# Parameters Identification for Lithium-Ion Battery Models Using the Levenberg–Marquardt Algorithm

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## Abstract

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## 1. Introduction

- We developed and implemented a new robust framework for model validation and parameter identification for lithium-ion batteries, leveraging a hybrid optimization approach that combines the Gauss–Newton algorithm and gradient descent technique, the so-called Levenberg–Marquardt algorithm.
- This framework effectively balances the precision of Gauss–Newton with the robustness of gradient descent, making it particularly valuable for parameter identification problems.
- This framework has been verified using experimental measurements on the INR 18650-20R battery, conducted by the Center for Advanced Life Cycle Engineering (CALCE) battery group at the University of Maryland.
- This work presented a comprehensive comparative study between various types of models, specifically first-, second-, and third-order models.

## 2. Battery Modeling

- First-Order Model ($N=1$):$${V}_{t}=\mathrm{OCV}-I\left(t\right){R}_{0}-{V}_{1}$$$${V}_{1}\left(t\right)={V}_{1}\left(0\right){e}^{-{\displaystyle \frac{t}{{R}_{1}{C}_{1}}}}+\left(1-{e}^{-{\displaystyle \frac{t}{{R}_{1}{C}_{1}}}}\right)\times I\left(t\right){R}_{1}$$
- Second-Order Model ($N=2$):$${V}_{t}=\mathrm{OCV}-I\left(t\right){R}_{0}-{V}_{1}-{V}_{2}$$$${V}_{1}\left(t\right)={V}_{1}\left(0\right){e}^{-{\displaystyle \frac{t}{{R}_{1}{C}_{1}}}}+\left(1-{e}^{-{\displaystyle \frac{t}{{R}_{1}{C}_{1}}}}\right)\times I\left(t\right){R}_{1}$$$${V}_{2}\left(t\right)={V}_{2}\left(0\right){e}^{-{\displaystyle \frac{t}{{R}_{2}{C}_{2}}}}+\left(1-{e}^{-{\displaystyle \frac{t}{{R}_{2}{C}_{2}}}}\right)\times I\left(t\right){R}_{2}$$
- Third-Order Model ($N=3$):$${V}_{t}=\mathrm{OCV}-I\left(t\right){R}_{0}-{V}_{1}-{V}_{2}-{V}_{3}$$$${V}_{1}\left(t\right)={V}_{1}\left(0\right){e}^{-{\displaystyle \frac{t}{{R}_{1}{C}_{1}}}}+\left(1-{e}^{-{\displaystyle \frac{t}{{R}_{1}{C}_{1}}}}\right)\times I\left(t\right){R}_{1}$$$${V}_{2}\left(t\right)={V}_{2}\left(0\right){e}^{-{\displaystyle \frac{t}{{R}_{2}{C}_{2}}}}+\left(1-{e}^{-{\displaystyle \frac{t}{{R}_{2}{C}_{2}}}}\right)\times I\left(t\right){R}_{2}$$$${V}_{3}\left(t\right)={V}_{3}\left(0\right){e}^{-{\displaystyle \frac{t}{{R}_{3}{C}_{3}}}}+\left(1-{e}^{-{\displaystyle \frac{t}{{R}_{3}{C}_{3}}}}\right)\times I\left(t\right){R}_{3}$$

## 3. Methodology

- Model Verification: This step involves validating the battery model by comparing the simulation results to lab experimental measurements. The model will be verified if it matches accurately with simulation data, and ensures that the model parameters are accurately tuned.
- Model Parameter Identification: In this step, the LMA is utilized, to obtain and optimize the model parameters. If the simulation results match well with the experimental lab measurements, the model is verified, and there is no need for this step.

#### 3.1. Levenberg–Marquardt Algorithm (LMA)

#### 3.2. Evaluation Criteria

#### 3.3. Parameter Extraction Process

## 4. Experimental Methodology

#### 4.1. Experimental Setup

- An incremental OCV test using negative pulse discharge (PD) or positive pulse charge (PC) tests:
- C-rate: The pulse charge–discharge tests were conducted using a current corresponding to a C-rate of 0.5C [64]. The C-rate is the rate at which a battery is charged or discharged [66,67]. For example, in our experiments, the rated capacity was 2 Ah, and a C-rate of 0.5C meant the battery was charged or discharged at half the usual rate, taking two hours to complete. The corresponding charging or discharging current would be 1 A.
- Temperature: PD and PC tests were performed at a controlled temperature of 25 °C to ensure consistent and reliable results.

- Estimation method development: Using PD test results, an estimation method for OCV–SOC was developed.
- Method validation: The developed method was validated using results from the PC test.

#### 4.2. Assumptions and Limitations

- Nonlinear model behavior: The proposed ECM is inherently nonlinear, with parameters that vary in a stepwise manner with the SOC. This assumption helps the model accurately capture the dynamic behavior of the battery across different SOC ranges.
- Temperature control: The model assumes that the battery operates under a controlled temperature environment (25 °C) during both the charging and discharging phases. This minimizes the impact of temperature fluctuations on the accuracy of the model’s parameters.
- Constant discharge rates: The model is based on the assumption of a constant discharge rate during parameter identification. Consistency in discharge rates is crucial for maintaining the validity of the identified parameters.

- Applicability to battery packs: While the model has been validated for a single cell, extending the proposed method to battery packs introduces challenges, such as managing inter-cell variations, thermal management, and balancing issues. Future work will focus on refining the model to address these complexities.
- Operating conditions: The model’s performance might be affected by operating conditions not covered in this study, such as extreme temperatures or varying discharge rates. These factors can introduce non-linearity that is not accounted for by the current model configuration.
- SOC step-wise variations: The assumption of step-wise changes in model parameters with SOC, while useful for single-cell analysis, may require refinement when applied to battery packs where SOC variations are more gradual and influenced by inter-cell differences.
- Limited experimental data: The parameter identification and model validation were based on a controlled set of experiments. Expanding the model’s applicability would require additional data collection under a broader range of conditions, including different battery chemistries, sizes, operating environments, and varying C-rates.

#### 4.3. Experimental Data

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AFFRLS | adaptive forgetting factor recursive least squares |

BMS | battery management system |

CALCE | Center for Advanced Life Cycle Engineering |

ECM | equivalent circuit model |

EV | electric vehicle |

LIBs | lithium-ion batteries |

LMA | Levenberg–Marquardt algorithm |

MAE | mean absolute error |

OCV | open-circuit voltage |

PC | pulse charge |

PD | pulse discharge |

PNGV | partnership for a new generation of vehicle |

RC | resistor capacitor |

RMSE | root mean square error |

RLS | recursive least squares |

RTLS | recursive total least squares |

SOC | state of charge |

SOH | state of health |

VRLS | variable recursive least squares |

ML | machine learning |

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**Figure 1.**ECM of the Li-ion battery model that consists of N pairs of resistors and capacitors connected in parallel, using Thevenin’s method.

**Figure 3.**Experimental discharging current [64].

**Figure 4.**Experimental charging current [64].

**Figure 5.**Experimental setup for battery tests [69].

**Figure 7.**The simulation and experimental comparison results of the ${V}_{t}$ described by the first-order RC equivalent circuit model during the discharge phase.

**Figure 8.**The simulation and experimental comparison results of the ${V}_{t}$ described by the second-order RC equivalent circuit model during the discharge phase.

**Figure 9.**The simulation and experimental comparison results of the ${V}_{t}$ described by the third-order RC equivalent circuit model during the discharge phase.

**Figure 10.**Terminal voltage prediction for the first-order model during the pulse charging validation experiment.

**Figure 11.**Terminal voltage prediction for the second-order model during the pulse charging validation experiment.

**Figure 12.**Terminal voltage prediction for the third-order model during the pulse charging validation experiment.

Battery (Parameters) | Specifications (Value) |
---|---|

Capacity rating | 2000 mAh |

Cell chemistry | (LiNiMnCo)/Graphite |

Weight (without safety circuit) | 45 g |

Diameter | 18.3 mm |

Length | 64.85 mm |

**Table 2.**The identified parameters of the INR 18650-20R battery for the first-order RC equivalent circuit model.

SOC (%) | OCV (V) | R0 (Ω) | R1 (Ω) | ${\mathit{\tau}}_{1}$ (S) |
---|---|---|---|---|

100 | 3.34461 | 0.01012 | 0.00127 | 29.54095 |

90 | 3.55692 | 0.12171 | 0.02887 | 514.26424 |

80 | 3.59764 | 0.11864 | 0.02905 | 417.01342 |

70 | 3.62424 | 0.11946 | 0.02609 | 630.70774 |

60 | 3.66142 | 0.11910 | 0.00903 | 647.54580 |

50 | 3.74990 | 0.13143 | 0.04547 | 871.84257 |

40 | 3.83603 | 0.11698 | 0.03373 | 120.76715 |

30 | 3.93572 | 0.11904 | 0.02057 | 166.97654 |

20 | 4.04439 | 0.11716 | 0.01272 | 197.77709 |

10 | 4.17219 | 0.10940 | 0.01502 | 52.25380 |

**Table 3.**The identified parameters of the INR 18650-20R battery for the second-order RC equivalent circuit model.

SOC (%) | OCV (V) | R0 (Ω) | R1 (Ω) | ${\mathit{\tau}}_{1}$ (S) | R2 (Ω) | ${\mathit{\tau}}_{2}$ (S) |
---|---|---|---|---|---|---|

100 | 3.34503 | 0.01012 | 0.00127 | 29.55413 | 0.00050 | 99.99188 |

90 | 3.55696 | 0.12154 | 0.02862 | 524.85038 | 0.00052 | 99.02569 |

80 | 3.59765 | 0.11845 | 0.02878 | 422.61110 | 0.00052 | 98.71502 |

70 | 3.62425 | 0.11921 | 0.02594 | 645.84416 | 0.00053 | 99.66802 |

60 | 3.66147 | 0.11884 | 0.00885 | 663.15801 | 0.00052 | 98.75966 |

50 | 3.74995 | 0.13118 | 0.04523 | 880.13736 | 0.00054 | 98.72557 |

40 | 3.83607 | 0.11687 | 0.03345 | 120.96445 | 0.00050 | 99.34871 |

30 | 3.93573 | 0.11891 | 0.01992 | 163.96142 | 0.00050 | 99.12357 |

20 | 4.04442 | 0.11680 | 0.01263 | 195.55564 | 0.00050 | 99.04769 |

10 | 4.17220 | 0.10950 | 0.01462 | 53.27939 | 0.00049 | 98.63103 |

**Table 4.**The identified parameters of the INR 18650-20R battery for the third-order RC equivalent circuit model.

SOC (%) | OCV (V) | R0 (Ω) | R1 (Ω) | ${\mathit{\tau}}_{1}$ (S) | R2 (Ω) | ${\mathit{\tau}}_{2}$ (S) | R3 (Ω) | ${\mathit{\tau}}_{3}$ (S) |
---|---|---|---|---|---|---|---|---|

100 | 3.34464 | 0.01012 | 0.00121 | 30.01695 | 0.00047 | 99.03650 | 0.00049 | 501.81983 |

90 | 3.55697 | 0.12121 | 0.02764 | 547.68557 | 0.00130 | 58.65421 | 0.00051 | 483.07006 |

80 | 3.59769 | 0.11794 | 0.02852 | 438.47975 | 0.00148 | 47.90548 | 0.00048 | 552.70953 |

70 | 3.62434 | 0.11903 | 0.02712 | 775.23616 | 0.00162 | 94.37027 | 0.00063 | 374.92226 |

60 | 3.66163 | 0.11848 | 0.00753 | 805.27240 | 0.00148 | 58.48467 | 0.00059 | 187.29149 |

50 | 3.74997 | 0.13007 | 0.04306 | 922.65085 | 0.00180 | 103.97069 | 0.00069 | 288.77664 |

40 | 3.83622 | 0.11585 | 0.03351 | 122.17773 | 0.00098 | 77.93419 | 0.00057 | 120.80545 |

30 | 3.93577 | 0.11797 | 0.01914 | 157.44710 | 0.00091 | 73.59992 | 0.00054 | 203.96416 |

20 | 4.04447 | 0.11585 | 0.01142 | 196.78995 | 0.00084 | 70.77812 | 0.00053 | 180.80653 |

10 | 4.17227 | 0.10830 | 0.01433 | 56.73491 | 0.00049 | 98.95913 | 0.00057 | 187.58836 |

**Table 5.**A comparative analysis of the different model orders during the identification and verification phases.

Model | Experimental Phase | RMSE | MAE | Identification Time [Minutes] |
---|---|---|---|---|

First-order model | Identification | $3.25\times {10}^{-3}$ | $1.20\times {10}^{-3}$ | 182 |

Verification | $4.79\times {10}^{-3}$ | $3.50\times {10}^{-3}$ | ||

Second-order model | Identification | $3.12\times {10}^{-3}$ | $1.10\times {10}^{-3}$ | 245 |

Verification | $4.69\times {10}^{-3}$ | $3.40\times {10}^{-3}$ | ||

Third-order model | Identification | $2.99\times {10}^{-3}$ | $1.10\times {10}^{-3}$ | 593 |

Verification | $4.53\times {10}^{-3}$ | $3.30\times {10}^{-3}$ |

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© 2024 by the authors. Published by MDPI on behalf of the World Electric Vehicle Association. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Alshawabkeh, A.; Matar, M.; Almutairy, F.
Parameters Identification for Lithium-Ion Battery Models Using the Levenberg–Marquardt Algorithm. *World Electr. Veh. J.* **2024**, *15*, 406.
https://doi.org/10.3390/wevj15090406

**AMA Style**

Alshawabkeh A, Matar M, Almutairy F.
Parameters Identification for Lithium-Ion Battery Models Using the Levenberg–Marquardt Algorithm. *World Electric Vehicle Journal*. 2024; 15(9):406.
https://doi.org/10.3390/wevj15090406

**Chicago/Turabian Style**

Alshawabkeh, Ashraf, Mustafa Matar, and Fayha Almutairy.
2024. "Parameters Identification for Lithium-Ion Battery Models Using the Levenberg–Marquardt Algorithm" *World Electric Vehicle Journal* 15, no. 9: 406.
https://doi.org/10.3390/wevj15090406