Model reduction of fractional impedance spectra for time-frequency analysis of batteries, fuel cells, and supercapacitors

Abstract

•Formularized model reduction of fractional impedance spectra for batteries/ supercapacitors/fuel cells.

•Gained an insight into the characteristic time constant evolution for the joint time–frequency analysis.

•Validated the thorough model reduction of fractional impedance spectra by numerical simulations.

•Enhanced the reliability of joint time–frequency analysis for electrochemical energy devices.

Keywords

battery, fuel cell, supercapacitor / fractional impedance spectroscopy / model reduction / time-frequency analysis

References

[1] Duan J, Tang X, Dai H, et al. Building safe lithium-ion batteries for electric vehicles: a review. Electrochem Energy Rev. 2020; 3 (1): 1- 42.

[2] Wu F, Liu M, Li Y, et al. High-mass-loading electrodes for advanced secondary batteries and supercapacitors. Electrochem Energy Rev. 2021; 4 (2): 382- 446.

[3] Krishnamoorthy K, Pazhamalai P, Manoharan S, Liyakath Ali NUH, Kim S-J. Recent trends, challenges, and perspectives in piezoelectric-driven self-chargeable electrochemical supercapacitors. Carbon Energy. 2022; 4 (5): 833- 855.

[4] Chen X, Liu C, Fang Y, et al. Understanding of the sodium storage mechanism in hard carbon anodes. Carbon Energy. 2022; 4 (6): 1133- 1150.

[5] Huang Q-A, Hui R, Wang B, Zhang J. A review of AC impedance modeling and validation in SOFC diagnosis. Electrochim Acta. 2007; 52 (28): 8144- 8164.

[6] Guo Y, Pan F, Chen W, et al. The controllable design of catalyst inks to enhance PEMFC performance: a review. Electrochem Energy Rev. 2021; 4 (1): 67- 100.

[7] Li Z, Li M, Zhu Z. Perovskite cathode materials for low-temperature solid oxide fuel cells: fundamentals to optimization. Electrochem Energy Rev. 2022; 5 (2): 263- 311.

[8] Or T, Gourley SWD, Kaliyappan K, Yu A, Chen Z. Recycling of mixed cathode lithium-ion batteries for electric vehicles: current status and future outlook. Carbon Energy. 2020; 2 (1): 6- 43.

[9] Pham TN, Park D, Lee Y, et al. Combination-based nanomaterial designs in single and double dimensions for improved electrodes in lithium ion-batteries and faradaic supercapacitors. J Energy Chem. 2019; 38: 119- 146.

[10] Liu Y, Li W, Xia Y. Recent progress in polyanionic anode materials for Li (Na)-ion batteries. Electrochem Energy Rev. 2021; 4 (3): 447- 472.

[11] Wang L, Qiu J, Wang X, et al. Insights for understanding multiscale degradation of LiFePO₄ cathodes. eScience. 2022; 2 (2): 125- 137.

[12] Huang Q-A, Li Y, Tsay K-C, et al. Multi-scale impedance model for supercapacitor porous electrodes: theoretical prediction and experimental validation. J Power Sources. 2018; 400: 69- 86.

[13] Li W, Huang Q-A, Li Y, et al. Capacitive energy storage from single pore to porous electrode identified by frequency response analysis. J Energy Chem. 2023; 77: 384- 405.

[14] Wu H, Feng C, Zhang L, Zhang J, Wilkinson DP. Nonnoble metal electrocatalysts for the hydrogen evolution reaction in water electrolysis. Electrochem Energy Rev. 2021; 4 (3): 473- 507.

[15] Huck M, Sauer D-U. Modeling transient processes in lead-acid batteries in the time domain. J Energy Storage. 2020; 29: 101430.

[16] Denisov E, Nigmatullin R, Evdokimov Y, Timergalina G. Lithium battery transient response as a diagnostic tool. J Electron Mater. 2018; 47 (8): 4493- 4501.

[17] Devan S, Subramanian VR, White RE. Transient analysis of a porous electrode. J Electrochem Soc. 2005; 152 (5): A947- A955.

[18] Huang Q-A, Liu M, Liu M. Impedance spectroscopy study of an SDC-based SOFC with high open circuit voltage. Electrochim Acta. 2015; 177: 227- 236.

[19] Huang Q-A, Shen Y, Huang Y, Zhang L, Zhang J. Impedance characteristics and diagnoses of automotive lithium-ion batteries at 7.5% to 93.0% state of charge. Electrochim Acta. 2016; 219: 751- 765.

[20] Mei B-A, Munteshari O, Lau J, Dunn B, Pilon L. Physical interpretations of Nyquist plots for EDLC electrodes and devices. J Phys Chem C. 2018; 122 (1): 194- 206.

[21] Song Y-W, Peng Y-Q, Zhao M, et al. Understanding the impedance response of lithium polysulfide symmetric cells. Small Sci. 2021; 1 (11): 2100042.

[22] Shie Qian Q, Dapang Chen C. Joint time-frequency analysis. IEEE Signal Process Mag. 1999; 16 (2): 52- 67.

[23] Nyikos L, Pajkossy T. Electrochemistry at fractal interfaces: the coupling of AC and DC behaviour at irregular electrodes. Electrochim Acta. 1990; 35 (10): 1567- 1572.

[24] Deng C, Lu W. Consistent diffusivity measurement between galvanostatic intermittent titration technique and electrochemical impedance spectroscopy. J Power Sources. 2020; 473: 228613.

[25] Schwan HP, Onaral B. Linear and nonlinear properties of platinum electrode polarisation III: equivalence of frequency-and time-domain behaviour. Med Biol Eng Comput. 1985; 23 (1): 28- 32.

[26] Han X, Guo D, Feng X, Lu L, Ouyang M. Equivalence of time and frequency domain modeling for lithium ion batteries. In: Gao F, Maillard-Salin C, eds. 2021 IEEE 30th International Symposium on Industrial Electronics (ISIE). IEEE; 2021: 1- 4.

[27] Stolz L, Gaberšček M, Winter M, Kasnatscheew J. Different efforts but similar insights in battery R&D: electrochemical impedance spectroscopy vs galvanostatic (constant current) technique. Chem Mater. 2022; 34 (23): 10272- 10278.

[28] Bisquert J, Janssen M. From frequency domain to time transient methods for halide perovskite solar cells: the connections of IMPS, IMVS, TPC, and TPV. J Phys Chem Lett. 2021; 12 (33): 7964- 7971.

[29] Teo L, Subramanian VR, Schwartz DT. Dynamic electrochemical impedance spectroscopy of lithium-ion batteries: revealing underlying physics through efficient joint timefrequency modeling. J Electrochem Soc. 2021; 168 (1): 010526.

[30] Zheng Y, Shi Z, Guo D, Dai H, Han X. A simplification of the time-domain equivalent circuit model for lithium-ion batteries based on low-frequency electrochemical impedance spectra. J Power Sources. 2021; 489: 229505.

[31] Schröer P, Khoshbakht E, Nemeth T, Kuipers M, Zappen H, Sauer DU. Adaptive modeling in the frequency and time domain of high-power lithium titanate oxide cells in battery management systems. J Energy Storage. 2020; 32: 101966.

[32] Wildfeuer L, Gieler P, Karger A. Combining the distribution of relaxation times from EIS and time-domain data for parameterizing equivalent circuit models of lithium-ion batteries. Batteries. 2021; 7 (3): 52.

[33] Zhao Y, Jossen A. Comparative study of parameter identification with frequency and time domain fitting using a physics-based battery model. Batteries. 2022; 8 (11): 222.

