Time-varying model of self-discharge in a double layer supercapacitor with blocking layer

One of the main shortcomings of the supercapacitors is related to self-discharge phenomenon at open circuit conditions due to the existence of an internal leakage current. Using two thin insulating blocking layers on the contacts-electrolyte interface is a promising structural approach to tackle this problem. A single-branch equivalent circuit is presented to model the self-discharge behavior of a supercapacitor with 1.5 nm of PolyPhenylene Oxide as a blocking layer. A variable resistance in parallel with the equivalent capacitance is considered to model the leakage current and self-discharge procedure. It has been shown that considering a constant parallel resistance could not model the self-discharge appropriately. Hence, two different approaches are proposed to derive the time-varying parallel resistance in the model. The first approach results in an accurate continuous timevarying parallel resistance based on the equality of voltage from numerical solution to the first-order nonlinear dynamics of Tafel equation and natural response of the time-varying RC-circuit. In the second approach, an optimal number of different exponential functions are fitted to the experimental measurements from the self-discharge phenomenon based on the weighted linear regression analysis for some time intervals during one hour. In this method, optimal discrete parallel resistance values are obtained with a trade-off between model accuracy and its simplicity. It is shown that both the continuous and discrete models are accurate while the discrete model has better performance and more beneficial. Nevertheless, a relatively high sampling rate is needed for the initial time interval in which the self-discharge experiences a faster variation.