Abstract
•SCs equivalent circuit models have been widely applied because of their accuracy.•Mathematical analysis that enables calculation of SCs variables has been provided.•The mathematical expressions for the SCs variables are transcendental.•Analytical solutions for the charging and discharging equations are derived and solved.•Proposed solutions contribute to sound modeling of SC in constant power applications.
Supercapacitor (SC) is one of the most trending energy storage solutions. The SCs equivalent circuit models have been extensively applied to energy management because of their accuracy and simplicity. However, there is a difficulty in solving the conventional differential equation-based model used to characterize the electrical behavior of SCs operating in constant power applications. Hence, numerical or metaheuristic techniques have been used in the literature to derive SC's internal voltage at any time. In this work, a thorough mathematical analysis that enables a precise calculation of the electrical variables implied in the charge/discharge processes of SCs operated at constant power as a function of time is presented. First, the transcendental discharge voltage expression of SCs operating at constant power is formulated, and then it was solved using the Special Trans Function theory (STFT). The precision of calculation of the method used for solving the transcendental expression is presented and discussed. Second, the transcendental voltage of charging expression of SCs operating at constant power as a function of time is also formulated, and then it was solved using Lambert W equation. Third, a comparison of different methods for solving mentioned equations is presented. Fourth, the electrical variables involved in the charge/discharge processes of SCs – voltage, current, power, energy, state of charge, and power loss are investigated. Furthermore, the results obtained for the variation of parameters and their operating conditions demonstrate the proposed equations’ applicability and accuracy. Finally, the results obtained validate that the closed-form expressions suggested in this paper are accurate and straightforward, which can contribute to proper modeling, investigation, sizing, regulation, and control of constant power SCs in modern energy systems.
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