This post follows the method proposed in “Power System Stability and Control” by Prabha S. Kundur and Om P. Malik.

In p.u.

We are looking for a representation :

\[\Delta \dot{x}_i = A_i \Delta x_i + B_i \Delta v\] \[\Delta i_i = C_i \Delta x_i + D_i \Delta v\]

Where :

\[\Delta v = \begin{bmatrix}\Delta v_{d} \\\Delta v_{q} \end{bmatrix}\] \[\Delta i_i = \begin{bmatrix}\Delta i_{d} \\\Delta i_{q} \end{bmatrix}\]

In order to obtain the state-space representation matrices \(A_i\) , \(B_i\) , \(C_i\) and \(D_i\) , we consider the stator voltage equations and the equations of motion.

The stator voltage equations

\[v_d= -\psi_q -R_a i_d\] \[v_q = \psi_d -R_a i_q\]

Where \(\psi_d = -L_di_d + L_{ad} i_{fd}\) and \(\psi_q = -L_q i_q\) .

If we linearize the equations, considering \(i_d\) and \(i_q\) as the only inputs ( \(i_{fd} = cte \rightarrow \Delta i_{fd} = 0\) ):

\[\Delta v_d = \frac {\delta v_d}{\delta iq}|_{t=0} \Delta i_q + \frac {\delta v_d}{\delta id}|_{t=0} \Delta i_d\] \[\Delta v_q = \frac {\delta v_q}{\delta iq}|_{t=0} \Delta i_q + \frac {\delta v_q}{\delta id}|_{t=0} \Delta i_d\]

Considering

\[L_d = L_q \rightarrow \begin{bmatrix}\Delta v_{d} \\\Delta v_{q} \end{bmatrix} = \begin{bmatrix}-R_a && L\\\ -L && -R_a\end{bmatrix}\begin{bmatrix}\Delta i_{d} \\ \Delta i_{q} \end{bmatrix}\] \[\rightarrow \begin{bmatrix}\Delta i_{d} \\\Delta i_{q} \end{bmatrix} = \begin{bmatrix}-R_a && L \\\ -L && -R_a\end{bmatrix}^{-1} \begin{bmatrix}\Delta v_{d} \\\Delta v_{q} \end{bmatrix}\]

Hence :

\[C_i = - \begin{bmatrix}0 && 0\\\ 0 &&0\end{bmatrix}\] \[D_i = \begin{bmatrix}-R_a && L \\\ -L && -R_a\end{bmatrix}^{-1}\]

The equations of motion

\(\Delta \delta = \frac {1}{\omega_0} \frac{d \Delta \delta}{dt}\)

The swing equation: \(T_a = T_m - T_e\)

It is often desirable to include a component of damping torque \(K_D\) , not accounted for the calculation of \(T_e\) , separately.

\[2H \frac{d \Delta \omega_r}{dt} = \Delta T_m - \Delta T_e - K_D \Delta \omega_r\]

If we linearize the previous equation :

\[\frac{d \Delta \omega_r}{dt} =\frac {1}{2H}[\Delta T_m - \Delta T_e - K_D \Delta \omega_r]\]

\(\Delta T_m = 0\) : as we consider as inputs \(i_d\) and \(i_q\) .
Air-gap torque : \(T_e = \psi_d i_q - \psi_q i_d = (-L_di_d + L_{ad} i_{fd})i_q - L_q i_q i_d\)
- If \(L_d = L_q\) and \(L_{ad} i_{fd} = e_0 \rightarrow T_e = e_0 i_q \rightarrow \Delta T_e = e_0 \Delta i_q\)

We arrange the equations in the form of matrices and we get :

\[\begin{bmatrix}\Delta \dot\omega_r \\ \Delta \dot\delta \end{bmatrix} = \begin{bmatrix} \frac {- K_D}{2H} && 0 \\\ \omega_0 && 0\end{bmatrix}\begin{bmatrix}\Delta \omega_r \\ \Delta \delta \end{bmatrix} + \begin{bmatrix} 0 && -\frac{e_{0}}{2H} \\\ 0 && 0\end{bmatrix} \begin{bmatrix}\Delta i_{d} \\\Delta i_{q} \end{bmatrix}\]

Taking into account the stator voltage equations :

\[\Delta i_i = C_i \Delta x_i + D_i \Delta v\] \[A_i = \begin{bmatrix} \frac {- K_D}{2H} && 0 \\\ \omega_0 && 0\end{bmatrix} + \begin{bmatrix} 0 && -\frac{e_{0}}{2H} \\\ 0 && 0\end{bmatrix} C_i\] \[B_i =\begin{bmatrix} 0 && -\frac{e_{0}}{2H} \\\ 0 && 0\end{bmatrix} D_i\]