# Accurate Real Time On-Line Estimation of State-of-Health and Remaining Useful Life of Li ion Batteries

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## Abstract

**:**

## 1. Introduction

_{m}(the remaining charge in LiB) can be a lot lower and thus 20% SoC that are ready for re-charge could actually be much lower. Such inaccuracy or uncertainty in SoH estimation can also lead to being over conservative on the user’s part that increase the SoC cut off for battery charging, and lead to a higher number of charge cycles than necessary [6]. Both situations can accelerate the battery’s degradation.

## 2. Materials and Methods

_{m}). This Q

_{m}is used to compute the corresponding SOH using Equation (1) where Q

_{max(fresh)}is the Q

_{m}after the first discharge cycle, and Q

_{max(aged)}is the Q

_{m}after the subsequent cycles. The SoH computed is termed as Experimental SoH when the Q

_{m}s are determined using ECBE model.

_{1}accounts for the capacity losses that increase rapidly during the conditions of cycling at high temperature, and k

_{2}is a factor to account for capacity losses under the normal conditions of cycling. k

_{3}accounts for the capacity loss due to C-rate [21].

_{1}, k

_{2}, and k

_{3}) can be extracted by substituting the Experimental SoH obtained from Equation (1) into Equation (2) at 3 different cycles (the exact cycles can be seen in Table 3 later in order to ensure the largest cycle chosen among the 3 cycles is half of the cycle at which the SoH of the LiB is around 80%) as shown in Figure 2. The extracted parameters are then used to calculate the Estimated SoH in this work. Our experimental procedure is depicted in Figure 3. There are large fluctuations or non-linearity observed in Qm over the first few cycles, and this non-linearity is higher at lower temperatures, which is also observed by other researchers [14], thus the first 50 cycles are not used for the k values extraction.

## 3. Results

#### 3.1. Effect of C-Rate at Constant Temperature Conditions

#### 3.1.1. SOH Estimation for Batteries under Different Discharge C-Rate at 55 °C

_{1}is likely due to fitting approximation error and hence they are set to zero. In fact, k

_{1}value should be zero as our ambient temperature is not high as expected from the work by [21].

#### 3.1.2. SoH Estimation for Batteries at Different C-Rates under 25 °C Ambient

#### 3.2. Effect of Temperature

#### 3.2.1. SOH Estimation for Batteries at 25 °C using 55 °C Battery’s Parameter Values (55 °C to 25 °C Case)

#### 3.2.2. SOH Estimation for Batteries at 55 °C using 25 °C Battery’s Parameter Values (25 °C to 55 °C Case)

^{2}factorial design analysis, we obtain the equations for k

_{2}and k

_{3}as given in Equations (3) and (4). k

_{1}is zero as our experiments do not involve high temperatures.

_{2}and k

_{3}.

_{2}= 0.000287 − 0.000115 A − 0.000080 B − 0.000032 A × B

_{3}= 0.003557 + 0.002207 A + 0.002843 B + 0.001493 A × B

_{1}is always zero for the case of LiB

## 4. Applications of the SECF Model

_{2}= 0.00028, k

_{3}= 0.003557, and Q

_{max(fresh)}= 1.9463 Ah. Upon solving it, the n value is 689, which will be the cycles where the SoH reaches 80%, and thus the useful life of the LiB cell at different operating condition can be determined.

_{2}and k

_{3}values of a LiB as in this work, and assuming the LiB is used in 1 C discharge rate and has undergone 100 cycles at 30 °C, the LiB is now to be used in 1.5 C discharge rate at the same temperature of 30 °C. If this LiB has been through 20 cycles under 1.5 C discharge rate, we can use our SECF model to compute the RUL of the LiB under this 1.5 C discharge rate as follows.

_{2}and k

_{3}as computed using Equations (3) and (4), we obtain the SoH after 100 cycles to be 96.25%, using Equation (2). This is the SoH of the LiB at the start of the next application of 1.5 C. The equivalent cycle for the LiB to reach this SoH at 1.5 C and 30 °C can be computed from the model by having the B = −0.5. A will not change as the temperature remains the same. With this new B, we have new values of k

_{2}and k

_{3}and the equivalent cycle is found to be 88 cycles (N

_{equivalent}).

_{total.}From our SECF model, the value is found to be 769 cycles. Since the LiB has been through 20 cycles under 1.5 C, the RUL of the LiB will be N

_{total}− N

_{equivalent}− 20 = 769 − 88 − 20 = 661 cycles.

## 5. Future Works

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Variation of terminal voltage recorded during discharge period for 1 and 3 C-rate at different temperature.

**Figure 2.**State-of-health (SoH) estimation results from ECBE model (black curve) and semi-empirical fading model (red curve) for battery tested at 55 °C and 1 C discharge current. Green circle represents the points used to find the k values.

**Figure 5.**The estimation error between different discharge current under 55 °C. % value on top of the bar represents the % error in estimation.

**Figure 6.**The estimation error between 1 and 3 C-rate at 25 °C. The % value on top of each bar represent the % error of estimation.

