MaplesoftBattery

LiIon $—$ Electrochemical model of a lithium-ion battery

Description

The LiIon component models a lithium-ion battery using order-reduced equations derived from John Newman’s works on porous-electrode theory [1-3]. The following figure shows the basic anatomy of a lithium-ion cell, which has four main components: the negative composite electrode connected to the negative terminal of the cell, the positive electrode connected to the positive terminal of the cell, the separator, and the electrolyte. The chemistries of the positive and negative electrodes are independently selectable and define the electrochemical and thermal behaviors of the battery.

Main chemical reactions (assuming ${\mathrm{Li}}_{y}{\mathrm{CoO}}_{2}$ cathode and ${\mathrm{Li}}_{x}{C}_{6}$ anode).

Cathode: ${\mathrm{Li}}_{1-y}{\mathrm{CoO}}_{2}+y{\mathrm{Li}}^{+}+y{e}^{-}\to {\mathrm{LiCoO}}_{2}$

Anode: ${\mathrm{Li}}_{y}{C}_{6}\to {C}_{6}+y{\mathrm{Li}}^{+}+y{e}^{-}$

During battery operation, the position lithium ions (${\mathrm{Li}}^{+}$) travel between the two electrodes via diffusion and ionic conduction through the porous separator and the surface of the active material particles where they undergo electrochemical reactions. This process is called intercalation.

Electrochemical Behavior

 Transport in solid phase The following partial differential equation (PDE) describes the solid phase ${\mathrm{Li}}^{+}$ concentration in a single spherical active material particle in solid phase: $\frac{\partial {c}_{s}}{\partial t}=\frac{{\mathrm{D}}_{s}}{{r}^{2}}\frac{\partial }{\partial r}\left({r}^{2}\frac{\partial {c}_{s}}{\partial r}\right)$ where ${\mathrm{D}}_{s}$ is the ${\mathrm{Li}}^{+}$ diffusion coefficient in the intercalation particle of the electrodes.

Transport in electrolyte

The ${\mathrm{Li}}^{+}$ concentration in the electrolyte phase changes due to the changes in the gradient diffusive flow of ${\mathrm{Li}}^{+}$ ions and is described by the following PDE:

$\epsilon \frac{\partial {c}_{e}}{\partial t}=\frac{\partial }{\partial x}\left({\mathrm{D}}_{\mathrm{eff}}\frac{\partial {c}_{e}}{\partial x}\right)+a\left(1+{t}^{+}\right)j$

where

 $\mathrm{\epsilon }$ is the volume fraction,
 ${\mathrm{D}}_{\mathrm{eff}}$ is the ${\mathrm{Li}}^{+}$ diffusion coefficient in the electrolyte,
 $a=\frac{3}{{R}_{s}}\left(1-\mathrm{\epsilon }-{\mathrm{\epsilon }}_{f}\right)$  is the specific surface area of electrode,
 ${R}_{s}$ is the radius of intercalation of electrode
 ${\mathrm{\epsilon }}_{f}$  is the volume fraction of fillers
 ${t}^{+}$ is the ${\mathrm{Li}}^{+}$ transference constant in the electrolyte, and
 $j$ is the wall-flux of ${\mathrm{Li}}^{+}$ on the intercalation particle of electrode.

Electrical potentials

Charge conservation in the solid phase of each electrode is described by Ohm’s law:

${\mathrm{\sigma }}_{\mathrm{eff}}\frac{{{\partial }}^{2}}{{\partial }{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Phi }}_{s}=aFj$

In the electrolyte phase, the electrical potential is described by combining Kirchhoff’s law and Ohm’s law:

$-{\mathrm{\sigma }}_{\mathrm{eff}}\left(\frac{\partial {\mathrm{\Phi }}_{s}}{\partial x}\right)-{\mathrm{\kappa }}_{\mathrm{eff}}\left(\frac{\partial {\mathrm{\Phi }}_{e}}{\partial x}\right)+\frac{2{\mathrm{\kappa }}_{\mathrm{eff}}RT}{F}\left(1-{t}^{+}\right)\frac{\partial \mathrm{ln}\left({c}_{e}\right)}{\partial x}=J$

where

 ${\mathrm{\sigma }}_{\mathrm{eff}}=\mathrm{\sigma }\left(1-\mathrm{\epsilon }-{\mathrm{\epsilon }}_{\mathrm{eff}}\right)$ is the effective electronic conductivity,
 $\mathrm{\sigma }$ is the electronic conductivity in solid phase,
 ${\mathrm{\kappa }}_{\mathrm{eff}}$ is the effective ionic conductivity of the electrolyte, and
 $J$ is the applied current density.

