Lithium-Ion Battery Base Model in 1D

This is a template base model containing the physics, geometry, and mesh of a lithium-ion battery.

The model is defined using the Lithium-Ion Battery interface, based on the Doyle–Fuller–Newman framework (Ref. 1). For a general introduction to the Lithium-Ion Battery interface, the user is recommended to first run the tutorial 1D Isothermal Lithium-Ion Battery, which is set up in a similar way.

This base model introduces a number of lithiation parameters which are used to define the relative balancing of the negative and positive electrodes, defining the degrees of lithiations in the individual electrodes at 0% and 100 % cell state of charge (SOC). This allows for a global SOC variable to be defined in the model, and for defining the initial lithiation levels of the electrodes based on an initial SOC parameter value. In addition, a cell cross-sectional capacity parameter is used to define the electrode thicknesses.

The following tutorials, available in the Battery Design application library, make use of this base model:

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1D Lithium-Ion Battery Drive-Cycle Monitoring, which demonstrates how to run the model using a time-dependent current load curve

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Lithium-Ion Battery Rate Capability, which uses the base model to perform a range of discharge simulations at different rates. By modifying the cross-sectional capacity of the cell, an energy-optimized cell is compared to a power-optimized cell in a Ragone plot.

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Lithium-Ion Battery Internal Resistance, which simulates a hybride pulse power characterization (HPPC) test and makes a full analysis of the different roots of the various voltage losses in the cell.

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Diffusion-Induced Stress in a Lithium-Ion Battery, which adds stresses and strains to the particles and in the negative electrode, and analyzes the mechanical stress that the particles are subjected to during a current load.

Model Definition

Geometry

The model is defined in 1D along the through-layer direction between the metal current collector foils of a lithium jelly roll. The geometry thus consists of one negative porous electrode, one separator and one positive porous electrode domain.

Given the 1D geometry, the capacity of the cell is defined per cross-sectional area as Ah/m2, and the current boundary condition for the physical model is expressed as a current density (A/m2). Conversion to and from total capacity and current (Ah and A) from and to areal capacity and current density (Ah/m2 and A/m2) can be done based on the layer thicknesses and an assumed active jelly roll volume. This is exemplified in the Lithium-Ion Battery Internal Resistance tutorial.

Materials

The battery model consists of the following materials:

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Negative electrode: Graphite (MCMB LixC6)

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Positive electrode: NMC 111 (Li1/3Mn1/3Co1/3O2)

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Electrolyte: 1.0 M LiPF6 in 3:7 EC:EMC

These materials are available from the Battery Material Library.

Physical Model

The Lithium-Ion Battery interface, used for defining the model, accounts for:

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Electronic conduction in the electrodes

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Ionic charge transport in the electrodes and electrolyte/separator

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Material transport in the electrolyte, allowing for the introduction of the effects of concentration on the ionic conductivity and concentration overpotentials

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Material transport within the spherical particles that form the electrodes, defined on an extra (pseudo) dimension

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Butler–Volmer electrode kinetics using experimentally measured discharge curves for the equilibrium (half cell) potentials.

Cell State of Charge Definition Based on the Electrode Degrees of Lithiation

The Lithium-Ion Battery interface defines and solves for the solid lithium concentration in the electrode particles, individually for each electrode. Based on the lithium concentrations, the degrees of lithiation, sol (dimensionless), then relates the solid concentration levels to the maximum concentrations of lithium atoms that the electrode materials can host.

(1)

The individual degrees of lithiation for each electrode then in turns determine the local equilibrium potential for intercalation (the half cell potentials). During charge of a lithium- ion battery, cs in the negative electrode will increase, whereas cs in the positive electrode will decrease, with the rate being proportional to the battery current.

At open circuit of a fully relaxed cell, the cell voltage EOCV,cell can be expressed as

(2)

where Eeq,pos and Eeq,neg (V) are the equilibrium potentials, expressed as functions of the degrees of lithiation solpos and solneg (1) of the positive and negative electrode materials, respectively, based on the average cs variables of the corresponding electrode.

Unless the degrees of lithiation in the individual electrodes are coupled in some way there exists an infinite number of combinations for values of solpos and solneg to fulfill the above equation. For an assembled battery cell however, a change in solpos will always result in a corresponding change in solneg and vice versa since the total amount of lithium in the cell conserved during cycling, resulting in a unique solution to the above equation.

In contrast to the individual electrode degrees of intercalation, which depend on the local concentrations of lithium, the state of charge (SOC) of a battery cell is typically defined on a global level as being proportional to the amount of charge passed when cycling between two corresponding open circuit voltage limits, the lower voltage limit corresponding to 0% SOC and the upper voltage limit to 100% SOC.

To define the conversion between the global SOC and the local sol variables, the model makes use of the four lithiation parameters solpos,100, solneg,100, solpos,0 and solneg,0 for the degrees of lithiation in the positive and negative electrodes at 100 and 0% SOC, respectively. Generally, as a result of the negative electrode being lithiated during charge, and the positive electrode being delithiated, solneg,100 > solneg,0 and solpos,0 > solpos,100.

The four lithiation parameters will determine the open circuit voltage at 0% and 100% SOC of the cell according to

(3)

What cell voltage, and in term what individual electrode voltages, should correspond a 0% and a 100% cell state of charge (SOC) is to a high degree an engineering decision, related mainly to durability constraints, since a too low cell voltage could result in, for instance, corrosion of the negative current collector, and a too high cell voltage could result in excessive SEI formation and/or lithium plating on the negative electrode, or degradation of and/or electrolyte decomposition (gassing) on the positive electrode. As discussed in the next section, the lithiation parameters also determine the relative balancing of the host capacities and resulting electrode thicknesses of the cell.

The state-of-charge of the battery cell may now be defined either by the lithiation level in the negative or positive electrode as

(4)

It should be noted that these definitions of SOC will only be equal and consistent as long as no parasitic (aging) side reactions occur in the cell, resulting in loss of either cyclable lithium and/or loss of electrode material available for lithium intercalation.

Conversely, we may express the degrees of lithiation based on the SOC as

(5)

The above expression is used to set the initial solid concentration levels in the electrodes in the model, based on an initial SOC parameter.

Electrode Thicknesses Based on the Cell Capacity

The parameter qcell (Ah/m2) for the cross-sectional cell capacity of the battery when cycling between 0 and 100% SOC, together with the four degree of lithiation parameters described in the previous section, are used to define the electrode thicknesses.

We first define the host capacity of each electrode as

(6)