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.

Diagrama Mermaid

Building a multimachine linear system

Transformación de marco de referencia

Reference frame tranformation and associated angles between axes and variables

\[i_{dq} = R i_{IR} \rightarrow \begin{bmatrix} i_{d} \\ i_{q} \end{bmatrix} = \begin{bmatrix}-cos(\delta) && sin(\delta) \\ sin(\delta) && cos(\delta)\end{bmatrix} \begin{bmatrix} i_{I} \\i_{R} \end{bmatrix}\]

  • Linearizing: \(\Delta i_{dq} = {\frac{dR}{d \delta}}|_{\delta = \delta_0}i_{IR0} \Delta x+R \Delta i_{IR} = {\frac{dR}{d \delta}}|_{\delta = \delta_0}{R^{-1}i_{dq}}_{0} \Delta x+R \Delta i_{IR} = t_i \Delta x + R \Delta i_{IR}\)

  • Defining : \(t_i = {\frac{dR}{d \delta}}|_{\delta = \delta_0}i_{IR0} =\begin{bmatrix}sin(\delta) && cos(\delta) \\ cos(\delta) && -sin(\delta)\end{bmatrix}\begin{bmatrix}-cos(\delta) && sin(\delta) \\ sin(\delta) && cos(\delta)\end{bmatrix} i_{dq0} = \begin{bmatrix}0 && i_{q0} && 0 && \cdots && 0 \\ 0 && -i_{d0} && 0 && \cdots && 0 \end{bmatrix}\) ; \([t_i] = 2 \times n\) , being \(n\) the number of state variables.

\[v_{dq} = R v_{IR} \rightarrow \begin{bmatrix} v_{d} \\ v_{q} \end{bmatrix} = \begin{bmatrix}-cos(\delta) && sin(\delta) \\ sin(\delta) && cos(\delta)\end{bmatrix} \begin{bmatrix} v_{I} \\ v_{R} \end{bmatrix}\]
  • Linearizing : \(\Delta v_{dq} = {\frac{dR}{d \delta}}|_{\delta = \delta_0}v_{IR0} \Delta x+R \Delta v_{IR} = {\frac{dR}{d \delta}}|_{\delta = \delta_0}{R^{-1}v_{dq}}_{0} \Delta x+R \Delta v_{IR} = t_v \Delta x + R \Delta v_{IR}\)
  • Defining: \(t_v = {\frac{dR}{d \delta}}|_{\delta = \delta_0}v_{IR0} =\begin{bmatrix}sin(\delta) && cos(\delta) \\ cos(\delta) && -sin(\delta)\end{bmatrix}\begin{bmatrix}-cos(\delta) && sin(\delta) \\ sin(\delta) && cos(\delta)\end{bmatrix} v_{dq0} = \begin{bmatrix}0 && v_{q0} && 0 && \cdots && 0 \\ 0 && -v_{d0} && 0 && \cdots && 0 \end{bmatrix}\) ; \([t_v] = 2 \times n\) , being \(n\) the number of state variables.
  • State space equations : \(\Delta \dot{x}_i = A_i \Delta x_i + B_i \Delta v_{dq} \rightarrow \Delta \dot{x}_i = A_i \Delta x_i + B_i (t_v \Delta x + R \Delta v_{IR})\) ; \(\Delta i_{dq} = C_i \Delta x_i + D_i \Delta v_{dq} \rightarrow t_i \Delta x + R \Delta i_{IR} = C_i \Delta x_i + D_i (t_v \Delta x + R \Delta v_{IR})\)
\[A = A_i + B_i t_v\] \[B = B_i R\] \[C = R^{-1}(C_i + D_i t_v - t_i)\] \[D = R^{-1}D_iR\]