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arxiv: 1907.02775 · v1 · pith:LXBXB7DWnew · submitted 2019-07-05 · ⚛️ physics.app-ph

Asymptotic reduction, solution, and homogenisation of a thermo-electrochemical model for a lithium-ion battery

Matthew G. Hennessy , Iain R. Moyles This is my paper

Pith reviewed 2026-05-25 01:55 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords lithium-ion batterythermo-electrochemical modelasymptotic analysishomogenisationthermal runawayvolume averagingP2D model
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0 comments X

The pith

Asymptotic reduction of a lithium-ion battery model yields accurate solutions up to 2C rates and shows no thermal runaway occurs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a volume-averaged thermo-electrochemical model for lithium-ion batteries and applies scaling analysis to show that heat from electrochemical reactions dominates. Matched asymptotic expansions then produce explicit solutions for constant current, oscillating current, and constant potential operation. These reduced solutions agree closely with full thermal P2D simulations at discharge rates up to 2C. Homogenisation extends the approach to a battery pack of many cells, where the model often admits analytic solutions and demonstrates that the cell potential simply shifts toward the open-circuit value rather than runaway heating.

Core claim

Scaling analysis identifies reaction heat as the leading source, allowing matched asymptotic expansions to solve the volume-averaged model for constant current, oscillating current, and constant potential cases. The resulting solutions match numerical solutions of the thermal P2D model up to 2C rates. Homogenisation then produces a multi-cell battery model that remains free of thermal runaway because the cell potential is driven closer to the open-circuit potential, and the model admits analytic solutions in many operating regimes.

What carries the argument

Matched asymptotic expansions on the volume-averaged thermo-electrochemical equations, followed by homogenisation across multiple cells.

If this is right

  • The volume-averaged model remains accurate for (dis)charge rates up to 2C and gives reasonable results at 4C.
  • Thermal runaway is absent; the cell potential is driven closer to the open-circuit potential instead.
  • The homogenised multi-cell model admits analytic solutions in many cases.
  • The reduced model is suitable for on-board thermal management systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The analytic tractability of the homogenised model suggests it could be embedded directly in real-time battery control algorithms.
  • The absence of runaway under Arrhenius kinetics may extend to other electrochemical systems where reaction heat dominates.
  • Higher-rate regimes where the reduction breaks down could be recovered by retaining additional terms identified in the scaling.

Load-bearing premise

Electrochemical reactions are assumed to be the dominant source of heat generation.

What would settle it

Numerical or experimental solutions of the full thermal P2D model that deviate strongly from the asymptotic predictions at discharge rates of 1C or 2C.

Figures

Figures reproduced from arXiv: 1907.02775 by Iain R. Moyles, Matthew G. Hennessy.

Figure 1
Figure 1. Figure 1: Schematic of a lithium-ion cell consisting of positive and negative electrodes and a separator. During discharge [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulations of galvanostatic discharge processes at various C-rates [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the C-rate I and the cell temperature T during a potentiostatic discharge. The asymptotic solutions in panels (a)–(b) are constructed using only the leading-order contributions; in panels (c)–(d) the leading- and first-order contributions are considered. The resting potential of the cell is ∆V = 3.47 V. The legends apply to all of the panels. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the C-rate I and the cell temperature T during a potentiostatic discharge. The asymptotic solutions in panels (a)–(b) are constructed using only the leading-order contributions; in panels (c)–(d) the leading- and first-order contributions are considered. The resting potential of the cell is ∆V = 3.33 V. The legends apply to all of the panels. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the cell potential and temperature for an oscillating C-rate of the form [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic diagram of an LIB consisting of several repeating identical cells connected by metal current collectors [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the maximum temperature in a battery pack at a discharge rate of 1C. Time as been normalised by [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Temperature increase and cell potential in a battery pack consisting of 60 cells at various discharge rates. The [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
read the original abstract

