(356e) Dynamic Optimization of Lithium-Ion Batteries – Current Profiles for Improved Utilization

Dynamic optimization is defined as the minimization or maximization of an objective function without violating given dynamic constraints. In general dynamic optimization for any system can be defined as¹

where the vectors represent the differential state variables z(t), algebraic variables y(t), control variables u(t), and all vectors of parameters p. There are different methods to solve constrained optimization problems. Typical methods for optimization include (1) Pontyragin's principle (2) Control Vector Iteration (3) Control Vector Parametrization (4) Simultaneous Nonlinear Programming. Irrespective of the method used the most efficient and stable model for the problem with minimal number of state variables is advantageous for obtaining optimal profiles and for estimating uncertainties. The objective function for the dynamic optimization could be identifying how should the charge/discharge protocol change for (1) improved utilization (individual or total) at the end of discharge (2) decreased ohmic loss in a particular electrode (3) improved cycle life with mechanism for capacity fade (4) expected charge/discharge behavior (5) ideal thermal behavior etc.

While there have been excellent progress made in the applied mathematics and systems communities in the development of reduced-order models (such as proper orthogonal decomposition), efficient solvers and schemes for batteries offer unique challenges. We have been developing efficient electrochemical engineering models for lithium-ion batteries by applying a wide variety of applied mathematical concepts guided by our extensive experience in electrochemistry and insights obtained by experiments and collaborations.² High-fidelity low-cost models with lower number of state variables provide for and enable the latest systems engineering approaches (Bayesian estimation, dynamic optimization, robust design, etc) to design operational scenarios that meet specified performance objectives. Our experience with various numerical schemes has resulted in an efficient mathematical approach that combines volume averaging with polynomial representations to yield highly-efficient schemes for models governed by large systems of differential algebraic equations that describe battery models. The constraints on the control variable (charge/discharge protocol), and the state variables significantly add to the complexity of the optimization routine. Though first principles-based models have been discussed in detail in the literature; as of today, literature on dynamic optimization of physics-based lithium-ion batteries models is non-existent (Linearized models were analyzed in the frequency domain by K. Smith,³ which cannot be used in time-domain or for varying parameters).

In a previous paper⁴, simple nonlinear diffusion equation for a planar electrode was modeled and the current profile was optimized with time to improve utilization at the end of the discharge. While these results are interesting, it is not applicable for a practical situation because of the physics ignored in the model.

In our talk, we will present results on dynamic optimization of lithium-ion batteries using reformulated physics based models.² The full-order model involves more than 1000 state variables (after discretization) and is not ideal for dynamic optimization. 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 cycles.² Further, the model can be used to quantify the effects of model uncertainties and variations in the design parameters on the battery performance. The nonlinear nature of the models motivates the use of the Markov Chain Monte Carlo approach to quantify the uncertainties in the parameter estimates. The reformulated model facilitates such a rigorous approach. The process of optimization and analysis is summarized in figure 1. As a first step, we will present results for optimal current profile to improve utilization in each electrode with different constraints on applied current (for example, what is an optimal profile between 0.1 C to 2C to draw maximum energy in a specified time or for a  particular cut-off voltage).

Figure 1: Process of optimization and model analysis.

Acknowledgements

The authors are thankful for the partial financial support of this work by the National Science Foundation (CBET ? 0828002), U.S. Army Communications-Electronics Research, Development and
Engineering Center (W909MY-06-C-0040), Oronzio de Nora Industrial Electrochemistry Postdoctoral Fellowship of The Electrochemical Society, and the
United States government.

References

1.        S. Kameswaran, and L.T. Biegler, Computers and Chemical Engineering, 30, 1560 (2006).

2.        V. R. Subramanian, V. Boovaragavan, V.  Ramadesigan, and  M. Arabandi, J. Electrochem. Soc., 156, A260 (2009).

3.        K. Smith, C. D. Rahn, and C. Y. Wang, J.  Dynamic Systems, Measurement & Control, 130, 11012-1 (2008).

4.        V. Boovaragavan, and V. R. Subramanian, J. Power Sources, 173, 1006 (2007).