Research Papers

Physics-Based Model of a Valve-Regulated Lead-Acid Battery and an Equivalent Circuit

[+] Author and Article Information
Zheng Shen

e-mail: zus120@psu.edu

Christopher D. Rahn

Fellow ASME
e-mail: cdrahn@psu.edu
Department of Mechanical and Nuclear Engineering,
Mechatronics Research Laboratory,
The Pennsylvania State University,
University Park, PA 16802

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 17, 2012; final manuscript received February 15, 2013; published online May 16, 2013. Assoc. Editor: Luis Alvarez.

J. Dyn. Sys., Meas., Control 135(4), 041011 (May 16, 2013) (7 pages) Paper No: DS-12-1143; doi: 10.1115/1.4023765 History: Received May 17, 2012; Revised February 15, 2013

Lead (Pb)-acid batteries are a low-cost power source for applications ranging from hybrid and electric vehicles (HEVs) to large-scale energy storage. Efficient simulation, design, and management systems require the development of low order but accurate models. In this paper we develop a reduced-order Pb-acid battery model from first principles using linearization and the Ritz discretization method. The model, even with a low-order discretization, accurately predicts the voltage response to a dynamic pulse current input and outputs spatially distributed variables of interest. A dynamic averaged model is developed from the Ritz model and realized by an equivalent circuit. The circuit resistances and capacitances depend on electrochemical parameters, linking the equivalent circuit model to the underlying electrochemistry of the first principles model.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Bode, H., 1977, Lead-Acid Batteries, Wiley, New York.
Hejabi, M., Oweisi, A., and Gharib, N., 2006, “Modeling of Kinetic Behavior of the Lead Dioxide Electrode in a Lead-Acid Battery by Means of Electrochemical Impedance Spectroscopy,” J. Power Sources, 158(2), pp. 944–948. [CrossRef]
Srinivasan, V., Wang, G. Q., and Wang, C. Y., 2003, “Mathematical Modeling of Current-Interrupt and Pulse Operation of Valve-Regulated Lead Acid Cells,” J. Electrochem. Soc., 150(3), pp. A316–A325. [CrossRef]
Vasebi, A., Partovibakhsh, M., and Bathaee, S. M. T., 2007, “A Novel Combined Battery Model for State-of-Charge Estimation in Lead-Acid Batteries Based on Extended Kalman Filter for Hybrid Electric Vehicle Applications,” J. Power Sources, 174(1), pp. 30–40. [CrossRef]
Salameh, Z., Casacca, M., and Lynch, W., 1992, “A Mathematical Model for Lead-Acid Batteries,” IEEE Trans. Energy Conversion, 7(1), pp. 93–98. [CrossRef]
Ceraolo, M., 2000, “New Dynamical Models of Lead-Acid Batteries,” IEEE Trans. Power Syst., 15(4), pp. 1184–1190. [CrossRef]
Huggins, R., 1999, “General Equivalent Circuit of Batteries and Fuel Cells,” Ionics, 5, pp. 269–274. [CrossRef]
Durr, M., Cruden, A., Gair, S., and McDonald, J., 2006, “Dynamic Model of a Lead Acid Battery for Use in a Domestic Fuel Cell System,” J. Power Sources, 161(2), pp. 1400–1411. [CrossRef]
Bernardi, D. M., and Carpenter, M. K., 1995, “A Mathematical Model of the Oxygen-Recombination Lead-Acid Cell,” J. Electrochem. Soc., 142(8), pp. 2631–2642. [CrossRef]
Gu, W. B., Wang, C. Y., and Liaw, B. Y., 1997, “Numerical Modeling of Coupled Electrochemical and Transport Processes in Lead-Acid Batteries,” J. Electrochem. Soc., 144(6), pp. 2053–2061. [CrossRef]
Gu, W. B., Wang, G. Q., and Wang, C. Y., 2002, “Modeling the Overcharge Process of VRLA Batteries,” J. Power Sources, 108, pp. 174–184. [CrossRef]
Newman, J., and Tiedemann, W., 1975, “Porous-Electrode Theory With Battery Applications,” AIChE J., 21(1), pp. 25–41. [CrossRef]
Gu, H., Nguyen, T. V., and White, R. E., 1987, “A Mathematical Model of a Lead-Acid Cell,” J. Electrochem. Soc., 134(12), pp. 2953–2960. [CrossRef]
Gebhart, B., 1993, Heat Conduction and Mass Diffusion, McGraw-Hill, New York.
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]
Subramanian, V. R., Ritter, J. A., and White, R. E., 2001, “Approximate Solutions for Galvanostatic Discharge of Spherical Particles I. Constant Diffusion Coefficient,” J. Electrochem. Soc., 148(11), pp. E444–E449. [CrossRef]
Ritz, W., 1909, “Uber Eine Neue Methode zur Losung Gewisser Variationsprobleme der Mathematischen Physik,” J. Reine Angew. Math., 135, pp. 1–61.
Reddy, J., and Gartling, D., 2010, The Finite Element Method in Heat Transfer and Fluid Dynamics, 3rd ed., Taylor and Francis, London.
Hill, J. M., and Dewynne, J. N., 1987, Heat Conduction, Blackwell Scientific, Oxford.
Ozisik, M. N., 1980, Heat Conduction, John Wiley and Sons, New York.
Shamash, Y., 1975, “Linear System Reduction Using Pade Approximation to Allow Retention of Dominant Modes,” Int. J. Control, 21(2), pp. 257–272. [CrossRef]
Golub, G., and Van Loan, C., 1996, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD.
Shi, Y., Prasad, G., Shen, Z., and Rahn, C., 2011, “Discretization Methods for Battery Systems Modeling,” American Control Conference.
Ge, S., and Sun, Z., 2005, Switched Linear Systems: Control and Design, Springer, New York.
Lin, H., and Antsaklis, P., 2009, “Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results,” IEEE Trans. Auto. Control, 54(2), pp. 308–322. [CrossRef]
Bard, A. J., and Faulkner, L. R., 2001, Electrochemical Methods: Fundamentals and Applications, John Wiley and Sons, New York.
Shi, Y., Ferone, C., Rao, C., and Rahn, C., 2012, “Nondestructive Forensic Pathology of Lead-Acid Batteries,” American Control Conference.


Grahic Jump Location
Fig. 1

Schematic diagram of a lead-acid cell

Grahic Jump Location
Fig. 2

Experimental and simulated time response to a pulse charge/discharge current input: (a) voltage output for the switched linear models with N = 8 (blue, solid) and N = 1 (green, dashed), charge model (red, dash-dotted), discharge model (yellow, dotted), and experiment (black, solid). (b) Input current.

Grahic Jump Location
Fig. 3

Spatial distributions in response to current input from Fig. 2(b) at t = 200 s (blue, solid), 300 s (green, dashed), 400 s (yellow, dash-dotted), and 500 s (red, dotted): (a) acid concentration c(x, t), (b) electrolyte potential ϕe (x, t), and (c) solid-phase potential ϕs (x, t)

Grahic Jump Location
Fig. 4

Equivalent circuit model



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