We study two thermo-electrochemical models for lithium-ion batteries. The first is based on volume averaging the electrode microstructure whereas the second is based on the pseudo-two-dimensional (P2D) approach which treats the electrode as a collection of spherical particles. A scaling analysis is used to reduce the volume-averaged model and show that the electrochemical reactions are the dominant source of heat. Matched asymptotic expansions are used to compute solutions of the volume-averaged model for the cases of constant applied current, oscillating applied current, and constant cell potential. The asymptotic and numerical solutions of the volume-averaged model are in remarkable agreement with numerical solutions of the thermal P2D model for (dis)charge rates up to 2C, and reasonable agreement is found at 4C. Homogenisation is then used to derive a thermal model for a battery consisting of several connected lithium-ion cells. Despite accounting for the Arrhenius dependence of the reaction coefficients, we show that thermal runaway does not occur in the model. Instead, the cell potential is simply pushed closer to the open-circuit potential. We also show that in many cases, the homogenised battery model can be solved analytically, making it ideal for use in on-board thermal management systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops two thermo-electrochemical models for lithium-ion batteries (volume-averaged and thermal P2D). Scaling analysis reduces the volume-averaged model by identifying electrochemical reactions as the dominant heat source. Matched asymptotic expansions then yield solutions for constant current, oscillating current, and constant-potential operation. These reduced solutions show remarkable agreement with full thermal P2D numerics up to 2C (reasonable at 4C). Homogenization produces a multi-cell battery model in which, despite Arrhenius temperature dependence of reaction rates, thermal runaway is absent; cell potential instead approaches the open-circuit value. Many cases of the homogenized model admit closed-form solutions.

Significance. If the scaling reduction and agreement hold, the work supplies a systematic asymptotic framework that delivers computationally cheap models with analytical solutions in multiple regimes, directly relevant to on-board thermal management. The explicit demonstration that Arrhenius dependence alone does not trigger runaway in this reduced setting is a concrete, falsifiable insight. The provision of both asymptotic derivations and numerical cross-validation is a methodological strength.

major comments (2)
  1. [Scaling analysis section] Scaling analysis section: the reduction rests on the claim that electrochemical reaction heat dominates all other sources. The manuscript performs the scaling but supplies no independent order-of-magnitude bounds or parameter sweeps comparing reaction heat to ohmic, reversible, or entropic contributions across the full C-rate range (0–4C). Because this dominance assumption directly justifies both the reduced equations and the subsequent no-runaway conclusion, explicit verification is required.
  2. [Comparison with thermal P2D section] Comparison with thermal P2D section: the central claim of “remarkable agreement” up to 2C and “reasonable agreement” at 4C is stated qualitatively. No L² errors, maximum pointwise deviations, or tabulated discrepancy measures are reported between the asymptotic solutions and the reference numerics. Without such quantification the strength of the validation cannot be assessed precisely.
minor comments (2)
  1. [Homogenization section] Notation for the homogenized multi-cell model should be introduced with a clear table of symbols, especially the effective thermal conductivity and inter-cell coupling terms.
  2. [Figures] Figure captions for the asymptotic–numerical comparisons should state the exact C-rates, initial conditions, and whether the plotted quantities are dimensional or nondimensional.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our work and for the constructive major comments. We will revise the manuscript to incorporate explicit verification of the scaling assumptions via parameter sweeps and to provide quantitative error metrics for the model comparisons.

read point-by-point responses

  1. Referee: Scaling analysis section: the reduction rests on the claim that electrochemical reaction heat dominates all other sources. The manuscript performs the scaling but supplies no independent order-of-magnitude bounds or parameter sweeps comparing reaction heat to ohmic, reversible, or entropic contributions across the full C-rate range (0–4C). Because this dominance assumption directly justifies both the reduced equations and the subsequent no-runaway conclusion, explicit verification is required.

    Authors: We agree with the referee that additional verification would strengthen the paper. In the revised manuscript, we will add a new figure or table showing parameter sweeps over C-rates 0-4C, comparing the magnitudes of reaction heat, ohmic heating, reversible heat, and entropic contributions using the model parameters. This will provide independent confirmation of the dominance assumption used in the scaling analysis. revision: yes

  2. Referee: Comparison with thermal P2D section: the central claim of “remarkable agreement” up to 2C and “reasonable agreement” at 4C is stated qualitatively. No L² errors, maximum pointwise deviations, or tabulated discrepancy measures are reported between the asymptotic solutions and the reference numerics. Without such quantification the strength of the validation cannot be assessed precisely.

