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Equivalent Circuit Lithium Ion

Equivalent-circuit model of a lithium-ion battery

Degradation Parameters

Basic Thermal Parameters

Electrode Chemistry Parameters

General Parameters

References

Description

The EquivCircuit.LiIon component is an equivalent-circuit model of a lithium-ion battery; see the following figure.

$R_{0} = expoly (R_{out}, soc)$

$R_{1} = expoly (R_{tc1}, soc)$

$R_{2} = expoly (R_{tc2}, soc)$

$R_{1} C_{1} = expoly (T_{tc1}, soc)$

$R_{2} C_{2} = expoly (T_{tc2}, soc)$

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

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

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

$P_{cell} = i_{cell} T_{cell} (\frac{d U_{p}}{d T} - \frac{d U_{n}}{d T}) + i_{cell} (v_{cell} - v_{oc})$

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

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

	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 a cell can either be a fixed value, $R_{cell}$ , or be controlled via an input signal, $R_{in}$ , if the use resistance input box is checked. This resistance is in addition to the resistance of the equivalent circuit.

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 either terminated and an error message is raised, or, if the allow overdischarge parameter is true, a warning is generated. A similar effect occurs if the battery is fully charged so that the state of charge reaches one; the simulation is terminated unless allow overcharge is true.

The parameter ${SOC}_{0}$ assigns the initial state-of charge of the battery.

Connections

Name	Type	Description	Modelica ID
$p$	Electrical	Positive pin	p
$n$	Electrical	Negative pin	n
$soc$	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 cell resistance input is true	Rin

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

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
${SOC}_{\min}$	$0.02$		Minimum allowable state-of-charge	SOCmin
$R_{cell}$	$0.005$	$Ω$	Fixed cell resistance, if use cell resistance input is false	Rcell
allow overcharge	false		True allows simulation to continue with $1 < SoC$	allow_overcharge
allow overdischarge	false		True allows simulation to continue with $SoC < {SoC}_{\min}$	allow_overdischarge
use capacity input	false		True allows enables the $C_{in}$ input port	useCapacityInput
use cell resistance input	false		True allows enables the $R_{in}$ input port	useResistInput

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 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
$m_{cell}$	$0.014$	$kg$	Mass of one cell	mcell
$h$	$100$	$\frac{W}{m^{2} K}$	Surface coefficient of heat transfer; used with convection heat model	h
$A_{cell}$	$0.0014$	$m^{2}$	Surface area of one cell; used with convection heat model	Acell
$T_{amb}$	$298.15$	$K$	Ambient temperature; used with convection heat model	Tamb

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

General Parameters

Name	Default	Units	Description	Modelica ID
$R_{out}$		$Ω$	expoly array for series resistance	Rout
$R_{tc1}$		$Ω$	expoly array for short time-constant resistance	Rtc1
$T_{tc1}$		$s$	expoly array for short time-constant duration	Ttc1
$R_{tc2}$		$Ω$	expoly array for long time-constant resistance	Rtc2
$T_{tc2}$		$s$	expoly array for long time-constant duration	Ttc2

An exponential-polynomial (expoly) is a polynomial with an exponential term included. Its coefficients are given by a one-dimensional array, $k$ , such that $&ExponentialE;xpoly (k, soc) = k_{1} &ExponentialE;xp (k_{2} soc) + k_{3} + k_{4} soc + k_{5} {soc}^{2} + \dots$ .

References

[1] Chen, M. and Rincón-Mora, G.A., Accurate electrical battery model capable of predicting runtime and I-V performance, IEEE Transactions of Energy Conversion, Vol. 21, No. 2, 2006.

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