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Battery discharge follows a universal power-efficiency parabola with maximum power at half efficiency, and the optimal multistage schedule is simply max(load minimum, shared baseline current).

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2026-07-12 04:27 UTC pith:62KSSLD2

load-bearing objection Clean Ohmic FTT baseline for discharging plus a compact, load-aware KKT schedule that is not just the reverse of their charging paper.

arxiv 2607.03157 v1 pith:62KSSLD2 submitted 2026-07-03 cond-mat.stat-mech physics.class-ph

Finite-Time Thermodynamics of Battery Discharging: Power-Efficiency Trade-Off and Optimization

Rui-Han Liu , Yun-Qian Lin , Yu-Han Ma This is my paper
classification cond-mat.stat-mech physics.class-ph
keywords battery management systembattery dischargingfinite-time thermodynamicspower-efficiency trade-offmultistage constant-current dischargingPareto frontOhmic dissipation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Discharging a battery always costs heat inside the cell. The paper shows that this cost forces a universal trade-off: output power versus efficiency always lies under the same parabola P proportional to η(1−η), so the efficiency that maximises power is exactly one half. That half-efficiency point is the electrochemical twin of the half-Carnot limit familiar from finite-time heat engines. From the same physics the authors derive a practical multistage schedule that must satisfy both instantaneous load demands and a global deadline. The mathematically optimal currents turn out to be extremely simple: run each stage at the lowest current the load will accept, or raise every unconstrained stage to one common baseline fixed by the deadline. The resulting heat-versus-time curve is a Pareto front whose “knee” moves outward as internal resistance grows. The construction therefore supplies a model-independent thermodynamic floor that any battery-management scheduler can use as a baseline before adding temperature, state-of-charge, or ageing corrections.

Core claim

Across three models of increasing realism—constant electromotive force, finite-capacitance RC dynamics, and active constant-current control—battery discharge obeys the identical parabolic envelope P ∝ η(1−η) with efficiency at maximum power fixed at exactly ½. When the same Ohmic dissipation is optimised under stage-wise load lower bounds and a global deadline, the Karush–Kuhn–Tucker conditions yield the closed-form policy I_i^* = max(I_i^−, I_0), where I_0 is the single free parameter set by the deadline equality.

What carries the argument

The Ohmic identity Q = r I Δq together with the KKT stationarity condition that forces all unconstrained stage currents to a common value I_0 = √(λ/r). Complementary slackness then pins demand-limited stages to their minimum admissible currents, producing the compact rule I_i^* = max(I_i^−, I_0).

Load-bearing premise

Internal resistance and capacitance are treated as fixed numbers that do not change with temperature, state of charge, or current; the entire parabola and the optimal schedule rest on pure I-squared-R heating.

What would settle it

Measure average power and efficiency while discharging a real cell at several constant currents over the same voltage window; if the data deviate systematically from the predicted parabola or if the efficiency at peak power is not ½, the claimed universality fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any battery-management scheduler can adopt the closed-form rule I_i = max(load minimum, shared baseline) without solving a new optimisation at every step.
  • The dissipation–time Pareto front supplies a quantitative floor against which more elaborate electrochemical models can be benchmarked.
  • Raising internal resistance both elevates the entire heat surface and pushes the knee of the trade-off toward longer discharge times, giving a direct thermodynamic signature of ageing.
  • Active constant-current control always dissipates less heat than passive resistive discharge over the same voltage window and total time.

Where Pith is reading between the lines

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

  • Because the same half-efficiency bound appears for both charging and discharging, a single thermodynamic length argument may govern the entire charge–discharge cycle once temperature dependence is restored.
  • The dual upper-bound form that appears under an efficiency floor (I_i = min(I_c, I_η)) suggests that power and efficiency constraints can be toggled by simply swapping max for min inside the same code base.
  • If real cells obey the predicted Q–τ surface even approximately, online BMS algorithms could estimate internal resistance from the location of the observed knee without extra sensors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The manuscript applies finite-time thermodynamics to battery discharging. Across three models (constant EMF, finite-capacitance RC with fixed power demand, and constant-current discharge), it derives a universal parabolic power–efficiency envelope P ∝ η(1 − η) with efficiency at maximum power exactly η_EMP = 1/2. It then formulates multistage constant-current discharging (MSCD) under stagewise load lower bounds and a global deadline, solves the problem via KKT conditions, and obtains the compact policy I_i^* = max(I_i^−, I_0) with I_0 fixed by the deadline equality. Complementary efficiency-upper-bound and load-lower-bound feasibility maps, a Q–τ Pareto front, and a three-dimensional surface over internal resistance are constructed, with a numerical PHEV-profile example illustrating heat reduction relative to a peak-current benchmark.

