  The issue of synchronization in the power grid is receiving renewed
attention, as new energy sources with different dynamics enter the picture.
Global metrics have been proposed to evaluate performance and analyzed under
highly simplified assumptions. In this paper, we extend this approach to more
realistic network scenarios and more closely connect it with metrics used in
power engineering practice. In particular, our analysis covers networks with
generators of heterogeneous ratings and richer dynamic models of machines.
Under a suitable proportionality assumption in the parameters, we show that the
step response of bus frequencies can be decomposed in two components. The first
component is a {system-wide frequency} that captures the aggregate grid
behavior, and the residual component represents the individual bus frequency
deviations from the aggregate. Using this decomposition, we define --and
compute in closed form-- several metrics that capture dynamic behaviors that
are of relevance for power engineers. In particular, using the \emph{system
frequency}, we define industry-style metrics (Nadir, RoCoF) that are evaluated
through a representative machine. We further use the norm of the residual
component to define a \emph{synchronization cost} that can appropriately
quantify inter-area oscillations. Finally, we employ robustness analysis tools
to evaluate deviations from our proportionality assumption. We show that the
system frequency still captures the grid steady-state deviation, and becomes an
accurate reduced-order model of the grid as the network connectivity grows.
Simulation studies with practically relevant data are included to validate the
theory and further illustrate the impact of network structure and parameters on
synchronization. Our analysis gives conclusions of practical interest,
sometimes challenging the conventional wisdom in the field.
