Zusammenfassung
Es werden verteilt-parametrische Modelle hergeleitet, die das nichtlineare fraktionale Verhalten von Lithium-Ionen-Zellen beschreiben. Die vorgestellten Modelle interpolieren lineare verteilt-parametrische Kleinsignalmodelle, die sich als diffusive Realisierungen fraktionaler Übertragungsfunktionen in den Arbeitspunkten ergeben. Diese Übertragungsfunktionen werden mittels Impedanzspektroskopie parametriert. Zur endlich-dimensionalen Approximation der vorgestellten Modelle werden Krylov-Unterraum-Verfahren und Finite-Elemente-Ansätze genutzt. Diese nichtlinearen konzentriert-parametrischen Modelle werden in Simulationsstudien miteinander verglichen und anhand experimenteller Daten validiert.
Abstract
Distributed-parameter models describing the nonlinear fractional behavior of lithium-ion cells are derived. The proposed models interpolate linear distributed-parameter models which are obtained as diffusive realizations of fractional transfer functions describing the small signal behavior at several stationary operating points. The linear models are parameterized by means of so-called impedance spectroscopy. Finite-dimensional approximations of the derived models are obtained either by the Krylov subspace method or by the finite element method. The obtained nonlinear lumped-parameter models are compared in simulation scenarios and validated on the basis of experimental data.
Über die Autoren
Lukas Tappeiner ist wissenschaftlicher Mitarbeiter am Institut für Automatisierungs- und Regelungstechnik an der UMIT TIROL in Hall in Tirol. Hauptarbeitsgebiete: Modellierung von Batteriesystemen, Modellreduktion.
Marc Oldenburger ist Entwicklungsingenieur bei der Mercedes-Benz AG. Hauptarbeitsgebiete: Modellierung und Simulation von Batteriesystemen.
Benjamin Bedürftig ist wissenschaftlicher Mitarbeiter am Institut für Automatisierungstechnik (IFAT): Lehrstuhl für Systemtheorie und Regelungstechnik. Hauptarbeitsgebiete: Modellierung der Dynamik und Alterung von Batteriesystemen.
Frank Woittennek leitet das Institut für Automatisierungs- und Regelungstechnik an der UMIT TIROL in Hall in Tirol. Hauptarbeitsgebiete: Regelungs- und Beobachterentwurf für lineare und nichtlineare unendlichdimensionale Systeme, modellbasierte Regelung mechatronischer Systeme.
Literatur
1. M. Abouzari, F. Berkemeier, G. Schmitz and D. Wilmer, “On the physical interpretation of constant phase elements”, Solid State Ionics, Jg. 180, Nr. 14, S. 922—927, 2009.10.1016/j.ssi.2009.04.002Search in Google Scholar
2. J. Audounet, D. Matignon and G. Montseny, “Semi-linear diffusive representations for nonlinear fractional differential systems”, in Nonlinear control in the Year 2000, London: Springer London, 2001, S. 73—82.10.1007/BFb0110208Search in Google Scholar
3. K. Biswas Sen and P. K. Dutta, “Realization of a constant phase element and its performance study in a differentiator circuit”, IEEE Transactions on Circuits and Systems II: Express Briefs, Jg. 53, Nr. 9, S. 802—806, 2006.10.1109/TCSII.2006.879102Search in Google Scholar
4. S. Buller, “Impedance based simulation models for energy storage devices in advanced automotive power systems”, Diss., Aachen, 2003.Search in Google Scholar
5. R. F. Curtain and H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory. New York: Springer-Verlag New York, 1995.10.1007/978-1-4612-4224-6Search in Google Scholar
6. M. Eckert, “Modellbasierte Identifikation fraktionaler Systeme und ihre Anwendung auf die Lithium-Ionen-Zelle”, Diss., Karlsruher Institut für Technologie (KIT), 2017.Search in Google Scholar
7. S. Gantenbein, M. Weiss and E. Ivers-Tiffée, “Impedance based time-domain modeling of lithium-ion batteries: Part I”, Journal of Power Sources, Jg. 379, S. 317—327, 2018.10.1016/j.jpowsour.2018.01.043Search in Google Scholar
8. M. Hannan, M. Lipu, A. Hussain and A. Mohamed, “A review of lithium-ion battery state of charge estimation and management system in electric vehicle applications: Challenges and recommendations”, Renewable and Sustainable Energy Reviews, Jg. 78, S. 834—854, 2017.10.1016/j.rser.2017.05.001Search in Google Scholar
9. K. Hariharan and V. Kumar, “A nonlinear equivalent circuit model for lithium ion cells”, Journal of Power Sources, Jg. 222, S. 210–217, 2013.10.1016/j.jpowsour.2012.08.090Search in Google Scholar
10. C. Harkort and J. Deutscher, “Krylov subspace methods for linear infinite-dimensional systems”, IEEE Transactions on Automatic Control, Jg. 56, Nr. 2, S. 441–447, 2011.10.1109/TAC.2010.2090063Search in Google Scholar
11. C. Hernandez-Jaimes, J. Vazquez-Arenas, J. Vernon-Carter and J. Alvarez-Ramirez, “A nonlinear Cole-Cole model for large-amplitude electrochemical impedance spectroscopy”, Chemical Engineering Science, Jg. 137, S. 1–8, 2015.10.1016/j.ces.2015.06.015Search in Google Scholar