Butler-Volmer kinetics

The Butler-Volmer equation describes the relationship between the current density, concentrations, and over-potential:

$j=k{\left({c}_{s,\mathrm{max}}-{c}_{s,\mathrm{surf}}\right)}^{0.5}{\left({c}_{s,\mathrm{surf}}\right)}^{0.5}{\left({c}_{e}\right)}^{0.5}\left(\mathrm{exp}\left(0.5\frac{F\mathrm{\mu }}{RT}\right)-\mathrm{exp}\left(-0.5\frac{F\mathrm{\mu }}{RT}\right)\right)$

where

 $k$ is the reaction rate constant,
 $\mathrm{\mu }={\mathrm{\Phi }}_{s}-{\mathrm{\Phi }}_{e}-U$ is the over-potential of the intercalation reaction,
 ${c}_{s,\mathrm{max}}$ is maximum concentration of ${\mathrm{Li}}^{+}$ ions in the intercalation particles of the electrode,
 ${c}_{s,\mathrm{surf}}$ is the concentration of of ${\mathrm{Li}}^{+}$ ions on the surface of the intercalation particles of the electrode, and
 $U$ is the open-circuit potential for the electrode material.

The open-circuit potential for each cathode and anode material has been curve-fitted based on experimental measurements.

An example of the open-circuit potentials for ${Li}_{y}Co{O}_{2}$ cathode and ${Li}_{x}{C}_{6}$ anode, curve-fitted from experiment measurement, are shown in the following figure:

The gradual decay, with use, of a cell's capacity and increase of its resistance is modeled by enabling the include degradation effects boolean parameter. Enabling this feature adds a state-of-health (soh) output to the model. This signal is 1 when the cell has no decay and 0 when is completely decayed.

The soh output is given by

$\mathrm{soh}={\left(1-\frac{s}{{R}_{s}}\right)}^{3}$

where

 $s$ is thickness of the solid-electrolyte interface (SEI),
 ${R}_{s}$ is radius of the particles of active material in the SEI.

The decay of the capacity is

$C={C}_{\mathrm{max}}\mathrm{soh}$

where

 $C$ is the effective capacity, and
 ${C}_{\mathrm{max}}$ is the specified capacity equal to either the parameter $\mathrm{CA}$ or the input ${C}_{\mathrm{in}}$.

${R}_{\mathrm{sei}}=\frac{s}{\mathrm{\kappa }}$

with $\mathrm{\kappa }$ a parameter of the model.

The following equations govern the increase in the thickness of the SEI layer ($s$).

$k={A}_{e}\mathrm{exp}\left(-\frac{{E}_{a}}{RT}\right)$

$\frac{\mathrm{ds}}{\mathrm{dt}}=\left\{\begin{array}{cc}\frac{kcM}{\left(1+\frac{ks}{{\mathrm{D}}_{\mathrm{diff}}}\right){\mathrm{\rho }}_{\mathrm{sei}}}& \mathrm{charging}\\ 0& \mathrm{otherwise}\end{array}$

Thermal Effects

Select the thermal model of the battery from the heat model drop-down list.  The available models are: isothermal, external port, and convection.

