# Modification of Cycle Life Model for Normal Aging Trajectory Prediction of Lithium-Ion Batteries at Different Temperatures and Discharge Current Rates

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## Abstract

**:**

## 1. Introduction

_{0.5}Co

_{0.2}Mn

_{0.3}]O

_{2}(NCM) batteries have been widely used in electric vehicles (EVs) and hybrid electric vehicles (HEVs) due to their extended cycle life and high energy density [1,2]. Battery life, which is of significance for reliable and economical operation of electric vehicles, usually includes calendar life and cycle life [3]. The degradation of battery consists of normal degradation and accelerated aging. The normal degradation is caused by the continuous growth of solid electrolyte interface (SEI). The trend of normal degradation is approximately linear [4]. The degradation of battery may lead to safety problems and reduce ranges of EVs [5,6,7]. Therefore, it is necessary to predict the battery ageing trajectory accurately.

^{1/2}) and Arrhenius kinetics. The common form of battery cycle life model is the cycle number to the power of 1/2 based on the SEI growth phenomenon [17,18]. In addition to the power of 1/2, the double exponential cycle life model is another common life model which describes the time dependency [19]. The cycle life model agreed well with the experimental data but featured a much higher power law factor (1.3) compared to the value of 0.5 associated with the general t

^{1/2}rule based on the rate of solid electrolyte interphase (SEI) growth [18]. Arrhenius equation can be used to improve the extrapolation ability of the life model under various temperatures [20]. A life model which describes capacity fade and resistance increase as a function of the influencing stress factors and battery charge throughput was proposed [21]. John Wang et al. [22] proposed a cycle life model with an Arrhenius correlation and the exponential dependence of the current rate. Johannes Schmalstieg et al. [23] created a cycle life model taking cycle depths and mean state of charge (SOC) into account. Pecht Michael et al. [24] developed an empirical accelerated degradation model that can describe the two-stage capacity fade process. The double exponential cycle life model was improved for the accelerated aging [25]. Stefan Kabitz et al. [17] built a life model by combining a calendar life model and a cycle life model. In addition to the empirical model considering the impact of the cycle time and the operation conditions, researchers also built the life model based on the impedance spectroscopy [26], physics-based single-particle model [27], coulombic efficiency [28] and lithium plating [29].

## 2. Cycle Life Experiments and Results

#### 2.1. Cycle Life Test

#### 2.2. Cycle Life Test Results

_{re}is calculated by Formula (1). The cumulative discharge ampere hour is calculated by Formula (2)

_{n}is the actual capacity at nth cycle $A{h}_{\mathrm{throughput}}$. Q

_{start}is the capacity at the initial cycle test. $A{h}_{\mathrm{throughput}}$ is the cumulative discharge ampere-hour from the first cycle to the Nth cycle. n is the cycle number. $A{h}_{\mathrm{dis},\text{}n}$ is the discharge ampere-hour at the nth cycle.

_{throughput}at 0.5 C and 1 C, while the capacity decay is slower at 1.5 C and 2 C. It is noted that there are many obvious data noises for battery cycle life tests. The dada noises are caused by the test error and the capacity recovery during RPT. The capacity recovery phenomenon may influence the battery life prediction [34,36]. Figure 1b shows the capacity retention of Battery II with Ah

_{throughput}at 25–45 °C and at 1–2 C. The Battery II experienced obvious accelerated aging during cycling. The aging process of Battery II included normal aging stage and accelerated aging stage. During the normal fade stage, the capacity of Battery II declined faster at 35 °C and 45 °C than that at 25 °C. However, the accelerated aging point appeared earlier at 25 °C and 45 °C than the Battery II at 35 °C. The cycle life of Battery II was longest at 35 °C. For the discharge current rate under 1–2 C, the capacity degradation speeds of Battery II were similar during the normal stage. The accelerated aging point arose earlier at 2 C rate.

