Modeling of an EDLC with Fractional Transfer Functions Using Mittag-Leffler Equations

Abstract

Electrochemical double-layer capacitors (EDLC), also known as supercapacitors or ultracapacitors, are devices in which diffusion phenomena play an important role. For this reason, their modeling using integer-order differential equations does not yield satisfactory results. The higher the temporal intervals are, the more problems and errors there will be when using integer-order differential equations. In this paper, a simple model of a real capacitor formed by an ideal capacitor and two parasitic resistors, one in series and the second in parallel, is used. The proposed model is based on the ideal capacitor, adding a fractional behavior to its capacity. The transfer function obtained is simple but contains elements in fractional derivatives, which makes its resolution in the time domain difficult. The temporal response has been obtained through the Mittag-Leffler equations being adapted to any EDLC input signal. Different charge and discharge signals have been tested on the EDLC allowing modeling of this device in the charge, rest, and discharge stages. The obtained parameters are few but identify with high precision the charge, rest, and discharge processes in these devices.

Comparative analysis to determine the accuracy of fractional derivatives in modeling supercapacitors

“…Although (1), (4), and (6) are linear operators and they posses some fine properties, nevertheless, they do not inherit all the operational rules from the typical integer-order derivative, such as product rule, quotient rule, chain rule, and semigroup property. These inconsistencies lead to the development of the local conformable derivative whose mostly properties coincide with the classical integer-order derivative.…”

“…Due to the porous dispersive nature of their material, carbon‐carbon, SCs are devices in which diffusion phenomena play an important role. For this reason, their modeling using integer‐order differential equations does not yield satisfactory results . In the electric power field, the current through a capacitor is proportional to the fractional order integral for electric current …”

“…For this reason, their modeling using integer-order differential equations does not yield satisfactory results. 4 In the electric power field, the current through a capacitor is proportional to the fractional order integral for electric current. 5 In Allagui et al, 6 a fractional-order energy equation was obtained allowing to quantify the energy stored in an SC, when it is loaded by means of a linear voltage ramp.…”

Various Analytical Models for Supercapacitors: A Mathematical Study

“…A similar model was proposed in [39] and analyzed using integer-order calculus. Later, this model was studied with fractional-order calculus to account for the self-discharge phenomenon more accurately [40]…”

Ensayo experimental con supercondensadores para su utilización como sistema de almacenamiento de energía

“…El modelo eléctrico equivalente de estos dispositivos no es simple y numerosos trabajos se han desarrollados para identificar los parámetros más relevantes del comportamiento del mismo [5][6] [7]. Sin embargo, las contribuciones de modelos para el almacenamiento energético son menores y mayoritariamente son sistemas eléctricos compuestos por resistencia y condensador únicamente [8].…”