# Second-Life Batteries Modeling for Performance Tracking in a Mobile Charging Station

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## Abstract

**:**

## 1. Introduction

#### 1.1. Contributions

**1. Description of a new possible application for second life batteries**

**2. Experimental evaluation of second life battery performance**

**3. Modeling second life battery performance**

**4. Open access publishing of the experimental data and software**

#### 1.2. Layout

## 2. Second-Life Cell Performance Assessment

#### 2.1. Experimental Setup

_{2}eq and cost 53 € [36]. In order to minimize the cost of battery research, the data has been made open access. The authors hope that this data will benefit future research and allow a reduction in the number of experiments conducted.

#### 2.2. Capacity Measurement

#### 2.3. Resistance Measurement

#### 2.4. Performance Comparison of a Fresh and Second Life Cell

## 3. Second-Life Cell Modeling

#### 3.1. Model Definition

- ${U}_{cell}$: the voltage response of the cell (V)
- ${U}_{oc}$: the open circuit voltage (V)
- ${U}_{R0}$: the ohmic drop (V)
- ${U}_{CPE}$: the voltage dynamic modeled by the constant phase element (V)

- ${U}_{cell}$: the voltage response of the cell (V)
- ${U}_{oc}$: the open circuit voltage (V)
- ${U}_{R0}$: the ohmic drop (V)
- ${U}_{RiCi}$: the voltage dynamic modeled by i-th RC circuit (V)

#### 3.2. Static Behavior Identification

#### 3.3. Dynamic Behavior Identification

## 4. Models Accuracy Assessment

#### 4.1. Accuracy on a WLTC Profile

#### 4.1.1. Influence of the Model Choice over Models Accuracy

#### 4.1.2. Influence of the Temperature over Models Accuracy

#### 4.2. Accuracy on A Mobile Charging Station Profile

#### Influence of the Model Choice over Models Accuracy

#### 4.3. Computation Time and Error Assessment

^{®}Core™ i7-8650U CPU and a clock speed of 1.90 GHz.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BOL | Beginning of Life |

CC | Constant Current |

CPE | Constant Phase Element |

CV | Constant Voltage |

DoD | Depth of discharge |

EIS | Electrochemical Impedance Spectroscopy |

ICA | Incremental Capacity Analysis |

NMC | Nickel Manganese Cobalt |

OCV | Open Circuit Voltage |

RMSE | Root Mean Square Error |

SoH | Stater of Health |

WLTC | Worldwide Harmonized Light Vehicles Test Cycle |

## Appendix A. Equations

#### Appendix A.1. Capacity

- $Q\left(t\right)$: the discharge capacity measured (Ah)
- $t-{t}_{0}$: the duration of a discharge cycle (s)
- i: the discharge current applied to the cell (A)

#### Appendix A.2. State of Health Capacity

- $So{H}_{Q}$: the state of health “capacity” (%)
- $Q\left(t\right)$: the capacity measured at a given time (Ah)
- ${Q}_{BOL}$: the capacity measured at the beginning of life (BOL) (Ah)

#### Appendix A.3. Depth of Discharge

- $DoD$: the depth of discharge of the cell (%)
- $Do{D}_{t0}$: the initial cell depth of discharge (%)
- ${Q}_{BOL}$: the capacity measured at the beginning of life (BOL) (Ah)
- $i\left(t\right)$: the current at a given time (A) (Current is negative in discharge)

#### Appendix A.4. Pulse Resistance

- ${R}_{\Delta t}\left(t\right)$: the resistance measured at a given time ($\Omega $)
- $\Delta t$: the time after the pulse beginning (s)
- $\Delta V$: the voltage variation after $\Delta t$ (V)
- $\Delta I$: the current pulse amplitude (A)

#### Appendix A.5. State of Health Resistance

- $So{H}_{R}$: the state of health “resistance” (%)
- $R\left(t\right)$: the resistance measured at a given time (m$\Omega $)
- ${R}_{BOL}$: the resistance measured at the beginning of life (BOL) (m$\Omega $)

