# A Health Indicator for the Online Lifetime Estimation of an Electric Vehicle Power Li-Ion Battery

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. ERL-Based Lifetime Estimator

#### 2.1. ERL Extraction

#### 2.2. Box–Cox Transformation

**y**, the transformation parameter λ, and the transformed data

**y**

^{(λ)}. The specific transformation equation is as follows:

_{0}, β

_{1}, …, β

_{q}are the coefficients, q is the number of independent variables, and ε

_{i}are independent random errors and satisfy ${\epsilon}_{\mathrm{i}}\sim N(0,{\sigma}^{2}),i=1,2,\cdots ,n$.

**y**

^{(λ)}~ N(

**Xβ**, σ

^{2}

**I**), where

**X**is a design matrix with

**X**= (

**X**

_{1},

**X**

_{2}, …,

**X**

_{n})

^{T},

**X**

_{i}= (1, x

_{i1}, x

_{i2}, …, x

_{iq}),

**β**= (β

_{0}, β

_{1}, …, β

_{q})

^{T}and the model parameters are (λ, β, σ

^{2}). Construct the likelihood function L(λ, β, σ

^{2}|

**y**,

**X**) as follows:

**y**

^{(λ)}) is the density of

**y**

^{(λ)}and J(λ,

**y**) is the Jacobian of

**y**transformed into

**y**

^{(λ)}, namely $J(\lambda ,\mathit{y})={\displaystyle \prod _{i=1}^{n}{\mathit{y}}_{i}^{\lambda -1}}$.

**β**, σ

^{2}) likelihood equation used to estimate the observed

**y**

^{(λ)}for each fixed λ. Therefore, the maximum likelihood estimation formula of (

**β**, σ

^{2}) can be obtained as follows:

**β**, σ

^{2}):

- (a)
- Choose an initial value of λ within a suitable range (such as [–5,5]).
- (b)
- Substitute the initial λ to calculate the corresponding g(λ).
- (c)
- Calculate all g(λ) corresponding to the remaining λ in turn.
- (d)
- Plot the correlation curve of g(λ) and λ.
- (e)
- Select the λ that maximizes g(λ).

## 3. Experimental Section

_{2}(NCM) batteries were employed to verify the performance of the proposed method, and the specifications are shown in Table 1. The test bench, shown in Figure 2, consists of a Neware BTS-4000-5V/10A machine (Shenzhen, Guangdong, China) with a sampling frequency of 10 Hz and a sampling terminal voltage accuracy of 0.1% [25]. The batteries were kept in a thermostat at 25 °C.

## 4. Results and Discussion

#### 4.1. Estimation Dispersion of ERL

#### 4.2. Optimization of ERL

**X**= [1, ERL],

**β**= [β

_{0}, β

_{1}]

^{T}, and

**C**

^{(λ)}can be transformed as follows:

#### 4.3. Estimation Result

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Relationship between excitation response level (ERL) and capacity: (

**a**) a dynamic stress test (DST) profile; (

**b**) a constant current (CC) profile.

Type | Size | Terminal Voltage (V) | Operating Voltage (V) | Capacity (Ah) | Maximum Discharge Rate (C) |
---|---|---|---|---|---|

Li(NiCoMn)O_{2} (NCM) | 21,700 | 3.7 | 2.7–4.2 | 4 | 4 |

Battery | Depth of Discharge (DOD) | State of Charge (SOC) Ranges |
---|---|---|

# 1 | 50% | 25–75% |

# 2 | 50% | 35–85% |

# 3 | 40% | 30–70% |

Error | IR-Based Method | IC-Based Method | ERL-Based Method |
---|---|---|---|

Peak error | 18.58% | 15.84% | 5.89% |

Average error | 4.90% | 5.02% | 2.95% |

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## Share and Cite

**MDPI and ACS Style**

Yu, B.; Qiu, H.; Weng, L.; Huo, K.; Liu, S.; Liu, H.
A Health Indicator for the Online Lifetime Estimation of an Electric Vehicle Power Li-Ion Battery. *World Electr. Veh. J.* **2020**, *11*, 59.
https://doi.org/10.3390/wevj11030059

**AMA Style**

Yu B, Qiu H, Weng L, Huo K, Liu S, Liu H.
A Health Indicator for the Online Lifetime Estimation of an Electric Vehicle Power Li-Ion Battery. *World Electric Vehicle Journal*. 2020; 11(3):59.
https://doi.org/10.3390/wevj11030059

**Chicago/Turabian Style**

Yu, Bin, Haifeng Qiu, Liguo Weng, Kailong Huo, Shiqi Liu, and Haolu Liu.
2020. "A Health Indicator for the Online Lifetime Estimation of an Electric Vehicle Power Li-Ion Battery" *World Electric Vehicle Journal* 11, no. 3: 59.
https://doi.org/10.3390/wevj11030059