A New State of Charge Estimation Algorithm for Lithium-Ion Batteries Based on the Fractional Unscented Kalman Filter
Abstract
:1. Introduction
2. Battery Modeling
2.1. Integer Second-Order RC Model
2.2. The Definition of Fractional Capacitor
2.3. The Definition and Properties of Fractional-Order Calculus
2.4. Fractional Second-Order RC Model
3. Fractional Unscented Kalman Filter
3.1. The Observability of the Battery Model
3.2. Details of Fractional Unscented Kalman Filter
4. Model Parameter Identification
4.1. Test Bench
- When the battery capacity is measured with 1 C, 2 C and 3 C discharge rates. The corresponding results are 23.587 Ah, 23.294 Ah and 22.460 Ah, respectively. It is shown that the discharge capacity at higher rates is lower than that at low rates and it is different from the nominal value 24 Ah. For the non-fresh lithium battery pack, it is acceptable that the discharge capacity is slightly less than 24 Ah of the nominal capacity. Hence, the rated capacity of the battery is considered as 23.587 Ah instead of 24 Ah.
- Since the measurement error of the current and voltage of the test platform is less than 0.1% of full scale, it is feasible to assume that the battery value can be obtained by this high-precision battery testing system BTS-4000. Hence, the obtained is used as a true value although some sensor errors exist.
- For convenience, the integer order model can be obtained by setting the orders of the fractional capacitors . The physical meanings of all the seven parameters of the battery second-order RC model are identical no matter whether the model is a fractional order model or an integer order model.
4.2. The Identification for Resistor
4.3. RC Loop Identification
5. Experimental Verification
5.1. Pulse Characterization Experiment
5.2. Static Discharge Experiment
5.3. Dynamic Discharge Experiment
- The fractional second-order RC model and FUKF algorithm track the measured SOC and terminal voltage well. The SOC error and the battery terminal voltage error are 11.55% bound and 0.2712 V bound, respectively. On the contrast, the SOC error and the battery terminal voltage error of the integer model and UKF are 14.20% bound and 0.3568 V bound, respectively. The mean errors of SOC estimation based on FUKF algorithm and UKF algorithm are also given as 2.88% and 5.92%, respectively. Although the maximal error of the SOC estimation based on FUKF algorithm is quite large, the mean error is mild enough. Thus, for most time, FUKF algorithm presents extraordinary precision on SOC estimation.
- In both static and dynamic experiments, the result curves of fractional order model and integer order model can both converge to the measurement curve even though the initial values are incorrect and the curve of fractional order model recovers the initial SOC error on a larger scale and with less time according to the enlarged drawings. This demonstrates better stability and robustness of FUKF.
- In both static and dynamic discharge experiment, both UKF and FUKF show a convergent performance. However, when the remaining capacity of the battery is little, due to the enlarged battery polarization effect, the battery model parameter estimation becomes inaccurate. This leads to the error of the space state estimation. UKF loses its convergence and performs divergently. Nevertheless, the fractional second-order RC model and FUKF algorithm could still make precise estimation of the battery parameters and track the actual value of .
- Compared with the integer model, the fractional orders of the capacitors and are more capable of reducing the performance indicator in Equation (26) with least square method, because the fractional order model and fractional parameter identification reflect the system performance more precisely.
- It is obvious that some sharp peaks of the terminal voltage error appear at the beginning and the end of the discharge process in Figure 9d. That is because the battery model parameters suddenly change at the instant the battery current pulse appears or disappears. This phenomenon is caused by the battery polarization effect. The fractional order model and FUKF exhibit a more accurate performance on the suppression of sharp peak error. In dynamic operation condition, the error of the terminal voltage estimation is more severe than in the static operation condition, because the battery polarization effect is more obvious in the rapidly changing operating conditions. This effect causes the dynamics which cannot be modeled. However, FUKF is still effective with this phenomenon.