#### Appendix A.6. Resistance Dependence to Depth of Discharge

- $R\left(DoD\right)$: the resistance for a given depth of discharge ($\Omega $)
- ${r}_{i}$: the polynomial coefficient of the model ($\Omega $)

#### Appendix A.7. C1 Dependence to Depth of Discharge

- $C1\left(DoD\right)$: the resistance for a given depth of discharge (F)
- ${c}_{i}$: the polynomial coefficient of the model (F)

#### Appendix A.8. RC Impedance

- ${Z}_{RC}\left(jw\right)$: the RC impedance ($\Omega $)
- R: the resistance($\Omega $)
- R: the capacitor (F)

#### Appendix A.9. R1C1 Pulse Response Voltage

- ${U}_{R1C1}$: the voltage polarization caused by ${R}_{1}{C}_{1}$ (V)
- ${\tau}_{1}$: the polarization time constant of the fastest dynamic phenomena (s)

#### Appendix A.10. R2C2 Pulse Response Voltage

- ${U}_{R2C2}$: the voltage polarization caused by ${R}_{2}{C}_{2}$ (V)
- ${\tau}_{2}$: the polarization time constant of the fastest dynamic phenomena (s)

#### Appendix A.11. CPE Impedance

- ${Z}_{CPE}\left(jw\right)$: the CPE impedance ($\Omega $)
- $Q\left(DoD\right)$: the main CPE element for a given depth of discharge (${\Omega}^{-1}{s}^{\alpha}$)
- $\alpha $: the depression factor of the CPE (-)

#### Appendix A.12. CPE Voltage

- ${U}_{CPE}$: the voltage polarization caused by the CPE element (V)
- ${t}_{1}$: the time at the beginning of the pulse (s)
- ${t}_{2}$: the time at the end of the pulse (s)
- $\alpha $: the depression factor of the CPE (-)
- Q: the main CPE element (${\Omega}^{-1}{s}^{\alpha}$)
- $\Gamma $: the Gamma function

#### Appendix A.13. Gamma Function

- $\Gamma $: the Gamma function
- t: the time (s)
- $\alpha $: the depression factor of the CPE (-)

## Appendix B. Models Parameters

DoD | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

OCV | 4.150 | 4.099 | 4.080 | 4.069 | 4.053 | 4.046 | 4.038 | 4.029 | 4.021 |

4.150 | 4.105 | 4.089 | 4.082 | 4.075 | 4.067 | 4.058 | 4.049 | 4.039 | |

4.150 | 4.120 | 4.109 | 4.094 | 4.084 | 4.073 | 4.061 | 4.053 | 4.042 |

DoD | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 4.011 | 4.001 | 3.991 | 3.981 | 3.969 | 3.959 | 3.948 | 3.936 | 3.925 | 3.913 |

4.029 | 4.019 | 4.008 | 3.997 | 3.986 | 3.975 | 3.964 | 3.952 | 3.940 | 3.929 | |

4.023 | 4.009 | 3.997 | 3.990 | 3.982 | 3.974 | 3.965 | 3.956 | 3.947 | 3.937 |

DoD | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.902 | 3.890 | 3.879 | 3.867 | 3.856 | 3.845 | 3.834 | 3.823 | 3.813 | 3.803 |

3.917 | 3.905 | 3.894 | 3.882 | 3.871 | 3.859 | 3.849 | 3.838 | 3.827 | 3.817 | |

3.927 | 3.916 | 3.906 | 3.895 | 3.885 | 3.874 | 3.863 | 3.853 | 3.842 | 3.831 |

DoD | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.792 | 3.783 | 3.773 | 3.764 | 3.755 | 3.745 | 3.738 | 3.730 | 3.723 | 3.716 |

3.807 | 3.797 | 3.788 | 3.778 | 3.769 | 3.761 | 3.752 | 3.744 | 3.737 | 3.729 | |

3.821 | 3.811 | 3.801 | 3.791 | 3.781 | 3.772 | 3.763 | 3.754 | 3.746 | 3.738 |

DoD | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.709 | 3.702 | 3.696 | 3.690 | 3.685 | 3.679 | 3.674 | 3.669 | 3.665 | 3.661 |

