Data-Driven Voltage Analysis of an Electric Power Grid via Delay Embedding and Extended Dynamic Mode Decomposition
- 193 Downloads
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.
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.
- 2.Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. Benjamin, Amsterdam (1968)Google Scholar
- 3.Bishop, C.M.: Pattern Recognition and Machine Larning. Springer, New York (2006)Google Scholar
- 5.Das, S., Giannakis, D.: Delay-coordinate maps and the spectra of Koopman operators (2017). arXiv:1706.08544v6
- 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.Kamb, M., Kaiser, E., Brunton, S.L., Kutz, J.N.: Time-delay observables for Koopman: theory and applications (2018). arXiv:1810.01479v1
- 11.Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press, Cambridge (2003)Google Scholar
- 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
- 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.Susuki, Y., Mezić, I.: Invariant sets in quasiperiodically forced dynamical systems (2019). arXiv:1808.08340
- 26.Susuki, Y., Sako, K., Hikihara, T.: On the spectral equivalence of Koopman operators through delay embedding (2017). arXiv:1706.1006
- 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