## 3 Stability Analysis and Improvement in the Multi-Converter Grid-Tied System
### Stability Analysis of the Multi-Converter Grid-Tied System
When the value of the lowest point \(M\) in the function \(h(f)\) is greater than 0, the trajectories of the Gerschgorin discs do not enter the forbidden zone and the system is stable. \[h(f)=\max(h_{\mathrm{r}}(f),h_{\mathrm{c}}(f)) \tag{19}\]  
where  
\[h_{\mathrm{r}}(f)=\left\{ \begin{array}{l}\min(|\mathrm{Im}(L_{ii})|-R_{ \mathrm{ri}}(\mathbf{L})),\mathrm{Re}(L_{ii})\leq-1\\ \min(|L_{ii}+1|-R_{\mathrm{ri}}(\mathbf{L})),\mathrm{Re}(L_{ii})>-1\end{array} \right. \tag{20}\]  
\[h_{\mathrm{c}}(f)=\left\{ \begin{array}{l}\min(|\mathrm{Im}(L_{jj})|-R_{ \mathrm{c}j}(\mathbf{L})),\mathrm{Re}(L_{jj})\leq-1\\ \min(|L_{jj}+1|-R_{\mathrm{c}j}(\mathbf{L}),\mathrm{Re}(L_{jj})>-1\end{array} \right. \tag{21}\]  
According to GNC, when the system is unstable, the farther the intersection point of the Nyquist curve of the \(\mathbf{L}\)(s) with the real axis is away from the point \((-1,j0)\), the worse the system stability is. \(h(f)\) only considers the shortest distance from the boundary of the Gerschgorin discs to the range \([-\infty,-1]\) when \(\mathrm{Re}(L_{ii(j)})\leq-1\), without considering the distance between the fastest intersection of the Gerschgorin disc with the real axis and the point \((-1,j0)\). Figure 5 shows that the Gerschgorin discs correspond to the lowest points \(M\) of the \(h(f)\) function under different parameters. The function value \(M_{2}\) under parameter 2 is greater than \(M_{1}\) under parameter 1. However, the intersection point between the Gerschgorin disc and the real axis is farther from \((-1,j0)\) under parameter 2 due to the change in the Gerschgorin disc's radius after parameter optimization, and the system stability is worse. In this scenario, the theoretical direction of parameter optimization is opposite to the actual correction direction when performing subsequent parameter sensitivity analysis on the system. Based on the existing stability evaluation function, this article incorporates the distance between the intersection point and \((-1,j0)\) when the Gerschgorin discs intersect with the real axis and \(\mathrm{Re}(L_{ii(j)})\leq-1\), as illustrated by the green double arrow line in Figure 6.  
Figure 5: Scenarios of misjudgment in the stability evaluation function \(h(f)\). Figure 6: The geometric schematic diagram of system stability for new judgment condition. Combining the geometric conditions for system stability defined by Equation (18) and the proposed stability evaluation criterion introduced in Figure 6, a more comprehensive stability evaluation function for the multi-converter grid-tied system can be formulated as follows. \[g(f)=\max(g_{\mathrm{r}}(f),g_{\mathrm{c}}(f)) \tag{22}\]  
where  
\[g_{\mathrm{r}}(f)=\left\{ \begin{array}{l}\min[|\mathrm{Im}(L_{ii})|-R_{ri}( \boldsymbol{L}),\mathrm{Re}(L_{ii})+1-\sqrt{R_{ri}(\boldsymbol{L})^{2}-| \mathrm{Im}(L_{ii})|^{2}}],\ \mathrm{Re}(L_{ii})\leq-1\upkappa[|\mathrm{Im}(L_{ii})|-R_{ri}( \boldsymbol{L})\leq 0]\\ \min[|\mathrm{Im}(L_{ii})|-R_{ri}(\boldsymbol{L})],\ \mathrm{Re}(L_{ii})\leq-1 \upkappa[|\mathrm{Im}(L_{ii})|-R_{ri}(\boldsymbol{L})>0]\\ \min[|L_{ii}+1|-R_{ri}(\boldsymbol{L})],\mathrm{Re}(L_{ii})>-1\end{array} \right. \tag{23}\]  
\[g_{\mathrm{c}}(f)=\left\{ \begin{array}{l}\min[|\mathrm{Im}(L_{jj})|-R_{cj}( \boldsymbol{L}),\mathrm{Re}(L_{jj})+1-\sqrt{R_{cj}(\boldsymbol{L})^{2}-| \mathrm{Im}(L_{jj})|^{2}}],\ \mathrm{Re}(L_{ii})\leq-1\upkappa[|\mathrm{Im}(L_{jj})|-R_{cj}( \boldsymbol{L})\leq 0]\\ \min[|\mathrm{Im}(L_{jj})|-R_{cj}(\boldsymbol{L})],\ \mathrm{Re}(L_{jj})\leq-1 \upkappa[|\mathrm{Im}(L_{jj})|-R_{cj}(\boldsymbol{L})>0]\\ \min[|\mathrm{Im}(L_{jj})|-R_{cj}(\boldsymbol{L})],\ \mathrm{Re}(L_{jj})>-1\end{array} \right. \tag{24}\]  
The rules for determining system stability based on the stability evaluation function are illustrated in Figure 7. When the function value of the lowest point \(M\) of \(g(f)\) is greater than 0, it means that the trajectory of the Gerschgorin discs does not enter the forbidden zone, and the system is stable. Otherwise, the system is unstable.