## 3. Normal Aging Trajectory Acquisition Based on Wavelet Transform

#### 3.1. Wavelet Transform Method

#### 3.2. Normal Aging Trajectory Acquisition Results

## 4. Modified Life Model for Normal Aging Trajectory Prediction

#### 4.1. Introduction of the Empirical Models

#### 4.2. Influence of Life Model Structure on the Normal Aging Trajectory Prediction

#### 4.3. Sensitivity Analysis of the Life Model Parameters for Normal Aging Trajectory Prediction

#### 4.3.1. Sensitivity Analysis Method

- Set an appropriate variation range of the model parameter. The common values of the model parameters are calculated by the PSO algorithm. The variation range is set to be ±20% of the common values.
- Generate 700 numbers with the uniform distribution within the variation range for each parameter.
- Calculate the common output capacity retention series ${Q}_{b,i}$ using the empirical model with the common parameter values, where i is the condition number in the dataset. Then, calculate the distribution output capacity retention series ${Q}_{\mathrm{dis},i,n}$ using the same model with the generated parameter values, where n is the generated number. Then, the relative sensitivity criteria under different operating conditions of each model parameter can be calculated using Equation (17):$${C}_{i}={{\displaystyle \sum}}_{n=1}^{700}{\left({Q}_{b,i}-{Q}_{\mathrm{dis},i,n}\right)}^{2}/{Q}_{b,i}$$
- Obtain the overall parameter sensitivity under all conditions. The parameter sensitivity index value ${S}_{k}$ can be defined as the sum of the relative sensitivity criteria at various battery operating conditions, k is the parameter number and m is the number of the conditions in the Dataset I.$${S}_{k}={{\displaystyle \sum}}_{i=1}^{m}{C}_{i}$$

#### 4.3.2. MPSA Analysis Results and Discussion

_{a}has low sensitivity in the single-factor model. The accuracy of Model 14 is attributed to the improvement of the pre-exponential factor and the active energy by considering the discharge current rate. There is only a highly sensitive parameter in Model 14, 15, 16 and 17. So, the number of the highly sensitive parameters is not the main reason for the poor fitting and extrapolation ability of Model 13, 15, 16 and 17. The reason why the couple-factor life model 15, 16 and 17 is not accurate is that the form of their pre-exponential factor and index about discharge current rate are not appropriate. In order to get a more accurate model, the improvement of Model 14 can be summarized as follows: (1) reducing or improving the constant parameter, (2) improving the index and pre-exponential factor by considering both temperature and discharge current rate and (3) simplifying or improving the active energy due to its low sensitivity. In (14), the index can be improved by considering the temperature or the discharge current rate, so that the index sensitivity can be reduced.

#### 4.4. Modified Life Model for Normal Aging Trajectory Prediction

_{1}, a

_{2}, a

_{3,}a

_{4}and a

_{5}are the model parameters.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Model | Model Parameter |
---|---|

$1.\text{}{Q}_{loss}=1-({a}_{1}\ast {n}^{0.5}+{a}_{2}$) | n is the cycle number; a_{1}, a_{2} are the model parameters. |

$2.\text{}C=1-{a}_{1}\xb7{n}^{0.5}$ | n is the cycle number; a_{1} is the model parameter. |

$3.\text{}C=1-{a}_{1}\xb7{n}^{0.5}-{a}_{2}\xb7n$ | n is the cycle number; a_{1}, a_{2} are the model parameters. |

$4.\text{}{Q}_{loss}={a}_{1}\xb7{n}^{0.5}+{a}_{2}\xb7n+{a}_{3}$ | n is the cycle number; a_{1}, a_{2}, a_{3} are the model parameters. |

$5.\text{}{Q}_{loss}={a}_{1}\xb7{n}^{0.5}+{a}_{2}\xb7\left(n-{N}_{0}\right)+{a}_{3}$ | n is the cycle number; a_{1}, a_{2}, a_{3}, N_{0} are the model parameters. |

