COMSOL 6.3 - Single-Particle Modeling of Lithium-Ion Batteries

Single-Particle Modeling of Lithium-Ion Batteries

The single-particle approach (Ref. 1 and Ref. 2) is a way to simplify the intercalating porous electrode formulation in the traditional Doyle–Fuller–Newman (DFN) model (see the 1D Isothermal Lithium-Ion Battery model example). In the single-particle model (SPM), the local potential and concentration gradients in the solution (electrolyte) phase are ignored, and instead the electrolyte potential losses are accounted for using a lumped solution resistance term. At the same time, the potential gradients in the solid phase of the electrodes are also neglected. As for the DFN model, the single-particle formulation accounts for solid diffusion in the electrode particles and the intercalation reaction kinetics. However, all particles are assumed to be identical, and since the reaction current distribution across the porous electrodes is assumed to be uniform, all particles are subjected to the same conditions, with the result that all particles may be treated as one “single” particle.

The assumptions of the single-particle formulation are typically valid for low to medium applied current densities. Additionally, the validity of the assumptions and the applicability of the model also depends on the parameters values and electrode-electrolyte chemistry used in the model. Typically the assumptions are applicable to thin, or highly conductive, electrodes.

The single-particle model formulation is typically used to reduce computational load in, for instance, capacity fade modeling, battery pack simulations, control systems, or parameter estimation applications.

As mentioned above, the traditional SPM model defines the electrolyte as a lumped resistor. However, it is also possible to keep the electrolyte transport model of the DFN model, solving for both the salt concentration and potential in the electrolyte phase, and combine it with the single-particle approach for the electrodes. This extension is often called SPMe in scientific literature.

This tutorial example demonstrates how to use the Thin Porous Electrode nodes in the Lithium-Ion Battery interface for defining SPMe and SPM models, as well as the Two Electrodes option in the Lumped Battery interface for defining a SPM model.

First a DFN model is defined in 1D, and the cell voltage response to a charge–discharge load cycle is computed. The voltage profile is then compared to the different single-particle models.

Model Definition

Doyle–Fuller–Newman base model

The DFN model is based on the Lithium-Ion Battery Base Model in 1D. The model is defined using the interface, using two and one domain node. (Compared with the Lithium-Ion Battery Base Model in 1D, the electrode thicknesses are increased somewhat in order to see larger differences between the DFN and single-particle models.)

Single-Particle model with Electrolyte salt Transport

The SPMe makes use of a single domain node in the interface, and uses the same charge balance model as the DFN model. The two single-particle electrodes are defined using the boundary nodes.

The length of the single domain Lel (m) in the SPMe model is defined as

(1)

where Lsep (m), Lneg (m), and Lpos (m) are the lengths of the separator, negative electrode and positive electrode, respectively.

Apart from the size of the single domain, the SPMe is parameterized in the same way as the DFN model.

Single-particle model with constant electrolyte resistance

The SPM defined in the tutorial uses the same setup as the SPMe, but with the charge balance model set to using a user-defined effective electrolyte conductivity. This excludes solving for the electrolyte salt concentration in the single separator domain.

The lumped solution resistance R (Ω) of the SPM is defined as

(2)

where EIR,1C (V) is the electrolyte voltage drop when operating the cell at a 1C current, and I1C (A) is the 1C current.

The effective conductivity σeff (S/m) used in the SPM model is then defined as

(3)

where Acell (m2) is the cross-sectional area of the cell.

In the tutorial, the computational solution of the SPMe is used to estimate EIR,1C. The resulting effective conductivity of the SPM is slightly lower than for the SPMe due to the impact of concentration polarization effects in the electrolyte.

Due to the simplified equation formulation, a linear discretization may be used in combination with one mesh element only.

Single-particle model using the Lumped battery Interface

A second SPM (Lumped) model is also defined in the tutorial, using the model option in the interface.

The SPM (Lumped) model is mathematically equivalent to the SPM. There is a difference however with regards to what parameters are used to define the two models. Whereas the Lithium-Ion Battery interface makes use of locally defined parameters in the geometry, the Lumped Battery interface make use of lumped zero-dimensional parameters.

The host capacity Qhost (C) of each electrode in the SPM (Lumped) model is defined as

(4)

where is ε (1) the electrode phase volume fraction, cs,max (mol/m3) the maximum concentration of intercalated lithium and L the electrode thickness.

For the solid intercalation, the diffusion time constants τ (s) of the two electrodes are defined as

(5)

where rp (m) is the electrode particle radius and D (m2/s) the diffusivity.

