Abstract

Power grids are undergoing major changes, shifting from a few large producers to smart grids built upon renewable energies. Mathematical models for power grid dynamics have to be adapted to capture when dynamic nodes can achieve synchronization to a common grid frequency on complex network topologies. In this paper we study a second-order rotator model in the large network limit. We merge the recent theory of random graph limits for complex networks with approaches to first-order systems on graphons. We prove that there exists a well-posed continuum limit integral equation approximating a large finite-dimensional synchronization model problem from power grid network dynamics. Then we analyze the linear stability of synchronized solutions and prove linear stability. However, on small-world networks we demonstrate that there are topological parameters moving the spectrum arbitrarily close to the imaginary axis leading to potential instability on finite time scales.

Keywords

  1. power networks
  2. graphs
  3. graphons
  4. linear stability
  5. coupled oscillators

MSC codes

  1. 05C82
  2. 05C90
  3. 34C15
  4. 37C75
  5. 45J05

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Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1271 - 1292
ISSN (online): 1095-712X

History

Submitted: 11 July 2018
Accepted: 23 April 2019
Published online: 9 July 2019

Keywords

  1. power networks
  2. graphs
  3. graphons
  4. linear stability
  5. coupled oscillators

MSC codes

  1. 05C82
  2. 05C90
  3. 34C15
  4. 37C75
  5. 45J05

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659
Volkswagen Foundation https://doi.org/10.13039/501100001663

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