# A Health Assessment Method for Lithium-Ion Batteries Based on Evidence Reasoning Rules with Dynamic Reference Values

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## Abstract

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## 1. Introduction

- (1)
- An ER-DRV model is proposed to evaluate the health state of lithium-ion batteries.
- (2)
- In order to enhance the accuracy of the ER-DRV model, an improved WOA optimization method is proposed.
- (3)
- The perturbation analysis method is introduced into the ER-DRV model to explore its robustness.

## 2. Problem Formulation and Construction of the ER-DRV Model

#### 2.1. Problem Formulation

**Problem 1:**Construction of the lithium-ion battery health assessment model. When evaluating the health state of lithium batteries, the parameters in the assessment model cannot be accurately set due to various external environmental influences, which affects the accuracy of the assessment model. Real-time reliability, real-time weight, and indicator level reference values are important parameters in the model, which greatly affect the aggregation results of ER rules and the accuracy of the model. Therefore, we propose the following model to integrate multiple indicators and generate a comprehensive health assessment result $D(k)$ [19,24]:

**Problem 2:**Construction of the improved whale optimization algorithm (IWOA) optimization model. Some parameters of the health assessment model are provided by experts, such as the level reference values of indicators, and this may reduce the accuracy of the model. Therefore, the IWOA optimization model under non-perturbation conditions $W(\xb7)$ and the IWOA optimization model under perturbation conditions ${W}^{\prime}(\xb7)$ are proposed and used to adjust the level reference values, thus improving the accuracy of the health assessment model. The two kinds of IWOA optimization models are as follows:

**Problem 3:**Construction of a perturbation analysis method for investigating the robustness of the health assessment model. Lithium batteries are affected by various perturbations during their operation, leading to fluctuations in their health statuses. Therefore, a perturbation analysis is introduced to investigate the robustness of the model. The following method is used to perform health assessment results ${D}^{\prime}(k)$ for lithium-ion batteries under perturbations [19,25]:

#### 2.2. Construction of the ER-DRV Health Assessment Model

## 3. The Inference and Optimization Process of the ER-DRV Model

#### 3.1. Indicator Reliability

#### 3.2. Indicator Weight

#### 3.3. Indicator Standardization

#### 3.4. Fusion Method

#### 3.5. Optimization of the ER-DRV Model Parameters

#### 3.5.1. Improved Whale Optimization Algorithm (IWOA)

**Strategy 1:**Improve the convergence factor.

**Strategy 2:**Disturbances are introduced.

#### 3.5.2. The Optimization Process of the ER-DRV Model’s Parameters

**Step 1 (Initialization):**The variable $n$ represents the initial population of whales. The number of iterations of the search algorithm is denoted by the variable $t$. The maximum number of iterations allowed is denoted by the variable ${T}_{\mathrm{max}}$. The dimensions of the search space, which represent the number of variables in the optimization problem, are denoted by the variable $d$.

**Step 2 (Sample operations):**Randomly generate the location of each search agent $Y(t)$.

**Step 3 (Determine fitness):**The objective of this article is to enhance the assessment accuracy of the model. To achieve this, an objective function for optimization was developed and is explained as follows:

**Step 4 (Constraint operation):**Combining indicator data and expert knowledge, the upper and lower limits of constraint conditions were set.

**Step 5 (Exploration and exploitation):**When $p<0.5$ and $\left|A\right|<1$, humpback whales can capture prey and surround them. Because the exact location of the optimal solution in the search space is uncertain, the WOA operates under the assumption that the current best candidate solution represents the target prey or is in close proximity to the optimal solution. This assumption helps guide the search toward potentially better solutions and reduces the chances of becoming stuck in suboptimal regions. Once the best search agent is determined, other search agents strive to adjust their positions to match that of the best search agent. This process is described by the following two formulas:

## 4. Perturbation Analysis

#### 4.1. Definition of Perturbation

#### 4.2. The Reasoning Process of Perturbation Analysis

- Step 1: Based on the ER rule in (12)–(14), the first $(L-M)$ pieces of evidence without perturbation are fused with weight and reliability, and then $e(L-M)$ is calculated using (15).
- Step 2: Based on the ER rule, the previous assessment result $e(L-M)$ is mixed with the first perturbation piece of evidence to obtain $e(L-M+1)$. Subsequently, the expected utility $u({X}_{1})$ and perturbed expected utility $u({X}_{1}+\tau \Delta {x}_{1})$ are calculated using (16).
- Step 3: Repeat the ER rule in step 1 and combine $e(L-M+1)$ in step 2 with the remaining $(M-1)$ pieces of evidence to obtain the expected utility.

