# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 1 Introduction
Figure 1: \(Cont.\)  
A more sophisticated SC model than the previous one is depicted in Figure 1b, where the capacitance is not constant but varies linearly with the internal voltage [24; 25; 26; 27]. This model accurately emulates the behavior of the SC when it is rapidly charged or discharged (short-term characterization), but its accuracy may be insufficient to analyze the cell response when it is charged or discharged slowly (medium- or long-term characterization), due to the charge redistribution phenomena that occur inside the SC [28]. To enhance the precision of this model, additional RC branches are frequently connected in parallel, typically of constant capacity, with time constants that are considerably larger than those of the primary branch [29; 30; 31]. Two-branch (Figure 1c) or three-branch (Figure 1d) models can be employed, and even more branches can be added. As in the constant capacitance model, the leakage resistance, R\({}_{\rm L}\), can be included if self-discharge requires to be analyzed. The models with two or more branches, despite their ability to accurately simulate the behavior of the SC in all situations, present a disadvantage in that they are complex to express mathematically in terms of the main operating variables, such as current and voltage. Consequently, these models are often considered using electrical circuit simulation software. Other models that accurately reproduce the dynamic behavior of SCs are transmission line models (Figure 1e). However, working with them requires numerical simulation tools applied to a model carefully adjusted to the characteristics of the cell. Unfortunately, this adjustment, which involves measuring all the components using complex estimation methods, limits their use to very specific studies. In fact, the high accuracy provided by these models is rarely needed in industrial applications. Among the models based on electrical circuits that have been presented by other researchers, two are particularly suitable to be used in most industrial applications, combining simplicity and good accuracy. The one-branch variable-capacity model (Figure 1b) is valid for the study of short-term behavior, while the two-branch model shown in Figure 1c is suitable for characterizing the behavior from short-term to medium- to long-term operation. Both models do not require the calculation of many components, as in more complex approaches, and provide sufficiently accurate results for most studies requiring the analysis of SCs. However, manufacturers do not provide the values of all the parameters of these models, so estimation procedures from laboratory tests have been developed in previous studies. These methods typically use charging or discharging the cell with a very high constant current source. Figure 1: Some of the most widely used RC circuit-based SC models. The main operating modes of SCs can be summarized as follows: constant current charge/discharge, constant power, constant voltage real source, or constant resistance discharge. Although the first two are widely used, the latter also appear in many industrial applications. For example, when charging an SC, it is common to do so in two stages. The first one is carried out at a constant current of a high value. To avoid damaging the SC, the external voltage of the cell is controlled. When the rated value of the cell is reached externally, U\({}_{\text{N}}\), the internal voltage will not yet have reached this rated voltage due to the voltage drop across its series resistor, so the cell will not be 100% charged [32]. Therefore, in the second phase, the charge is completed using a real constant voltage source, whose no-load voltage is equal to U\({}_{\text{N}}\). In this way, full charge is achieved with high efficiency and in a relatively short time. This method is known as the constant-current, constant voltage method (CC/CV) [33; 34; 35]. Recent works have explored the possibility of installing SCs on solar panels, so that they have their own built-in energy storage system.