(374g) Expediting the Numerical Simulation of Lithium-Ion Battery Models

Lithium-ion batteries are important power sources for consumer electronics, and they provide excellent energy to weight ratios, no memory effect, and low loss of charge when idle.  To maintain battery health, mathematical models are valuable to monitor performance, make control decisions, and to determine optimal operation of these devices [1-12].  Many of these models are of the dynamic, distributed-parameter type and require the solution of a large system of differential-algebraic equations (DAEs).

Although parameter estimation of rigorous battery models has been performed [13], there are computational difficulties associated with dealing with these models, mainly the high CPU and storage requirements. Thus, model-based decisions may not be available on a timely basis. 

While reformulated, reduced-order battery models [13-14] will probably be the eventual choice for most battery control applications, a finite difference or finite element based approach can also play a role, especially for comparison of accuracy.  Thus numerical techniques that facilitate their solution are extremely valuable in-lieu of the fact that the finite-difference approach remains the most common among the researchers. While reformulated models explore apriori efficient manipulation and analytical solution of dependent variables, this talk explores the efficient numerical simulation of finite-difference schemes by employing efficient Jacobian restructuring and heuristic initialization schemes.

In this paper, we present numerical methods that involve both judicious restructuring and rapid initialization of the large system of DAEs.  The restructuring involves re-ordering dependent variables, yielding a banded Jacobian.  The initialization procedure involves a very rapid temporal switching that eliminates discontinuities that lead to inconsistent initial conditions.  Such conditions can severely strain the capabilities of DAE solvers such as DASSL. These techniques allow computations to be completed within 1 to 4 seconds using a Intel 2.66 GHz duo-core processor, even when as many as 150 node points are used.

Parameters such as the kinetic constants for electrode chemistry, electrolyte diffusion coefficients, and intraparticle diffusion coefficients can be rapidly estimated using modern global optimization methods, such as those advocated by the groups of Floudas and Stadtherr [15-17], within 30 seconds or so.  Once initial estimation is complete, subsequent updating of parametric values proceeds more rapidly, as the system is likely to lie within the region of quadratic convergence. 

Our analysis concludes that while reformulated models with polynomial representation provides 2 orders of magnitude in computation speed-up, the approach presented here provides an order or magnitude speed-up. In addition, finite difference approach provides a tool for benchmarking reformulate models and the overhead involved in developing this model is insignificant compared to developing reformulated models which require significant mathematical analysis.

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), and the United States government.

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