 Isothermal The isothermal model sets the cell temperature to a constant parameter, ${T}_{\mathrm{iso}}$.
 External Port The external port model adds a thermal port to the battery model. The temperature of the heat port is the cell temperature. The parameters ${m}_{\mathrm{cell}}$ and ${c}_{p}$ become available and are used in the heat equation ${m}_{\mathrm{cell}}{c}_{p}\frac{\mathrm{d}{T}_{\mathrm{cell}}}{\mathrm{d}t}={P}_{\mathrm{cell}}-{Q}_{\mathrm{cell}}$ ${Q}_{\mathrm{flow}}={n}_{\mathrm{cell}}{Q}_{\mathrm{cell}}$ ${P}_{\mathrm{cell}}={i}_{\mathrm{cell}}^{2}{R}_{\mathrm{cell}}+{i}_{\mathrm{cell}}{T}_{\mathrm{cell}}\left(\frac{\mathrm{d}{U}_{p}}{\mathrm{d}T}-\frac{\mathrm{d}{U}_{n}}{\mathrm{d}T}\right)+{i}_{\mathrm{cell}}\left({\mathrm{\mu }}_{p}-{\mathrm{\mu }}_{n}\right)$ where ${P}_{\mathrm{cell}}$ is the heat generated in each cell, including chemical reactions and ohmic resistive losses, ${Q}_{\mathrm{cell}}$ is the heat flow out of each cell, and ${Q}_{\mathrm{flow}}$ is the heat flow out of the external port.
 Convection The convection model assumes the heat dissipation from each cell is due to uniform convection from the surface to an ambient temperature. The parameters ${m}_{\mathrm{cell}}$, ${c}_{p}$, ${A}_{\mathrm{cell}}$, $h$, and ${T}_{\mathrm{amb}}$ become available, as does an output signal port that gives the cell temperature in Kelvin. The heat equation is the same as the heat equation for the external port, with ${Q}_{\mathrm{cell}}$ given by ${Q}_{\mathrm{cell}}=h{A}_{\mathrm{cell}}\left({T}_{\mathrm{cell}}-{T}_{\mathrm{amb}}\right)$
 Arrhenius equations For all thermal models, the Arrhenius equations model the effect of cell temperature on the chemical reaction. ${\mathrm{D}}_{s,x}={\mathrm{D}}_{s,x,\mathrm{ref}}\mathrm{exp}\left(\frac{{E}_{\mathrm{dx},p}}{R}\left(\frac{1}{{T}_{\mathrm{ref}}}-\frac{1}{{T}_{\mathrm{cell}}}\right)\right)$ ${\mathrm{D}}_{\mathrm{eff},x}={\mathrm{D}}_{e}{\mathrm{\epsilon }}_{x}^{\mathrm{brugg}}\mathrm{exp}\left(\frac{{E}_{\mathrm{de},x}}{R}\left(\frac{1}{{T}_{\mathrm{ref}}}-\frac{1}{{T}_{\mathrm{cell}}}\right)\right)$ with $x\in \left\{p,s\right\}$.
 State of Charge A signal output, soc, gives the state-of-charge of the battery, with 0 being fully discharged and 1 being fully charged. The parameter ${\mathrm{SOC}}_{\mathrm{min}}$ sets the minimum allowable state-of-charge; if the battery is discharged past this level, the simulation is terminated and an error message is raised. This prevents the battery model from reaching non-physical conditions. A similar effect occurs if the battery is fully charged so that the state of charge reaches one. The parameter ${\mathrm{SOC}}_{0}$ assigns the initial state-of charge of the battery.
 Capacity The capacity of the battery can either be a fixed value, $\mathrm{CA}$, or be controlled via an input signal, ${C}_{\mathrm{in}}$, if the use capacity input box is checked.
 Resistance The resistance of each cell can either be a fixed value, ${R}_{\mathrm{cell}}$, or be controlled via an input signal, ${R}_{\mathrm{in}}$, if the use cell resistance input box is checked.

