# Combining the Distribution of Relaxation Times from EIS and Time-Domain Data for Parameterizing Equivalent Circuit Models of Lithium-Ion Batteries

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. State of the Art

#### 1.3. Contributions

- Evaluation of frequency- and time-domain measurements:To capture the full dynamic behavior of a cell, both EIS and TDM measurements have been considered in the ECM parameterization process and their DRT computed. Respective measurements in the frequency-domain as well as in the time-domain were performed (Section 2).
- Optimized calculation of the DRT via Tikhonov regularization:Since an accurate DRT is essential for the proposed parameterization process, the L-curve criterion was employed to optimize the calculation of the DRT via Tikhonov regularization. Moreover, the impact of different regularization terms on the calculated DRT was investigated (Section 3).
- Process for direct parameterization of RC elements from the DRT:Instead of using only partial information from the DRT to supplement a conventional fitting algorithm, the full potential of the DRT was exploited by a parameterization process which determines RC parameters directly from the DRT (Section 4).
- Analysis of various data merging approaches:Different methods of combining EIS and TDM data in the parameterization process are presented in this work. One of those methods includes a new way of calculating the DRT using TDM and EIS simultaneously, uniting the steps of DRT calculation and merging frequency- and time-domain data (Section 5).

## 2. Experimental

#### 2.1. EIS

#### 2.2. Time-Domain Measurements (TDMs)

#### 2.3. Open-Circuit Voltage

## 3. Calculation of the DRT

#### 3.1. DRT Calculated from EIS (EIS-DRT)

#### 3.2. DRT Calculated from Time-Domain Measurements (TDM-DRT)

#### 3.3. Tikhonov Regularization

- The standard form of the Tikhonov regularization is the regularization with a squared norm of the solution itself. Hence, it penalizes high magnitudes of the solution and therefore favors solutions with smaller peaks. $\mathit{L}$ has to be set to be the identity matrix $\mathit{I}$ such that $\mathit{Lx}=\mathit{x}$ [29].
- The regularization with the squared norm of the solution’s first derivative penalizes high gradients in the solution and therefore favors solutions with moderate slopes. For this, $\mathit{L}$ has to be chosen such that $\mathit{Lx}=\frac{dx}{dln\tau}$, as shown in Appendix C [27,29].
- The regularization with the squared norm of the solution’s second derivative penalizes high curvatures in the solution and therefore favors flat and smooth solutions. For this, $\mathit{L}$ has to be chosen such that $\mathit{Lx}=\frac{{d}^{2}x}{d{(ln\tau )}^{2}}$, as shown in Appendix C [20,21,25,29].

#### 3.4. L-Curve Method for Optimized Determination of the Regularization Factor $\lambda $

#### 3.5. Defining the Vector of Relaxation Times

## 4. ECM Parameterization

#### 4.1. Employed ECM

#### 4.2. DRT-Fitting: Direct RC Parameterization from the DRT

#### 4.3. Comparison to Conventional EIS-Fitting

## 5. Combination of EIS and TDM

#### 5.1. Merging Approaches

- 1.
**Separate DRT-fitting (blue):**In our first approach, EIS and TDM are used individually to parameterize different RC elements, the allocation of which is defined a priori. With this assumption, EIS and TDM are processed independent from each other. The ohmic resistance and the first RC elements (two in our case) are parameterized purely based on the EIS-DRT, whereas the remaining RC elements (two in our case) are determined from TDM-DRT only. Therefore, relaxation times higher than $\widehat{\tau}$, with $\widehat{\tau}$ being the highest relaxation time with a local minimum in the EIS-DRT, are ignored in the EIS-DRT. Likewise, the TDM-DRT is only evaluated at $\tau >\widehat{\tau}$. Depending on the sampling frequency, it is difficult to capture high-frequent loss processes during TDM, where meaningless oscillations can occur at low relaxation times close to $\widehat{\tau}$ of the TDM-DRT. This motivates an interconnected approach.- 2.
**Interconnected DRT-fitting (orange):**The goal of our second approach is to use information of EIS measurements in a more intertwined calculation of the TDM-DRT to avoid the described problems of meaningless oscillations at low relaxation times. Since the DRT equals a series of an infinite number of RC elements, the voltage relaxation which is caused by the high-frequency processes of the cell can be calculated from the EIS-DRT:$${u}_{\mathrm{EIS}}(t>({t}_{0}+{T}_{\mathrm{p}}))={\int}_{-\infty}^{\infty}{\gamma}_{\mathrm{EIS}}(ln\tau )\phantom{\rule{0.277778em}{0ex}}{I}_{\mathrm{p}}\phantom{\rule{0.277778em}{0ex}}[exp\left(\frac{{t}_{0}+{T}_{\mathrm{p}}-t}{\tau}\right)-exp(-\frac{{t}_{0}-t}{\tau})]\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}ln\tau .$$This virtual voltage response is subtracted from the measured voltage of TDM. Since the EIS-DRT is only used to subtract the high-frequency behavior of the cell, the diffusive branch is removed from the impedance spectrum before evaluating Equation (13). Afterwards, the remaining voltage response is used to calculate the TDM-DRT according to Equation (7). Adding EIS-DRT and TDM-DRT finally yields a DRT of the complete system, which is used for the parameterization of all three RC elements. A disadvantage of this rather complex approach is that errors within calculation of the EIS-DRT propagate into the next step and will influence the voltage signal and thus also the TDM-DRT.- 3.
**Combined DRT-fitting (green):**Based on the disadvantages of the first two approaches, we developed a combined DRT-fitting, where the calculation of EIS-DRT and TDM-DRT is performed simultaneously in one single step. This makes decision of a border time constant $\widehat{\tau}$ superfluous and avoids complex interconnected fitting. Defining one system of equations in order to find $\gamma (ln\tau )$, such that it best fits both measurements, directly results in a DRT that displays the complete dynamic behavior of the measured cell from high to low frequencies. This can be implemented by merging Equations (3) and (7) to$$\underset{{U}_{\mathrm{OCV}},{R}_{0},\mathit{R}}{min}\phantom{\rule{0.277778em}{0ex}}\u2225\sqrt{\mathit{w}}\left[\begin{array}{ccc}{\mathit{A}}_{\mathrm{Z}}^{\prime}& \mathbf{0}& \mathbf{1}\\ {\mathit{A}}_{\mathrm{Z}}^{\u2033}& \mathbf{0}& \mathbf{0}\\ {\mathit{A}}_{\mathrm{u}}& \mathbf{1}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathit{R}\\ {U}_{\mathrm{OCV}}\\ {R}_{0}\end{array}\right]\phantom{\rule{0.277778em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}{\sqrt{\mathit{w}}\left[\begin{array}{c}{\mathit{Z}}_{\mathrm{exp}}^{\prime}\\ {\mathit{Z}}_{\mathrm{exp}}^{\u2033}\\ {\mathit{u}}_{\mathrm{exp}}\end{array}\right]\u2225}_{2}^{2}$$$$\begin{array}{cccc}\hfill \underset{\mathit{x}}{min}\parallel \phantom{\rule{0.277778em}{0ex}}{\mathit{B}}_{\mathrm{comb}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathit{x}\hfill & \hfill \phantom{\rule{0.277778em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathit{D}}_{\mathrm{exp}}{\phantom{\rule{0.277778em}{0ex}}\parallel}_{2}^{2}\hfill \end{array}$$This equals a summation of the residuals of EIS-DRT and TDM-DRT. The diagonal Matrix $\mathit{w}$ applies a weighting between EIS and TDM data to achieve an equal influence of both measurements, the derivation of which can be found in Appendix D. Minimizing Equation (15) yields the DRT of the whole system as well as ${R}_{0}$ and ${U}_{OCV}$ and can thus be used to parameterize all ECM parameters at once.

