Fractional Energy: A Theoretical Characterization of the State of Charge of the Ultracapacitor Modeled as a Constant Phase Element

Based on the Infinite State Representation (ISR) of the Riemann–Liouville integral, the energy stored in a fractional-order integrator is revisited, together with the energy dissipated through Joule losses. Using an idealized ultracapacitor model based on the Constant Phase Element (CPE), i.e., a fractional-order capacitor, theoretical expressions for the stored and dissipated energies during current charging of the CPE are derived. Numerical simulation of the fractional integrator over a frequency interval {ωmin, ωmax} validates a realistic CPE model, in which low-frequency modes correspond to energy storage, while high-frequency modes account for self-discharge and the origin of dissipated energy. This theoretical study leads to the definition of a new ultracapacitor model composed of an internal resistor and the previous realistic CPE, whose frequency-distributed representation enables prediction of the internal state variables and, consequently, the State of Charge.