$6.\text{}C={a}_{1}+{a}_{2}\xb7n+{a}_{3}\xb7{n}^{2}+{a}_{4}\xb7{n}^{3}$ | n is the cycle number; a_{1}, a_{2}, a_{3}, a_{4} are the model parameters. |

$7.\text{}C={a}_{1}+{a}_{2}\xb7{n}^{0.5}$ | n is the cycle number; a_{1}, a_{2} are the model parameters. |

$8.\text{}{Q}_{loss}=1-({a}_{1}\ast \mathrm{exp}\left({a}_{2}\ast n\right)+{a}_{3}\ast \mathrm{exp}({a}_{4}\ast n)$) | n is the cycle number; a_{1}, a_{2}, a_{3}, a_{4} are the model parameters. |

$9.\text{}C=A\xb7\mathrm{exp}\left(\frac{n}{{t}_{1}}\right)+{y}_{0}$ | n is the cycle number; A, t_{1}, y_{0} are the model parameters. |

$10.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{-{E}_{a}}{R\xb7T}\right)\xb7{n}^{z}$ | n is the cycle number; B and z are the model parameters; T is the absolute temperature; E_{a} is the active energy; and R is the universal gas constant. |

$11.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{-{E}_{a}}{R\xb7T}\right)\xb7A{h}^{z}$ | Ah is the cumulative discharge capacity; B and z are the model parameters; T is the absolute temperature; E_{a} is the active energy; and R is the universal gas constant. |

$12.\text{}C=1-{d}_{Tref}\xb7{\alpha}^{\left(\frac{T-{T}_{ref}}{10}\right)}\xb7{n}^{0.5}$ | ${T}_{\mathit{ref}}$ are model parameters; T is absolute temperature; and the T_{ref} is 40 °C in literature. |

$13.\text{}C=A\left(I,T\right)\xb7{n}^{B\left(I,T\right)}$ $A\left(I,T\right)=a\xb7\mathrm{exp}\left(\frac{\alpha}{T}\right)+b\xb7{I}^{\beta}+c$ $B\left(I,T\right)=l\xb7\mathrm{exp}\left(\frac{\lambda}{T}\right)+m\xb7{I}^{n}+f$ | $I\mathrm{is}\text{}\mathrm{discharge}\text{}\mathrm{current}\text{}\mathrm{rate};\text{}T\text{}\mathrm{is}\text{}\mathrm{absolute}\text{}\mathrm{temperature};\text{}\mathrm{and}\text{}\alpha ,\beta ,l,\lambda ,f,c$are the model parameters. |

$14.\text{}{Q}_{loss}=\alpha \xb7\mathrm{exp}\left(\frac{{k}_{3}\xb7{C}_{rate}\text{}+\text{}{k}_{4}}{R\xb7T}\right)\xb7{C}_{rate}{}^{\beta}\xb7{n}^{\eta}$ | $\alpha \beta \eta {k}_{3},{k}_{4}\text{}\mathrm{are}\text{}\mathrm{model}\text{}\mathrm{parameters};\text{}R\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{universal}\text{}\mathrm{gas}\text{}\mathrm{constant};\text{}n\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{cycle}\text{}\mathrm{number};\text{}T\text{}\mathrm{is}\text{}\mathrm{absolute}\text{}\mathrm{temperature};\text{}\mathrm{and}\text{}{C}_{rate}$ is the discharge current rate. |

$15.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{a\text{}+\text{}b\xb7{C}_{rate}}{R\xb7T}\right)\xb7{n}^{0.55}$ | $a,\text{}b,\text{}B\text{}\mathrm{are}\text{}\mathrm{model}\text{}\mathrm{parameters};\text{}R\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{universal}\text{}\mathrm{gas}\text{}\mathrm{constant};\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{cycle}\text{}\mathrm{number};\text{}T\text{}\mathrm{is}\text{}\mathrm{absolute}\text{}\mathrm{temperature};\text{}\mathrm{and}\text{}{C}_{rate}$ is the discharge current rate. |

