## 1 Introduction
The thermal resistance and capacitance required to build the thermal model of the SC are provided by manufacturers as well, thus allowing its implementation. The main operation modes of the SCs are the constant-current and constant-power charge and discharge. In any operation mode, both sources and loads may act like constant-power systems in a large number of power conversion systems (e.g., those involving Variable Speed Drives and DC-DC converters [32; 33]). On the contrary, charging processes conducted by means of real voltage sources are not common in industrial applications because of its poor efficiency. As demonstrated by [PERSON] et al. [39], efficiency cannot surpass the 50% when the SC starts charging at a nil voltage. Finally, [PERSON] et al. [40] referred to various applications where SCs can be discharged through a constant resistor, as is the case of heaters and speakers. Several studies have focused on studying SCs in constant-power applications. For instance, the analysis of the efficiency and losses of SCs operating at constant power is presented in [41]. The traditional RC-series model considered in [42; 43; 44] has also been used by other researchers to compute the autonomy time of SCs discharging at constant power. In this regard, an approximate method based on an energy balance is proposed in [33] and an accurate equation expressed as a function of the internal voltage is obtained in [32]. Moreover, [PERSON] gave an alternative formula that produces excellent results in [45]. [PERSON] et al. presented in [46] a complete analysis of the main electrical variables involved in the constant-power charge and discharge operation that enables the expression of any of those variables as a function of the rest. The thermal behavior of SCs is of paramount importance when it comes to the design of an energy storage system, its installation, operation, service, and lifetime. In this respect, it is crucial to analyze the temperature the SCs may reach when being subjected to a succession of charging and discharging processes. If temperature is accurately calculated, the most suitable and efficient refrigeration system can be selected. The SCs array exact dimensioning will be possible, and its lifespan will also be assessed, since it depends on the voltage range and average temperature of the individual cells during operation [47; 48; 49; 50; 51; 52; 53]. Whereas the analysis of the temperature of the cells when SCs are charged or discharged at constant current is well documented [54; 55], the number of references that can be found on the subject for constant-power operation is scarce. A simplified study based on the same exponential behavior as the one considered in the constant-current analysis is undertook in [43]. This study is only acceptable when the thermal capacitance is particularly high. [PERSON] et al. [56] developed a mathematical expression to assess the instantaneous temperature of the cells when they operate at constant power. Such an expression is valid for any value of the thermal capacitance; however, the obtained results are not accurate under high power demand conditions due to the various simplifications that are made. This paper aims at providing a formula free of such simplifications and, hence, able to render an improved evolution of the temperature of the cells over time, as well as the one of any other electrical variables intervening in the constant-power charging and discharging processes. This extension of the theoretical background can be useful to implement simple tools for the pre-design of the cooling system of SCs arrays, with a great potential to simplify the conduction of final experimental tests. It is important to point out that, even if the power profile imposed to the SC array is not constant, as in the example shown in Figure 1, it can always be discretized to a stepwise function, either using a fixed or variable time step, \(\mathfrak{t}_{i}\).