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A review of fractional-order techniques applied to lithium-ionbatteries, lead-acid batteries, and supercapacitors
Downloaded from: https://research.chalmers.se, 2020-10-15 00:13 UTC
Citation for the original published paper (version of record):Zou, C., Zhang, L., Hu, X. et al (2018)A review of fractional-order techniques applied to lithium-ion batteries, lead-acid batteries,and supercapacitorsJournal of Power Sources, 390: 286-296http://dx.doi.org/10.1016/j.jpowsour.2018.04.033
N.B. When citing this work, cite the original published paper.
research.chalmers.se offers the possibility of retrieving research publications produced at Chalmers University of Technology.It covers all kind of research output: articles, dissertations, conference papers, reports etc. since 2004.research.chalmers.se is administrated and maintained by Chalmers Library
(article starts on next page)
A review of fractional-order techniques applied to lithium-ion batteries, lead-acidbatteries, and supercapacitors
Changfu Zoua,∗, Lei Zhangb,∗, Xiaosong Huc,∗∗, Zhenpo Wangb, Torsten Wika, Michael Pechtd
aDepartment of Electrical Engineering, Chalmers University of Technology, Gothenburg 41296, SwedenbNational Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology, Beijing 100081, China
cState Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400044, ChinadCenter for Advanced Life Cycle Engineering, University of Maryland, College Park, MD 20742, USA
Abstract
Electrochemical energy storage systems play an important role in diverse applications, such as electrified transportation and inte-gration of renewable energy with the electrical grid. To facilitate model-based management for extracting full system potentials,proper mathematical models are imperative. Due to extra degrees of freedom brought by differentiation derivatives, fractional-ordermodels may be able to better describe the dynamic behaviors of electrochemical systems. This paper provides a critical overviewof fractional-order techniques for managing lithium-ion batteries, lead-acid batteries, and supercapacitors. Starting with the basicconcepts and technical tools from fractional-order calculus, the modeling principles for these energy systems are presented by iden-tifying disperse dynamic processes and using electrochemical impedance spectroscopy. Available battery/supercapacitor modelsare comprehensively reviewed, and the advantages of fractional types are discussed. Two case studies demonstrate the accuracy andcomputational efficiency of fractional-order models. These models offer 15-30% higher accuracy than their integer-order analogues,but have reasonable complexity. Consequently, fractional-order models can be good candidates for the development of advancedbattery/supercapacitor management systems. Finally, the main technical challenges facing electrochemical energy storage systemmodeling, state estimation, and control in the fractional-order domain, as well as future research directions, are highlighted.
Keywords: Batteries, electrochemical energy storage systems, energy management, fractional-order models, supercapacitors
1. Introduction
Transportation electrification and grid integration of renew-able energy sources constitute two renewed research efforts toreduce dependence on fossil fuels and mitigate global warming[1]. Market penetration of electrified vehicles (EVs) can helpmeet these goals if it is coupled with decarbonized electricity,for example, solar and wind power [2]. Electrochemical energystorage systems (EESSs) play a critical role in both EVs andrenewable energy integration applications. They serve as en-ergy sources to provide power supply and/or energy buffers toimprove efficiency and the overall economy.
Rechargeable batteries and supercapacitors are typicalEESSs that share a similar structure–both of them store andconvert energy through diffusion and migration of ions. Eachbattery or supercapacitor cell is composed of positive and neg-ative electrodes separated by an enabling separator that allowsion transfer but prevents electron conduction. Electrodes andtheir separators are often immersed in an electrolyte solutionthat contains mobile ionic species [3]. Among a number of
∗C. Zou and L. Zhang equally contributed to this work.∗∗Corresponding author.
different energy storage technologies, lithium-ion (Li-ion) bat-teries have currently been accepted as the leading candidate forcommercial EESSs because of their superiority, especially involumetric and gravimetric energy densities [4, 5]. However,each EESS has unique features and characteristics, and maybe well suited for particular applications. For example, whilelead-acid batteries are primarily used in cases where cost, reli-ability, and abuse tolerance are crucial [6], supercapacitors arepreferred in devices that require high power density and longcycling lifetime [7].
EESSs must be safe and highly tolerant of high/low temper-atures. They must also be cost-effective and provide large en-ergy/power density and long cycle life. To pursue these objec-tives, model-based state estimation/monitoring techniques andenergy management schemes have been extensively studied inthe literature, e.g., [8, 9]. A common requirement of these tasksis to construct accurate yet simple mathematical models that areadaptable to thermal and aging phenomena inherent in EESSs.
Considerable research efforts have been dedicated to math-ematically modeling EESS dynamics and have resulted inphysics-based, equivalent circuit, and data-driven models [10,11, 12]. Based on differentiation orders, these models can gen-erally be assorted into integer- and fractional- order models.Integer-order models dominate the research and engineeringapplication of electrochemical energy storage. Hu et al. [13]compared commonly used equivalent circuit models of Li-ion
Preprint submitted to Journal of Power Sources February 23, 2020
batteries in terms of accuracy, complexity, and robustness un-der vehicle driving cycles. Doyle et al. [14] and Zou et al. [15]formulated electrochemical models governed by a set of par-tial or ordinary differential equations for Li-ion cells. Zhang etal. [16] and Drummond et al. [17] discussed electrochemicalmodels for supercapacitors. However, it has been incremen-tally recognized, such as by Freeborn et al. [18], that EESSsexhibit some mathematical characteristics in accordance withfractional-order systems. This fact solicits increased interestand endeavors to come up with novel EESS models in the do-main of fractional calculus. As a result, the fractional-ordermodeling methodology may not only improve prediction ac-curacy but also preserve some physical meanings underlyingmodel parameters.
This paper provides a comprehensive review of fractional-order techniques for typical EESSs, including Li-ion bat-teries, lead-acid batteries, and supercapacitors. Section 2presents the mathematical fundamentals of fractional-order cal-culus. Section 3 introduces the common dynamic processes ofEESSs and electrochemical impedance spectroscopy to eluci-date the principles of fractional-order modeling. Available bat-tery/supercapacitor models are sequentially surveyed, grouped,and characterized. After analyzing parameter identificationtechniques in Section 4, the accuracy and computational re-quirement of fractional-order models (FOMs) are quantitativelyinvestigated via case studies in Section 5. Section 6 highlightsthe main technical challenges facing FOM-based managementfor EESSs, including modeling of coupled system dynamics,state estimation, and charge/discharge control, followed by con-cluding summaries in Section 7.
