Online Parameter Identification of Lithium-Ion Batteries Using a Novel Multiple Forgetting Factor Recursive Least Square Algorithm
Abstract
:1. Introduction
2. Battery Modeling and State of Charge Estimation Scheme
2.1. Model Selection of the Battery
2.2. Nonlinear Kalman Filter Based SOC Estimation
3. A Novel Multiple Forgetting Factor Recursive Least Square Method
3.1. Single Forgetting Factoc Recursive Least Square Algorithm
3.2. Introducing the Multiple Forgetting Factors Scheme
3.3. The Inplementation of MFFRLS on Batteriey Parameter Identification
3.4. The Optimization of Forgeeting Factors
3.5. Joint Algorithm of Online Parameter Identification and SOC Estimation
4. Experimental Verifications
4.1. A Brief Introduction of the Test Bench
4.2. Battery Characterization
4.3. NEDC Test under Constant Temperature
4.4. NEDC Test under Variable Temperature
4.5. Test of Multiple Cycles under Variable Temperature
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Acronyms | Symbols | ||
1-RC | First-order resistor-capacitor | Parameters to be identified | |
2-RC | Second-order resistor-capacitor | Nominal Capacity | |
ARX | Auto regressive exogenous | Polarization capacity | |
BMS | Battery management system | Forgetting map | |
CC-CV | Constant current-constant voltage | Fitness function | |
CKF | Cubature Kalman filter | Load current | |
DST | Dynamic Street Test | Identical matrix | |
ECM | Equivalent circuit model | Time step index | |
EKF | Extended Kalman filter | Coefficient of polynomial function | |
EV | Electric vehicle | Number of observations | |
FUDS | Federal Urban Driving Schedule | Positive definite matrix | |
LS | Least squares | Dimension of parameters | |
MFF | Multiple forgetting factors | SISO system constants | |
MFFRLS | Multiple forgetting factors recursive least square | Covariance vector | |
NEDC | New European Driving Cycle | Internal resistance | |
NEVs | New energy vehicles | Polarization resistance | |
OCV | Open circuit voltage | Complex variable | |
PC | Personal computer | Time | |
PNGV | Partnership for a New Generation of Vehicles | Sampling time | |
PSO | Particle swarm optimization | Open circuit voltage | |
RC | Resistor-capacitor | Terminal voltage | |
RELS | Recursive extended least square | Least square error function | |
RLS | Recursive least square | Output of SISO system | |
RMSE | Root mean square error | Discretization variable | |
RTLS | Recursive total least square | Greek symbols | |
SISO | Single input single output | White noise | |
SOC | State of charge | Filtering constant | |
SOH | State of health | Parameter vector | |
SFFRLS | Single forgetting factor recursive least square | Forgetting factor | |
U-D | Unit upper triangular factor-diagonal factor | Data vector | |
UDDS | Urban Dynamometer Driving Schedule | ||
UKF | Unscented Kalman filter | ||
Subscript | Superscript | ||
Minimum value | Filtered value | ||
Maximum value |
References
- Zheng, Y.; Ouyang, M.; Han, X.; Lu, L.; Li, J. Investigating the error sources of the online state of charge estimation methods for lithium-ion batteries in electric vehicles. J. Power Sources 2018, 377, 161–188. [Google Scholar] [CrossRef]
