# Optimized Process Parameters for a Reproducible Distribution of Relaxation Times Analysis of Electrochemical Systems

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## Abstract

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## 1. Introduction

- the optimization algorithm,
- the error function,
- the type of data used, i.e., using real or imaginary parts of the complex impedance,
- measurement parameters, e.g., current or frequency range,
- regularization parameter,
- number of time constants,
- the pre- and post-processing routine,
- the minimum and maximum time constants for the DRT.

## 2. Deriving DRT from EIS Data

- The total polarization resistance of the system gets apparent by ${R}_{\mathrm{pol}}$.
- Systems with impedances in different orders of magnitude can be compared easily.

#### 2.1. Dealing with Artefacts at the Boundaries of the Measured Frequency Range

#### 2.2. Calculation of g($\tau $)

- Using complex values:$$\begin{array}{c}\hfill \mathbf{A}=\left[\begin{array}{cccc}{A}_{1,1}& {A}_{1,2}& \cdots & {A}_{1,n}\\ {A}_{2,1}& {A}_{2,2}& \cdots & {A}_{2,n}\\ \vdots & \vdots & \ddots & \vdots \\ {A}_{m,1}& {A}_{m,2}& \cdots & {A}_{m,n}\end{array}\right],\phantom{\rule{1.em}{0ex}}\mathbf{b}=\left[\begin{array}{c}{Z}_{\mathrm{meas},1}\\ {Z}_{\mathrm{meas},2}\\ \vdots \\ {Z}_{\mathrm{meas},m}\end{array}\right].\end{array}$$In this case, a solver is needed which is able to handle complex values as well as constraints.
- Using the real parts only:$$\begin{array}{c}\hfill \mathbf{A}=\left[\begin{array}{ccc}Re\left({A}_{1,1}\right)& \cdots & Re\left({A}_{1,n}\right)\\ \vdots & \ddots & \vdots \\ Re\left({A}_{m,1}\right)& \cdots & Re\left({A}_{m,n}\right)\end{array}\right],\phantom{\rule{1.em}{0ex}}\mathbf{b}=\left[\begin{array}{c}Re\left({Z}_{\mathrm{meas},1}\right)\\ \vdots \\ Re\left({Z}_{\mathrm{meas},m}\right)\end{array}\right].\end{array}$$Ignoring imaginary parts is legitimate due to the Kramers–Kronig relationship [27,28], as long as its conditions are fulfilled. It can be assumed that a properly measured spectrum complies with these. The linear Kramers–Kronig test is one option to prove the Kramers–Kronig validity of the examined impedance data.
- Using the imaginary parts only: equivalent to Equation (13) substituting real parts by imaginary parts.
- Using both real and imaginary parts:$$\begin{array}{c}\hfill \mathbf{A}=\left[\begin{array}{ccc}Re\left({A}_{1,1}\right)& \cdots & Re\left({A}_{1,n}\right)\\ \vdots & \ddots & \vdots \\ Re\left({A}_{m,1}\right)& \cdots & Re\left({A}_{m,n}\right)\\ Im\left({A}_{1,1}\right)& \cdots & Im\left({A}_{1,n}\right)\\ \vdots & \ddots & \vdots \\ Im\left({A}_{m,1}\right)& \cdots & Im\left({A}_{m,n}\right)\end{array}\right],\phantom{\rule{1.em}{0ex}}\mathbf{b}=\left[\begin{array}{c}Re\left({Z}_{\mathrm{meas},1}\right)\\ \vdots \\ Re\left({Z}_{\mathrm{meas},m}\right)\\ Im\left({Z}_{\mathrm{meas},1}\right)\\ \vdots \\ Im\left({Z}_{\mathrm{meas},m}\right)\end{array}\right],\end{array}$$

#### 2.3. Pre-Processing of Measurement Data

#### 2.4. Post-Processing of Result

## 3. Analysis of DRT Process Parameters

#### 3.1. Regularization Parameter and Number of Time Constants

#### 3.2. Setup of A and b

#### 3.3. Impact of Optimization Function and Solving Algorithm

- Convergence towards different local minima. This can be resolved using proper initial values and analyzing the reconstructed impedance spectrum.
- Different optimization functions might lead to differing results, e.g., by using relative or absolute errors.
- Treatment of non-negativity constraint. Various algorithms handle constraints differently which can influence the result.