2. Mathematical fundamentals
This section exhibits the mathematical fundamentals offractional-order calculus (FOC) to facilitate the understandingof concepts and technical tools used for modeling electrochem-ical energy systems. In particular, the definitions of impedanceand fractional-order derivatives and the FOM’s state-space rep-resentation will be discussed. A thorough exposition of FOCcan be found in textbooks on fractional-order system model-ing, analysis, and applications [19, 20, 21] and related surveyarticles [22, 23].
Frequency-domain electric impedance. In the frequency do-main, a general impedance, Z, in electrical circuits may be de-fined by the following proportional relation
Z ∝ ( jω)α, for α ∈ [−1, 1], ω ∈ R, (1)
where j is the imaginary number and ω is the radian frequency.The conventional equivalent circuit elements, including purecapacitors, resistors, and inductors, are special cases of Z, cor-responding to α = 1, 0, and −1, respectively.
As initially proposed by Cole and Cole [24], a fractional-order capacitive element can be characterized by the impedancein (1) as
ZCPE =1
Cα( jω)α, for α ∈ (0, 1), (2)
where the exponent α is a fractional-order and Cα is a constantand is called a pseudo-capacitance with the dimension F·secα−1
[25]. ZCPE has a constant phase angle at απ/2 [26] and is of-ten called a constant phase element (CPE). In comparison, thephase shift for pure capacitors is π/2.
Fractional-order derivatives. The fractional-order opera-tor for the CPE in (2) is mathematically defined by 0D
αt (·) =
dα(·)/dtα. An equation with 0Dαt describes dynamic processes
with infinite dimension. To facilitate analysis and numericalimplementation, three different definitions, namely, Riemann-Liouville (RL), Caputo, and Grunwald-Letnikov (GL) frac-tional derivatives are often utilized for such an operator [19].For instance, the GL fractional derivative takes explicitly theform
0Dαt f (t) = lim
T→0
1Tα
bt/T c∑k=0
(−1)k〈α, k〉 f (t − kT ), (3)
where T is the sampling time interval, bt/T c is the maximum in-teger lower than t/T , and 〈α, k〉 represents the Newton binomialterm defined as
〈α, k〉 =Γ(α + 1)
Γ(k + 1) · Γ(α − k + 1), (4)
where Γ(·) is the gamma function with the definition of
Γ(α) =
∫ ∞0
ξα−1e−ξdξ. (5)
For simplicity, 0Dαt f (t) is written as Dα f (t) in the sequel. A
comprehensive description of these definitions as well as theirpeculiarities has been presented in [19, 27].
Unlike their integer alternatives, fractional derivatives arenot local operators because they take into account the entirepast trajectory of f (·) over the interval [0, t], as seen in (3).This is the so-called long memory property of fractional deriva-tives. However, this property significantly increases the com-putational burden for engineering applications of FOMs, par-ticularly for real-time model-based optimization and control.To improve implementation efficiency, a short memory princi-ple was therefore proposed by Podlubny [28] to approximate(3) with high-order difference equations, which consider onlyrecently past information in the state propagation. This ap-proach has been shown to be effective in a number of examplesin fractional-order modeling of Li-ion batteries [29] and super-capacitors [30]. Indeed, there is in general a trade-off betweenmodeling accuracy and computational complexity around thememory length.
System representation and types. The state-space representa-tion of a general fractional-order system can be written in thefollowing form
Dαx(t) = f (t, x(t), u(t)), (6a)
y(t) = h(t, x(t), u(t)), (6b)
where x := [x1, · · · , xn] is the state vector, Dαx :=[Dα1 x1, · · · ,D
αn xn], and u, y are separately the system input andoutput vectors. This representation is the same as integer-order
2
system representations except for the fractional derivative onthe left-hand side of (6a). If α1, · · · , αn are all positive inte-ger multiples of a real number γ, then (6) is a commensuratefractional-order system of order γ; otherwise, it is said to beincommensurate, with more degrees of freedom to fit systemdynamics [19, 31, 32].
3. EESS modeling
Before reviewing fractional models for different EESSs, themodeling principles are first explained by analyzing system dy-namic processes and electrochemical impedance spectroscopy.
3.1. Modeling principlesDynamic processes. A common feature of EESSs is that
multiple dynamic processes occur simultaneously during op-eration and inherently exhibit different time scales. The fastestdynamic process is dedicated to the movement of charge car-riers through the electrolyte and current collectors to the ex-ternal circuit. Along with the decreasing direction of the fre-quency spectra, there are electrochemical double-layer effectsand charge-transfer reactions. This is followed by the solid-phase ion diffusion in batteries and the stray inductance of cur-rent collectors and porous electrodes in supercapacitors. AllEESSs suffer from persistent, irreversible aging phenomenaduring static storage or cycling operation, which is the slow-est dynamic process. These dynamic processes in general arecoupled. For example, the stressed electrochemical reactionsexpedite system degradation, and in turn, the aging reaction in-fluences charge/discharge performances.
An explicit identification of multiple time scales from Li-ionbattery dynamics was conducted in [15, 33]. Therein, tech-niques based on singular perturbations and the averaging theorywere proposed to systematically separate the dynamics. Theprocesses that occur in a typical supercapacitor have also beenexplained via frequency-domain impedance analysis, e.g., in[34, 35].
Electrochemical impedance spectroscopy. Electrochemicalimpedance spectroscopy (EIS) is a powerful tool to investi-gate the behavior and properties of EESSs in a non-destructivemanner [36, 37, 38]. The principle of EIS analysis is to dis-tinguish the above different physical processes by characteriz-ing the impedance over wide frequency ranges. Specifically,the impedance Z of an electrochemical system around somesteady or quasi-steady state can be determined using the fol-lowing two-step procedure:
(i) Apply a sequence of small AC currents, which can be ex-pressed as I(t) = |I|e j(ωt+φI ) if they are sinusoidal signals,to excite the system and then measure its voltage response,V(t) = |V |e j(ωt+φV ).
(ii) Collect current and voltage data and evaluate theimpedance by dividing the voltages by its correspondingcurrents, namely Z = |V |e j(φV−φI )/|I|.
The characteristics of the impedance spectra can provide in-sights into electrochemical systems and then be used to developmathematical models for predicting system dynamics.
Equivalent circuit modeling principles. Equivalent circuitmodeling can be motivated by EIS, with the goal to fit ex-perimentally measured impedance data using circuit elements.Lumped resistors, capacitors, inductors, and voltage sources aretypical elements used in conventional equivalent circuit mod-els (ECMs) [13, 39, 40]. To improve modeling fidelity, a CPEwas proposed in [24] with the mathematical definition givenby (2). With CPEs, the plate hypothesis underlying the realelectrodes can be relaxed, and non-uniform boundary and dis-tributed intercalation/de-intercalation processes within porouselectrodes can be described. The obtained models incorporatingone or more CPEs are often referred to as FOMs, which havebeen used to mimic dynamic behaviors of EESSs [41, 42, 29].