- Chen, X.; Shen, W.; Cao, Z.; Kapoor, A. A novel approach for state of charge estimation based on adaptive switching gain sliding mode observer in electric vehicles. J. Power Sources 2014, 246, 667–678. [Google Scholar] [CrossRef]
- Zhang, R.; Xia, B.; Li, B.; Cao, L.; Lai, Y.; Zheng, W.; Wang, H.; Wang, W. State of the Art of Lithium-Ion Battery SOC Estimation for Electrical Vehicles. Energies 2018, 11, 1820. [Google Scholar] [CrossRef]
- Lu, L.; Han, X.; Li, J.; Hua, J.; Ouyang, M. A review on the key issues for lithium-ion battery management in electric vehicles. J. Power Sources 2013, 226, 272–288. [Google Scholar] [CrossRef]
- Panchal, S.; Mathew, M.; Fraser, R.; Fowler, M. Electrochemical thermal modeling and experimental measurements of 18650 cylindrical lithium-ion battery during discharge cycle for an EV. Appl. Therm. Eng. 2018, 135, 123–132. [Google Scholar] [CrossRef]
- Tang, X.; Zou, C.; Yao, K.; Chen, G.; Liu, B.; He, Z.; Gao, F. A fast estimation algorithm for lithium-ion battery state of health. J. Power Sources 2018, 396, 453–458. [Google Scholar] [CrossRef]
- Xia, B.; Chen, C.; Tian, Y.; Sun, W.; Xu, Z.; Zheng, W. A novel method for state of charge estimation of lithium-ion batteries using a nonlinear observer. J. Power Sources 2014, 270, 359–366. [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]
- Tong, S.; Klein, M.P.; Park, J.W. On-line optimization of battery open circuit voltage for improved state-of-charge and state-of-health estimation. J. Power Sources 2015, 293, 416–428. [Google Scholar] [CrossRef]
- Xiong, R.; He, H.; Sun, F.; Zhao, K. Evaluation on State of Charge Estimation of Batteries With Adaptive Extended Kalman Filter by Experiment Approach. IEEE Trans. Veh. Technol. 2013, 62, 108–117. [Google Scholar] [CrossRef]
- Plett, G.L. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs. J. Power Sources 2004, 134, 277–292. [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]
- He, W.; Williard, N.; Chen, C.; Pecht, M. State of charge estimation for electric vehicle batteries using unscented kalman filtering. Microelectron. Reliab. 2013, 53, 840–847. [Google Scholar] [CrossRef]
- Xia, B.; Wang, H.; Tian, Y.; Wang, M.; Sun, W.; Xu, Z. State of Charge Estimation of Lithium-Ion Batteries Using an Adaptive Cubature Kalman Filter. Energies 2015, 8, 5916–5936. [Google Scholar] [CrossRef] [Green Version]
- Xia, B.; Wang, H.; Wang, M.; Sun, W.; Xu, Z.; Lai, Y. A New Method for State of Charge Estimation of Lithium-Ion Battery Based on Strong Tracking Cubature Kalman Filter. Energies 2015, 8, 13458–13472. [Google Scholar] [CrossRef] [Green Version]
- Xia, B.; Sun, Z.; Zhang, R.; Lao, Z. A Cubature Particle Filter Algorithm to Estimate the State of the Charge of Lithium-Ion Batteries Based on a Second-Order Equivalent Circuit Model. Energies 2017, 10, 457. [Google Scholar] [CrossRef]
- Kim, I.-S. The novel state of charge estimation method for lithium battery using sliding mode observer. J. Power Sources 2006, 163, 584–590. [Google Scholar] [CrossRef]
- Tian, Y.; Chen, C.; Xia, B.; Sun, W.; Xu, Z.; Zheng, W. An Adaptive Gain Nonlinear Observer for State of Charge Estimation of Lithium-Ion Batteries in Electric Vehicles. Energies 2014, 7, 5995–6012. [Google Scholar] [CrossRef] [Green Version]
- Lin, C.; Mu, H.; Xiong, R.; Shen, W. A novel multi-model probability battery state of charge estimation approach for electric vehicles using H-infinity algorithm. Appl. Energy 2016, 166, 76–83. [Google Scholar] [CrossRef]
