## 3 Modeling and Linearization of the System
### Linearization
\right]\left[ \begin{array}{c} \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\end{array} \right]\left[ \begin{array}{c} \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\end{array} \right]\left[ \begin{array}{c} \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\end{array} \right]\left[ \begin{array}{c} \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\end{array} \right]\left[ \begin{array}{c} \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\end{array} \right] \tag{45}\]  
\[D_{2_{i}}=\left[\begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0\\ \cos(\theta_{i}^{\prime})&\cos(\theta_{i}^{\prime}+2\pi/3)&\cos(\theta_{i}  
\[ \begin{bmatrix}Z_{a\delta}\\ Z_{b\delta}\\ Z_{c\delta}\\ Z_{d\delta}\\ Z_{a}^{tag}\\ Z_{b\frac{tag}{2}}\\ Z_{b\frac{tag}{2}}\\ Z_{c\delta}^{tag}\\ Z_{c}^{tag}\\ Z_{d}^{tag}\\ Z_{d}^{tag}\end{bmatrix}  = \begin{bmatrix}\sin(\delta)\cos(\frac{2\pi}{3})+\cos(\delta)\sin( \frac{2\pi}{3})\\ \sin(\delta)\sin(\frac{2\pi}{3})-\cos(\delta)\cos(\frac{2\pi}{3})\\ \sin(\delta)\cos(\frac{2\pi}{3})-\cos(\delta)\sin(\frac{2\pi}{3})\\ \sin(\delta)\sin(\frac{2\pi}{3})+\cos(\delta)\cos(\frac{2\pi}{3})\\ \sin(\delta)\sin(\frac{2\pi}{3})+\cos(\delta)\cos(\frac{2\pi}{3})\\ \cos(\frac{2\pi}{3})+X^{tag}\sin(\frac{2\pi}{3})\\ \sin(\frac{2\pi}{3})-X^{tag}\cos(\frac{2\pi}{3})\\ \cos(\frac{2\pi}{3})-X^{tag}\sin(\frac{2\pi}{3})\\ \sin(\frac{2\pi}{3})+X^{tag}\cos(\frac{2\pi}{3})\end{bmatrix}  \tag{47}\]  
The matrices of all the systems are formed in (43). The network equations for the generation bars are shown below:  
\[0 = \begin{bmatrix}C_{2_{(1)}}&&\\ &\ddots&&\\ &&C_{2_{(m)}}\end{bmatrix} \cdot \begin{bmatrix}\Delta x_{1}\\ \vdots\\ \Delta x_{m}\end{bmatrix} + \begin{bmatrix}D_{3_{(1)}}&&\\ &\ddots&\\ &&D_{3_{(m)}}\end{bmatrix} \cdot \begin{bmatrix}\Delta I_{g1}\\ \vdots\\ \Delta I_{gm}\end{bmatrix}  \tag{48}\] \[+ \begin{bmatrix}D_{4_{(1,1)}}&\cdots&D_{4_{(1,m)}}\\ \vdots&\ddots&\vdots\\ D_{4_{(m,1)}}&\cdots&D_{4_{(m,m)}}\end{bmatrix} \cdot \begin{bmatrix}\Delta V _{g(1)}\\ \vdots\\ \Delta V_{g(m)}\end{bmatrix} + \begin{bmatrix}D_{5_{(1,m+1)}}&\cdots&D_{5_{(1,m) }}\\ \vdots&\ddots&\vdots\\ D_{5_{(m,m+1)}}&\cdots&D_{5_{(m,m)}}\end{bmatrix} \cdot \begin{bmatrix}\Delta V _{l(m+1)}\\ \vdots\\ \Delta V_{l(n)}\end{bmatrix} \]  
where:  
\[C_{2_{i}}= \begin{bmatrix}0\end{bmatrix}  \tag{49}\]  
\[D_{3_{i}}= \begin{bmatrix}0&0&V_{i0}^{a}\cos(\theta_{i0}^{a})&V_{i0}^{a}\sin( \theta_{i0}^{a})&0&0&0&0&0\\ 0&0&V_{i0}^{a}\sin(\theta_{i0}^{a})&-V_{i0}^{a}\cos(\theta_{i0}^{a})&0&0&0&0&0 \\ 0&0&0&0&V_{i0}^{b}\cos(\theta_{i0}^{b})&V_{i0}^{b}\sin(\theta_{i0}^{b})&0&0&0 \\ 0&0&0&0&V_{i0}^{b}\sin(\theta_{i0}^{b})&-V_{i0}^{b}\cos(\theta_{i0}^{b})&0&0&0 \\ 0&0&0&0&0&0&V_{i0}^{c}\cos(\theta_{i0}^{c})&V_{i0}^{c}\sin(\theta_{i0}^{c})\\ 0&0&0&0&0&0&V_{i0}^{c}\sin(\theta_{i0}^{c})&-V_{i0}^{c}\cos(\theta_{i0}^{c}) \\ \end{bmatrix} \cdot \begin{bmatrix}\Delta I_{d_{i}}\\ \Delta I_{d_{i}}\\ \Delta I_{d_{i}}\\ \Delta I_{d_{i}}\\ \Delta I_{d_{i}}^{tag}\\ \Delta I_{d_{i}}^{tag}\\ \end{bmatrix}  \tag{50}\]  
Following the same logic, the network equations for the charge bars correspond to:  
\[0= \begin{bmatrix}D_{6_{(m+1,1)}}&\cdots&D_{6_{(m+1,m)}}\\ \vdots&\ddots&\vdots\\ D_{6_{(m,1)}}&\cdots&D_{6_{(m,m)}}\end{bmatrix} \cdot \begin{bmatrix}\Delta V _{g(m+1)}\\ \vdots\\ \Delta V_{g(n)}\end{bmatrix} + \begin{bmatrix}D_{7_{(m+1,m+1)}}&\cdots&D_{7_{(m+1, m)}}\\ \vdots&\ddots&\vdots\\ D_{7_{(m,m+1)}}&\cdots&D_{7_{(m,m)}}\end{bmatrix} \cdot \begin{bmatrix}\Delta V _{l(m+1)}\\ \vdots\\ \Delta V_{l(n)}\end{bmatrix}  \tag{51}\]  
where the \(A_{sys}\) of Equation (3) has the following form:  
\[A_{sys}=A^{{}^{\prime}}-B^{{}^{\prime}}D^{{}^{\prime-1}}C^{{}^{\prime}} \tag{52}\]  
considering:  
\[A^{{}^{\prime}}= [A_{1}-B_{1}D_{1}^{-1}C_{1}] \tag{53}\] \[B^{{}^{\prime}}= \begin{bmatrix}B_{2}-B_{1}D_{1}^{-1}]D_{2}&0\end{bmatrix} \] (54) \[C^{{}^{\prime}}= \begin{bmatrix}-D_{3}^{-1}D_{1}^{-1}C_{1}\\ 0\end{bmatrix}