In the context of circuit approaches, the development ofFOMs for a battery or supercapacitor, in essence, consists ofselecting CPEs together with other circuit elements and then ap-propriately organizing them in a circuit. Usually, there are somemodeling criteria, depending on specific applications, such asaccuracy and complexity. Different techniques available for Li-ion batteries, lead-acid batteries, and supercapacitors are sur-veyed in the following subsections.
3.2. Lithium-ion battery models
Li-ion batteries were first commercially developed by Sonyin the early 1990s and have experienced remarkable advancesover recent years [6, 43]. Their prosperity is largely driven bydemands for portable electronic devices, smart grids, and EVs.With continuous performance improvements and cost reduc-tions, the deployment of Li-ion batteries is predicted to increaserapidly in the near future [44] and thus is primed to dominatethe energy storage market.
Li-ion batteries, with a cost of US$250 per kilowatt-hour [45]and energy efficiency at 200-250 watt-hours per kilogram [46],however, are expensive compared to fossil fuels like petrol anddiesel. In addition, battery state of health (SoH), reliability, andsafety are critical concerns that need to be addressed over a bat-tery’s entire lifespan. A battery management system (BMS) candeal with these economic and performance concerns becauseit synthesizes from hardware and software to monitor, control,and diagnose the battery pack and individual cells [10, 12]. Ex-plicitly, its functionality may involve state estimation, thermalmanagement, charge/discharge control [47], and cell balancing[48]. To realize these functions, an important first step is toestablish reliable and numerically efficient battery models.
For a typical Li-ion cell, the Nyquist plot of its impedancespectrum is illustrated in Fig. 1. The impedance spectrum canbe divided into three sections according to frequency. The high-frequency tail is typically interpreted as the ohmic resistanceof inductive components, such as current collectors and test ca-bles. The low-frequency straight line is mainly invoked by elec-trochemical double-layer and charge-transfer reactions. Thiscan be captured by a CPE, usually referred to as a Warburgelement. The mid-frequency semi-ellipse stems from lithiumdiffusion within solid electrodes and can potentially be mod-eled by some tandem fractional-order networks, each of whichis constituted by a CPE in parallel with a resistor [38].
3
18,650 cell by A123 Systems, measured with a small signal ac cur-rent of 100 mA, in a frequency range of 10 mHz to 5 kHz, at 50%state of charge (SOC) and ambient temperature. The differentprocesses with typical frequency ranges are annotated in the figure.
For a simplified description of the complex physical andchemical processes occurring within electrochemical systems suchas batteries or fuel cells, electrical equivalent circuit models (ECMs)can be used to model the small signal dynamic behaviour [3,4,12].One of the most widely used equivalent circuit representations ofLi-ion batteries is based on the Randles circuit [4], which consists ofan ohmic resistor to represent the conduction of charge carriersthrough electrolyte and metallic conductors, and a resistor in par-allel with a capacitor to represent the charge transfer resistance anddouble layer capacitance, respectively. However, these idealisedcircuit components are inadequate to model electrochemicalcharge transfer and double layer capacitance in real systems due tothe spatial distribution of these processes. For a better represen-tation of distributed electrode processes in real systems, Cole andCole [3] proposed an equivalent circuit element with constantphase, independent of the frequency: the so-called constant phaseelement (CPE), which has been adopted for Li-ion ECMs (e.g. Ref.[13]). The impedance of the CPE, ZCPE, is expressed as:
ZCPEðuÞ ¼1
QðjuÞa(1)
where Q is a constant, j is the imaginary number, u is the radianfrequency and a (0> a> 1) is the exponent value, which is 1 for anideal capacitor. In the fractional case, the dimension of Q isF cm$2 sa$1 [14,15]. In order to model the impedance for all high- tomid-frequency processes in a Li-ion cell, the CPE is combined inparallel with a resistor representing charge transfer resistance (R),and in series with another resistor representing ohmic contribu-tions (R∞), which results in:
ZðuÞ ¼ R∞ þR
RQðjuÞa þ 1(2)
where, Z denotes the cell impedance. If we replace ju with theLaplace operator s in (2), then the ECM expressed as a transferfunction is:
ZðsÞ ¼ VðsÞIðsÞ
¼ R∞ þ RRQsa þ 1
(3)
where V(s) and I(s) are the Laplace transforms of the cell voltageand current signals. Fig. 2 shows the equivalent circuit of the EISmodel (3).
Although relatively simple, this model is sufficient to representthe basic dynamic behaviour of a large number of electrochemicalsystems [16]. Themodel could be extended to include further R-CPEparallel pairs depending on the behaviour and requirements of theelectrochemical system. To model the Warburg element, a singleCPE in series with R∞ could be added. However, in order todemonstrate the time-domain fitting technique without over-fitting, we focus on using this model to solve the followingproblem:
Problem statement: Consider an electrochemical system withan EIS model given by (3). The problem is to estimate the values ofR∞, R, Q and a from suitably excited current and voltage, i(t) andv(t), such that the frequency response of Z(s) fits the measuredimpedance spectra within the frequency range of interest [uL, uH].It is assumed that i(t) and v(t) are discrete time sampled data withsample time Ts, from Ref. t¼ 0 to t¼ tf¼ KTs and K2Zþ.
For many applications, the ECM parameters of (3) are typicallyestimated in the frequency domain by complex non-linear leastsquares (CNLS) regression [17,18]. This approach works well inthe lab, since low noise frequency domain measurements aredirectly available by exciting the cell with a sinusoidal excitationat various frequencies. However, this is impractical for conditionmonitoring in real applications, since it is relatively slow. Instead,a wide range of frequencies may be used to excite the cell at thesame time, for example a pseudo-random binary sequence,broadband noise, or multisine excitation [19], allowing a muchfaster measurement, and the possibility to use passively occur-ring noise as the excitation source. In this scenario, before fitting,time-domain data must be converted to the frequency domainusing Fourier transforms. Since the excitation and response dataare often non-periodic, a window function must be used tosmooth the spectral estimation, however this introduces a bias tothe estimation. It is therefore preferable outside the lab forcondition monitoring purposes to estimate the model parame-ters of an EIS model such as (3) directly from time domainmeasurements of voltage and current, making use of all availabledata and without introducing bias.