- Zhu, Q.; Li, L.; Hu, X.; Xiong, N.; Hu, G.-D. H∞-Based Nonlinear Observer Design for State of Charge Estimation of Lithium-Ion Battery With Polynomial Parameters. IEEE Trans. Veh. Technol. 2017, 66, 10853–10865. [Google Scholar] [CrossRef]
- Marongiu, A.; Nußbaum, F.G.W.; Waag, W.; Garmendia, M.; Sauer, D.U. Comprehensive study of the influence of aging on the hysteresis behavior of a lithium iron phosphate cathode-based lithium ion battery—An experimental investigation of the hysteresis. Appl. Energy 2016, 171, 629–645. [Google Scholar] [CrossRef]
- Zhang, R.; Xia, B.; Li, B.; Lai, Y.; Zheng, W.; Wang, H.; Wang, W.; Wang, M. Study on the Characteristics of a High Capacity Nickel Manganese Cobalt Oxide (NMC) Lithium-Ion Battery—An Experimental Investigation. Energies 2018, 11, 2275. [Google Scholar] [CrossRef]
- Yu, Q.; Xiong, R.; Lin, C.; Shen, W.; Deng, J. Lithium-Ion Battery Parameters and State-of-Charge Joint Estimation Based on H-Infinity and Unscented Kalman Filters. IEEE Trans. Veh. Technol. 2017, 66, 8693–8701. [Google Scholar] [CrossRef]
- Tang, X.; Wang, Y.; Zou, C.; Yao, K.; Xia, Y.; Gao, F. A novel framework for Lithium-ion battery modeling considering uncertainties of temperature and aging. Energy Convers. Manag. 2019, 180, 162–170. [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] [Green Version]
- Xiong, R.; Sun, F.; Chen, Z.; He, H. A data-driven multi-scale extended Kalman filtering based parameter and state estimation approach of lithium-ion olymer battery in electric vehicles. Appl. Energy 2014, 113, 463–476. [Google Scholar] [CrossRef]
- Giordano, G.; Klass, V.; Behm, M.; Lindbergh, G.; Sjoberg, J. Model-Based Lithium-Ion Battery Resistance Estimation from Electric Vehicle Operating Data. IEEE Trans. Veh. Technol. 2018, 67, 3720–3728. [Google Scholar] [CrossRef]
- Zhang, J.; Wei, Y.; Qi, H. State of charge estimation of LiFePO4 batteries based on online parameter identification. Appl. Math. Model. 2016, 40, 6040–6050. [Google Scholar] [CrossRef]
- Wei, Z.; Zou, C.; Leng, F.; Soong, B.H.; Tseng, K.-J. Online Model Identification and State-of-Charge Estimate for Lithium-Ion Battery With a Recursive Total Least Squares-Based Observer. IEEE Trans. Ind. Electron. 2018, 65, 1336–1346. [Google Scholar] [CrossRef]
- Wei, Z.; Meng, S.; Xiong, B.; Ji, D.; Tseng, K.J. Enhanced online model identification and state of charge estimation for lithium-ion battery with a FBCRLS based observer. Appl. Energy 2016, 181, 332–341. [Google Scholar] [CrossRef]
- Rahimi-Eichi, H.; Baronti, F.; Chow, M.Y. Modeling and Online Parameter Identification of Li-Polymer Battery Cells for SOC estimation. In Proceedings of the 2012 IEEE International Symposium on Industrial Electronics, Hangzhou, China, 28–31 May 2012; pp. 1336–1341. [Google Scholar]
- Feng, T.; Yang, L.; Zhao, X.; Zhang, H.; Qiang, J. Online identification of lithium-ion battery parameters based on an improved equivalent-circuit model and its implementation on battery state-of-power prediction. J. Power Sources 2015, 281, 192–203. [Google Scholar] [CrossRef]
- Dai, H.; Xu, T.; Zhu, L.; Wei, X.; Sun, Z. Adaptive model parameter identification for large capacity Li-ion batteries on separated time scales. Appl. Energy 2016, 184, 119–131. [Google Scholar] [CrossRef]
