The first attempt to improve battery performance through model-based design optimization was made by W. Tiedemann and J. Newman in 1975¹. They varied the electrode thickness and porosity and then observed the effects on the effective capacity. Newman later applied the reaction-zone model to maximize the specific energy of the system using the same varying and observing method². V. Ramadesigan et al.³developed optimal spatially varying porosity for porous electrodes by considering ohmic resistance model in the electrode. They minimized the ohmic resistance in the positive electrode by varying porosity using control vector parameterization (CVP) method. They also introduced the idea of using graded electrode to further improve the performace.

Since the introduction of the pseudo-2D (P2D) model by the Newman group, many efforts have been made in utilizing the P2D model in battery design. W. Du et al. applied an optimization framework based on surrogate method⁴. S. Golmon et al. performed the multi-objective and multi-design-parameter optimization problem with adjoint sensitivity analysis⁵ incorporating mechanical stress-strain relationship. S. De and his coworkers⁶ used a reformulated model, to perform simultaneous optimization of multiple design parameters. N. Xue et al. ⁷ applied the gradient-based algorithm framework to optimize the cell design. More recently, Y. Dai and V. Srinivasan⁸used a gradient-free direct search method to maximize the specific energy. Z. Du et al. studies the effects of various design parameters on thick electrode performance⁹.

Most of the aforementioned optimization research, used the sequential optimization approach, also known as CVP. CVP takes the differential algebraic equation (DAE) system describing the battery physics, applies a certain nonlinear programming (NLP) solver to them, discretizes only the control variables, and solves the model. The only exception being the effort by Golmon et al which used an indirect approach. Note that as of today, global optimization cannot be guaranteed when CVP type approaches are used with P2D type models.

In general optimization approaches can be classified as indirect and direct approach. In the direct approach there are two main categories – sequential and simultaneous approach.

The simultaneous approach¹⁰, on the other hand, discretizes both the control variables and the design variables before solving the problem. When used for optimization, the DAE system will only be solved once at the optimal point, compared to the repeated numerical integration needed for sequential optimization. By using high order discretization scheme on both the state and the design variables, simultaneous optimization approach results in faster determination of the optimum with fewer time steps. Furthermore, this approach offers more flexibility over constraints on the state variables. Since the state variables are also discretized, it is possible to apply equality/inequality constraints on their internal values.

In this work, we will first apply the simultaneous optimization approach to a resistance model similar to the one mentioned earlier³, and compare its performance with the sequential approach. We will also explore the application of the same optimization approach to the P2D model, and identify the challenges. It is viewed as a controversial topic in the literature wherein different authors disagree on the relevance of graded porosity for improved performance⁸. By using a simultaneous optimization approach and guaranteeing global optimization for convex problem statements, we hope to provide additional insights.

Acknowledgements

The authors are thankful for the financial support by the Clean Energy Institute at the University of Washington and the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium).

References

1. W. Tiedemann and J. Newman, J. Electrochem. Soc., 122, 1482–1485 (1975)

2. J. Newman, J. Electrochem. Soc., 142, 97 (1995).

3. V. Ramadesigan, R. N. Methekar, F. Latinwo, R. D. Braatz, and V. R. Subramanian, J. Electrochem. Soc., 157, A1328 (2010).

4. W. Du, A. Gupta, X. Zhang, A. M. Sastry, and W. Shyy, Int. J. Heat Mass Transf.,53, 3552–3561 (2010)

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