Direct time domain fitting of (3) is challenging, because theexponent a is typically a non-integer due to the CPE. This is termeda ‘fractional order system’ and since the Laplace operator s repre-sents differentiation, this is equivalent to a dynamic system with anon-integer derivative operator.
In order to address this challenge, CPEs for Li-ion ECMs aregenerally either approximated by ideal capacitors [20], or by aseries connection of numerous R-C pairs [21e23], or a distribu-tion of relaxation times [24]. The former has found widespreadapplication in ECMs used for battery management systems (BMS)of Li-ion batteries due to its simplicity and computational effi-ciency [25]. This approach may be sufficient to simulate Li-ioncells under load to moderate degrees of accuracy [26] at lowsampling frequencies. However, it does not allow for a preciserepresentation of Li-ion dynamics in both the time and frequency
Fig. 1. Nyquist plot of impedance spectrum of a Li-ion cell.
Fig. 2. Battery EIS equivalent circuit model (2).
S.M.M. Alavi et al. / Journal of Power Sources 288 (2015) 345e352346
Im
(Z)[
m
]
!<1Hz!>1kHz
OhmicResistance
Diffusion inSolid Electrodes
Electrochemical Double Layer and Charge Transfer Reaction
1Hz<!<1kHz
Re(Z)[m]
Figure 1: Nyquist plot of impedance spectrum measured from a typical Li-ioncell (modified from [49]).
The phase shift of a fractional-order capacitor, απ/2, is calleda phasance, a term introduced by Jean [26]. The phasanceis an important characteristic parameter of the Nyquist plot inFig. 1. In particular, the phasance of a Warburg element rep-resents the slope of the low-frequency straight line, while for aCPE-resistor network, it is related to the shape of the depressedsemicircle. In contrast, the phasance of pure capacitors is fixedto be π/2, which cannot well capture the Nyquist plot’s charac-teristics. A thorough explanation of the phasance concept andmathematics behind it can be found in [26].
In integer-order models (IOMs) of Li-ion batteries, CPEs areeither approximated by ideal capacitors [50], or a number ofresistor-capacitor (RC) networks [13, 51], or relaxation times[52]. The first option, as exemplified in Fig. 2(a), is exten-sively applied to battery management due to the simplicity inits parameterization and implementation. The resulting mod-els may capture a battery’s behavior to a moderate degree ofaccuracy within a limited range of operating conditions [53].However, they are commonly not capable of predicting bat-tery dynamics in both the time and frequency domains overthe entire operating range. For the latter approaches shown inFig. 2(b), in general, a greater number of RC networks are re-quired for wider frequency bands. Specifically, the first-orderRC model tries to mimic ohmic resistance and charge-transferreactions, corresponding to the high-frequency tail and mid-frequency range, but does not describe the diffusion behaviorin the low-frequency straight line. For the second-order RCmodel, the ohmic resistance and diffusion behavior can be sim-ulated, but the characteristic frequency of the charge-transferprocess corresponding to the maximum imaginary part of theimpedance cannot be considered [54]. With the third or higherorder RC models, all three processes can be produced. How-ever, these models have a large set of parameters. This not onlycomplicates the model mathematical structure associated withcomputational burden, but also increases the efforts for systemcalibration and the risk of over-fitting. In addition, as Wester-hoff et al. [54] demonstrated, even though five RC networks areutilized, the IOMs cannot capture the phase at zero very accu-
· · ·Cn
Rn
R1
C1
C1
R1
(a)
(b)
R1
R1R2
C2
-
+
V
-
+
V
Figure 2: Integer-order equivalent circuit models with different orders.
rately.Fractional-order electrical models. To address the above
problems, fractional calculus has recently been explored forLi-ion battery applications. By simply replacing the ideal ca-pacitor in the first-order RC model to a fractional element, aninfinite-dimensional model was developed for Li-ion cells in[55]. The obtained FOM is presented in Fig. 3(a). To facilitatenumerical calculation, these authors adopted the Oustaloup re-cursive approximation from [56] to transfer the fractional equa-tions to ordinary difference equations. In this approximation,the lower and upper frequency bounds will impact poles and ze-ros of the model’s transfer function, and then affect the model’saccuracy [57]. By using experimental data from time and fre-quency domains, Alavi et al. [49] found that this model can re-produce a Li-ion battery’s behavior better than its integer coun-terpoint, thanks to an additional degree of freedom, namely thefractionation order. Waag et al. [58] utilized this model to de-scribe the current-voltage response at dynamic loads measuredfrom EVs. Therein, the FOM with one CPE was found to beequivalent to an IOM with five RC networks. To achieve higheraccuracy, Wang et al. [59] presented an FOM by adding a War-burg element (W) in series with the charge-transfer resistor (R1)(see Fig. 3(b)). By using the GL fractional derivative, this FOMhas demonstrated a high fidelity to experimental data. Liao etal. [60] exploited this model to study the electrochemical be-havior of a lithium iron phosphate/hard carbon cell. With thesame circuit elements but a different structure, the model shownin Fig. 3(c) was used by Xu et al. [29] to describe Li-ion batterydynamics. The model presented in Fig. 3(d) with two CPEs hasalso been employed, e.g., in [61], and is expected to be morerobust to uncertainties.
All the above works tend to confirm that the fractional mod-eling approach is capable of accurately predicting Li-ion cellelectrical dynamics. Consequently, these models have attractedincreased interest in model-based battery management. Xu etal. [29] synthesized an FOM and fractional Kalman filter to es-timate the state of charge of a Li-ion battery. Zou et al. [62] pro-
4
-
+ R1
V
(a)
(b)
R1
R1R1 W
R1R1
W
-
+
V
-
+
V
-
+ R1
V
R1
(c)
(d)
R2
CPE1
CPE2
CPE1
CPE1
CPE1
Figure 3: Typical fractional-order circuit models for a Li-ion cell.
posed a nonlinear fractional estimation algorithm with provablestability and robustness and then applied it to monitor batterystates. Other algorithms for Li-ion battery applications basedon fractional electrical models can be found in [63, 64, 65] andreferences therein.
Fractional-order physics-based models. Alternatively toequivalent circuit models, Li-ion battery models can be estab-lished from first principles that describe electrochemical reac-tions and the Li-ion intercalation/de-intercalation process. Theinitial electrochemical model for a Li-ion cell was proposedby Doyle et al. [14] using concentrated solution theory. Thismodel was extended by Zou et al. [15, 66] to incorporateelectrical, thermal, and aging dynamics, and was constitutedby a number of coupled nonlinear partial differential equations(PDEs). These authors then reformulated this complete batterymodel in a Hilbert space to precisely characterize its mathemat-ical structure. As a result, the singularly perturbed structureunderlying the battery model is uncovered. This enables theuse of available singular perturbation theory for timescale sep-aration of battery dynamics, leading to a family of simplifiedPDE models.
