# A Julia Module for Polynomial Optimization with Complex Variables applied to Optimal Power Flow

## TL;DR

This paper introduces a Julia module that directly handles polynomial optimization problems with complex variables, specifically applied to power system optimization tasks like optimal power flow, preserving the problem's original complex structure.

## Contribution

The novel Julia module allows direct representation of complex polynomial problems, improving modeling flexibility for power systems optimization.

## Key findings

- Effective application to AC optimal power flow
- Successful modeling of security-constrained power flow
- Demonstrated versatility for various power system problems

## Abstract

Many optimization problems in power transmission networks can be formulated as polynomial problems with complex variables. A polynomial optimization problem with complex variables consists in optimizing a real-valued polynomial whose variables and coefficients are complex numbers subject to some complex polynomial equality or inequality constraints. These problems are usually directly converted to real variables, either using the polar form or the rectangular form. In this work, we propose a Julia module allowing the representation of polynomial problems in their original complex formulation. This module is applied to power systems optimization and its generic design enables the description of several variants of power system problems. Results for the Optimal Power Flow in Alternating Current problem and for the Preventive-Security Constrained Optimal Power Flow problem are presented.