OTCexpoly

One time-constant expoly-based equivalent-circuit model of a battery

Description

The EquivCircuit.OTCexpoly component is an equivalent-circuit model of a generic battery.  The open-circuit voltage depends on the state-of-charge (SoC) based on a polynomial with an exponential term. The transient response is modeled by an SoC-dependent resistor-capacitor network.

$\mathrm{Voc}=\mathrm{expoly}\left({V}_{\mathrm{oc}},\mathrm{soc}\right)$

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

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

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

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 ${m}_{\mathrm{cell}}{c}_{p}\frac{\mathrm{d}{T}_{\mathrm{cell}}}{\mathrm{d}t}={P}_{\mathrm{cell}}-{Q}_{\mathrm{cell}}$ ${Q}_{\mathrm{flow}}={n}_{\mathrm{cell}}{Q}_{\mathrm{cell}}$ ${P}_{\mathrm{cell}}={i}_{\mathrm{cell}}\left({v}_{\mathrm{cell}}-{v}_{\mathrm{oc}}\right)$ 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 a cell 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 a cell can either be a fixed value, ${R}_{\mathrm{cell}}$, or be controlled via an input signal, ${R}_{\mathrm{in}}$, 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 ${\mathrm{SOC}}_{\mathrm{min}}$ 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 ${\mathrm{SOC}}_{0}$ 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 $\mathrm{soc}$ 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 cell resistance input is true Rin

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

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

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

General Parameters

 Name Default Units Description Modelica ID ${V}_{\mathrm{oc}}$ $V$ expoly array for open-circuit voltage Voc ${R}_{\mathrm{out}}$ $\mathrm{\Omega }$ expoly array for series resistance Rout ${R}_{\mathrm{tc}}$ $\mathrm{\Omega }$ expoly array for time-constant resistance Rtc ${T}_{\mathrm{tc}}$ $s$ expoly array for time-constant duration Ttc

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.