# State-of-Charge Estimation of Lithium-Ion Batteries Based on Dual-Coefficient Tracking Improved Square-Root Unscented Kalman Filter

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equivalent Circuit Model and State Space Equation of Lithium-Ion Batteries

#### 2.1. Equivalent Circuit Model of the Batteries

#### 2.2. State Space Equation of the Battery Model

## 3. SOC Estimation Based on the Dual-Coefficient Tracking ISRUKF

#### 3.1. Improved Square-Root Unscented Kalman Filter (ISRUKF)

#### 3.1.1. Standard SRUKF

- (1)
- Initialize the mean (x
_{0}) and the state covariance square root (${S}_{0}$) of the system state:

^{T}is the matrix transpose operation.

- (2)
- Assign weights and obtain sampling points:

- (3)
- Time update for the system states:
- a.
- Update the sample point:$${x}_{i,k|k-1}=f({x}_{i,k-1})$$
- b.
- Estimate the system state:$${\stackrel{\wedge}{x}}_{k|k-1}=\sum _{i=0}^{2n}{w}_{i}^{m}{x}_{i,k|k-1}+{q}_{k}$$
- c.
- Update the covariance of the estimated state:$${S}_{k|k-1}^{*}=qr\left\{\left[\sqrt{{w}_{i}^{c}}\left({x}_{i=1:2n,k|k-1}-{\stackrel{\wedge}{x}}_{k|k-1}\right),\sqrt{{Q}_{k-1}}\right]\right\},i=\mathrm{1,2},\dots ,2n$$$${S}_{k|k-1}=cholupdate\{{S}_{k|k-1}^{*},{x}_{i,k|k-1}-{\stackrel{\wedge}{x}}_{k|k-1},{w}_{0}^{c}\}$$
_{k|k−1}is the state covariance square-root matrix; and $cholupdate$($\xb7$) is the Cholesky decomposition.

- (4)
- Measurement update:
- a.
- Attain the measurement:$${y}_{i,k|k-1}=g({x}_{i,k-1})$$
- b.
- Update the measurement:$${\stackrel{\wedge}{y}}_{k|k-1}={\sum}_{i=0}^{2n}{w}_{i}^{m}{y}_{i,k|k-1}+{r}_{k}$$

- (5)
- Calculate the SRUKF gain matrix L:$${L}_{k}={P}_{xy}/{S}_{z}{S}_{z}^{T}$$$${P}_{xy}=\sum _{i=0}^{2n}{w}_{i}^{c}({x}_{i,k|k-1}-{\stackrel{\wedge}{x}}_{k|k-1}){({y}_{i,k|k-1}-{\stackrel{\wedge}{y}}_{k|k-1})}^{T}$$$${S}_{z}^{*}=qr\{[\sqrt{{w}_{i}^{c}}({\stackrel{\wedge}{y}}_{i=1:2n,k|k-1}-{\stackrel{\wedge}{y}}_{k|k-1}),\sqrt{{R}_{k-1}}]\}$$$${S}_{z}=cholupdate\{{S}_{z}^{*},{y}_{i,k|k-1}-{\stackrel{\wedge}{y}}_{k|k-1},{w}_{0}^{c}\}$$

- (6)
- Measurement correction:
- a.
- Update the estimated state:$${\stackrel{\wedge}{x}}_{k|k}={\stackrel{\wedge}{x}}_{k|k-1}+{L}_{k}({y}_{k}-{\stackrel{\wedge}{y}}_{k|k-1})$$
- b.
- Update the propagated covariance:$${S}_{k|k}=cholupdate\{{S}_{k|k-1},{L}_{k}{S}_{z},-1\}$$

#### 3.1.2. The ISRUKF Based on the QR Decomposition Method

_{k/k−1}and the state covariance square-root matrix ${S}_{k|k-1}$ is presented as

_{k}is the upper triangular matrix, and ${r}_{k}$ is the orthogonal matrix.

#### 3.2. The Dual-Coefficient Tracker Based on the Strong Tracking Filter

#### 3.2.1. Strong Tracking Filter

#### 3.2.2. The State Noise Tracking Coefficient

- (1)
- Initialize the ${m}_{k}$ and its dimension to n columns.

