Modeling electrochemical double layer capacitor, from classical to fractional impedance

Martin, R.; Quintana, Jose J.; Ramos, Alejandro; Nuez, Iván de la

doi:10.1109/melcon.2008.4618411

Cited by 55 publications

(44 citation statements)

References 32 publications

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“…ctional calculus, as was demonstrated in [4,5]. tly, the case when order is time-varying, begun to be xtensively.…”

Section: ) Hementioning

confidence: 94%

“…The matrices of equivalen to (5) [19,20] and [21]. a numerical solution of m in a state-space form as well as time-variant re also valid for system t types of variable orders he fractional variable order state-space system was physically build and the experimental results were compared with numerical implementations.…”

Section: Definitions Of Variable Order Operatorsmentioning

confidence: 99%

“…Fractional calculus has been found a convenient tool to model behavior of many materials and systems, particularly those involving diffusion processes. For example, ultracapacitors can be modeled more efficiently using fractional calculus, as was demonstrated in [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…For example, ultracapacitors can be modeled more efficiently using fractional calculus, as was demonstrated in [4,5].…”

Section: Introductionmentioning

confidence: 99%

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Numerical solution of fractional variable order linear control system in state-space form

Malesza

Macias

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The aim of this paper is to introduce a matrix approach for approximate solving of non-commensurate fractional variable order linear control systems in state-space form. The approach is based on switching schemes that realize variable order derivatives. The obtained numerical solution is compared with simulation and analog model results. In our paper, a method of finding a numerical solution of fractional variable order control system in a state-space form is introduced, both for time-invariant and time-variant case. Moreover, the obtained results are also valid for system of differential equations with different types of variable order derivatives. To validate our approach the fractional variable order state-space system was physically built and the experimental results were compared with numerical implementations.The paper is organized as follows. At the beginning, in Section 2, the few types of fractional variable order derivatives are recalled, together with their discrete approximations and matrix forms. In Section 3 the solution of linear control system in statespace form for time-variant and time-invariant non-commensurate fractional variable order system is presented. An analog model of particular type of fractional variable order state-space system is introduced in Section 4. The experimental and numerical results are presented in Section 5. Finally, Section 6 summarizes the main results. Fractional variable order operatorsBelow, we recall the already known different types of fractional constant and variable order derivatives and differences. In our paper, a method of finding a numerical solution of actional variable order control system in a state-space form introduced, both for time-invariant as well as time-variant der state-space system was physically build and the experimental results were compared with numerical implementations. The paper is organized as follows. At the beginning, in Sect. 2, the few types of fractional variable order derivatives are recalled, together with their discrete approximations and matrix forms. In Sect. 3 the solution of linear control system in state-space form for time-variant and time-invariant noncommensurate fractional variable order system is presented. An analog model of particular type of fractional variable order state-space system is introduced in Sect. 4. The experimental and numerical results are collected in Sect. 5. Finally, Sect. 6 summarizes the main results. Fractional variable order operatorsBelow, we recall the already known different types of fractional constant and variable order derivatives and differences. Definitions of variable order operatorsThe following fractional constant order difference of Grünwald-Letnikov type will be used as a base of generalization onto variable orderwhere α ∈ R, l = 0, . . . , k, and h > 0 is a sample time. We will consider the following four types of fractional variable order derivatives and their discrete approximations (differences). We admit the order is changing in time, i.e., α(t) ∈ R for t > 0; and in ...

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“…ctional calculus, as was demonstrated in [4,5]. tly, the case when order is time-varying, begun to be xtensively.…”

Section: ) Hementioning

confidence: 94%

Section: Definitions Of Variable Order Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For example, ultracapacitors can be modeled more efficiently using fractional calculus, as was demonstrated in [4,5].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical solution of fractional variable order linear control system in state-space form

Malesza

Macias

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…For systems containing supercapacitors, these differences are mainly due to their electrochemical construction [8]. Besides supercapacitors, many new practical realizations of the fractional-order capacitor C α have been developed, among others using the equivalent RC ladder structures, some dielectrics (e.g., LiN 2 H 5 SO 4 ), fractal systems [4,14,27] and many others. For the description of lossy coils with soft ferromagnetic cores, fractional-order models have been used too [28,29].…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Transient State in a Series Circuit of the Class $$RL_{\beta }C_{\alpha }$$ R L β C α

Jakubowska¹,

Walczak²

2016

Circuits Syst Signal Process

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Results of analysis of transient states in a series circuit of the class RL β C α , supplied by an ideal voltage source, have been described in the paper. This circuit consists of a coil L β and a supercapacitor C α described by fractional-order differential equations. A method for determining the current and voltage waveforms in the analyzed circuit, based on the decomposition of rational functions into partial fractions, has been described. This method allows to determine transient waveform shapes in the system for any kind of voltage excitation. Two cases of the problem solutions have been considered. The first case concerns a situation where poles of rational functions are real, and the second where rational functions have complex poles. Effective relations enabling the determination of transient waveforms in a closed form have been given. Analytical formulae describing transient state waveforms in the system for different types of voltage excitations: constant, monoharmonic, periodic and arbitrary being an element of a Hilbert space, have been determined, too. The obtained results have been illustrated by an example.

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Determination of supercapacitor parameters based on fractional differential equations

Hidalgo‐Reyes

Gómez‐Aguilar

Escobar-Jiménez

et al. 2019

Circuit Theory & Apps

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In this paper, we estimate the parameter values of a fractional-order model of supercapacitors involving fractional derivatives of Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu and fractional conformable derivative in the Liouville-Caputo sense. We present the exact solution of the considered model using the properties of the Laplace transform operator together with the convolution theorem. They developed numerical simulations using each one of the fractional derivatives; the results were compared graphically with experimental data obtained from different supercapacitors using standard laboratory equipment. The nonlocal parameters involved in the equivalent electrical circuit for the supercapacitor model are recalculated for each fractional derivative using a particle swarm optimization algorithm for generating optimal solutions. KEYWORDS exponential decay-law, fractional calculus, fractional conformable derivative, Mittag-Leffler function, power-law, supercapacitor model Int J Circ Theor Appl. 2019;47:1225-1253.wileyonlinelibrary.com/journal/cta

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Modeling electrochemical double layer capacitor, from classical to fractional impedance

Cited by 55 publications

References 32 publications

Numerical solution of fractional variable order linear control system in state-space form

Numerical solution of fractional variable order linear control system in state-space form

Analysis of the Transient State in a Series Circuit of the Class $$RL_{\beta }C_{\alpha }$$ R L β C α

Determination of supercapacitor parameters based on fractional differential equations

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