  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.

Optimization and Control