# Enhancing Supercapacitor Simulation Accuracy Through a Novel Hybrid Modeling Approach
## 3 HyMASC Method
### Parameter Identification and Validation  
For the above presented model, all the parameters that define the model must be determined. A two-step procedure is used for this process.  
_Step 1--Parameter estimation_: In the first step, the initial values for each of the following parameters, R1, C0, Cv, R2, C2, R3, and C3, are determined. For this, [PERSON]'s method is employed [30]. The method uses a set of experimental data that describes the charging and relaxation curve. The charging curve presents the Vsc variation from 0 V to the rated voltage of the modeled SC for a predefined constant input current Isc. The relaxation curve describes the Vsc variation after the charging phase is completed and the input current Isc is set to zero.  
The quality of the model parameter estimation depends highly on the DAQ system used. The precision and performance of the DAQ influence the data acquisition accuracy and the charging curve analysis (evaluation of the rising time in different segments of the curve), which varies depending on the applied current for charging. To determine the parameters using the electrical behavior of an SC, [PERSON] described a series of events using different points along the measured voltage curve. With each event, the equivalent model parameters are calculated using equations that describe the SC internal charge distribution as described below:  
* The fast branch's parameters are determined from the charging phase of the SC. For this, a fully discharged SC is charged with a constant current I\({}_{\text{sc}}\) until it reaches the rated voltage. It is considered that the initial charge is stored in the first branch. These parameters are calculated based on the rise and fall time of the applied current. This takes place in a matter of milliseconds, which is considerably smaller compared with the timestamps of the other two branches.
* The delayed branch's parameters are calculated based on the internal equivalent current that flows through this branch after the current was applied and the SC reached its rated voltage. This leads to charge redistribution within this branch with a time constant larger than the previous one.
* The long-term branch's parameters are determined based on the charge redistribution through the internal current that flows from the other branches. At the beginning of this stage, the long-term capacitor is fully discharged. All the above-mentioned steps follow the proposed method presented in [30].  
_Step 2--Parameter optimization_: In the second step, the obtained values are introduced in the Simscape model (Figure 7), and optimized values are calculated using the MATLAB Parameter Estimator module. In the process, experimental data are used. The data should describe a charging and relaxation curve, or a complete functioning cycle that comprises a charging, relaxation, and discharging curve. An example of an experimental stand that can be used to record the experimental data is presented in Figure 8. The system comprises a current source (used during charging phase), a current load (used during discharge phase), the supercapacitor that is modelled, a DAQ system, and a PC used for storing and analyzing the data. The experimental setup facilitates the measurement of the charging, relaxation, and discharging curves of the SC.  
Figure 7: Simulink model for the supercapacitor.  
The experimental data, together with the initial values (obtained in the previous step) for the parameters, are introduced in the MATLAB/Simscape model. Using the MATLAB Parameter Estimator application, an iterative process based on the nonlinear least-squares method is implemented. The algorithm searches for optimized values of R\({}_{1}\), C\({}_{0}\), C\({}_{v}\), R\({}_{2}\), C\({}_{2}\), R\({}_{3}\), and C\({}_{3}\) parameters that better fit the response curves of the model in accordance with the experimental data. To overcome the downside regarding MATLAB estimation's multiple solutions, an improvement is proposed by conditioning the parameters' range of values.