However, because the obtained PDE-based models arestill too computationally expensive for real-time implementa-tion, here is an incentive to perform model-order reductions.Sabatier et al. [67] proposed a fractional electrochemical model
by simplifying a PDE-based electrochemical model. Startingfrom this model, Sabatier et al. [68] developed a fractional sin-gle electrode model by gradually introducing assumptions onbattery physical and chemical properties. Alternatively, accord-ing to Li et al. [69], a simplified physics-based model can beestablished with fractional-order transfer functions to describesolid-phase lithium diffusion. In addition, Li et al. [70] pre-sented an electrochemistry-based impedance model to describelithium diffusion in the electrodes, charge-transfer reactions atthe solid-electrolyte interphase (SEI), double-layer effects, andresistance/capacitance changes associated with the anode SEIfilm growth. In such a model, electrical elements are used tomimic these internal electrochemical processes, so that their in-teractions with external current/voltage measurements can beeasily understood.
The above simplified models have several intriguing at-tributes. They largely mitigate the computational burden com-monly upon high-order physics-based models, but still capturethe key battery characteristics. Furthermore, equipped withfractional-order differentiation, the models contain only a fewparameters that have the potential to maintain physical mean-ings [70, 71]. Indeed, physically meaningful parameters arehelpful for various model-based applications, such as in SoHestimation, lifetime prediction, and optimal fast charging con-trol.
Fractional-order thermal models. Temperature plays an in-fluential role in a battery’s dynamic performance and SoH[72, 73], and effectively modeling the thermal behavior canfacilitate advanced temperature management. On the basis ofmodel-based control algorithms, the temperature can be manip-ulated in a proper range to ensure safe and efficient utilization.The available models are often built on a lumped-parameter en-ergy balance, in which the cell temperature is assumed to bespatially uniform [74, 75]. However, when the heat convectionat the cell surface is faster than the heat conduction inside, theestablished model can appreciably deviate from its real batterysystem. Aoki et al. [76] proposed a general fractional-ordermodel to approximate transient temperatures. Reyes-Marambioet al. [77] then introduced this idea to model air-cooled cylin-drical Li-ion batteries. This model has demonstrated a high pre-dictive capability against experimental results and is intended tobe used in thermal control strategy design of battery cells andmodules.
3.3. Lead-acid battery modelsLead-acid batteries, invented in 1859, have matured to be the
most extensively used rechargeable battery technology. Theirlow cost and high reliability [78] make them competent inlarge-scale applications, such as uninterrupted power supplyand power quality regulation.
Similarly to Li-ion batteries, fractional-order modeling hasalso seen considerable applications in lead-acid batteries anddemonstrated attractive modeling performance. Garcia et al.[79] presented a diffusive model to delineate the dynamic be-haviors of lead-acid batteries using a fractional-order opera-tor. Following this, Lin et al. [80] proposed a frameworkof fractional-order modeling for diffuse processes, which was
5
then used to simulate the dynamics of lead-acid batteries. Theoutput-error technique was employed by them to derive themodel parameters based on classical input/output data of testcells. To characterize the crankability of lead-acid batteries,Sabatier et al. [81] and Cugnet et al. [82] introduced simpli-fied fractional-order models on the basis of conventional ECMs.These models are capable of capturing battery dynamics at afrequency range of 8-30 Hz in which batteries typically operatefor engine cranking.
3.4. Supercapacitor modelsSupercapacitors have emerged as a promising energy storage
source particularly suitable for storing and supplying high en-ergy in short periods of time, for instance, in vehicle accelera-tion and regenerative braking conditions [7]. This phenomenonis functionally attributed to their advantageous performance,such as large power density, high temperature tolerance, andexcellent cyclability [11]. Fundamentally, these merits stemfrom the highly reversible ions adoption mechanism that is non-faradaic and void of ions diffusion in the bulk of the high-conductivity electrodes. Nevertheless, the poor energy densityconstitutes the main bottleneck for applicability and has beenthe focus of intensive research. In the past decade, substantialprogress has been made to improve the performance of super-capacitors through continuous investigation of efficient storagemechanisms and potentially enhanced electrode and electrolytematerials. Nowadays, cost-effective and reliable supercapaci-tors with an energy density up to 10 Wh/kg are commerciallyavailable [83]. In practice, supercapacitors can complement andeven substitute for some high-energy EESSs (e.g., rechargeablebatteries) for power sinking and sourcing. In cell design, ongo-ing research and development are mainly directed towards in-creasing the energy density, which is now approximately 1/30-1/20 the energy density of state-of-the-art Li-ion batteries [84],and towards further lowering the cost.
An efficient management system is often required to mea-sure, monitor, and control supercapacitor systems that are usu-ally composed of a number of individual cells in series-parallelconnections. The management system’s functionality includesbut is not limited to cell balance, state estimation, safety su-pervision, and fault detection and isolation. To realize thesefunctions, the fundamental step is to build reliable and accuratemodels. Many mathematical models for supercapacitors havebeen presented to simulate the system behaviors. These can beroughly sorted into three categories: integer-order electrochem-ical models, conventional ECMs, and fractional ECMs.
The initial electrochemical models describe a supercapaci-tor’s internal electrochemical reactions based on first principles.This modeling methodology and its application for supercapac-itor management are still an active research topic. These modelsretain a high model precision but suffer from a heavy computa-tional burden caused by coupled PDEs [17, 85]. Furthermore,parameterization of high-order electrochemical models is tech-nically challenging due to potential identifiability issues andpersistent system aging phenomena. Conventional ECMs em-ploy basic electrical circuit elements to represent the superca-pacitor dynamics, with varied modeling performance, depend-
R1
C1(a)
R1+ R2
C2
-
V
R1+ -
V
(b)
R1 W
CPE1
C
Figure 4: Integer- and fractional-order electrical circuit models for a superca-pacitor cell.
ing on circuit topologies. See Fig. 4(a) for a popular exampleof conventional ECMs for a supercapacitor cell.