3.722 | 3.716 | 3.709 | 3.703 | 3.698 | 3.692 | 3.687 | 3.682 | 3.678 | 3.674 | |

3.730 | 3.723 | 3.716 | 3.709 | 3.702 | 3.696 | 3.690 | 3.685 | 3.680 | 3.675 |

DoD | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.658 | 3.654 | 3.651 | 3.647 | 3.644 | 3.641 | 3.639 | 3.636 | 3.633 | 3.630 |

3.669 | 3.666 | 3.663 | 3.659 | 3.656 | 3.653 | 3.651 | 3.648 | 3.645 | 3.643 | |

3.671 | 3.667 | 3.663 | 3.659 | 3.656 | 3.653 | 3.649 | 3.647 | 3.644 | 3.642 |

DoD | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.628 | 3.625 | 3.622 | 3.619 | 3.616 | 3.612 | 3.608 | 3.604 | 3.599 | 3.595 |

3.640 | 3.637 | 3.635 | 3.632 | 3.629 | 3.626 | 3.623 | 3.619 | 3.615 | 3.611 | |

3.639 | 3.637 | 3.634 | 3.632 | 3.629 | 3.627 | 3.624 | 3.621 | 3.618 | 3.615 |

DoD | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.589 | 3.583 | 3.577 | 3.569 | 3.561 | 3.552 | 3.543 | 3.532 | 3.520 | 3.508 |

3.606 | 3.601 | 3.596 | 3.589 | 3.583 | 3.576 | 3.568 | 3.559 | 3.549 | 3.538 | |

3.611 | 3.607 | 3.602 | 3.597 | 3.591 | 3.585 | 3.577 | 3.569 | 3.561 | 3.551 |

DoD | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 |
---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.494 | 3.479 | 3.463 | 3.445 | 3.426 | 3.405 | 3.383 | 3.359 | 3.333 | 3.305 |

3.527 | 3.514 | 3.500 | 3.486 | 3.469 | 3.452 | 3.433 | 3.413 | 3.390 | 3.367 | |

3.540 | 3.529 | 3.516 | 3.502 | 3.486 | 3.469 | 3.451 | 3.431 | 3.410 | 3.387 |

DoD | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
---|---|---|---|---|---|---|---|---|---|---|---|

OCV | 3.275 | 3.244 | 3.209 | 3.173 | 3.135 | 3.094 | 3.050 | 3.004 | 2.955 | 2.903 | 2.700 |

3.342 | 3.315 | 3.286 | 3.255 | 3.222 | 3.187 | 3.150 | 3.110 | 3.068 | 2.944 | 2.700 | |

3.362 | 3.335 | 3.306 | 3.275 | 3.241 | 3.205 | 3.167 | 3.127 | 3.083 | 2.977 | 2.700 |

R0 Discharge | |||
---|---|---|---|

Temperature | ${\mathbf{r}}_{\mathbf{0},\mathbf{2},\mathbf{dch}}$ | ${\mathbf{r}}_{\mathbf{0},\mathbf{1},\mathbf{dch}}$ | ${\mathbf{r}}_{\mathbf{0},\mathbf{0},\mathbf{dch}}$ |

0 °C | $5.746\times {10}^{-4}$ | $-4.720\times {10}^{-3}$ | $1.157\times {10}^{-2}$ |

25 °C | $1.259\times {10}^{-4}$ | $8.614\times {10}^{-4}$ | $5.356\times {10}^{-3}$ |

40 °C | $-1.501\times {10}^{-4}$ | $9.962\times {10}^{-4}$ | $-9.188\times {10}^{-5}$ |

R0 Charge | |||
---|---|---|---|

Temperature | ${\mathbf{r}}_{\mathbf{0},\mathbf{2},\mathbf{ch}}$ | ${\mathbf{r}}_{\mathbf{0},\mathbf{1},\mathbf{ch}}$ | ${\mathbf{r}}_{\mathbf{0},\mathbf{0},\mathbf{ch}}$ |