## 5. Case Study

#### 5.1. Research Background

#### 5.2. Health State Assessment of Lithium-Ion Batteries Based on ER Rules

#### 5.3. Health State Assessment of Lithium-Ion Batteries Using the ER-DRV Model

#### 5.4. Robustness Analysis of the ER-DRV Model

#### 5.5. Additional Experiment

#### 5.5.1. Health State Assessment

#### 5.5.2. Robustness Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Convergence factor and disturbances. (

**a**) Change in the convergence factor. (

**b**) The comparison of disturbances at different relative error levels.

**Figure 7.**Health assessment results: (

**a**) belief degree and (

**b**) a comparison between the actual results and predicted results.

**Figure 9.**Health assessment results after IWOA optimization: (

**a**) belief degree and (

**b**) comparison between the actual results and predicted results.

**Figure 16.**Health assessment results: (

**a**) belief degree before the IWOA and (

**b**) belief degree after the IWOA.

Indicators (Hour) | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) |
---|---|---|---|

CCCT | 1.7224 | 1.5523 | 1.4023 |

CVCT | 1.4564 | 1.2607 | 1.1434 |

DT | 0.2473 | 0.2014 | 0.1735 |

Performance Status | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) |
---|---|---|---|

Predicted results | 1–53 | 54–158 | 159–214 |

Actual results | 1–53 | 54–122 | 123–214 |

Indicators (Hour) | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) |
---|---|---|---|

CCCT | 2.0279 | 1.6878 | 1.4680 |

CVCT | 1.8975 | 1.3992 | 1.2084 |

DT | 0.2210 | 0.2000 | 0.1800 |

Accuracy of Assessment | $\mathit{n}$ = 20 | $\mathit{n}$ = 30 | $\mathit{n}$ = 50 | $\mathit{n}$ = 70 |
---|---|---|---|---|

$t$ = 10 | 92.52% | 93.46% | 94.86% | 95.79% |

$t$ = 30 | 93.46% | 94.86% | 95.33% | 95.79% |

$t$ = 50 | 95.33% | 95.33% | 95.33% | 96.73% |

$t$ = 100 | 95.33% | 95.79% | 95.79% | 95.33% |

Accuracy of Assessment | $\mathit{n}$ = 20 | $\mathit{n}$ = 30 | $\mathit{n}$ = 50 | $\mathit{n}$ = 70 |
---|---|---|---|---|

$t$ = 10 | 97.66% | 98.06% | 99.07% | 99.07% |

$t$ = 30 | 98.13% | 99.07% | 99.53% | 99.13% |

$t$ = 50 | 97.66% | 99.07% | 99.53% | 99.53% |

$t$ = 100 | 99.53% | 99.53% | 99.53% | 99.53% |

NO. | Model | Accuracy of Assessment |
---|---|---|

1 | ER | 83.18% |

2 | ER-WOA | 95.33% |

3 | ER-DRV | 99.53% |

4 | Decision Tree | 96.24% |

5 | MLP | 85.94% |

6 | SVM | 84.51% |

Observed Indicators (Hour) | Perturbation Intensities | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) | Perturbation Intensities | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) |
---|---|---|---|---|---|---|---|---|

