# State of Charge Estimation for Lithium-Ion Battery with a Temperature-Compensated Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiments

_{x}Co

_{y}Mn

_{z}O

_{2}cell with a nominal voltage 3.6 V and nominal capacity 2.5 Ah; (2) a thermal chamber with temperature control deviation less than 1 °C; (3) a battery test system (Neware CT-4008, manufactured in Shenzhen, China, voltage measure range 0.025–5 V, current measure range 0.1–30 A. The measurement deviations of the current and voltage sensors are within 0.1%); (4) a PC installed Neware software for battery load control and data acquisition (the voltage, current, and surface temperature of the cell are measured).

#### 2.1. SOC-OCV Test

#### 2.2. Model Identification Test

#### 2.3. Model Validation Test

## 3. Battery Modeling and Identification

#### 3.1. Battery Model

#### 3.2. Model Parameters Identification

_{∞}, because capacity interpolated from battery temperature will cause a large error. For example, the cell is discharged with 2 C rate at 0 °C from SOC 1 to the cut-off voltage. Cell temperature reaches 15 °C when the discharged ampere-hour reaches 1.5 Ah. If the capacity is inferred from battery temperature by Equation (2), it will be around 2.4 Ah. However, the experiment result is 2.173 Ah. Therefore, it can be concluded that the battery capacity won’t change significantly induced by the rapid rising of the temperature during large discharge rate at low temperature. The diffusion in the battery is slow so that the effect of temperature on the battery takes a long time to appear. Therefore, predicting the battery capacity from the ambient temperature is more reasonable.

_{L}t (when the battery is discharged in a narrow SOC region, the linear hypothesis of ${U}_{oc}$ is valid). Where the ${U}_{oc}(t={0}^{-})$ is the OCV just before the discharge which can be measured, $k$ is a constant considering the OCV decline caused by discharge. Therefore, Equation (3) is changed into Equation (4), and the latter can be directly fitted to identify the ${R}_{o}$, ${R}_{p}$ and $\tau $($\tau ={R}_{p}{C}_{p}$ is the time constant).

## 4. EKF-Based SOC Estimation Approach

#### 4.1. SOC Definition

#### 4.2. EKF Algorithm

- (1)
- InitializationAssign the initial state estimate ${\widehat{x}}_{0|0}$, error covariance ${P}_{0|0}$, $Q$ and $R$.
- (2)
- Prediction$$\{\begin{array}{l}{\widehat{x}}_{k|k-1}={A}_{k-1}{\widehat{x}}_{k-1|k-1}+{B}_{k-1}{u}_{k-1}\\ {P}_{k|k-1}={A}_{k-1}{P}_{k-1|k-1}{A}_{k-1}^{T}+{Q}_{k-1}\end{array}$$
- (2)
- Correction$$\{\begin{array}{l}{K}_{k}={P}_{k|k-1}{C}_{k}^{T}{({C}_{k}{P}_{k|k-1}{C}_{k}^{T}+{R}_{k})}^{-1}\\ {\widehat{x}}_{k|k}={\widehat{x}}_{k|k-1}+{K}_{k}\left[{y}_{k}-g\left({\widehat{x}}_{k|k-1},{u}_{k}\right)\right]\\ {P}_{k|k}=\left(I-{K}_{k}{C}_{k}\right){P}_{k|k-1}\end{array}$$

#### 4.3. SOC Estimation with EKF

## 5. Validation and Improvement of the Estimator

#### 5.1. Constant Discharge Validation

#### 5.2. DST Validation

**Q**is [0.1 0; 0 0.002], measurement noise covariance

**R**is 0.5). (3) The cell nonlinear behavior in low SOC region is strong so that the correction proposed above is invalid. (4) The correction is obtained by calibration from constant current discharge tests, and it may be invalid in the dynamic current condition of the DST cycle.

**R**, has a strong influence on the Kalman gain. When

**R**is large, the estimation mainly depends on the process model, otherwise the estimation mainly depending on the measurement model. We adjust the