    Authors: We accept that quantitative error measures are necessary for a rigorous validation. We will include in the revised version calculations of L² norms, maximum absolute errors, and relative errors for key variables (temperature, concentrations, potentials) at different C-rates, presented in a table comparing the asymptotic solutions to the full numerical solutions of the thermal P2D model. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling and asymptotics are applied to established models without self-referential reduction.

full rationale

The paper performs a scaling analysis on the volume-averaged thermo-electrochemical model to identify dominant heat generation terms from the governing equations, followed by matched asymptotic expansions for constant/oscillating current and constant-potential cases. These are then compared to numerical solutions of the independent thermal P2D model, with homogenisation applied to multi-cell batteries. No quoted step reduces a prediction or central result to a fitted input by construction, nor does any load-bearing premise rest on a self-citation chain or imported uniqueness theorem. The Arrhenius dependence and no-runaway conclusion emerge from solving the reduced system rather than being presupposed. The derivation chain is self-contained against the input models.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract provides limited information; no explicit free parameters or invented entities are detailed. The scaling analysis invokes a domain assumption about dominant heat sources.

axioms (2)
  • domain assumption Electrochemical reactions are the dominant source of heat in the volume-averaged model
    Invoked via scaling analysis to justify model reduction.
  • standard math Matched asymptotic expansions apply to the reduced model equations
    Used to obtain solutions for constant current, oscillating current, and constant potential cases.

pith-pipeline@v0.9.0 · 5752 in / 1394 out tokens · 31929 ms · 2026-05-25T01:55:09.130706+00:00 · methodology

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Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages

  1. [1]

    Tarascon, M

    J.-M. Tarascon, M. Armand, Issues and challenges facing rechargeable lithium batteries, Nature 414 (6861) (2001) 359–367

    work page 2001

  2. [2]

    Abada, G

    S. Abada, G. Marlair, A. Lecocq, M. Petit, V. Sauvant-Moynot, F. Huet, Safety focused modeling of lithium-ion batteries: A review, Journal of Power Sources 306 (2016) 178–192

    work page 2016

  3. [3]

    Waldmann, M

    T. Waldmann, M. Wilka, M. Kasper, M. Fleischhammer, M. Wohlfahrt-Mehrens, Temperature dependent ageing mechanisms in Lithium-ion batteries–A Post-Mortem study, Journal of Power Sources 262 (2014) 129–135

    work page 2014

  4. [4]

    Ramadass, B

    P. Ramadass, B. Haran, R. White, B. N. Popov, Capacity fade of Sony 18650 cells cycled at elevated temperatures: Part I. Cycling performance, Journal of Power Sources 112 (2) (2002) 606–613

    work page 2002

  5. [5]

    Ramadass, B

    P. Ramadass, B. Haran, R. White, B. N. Popov, Capacity fade of Sony 18650 cells cycled at elevated temperatures: Part II. Capacity fade analysis, Journal of Power Sources 112 (2) (2002) 614–620

    work page 2002

  6. [6]

    T. M. Bandhauer, S. Garimella, T. F. Fuller, A critical review of thermal issues in lithium-ion batteries, Journal of the Electrochemical Society 158 (3) (2011) R1–R25

    work page 2011

  7. [7]

    J. Wen, Y. Yu, C. Chen, A review on lithium-ion batteries safety issues: existing problems and possible solutions, Materials Express 2 (3) (2012) 197–212

    work page 2012

  8. [8]

    Lisbona, T

    D. Lisbona, T. Snee, A review of hazards associated with primary lithium and lithium-ion batteries, Process Safety and Environmental Protection 89 (6) (2011) 434–442

    work page 2011

  9. [9]

