Analysis on pulse charging–discharging strategies for improving capacity retention rates of lithium-ion batteries

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The capacity fade of lithium-ion batteries (LIBs) are intimately dependent upon charging–discharging strategies. In this work, a pseudo-two-dimensional model coupled with thermal effects was developed to investigate the effects of pulse current charging–discharging strategies on the capacity fade for LIBs, in which the growth of solid electrolyte interphase (SEI) and the lithium ion migration process are highlighted. LiFePO4/graphite (LFP) batteries were taken as samples. The capacity fading processes of the LFP batteries under different pulse current charging–discharging strategies were studied by numerical simulations. The impacts of the growth of the SEI layer at the negative electrode on the capacity degradation were studied by using eleven charging–discharging cases. The results show that, compared with the constant current-constant voltage (CC-CV) strategy, the pulse current-constant voltage (PC-CV) strategies can effectively alleviate the capacity fade of the LFP batteries due to the relaxation process in the pulse charging process. Under the PC-CV strategies, the relaxation duration and the peak current were altered. The relaxation process renders the decrease of the current density of side reactions in the batteries, which is conducive to the recovery of the aged batteries, thus improving the capacity retention rates of the batteries. The advantages of the PC-CV strategies were demonstrated in this work. The results will provide theoretical guidance for the rapid charging–discharging strategies of LIBs.

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A c :

area of the electrode (m2).

c 1 :

concentration of lithium ions in the active material particles (mol m−3).

c 1, max :

maximum concentration of lithium ions in the active material particles (mol m−3).

c 2 :

concentration of lithium ions in the electrolyte (mol m−3).

c EC :

concentration of EC (mol m−3).

c int :

initial concentration of lithium ions in the electrolyte (mol m−3).

c p :

specific heat capacity (J kg−1 K−1).

C rate :

charge/discharge rate.

D 1 :

diffusion coefficient of lithium ions in the active material (m2 s−1).

D 2 :

diffusion coefficient of electrolyte (m2 s−1).

D EC :

diffusion coefficient of EC in SEI layer (m2 s−1).

D p :

adjusted diffusion coefficient of electrolyte (m2 s−1).

E :

activation energy (kJ mol−1).

f :

average molar activity coefficient.

F :

Faraday’s constant (C mol−1).

i 2 :

ionic current density in the electrolyte (A m−2).

i a :

average current density (A m−2).

i app :

current density in positive(A m−2).

i loc :

local current density (A m−2).

i p :

peak current density (A m−2).

i SEI :

side reaction current density (A m−2).

ka,kc :

rate constants for reduction and oxidation reactions (m s−1).

k n :

reaction rates of intercalation/deintercalation at the negative and positive electrode.

k SEI :

rate constants for the side reaction (m s−1).

L :

thickness (μm).


molar mass of SEI layer (kg mol−1).

q :

volumetric heat generation (W m−3).

q act :

polarization volumetric heat generation (W m−3).

q ohm :

ohmic volumetric heat generation (W m−3).

q rea :

reaction volumetric heat generation (W m−3).

r :

radius distance (m).

r s :

radius of electrode particles (m).

R :

gas constant (J mol−1 K−1).


resistance across the SEI layer (Ω).


state of charge.

S a :

specific surface area (m−1).

t :


t off :

relaxation duration (s).

t on :

duration of pulse (s).

t pc :

period of pulse (s).

\( {t}_{+}^0 \) :

transference number of Li+.

T :

temperature (K).

u :

growth rate of SEI (m s−1).

U :

open circuit potential of the electrode (V).

U side :

open circuit potential of side reaction(V).

\( \frac{\partial U}{\partial T} \) :

entropy change

x :


α :

duty cycle.

β :

transfer coefficient.

β SEI :

transfer coefficient of side reaction.

δ SEI :

thickness of SEI (nm).

ε 1 :

volume fraction of active material.

ε 2 :

volume fraction of electrolyte.

η :

over potential (V).

η side :

over potential of side reaction(V).

θ :

dimensionless time.

λ :

thermal conductivity (W m−1 K−1).

ξ :

capacity retention rate.

ρ :

density (kg m−3).

σ 1 eff :

effective electronic conductivity in solid phase material (S m−1).

σ 2 eff :

effective ionic conductivity of electrolyte (S m−1).

ϕ 1 :

solid phase potential (V).

ϕ 2 :

electrolyte phase potential (V).


initial or equilibrium state.


solid phase.


liquid phase.

amb :

ambient temperature

des :




n :

negative electrode.

p :

positive electrode.

ref :

reference value.

surf :

surface of active material particles

S :



solid electrolyte interphase.

eff :

effective value.


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Appendix 1 Fundamental data of 2.5 Ah cylindrical LFP battery and the related parameters for the SEI layer

The model battery used in this work was 2.5 Ah cylindrical LiFePO4 battery. The fundamental data of the batteries and the related parameters in the simulations are listed in Table 4.

