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Hard limitations of polynomial approximations for reduced-order models of lithium-ion cells

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Abstract

Electrochemical models are a widespread framework to simulate lithium-ion (Li-ion) batteries and to design their battery management algorithms. To obtain numerically efficient models, the polynomial approximation (PA) is one of many order reduction techniques applied to the solid-state lithium-ion (SSLi+) diffusion equation, the core of the so-called pseudo-two-dimensional electrochemical model (P2DM). Although the validity of PA is constrained to slow-varying current scenarios, many algorithms reported in the literature to estimate the state-of-charge of Li-ion cells rely on low-order PA-based dynamic models (PADMs) derived from the P2DM. Moreover, assuming that most properties of their low-order counterparts are inherited, some authors suggest that PADMs of arbitrary high order can be used to provide very accurate approximations of the state-of-charge, even under complex operating conditions. Nevertheless, to authors knowledge, in the open literature there is not a proper analysis to support such assertion. In this paper, by introducing a systematic method to derive PADMs of arbitrary order, the ability of high-order PADMs to reproduce the average and surface concentrations described by the SSLi+ diffusion equation is investigated in time and frequency domains, with the aid of classic control systems theory techniques. The main result shows that PADMs of order greater than 2 are structurally fragile as non-minimum-phase zeros as well as spurious unstable modes are induced by the PA technique, implying that high-order PADMs are not suitable for simulation or estimation purposes because of their weak internal structure.

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Notes

  1. The C rate of a current (charging or discharging) in \(h^{-1}\) units is defined as the ratio of the electric current in A over the nominal capacity \(C_{nom}\) of the cell in A\(\cdot\)h.

  2. This difference of 2 dB is established arbitrarily as a quantitative criterion to evaluate the accuracy of the PADMs.

  3. Here order refers to truncation error of the numerical method.

  4. Readers interested in simulation times for the complete P2DM can consult, for example, [37] and [38].

References

  1. Armand M, Tarascon JM (2008) Building better batteries. Nature 451:652–657

    CAS  PubMed  Google Scholar 

  2. Chaturvedi N, Klein R, Christensen J, Ahmed J, Kojic A (2010) Algorithms for advanced battery management systems—modeling, estimation, and control challenges for lithium-ion batteries. IEEE Control Syst Mag 30:49–68

    Google Scholar 

  3. Dao TS, Vyasarayani CP, McPhee J (2012) Simplification and order reduction of lithium-ion battery model based on porous-electrode theory. J Power Sources 198:329–337

    CAS  Google Scholar 

  4. Dey S, Ayalew B, Pisu P (2015) Nonlinear robust observers for state-of-charge estimation of lithium-ion cells based on a reduced electrochemical model. IEEE Trans Control Syst Technol 23:1935–1942

    Google Scholar 

  5. Di Domenico D, Stefanopoulou A (2010) Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter. J Dyn Syst Meas Control 132:061302-1–061302-11. https://doi.org/10.1115/1.4002475

    Article  Google Scholar 

  6. Doyle M, Fuller T, Newman J (1993) Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. J Electrochem Soc 140:1526–1533

    CAS  Google Scholar 

  7. Doyle M, Newman J (1996) Comparison of modeling predictions with experimental data from plastic lithium ion cells. J Electrochem Soc 143:1890–1903

    Google Scholar 

  8. Fang H, Wang Y, Sahinoglu Z, Wada T, Hara S (2014) State of charge estimation for lithium-ion batteries: an adaptive approach. Control Eng Pract 25:45–54

    Google Scholar 

  9. Fang H, Zhao X, Wang Y, Sahinoglu Z, Wada T, Hara S, de Callafon RA (2014) Improved adaptive state-of-charge estimation for batteries using a multi-model approach. J Power Sources 254:258–267

    CAS  Google Scholar 

  10. Forman JC, Bashash S, Stein JL, Fathy HK (2011) Reduction of an electrochemistry-based li-ion battery model via quasi-linearization and Padé approximation. J Electrochem Soc 158:A93–A101

    CAS  Google Scholar 

  11. Fuller T, Doyle M, Newman J (1994) Simulation and optimization of the dual lithium ion insertion cell. J Electrochem Soc 141:1–10

    CAS  Google Scholar 

  12. Guo M, Sikha G, White RE (2011) Single-particle model for a lithium-ion cell: thermal behavior. J Electrochem Soc 158:A122–A132

    CAS  Google Scholar 

  13. Han X, Ouyang M, Lu L, Li J (2015) Simplification of physics-based electrochemical model for lithium ion battery on electric vehicle. Part I: diffusion simplification and single particle model. J Power Sources 278:802–813

    CAS  Google Scholar 

  14. Han X, Ouyang M, Lu L, Li J (2015) Simplification of physics-based electrochemical model for lithium ion battery on electric vehicle. Part II: pseudo-two-dimensional model simplification and state of charge estimation. J Power Sources 278:814–825