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- Qiu, S.; Chen, Z.; Masrur, M.A.; Murphey, Y.L. Battery hysteresis modeling for state of charge estimation based on Extended Kalman Filter. In Proceedings of the 2011 6th IEEE Conference on Industrial Electronics and Applications (ICIEA), Beijing, China, 21–23 June 2011; pp. 184–189. [Google Scholar]
- Li, J.; Lai, Q.; Wang, L.; Lyu, C.; Wang, H. A method for SOC estimation based on simplified mechanistic model for LiFePO4 battery. Energy 2016, 114, 1266–1276. [Google Scholar] [CrossRef]
- Ma, Y.; Zhou, X.; Li, B.; Chen, H. Fractional modeling and SOC estimation of lithium-ion battery. IEEE/CAA J. Autom. Sin. 2016, 3, 281–287. [Google Scholar]
- Doucette, R.T.; McCulloch, M.D. Modeling the prospects of plug-in hybrid electric vehicles to reduce CO2 emissions. Appl. Energy 2011, 88, 2315–2323. [Google Scholar] [CrossRef]
- He, H.; Xiong, R.; Guo, H. Online estimation of model parameters and state-of-charge of LiFePO4 batteries in electric vehicles. Appl. Energy 2012, 89, 413–420. [Google Scholar] [CrossRef]
- Aylor, J.H.; Thieme, A.; Johnso, B.W. A battery state-of-charge indicator for electric wheelchairs. IEEE Trans. Ind. Electron. 1992, 39, 398–409. [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]
- Zhang, F.; Liu, G.J.; Fang, L.J.; Wang, H.G. Estimation of Battery State of Charge With H-infinity Observer: Applied to a Robot for Inspecting Power Transmission Lines. IEEE Trans. Ind. Electron. 2012, 59, 1086–1095. [Google Scholar] [CrossRef]
- Belhani, A.; M’Sirdi, N.K.; Naamane, A. Adaptive Sliding Mode Observer for Estimation of State of Charge. Energy Procedia 2013, 42, 377–386. [Google Scholar] [CrossRef]
- Xia, B.; Chen, C.; Tian, Y.; Wang, M.; Sun, W.; Xu, Z. State of charge estimation of lithium-ion batteries based on an improved parameter identification method. Energy 2015, 90, 1426–1434. [Google Scholar] [CrossRef]
- Dai, H.; Guo, P.; Wei, X.; Sun, Z.; Wang, J. ANFIS (adaptive neuro-fuzzy inference system) based online SOC (State of Charge) correction considering cell divergence for the EV (electric vehicle) traction batteries. Energy 2015, 80, 350–360. [Google Scholar] [CrossRef]
- Yao, L.W.; Aziz, J.A.; Idris, N.R.N. State-of-charge estimation for lithium-ion battery using Busse’s adaptive unscented Kalman filter. In Proceedings of the 2015 IEEE Conference onEnergy Conversion (CENCON), Johor Bahru, Malaysia, 19–20 October 2015; pp. 227–232. [Google Scholar]
- Sun, F.; Hu, X.; Zou, Y.; Li, S. Adaptive unscented Kalman filtering for state of charge estimation of a lithium-ion battery for electric vehicles. Energy 2011, 36, 3531–3540. [Google Scholar] [CrossRef]
- Julier, S.J.; Uhlmann, J.K. New extension of the Kalman filter to nonlinear systems. In Proceedings of the International Society for Optics and Photonics, Orlando, FL, USA, 21–25 April 1997; pp. 182–193. [Google Scholar]
- 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]
- 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]
- D’Alfonso, L.; Lucia, W.; Muraca, P.; Pugliese, P. Mobile robot localization via EKF and UKF: A comparison based on real data. Robot. Auton. Syst. 2015, 74, 122–127. [Google Scholar] [CrossRef]
- Miyabayashi, K.; Tonomura, O.; Kano, M.; Hasebe, S. Comparative study of state estimation of tubular microreactors using ukf and ekf. IFAC Proc. Vol. 2012, 45, 513–518. [Google Scholar] [CrossRef]
- He, Z.; Chen, D.; Pan, C.; Chen, L.; Wang, S. State of charge estimation of power Li-ion batteries using a hybrid estimation algorithm based on UKF. Electrochim. Acta 2016, 211, 101–109. [Google Scholar]
