When modeling physical phenomena, the representative system of equations often includes a combination of ordinary differential equations (ODEs), partial differential equations (PDEs), and algebraic equations (AEs). These phenomena often use PDEs as governing equations of the system that vary in both space and time. When discretizing these PDEs spatially, the system becomes a set of ODEs and AEs. The resulting set of equations is known as a system of differential algebraic equations (DAEs). In physical systems, ODEs will typically represent most of the governing equations, while AEs act as constraints applied to the system ensuring that the solution accurately reflects the physical possibilities (e.g. conservation laws, boundary conditions, etc.). Systems of DAEs have been used to model real-world environments across a range of problems including large industrial processes, predator-prey eco-systems, electric power systems, and electrochemical environments. Computing solutions for systems of DAEs is important across a variety of fields. These systems combine algebraic states that act as constraints (nonlinear) on a set of ODEs, making the system more difficult to solve.

… In this paper, we show how explicit ODE solvers (such as Maple’s rkf45, a fourth-fifth order Runge–Kutta solver) can be used to solve nonlinear DAEs starting from inconsistent algebraic states.

—Lawder et al.

Dr. Subramanian points out that a major difficulty solvers have with battery models is the unknown values for certain battery states such as state of charge (SOC), state of health (SOH), or individual electrode potential at the start of a simulation. When trying to solve a system of differential algebraic equations (DAEs), which are present in most electrochemically based battery models, normal solvers will fail if the initial conditions for all the state variables are inconsistent. This causes issues in Battery Management Systems (BMS).

The MAPLE approach corrects all of the algebraic variables to a set of consistent conditions. Additionally, once the consistent values are found, the proposed approach uses the same solver for the simulation of the model. This means there is only one solver call for both initialization and simulation.

The MAPLE team applied this approach successfully to a nickel electrode model, single particle model, porous electrode model for lithium-ion batteries and many other DAE systems.

Dr. Subramanian claims that the proposed approach is the most robust approach for solving index -1 DAEs as of today.

If there is a better approach for robustness and efficiency, we would like to know of the same. In addition, if the proposed approach fails for any index-1 DAE or battery model, we would like to be alerted about the same.

—Dr. Venkat Subramanian

The United States Department of Energy (DOE) supported the MAPLE work though the Advanced Research Projects Agency – Energy (ARPA-E), award #DE-AR0000275.

Resources

• Matthew T. Lawder, Venkatasailanathan Ramadesigan, Bharatkumar Suthar, Venkat R. Subramanian (2015) “Extending explicit and linearly implicit ODE solvers for index-1 DAEs,” Computers & Chemical Engineering, Volume 82, Pages 283-292 doi: 10.1016/j.compchemeng.2015.07.002.