#### 3.4. Regularization Matrix

## 4. Analysis of Measurement Data

- The resistance increases for high and low values reaching a minimum at medium $SOC$s. Figure 9 underlines this observation. This is in accordance with the Butler–Volmer kinetics, where the exchange current density is small at high and low $SOC$s [14,41]. To differentiate the process from the cathodic reaction, Figure 9 shows that the resistance is highest at low SOCs, whereas the increase is only shallow at 80% and 100%. It has to be considered that in commercial cells, the anode is over-dimensioned so that lithium plating is avoided by restricting high lithium concentrations in the anode. This leads to an disproportionate distribution of the anode’s degree of lithiation. Thus, its SOC is not well-proportionate but shifted towards smaller concentrations. This leads to the conclusion that peak 3 is caused by the anodic charge transfer reaction.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Ivers-Tiffée, E.; Weber, A. Evaluation of electrochemical impedance spectra by the distribution of relaxation times. J. Ceram. Soc. Jpn.
**2017**, 125, 193–201. [Google Scholar] [CrossRef] [Green Version] - Illig, J.; Schmidt, J.P.; Weiss, M.; Weber, A.; Ivers-Tiffée, E. Understanding the impedance spectrum of 18650 LiFePO4-cells. J. Power Sources
**2013**, 239, 670–679. [Google Scholar] [CrossRef] - Barsoukov, E. Kinetics of lithium intercalation into carbon anodes: In situ impedance investigation of thickness and potential dependence. Solid State Ionics
**1999**, 116, 249–261. [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] - Costard, J.; Ender, M.; Weiss, M.; Ivers-Tiffée, E. Three-Electrode Setups for Lithium-Ion Batteries: II. Experimental Study of Different Reference Electrode Designs and Their Implications for Half-Cell Impedance Spectra. J. Electrochem. Soc.
**2016**, 164, A80–A87. [Google Scholar] [CrossRef] - Graves, C.; Ebbesen, S.D.; Mogensen, M. Co-electrolysis of CO
_{2}and H_{2}O in solid oxide cells: Performance and durability. Solid State Ionics**2011**, 192, 398–403. [Google Scholar] [CrossRef] - Weiß, A.; Schindler, S.; Galbiati, S.; Danzer, M.A.; Zeis, R. Distribution of Relaxation Times Analysis of High-Temperature PEM Fuel Cell Impedance Spectra. Electrochim. Acta
**2017**, 230, 391–398. [Google Scholar] [CrossRef] - Hanft, D.; Exner, J.; Moos, R. Thick-films of garnet-type lithium ion conductor prepared by the Aerosol Deposition Method: The role of morphology and annealing treatment on the ionic conductivity. J. Power Sources
**2017**, 361, 61–69. [Google Scholar] [CrossRef] - Exner, J.; Fuierer, P.; Moos, R. Aerosol deposition of (Cu,Ti) substituted bismuth vanadate films. Thin Solid Film.
**2014**, 573, 185–190. [Google Scholar] [CrossRef] - Thomas, B.J.; Ward, L.C.; Cornish, B.H. Bioimpedance spectrometry in the determination of body water compartments: Accuracy and clinical significance. Appl. Radiat. Isot.
**1998**, 49, 447–455. [Google Scholar] [CrossRef] - Schindler, S.; Danzer, M.A. Influence of cell design on impedance characteristics of cylindrical lithium-ion cells: A model-based assessment from electrode to cell level. J. Energy Storage
**2017**, 12, 157–166. [Google Scholar] [CrossRef] - Schönleber, M.; Uhlmann, C.; Braun, P.; Weber, A.; Ivers-Tiffée, E. A Consistent Derivation of the Impedance of a Lithium-Ion Battery Electrode and its Dependency on the State-of-Charge. Electrochim. Acta
**2017**, 243, 250–259. [Google Scholar] [CrossRef] - Schindler, S.; Bauer, M.; Petzl, M.; Danzer, M.A. Voltage relaxation and impedance spectroscopy as in-operando methods for the detection of lithium plating on graphitic anodes in commercial lithium-ion cells. J. Power Sources
**2016**, 304, 170–180. [Google Scholar] [CrossRef] - Kindermann, F.M.; Noel, A.; Erhard, S.V.; Jossen, A. Long-term equalization effects in Li-ion batteries due to local state of charge inhomogeneities and their impact on impedance measurements. Electrochim. Acta
**2015**, 185, 107–116. [Google Scholar] [CrossRef] - Schichlein, H.; Müller, A.C.; Voigts, M.; Krügel, A.; Ivers-Tiffée, E. Deconvolution of electrochemical impedance spectra for the identification of electrode reaction mechanisms in solid oxide fuel cells. J. Appl. Electrochem.
**2002**, 32, 875–882. [Google Scholar] [CrossRef] - Sumi, H.; Yamaguchi, T.; Hamamoto, K.; Suzuki, T.; Fujishiro, Y.; Matsui, T.; Eguchi, K. AC impedance characteristics for anode-supported microtubular solid oxide fuel cells. Electrochim. Acta
**2012**, 67, 159–165. [Google Scholar] [CrossRef] - Schmidt, J.P.; Berg, P.; Schönleber, M.; Weber, A.; Ivers-Tiffée, E. The distribution of relaxation times as basis for generalized time-domain models for Li-ion batteries. J. Power Sources
**2013**, 221, 70–77. [Google Scholar] [CrossRef] - Boukamp, B.A. A Linear Kronig-Kramers Transform Test for Immittance Data Validation. J. Electrochem. Soc.