An advantage of using the Lumped Battery interface is that the model may be defined in 0D, so that no geometry needs to be defined.

Another difference between the SPM and SPM (Lumped) models is the handling of the Butler–Volmer kinetics-based activation overpotentials. In the SPM (Lumped) model, the activation overpotential is defined using an analytical function of the form

(6)

where η is the activation overpotential (V) and I0 is the exchange current (A). This allows for decoupling of the electrode and electrolyte phase potentials in the equation system and removes the need to solve for the potentials as dependent variables in the model. This lowers the computational complexity of the model. The above equation for the activation overpotential is however only valid for the case of symmetric transfer coefficients and one single electrode reaction.

Load Cycle

A load cycle, starting from 75% SOC, discharging at 1C for 2000 s, resting for 300 s, and then recharging at 1C for 2000 s, is defined using the interface.

Components and Studies

The DFN, SPMe, SPM, and SPM(Lumped) models are defined in separate components, using separate geometries and meshes.

The Events interface defining the load cycle, is defined in Component 1, but is used by all models.

Each model is solved for in an individual time-dependent study, simulating the same charge-discharge cycle for 5000 s.

Results and Discussion

Figure 2 shows the cell voltage vs time for all four models. The discrepancies between the models are generally relatively small, with the largest deviations seen between the DFN and the other models. These differences are due changes in the current distribution within the porous electrodes of the DFN models, something the SPMe and SPM models are unable to capture.

Figure 1: Cell voltage comparison for all four models.

Figure 2 shows a closeup of the cell voltages at the initiation of the charge pulse. The SPMe differs from the SPM models only during the first 20 s after initiation of the charge, during which a steady-state concentration profile in the electrolyte establishes.

Figure 2: Cell voltage comparison for all four models (closeup).

Table 1 shows the number of degrees of freedoms (DOFs) and solution time as reported in the solver log when solving the four different models (the solution time may vary depending on computer). The lower amount of DOFs and shorter solution time for SPM (Lumped) compared to SPM is related to the treatment of the nonlinear activation overpotential, which allows for decoupling of the electrolyte and electrode potentials.

Table 1: Degrees of freedom and solution times for the different models.
Model	DOFs	Solver Time(s)
DFN	675	5
SPMe	170	4
SPM	26	3
SPM (Lumped)	23	2

References

1. S. Santhanagopalan, Q. Guo, P. Ramadass, and R.E. White, “Review of Models for Predicting the Cycling Performance of Lithium Ion Batteries,” J. Power Sources, vol. 156, no. 2, pp. 620–628, 2006.

2. M. Guo, G. Sikha, and R.E. White, “Single Particle Model for a Lithium Ion Cell: Thermal Behavior,” J. Electrochem. Soc., vol. 158, no. 2, pp. A122–A132, 2011.

Battery_Design_Module/Lithium-Ion_Batteries,_Performance/lib_single_particle

Modeling Instructions

Application Libraries

From the menu, choose .

In the window, select > > in the tree.

Click

First define an -based load cycle for the Doyle-Fuller-Newman (DFN) model. We will compare the probe cell voltage output for the DFN model to the various single-particle models defined later.

Component 1 - DFN

In the window, click .

In the window for , type Component 1 - DFN in the text field.

Add Physics

In the toolbar, click

and choose .

Go to the window.

In the tree, select > > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Events (ev)

Explicit Event List 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the u text field, type C_rate.

In the text field, type C rate.

In the u0 text field, type -1.

Locate the section. Click

Click

In the table, enter the following settings:


Time (s)	Reinitialization expression
2000[s]	0
2300[s]	1
3300[s]	0

Lithium-Ion Battery (liion)

Electrode Current Density 1

In the window, expand the > node, then click .

In the window for , locate the section.

In the in,s text field, type I_1C*C_rate.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
L_pos	60[um]	6E-5 m	Positive electrode thickness
soc_init	0.75	0.75	Initial SOC

Definitions (comp1)

Point Probe - DFN

In the window, expand the > node, then click .

In the window for , type Point Probe - DFN in the text field.

Locate the section. In the text field, type Cell Voltage - DFN.

Study 1

Step 2: Time Dependent

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the text field, type 0 5000.

In the window, click .

In the window for , type Study 1 - DFN in the text field.

Locate the section. Clear the checkbox.

In the toolbar, click

Results

Probe Plot Group 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Component 1 - DFN (comp1)

In the window, collapse the node.