$16.\text{}{Q}_{loss}=\left(a\xb7{T}^{2}+b\xb7T+c\right)\xb7\mathrm{exp}\left(\left(d\xb7T+e\right)\xb7{I}_{rate}\right)\xb7n$ | T is the absolute temperature; n is the cycle number; I_{rate} is the discharge current rate; and a, b, c, d, e are the model parameters. |

$17.\text{}{Q}_{loss}=\left(a\xb7\mathrm{exp}\left(b\xb7I\right)+c\xb7\mathrm{exp}\left(d\xb7I\right)\right)\xb7\mathrm{exp}\left(\frac{e\text{}+\text{}f\xb7I}{R\xb7T}\right)\xb7A{h}^{0.55}$ | I is the discharge current rate; R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; and a, b, c, d, e, f are model parameters. |

$18.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7\mathrm{exp}\left(\frac{{a}_{2}}{R\xb7T}\right)\xb7A{h}^{{C}^{{a}_{3}}+{a}_{4}}$ | C is the discharge current rate; R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; a_{1}, a_{2}, a_{3}, a_{4} are the model parameters; -a_{2} represents the active energy; |

$19.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7\mathrm{exp}\left(\frac{{a}_{2}\ast {T}^{2}+{a}_{3}}{R\xb7T}\right)\xb7A{h}^{({C}^{{a}_{4}}+{a}_{5})}$ | R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; and a_{1}, a_{2}, a_{3}, a_{4}, a_{5} are the model parameters. |

$20.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7\mathrm{exp}\left(\frac{{a}_{2}\xb7C+{a}_{3}}{R\xb7T}\right)\xb7{C}^{{a}_{4}}\xb7A{h}^{{a}_{5}}+{a}_{6}\xb7\mathrm{exp}\left(\frac{{a}_{7}}{R\xb7T}\right)\xb7A{h}^{{a}_{8}}$ | C is the discharge current rate; R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; and a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7},a_{8} are the model parameters. |

$21.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7{C}^{\beta}\xb7\mathrm{exp}\left(\frac{{a}_{2}}{R\xb7T}\right)\xb7A{h}^{{a}_{3}}$ | C is the discharge current rate; R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; a_{1}, a_{2}, a_{3}, β are the model parameters; -a_{2} represents the active energy; |

$22.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7{C}^{\beta}\xb7\mathrm{exp}\left(\frac{{a}_{2}\xb7C+{a}_{3}}{R\xb7T}\right)\xb7A{h}^{\left({a}_{4}\xb7C+{a}_{5}\right)}$ | C is the discharge current rate; R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; and a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, β are the model parameters. |

$23.\text{}{Q}_{loss}={a}_{1}\xb7{C}^{{a}_{2}}\xb7\mathrm{exp}\left(\frac{{a}_{3}}{R\xb7T}\right)\xb7A{h}^{\left({a}_{4}\xb7C+{a}_{5}\right)}$ | C is the discharge current rate; R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; and a_{1}, a_{2}, a_{3}, a_{4}, a_{5} are the model parameters. -a_{3} represents the active energy |

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**Figure 2.**Comparison of the maximal available capacity and the cycle capacity of Battery I at different temperatures: (

