Welcome to the WOLF-I project page. This project focuses on studying and analyzing the role of IBRs in the damping of inter-area oscillations. Our goal is to understand the dynamics of these oscillations and develop strategies and tools to mitigate their impact on power systems.

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 rotor circuit equations

\[p \psi_{fd} = \frac{\omega_0 R_{fd}}{X_{ad}} E_{fd} - \omega_0 R_{fd} i_{fd}\] \[p \psi_{1d} = -\omega_0 R_{1d} i_{1d}\] \[p \psi_{1q} = -\omega_0 R_{1q} i_{1q}\] \[p \psi_{2q} = -\omega_0 R_{2q} i_{2q}\]

The rotor currents are given by : \(i_{fd} = \frac{1}{X_{fd}}(\psi_{fd} - \psi_{ad})\)
\[i_{1d} = \frac{1}{X_{1d}}(\psi_{1d} - \psi_{ad})\] \[i_{1q} = \frac{1}{X_{1q}}(\psi_{1q} - \psi_{aq})\] \[i_{2q} = \frac{1}{X_{2q}}(\psi_{2q} - \psi_{aq})\]
The d- and q-axis mutual flux linkages are given by: \(\psi_{ad} = -X_{ad}i_d + X_{ad}i_{fd}+X_{ad}i_{1d} = xaux_d (-i_d + \frac{\psi_{fd}}{X_{fd}}+\frac{\psi_{1d}}{X_{1d}})\)
\[\psi_{aq} = -X_{aq}i_q + X_{aq}i_{1q}+X_{aq}i_{2q} = xaux_q (-i_q + \frac{\psi_{1q}}{X_{1q}}+\frac{\psi_{2q}}{X_{2q}})\] \[xaux_d = \frac{1}{\frac{1}{X_{ad}} + \frac{1}{X_{fd}}+ \frac{1}{X_{1d}}}\] \[xaux_q = \frac{1}{\frac{1}{X_{aq}} + \frac{1}{X_{1q}}+ \frac{1}{X_{2q}}}\]
Linearizing the previous equations and considering \(\Delta E_{fd} = 0\) :
\[\Delta \psi_{ad} = xaux_d (-\Delta i_d + \frac{\Delta \psi_{fd}}{X_{fd}}+\frac{\Delta \psi_{1d}}{X_{1d}})\] \[\Delta \psi_{aq} = xaux_q (-\Delta i_q + \frac{\Delta \psi_{1q}}{X_{1q}}+\frac{\Delta \psi_{2q}}{X_{2q}})\] \[\Delta i_{fd} = \frac{1}{X_{fd}}(\Delta \psi_{fd} - \Delta \psi_{ad}) = \frac{1}{X_{fd}}(\Delta \psi_{fd} - xaux_d (-\Delta i_d + \frac{\Delta \psi_{fd}}{X_{fd}}+\frac{\Delta \psi_{1d}}{X_{1d}}))\] \[\Delta i_{1d} = \frac{1}{X_{1d}}(\Delta \psi_{1d} - \Delta \psi_{ad}) = \frac{1}{X_{1d}}(\Delta \psi_ {1d} - xaux_d (-\Delta i_d + \frac{\Delta \psi_{fd}}{X_{fd}}+\frac{\Delta \psi_{1d}}{X_{1d}}))\] \[\Delta i_{1q} = \frac{1}{X_{1q}}(\Delta \psi_{1q} - \Delta \psi_{aq}) = \frac{1}{X_{1q}}(\Delta \psi_{1q} - xaux_q (-\Delta i_q + \frac{\Delta \psi_{1q}}{X_{1q}}+\frac{\Delta \psi_{2q}}{X_{2q}}))\] \[\Delta i_{2q} = \frac{1}{X_{2q}}(\Delta \psi_{2q} - \Delta \psi_{aq}) = \frac{1}{X_{2q}}(\Delta \psi_{2q} - xaux_q (-\Delta i_q + \frac{\Delta \psi_{1q}}{X_{1q}}+\frac{\Delta \psi_{2q}}{X_{2q}}))\] \[\Delta \dot \psi_{fd} = \frac{\omega_0 R_{fd}}{X_{ad}} \Delta E_{fd} - \omega_0 R_{fd} \Delta i_{fd} = -\omega_0 R_{fd} \frac{1}{X_{fd}}(\Delta \psi_{fd} - xaux_d (-\Delta i_d + \frac{\Delta \psi_{fd}}{X_{fd}}+\frac{\Delta \psi_{1d}}{X_{1d}}))\] \[\Delta \dot \psi_{1d} = -\omega_0 R_{1d} \Delta i_{1d} = -\omega_0 R_{1d} \frac{1}{X_{1d}}(\Delta \psi_{1d} - xaux_d (-\Delta i_d + \frac{\Delta \psi_{fd}}{X_{fd}}+\frac{\Delta \psi_{1d}}{X_{1d}}))\] \[\Delta \dot \psi_{1q} = -\omega_0 R_{1q} \Delta i_{1q} = -\omega_0 R_{1q} \frac{1}{X_{1q}}(\Delta \psi_{1q} - xaux_q (-\Delta i_q + \frac{\Delta \psi_{1q}}{X_{1q}}+\frac{\Delta \psi_{2q}}{X_{2q}}))\] \[\Delta \dot \psi_{2q} = -\omega_0 R_{2q} \Delta i_{2q} = -\omega_0 R_{2q}\frac{1}{X_{2q}}(\Delta \psi_{2q} - xaux_q (-\Delta i_q + \frac{\Delta \psi_{1q}}{X_{1q}}+\frac{\Delta \psi_{2q}}{X_{2q}}))\]