    R. Zhao, S. Zhang, J. Liu, J. Gu, A review of thermal performance improving methods of lithium ion battery: electrode modification and thermal management system, Journal of Power Sources 299 (2015) 557–577

    work page 2015

  10. [10]

    Musk, Model S fire, https://www.tesla.com/en_CA/blog/model-s-fire?redirect=no

    E. Musk, Model S fire, https://www.tesla.com/en_CA/blog/model-s-fire?redirect=no. Accessed June 24, 2019. (2013)

    work page 2019

  11. [11]

    Rawlinson, Vehicle battery pack ballistic shield, https://worldwide.espacenet.com/publicationDetails/biblio? CC=US&NR=8286743&KC=&FT=E&locale=en_EP (2012)

    P. Rawlinson, Vehicle battery pack ballistic shield, https://worldwide.espacenet.com/publicationDetails/biblio? CC=US&NR=8286743&KC=&FT=E&locale=en_EP (2012)

    work page 2012

  12. [12]

    Seaman, T.-S

    A. Seaman, T.-S. Dao, J. McPhee, A survey of mathematics-based equivalent-circuit and electrochemical battery models for hybrid and electric vehicle simulation, Journal of Power Sources 256 (2014) 410–423

    work page 2014

  13. [13]

    R. B. Smith, E. Khoo, M. Z. Bazant, Intercalation kinetics in multiphase-layered materials, The Journal of Physical Chemistry C 121 (23) (2017) 12505–12523

    work page 2017

  14. [14]

    Dargaville, T

    S. Dargaville, T. W. Farrell, Predicting active material utilization in LiFePO4 electrodes using a multiscale mathe- matical model, Journal of the Electrochemical Society 157 (7) (2010) A830–A840

    work page 2010

  15. [15]

    Hosseinzadeh, J

    E. Hosseinzadeh, J. Marco, P. Jennings, The impact of multi-layered porosity distribution on the performance of a lithium ion battery, Applied Mathematical Modelling 61 (2018) 107–123

    work page 2018

  16. [16]

    Chakraborty, C

    J. Chakraborty, C. P. Please, A. Goriely, S. J. Chapman, Combining mechanical and chemical effects in the deformation and failure of a cylindrical electrode particle in a Li-ion battery, International Journal of Solids and Structures 54 (2015) 66–81

    work page 2015

  17. [17]

    J. M. Foster, S. J. Chapman, G. Richardson, B. Protas, A mathematical model for mechanically-induced deterioration of the binder in lithium-ion electrodes, SIAM Journal on Applied Mathematics 77 (6) (2017) 2172–2198

    work page 2017

  18. [18]

    R. B. Smith, M. Z. Bazant, Multiphase porous electrode theory, Journal of the Electrochemical Society 164 (11) (2017) E3291–E3310

    work page 2017

  19. [19]

    T. R. Ferguson, M. Z. Bazant, Nonequilibrium thermodynamics of porous electrodes, Journal of the Electrochemical Society 159 (12) (2012) A1967–A1985

    work page 2012

  20. [20]

    T. R. Ferguson, M. Z. Bazant, Phase transformation dynamics in porous battery electrodes, Electrochimica Acta 146 (2014) 89–97

    work page 2014

  21. [21]

    Orvananos, T

    B. Orvananos, T. R. Ferguson, H.-C. Yu, M. Z. Bazant, K. Thornton, Particle-level modeling of the charge-discharge behavior of nanoparticulate phase-separating li-ion battery electrodes, Journal of the Electrochemical Society 161 (4) (2014) A535–A546

    work page 2014

  22. [22]

    J. Li, Y. Cheng, M. Jia, Y. Tang, Y. Lin, Z. Zhang, Y. Liu, An electrochemical–thermal model based on dynamic responses for lithium iron phosphate battery, Journal of Power Sources 255 (2014) 130–143

    work page 2014

  23. [23]

    J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-s. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, W. C. Chueh, Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles, Science 353 (6299) (2016) 566–571

    work page 2016

  24. [24]

    F. Font, B. Protas, G. Richardson, J. M. Foster, Binder migration during drying of lithium-ion battery electrodes: Modelling and comparison to experiment, Journal of Power Sources 393 (2018) 177–185. 38

    work page 2018

  25. [25]