**a**) 25 °C, (

**b**) 35 °C, (

**c**) 45 °C and (

**d**) 50 °C.

**Figure 3.**The denoising results with the wavelet transform: the per-cycle degradation speed of Dataset I at (

**a**) 25 °C and (

**b**) 35 °C; the removal of the capacity drop of Dataset II at (

**c**) 25 °C and (

**d**) 35 °C.

**Figure 4.**Prediction RMSE of the non-factor models for normal aging: (

**a**) on the train dataset of Dataset I, (

**b**) on the train dataset of Dataset II, (

**c**) on the test dataset of Dataset I and (

**d**) on the test dataset of Dataset II.

**Figure 5.**Prediction RMSE of the single-factor models for normal aging: (

**a**) on the train dataset of Dataset I, (

**b**) on the train dataset of Dataset II, (

**c**) on the test dataset of Dataset I and (

**d**) on the test dataset of Dataset II.

**Figure 6.**Prediction RMSE of the coupling-factor models for normal aging: (

**a**) on the train dataset of Dataset I, (

**b**) on the train dataset of Dataset II, (

**c**) on the test dataset of Dataset I and (

**d**) on the test dataset of Dataset II.

**Figure 7.**Prediction RMSE of the non-factor models for normal aging: (

**a**) on the train dataset of Dataset I, (

**b**) on the train dataset of Dataset II, (

**c**) on the test dataset of Dataset I and (

**d**) on the test dataset of Dataset II.

**Figure 8.**Accuracy validation results of the final improved model 23 under two conditions: (

**a**) 25 °C, 1 C, and (

**b**) 35 °C, 1 C.

Item | Battery I | Battery II |
---|---|---|

Cathode material | Li[Ni_{0.5}Co_{0.2}Mn_{0.3}]O_{2} | Li[Ni_{0.6}Co_{0.2}Mn_{0.2}]O_{2} |

Anode material | Graphite | Graphite |

Nominal capacity | 114 Ah | 36 Ah |

Charging cut-off voltage | 4.25 V | 4.15 V |

Discharging cut-off voltage | 2.8 V | 2.5 V |

Shape | prismatic | pouch |

Condition Number | Cycle Temperature | Discharge Current Rate |
---|---|---|

1 | 25 °C | 1 C |

2 | 35 °C | 1 C |

3 | 45 °C | 1 C |

4 | 50 °C | 1 C |

5 | 55 °C | 1 C |

6 | 25 °C | 0.5 C |

7 | 25 °C | 1 C |

8 | 25 °C | 1.5 C |

9 | 25 °C | 2 C |

Condition Number | Cycle Temperature | Discharge Current Rate |
---|---|---|

10 | 25 °C | 1 C |

11 | 35 °C | 1 C |

12 | 45 °C | 1 C |

13 | 25 °C | 1 C |

14 | 25 °C | 1.5 C |

15 | 25 °C | 2 C |

16 | 35 °C | 1.5 C |

17 | 35 °C | 2 C |

Wavelet Function | Decomposed Number | Denoising Method | Threshold Estimation Method | $\mathsf{\Delta}\mathit{v}$ |
---|---|---|---|---|

Haar | 3 | s | minimaxi | 58–78% |

Haar | 3 | h | minimaxi | 52–77% |

Haar | 4 | s | minimaxi | 68–87% |

db4 | 3 | s | minimaxi | 54–70% |

db4 | 4 | s | minimaxi | 66–87% |

db4 | 3 | s | heursure | 43–70% |

db4 | 4 | s | heursure | 43–87% |

db4 | 4 | h | heursure | 11–86% |

db4 | 5 | s | heursure | 48–89% |

db4 | 5 | h | heursure | 20–84% |

sym4 | 4 | s | heursure | 44–87% |

sym4 | 3 | s | sqtwolog | 65–80% |

Model | Model Parameter |
---|---|

$1.\text{}{Q}_{loss}=1-({a}_{1}\ast {n}^{0.5}+{a}_{2}$) | n is the cycle number; a_{1}, a_{2} are the model parameters. |

$2.\text{}C=1-{a}_{1}\xb7{n}^{0.5}$ | n is the cycle number; a_{1} is the model parameter. |

$3.\text{}C=1-{a}_{1}\xb7{n}^{0.5}-{a}_{2}\xb7n$ | n is the cycle number; a_{1}, a_{2} are the model parameters. |

$4.\text{}{Q}_{loss}={a}_{1}\xb7{n}^{0.5}+{a}_{2}\xb7n+{a}_{3}$ | n is the cycle number; a_{1}, a_{2}, a_{3} are the model parameters. |

