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Aging Analysis of a Lumped Battery Model
Introduction
Aging in batteries occurs due to multiple complex degradation phenomena and side reactions that occur simultaneously at different places in the battery. This tutorial demonstrates the Lumped Battery interface for modeling capacity loss in a lithium-ion battery.
A set of lumped parameters are used to describe the capacity loss that occurs due to parasitic reactions in the battery. The lumped modeling approach used here does not require knowledge of the internal structure, design of the battery electrodes, or choice of materials. Therefore, any aging model will be empirical and cannot distinguish between different degradation phenomena. Typically, capacity loss and aging may be affected by the battery voltage, capacity throughput, aging history and temperature.
The aging analysis presented in this tutorial includes calendar life and cycle life studies.
Model Definition
The cell model is created using the Lumped Battery interface. The interface requires inputs such as the battery capacity, initial state of charge (SOC), and an open circuit voltage versus SOC curve, and consists of lumped parameters that represent the ohmic, activation, and concentration overpotential contributions. A detailed description on how to optimize the parameters of the lumped model against experimental data can be found in the Application Libraries example Parameter Estimation of a Time-Dependent Lumped Battery Model.
The capacity fade that occurs in the battery due to parasitic reactions is modeled using the Capacity Loss node. The loss kinetics is specified using the built-in expression available in this node. The expression calculates a loss current based on a calendar aging time constant that defines the rate of the parasitic reactions, and dimensionless aging factors dependent on voltage, current, aging history and temperature. Finally, the loss current is used to calculate the accumulated capacity loss corresponding to the parasitic reactions.
In this tutorial, representative values have been used for the voltage and capacity loss lumped parameters. As an alternative, these parameters could have been obtained by parameter estimation (using an optimization solver) against available experimental data for calendar life and cycle life aging.
The temperature is set to 298.15 K in both studies.
Calendar life analysis
A calendar life analysis involves aging the battery at open circuit at constant SOC. The potentiostatic mode of operation is used to maintain the battery at a particular SOC. Calendar life aging analysis requires that aging factors dependent on voltage and aging history are included. The aging factor dependent on voltage relates to change in the parasitic reaction rate for different values of the battery voltage or SOC, and would correspond either to a parasitic electrochemical reduction reaction occurring on the negative electrode, or a parasitic oxidation reaction occurring on the positive electrode. The rate of capacity fade may decrease as a result of products formed by the parasitic reactions, for example, by the formation of a mass-transport limiting film on the electrode particles. A decelerating aging rate is defined using the aging factor dependent on aging history. The calendar life aging study sets up a parametric sweep over three different applied battery voltages corresponding to the particular states-of-charge (25%, 50%, and 100% SOC). The battery is aged for a period of two years in the calendar life study.
Cycle Life Analysis
A cycle life analysis of the battery is performed in the second part of the tutorial, where the battery is aged using a constant 1C charge–discharge cycling scheme.
The Charge–discharge cycling mode of operation of the Lumped Battery interface is used to set up the 1C cycling scheme, which starts with a 1C discharge until the cell reaches a minimum voltage of 3.2 V, followed by a 500 s rest period, a 1C charge until the cell attains a maximum voltage of 4.15 V, and finally a 500 s rest period.
Figure 1 shows the cell potential and current corresponding to the 1C charge–discharge cycling scheme. The corresponding cell state-of-charge variation is shown in Figure 2.
Figure 1: Cell potential and current corresponding to the 1C charge–discharge cycling scheme.
Figure 2: Cell state of charge and current corresponding to the 1C charge–discharge cycling scheme.
In the cycle life aging analysis, the aging factor dependent on current is also included, in addition to the aging factors dependent on voltage and aging history. For many battery systems it is often observed that the lifetime is closely related to the amount of cycled equivalent full cycles (capacity throughput) and hence the aging factor dependent on current is used to define the additional capacity loss induced by cycling. The battery is aged for a period of 1 year in the cycle life study.
Results and Discussion
Figure 3 shows the cell state of health (SOH, relative cell capacity) variation with time, for the calendar life and cycle life studies.
Figure 3: Cell state-of-health variation with time, for the calendar life and cycle life studies.
In many lithium-ion battery systems, it is seen that high SOC values (typically resulting in high battery voltage) accelerate capacity loss. The same is observed in Figure 3, where the capacity loss is seen to be higher for calendar aging at higher SOC values. Additionally, the capacity loss is seen to be accelerated during the 1C cycle life aging.
The behavior seen in Figure 3 is similar to that typically observed for many battery systems. In this tutorial, representative values have been chosen for the lumped parameters describing capacity loss. Alternatively, these parameters can be obtained by parameter estimation (using an optimization solver) against available experimental data for calendar life and cycle life aging. Subsequently, the cell model using the optimized parameters can be used for capacity loss prediction of batteries aged using more complex cycling schemes.
