  We study nonlinear power systems consisting of generators, generator buses,
and non-generator buses. First, looking at a generator and its bus' variables
jointly, we introduce a synchronization concept for a pair of such joint
generators and buses. We show that this concept is related to graph symmetry.
Next, we extend, in two ways, the synchronization from a pair to a partition of
all generators in the networks and show that they are related to either graph
symmetry or equitable partitions. Finally, we show how an exact reduced model
can be obtained by aggregating the generators and associated buses in the
network when the original system is synchronized with respect to a partition,
provided that the initial condition respects the partition. Additionally, the
aggregation-based reduced model is again a power system.
