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The DC Power Flow Approximation

Many network matrices in PowerNetworkMatrices.jl rely on the DC power flow approximation. It linearizes the AC power flow equations into a purely algebraic relationship between active-power injections and voltage angles, which is what makes the sensitivity matrices (PTDF, LODF) fast to build and cheap to reason about.

Assumptions

  1. Voltage magnitude: all bus voltages are approximately 1.0 per unit
  2. Small angles: voltage angle differences are small (< 15°), so $\sin(\theta_i - \theta_j) \approx \theta_i - \theta_j$
  3. Resistance: line resistance is negligible compared to reactance
  4. Active power: only active power flows are considered

Under these assumptions the branch flow becomes a linear function of the angle difference, $P_{ij} \approx (\theta_i - \theta_j)/X_{ij}$, and the whole network collapses to the susceptance-weighted graph Laplacian that the BA_Matrix and ABA_Matrix encode.

When the DC approximation works well

  • Transmission systems (high voltage)
  • Normal operating conditions
  • Security and market analysis
  • Planning studies

When to be cautious

  • Distribution systems (high R/X ratios)
  • Large angle differences
  • Voltage-constrained systems
  • Detailed reactive power analysis

References

For the DC power-flow approximation and its assumptions see B. Stott, J. Jardim, and O. Alsaç, "DC Power Flow Revisited," IEEE Transactions on Power Systems, vol. 24, no. 3, pp. 1290–1300, 2009; and A. J. Wood, B. F. Wollenberg, and G. B. Sheblé, Power Generation, Operation, and Control, 3rd ed., Wiley, 2013. See also https://en.wikipedia.org/wiki/Power-flow_study.