- (2)
- ${P}_{k|k-1}^{(\mathrm{i})}$ is set as an element in the $\mathrm{i}th$ row and $\mathrm{i}th$ column of the state covariance square-root matrix at time k. We set the mean of the elements in m adjacent square-root matrices as a parameter and definite A and B as the pairwise comparisons between adjacent parameters, which can be described as

- (3)
- Attain the updated ${m}_{k}$. If the outcome does not match outcome (II) or outcome (V), the ${m}_{k}$ should not be modified. If the calculated result matches outcome (II) or outcome (V), the ${m}_{k}$ should be reassigned. If the value of ${Q}_{0}$ is higher than 2% of the ${U}_{OC}$, the ${Q}_{0}$ is judged to be too large, and the state noise tracking coefficient should be shrunk, and then the value ${m}_{k}=[1-\gamma ,1-\gamma ,1-\gamma ,...,1-\gamma ]$ should be assigned. If the ${Q}_{0}$ value is less than 2% of the ${U}_{OC}$, the ${Q}_{0}$ is judged to be too small, and the state noise tracking coefficient should be amplified, and then the value ${m}_{k}=[1+\gamma ,1+\gamma ,1+\gamma ,...,1+\gamma ]$ should be assigned, where $\gamma $ is the correction coefficient.

- (4)
- According to ${m}_{k}$ and Formula (39), we can get the state noise tracking coefficient ${\delta}_{k}$.

- (5)
- The ${\delta}_{k}$ and ${\lambda}_{k}$ are used as inputs to form the dual-coefficient tracker.

- (6)
- The dual-tracking coefficient ${\lambda}_{k}$ and ${\delta}_{k}$ are put into the ISRUKF to calculate the updated state covariance square-root matrix ${S}_{k|k-1}^{c}$, which can be illustrated as$$\left\{\right)separators="|">\begin{array}{c}{r}_{k}=qr\{[{\lambda}_{k}\sqrt{{w}_{i}^{c}}({x}_{i=1:2n,k|k-1}-{\stackrel{\wedge}{x}}_{k|k-1}),{\delta}_{k}\sqrt{{Q}_{k-1}}{]}^{T}\}\\ {S}_{k|k-1}^{c}={r}_{k}^{T}\end{array}$$

#### 3.3. Battery SOC Estimation Procedure Based on Dual-Coefficient Tracking ISRUKF

_{0}) and the state covariance square root (${S}_{0}$) by Formula (13), and initialize the ${\lambda}_{0}$ and ${\delta}_{0}$;

## 4. Simulation and Experimental Results and Analysis

#### 4.1. Test Platform and Experiment Parameters

#### 4.2. Comparison of SOC Estimation Accuracy Using the Developed ISRUKF, SRUKF, and UKF with Different Covariance ${Q}_{k}$

#### 4.3. Comparison of SOC Estimation Results Using Different ISRUKFs, the ISRUKF with Standard STF and the Developed ISRUKF