0 °C | $6.953\times {10}^{-4}$ | $-5.542\times {10}^{-3}$ | $1.311\times {10}^{-2}$ |

25 °C | $-5.455\times {10}^{-5}$ | $2.307\times {10}^{-2}$ | $5.995\times {10}^{-3}$ |

40 °C | $2.273\times {10}^{-4}$ | $-1.776\times {10}^{-3}$ | $5.070\times {10}^{-3}$ |

R1 | |||
---|---|---|---|

Temperature | ${\mathbf{r}}_{\mathbf{1},\mathbf{2}}$ | ${\mathbf{r}}_{\mathbf{1},\mathbf{1}}$ | ${\mathbf{r}}_{\mathbf{1},\mathbf{0}}$ |

0 °C | $2.037\times {10}^{-4}$ | $-1.697\times {10}^{-3}$ | $4.208\times {10}^{-3}$ |

25 °C | $5.111\times {10}^{-5}$ | $-5.698\times {10}^{-4}$ | $1.847\times {10}^{-4}$ |

40 °C | $6.204\times {10}^{-5}$ | $-4.265\times {10}^{-4}$ | $1.405\times {10}^{-3}$ |

R2 | |||
---|---|---|---|

Temperature | ${\mathbf{r}}_{\mathbf{2},\mathbf{2}}$ | ${\mathbf{r}}_{\mathbf{2},\mathbf{1}}$ | ${\mathbf{r}}_{\mathbf{2},\mathbf{0}}$ |

0 °C | $5.399\times {10}^{-6}$ | $-2.517\times {10}^{-5}$ | $2.366\times {10}^{-5}$ |

25 °C | $2.813\times {10}^{-6}$ | $-1.355\times {10}^{-5}$ | $1.555\times {10}^{-5}$ |

40 °C | $5.994\times {10}^{-7}$ | $-3.216\times {10}^{-6}$ | $6.739\times {10}^{-6}$ |

C1 | C2 | ||||
---|---|---|---|---|---|

Temperature | ${\mathbf{c}}_{\mathbf{1},\mathbf{3}}$ | ${\mathbf{c}}_{\mathbf{1},\mathbf{2}}$ | ${\mathbf{c}}_{\mathbf{1},\mathbf{1}}$ | ${\mathbf{c}}_{\mathbf{1},\mathbf{0}}$ | $\mathbf{C}\mathbf{2}$ |

0 °C | $-3.094$ | 547 | $-3.119\times {10}^{4}$ | $6.601\times {10}^{5}$ | 58,422 |

25 °C | $-2.517$ | $380.3$ | $-1.765\times {10}^{4}$ | $4.053\times {10}^{5}$ | 62,841 |

40 °C | $-1.610$ | $178.7$ | $-3902$ | $1.703\times {10}^{5}$ | 95,467 |

Temperature | Q | $\mathit{\alpha}$ |
---|---|---|

0 °C | 3873 | $0.1$ |

25 °C | 4852 | $0.1$ |

40 °C | 5812 | $0.1$ |

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**Figure 3.**Voltage profile of the test. Step 1 is a capacity test, step 2 is the impedance measurement, step 3 is the low-current measurement and step 4 is the validation cycle.

**Figure 5.**Discharge resistance of a second life cell at 0 (blue points), 25 (green squares) and 40 °C (red diamonds). Measurement is made at −1C and a 5 s resistance is considered.

**Figure 8.**Experimental OCV-DoD relationship curves for second life battery at 0 °C (blue line), 25 °C (green line) and 40 °C (red line).

**Figure 9.**Experimental result and model response (lines) for 0 s discharge resistance versus depth of discharge and temperature 0 (blue), 25 (green) and 40 °C (red).

**Figure 10.**(

**a**) Power, (

**b**) state of charge, (

**c**) voltage and (

**d**) current evolution during a Worldwide Harmonized Light Vehicles Test Cycle profile.

**Figure 11.**(

**a**) Experimental data (blue line) compared to simulation results for the CPE model (solid line) and the RC model (dashed line) at 25 °C. (

**b**) Zoom around 5 min.

**Figure 12.**Error distribution functions for the CPE model (solid line) and the RC model (dashed line) emulating a cell voltage response used with a WLTC profile at 25 °C.