CCCT | 0.00150 | 2.1424 | 1.5926 | 1.1193 | 0.00155 | 2.2000 | 1.6900 | 1.4700 |

CVCT | 1.6234 | 1.3368 | 1.2092 | 1.4661 | 1.4100 | 1.2100 | ||

DT | 0.2200 | 0.2162 | 0.1890 | 0.2203 | 0.1988 | 0.1873 | ||

CCCT | 0.00160 | 2.2000 | 1.6723 | 1.4700 | 0.00165 | 2.2000 | 1.6258 | 1.2068 |

CVCT | 1.4565 | 1.3027 | 0.6707 | 1.4200 | 1.3071 | 1.2100 | ||

DT | 0.2216 | 0.2134 | 0.1890 | 0.2207 | 0.2150 | 0.1890 | ||

CCCT | 0.00170 | 1.8440 | 1.5798 | 1.2009 | 0.00175 | 1.8336 | 1.5573 | 1.4678 |

CVCT | 1.4200 | 1.3839 | 1.1566 | 1.6988 | 1.4096 | 1.0992 | ||

DT | 0.2217 | 0.2133 | 0.1890 | 0.2218 | 0.2114 | 0.1888 |

NO. | Perturbation Intensities | Accuracy |
---|---|---|

1 | 0.00150 | 98.79% |

2 | 0.00155 | 98.13% |

3 | 0.00160 | 97.66% |

4 | 0.00165 | 97.66% |

5 | 0.00170 | 97.66% |

6 | 0.00175 | 97.66% |

Indicators (Hour) | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) |
---|---|---|---|

CCCT | 0.9190 | 0.6479 | 0.4120 |

CVCT | 1.6852 | 2.1058 | 2.3990 |

Indicators (Hour) | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) |
---|---|---|---|

CCCT | 0.9350 | 0.6520 | 0.5421 |

CVCT | 1.7910 | 1.9921 | 2.2113 |

NO. | Model | Accuracy of Assessment |
---|---|---|

1 | ER | 84.87% |

2 | ER-WOA | 96.71% |

3 | ER-DRV | 98.03% |

4 | Decision Tree | 97.35% |

5 | MLP | 95.36% |

6 | SVM | 94.04% |

Observed Indicators (Hour) | Perturbation Intensities | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) | Perturbation Intensities | ${\mathit{H}}_{1}$ (Good) | ${\mathit{H}}_{2}$ (Normal) | ${\mathit{H}}_{3}$ (Poor) |
---|---|---|---|---|---|---|---|---|

CCCT | 0.00150 | 0.9200 | 0.6799 | 0.5827 | 0.00155 | 0.9400 | 0.6600 | 0.5500 |

CVCT | 1.7500 | 2.1167 | 2.1325 | 1.8300 | 2.1512 | 2.1823 | ||

CCCT | 0.00160 | 0.9428 | 0.6785 | 0.5420 | 0.00165 | 0.9425 | 0.6714 | 0.5539 |

CVCT | 1.7535 | 2.1230 | 2.1680 | 1.8385 | 2.1552 | 2.1822 |

NO. | Perturbation Intensities | Accuracy |
---|---|---|

1 | 0.00150 | 96.05% |

2 | 0.00155 | 96.05% |

3 | 0.00160 | 96.05% |

4 | 0.00165 | 96.05% |

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## Share and Cite

**MDPI and ACS Style**

Yang, Z.; Zhao, X.; Zhang, H.
A Health Assessment Method for Lithium-Ion Batteries Based on Evidence Reasoning Rules with Dynamic Reference Values. *Batteries* **2024**, *10*, 26.
https://doi.org/10.3390/batteries10010026

**AMA Style**

Yang Z, Zhao X, Zhang H.
A Health Assessment Method for Lithium-Ion Batteries Based on Evidence Reasoning Rules with Dynamic Reference Values. *Batteries*. 2024; 10(1):26.
https://doi.org/10.3390/batteries10010026

**Chicago/Turabian Style**

Yang, Zijiang, Xiaofeng Zhao, and Hongquan Zhang.
2024. "A Health Assessment Method for Lithium-Ion Batteries Based on Evidence Reasoning Rules with Dynamic Reference Values" *Batteries* 10, no. 1: 26.
https://doi.org/10.3390/batteries10010026