Time delay in the charge/discharge of fractional-order capacitive energy storage devices

Electrical energy storage devices exhibit dispersive properties that control their charge and discharge processes. To get a deeper understanding of these anomalous phenomena, it is essential to go beyond static viewpoints of circuit theory in order to accurately characterize the complex interplay of internal mechanisms. Specifically, the (dis)charging time of resistive-capacitive networks is commonly estimated as four times the product of the The<acute accent>venin resistance and the capacitance itself by assuming ideal exponential relaxations in spite of the intrinsic fractional dynamics of the real energy storage materials, leading to inaccurate and erroneous characterization protocols. The purpose of this work is to provide recommended practices to find the steady-state operation of such type of devices from time-domain data with a decelerated behavior of the Mittag-Leffler function at long time scales, introducing the concept of "incremental capacitance" in the transition from ideal to fractional-order capacitor and thus, an estimation of the charge/discharge time delay. Our theoretical analysis is validated by providing a representative example of experimental application, based on an electrochemical power source, such as supercapacitors under switching-type operation. We hope to bring such study to the attention of multidisciplinary readers, both from academia and industry, focused on energy storage device research.