(143c) Towards Model-Based Optimal Design of Lithium-Ion Batteries

Prof. Newman and his group have applied macroscopic models to optimize the electrode thickness or porosity.¹ These studies have been performed by comparing the Ragone plots for different design parameters. A single curve in a Ragone plot may involve hundreds of simulations wherein the applied current is varied over a wide range of magnitude. Ragone plots for different configurations are obtained by changing the design parameters (e.g., thickness) one at a time, and by keeping the other parameters constant. This process of generating a Ragone plot is quite tedious and typically Ragone curves reported in the literature are not smooth because of computational constraints.

To our knowledge, the literature does not report the application of such first-principles models to the global optimization of multiple battery design parameters. Also, batteries are typically designed only to optimize the performance at cycle one of the battery, whereas in practice most of the battery's operation occurs under significantly degraded conditions. The reformulated model is sufficiently computationally efficient to enable the simultaneous optimal design of multiple parameters over any number of cycles by including the model for capacity fade by tracking parameters with cycles². Further, the model can be used to quantify the effects of model uncertainties and variations in the design parameters on the battery performance. As an example of such robustness analysis, the utilization averaged over 1000 cycles are reported in Fig. 1 for the battery design obtained by (1) simultaneous optimization of the applied current density (I) and thickness of the separator and the two electrodes (l_s, l_n, l_p) for cycle 1, and (2) variations in these four design parameters. The battery design optimized for cycle 1 does not maximize the cycle-averaged utilization.

Fig. 1. Utilization averaged over cycle 25, 500, and 1000 for a 3-level 4-factor factorial design. The title of each plot indicates the deviation in the design variables I and l_s from their values optimized for cycle 1. Circles, stars, and dots are for the l_n value optimized for cycle 1 and ±20% of that value, respectively.

We are also investigating the optimal design of distributions of properties, which cannot be reasonably handled by one-at-a-time optimization. In particular, we are optimizing porosity distributions and particle size distributions across the electrode, in addition to the standard design parameters considered in Fig. 1.

                Performing global optimization for design is a computationally challenging task. Porosity affects transport and kinetic parameters in the electrode. When porosity changes as a function of distance, the transport and kinetic parameters change as a function of distance based on Bruggman coefficients. When concentration gradients and solid-phase diffusion limitations are ignored, the model formulation for minimizing the ohmic drop governing the porous electrode given by porous electrode theory can be represented as

The ?unknowns? in this optimization problem are the differential state variables z(t), algebraic variables y(t), control variables u(t), and vector of parameters p.

Even for this simple porous electrode theory with Butler-Volmer kinetics, obtaining a globally optimal profile for porosity distribution is nontrival. Typical methods for optimization include (1) Pontyragin's principle (2) CVI (3) CVP (4) Simultaneous Nonlinear Programming. Optimal profiles for porosity distribution for a base set of parameters² is given in Figure 2. The design profile obtained will then be fed to the pseudo-2D model with time and x as independent variables.

Fig. 2. Porosity variation across the electrode  

The objective function for the optimization could be (1) Improved utilization (individual or total) (2) Improved cycle life with mechanism for capacity fade (3) Ideal thermal behavior etc. It is worth noting that the intent of this paper is not to state that the pseudo-2D model is sufficient, however the profiles obtained with the electrochemical engineering model can be fed as inputs to detailed microscale, multiscale models that include stress relationships, molecular models, etc to obtain meaningful material design characteristics.²

Acknowledgements

The authors are thankful for the financial support of this work by the NSF (CBET ? 0828002), U.S. Army CERDEC (W909MY-06-C-0040), Oronzio de Nora Industrial Electrochemistry Postdoctoral Fellowship of ECS, and the
United States government.

References

1. T. F. Fuller, M. Doyle and J. Newman, J. Electrochem. Soc., 141, 982 (1994).
2. V. R. Subramanian, V. Boovaragavan, V.  Ramadesigan, and  M. Arabandi, J. Electrochem. Soc., 156, A260 (2009).