# Project Overview

## A Combinatorial and Geometric Approach to Studying Electrical Networks

### Department(s)

Mathematics

### Abstract

An electric power network is a network of electrical components for generating, transferring, and consuming electric energy. Power networks are of fundamental importance to every aspect of modern life, yet they still lack global stability and are inherently fragile in the sense that sufficiently large disturbances may cause the loss of stability triggering cascading failures. At the heart of stability analysis lies the mathematical problem of understanding power-flow equations: systems of nonlinear equations that describe the intricate balancing conditions of electric power in power system. Despite many decades of active research, the rigorous analysis of power-flow equations is still a difficult and computationally intensive task. Part of the difficulty lies in the nonlinearity of the equations, and, as a result, there is generally more than one solution. Computational methods exist for finding the solutions, but these often rely on knowing how many solutions there are in the first place.

To count solutions to equations, methods from discrete mathematics -- namely, graph theory and polyhedral geometry -- have proven very useful. In this project, students will closely study the discrete side of this research area, examining the connections between the polyhedra and the underlying power networks.

### Student Qualifications

Grade of B+ or better in MATH 250. Recommended, but not required: coding experience; a grade of B+ or better in MATH 310, MATH 360, or COSC 290, or other experience in discrete mathematics.

### Number of Student Researchers

2 students

### Project Length

8 weeks
Applications open on 01/03/2022 and close on 02/04/2022