Analysis of a mathematical model of a battery

Lithium-ion cells are the most diffused technology for energy storage. Their application spans from portable devices to satellites, thanks to their high energy density and specific energy. The high diffusion of electric vehicles goes hand in hand with the need for high range, fast charge and long life for the battery pack. This concerns both usability and sustainability. While discharging operation cannot be controlled, because depending on the amount of torque requested by the driver, the charging phase can be addressed to minimize charging time or aging, or even selecting the best trade-off of the two. For a safe, reliable and efficient operation of the battery pack, the battery management system (BMS) monitors and controls the operation of each cell. To predict the behavior of a cell and apply the best control strategy, a model that describes the electrical operation is necessary. Two types of models are presented. One is based on the dynamics of lithium atoms across the two electrodes, thus it is an accurate description of the real cell. The high conceptual and computational complexity makes the physics based model (PBM) intractable in this work because of the poor availability of data. The equivalent circuit model (ECM) is instead built from the analysis of variables at the terminals of the cell, namely current and voltage, and related parameters such as resistances and capacitors. The model can be improved by integrating a thermal model and an aging model. The heat developed by the current flowing through the cell is responsible for the temperature rise. The temperature variation affects the parameters that describe the cell, thus the state of the cell is continuously updated. Moreover, extreme temperatures accelerate undesired reactions, making it necessary to provide the battery pack with a cooling system. High current intensity for fast charging is responsible also for the aging of the cell. It leads to lithium loss, hence to a reduction in the capacity. Although most present electric vehicles are provided with pouch cells, the three coupled models are built considering an A123 cell. The choice comes from the lack of data about pouch cells, while analyses on the selected cylindrical cell are available in some papers. Nevertheless, the model can be applied to any cell for which electrical, thermal and cycling data are accessible.