A Mass and Charge Conserving Tanks-in-series Model for Lithium-Ion Batteries
Authors: Subramaniam Akshay , Kolluri Suryanarayana , Parke Caitlin D. , Pathak Manan , Santhanagopalan Shriram , Subramanian Venkat R. 
Published: February 2020

The incorporation of physics-based lithium-ion battery models in real-time optimization, control and estimation has the potential to substantially facilitate their efficient and safe operation.1,2 In addition to improved prediction of battery states, the access to predictions of variables internal to battery electrochemistry allows the formulation of more complete optimization and control problems than would be possible by simple circuit models.3 Various approaches have been employed to enable this integration by reducing the computational demands of electrochemical models, particularly for the macro-homogeneous ‘P2D’ models of Newman and co-workers.4 These may be broadly divided into two categories: simplifications of the full model subject to limiting assumptions,5–7 or reformulation techniques which exploit the underlying structure of model equations to achieve fast simulation.1,8,9 The single particle model (SPM) visualizes the electrodes in a lithium-ion battery as two spherical particles and considers the electrode reactions and lithium transport through the active particle. Importantly however, it neglects potential and concentration variations in the electrolyte phase, which limits its predictive ability to low-current scenarios. Reformulated P2D models offer the desired computational performance without sacrificing physical detail, but their utility is dependent on the precise estimation of a large number of parameters from experimental data.10

This talk focuses on the development of a lumped physics-based model for lithium-ion batteries. The governing equations of the P2D model are integrated over the volume of each region in a cathode-separator-anode representation. This results in a set of volume-averaged quantities and relations, with the temporal evolution of each averaged variable expressed as an overall balance containing internal source terms and interfacial fluxes, which need to be approximated. The averaged porous domains may thus be regarded as three ‘tanks-in-series’. The model equations are simulated, and the electrochemical variable predictions compared against both the full P2D model and SPM over a range of parameter combinations and operating conditions. Various approximations for the interfacial fluxes are also evaluated in terms of their impact on cell-level predictions. The model is also extended to incorporate thermal effects via volume-averaged energy balances. Model predictions for different galvanostatic discharge rates suggest that the lumped model is of intermediate sophistication between the SPM and the P2D models. This is most likely because it incorporates liquid phase transport in an average sense, in contrast to SPM, which ignores them entirely. The lumped model is conservative, and possesses the computational simplicity of SPM, while also incorporating elements from the P2D model, but with a smaller number of parameters, making it easier to parameterize and potentially facilitating use in real-time prediction and control. In addition, tank models have been extensively researched in control theory, enabling the extension of these concepts to battery systems through lumped models. The ‘tanks-in-series’ averaging procedure described herein may be generalized to any electrochemical battery model.

Acknowledgements

The authors are also thankful for financial support from the Battery500 Consortium and the Advanced Research Projects Agency (ARPA-E). The authors acknowledge financial support from the Department of Chemical Engineering and the Clean Energy Institute at the University of Washington.

References

A. M. Bizeray, S. Zhao, S. R. Duncan, and D. A. Howey, J. Power Sources, 296, 400 (2015).

V. Ramadesigan, P. W. C. Northrop, S. De, S. Santhanagopalan, R. D. Braatz, and V. R. Subramanian, J. Electrochem. Soc., 159, R31 (2012).

M. Pathak, D. Sonawane, S. Santhanagopalan, R. D. Braatz, and V. R. Subramanian, ECS Trans., 75, 51 (2017).

M. Doyle, T. J. Fuller, and J. Newman, J. Electrochem. Soc., 140, 1526 (1993).

S. Santhanagopalan and R. E. White, J. Power Sources, 161, 1346 (2006).

B. S. Haran, B. N. Popov, and R. E. White, J. Power Sources, 75, 56 (1998).

M. Guo, G. Sikha, and R. E. White, J. Electrochem. Soc., 158, A122 (2011).

P. W. C. Northrop, V. Ramadesigan, S. De, and V. R. Subramanian, J. Electrochem. Soc., 158, A1461 (2011).

P. W. C. Northrop, M. Pathak, D. Rife, S. De, S. Santhanagopalan, and V. R. Subramanian, J. Electrochem. Soc., 162, A940 (2015).

V. Ramadesigan, K. Chen, N. A. Burns, V. Boovaragavan, R. D. Braatz, and V. R. Subramanian, J. Electrochem. Soc., 158, A1048 (2011).