- Monje, C.A.; Chen, Y.Q.; Vinagre, B.M.; Xue, D.; Feliu-Batlle, V. Fractional-Order Systems and Controls: Fundamentals and Applications; Springer Science & Business Media: Berlin, Germany, 2010. [Google Scholar]
- Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Academic Press: Cambridge, MA, USA, 1998. [Google Scholar]
- Ortigueira, M.D.; Machado, J.A.T. Fractional calculus applications in signals and systems. Signal Process. 2006, 86, 2503–2504. [Google Scholar] [CrossRef]
- Petras, I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation; Springer Science & Business Media: Berlin, Germany, 2011. [Google Scholar]
- Zhang, L.; Hu, X.; Wang, Z.; Sun, F.; Dorrell, D.G. Fractional-order modeling and State-of-Charge estimation for ultracapacitors. J. Power Sources 2016, 314, 28–34. [Google Scholar] [CrossRef]
- Liu, C.; Liu, W.; Wang, L.; Hu, G.; Ma, L.; Ren, B. A new method of modeling and state of charge estimation of the battery. J. Power Sources 2016, 320, 1–12. [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]
- Hu, Y.; Yurkovich, S.; Guezennec, Y.; Guezennec, Y. Electro-thermal battery model identification for automotive applications. J. Power Sources 2011, 196, 449–457. [Google Scholar] [CrossRef]
- Westerlund, S.; Ekstam, L. Capacitor theory. IEEE Trans. Dielectr. Electr. Insul. 1994, 1, 826–839. [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]
- Caponetto, R. Fractional Order Systems: Modeling and Control Applications; World Scientific: Singapore, 2010. [Google Scholar]
- Simon, D. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Park, S. Fixed point theorems for better admissible multimaps on almost convex sets. J. Math. Anal. Appl. 2007, 329, 690–702. [Google Scholar] [CrossRef]
- Nosrati, K.; Rostami, A.S.; Azemi, A.; Pariz, N. Unscented Kalman Filter Applied to noisy synchronization of Rossler chaotic system. In Proceedings of the 2011 3rd International Conference on Advanced Computer Control (ICACC), Harbin, China, 18–20 January 2011; pp. 378–383. [Google Scholar]
- Haykin, S. Kalman Filtering and Neural Networks; Wiley: New York, NY, USA, 2001. [Google Scholar]
- Unterrieder, C.; Zhang, C.; Lunglmayr, M.; Priewasser, R.; Marsili, S.; Huemer, M. Battery state-of-charge estimation using approximate least squares. J. Power Sources 2015, 278, 274–286. [Google Scholar] [CrossRef]
- Radwan, A.G.; Elwakil, A.S.; Soliman, A.M. On the generalization of second-order filters to the fractional-order domain. J. Circuits Syst. Comput. 2009, 18, 361–386. [Google Scholar] [CrossRef]
Method | SOC Error | Maximum Terminal Voltage Error | |
---|---|---|---|
Maximum Error | Mean Error | ||
FUKF | 5.36% | 2.56% | 0.0296 V |
UKF | 9.47% | 3.65% | 0.0328 V |
Method | SOC Error | Maximum Terminal Voltage Error | |
---|---|---|---|
Maximum Error | Mean Error | ||
FUKF | 11.55% | 2.88% | 0.2712 V |
UKF | 14.20% | 5.92% | 0.3568 V |
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Chen, Y.; Huang, D.; Zhu, Q.; Liu, W.; Liu, C.; Xiong, N. A New State of Charge Estimation Algorithm for Lithium-Ion Batteries Based on the Fractional Unscented Kalman Filter. Energies 2017, 10, 1313. https://doi.org/10.3390/en10091313
Chen Y, Huang D, Zhu Q, Liu W, Liu C, Xiong N. A New State of Charge Estimation Algorithm for Lithium-Ion Batteries Based on the Fractional Unscented Kalman Filter. Energies. 2017; 10(9):1313. https://doi.org/10.3390/en10091313
Chicago/Turabian StyleChen, Yixing, Deqing Huang, Qiao Zhu, Weiqun Liu, Congzhi Liu, and Neng Xiong. 2017. "A New State of Charge Estimation Algorithm for Lithium-Ion Batteries Based on the Fractional Unscented Kalman Filter" Energies 10, no. 9: 1313. https://doi.org/10.3390/en10091313