## 3 Modeling and Linearization of the System
### Linearization  
The equivalent model for the small signal stability analysis is obtained by linearizing the differential and algebraic equations based on the operation points obtained from the power flow. For an \(n\) bar and \(m\) machine system \((m<n)\), the following system matrices are defined. The variables of the general system are displayed in Equation (39) where \"\(x_{i}\)\" contains the state  
\[B_{2_{i}}=\left[ \begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ -\frac{K_{i}}{3T_{A_{i}}}&-\frac{K_{i}}{3T_{A_{i}}}&-\frac{K_{i}}{3T_{A_{i}}}&0 &0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{array} \right].\left[ \begin{array}{c}\Delta V_{i}^{a}\\ \Delta V_{i}^{b}\\ \Delta V_{i}^{c}\\ \Delta\theta_{i}^{a}\\ \Delta\theta_{i}^{b}\\ \Delta\theta_{i}^{c}\end{array} \right] \tag{42}\]  
To form the corresponding matrices of all the system, diagonal matrices are formed:  
\[A_{1}=\left[ \begin{array}{cccc}A_{1_{(1)}}&&&\\ &\ddots&\\ &&A_{1_{(m)}}\end{array} \right] \tag{43}\]  
For the stator equations:  
\[C_{\lambda_{i}}=\left[ \begin{array}{cccccc}0&-\left[\cos(\delta_{0})|I_{R_{ \alpha_{i}}}^{\prime}+\cos(\frac{2\pi}{3})(I_{R_{\alpha_{i}}}^{\prime}+I_{R_{ \alpha_{i}}}^{\prime})-\sin(\frac{2\pi}{3})(I_{R_{\alpha_{i}}}^{\prime}-I_{R_ {\alpha_{i}}}^{\prime})\right]&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ \end{array} \right]\left[ \begin{array}{c}\Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \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}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \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}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\\ \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}\\ \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}\\ \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}\\ \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}\\ \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}\\ \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}\\ \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}\\ \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}\\ \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}\\ \Delta F_{R_{\alpha_{i}}}^{\prime}\end{array}