**Charging behavior of an Electric Vehicles battery**

To study the EV charging behavior or design the battery management system, we need to understand how the batteries of EVs are charged and discharged. There are four kinds of batteries that are commonly used in electric vehicles,

Lead-Acid, Li-Ion, NiMH and Ni Cd. The typical discharging behavior curve of a battery is presented in Figure 2.4. The terminal voltage of the battery versus the state of charge (SOC) shows nonlinear behavior. When the battery capacity is almost full, the voltage drops quickly to a steady level as the battery discharges. Then the battery enters a linear range. When the capacity is as low as 20%, the battery enters the other nonlinear zone, in which the terminal voltage drops exponentially.

The modeling of battery has been studied extensively. The most important

articles in the field date back to 1960s. Shepherd first proposed an equation description of battery discharge and the method to fit the parameters. If all factors except polarization is ignored, then the battery voltage is defined as

where is the battery voltage, the battery constant voltage, K the polarization resistance, Q the battery capacity, i is the battery current, and the actual battery charge (state of charging). To include the internal resistance, the voltage equation becomes

where R is the internal resistance. The above formula can describe the linear zone well. However, the initial drop at the beginning of a battery discharge is not included. Thus an extra exponential term A exp(−B × it) is added to correct for the difference, where A and B are empirical constants. The final equation is as follows.

Tremblay & Dessaint extend Shepherd’s model and propose detailed models for four different kinds of batteries taking into account the open circuit voltage as a function of SOC. The models are summarized as follows.

• Lead-Acid

where is the battery voltage, the battery constant voltage, K the polarization resistance, Q the battery capacity, actual battery charge (state of charging), i battery current, filtered current, and Exp(t) satisfies

where u(t) is the charge and discharge mode. u(t) = 1 is the charge mode and u(t) = 0 is the discharge mode.

• Li-Ion

• NiMH and NiCd

These battery models describe the charging/discharging curves. Using these detailed models of battery charging, one can carefully investigate the charging behavior of an individual EV and the impact on the grid. However, the detailed model of batteries does not help when considering the large scale of charging.