## 3 Stability Analysis and Improvement in the Multi-Converter Grid-Tied System  
### Stability Analysis of the Multi-Converter Grid-Tied System  
GNC [21; 22] is the traditional method to analyze the stability of the multi-converter grid-tied system. If the Nyquist curves for the eigenvalues of \(\mathbf{L}(s)\) do not enclose the \((-1\), j0) point, the system is stable. However, this method requires the complex eigenvalue decomposition of \(\mathbf{L}(s)\), which makes this method inefficient and impractical. Figure 3: Parallel operation of multiple VSCs. To determine the system stability, it is not necessary to obtain the exact eigenvalues of the \(\mathbf{L}(s)\). Instead, it is sufficient to assess the rough trajectory range. Therefore, the Gerschgorin disc theorem can be used to estimate the position of \(\mathbf{L}(s)\) eigenvalues, which can simplify the judgment of system stability. Gerschgorin disc theorem [23]:  
Let \(\mathbf{A}=(a_{\bar{\imath}\bar{\jmath}})\mathbb{\subset}C^{n\times n}\), where \(a_{\bar{\imath}\bar{\jmath}}\) represents the element in the \(i^{\text{th}}\) row and \(j^{\text{th}}\) column of matrix \(\mathbf{A}\), and \(\mathbb{\subset}\) represents the set of complex numbers. Gerschgorin disc is a circle with \(a_{\bar{\imath}\bar{\imath}}\) as its center and the sum of the modulus values of non-diagonal elements as its radius. According to the Gerschgorin disc theorem, the eigenvalues \(\lambda_{i}\) of \(\mathbf{L}(s)\) are located in the union of \(n\) row Gerschgorin discs \(O_{\text{r}}\) and \(n\) column Gerschgorin discs \(O_{\text{c}}\). The accuracy of the eigenvalue estimation of \(\mathbf{L}(s)\) can be improved by the fact that any eigenvalue of \(\mathbf{L}(s)\) is located at the part \(T(\mathbf{L})\) intersecting the set of \(O_{\text{r}}\) and \(O_{\text{c}}\):  
\[T(\mathbf{L})=O_{\text{r}}\cap O_{\text{c}} \tag{15}\]  
\[O_{\text{r}}:\mathop{\cup}\limits_{i=1}^{2}\left\{z||z-L_{i\bar{\imath}}|\, \leq R_{\text{ri}}(\mathbf{L})=\,\,\,\mathop{\sum}\limits_{ \begin{array}{c}j=1 \\ j\
eq i\end{array} }^{2}\left|L_{i\bar{\jmath}}\right|\right\} \tag{16}\]  
\[O_{\text{c}}:\mathop{\cup}\limits_{j=1}^{2}\left\{z||z-L_{j\bar{\jmath}}|\, \leq R_{\text{c}j}(\mathbf{L})=\,\,\,\mathop{\sum}\limits_{ \begin{array}{c}i=1\\ i\
eq j\end{array} }^{2}\left|L_{i\bar{\jmath}}\right|\right\} \tag{17}\]  
where \(z=(x,y)\) represents a set of points in the complex plane. \(R_{\text{ri}}(\mathbf{L})\) and \(R_{\text{c}j}(\mathbf{A})\) represent the radius of the \(i^{\text{th}}\) row Gerschgorin disc and \(j^{\text{th}}\) column Gerschgorin disc for \(\mathbf{L}(s)\). According to GNC, a criterion for system stability based on the Gerschgorin disc theorem can be derived. When the trajectory of \(T(\mathbf{L})\) does not enclose the point \((-1,\bar{\jmath}0)\), the curves of the eigenvalues of \(\mathbf{L}(s)\) do not enclose the point \((-1,\bar{\jmath}0)\) and the system is stable. In other words, when the trajectories of the row Gerschgorin discs or the column Gerschgorin discs do not enter the region starting from the point \((-1,\bar{\jmath}0)\) and extending to the left, the system is stable. Here, the region starting from the point \((-1,\bar{\jmath}0)\) and extending to the left is defined as the forbidden zone \(P\). The geometric condition for the stability of a multi-converter grid-tied system according to the Gerschgorin disc theorem is as follows:  
\[(O_{\text{r}}\cap P)\cup(O_{\text{c}}\cap P)=\varnothing \tag{18}\]  
The shortest distance from the Gerschgorin disc boundary to \([-\infty,\,-1]\) (pink lines) is used as the stability criterion, which is illustrated in Figure 4[24]. Figure 4: The geometric schematic diagram of system stability judgment conditions. Combining Equation (18) and Figure 4, a stability evaluation function \(h(f)\) for a multi-converter grid-tied system can be constructed. The expression of \(h(f)\) is given by Equation (19).