# Convex Hull of the Quadratic Branch AC Power Flow Equations and Its Application in Radial Distribution Networks

## TL;DR

This paper develops a tight convex relaxation for the quadratic equations in AC power flow models using convex hull formulations, and applies it to optimize distributed energy storage in radial distribution networks with high PV penetration.

## Contribution

It introduces the convex hull of quadratic branch flow equations and extends this to the DES optimal scheduling problem in radial networks.

## Key findings

- Convex hull formulation is analytically proved and geometrically validated.
- The approach yields a tight convex relaxation for the DES scheduling problem.
- Tests on radial systems demonstrate improved solution quality and computational efficiency.

## Abstract

A branch flow model (BFM) is used to formulate the AC power flow in general networks. For each branch/line, the BFM contains a nonconvex quadratic equality. A mathematical formulation of its convex hull is proposed, which is the tightest convex relaxation of this quadratic equation. The convex hull formulation consists of a second order cone inequality and a linear inequality within the physical bounds of power flows. The convex hull formulation is analytically proved and geometrically validated. An optimal scheduling problem of distributed energy storage (DES) in radial distribution systems with high penetration of photovoltaic resources is investigated in this paper. To capture the performance of both the battery and converter, a second-order DES model is proposed. Following the convex hull of the quadratic branch flow equation, the convex hull formulation of the nonconvex constraint in the DES model is also derived. The proposed convex hull models are used to generate a tight convex relaxation of the DES optimal scheduling (DESOS) problem. The proposed approach is tested on several radial systems. A discussion on the extension to meshed networks is provided.

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Source: https://tomesphere.com/paper/1701.07146