  Existing methods to determine the stability of a power system to small
perturbations are based on eigenvalue analysis and focus on the asymptotic
(long-term) behavior of the power grid. During the preasymptotic (short-term)
transient, however, the system can exhibit large growth that is not explained
by eigenvalues alone. In this paper we propose a new framework to determine the
maximum (optimal) preasymptotic growth using the singular value decomposition.
The approach is tailored to the analysis of quantities of interest in power
system dynamics, such as the set of rotor speed deviations. Matrix-free
techniques are developed to avoid the explicit formation of dense matrices and
enable the analysis of large-scale systems without reaching memory bounds.
Extensive results carried out from small to very large-scale systems (e.g.,
70k-bus system) verify the theoretical aspects of the technique.
