## 1 Introduction  
The high capacitance and low series-resistance of supercapacitors (SCs) provide them with a significant specific power, high charging and discharging currents, and a notable efficiency. Therefore, SCs are increasingly used in a wide range of industrial pieces of equipment as the principal energy storage system or, more frequently, as a support to devices with a higher energy storage capacity, albeit dynamically poorer because of its slower response, as in the case of batteries and fuel cells [1, 2, 3, 4, 5]. Together with them, SCs make up hybrid energy storage units installed in electric vehicles [6, 7, 8, 9], renewable energy systems [10, 11, 12, 13], microgrids [14, 15], and public transport vehicles [16]. This hybrid combination presents particular uses as well, such as public works machinery [17], military [18] and recreational vehicles [19], and even maritime transport [20]. Diverse models to analyze the behavior of SCs have been already developed. [PERSON] et al. [21] classified such models into four categories: electrochemical, equivalent-circuit-based, neural network, and fractional-order models. Although neural network models, such as the one developed by [PERSON] et al. [22], remarkably capture the dynamic features of the cell, complex calculations are needed and variables that must be experimentally measured are frequently required. Moreover, several of these variables have no physical meaning and cannot be easily incorporated into an electrical model. On another note, fractional-order models were introduced to improve the accuracy of the reproduction of the behavior of SCs [23] and to identify their parameters [24, 25]. The order of the differential equations defining them is a non-integer value, unlike that of ordinary equations. Furthermore, a series-connected component (called a Warburg-like element) is added, whereas it is absent from the other aforementioned models. Equivalent-circuit-based models are the most widely chosen due to their simplicity and the use of ordinary differential equations. Numerous variants of these models have already been developed and tested. Models with multiple parallel-connected RC branches provide results quite similar to those given by an actual SC. The number of branches in parallel does not tend to be higher than three, and each branch is characterized by having a specific time constant. Their dynamics are notable, especially at voltages higher than 40% of the rated value, and their accuracy can be improved by incorporating further branches, as stated by [PERSON] et al. [26]. Aside from this, the physical structure and dynamic response of SCs is notably improved when using transmission-line RC models. However, they require intricate analytical formulas, and their parameters are not provided by the manufacturers. Other RC models do not consider the capacitance to be constant, but variable with the internal voltage [27; 28]. Although the results provided by such models are good, their complexity makes it difficult to build analytical equations for the operation at constant power, they are, thus, suitable and commonly used for numerical simulation. When the study is aimed at the self-discharge processes, a parallel-connected resistor is incorporated. This resistor is assumed to be constant by [PERSON] et al. [29] or treated as variable by [PERSON] et al. [30]. Despite the huge number of possibilities that can be found in the previous research, the RC-series model is most widely used. Both the resistance and the capacitance are provided by manufacturers and can be also obtained by means of standard experimental tests [31]. The constant-parameter RC-series model [32; 33; 34] renders reasonably simple analytical formulas when analyzing the SCs charge and discharge at constant power. Moreover, the well-known analogy between electric and thermal circuits is used in this model to assess the temperature of the SC [35; 36; 37; 38].