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.

In p.u. We are looking for a representation :

Δ {\dot{x}}_{i} = A_{i} Δ x_{i} + B_{i} Δ v

Δ i_{i} = C_{i} Δ x_{i} + D_{i} Δ v

Where : $Δ v = [\begin{matrix} Δ v_{d} \\ Δ v_{q} \end{matrix}]$ ; $Δ i_{i} = [\begin{matrix} Δ i_{d} \\ Δ i_{q} \end{matrix}]$ . 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} = - ψ_{q} - R_{a} i_{d}

v_{q} = ψ_{d} - R_{a} i_{q}

Where $ψ_{d} = - L_{d} i_{d} + L_{a d} i_{f d}$ and $ψ_{q} = - L_{q} i_{q}$ .
If we linearize the equations, considering $i_{d}$ and $i_{q}$ as the only inputs ( $i_{f d} = c t e \to Δ i_{f d} = 0$ ):

Δ v_{d} = \frac{δ v_{d}}{δ i q} |_{t = 0} Δ i_{q} + \frac{δ v_{d}}{δ i d} |_{t = 0} Δ i_{d}

Δ v_{q} = \frac{δ v_{q}}{δ i q} |_{t = 0} Δ i_{q} + \frac{δ v_{q}}{δ i d} |_{t = 0} Δ i_{d}

Considering $L_{d} = L_{q} \to [\begin{matrix} Δ v_{d} \\ Δ v_{q} \end{matrix}] = [\begin{matrix} - R_{a} & L \\ - L & - R_{a} \end{matrix}] [\begin{matrix} Δ i_{d} \\ Δ i_{q} \end{matrix}]$
- $\to [\begin{matrix} Δ i_{d} \\ Δ i_{q} \end{matrix}] = {[\begin{matrix} - R_{a} & L \\ - L & - R_{a} \end{matrix}]}^{- 1} [\begin{matrix} Δ v_{d} \\ Δ v_{q} \end{matrix}]$
- Hence :

C_{i} = - [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]

D_{i} = {[\begin{matrix} - R_{a} & L \\ - L & - R_{a} \end{matrix}]}^{- 1}

The equations of motion

$Δ δ = \frac{1}{ω_{0}} \frac{d Δ δ}{d t}$

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.

2 H \frac{d Δ ω_{r}}{d t} = Δ T_{m} - Δ T_{e} - K_{D} Δ ω_{r}

If we linearize the previous equation : $\frac{d Δ ω_{r}}{d t} = \frac{1}{2 H} [Δ T_{m} - Δ T_{e} - K_{D} Δ ω_{r}]$
- $Δ T_{m} = 0$ : as we consider as inputs $i_{d}$ and $i_{q}$ .
- Air-gap torque : $T_{e} = ψ_{d} i_{q} - ψ_{q} i_{d} = (- L_{d} i_{d} + L_{a d} i_{f d}) i_{q} - L_{q} i_{q} i_{d}$
  - If $L_{d} = L_{q}$ and $L_{a d} i_{f d} = e_{0} \to T_{e} = e_{0} i_{q} \to Δ T_{e} = e_{0} Δ i_{q}$
We arrange the equations in the form of matrices and we get :
$[\begin{matrix} Δ {\dot{ω}}_{r} \\ Δ \dot{δ} \end{matrix}] = [\begin{matrix} \frac{- K_{D}}{2 H} & 0 \\ ω_{0} & 0 \end{matrix}] [\begin{matrix} Δ ω_{r} \\ Δ δ \end{matrix}] + [\begin{matrix} 0 & - \frac{e_{0}}{2 H} \\ 0 & 0 \end{matrix}] [\begin{matrix} Δ i_{d} \\ Δ i_{q} \end{matrix}]$
Taking into account the stator voltage equations : $Δ i_{i} = C_{i} Δ x_{i} + D_{i} Δ v$

A_{i} = [\begin{matrix} \frac{- K_{D}}{2 H} & 0 \\ ω_{0} & 0 \end{matrix}] + [\begin{matrix} 0 & - \frac{e_{0}}{2 H} \\ 0 & 0 \end{matrix}] C_{i}

B_{i} = [\begin{matrix} 0 & - \frac{e_{0}}{2 H} \\ 0 & 0 \end{matrix}] D_{i}