The next step is to define a model using single particles for the electrodes, but with the same electrolyte transport model as in the DFN model. The new model is referred to as SPMe.

Global Definitions

DFN Parameters

In the window, under click .

In the window for , type DFN Parameters in the text field.

Lumped Parameters

In the toolbar, click

and choose > .

Load some additional lumped parameters from a text file. These parameters will be used to define the simplified single-particle models.

In the window for , type Lumped Parameters in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file lib_single_particle_parameters.txt.

Root

Define the SMPe model in a separate component, using a different geometry and mesh than the DFN model.

Add Component

In the window, right-click the root node and choose > .

Component 2 - SPMe

In the window for , type Component 2 - SPMe in the text field.

Geometry 2

Interval 1 (i1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Coordinates (m)
0
L_el

The length of the domain, defined in the parameter file you imported before, is longer than the separator domain length in the DFN model.

Add Physics

In the toolbar, click

and choose .

Go to the window.

In the tree, select > > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Lithium-Ion Battery 2 (liion2)

In the window for , locate the section.

In the Ac text field, type A_cell.

Separator 1

The settings for the node are identical to the DFN model.

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the εl text field, type epsl_sep.

Thin Porous Electrode 1

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

In the dpe text field, type L_neg.

In the εs text field, type epss_neg.

Particle Intercalation 1

The settings for the and child nodes are identical to the corresponding settings in the DFN model.

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the rp text field, type rp_neg.

Porous Electrode Reaction 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the i0,ref(T) text field, type i0_ref_neg.

Thin Porous Electrode 2

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

In the dpe text field, type L_pos.

In the εs text field, type epss_pos.

Locate the section. From the list, choose .

The total current will be specified later as a multiple of the C-rate current variable liion2.I_1C_cell. This variable and the option to specify a C-rate multiple value will be available once we have enabled the node later.

Particle Intercalation 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the rp text field, type rp_pos.

Porous Electrode Reaction 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the i0,ref(T) text field, type i0_ref_pos.

In the window, click .

In the window for , locate the section.

Select the checkbox.

SOC and Initial Charge Distribution 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the SOC0 text field, type soc_init.

Negative Electrode Boundary Selection 1

In the window, click .

Select Boundary 1 only.

Positive Electrode Boundary Selection 1

In the window, click .

Select Boundary 2 only.

Thin Porous Electrode 2

In the window, under > click .

In the window for , locate the section.

From the list, choose .

In the Crate text field, type comp1.C_rate.

Definitions (comp2)

Global Variable Probe - SPMe

In the toolbar, click

and choose .

In the window for , type Global Variable Probe - SPMe in the text field.

Click in the upper-right corner of the section. From the menu, choose > > .

Locate the section.

Select the checkbox. In the associated text field, type Cell Voltage - SPMe.

Click to expand the section. Click

From the list, choose .

Also add an integration operator. This operator will be used later when computing the total electrolyte potential drop.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 1 only.

In the text field, type intop_neg.

Add Study

In the toolbar, click

and choose .

Go to the window.

Find the subsection. In the tree, select > > .

Find the subsection. In the table, clear the checkbox for .

Find the subsection. Right-click and choose .

In the window, click the root node.

In the toolbar, click

to close the window.

Study 2

Step 2: Time Dependent

In the window for , locate the section.

In the text field, type 0 5000.

Click to expand the section. From the list, choose .

In the list, select .

Under , click

In the window, click .

In the window for , type Study 2 - SPMe in the text field.

Locate the section. Clear the checkbox.

In the toolbar, click

Results

Probe Plot Group 1

The probe plot should now show a comparison between the DFN and SPMe cell voltage outputs.

In the window, under click .

In the toolbar, click

Electrolyte domain potential drop in SPMe model

The next step is to create a single particle model where we treat the electrolyte transport as a constant resistor, which will be called SPM. In order to compute the corresponding resistivity, we evaluate the total electrolyte potential drop of the SPMe model as follows:

In the toolbar, click

In the window for , type Electrolyte domain potential drop in SPMe model in the text field.

Locate the section. From the list, choose .

Point Graph 1

Right-click and choose .

Select Boundary 2 only.

In the window for , locate the section.

In the text field, type phil2-intop_neg(phil2).

Electrolyte domain potential drop in SPMe model

In the window, click .

In the toolbar, click

Note that the voltage drop at the end of the 1C current loads is about 45 mV.

Component 2 - SPMe (comp2)

Make a copy of the whole SPMe model and paste it into a new component. We will use the new component to create the SPM model.

