COMSOL 6.4 - Minimizing the Charging Time of a Lithium-Ion Battery

Minimizing the Charging Time of a Lithium-Ion Battery

Optimizing a fast-charging curve for a battery is a multiple-objective problem. At the same time as one strives to minimize the total charging time, the applied high charge rates must not result in excessive aging of the battery. In a typical fast charge cycle for a lithium-ion battery, the initially high charging current is continuously lowered in order not to induce lithium plating or accelerated solid electrolyte interphase (SEI) formation.

This model extends the Lithium-Ion Battery Base Model in 1D in the Battery Design Module Application Library with an optimization study, with the objective to minimize the required charging time for charging the battery from 0 to 90% state of charge (SOC). This is achieved by fitting the shape of the charging profile using a control function, subject to a constraint for the maximum allowed degradation during the charge cycle.

Model Definition

Charge Curve Definition and the Control Function

The C rate of a battery is proportional to the battery current, where a C rate of 1 equals the current required to fully charge, or discharge, the battery in 1 h. A total current condition is used as boundary condition for the battery model, with the applied current defined as

(1)

Here, Qcell is the battery capacity as defined by the Lithium-Ion Battery interface.

The charging function C is defined using a Control Function, based on a five-segment piecewise Bernstein polynomial, resulting in seven control variables (fitting parameters). The argument for the control function, ranging between 0 and 1, represents a dimensionless time

, scaled so that the value 0 represents the time at the beginning of the charge cycle and 1 the final time when the battery has reached the target SOC of 90%. By this definition, the area of the control function equals the average C-rate over the whole charge cycle.

(2)

and the time scale, tscale, can be computed as

(3)

A stop expression is enabled in optimization study in order to stop the time-dependent solver when the battery has reached 90% SOC (at t = tscale).

The optimization solver is set to maximize the value of the variable Cavg.

Constraining the parasitic reactions

The amount of degradation during the charging cycle is computed using a generalized volumetric Butler–Volmer expression in the negative graphite electrode of the following form.

(4)

where the overpotential η is defined as

(5)

The volumetric current is defined to be zero for positive (anodic) overpotentials by the expression

(6)

The above parasitic aging expression was chosen for tutorial purposes, but could represent, for instance, irreversible SEI formation with an onset potential of 25 mV, or lithium plating (with an added safety margin of 25 mV).

Integrating the volumetric current density in the negative electrode over the course of the whole simulation gives a measure of the total amount of capacity lost due to the parasitic reaction

(7)

During optimization, charging profiles that result in a degradation larger than 10% over 1000 cycles are prohibited by defining a constraint variable as

(8)

The above variable is used as a constraint expression optimization solver, with an upper bound of 1.

Results and Discussion

Figure 1 compares the initial loading cycle with the optimized one.

Figure 1: The optimized charge profile is plotted together with the electroplating current.

The optimization reduces the charging time by a factor of 2 relative to the initial charging profile, but this comes at the cost of a nonzero electroplating current. The relationship between these two objectives is plotted in Figure 2, where the charging profile has been optimized for different values of the maximum number of cycles. The figure shows that it is possible to achieve a tenfold improvement in longevity for the battery at the cost of a 20% increase in the charging time.

Figure 2: The maximum number of cycles is plotted as a function of the charging time.

Optimization_Module/Optimal_Control/lib_charge_rate_optimization

Modeling Instructions

This example starts from an existing model from the Battery Design Module Application Library.

Application Libraries

From the menu, choose .

In the window, select > > in the tree.

Click

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	70[um]	7E-5 m	Positive electrode thickness

Optimization Parameters

In the toolbar, click

and choose > .

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

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
end_soc	0.9	0.9	State of charge at end of charge
t_1C	end_soc[h]	3240 s	Time required to reach end soc in 1 h
vol_lb	25[mV]	0.025 V	Lower voltage bound
maxCycles	1000[1]	1000	Maximum number of cycles
maxDegradation	10[%]	0.1	Maximum degradation
t	0[s]	0 s	Time
cfunc0	1	1	Initial control function

Use a to optimize the nondimensional charging rate.

Definitions (comp1)

Control Function 1 (cfunc1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the fmax text field, type 4.

In the c0 text field, type 1.

Locate the section. From the list, choose .

Locate the section. In the c0 text field, type cfunc0.

Locate the section. Click

In the dialog, select > in the tree.

Click .

Variables 2 - Negative Electrode

In the window, under right-click and choose .

In the window for , type Variables 2 - Negative Electrode in the text field.

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
eta_aging	phis-phil-vol_lb	V	Overpotential of plating reaction (with added 25 mV margin offset)
i_aging	if(eta_aging<0,1e4[A/m^3](cl[1/M])^0.5(exp(0.5eta_agingF_const/(R_constT))-exp(-0.5eta_agingF_const/(R_constT))),0)	A/m³	Parasitic aging current density to use in constraint variable

Domain Probe 1 (dom1)

In the toolbar, click

and choose .

In the window for , type total_aging_current in the text field.

Locate the section. From the list, choose .

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

Locate the section. In the text field, type -i_aging*A_cell.

Lithium-Ion Battery (liion)

Electrode Current 1

In the window, expand the node.

Right-click > and choose > .

Select Boundary 4 only.

In the window for , locate the section.

From the list, choose .

In the Crate text field, type cfunc1(t*avg_C_rate/t_1C).

Using cfunc1(t*avg_C_rate/t_1C) instead of cfunc1(t) ensures that the entire range of the is utilized.

Load Cycle 1

In the window, right-click and choose .

Particle Intercalation 1

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

In the window for , click to expand the section.

Select the checkbox to reduce the computational time.

Particle Intercalation 1

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

In the window for , locate the section.

Select the checkbox.

Component 1 (comp1)

Add an ODE to compute the parasitic aging charge as the integral of the parasitic aging current.

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.

Global ODEs and DAEs (ge)

Global Equations 1 (ODE1)

In the window for , locate the section.

In the table, enter the following settings:


Name	f(u,ut,utt,t) (1)	Initial value (u_0) (1)	Initial value (ut_0) (1/s)	Description
Q_aged	Q_agedt-total_aging_current	0	0	Integrated charge of aging reactions