Fractional ECMs incorporate circuit elements, such asWarburg elements and fractional-order capacitors shown inFig. 4(b), to delineate the electrical response of supercapacitors.Using the same principle as Li-ion batteries, fractional superca-pacitor models armed with electrical elements at the phasanceof απ/2 often are better able to fit experimental data using a fewparameters, in contrast to their integral-order counterparts. Thismay efficiently ease the computation intensity and render on-line implementation of model-based algorithms applicable. Forexample, Riu et al. [86] introduced a half-order supercapaci-tor model and demonstrated high accuracy in representing thesystem dynamics . In addition, Martynyuk and Ortigueira [87]utilized a least-squares fitting method to extract parameters ofa fractional-order model based on impedance data. Bertrand etal. [88] and [89] synthesized a fractional-order nonlinear modelon the basis of frequency analysis. In a similar fashion, Martınet al. [90] proposed a Havriliak-Negami function-based modelthat is able to predict a supercapacitor’s static and dynamicalbehaviors throughout the spectrum.
4. System identification
The usefulness of mathematical models for EESSs highly re-lies on their parameters. Therefore, parameter identificationis a prerequisite to performing model-based simulation, esti-mation, and control algorithms. The identification of a gen-eral fractional-order system was initially conducted by Le Layin his Ph.D thesis [91]. However, for general nonlinear-in-the-parameters FOMs, no formal identification algorithms canprovide provable convergence. To determine which parame-ters are identifiable for given external excitations, Zhou et al.[92] conducted a sensitivity analysis for equivalent circuit com-ponent coefficients and fractional-order values. Most of thesubsequent approaches are dedicated to generalizing standardmethods, which are used in integer-order systems, to fractional-
6
order systems. These can be classified into time-domain andfrequency-domain methods.
Least-squares (LSQ) estimation techniques have been widelyused to identify fractional-order systems in the frequency do-main. For example, an output-error identification algorithmbased on LSQ was introduced for an inverse heat conductionproblem [93]. Sabatier et al. [94] adopted this approach to esti-mate internal states of lead-acid batteries and then demonstratedits effectiveness in a laboratory environment, where measure-ments are usually less noisy than real-world applications. Al-though it is possible to artificially generate broadband noises,data measured in the time domain needs to be converted to thefrequency domain using Fourier transforms. However, this pro-cess will inevitably result in biased measurements than can de-grade estimation accuracy.
It is therefore preferred to identify FOMs directly from time-domain measurements. Recently, this research topic has at-tracted considerable interest. Before identification of a frac-tional Li-ion cell model, Zhou et al. [92] adopted a statistical,multi-parametric method to analyze each parameter’s sensitiv-ity. Global optimizers, such as genetic algorithm and particleswarm optimization, have been exploited to calibrate fractionalnonlinear battery models in [62, 59] and fractional nonlinear su-percapacitor models in [30]. To estimate the orders and parame-ters in an incommensurate fractional-order chaotic system, Zhuet al. [95] proposed a switching differential evolution scheme,where the switching population size is adjusted dynamically.Motivated by this proposal, Lai et al. [96] employed a sequen-tial parameter identification method for a fractional-order Duff-ing system based on a differential evolution scheme. These au-thors then demonstrated an improved convergence of the pro-posed algorithm via numerical implementation. To provideguaranteed error convergence for parameter estimates in finitetime, Liu et al. [97] extended a modulating function method foronline identification of general linear fractional-order systems.What is interesting with this approach is that it does not requireinitial conditions and fractional derivatives of the output.
To improve the robustness against measurement noise, Victoret al. [98] developed the instrumental variable state variable fil-ter (IVSVF) and its simplified version for unbiased estimationof fractional-order systems. These methodologies have beensuccessfully applied to solve related problems in the field ofEESSs. Alavi et al. [49] combined Victor’s IVSVF method anda gradient-based optimization to identify parameters of elec-trochemical impedance models. Allafi et al. [99] applied asimplified refined IVSVF to identify a fractional transfer func-tion (FTF) model of a Li-ion battery. In contrast to the in-strumental variable method, Jacob et al. [100] recently pro-posed a Bayesian approach to identify the parameters of genericfractional-order systems and then applied this approach to bat-tery models.
In addition to the employed technical methods, the identifi-cation results can be affected by various factors such as inputsignal and measurement noise. Dzielinski et al. [101] proposeda fractional-order model for supercapacitors and parameterizedit using time-domain data collected through a constant-currentcharging test. Freeborn et al. [102] calculated the impedance
parameters of a fractional-order supercapacitor model by thevoltage step response. Nonetheless, the model precision maybe significantly curtailed when exposed to real-time loadingconditions, where the current direction, temperature, and SoCcan change rapidly, leading to parameter variations. To addressthis, Gabano et al. [103] used a cubic spine interpolation tech-nique to derive a fractional continuous linear-parameter-varyingmodel based on locally identified linear-time-invariant frac-tional impedance models. According to Jacob et al. [100], thememory capability could affect the performance of parameteridentification for FOMs, as demanding computation is requiredby their non-Markovian model setting. These authors also in-vestigated the effects of data length, magnitude of input signals,parameter initialization, and measurement noise on identifyinga non-commensurate fractional-order battery model.
All the above approaches aim to address the identifiabilityproblem practically, accounting for information such as noise,bias, and signal quality. As a different concept, structural iden-tifiability is a tool to study the identifiability of model param-eters without data. In other words, the input/output data is as-sumed to be sufficiently rich. Based on this, Alavi et al. [104]performed a structural identifiability analysis for both commen-surate and non-commensurate models based on the concept ofcoefficient maps. After applying the theoretical result to batterysystems, they then could prove that fractional circuit batterymodels with finite numbers of CPEs are structurally identifi-able. These results provide fundamental insights and can guidethe design and implementation of practical identifiability algo-rithms.
5. Quantitative evaluation of model complexity and accu-racy
As discussed in the previous sections, the high-order electro-chemical models and conventional circuit models are more orless restricted by their particular attributes and, consequently,may not be the most suitable options for the next-generationmanagement systems of EESSs. In contrast, the fractional mod-eling approach is very appealing because FOMs are structurallysimpler and computationally cheaper than the original electro-chemical models and can be more accurate than conventionalcircuit models. In addition to these general comments, a quan-titative evaluation of FOM performance under different operat-ing conditions is preferred. This section investigates FOMs viatwo case studies, with a special focus placed on model accuracyand computational efficiency.
