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Spontaneous synchrony in power-grid networks

Nature Physics volume 9, pages 191–197 (2013)Cite this article

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An Author Correction to this article was published on 07 December 2018

Abstract

An imperative condition for the functioning of a power-grid network is that its power generators remain synchronized. Disturbances can prompt desynchronization, which is a process that has been involved in large power outages. Here we derive a condition under which the desired synchronous state of a power grid is stable, and use this condition to identify tunable parameters of the generators that are determinants of spontaneous synchronization. Our analysis gives rise to an approach to specify parameter assignments that can enhance synchronization of any given network, which we demonstrate for a selection of both test systems and real power grids. These findings may be used to optimize stability and help devise new control schemes, thus offering an additional layer of protection and contributing to the development of smart grids that can recover from failures in real time.

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Figure 1: Physical versus effective network for the power grid of Northern Italy.
Figure 2: Stability of synchronous states for systems with βi = β.
Figure 3: Enhancement of the stability of synchronous states.

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References

  1. Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).

    Article  ADS  Google Scholar 

  2. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  3. Strogatz, S. H. Exploring complex networks. Nature 410, 268–276 (2001).

    Article  ADS  MATH  Google Scholar 

  4. Abrams, D. M. & Strogatz, S. H. Chimera states for coupled oscillators. Phys. Rev. Lett. 93, 174102 (2004).

    Article  ADS  Google Scholar 

  5. Ott, E. & Antonsen, T. M. Low dimensional behaviour of large systems of globally coupled oscillators. Chaos 18, 037113 (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Nishikawa, T. & Motter, A. E. Network synchronization landscape reveals compensatory structures, quantization, and the positive effect of negative interactions. Proc. Natl Acad. Sci. USA 107, 10342–10347 (2010).

    Article  ADS  Google Scholar 

  7. Hagerstrom, A. M. et al. Experimental observation of chimeras in coupled-map lattices. Nature Phys. 8, 658–661 (2012).

    Article  ADS  Google Scholar 

  8. Tinsley, M. R., Nkomo, S. & Showalter, K. Chimera and phase-cluster states in populations of coupled chemical oscillators. Nature Phys. 8, 662–665 (2012).

    Article  ADS  Google Scholar 

  9. Assenza, S., Gutiérrez, R., Gómez-Gardeñes, J., Latora, V. & Boccaletti, S. Emergence of structural patterns out of synchronization in networks with competitive interactions. Sci. Rep. 1, 99 (2011).

    Article  ADS  Google Scholar 

  10. Ravoori, B. et al. Robustness of optimal synchronization in real networks. Phys. Rev. Lett. 107, 034102 (2011).

    Article  ADS  Google Scholar 

  11. Hunt, D., Korniss, G. & Szymanski, B. K. Network synchronization in a noisy environment with time delays: Fundamental limits and trade-offs. Phys. Rev. Lett. 105, 068701 (2010).

    Article  ADS  Google Scholar 

  12. Sun, J., Bollt, E. M. & Nishikawa, T. Master stability functions for coupled nearly identical dynamical systems. Europhys. Lett. 85, 60011 (2009).

    Article  ADS  Google Scholar 

  13. Yu, W., Chen, G. & Lue, J. On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  14. Kiss, I. Z., Rusin, C. G., Kori, H. & Hudson, J. L. Engineering complex dynamical structures: Sequential patterns and desynchronization. Science 316, 1886–1889 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Restrepo, J. G., Ott, E. & Hunt, B. R. The emergence of coherence in complex networks of heterogeneous dynamical systems. Phys. Rev. Lett. 96, 254103 (2006).

    Article  ADS  Google Scholar 

  16. Strogatz, S. H., Abrams, D. M., McRobie, A., Eckhardt, B. & Ott, E. Crowd synchrony on the Millennium Bridge. Nature 438, 43–44 (2005).

    Article  ADS  Google Scholar 

  17. Néda, Z., Ravasz, E., Brechet, Y., Vicsek, T. & Barabási, A-L. The sound of many hands clapping. Nature 403, 849–850 (2000).

    Article  ADS  Google Scholar 

  18. Gellings, C. W. & Yeagee, K. E. Transforming the electric infrastructure. Phys. Today 57, 45–51 (2004).

    Article  ADS  Google Scholar 

  19. Strogatz, S. H. SYNC: The Emerging Science of Spontaneous Order (Hyperion, 2003).

    Google Scholar 

  20. Lozano, S., Buzna, L. & Dı´az-Guilera, A. Role of network topology in the synchronization of power systems. Eur. Phys. J. B 85, 1–8 (2012).

    Article  Google Scholar 

  21. Rohden, M., Sorge, A., Timme, M. & Witthaut, D. Self-organized synchronization in decentralized power grids. Phys. Rev. Lett. 109, 064101 (2012).

    Article  ADS  Google Scholar 

  22. Susuki, Y. & Mezić, I. Nonlinear Koopman modes and coherency identification of coupled swing dynamics. IEEE T. Power Syst. 26, 1894–1904 (2011).

    Article  Google Scholar 

  23. Susuki, Y., Mezić, I. & Hikihara, T. Global swing instability in the New England power grid model. Proc. 2009 American Control Conf. 3446–3451 (IEEE, 2009).

    Chapter  Google Scholar 

  24. Parrilo, P. Model reduction for analysis of cascading failures in power systems. Proc. 1999 American Control Conf. 4208–4212 (IEEE, 1999).

