  The paper introduces the concept of complex frequency. The imaginary part of
the complex frequency is the variation with respect of a synchronous reference
of the local bus frequency as commonly defined in power system studies. The
real part is defined based on the variation of the voltage magnitude. The
latter term is crucial for the correct interpretation and analysis of the
variation of the frequency at each bus of the network. The paper also develops
a set of differential equations that describe the link between complex powers
and complex frequencies at network buses in transient conditions. No
simplifications are assumed except for constant elements of the network
admittance matrix. A variety of analytical and numerical examples show the
applications and potentials of the proposed concept.
