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Unified algebraic deviation of distribution factors in linear power flow

Joost van Dijk, Nico Westerbeck, Lars Schewe, Andrea Benigni, Dirk Witthaut

TL;DR

This paper develops a unified algebraic framework for distribution factors in linear power flow, tying together line modifications, outages, closings, and bus merges/splits. By representing topology changes as low-rank updates to the grounded Laplacian and applying the Woodbury matrix identity, it derives compact, update-friendly expressions for $PTDF$ and $LODF$ that generalize to multiple simultaneous modifications. The approach enables efficient $n-1$ security analyses and islanding detection, and it integrates phase-shifting transformers via equivalent injections or Phase Shifter Distribution Factors. The resulting methodology is flexible, scalable to complex topology reconfigurations, and poised to enhance topology optimization and real-time grid operation, with potential extensions to reactive power and voltage stability.

Abstract

Distribution factors are indispensable tools in the design and analysis of power transmission grids. Recently, they received a renewed interest in the field of topology optimization, leading to the definition of bus merge and bus split distribution factors. In this article, we introduce a unified derivation of the most relevant distribution factors based on matrix algebraic manipulations. This approach facilitates the generalization to more complex grid modification, in particular simultaneous switching events or bus splits.

Unified algebraic deviation of distribution factors in linear power flow

TL;DR

This paper develops a unified algebraic framework for distribution factors in linear power flow, tying together line modifications, outages, closings, and bus merges/splits. By representing topology changes as low-rank updates to the grounded Laplacian and applying the Woodbury matrix identity, it derives compact, update-friendly expressions for and that generalize to multiple simultaneous modifications. The approach enables efficient security analyses and islanding detection, and it integrates phase-shifting transformers via equivalent injections or Phase Shifter Distribution Factors. The resulting methodology is flexible, scalable to complex topology reconfigurations, and poised to enhance topology optimization and real-time grid operation, with potential extensions to reactive power and voltage stability.

Abstract

Distribution factors are indispensable tools in the design and analysis of power transmission grids. Recently, they received a renewed interest in the field of topology optimization, leading to the definition of bus merge and bus split distribution factors. In this article, we introduce a unified derivation of the most relevant distribution factors based on matrix algebraic manipulations. This approach facilitates the generalization to more complex grid modification, in particular simultaneous switching events or bus splits.

Paper Structure

This paper contains 15 sections, 3 theorems, 58 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Literature overview
  3. Background and Notation
  4. Single branch modifications
  5. Branch modifications and line outages
  6. Phase shifting transformers
  7. Bus Merge Distribution Factors
  8. Bus Split Distribution Factors
  9. Alternative treatment of bus splits
  10. Detecting islanding
  11. Multiple branch modifications
  12. Finite topology modifications and line outages
  13. Bus Merge Distribution Factors
  14. Bus Split Distribution Factors
  15. Summary and Outlook

Key Result

Lemma 1

The inverse grounded Laplacian of the grid with open coupler is given by

Figures (1)

  • Figure 1: A bus split in an elementary test grid. Bus number 5 is split into the two buses 5 and 7 (marked in red color). The three columns show the different grid configurations used in the derivation of the bus split distribution factors: (a) the 'merged' configuration before the split, (b) the 'closed' configuration before the split, and (c) the 'open' configuration after the split. The bottom row shows the inverse of the grounded Laplacian in the three different configurations ${\boldsymbol{B}}_{m,c,o}^{-1}$. We use the test grid case6ww from the software Matpower 8.0 zimmerman2010matpower which is based on an example in wood2013power.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof