# Hybrid Modeling of Lithium-Ion Battery: Physics-Informed Neural Network for Battery State Estimation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Adopting a hybrid approach that integrates relatively limited training data with knowledge of the internal phenomena of LIBs is a pivotal step toward achieving a significant breakthrough in the field of battery modeling**.

- Development of PINN for estimating the SOC and SOH of three LIB cells operating at different temperature ranges;
- PINN implementation is tested in different Python packages in order to verify the transferability of the methodology in different platforms. This allows for the wider adoption and practical application of the model in diverse settings;
- The model incorporates the governing equation of Fick’s law of diffusion behavior of solid-phase Li-ions to train the NN. This improves the accuracy of NN training and enhances the model’s predictive power.

## 2. State of the Research: Hybrid Modeling of Lithium-Ion Batteries

## 3. Formulation of the Problem

- Fick’s law of diffusion determined the solid-state Li-ion concentration (c
_{s}) in the electrodes; - The law of charge conservation determines the liquid-phase Li-ion concentration (c
_{e}) in the electrolyte and in the separator; - Ohm’s law determines the solid-state potential (ϕ
_{s}) in the electrodes; - Kirchhoff’s and Ohm’s laws are used to calculate the liquid-phase potential (ϕ
_{e}) in the electrolyte and in the separator; - The Butler–Volmer kinetics equation describes the flux density of Li-ions (j).

**Figure 1.**Illustration of a single particle model adapted from [41]. Particle represents the concentration gradient due to solid–phase diffusion. ${R}_{p}^{+}$ and ${R}_{p}^{-}$ are the particle radius and ${c}_{s}^{+}\left(r,t\right)$ and ${c}_{s}^{-}\left(r,t\right)$ are the solid–phase Li-ion concentration of the cathode and anode, respectively. $V\left(t\right)$ is the terminal voltage across cell.

_{x}C

_{6}→ C

_{6}+ xLi

^{+}+ xe

^{−};

_{1−x}Ni

_{0.6}Co

_{0.2}Mn

_{0.2}O

_{2}+ xLi

^{+}+ xe

^{−}→ LiNi

_{0.6}Co

_{0.2}Mn

_{0.2}O

_{2};

_{1−x}Ni

_{0.6}Co

_{0.2}Mn

_{0.2}O

_{2}+ Li

_{x}C

_{6}→ C

_{6}+ LiNi

_{0.6}Co

_{0.2}Mn

_{0.2}O

_{2};

## 4. Architecture and Methodology

#### 4.1. Experimental Data

#### 4.2. Data Preparation

#### 4.3. Architecture

#### 4.4. Training, Validation and Testing Data

^{−5}is set to ensure that the training stops when the loss function improvement falls below the given threshold.

^{−3}and the final learning rate to 1/100 of the initial learning rate. ‘ExponentialDecay’ learning rate scheduler is then used to gradually reduce the learning rate over time. In order to prevent overfitting, the ‘EarlyStopping’ callback function is used to monitor the training process. Adaptive weights are used to dynamically adjust the weights assigned to each loss term in the total loss function during training. This is implemented using the GradNorm method [56]. This is done in order to address the issue of gradient pathology, where loss terms with higher derivatives tend to dominate the total gradient vector and negatively affect the accuracy of the solution.

## 5. Results and Discussion

_{p}, the rate of change in Li-ion concentration depends on the molar flux density, as per the maximum boundary condition (Equation (6)).

#### 5.1. State of Charge Estimation

#### 5.2. State of Health Estimation

^{+}in the anode, as per Equations (16) and (17). Figure 9 shows the measured and predicted SOH plots for cell number 5 and cell number 7. The RMSE of SOH estimation for cell 5 is 2.381% and for cell 7 is 1.176%. The change in temperature for cell cycling does not majorly affect the PINN performance for SOH estimation.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Collocation points across the training domain, distributed randomly across the spatio-temporal domain and at the initial and boundary conditions.

**Figure 3.**Capacity check−up cycle followed by two sets of WLTP driving cycle. (

**a**) Constant current and constant voltage charge and constant current discharge during the check−up cycle. Cells are subjected to WLTP cycles until the SOC reaches 5%; (

**b**) temperature variations during cycling and check−ups. Purple curve is the cell temperature; blue curve is the temperature at the negative terminal; black curve is the temperature at the positive terminal.

**Figure 5.**(

**Left**) Change in the Li

^{+}concentration across the anode particle radius at the beginning of the CC discharge cycle. (

**Right**) Change in the Li

^{+}concentration across the anode particle radius at the end of the CC discharge cycle.

**Figure 6.**(

**Left**) Change in the Li

^{+}concentration across the anode particle radius at the beginning of the CCCV charge cycle. (

**Right**) Change in the Li

^{+}concentration across the anode particle radius at the end of the CC discharge cycle.

**Figure 8.**SOC calculated from the estimation of lithium concentration in the anode: (

**a**) Discharge cycle of cell 3 during the 1st check-up cycle; (

**b**) charge cycle of cell 3 during the 1st check-up cycle.

**Figure 9.**(

**a**) SOH estimation of cell 5 across 1100 cycles; (

**b**) SOH estimation of cell 5 across 1000 cycles.

**Figure 10.**SOH estimation of cell 3 across 1300 cycles trained until the 8th check-up cycle and tested afterwards until the 13th check-up cycle. Comparison of SOH trained solely based on data, solely based on physics, and a combination of the two.

Symbol | Value | Unit | Denotation |
---|---|---|---|

C_{nominal} | 51 | Ah | Nominal Cell Capacity |

A | 1.843 | m^{2} | Electrode surface area |

x_{0%} | 0.028 | Stoichiometry anode (SOC = 0%) | |

x_{100%} | 0.794 | Stoichiometry anode (SOC = 100%) | |

y_{0%} | 0.914 | Stoichiometry cathode (SOC = 0%) | |

y_{100%} | 0.344 | Stoichiometry cathode (SOC = 100%) | |

L^{+} | 55.5 × 10^{−6} | m | Cathode thickness |

L^{−} | 67.3 × 10^{−6} | m | Anode thickness |

R^{+} | 3.28 × 10^{−6} | m | Cathode particle radius |

R^{−} | 6.72 × 10^{−6} | m | Anode particle radius |

ε^{+} | 0.755 | Active material volume fraction cathode | |

ε− | 0.7 | Active material volume fraction anode | |

${c}_{s,max}^{i}$^{1} | 36,129.55 | mol/m^{3} | Maximum theoretical solid-phase concentration of Li^{+} ions in the electrodes |

I | −17 to 17 | A | Charge and discharge current input |

t | $f\left(I\right)$ | s | Time corresponding to the input current |

^{1}i = p for cathode and i = n for anode.

PINN | Only Data | Only Physics | |
---|---|---|---|

RMSE | 1.32% | 3.48% | 3.94% |

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## Share and Cite

**MDPI and ACS Style**

Singh, S.; Ebongue, Y.E.; Rezaei, S.; Birke, K.P.
Hybrid Modeling of Lithium-Ion Battery: Physics-Informed Neural Network for Battery State Estimation. *Batteries* **2023**, *9*, 301.
https://doi.org/10.3390/batteries9060301

**AMA Style**

Singh S, Ebongue YE, Rezaei S, Birke KP.
Hybrid Modeling of Lithium-Ion Battery: Physics-Informed Neural Network for Battery State Estimation. *Batteries*. 2023; 9(6):301.
https://doi.org/10.3390/batteries9060301

**Chicago/Turabian Style**

Singh, Soumya, Yvonne Eboumbou Ebongue, Shahed Rezaei, and Kai Peter Birke.
2023. "Hybrid Modeling of Lithium-Ion Battery: Physics-Informed Neural Network for Battery State Estimation" *Batteries* 9, no. 6: 301.
https://doi.org/10.3390/batteries9060301