    Google Scholar 

  25. Dörfler, F. & Bullo, F. Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators. Proc. 2010 American Control Conf. 930–937 (IEEE, 2010).

    Chapter  Google Scholar 

  26. Dörfler, F. & Bullo, F. On the critical coupling for Kuramoto oscillators. SIAM J. Appl. Dyn. Syst. 10, 1070–1099 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  27. Dörfler, F. & Bullo, F. Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators. SIAM J. Control Optim. 50, 1616–1642 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  28. NERC System Disturbances Reports 1992–2009 (North American Electric Reliability Corporation, http://www.nerc.com).

  29. Grainger, J. J. & Stevenson, W. D. Jr Power System Analysis (McGraw-Hill, 2004).

    Google Scholar 

  30. Anderson, P. M. & Fouad, A. A. Power System Control and Stability 2nd edn (IEEE Press-Wiley Interscience, 2003).

    Google Scholar 

  31. Dörfler, F. & Bullo, F. Spectral analysis of synchronization in a lossless structure-preserving power network model. Proc. First IEEE Int. Conf. Smart Grid Communications 179–184 (IEEE, 2010).

    Google Scholar 

  32. Nishikawa, T., Motter, A. E., Lai, Y-C. & Hoppensteadt, F. C. Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize? Phys. Rev. Lett. 91, 014101 (2003).

    Article  ADS  Google Scholar 

  33. Motter, A. E., Zhou, C. S. & Kurths, J. Enhancing complex-network synchronization. Europhys. Lett. 69, 334–340 (2005).

    Article  ADS  Google Scholar 

  34. Pecora, L. M. & Carroll, T. L. Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998).

    Article  ADS  Google Scholar 

  35. Fink, K. S., Johnson, G., Carroll, T., Mar, D. & Pecora, L. Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays. Phys. Rev. E 61, 5080–5090 (2000).

    Article  ADS  Google Scholar 

  36. Gooi, H. B., Hill, E. F., Mobarak, M. A., Thorne, D. H. & Lee, T. H. Coordinated multi-machine stabilizer settings without eigenvalue drift. IEEE T. Power Ap. Syst. 100, 3879–3887 (1981).

    Article  Google Scholar 

  37. Dobson, I. et al. Avoiding and Suppressing Oscillations PSerc Publication 00–01 (Univ. of Wisconsin, 1999).

    Google Scholar 

  38. Zhang, P., Chen, J. & Shao, M. Phasor Measurement Unit (PMU) Implementation and Applications (Electric Power Research Institute, 2007).

    Google Scholar 

  39. Rinaldi, S. M., Peerenboom, J. P. & Kelly, T. K. Identifying, understanding, and analysing critical infrastructure interdependencies. IEEE Contr. Syst. Mag. 21, 11–25 (2001).

    Article  Google Scholar 

  40. Brede, M. Synchrony-optimized networks of non-identical Kuramoto oscillators. Phys. Lett. A 372, 2618–2622 (2008).

    Article  ADS  MATH  Google Scholar 

  41. Carareto, R., Orsatti, F. M. & Piqueira, J. R. C. Optimized network structure for full-synchronization. Commun. Nonlinear Sci. 14, 2536–2541 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  42. Buzna, L., Lozano, S. & DĂ­az-Guilera, A. Synchronization in symmetric bipolar population networks. Phys. Rev. E 80, 066120 (2009).

    Article  ADS  Google Scholar 

  43. Kelly, D. & Gottwald, G. A. On the topology of synchrony optimized networks of a Kuramoto-model with non-identical oscillators. Chaos 21, 025110 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Gómez-Gardeñes, J., Gómez, S., Arenas, A. & Moreno, Y. Explosive synchronization transitions in scale-free networks. Phys. Rev. Lett. 106, 128701 (2011).

    Article  ADS  Google Scholar 

  45. Garlaschelli, D., Capocci, A. & Caldarelli, G. Self-organized network evolution coupled to extremal dynamics. Nature Phys. 3, 813–817 (2007).

    Article  ADS  MATH  Google Scholar 

  46. Milano, F. Power Systems Analysis Toolbox (Univ. Castilla, 2007).

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Acknowledgements

The authors thank F. Milano for providing power-grid data, E. Mallada for sharing unpublished simulation details, and F. Dörfler for insightful discussions. This work was supported by the NSF under grants DMS-1057128 and DMS-0709212, the LANL LDRD project Optimization and Control Theory for Smart Grids, and a Northwestern-Argonne Early Career Investigator Award for Energy Research to A.E.M.

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Authors and Affiliations

  1. Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA

    Adilson E. Motter & Takashi Nishikawa

  2. Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA

    Adilson E. Motter

  3. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, California 94305, USA

    Seth A. Myers

  4. Information Sciences Group, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA

    Marian Anghel

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  1. Adilson E. Motter
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Contributions

All authors contributed to the design of the research and analytical calculations. S.A.M., M.A. and T.N. performed the numerical simulations. A.E.M. and T.N. wrote the paper, and A.E.M. supervised the project.

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Correspondence to Adilson E. Motter.

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Motter, A., Myers, S., Anghel, M. et al. Spontaneous synchrony in power-grid networks. Nature Phys 9, 191–197 (2013). https://doi.org/10.1038/nphys2535

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