The stator voltage equations

\[v_d = -R_a i_d + X_l i_q - \psi_{ad}\] \[v_q = -R_a i_q - X_l i_d + \psi_{aq}\]

Linearizing the previous equations:

\(\Delta v_d = -R_a \Delta i_d + X_l \Delta i_q - \Delta \psi_{aq} = -R_a \Delta i_d + X_l \Delta i_q - xaux_q (-\Delta i_q + \frac{\Delta \psi_{1q}}{X_{1q}}+\frac{\Delta \psi_{2q}}{X_{2q}})\) \(\Delta v_q = -R_a \Delta i_q - X_l \Delta i_d + \Delta \psi_{ad} = -R_a \Delta i_q - X_l \Delta i_d + xaux_d (-\Delta i_d + \frac{\Delta \psi_{fd}}{X_{fd}}+\frac{\Delta \psi_{1d}}{X_{1d}})\)

The swing equation

\[p \omega_r = \frac{1}{2H}(T_m - T_e)\] \[T_e = \psi_d i_q - \psi_q i_d = \psi_{ad} i_q - \psi_{aq} i_d\]

Linearizing the previous equations and adding a term to account for the damping:
\[\Delta T_e = i_{q0} \Delta \psi_{ad} + \psi_{ad0} \Delta i_q - i_{d0} \Delta \psi_{aq} - \psi_{aq0} \Delta i_d = i_{q0}xaux_d (-\Delta i_d + \frac{\Delta \psi_{fd}}{X_{fd}}+\frac{\Delta \psi_{1d}}{X_{1d}})+ \psi_{ad0} \Delta i_q - i_{d0}xaux_q (-\Delta i_q + \frac{\Delta \psi_{1q}}{X_{1q}}+\frac{\Delta \psi_{2q}}{X_{2q}}) - \psi_{aq0} \Delta i_d\] \[\Delta \dot \omega_r = \frac{1}{2H}(\Delta T_m - \Delta T_e - K_D \Delta \omega_r)\]

