  Electrochemical capacitors are a class of energy devices in which complex
mechanisms of accumulation and dissipation of electric energy take place when
connected to a charging or discharging power system. Reliably modeling their
frequency-domain and time-domain behaviors is crucial for their proper design
and integration in engineering applications, knowing that electrochemical
capacitors in general exhibit anomalous tendency that cannot be adequately
captured with traditional integer-order-based models. In this study we first
review some of the widely used fractional-oder models for the description of
impedance and relaxation functions of dissipative resistive-capacitive system,
namely the Cole-Cole, Davidson-Cole, and Havriliak-Negami models. We then
propose and derive new q-deformed models based on modified evolution equations
for the charge or voltage when the device is discharged into a parallel
resistive load. We verify our results on anomalous spectral impedance response
and time-domain relaxation data for voltage and charge obtained from a
commercial supercapacitor.
