  A recent trend in control of power systems has sought to quantify the
synchronization dynamics in terms of a global performance metric, compute it
under very simplified assumptions, and use it to gain insight on the role of
system parameters, in particular, inertia. In this paper, we wish to extend
this approach to more realistic scenarios, by incorporating the heterogeneity
of machine ratings, more complete machine models, and also to more closely map
it to classical power engineering notions such as Nadir, Rate of Change of
Frequency (RoCoF), and inter-area oscillations.
  We consider the system response to a step change in power excitation, and
define the system frequency as a weighted average of generator frequencies
(with weights proportional to each machine's rating); we characterize Nadir and
RoCoF by the $L_\infty$ norm of the system frequency and its derivative,
respectively, and inter-areas oscillations by the $L_2$ norm of the error of
the vector of bus frequencies w.r.t. the system frequency.
  For machine models where the dynamic parameters (inertia, damping, etc.) are
proportional to rating, we analytically compute these norms and use them to
show that the role of inertia is more nuanced than in the conventional wisdom.
With the classical swing dynamics, inertia constant plays a secondary role in
performance. It is only when the turbine dynamics are introduced that the
benefits of inertia become more prominent.
