The mathematics of battery efficiency

In many ways, the holy grail of energy efficiency right now is battery technology. There are massive efforts underway to boost battery energy density (energy stored per unit mass), extend battery life, and improve overall charge-discharge efficiencies to make batteries less expensive and more reliable. While a lot of progress has taken place in the last decade, certain industries still have a long way to go.

In particular, the electric-vehicle industry faces pressures to upgrade energy densities as a way to reduce costs. The cost-benefit arguments for consumers to move from conventional internal combustion vehicles to electric or hybrid-electric vehicles are, even now, marginal. According to several studies, the electric vehicle has not lived up to its promise economically because the cost-benefit model is extremely sensitive to two factors: the cost of energy storage and the cost of fossil fuel.

Since 9/11, the cost of crude oil has steadily risen from around $20/barrel to the neighborhood of $100/barrel 10 years later. This has driven more investment into alternatives such as electric vehicles. However, the last year has actually seen a sharp decline in oil prices (with a brief spike during the Libyan uprising). The lesson is that relying on oil prices to support the electric vehicle business model is untenable.

As faculty co-chair of Energy Technology Innovation Policy research at Harvard Kennedy School Henry Lee puts it: “The industry will need improvements in battery technology and reductions in battery costs for electric vehicles to meet their potential. Such improvements will require continued government support of battery research and development and higher gasoline prices either through government action in the form of taxes, a cap-and-trade program, or through the market.”

So, the race is on to produce better, lower-cost energy storage products (For the sake of brevity, the products will be called “batteries,” but there are other forms of energy storage such as hydrogen fuel cells and supercapacitors.) as well as improve the circuitry and electromechanical components that generate power to charge them and draw power to drive the electric vehicle.

Battery modeling: back to basics

One constant of battery research is the need to consider fundamental physical concepts when designing new batteries. To facilitate this, the industry increasingly uses math-based modeling techniques that let engineers accurately describe the behavior and the constraints on a system, in physical terms. The model equations then help in developing, testing, and refining designs quickly and without building physical prototypes. Hence, it's essential to have a good virtual model which properly reflects the battery behavior and physical interactions with all the other components.

Another important part of battery research relies on having a high-fidelity dynamic model of the rest of the vehicle. This model reflects the loading conditions that the battery sees during many different drive cycles. It also provides considerable insight into the two-way demands placed on both the battery and the rest of the vehicle. Because the battery plays such a vital role in an EV, it is essential that these interactions be captured.

In a recent case study, a research team at the University of Waterloo, Ontario, developed high-fidelity models of hybrid-electric and electric vehicles, including the batteries. Headed by the NSERC-Toyota-Maplesoft Industrial Research Chair for Mathematics-based Modeling and Design John McPhee, the group did the work with MapleSim, multi-domain physical modeling and simulation software from Maplesoft. They say the symbolic approach in MapleSim is an effective way to develop simulation models of sufficient fidelity without sacrificing real-time performance for hardware in the loop (HIL) testing, a critical part of the design and development process.

Two of the group's projects illustrate the approach: creating a battery electric vehicle (BEV) model and creating a hybrid electric vehicle (HEV) model.

Battery electric vehicle model

Lithium-ion (and other lithium-chemistry) batteries are becoming widely used for EVs because they are light and provide more power than other common types of batteries of the same size and weight. Batteries in vehicles see periods of high current draw, periods of recharging, and large temperature variations, which can significantly affect their performance and lifespan.

To capture these effects, McPhee's team needed a model of lithium-ion battery chemistry that would reflect a wide state-of-charge range, widely varying currents, and behavior at various temperatures. Starting with the electric circuit battery model of Chen and Rincón-Mora, they implemented the components in MapleSim, using a custom function component to represent the nonlinear relationship between the state of charge and the electrical components.

The group modified the battery equations to simulate a battery pack comprising series and parallel combinations of single cells. Next, they developed a power controller model to connect the battery pack to a motor. They next incorporated a one-dimensional vehicle model into the model. The simple vehicle model drives on an inclined plane. The plane is, in turn, controlled by a terrain model. A drive-cycle model was included to control the vehicle speed. The resulting differential equations, generated by MapleSim, were simplified symbolically and then simulated numerically.

The group simulated a variety of driving conditions such as hard and gentle acceleration and driving up and down hills. The results were physically consistent and clearly demonstrated the tight coupling between the battery and the movement of the vehicle. This model will form the basis for a more comprehensive vehicle model, which will include a more sophisticated power controller and more complex motor, terrain, and drive-cycle models.

Hybrid-electric vehicle model

The team used MapleSim to develop a multi-domain model of a series HEV, including an automatically generated optimized set of governing equations. The HEV model consists of a mean-value internal combustion engine (ICE), dc motors driven by a chemistry-based Ni-MH battery pack, and a multibody vehicle model. The model employs a Ni-MH battery because of its widespread use in hybrid-electric vehicles.

The group used a chemistry-based modeling approach that captures the chemical and electrochemical processes inside the battery. With this modeling approach, they could modify the physical parameters of the battery as needed to meet overall design requirements for the vehicle. They modeled the battery by placing the governing equations of the battery processes directly inside MapleSim custom components.

MapleSim generated an optimized set of governing equations for the entire HEV system, which combined mechanical, electrical, chemical, and hydraulic domains. Simulations were then used to demonstrate how the HEV system performed. Simulation results showed that the model is viable. MapleSim's lossless symbolic techniques for producing an optimal set of equations also significantly reduced the number of governing equations, resulting in a computationally efficient system. This HEV model can be used for design, control, and prediction of vehicle handling performance under different driving scenarios. The model can also aid in sensitivity analysis, model reduction, and real-time applications such as hardware-in-the-loop (HIL) simulations.

Development of a high-fidelity battery model is only part of the process. To model the loads applied to the battery, the group connects the battery model to a range of electromechanical system models. These models include the those for IC engine (in the case of a hybrid), those for power generation and motors, as well as those representing the power electronic circuitry such as ac-dc converters, inverters, and power drives for the motors. The modeling effort takes things a step further in that the drives connect to a three-demensional chassis model to obtain a full-vehicle model. This full-vehicle model can be taken through numerous test drive cycles to get a more holistic view of the loads on the battery.

The next step in these projects is to consider how the power electronics affects the battery. Currently, these effects are simulated using “mean value” models, but work has begun on incorporating the three-phase switched networks in detail to investigate things with more precision.

MapleSim already has many built-in electronic components (transistors, thyristors, diodes, C-MOS, and MOSFETs) and there is a growing collection of power electronics sub-systems that research groups like McPhee's have submitted to the Maplesoft Application Center, such as PWM controllers, IGBT inverters and power amplifiers.