Research Papers

Reduced Order Impedance Models of Lithium Ion Batteries

[+] Author and Article Information
Githin K. Prasad

Research Assistant
Department of Mechanical and
Nuclear Engineering,
Pennsylvania State University,
University Park, PA 16802
e-mail: gkp104@psu.edu

Christopher D. Rahn

Department of Mechanical and
Nuclear Engineering,
Pennsylvania State University,
University Park, PA 16802
e-mail: cdrahn@psu.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 24, 2012; final manuscript received January 28, 2014; published online April 8, 2014. Assoc. Editor: Evangelos Papadopoulos.

J. Dyn. Sys., Meas., Control 136(4), 041012 (Apr 08, 2014) (8 pages) Paper No: DS-12-1317; doi: 10.1115/1.4026652 History: Received September 24, 2012; Revised January 28, 2014

This paper develops reduced order, linear models of lithium ion batteries that can be used for model-based power train simulation, design, estimation, and controlling in hybrid and electric vehicles (HEV). First, a reduced order model is derived from the fundamental governing electrochemical charge and Li+ conservation equations that are linearized at the operating state of charge and low current density. The equations are solved using analytical and numerical techniques to produce the transcendental impedance or transfer function from input current to output voltage. This model is then reduced to a low order state space model using a system identification technique based on least squares optimization. Given the prescribed current, the model predicts voltage and other variables such as electrolyte and electrode surface concentration distributions. The second model is developed by neglecting electrolyte diffusion and modeling each electrode with a single active material particle. The transcendental particle transfer functions are discretized using a Padé Approximation. The explicit form of the single particle model impedance can be realized by an equivalent circuit with resistances and capacitances related to the cell parameters. Both models are then tuned to match experimental electrochemical impedance spectroscopy (EIS) and pulse current-voltage data.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Plett, G. L., 2004, “Extended Kalman Filtering for Battery Management Systems of LiPB-Based HEV Battery Packs: Part 1. Background,” J. Power Sources, 134(2), pp. 252–261. [CrossRef]
Plett, G. L., 2004, “Extended Kalman Filtering for Battery Management Systems of LiPB-Based HEV Battery Packs: Part 2. Modeling and Identification,” J. Power Sources, 134(2), pp. 262–276. [CrossRef]
Plett, G. L., 2004, “Extended Kalman Filtering for Battery Management Systems of LiPB-Based HEV Battery Packs: Part 3. State and Parameter Estimation,” J. Power Sources, 134(2), pp. 277–292. [CrossRef]
Verbrugge, M. W., and Conell, R. S., 2002, “Electrochemical and Thermal Characterization of Battery Modules Commensurate With Electric Vehicle Integration,” J. Electrochem. Soc., 149(1), pp. A45–A53. [CrossRef]
Verbrugge, M. W., and Conell, R. S., 2007, “Electrochemical Characterization of High-Power Lithium Ion Batteries Using Triangular Voltage and Current Excitation Sources,” J. Power Sources, 174(1), pp. 2–8. [CrossRef]
Chen, M., and Rincon-Mora, G. A., 2006, “Accurate Electrical Battery Model Capable of Predicting Runtime and IV Performance,” IEEE Trans. Energy Convers., 21(2), pp. 504–511. [CrossRef]
Schweighofer, B., Raab, K. M., and Brasseur, G., 2003, “Modeling of High Power Automotive Batteries by the Use of an Automated Test System,” IEEE Trans. Instrum. Meas., 52(4), pp. 1087–1091. [CrossRef]
Moss, P. L., Au, G., Plichta, E. J., and Zheng, J. P., 2008, “An Electrical Circuit for Modeling the Dynamic Response of Li-Ion Polymer Batteries,” J. Electrochem. Soc., 155(12), pp. A986–A994. [CrossRef]
Hu, X., Li, S., and Peng, H., 2012, “A Comparative Study of Equivalent Circuit Models for Li-Ion Batteries,” J. Power Sources, 198, pp. 359–367. [CrossRef]
Dong, T. K., Kirchev, A., Mattera, F., Kowal, J., and Bultel, Y., 2011, “Dynamic Modeling of Li-Ion Batteries Using an Equivalent Electrical Circuit,” J. Electrochem. Soc., 158(3), pp. A326–A336. [CrossRef]
