# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 3 Methods
### Least Squares Method (LSM) Using the External Voltage, u\({}_{\rm co}\)(t)
The value of R is first calculated, and its final expression, after some operations, is as follows:  
\[\rm{R=\frac{\gamma}{\beta}\cdot\frac{R_{c}\cdot\left(U_{0}-E\right)}{W_{0} \left(\frac{\gamma}{\beta}\cdot\exp\left(-\frac{1}{\beta}\cdot(\alpha+\gamma \cdot E)\right)\right)}-R_{c}}. \tag{35}\]  
Once the value of R has been obtained, the values of \(\rm{C_{0}}\) and \(\rm{k_{c}}\) are obtained by using the following formulas:  
\[\rm{k_{c}}=\frac{-\gamma\cdot R_{c}}{2\cdot\left(R_{c}+R\right)^{2}}. \tag{36}\]  
\[\rm{C_{0}}=\frac{-\beta}{\left(R_{c}+R\right)}-2\cdot\rm{k_{c}}\cdot E. \tag{37}\]  
For this method to yield accurate results, the charging or discharging process must be completed in a relatively short period of time. This is necessary to prevent the slower dynamics of the secondary branch of the SC from influencing the primary branch. Therefore, it is advisable to use a real voltage source with a low internal resistance \(\rm{R_{c}}\). The voltage of this source may be higher than the rated voltage of the SC, provided that the charging is stopped before the SC exceeds its rated value. If it is not desired that the source initially injects a too high a current, it is sufficient to connect a resistor of a known value in series with the SC to limit it. This avoids the possibility of damaging the voltage source. However, it is necessary to find a commitment between the current limitation and the charging time, as the latter should not be slowed down too much. If the tested SC has a low-rated capacitance and during the test, it is observed that it has reached the steady state, a shorter charging time must be taken to ensure that the first branch is far from reaching the value of the voltage source E. This method is applicable to both a charge (E > U\({}_{0}\)) and a discharge (E < U\({}_{0}\)), as well as to a discharge through a constant resistance (E = 0). The 3 PM method has the advantage that it only needs three measurements of the voltage u\({}_{\text{co}}\)(t) and the three parameters are very easily obtained using simple mathematical functions. The LSM method is more accurate, as it estimates the three parameters using the whole register of u\({}_{\text{co}}\)(t), thus getting the best fit, but for the calculation of the resistance, it is necessary to use the Lambert-W function. The two methods described have the following advantages over other methods in which an ideal current source is used to carry out the charging or discharging:  
1. These methods can be used directly in those applications where the SC is to operate connected to an RVS or be discharged through a constant resistance, which makes it possible to estimate the parameters during normal operation of the SC. 2. Only the cell terminal voltage, u\({}_{\text{co}}\)(t), needs to be measured. It is not necessary to obtain any other signal, such as the internal voltage or any of its derivatives, nor is it necessary to adjust those functions by means of polynomials or Taylor series, as is performed in other methods [30], which adds complexity and may cause additional errors. 3. The three parameters (R, C\({}_{0}\), and k\({}_{\text{c}}\)) can be obtained directly by both methods (3 PM and LSM) in a single step. However, in other approaches, like the one proposed by [PERSON] et al. [30], C\({}_{0}\) and k\({}_{\text{c}}\) values must be obtained in the first stage using a discharge of SC with a high constant current. From this, in the second phase, the value of the resistance R.
4. When an ideal constant current source is used, a high value of current is required to measure the voltage drop across the internal resistance, \(\Delta\)V = R\(\cdot\)I. For example, [PERSON] [28] used 33 A current source to estimate the parameters of a 600 F cell. It must be considered that the higher the capacitance of an SC, the lower its internal resistance R. [PERSON] et al. [30] used 120 A discharge current in their method for a 3000 F cell.