The role of symmetries on the stability of power grids

Abstract

The stability of power grids principally depends on the synchronization of multiple generating units. So far, the relationship between grid stability and symmetries remains unclear. This article fills in the research gap by innovatively using the mathematical tool of group theory. We find that grid symmetries involve two aspects-network and component symmetries, which reflect the homogeneities of network structures and generating units, respectively. As disclosed, the stability of power grids has a tight bearing on network and component symmetries, where corresponding automorphism group and symmetric subgroup orders reflect their levels of symmetries, respectively. In the perspective of network symmetries, where grid stability and synchronization speed are dictated by the algebraic connectivity, the stability ranking list of grids with four generators is K4 > C4 > K13 > R4 > P4 > X4, which fully agrees with that of network symmetries. In terms of component symmetries, the stability of power grids largely benefits from improved component symmetries yet with R4 as an exceptional case, where it is recommended to avoid the centralized effort of a single generator. These conclusions have significant implications for the control of modern power grids with distributed resources and controllable generators, including energy storage and renewable power generators interfaced by grid-following/-forming converters.