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Data-Driven Voltage Analysis of an Electric Power Grid via Delay Embedding and Extended Dynamic Mode Decomposition

  • Yoshihiko SusukiEmail author
  • Kyoichi Sako
Chapter
  • 193 Downloads
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 484)

Abstract

Data-driven analysis of dynamic performance has attracted a lot of interest in the modern power grid with highly accurate measurement technologies such as Synchrophasor. In this chapter, we report the research effort to do this for voltage dynamics in a rudimentary model of power grids. We use the combination of the so-called delay embedding with the extended dynamic mode decomposition, which is a technique for approximating the Koopman operator directly from time-series data. This combination enables us to approximate the spectral properties of the Koopman operator through voltage–amplitude measurement of a single bus. The spectral properties clarify not only the local information on the voltage dynamics such as frequency/damping in transient responses but also global one such as the flow structure of the underlying mathematical model.

Notes

Acknowledgements

The authors greatly appreciate Professor Takashi Hikihara (Kyoto University) for valuable discussion on this work and Professor Igor Mezić (University of California, Santa Barbara) for his careful reading of the draft of this chapter with valuable comments, which are cited in the remark of the appendix. During part of the work on this chapter, the authors were at Laboratory of Advanced Electrical Systems Theory, Department of Electrical Engineering, Kyoto University, Kyoto, Japan. The work is supported in part by JST-CREST program #JP-MJCR15K3 and JSPS-KAKEN #15H03964.

References

  1. 1.
    Arbabi, H., Mezić, I.: Ergodic theory, dynamic mode decomposition and computation of spectral properties of the Koopman operator. SIAM J. Appl. Dyn. Syst. 16(4), 2096–2126 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. Benjamin, Amsterdam (1968)Google Scholar
  3. 3.
    Bishop, C.M.: Pattern Recognition and Machine Larning. Springer, New York (2006)Google Scholar
  4. 4.
    Budišić, M., Mohr, R., Mezić, I.: Applied Koopmanism. CHAOS 22(4), 047, 510 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Das, S., Giannakis, D.: Delay-coordinate maps and the spectra of Koopman operators (2017). arXiv:1706.08544v6
  6. 6.
    Dasgupta, S., Paramasvam, M., Vaidya, U., Ajjarrapu, V.: Real-time monitoring of short-term voltage stability using PMU data. IEEE Trans. Power Syst. 28(4), 3702–3711 (2013)CrossRefGoogle Scholar
  7. 7.
    De La Ree, J., Centeno, V., Thorp, J.S., Phadke, A.G.: Synchronized phasor measurement applications in power systems. IEEE Trans. Smart Grid 1(1), 20–27 (2010)CrossRefGoogle Scholar
  8. 8.
    Dobson, I., Chiang, H.D.: Towards a theory of voltage collapse in electric power systems. Syst. Control Lett. 13, 253–262 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Giannakis, D., Slawinska, J., Zhao, Z.: Spatiotemporal feature extraction with data-driven Koopman operators. In: JMLR: Workshop and Conference Proceedings, vol. 44, pp. 103–115 (2015)Google Scholar
  10. 10.
    Kamb, M., Kaiser, E., Brunton, S.L., Kutz, J.N.: Time-delay observables for Koopman: theory and applications (2018). arXiv:1810.01479v1
  11. 11.
    Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press, Cambridge (2003)Google Scholar
  12. 12.
    Mauroy, A., Mezić, I.: On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics. CHAOS 22(3), 033, 112 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mauroy, A., Mezić, I., Moehlis, J.: Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics. Phys. D 261, 19–30 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41, 309–325 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mezić, I.: On applications of the spectral theory of the Koopman operator in dynamical systems and control theory. In: Proceedings of 2015 IEEE 54th Annual Conference on Decision and Control, pp. 7034–7041. Osaka, Japan (2015)Google Scholar
  17. 17.
    Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Phys. D 197(1–2), 101–133 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mezić, I., Wiggins, S.: A method for visualization of invariant sets of dynamical systems based on the ergodic partition. CHAOS 9(1), 213–218 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Raak, F., Susuki, Y., Hikihara, T.: Data-driven partitioning of power networks via Koopman mode analysis. IEEE Trans. Power Syst. 31(4), 2799–2807 (2016)CrossRefGoogle Scholar
  20. 20.
    Raak, F., Susuki, Y., Tsuboki, K., Kato, M., Hikihara, T.: Quantifying smoothing effects of wind power via Koopman mode decomposition: a numerical test of wind speed predictions in Japan. Nonlinear Theory Its Appl., IEICE 8(4), 342–357 (2017)CrossRefGoogle Scholar
  21. 21.
    Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Susuki, Y., Mezić, I.: A Prony approximation of Koopman mode decomposition. In: Proceedings of IEEE Conference on Decision and Control, pp. 7022–7727 (2015)Google Scholar
  23. 23.
    Susuki, Y., Mezić, I.: Invariant sets in quasiperiodically forced dynamical systems (2019). arXiv:1808.08340
  24. 24.
    Susuki, Y., Mezić, I., Raak, F., Hikihara, T.: Applied Koopman operator theory for power systems technology. Nonlinear Theory Its Appl., IEICE 7(4), 430–459 (2016)CrossRefGoogle Scholar
  25. 25.
    Susuki, Y., Sako, K.: Data-based voltage analysis of power systems via delay-embedding and extended dynamic mode decomposition. IFAC-PapersOnLine 51, 221–226 (2018)CrossRefGoogle Scholar
  26. 26.
    Susuki, Y., Sako, K., Hikihara, T.: On the spectral equivalence of Koopman operators through delay embedding (2017). arXiv:1706.1006
  27. 27.
    Takens, F.: Detecting strange attractors in turbulence. In: D. Land, L.S. Young (eds.) Dynamical Systems and Turbulence. Lecture Notes in Mathematics, vol. 898. Springer, Berlin (1981)Google Scholar
  28. 28.
    Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25, 1307–1346 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yan, J., Liu, C.C., Vaidya, U.: PMU-based monitoring of rotor angle dynamics. IEEE Trans. Power Syst. 26(4), 2125–2133 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Electrical and Information SystemsOsaka Prefecture UniversityOsakaJapan
  2. 2.Department of Electrical EngineeringKyoto UniversityKyotoJapan
  3. 3.Nidec CorporationKyotoJapan

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