Power System Small Signal Stability Analysis and Control

For i = m, Equation (5.34) gives

$\begin{array}{c}\hfill 0=\cdots +\left(-I_{d_i}{V}_{i}cos\left({\delta }_i-{\theta }_i\right)+I_{q_i}{V}_{i}sin\left({\delta }_i-{\theta }_i\right)+\left[V_i{\displaystyle \sum _{k=1\ne i}^n{V}_k{Y}_{\mathit{ik}}sin\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)}\right]\right)\mathrm{\Delta }{\theta }_m\hfill \\ \hfill +\left(I_{d_i}sin\left({\delta }_i-{\theta }_i\right)+I_{q_i}cos\left({\delta }_i-{\theta }_i\right)-\left[{\displaystyle \sum _{k=1}^n{V}_k{Y}_{\mathit{ik}}cos\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)}\right]+\frac{\partial P_{{\mathrm{L}}_i}\left(V_i\right)}{\partial V_i}\right)\mathrm{\Delta }{V}_m\hfill \\ -V_i{Y}_{\mathit{ik}}cos\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)\mathrm{\Delta }{V}_1-V_i{Y}_{\mathit{ik}}cos({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}})\mathrm{\Delta }{V}_2-\cdots \hfill \\ -V_i{Y}_{\mathit{ik}}cos\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)\mathrm{\Delta }{V}_m-V_i{Y}_{\mathit{ik}}cos({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}})\mathrm{\Delta }{V}_{m+1}-\cdots \hfill \\ -V_i{Y}_{\mathit{ik}}cos\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)\mathrm{\Delta }{V}_n-V_i{V}_k{Y}_{\mathit{ik}}sin({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}})\mathrm{\Delta }{\theta }_1\hfill \\ -V_i{V}_k{Y}_{\mathit{ik}}sin\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)\mathrm{\Delta }{\theta }_2\cdots -V_i{V}_k{Y}_{\mathit{ik}}sin({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}})\mathrm{\Delta }{\theta }_{m-1}\hfill \\ -V_i{V}_k{Y}_{\mathit{ik}}sin\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)\mathrm{\Delta }{\theta }_{m+1}\cdots -V_i{V}_k{Y}_{\mathit{ik}}sin({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}})\mathrm{\Delta }{\theta }_n\hfill \end{array}$

  (5.45)

Rearranging the terms according to the variables $\mathrm{\Delta }{V}_{{\mathrm{g}}_1}\dots \mathrm{\Delta }{V}_{{\mathrm{g}}_m}$ and $\mathrm{\Delta }{V}_{{\mathrm{g}}_{m+1}}\dots \mathrm{\Delta }{V}_{{\mathrm{g}}_n}$, Equation (5.45) can be written as

$\begin{array}{l}\hfill 0=\cdots +[-V_i{V}_k{Y}_{\mathit{ik}}sin\left({\theta }_i-{\theta }_k-{\alpha }_{\mathit{ik}}\right)-V_i{Y}_{\mathit{ik\; ...}}\hfill \end{array}$