**Figure 13.**Influence of temperature over RC model ((

**left**), dashed line) and CPE model ((

**right**), solid line) accuracy. Blue lines are at 0 °C, green lines at 25 °C and red lines at 40 °C.

**Figure 14.**(

**a**) Power, (

**b**) state of charge, (

**c**) voltage and (

**d**) current evolution during a mobile charging station usage profile.

**Figure 15.**Influence of model choice over accuracy at 25 °C. Left, voltage profile; right error distribution (blue line = measurements, green solid line = CPE model, green dashed line = RC model).

**Table 1.**Samsung SDI 94Ah main characteristics. Data from [35].

Characteristics | Values |
---|---|

Format | Prismatic |

Nominal capacity [Ah] | 94 |

Positive electrode material | NMC |

Negative electrode material | Graphite |

Nominal voltage [V] | 3.68 |

Maximal voltage [V] | 4.15 |

Minimal voltage [V] | 2.7 |

Specific energy [Wh/kg] | 165 |

Size L × W × H [mm] | 173 × 125 × 45 |

Weight [kg] | 2.06 |

Step | Test | Estimated Duration (h) |
---|---|---|

1 | Capacity test | 18 |

2 | Impedance test | 8 |

3 | Pseudo OCV test | 42 |

4 | Mobile charging station cycles | 12 |

**Table 3.**Performance comparison between a fresh cell (datasheet) and a second life cell (experiments). “-” means that the state of health could not be calculated.

Temperature | Fresh Cell | Second Life Cell | State of Health | |
---|---|---|---|---|

Capacity (CC-CV, −1C) | 0 °C | No data | 90.5 Ah | - |

25 °C | 95.2 Ah | 92.4 Ah | 97% | |

40 °C | 95.7 Ah | 93.3 Ah | 97% | |

Resistance (5 s, DoD 50%, −1C) | 0 °C | No data | 2.3 m$\Omega $ | - |

25 °C | 0.75 m$\Omega $ | 1.3 m$\Omega $ | 27% | |

40 °C | No data | 1.6 m$\Omega $ | - |

**Table 4.**Accuracy and computation time for different temperatures using the mobile charging station and WLTC profiles.

WLTC | CPE Model | RC Model | ||

Accuracy(RMSE) | 0 °C | 17.5 mV | 14.8 mV | |

25 °C | 10.5 mV | 12.9 mV | ||

40 °C | 15.9 mV | 19.2 mV | ||

Computation time | 0 °C | 11.2 s | 3 s | |

25 °C | 8.7 s | 3.1 s | ||

40 °C | 9.9 s | 3.3 s | ||

Charger | CPE model | RC model | ||

Accuracy(RMSE) | 0 °C | 32.6 mV | 35.4 mV | |

25 °C | 29.5 mV | 27.2 mV | ||

Computation time | 0 °C | 71.9 s | 35.6 s | |

25 °C | 80.8 s | 35.1 s |

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## Share and Cite

**MDPI and ACS Style**

Hassini, M.; Redondo-Iglesias, E.; Venet, P.
Second-Life Batteries Modeling for Performance Tracking in a Mobile Charging Station. *World Electr. Veh. J.* **2023**, *14*, 94.
https://doi.org/10.3390/wevj14040094

**AMA Style**

Hassini M, Redondo-Iglesias E, Venet P.
Second-Life Batteries Modeling for Performance Tracking in a Mobile Charging Station. *World Electric Vehicle Journal*. 2023; 14(4):94.
https://doi.org/10.3390/wevj14040094

**Chicago/Turabian Style**

Hassini, Marwan, Eduardo Redondo-Iglesias, and Pascal Venet.
2023. "Second-Life Batteries Modeling for Performance Tracking in a Mobile Charging Station" *World Electric Vehicle Journal* 14, no. 4: 94.
https://doi.org/10.3390/wevj14040094