Admittance Matrix Concentration Inequalities for Understanding Uncertain Power Networks

TL;DR

This paper develops probabilistic bounds for the spectrum of the admittance matrix in uncertain power networks, aiding in understanding the impact of parameter variability on power flow approximations.

Contribution

It introduces a novel application of concentration inequalities to derive bounds on the admittance matrix spectrum under uncertainty, linking spectral behavior to network criticality.

Findings

Bounds accurately predict spectral perturbations in IEEE test networks.

Spectral bounds scale with nodal criticality, highlighting critical nodes.

The approach provides conservative but useful error estimates for power flow models.

Abstract

This paper presents conservative probabilistic bounds for the spectrum of the admittance matrix and classical linear power flow models under uncertain network parameters; for example, probabilistic line contingencies. Our proposed approach imports tools from probability theory, such as concentration inequalities for random matrices. This provides a theoretical framework for understanding error bounds of common approximations of the AC power flow equations under parameter uncertainty, including the DC and LinDistFlow approximations. Additionally, we show that the upper bounds scale as functions of nodal criticality. This network-theoretic quantity captures how uncertainty concentrates at critical nodes for use in contingency analysis. We validate these bounds on IEEE test networks, demonstrating that they correctly capture the scaling behavior of spectral perturbations up to conservative constants.