Significance. If the results hold under the stated Ohmic idealization, the paper supplies a clean, analytically closed thermodynamic baseline for the scheduling layer of battery management systems. The parabolic envelope and half-reversible EMP are elementary but universal consequences of pure Joule dissipation; the KKT policy is compact, falsifiable, and directly dual to the authors’ earlier charging work. Explicit derivation of stationarity and complementary slackness, the unconstrained Cauchy–Schwarz bound, and the dual efficiency-constrained form are strengths that make the claims reproducible without numerical black boxes. The constant-r,C assumption is the principal idealization, but it is flagged for future work and does not undermine the baseline analytic results.

minor comments (5)
  1. In Sec. 2.3 the mean voltage is written V ≡ (V0 + Vf)/2, while later stages use Vi; a single consistent notation for stage-end versus process-averaged voltage would reduce momentary ambiguity when reading Eqs. (11)–(12) against (22).
  2. Figure 4 caption and surrounding text refer to “accumulated Joule heat Qdiss(t)”; the main text uses Q. Aligning the symbol would improve consistency.
  3. The numerical example (Sec. 3.2) cites r = 0.08 Ω and C = 1.103 × 10^4 F without stating the precise literature source for the effective capacitance; a short parenthetical or reference would aid reproducibility.
  4. In the feasibility maps (Fig. 5) the axes labels “max_i P_req_i / P_max,i” and “τ_P_min / τ_max” are clear, but a one-sentence reminder in the caption that the relative load shape is held fixed while only the overall amplitude is scaled would help readers who skip Sec. 3.3.
  5. A few typographical inconsistencies appear (e.g., “efficiency” with ligature artifacts in the abstract and keywords, and occasional spacing around “I_i^*”). These are purely cosmetic.

Circularity Check

0 steps flagged

No significant circularity: parabolic envelope and KKT policy are derived directly from Ohmic power balance and standard constrained optimization.

full rationale

The central results follow by direct algebra from the stated circuit models and by standard KKT analysis of an explicitly written objective plus constraints; nothing is forced by construction from a fitted parameter or from a self-citation that itself encodes the target claim. In Sec. 2 the three models begin from the instantaneous power balance (chemical power = load power + I^{2}r) and the definition η = P/(VI); eliminating I immediately produces the parabola P ∝ η(1-η) and the stationary point η_EMP = 1/2 (Eqs. 1–4, 5–9, 11–12). The same algebraic identity reappears for constant-current discharge once the mean open-circuit voltage is substituted. Section 3 formulates the MSCD problem as minimize ∑ r Δq_i I_i subject to ∑ Δq_i/I_i ≤ τ_max and I_i ≥ I_i^-, writes the Lagrangian, and obtains the stationarity and complementary-slackness conditions that yield I_i^* = max(I_i^-, I_0) with I_0 fixed by the deadline equality (Eqs. 24–27). The numerical illustration uses literature values of r, C and a published PHEV power profile; no free parameter is adjusted to reproduce a target efficiency or heat. The sole self-citation to the authors’ prior charging paper [42] is used only for contrast (MSCC versus MSCD) and is not an input that forces the discharging optimum. The constant-r,C idealization is an explicit modeling assumption flagged for future relaxation, not a circular definition. Consequently the derivation chain is self-contained against its own premises.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central analytic claims rest only on standard circuit laws, Ohmic dissipation, and convex optimization. Free parameters appear solely in the illustrative numerical example and do not enter the derivation of the parabola or the KKT policy. No new physical entities are postulated.