$5.\text{}{Q}_{loss}={a}_{1}\xb7{n}^{0.5}+{a}_{2}\xb7\left(n-{N}_{0}\right)+{a}_{3}$ | n is the cycle number; a_{1}, a_{2}, a_{3}, N_{0} are the model parameters. |

$6.\text{}C={a}_{1}+{a}_{2}\xb7n+{a}_{3}\xb7{n}^{2}+{a}_{4}\xb7{n}^{3}$ | n is the cycle number; a_{1}, a_{2}, a_{3}, a_{4} are the model parameters. |

$7.\text{}C={a}_{1}+{a}_{2}\xb7{n}^{0.5}$ | n is the cycle number; a_{1}, a_{2} are the model parameters. |

$8.\text{}{Q}_{loss}=1-({a}_{1}\ast \mathrm{exp}\left({a}_{2}\ast n\right)+{a}_{3}\ast \mathrm{exp}({a}_{4}\ast n)$) | n is the cycle number; a_{1}, a_{2}, a_{3}, a_{4} are the model parameters. |

$9.\text{}C=A\xb7\mathrm{exp}\left(\frac{n}{{t}_{1}}\right)+{y}_{0}$ | n is the cycle number; A, t_{1}, y_{0} are the model parameters. |

$10.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{-{E}_{a}}{R\xb7T}\right)\xb7{n}^{z}$ | n is the cycle number; B and z are the model parameters; T is the absolute temperature; E_{a} is the active energy; and R is the universal gas constant. |

$11.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{-{E}_{a}}{R\xb7T}\right)\xb7A{h}^{z}$ | Ah is the cumulative discharge capacity; B and z are the model parameters; T is the absolute temperature; E_{a} is the active energy; and R is the universal gas constant. |

$12.\text{}C=1-{d}_{Tref}\xb7{\alpha}^{\left(\frac{T-{T}_{ref}}{10}\right)}\xb7{n}^{0.5}$ | T_{ref} is the reference temperature; n is the cycle number; d_{Tref} and α are model parameters; T is absolute temperature; and the T_{ref} is 40 °C in literature. |

$13.\text{}C=A\left(I,T\right)\xb7{n}^{B\left(I,T\right)}$ $A\left(I,T\right)=a\xb7\mathrm{exp}\left(\frac{\alpha}{T}\right)+b\xb7{I}^{\beta}+c$ $B\left(I,T\right)=l\xb7\mathrm{exp}\left(\frac{\lambda}{T}\right)+m\xb7{I}^{n}+f$ | $I\mathrm{is}\text{}\mathrm{discharge}\text{}\mathrm{current}\text{}\mathrm{rate};\text{}T\text{}\mathrm{is}\text{}\mathrm{absolute}\text{}\mathrm{temperature};\text{}\mathrm{and}\text{}\alpha ,\beta ,l,\lambda ,f,c$are the model parameters. |

$14.\text{}{Q}_{loss}=\alpha \xb7\mathrm{exp}\left(\frac{{k}_{3}\xb7{C}_{rate}+{k}_{4}}{R\xb7T}\right)\xb7{C}_{rate}{}^{\beta}\xb7{n}^{\eta}$ | $\alpha ,\text{}\beta ,\text{}\eta ,\text{}{k}_{3},\text{}{k}_{4}\text{}\mathrm{are}\text{}\mathrm{model}\text{}\mathrm{parameters};\text{}R\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{universal}\text{}\mathrm{gas}\text{}\mathrm{constant};n\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{cycle}\text{}\mathrm{number};\text{}T\text{}\mathrm{is}\text{}\mathrm{absolute}\text{}\mathrm{temperature};\text{}\mathrm{and}\text{}{C}_{rate}$ is the discharge current rate. |

$15.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{a+b\xb7{C}_{rate}}{R\xb7T}\right)\xb7{n}^{0.55}$ | $a,\text{}b,\text{}B,\text{}\mathrm{are}\text{}\mathrm{model}\text{}\mathrm{parameters};\text{}R\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{universal}\text{}\mathrm{gas}\text{}\mathrm{constant};\text{}n\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{cycle}\text{}\mathrm{number};\text{}T\text{}\mathrm{is}\text{}\mathrm{absolute}\text{}\mathrm{temperature};\text{}\mathrm{and}\text{}{C}_{rate}$ is the discharge current rate. |