- Schwunk, S.; Straub, S.; Jung, M. Online Parameterization of a Function Describing the Open-Circuit Voltage by a Least Square Method with Adaptive Forgetting Factor. J. Electrochem. Soc. 2013, 160, A2155–A2159. [Google Scholar] [CrossRef]
- Xiong, R.; He, H.; Sun, F.; Zhao, K. Online Estimation of Peak Power Capability of Li-Ion Batteries in Electric Vehicles by a Hardware-in-Loop Approach. Energies 2012, 5, 1455–1469. [Google Scholar] [CrossRef] [Green Version]
- Verbrugge, M. Adaptive, multi-parameter battery state estimator with optimized time-weighting factors. J. Appl. Electrochem. 2007, 37, 605–616. [Google Scholar] [CrossRef]
- Duong, V.-H.; Bastawrous, H.A.; Lim, K.; See, K.W.; Zhang, P.; Dou, S.X. Online state of charge and model parameters estimation of the LiFePO4 battery in electric vehicles using multiple adaptive forgetting factors recursive least-squares. J. Power Sources 2015, 296, 215–224. [Google Scholar] [CrossRef]
- Rijanto, E.; Rozaqi, L.; Nugroho, A.; Kanarachos, S. RLS with Optimum Multiple Adaptive Forgetting Factors for SoC and SoH Estimation of Li-Ion Battery. In Proceedings of the 2017 5th International Conference on Instrumentation, Control, and Automation (ICA), Yogyakarta, Indonesia, 9–11 August 2017; pp. 73–77. [Google Scholar]
- Vahidi, A.; Stefanopoulou, A.; Peng, H. Recursive least squares with forgetting for online estimation of vehicle mass and road grade: Theory and experiments. Veh. Syst. Dyn. 2005, 43, 31–55. [Google Scholar] [CrossRef]
- Uosaki, K.; Yotsuya, M.; Hatanaka, T. Adaptive identification of non-stationary systems with multiple forgetting factors. In Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 11–13 December 1996; pp. 851–856. [Google Scholar]
- Hardier, G. An Extended U-D Algorithm with Multiple Forgetting Factors for RLS Estimation of Model Parameters. IFAC-PapersOnLine 2015, 48, 200–207. [Google Scholar] [CrossRef]
- Saelid, S.; Foss, B. Adaptive controllers with a vector variable forgetting factor. In Proceedings of the 22nd IEEE Conference on Decision and Control, San Antonio, TX, USA, 14–16 December 1983; pp. 1488–1494. [Google Scholar]
- Li, J.; Wang, L.; Lyu, C.; Liu, E.; Xing, Y.; Pecht, M. A parameter estimation method for a simplified electrochemical model for Li-ion batteries. Electrochim. Acta 2018, 275, 50–58. [Google Scholar] [CrossRef]
- Panchal, S.; McGrory, J.; Kong, J.; Fraser, R.; Fowler, M.; Dincer, I.; Agelin-Chaab, M. Cycling degradation testing and analysis of a LiFePO4 battery at actual conditions. Int. J. Energy Res. 2017, 41, 2565–2575. [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]
- Fraccaroli, F.; Peruffo, A.; Zorzi, M. A New Recursive Least Squares Method with Multiple Forgetting Schemes. In Proceedings of the IEEE Conference on Decision and Control (CDC), Osaka, Japan, 15–18 December 2015; pp. 3367–3372. [Google Scholar]
- Tian, Y.; Li, D.; Tian, J.; Xia, B. State of charge estimation of lithium-ion batteries using an optimal adaptive gain nonlinear observer. Electrochim. Acta 2017, 225, 225–234. [Google Scholar] [CrossRef]
- Lao, Z.; Xia, B.; Wang, W.; Sun, W.; Lai, Y.; Wang, M. A Novel Method for Lithium-Ion Battery Online Parameter Identification Based on Variable Forgetting Factor Recursive Least Squares. Energies 2018, 11, 1358. [Google Scholar] [CrossRef]
