Complex Frequency

TL;DR

This paper introduces the concept of complex frequency, combining voltage magnitude variation and local bus frequency change, with differential equations linking complex powers and frequencies in power networks.

Contribution

It defines complex frequency in power systems and develops differential equations describing its relation to complex power and voltage variations during transients.

Findings

Complex frequency captures both voltage and frequency variations.

Differential equations link complex power and complex frequency.

Applications demonstrate the usefulness of the concept.

Abstract

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.