This paper presents an effective first step in the mathematical reformulation of physics-based lithium-ion battery models to improve computational efficiency. While the additional steps listed elsewhere ͓Electrochem. Solid-State Lett., 10, A225 ͑2007͔͒ can be carried out to expedite the computation, the method described here is an effective first step toward efficient reformulation of lithium-ion battery models to expedite computation. The battery model used for the simulation is derived from the first principles as an isothermal pseudo-two-dimensional model with volume-averaged equations for the solid phase and with incorporation of concentrated solution theory, porous electrode theory, and with due consideration to the variations in electronic/ionic conductivities and diffusivities. The nature of the model and the structure of the governing equations are exploited to facilitate model reformulation, yielding efficient and accurate numerical computations.Mathematical modeling of lithium-ion batteries involves the specification of the dependent variables of interest ͑e.g., solutionphase concentration͒ and the first principles based derivation of governing equations for these dependent variables ͑based on the physics of the battery system͒ with specification of boundary/initial conditions and nonlinear expressions for transport/kinetic parameters. Doyle et al. 1 developed a model for a lithium-ion sandwich that consists of a porous electrode, separator, and a current collector. This model is based on the concentrated solution theory. 2 This important effort paved the way for a number of similar models, because it is general enough to incorporate further developments in a battery system. 3-13 Reviews of models for lithium-ion batteries can be found elsewhere in the literature. [10][11][12] Table I depicts a pseudotwo-dimensional isothermal model for a lithium-ion battery which has been converted to a one-dimensional ͑1D͒ model using approximations for solid-state diffusion. 14-16 Table II presents the various expressions used in the model. The parameters used for the simulation are given in Table III. For analysis and control of lithium-ion batteries in hybrid environments ͑with a fuel cell, capacitor, or electrical components͒, there is a need to simulate state of charge, state of health, and other parameters of lithium-ion batteries in milliseconds. Rigorous physics-based models take a few seconds up to a few minutes to simulate discharge curves, depending on the solvers, routines, computers, etc. Circuit-based or empirical models ͑based on the past data͒ can be simulated in milliseconds. However, these models fail at various operating conditions, and use of these models might cause abuse or under-utilization of electrochemical power sources. This paper presents the mathematical analysis for reformulation of physics-based models. Lithium-Ion Battery Model ComplexitiesSimulation of lithium-ion battery models requires simultaneous evaluation of concentration and potential fields, in both solid as well as liquid phas...

Transport and kinetic parameters of lithium-ion batteries are estimated using a first-principles electrochemical engineering model based on porous electrode theory (1, 2). A full-order model reformulated using advanced mathematical techniques (3, 4) was used for the simulations. Since batteries and other power sources are used in hybrid environments, with devices with time constants less than a second (like a super capacitor or an induction motor), parameter estimation algorithms were developed with high computational efficiency. As a complement to approaches to mathematically model capacity fade that require detailed understanding of each mechanism (5), capacity fade was accurately and efficiently predicted for future cycles by extrapolating the change in effective transport and kinetic parameters with cycle number (N), for a battery under controlled experimental conditions. Parameter estimation using mathematical reformulation (4) was more efficient and robust than full-order models based on the traditional finite difference approach.

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