Fractionalorder models of supercapacitors, batteries and fuel cells: a survey
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Abstract
This paper surveys fractionalorder electric circuit models that have been reported in the literature to best fit experimentally collected impedance data from energy storage and generation elements, including supercapacitors, batteries, and fuel cells. In all surveyed models, the employment of fractionalorder capacitors, also known as constant phase elements, is imperative not only to the accuracy of the model but to reflect the physical electrochemical properties of the device.
Keywords
Impedance spectroscopy Constant phase element Supercapacitors Batteries Fuel cellsIntroduction
Modeling of energy storage and generation elements used in hybrid and renewable energy sources is crucial for future development of these sources [1]. Of specific importance are supercapacitors, batteries and fuel cells [2]. A survey of published fractionalorder models has been carried out and is presented in this work. These models provide best fit to experimentally measured impedances and/or transient responses and employ one or more constant phase element (CPEs) [3]; also known as the fractionalorder capacitor [4].
From a circuit theory perspective, it is possible to define a general frequencydomain electrical impedance which is proportional to \(s^{\alpha };s=j\omega\). This device is known as a fractance [5] from which classical circuit elements are special cases of the general device when the order \(\alpha\) is \(1\), 0, and 1 for a capacitor, resistor, and inductor, respectively. A CPE (or fractionalorder capacitor) is characterized by the impedance \(Z_{CPE}=(1/C_{\alpha })s^{\alpha }\); where \(C_{\alpha }\) is termed pseudocapacitance with units F/sec\(^{(1\alpha )}\), and \(\alpha\) is the order. These units were originally proposed in [6]. A CPE has a phase angle, \(\phi _{CPE}=\alpha \pi /2\) which is constant, independent of frequency and dependent only on the order \(\alpha\). While \(\alpha \in \mathfrak {R}\) is mathematically possible, the values from experimentally collected data of CPEs are in the range of \(0<\alpha \le 1\); therefore, it has also become known as a fractionalorder capacitor.
A number of recent surveys and special issues have focused on the applications of fractionalorder circuits and systems in general [7] and particularly in industrial automation [8]. Meanwhile, energy storage components and particularly electrochemical doublelayer capacitors [9] [fabricated to provide high and very high capacitances (supercapacitors)], as well as batteries are critical components in hybrid energy systems which require accurate models for these devices [10]. Although there exists a considerable mass of research on the modeling of these devices, a survey focusing on fractionalorder models has not been conducted. It is the purpose of this paper to fill this gap.
Supercapacitor models

Impedance modeling of nickel finer mesh for simulation and design of elements in pulse power systems [18];

Modeling high power performance to characterize supercapacitor capabilities [19];

Modeling the impedance of supercapacitors over finite frequency bands to reduce the required model parameters [20, 21, 22];

Modeling the transient characteristics from very small (seconds) [23] to very large (months) timescales [24];

Measurement of fractional characteristics from transient behavior [25, 26];

Modeling and control of supercapacitor systems using fractional statespace models [27] and fractional linear systems [28, 29];

Implementation in a buck–boost converter for power electronics [30].
Battery models

Modeling the impedance of leadacid [32], nickel/metal hydride [33], and lithiumion batteries [34, 35] in finite frequency bands;

Determining the relationship between fractionalorder model parameters and stateofcharge of nickelcadmium [36], lithiumion [37, 38, 39], alkaline [40], leadacid [41] batteries;

Designing highpower batteries based on the fractional contribution of the elements to total polarization [42];

Stateofhealth estimations based on parameters measured during the cranking function of leadacid batteries in vehicles [43, 44].
Fuel cell models

Monitoring the stateofhealth of a PEMFC with respect to the water content of the membrane electrode assembly [46, 47];

Analyzing the reaction kinetics and interfacial characteristics of an anode in a DMFC [48];

Characterizing the output power dynamics of a SOFC for management by a control system [49];

Monitoring the anode colonization by electrodereducing microorganisms in an MFC [50].
Conclusion
We have surveyed published fractionalorder circuit models that best fit experimentally collected impedance data of supercapacitors, batteries, and fuel cells. All surveyed models rely on using at least one fractionalorder capacitor in an attempt to accurately capture the underlying electrochemical dynamics. This survey should also serve to bridge the gap between the wide employment of fractionalorder impedances in biochemistry and their much less employment in circuit design.
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