    CAS  Google Scholar 

  15. Hannan M, Lipu M, Mohamed A (2017) A review of lithium-ion battery state of charge estimation and management system in electric vehicle applications: challenges and recommendations. Renew Sustain Energy Rev 78:834–854

    Google Scholar 

  16. Haran B, Popov B, White R (1998) Determination of the hydrogen diffusion coefficient in metal hydrides by impedance spectroscopy. J Power Sources 75:56–63

    CAS  Google Scholar 

  17. Hu X, Stanton S, Cai L, White R (2012) A linear time-invariant model for solid-phase diffusion in physics-based lithium ion cell models. J Power Souces 214:40–50

    CAS  Google Scholar 

  18. Hu X, Stanton S, Cai L, White R (2012) Model order reduction for solid-phase diffusion in physics-based lithium ion cell models. J Power Sources 218:212–220

    CAS  Google Scholar 

  19. Jacobsen T, West K (1995) Diffusion impedance in planar, cylindrical and spherical geometry. Electrochim Acta 40:255–262

    CAS  Google Scholar 

  20. Jokar A, Rajabloo B, Désilets M, Lacroix M (2016) Review of simplified pseudo-two-dimensional models of lithium-ion batteries. J Power Sources 327:44–55

    CAS  Google Scholar 

  21. Karden E, Ploumen S, Fricke B, Miller T, Snyder K (2007) Energy storage devices for future hybrid electric vehicles. J Power Sources 168:2–11

    CAS  Google Scholar 

  22. Kemper P, Kum D (2013) Extended single particle model of li-ion batteries towards high current applications. In: Proceedings of the 2013 IEEE vehicle power and propulsion conference (VPPC), Beijing, pp 158–163

  23. Klein R, Chaturvedi NA, Christensen J, Ahmed J, Findeisen R, Kojic A (2010) State estimation of a reduced electrochemical model of a lithium-ion battery. In: Proceedings of the American control conference, Baltimore, pp 289–301

  24. Klein R, Chaturvedi NA, Christensen J, Ahmed J, Findeisen R, Kojic A (2013) Electrochemical model based observer design for a lithium-ion battery. IEEE Trans Control Syst Technol 21:289–301

    Google Scholar 

  25. Lee JL, Chemistruck A, Plett GL (2012) One-dimensional physics-based reduced-order model of lithium-ion dynamics. J Power Sources 220:430–448

    CAS  Google Scholar 

  26. Lee T, Filipi Z (2011) Electrochemical Li-Ion battery modeling for control design with optimal uneven discretization. In: Proceedings of the ASME dynamic systems and control conference, Arlington, pp 493–500

  27. Li X, Fan G, Rizzoni G, Canova M, Zhu C, Wei G (2016) A simplified multi-particle model for lithium ion batteries via a predictor-corrector strategy and quasi-linearization. Energy 116:154–169

    CAS  Google Scholar 

  28. Lukic S, Cao J, Bansal R, Rodriguez F, Emadi A (2008) Energy storage systems for automotive applications. IEEE Trans Ind Electron 55:2258–2267

    Google Scholar 

  29. Ma Y, Ru J, Yin M, Chen H, Zheng W (2016) Electrochemical modeling and parameter identification based on bacterial foraging optimization algorithm for lithium-ion batteries. J Appl Electrochem 46:1119–1131

    CAS  Google Scholar 

  30. Majdabadi MM, Farhad S, Farkhondeh M, Fraser RA, Fowler M (2015) Simplified electrochemical multi-particle model for LiFePO4 cathodes in lithium-ion batteries. J Power Sources 275:633–643

    Google Scholar 

  31. Moura S, Chaturvedi N, Krstić M (2012) PDE estimation techniques for advanced battery management systems—Part II: SOH identification. In: Proceedings of American control conference, Quebec, pp 566–571

  32. Perez H, Dey S, Hu X, Moura S (2017) Optimal charging of li-ion batteries via a single particle model with electrolyte and thermal dynamics. J Electrochem Soc 164:A1679–A1687

    CAS  Google Scholar 

  33. Plett GL (2015) Battery modeling, Battery Managemet Systems, vol I. Artech House, Norwood

    Google Scholar 

  34. Rahimian S, Rayman S, White R (2013) Extension of physics-based single particle model for higher charge-discharge rates. J Power Souces 224:180–194

    Google Scholar 

  35. Rahman M, Anwar S, Izadian A (2016) Electrochemical model parameter identification of a lithium-ion battery using particle swarm optimization method. J Power Sources 307:86–97

    CAS  Google Scholar 

  36. Ramachandran S, Khandelwal A, Hariharan KS, Kim BC, Kim KY (2016) Rapid analysis of charging profiles of lithium ion batteries using a hybrid simplified electrochemical model. J Electrochem Soc 163:A1101–A1111

