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Electrochemical Lithium Ion

Electrochemical model of a lithium-ion battery

Description

Variables

Connections

Electrode Chemistry Parameters

Degradation Parameters

Basic Parameters

Basic Thermal Parameters

Detailed Parameters

Detailed Thermal Parameters

References

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 ${Li}_{y} {CoO}_{2}$ cathode and ${Li}_{x} C_{6}$ anode).

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

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

During battery operation, the position lithium ions ( ${Li}^{+}$ ) 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 ${Li}^{+}$ concentration in a single spherical active material particle in solid phase: $\frac{\partial c_{s}}{\partial t} = \frac{D_{s}}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial c_{s}}{\partial r})$ where $D_{s}$ is the ${Li}^{+}$ diffusion coefficient in the intercalation particle of the electrodes.

Transport in electrolyte

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

$ε \frac{\partial c_{e}}{\partial t} = \frac{\partial}{\partial x} (D_{eff} \frac{\partial c_{e}}{\partial x}) + a (1 + t^{+}) j$

where

$ε$ is the volume fraction,

$D_{eff}$ is the ${Li}^{+}$ diffusion coefficient in the electrolyte,

$a = \frac{3}{R_{s}} (1 - ε - ε_{f})$ is the specific surface area of electrode,

$R_{s}$ is the radius of intercalation of electrode

$ε_{f}$ is the volume fraction of fillers

$t^{+}$ is the ${Li}^{+}$ transference constant in the electrolyte, and

$j$ is the wall-flux of ${Li}^{+}$ on the intercalation particle of electrode.

Electrical potentials

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

$σ_{eff} (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} Φ_{s}) = a F j$

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

$- σ_{eff} (\frac{\partial Φ_{s}}{\partial x}) - κ_{eff} (\frac{\partial Φ_{e}}{\partial x}) + \frac{2 κ_{eff} R T}{F} (1 - t^{+}) \frac{\partial \ln (c_{e})}{\partial x} = J$

where

$σ_{eff} = σ (1 - ε - ε_{eff})$ is the effective electronic conductivity,

$σ$ is the electronic conductivity in solid phase,

$κ_{eff}$ 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:

$j = k {(c_{s, \max} - c_{s, surf})}^{0.5} {(c_{s, surf})}^{0.5} {(c_{e})}^{0.5} (\exp (0.5 \frac{F μ}{R T}) - \exp (−0.5 \frac{F μ}{R T}))$

where

$k$ is the reaction rate constant,

$μ = Φ_{s} - Φ_{e} - U$ is the over-potential of the intercalation reaction,

$c_{s, \max}$ is maximum concentration of ${Li}^{+}$ ions in the intercalation particles of the electrode,

$c_{s, surf}$ is the concentration of of ${Li}^{+}$ 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:

Degradation

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

$soh = {(1 - \frac{s}{R_{s}})}^{3}$

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

$C = C_{\max} soh$

where

$C$ is the effective capacity, and

$C_{\max}$ is the specified capacity equal to either the parameter $CA$ or the input $C_{in}$ .

The additional series resistance added to a cell is

$R_{sei} = \frac{s}{κ}$

with $κ$ a parameter of the model.

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

$k = A_{e} \exp (- \frac{E_{a}}{R T})$

$\frac{ds}{dt} = {\begin{array}{c} \frac{k c M}{(1 + \frac{k s}{D_{diff}}) ρ_{sei}} & charging \\ 0 & otherwise \end{array}$

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, $T_{iso}$ .

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 $m_{cell}$ and $c_{p}$ become available and are used in the heat equation

$m_{cell} c_{p} \frac{d T_{cell}}{d t} = P_{cell} - Q_{cell}$

$Q_{flow} = n_{cell} Q_{cell}$

$P_{cell} = i_{cell}^{2} R_{cell} + i_{cell} T_{cell} (\frac{d U_{p}}{d T} - \frac{d U_{n}}{d T}) + i_{cell} (μ_{p} - μ_{n})$

where $P_{cell}$ is the heat generated in each cell, including chemical reactions and ohmic resistive losses, $Q_{cell}$ is the heat flow out of each cell, and $Q_{flow}$ 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 $m_{cell}$ , $c_{p}$ , $A_{cell}$ , $h$ , and $T_{amb}$ 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 $Q_{cell}$ given by

$Q_{cell} = h A_{cell} (T_{cell} - T_{amb})$

	Arrhenius equations
	For all thermal models, the Arrhenius equations model the effect of cell temperature on the chemical reaction. $D_{s, x} = D_{s, x, ref} \exp (\frac{E_{dx, p}}{R} (\frac{1}{T_{ref}} - \frac{1}{T_{cell}}))$ $D_{eff, x} = D_{e} ε_{x}^{brugg} \exp (\frac{E_{de, x}}{R} (\frac{1}{T_{ref}} - \frac{1}{T_{cell}}))$ with $x \in \{p, s\}$ .

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 ${SOC}_{\min}$ 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 ${SOC}_{0}$ assigns the initial state-of charge of the battery.

	Capacity
	The capacity of a cell can either be a fixed value, $CA$ , or be controlled via an input signal, $C_{in}$ , if the use capacity input box is checked.

	Resistance
	The resistance of each cell can either be a fixed value, $R_{cell}$ , or be controlled via an input signal, $R_{in}$ , if the use cell resistance input box is checked.