#### 4.4. SOC Estimation by the Developed ISRUKF with Different Covariance ${Q}_{k}$

#### 4.5. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

SOC | state of charge (-) |

KF | Kalman filter (-) |

EKF | extended Kalman filter (-) |

UKF | unscented Kalman filter (-) |

SRUKF | square-root unscented Kalman filter (-) |

AUKF | adaptive unscented Kalman filter (-) |

STF | strong tracking filter (-) |

OCV | open-circuit voltage (V) |

ECM | equivalent circuit model (-) |

U_{OC} | open-circuit voltage of the batteries (V) |

R_{0} | internal resistance of the batteries (Ω) |

R_{L} | concentration resistance of the batteries (Ω) |

C_{L} | concentration capacitance of the batteries (F) |

R_{S} | electrochemical resistance (Ω) |

C_{S} | electrochemical capacitance (F) |

I_{t} | current of the batteries (A) |

U_{t} | terminal voltage of the batteries (V) |

a_{0}~a_{5} | coefficient of the equation (-) |

b_{0}~b_{5} | coefficient of the equation (-) |

c_{0}~c_{2} | coefficient of the equation (-) |

d_{0}~d_{2} | coefficient of the equation (-) |

e_{0}~e_{2} | coefficient of the equation (-) |

f_{0}~f_{2} | coefficient of the equation (-) |

SOC_{0} | initial value of SOC (-) |

$\eta $ | coulomb efficiency (-) |

Q_{N} | nominal capacity of the batteries (Wh) |

U_{S} | electrochemical voltage of the batteries (V) |

U_{L} | concentration voltage of the batteries (V) |

$\Delta t$ | sampling time (s) |

${\tau}_{1}$ | time parameter (s) |

${\tau}_{2}$ | time parameter (s) |

w_{k} | state noise (-) |

v_{k} | measurement noise (-) |

x_{k} | system state vector (-) |

y_{k} | system measurement vector (-) |

u_{k} | system input vector (V) |

f(·) | nonlinear state model (-) |

g(·) | nonlinear measurement model (-) |

r | mean of the measurement noise (-) |

q | mean of the state noise (-) |

R | covariance value of measurement noise (-) |

Q | covariance value of state noise (-) |

x_{0} | initial mean (-) |

S_{0} | initial state covariance square root (-) |

E(·) | expectation mean value (-) |

(·)^{T} | matrix transpose operation (-) |

n | dimension of the state vector (-) |

$\alpha $ | scaling parameter (-) |

h | column factor (-) |

w^{m} | variance weight factor (-) |

w^{c} | mean weight factor (-) |

$\beta $ | error magnitude of the higher-order term (-) |

${S}_{k|k-1}^{*}$ | updated state calculation value (-) |

$qr$($\xb7$) | QR decomposition (-) |

S_{k|k−1} | state covariance square-root matrix (-) |

$cholupdate$($\xb7$) | Cholesky decomposition (-) |

P_{xy} | mutual covariance (-) |

${S}_{z}^{*}$ | updated calculation measurement |

S_{z} | measurement covariance square-root matrix (-) |

S_{k|k} | state covariance square-root optimal estimation matrix (-) |

P_{k|k−1} | state covariance matrix (-) |

q_{k} | upper triangular matrix (-) |

r_{k} | orthogonal matrix (-) |

e | identity matrix of order n (-) |

H_{k} | coefficient matrix of the measurement function (-) |

${\lambda}_{k}$ | fading factor (-) |

trace($\xb7$) | trace of the corresponding matrix (-) |

C_{k} | covariance of the residual sequence of outputs (-) |

$\rho $ | forgetting factor (-) |

${\epsilon}_{k}$ | output residual sequence |

${S}_{k|k-1}^{c}$ | corrected state covariance square-root matrix (-) |

${\delta}_{k}$ | state noise tracking coefficient (-) |

## References

- Renxin, X.; Yanwen, H.; Wei, Z.; Zhaohui, C. A novel approach to estimate the state of charge for lithium-ion battery under different temperatures incorporating open circuit voltage online identification. J. Energy Storage
**2023**, 67, 107509. [Google Scholar] [CrossRef] - Meng, J.; Yue, M.; Diallo, D. Nonlinear extension of battery constrained predictive charging control with transmission of Jacobian matrix. Int. J. Electr. Power Energy Syst.
**2023**, 146, 108762. [Google Scholar] [CrossRef] - Qiao, J.; Wang, S.; Yu, C.; Yang, X.; Fernandez, C. A chaotic firefly-Particle filtering method of dynamic migration modeling for the state-of-charge and state-of-health co-estimation of a lithium-ion battery performance. Energy
**2023**, 263, 126164. [Google Scholar] [CrossRef] - Yu, Q.; Liu, Y.; Long, S.; Jin, X.; Li, J.; Shen, W. A Branch Current Estimation and Correction Method for a Parallel Connected Battery System Based on Dual BP Neural Networks. Green Energy Intell. Transp.
**2022**, 1, 100029. [Google Scholar] [CrossRef] - Peng, S.; Zhu, L.; Dou, Z.; Liu, D.; Yang, R.; Pecht, M. Method of Site Selection and Capacity Setting for Battery Energy Storage System in Distribution Networks with Renewable Energy Sources. Energies
**2023**, 16, 3899. [Google Scholar] [CrossRef] - Chen, C.; Xiong, R.; Yang, R.; Li, H. A novel data-driven method for mining battery open-circuit voltage characterization. J. Green Energy Intell. Transp.
**2022**, 1, 100001. [Google Scholar] [CrossRef] - Chen, B.; Jiang, H.; Chen, X.; Li, H. Robust state-of-charge estimation for lithium-ion batteries based on an improved gas-liquid dynamics model. Energy
**2022**, 238, 122008. [Google Scholar] [CrossRef] - Xia, L.; Wang, S.; Yu, C.; Fan, Y.; Li, B.; Xie, Y. Joint estimation of the state-of-energy and state-of-charge of lithium-ion batteries under a wide temperature range based on the fusion modeling and online parameter prediction. J. Energy Storage
**2022**, 52, 105010. [Google Scholar] [CrossRef] - Wassiliadis, N.; Kriegler, J.; Gamra, K.A.; Lienkamp, M. Model-based health-aware fast charging to mitigate the risk of lithium plating and prolong the cycle life of lithium-ion batteries in electric vehicles. J. Power Sources
**2023**, 561, 232586. [Google Scholar] [CrossRef] - Liu, Z.; Zhao, Z.; Qiu, Y.; Jing, B.; Yang, C. State of charge estimation for Li-ion batteries based on iterative Kalman filter with adaptive maximum correntropy criterion. J. Power Sources
**2023**, 580, 233282. [Google Scholar] [CrossRef] - Zhao, X.; Jung, S.; Wang, B.; Xuan, D. State of charge estimation of lithium-ion battery based on improved adaptive boosting algorithm. J. Energy Storage
**2023**, 71, 108047. [Google Scholar] [CrossRef] - Li, X.; Wang, Z.; Zhang, L. Co-estimation of capacity and state-of-charge for lithium-ion batteries in electric vehicles. Energy
**2019**, 174, 33–44. [Google Scholar] [CrossRef] - Yu, Q.; Huang, Y.; Tang, A.; Wang, C.; Shen, W. OCV-SOC-Temperature Relationship Construction and State of Charge Estimation for a Series–Parallel Lithium-Ion Battery Pack. IEEE Trans. Intell. Transp. Syst.