Notes About the COMSOL Implementation
When computing the studies in the model file available in the Application Library, ‘Study 1: Calendar Life’ requires that the operation mode is set to Potentiostatic at the Lumped Battery interface level, and ‘Study 2: Single Load Cycle’ and ‘Study 3: Cycle Life’ require that the operation mode is set to Charge–discharge cycling at the Lumped Battery interface level.
Reference
1. H. Ekström and G. Lindbergh “A model for predicting capacity fade due to SEI formation in a commercial Graphite/LiFePO4 cell,” J. Electrochemical Society, vol. 162, pp. A1003–A1007, 2015.
Application Library path: Battery_Design_Module/Lithium-Ion_Batteries,_Aging_and_Abuse/lumped_li_battery_capacity_loss
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  0D.
2
In the Select Physics tree, select Electrochemistry > Batteries > Lumped Battery (lb).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Parameters 1
Import the model parameters from a text file.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file lumped_li_battery_capacity_loss_parameters.txt.
Definitions
Create an interpolation polynomial for the battery open circuit voltage from a text file. The function will be used later to specify the operating voltage, and the open circuit potential of the battery.
Interpolation 1 - E_OCP
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, type Interpolation 1 - E_OCP in the Label text field.
3
Locate the Definition section. In the Function name text field, type E_OCP.
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Click  Load from File.
5
Browse to the model’s Application Libraries folder and double-click the file lumped_li_battery_capacity_loss_E_OCP_data.txt.
Lumped Battery (lb)
You will now start defining the battery model. A calendar life analysis of the battery is performed in the first part of the tutorial, where the battery is aged at open circuit at constant state of charge. Potentiostatic mode of operation is used for maintaining the battery at a particular state of charge.
1
In the Model Builder window, under Component 1 (comp1) click Lumped Battery (lb).
2
In the Settings window for Lumped Battery, locate the Operation Mode section.
3
From the Operation mode list, choose Potentiostatic.
4
In the Eapp text field, type E_OCP(SOC_0).
5
Locate the Initial Capacity section. In the Qcell,0 text field, type Q_cell0.
6
Locate the Initial Cell Charge Distribution section. In the SOCcell,0 text field, type SOC_0.
Q_cell0 and SOC_0 were defined in the parameter text file you imported before.
Cell Equilibrium Potential 1
1
In the Model Builder window, under Component 1 (comp1) > Lumped Battery (lb) click Cell Equilibrium Potential 1.
2
In the Settings window for Cell Equilibrium Potential, locate the Open Circuit Voltage section.
3
From the Open circuit voltage input list, choose From definitions.
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From the EOCV,ref list, choose Interpolation 1 - E_OCP (E_OCP).
Note that in this node you may also add data for the temperature derivative of open circuit voltage, that is used to calculate the temperature dependence of the open circuit voltage. Additionally, this data is used in the calculation of the reversible (entropic) contribution and heat of mixing contribution to the total heat source. However, this data is not needed in this model.
Voltage Losses 1
Specify the lumped parameter values for the voltage losses.
1
In the Model Builder window, click Voltage Losses 1.
2
In the Settings window for Voltage Losses, locate the Model Input section.
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In the T text field, type T.
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Locate the Ohmic Overpotential section. In the ηIR,1C text field, type eta_IR_1C.
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Locate the Activation Overpotential section. In the J0 text field, type J0.
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Locate the Concentration Overpotential section. Select the Include concentration overpotential checkbox.
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In the τ text field, type tau.
Capacity Loss 1
Add a Capacity Loss node to define the accumulated capacity loss in the battery corresponding to parasitic reactions. The loss kinetics is specified using the Built in option that calculates a loss current based on a Calendar aging time constant that defines the rate of the parasitic reactions, and dimensionless aging factors dependent on Voltage, Current, Aging history and Temperature, respectively.
1
In the Physics toolbar, click  Global and choose Capacity Loss.
The calendar life aging study requires that aging factors dependent on Voltage and Aging history are included.
2
In the Settings window for Capacity Loss, locate the Model Input section.
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In the T text field, type T.
4
Locate the Capacity Loss section. In the τloss text field, type tau_loss.
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Select the Voltage checkbox.
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In the Eoffset text field, type E_offset.
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In the α text field, type alpha.
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Select the Aging history checkbox.
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In the G text field, type G.
Study 1: Calendar Life
The first study performs a calendar life aging analysis of the battery, where the battery is aged at open circuit at constant state of charge. Set up a parametric sweep for different applied voltages corresponding to particular states-of-charge. The battery is aged for two years.
1
In the Model Builder window, click Study 1.
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In the Settings window for Study, type Study 1: Calendar Life in the Label text field.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
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In the Settings window for Parametric Sweep, locate the Study Settings section.
