The need for modelling
The primary reason why one is interested in having a model of a battery is it allows us to make informed decisions, optimise or control a battery to deliver on its intended applications. For instance, we may be interested in knowing how different material choices, electrode compositions or cell packaging, will influence the electrochemical and thermal performance of the cell, a model can serve this purpose. Similarly, there are occasions when we want to know if and when a certain event might occur, e.g., when will the cell exceed a particular temperature limit, how much power can the cell deliver or admit, for a given time, before it reaches the cut-off voltage? Or, we might be interested in knowing what the internal states or parameters of the battery are when we only have access to voltage, current and the surface temperature of the cell.
The challenges of modelling
Developing effective battery models that can answer various application needs are however far from trivial. This is because batteries exhibit phenomena that span over multiple length (particles, electrodes, cell) and time scales (milli-seconds, to months) and changes over the usage operating conditions such as, ambient temperature, state-of-charge and state-of-health.
Of all the phenomena a battery can exhibit, the terminal voltage is a key response that is modelled frequently and discussed here in a bit more detail. There are three general contributing factors towards the terminal voltage of a battery which are the thermodynamics, the mass and charge transport effects, and the electrochemical kinetics. The transport effects and kinetics are losses that the battery has to overcome and the thermodynamic quantity is the open-circuit voltage (OCV) of the cell (more precisely that of the individual electrodes).
Of these contributions the OCV dominates the cell’s terminal voltage (lithium-ion batteries are high efficient devices). The table below shows that more than 90% of a typical terminal voltage can be described by the OCV when operating between 20% to 100% state-of-charge (SoC), while the losses increase at the lower (<20%) SoCs and the OCV contributes to around 75% of the terminal voltage.
|Contribution of an
towards the terminal
voltage at different
|100% to 20% SoC
This is an important feature to note, since the OCV is an experimentally determined quantity and is often not modelled, it is available as an OCV vs SoC look-up-table. The modelling effort therefore firmly sits in accounting for the 10% to 25% of losses that contribute to the terminal voltage. Various modelling approaches are available to account for these losses, of which one is briefly mentioned in the following section.
An equivalent circuit approach
By conducting experiments to measure the battery voltage at various SoCs and temperatures it is possible to develop phenomenological models that relate the applied current and the voltage. An equivalent circuit model (ECM) is one such phenomenological model most widely used in industry to simulate the voltage response for subsequent Battery Management System control and state estimation.
The figure above is an example of a first-order ECM that accounts for the OCV (thermodynamics) and losses modelled with electrical resistor and capacitor elements. The values of these circuit elements however depend on the temperature (T) and SoC of operation (z), enabling the model to be used over the operating conditions of the application. Values of these functions are determined by conducting specific experiments, such a pulse-power or multisine test , and running parameter estimation routines to determine the corresponding circuit element value. The figure below is an example, demonstrating how the Ohmic resistance (R0), decreases with increasing ambient temperature and increases at low SoCs. The SoC and temperature dependence of the other circuit elements are determined in similar manner, providing the complete ECM.
Other approaches and conclusions
It is possible to develop similar phenomenological models to simulate the cell thermal response. Known as thermal circuit models they are coupled with ECMs (described above) where the losses act as a heat source term and the simulated temperature simultaneously feeds back into the ECM to facilitate the parameter temperature dependence.
While phenomenological models form one approach, there is a wide spectrum of modelling methodologies, driven by the model requirements, that is used nowadays. Purely data-driven methods are widely employed to predict a battery’s state-of-health and remaining useful life prediction. Such methods however rely on access to high quality data  and are restricted in their generalisability.
To account for the physical geometry and chemical composition of a battery, a physical model (often termed as a physics-based model) is necessary. The Doyle Fuller Newman (DFN) model , in this regards, is the most well established and widely used model. It models the dynamics occurring at each electrode and electrolyte and allows ageing to be coupled systematically. However a large (>30) number of parameters are required for the model and determining them for any commercial Li-ion cell is challenging .
Models have to be simplified or restricted, when dealing with voltage, current and temperature data retrieved from batteries once deployed into their respective applications (field data). This restriction is driven by the accuracy and frequency of the measurements made. Hybrid data-driven and physics based models can offer potential benefits in this regards .
The field of battery cell modelling is a rich and active domain with many open-ended challenges (some of which will be further discussed in this platform). This can be noted from the number modelling software tools available as one model to fit all is not possible.
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