5.1. Case study 1 – battery models
The effectiveness of fractional-order modeling techniques isfirst examined on battery cells. The model in Fig. 3(a) is ex-emplified for this purpose and its governing equations are pre-sented in (A.1)-(A.3), where the CPE1 is subject to fractionalorder of γ. The GL definition is implemented for fractionalderivatives with the memory length limited to 5. As bench-marks, the first-, second-, and third-order RC models from Fig.2 were also studied and denoted as IOM1, IOM2, and IOM3,
Figure 5: Current signals to test Li-ion battery models.
respectively. Experiments are conducted on a lithium nickel-manganese-cobalt oxide (LiNMC) cell in the type of cylindrical18650 with a rated capacity of 0.9264 ampere-hours (Ah). Thecurrent signals were adopted from dynamic stress test (DST),hybrid pulse (HP) test, and Federal Urban Driving Schedule(FUDS) test, as plotted in Fig. 5. These battery models werecalibrated using the corresponding voltage measurements underthe FUDS test and then validated against the other two tests.
The parameters to be identified include resistance, capaci-tance, differentiation order, and the open-circuit voltage (OCV)curve for different models. In particular, the OCV curve is mod-
Table 2: RMS errors [mV] in identification and validation of Li-ion batterymodels using experimental data.
eled as a fourth-order polynomial function, according to theobservation in [105]. Then, a constrained nonlinear optimiza-tion problem can be formulated with the objective to minimizethe difference between measurements and model predicted volt-ages. To approach the globally optimal solution, optimiza-tion problems are implemented for multiple times with varioussets of initial conditions. The obtained solutions from particleswarm optimization for different models of the considered bat-tery cell are given in Table 1. For model fidelity assessment, theroot-mean-square (RMS) error and percentage relative error inpredicting the terminal voltages are adopted. In particular, thepercentage error is defined as
Percentage Error(k) :=Vmod(k) − Vexprt(k)
maxVexprt(k)× 100 (7)
where k ∈ 1, 2, · · · , Vmod is the terminal voltage from batterymodels, and Vexprt is the voltage measurement.
The FOM and its integer alternatives are first compared interms of their capability in predicting battery voltage behav-iors. The evolution profiles of voltage and modeling error aredepicted in Fig. 6. The FOM follows its true voltage trajectoriesbetter than IOM1 and IOM2 under both the HP and DST tests.It can also be found that IOM1 cannot well match the mea-sured data, particularly when large currents are applied. Thissimulation result is consistent with the analysis in Section 3.2.Namely, the first-order RC model is unable to accurately de-scribe lithium diffusion dynamics. The identification and mod-eling errors for different models are presented in Table 2. Com-pared with IOM1, FOM can improve the modeling accuracy by33.4% and 15.1% under the HP and DST tests, respectively. Atthe same time, it outperforms IOM2 under both tests. Further-more, under the DST condition, FOM has even better resultsthan IOM3 but with fewer parameters.
However, the high accuracy of the fractional-order modelingapproach is brought about at some sacrifice of computationalefficiency. Simulations of the three models with the same spec-ifications, in terms of input signal and sampling time, were con-ducted in a Matlab m-file environment. While the FOM takes1.6 microseconds (µs) on average to implement one samplingstep, the two IOMs take less than 0.3 µs. Such a computa-tional requirement from the FOM may or may not be an issuefor real-time model-based algorithms, depending on the batteryapplications.
It is worth mentioning that the obtained characteristic dataon accuracy and computation can be influenced by factors suchas the definition of fractional-order derivative, memory length,and operating conditions. Other related studies of either modelcomplexity or computational efficiency have been conducted in
8
0 1 2 3 4 5 6 7 8 9
104
3.6
3.8
4
4.2
0 1 2 3 4 5 6 7 8 9
104
-1
0
1
0 0.5 1 1.5 2
105
3.6
3.8
4
4.2
0 0.5 1 1.5 2
105
-1
-0.5
0
0.5
1
Figure 6: Comparison results of different models for a Li-ion battery againstexperimental data. (a) and (b) are voltage profiles and percentage errors underthe HP test. (c) and (d) are results under the DST test.
[49, 55, 59, 106], where similar conclusions were obtained.
5.2. Case study 2 – Supercapacitor
The performance of fractional-order techniques is also eval-uated on supercapacitor cells with a nominal capacity of 3000F and a rated voltage of 2.7 V. The model in Fig. 4(b) withdynamic equations (A.4)-(A.7) is used here to demonstrate theidea. A widely used dynamic model, as in Fig. 4(a) (see [107]for its explicit formulation), is considered as a modeling bench-mark. As the same as Section 5.1, the FUDS profile from isadapted here to excite supercapacitors for generating a datasetfor model parameterization and the DST test is used for modelvalidation. After scaling, the corresponding current signals ap-plied to the supercapacitors are illustrated in Fig. 7. All the op-erations are carried out within a thermal chamber with the tem-perature fixed at 0, 20, and 40 oC, respectively. By deployingthe genetic algorithm to globally minimize the squared model-plant error, the FOM and IOM can be parameterized. Param-eter identification results corresponding to 20oC are given inTable 3.
An EIS test was performed under a wide range of frequen-cies, varying from 0.1Hz to 100Hz. Note that this covers mostworking conditions of supercapacitor energy storage systems.For a comprehensive description of the experimental setup,
0 50 100 150 200 250 300 350 400
-40
-20
0
20
0 100 200 300 400 500 600 700 800
-100
-50
0
50
Figure 7: Current signals to test supercapacitor models.
readers are referred to our previous work [34]. Comparisonresults of both models against the EIS dataset are presented inFig. 8.
In general, the FOM is able to describe the supercapacitorimpedance better than its integer-order comparative, across theconsidered spectrum and over different temperatures. Such su-periority becomes more apparent at reduced temperatures andlow frequencies. The reasons are mainly twofold: the IOM can-not well capture the mass transfer effect at low temperatures;the charge-transfer polarization voltage is extremely small athigh frequencies but significantly increases as the frequency de-creases [108].
Quantitatively, the RMS errors are calculated to differentiatethe modeling performance of the FOM and IOM. The FOM hasan RMS error of 0.084 Ω for the above tests, in comparison withan RMS error of 0.105 Ω for the IOM. That is, the FOM-basedtechnique offers a 20% rise in accuracy. Indeed, this advantagemay vary for different test protocols and with different numer-ical specifications in implementing the fractional derivatives.For example, the modeling accuracy of FOMs can in general beimproved at a large memory length. However, systematicallyinvestigating their effects is beyond the scope of this reviewwork. The advantages of FOM over IOM for supercapacitorshave also been demonstrated and confirmed in other publica-tions, such as [109, 110].
