) and \(g(f)\) functions are constructed according to Equations (19) and (22) and plotted in Figure 9,b, respectively. From Figure 9, it can be observed that the function values of the lowest points \(M\) for both \(h(f)\) and \(g(f)\) are less than 0, and system A is unstable. Figure 8: Flowchart of parameter optimization for a multi-converter grid-tied system. When system A is unstable, parameter sensitivity analysis is required to identify the stability-weakest parameters according to Equation (25). Existing research has shown that the proportional parameters of the current loop [25] and PLL [26] in VSC have a greater impact on system stability compared to the integral parameters. Therefore, this article focuses on analyzing the impact of the proportional parameters in different VSCs on system A stability. The sensitivity results for each parameter are presented in Table 1.  
The sign of parameter sensitivity determines the direction of parameter optimization, according to parameter sensitivity optimization criterion 1. The sensitivity calculation results for the \(h(f)\) and \(g(f)\) functions exhibit opposite signs: positive values of parameter sensitivity for \(h(f)\) indicate an increase in the parameter, while negative values of parameter sensitivity for \(g(f)\) suggest a decrease in the parameter. The analytical conclusions of the traditional method and the method proposed in this paper are contradictory. The distance between the Nyquist curves of \(L(\mathrm{s})\) and the \((-1,j0)\) point can be used to evaluate the stability of system A. It can verify which method's parameter optimization direction is correct. Taking the parameter \(k_{\mathrm{ip1}}\) as an example, Figure 10 illustrates the changes in the Nyquist curves of system A. Figure 10 only displays the curves of the eigenvalues that have a substantial impact on the system. When \(k_{\mathrm{ip1}}\) increases, the Nyquist curves of system A move away from the \((-1,j0)\) point, and system A stability deteriorates. Meanwhile, when \(k_{\mathrm{ip1}}\) decreases, the Nyquist curves move closer to the \((-1,j0)\) point, and system A stability is improved. There is a misjudgment about the optimization direction of the \(h(f)\) function. It theoretically demonstrates that the method proposed in this paper can overcome the defect of traditional methods caused by the misjudgment of parameter optimization direction due to the change in the Gerschgorin disc's radius after parameter optimization. Following the second rule of parameter optimization, parameters with larger absolute sensitivity values have a greater impact on the system stability. By sorting the parameters based on the absolute values of their parameter sensitivities, the influence of each VSC parameter on system stability can be determined as follows: \(k_{\mathrm{ip1}}>k_{\mathrm{ip3}}>k_{\mathrm{ip2}}>k_{\mathrm{pllp2}}>k_{ \mathrm{pllp3}}>k_{\mathrm{pllp1}}\). When parameters change, the change degree of Nyquist curves can be used to judge the influence of parameters on system A stability, to verify the correctness of parameter sensitivity analysis. Figure 11 illustrates the Nyquist curves of system A when one control parameter of VSC is altered \(-5.000\times 10^{-4}\) while keeping other parameters constant. It depicts that when all parameters are reduced, the Nyquist curves of system A approach the point \((-1,j0)\), and the system stability is improved.