Hu, Y., Yurkovich, S., Guezennec, Y., and Yurkovich, B. J., 2011, “Electro-Thermal Battery Model Identification for Automotive Applications,” J. Power Sources, 196, pp. 449–457. [CrossRef]
Doyle, M., Fuller, T., and Newman, J., 1993, “Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell,” J. Electrochem. Soc., 140, pp. 1526–1533. [CrossRef]
Fuller, T., Doyle, M., and Newman, J., 1994, “Simulation and Optimization of the Dual Lithium Ion Insertion Cell,” J. Electrochem. Soc., 141, pp. 1–10. [CrossRef]
Smith, K. A., Rahn, C. D., and Wang, C.-Y., 2007, “Control Oriented 1D Electrochemical Model of Lithium Ion Battery,” Energy Convers. Manage., 48, pp. 2565–2578. [CrossRef]
Domenico, D., Di Fiengo, G., and Stefanopoulou, A., 2008, “Lithium-Ion Battery State of Charge Estimation With a Kalman Filter Based on a Electrochemical Model,” Proceedings of IEEE International Conference on Control Applications CCA, pp. 702–707.
Domenico, D. D., Stefanopoulou, A., and Fiengo, G., 2010, “Lithium-Ion Battery State of Charge and Critical Surface Charge Estimation Using an Electrochemical Model-Based Extended Kalman Filter,” ASME J. Dyn. Syst., Meas., Control, 132(6), p. 061302. [CrossRef]
Speltino, C., Domenico, D. D., Fiengo, G., and Stefanopoulou, A. G., 2009, “Experimental Identification and Validation of an Electrochemical Model of a Lithium-Ion Battery,” Proceedings of the American Control Conference 2009, St. Louis, MO.
Haran, B. S., Popov, B. N., and White, R. E., 1998, “Determination of the Hydrogen Diffusion Coefficient in Metal Hydrides by Impedance Spectroscopy,” J. Power Sources, 75(1), pp. 56–63. [CrossRef]
Ning, G., and Popov, B. N., 2004, “Cycle Life Modeling of Lithium-Ion Batteries,” J. Electrochem. Soc., 151(10), pp. A1584–A1591. [CrossRef]
Santhanagopalan, S., Guo, Q., Ramadass, P., and White, R. E., 2006, “Review of Models for Predicting the Cycling Performance of Lithium Ion Batteries,” J. Power Sources, 156(2), pp. 620–628. [CrossRef]
Chaturvedi, N., Klein, R., Christensen, J., Ahmed, J., and Kojic, A., 2010, “Algorithms for Advanced Battery-Management Systems: Modeling, Estimation, and Control Challenges for Lithium-Ion Batteries,” IEEE Control Syst. Mag., 30(3), pp. 40–68. [CrossRef]
Marcicki, J., Canova, M., Conlisk, A. T., and Rizzoni, G., 2013, “Design and Parameterization Analysis of a Reduced-Order Electrochemical Model of Graphite/LiFePO4 Cells for SOC/SOH Estimation,” J. Power Sources, 237, pp. 310–324. [CrossRef]
Gomadam, P. M., Weidner, J. W., Dougal, R. A., and White, R. E., 2002, “Mathematical Modeling of Lithium-Ion and Nickel Battery Systems,” J. Power Sources, 110, pp. 267–284. [CrossRef]
Forman, J. C., Bashash, S., Stein, J. L., and Fathy, H. K., 2011, “Reduction of an Electrochemistry-Based Li-Ion Battery Model Via Quasi-Linearization and Padé Approximation,” J. Electrochem. Soc., 158(2), pp. A93–A101. [CrossRef]
Jacobsen, T., and West, G., 1995, “Diffusion Impedance in Planar, Cylindrical, and Spherical Geometry,” Electrochim. Acta, 40(2), pp. 255–262. [CrossRef]
Shi, Y., Prasad, G., Shen, Z., and Rahn, C. D., 2011, “Discretization Methods for Battery Systems Modeling,” Proceedings of the American Control Conference 2011, San Francisco, CA.


Grahic Jump Location
Fig. 1

1D schematic of a lithium ion battery

Grahic Jump Location
Fig. 2

Impedance frequency response at 60% SOC: transcendental transfer function (dotted), reduced order model (dash-dotted), padé approximated single particle model (dashed), and experimental EIS (solid)

Grahic Jump Location
Fig. 3

Electrolyte concentration distribution, cex,t, time response: 5C discharge from 60% SOC at various times

Grahic Jump Location
Fig. 4

Solid phase surface concentration distribution, cs,e(x,t), time response: 5C discharge from 60% SOC at various times

Grahic Jump Location
Fig. 5

Current density distribution, j(x,t), time response: 5C discharge from 60% SOC at various times

Grahic Jump Location
Fig. 7

Equivalent circuit model of Padé approximated single particle model

Grahic Jump Location
Fig. 6

Experimental (solid), single particle model (dashed) and reduced order model (dashed) pulse charge/discharge time response at 60% SOC




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In