# Linear Optimal Power Flow Using Cycle Flows

## TL;DR

This paper introduces a new cycle flow formulation for linear optimal power flow problems, significantly improving computational efficiency especially in large, decentralized networks with renewable energy sources.

## Contribution

The paper presents a novel cycle flow-based formulation for LOPF that outperforms traditional angle-based methods in computational speed, especially in complex, large-scale networks.

## Key findings

- Cycle flow formulation solves up to 7 times faster than angle formulation.
- Existing cycle-based methods can be up to 20 times faster.
- Speed-up increases with network size and decentralization.

## Abstract

Linear optimal power flow (LOPF) algorithms use a linearization of the alternating current (AC) load flow equations to optimize generator dispatch in a network subject to the loading constraints of the network branches. Common algorithms use the voltage angles at the buses as optimization variables, but alternatives can be computationally advantageous. In this article we provide a review of existing methods and describe a new formulation that expresses the loading constraints directly in terms of the flows themselves, using a decomposition of the network graph into a spanning tree and closed cycles. We provide a comprehensive study of the computational performance of the various formulations, in settings that include computationally challenging applications such as multi-period LOPF with storage dispatch and generation capacity expansion. We show that the new formulation of the LOPF solves up to 7 times faster than the angle formulation using a commercial linear programming solver, while another existing cycle-based formulation solves up to 20 times faster, with an average speed-up of factor 3 for the standard networks considered here. If generation capacities are also optimized, the average speed-up rises to a factor of 12, reaching up to factor 213 in a particular instance. The speed-up is largest for networks with many buses and decentral generators throughout the network, which is highly relevant given the rise of distributed renewable generation and the computational challenge of operation and planning in such networks.