Implement generic supercapacitor model

Electric Drives/Extra Sources

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 (m^{2}) |

c | Molar concentration (mol/m^{3}) equal
to c = 1/(8N_{A}r^{3}) |

r | Molecular radius (m) |

F | Faraday constant |

i_{sc} | Supercapacitor current (A) |

V_{sc} | Supercapacitor voltage (V) |

C_{T} | Total capacitance (F) |

R_{sc} | Total resistance (ohms) |

N_{e} | Number of layers of electrodes |

N_{A} | Avogadro constant |

N_{p} | Number of parallel supercapacitors |

N_{s} | Number of series supercapacitors |

Q_{T} | Electric charge (C) |

R | Ideal gas constant |

d | Molecular radius |

T | Operating temperature (K) |

ε | Permittivity of material |

ε_{0} | Permittivity of free space |

**Rated capacitance (F)**Specify the nominal capacitance of the supercapacitor, in farad. Default is

`99.5`

.**Equivalent DC series resistance (Ohms)**Specify the internal resistance of the supercapacitor, in ohms. Default is

`8.9e-3`

.**Rated voltage (V)**Specify the rated voltage of the supercapacitor, in volts. Typical rated voltage is equal to 2.7 V. Default is

`48`

.**Number of series capacitors**Specify the number of series capacitors to be represented. Default is

`18`

.**Number of parallel capacitors**Specify the number of parallel capacitors to be represented. Default is

`1`

.**Initial voltage (V)**Specify the initial voltage of the supercapacitor, in volts. Default is

`0`

.**Operating temperature (celsius)**Specify the operating temperature of the supercapacitor. The nominal temperature is 25° C. Default is

`25`

.

**Use predetermined parameters**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. Default is cleared

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.**Estimate using test data**When this check box is selected, you provide test data required for the estimation of the Stern model parameters. Default is cleared. This parameter is available only if the Optimization Toolbox™ 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.**Number of layers**Specify the number of layers related to the Stern model. Default is

`1`

.**Molecular of radius (m)**Specify the molecular radius related to the Stern model, in meters. Default is

`1e-9`

.**Permittivity of electrolyte material (F/m)**Specify the permittivity of the electrolyte material, in farad/meter. Default is

`6.0208e-10`

.**Charge current (A)**Specify the charge current during a constant current charge test, in amperes. Default is

`10`

.**Voltage @ 0 s, 20 s, and 60 s [V_0, V_2, V_3] (V)**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. Default is`[0.161 2.7 7.8]`

.

**Simulate self-discharge**When this check box is selected, you provide test data required for modeling the self-discharge phenomenon. Default is selected.

**Current prior open-circuit (A)**Specify the current prior to an open-circuit event, in amperes. Default is

`10`

.**Voltage @ 0 s, 10 s, 100 s, and 1000 s [V_oc, V_3, V_4, V_5] (V)**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. Default is`[48 47.8 47.06 44.65]`

.**Plot charge characteristics**When this check box is selected, the block plots a figure containing the charge curves at the specified charge currents and time units. Default is cleared.

**Charge current [i_1, i_2, i_3, ...] (A)**Specify the charge currents, in amperes, used to plot the charge characteristics. Default is

`[10 20 100 500]`

.**Time units**Specify the time units (seconds, minutes, hours) used to plot the charge characteristics. Default is

`sec`

.

`m`

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_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.

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