MaplesoftBattery

EquivCircuit.LiIon $—$ Equivalent-circuit model of a lithium-ion battery

Description

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

${R}_{0}=\mathrm{expoly}\left({R}_{\mathrm{out}},\mathrm{soc}\right)$

${R}_{1}=\mathrm{expoly}\left({R}_{\mathrm{tc1}},\mathrm{soc}\right)$

${R}_{2}=\mathrm{expoly}\left({R}_{\mathrm{tc2}},\mathrm{soc}\right)$

${R}_{1}{C}_{1}=\mathrm{expoly}\left({T}_{\mathrm{tc1}},\mathrm{soc}\right)$

${R}_{2}{C}_{2}=\mathrm{expoly}\left({T}_{\mathrm{tc2}},\mathrm{soc}\right)$

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 ${m}_{\mathrm{cell}}{c}_{p}\frac{\mathrm{d}{T}_{\mathrm{cell}}}{\mathrm{d}t}={P}_{\mathrm{cell}}-{Q}_{\mathrm{cell}}$ ${Q}_{\mathrm{flow}}={n}_{\mathrm{cell}}{Q}_{\mathrm{cell}}$ ${P}_{\mathrm{cell}}={i}_{\mathrm{cell}}{T}_{\mathrm{cell}}\left(\frac{\mathrm{d}{U}_{p}}{\mathrm{d}T}-\frac{\mathrm{d}{U}_{n}}{\mathrm{d}T}\right)+{i}_{\mathrm{cell}}\left({v}_{\mathrm{cell}}-{v}_{\mathrm{oc}}\right)$ 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 ${Q}_{\mathrm{cell}}=h{A}_{\mathrm{cell}}\left({T}_{\mathrm{cell}}-{T}_{\mathrm{amb}}\right)$
 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.
 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.

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

 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 ${R}_{\mathrm{cell}}$ $0.005$ $\mathrm{\Omega }$ Fixed cell resistance, if 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 ${m}_{\mathrm{cell}}$ $0.014$ $\mathrm{kg}$ Mass of one cell mcell $h$ $100$ $\frac{W}{{m}^{2}K}$ Surface coefficient of heat transfer; used with convection heat model h ${A}_{\mathrm{cell}}$ $0.0014$ ${m}^{2}$ 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

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

General Parameters

 Name Default Units Description Modelica ID ${R}_{\mathrm{out}}$ $\left[0.11,-50,0.0075\right]$ expoly array for series resistance Rout ${R}_{\mathrm{tc1}}$ $\left[0.05,-29,0.0074\right]$ expoly array for short time-constant resistance Rtc1 ${T}_{\mathrm{tc1}}$ $\left[3.5,-10,10.5\right]$ expoly array for short time-constant duration Ttc1 ${R}_{\mathrm{tc2}}$ $\left[1,-150,0.008\right]$ expoly array for long time-constant resistance Rtc2 ${T}_{\mathrm{tc2}}$ $\left[-500,-20,710\right]$ 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 $ⅇxpoly\left(k,\mathrm{soc}\right)={k}_{1}ⅇxp\left({k}_{2}\mathrm{soc}\right)+{k}_{3}+{k}_{4}\mathrm{soc}+{k}_{5}{\mathrm{soc}}^{2}+\cdots$.

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.