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

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.