Electrochemical doublelayer capacitors (EDLC), also known as supercapacitors or ultracapacitors, are devices in which diffusion phenomena play an important role. For this reason, their modeling using integerorder differential equations does not yield satisfactory results. The higher the temporal intervals are, the more problems and errors there will be when using integerorder differential equations. In this paper, a simple model of a real capacitor formed by an ideal capacitor and two parasitic resistors, one in series and the second in parallel, is used. The proposed model is based on the ideal capacitor, adding a fractional behavior to its capacity. The transfer function obtained is simple but contains elements in fractional derivatives, which makes its resolution in the time domain difficult. The temporal response has been obtained through the MittagLeffler equations being adapted to any EDLC input signal. Different charge and discharge signals have been tested on the EDLC allowing modeling of this device in the charge, rest, and discharge stages. The obtained parameters are few but identify with high precision the charge, rest, and discharge processes in these devices.
In recent years, the growing demand for new electrical energy storing systems has led to a remarkable development of electrochemical doublelayer capacitors (EDLC). The EDLC are devices capable of storing energy and are characterized by their very rapid response during charge and discharge cycles, which allows them to provide high power and to hold a high number of charge and discharge cycles.
The complementary qualities of EDLC and batteries have allowed the generation of numerous hybrid applications for energy recovery or storage systems. Energy storage in EDLC is not supported by chemical processes. Moreover, supercapacitors have a high life cycle and do not need any maintenance. From the above it can be concluded that because of their characteristics these devices have raised great interesting expectations [
It is a usual procedure that the dynamics of real systems are modeled by differential equations. In most cases, the differential equations are based on conventional derivatives, yielding sufficiently accurate mathematical models. However, there are a variety of systems or phenomena in which mathematical models based on ordinary differential equations do not provide satisfactory solutions. One of these elements is the EDLC. It is this case, when the application of models based on fractional derivatives, studied in the field of fractional calculus, is very useful. These equations are called fractional differential equations.
In multiple fields of physics, there are numerous examples in which a differential equation is used as fractional modeling tool [
In this paper, a dynamic fractional model for EDLC is developed, based on a differential equation with fractional derivatives. This equation has been solved by means of the Laplace transform, and in order to obtain the time domain solution used the MittagLeffler function
This paper is organized as follows: Section
In this section is exposed a brief mathematical background of the fractional calculus [
RiemannLiouville definition expresses the fractional derivative as a time convolution integral
Caputo definition of a fractional derivative of a function is
GrunwaldLetnikov definition is a numerical form of the fractional derivative
An interesting property of the fractional derivative operator is its Laplace transform
The twoparameter function of the MittagLeffler type
Electric modeling of supercapacitors is an active line of investigation with numerous contributions in the last few years [
Due to diffusion processes resulting after the charging phase. The evolution of the processes occurring inside the EDLC generates a nonreversing thermal dissipation which makes the assessment of losses in the device possible. This process would occur in both the EDLC charge and discharge phases.
The model of EDLC has been considered as single input (current), single output (voltage) system. The electric model used comes from the model applied to ideal capacitors using integerorder differential equations. Subsequently, a model based on fractional order differential equations is proposed.
The mathematical equation, relating the current and the voltage at the terminals of a real capacitor, can be deduced from an electric circuit made of an ideal capacitor and two parasitic resistors [
Naming the series resistor as
The model obtained for real capacitor (
Many papers [
Equivalent circuit of a real capacitor.
In this section the solution to (
Multiplying transfer function (
The response can be extended to any input. For this purpose, the superposition of steps displaced by the sampling period of the input signal has been used. Thus, for the first input signal, the temporal response matches the response obtained by (
The experimental data have been obtained in the laboratory making a test circuit and recording the electrical variables of the EDLC. The analyzed EDLC, manufactured by ELNA, has a capacity of 4.7 F and a voltage of 2.5 V.
Figure
Schematic diagram of the experimental circuit.
Several tests have been performed to the EDLC in order to monitor voltage and current signals. Because EDLC have memory effect, just before each test they have been shortcircuited for 24 hours to ensure zero initial conditions.
The tests were performed as follows. In a first phase lasting a few minutes, was charged at a constant current between 20 mA and 0.9 A until the voltage reached 2.4 V; in the second phase, the current is cut off and left to rest for 8 hours to allow the voltage stabilization, and finally in the third phase is discharged through a resistance of 5 ohm until it was fully discharged.
Although the sampling period of voltage and current has been recorded at 0.1 s; sfor identification purposes it has been considered a 5 s sampling time. Figure
Chargerestdischarge at 20 mA.
The coefficients for the proposed model have been defined according to transfer function (
For identification the MATLAB simulation software was used. The function created has as input parameters
The index chosen in this paper is the standard deviation defined as
The EDLC identification has been carried out calculating the parameters
The data obtained for the charge and rest phases are shown in Table
Charge + rest identification parameters.
Test 







1  0.02 A  0.941 

0.239  0.202 

2  0.1 A  0.952 

0.261  0.187 

3  0.5 A  0.950 

0.267  0.170 

4  0.5 A  0.950 

0.250  0.169 

5  0.9 A  0.953 

0.228  0.164 

Considering relation (
Charge + rest electrical parameters.
Test 






1  0.02 A  0.941  4.95  0.239 

2  0.1 A  0.952  5.35  0.261 

3  0.5 A  0.950  5.89  0.267 

4  0.5 A  0.950  5.91  0.250 

5  0.9 A  0.953  6.08  0.228 

In order to compare the adjustment with traditional model (
Parameters calculated for test number 5 setting
Test 





1  1.000  9.00  1.262 

Fitting in the resting phase using integer derivatives.
Operating in a similar way, the parameters for the discharge through the 5 Ω resistor are in Table
Discharge identification parameters.
Test 






1  0.975 

0.235  0.187 

2  0.981 

0.244  0.185 

3  0.978 

0.267  0.192 

4  0.976 

0.251  0.202 

5  0.982 

0.240  0.171 

Of which the following electric parameters are obtained in Table
Discharge electrical parameters.
Test 





1  0.975  5.34  0.235 

2  0.981  5.40  0.244 

3  0.978  5.21  0.267 

4  0.976  4.95  0.251 

5  0.982  5.85  0.240 

The following figures show graphically the fit of the proposed model with experimental data. Figure
Experimental data of test number 5 versus proposed model.
Detail of fitting in charging phase.
Fitting in the resting phase using the proposed model.
Fitting in the discharging phase.
The selfdischarge adjustment of EDLC after charging and after long time resting phases is a complex phenomenon which is not easily applicable to modeling through integerorder transfer functions. In this work the EDLC has been modeled making use of the fractional derivative. Taking the electric circuit in Figure
First of all, considering Figure
Unlike the traditional capacitor model, which is defined by the value of two resistors and the capacity, the proposed model adds a new parameter, that is, the fractional index
Although constants in Tables
In all cases the adjustment has been very precise, with the typical deviation being less than 0.03.
The constancy of
At last, the most significant difference in terms of charge and discharge is given by the value of the fractional index
This work has consisted of EDLC modeling by implementing a simple fractional model allowing a very satisfactory response for long time periods. Traditional models based on transfer functions with integer coefficients yield good results in those cases when there is no selfdischarge phenomenon. Applying fractional mathematics implies an additional parameter to the already necessary parameters in order to define the basic model shown in Figure
Thanks to the proposed transfer function and the MittagLeffler functions, it is possible to estimate the voltage at the EDLC terminals after a long time, taking into account the charging current and the way it has been charged.