free parameters (3)
  • internal resistance r (numerical example) = 0.08 Ω (baseline)
    Fixed at 0.08 Ω for the five-stage PHEV illustration and swept over [0.04, 0.10] Ω for the surface; chosen from literature ranges, not fitted to new data, but still a free modeling choice that sets absolute heat scale.
  • effective capacitance C = 1.103e4 F
    Set to 1.103×10^4 F to match 3.68 Ah over 4.2–3.0 V; determines absolute times and heat but cancels in normalized efficiency.
  • stage voltage grid and load powers = P_req = {35.71, 32.14, 20.09, 16.07, 8.04} W
    Voltage breakpoints {4.2, 4.0, 3.75, 3.5, 3.25, 3.0} V and BSF-scaled powers from the DOE/INL profile are hand-chosen for the example; they define the numerical I_i^- and the Pareto knee but not the analytic policy form.
axioms (5)
  • domain assumption Dissipation is purely Ohmic: instantaneous heat power = I² r with constant r.
    Used from Eq. (1) onward; produces the quadratic term that yields the parabola and the linear heat sum Q = Σ r Δq_i I_i.
  • domain assumption Kirchhoff voltage law and charge conservation on an equivalent-circuit battery (constant EMF or finite capacitor C).
    Sections 2.1–2.3; standard circuit modeling of batteries.
  • standard math Karush–Kuhn–Tucker stationarity and complementary slackness characterize the global optimum of the convex MSCD program.
    Section 3.2, Lagrangian (24) and stationarity (25); standard convex optimization.
  • domain assumption Stagewise load power P_req_i is known a priori and constant within each stage.
    Defines the lower bounds I_i^- via Eq. (22); required for the closed-form max policy.
  • standard math Unconstrained minimum-dissipation bound follows from Cauchy–Schwarz (or thermodynamic length).
    Section 3.1; used only as a baseline, not for the constrained optimum.

pith-pipeline@v1.1.0-grok45 · 18027 in / 3212 out tokens · 32959 ms · 2026-07-12T04:27:51.565767+00:00 · methodology

0 comments
read the original abstract

Battery discharging is governed by a fundamental trade-off between output power and energy conversion efficiency due to internal dissipation. In this paper, we demonstrate that such a trade-off universally yields a parabolic envelope $P\propto\eta(1-\eta)$. The efficiency at maximum power is exactly one half, mirroring the well-known half-Carnot limit in finite-time thermodynamics. To extend this bound into practical operational rules, we formulate a multistage constant-current discharging (MSCD) schedule subject to simultaneous real-time load demands and a global discharging deadline. Analytical resolution via the Karush--Kuhn--Tucker conditions reveals a remarkably compact optimal policy: $I_{i}^{\star}=\max(I_{i}^{-},I_{0})$. Under this rule, stages limited by external demand run exactly at their minimum required currents, while all remaining stages are elevated to a uniform baseline $I_{0}$ fixed by the deadline constraint. By tracing the dissipation--time Pareto front, we quantify how internal resistance shifts the operational boundaries and sharpens the trade-off corner. This analysis establishes a rigorous thermodynamic baseline for the scheduling layer of battery management systems, offering natural extensions to nonlinear models incorporating temperature and state-of-charge dependencies.

Figures

Figures reproduced from arXiv: 2607.03157 by Rui-Han Liu, Yu-Han Ma, Yun-Qian Lin.

Figure 1
Figure 1. Figure 1: Equivalent-circuit schematics for the discharge models. (a) the constant electromotive force model, with an ideal source in series with internal resistance r driving a resistive load RL. Middle: the finite-capacitance model, where a capacitor with internal resistance r is coupled to a DC-DC converter that regulates the cell current I(t). (b) the prescribed external conditions used in the analysis, namely f… view at source ↗
Figure 2
Figure 2. Figure 2: plots Eq. (12) with a series of cut-off ratios Vf /V0. The curves exhibit the same parabolic dependence on η: the curves vanish at η = 0 and 1, and peak at η = 1/2. As Vf /V0 increases, the curve is lifted due to the larger vertical scale V 2 /r. Therefore, a shallower discharge permits a higher average power at fixed efficiency, which is consistent with the per￾formance improvement observed in pulsed-disc… view at source ↗
Figure 3
Figure 3. Figure 3: Heat ratio f(x) = Qdiss,active/Qdiss,passive is the function of the voltage ratio x = Vf /V0. The green region is the saving fraction of dissipation 1 − f. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of two discharging protocols under a stepwise dynamic load. (a) Load￾power demand Pload(t). (b) The discharging current applied in MSCD (green solid line) and CC discharge (red dashed line). (c) Accumulated Joule heat Qdiss(t) during discharging. As a practical five-stage example, we use the positive discharge-power steps of the DOE/INL PHEV charge-depleting cycle-life test profile [60]. The sys… view at source ↗
Figure 5
Figure 5. Figure 5: Feasibility maps for the two stagewise bounds in MSCD. (a) A minimum instantaneous efficiency η0 sets current upper bounds I η i . The solid and dashed curves denote τmax = τ η min and Ic = mini I η i , respectively. They divide the plane into the infeasible region (I), bounded￾MSCD region (II), and CC region (III). (b) In the load-power-constrained case, prescribed stage powers set current lower bounds I … view at source ↗
Figure 6
Figure 6. Figure 6: Three-dimensional Pareto surface QMSCD(τ, r) obtained by sweeping the deadline for r ∈ [40, 100] mΩ, using P req i = {35.71, 32.14, 20.09, 16.07, 8.04}W. Color encodes Qdiss. The blue ridge marks the baseline r = 80 mΩ, and the orange dashed locus traces the corner of each r-slice. 4 Summary and Outlook In this paper, we investigated battery discharging from a finite-time thermodynamic perspec￾tive. By eva… view at source ↗

