Batteries are energy storage devices that facilitate the conversion, or transduction, of chemical energy into electrical energy, and vice versa. They consist of a pair of electrodes (anode and cathode) immersed in an electrolyte and sometimes separated by a separator. The chemical driving force across the cell is due to the difference in the chemical potentials of its two electrodes, which is determined by the difference between the standard Gibbs free energies the products of the reaction and the reactants. However, the theoretical open circuit voltage, Eo is not available during use. This is due to the various factors like Ohmic loss, work function at the solid electrolyte interface, resistance to ion transfer through the electrolyte, etc.
The output current plays a big role in determining the losses inside a battery and is an important parameter to consider when analyzing battery performance. The term most often used to indicate the rate at which a battery is discharged is the C-Rate. The discharge rate of a battery is expressed as C/r, where r is the number of hours required to completely discharge its nominal capacity. So, a 2 Ah battery discharging at a rate of C/10 or 0.2 A would last for 10 hours. The terminal voltage of a battery, as also the charge delivered, can vary appreciably with changes in the C-Rate. Furthermore, the amount of energy supplied, related to the area under the discharge curve, is also strongly C-Rate dependent.
Modeling a Li-ion battery from the first principles of the internal electrochemical reactions can be very tedious and computationally intractable. Hence, represent the various losses inside a battery, like the IR drop, activation polarization and concentration polarization, as impedances in a lumped parameter model (shown below). The IR drop due to the electrolyte resistance is denoted as RE. The activation polarization is modeled as a charge transfer resistance RCT and a dual layer capacitance CDL in parallel, while the concentration polarization effect is encapsulated as the Warburg impedance RW.
Although this lumped parameter model is generic for most batteries, further fine grained modeling requires analyzing the dynamic characteristics of the specific chemistry involved. Keeping this in mind, the following models have been developed primarily for Li-ion and Li-polymer cells.
Since the impedance parameters are essentially representations of electrochemical reactions and transport processes inside the battery, they are strongly affected by the internal temperature of the battery, the current load and the ionic concentrations of the reactants. We postulate that as discharge progresses the heat generated by the reactions and the current flow causes the internal temperature to go up, effectively increasing the mobility of the ions in the electrolyte, thus decreasing RW. Increasing temperature increases the self-discharge rate, effectively increasing the electrolyte resistance RE of the battery. Also, the increase in temperature results in faster consumption of the cell reactants causing them to be used up rapidly near the end of the discharge resulting in an increase in RCT and a sharp drop in the cell voltage. The end-of-discharge (EOD) is reached when the output voltage hits the minimum safe voltage threshold, EEOD, of the cell. For a cell current of I, the output voltage E is given by:
|E = Eo – I(RE + RCT + RW).||            (1)|
The variations in Eo with internal temperature are not explicitly modeled, but accounted for by the adaptive powers of the prognosis framework. For the empirical charge depletion model considered here, we express the output voltage in terms of the effects of the changes in the internal parameters:
|E(t) = Eo – ΔEsd(t) – ΔErd(t) – ΔEmt(t),||            (2)|
where, t is the time variable during a discharge cycle, ΔEsd is the drop due to self-discharge, ΔErd is the drop due to cell reactant depletion and ΔEmt denotes the voltage drop due to internal resistance to mass transfer (diffusion of ions). These individual effects are modeled as shown in the figure below.
In order to effectively determine the EOL, we need to understand how the different operational modes, namely charge, discharge and rest, influence the charge capacity, C. The combined effect of charge and discharge cycles is captured by the Coulombic efficiency factor ηC, defined as the fraction of the prior charge capacity that is available during the following discharge cycle. This depends upon a number of factors, especially current and depth of discharge in each cycle. The temperature at which batteries are stored and operated under also has a significant effect on ηC. The remaining factor that needs to be accounted for is the self-recharge during rest. In any battery, reaction products build up around the electrodes and slow down the reaction. By letting the battery rest, the reaction products have a chance to dissipate, thus increasing the available capacity for the next cycle. For our empirical model, we represent this self-recharge as an exponential process, as suggested by data (sample shown in the figure below).