  Stability of power networks is an increasingly important topic because of the
high penetration of renewable distributed generation units. This requires the
development of advanced (typically model-based) techniques for the analysis and
controller design of power networks. Although there are widely accepted
reduced-order models to describe the dynamic behavior of power networks, they
are commonly presented without details about the reduction procedure, hampering
the understanding of the physical phenomena behind them. The present paper aims
to provide a modular model derivation of multi-machine power networks. Starting
from first-principle fundamental physics, we present detailed dynamical models
of synchronous machines and clearly state the underlying assumptions which lead
to some of the standard reduced-order multi-machine models, including the
classical second-order swing equations. In addition, the energy functions for
the reduced-order multi-machine models are derived, which allows to represent
the multi-machine systems as port-Hamiltonian systems. Moreover, the systems
are proven to be passive with respect to its steady states, which permits for a
power-preserving interconnection with other passive components, including
passive controllers. As a result, the corresponding energy function or
Hamiltonian can be used to provide a rigorous stability analysis of advanced
models for the power network without having to linearize the system.
