Supercapacitors in Constant-Power Applications: Mathematical Analysis for the Calculation of Temperature

Abstract

A set of analytical equations for the calculation of the temperature in supercapacitors operating in constant-power applications is presented in this paper. Although the main operation modes of supercapacitors are constant-current and constant-power charge and discharge, this study was focused on the latter, since both sources and loads act as constant-power systems in a wide range of power conversion facilities. The starting point of this study is the classical supercapacitor model based on electrical and thermal parameters provided by manufacturers or also obtained by experimental means. The proposed mathematical analysis is based on the so-called incomplete gamma function that presents two major advantages over previously existing methods. Firstly, it is not necessary to solve any differential equations system by means of numerical methods, which reduces the required computational effort. Secondly, no simplifications to relief the calculations are made in the computation of any variable. The new formulation renders valid solutions even for high-power demand situations. Moreover, the temperature of the supercapacitor can be expressed as a function of time or any other electrical variable in the charging and discharging processes. Therefore, the proposed formulas are especially remarkable for the electrical and thermal dimensioning of supercapacitors.

Supercapacitor Constant-Current and Constant-Power Charging and Discharging Comparison under Equal Boundary Conditions for DC Microgrid Application

“…It should be noted that, even though there is a discharge, the Equation (32) assumes that the power P is positive. The formulation of the problem statement for the given situation would be how long it would take the SC circuit voltage to decrease from V C2 to V C1 if a consumer, connected to this SC circuit, consumes a constant power P. However, the next option of interest is to calculate the constant power P that must be consumed by a consumer to decrease the SC circuit voltage from V C2 to V C1 during time duration t. As with Equation ( 21), the same conclusions can be drawn that the formula for calculating the power P cannot be derived directly from Equation (32), so the same two options exist with the use of a mathematical computing program or the use of a pre-calculated table of time t and P values.…”

“…The formula for calculating the discharging time t is derived by integrating the Equation (36), if the voltage V C changes from V C2 to V C1 over the time duration t under the condition that V C2 > V C1 , resulting in the expression [30,32]:…”

“…For example, refs. [30][31][32] describe the process of SC constant-power discharging by calculating the electrical parameters-current, internal voltage, external voltage etc.-using the Lambert W function in real time. The research was performed with theoretical calculations and simulations based on an RC circuit.…”

Sizing Methodology of a Fast Charger for Public Service Electric Vehicles Based on Supercapacitors

“…For this purpose, the RC series model of SCs will be used. This model is the most widely used for sizing [22][23][24][25][26][27] and provides simple analytical expressions when studying constant power and charging or discharging processes [28][29][30]. The SC resistance and capacitance values are provided by the manufacturers from standardized tests.…”

A Novel Numerical Approach in Solving Fractional Neutral Pantograph Equations via the ARA Integral Transform

“…For example, fractional derivatives have been used to describe chemical processes, mathematical biology, and many other problems in physics and engineering. More specific information about differential integral equations and fractional calculus can be found in [1][2][3][4][5][6][7][8][9][10][11][12].…”