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Click  Add.
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In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
SOC_0 (Initial state of charge)
0.25 0.5 1
 
Step 1: Time Dependent
1
In the Model Builder window, click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,10[d],2[a]).
4
In the Model Builder window, click Study 1: Calendar Life.
5
In the Settings window for Study, locate the Study Settings section.
6
Clear the Generate default plots checkbox.
7
In the Study toolbar, click  Compute.
Results
Proceed as follows to create a plot (Figure 3) for the cell state-of-health variation with time, at the different state of charge values.
Cell State-of-Health
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Cell State-of-Health in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1: Calendar Life/Parametric Solutions 1 (sol2).
Global 1
1
Right-click Cell State-of-Health and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Lumped Battery > lb.SOH_cell - Cell state of health - 1.
3
Locate the x-Axis Data section. From the Unit list, choose a.
4
Click to expand the Legends section. From the Legends list, choose Evaluated.
5
In the Legend text field, type eval(SOC_0*100)% SOC.
Cell State-of-Health
1
In the Model Builder window, click Cell State-of-Health.
2
In the Settings window for 1D Plot Group, click to expand the Title section.
3
From the Title type list, choose None.
4
Locate the Legend section. From the Position list, choose Lower left.
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In the Cell State-of-Health toolbar, click  Plot.
Lumped Battery (lb)
A cycle life aging analysis of the battery is performed in the second part of the tutorial, where the battery is aged using a constant 1C charge-discharge cycling scheme. Charge-discharge cycling mode of operation is used to set up the 1C cycling scheme (refer to Figure 1 and Figure 2). Note that lb.I_1C_cell is the 1C current variable already available in the interface.
1
In the Model Builder window, under Component 1 (comp1) click Lumped Battery (lb).
2
In the Settings window for Lumped Battery, locate the Operation Mode section.
3
From the Operation mode list, choose Charge-discharge cycling.
4
From the list, choose C-rate multiple.
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In the Vmin text field, type V_min.
6
Select the Include rest period checkbox.
7
In the trest,dch text field, type t_rest.
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From the list, choose C-rate multiple.
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In the Vmax text field, type V_max.
10
Select the Include rest period checkbox.
11
In the trest,ch text field, type t_rest.
Capacity Loss 1
The aging factor dependent on Current is also included for the cycle life study.
1
In the Model Builder window, under Component 1 (comp1) > Lumped Battery (lb) click Capacity Loss 1.
2
In the Settings window for Capacity Loss, locate the Capacity Loss section.
3
Select the Current checkbox.
4
In the H text field, type H.
Root
Add a study to first simulate a single cycle of the load that will be used in the cycle life aging analysis. Inspect the plots for the Cell Potential and Load (Figure 1) and Cell State of Charge (Figure 2) for a single cycle.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2: Single Load Cycle
In the Settings window for Study, type Study 2: Single Load Cycle in the Label text field.
Step 1: Time Dependent
1
In the Model Builder window, under Study 2: Single Load Cycle click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,1,7800).
4
In the Study toolbar, click  Compute.
Results
Cell Potential and Load (Single Load Cycle)
1
In the Settings window for 1D Plot Group, type Cell Potential and Load (Single Load Cycle) in the Label text field.
2
Click to expand the Title section. From the Title type list, choose None.
3
Locate the Legend section. From the Position list, choose Lower right.
Cell State of Charge (Single Load Cycle)
1
In the Model Builder window, under Results click Cell State of Charge (lb).
2
In the Settings window for 1D Plot Group, type Cell State of Charge (Single Load Cycle) in the Label text field.
3
Click to expand the Title section. From the Title type list, choose None.
4
Locate the Legend section. From the Position list, choose Lower right.
Root
Finally, add a third study to perform the cycle life aging analysis of the battery. The battery is aged for one year.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3: Cycle Life
In the Settings window for Study, type Study 3: Cycle Life in the Label text field.
Step 1: Time Dependent
1
In the Model Builder window, under Study 3: Cycle Life click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,10[d],1[a]).
4
In the Model Builder window, click Study 3: Cycle Life.
5
In the Settings window for Study, locate the Study Settings section.
6
Clear the Generate default plots checkbox.
7
In the Study toolbar, click  Compute.
Results
Cell State-of-Health
Proceed as follows to include the cycle life data in the Cell State-of-Health plot (Figure 3).
Global 2
1
In the Model Builder window, under Results > Cell State-of-Health right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Study 3: Cycle Life/Solution 7 (sol7).
4
Click to expand the Legends section. From the Legends list, choose Manual.
5
In the table, enter the following settings:
Legends
1 C cycling
Cell State-of-Health
1
In the Model Builder window, click Cell State-of-Health.
2
In the Cell State-of-Health toolbar, click  Plot.