**2023**, 24, 6362–6371. [Google Scholar] [CrossRef] - Wang, C.; Zhang, X.; Yun, X.; Fan, X. A novel hybrid machine learning coulomb counting technique for state of charge estimation of lithium-ion batteries. J. Energy Storage
**2023**, 63, 107081. [Google Scholar] [CrossRef] - Li, K.; Gao, X.; Liu, C.; Chang, C.; Li, X. A novel Co-estimation framework of state-of-charge, state-of-power and capacity for lithium-ion batteries using multi-parameters fusion method. Energy
**2023**, 269, 126820. [Google Scholar] [CrossRef] - Zhu, R.; Duan, B.; Zhang, J.; Zhang, Q.; Zhang, C. Co-estimation of model parameters and state-of-charge for lithium-ion batteries with recursive restricted total least squares and unscented Kalman filter. Appl. Energy
**2020**, 277, 115494. [Google Scholar] [CrossRef] - Xia, B.; Cui, D.; Sun, Z.; Lao, Z.; Zhang, R.; Wang, W.; Sun, W.; Lai, Y.; Wang, M. State of charge estimation of lithium-ion batteries using optimized Levenberg-Marquardt wavelet neural network. Energy
**2018**, 153, 694–705. [Google Scholar] [CrossRef] - Song, Q.; Wang, S.; Xu, W.; Shao, Y.; Fernandez, C. A novel joint support vector machine-cubature Kalman filtering method for adaptive state of charge prediction of lithium-ion batteries. Int. J. Electrochem. Sci.
**2021**, 16, 210823. [Google Scholar] [CrossRef] - Tagade, P.; Hariharan, K.S.; Gambhire, P.; Kolake, S.M.; Song, T.; Oh, D.; Yeo, T.; Doo, S. Recursive Bayesian filtering framework for lithium-ion cell state estimation. J. Power Sources
**2016**, 306, 274–288. [Google Scholar] [CrossRef] - Zhu, W.; Guo, B.; Li, Y.; Yang, Y.; Xie, C.; Jin, J.; Gooi, H.B. Uncertainty quantification of proton-exchange-membrane fuel cells degradation prediction based on Bayesian-Gated Recurrent Unit. eTransportation
**2023**, 16, 100230. [Google Scholar] [CrossRef] - Jiang, B.; Dai, H.; Wei, X.; Xu, T. Joint estimation of lithium-ion battery state of charge and capacity within an adaptive variable multi-timescale framework considering current measurement offset. Appl. Energy
**2019**, 253, 113619. [Google Scholar] [CrossRef] - Plett, G.L. Extended Kalman filtering for batteries management systems of LiPB-based HEV batteries packs: Part 2. Modeling and identification. J. Power Sources
**2004**, 134, 262–276. [Google Scholar] [CrossRef] - Chai, H.; Gao, Z.; Jiao, Z.; Yang, C. State of charge estimation for lithium-ion batteries based on an adaptive fractional-order cubature Kalman filter with initial value compensation. J. Energy Storage
**2023**, 68, 107544. [Google Scholar] [CrossRef] - Yang, F.; Xing, Y.; Wang, D.; Tsui, K.L. A comparative study of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dynamic loading profile. Appl. Energy
**2016**, 164, 387–399. [Google Scholar] [CrossRef] - Liu, S.; Deng, D.; Wang, S.; Luo, W.; Takyi-Aninakwa, P.; Qiao, J.; Li, S.; Jin, S.; Hu, C. Dynamic adaptive square-root unscented Kalman filter and rectangular window recursive least square method for the accurate state of charge estimation of lithium-ion batteries. J. Energy Storage
**2023**, 67, 107603. [Google Scholar] [CrossRef] - Aung, H.; Low, K.S. Temperature dependent state-of-charge estimation of lithium ion battery using dual spherical unscented Kalman filter. IET Power Electron.
**2015**, 8, 2026–2033. [Google Scholar] [CrossRef] - Menegaz, H.M.T.; Ishihara, J.Y. Unscented and square-root unscented Kalman filters for quaternionic systems. Int. J. Robust Nonlinear Control
**2018**, 28, 4500–4527. [Google Scholar] [CrossRef] - Wei, W.; Gao, S.; Zhong, Y.; Gu, C.; Hu, G. Adaptive square-root unscented particle filtering algorithm for dynamic navigation. Sensors
**2018**, 18, 2337. [Google Scholar] [CrossRef] [Green Version] - Asl, R.M.; Hagh, Y.S.; Simani, S.; Handroos, H. Adaptive square-root unscented Kalman filter: An experimental study of hydraulic actuator state estimation. Mech. Syst. Signal Process.
**2019**, 132, 670–691. [Google Scholar] - Zhu, Q.; Xu, M.; Liu, W.; Zheng, M. A state of charge estimation method for lithium-ion batteries based on fractional order adaptive extended kalman filter. Energy
**2019**, 187, 115880. [Google Scholar] [CrossRef] - Duan, L.; Zhang, X.; Jiang, Z.; Gong, Q.; Wang, Y.; Ao, X. State of charge estimation of lithium-ion batteries based on second-order adaptive extended Kalman filter with correspondence analysis. Energy
**2023**, 280, 128159. [Google Scholar] [CrossRef] - Peng, S.; Chen, C.; Shi, H.; Yao, Z. State of charge estimation of battery energy storage systems based on adaptive unscented Kalman filter with a noise statistics estimator. IEEE Access
**2017**, 5, 13202–13212. [Google Scholar] [CrossRef] - Xu, X.; Tang, S.; Ren, H.; Han, X.; Wu, Y.; Lu, L.; Feng, X.; Yu, C.; Xie, J.; Ouyang, M.; et al. Joint state estimation of lithium-ion batteries combining improved equivalent circuit model with electrochemical mechanism and diffusion process. J. Energy Storage
**2022**, 56, 106135. [Google Scholar] [CrossRef]