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Works this paper leans on

70 extracted references · 2 canonical work pages

  1. [1]

    W. Liu, T. Placke, and K. Chau, Energy Rep. 8, 4058 (2022)

    2022

  2. [2]

    R. R. Kumar, C. Bharatiraja, K. Udhayakumar, S. Devakirubakaran, K. S. Sekar, and L. Mihet-Popa, IEEE Access 11, 105761 (2023) . 13

    2023

  3. [3]

    R. Guo, L. Lu, M. Ouyang, and X. Feng, Sci. Rep. 6, 30248 (2016)

    2016

  4. [4]

    Maleki and J

    H. Maleki and J. N. Howard, J. Power Sources 160, 1395 (2006)

    2006

  5. [5]

    C. P. Grey and D. S. Hall, Nat. Commun. 11, 6279 (2020)

    2020

  6. [6]

    Usiskin, Y

    R. Usiskin, Y. Lu, J. Popovic, M. Law, P. Balaya, Y.-S. Hu, and J. Maier, Nat. Rev. Mater. 6, 1020 (2021)

    2021

  7. [7]

    Janek and W

    J. Janek and W. G. Zeier, Nat. Energy 8, 230 (2023)

    2023

  8. [8]

    Y. Liu, Y. Zhu, and Y. Cui, Nat. Energy 4, 540 (2019)

    2019

  9. [9]

    Borah, F

    R. Borah, F. Hughson, J. Johnston, and T. Nann, Mater. Today Adv. 6, 100046 (2020)

    2020

  10. [10]

    J. He, J. Meng, and Y. Huang, J. Power Sources 570, 232965 (2023)

    2023

  11. [11]

    Z. He, Y. Huang, H. Liu, Z. Geng, Y. Li, S. Li, W. Deng, G. Zou, H. Hou, and X. Ji, Nano Energy 129, 109996 (2024)

    2024

  12. [12]

    T. F. Fuller, M. Doyle, and J. Newman, J. Electrochem. Soc. 141, 982 (1994)

    1994

  13. [13]

    T. T. Vo, X. Chen, W. Shen, and A. Kapoor, J. Power Sources 273, 413 (2015)

    2015

  14. [14]

    Q. Lin, J. Wang, R. Xiong, W. Shen, and H. He, Energy 183, 220 (2019)

    2019

  15. [15]

    Lelie, T

    M. Lelie, T. Braun, M. Knips, H. Nordmann, F. Ringbeck, H. Zappen, and D. U. Sauer, Appl. Sci. 8, 10.3390/app8040534 (2018)

    work page doi:10.3390/app8040534 2018

  16. [16]

    Hu and Y.-Y

    Y. Hu and Y.-Y. Wang, IEEE Trans. Control Syst. Technol. 23, 1180 (2015)

    2015

  17. [17]

    F. L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975)

    1975

  18. [18]

    Andresen, P

    B. Andresen, P. Salamon, and R. S. Berry, Phys. Today 37, 62 (1984)

    1984

  19. [19]

    Bejan, Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes (CRC Press, Boca Raton, 1996)

    A. Bejan, Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes (CRC Press, Boca Raton, 1996)

    1996

  20. [20]