**R**in SOC lower than 0.2 by changing the original value 0.5 into 40 (this value acquired by trial and error). The result is shown in Figure 15. The MAE is below 4%. The measurement noise variation is a simple way to improve the estimation accuracy. However, if the measurement noise is too strong, a prompt correction of the initial SOC error will not be achieved. Therefore, it is preferred to set a different noise parameter in different SOC region to maximize the estimation accuracy.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Xing, Y.J.; He, W.; Pecht, M.; Tsui, K.L. State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient temperatures. Appl. Energy
**2014**, 113, 106–115. [Google Scholar] [CrossRef] - Waag, W.; Fleischer, C.; Sauer, D.U. Critical review of the methods for monitoring of lithium-ion batteries in electric and hybrid vehicles. J. Power Sources
**2014**, 258, 321–339. [Google Scholar] [CrossRef] - Ng, K.S.; Moo, C.-S.; Chen, Y.-P.; Hsieh, Y.-C. Enhanced coulomb counting method for estimating state-of-charge and state-of-health of lithium-ion batteries. Appl. Energy
**2009**, 86, 1506–1511. [Google Scholar] [CrossRef] - Kang, L.; Zhao, X.; Ma, J. A new neural network model for the state-of-charge estimation in the battery degradation process. Appl. Energy
**2014**, 121, 20–27. [Google Scholar] [CrossRef] - Sheikhan, M.; Pardis, R.; Gharavian, D. State of charge neural computational models for high energy density batteries in electric vehicles. Neural Comput. Appl.
**2013**, 22, 1171–1180. [Google Scholar] [CrossRef] - Antón, J.C.Á.; Nieto, P.J.G.; Viejo, C.B.; Vilán, J.A.V. Support vector machines used to estimate the battery state of charge. IEEE Trans. Power Electron.
**2013**, 28, 5919–5926. [Google Scholar] [CrossRef] - Andre, D.; Appel, C.; Soczka-Guth, T.; Sauer, D.U. Advanced mathematical methods of SOC and SOH estimation for lithium-ion batteries. J. Power Sources
**2013**, 224, 20–27. [Google Scholar] [CrossRef] - Plett, G.L. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs—Part 1. Background. J. Power Sources
**2004**, 134, 252–261. [Google Scholar] [CrossRef] - Plett, G.L. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs—Part 2. Modeling and identification. J. Power Sources
**2004**, 134, 262–276. [Google Scholar] [CrossRef] - Plett, G.L. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs—Part 3. State and parameter estimation. J. Power Sources
**2004**, 134, 277–292. [Google Scholar] [CrossRef] - He, Z.; Gao, M.; Wang, C.; Wang, L.; Liu, Y. Adaptive state of charge estimation for li-ion batteries based on an unscented Kalman filter with an enhanced battery model. Energies
**2013**, 6, 4134–4151. [Google Scholar] [CrossRef] - He, Y.; Liu, X.T.; Zhang, C.B.; Chen, Z.H. A new model for state-of-charge (SOC) estimation for high-power li-ion batteries. Appl. Energy
**2013**, 101, 808–814. [Google Scholar] [CrossRef] - Plett, G.L. Sigma-point Kalman filtering for battery management systems of LiPB-based HEV battery packs—Part 1: Introduction and state estimation. J. Power Sources
**2006**, 161, 1356–1368. [Google Scholar] [CrossRef] - Plett, G.L. Sigma-point Kalman filtering for battery management systems of LiPB-based HEV battery packs—Part 2: Simultaneous state and parameter estimation. J. Power Sources
**2006**, 161, 1369–1384. [Google Scholar] [CrossRef] - He, H.W.; Zhang, Y.Z.; Xiong, R.; Wang, C. A novel Gaussian model based battery state estimation approach: State-of-energy. Appl. Energy
**2015**, 151, 41–48. [Google Scholar] [CrossRef] - Sepasi, S.; Roose, L.; Matsuura, M. Extended kalman filter with a fuzzy method for accurate battery pack state of charge estimation. Energies
**2015**, 8, 5217–5233. [Google Scholar] [CrossRef] - Sepasi, S.; Ghorbani, R.; Liaw, B.Y. A novel on-board state-of-charge estimation method for aged Li-ion batteries based on model adaptive extended kalman filter. J. Power Sources
**2014**, 245, 337–344. [Google Scholar] [CrossRef] - He, H.; Xiong, R.; Peng, J. Real-time estimation of battery state-of-charge with unscented Kalman filter and RTOS μCOS-II platform. Appl. Energy
**2016**, 162, 1410–1418. [Google Scholar] [CrossRef] - Tulsyan, A.; Tsai, Y.; Gopaluni, R.B.; Braatz, R.D. State-of-charge estimation in lithium-ion batteries: A particle filter approach. J. Power Sources
**2016**, 331, 208–223. [Google Scholar] [CrossRef] - Zhang, Y.; Xiong, R.; He, H.; Shen, W. Lithium-ion battery pack state of charge and state of energy estimation algorithms using a hardware-in-the-loop validation. IEEE Trans. Power Electron.
**2017**, 32, 4421–4431. [Google Scholar] [CrossRef] - Liu, X.; Chen, Z.; Zhang, C.; Wu, J. A novel temperature-compensated model for power Li-ion batteries with dual-particle-filter state of charge estimation. Appl. Energy
**2014**, 123, 263–272. [Google Scholar] [CrossRef] - Hu, X.; Li, S.; Peng, H.; Sun, F. Robustness analysis of state-of-charge estimation methods for two types of Li-ion batteries. J. Power Sources
**2012**, 217, 209–219. [Google Scholar] [CrossRef] - Tian, Y.; Xia, B.; Sun, W.; Xu, Z.; Zheng, W. A modified model based state of charge estimation of power lithium-ion batteries using unscented Kalman filter. J. Power Sources
**2014**, 270, 619–626. [Google Scholar] [CrossRef] - Dey, S.; Ayalew, B.; Pisu, P. Nonlinear adaptive observer for a lithium-ion battery cell based on coupled electrochemical-thermal model. J. Dyn. Syst. Meas. Control
**2015**, 137, 111005. [Google Scholar] [CrossRef] - Dey, S.; Mohon, S.; Pisu, P.; Ayalew, B.; Onori, S. Online state and parameter estimation of battery-double layer capacitor hybrid energy storage system. In Proceedings of the 2015 IEEE 54th Annual Conference on Decision and Control, Osaka, Japan, 15–18 December 2015. [Google Scholar]
- Tanim, T.R.; Rahn, C.D.; Wang, C.Y. State of charge estimation of a lithium ion cell based on a temperature dependent and electrolyte enhanced single particle model. Energy
**2015**, 80, 731–739. [Google Scholar] [CrossRef] - Feng, F.; Lu, R.; Wei, G.; Zhu, C.; Sciubba, E. Online estimation of model parameters and state of charge of LiFePO
_{4}batteries using a novel open-circuit voltage at various ambient temperatures. Energies**2015**, 8, 2950–2976. [Google Scholar] [CrossRef] - Verbrugge, M.; Tate, E. Adaptive state of charge algorithm for nickel metal hydride batteries including hysteresis phenomena. J. Power Sources
**2004**, 126, 236–249. [Google Scholar] [CrossRef] - He, H.; Zhang, X.; Xiong, R.; Xu, Y.; Guo, H. Online model-based estimation of state-of-charge and open-circuit voltage of lithium-ion batteries in electric vehicles. Energy
**2012**, 39, 310–318. [Google Scholar] [CrossRef] - Hu, X.; Li, S.; Peng, H. A comparative study of equivalent circuit models for li-ion batteries. J. Power Sources
**2012**, 198, 359–367. [Google Scholar] [CrossRef] - Xiong, R.; Gong, X.; Mi, C.C.; Sun, F. A robust state-of-charge estimator for multiple types of lithium-ion batteries using adaptive extended Kalman filter. J. Power Sources
**2013**, 243, 805–816. [Google Scholar] [CrossRef] - Xiong, R.; Sun, F.; Gong, X.; Gao, C. A data-driven based adaptive state of charge estimator of lithium-ion polymer battery used in electric vehicles. Appl. Energy
**2014**, 113, 1421–1433. [Google Scholar] [CrossRef] - Lee, J.; Nam, O.; Cho, B.H. Li-ion battery SOC estimation method based on the reduced order extended Kalman filtering. J. Power Sources
**2007**, 174, 9–15. [Google Scholar] [CrossRef]