**Figure 6.**SOC estimation results using three different algorithms when the ${Q}_{k}$ is set as 0.005.

**Figure 7.**SOC estimation results using three different algorithms when the ${Q}_{k}$ is set as 0.01.

Normal Voltage | 3.7 V | Battery Capacity | 860 mAh | ||
---|---|---|---|---|---|

Upper Cut-Off Voltage | 4.2 V | Lower Cut-Off Voltage | 3.2 V | ||

a_{0} | −0.915 | a_{1} | 40.867 | a_{2} | 3.632 |

a_{3} | 0.537 | a_{4} | 0.499 | a_{5} | 0.522 |

b_{0} | 0.1463 | b_{1} | 30.27 | b_{2} | 0.1037 |

b_{3} | 0.0584 | b_{4} | 0.1747 | b_{5} | 0.1288 |

c_{0} | 0.1063 | c_{1} | 62.49 | c_{2} | 0.0437 |

d_{0} | −200 | d_{1} | 138 | d_{2} | 300 |

e_{0} | 0.0712 | e_{1} | 61.4 | e_{2} | 0.0288 |

f_{0} | −3083 | f_{1} | 180 | f_{2} | 5088 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Peng, S.; Zhang, A.; Liu, D.; Cheng, M.; Kan, J.; Pecht, M.
State-of-Charge Estimation of Lithium-Ion Batteries Based on Dual-Coefficient Tracking Improved Square-Root Unscented Kalman Filter. *Batteries* **2023**, *9*, 392.
https://doi.org/10.3390/batteries9080392

**AMA Style**

Peng S, Zhang A, Liu D, Cheng M, Kan J, Pecht M.
State-of-Charge Estimation of Lithium-Ion Batteries Based on Dual-Coefficient Tracking Improved Square-Root Unscented Kalman Filter. *Batteries*. 2023; 9(8):392.
https://doi.org/10.3390/batteries9080392

**Chicago/Turabian Style**

Peng, Simin, Ao Zhang, Dandan Liu, Mengzeng Cheng, Jiarong Kan, and Michael Pecht.
2023. "State-of-Charge Estimation of Lithium-Ion Batteries Based on Dual-Coefficient Tracking Improved Square-Root Unscented Kalman Filter" *Batteries* 9, no. 8: 392.
https://doi.org/10.3390/batteries9080392