Fractional-order models: The case study of the supercapacitor capacitance measurement

Abstract

Abstract. At the beginning of the paper, the fractional calculus is briefly presented. Then, the models of dielectric relaxation in supercapacitors are described. On the basis of the Cole-Cole model, a fractional-order model of supercapacitor impedance is formulated. The frequency characteristics of selected supercapacitors and their voltage response to a current step are assumed as a basis for the analysis of their dynamics. An example of the fractional dynamic model application was used for the critical assessment of the IEC standard recommendation on the conditions of supercapacitor capacitance measurements. The presented study shows some imperfections of the IEC standard recommendations, which probably result from the use of an inaccurate dynamics model. At the end of the paper, the authors propose a solution to this problem by changing the measurement conditions and introducing a concept of dynamic capacitance. The conclusions of the paper indicate that the models of fractional-order dynamics may be useful not only for the control purposes but also in other domains. At the beginning, after some information about fractional calculus, the models of dielectric relaxation in supercapacitors are described. On this basis, a fractional-order model of supercapacitor impedance is formulated. In accordance with the recommendation of the IEC standard concerning the requirements for supercapacitor capacitance measurement [21], the analyses of frequency characteristics of selected supercapacitor samples and their voltage responses to current steps were made. The study shows the imperfection of the IEC standard recommendations, which probably results from the use of an inaccurate dynamics model. At the end of the paper, the authors propose a solution to this problem by changing the measurement conditions and by introducing a concept of dynamic capacitance. Fractional-order dynamic modelsAs mentioned, most dynamic systems are described by the integer order models. The dynamics is described in many ways. One of the widely used methods is to employ the differential equation of general form: Abstract. At the beginning of the paper the fractional calculus is briefly presented. Then the models of dielectric relaxation in supercapacitors are described. On the basis of the Cole-Cole model a fractional-order model of supercapacitor impedance is formulated. The frequency characteristics of selected supercapacitors and their voltage response to a current step are assumed as basis to the analysis of their dynamics. An example of the fractional dynamic model application was used for a critical assessment of the IEC standard recommendation on the conditions of supercapacitor capacitance measurements. The presented study shows some imperfections of the IEC standard recommendations , which probably result from the use of inaccurate dynamics model. At the end of the paper the authors propose a solution to this problem by changing the measurement conditions and introducing a concept of dynamic capacitance. The conclusions of the paper ind...

Application of supercapacitors in electric traction storage systems

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Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System

“…Apart from classic examples like the modeling of viscoelastic materials [ 4 ] and the diffusion process [ 5 , 6 ], it was also found to be useful in economics for modeling of financial systems and economic growth [ 7 , 8 ], in medicine and biomedical engineering for disease analysis, drug modeling and administration, and signal acquisition [ 9 , 10 , 11 , 12 , 13 ], in computer science [ 14 ] for the development of neural networks with memory effects, and other applications with time-varying values of fractional-orders [ 15 ]. Only in the field of electrical engineering do some recent advances involve mathematical descriptions of supercapacitors [ 16 , 17 ], lithium-ion batteries with nonlinear capacities [ 18 ], and nonlinear coils in a ferroresonant circuit [ 19 ]. In control engineering, the algorithms of fractional-order controllers have received much interest, in particular, the fractional-order PID controller [ 20 , 21 , 22 ].…”

Novel Time Method of Identification of Fractional Model Parameters of Supercapacitor

“…The literature presents numerous descriptions of SC impedance models [3][4][5], among others the fractional order models [6]. The fractional models of SC [7][8][9][10][11] can be associated with physical phenomena in the electric double layer [12] and at the same accuracy of dynamics description are less complex than the models of integer order [5,13]. Due to the smaller number of parameters and their relationships with physical phenomena, they are easier to identify.…”