**Figure 10.**SOC estimation result (25 °C, 1/3 C): (

**a**) Estimated SOC and true SOC; (

**b**) SOC estimation error.

**Figure 11.**SOC estimation result (25 °C, 1 C): (

**a**) Estimated SOC and true SOC; (

**b**) SOC estimation error.

**Figure 12.**SOC estimation result (25 °C, 2 C): (

**a**) Estimated SOC and true SOC; (

**b**) SOC estimation error.

**Figure 13.**SOC estimation result (25 °C, 2 C, resistance correction): (

**a**) Estimated SOC and true SOC; (

**b**) SOC estimation error.

**Figure 14.**SOC estimation result (DST, 20 °C): (

**a**) Estimated SOC and true SOC; (

**b**) SOC estimation error.

**Figure 15.**SOC estimation result (DST, 20 °C, measurement noise adjustment): (

**a**) Estimated SOC and true SOC; (

**b**) SOC estimation error.

Temperature/°C | 1/3 C Intermittent Discharge Capacity/Ah | 1/3 C Continuous Discharge Capacity/Ah |
---|---|---|

0 | 2.288 | 2.207 |

12.5 | 2.421 | 2.391 |

25 | 2.573 | 2.527 |

45 | 2.676 | 2.643 |

Discharge Rate | Temperature/°C | |||
---|---|---|---|---|

0 | 12.5 | 25 | 45 | |

1/3 C | 1 | 1 | 1 | 1 |

1 C | 0.85 | 1 | 1 | 1 |

2 C | 0.8 | 0.93 | 0.72 | 0.85 |

Temperature/°C | MAE without Correction | MAE with Correction |
---|---|---|

0 | 7.5 | 3.3 |

12.5 | 5.8 | 4.5 |

25 | 7.2 | 2.6 |

45 | 4.3 | 3.8 |

© 2017 by the authors. 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, S.; Deng, C.; Zhang, Y.; He, Y.
State of Charge Estimation for Lithium-Ion Battery with a Temperature-Compensated Model. *Energies* **2017**, *10*, 1560.
https://doi.org/10.3390/en10101560

**AMA Style**

Yang S, Deng C, Zhang Y, He Y.
State of Charge Estimation for Lithium-Ion Battery with a Temperature-Compensated Model. *Energies*. 2017; 10(10):1560.
https://doi.org/10.3390/en10101560

**Chicago/Turabian Style**

Yang, Shichun, Cheng Deng, Yulong Zhang, and Yongling He.
2017. "State of Charge Estimation for Lithium-Ion Battery with a Temperature-Compensated Model" *Energies* 10, no. 10: 1560.
https://doi.org/10.3390/en10101560