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}
A_{11} & A_{12} \\
A_{21} & A_{22} \\
\end{bmatrix}
\times
\begin{bmatrix}
\Delta \theta\\
\Delta |V|\\
\end{bmatrix}
=
\begin{bmatrix}
Rhs_1\\
Rhs_2\\
\end{bmatrix}

Where:

  • A_{11} = -Im\left(Y_{series}[pqpv, pqpv]\right)
  • A_{12} = Re\left(Y_{bus}[pqpv, pq]\right)
  • A_{21} = -Im\left(Y_{series}[pq, pqpv]\right)
  • A_{22} = -Re\left(Y_{bus}[pq, pq]\right)
  • Rhs_1 = P[pqpv]
  • Rhs_2 = Q[pq]

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.

Solving the vector [\Delta \theta + 0, \Delta |V| + 1] we get \theta for the pq and pv nodes and |V| for the pq nodes.

For equivalence with the paper:

  • -B' = -Im(Y_{series}[pqpv, pqpv])
  • G = Re(Y_{bus}[pqpv, pq])
  • -G' = -Im(Y_{series}[pq, pqpv])
  • -B = -Re(Y_{bus}[pq, pq])
[1]Rossoni, P. / Moreti da Rosa, W. / Antonio Belati, E., Linearized AC Load Flow Applied to Analysis in Electric Power Systems, IEEE Latin America Transactions, 14, 9; 4048-4053, 2016