    A. K. Sethurajan, S. A. Krachkovskiy, I. C. Halalay, G. R. Goward, B. Protas, Accurate characterization of ion transport properties in binary symmetric electrolytes using in situ nmr imaging and inverse modeling, The Journal of Physical Chemistry B 119 (37) (2015) 12238–12248

    work page 2015

  26. [26]

    S. A. Krachkovskiy, J. M. Foster, J. D. Bazak, B. J. Balcom, G. R. Goward, Operando mapping of Li concentration profiles and phase transformations in graphite electrodes by MRI and NMR, The Journal of Physical Chemistry C 122 (38) (2018) 21784–21791

    work page 2018

  27. [27]

    J. S. Newman, C. W. Tobias, Theoretical analysis of current distribution in porous electrodes, Journal of the Electro- chemical Society 109 (12) (1962) 1183–1191

    work page 1962

  28. [28]

    Newman, W

    J. Newman, W. Tiedemann, Porous-electrode theory with battery applications, AIChE Journal 21 (1) (1975) 25–41

    work page 1975

  29. [29]

    Doyle, T

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

    work page 1993

  30. [30]

    Jokar, B

    A. Jokar, B. Rajabloo, M. D´ esilets, M. Lacroix, Review of simplified Pseudo-two-Dimensional models of lithium-ion batteries, Journal of Power Sources 327 (2016) 44–55

    work page 2016

  31. [31]

    Santhanagopalan, Q

    S. Santhanagopalan, Q. Guo, P. Ramadass, R. E. White, Review of models for predicting the cycling performance of lithium ion batteries, Journal of Power Sources 156 (2) (2006) 620–628

    work page 2006

  32. [32]

    C. R. Pals, J. Newman, Thermal modeling of the lithium/polymer battery i. discharge behavior of a single cell, Journal of the Electrochemical Society 142 (10) (1995) 3274–3281

    work page 1995

  33. [33]

    W. Gu, C. Wang, Thermal-electrochemical modeling of battery systems, Journal of the Electrochemical Society 147 (8) (2000) 2910–2922

    work page 2000

  34. [34]

    Kumaresan, G

    K. Kumaresan, G. Sikha, R. E. White, Thermal model for a Li-ion cell, Journal of the Electrochemical Society 155 (2) (2008) A164–A171

    work page 2008

  35. [35]

    R. Zhao, J. Liu, J. Gu, Simulation and experimental study on lithium ion battery short circuit, Applied Energy 173 (2016) 29–39

    work page 2016

  36. [36]

    C. H. Lee, S. J. Bae, M. Jang, A study on effect of lithium ion battery design variables upon features of thermal- runaway using mathematical model and simulation, Journal of Power Sources 293 (2015) 498–510

    work page 2015

  37. [37]

    T. G. Zavalis, M. Behm, G. Lindbergh, Investigation of short-circuit scenarios in a lithium-ion battery cell, Journal of the Electrochemical Society 159 (6) (2012) A848–A859

    work page 2012

  38. [38]

    X. Feng, L. Lu, M. Ouyang, J. Li, X. He, A 3d thermal runaway propagation model for a large format lithium ion battery module, Energy 115 (2016) 194–208

    work page 2016

  39. [39]

    Melcher, C

    A. Melcher, C. Ziebert, M. Rohde, H. J. Seifert, Modeling and simulation the thermal runaway behavior of cylindrical li-ion cellscomputing of critical parameter, Energies 9 (4) (2016) 292

    work page 2016

  40. [40]

    Q. Wang, P. Ping, X. Zhao, G. Chu, J. Sun, C. Chen, Thermal runaway caused fire and explosion of lithium ion battery, Journal of Power Sources 208 (2012) 210–224

    work page 2012

  41. [41]

    G.-H. Kim, A. Pesaran, R. Spotnitz, A three-dimensional thermal abuse model for lithium-ion cells, Journal of Power Sources 170 (2) (2007) 476–489

    work page 2007

  42. [42]