In the window, right-click and choose .

Component 2 - SPMe 1 (comp3)

In the window, right-click the root node and choose .

Component 2 - SPMe 1 (comp3), Definitions (comp3), Geometry 2, Lithium-Ion Battery 2 (liion3), Mesh 2

In the dialog, click .

In the copy operation the name of the llion2 interface was pasted as liion3. As a result of this, we will have to make updates various variable names in order to get the SPM model to work properly.

Component 2 - SPMe (comp2)

In the window, collapse the node.

Component 3 - SPM

In the window, click .

In the window for , type Component 3 - SPM in the text field.

Definitions (comp3)

Global Variable Probe - SPM

In the window, under > click .

In the window for , type Global Variable Probe - SPM in the text field.

Locate the section. In the text field, type liion3.phis_tpce2.

In the text field, type Cell Voltage - SPM.

Locate the section. Click

Variables 2

In the window, right-click and choose .

As conductivity value we will use a constant effective conductivity value based on the average electrolyte potential drop value we estimated in the SPMe model. Define this as a variable here for later use when defining the domain.

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
sigmal_eff	liion3.I_1C_cellL_el/(E_IR_1CA_cell)	S/m

E_IR_1C equals 27 mV in the parameter list.

Lithium-Ion Battery 3

The difference between the SPMe and the SPM model lies in the electrolyte transport formulation. The SPM model models the electrolyte as a resistor. This can be accomplished by using the model in the interface.

In the window, under click .

In the window for , type Lithium-Ion Battery 3 in the text field.

Locate the section. From the list, choose .

Click to expand the section. Since there are no source terms present in the single separator domain in the SPM model, it suffices to use a linear discretization of the electrolyte potential. This saves computational resources.

From the list, choose .

Separator 1

In the window, expand the > node, then click .

In the window for , locate the section.

From the σl list, choose . In the associated text field, type sigmal_eff.

Locate the section. From the list, choose .

Mesh 2

Edge 1

In the toolbar, click

Distribution 1

Right-click and choose .

Since there are no source terms present in the single domain in the SPM model, it suffices to use a single mesh element. This also saves computational resources.

In the window for , locate the section.

In the text field, type 1.

Add Study

In the toolbar, click

and choose .

Go to the window.

Find the subsection. In the tree, select > > .

Find the subsection. In the table, clear the checkboxes for and .

Click the button in the window toolbar.

In the window, click the root node.

In the toolbar, click

to close the window.

Study 3

Step 2: Time Dependent

In the window for , locate the section.

In the text field, type 0 5000.

Locate the section. From the list, choose .

In the list, choose and .

Under , click

In the window, click .

In the window for , type Study 3 - SPM in the text field.

Locate the section. Clear the checkbox.

In the toolbar, click

Results

Probe Plot Group 1

In the window, under click .

In the toolbar, click

Component 3 - SPM (comp3)

In the window, collapse the node.

It is also possible to create a SPM model by using the interface, we will now proceed to make such a model, which we will call SPM (Lumped).

Root

By the use of the interface, no geometry needs to be defined, and the model may hence be created in 0D.

Add Component

In the window, right-click the root node and choose > .

Add Physics

In the toolbar, click

to open the window.

In the window for , type Component 4 - SPM (Lumped) in the text field.

Go to the window.

In the tree, select > > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Lumped Battery (lb)

In the window for , locate the section.

From the list, choose .

In the Crate text field, type comp1.C_rate.

The SPM (Lumped) model makes use of a number of lumped parameters, defined in the parameters file you imported earlier.

Locate the section. In the Qhost,neg,0 text field, type Q_host_neg.

In the Qhost,pos,0 text field, type Q_host_pos.

Locate the section. In the SOCcell,0 text field, type soc_init.

Negative Equilibrium Potential 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the T text field, type T.

Positive Equilibrium Potential 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the T text field, type T.

Definitions (comp4)

Also add some variable expressions for the lumped exchange current densities as follows:

Variables 3

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
J0_neg	(lb.DOC_neg_surface(1-lb.DOC_neg_surface))^0.52i0_ref_neg(3/rp_neg)/(cs_max_neg*F_const/1[h])		Dimensionless exchange current density, negative electrode
J0_pos	(lb.DOC_pos_surface(1-lb.DOC_pos_surface))^0.52i0_ref_pos(3/rp_pos)/(cs_max_pos*F_const/1[h])		Dimensionless exchange current density, positive electrode