Variables

 Name Units Description Modelica ID ${T}_{\mathrm{cell}}$ $K$ Internal temperature of battery Tcell $i$ $A$ Current into battery i $v$ $V$ Voltage across battery v

Connections

 Name Type Description Modelica ID $p$ Electrical Positive pin p $n$ Electrical Negative pin n $\mathrm{soh}$ Real output State of health [0..1]; available when include degradation effects is enabled soh $\mathrm{SOC}$ Real output State of charge [0..1] SOC ${C}_{\mathrm{in}}$ Real input Sets capacity of cell, in ampere hours; available when use capacity input is true Cin ${R}_{\mathrm{in}}$ Real input Sets resistance of cell, in Ohms; available when use resistance input is true Rin ${T}_{\mathrm{out}}$ Real output Temperature of cell, in Kelvin; available with convection heat model Tout $\mathrm{heatPort}$ Thermal Thermal connection; available with external port heat model heatPort

Electrode Chemistry Parameters

 Name Default Units Description Modelica ID ${\mathrm{chem}}^{+}$ LiCoO2 Chemistry of the positive electrode chem_pos ${\mathrm{chem}}^{-}$ Graphite Chemistry of the negative electrode chem_neg

The chem_pos and chem_neg parameters select the chemistry of the positive and negative electrodes, respectively. They are of types MaplesoftBattery.Selector.Chemistry.Positive and MaplesoftBattery.Selector.Chemistry.Negative. The selection affects the variation in the open-circuit electrode potential and the chemical reaction rate versus the concentration of lithium ions in the intercalation particles of the electrode.

If the Use input option is selected for either the positive or negative electrode, a vector input port appears next to the corresponding electrode. The port takes two real signals, $U$ and $S$, where $U$ specifies the potential in volts at the electrode and $S$ specifies the entropy in $\frac{J}{\mathrm{mol}K}$.

If any of the chem_pos materials $LiNi{O}_{2}$, $LiTi{S}_{2}$, $Li{V}_{2}{O}_{5}$, $LiW{O}_{3}$, or $NaCo{O}_{2}$ is selected, the isothermal model is used.

Supported positive electrode materials

 Chemical composition Chemical name Common name $LiCo{O}_{2}$ Lithium Cobalt Oxide LCO $LiFⅇP{O}_{4}$ Lithium Iron Phosphate LFP $Li{Mn}_{2}{O}_{4}$ Lithium Manganese Oxide LMO $Li{Mn}_{2}{O}_{4}$ - low plateau Lithium Manganese Oxide ${Li}_{1.156}{Mn}_{1.844}{O}_{4}$ Lithium Manganese Oxide $Li{Ni}_{0.8}{Co}_{0.15}{Al}_{0.05}{O}_{2}$ Lithium Nickel Cobalt Aluminum Oxide NCA $Li{Ni}_{0.8}{Co}_{0.2}{O}_{2}$ Lithium Nickel Cobalt Oxide $Li{Ni}_{0.7}{Co}_{0.3}{O}_{2}$ Lithium Nickel Cobalt Oxide $Li{Ni}_{0.33}{Mn}_{0.33}{Co}_{0.33}{O}_{2}$ Lithium Nickel Manganese Cobalt Oxide NMC $LiNi{O}_{2}$ Lithium Nickel Oxide $LiTi{S}_{2}$ Lithium Titanium Sulphide $Li{V}_{2}{O}_{5}$ Lithium Vanadium Oxide $LiW{O}_{3}$ Lithium Tungsten Oxide $NaCo{O}_{2}$ Sodium Cobalt Oxide

Supported negative electrode materials

 Chemical composition Chemical name Common name $Li{C}_{6}$ Lithium Carbide Graphite $LiTi{O}_{2}$ Lithium Titanium Oxide ${Li}_{2}{Ti}_{5}{O}_{12}$ Lithium Titanate LTO

 Name Default Units Description Modelica ID ${A}_{e}$ 1.2 $\frac{m}{s}$ Factor for reaction rate equation Ae ${\mathrm{D}}_{0}$ $1.8·{10}^{-19}$ $\frac{{m}^{2}}{s}$ Diffusion coefficient at standard conditions D0 ${E}_{a}$ 10000 $\frac{J}{\mathrm{mol}}$ Activation energy Ea $M$ 0.026 $\frac{\mathrm{kg}}{\mathrm{mol}}$ Molar mass of SEI layer M ${R}_{s}$ $2·{10}^{-6}$ $m$ Radius of particles of active material in anode Rs ${\mathrm{SoH}}_{0}$ 1 Initial state-of-health: $0\le {\mathrm{SoH}}_{0}\le 1$ SoH0 $c$ 5000 $\frac{\mathrm{mol}}{{m}^{3}}$ Molar concentration of electrolyte c $\mathrm{\kappa }$ 0.001 $\frac{m}{\mathrm{\Omega }}$ Specific conductivity coefficient kappa ${\mathrm{\rho }}_{\mathrm{sei}}$ 2600 $\frac{\mathrm{kg}}{{m}^{3}}$ Density of SEI layer rho_sei

Basic Parameters

 Name Default Units Description Modelica ID ${N}_{\mathrm{cell}}$ $1$ Number of cells, connected in series ncell $\mathrm{CA}$ $1$ $\mathrm{A·h}$ Capacity of cell; available when use capacity input is false C ${\mathrm{SOC}}_{0}$ $1$ Initial state-of-charge [0..1] SOC0 ${\mathrm{SOC}}_{\mathrm{min}}$ $0.01$ Minimum allowable state-of-charge SOCmin ${R}_{\mathrm{cell}}$ $0.005$ $\mathrm{\Omega }$ Series resistance of each cell; available when use cell resistance input is false Rcell

Basic Thermal Parameters

 Name Default Units Description Modelica ID ${T}_{\mathrm{iso}}$ $298.15$ $K$ Constant cell temperature; used with isothermal heat model Tiso ${c}_{p}$ $750$ $\frac{J}{\mathrm{kg}K}$ Specific heat capacity of cell cp ${m}_{\mathrm{cell}}$ $0.55$ $\mathrm{kg}$ Mass of one cell mcell $h$ $100$ $\frac{W}{{m}^{2}K}$ Surface coefficient of heat transfer; used with convection heat model h ${A}_{\mathrm{cell}}$ $0.0085$ ${m}^{2}$ Surface area of one cell; used with convection heat model Acell ${T}_{\mathrm{amb}}$ $298.15$ $K$ Ambient temperature; used with convection heat model Tamb

Detailed Parameters

 Name Default Units Description Modelica ID ${\mathrm{D}}_{e}$ $7.5·{10}^{-11}$ $\frac{{m}^{2}}{s}$ Electrolyte diffusion coefficient De ${\mathrm{D}}_{s,n,\mathrm{ref}}$ $3.9·{10}^{-14}$ $\frac{{m}^{2}}{s}$ Lithium-ion diffusion coefficient in the intercalation particles of the negative electrode Dsnref ${\mathrm{D}}_{s,p,\mathrm{ref}}$ $1.0·{10}^{-14}$ $\frac{{m}^{2}}{s}$ Lithium-ion diffusion coefficient in the intercalation particles of the positive electrode Dspref ${L}_{n}$ $8.8·{10}^{-5}$ $m$ Thickness of negative electrode Ln ${L}_{p}$ $8.0·{10}^{-5}$ $m$ Thickness of positive electrode Lp ${L}_{s}$ $2.5·{10}^{-5}$ $m$ Thickness of separator Ls ${R}_{s,n}$ $2·{10}^{-6}$ $m$ Radius of intercalation particles at negative electrode Rsn ${R}_{s,p}$ $2·{10}^{-6}$ $m$ Radius of intercalation particles at positive electrode Rsp $\mathrm{brugg}$ 1.5 Bruggeman's constant brugg ${c}_{\mathrm{e0}}$ 5000 $\frac{\mathrm{mol}}{{m}^{3}}$ Initial concentration of Li in electrolyte Ce0 ${c}_{s,n,\mathrm{max}}$ 30555 $\frac{\mathrm{mol}}{{m}^{3}}$ Maximum concentration of Li at the anode Csnmax ${c}_{s,p,\mathrm{max}}$ 51554 $\frac{\mathrm{mol}}{{m}^{3}}$ Maximum concentration of Li at the cathode Cspmax ${\mathrm{\epsilon }}_{f,n}$ $0.0326$ Volumetric fraction of negative electrode fillers efn ${\mathrm{\epsilon }}_{f,p}$ $0.0250$ Volumetric fraction of positive electrode fillers efp ${\mathrm{\epsilon }}_{n}$ $0.485$ Porosity of negative electrode en ${\mathrm{\epsilon }}_{p}$ $0.385$ Porosity of positive electrode ep ${\mathrm{\epsilon }}_{s}$ $0.724$ Porosity of separator electrode es ${k}_{n}$ $5.0307·{10}^{-11}$ $\frac{\mathrm{mol}}{{\left(\frac{\mathrm{mol}}{{m}^{3}}\right)}^{3}{2}}}$ Intercalation/deintercalation reaction-rate constant at the negative electrode Kn ${k}_{p}$ $2.334·{10}^{-11}$ $\frac{\mathrm{mol}}{{\left(\frac{\mathrm{mol}}{{m}^{3}}\right)}^{3}{2}}}$ Intercalation/deintercalation reaction-rate constant at the positive electrode Kp ${\mathrm{\sigma }}_{n}$ 100 $\frac{S}{m}$ Conductivity of solid phase of negative electrode sigman ${t}^{+}$ 0.363 LiOn transference number in the electrolyte Tplus