**1995**, 142, 1885. [Google Scholar] [CrossRef] [Green Version] - Schönleber, M.; Klotz, D.; Ivers-Tiffée, E. A Method for Improving the Robustness of linear Kramers-Kronig Validity Tests. Electrochim. Acta
**2014**, 131, 20–27. [Google Scholar] [CrossRef] - Sonn, V.; Leonide, A.; Ivers-Tiffée, E. Combined Deconvolution and CNLS Fitting Approach Applied on the Impedance Response of Technical Ni/8YSZ Cermet Electrodes. J. Electrochem. Soc.
**2008**, 155, B675. [Google Scholar] [CrossRef] - Tröltzsch, U.; Kanoun, O.; Tränkler, H.R. Characterizing aging effects of lithium ion batteries by impedance spectroscopy. Electrochim. Acta
**2006**, 51, 1664–1672. [Google Scholar] [CrossRef] - Nara, H.; Mukoyama, D.; Yokoshima, T.; Momma, T.; Osaka, T. Impedance Analysis with Transmission Line Model for Reaction Distribution in a Pouch Type Lithium-Ion Battery by Using Micro Reference Electrode. J. Electrochem. Soc.
**2015**, 163, A434–A441. [Google Scholar] [CrossRef] [Green Version] - Tikhonov, A.N. Numerical Methods for the Solution of Ill-Posed Problems; Springer: Dordrecht, The Netherlands, 2010. [Google Scholar]
- Weese, J. A reliable and fast method for the solution of Fredholm integral equations of the first kind based on Tikhonov regularization. Comput. Phys. Commun.
**1992**, 69, 99–111. [Google Scholar] [CrossRef] - Ciucci, F.; Chen, C. Analysis of Electrochemical Impedance Spectroscopy Data Using the Distribution of Relaxation Times: A Bayesian and Hierarchical Bayesian Approach. Electrochim. Acta
**2015**, 167, 439–454. [Google Scholar] [CrossRef] - Wan, T.H.; Saccoccio, M.; Chen, C.; Ciucci, F. Influence of the Discretization Methods on the Distribution of Relaxation Times Deconvolution: Implementing Radial Basis Functions with DRTtools. Electrochim. Acta
**2015**, 184, 483–499. [Google Scholar] [CrossRef] - de L. Kronig, R. On the Theory of Dispersion of X-rays. J. Opt. Soc. Am.
**1926**, 12, 547. [Google Scholar] [CrossRef] - Kramers, M.H.A. La diffusion de la lumière par les atomes. Trans. Volta Centenary Congr.
**1927**, 2, 545–557. [Google Scholar] - Lawson, C.L.; Hanson, R.J. Solving least squares problems. In Classics in Applied Mathematics; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1995; Volume 15. [Google Scholar]
- Kaipio, J.; Somersalo, E. Statistical and computational inverse problems. In Applied Mathematical Sciences; Springer: New York, NY, USA, 2010; Volume 160. [Google Scholar]
- Wu, L. A Parameter Choice Method for Tikhonov Regularization. Electron. Trans. Numer. Anal.
**2003**, 16, 107–128. [Google Scholar] - Gavrilyuk, A.L.; Osinkin, D.A.; Bronin, D.I. The use of Tikhonov regularization method for calculating the distribution function of relaxation times in impedance spectroscopy. Russ. J. Electrochem.
**2017**, 53, 575–588. [Google Scholar] [CrossRef] - Davies, A.R.; Anderssen, R.S. Optimisation in the regularisation ill-posed problems. J. Aust. Math. Soc. Ser. B Appl. Math.
**1986**, 28, 114. [Google Scholar] [CrossRef] [Green Version] - Saccoccio, M.; Wan, T.H.; Chen, C.; Ciucci, F. Optimal Regularization in Distribution of Relaxation Times applied to Electrochemical Impedance Spectroscopy: Ridge and Lasso Regression Methods—A Theoretical and Experimental Study. Electrochim. Acta
**2014**, 147, 470–482. [Google Scholar] [CrossRef] - Hansen, P.C. Analysis of Discrete Ill-Posed Problems by Means of the L-Curve. SIAM Rev.
**1992**, 34, 561–580. [Google Scholar] [CrossRef] - Hansen, P.C.; O’Leary, D.P. The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems. SIAM J. Sci. Comput.
**1993**, 14, 1487–1503. [Google Scholar] [CrossRef] [Green Version] - Byrd, R.H.; Gilbert, J.C.; Nocedal, J. A trust region method based on interior point techniques for nonlinear programming. Math. Program.
**2000**, 89, 149–185. [Google Scholar] [CrossRef] [Green Version] - Boukamp, B.A. Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg. Electrochim. Acta
**2017**, 252, 154–163. [Google Scholar] [CrossRef] - Levi, M.D.; Aurbach, D. Simultaneous Measurements and Modeling of the Electrochemical Impedance and the Cyclic Voltammetric Characteristics of Graphite Electrodes Doped with Lithium. J. Phys. Chem. B
**1997**, 101, 4630–4640. [Google Scholar] [CrossRef] - Piao, T. Intercalation of Lithium Ions into Graphite Electrodes Studied by AC Impedance Measurements. J. Electrochem. Soc.
**1999**, 146, 2794. [Google Scholar] [CrossRef] [Green Version] - Fuller, T.F. Simulation and Optimization of the Dual Lithium Ion Insertion Cell. J. Electrochem. Soc.
**1994**, 141, 1. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Impedance spectrum of two RC elements connected in series (

**a**) appearing as a single, asymmetric semi-circle in the Nyquist plot; the distribution of relaxation times (DRT) (

**b**) reveals both processes involved.

**Figure 3.**DRT of an RC-element (${R}_{1}=5\mathrm{m}\Omega $, ${C}_{1}=0,1\mathrm{F}$) and a ZARC-element (${R}_{2}=7\mathrm{m}\Omega $, ${C}_{2}=0,71\mathrm{F}$, $\phi =0.8$) in series, calculated at five different $\lambda $ logarithmically equally-distributed between 0.001 and 10.

**Figure 4.**The sum of squared errors ($sse$) of the reconstructed spectra for differing n and a variation of $\lambda $ between 0.01. and 2 (

**a**) and a magnified detail for small $\lambda $ (

**b**), ${g}_{k}$ for $\lambda =0.1$ and varied n (

**c**) and $c\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{g}_{k}$ (

**d**), where c is the multiple of m for every line respectively.

**Figure 5.**Condition number of the regularized matrix $\mathit{A}$ for 100 to 500 time constants between 10 $\mathrm{m}$$\mathrm{Hz}$ and 10 $\mathrm{k}$$\mathrm{Hz}$ and $\lambda $ from Figure 3. Condition number for non-regularized matrix between 1.1 × 10${}^{19}$ and 8.2 × 10${}^{19}$.

