A full model for simulation of electrochemical cells including complex behavior
Introduction
The modeling of electrochemical cells, secondary batteries or electrochemical energy storage devices in general has been widely studied, there being a considerable number of previously published works available [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. The large number of works carried out around this topic is partly motivated by the interest aroused by electric and hybrid vehicles. For this reason, many of these works have been developed as a part of simulation models for these vehicles [1], [2], [3], [4], [5], [6].
These works are sometimes based on empirical relationships, and other times on a detailed description of the physical and chemical processes that take place in the cell, and even on the development of equivalent circuits [6], [7], [8]. In order to develop these models various techniques have been used, such as lookup tables, lumped parameters or bond graphs.
Taking the first works published as a starting point, different extensions and modifications of increasing complexity and precision have gradually been proposed. In some cases [4], [9], [10], [11] modifications have been made to previously published works [3], [12], while in other cases, it has been the authors themselves [13] who have made suggestions for improving their own models [10].
There is also a wide variety depending on the goals. There are isothermal models, for instance, to describe electrochemical cell behavior under normal charge or discharge conditions, models that seek to get as close as possible to the limit current conditions [13], thermal models that include the effect of temperature [1], [7], [8], [14], [15], and even age on cell performance [1], models that study the recombination of gases during the charge process, particularly in lead-acid batteries (VRLA) [10], [13], [16], [17], or models specially developed to describe cell behavior during a pulse charge/discharge operation [17], [18].
A highly interesting alternative for developing these types of models is to use the bond graph technique. This technique allows for the development of models that may belong to different physical domains. This is especially useful to integrate the models developed into other more complex and multidisciplinary ones as, for example, a complete hybrid vehicle model, with all the components like engine, clutch, gear box, chassis, and power system, including the battery and the control system.
In this paper, based on two previous ones by the authors [21], [26], a model for an electrochemical cell has been developed. The model has been applied to the particular case of the Pb,PbSO4|H2SO4 (aq)|PbO2,Pb cell. This kind of cell forms the basis of lead-acid batteries that are widely used in the automotive industry as well as traction batteries in electric or hybrid vehicles.
The main goal of this work has been to obtain a model capable of reproducing cell behavior for both charge and discharge conditions. Therefore, an initial isothermal model has been constructed. In many cases, this isothermal model may be sufficient to provide an adequate representation of cell behavior. However, in those cases where high charge currents are involved, the thermal effects may be considerable. A thermal model has also been developed for this case, aimed at providing an adequate representation of these effects.
Section snippets
Equilibrium potential and open circuit voltage
Let us consider a cell such as shown in Fig. 1. It consists of a lead electrode (Pb) and another of lead dioxide (PbO2) submerged in an H2SO4 solution. When the two electrodes come together (Fig. 1a), an electric current flows from one to the other due to a potential difference between the two electrodes. The reaction giving rise to this electromotive force is:PbO2 + Pb + 2H2SO4 → 2PbSO4 + 2H2O
This reaction can be split into two parts. One that takes place in the left electrode:Pb + SO42− → PbSO4 + 2e−
and
Simulation model
The simulation model has been developed using block diagrams in Simulink, and implementing the equations and dynamic model of the battery in this program. In order to understand the composition of the model, a model with its equivalents in electrical components has been developed that represent the different mechanisms for accumulating potential energy (compliances or capacitors), kinetic (inertances or inductions) or energy dissipation (resistances or resistors). The structure of the model and
Simulation results
The model developed has been used to represent several situations in order to observe their behavior, particularly with regard to thermal effects, as well as, for contrasting the results with some of those found in literature. Annex I contains the values taken into account in the simulation as well as the references from which they were taken.
Firstly, the charge process of a six element cell coupled in series (corresponding to a standard 12 V. battery) was simulated for two different charge
Model validation
A model of these characteristics is highly dependent on a large number of parameters, not only of an electrochemical nature, but also of a constructive nature (initial electrolyte volume and density, size and internal surface area of the electrode, thermal characteristics of the container material, its surface area in contact with the air, its thickness, …). It is, therefore, complicated to validate the model from either experimental or theoretical results found in literature, as the cell or
Conclusions
This work has presented a model of an electrochemical cell including complex behavior, in order to show the different behavior presented in the charging–discharging process of an electrochemical cell.
The charging process for a standard 12 V battery for two different voltage charges was simulated using both the isothermal model and the thermal model. With the latter, it can be seen that when the charge currents are high, the thermal effects are considerable, and at the end of the charge process
Acknowledgements
This work belongs to project TRA99-0919 and DPI2002-2198 and has been developed under the support of the Spanish Administration in the programs of the Ministry of Science and Technology related to Research in Transportation and Design and Industrial Production.
References (26)
J. Power Sources
(1993)- et al.
Electrochim. Ada
(1993) J. Franklin Inst.
(1990)Simul. Pract. Theory
(1999)- et al.
J. Power Sources
(1994) - et al.
- et al.
- et al.
- et al.
Tutorials Electrochem. Eng.-Math. Model.
(1999) - et al.