## 1 Introduction  
Improving the efficiency of energy recovery processes in relevant applications, such as the regenerative braking in vehicles, elevators and cranes, requires energy storage devices that enable faster charging and discharging compared to existing lithium-ion batteries, which are the highest energy density batteries currently available on the market [1]. Capacitors are also able to charge and discharge much faster than existing batteries, performing hundreds of thousands of charge/discharge cycles without significant loss of capacity. However, the energy storage density of capacitors is generally significantly lower compared to batteries. Lead batteries include three essential elements: sulfuric acid, used as an electrolyte, and lead and lead dioxide, used as a negative and a positive electrode. Each cell is able to supply a voltage of about 2 volts, while the current is a function of the electrode surface. The electrode surface is thus the characterizing element of this device, and several studies can be found in the literature, discussing possible solutions to obtain electrodes with a high active surface, thus improving the reactions occurring at the electrode-electrolyte interface, in order to ensure electrical continuity and an increased number of charge/discharge cycles. However, due to the low specific energy storage of these devices (about 30-40 Whg\({}^{-1}\)), compared to other batteries, and considering that current lead-acid batteries work in optimal conditions at a maximum discharge current (C-rate) of C/5 [2], their application is limited in advanced systems. In addition, due to the high atomic weight of lead, such batteries are characterized by a low specific energy and, consequently, by a high battery weight. Another relevant drawback is related to the low utilization degree of active pastes. In fact, during the charge/discharge cycles, the active pastes are subject to the desulphation-sulphation reactions. In particular, during discharge, there is the formation of large and non-conductive PbSO\({}_{4}\) crystals, both in the positive and negative plate, which progressively reduce the active surface [3]. Moreover, the continuous volume variations due to the sulphation/desulphation reactions cause the mechanical instability of the plates, which causes a progressive deterioration and a gradual loss of capacity [4]. Lead-acid batteries are thus less competitive than other rechargeable batteries. Different studies have been conducted over the years to improve the performance of lead batteries, to increase the number of life cycles, to increase their energy density and to decrease sulphation phenomena. The approaches currently employed in such regard involve the use of additives, such as different types of carbon powder, carbon nanotubes, titanium dioxide, glass fibers, silicon dioxide and aluminum oxide [5], carboxes [6] and graphene [7]. [PERSON] et al. [8] studied the effect of adding TiO\({}_{2}\) in the positive active material (PAM) and carbon nanostructures on the negative active material (NAM), obtaining small tetrabasic sulfate crystals that increase the exposed active surface area and thus achieve longer battery life and capacity by retarding sulfate formation. [PERSON] et al. [9] also achieved improved cycle life (up to 3600 cycles) and increased the overvoltage of hydrogen evolution under high-rate partial state-of-charge conditions (HRPSoC) by adding ZnO and nanocarbon in the NAM. Another peculiarity of these batteries is that their performance is highly dependent on temperature. In their studies, in fact, the authors achieved an excellent efficiency at high temperature due to kinetic reasons, but nevertheless, with each 8 \({}^{\circ}\)C increase in temperature, they reported a reduction of the battery life by up to half [10].