# Supercapacitor

Implement generic supercapacitor model

Libraries:
Simscape / Electrical / Specialized Power Systems / Sources

## Description

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

`${Q}_{T}=\int {i}_{SC}dt$`

To represent the self-discharge phenomenon, the supercapacitor electric charge is modified as follows (when iSC = 0):

`${Q}_{T}=\int {i}_{self_dis}dt$`

where

`${i}_{self_dis}=\left\{\begin{array}{l}\frac{{C}_{T}{\alpha }_{1}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& t-{t}_{oc}\le {t}_{3}\end{array}\end{array}\\ \frac{{C}_{T}{\alpha }_{2}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& {t}_{3}\prec t-{t}_{oc}\le {t}_{4}\end{array}\end{array}\\ \frac{{C}_{T}{\alpha }_{3}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& t-{t}_{oc}\succ {t}_{4}\end{array}\end{array}\end{array}$`

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:

VariableDescription
AiInterfacial area between electrodes and electrolyte (m2)
cMolar concentration (mol/m3) equal to c = 1/(8NAr3)
iscSupercapacitor current (A)
VscSupercapacitor voltage (V)
CTTotal capacitance (F)
RscTotal resistance (ohms)
NeNumber of layers of electrodes
Np Number of parallel supercapacitors
NsNumber of series supercapacitors
QTElectric charge (C)
RIdeal gas constant
TOperating temperature (K)
εPermittivity of material
ε0Permittivity of free space

### Examples

The `parallel_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.

## Assumptions and Limitations

• 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.

## Ports

### Conserving

expand all

Specialized electrical conserving port associated with the supercapacitor positive terminal.

Specialized electrical conserving port associated with the supercapacitor negative terminal.

### Output

expand all

Measurement signals. You can demultiplex these signals using the Bus Selector block.

SignalDefinitionUnitsSymbol
1The supercapacitor currentA`Current`
2The supercapacitor voltageV`Voltage`
3The 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:

`$SOC=\frac{\underset{0}{\overset{t}{Qinit-\int i\left(\tau \right)d\tau }}}{{Q}_{T}}×100$`

## Parameters

expand all

To edit block parameters interactively, use the Property Inspector. From the Simulink® Toolstrip, on the Simulation tab, in the Prepare gallery, select .

### Parameters

Nominal capacitance of the supercapacitor, in farad.

Internal resistance of the supercapacitor, in ohms.

Rated voltage of the supercapacitor, in volts. Typical rated voltage is equal to 2.7 V.

Number of series capacitors to be represented.

Number of parallel capacitors to be represented.

Initial voltage of the supercapacitor, in volts.

Operating temperature of the supercapacitor. The nominal temperature is 25° C.

### Stern

Option to load 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.

#### Dependencies

To enable this parameter, clear the Estimate using test data (requires the Optimization Toolbox) parameter.

Option to provide test data required for the estimation of the Stern model parameters. This parameter is available only if the Optimization Toolbox™ is installed.

#### Dependencies

To enable this parameter, clear the Use predetermined parameters parameter.

Number of layers related to the Stern model.

#### Dependencies

To enable this parameter, clear the Use predetermined parameters and Estimate using test data (requires the Optimization Toolbox) parameters.

Molecular radius related to the Stern model, in meters.

#### Dependencies

To enable this parameter, clear the Use predetermined parameters and Estimate using test data (requires the Optimization Toolbox) parameters.

Permittivity of the electrolyte material, in farad/meter.

#### Dependencies

To enable this parameter, clear the Use predetermined parameters and Estimate using test data (requires the Optimization Toolbox) parameters.

Charge current during a constant current charge test, in amperes.

#### Dependencies

To enable this parameter, select the Estimate using test data (requires the Optimization Toolbox) parameter.

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.

#### Dependencies

To enable this parameter, select the Estimate using test data (requires the Optimization Toolbox) parameter.

### Self-discharge

Option to provide test data required for modeling the self-discharge phenomenon.

Current prior to an open-circuit event, in amperes.

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.

Option to plot a figure containing the charge curves at the specified charge currents and time units.

Charge currents, in amperes, used to plot the charge characteristics.

Time units (seconds, minutes, hours) used to plot the charge characteristics.

## References

[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.

## Version History

Introduced in R2013a