# Enhancing Supercapacitor Simulation Accuracy Through a Novel Hybrid Modeling Approach
## 1 Introduction  
Supercapacitors (SCs), often termed ultracapacitors or electric double-layer capacitors (EDLCs), are electrochemical energy storage systems (ESSs) that are characterized by a fast energy load and release due to their high-power density. Their specific physical and electrochemical properties allow for rapid movement of charge without the need for chemical reactions that typically limit the charge/discharge speed of batteries. This makes them superior for applications requiring quick bursts of energy compared to traditional battery storage systems [1, 2, 3, 4, 5]. One disadvantage of supercapacitors is their relatively low-energy density when compared to batteries. This makes them less effective for long-duration energy demands. One solution to this limitation is the integration of supercapacitors with batteries into Hybrid Energy Storage Systems (HESSs). The new system takes advantage of both technologies. As can be seen in the literature [6, 7, 8, 9, 10, 11], this approach has become more and more popular. HESSs offer a promising solution to address the limitations of existing ESSs and, at the same time, support the growing energy demands [12, 11]. There are multiple applications where HESSs are used. One important field where HESSs are more and more present is the automotive industry (ex. electric vehicles (EVs) and hybrid electric vehicles (HEVs)) [9; 13]. In these applications, SCs are employed to supply power during acceleration phases or to store the regenerated energy during braking. The use of SCs contributes to a better vehicle response time, improved peak power management, and ensures a longer lifespan for the HESSs. For these types of applications, an increase of 9% to 11% in the ESS system efficiency with a positive impact on the ESS life span and potential range increase for the EV/HEV was noted [14; 15]. Another important field that implements HESS systems is electrical power generation and distribution. HESSs are used, for example, in autonomous renewable energy systems, microgrids, or large-scale administration of wind and solar power parks [16; 17; 18; 19; 20]. In these systems, HESSs are used to reduce the negative influence of variation in energy production of the power sources and load demand [4]. For example, in the case of uninterruptable power supply HESS, the SCs are used to provide complete power delivery during brief grid outages. At the same time, HESS reduces the negative effect on the batteries due to the peak power applied during backup time, extending, in this way, the overall energy storage system's lifetime [21]. In the design process of the HESSs, an important step is the development of its virtual prototype. The prototype integrates the models of all HESS components, including the SC models. The HESS virtual prototype is used to create simulations, which are important tools that allow researchers and engineers to explore multiple operating scenarios, test novel energy management strategies, and optimize system performance [21; 22; 23]. The accuracy of the simulation results and the HESS efficiency and efficacy performance are directly connected with the model quality of each component [6; 12; 24]. An accurate model of the supercapacitor as a part of a HESS can bring improvements in terms of choosing the most suitable energy management technique, including energy consumption, optimal and safe operating temperatures, an increase in the energy storage units' lifespan, and overall system performance [6; 12; 24]. Furthermore, the models can be used to develop digital twins of the system, allowing, in this way, the implementation of online detection and prognosis functionalities, system/model parameter estimations, and real-time monitoring [25]. As can be seen in the literature [21; 26; 27], there are various modeling approaches that were proposed to model supercapacitors. These methods include electrochemical models, equivalent circuit models, data-driven models using machine-learning algorithms, or fractional order models [19; 28].