New conservative numerical schemes in mass conservation in solid particles of Lithium-ion battery for efficient simulation

Lithium-ion batteries have been used prevalently as an energy storage system in electric and hybrid electric vehicles (HEVs) due to no memory effect, low self-discharge and high energy/power density. To design a more reliable battery with higher power/energy density, in depth understanding of phenomena occurred inside the battery is required. By using modeling and simulation, these can be handled in a low cost with less restriction against experimental methods. But simulation time is one of the most barriers to use it in applications which require huge amount of data and in battery management systems. Common methods used to solve mass transfer problem in Li-ion battery particles often require significant computational effort which makes the P2D model [1] too slow for optimization, uncertainty quantification and control purposes. Various methods have been employed to overcome these difficulties such as approximate methods including diffusion length method [2], Duhamel’s superposition integral [3], polynomial approximation [4], PSS method [5], finite element method [6], finite difference method [7], and finite volume method. Some methods lose their validity under some situation such as non-constant diffusion coefficient and high rate charge/discharge. These methods show less accuracy respect to full order model and sometimes they provide nonphysical results. Since the value of Li concentration at the surface of particle determines the electrochemical reaction rate, accuracy of simulation results highly depend on its accurate value. Moreover, at the surface of particle, concentration gradient is high, so to decrease computational nodes in whole domain, non-uniform mesh spacing scheme is more convenient. To this end, the vertex based finite volume method is chosen to directly involve concentration gradient at the particle surface with variable mesh spacing. With this motivation, a new numerical discretization method is derived for spherical diffusion equation. First original PDE is integrated along spherical radius to obtain a coupled system of 51 ODEs, then the integrand is interpolated by second degree Lagrange polynomial which provides 3rd-order approximation of integral. Despite of high order accuracy of the method, it is remained tri-diagonal. For solving system of ordinary differential equations the Crank-Nicolson method is used which is A-stable finite-difference scheme with the second order accuracy. This new numerical scheme is proposed to solve P2D mathematical model of a Li-ion battery. As shown in following figure, the results indicate good agreement as compared to Khaleghi et al. study [8] with significant time reduction.