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Implement generic supercapacitor model

The Supercapacitor block implements a generic model parameterized to represent most popular types of supercapacitors. The block implements the following Stern equation and Tafel equation:

Variable | Description |
---|---|

A | Interfacial area between electrodes and electrolyte (m2) |

c | Molar concentration (mol m −3) equal to c = 0.86/(8NAr3) |

F | Faraday constant |

i | Current density (Am−2) |

if | Leakage current (A) |

i0 | Exchange current density i0 = if/A (Am−2) |

k | Stefan-Boltzmann constant |

N | Number of layers of electrodes |

NA | Avogadro constant |

Np | Number of parallel supercapacitors |

Ns | Number of series supercapacitors |

Q | Electric charge (C) |

R | Ideal gas constant |

r | Molecular radius equal x2 |

x2 | Helmholtz layer length (m) |

α | Charge transfer coefficient, Tafel equation (0<alpha<1) |

ε | Permittivity of material |

ε0 | Permittivity of free space |

ΔV | Over potential |

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

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

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

**Surge voltage (V)**Specify the surge voltage or maximum voltage of the supercapacitor. Surge voltage corresponds to the supercapacitor voltage when internal electrolyte becomes gas.

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

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

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

**Leakage current (A)**Specify the leakage current of the supercapacitor, in amperes.

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

**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 4.57% for self discharge and maximum error of 1.2% for charge and discharge when using the predetermined parameters.

When this check box is selected, the

**Number of layer**,**Molecular radius (m)**,**Overpotential (V)**,**Charge transfer coefficient alpha**parameters appear dimmed.**Number of layers**Specify the number of layers related to the Stern model.

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

**Over-potential (V)**Specify the over-potential related to Tafel equation, in volts.

**Charge transfer coefficient (R)**Specify the charge transfer coefficient related to Tafel equation as a value between 0 and 1.

**Plot capacitor vs time**When this check box is selected, the block plots a figure containing three graphs. The first graph represents capacitance as a function of time, the second graph represents voltage as a function of time, and the third represents current as a function of time.

**Charge current (A)**Specify the charge current, in ampheres, used in the plot Capacitor vs Time and shown in the third graph.

**Time (s)**Specify the maximum time value of the plot, in seconds.

`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:

Internal resistance and capacitance are assumed constant during the charge and the discharge cycles.

The model does not take into account temperature effect.

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``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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