Initialization | Step 1: Initial value of , , and , Step 2: Weight calculation 1 |
Prediction | Step 3: Create sigma points 2 Step 4: Time update for estimated system |
Correction | Step 5: Prior estimation of system output |
Step 6: UKF gain calculation Step 7: Update of system states and state covariance |
Initialization | Step 1: Define the number of particles , maximum iteration , initial value and limitation of velocity and position , personal best position of each particle , global best position of particles , inertia weight limitation and adjustment weight . |
Updating | Step 2: Update velocity and position of each particle using equations below 1, if the calculated value exceeds the limitation, then it will be replaced by the corresponding limit. Step 3: Calculate the fitness function of each particle, update the personal best position and global best position by |
Iteration and exit | Step 4: If , then exit and display the results, otherwise and return to step 2 |
Algorithms | Forgetting Factors |
---|---|
SFFRLS | 0.9689 |
MFFRLS | [0.9272, 0.9054, 0.9062] |
Coefficients | |||||||
---|---|---|---|---|---|---|---|
Values | 3.4453 | 0.8606 | −0.0442 | −8.6603 | 30.2828 | −36.1564 | 14.4612 |
SOC (%) | 98 | 95 | 90 | 80 | 70 | 60 | 50 | 40 |
0.0367 | 0.0369 | 0.0373 | 0.0368 | 0.0363 | 0.0359 | 0.0360 | 0.0364 | |
SOC (%) | 30 | 20 | 10 | 8 | 5 | 3 | 1 | 0 |
0.0367 | 0.0374 | 0.0393 | 0.0410 | 0.0430 | 0.0456 | 0.0502 | 0.0522 |
Parameters | |||
---|---|---|---|
Values | 0.0367 | 0.0183 | 3768 |
Algorithms | UKF | SFFRLS-UKF | MFFRLS-UKF |
---|---|---|---|
Mean error | 0.0120 | 0.0061 | 0.0060 |
Max error | 0.0190 | 0.0085 | 0.0080 |
RMSE | 0.0131 | 0.0063 | 0.0062 |
Algorithms | UKF | SFFRLS-UKF | MFFRLS-UKF |
---|---|---|---|
Mean error | 0.0440 | 0.0082 | 0.0029 |
Max error | 0.0600 | 0.0123 | 0.0047 |
RMSE | 0.0477 | 0.0088 | 0.0030 |
Tests | Multiple Cycle 1 | Multiple Cycle 2 | ||||
---|---|---|---|---|---|---|
Algorithms | UKF | SFFRLS-UKF | MFFRLS-UKF | UKF | SFFRLS-UKF | MFFRLS-UKF |
Mean error | 0.0528 | 0.0111 | 0.0076 | 0.0558 | 0.0126 | 0.0085 |
Max error | 0.0732 | 0.0160 | 0.0090 | 0.0827 | 0.0166 | 0.0103 |
RMSE | 0.0583 | 0.0116 | 0.0076 | 0.0623 | 0.0132 | 0.0086 |
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Xia, B.; Huang, R.; Lao, Z.; Zhang, R.; Lai, Y.; Zheng, W.; Wang, H.; Wang, W.; Wang, M. Online Parameter Identification of Lithium-Ion Batteries Using a Novel Multiple Forgetting Factor Recursive Least Square Algorithm. Energies 2018, 11, 3180. https://doi.org/10.3390/en11113180
Xia B, Huang R, Lao Z, Zhang R, Lai Y, Zheng W, Wang H, Wang W, Wang M. Online Parameter Identification of Lithium-Ion Batteries Using a Novel Multiple Forgetting Factor Recursive Least Square Algorithm. Energies. 2018; 11(11):3180. https://doi.org/10.3390/en11113180
Chicago/Turabian StyleXia, Bizhong, Rui Huang, Zizhou Lao, Ruifeng Zhang, Yongzhi Lai, Weiwei Zheng, Huawen Wang, Wei Wang, and Mingwang Wang. 2018. "Online Parameter Identification of Lithium-Ion Batteries Using a Novel Multiple Forgetting Factor Recursive Least Square Algorithm" Energies 11, no. 11: 3180. https://doi.org/10.3390/en11113180