  We investigate the total resistive losses incurred in returning a power
network of identical generators to a synchronous state following a transient
stability event or in maintaining this state in the presence of persistent
stochastic disturbances. We formulate this cost as the input-output $H^2$ norm
of a linear dynamical system with distributed disturbances. We derive an
expression for the total resistive losses that scales with the size of the
network as well as properties of the generators and power lines, but is
independent of the network topology. This topologically invariant scaling of
what we term the price of synchrony is in contrast to typical power system
stability notions like rate of convergence or the region of attraction for
rotor-angle stability. Our result indicates that highly connected power
networks, whilst desirable for higher phase synchrony, do not offer an
advantage in terms of the total resistive power losses needed to achieve this
synchrony. Furthermore, if power flow is the mechanism used to achieve
synchrony in highly-distributed-generation networks, the cost increases
unboundedly with the number of generators.