**Table 1.**Samsung 18,650 battery specification provided by manufacturer [8].

Battery Characteristics | |
---|---|

Type | Cylindrical |

Chemical system | NMC |

Nominal voltage | 3.62 V |

Typical capacity | 2150 mAh |

Cut-off voltage Charging | 4.2 V |

Discharging | 2.75 V |

Dimensions(mm) | 18.4 × 65 |

Approx. weight | 44.5 g |

Cell Name | Test | Temperature (°C) | Discharge Current |
---|---|---|---|

A | 1 | 55 | 1 C |

B | 2 | 55 | 3 C |

C | 3 | 25 | 1 C |

D | 4 | 25 | 3 C |

**Table 3.**k values of the semi-empirical capacity fading (SECF) model obtained at four test conditions.

Temperature (°C) | C-Rate | 3 Cycles for the Extraction of k Values | k_{1} | k_{2} | k_{3} |
---|---|---|---|---|---|

25 | 1 | 100, 200, 300 | 0 | 0.000283 | 0.0027 |

25 | 3 | 100, 200, 300 | 0 | 0.0000599 | 0.0101 |

55 | 1 | 100, 200, 300 | 0 | 0.000354 | 0 |

55 | 3 | 75, 125, 250 | 0 | 0.00045 | 1.43 × 10^{−3} |

Cycle Number | 1 C | 3 C |
---|---|---|

0 | 100 | 100 |

100 | 96.86 | 95.24 |

200 | 93.55 | 91.29 |

300 | 90.21 | 87.68 |

400 | 87.46 | 82.98 |

500 | 83.88 | 79.67 |

600 | 80.28 | - |

${\mathrm{k}}_{1}$ | ${\mathrm{k}}_{2}$ | ${\mathrm{k}}_{3}$ | |
---|---|---|---|

1 C-rate of 55 °C (A) | $-1.61478\times {10}^{-7}$ | $3.5497\times {10}^{-4}$ | $0$ |

3 C-rate of 55 °C (B) | $-3.7322\times {10}^{-7}$ | $4.5086\times {10}^{-4}$ | $1.43376\times {10}^{-3}$ |

**Table 6.**Percentage estimation error of SoH for a different set of three cycles in the k’s values extraction.

Cycles Selected for k’s Values Extraction | % Estimation Error |
---|---|

100, 200, 300 | 3.14 |

125, 225, 325 | 3.68 |

150, 250, 350 | 3.52 |

Cycle Number | 1 C | 3 C |
---|---|---|

1 | 100% | 100% |

100 | 97.38% | 96.28% |

200 | 94.91% | 93.00% |

300 | 94.26% | 91.30% |

400 | 93.14% | 88.21% |

500 | 92.48% | 84.48% |

600 | 91.88% | 83.80% |

700 | 90.81% | 80.41% |

800 | 89.76% | - |

**Table 8.**The k-values of 1 and 3 C-rate respectively at 25 °C. Negative value of k

_{1}is set to zero.

k-Values | ${\mathrm{k}}_{1}$ | ${\mathrm{k}}_{2}$ | ${\mathrm{k}}_{3}$ |
---|---|---|---|

1 C-rate of 25 °C (C) | $-8.9785\times {10}^{-8}$ | $2.8312\times {10}^{-4}$ | $0.0027$ |

3 C-rate of 25 °C (D) | $8.3792\times {10}^{-7}$ | $5.9967\times {10}^{-5}$ | 0.0101 |

1 C Rate | 3 C Rate | |
---|---|---|

25 °C Ambient | 34.86 °C | 42.32 °C |

55 °C Ambient | 58 °C | 63 °C |

Test Condition | SoH (Using ECBE) | % Error in SoH Estimation Using Its Own k Values | % Error in SoH Estimation with k Values Computed from Equations (3) and (4) |
---|---|---|---|

25 °C_1 C | 89.76(after 800 cycles) | 2.7 | 0.89 |

25 °C_3 C | 80.41(after 300 cycles) | 2.51 | 0.77 |

55 °C_1 C | 80.29(after 600 cycles) | 0.67 | 0.59 |

55 °C_3 C | 79.68(after 500 cycles) | 0.68 | 0.41 |

Test Condition | SoH (Using ECBE) | % Error in SoH Estimation | % Error in SoH Estimation with Generalized k Values |
---|---|---|---|

55 °C_0.5 C | 80.27(After 600 cycles) | 0.97 | 0.84 |

55 °C_5 C | 76.08(After 200 cycles) | 5.47 | 0.61 |

**Table 12.**Verification of the model for additional three more batteries tested at 55 °C and 5 C discharging current.

Sample # | SoH (Using ECBE) | SoH Estimation Using Our Generalized k Values | % Error in SoH Estimation with Generalized k Values |
---|---|---|---|

1 | 89.06(After 100 cycles) | 88.24 | 0.92 |

2 | 91.39(After 100 cycles) | 90.72 | 0.73 |

3 | 90.66(After 100 cycles) | 89.94 | 0.79 |

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**MDPI and ACS Style**

Tan, C.M.; Singh, P.; Chen, C. Accurate Real Time On-Line Estimation of State-of-Health and Remaining Useful Life of Li ion Batteries. *Appl. Sci.* **2020**, *10*, 7836.
https://doi.org/10.3390/app10217836

**AMA Style**

Tan CM, Singh P, Chen C. Accurate Real Time On-Line Estimation of State-of-Health and Remaining Useful Life of Li ion Batteries. *Applied Sciences*. 2020; 10(21):7836.
https://doi.org/10.3390/app10217836

**Chicago/Turabian Style**

Tan, Cher Ming, Preetpal Singh, and Che Chen. 2020. "Accurate Real Time On-Line Estimation of State-of-Health and Remaining Useful Life of Li ion Batteries" *Applied Sciences* 10, no. 21: 7836.
https://doi.org/10.3390/app10217836