Notes on BIM and BFM Optimal Power Flow With Parallel Lines and Total Current Limits

TL;DR

This paper compares BIM and BFM models for optimal power flow in meshed networks, introduces new constraints to improve relaxations, and demonstrates their effectiveness on benchmark test cases.

Contribution

It shows BIM's dominance over BFM in meshed networks and proposes novel constraints to enhance second-order cone relaxations.

Findings

BIM dominates BFM in meshed networks with parallel lines.

New constraints tighten the relaxation gap in benchmark tests.

Proposed methods show limited but consistent improvements.

Abstract

The second-order cone relaxation of the branch flow model (BFM) and bus injection model (BIM) variants of optimal power flow are well-known to be equivalent for radial networks. In this work we show that in meshed networks with parallel lines, BIM dominates BFM, and propose novel constraints to make them equivalent in general. Furthermore, we develop an improvement to the second-order cone relaxations of optimal power flow, adding novel and valid linear constraints on the lifted current expressions. We develop two simple test cases to highlight the advantages of the proposed constraints. These novel constraints tighten the second-order cone relaxation gap on test cases in the `PG Lib' optimal power flow benchmark library, albeit generally in limited fashion.