Linear AC Power Flow

Following the formulation presented in Linearized AC Load Flow Applied to Analysis in Electric Power Systems [1], we obtain a way to solve circuits in one shot (without iterations) and with quite positive results for a linear approximation.

$\begin{bmatrix} -Bs_{pqpv, pqpv} & G_{pqpv, pq} \\ -Gs_{pq, pqpv} & -B_{pq, pq} \\ \end{bmatrix} \times \begin{bmatrix} \Delta \theta_{pqpv} \\ \Delta |V|_{pq}\\ \end{bmatrix} = \begin{bmatrix} P_{pqpv}\\ Q_{pq}\\ \end{bmatrix}$

Where:

$G = Re\left(Y_{bus}\right)$
$B = Re\left(Y_{bus}\right)$
$Gs = Im\left(Y_{series}\right)$
$Bs = Im\left(Y_{series}\right)$

Here, $Y_{bus}$ is the normal circuit admittance matrix and $Y_{series}$ is the admittance matrix formed with only series elements of the $\pi$ model, this is neglecting all the shunt admittances.

After the voltage delta computations, we obtain the final voltage vector by:

Copy the initial voltage: $V = V_0$ . This copies the slack values.
Set the voltage module values for the pq nodes: $|V|_{pq} = 1 - \Delta |V|_{pq}$
Copy the voltage module values for the pq nodes: $|V|_{pv} = |V_0|_{pv}$
Set the voltage angle for the pq and pv nodes: $\theta_{pqpv} = \Delta \theta_{pqpv}$

This last part has not been explained in the paper but is is necessary for the adequate performance of the method. For equivalence with the paper [1]:

$-B' = -Bs$
$-G' = -Gs$