12. J. Huang, J. Zhang, Z. Li, S. Song, und N. Wu, “Exploring differences between charge and discharge of LiMn2O4/Li half-cell with dynamic electrochemical impedance spectroscopy”, Electrochimica Acta, Jg. 131, S. 228–235, 2014, Electrochemical Impedance Spectroscopy.10.1016/j.electacta.2014.02.030Search in Google Scholar
13. J. Illig, “Physically based Impedance Modelling of Lithium-Ion Cells”, Diss., 2014.Search in Google Scholar
14. A. Jokar, B. Rajabloo, M. Désilets and M. Lacroix, “Review of simplified pseudo-two-dimensional models of lithium-ion batteries”, Journal of Power Sources, Jg. 327, S. 44–55, 2016.10.1016/j.jpowsour.2016.07.036Search in Google Scholar
15. U. Krewer, F. Roder, E. Harinath, R. D. Braatz, B. Bedürftig and R. Findeisen, “Review—dynamic models of li-ion batteries for diagnosis and operation: a review and perspective”, Journal of The Electrochemical Society, Jg. 165, Nr. 16, S. A3656–A3673, 2018.10.1149/2.1061814jesSearch in Google Scholar
16. M. Kupper, A. Creutz, O. Stark, S. Krebs and S. Hohmann, “Current and state of charge estimation of lithium-ion battery packs using distributed fractional extended Kalman filters”, in 2019 IEEE Conference on Control Technology and Applications (CCTA), 2019, S. 462469.10.1109/CCTA.2019.8920558Search in Google Scholar
17. S. Liu, J. Jiang, W. Shi, Z. Ma, L. Y. Wang and H. Guo, “Butler-Volmer-equation-based electrical model for high-power lithium titanate batteries used in electric vehicles”, IEEE Transactions on Industrial Electronics, Jg. 62, Nr. 12, S. 7557–7568, 2015.10.1109/TIE.2015.2449776Search in Google Scholar
18. M. Lucu, E. Martinez-Laserna, I. Gandiaga, K. Liu, H. Camblong, W. Widanage and J. Marco, “Data-driven non-parametric Li-ion battery ageing model aiming at learning from real operation data – Part A: Storage operation”, Journal of Energy Storage, Jg. 30, 101409, 2020.10.1016/j.est.2020.101409Search in Google Scholar
19. J. Macdonald, “New aspects of some small-signal ac frequency response functions”, Solid State Ionics, Jg. 15, S. 159–161, 1985.10.1016/0167-2738(85)90095-5Search in Google Scholar
20. G. Montseny, “Diffusive representation of pseudodifferential time-operators”, ESAIM: Proc., Jg. 5, 159175, 1998.10.1051/proc:1998005Search in Google Scholar
21. B. Moore, “Principal component analysis in linear systems: controllability, observability, and model reduction”, IEEE Trans. Automat. Control, Jg. 26, 17–32, 1 1981.10.1109/TAC.1981.1102568Search in Google Scholar
22. P. L. Moss, G. Au, E. J. Plichta and J. P. Zheng, “An electrical circuit for modeling the dynamic response of Li-ion polymer batteries”, Journal of The Electrochemical Society, 155, 12, A986, 2008.10.1149/1.2999375Search in Google Scholar
23. H. Mu, R. Xiong, H. Zheng, Y. Chang and Z. Chen, “A novel fractional order model based state-of-charge estimation method for lithium-ion battery”, Applied Energy, Jg. 207, S. 384–393, 2017.10.1016/j.apenergy.2017.07.003Search in Google Scholar
24. M. Oldenburger, B. Bedürftig, A. Gruhle and E. Richter, “A new approach to measure the non-linear Butler-Volmer behavior of electrochemical systems in the time domain”, Journal of Energy Storage, Jg. 14, S. 16–21, 2017.10.1016/j.est.2017.09.002Search in Google Scholar
25. T. Osaka, D. Mukoyama and H. Nara, “Review—development of diagnostic process for commercially available batteries, especially lithium ion battery, by electrochemical impedance spectroscopy”, Journal of The Electrochemical Society, Jg. 162, Nr. 14, S. A2529–A2537, 2015.10.1149/2.0141514jesSearch in Google Scholar
26. A. Oustaloup, F. Levron, B. Mathieu and F. Nanot, “Frequency-band complex noninteger differentiator: characterization and synthesis”, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Jg. 47, Nr. 1, S. 25–39, 2000.10.1109/81.817385Search in Google Scholar
27. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering. London: Academic Press, 1999.Search in Google Scholar
28. T. Poinot and J.-C. Trigeassou, “A method for modelling and simulation of fractional systems”, Signal Processing, Jg. 83, Nr. 11, S. 2319–2333, 2003.10.1016/S0165-1684(03)00185-3Search in Google Scholar
29. J. Reddy, An Introduction to the Finite Element Method, McGraw-Hill Series in Mechanical Engineering. McGraw-Hill, 2006.Search in Google Scholar
30. S. Shen, M. Sadoughi, X. Chen, M. Hong and C. Hu, “A deep learning method for online capacity estimation of lithium-ion batteries”, Journal of Energy Storage, Jg. 25, 100817, 2019.10.1016/j.est.2019.100817Search in Google Scholar
31. M. Torchio, L. Magni, R. B. Gopaluni, R. D. Braatz and D. M. Raimondo, “LIONSIMBA: a Matlab framework based on a finite volume model suitable for li-ion battery design, simulation, and control”, Journal of The Electrochemical Society, Jg. 163, Nr. 7, S. A1192–A1205, 2016.10.1149/2.0291607jesSearch in Google Scholar
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