# System-Theoretic Performance Metrics for Low-Inertia Stability of Power Networks

## TL;DR

This paper provides a theoretical analysis of various system-theoretic performance metrics for low-inertia power grids, revealing how these metrics depend on system parameters and can behave differently.

## Contribution

It derives explicit expressions for eigenvalue damping ratios and $\\mathcal{H}_2$ and $\mathcal{H}_\infty$ norms, comparing their behavior in low-inertia power networks.

## Key findings

- $\mathcal{H}_2$ and $\mathcal{H}_\infty$ norms can behave differently as inertia varies.
- Performance metrics depend on inertia, damping, and topology.
- Low-inertia performance assessment requires careful metric selection.

## Abstract

As bulk synchronous generators in the power grid are replaced by distributed generation interfaced through power electronics, inertia is removed from the system, prompting concerns over grid stability. Different metrics are available for quantifying grid stability and performance; however, no theoretical results are available comparing and contrasting these metrics. This paper presents a rigorous system-theoretic study of performance metrics for low-inertia stability. For networks with uniform parameters, we derive explicit expressions for the eigenvalue damping ratios, and for the $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ norms of the linearized swing dynamics, from external power disturbances to different phase/frequency performance outputs.These expressions show the dependence of system performance on inertia constants, damping constants, and on the grid topology. Surprisingly, we find that the $\mathcal{H}_2$ and $\mathcal{H}_{\infty}$ norms can display contradictory behavior as functions of the system inertia, indicating that low-inertia performance depends strongly on the chosen performance metric.