Variables

Name	Units	Description	Modelica ID
$T_{cell}$	$K$	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
$soh$	Real output	State of health [0..1]; available when include degradation effects is enabled	soh
$SOC$	Real output	State of charge [0..1]	SOC
$C_{in}$	Real input	Sets capacity of cell, in ampere hours; available when use capacity input is true	Cin
$R_{in}$	Real input	Sets resistance of cell, in Ohms; available when use resistance input is true	Rin
$T_{out}$	Real output	Temperature of cell, in Kelvin; available with convection heat model	Tout
$heatPort$	Thermal	Thermal connection; available with external port heat model	heatPort

Electrode Chemistry Parameters

Name	Default	Units	Description	Modelica ID
${chem}^{+}$	LiCoO2		Chemistry of the positive electrode	chem_pos
${chem}^{-}$	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 $\frac{J}{mol K}$ .

If any of the chem_pos materials $Lⅈ Nⅈ O_{2}$ , $Lⅈ Tⅈ S_{2}$ , $Lⅈ V_{2} O_{5}$ , $Lⅈ W O_{3}$ , or $Na Co O_{2}$ is selected, the isothermal model is used.

If the Use interpolation table option is selected for either the positive or negative electrode, a 2-D table defines the electrode potential and entropy in terms of the state-of-charge. The mode option selects whether the table is defined by an attachment, a file, or inline. The table has three columns:

The first column is the state-of-charge (soc), a real number between 0 and 1.

The second column is the electrode potential ( $U$ ) in volts.

The third column is the electrode entropy ( $S$ ) in $\frac{J}{mol K}$ .

Supported positive electrode materials

Chemical composition	Chemical name	Common name
$Lⅈ Co O_{2}$	Lithium Cobalt Oxide	LCO
$Lⅈ F&ExponentialE; P O_{4}$	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
$Lⅈ Nⅈ O_{2}$	Lithium Nickel Oxide
$Lⅈ Tⅈ S_{2}$	Lithium Titanium Sulphide
$Lⅈ V_{2} O_{5}$	Lithium Vanadium Oxide
$Lⅈ W O_{3}$	Lithium Tungsten Oxide
$Na Co O_{2}$	Sodium Cobalt Oxide

Supported negative electrode materials

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

Degradation Parameters

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

Basic Parameters

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

Basic Thermal Parameters

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

Detailed Parameters

Name	Default	Units	Description	Modelica ID
$D_{e}$	$7.5 \cdot 10^{−11}$	$\frac{m^{2}}{s}$	Electrolyte diffusion coefficient	De
$D_{s, n, ref}$	$3.9 \cdot 10^{−14}$	$\frac{m^{2}}{s}$	Lithium-ion diffusion coefficient in the intercalation particles of the negative electrode	Dsnref
$D_{s, p, ref}$	$1.0 \cdot 10^{−14}$	$\frac{m^{2}}{s}$	Lithium-ion diffusion coefficient in the intercalation particles of the positive electrode	Dspref
$L_{n}$	$8.8 \cdot 10^{−5}$	$m$	Thickness of negative electrode	Ln
$L_{p}$	$8.0 \cdot 10^{−5}$	$m$	Thickness of positive electrode	Lp
$L_{s}$	$2.5 \cdot 10^{−5}$	$m$	Thickness of separator	Ls
$R_{s, n}$	$2 \cdot 10^{−6}$	$m$	Radius of intercalation particles at negative electrode	Rsn
$R_{s, p}$	$2 \cdot 10^{−6}$	$m$	Radius of intercalation particles at positive electrode	Rsp
$brugg$	1.5		Bruggeman's constant	brugg
$c_{e0}$	5000	$\frac{mol}{m^{3}}$	Initial concentration of Li in electrolyte	Ce0
$c_{s, n, \max}$	30555	$\frac{mol}{m^{3}}$	Maximum concentration of Li at the anode	Csnmax
$c_{s, p, \max}$	51554	$\frac{mol}{m^{3}}$	Maximum concentration of Li at the cathode	Cspmax
$ε_{f, n}$	$0.0326$		Volumetric fraction of negative electrode fillers	efn
$ε_{f, p}$	$0.0250$		Volumetric fraction of positive electrode fillers	efp
$ε_{n}$	$0.485$		Porosity of negative electrode	en
$ε_{p}$	$0.385$		Porosity of positive electrode	ep
$ε_{s}$	$0.724$		Porosity of separator electrode	es
$k_{n}$	$5.0307 \cdot 10^{−11}$	$\frac{mol}{{(\frac{mol}{m^{3}})}^{\frac{3}{2}}}$	Intercalation/deintercalation reaction-rate constant at the negative electrode	Kn
$k_{p}$	$2.334 \cdot 10^{−11}$	$\frac{mol}{{(\frac{mol}{m^{3}})}^{\frac{3}{2}}}$	Intercalation/deintercalation reaction-rate constant at the positive electrode	Kp
$σ_{n}$	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
$E_{de, n}$	10000	$\frac{J}{mol}$	Activation energy for electrolyte phase diffusion, De, of the negative electrode	Eden
$E_{de, p}$	10000	$\frac{J}{mol}$	Activation energy for electrolyte phase diffusion, De, of the positive electrode	Edep
$E_{de, s}$	10000	$\frac{J}{mol}$	Activation energy for electrolyte phase diffusion, De, of the separator	Edes
$E_{ds, n}$	50000	$\frac{J}{mol}$	Activation energy for solid phase Li diffusion coefficient, Ds, of the negative electrode	Edsn
$E_{ds, p}$	25000	$\frac{J}{mol}$	Activation energy for solid phase Li diffusion coefficient, Dp, of the positive electrode	Edsp
$E_{k, n}$	20000	$\frac{J}{mol}$	Activation energy for ionic conductivity of electrolyte solution, κ, of the negative electrode	Ekn
$E_{k, p}$	20000	$\frac{J}{mol}$	Activation energy for ionic conductivity of electrolyte solution, κ, of the positive electrode	Ekp
$E_{k, s}$	20000	$\frac{J}{mol}$	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.

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