Arranging the previous equations to assemble the state space equations:

\[a_{11} = -\frac{K_D}{2H}\] \[a_{12} = 0\] \[a_{13} = - \frac{1}{2H} \frac{i_{q0}xaux_d}{X_{fd}}\] \[a_{14} = -\frac{1}{2H}\frac{i_{q0}xaux_d}{X_{1d}}\] \[a_{15} = -\frac{1}{2H} \frac {-i_{d0} xaux_q}{X_{1q}}\] \[a_{16} = -\frac{1}{2H} \frac{-i_{d0}xaux_q}{X_{2q}}\] \[a_{21} = \omega_0\] \[a_{22} = 0\] \[a_{23} = 0\] \[a_{24} = 0\] \[a_{25} = 0\] \[a_{26} = 0\] \[a_{31} = 0\] \[a_{32} = 0\] \[a_{33} = -\omega_0 R_{fd} \frac{1}{X_{fd}}(1-\frac{xaux_d}{X_{fd}})\] \[a_{34} = -\omega_0 R_{fd} \frac{1}{X_{fd}}(-\frac{xaux_d}{X_{1d}})\] \[a_{35} = 0\] \[a_{36} = 0\] \[a_{41} = 0\] \[a_{42} = 0\] \[a_{43} = -\omega_0 R_{1d} \frac{1}{X_{1d}}(- \frac{xaux_d}{X_{fd}})\] \[a_{44} = -\omega_0 R_{1d} \frac{1}{X_{1d}}(1- \frac{xaux_d}{X_{1d}})\] \[a_{45} = 0\] \[a_{46} =0\] \[a_{51} =0\] \[a_{52} = 0\] \[a_{53} = 0\] \[a_{54} = 0\] \[a_{55} = -\omega_0 R_{1q} \frac{1}{X_{1q}}(1 - \frac{xaux_q}{x_{1q}})\] \[a_{56} = -\omega_0 R_{1q} \frac{1}{X_{1q}}(- \frac{xaux_q}{x_{2q}})\] \[a_{61} = 0\] \[a_{62} =0\] \[a_{63} = 0\] \[a_{64} = 0\] \[a_{65} = -\omega_0 R_{2q}\frac{1}{X_{2q}} (\frac{-xaux_q}{X_{1q}})\] \[a_{66} = -\omega_0 R_{2q}\frac{1}{X_{2q}} (1 - \frac{xaux_q}{X_{2q}})\] \[b_{11} = \frac{-1}{2H} (-\psi_{aq0} - i_{q0}xaux_d)\] \[b_{12} = \frac{-1}{2H} (\psi_{ad0} + i_{d0}xaux_q)\] \[b_{21} = 0\] \[b_{22} = 0\] \[b_{31} = -\omega_0 R_{fd} \frac{xaux_d}{X_{fd}}\] \[b_{32} = 0\] \[b_{41} = -\omega_0 R_{1d} \frac{xaux_d}{X_{1d}}\] \[b_{42} = 0\] \[b_{51} = 0\] \[b_{52} = -\omega_0 R_{1q} \frac{xaux_q}{X_{1q}}\] \[b_{61} = 0\] \[b_{62} = -\omega_0 R_{2q} \frac{xaux_q}{X_{2q}}\] \[c_{11} = 0\] \[c_{12} = 0\] \[c_{13} = 0\] \[c_{14} = 0\] \[c_{15} = \frac{- xaux_q}{x_{1q}}\] \[c_{16} = \frac{-xaux_q}{x_{2q}}\] \[c_{21} = 0\] \[c_{22} = 0\] \[c_{23} = \frac{xaux_d}{X_{fd}}\] \[c_{24} = \frac{xaux_d}{X_{1d}}\] \[c_{25} = 0\] \[c_{26} = 0\] \[d_{11} = -R_a\] \[d_{12} = X_l + xaux_q\] \[d_{21} = -X_l - xaux_d\] \[d_{22} = -R_a\]