  Supercapacitors are mostly recognized for their high power density
capabilities and fast response time when compared to secondary batteries.
However, computing their power in response to a given excitation using the
standard formul{\ae} of capacitors is misleading and erroneous because
supercapacitors are actually non-ideal capacitive devices that cannot be
characterized with a single constant capacitance. In this study we show how to
estimate accurately the time-domain power and energy of supercapacitors in
response to any excitation signal represented in terms of its Fourier series
coefficients with the sole knowledge of the frequency-domain impedance
parameters of device. The presented theory is first verified and validated with
simulations conducted on an equivalent fifth-order RC circuit emulating the
behavior of a fractional circuit consisting of a resistance (Rs) in series with
a constant phase element (CPE) of fractional impedance ZCPE =
1/C{\alpha}s{\alpha}. Then we do the same for a commercial supercapacitor
modeled as an Rs-CPE circuit, and subjected to both a periodic triangular
voltage waveform and a random voltage excitation. The results are conclusive
and very promising for adopting the proposed procedure to estimate the power
and energy performance of supercapacitors in response to real-world charging
and discharging signals.
