The equivalent circuit model (ECM) is a common lumped-element model for Lithium-ion battery cells. The ECM simulates the terminal voltage dynamics of a Li-ion cell through an equivalent electrical network composed of passive elements, such as resistors and capacitors, and a voltage generator. The ECM is semi-mechanical and semi-empirical model. It is widely employed in several application fields, including computerized simulation, because of its simplicity, reliability, its low computational demand, its ease of characterization and implementation, and its structural flexibility. These features make the ECM suitable for real-time Battery Management System (BMS) tasks like State of Charge (SoC) estimation, State of Health (SoH) monitoring and battery thermal management.
The equivalent-circuit model is used to simulate the voltage at the cell terminals when an electric current is applied to discharge or recharge it. The most common circuital representation consists of three elements in series: a variable voltage source, representing the open-circuit voltage (OCV) of the cell, a resistor representing ohmic internal resistance of the cell and a set of resistor-capacitor (RC) parallels accounting for the dynamic voltage drops.
The open-circuit voltage of a Li-ion cell (or battery) is its terminal voltage in equilibrium conditions, i.e. measured when no load current is applied and after a long rest period. The open-circuit voltage is a decreasing nonlinear function of the state of charge, and its shape depends on the chemical composition of the anode (usually made of graphite) and cathode (LFP, NMC, NCA, LCO...) of the cell. The open-circuit voltage, represented in the circuit by a state of charge-driven voltage generator, is the major voltage contribution and is the most informative indicator of cell's state of charge.
The internal resistance, represented in the circuit by a simple resistor, is used to simulate the instantaneous voltage drops due to ohmic effects such as electrodes resistivity, electrolyte conductivity and contact resistance (e.g. solid-electrolyte interface (SEI) and collectors contact resistance).
Internal resistance is strongly influenced by several factors, such as:
One or more RC parallels are often added to the model to improve its accuracy in simulating dynamic voltage transients. The number of RC parallels is an arbitrary modeling choice: in general, a large number of RC parallels improves the accuracy of the model but complicates the identification process and increases the computational load, while a small number will result in a computationally light and easy-to-characterize model but less accurate in predicting cell voltage during transients. Commonly, one or two RC parallels are considered the optimal choices.
Rint model is the earliest proposed equivalent circuit model for Li-ion cells for the SoC estimation and the SoH monitoring of Li-ion cells. This model uses only one resistance that represents the internal Ohmic effect. The model is simple, however it can not represent the polarization impact of Lithium-ion cells. Thus, it has large error and it is less used in engineering applications.
Thevenin model adds a resistance-capacitance circuit to the series of Rint Model so that it can characterise the polarisation effect of the Lithium-Ion battery.
The ECM can be described by a state-space representation that has current () as input and voltage at the cell terminals () as output. Consider a generic ECM model with a number of RC parallels . The states of the model, (i.e., the variables that evolve over time via differential equations), are the state of charge () and the voltage drops across the RC parallels (). The state of charge is usually computed integrating the current drained/supplied by/to the battery through the formula known as Coulomb Counting:
where is the cell nominal capacity (expressed in ampere-hours). The voltage across each RC parallel is simulated as:
where and are, respectively, the polarization resistance and capacity. Finally, knowing the open-circuit voltage-state of charge relationship and the internal resistance , the cell terminal voltage can be computed as:
Experimental identification of the ECM involves the estimation of unknown parameters, especially the capacitance , the open-circuit voltage curve , and the passive components and ,. Commonly, identification is addressed in sequential steps.
Cell capacity is usually measured by fully discharging the cell at constant current. The capacity test is commonly carried out by discharging the cell completely (from upper voltage limit to lower voltage limit ) at the rated current of 0.5C/1C (that is, the current required, according to the manufacturer, to fully discharge it in two/one hours) and after a full charge (usually conducted via CC-CV charging strategy). Capacity can be computed as: .
There are two main experimental techniques for characterizing the open-circuit voltage:
The parameters that characterize the dynamic response, namely the ohmic resistance and the parameters of RC parallels ,, are usually identified experimentally in two different ways:
Some of the possible uses of ECM include: