# Introducing the Loewner Method as a Data-Driven and Regularization-Free Approach for the Distribution of Relaxation Times Analysis of Lithium-Ion Batteries

^{1}

^{2}

^{3}

^{4}

^{*}

*Batteries*)

## Abstract

**:**

## 1. Introduction

- Analysis of the LM through different ECMs with known time constants and gains;
- Detailed discussion of the correlation between model order, error, and distribution of gains for synthetic data;
- Investigation of the effects of noise on the distribution of gains;
- Application of the LM for process identification of LIB;
- Comparison of LM and gDRT.

## 2. Loewner Method

## 3. Application of Loewner Method for Process Identification

#### 3.1. Analysis of Different Equivalent Circuit Models

- Using the bend point of the singular value curve, considering all values with the highest gradient, leading to $k=8$;
- Introducing a tolerance limit, here exemplary ${10}^{-8}$ leading to $k=22$;
- Choosing the first model order in which the singular values are (nearly) zero.

#### 3.2. Analysis of Noise

#### 3.3. Measured Impedance Data

## 4. Comparison of Loewner Method and Generalized Distribution of Relaxation Times

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

**ReDesign**(Development of design guidelines for the recycling-oriented design of battery systems in context of the circular economy) which is funded by the Federal Ministry of Education and Research under grant number 03XP0318C. ReDesign is a part of the

**greenBatt**cluster.

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BOL | Begin of life |

CPE | Constant phase element |

DRT | Distribution of relaxation times |

ECM | Equivalent circuit model |

EIS | Electrochemical impedance spectroscopy |

EOL | End of life |

gDRT | Generalized distribution of relaxation times |

LIB | Lithium-ion battery |

LM | Loewner method |

LWF | Loewner framework |

PEMFC | Polymer electrolyte membrane fuel cell |

SEI | Solid electrolyte interphase |

SNR | Signal-to-noise ratio |

SOC | State of charge |

SVD | Singular value decomposition |

## Appendix A

**Figure A1.**Nyquist plot (

**a**), distribution of gains obtained by gDRT (

**b**), distribution of gains obtained by LWF (

**c**) of the measurement data at different temperatures.

**Figure A2.**Number of peaks of the distribution of gains determined with gDRT as function of the regularization parameter.

1a | 1b | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|

BOL | $27.30$$\mathrm{m}$$\mathsf{\Omega}$ | $-16.36$$\mathrm{m}$$\mathsf{\Omega}$ | $2.50$$\mathrm{m}$$\mathsf{\Omega}$ | $1.06$$\mathrm{m}$$\mathsf{\Omega}$ | $0.30$$\mathrm{m}$$\mathsf{\Omega}$ | $0.79$$\mathrm{m}$$\mathsf{\Omega}$ | $2.17$$\mathrm{m}$$\mathsf{\Omega}$ | $11.54$$\mathrm{m}$$\mathsf{\Omega}$ |

EOL | $29.96$$\mathrm{m}$$\mathsf{\Omega}$ | $-15.95$$\mathrm{m}$$\mathsf{\Omega}$ | $4.03$$\mathrm{m}$$\mathsf{\Omega}$ | $3.31$$\mathrm{m}$$\mathsf{\Omega}$ | $0.85$$\mathrm{m}$$\mathsf{\Omega}$ | $1.03$$\mathrm{m}$$\mathsf{\Omega}$ | $3.44$$\mathrm{m}$$\mathsf{\Omega}$ | $18.52$$\mathrm{m}$$\mathsf{\Omega}$ |

**Table A2.**Distribution of gains for the different processes of the BOL measurement obtained by gDRT.

$\mathit{\lambda}$ | I | II | III | IV | V | VI | VII |
---|---|---|---|---|---|---|---|

0.08 | $3.58$$\mathrm{m}$$\mathsf{\Omega}$ | $3.38$$\mathrm{m}$$\mathsf{\Omega}$ | $0.67$$\mathrm{m}$$\mathsf{\Omega}$ | $0.21$$\mathrm{m}$$\mathsf{\Omega}$ | $1.46$$\mathrm{m}$$\mathsf{\Omega}$ | $1.27$$\mathrm{m}$$\mathsf{\Omega}$ | $3.65$$\mathrm{m}$$\mathsf{\Omega}$ |

0.11 | $3.13$$\mathrm{m}$$\mathsf{\Omega}$ | $3.38$$\mathrm{m}$$\mathsf{\Omega}$ | $0.62$$\mathrm{m}$$\mathsf{\Omega}$ | $0.20$$\mathrm{m}$$\mathsf{\Omega}$ | $1.32$$\mathrm{m}$$\mathsf{\Omega}$ | $1.22$$\mathrm{m}$$\mathsf{\Omega}$ | $3.54$$\mathrm{m}$$\mathsf{\Omega}$ |

0.14 | $2.99$$\mathrm{m}$$\mathsf{\Omega}$ | $3.41$$\mathrm{m}$$\mathsf{\Omega}$ | $0.60$$\mathrm{m}$$\mathsf{\Omega}$ | $0.19$$\mathrm{m}$$\mathsf{\Omega}$ | $1.19$$\mathrm{m}$$\mathsf{\Omega}$ | $1.40$$\mathrm{m}$$\mathsf{\Omega}$ | $3.28$$\mathrm{m}$$\mathsf{\Omega}$ |

## References

- Rodat, S.; Sailler, S.; Druart, F.; Thivel, P.X.; Bultel, Y.; Ozil, P. EIS measurements in the diagnosis of the environment within a PEMFC stack. J. Appl. Electrochem.
**2009**, 40, 911–920. [Google Scholar] [CrossRef] - Fang, Q.; de Haart, U.; Schäfer, D.; Thaler, F.; Rangel-Hernandez, V.; Peters, R.; Blum, L. Degradation Analysis of an SOFC Short Stack Subject to 10,000 h of Operation. J. Electrochem. Soc.
**2020**, 167, 144508. [Google Scholar] [CrossRef] - Middlemiss, L.A.; Rennie, A.J.; Sayers, R.; West, A.R. Characterisation of batteries by electrochemical impedance spectroscopy. Energy Rep.
**2020**, 6, 232–241. [Google Scholar] [CrossRef] - Maheshwari, A.; Heck, M.; Santarelli, M. Cycle aging studies of lithium nickel manganese cobalt oxide-based batteries using electrochemical impedance spectroscopy. Electrochim. Acta
**2018**, 273, 335–348. [Google Scholar] [CrossRef] - Katzer, F.; Rüther, T.; Plank, C.; Roth, F.; Danzer, M.A. Analyses of polarisation effects and operando detection of lithium deposition in experimental half- and commercial full-cells. Electrochim. Acta
**2022**, 436, 141401. [Google Scholar] [CrossRef] - Danzer, M.A. Generalized Distribution of Relaxation Times Analysis for the Characterization of Impedance Spectra. Batteries
**2019**, 5, 53. [Google Scholar] [CrossRef] - Yoo, H.D.; Jang, J.H.; Ryu, J.H.; Park, Y.; Oh, S.M. Impedance analysis of porous carbon electrodes to predict rate capability of electric double-layer capacitors. J. Power Sources
**2014**, 267, 411–420. [Google Scholar] [CrossRef] - Schmidt, J.P.; Arnold, S.; Loges, A.; Werner, D.; Wetzel, T.; Ivers-Tiffée, E. Measurement of the internal cell temperature via impedance: Evaluation and application of a new method. J. Power Sources
**2013**, 243, 110–117. [Google Scholar] [CrossRef] - McGrogan, F.P.; Bishop, S.R.; Chiang, Y.M.; van Vliet, K.J. Connecting Particle Fracture with Electrochemical Impedance in LiXMn
_{2}O_{4}. J. Electrochem. Soc.**2017**, 164, A3709–A3717. [Google Scholar] [CrossRef] - Meddings, N.; Heinrich, M.; Overney, F.; Lee, J.S.; Ruiz, V.; Napolitano, E.; Seitz, S.; Hinds, G.; Raccichini, R.; Gaberšček, M.; et al. Application of electrochemical impedance spectroscopy to commercial Li-ion cells: A review. J. Power Sources
**2020**, 480, 228742. [Google Scholar] [CrossRef] - Rüther, T.; Plank, C.; Schamel, M.; Danzer, M.A. Detection of inhomogeneities in serially connected lithium-ion batteries. Appl. Energy
**2023**, 332, 120514. [Google Scholar] [CrossRef] - Carthy, K.M.; Gullapalli, H.; Ryan, K.M.; Kennedy, T. Review—Use of Impedance Spectroscopy for the Estimation of Li-ion Battery State of Charge, State of Health and Internal Temperature. J. Electrochem. Soc.
**2021**, 168, 080517. [Google Scholar] [CrossRef] - Rivera-Barrera, J.; Muñoz-Galeano, N.; Sarmiento-Maldonado, H. SoC Estimation for Lithium-ion Batteries: Review and Future Challenges. Electronics
**2017**, 6, 102. [Google Scholar] [CrossRef] - Galeotti, M.; Cinà, L.; Giammanco, C.; Cordiner, S.; Carlo, A.D. Performance analysis and SOH (state of health) evaluation of lithium polymer batteries through electrochemical impedance spectroscopy. Energy
**2015**, 89, 678–686. [Google Scholar] [CrossRef] - Pan, Y.; Ren, D.; Han, X.; Lu, L.; Ouyang, M. Lithium Plating Detection Based on Electrochemical Impedance and Internal Resistance Analyses. Batteries
**2022**, 8, 206. [Google Scholar] [CrossRef] - Schmidt, J.P.; Adam, A.; Wandt, J. Time-Resolved and Robust Lithium Plating Detection for Automotive Lithium-Ion Cells with the Potential for Vehicle Application. Batteries
**2023**, 9, 97. [Google Scholar] [CrossRef] - Koseoglou, M.; Tsioumas, E.; Ferentinou, D.; Jabbour, N.; Papagiannis, D.; Mademlis, C. Lithium plating detection using dynamic electrochemical impedance spectroscopy in lithium-ion batteries. J. Power Sources
**2021**, 512, 230508. [Google Scholar] [CrossRef] - Gaddam, R.R.; Katzenmeier, L.; Lamprecht, X.; Bandarenka, A.S. Review on physical impedance models in modern battery research. Phys. Chem. Chem. Phys.
**2021**, 23, 12926–12944. [Google Scholar] [CrossRef] - Hahn, M.; Schindler, S.; Triebs, L.C.; Danzer, M.A. Optimized Process Parameters for a Reproducible Distribution of Relaxation Times Analysis of Electrochemical Systems. Batteries
**2019**, 5, 43. [Google Scholar] [CrossRef] - Zhao, Y.; Kücher, S.; Jossen, A. Investigation of the diffusion phenomena in lithium-ion batteries with distribution of relaxation times. Electrochim. Acta
**2022**, 432, 141174. [Google Scholar] [CrossRef] - Zhang, Q.; Wang, D.; Schaltz, E.; Stroe, D.I.; Gismero, A.; Yang, B. Degradation mechanism analysis and State-of-Health estimation for lithium-ion batteries based on distribution of relaxation times. J. Energy Storage
**2022**, 55, 105386. [Google Scholar] [CrossRef] - Iurilli, P.; Brivio, C.; Wood, V. Detection of Lithium-Ion Cells’ Degradation through Deconvolution of Electrochemical Impedance Spectroscopy with Distribution of Relaxation Time. Energy Technol.
**2022**, 10, 2200547. [Google Scholar] [CrossRef] - He, R.; He, Y.; Xie, W.; Guo, B.; Yang, S. Comparative analysis for commercial li-ion batteries degradation using the distribution of relaxation time method based on electrochemical impedance spectroscopy. Energy
**2023**, 263, 125972. [Google Scholar] [CrossRef] - Illig, J.; Ender, M.; Weber, A.; Ivers-Tiffée, E. Modeling graphite anodes with serial and transmission line models. J. Power Sources
**2015**, 282, 335–347. [Google Scholar] [CrossRef] - Chen, X.; Li, L.; Liu, M.; Huang, T.; Yu, A. Detection of lithium plating in lithium-ion batteries by distribution of relaxation times. J. Power Sources
**2021**, 496, 229867. [Google Scholar] [CrossRef] - Brown, D.E.; McShane, E.J.; Konz, Z.M.; Knudsen, K.B.; McCloskey, B.D. Detecting onset of lithium plating during fast charging of Li-ion batteries using operando electrochemical impedance spectroscopy. Cell Rep. Phys. Sci.
**2021**, 2, 100589. [Google Scholar] [CrossRef] - Bergmann, T.G.; Schlüter, N. Introducing Alternative Algorithms for the Determination of the Distribution of Relaxation Times. ChemPhysChem
**2022**, 23, e202200012. [Google Scholar] [CrossRef] - Mayo, A.J.; Antoulas, A.C. A framework for the solution of the generalized realization problem. Linear Algebra Its Appl.
**2007**, 425, 634–662. [Google Scholar] [CrossRef] - Patel, B. Application of Loewner Framework for Data-Driven Modeling and Diagnosis of Polymer Electrolyte Membrane Fuel Cells. Master’s Thesis, Otto von Guericke-University, Magdeburg, Germany, 2021. [Google Scholar]
- Sorrentino, A.; Gosea, I.V.; Patel, B.; Antoulas, A.C.; Vidakovic-Koch, T. Loewner Framework and Distribution of Relaxation Times of Electrochemical Systems: Solving Issues Through a Data-Driven Modeling Approach. SSRN Electron. J.
**2022**. [Google Scholar] [CrossRef] - Gosea, I.V.; Zivkovic, L.; Karachalios, D.S.; Antoulas, A.C.; Vidakovic-Koch, T. A data-based nonlinear frequency response approach based on the Loewner framework: Preliminary analysis. In Proceedings of the 12th IFAC Symposium on Nonlinear Control Systems, Canberra, Australia, 4–6 January 2023; Elsevier: Amsterdam, The Netherlands, 2023. [Google Scholar]
- Sorrentino, A.; Gosea, I.V.; Patel, B.; Antoulas, A.C.; Vidakovic-Koch, T. The Loewner Framework for Data-Driven Identification of Electrochemical Systems; MPI: Magdeburg, Germany, 2022. [Google Scholar]
- Antoulas, A.C.; Lefteriu, S.; Ionita, A.C. A tutorial introduction to the Loewner framework for model reduction; Computational Science & Engineering, Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2017. [Google Scholar]
- Gosea, I.V.; Poussot-Vassal, C.; Antoulas, A.C. Chapter 15: Data-driven modeling and control of large-scale dynamical systems in the Loewner framework: Methodology and applications. In Handbook of Numerical Analysis, Numerical Control: Part A; Trélat, E., Zuazua, E., Eds.; Elsevier: Amsterdam, The Netherlands, 2022; Volume 23, pp. 499–530. [Google Scholar] [CrossRef]
- Antoulas, A.C.; Beattie, C.A.; Gugercin, S. Interpolatory Methods for Model Reduction; Computational Science & Engineering, Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2020. [Google Scholar]
- Embree, M.; Ionita, A.C. Pseudospectra of Loewner matrix pencils. In Realization and Model Reduction of Dynamical Systems; Springer: Berlin/Heidelberg, Germany, 2022; pp. 59–78. [Google Scholar] [CrossRef]
- Drmač, Z.; Peherstorfer, B. Learning low-dimensional dynamical-system models from noisy frequency-response data with Loewner rational interpolation. In Realization and Model Reduction of Dynamical Systems; Springer: Berlin/Heidelberg, Germany, 2022; pp. 39–57. [Google Scholar]
- Gosea, I.V.; Zhang, Q.; Antoulas, A.C. Preserving the DAE structure in the Loewner model reduction and identification framework. Adv. Comput. Math.
**2020**, 46, 1–32. [Google Scholar] [CrossRef] - Wang, X.; Wei, X.; Zhu, J.; Dai, H.; Zheng, Y.; Xu, X.; Chen, Q. A review of modeling, acquisition, and application of lithium-ion battery impedance for onboard battery management. eTransportation
**2021**, 7, 100093. [Google Scholar] [CrossRef] - Deng, Z.; Zhang, Z.; Lai, Y.; Liu, J.; Li, J.; Liu, Y. Electrochemical Impedance Spectroscopy Study of a Lithium/Sulfur Battery: Modeling and Analysis of Capacity Fading. J. Electrochem. Soc.
**2013**, 160, A553–A558. [Google Scholar] [CrossRef] - Golub, G.H.; Van Loan, C.F. Matrix Computations, 3rd ed.; Johns Hopkins University Press: Baltimore, MD, USA, 1996. [Google Scholar]
- Plank, C.; Ruther, T.; Danzer, M.A. Detection of Non-Linearity and Non-Stationarity in Impedance Spectra using an Extended Kramers-Kronig Test without Overfitting. In Proceedings of the 2022 International Workshop on Impedance Spectroscopy (IWIS), Chemnitz, Germany, 27–30 September 2022. [Google Scholar] [CrossRef]
- Shafiei Sabet, P.; Sauer, D.U. Separation of predominant processes in electrochemical impedance spectra of lithium-ion batteries with nickel-manganese-cobalt cathodes. J. Power Sources
**2019**, 425, 121–129. [Google Scholar] [CrossRef] - Manikandan, B.; Ramar, V.; Yap, C.; Balaya, P. Investigation of physico-chemical processes in lithium-ion batteries by deconvolution of electrochemical impedance spectra. J. Power Sources
**2017**, 361, 300–309. [Google Scholar] [CrossRef] - Schlüter, N.; Ernst, S.; Schröder, U. Finding the Optimal Regularization Parameter in Distribution of Relaxation Times Analysis. ChemElectroChem
**2019**, 6, 6027–6037. [Google Scholar] [CrossRef] - Schlüter, N.; Ernst, S.; Schröder, U. Direct Access to the Optimal Regularization Parameter in Distribution of Relaxation Times Analysis. ChemElectroChem
**2020**, 7, 3445–3458. [Google Scholar] [CrossRef] - Paul, T.; Chi, P.W.; Wu, P.M.; Wu, M.K. Computation of distribution of relaxation times by Tikhonov regularization for Li ion batteries: Usage of L-curve method. Sci. Rep.
**2021**, 11, 12624. [Google Scholar] [CrossRef] - Hu, D.; Chen, L.; Tian, J.; Su, Y.; Li, N.; Chen, G.; Hu, Y.; Dou, Y.; Chen, S.; Wu, F. Research Progress of Lithium Plating on Graphite Anode in Lithium–Ion Batteries. Chin. J. Chem.
**2020**, 39, 165–173. [Google Scholar] [CrossRef] - Wang, C.; Appleby, A.J.; Little, F.E. Low-Temperature Characterization of Lithium-Ion Carbon Anodes via Microperturbation Measurement. J. Electrochem. Soc.
**2002**, 149, A754–A760. [Google Scholar] [CrossRef] - Zhang, S.; Xu, K.; Jow, T. Low temperature performance of graphite electrode in Li-ion cells. Electrochim. Acta
**2002**, 48, 241–246. [Google Scholar] [CrossRef] - Jow, T.R.; Delp, S.A.; Allen, J.L.; Jones, J.P.; Smart, M.C. Factors Limiting Li+ Charge Transfer Kinetics in Li-Ion Batteries. J. Electrochem. Soc.
**2018**, 165, A361–A367. [Google Scholar] [CrossRef]

**Figure 2.**ECM for two RC elements (

**a**), Nyquist plot of the impedance ${Z}_{\mathrm{RC}}$ (

**b**), SVD of the Loewner matrices (

**c**), DG (

**d**).

**Figure 3.**ECM for two RC elements and CPE (

**a**), Nyquist plot of the impedance ${Z}_{\mathrm{RC},\mathrm{CPE}}$ and LM with $k=22$ (

**b**), SVD of the Loewner matrices (

**c**), error of magnitude for different model orders (

**d**), DG for $k=8$ (

**e**) and $k=22$ (

**f**).

**Figure 4.**ECM for battery model (

**a**), Nyquist plot of the impedance ${Z}_{\mathrm{Battery}}$ (

**b**), SVD of the Loewner matrices (

**c**), DG (

**d**), and error of magnitude (

**e**).

**Figure 5.**Nyquist plot (

**a**), SVD (

**b**), error of magnitude (

**c**), distribution of gains (

**d**) for different level of noise. Distribution of gains for $SNR=45$ for different model orders (

**e**), zoom for the different model orders (

**f**).

**Figure 6.**Nyquist plot (

**a**), SVD (

**b**), distribution of gains (

**d**), and analyzed processes of the distribution of gains (

**c**) of the measurement data.

**Figure 7.**SVD (

**a**), Nyquist plot (

**b**), error of magnitude (

**c**), DG (

**d**), zoom of the distribution of gains (

**e**), distribution of gains (

**f**) for the BOL and EOL LIB.

Lowener Method | Generalized DRT |
---|---|

− Meta parameter needed (k) | − Meta parameter needed ($\lambda ,{n}_{\tau}$) |

+ Simple process identification | − Difficult process identification |

− Interpretation of resistive, capacitive, and resistive–inductive processes is challenging | + Interpretation of resistive, inductive, and resistive–inductive behavior possible |

+ Smaller polarization contributions interpretable | − Partial merging of peaks due to regularization |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rüther, T.; Gosea, I.V.; Jahn, L.; Antoulas, A.C.; Danzer, M.A.
Introducing the Loewner Method as a Data-Driven and Regularization-Free Approach for the Distribution of Relaxation Times Analysis of Lithium-Ion Batteries. *Batteries* **2023**, *9*, 132.
https://doi.org/10.3390/batteries9020132

**AMA Style**

Rüther T, Gosea IV, Jahn L, Antoulas AC, Danzer MA.
Introducing the Loewner Method as a Data-Driven and Regularization-Free Approach for the Distribution of Relaxation Times Analysis of Lithium-Ion Batteries. *Batteries*. 2023; 9(2):132.
https://doi.org/10.3390/batteries9020132

**Chicago/Turabian Style**

Rüther, Tom, Ion Victor Gosea, Leonard Jahn, Athanasios C. Antoulas, and Michael A. Danzer.
2023. "Introducing the Loewner Method as a Data-Driven and Regularization-Free Approach for the Distribution of Relaxation Times Analysis of Lithium-Ion Batteries" *Batteries* 9, no. 2: 132.
https://doi.org/10.3390/batteries9020132