Shriya V. Nagpal | Searcharxiv

arXiv2025

Synchronization in random networks of identical phase oscillators: A graphon approach

Networks of coupled nonlinear oscillators have been used to model circadian rhythms, flashing fireflies, Josephson junction arrays, high-voltage electric grids, and many other kinds of self-organizing systems. Recently, several authors have sought to understand how coupled oscillators behave when they interact according to a random graph. Here we consider interaction networks generated by a graphon model known as a $W$-random network, and examine the dynamics of an infinite number of identical phase oscillators. We show that with sufficient regularity on $W$, the solution to the dynamical system over a $W$-random network of size $n$ converges in the $L^{\infty}$ norm to the solution of the infinite graphon system, with high probability as $n\rightarrow\infty$. We leverage this convergence result to explore synchronization for two classes of identical phase oscillators on Erdős-Rényi random graphs. This result suggests a framework for studying synchronization properties in large but finite random networks.

math.DS↗

arXiv2023

Designing for Robustness in Electric Grids via a General Effective Resistance Measure

We propose a mathematical framework for designing robust networks of coupled phase-oscillators by leveraging a vulnerability measure proposed by Tyloo et. al that quantifies how much a small perturbation to a phase-oscillator's natural frequency impacts the system's global synchronized frequencies. Given a fixed complex network topology with specific governing dynamics, the proposed framework finds an optimal allocation of edge weights that minimizes such vulnerability measure(s) at the node(s) for which we expect perturbations to occur by solving a tractable semidefinite programming problem. We specify the mathematical model to high voltage electric grids where each node corresponds to a voltage phase angle associated with a bus and edges correspond to transmission lines. Edge weights are determined by the susceptance values along the transmission lines. In this application, frequency synchronization is increasingly challenged by the integration of renewable energy, yet is imperative to the grid's health and functionality. Our framework helps to alleviate this challenge by optimizing the placement of renewable generation and the susceptance values along the transmission lines.

math.OC↗