**Figure 6.**(

**a**) DRT of an RC- (${R}_{1}=5\mathrm{m}\Omega $, ${C}_{1}=0,1\mathrm{F}$) and a ZARC-element (${R}_{2}=7\mathrm{m}\Omega $, ${C}_{2}=0,71\mathrm{F}$, $\phi =0.8$) in series, $\lambda =0.1$, $n=3\phantom{\rule{0.166667em}{0ex}}m$. Algorithm 1 as pointed out within Section 2, Algorithm 2 using an interior-point method, (

**b**) using identity matrix, first and second derivative for regularization. (

**c**) Showing the regularization term representing the matrix according to (b).

**Figure 7.**(

**a**) Measured impedance spectra of commercial High Energy-NMC cell at varied $SOC$s (frequencies correspond to 100% $SOC$, (

**b**) non-normalized distribution of time constants, (

**c**) measured and reconstructed impedance spectrum for 20% $SOC$, (

**d**) absolute residual between measured and reconstructed impedance for 20% $SOC$.

**Figure 8.**Non-normalized DRT with peak analysis for (

**a**) 100% $SOC$, (

**b**) 80% $SOC$, (

**c**) 60% $SOC$, (

**d**) 40% $SOC$, (

**e**) 20% $SOC$.

**Figure 9.**Polarization resistance of (presumably anodic) charge transfer (peak 4) reaction and the value obtained by the DRT for the diffusion of both electrodes (which must not be interpreted as an absolute resistance value).

**Table 1.**$\lambda $ obtained from three different optimum criteria for the RC-ZARC-element as shown in Figure 3.

Criterion | $\mathit{\lambda}$ |
---|---|

Discrepancy | 0.13 |

Cross Validation | <1 × 10${}^{-5}$ |

L-Curve | 6 × 10${}^{-4}$ |

**Table 2.**The sum of squared errors ($sse$) of the reconstructed impedance spectrum using DRT varying $\mathit{A}$ and $\mathit{b}$. RC-ZARC elements. For the ZARC, $\phi =0.8$ is kept constant. Also, $\lambda =0.1$, $n=3\phantom{\rule{0.166667em}{0ex}}m$.

Parameters | ${\mathit{sse}}_{\mathbf{real}}$ | ${\mathit{sse}}_{\mathbf{imag}}$ | ${\mathit{sse}}_{\mathbf{real}\&\mathbf{imag}}$ |
---|---|---|---|

${R}_{1}=5\mathrm{m}\Omega $, ${C}_{1}=0,1\mathrm{F}$ | 4.57 × 10${}^{-9}$ | 3.42 × 10${}^{-9}$ | 2. 86 × 10${}^{-9}$ |

${R}_{2}=7\mathrm{m}\Omega $, ${C}_{2}=0,71\mathrm{F}$ | |||

${R}_{1}=5\mathsf{\mu}\Omega $, ${C}_{1}=0,1\mathrm{F}$ | 4.56 × 10${}^{-15}$ | 3. 42 × 10${}^{-15}$ | 2. 86 × 10${}^{-15}$ |

${R}_{2}=7\mathsf{\mu}\Omega $, ${C}_{2}=0,71\mathrm{F}$ | |||

${R}_{1}=5\Omega $, ${C}_{1}=0,1\mathrm{F}$ | 4. 57 × 10${}^{-3}$ | 3.42 × 10${}^{-3}$ | 2.86 × 10${}^{-3}$ |

${R}_{2}=7\Omega $, ${C}_{2}=0,71\mathrm{F}$ | |||

${R}_{1}=5\mathrm{m}\Omega $, ${C}_{1}=10\mathrm{F}$ | 4.52 × 10${}^{-3}$ | 3.48 × 10${}^{-3}$ | 2.83 × 10${}^{-3}$ |

${R}_{2}=7\mathrm{m}\Omega $, ${C}_{2}=0,71\mathrm{F}$ |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/batteries5020043

**AMA Style**

Hahn M, Schindler S, Triebs L-C, Danzer MA.
Optimized Process Parameters for a Reproducible Distribution of Relaxation Times Analysis of Electrochemical Systems. *Batteries*. 2019; 5(2):43.
https://doi.org/10.3390/batteries5020043

**Chicago/Turabian Style**

Hahn, Markus, Stefan Schindler, Lisa-Charlotte Triebs, and Michael A. Danzer.
2019. "Optimized Process Parameters for a Reproducible Distribution of Relaxation Times Analysis of Electrochemical Systems" *Batteries* 5, no. 2: 43.
https://doi.org/10.3390/batteries5020043