[34] Hasyim MR, Ma D, Rajagopalan R, Randall C. Prediction of charge-discharge and impedance characteristics of electric double-layer capacitors using porous electrode theory. J Electrochem Soc. 2017; 164 (13): A2899- A2913.

[35] Weydahl H, Møller-Holst S, Hagen G, Børresen B. Transient response of a proton exchange membrane fuel cell. J Power Sources. 2007; 171 (2): 321- 330.

[36] Huang Q-A, Bai Y, Wang L, et al. Time-frequency analysis of Li solid-phase diffusion in spherical active particles under typical discharge modes. J Energy Chem. 2022; 67: 209- 224.

[37] Li W, Huang Q-A, Yang C, et al. A fast measurement of Warburg-like impedance spectra with Morlet wavelet transform for electrochemical energy devices. Electrochim Acta. 2019; 322: 134760.

[38] Tian R, Park S-H, King PJ, et al. Quantifying the factors limiting rate performance in battery electrodes. Nat Commun. 2019; 10: 1933.

[39] Murbach MD, Schwartz DT. Analysis of li-ion battery electrochemical impedance spectroscopy data: an easy-to-implement approach for physics-based parameter estimation using an open-source tool. J Electrochem Soc. 2018; 165 (2): A297- A304.

[40] Doyle M, Fuller TF, Newman J. Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. J Electrochem Soc. 1993; 140 (6): 1526- 1533.

[41] Fuller TF, Doyle M, Newman J. Simulation and optimization of the dual lithium ion insertion cell. J Electrochem Soc. 1994; 141 (1): 1- 10.

[42] Wang J, Huang Q-A, Li W-H, WANG J, Zhang Q-C, Zhang J-J. Fundamentals of distribution of relaxation times for electrochemical impedance spectroscopy. J Electrochem. 2020; 26 (5): 607- 627.

[43] Lu Y, Zhao C-Z, Huang J-Q, Zhang Q. The timescale identification decoupling complicated kinetic processes in lithium batteries. Joule. 2022; 6 (6): 1172- 1198.

[44] Soni R, Robinson JB, Shearing PR, Brett DJL, Rettie AJE, Miller TS. Lithium-sulfur battery diagnostics through distribution of relaxation times analysis. Energy Storage Mater. 2022; 51: 97- 107.

[45] Moya AA. Low-frequency development approximations to the transmissive Warburg diffusion impedance. J Energy Storage. 2022; 55: 105632.

[46] ZHANG Y-L, XIN S, GUO Y-G, WAN L-J. Comments on internal alternating current resistance test standard and methods of lithium-ion batteries. J Electrochem. 2012; 18 (3): 205- 214.

[47] Xiong R, Tian J, Mu H, Wang C. A systematic model-based degradation behavior recognition and health monitoring method for lithium-ion batteries. Appl Energy. 2017; 207: 372- 383.

[48] Jiang B, Zhu J, Wang X, Wei X, Shang W, Dai H. A comparative study of different features extracted from electrochemical impedance spectroscopy in state of health estimation for lithium-ion batteries. Appl Energy. 2022; 322: 119502.

[49] Jia Y, Dong L, Yang G, et al. Parameter identification method for a fractional-order model of lithium-ion batteries considering electrolyte-phase diffusion. Batteries. 2022; 8 (8): 90.

[50] Tran NT, Vilathgamuwa M, Farrell T, Choi SS, Li Y, Teague J. A padé approximate model of lithium ion batteries. J Electrochem Soc. 2018; 165 (7): A1409- A1421.

[51] Rodríguez A, Plett GL, Trimboli MS. Comparing four modelorder reduction techniques, applied to lithium-ion battery-cell internal electrochemical transfer functions. eTransportation. 2019; 1: 100009.

[52] Mundy A, Plett GL. Reduced-order physics-based modeling and experimental parameter identification for non-Faradaic electrical double-layer capacitors. J Energy Storage. 2016; 7: 167- 180.

[53] Smith KA, Rahn CD, Wang CY. Model order reduction of 1D diffusion systems via residue grouping. J Dyn Syst Meas Control. 2008; 130 (1): 011012.

[54] Kapoulea S, Psychalinos C, Elwakil AS. Passive approximations of double-exponent fractional-order impedance functions. Int J Circuit Theory Appl. 2021; 49 (5): 1274- 1284.

[55] Mauracher P, Karden E. Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification. J Power Sources. 1997; 67 (1-2): 69- 84.

[56] Kuhn E, Forgez C, Lagonotte P, Friedrich G. Modelling NimH battery using Cauer and Foster structures. J Power Sources. 2006; 158 (2): 1490- 1497.

[57] Iftikhar MU, Riu D, Druart F, Rosini S, Bultel Y, Retière N. Dynamic modeling of proton exchange membrane fuel cell using non-integer derivatives. J Power Sources. 2006; 160 (2): 1170- 1182.

[58] Oustaloup A, Levron F, Mathieu B, Nanot FM. Frequencyband complex noninteger differentiator: characterization and synthesis. IEEE Trans Circuits Syst I Fundam Theory Appl. 2000; 47 (1): 25- 39.

[59] Shin D-H, Jeong J-B, Kim T-H, Kim H-J. Modeling of lithium battery cells for plug-in hybrid vehicles. J Power Electron. 2013; 13 (3): 429- 436.

[60] Vinagre BM, Podlubny I, Hernandez A, Feliu V. Some approximations of fractional order operators used in control theory and applications. Fract Calculus Appl Anal. 2000; 3 (3): 231- 248.

[61] Xue D, Zhao C, Chen Y. A modified approximation method of fractional order system. In: 2006 International conference on mechatronics and automation. IEEE; 2006: 1043- 1048.

[62] Crank J. The Mathematics of Diffusion. 2th ed. Oxford University Press; 1979.

[63] Berthier F, Diard J-P, Montella C. Numerical solution of coupled systems of ordinary and partial differential equations. Application to the study of electrochemical insertion reactions by linear sweep voltammetry. J Electroanal Chem. 2001; 502 (1-2): 126- 131.

[64] Montella C, Diard J-P. New approach of electrochemical systems dynamics in the time-domain under small-signal conditions. I. A family of algorithms based on numerical inversion of Laplace transforms. J Electroanal Chem. 2008; 623 (1): 29- 40.

[65] Wang JC. Realizations of generalized Warburg impedance with RC ladder networks and transmission lines. J Electrochem Soc. 1987; 134 (8): 1915- 1920.

[66] Chen Y, Petras I, Xue D. Fractional order control-a tutorial. In: Aghdam A, eds. 2009 American Control Conference. IEEE; 2009: 1397- 1411.

[67] Hikono S, Sugiura A, Kawaguchi T, Baba A, Maruta I, Adachi S. Continuous-time system identification of rechargeable battery in electric vehicles in consideration of temperature characteristics. In: 2015 IEEE Conference on Control Applications (CCA). IEEE; 2015: 1939- 1944.

[68] Tsirimokou G. A systematic procedure for deriving RC networks of fractional-order elements emulators using MATLAB. AEU Int J Electron Commun. 2017; 78: 7- 14.

[69] Matsuda K, Fujii H. H(infinity) optimized wave-absorbing control-analytical and experimental results. J Guid Control Dyn. 1993; 16 (6): 1146- 1153.

[70] Khanra M, Pal J, Biswas K. Rational approximation of fractional operator—a comparative study. In: 2010 International Conference on Power, Control and Embedded Systems. IEEE; 2010: 1- 5.

[71] Sun H, Charef A, Tsao YY, Onaral B. Analysis of polarization dynamics by singularity decomposition method. Ann Biomed Eng. 1992; 20 (3): 321- 335.

[72] Charef A, Sun HH, Tsao YY, Onaral B. Fractal system as represented by singularity function. IEEE Trans Autom Control. 1992; 37 (9): 1465- 1470.

[73] Buller S, Thele M, De Doncker RWAA, Karden E. Impedance-based simulation models of supercapacitors and Li-ion batteries for power electronic applications. IEEE Trans Ind Appl. 2005; 41 (3): 742- 747.

[74] Tanaka T, Ito S, Muramatsu M, et al. Accurate and versatile simulation of transient voltage profile of lithium-ion secondary battery employing internal equivalent electric circuit. Appl Energy. 2015; 143: 200- 210.

[75] Logerais P-O, Camara MA, Riou O, et al. Modeling of a supercapacitor with a multibranch circuit. Int J Hydrogen Energy. 2015; 40 (39): 13725- 13736.

[76] Zubieta L, Bonert R. Characterization of double-layer capacitors for power electronics applications. IEEE Trans Ind Appl. 2000; 36 (1): 199- 205.

[77] Devillers N, Jemei S, Péra M-C, Bienaimé D, Gustin F. Review of characterization methods for supercapacitor modelling. J Power Sources. 2014; 246: 596- 608.

[78] Qi Y, Huang B, Chuang KT. Dynamic modeling of solid oxide fuel cell: the effect of diffusion and inherent impedance. J Power Sources. 2005; 150: 32- 47.

[79] Comminges C, Fu QX, Zahid M, Steiner NY, Bucheli O. Monitoring the degradation of a solid oxide fuel cell stack during 10,000 h via electrochemical impedance spectroscopy. Electrochim Acta. 2012; 59: 367- 375.

[80] Krishna BT, Reddy KVVS. Active and passive realization of fractance device of order 1/2. Act Passive Electron Compon. 2008; 2008: 1- 5.

[81] Brezinski C. History of Continued Fractions and Padé Approximants. Springer Science & Business Media; 2012.

[82] Jones WB, Thron WJ. Continued fractions in numerical analysis. Appl Numer Math. 1988; 4 (2-4): 143- 230.

[83] Rogers LJ. On the representation of certain asymptotic series as convergent continued fractions. Proc London Math Soc. 1907; 2 (1): 72- 89.

[84] Khovanskii AN. The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory. P. Noordhoff Ltd; 1963.

[85] Oldenburger M, Bedürftig B, Gruhle A, et al. Investigation of the low frequency Warburg impedance of Li-ion cells by frequency domain measurements. J Energy Storage. 2019; 21: 272- 280.

[86] Lasia A. Electrochemical Impedance Spectroscopy and its Applications. Springer; 2014.

[87] Huang J. Diffusion impedance of electroactive materials, electrolytic solutions and porous electrodes: Warburg impedance and beyond. Electrochim Acta. 2018; 281: 170- 188.

[88] Warburg E. Ueber das Verhalten sogenannter unpolarisirbarer Elektroden gegen Wechselstrom. Ann Phys. 1899; 303 (3): 493- 499.

[89] Levi M. Application of finite-diffusion models for the interpretation of chronoamperometric and electrochemical impedance responses of thin lithium insertion V₂O₅ electrodes. Solid State Ionics. 2001; 143 (3-4): 309- 318.

[90] Boukamp BA, Rolle A. Use of a distribution function of relaxation times (DFRT) in impedance analysis of SOFC electrodes. Solid State Ionics. 2018; 314: 103- 111.

[91] Moss PL, Au G, Plichta EJ, Zheng JP. An electrical circuit for modeling the dynamic response of Li-ion polymer batteries. J Electrochem Soc. 2008; 155 (12): A986- A994.

[92] Orazem ME, Tribollet B. Electrochemical Impedance Spectroscopy. John Wiley & Sons; 2017.

[93] Wang J, Huang Q-A, Li W, et al. Insight into the origin of pseudo peaks decoded by the distribution of relaxation times/differential capacity method for electrochemical impedance spectroscopy. J Electroanal Chem. 2022; 910: 116176.

