# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 1 Introduction  
Supercapacitors (SCs) are energy storage devices used in many industrial applications due to their unique characteristics. These include their high specific power, which allows them to be charged and discharged very quickly. Their low internal resistance makes them to dissipate very little energy in the form of heat, which provides them with a high efficiency. Moreover, their useful life, under certain temperature conditions, can reach up to a million cycles, a value that is well above other storage systems, such as batteries, which makes them a good option in those applications that require a continuous exchange of energy. These devices have a wide operating temperature range, typically between \(-40\)\({}^{\circ}\)C and \(65\)\({}^{\circ}\)C. Between \(0\)\({}^{\circ}\)C and \(65\)\({}^{\circ}\)C, their electrical parameters (resistance and capacitance) are quite stable and vary very little. In addition, at the end of their useful life, they can be easily recycled since they do not contain polluting or toxic chemical elements. The main component of porous membranes is carbon, an abundant material that is also composable [1; 2; 3; 4; 5; 6; 7]. For all these reasons, these devices are a good option in many facilities. In any industrial application where SCs are used it is essential to have a perfect understanding of their performance in different situations. Therefore, it is necessary to develop equivalent circuit models that combine simplicity and good accuracy. There are a multitude of SC models in the literature, which can essentially be divided into four categories [8]: electrochemical models, equivalent-circuit-based models, neural-network-based models, and fractional-order models. The final two groups, despite their ability to accurately reproduce the dynamic characteristics of SCs, do have some disadvantages. Onesuch drawback is that they require a set of parameters that are not specified in the datasheets provided by manufacturers. In some cases, some of their variables do not even have a physical meaning and must be measured in laboratory tests using complex estimation algorithms. In addition, the accuracy they provide is rarely a necessary requirement for most industrial applications. This, together with their complexity, leads many researchers to opt for simpler solutions, even if they slightly penalize accuracy. In these cases, equivalent circuit models are usually chosen, as they can be used to obtain simple mathematical expressions for the different operating modes, producing results that, in the vast majority of cases, correctly characterize the real behavior of the SC. Moreover, these models can be employed in electrical circuit simulation environments. The most common circuit-based models are shown in Figure 1. The first one (Figure 1a) is the constant capacitance RC series model. This representation of the SC is the most widely used due to its extreme simplicity and the existence of standardized test protocols, from which manufacturers provide its parameters in catalogs or datasheets. Moreover, it allows analytical expressions to be obtained for the four main operating modes: constant voltage charge/discharge, constant current, constant power [9; 10; 11; 12], and constant resistance discharge, for both electrical and thermal analyses [13; 14; 15; 16]. For the aforementioned reasons and because its results are more than acceptable, it is mostly used in the sizing of SC banks [17; 18; 19; 20; 21; 22]. In studies of self-discharge, a leakage resistor (R\({}_{\rm L}\)) is connected in parallel with the capacitance. However, in the majority of other studies, this is typically considered to be negligible, given that its ohmic value is very high, and therefore, the current flowing through it is very low. Although many researchers consider R\({}_{\rm L}\) to be constant, [PERSON] et al. [23] have treated it as a variable resistance dependent on the internal voltage. However, there are no mathematical expressions that allow its analytical function to be determined by laboratory testing.