$16.\text{}{Q}_{loss}=\left(a\xb7{T}^{2}+b\xb7T+c\right)\xb7\mathrm{exp}\left(\left(d\xb7T+e\right)\xb7{I}_{rate}\right)\xb7n$ | T is the absolute temperature; n is the cycle number; I_{rate} is the discharge current rate; and a, b, c, d, e are the model parameters. |

$17.\text{}{Q}_{loss}=\left(a\xb7\mathrm{exp}\left(b\xb7I\right)+c\xb7\mathrm{exp}\left(d\xb7I\right)\right)\xb7\mathrm{exp}\left(\frac{e+f\xb7I}{R\xb7T}\right)\xb7A{h}^{0.55}$ | I is the discharge current rate; R is the universal gas constant; T is the absolute temperature; Ah is the cumulative discharge capacity; and a, b, c, d, e, f are model parameters. |

Model | Highly Sensitive Parameters |
---|---|

$1.\text{}{Q}_{loss}=1-({a}_{1}\ast {n}^{0.5}+{a}_{2}$) | ${a}_{1},{a}_{2}$ |

$2.\text{}C=1-{a}_{1}\xb7{n}^{0.5}$ | none |

$3.\text{}C=1-{a}_{1}\xb7{n}^{0.5}-{a}_{2}\xb7n$ | none |

$4.\text{}{Q}_{loss}={a}_{1}\xb7{n}^{0.5}+{a}_{2}\xb7n+{a}_{3}$ | none |

$5.\text{}{Q}_{loss}={a}_{1}\xb7{n}^{0.5}+{a}_{2}\xb7\left(n-{N}_{0}\right)+{a}_{3}$ | none |

$6.\text{}C={a}_{1}+{a}_{2}\xb7n+{a}_{3}\xb7{n}^{2}+{a}_{4}\xb7{n}^{3}$ | ${a}_{2},{a}_{3},{a}_{4}$ |

$7.\text{}C={a}_{1}+{a}_{2}\xb7{n}^{0.5}$ | ${a}_{1},{a}_{2}$ |

$8.\text{}{Q}_{loss}=1-({a}_{1}\ast \mathrm{exp}\left({a}_{2}\ast n\right)+{a}_{3}\ast \mathrm{exp}({a}_{4}\ast n)$) | ${a}_{3},{a}_{4}$ |

$9.\text{}C=A\xb7\mathrm{exp}\left(\frac{n}{{t}_{1}}\right)+{y}_{0}$ | $A,{y}_{0}$ |

$10.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{-{E}_{a}}{R\xb7T}\right)\xb7{n}^{z}$ | $B,z$ |

$11.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{-{E}_{a}}{R\xb7T}\right)\xb7A{h}^{z}$ | $B,z$ |

$12.\text{}C=1\text{}-\text{}{d}_{Tref}\xb7{\alpha}^{\left(\frac{T-{T}_{ref}}{10}\right)}\xb7{n}^{0.5}$ | ${T}_{ref}$ |

$13.\text{}C=A\left(I,T\right)\xb7{n}^{B\left(I,T\right)}$ $A\left(I,T\right)\text{}=\text{}a\xb7\mathrm{exp}\left(\frac{\alpha}{T}\right)\text{}+\text{}b\xb7{I}^{\beta}+c$ $B\left(I,T\right)=l\xb7\mathrm{exp}\left(\frac{\lambda}{T}\right)\text{}+\text{}m\xb7{I}^{n}+f$ | $a,b,c,l,m,f$ |

$14.\text{}{Q}_{loss}=\alpha \xb7\mathrm{exp}\left(\frac{{k}_{3}\xb7{C}_{rate}\text{}+\text{}{k}_{4}}{R\xb7T}\right)\xb7{C}_{rate}{}^{\beta}\xb7{n}^{\eta}$ | η |