[94] Fletcher S. Tables of degenerate electrical networks for use in the equivalent-circuit analysis of electrochemical systems. J Electrochem Soc. 1994; 141 (7): 1823- 1826.

[95] Fletcher S, Black VJ, Kirkpatrick I. A universal equivalent circuit for carbon-based supercapacitors. J Solid State Electrochem. 2014; 18 (5): 1377- 1387.

[96] Schönleber M, Ivers-Tiffée E. The distribution function of differential capacity as a new tool for analyzing the capacitive properties of lithium-ion batteries. Electrochem Commun. 2015; 61: 45- 48.

[97] Sekushin NA. Equivalent circuit of Warburg impedance. Russ J Electrochem. 2009; 45 (7): 828- 832.

[98] Shamash Y. Stable reduced-order models using Padé-type approximations. IEEE Trans Autom Control. 1974; 19 (5): 615- 616.

[99] Sinha AK, Pal J. Simulation based reduced order modelling using a clustering technique. Comput Electr Eng. 1990; 16 (3): 159- 169.

[100] Hutton M, Friedland B. Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans Autom Control. 1975; 20 (3): 329- 337.

[101] Zakian V. Simplification of linear time-invariant systems by moment approximants. Int J Control. 1973; 18 (3): 455- 460.

[102] Mukherjee S, Satakshi , Mittal RC. Model order reduction using response-matching technique. J Franklin Inst. 2005; 342 (5): 503- 519.

[103] Sinha NK, Kuszta B. Modelling and Identification of Dynamic Systems. Springer; 1983.

[104] Mukherjee S, Mishra RN. Order reduction of linear systems using an error minimization technique. J Franklin Inst. 1987; 323 (1): 23- 32.

[105] Parthasarathy R, John S. System reduction using Cauer continued fraction expansion about s = 0 and s = ∞ alternately. Electron Lett. 1978; 14 (8): 261- 262.

[106] Prasad R, Pal J. Use of continued fraction expansion for stable reduction of linear multivariable systems. J Inst Eng India Part E Electr Eng Div. 1991; 72: 43- 47.

[107] Krishnamurthy V, Seshadri V. Model reduction using the Routh stability criterion. IEEE Trans Autom Control. 1978; 23 (4): 729- 731.

[108] Chen TC, Chang CY, Han KW. Reduction of transfer functions by the stability-equation method. J the Franklin Inst. 1979; 308 (4): 389- 404.

[109] Chen TC, Chang CY, Han KW. Model reduction using the stability-equation method and the continued-fraction method. Int J Control. 1980; 32 (1): 81- 94.

[110] Chen TC, Chang CY, Han KW. Model reduction using the stability-equation method and the Padé approximation method. J Franklin Inst. 1980; 309 (6): 473- 490.

[111] Desai SR, Prasad R. A new approach to order reduction using stability equation and big bang big crunch optimization. Syst Sci Control Eng Open Access. 2013; 1 (1): 20- 27.

[112] Singh P, Dewangan PD. Model order reduction of fixed coefficient system using genetic algorithm. In: Kumar R, Dohare RK, Dubey H, Singh VP, eds. Applications of Advanced Computing in Systems. Springer; 2021: 107- 116.

[113] Oldham K, Myland J, Spanier J. An Atlas of Functions: With Equator, the Atlas Function Calculator. Springer; 2010.

[114] Chen Y, Petras I, Vinagre B. A list of Laplace and inverse Laplace transforms related to fractional order calculus, 2001. Accessed February 7, 2023.

[115] Kollmeyer P, Hackl A, Emadi A. Li-ion battery model performance for automotive drive cycles with current pulse and EIS parameterization. In: 2017 IEEE transportation electrification conference and expo (ITEC). IEEE; 2017: 486- 492.

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