Branch Flow Model: Relaxations and Convexification (Parts I, II)

TL;DR

This paper introduces a branch flow model for power networks, providing relaxation techniques and convexification methods that enable efficient and exact solutions for optimal power flow problems, especially in radial and certain mesh networks.

Contribution

It presents a novel branch flow model with two relaxation steps and a convexification approach using phase shifters, ensuring efficient and exact solutions for specific network types.

Findings

Relaxation steps are always exact for radial networks without load bounds.

Conic relaxation is always exact for mesh networks; angle relaxation may not be.

Convexification with phase shifters depends only on network topology.

Abstract

We propose a branch flow model for the anal- ysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.