Linear Approximations Provide Rigorous Conditions for Phase Locking in Oscillator Networks and Power Grids

arXiv (Cornell University)

Abstract

Synchronization is a fundamental collective phenomenon in many natural and engineered networks. The stable operation of oscillator networks, such as electric power grids, requires phase locking with phase differences across edges below a given threshold. A central question is whether a given network supports such a state. In this contribution, we extend a convex optimization formulation of the nonlinear fixed point equations of the Kuramoto model, that also describe lossless power grids. Based on this formulation, we derive rigorous and tight bounds on the error introduced by an arbitrary approximation of the original model, in particular linear approximations. The bounds yield edge-wise regions of trust around the approximation, within which the true nonlinear solution must lie. They further provide sufficient and necessary conditions for phase locking under prescribed phase difference limits. Numerical experiments show that the bounds are tight over a broad range of parameters.

Authors 6

1. Iva Bačić

   Forschungszentrum Jülich · University of Belgrade · Institute of Physics Belgrade

   Affiliation as printed

   Center of Excellence for Study of Complex Systems , Institute of Physics Belgrade , University of Belgrade , Belgrade , Serbia

   Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1) , Forschungszentrum Jülich , Jülich , Germany

2. Carsten Hartmann Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1) , Forschungszentrum Jülich , Jülich , Germany

   RWTH Aachen University , Aachen , Germany

3. Philipp Böttcher

   Forschungszentrum Jülich

   Affiliation as printed

   Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1) , Forschungszentrum Jülich , Jülich , Germany

4. David Gross

   University of Cologne

   Affiliation as printed

   Institute for Theoretical Physics , University of Cologne , Köln , Germany

5. Andrea Benigni Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1) , Forschungszentrum Jülich , Jülich , Germany

   RWTH Aachen University , Aachen , Germany

6. Dirk Witthaut

   Forschungszentrum Jülich · University of Cologne

   Affiliation as printed

   Institute for Theoretical Physics , University of Cologne , Köln , Germany

   Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1) , Forschungszentrum Jülich , Jülich , Germany