    K.-J. Lee, K. Smith, A. Pesaran, G.-H. Kim, Three dimensional thermal-, electrical-, and electrochemical-coupled model for cylindrical wound large format lithium-ion batteries, Journal of Power Sources 241 (2013) 20–32

    work page 2013

  43. [43]

    Z. An, L. Jia, L. Wei, C. Dang, Q. Peng, Investigation on lithium-ion battery electrochemical and thermal characteristic based on electrochemical-thermal coupled model, Applied Thermal Engineering 137 (2018) 792–807

    work page 2018

  44. [44]

    Smyshlyaev, M

    A. Smyshlyaev, M. Krstic, N. Chaturvedi, J. Ahmed, A. Kojic, Pde model for thermal dynamics of a large li-ion battery pack, in: Proceedings of the 2011 American Control Conference, IEEE, 2011, pp. 959–964

    work page 2011

  45. [45]

    L. Cai, R. E. White, Mathematical modeling of a lithium ion battery with thermal effects in comsol inc. multiphysics (mp) software, Journal of Power Sources 196 (14) (2011) 5985–5989

    work page 2011

  46. [46]

    Di Domenico, G

    D. Di Domenico, G. Fiengo, A. Stefanopoulou, Lithium-ion battery state of charge estimation with a Kalman filter based on a electrochemical model, in: Proceedings of the 17th IEEE International Conference on Control Applications, Ieee, 2008, pp. 702–707

    work page 2008

  47. [47]

    Speltino, D

    C. Speltino, D. Di Domenico, G. Fiengo, A. Stefanopoulou, Comparison of reduced order lithium-ion battery models for control applications, in: Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, IEEE, 2009, pp. 3276–3281

    work page 2009

  48. [48]

    C. Wang, W. Gu, B. Liaw, Micro-macroscopic coupled modeling of batteries and fuel cells i. model development, Journal of the Electrochemical Society 145 (10) (1998) 3407–3417

    work page 1998

  49. [49]

    V. R. Subramanian, V. D. Diwakar, D. Tapriyal, Efficient macro-micro scale coupled modeling of batteries, Journal of the Electrochemical Society 152 (10) (2005) A2002–A2008

    work page 2005

  50. [50]

    V. R. Subramanian, V. Boovaragavan, V. D. Diwakar, Toward real-time simulation of physics based lithium-ion battery models, Electrochemical and Solid-State Letters 10 (11) (2007) A255–A260. 39

    work page 2007

  51. [51]

    G. Fan, X. Li, M. Canova, A reduced-order electrochemical model of Li-ion batteries for control and estimation applications, IEEE Transactions on Vehicular Technology 67 (1) (2018) 76–91

    work page 2018

  52. [52]

    T.-S. Dao, C. P. Vyasarayani, J. McPhee, Simplification and order reduction of lithium-ion battery model based on porous-electrode theory, Journal of Power Sources 198 (2012) 329–337

    work page 2012

  53. [53]

    V. R. Subramanian, V. Boovaragavan, V. Ramadesigan, M. Arabandi, Mathematical model reformulation for lithium- ion battery simulations: Galvanostatic boundary conditions, Journal of the Electrochemical Society 156 (4) (2009) A260–A271

    work page 2009

  54. [54]

    K. A. Smith, C. D. Rahn, C.-Y. Wang, Model-based electrochemical estimation and constraint management for pulse operation of lithium ion batteries, IEEE Transactions on Control Systems Technology 18 (3) (2010) 654–663

    work page 2010

  55. [55]

    A. M. Bizeray, S. Zhao, S. R. Duncan, D. A. Howey, Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter, Journal of Power Sources 296 (2015) 400–412

    work page 2015

  56. [56]

    Marcicki, M

    J. Marcicki, M. Canova, A. T. Conlisk, G. Rizzoni, Design and parametrization analysis of a reduced-order electro- chemical model of graphite/LiFePO4 cells for SOC/SOH estimation, Journal of Power Sources 237 (2013) 310–324

    work page 2013

  57. [57]