Detailed Thermal Parameters

 Name Default Units Description Modelica ID ${E}_{\mathrm{de},n}$ 10000 $\frac{J}{\mathrm{mol}}$ Activation energy for electrolyte phase diffusion, De, of the negative electrode Eden ${E}_{\mathrm{de},p}$ 10000 $\frac{J}{\mathrm{mol}}$ Activation energy for electrolyte phase diffusion, De, of the positive electrode Edep ${E}_{\mathrm{de},s}$ 10000 $\frac{J}{\mathrm{mol}}$ Activation energy for electrolyte phase diffusion, De, of the separator Edes ${E}_{\mathrm{ds},n}$ 50000 $\frac{J}{\mathrm{mol}}$ Activation energy for solid phase Li diffusion coefficient, Ds, of the negative electrode Edsn ${E}_{\mathrm{ds},p}$ 25000 $\frac{J}{\mathrm{mol}}$ Activation energy for solid phase Li diffusion coefficient, Dp, of the positive electrode Edsp ${E}_{k,n}$ 20000 $\frac{J}{\mathrm{mol}}$ Activation energy for ionic conductivity of electrolyte solution, κ, of the negative electrode Ekn ${E}_{k,p}$ 20000 $\frac{J}{\mathrm{mol}}$ Activation energy for ionic conductivity of electrolyte solution, κ, of the positive electrode Ekp ${E}_{k,s}$ 20000 $\frac{J}{\mathrm{mol}}$ Activation energy for ionic conductivity of electrolyte solution, κ, of the separator Eks

References

 [1] Newman, J. and William, T., Porous-electrode theory with battery applications, AIChE Journal, Vol. 21, No. 1, pp.25-41, 1975.
 [2] Dao, T.-S., Vyasarayani, C.P., McPhee, J., Simplification and order reduction of lithium-ion battery model based on porous-electrode theory, Journal of Power Sources, Vol. 198, pp. 329-337, 2012.
 [3] Subramanian,V.R., Boovaragavan,V., and Diwakar, V.D., Toward real-time simulation of physics based lithium-ion battery models, Electrochemical and Solid-State Letters, Vol. 10, No. 11, pp. A255-A260, 2007.
 [4] Kumaresan, K., Sikha G., and White, R.E., Thermal model for a Li-ion cell, Journal of the Electrochemical Society, Vol. 155, No. 2, pp. A164-A171, 2008.
 [5] Newman, J. and William, T., Porous-electrode theory with battery applications, AIChE Journal, Vol. 21, No. 1, pp.25-41, 1975.
 [6] Viswanathan, V.V., Choi, D., Wang, D., Xu, W., Towne, S., Williford, R.E., Zhang, J.G., Liu, J., and Yang, Z., Effect of entropy change of lithium intercalation in cathodes and anodes on Li-ion battery thermal management, Journal of Power Sources, Vol. 195, No. 11, pp. 3720–3729, 2010.