    CAS  Google Scholar 

  37. Ramadesigan V, Boovaragavan V, Pirkle JCJ, Subramanian VR (2010) Efficient reformulation of solid-phase diffusion in physics-based lithium-ion battery models. J Electrochem Soc 157:A854–A860

    CAS  Google Scholar 

  38. Ramadesigan V, Northrop PW, De S, Santhanagopalan S, Braatz RD, Subramanian VR (2012) Modeling and simulation of lithium-ion batteries from a systems engineering perspective. J Electrochem Soc 159:R31–R45

    CAS  Google Scholar 

  39. Rice RG, Do DD (1995) Applied mathematics and modeling for chemical engineers. Wiley, New York

    Google Scholar 

  40. Scrosati B, Garche J (2010) Lithium batteries: status, prospects and future. J Power Sources 195:2419–2430

    CAS  Google Scholar 

  41. Seaman A, Dao TS, McPhee J (2014) A survey of mathematics-based equivalent-circuit and electrochemical battery models for hybrid and electric vehicle suimulation. J Power Sources 256:410–423

    CAS  Google Scholar 

  42. Skeel R, Berzins M (1990) Simultaneous block diagonalization of two real symmetric matrices. SIAM J Sci Stat Comput 11:1–32

    Google Scholar 

  43. Smith K, Rahn C, Wang C (2007) Control oriented 1D electrochemical model of lithium ion battery. Energy Convers Manag 48:2565–2578

    CAS  Google Scholar 

  44. Smith K, Wang CY (2006) Solid-state diffusion limitations on pulse operation of a lithium ion cell for hybrid electric vehicles. J Power Sources 161:628–639

    CAS  Google Scholar 

  45. Smith KA, Rahn CD, Wang CY (2008) Model order reduction of 1D diffusion systems via residue grouping. J Dyn Syst Meas Control 130:011012-1–011012-8

    Google Scholar 

  46. Subramanian VR, Boovaragavan V, Diwakar VD (2007) Towards real-time simulation of physics based lithium-ion battery models. Electrochem Solid State Lett 10:A255–A260

    CAS  Google Scholar 

  47. Subramanian VR, Boovaragavan V, Ramadesigan V, Arabandi M (2009) Mathematical modeling reformulation for lithium-ion battery simulations: galvanostatic boundary conditions. J Electrochem Soc 156:A260–A271

    CAS  Google Scholar 

  48. Subramanian VR, Ritter JA, White RE (2001) Approximate solutions for galvanostatic discharge of spherical particles I. Constant diffusion coefficient. J Electrochem Soc 148:E444–E449

    CAS  Google Scholar 

  49. Subramanian VR, Tikawar VD, Tapriyal D (2005) Efficient macro-micro scale coupled modeling of batteries. J Electrochem Soc 152:A2002–A2008

    CAS  Google Scholar 

  50. Waag W, Fleischer C, Sauer DU (2014) Critical review of the methods for monitoring of lithium-ion batteries in electric and hybrid vehicles. J Power Sources 258:321–339

    CAS  Google Scholar 

  51. Wang Y, Fang H, Sahinoglu Z, Wada T, Hara S (2015) Adaptive estimation of the state of charge of lithium-ion batteries: nonlinear geometric observer approach. IEEE Trans Control Syst Technol 23:948–962

    Google Scholar 

  52. Zeng Y, Albertus P, Klein R, Chaturvedi N, Kojic A, Bazant M, Christensen J (2013) Efficient conservative numerical schemes for 1D nonlinear spherical diffusion equations with applications in battery modeling. J Electrochem Soc 160:A1565–A1571

    CAS  Google Scholar 

  53. Zhang Q, White RE (2007) Comparison of approximate solution methods for the solid phase diffusion equation in a porous electrode model. J Power Sources 165:880–886

    CAS  Google Scholar 

  54. Zuo C, Manzie C, Nešić D (2016) A framework for simplification of PDE-based lithium-ion battery models. IEEE Trans Control Syst Technol 24:1594–1609

    Google Scholar 

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Acknowledgements

The authors gratefully thank the financial support from PAPIIT-UNAM, Grant IN104218; and CONACYT CVU: 557079, and CVU: 229948.

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  1. Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510, Coyoacán CDMX, Mexico

    Fernando A. Ortiz-Ricardez, Aldo Romero-Becerril & Luis Alvarez-Icaza

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Ortiz-Ricardez, F.A., Romero-Becerril, A. & Alvarez-Icaza, L. Hard limitations of polynomial approximations for reduced-order models of lithium-ion cells. J Appl Electrochem 50, 343–354 (2020). https://doi.org/10.1007/s10800-019-01395-y

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