    K. H. Hoffmann, J. M. Burzler, and S. Schubert, J. Non. Equilib. Thermodyn. 22, 311 (1997)

    1997

  21. [21]

    Andresen, Angewandte Chemie International Edition 50, 2690 (2011)

    B. Andresen, Angewandte Chemie International Edition 50, 2690 (2011)

    2011

  22. [22]

    Tu, Front

    Z.-C. Tu, Front. Phys. 16, 33202 (2021)

    2021

  23. [23]

    Y. Qiu, M. Nomura, Z. Zhang, et al. , Front. Phys. 20, 065500 (2025)

    2025

  24. [24]

    Zhao and Y.-H

    X.-H. Zhao and Y.-H. Ma, Entropy 27, 195 (2025)

    2025

  25. [25]

    Chen and Z

    L. Chen and Z. Yan, J. Chem. Phys. 90, 3740 (1989)

    1989

  26. [26]

    Van den Broeck, Phys

    C. Van den Broeck, Phys. Rev. Lett. 95, 190602 (2005)

    2005

  27. [27]

    Z. C. Tu, J. Phys. A: Math. Theor. 41, 312003 (2008)

    2008

  28. [28]

    Schmiedl and U

    T. Schmiedl and U. Seifert, Europhys. Lett. 81, 20003 (2008)

    2008

  29. [29]

    Esposito, R

    M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010)

    2010

  30. [30]

    Proesmans, B

    K. Proesmans, B. Cleuren, and C. Van den Broeck, Phys. Rev. Lett. 116, 220601 (2016) . 14

    2016

  31. [31]

    Shiraishi, K

    N. Shiraishi, K. Saito, and H. Tasaki, Phys. Rev. Lett. 117, 190601 (2016)

    2016

  32. [32]

    Pietzonka and U

    P. Pietzonka and U. Seifert, Phys. Rev. Lett. 120, 190602 (2018)

    2018

  33. [33]

    Y.-H. Ma, D. Xu, H. Dong, and C.-P. Sun, Phys. Rev. E 98, 042112 (2018)

    2018

  34. [34]

    Y.-H. Ma, C. P. Sun, and H. Dong, Commun. Theor. Phys. 73, 125101 (2021)

    2021

  35. [35]

    Fei, J.-F

    Z. Fei, J.-F. Chen, and Y.-H. Ma, Phys. Rev. A 105, 022609 (2022)

    2022

  36. [36]

    Yuan, Y.-H

    H. Yuan, Y.-H. Ma, and C. P. Sun, Phys. Rev. E 105, L022101 (2022)

    2022

  37. [37]

    Ma, J.-F

    Y.-H. Ma, J.-F. Chen, C. P. Sun, and H. Dong, Phys. Rev. E 106, 034112 (2022)

    2022

  38. [38]

    Zhou, Y.-H

    T.-J. Zhou, Y.-H. Ma, and C. P. Sun, Phys. Rev. Res. 6, 043001 (2024)

    2024

  39. [39]

    X.-H. Zhao, Z. C. Tu, and Y.-H. Ma, Phys. Rev. E 110, 034105 (2024)

    2024

  40. [40]

    Ma and C

    Y.-H. Ma and C. Fu, Phys. Rev. E 112, 054130 (2025)

    2025

  41. [41]

    Y. Wang, E. Lei, Y.-H. Ma, Z. C. Tu, and G. Li, Phys. Rev. E 112, 054124 (2025)

    2025

  42. [42]

    Lei, Y.-Q

    J.-R. Lei, Y.-Q. Lin, S.-G. Ou, and Y.-H. Ma, Front. Phys. 20, 042202 (2025)

    2025

  43. [43]

    P. H. L. Notten, J. H. G. Op het Veld, and J. R. G. van Beek, J. Power Sources 145, 89 (2005)

    2005

  44. [44]

    S. S. Zhang, J. Power Sources 161, 1385 (2006)

    2006

  45. [45]

    Lee, M.-Y

    C.-H. Lee, M.-Y. Chen, S.-H. Hsu, and J.-A. Jiang, J. Energy Storage 18, 528 (2018)

    2018

  46. [46]

    Banguero, A

    E. Banguero, A. Correcher, ￿. Pérez-Navarro, F. Morant, and A. Aristizabal, Energies 11, 10.3390/en11041021 (2018)

    work page doi:10.3390/en11041021 2018

  47. [47]

