# Enhancing Supercapacitor Simulation Accuracy Through a Novel Hybrid Modeling Approach
## 2 SC Modeling and Parameter Estimation Methods
### Supercapacitor Modeling Approaches
#### 2.1.2 Equivalent Circuit Modeling  
RC (Resistor-Capacitor) equivalent circuit modeling is a popular and effective method for simulating the electrical characteristics of SCs. The equivalent circuit models make use of parameterized RC networks that reproduce the electrical behavior of the SC. The literature reveals various approaches to modeling an SC based on the number of RC branches. [PERSON] [8] explored models with one, two, and multiple RC branches, emphasizing the distinctions in terms of accuracy and simulation time. [PERSON] [33] advocated for a three-RC model and proposed a parameter estimation method, underscoring the accuracy achieved through this technique. Additionally, [PERSON] [34] introduced a simplified two-RC model derived from [PERSON]'s model and method. The advantage of using this type of circuit relies on the possibility of describing their dynamics using ordinary differential equations (ODEs) [21; 26]. This ensures simplicity and facilitates the simulation of this type of model. Figure 1: Electrochemical characteristics and operating principle of supercapacitors. The simplest equivalent model uses a single RC branch to represent the behavior of the supercapacitor. In this way, the SC is modeled similarly to an electrolytic capacitor [18; 21]. The model consists of a series resistor representing the overall resistance R\({}_{\text{SC}}\), a leakage resistor R\({}_{\text{f}}\), and a capacitor representing the canonical capacitance C\({}_{\text{SC}}\)[10; 31]. All these components form the basis of a single-branch RC circuit for modeling a supercapacitor, as presented in Figure 2.  
Equations (1) and (2) describe the electrical behavior of the one RC branch model, whereas the V\({}_{\text{C}}\) is the voltage across the capacitor C\({}_{\text{SC}}\)[21]. \[\frac{\text{d}\text{V}_{\text{C}}}{\text{dt}}=-\frac{\text{V}_{\text{C}}}{ \text{C}\cdot\text{R}_{\text{f}}}+\frac{\text{I}_{\text{sc}}}{\text{C}} \tag{1}\]  
\[\text{V}_{\text{sc}}=\text{V}_{\text{C}}+\text{I}_{\text{sc}}\text{R}_{\text{ sc}} \tag{2}\]  
However, the basic capacitor does not account for the physical processes at the electrical double-layer interfaces or the fluctuation in capacitance as a function of voltage. For this reason, models with multiple RC branches were proposed for modeling the dynamics of SCs. Still, this representation offers an assessment of the overall behavior [29; 41]. To better describe the behavior of SCs, multibranch models are built by connecting multiple RC branches with different time constants, capturing both static and dynamic responses with greater accuracy than a single RC model [28]. These types of models typically consist of multiple RC groups arranged in parallel, resembling specific physical behaviors of SCs [37]. In this way, the accuracy of high-frequency applications is improved [13]. This type of model considers the phenomenon of load propagation that manifests itself in the voltage of the components immediately after sudden changes in the applied current [8; 29]. In an attempt to address a better account for the slow behavior of supercapacitors, this method represents the propagation of charges during the transient phase using a simplified transmission line model as shown in Figure 3.  
Figure 3: Multiple RC branches equivalent circuit. Figure 2: Single-RC equivalent circuit. The equations regarding multiple RC branches equivalent circuit models for the supercapacitor are shown below where C(V\({}_{\text{n}}\)) is the capacitance at the particular voltage V\({}_{\text{n}}\), and V is the current voltage [29].