$15.\text{}{Q}_{loss}=B\xb7\mathrm{exp}\left(\frac{a\text{}+\text{}b\xb7{C}_{rate}}{R\xb7T}\right)\xb7{n}^{0.55}$ | a |

$16.\text{}{Q}_{loss}=\left(a\xb7{T}^{2}+b\xb7T+c\right)\xb7\mathrm{exp}\left(\left(d\xb7T+e\right)\xb7{I}_{rate}\right)\xb7n$ | e |

$17.\text{}{Q}_{loss}=\left(a\xb7\mathrm{exp}\left(b\xb7I\right)+c\xb7\mathrm{exp}\left(d\xb7I\right)\right)\xb7\mathrm{exp}\left(\frac{e+f\xb7I}{R\xb7T}\right)\xb7A{h}^{0.55}$ | b |

Improved Models |
---|

$18.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7\mathrm{exp}\left(\frac{{a}_{2}}{R\xb7T}\right)\xb7A{h}^{\left({C}^{{a}_{3}}+{a}_{4}\right)}$ |

$19.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7\mathrm{exp}\left(\frac{{a}_{2}\text{}\ast \text{}{T}^{2}\text{}+\text{}{a}_{3}}{R\xb7T}\right)\xb7A{h}^{({C}^{{a}_{4}}+{a}_{5})}$ |

$20.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7\mathrm{exp}\left(\frac{{a}_{2}\text{}\xb7\text{}c\text{}+\text{}{a}_{3}}{R\xb7T}\right)\xb7{C}^{{a}_{4}}\xb7A{h}^{{a}_{5}}+{a}_{6}\xb7\mathrm{exp}\left(\frac{{a}_{7}}{R\xb7T}\right)\xb7A{h}^{{a}_{8}}$ |

$21.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7{C}^{\beta}\xb7\mathrm{exp}\left(\frac{{a}_{2}}{R\xb7T}\right)\xb7A{h}^{{a}_{3}}$ |

$22.\text{}{Q}_{\mathit{loss}}={a}_{1}\xb7{C}^{\beta}\xb7\mathrm{exp}\left(\frac{{a}_{2}\text{}\xb7\text{}C\text{}+\text{}{a}_{3}}{R\xb7T}\right)\xb7A{h}^{{a}_{4}\xb7C+{a}_{5}}$ |

Model | Dataset I | Dataset II | ||
---|---|---|---|---|

Train Dataset | Test Dataset | Train Dataset | Test Dataset | |

Model 14 | 1.12% | 1.22% | 1.12% | 1.19% |

Model 23 | 1.06% | 1.21% | 1.02% | 1.09% |

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## Share and Cite

**MDPI and ACS Style**

Jia, X.; Zhang, C.; Wang, L.; Zhang, W.; Zhang, L.
Modification of Cycle Life Model for Normal Aging Trajectory Prediction of Lithium-Ion Batteries at Different Temperatures and Discharge Current Rates. *World Electr. Veh. J.* **2022**, *13*, 59.
https://doi.org/10.3390/wevj13040059

**AMA Style**

Jia X, Zhang C, Wang L, Zhang W, Zhang L.
Modification of Cycle Life Model for Normal Aging Trajectory Prediction of Lithium-Ion Batteries at Different Temperatures and Discharge Current Rates. *World Electric Vehicle Journal*. 2022; 13(4):59.
https://doi.org/10.3390/wevj13040059

**Chicago/Turabian Style**

Jia, Xinyu, Caiping Zhang, Leyi Wang, Weige Zhang, and Linjing Zhang.
2022. "Modification of Cycle Life Model for Normal Aging Trajectory Prediction of Lithium-Ion Batteries at Different Temperatures and Discharge Current Rates" *World Electric Vehicle Journal* 13, no. 4: 59.
https://doi.org/10.3390/wevj13040059