# Tight-and-cheap conic relaxation for the AC optimal power flow problem

## TL;DR

This paper introduces a new conic relaxation method for the AC optimal power flow problem that balances solution tightness and computational efficiency, outperforming existing relaxations in large-scale tests.

## Contribution

It combines semidefinite relaxation with RLT to create a relaxation that is stronger than second-order cone but nearly as tight as semidefinite, with significantly improved computational speed.

## Key findings

- The new relaxation is nearly as tight as the standard semidefinite relaxation.
- It reduces solution time by up to an order of magnitude compared to chordal relaxation.
- Applicable to large-scale power networks with thousands of buses.

## Abstract

The classical alternating current optimal power flow problem is highly nonconvex and generally hard to solve. Convex relaxations, in particular semidefinite, second-order cone, convex quadratic, and linear relaxations, have recently attracted significant interest. The semidefinite relaxation is the strongest among them and is exact for many cases. However, the computational efficiency for solving large-scale semidefinite optimization is lower than for second-order cone optimization. We propose a conic relaxation obtained by combining semidefinite optimization with the reformulation-linearization technique, commonly known as RLT. The proposed relaxation is stronger than the second-order cone relaxation and nearly as tight as the standard semidefinite relaxation. Computational experiments using standard test cases with up to 6515 buses show that the time to solve the new conic relaxation is up to one order of magnitude lower than for the chordal relaxation, a semidefinite relaxation technique that exploits the sparsity of power networks.