As the price of oil rises and environmental concerns become more important, automotive companies are putting greater effort into electric and hybrid-electric vehicles. There is an increased focus on developing high efficiency, cost-effective electric vehicles whose performance will be competitive with gas-powered cars. The automotive industry is turning more and more to virtual prototyping for vehicle development as a way to significantly reduce development times and costs, and so it is essential that they have computationally efficient, high-fidelity battery models as part of their electric vehicle development.
Most current physics-based battery models are derived using porous electrode and concentrated solution theories which mathematically describe the electrical phenomena across a simplified 1-D spatial cell structure. For example, in a discharge cycle of a lithium-ion cell (as seen in Figure 1), the diffusion of Li+ ions leads to electrochemical reactions which results in ion flow in the electrolytic solution. The 1-D spatial cell structure can be modeled using a full order distributed battery model. In the full order model, electrochemical, diffusion, and transport processes are described using several dependent partial differential equations (PDEs). The solution to the full order model typically requires hours to calculate numerically in a computational fluid dynamics framework. The long simulation time results in a model that is not suitable for automotive applications, so a different approach is required.
Figure 1 - Anatomy of a Li-ion cell
Using MapleSim, batteries can be modeled and simulated with high accuracy and fast simulation times. These models provide critical information on battery performance and meet the requirements for real-time simulation of automotive vehicle batteries.
The symbolic capability of MapleSim and Maple allows for the development of a model of Li-ion batteries using Galerkin’s approximation approach. The first step of the Galerkin method is to choose a set of orthogonal basis functions that satisfy all the boundary conditions of the PDEs describing the Li+ concentrations and electrical potentials across the spatial dimension ‘x’
as shown in Figure 1
. Usually, a periodic or polynomial function can satisfy this requirement. An approximate solution can then be defined as a finite sum of basis functions, which can be substituted into the original PDEs to obtain an error function known as the residual
. In the Galerkin method, these residuals are orthogonal to the set of the basis functions; that is, the product of the residuals and the basis functions will equal zero when integrated over all space. This approach results in a set of differential algebraic equations that can be easily solved using a numerical solver.
All these steps are easily implemented in Maple through its strong symbolic computation capability and comprehensive library of algebraic algorithms. As an example, the steps showing the discretization of the Li-ion concentration in the liquid phase can be seen in Figure 2
Figure 2 - Discretization of Li-ion concentration in the liquid phase using Galerkin’s method
Figure 3 - Figure 3: Li-ion cells connected in series in MapleSim
To simulate these batteries, differential algebraic equations obtained using Galerkin’s method can easily be incorporated into a MapleSim custom component. Figure 3
shows a resulting battery pack consisting of four Li-ion cells connected in series being discharged by a pulse current. The physical parameters and state of charge of each cell in the battery pack can be adjusted for the specific needs of the project. The voltage curve for the entire pack and state of charge for each cell extracted from the simulation results of this model are shown in Figure 4
Figure 4 - Battery voltage and state of charge
The battery model can then be included as a component in a larger system model. Figure 5
shows a power-split hybrid-electric vehicle model in MapleSim. This complex system includes a 70-cell lithium-ion battery pack, a mean-value internal combustion engine, electrical motors and generators, a power-controller, a power-split device, and a 14 degree-of-freedom chassis with a differential gear box. The MapleSim model simulates very rapidly, thanks to the highly efficient battery model and high performance, high-fidelity simulations that the the symbolic technology behind MapleSim provides.
Figure 5 - Power-split hybrid-electric vehicle model
With MapleSim, high-fidelity battery models can be simulated at speeds that allow real-time simulation for hardware-in-the-loop testing. The flexibility of component-based modeling within MapleSim combined with the symbolic capabilities of Maple allows the development of highly accurate and customizable battery models. These new battery models can then be incorporated into larger models of electric vehicle systems, helping automobile manufactures develop higher performance electric cars while reducing development costs.
Contact Maplesoft to learn how MapleSim can be used in your projects.