    N. T. Tran, M. Vilathgamuwa, T. Farrell, S. S. Choi, Y. Li, J. Teague, A Pad´ e approximate model of lithium ion batteries, Journal of the Electrochemical Society 165 (7) (2018) A1409–A1421

    work page 2018

  58. [58]

    Forgez, D

    C. Forgez, D. V. Do, G. Friedrich, M. Morcrette, C. Delacourt, Thermal modeling of a cylindrical lifepo4/graphite lithium-ion battery, Journal of Power Sources 195 (9) (2010) 2961–2968

    work page 2010

  59. [59]

    B. Wu, V. Yufit, M. Marinescu, G. J. Offer, R. F. Martinez-Botas, N. P. Brandon, Coupled thermal–electrochemical modelling of uneven heat generation in lithium-ion battery packs, Journal of Power Sources 243 (2013) 544–554

    work page 2013

  60. [60]

    Al Hallaj, H

    S. Al Hallaj, H. Maleki, J.-S. Hong, J. R. Selman, Thermal modeling and design considerations of lithium-ion batteries, Journal of power sources 83 (1-2) (1999) 1–8

    work page 1999

  61. [61]

    Richardson, G

    G. Richardson, G. Denuault, C. Please, Multiscale modelling and analysis of lithium-ion battery charge and discharge, Journal of Engineering Mathematics 72 (1) (2012) 41–72

    work page 2012

  62. [62]

    Sulzer, S

    V. Sulzer, S. J. Chapman, C. P. Please, D. A. Howey, C. W. Monroe, Faster lead-acid battery simulations from porous-electrode theory: II. asymptotic analysis, arXiv preprint arXiv:1902.01774

    work page arXiv 1902

  63. [63]

    S. G. Marquis, V. Sulzer, R. Timms, C. P. Please, S. J. Chapman, An asymptotic derivation of a single particle model with electrolyte, arXiv preprint arXiv:1905.12553

    work page arXiv 1905

  64. [64]

    I. R. Moyles, M. G. Hennessy, T. G. Myers, B. R. Wetton, Asymptotic reduction of a porous electrode model for lithium-ion batteries, SIAM Journal on Applied Mathematics in press

    work page

  65. [65]

    Amiribavandpour, W

    P. Amiribavandpour, W. Shen, D. Mu, A. Kapoor, An improved theoretical electrochemical-thermal modelling of lithium-ion battery packs in electric vehicles, Journal of Power Sources 284 (2015) 328–338

    work page 2015

  66. [66]

    Newman, K

    J. Newman, K. E. Thomas-Alyea, Electrochemical Systems, John Wiley & Sons, 2004

    work page 2004

  67. [67]

    K. E. Thomas-Alyea, C. Jung, R. B. Smith, M. Z. Bazant, In situ observation and mathematical modeling of lithium distribution within graphite, Journal of the Electrochemical Society 164 (11) (2017) E3063–E3072

    work page 2017

  68. [68]

    Kaviany, Principles of heat transfer in porous media, Springer Science & Business Media, 2012

    M. Kaviany, Principles of heat transfer in porous media, Springer Science & Business Media, 2012

    work page 2012

  69. [69]

    Accessed February 26, 2018

    A123systems, Nanophosphate high power lithium ion cell anr26650m1a, https://www.rcgroups.com/forums/showatt.php?attachmentid=4155818. Accessed February 26, 2018. (2010)

    work page 2018

  70. [70]

    Biesheuvel, M

    P. Biesheuvel, M. Bazant, Nonlinear dynamics of capacitive charging and desalination by porous electrodes, Physical review E 81 (3) (2010) 031502

    work page 2010

  71. [71]

    S. J. Chapman, S. Mcburnie, Integral constraints in multiple-scales problems, European Journal of Applied Mathe- matics 26 (5) (2015) 595–614

    work page 2015

  72. [72]

    Srinivasan, J

    V. Srinivasan, J. Newman, Discharge model for the lithium iron-phosphate electrode, Journal of the Electrochemical Society 151 (10) (2004) A1517–A1529. 40

    work page 2004