# Impedance-Based Stability Analysis of Grid-Connected Inverters under the Unbalanced Grid Condition
## 3 The Simplified Stability Analysis Method Based on the GNC and Verification
### Stability Analysis Method of Inverters under the Unbalanced Grid Condition  
The block diagram of the DSOGI-PLL-based inverter under the unbalanced grid condition is plotted in Figure 6. From Figure 6, the transfer function of the forward path is the frequency-coupled admittance model of the inverter under the unbalanced grid condition, and the feedback path is the unbalanced grid impedance model. Figure 5: The block diagram of the DSOGI-PLL-based inverter under the balanced grid condition. The frequency-coupled impedance model under the unbalanced grid condition has been derived in the previous section. The transfer function of the forward path is the inverse matrix of the impedance matrix in Equation (16):  
\[\mathbf{Q}_{\mathrm{u}}= \begin{bmatrix}Y_{11}&Y_{12}&0&0\\ Y_{21}&Y_{22}&0&0\\ 0&0&Y_{33}&0\\ 0&0&0&Y_{44}\end{bmatrix}  \tag{20}\]  
where \(\mathbf{Q}_{\mathrm{u}}\) is the admittance matrix. According to the frequency relationship and the symmetric component method, the unbalanced grid impedance model can be derived. Therefore, the transfer function matrix of the feedback path is as follows:  
\[\mathbf{F}_{\mathrm{u}}= \begin{bmatrix}Z_{\mathrm{g11}}&0&Z_{\mathrm{g13}}&0\\ 0&Z_{\mathrm{g22}}&0&Z_{\mathrm{g24}}\\ Z_{\mathrm{g31}}&0&Z_{\mathrm{g33}}&0\\ 0&Z_{\mathrm{g42}}&0&Z_{\mathrm{g44}}\end{bmatrix}  \tag{21}\]  
where \(Z_{\mathrm{g11}}\)-\(Z_{\mathrm{g44}}\) represent grid impedance, respectively. After we obtain the transfer functions of the forward and feedback paths, the analysis of the system stability can be carried out according to the GNC. However, for a four-input four-output high-order system, the GNC is too complicated. In the Nyquist diagram, it is complex and not intuitive to analyze the trajectories of all eigenvalues and whether they encircle the origin. Therefore, we can perform some pre-processing on this fourth-order matrix to simplify the analysis process. Based on the Schur Complement, the determinant of a high-order matrix can be transformed into the product of determinants of two low-order matrices. Then, according to matrix theory, the determinant problem can be considered as the eigenvalue problem after some processing. Therefore, an eigenvalue problem of a high-order matrix can be transformed into an eigenvalue problem of two low-order matrices. Finally, the stability analysis of this high-order system can be carried out by analyzing the eigenvalues of two low-order matrices. First, the characteristic matrix \(\mathbf{Y}\) is written in the form of a block matrix as  
\[\mathbf{Y}=\mathbf{I}+\mathbf{Q}_{\mathrm{u}}\mathbf{F}_{\mathrm{u}}= \begin{bmatrix}\mathbf{A}&\bm {B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}  \tag{22}\]  
Figure 6: The block diagram of the DSOGI-PLL-based inverter under the unbalanced grid condition. where \(\mathbf{A}= \begin{bmatrix}1+Y_{11}Z_{\text{g}11}&Y_{12}Z_{\text{g}22}\\ Y_{21}Z_{\text{g}11}&1+Y_{22}Z_{\text{g}22}\end{bmatrix} \), \(\mathbf{B}= \begin{bmatrix}Y_{11}Z_{\text{g}13}&Y_{12}Z_{\text{g}24}\\ Y_{21}Z_{\text{g}13}&Y_{22}Z_{\text{g}24}\end{bmatrix} \), \(\mathbf{C}= \begin{bmatrix}Y_{33}Z_{\text{g}31}&0\\ 0&Y_{42}Z_{\text{g}42}\end{bmatrix} \), and \(\mathbf{D}= \begin{bmatrix}1+Y_{33}Z_{\text{g}33}&0\\ 0&1+Y_{44}Z_{\text{g}44}\end{bmatrix} \). Therefore, the determinant of matrix \(\mathbf{Y}\) is equal to the product of the determinant of matrix \(\mathbf{D}\) and \(\mathbf{A}-\mathbf{B}\mathbf{D}^{-1}\mathbf{C}\):  
\[\det(\mathbf{Y})=\det(\mathbf{D})\det\Bigl{(}\mathbf{A}-\mathbf{B}\mathbf{D}^{-1}\mathbf{C}\Bigr{)} \tag{23}\]  
According to matrix theory, when the determinant of matrix \(\mathbf{Y}\) is equal to the product of the determinants of matrices \(\mathbf{D}\) and \(\mathbf{A}-\mathbf{B}\mathbf{D}^{-1}\mathbf{C}\), the number of the eigenvalues of matrix \(\mathbf{Y}\) encircling the origin anticlockwise is equal to the sum of the number of the eigenvalues of matrices \(\mathbf{D}\) and \(\mathbf{A}-\mathbf{B}\mathbf{D}^{-1}\mathbf{C}\) encircling the origin anticlockwise.