6. Challenges and future prospects
Despite the advances in fractional modeling techniques, thedeployment of intelligent management algorithms of EESSsbased on FOMs still faces a number of technical challenges.Intensive studies on fractional modeling methodologies andmodel-based applications are mainly required in the followingareas:
System modeling. Fractional-order electrical models andthermal models have been individually established for batter-ies and supercapacitors. However, electrical and thermal dy-
Figure 8: Comparison of different supercapacitor models using electrochemicalimpedance spectroscopy.
namics intrinsically interact with each other in these electro-chemical systems [111, 112]. Thus, the coupling relationshipsbetween these dynamic processes are necessary to model. Fur-thermore, common to different EESSs is that the capacity andpower supply ability, which are often used to quantify the SoH,will inevitably experience an aging process. Such a process canbe accelerated dramatically if EESSs operate under inappropri-ate conditions [113]. The importance of SoH management be-comes crucial for safety-critical and cost-sensitive applications.Fractional-order modeling for the aging phenomena can be animportant step to effectively protect an EESS’s health and pro-long its lifetime. The last but not the least step is the model’sadaptivity. The available FOMs of batteries/supercapacitors areusually parameterized once and then are expected to play a roleover the whole lifespan. As observed from [62, 61], in this case,the models will mismatch their true system incrementally andfail at some stage. Therefore, adaptive FOMs need to be devel-oped in which model parameters can accommodate the effectsof system aging, ambient temperature, and SoC levels.
State estimation. SoC, SoH, state of energy (SoE), and stateof power (SoP) are the most important indicators of EESS in-ternal states. Accurate knowledge of these states is requiredin the pursuit of various objectives, for example, to ensurecharge/discharge safety, to satisfy end-user demands, to im-prove convenience, and to execute system-level energy man-agement. However, these states cannot be measured directlyduring on-board application using currently available sensingtechniques. This fact fundamentally motivates observer designsbased on measurements such as current, voltage, and cell sur-face temperature. A considerable number of integer-order esti-mation algorithms have been proposed to probe state/parameterbehavior inside EESSs, such as Luenberger observers andKalman filters. By extending these estimation approaches toFOMs, some initial work has recently been attempted to esti-mate the SoC of Li-ion cells [65, 114] and of supercapacitors[30]. In 2017, Li et al. [115] applied an adaptive fractional-order extended Kalman filter to the SoE estimation for Li-ionbatteries in EVs. In the context of lead-acid batteries, Cugnetet al. [82] pioneered a fractional resistance-estimator to indi-cate its crankability in starting a vehicle. Monitoring SoH andSoP in real time based on FOMs is crucial for safe and opti-mal utilization of EESSs but has not been comprehensively andsystematically studied yet. Meanwhile, given that several pro-cesses occur simultaneously in EESSs with different time con-stants, it would be desirable to have some dual fractional esti-mation algorithms in which the states could be estimated in sep-arate time scales. In addition, both the accuracy and resiliencyneed to be addressed in the presence of a range of uncertaintiesinherent in EESSs.
Charge/discharge control. The cycling operation of EESSsshould be meticulously managed. Electrical energy and powerneed to be delivered effectively and efficiently, and at the sametime, users’ requirements in charging time, vehicle transientacceleration, SoH, and/or the overall economy must be takeninto account. Usually, some or all of these factors are impor-tant yet compete with each other. Multi-objective control prob-lems may need to be considered to maintain an optimal trade-off among the selected objectives during the charge/dischargeprocess, such as health-aware fast charging and aging-adaptiveoptimal energy management. To do so, in-situ dynamic infor-mation for EESSs and its prediction into some future time in-terval from the FOMs can be critical.
Fractional automatic control can be explained as the rea-son that gives rise to a renewed interest in FOMs. In com-parison to integer-order proportional-integral-derivative (PID)controllers, fractional-order PID controllers have more tuningparameters within the embedded optimization algorithms andthus are able to achieve superior convergence and robustness
10
properties [116, 117]. The methodologies including CRONE(Commande Robuste d’Ordre Non Entier), H∞, and flatnesscontrol have been exhaustively reviewed in [27]. According to[118, 119], fractional calculus can also be integrated into slid-ing mode control (SMC) to obtain better performance. Thesefractional control theories and applications in other areas mayform a powerful tool to enhance EESSs’ dynamic performanceand extend the working life.
7. Conclusions
This paper provided an overview of the current develop-ment in mathematical models for lithium-ion batteries, lead-acid batteries, and supercapacitors, with a particular focus onfractional-order techniques. The review has illustrated the linksbetween fractional-order calculus, electrochemical impedancespectroscopy, and EESS dynamic characteristics. By survey-ing various available battery and supercapacitor applications,fractional-order models (FOMs) are shown to have been widelystudied, with attempts to capture system electrical, electro-chemical, and thermal dynamics. Such modeling mechanismsare capable of predicting system behaviors and have the poten-tial to maintain physically meaningful parameters. The advan-tages of model precision associated with computational com-plexity were further confirmed in this work via numerical casestudies on lithium-ion battery and supercapacitor cells. Toenable model usage, parameter identification techniques forFOMs were discussed, and the benefits of using time-domainmeasurements, instead of frequency-domain data, were pre-sented.
To enable further advances in battery and supercapaci-tor management, a research outlook for fractional modelingmethodology and model-based applications has been discussed.The research directions mainly include (1) the development ofsystem models that describe coupled electrochemical-thermaldynamics and are adaptive to factors such as system aging andtime-varying ambient conditions, (2) the design of estimationalgorithms to observe SoC, SoE, SoP, and SoH in real-time, and(3) the adoption of FOM-based controllers to improve chargeand discharge performance.
Appendix A. Fractional-order battery/supercapacitormodels
Governing equations of the Li-ion battery model in Fig. 3(a)can be formulated based on Kirchhoff’s current and voltagelaws:
D1S oC(t) =
η · I(t)3600Cn
(A.1)
DγV1(t) = −
V1(t)R1C1
+I(t)C1
(A.2)
V(t) = U(S oC(t)) + R∞I(t) + V1(t). (A.3)
Similarly, governing equations of the supercapacitor model inFig. 4(b) can be established:
D1S oC(t) =
η · I(t)3600Cn
(A.4)
DγV1(t) = −
V1(t)R1C1
+I(t)C1
(A.5)
DβV2(t) =
I(t)W
(A.6)
V(t) = R∞I(t) + V1(t) + V2(t) (A.7)
In the above two models, Cn is the nominal capacity in Ah andη is the coulombic efficiency.
Acknowledgements
This work of C. Zou and T. Wik was supported inpart by the Swedish Energy Agency under the Grant No.39786-1. This work of X. Hu was supported in partby Fundamental Research Funds for the Central Univer-sities, China (Project No.106112017CDJQJ338811, ProjectNo.106112016CDJXZ338822).
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