Table 4 The fundamental data of 2.5 Ah cylindrical LFP battery and the related parameters

Appendix 2 Parameters for the simulations of the performances during charging–discharging of the LFP batteries

In the simulations, the electrolyte conductivity σ2 and the thermodynamic factor ν, the diffusion coefficient in the solid phase D1, the reaction constant k, the entropy changes of the positive and negative electrodes are calculated by the following equations.

$$ {\displaystyle \begin{array}{l}{\sigma}_2=\Big(-10.5+0.074T-6.69\times {10}^{-5}{T}^2+6.68\times {10}^{-4}{c}_2-1.78\times {10}^{-5}{c}_2T\\ {}\kern1.8em +2.8\times {10}^{-8}{c}_2{T}^2+4.94\times {10}^{-7}{c}_2^2-8.86\times {10}^{-10}{c}_2^2T\Big){}^2{10}^{-4}{c}_2\end{array}} $$
$$ {\displaystyle \begin{array}{l}\nu =\left(1+\frac{\partial \ln {f}_{\pm }}{\partial \ln {c}_2}\right)\left(1-{t}_{+}^0\right)=0.601-0.024{\left(0.1{c}_2\right)}^{0.5}+\\ {}\kern1.5em 0.982\left[1-0.0052\left(T-294\right){\left(0.001{c}_2\right)}^{1.5}\right]\end{array}} $$
$$ {D}_{1,p}=\frac{1.18\times {10}^{-18}}{{\left(1+\overline{y}\right)}^{1.6}}\exp \left(\frac{E_{D_{1,p}}}{R}\left(\frac{1}{T_{ref}}-\frac{1}{T}\right)\right) $$
$$ {D}_{1,n}=3.9\times {10}^{-14}\exp \left(\frac{E_{D_{1,n}}}{R}\left(\frac{1}{T_{ref}}-\frac{1}{T}\right)\right) $$
$$ {D}_2={10}^{-8.43\left(\frac{54}{T-229-0.005{c}_2}\right)-2.2\times {10}^{-4}{c}_2} $$
$$ {k}_j={k}_{j,0}\exp \left[\frac{E_k}{R}\left(\frac{1}{T_{ref}}-\frac{1}{T}\right)\right] $$
$$ {\displaystyle \begin{array}{l}{U}_p=3.4323-0.8428\exp \left[-80.2493{\left(1-\overline{y}\right)}^{1.3198}\right]\\ {}\kern2em -3.2474\times {10}^{-6}\exp \left[20.2645{\left(1-\overline{y}\right)}^{3.8003}\right]\\ {}\kern2.25em +3.2482\times {10}^{-6}\exp \left[20.2646{\left(1-\overline{y}\right)}^{3.7995}\right]\end{array}} $$
$$ {\displaystyle \begin{array}{l}{U}_n=0.6379+0.5416\exp \left(-305.5309\overline{x}\right)+0.044\tanh \left(\frac{0.1958-\overline{x}}{0.1088}\right)\\ {}\kern1.75em -0.1978\tanh \left(\frac{\overline{x}-1.0571}{0.0854}\right)-0.6875\tanh \left(\frac{\overline{x}+0.0117}{0.0529}\right)\\ {}\kern1.75em -0.0175\tanh \left(\frac{\overline{x}-0.5692}{0.0875}\right)\end{array}} $$
$$ {\displaystyle \begin{array}{l}\frac{\partial {U}_p}{\partial T}=-0.35376{\overline{y}}^8+1.3902{\overline{y}}^7-2.2585{\overline{y}}^6+1.9635{\overline{y}}^5-0.98716{\overline{y}}^4\kern0.1em \\ {}\kern2.799999em +0.28857{\overline{y}}^3-0.046272{\overline{y}}^2+0.0032158\overline{y}-1.9186\times {10}^{-5}\end{array}} $$
$$ {\displaystyle \begin{array}{l}\frac{\partial {U}_n}{\partial T}=344.1347148\frac{\exp \left(-32.9633287\overline{x}+8.316711484\right)}{1+749.0756003\exp \left(-34.790996\overline{x}+8.887143624\right)}\\ {}\kern2.7em -0.8520278805\overline{x}+0.36229929{\overline{x}}^2+0.2698001697\end{array}} $$

The abovementioned parameters are mainly from the literature of the 2.3 Ah LiFePO4 battery [33]. Some parameters were adjusted by the following equations so that the capacities of both the positive and negative electrodes are 2.5 Ah.

$$ Q={\varepsilon}_1 FL{A}_{\mathrm{c}}{\mathrm{c}}_{s,\max}\left|{w}_{100\%}-{w}_{0\%}\right| $$

where Q is the battery capacity; ε1 is the volume fraction of the solid phase; F is Faraday constant; L is the thickness of the electrodes; Ac is the contact area; w0% and w100% are the minimum/maximum SOC, respectively.

In this work, it is assumed that in the process of pulse charging–discharging, the diffusion coefficient in the liquid phase D2 is calculated by Eq.(B.5), and diffusion coefficient in the liquid phase D2, PC is adjusted as

$$ {D}_{2, PC}={D}_2\left(2-2{c}_{2,\operatorname{int}}/{c}_2+{c}_{2,\operatorname{int}}^2/{c}_2^2\right) $$

where c2 is the lithium ion concentration in the electrolyte, and c2, int is the initial lithium ion concentration.

In this work, we also assumed that the side reactions for the growth of SEI layer are neglected during the relaxation. Then kSEI, 0 can be calculated by

$$ {k}_{SEI}=f(t){k}_{SEI,0} $$
$$ f(t)=\left\{\begin{array}{c}1\kern2.25em t={t}_{on}\\ {}0\kern2em t={t}_{off}\end{array}\right. $$

where kSEI, 0 is the rate constant of the side reaction; kSEI is the modified rate constant under the pulse charging–discharging strategies; f(t) is a modified coefficient.

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Lv, H., Huang, X. & Liu, Y. Analysis on pulse charging–discharging strategies for improving capacity retention rates of lithium-ion batteries. Ionics (2020) doi:10.1007/s11581-019-03404-8

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  • Lithium-ion battery
  • Charging–discharging strategy
  • Pulse current charging
  • Relaxation process
  • Capacity fade