**3. Description of an Energy Storage System Based on SCs**

Electric double layer capacitors (EDLCs) [30], commonly known as supercapacitors, are electrochemical capacitors composed of two conductive porous electrodes immersed in an electrolyte, between which a separator is placed. The electrodes are based on a sheet, usually made of aluminum, covered by activated carbon or carbon nanotubes. The electrolyte is the key element in determining internal resistance or equivalent series resistance (ESR). Nonaqueous solutions, such as acetonitrile or propylene carbonate, are often used because they support higher stress. The separator must allow the circulation of ions but avoid contact between the two electrodes. SCs, like conventional capacitors, store charge electrostatically and there is no charge transfer between the electrode and the electrolyte. EDLCs use double electrochemical layers to store energy. When a voltage is applied between the electrodes, the charge accumulates on the surface of the electrodes. Following the natural repulsion of oppositely charged, ions in the electrolyte solution spread through the separator into the pores of the oppositely charged electrode. These

double layers, together with an increase in surface area and a reduction in the distance between electrodes, allow SCs to achieve energy densities much higher than those of conventional capacitors [38]. Since there is no charge transfer between the electrolyte and electrode there are no changes in the chemical composition. For this reason, the storage of charge in EDLCs is reversible and is not associated with a significant loss of capacity with the number of charge and discharge cycles, which is not the case in electrochemical batteries.

SCs are especially suitable devices for high power applications with a relatively low energy capacity compared to batteries [39,40]. It has a lower cost than batteries in terms of power, a complete cycle efficiency of around 95%, the possibility of a high number of chargedischarge cycles (up to a million as per the datasheets) and a useful life of 20–25 years. The operating temperature range is between −40 and 60 ◦C, much wider than that of batteries, without significant impact on their response. The energy that a capacitor of any type is capable of storing is dependent on its capacitance and its voltage. The capacitance (C) is a constructive parameter related to the electrical charge that the device is capable of storing and that depends on constructive aspects such as the permeability of the dielectric, the area of the metallic plates and the distance between them. On the other hand, the maximum voltage between terminals depends on the insulation conditions of the dielectric that is between the two electrodes that make up the SC.

Regarding capacitance, it should be taken into account that EDLCs are not linear capacitors, but that the capacity depends on the voltage, as shown in Figure 2 [41,42]. In the case of cells, manufacturers provide values for capacitance and ESR. The capacitance value is only a mean value in the voltage operating range. For the basic cell upon which this study was based (BCAP3000 commercialized by Maxwell in the past) the maximum capacity value at the beginning of the operational life was above 3000 F, being 3000 F was the mean value. To model this evolution of the capacitance, an experimental loading process was performed in which measured capacitance was calculated in small voltage increments. Figure 2a shows the evolution of capacitance with voltage. It presents a quadratic relationship with the voltage as indicated in the following equation:

$$\mathcal{C}(V) = -95.756 \cdot V^2 + 613.58 \cdot V + 2216.6 \tag{1}$$

*V:* Voltage measured in the terminals of the SC (V); *C (V):* Capacitance as a function of the voltage (F).

**Figure 2.** (**a**) Evolution of the capacitance value as a function of the voltage in the BCAP3000 cell and (**b**) schematic of the electrical circuit used to model the behavior (voltage) of the supercapacitors (SCs).

An appropriate dimensioning of any storage system must take into account the power losses. The fundamental losses in the SCs are produced in the separator, in the positive and negative current collectors and in the positive and negative porous electrodes [40]. The total resistance of each of these parts is included in the ESR. It must be taken into account that the ESR value is not constant, but depends on the voltage in the cell, the temperature and the frequency of the current that passes through the cell. To model the losses in the SCs, which will be explained later, an electrical circuit like the one shown in Figure 2b is used [41].

Another difference between batteries and SCs is that, while in batteries the voltage is relatively constant regardless of the state of charge (SoC), in SCs the voltage is more or less linear with the SoC. For this reason, the integration of this type of ESS into industrial applications is usually done through DC/DC power converters. This characteristic means that the SCs cannot be fully discharged, since working at too low of a voltage would greatly penalize the performance of the power converter. For this reason, it is common to define a discharge limit up to half their maximum voltage, which means using <sup>3</sup> <sup>4</sup> parts of their theoretical energy, as can be seen in Equation (2).

$$E\_{\text{available}} = \frac{1}{2} \cdot \mathbb{C} \cdot \left( \mathcal{U}\_{MAX}^2 - \mathcal{U}\_{MIN}^2 \right) = \frac{1}{2} \cdot \mathbb{C} \cdot \left( \mathcal{U}\_{MAX}^2 - \left( \frac{\mathcal{U}\_{MAX}}{2} \right)^2 \right) = \frac{1}{2} \cdot \mathbb{C} \cdot \mathcal{U}\_{MAX}^2 \cdot \frac{3}{4} \tag{2}$$

*UMAX*: Maximum operating voltage measured on the SC (V); *UMIN*: Minimum operating voltage measured on the SC (V); *C*: Average value of SC capacitance (F).
