Lambert W Function Based Closed-Form Expressions Of Supercapacitor Electrical Variables In Constant Power Applications

The traditional model used to represent the electrical behavior of supercapacitors (SCs) operating at constant power leads to a well-known differential equation which allows to obtain the charge/discharge time of the device as a function of its internal voltage. However, the opposite is not true, i.e. it is necessary to resort to numerical methods to derive the internal voltage of the SC at any specific time. In this paper, new explicit expressions for the evolution of the electrical variables involved in the charge/discharge process of a SC bank operated at constant power are derived. The proposed formulation, which is based on the use of the Lambert W function, does not only allow a straightforward calculation of all the electrical variables as a function of time, but also sheds light on the direct relations between those variables. In the assumption of validity of the classic model, the results derived in this work can be considered exact, as no further approximations are made. The accuracy of the proposal is demonstrated by comparing the results derived from the new formulation with those yielded from the classical iterative resolution of the differential equations using numerical methods. The new closed-form expressions presented in this paper have the potential to simplify the sizing, regulation and control of power applications with embedded SC banks operated at constant power. (C) 2020 The Author(s). Published by Elsevier Ltd.