Thermal model of supercapacitors operating in constant power applications: New mathematical expressions for precise calculation of temperature change

Supercapacitors (SCs) are key enablers that can bridge the gap between batteries and conventional capacitors. Certainly, their different merits in high power density, rapid rates of charging/discharging, long life cycles, their ability to handle high levels of charge and discharge currents, and the small values of their internal parameters have drawn attention in modern power systems and energy applications. Unfortunately, although SCs have simple equivalent circuits, the analyses, testing, and mathematical description of these circuits have not been completely resolved. This paper deals with the presentation of new accurate mathematical formulations that allow the calculation of the temperature change of SCs engaged in constant power-based applications precisely in charging and discharging. To achieve this goal, firstly, a comparison is made between well-known mathematical expressions in literature to describe all electrical variables (current, internal voltage, external voltage, power loss, etc.). Secondly, since the discharge process can be accurately represented by a transcendent equation with two solutions, a fast iterative method for solving this equation is proposed and tested. Thirdly, a new mathematical expression is derived for the temperature change of SCs during both the charge and discharge processes without assuming or neglecting SC parameters. Further, a comparison of the proposed expressions with literature-known solutions, as well as numerical results, is presented and discussed. Fourthly, the results of SC temperature change during both charging and discharging are also compared with changing parameters and operating conditions. The findings and analyses confirm that the derived mathematical formulations, proposed in this work, are precise and very close to the numerical results that can facilitate the integrity of the investigation and control of SCs contributing to the work in constant power-based applications.