    Chiasserini and R

    C.-F. Chiasserini and R. Rao, IEEE J. Sel. Areas Commun. 19, 1235 (2001)

    2001

  48. [48]

    Masakure, A

    A. Masakure, A. Gill, and M. Singh, in Proceedings of the 2023 IEEE Asian Conference on Innovation in Technology (ASIANCON) (2023) pp. 1–5

    2023

  49. [49]

    Dubarry, A

    M. Dubarry, A. Devie, and K. McKenzie, J. Power Sources 358, 39 (2017)

    2017

  50. [50]

    Dehghani-Sanij, E

    A. Dehghani-Sanij, E. Tharumalingam, M. Dusseault, and R. Fraser, Renewable and Sus- tainable Energy Reviews 104, 192 (2019)

    2019

  51. [51]

    T. M. Bandhauer, S. Garimella, and T. F. Fuller, J. Electrochem. Soc. 158, R1 (2011)

    2011

  52. [52]

    Vetter, P

    J. Vetter, P. Novák, M. R. Wagner, C. Veit, K.-C. Möller, J. O. Besenhard, M. Winter, M. Wohlfahrt-Mehrens, C. Vogler, and A. Hammouche, J. Power Sources 147, 269 (2005)

    2005

  53. [53]

    Tomaszewska, Z

    A. Tomaszewska, Z. Chu, X. Feng, S. O’Kane, X. Liu, J. Chen, C. Ji, E. Endler, R. Li, L. Liu, et al. , eTransportation 1, 100011 (2019)

    2019

  54. [54]

    Park, Y.-W

    S.-J. Park, Y.-W. Song, B.-S. Kang, W.-J. Kim, Y.-J. Choi, C. Kim, and Y.-S. Hong, J. Energy Storage 59, 106477 (2023)

    2023

  55. [55]

    J. O. Besenhard, Handbook of battery materials (John Wiley & Sons, 2008)

    2008

  56. [56]

    H. Lv, X. Huang, and Y. Liu, Ionics 26, 1749 (2020)

    2020

  57. [57]

    Boyd and L

    S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cam- bridge, 2004). 15

    2004

  58. [58]

    D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, Belmont, MA, 1999)

    1999

  59. [59]

    Salamon and R

    P. Salamon and R. S. Berry, Phys. Rev. Lett. 51, 1127 (1983)

    1983

  60. [60]

    J. R. Belt, Battery Test Manual for Plug-In Hybrid Electric Vehicles , Tech. Rep. INL/EXT- 07-12536 (Idaho National Laboratory, 2010)

    2010

  61. [61]

    X. Hu, S. Li, and H. Peng, J. Power Sources 198, 359 (2012)

    2012

  62. [62]

    G. L. Plett, Battery Management Systems, Volume I: Battery Modeling (Artech House, Norwood, MA, 2015)

    2015

  63. [63]

    Miettinen, Nonlinear Multiobjective Optimization (Kluwer Academic Publishers, Boston, 1999)

    K. Miettinen, Nonlinear Multiobjective Optimization (Kluwer Academic Publishers, Boston, 1999)

    1999

  64. [64]

    Satopaa, J

    V. Satopaa, J. Albrecht, D. Irwin, and B. Raghavan, 31st International Conference on Distributed Computing Systems Workshops , 166 (2011)

    2011

  65. [65]

    Y. Wang, J. Tian, Z. Sun, L. Wang, R. Xu, M. Li, and Z. Chen, Renew. Sustain. Energy Rev. 131, 110015 (2020)

    2020

  66. [66]

    A. A. Pesaran, J. Power Sources 110, 377 (2002)

    2002

  67. [67]

    Q. Wang, B. Jiang, B. Li, and Y. Yan, Renew. Sustain. Energy Rev. 64, 106 (2016)

    2016

  68. [68]

    Newman and W

    J. Newman and W. Tiedemann, AIChE Journal 21, 25 (1975)

    1975

  69. [69]

    Doyle, T

    M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc. 140, 1526 (1993)

    1993

  70. [70]

    Sulzer, S

    V. Sulzer, S. G. Marquis, R. Timms, M. Robinson, and S. J. Chapman, J. Open Res. Softw. 9, 14 (2021) . 16

    2021