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Implement generic supercapacitor model
The Supercapacitor block implements a generic model parameterized to represent most popular types of supercapacitors. The figure shows the equivalent circuit of the supercapacitor:
The supercapacitor output voltage is expressed using a Stern equation as:
${V}_{SC}=\frac{{N}_{s}{Q}_{T}d}{{N}_{p}{N}_{e}\epsilon {\epsilon}_{0}{A}_{i}}+\frac{2{N}_{e}{N}_{s}RT}{F}{\mathrm{sinh}}^{-1}\left(\frac{{Q}_{T}}{{N}_{p}{N}_{e}{}^{2}{A}_{i}\sqrt{8RT\epsilon {\epsilon}_{0}c}}\right)-{R}_{SC}\cdot {i}_{SC}$
with
${Q}_{T}={\displaystyle \int {i}_{SC}dt}$
To represent the self-discharge phenomenon, the supercapacitor electric charge is modified as follows (when i_{SC} = 0):
${Q}_{T}={\displaystyle \int {i}_{self\_dis}dt}$
where
${i}_{self\_dis}=\{\begin{array}{l}\frac{{C}_{T}{\alpha}_{1}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& t-{t}_{oc}\le {t}_{3}\end{array}\end{array}\\ \frac{{C}_{T}{\alpha}_{2}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& {t}_{3}\prec t-{t}_{oc}\le {t}_{4}\end{array}\end{array}\\ \frac{{C}_{T}{\alpha}_{3}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& t-{t}_{oc}\succ {t}_{4}\end{array}\end{array}\end{array}$
The constants α1, α2, and α3 are the rates of change of the supercapacitor voltage during time intervals (toc, t3), (t3, t4), and (t4, t5) respectively, as shown in the figure:
Variable | Description |
---|---|
A_{i} | Interfacial area between electrodes and electrolyte (m2) |
c | Molar concentration (mol m −3) equal to c = 1/(8NAr3) |
F | Faraday constant |
i_{sc} | Supercapacitor current (A) |
Vsc | Supercapacitor voltage (V) |
C_{T} | Total capacitance (F) |
R_{sc} | Total resistance (ohms) |
N_{e} | Number of layers of electrodes |
NA | Avogadro constant |
Np | Number of parallel supercapacitors |
Ns | Number of series supercapacitors |
QT | Electric charge (C) |
R | Ideal gas constant |
d | Molecular radius |
T | Operating temperature (K) |
ε | Permittivity of material |
ε0 | Permittivity of free space |
Specify the nominal capacitance of the supercapacitor, in farad.
Specify the internal resistance of the supercapacitor, in ohms.
Specify the rated voltage of the supercapacitor, in volts. Typical rated voltage is equal to 2.7 V.
Specify the number of series capacitors to be represented.
Specify the number of parallel capacitors to be represented.
Specify the initial voltage of the supercapacitor, in volts.
Specify the operating temperature of the supercapacitor. The nominal temperature is 25° C.
When this check box is selected, loads predetermined parameters of the Stern model into the mask of the block. These parameter values have been determined from experimental tests, and they can be used as default values to represent a common supercapacitor. Experimental validation of the model has shown a maximum error of 2% for charge and discharge when using the predetermined parameters.
When this check box is selected, the Number of layers, Molecular radius (m), Permittivity of electrolyte material (F/m), and Estimate using test data parameters appear dimmed.
When this check box is selected, you provide test data required for the estimation of the Stern model parameters. This parameter is available only if the Optimization Toolbox™ of MATLAB^{®} is installed.
When this check box is selected, the Charge current (A) and Voltage @ 0 s, 20 s, and 60 s [V_0, V_2, V_3] (V) parameters are enabled. The Use predetermined parameters, Number of layers, Molecular radius (m), and Permittivity of electrolyte material (F/m) parameters appear dimmed.
Specify the number of layers related to the Stern model.
Specify the molecular radius related to the Stern model, in meters.
Specify the permittivity of the electrolyte material, in farad/meter.
Specify the charge current during a constant current charge test, in amperes.
Specify the supercapacitor voltage, in volts, at 0 s, 20 s, and 60 s, when the supercapacitor is charged with a constant current equal to the value provided in the Charge current (A) parameter.
When this check box is selected, you provide test data required for modeling the self-discharge phenomenon.
Specify the current prior to an open-circuit event, in amperes.
Specify the supercapacitor voltage, in volts, at 0 s, 10 s, 100 s, and at 1000 s, when the supercapacitor is open-circuit. The corresponding current prior to open-circuit is given in the Current prior open-circuit (A) parameter.
When this check box is selected, the block plots a figure containing the charge curves at the specified charge currents and time units.
Specify the charge currents, in amperes, used to plot the charge characteristics.
Specify the time units (seconds, minutes, hours) used to plot the charge characteristics.
Outputs a vector containing measurement signals. You can demultiplex these signals using the Bus Selector block.
Signal | Definition | Units | Symbol |
---|---|---|---|
1 | The supercapacitor current | A | Current |
2 | The supercapacitor voltage | V | Voltage |
3 | The state of charge (SOC), between 0 and 100 | % | SOC |
The SOC for a fully charged supercapacitor is 100% and for an empty supercapacitor is 0%. The SOC is calculated as:
$$SOC=\frac{\underset{0}{\overset{t}{Qinit-{\displaystyle \int i\left(\tau \right)d\tau}}}}{{Q}_{T}}\times 100$$
Internal resistance is assumed constant during the charge and the discharge cycles.
The model does not take into account temperature effect on the electrolyte material.
No aging effect is taken into account.
Charge redistribution is the same for all values of voltage.
The block does not model cell balancing.
Current through the supercapacitor is assumed to be continuous.
The parallel_battery_SC_boost_converterparallel_battery_SC_boost_converter example shows a simple hybridization of a supercapacitor with a battery. The supercapacitor is connected to a buck/boost converter and the battery is connected to a boost converter. The DC bus voltage is equal to 42V. The converters are doing power management. The battery power is limited by a rate limiter block, therefore the transient power is supplied to the DC bus by the supercapacitor.
[1] Oldham, K. B. "A Gouy-Chapman-Stern model of the double layer at a (metal)/(ionic liquid) interface." J. Electroanalytical Chem. Vol. 613, No. 2, 2008, pp. 131–38.
[2] Xu, N., and J. Riley. "Nonlinear analysis of a classical system: The double-layer capacitor." Electrochemistry Communications. Vol. 13, No. 10, 2011, pp. 1077–81.