## 1 Introduction
### Main Modes of Operation of the DLSCs  
When working with DLSCs, in most industrial applications, four modes of operation are usual: charging/discharging at a constant current, constant voltage, constant power, constant resistance or combinations of two or more of the above. A summary of the typical applications for each case, emphasizing the charging/discharging through a voltage source or discharging through a constant resistance, which is the most relevant to this study, are presented below. Full charging of an SC, using only a voltage source, is not usually performed in industrial applications, since, if starting from zero voltage, the charging efficiency cannot exceed 50% [19]. However, the higher the initial voltage of the SC, the higher the efficiency when charging. In fact, if the SC starts charging from a voltage that is 50% of that which provides the source, the charging efficiency rises to 75%. It is therefore common to split the charge into two stages. The first stage is at a constant and high current, until the SC is close to full charge. In the second stage, a voltage source is used, whose unloaded voltage is the SC's rated one. This ensures a full load up to the rated voltage of the SC with very high efficiency. This procedure is known as the constant current-constant voltage method (CC/CV) [20; 21; 22]. Constant current charging and discharging is also used by cell manufacturers to obtain the electrical parameters of the constant capacity RC series model, by applying standardized methods, and also to obtain the thermal parameters (resistance and thermal capacity). Recent studies have raised the possibility of integrating SCs into photovoltaic panels, so that the panel itself includes an energy storage system [23; 24; 25]. This new concept can have many applications for off-grid systems, such as uninterruptible power supply systems, as a backup in case of power failure in mobile phone towers or as off-grid lighting systems. In these applications, where DLSCs are directly connected to solar panels, over a wide range of voltages, during the charging process of the SCs, the panel can be considered as a real current source or as a real voltage source and, in both cases, it can be reduced to the latter. In many power conversion applications, both the load and the source can be modelled as a system operating at a constant power [26; 27; 28], for example, in emergency lighting systems and DC micro-grids. In other cases, the load can be represented as a constant value resistor [29; 30; 31]. [PERSON] et al. [29] proposed an instantaneous water heating system for domestic use, based on SCs, where a low voltage storage accumulator is needed, for safety reasons, but with the capacity to deliver the stored energy in a very short time. The aim is to heat the cold water remaining in the pipes very quickly to avoid waste. In this case, the SCs are discharged through a heating resistance that can be considered constant. The material used in their manufacturing is usually nichrome, an alloy whose resistivity is practically invariant in a wide range of temperatures. In addition, the heat produced will be quickly transferred to the water, for a very short time, causing the maximum temperature of the material to remain not too high. As the resistance of the heating system is much higher than the internal resistance of the SC bank itself, this application has a very high efficiency; practically all the energy extracted from the SC is dissipated in the heating resistor. A usual case where SCs can be considered to be discharged through a constant resistance appears when studying the phenomenon of self-discharge. The cells discharge their stored energy through their own leakage resistance, which can be considered constant and of a high value [32]. Another industrial application of SCs in which they are used as a capacitive voltage source is in welding guns [33; 34], where a bank of SCs replaces transformers.