WOLF-I Image 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.

In the upcoming blog posts, I will detail the systematic procedure for constructing the linear model of an electrical system. This process follows the method proposed in “Power System Stability and Control” by Prabha S. Kundur and Om P. Malik.

  • The linearized model of each dynamic device is expressed in the following form :
\[\Delta \dot{x}_i = A_i \Delta x_i + B_i \Delta v_i\] \[\Delta i_i = C_i \Delta x_i + D_i \Delta v_i\]
  • Where:
    • \(\Delta x_i\) : perturbed values of the individual device state variables.
    • \(\Delta i_i\) : perturbed value of the current injection into the network from the device.
    • \(\Delta v_i\) : perturbed value of the network bus voltages.
  • Such state equations for all the dynamic devices in the system may be combined into the form :
\[\Delta \dot{x} = A_D \Delta x + B_D \Delta v\] \[\Delta i = C_D \Delta x + D_D \Delta v\]
  • Where :
    • \(\Delta x\) : state vector of the complete system.
    • \(A_D\) , \(B_D\) , \(C_D\) and \(D_D\) : block diagonal matrices composed of \(A_i\) , \(B_i\) , \(C_i\) and \(D_i\) matrices respectively, associated with the individual devices.
  • The interconnecting transmission network is represented by the node equation : \(\Delta i = Y_N \Delta v\)
    • The elements of \(Y_N\) include the effects of nonlinear static loads.
  • Hence : \(C_D \Delta x + D_D \Delta v = Y_N \Delta v \rightarrow \Delta v = (Y_N - D_D)^{-1} C_D x \rightarrow \Delta \dot{x} = A_D x + B_D (Y_N - D_D)^{-1} C_D x = Ax\)
\[\rightarrow A = A_D + B_D (Y_N - D_D)^{-1} C_D\]