#### 5.2. Validation Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BMS | battery management system |

DRT | distribution of relaxation times |

DST | dynamic stress test |

DVA | differential voltage analysis |

ECM | equivalent circuit model |

EIS | electrochemical impedance spectoscropy |

GITT | galvanostatic intermittent titration technique |

LIB | lithium-ion battery |

NCA | nickel cobalt aluminium oxide |

OCV | open-circuit voltage |

pOCV | pseudo open-circuit voltage |

RMSE | root mean square error |

SOAP | state of available power |

SOC | state of charge |

SOH | state of health |

TDM | time-domain measurement |

## Appendix A. EIS-DRT

## Appendix B. TDM-DRT

## Appendix C. Regularization Terms

## Appendix D. Weighting of EIS and TDM Data

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**Figure 1.**Results of time-domain measurements for all SOC steps between 100% SOC and 0% SOC. (

**a**) $-1$ C discharge pulse for 10 $\mathrm{s}$. The voltage is normalized on the starting voltage for every SOC. (

**b**) Relaxation period of ${T}_{\mathrm{relax}}=60\mathrm{h}$. The voltage is normalized on the last data point of the relaxation period for every SOC. Note that the voltage was not totally relaxed after 60 $\mathrm{h}$ for low SOC.

**Figure 2.**(

**a**) Measured pseudo-OCV curve during discharging and subsequent charging with C/50 at 20 °C and the corresponding mean curve. Additionally, the relaxed voltage ${U}_{OCV,TDM}$, which was measured right before the discharge pulse ${I}_{P}$ after three weeks of relaxation, are plotted. (

**b**) Difference between the charge and discharge pseudo-OCV and difference between mean pseudo-OCV and ${U}_{OCV,TDM}$ (

**c**) Differential voltage of the pseudo-OCV curves. (

**d**) Difference between the differential voltage of the discharge and the mean pseudo-OCV curve.

**Figure 3.**DRT calculated from EIS and TDM at 50% SOC for different regularization terms, respectively. The EIS-DRT reveals the four processes P1–P4 and the TDM-DRT reveals the four processes P2–P5.

**Figure 4.**Flow chart of the proposed DRT-fitting for automated parameterization of RC elements from the DRT.

**Figure 5.**

**Left:**Impedance spectra from EIS measurements and impedance spectra estimated by the proposed DRT-fitting and benchmark EIS-fitting for different SOC at 20 ${}^{\circ}\mathrm{C}$. Impedance spectra are shifted in the y-direction according to the SOC for the sake of better visibility.

**Right:**Resulting ECM parameters over SOC for DRT-fitting and EIS-fitting.

**Figure 6.**Flow chart of different approaches for combining EIS and TDM before the DRT-fitting is performed to parameterize the RC elements of an ECM. Blue: Separate DRT-fitting. Orange: Interconnected DRT-fitting. Green: Combined DRT-fitting.

**Figure 7.**Simulation results of the combined DRT-fitting during the validation cycle. (

**a**) Current profile. (

**b**) Measured and simulated voltage. (

**c**) Model error ${U}_{\mathrm{error}}$ = ${U}_{\mathrm{sim}}$−${U}_{\mathrm{exp}}$.

EIS-Data | TDM-Data | |||
---|---|---|---|---|

${\mathbf{\tau}}_{\mathbf{min}}$ | ${\mathbf{\tau}}_{\mathbf{max}}$ | ${\mathbf{\tau}}_{\mathbf{min}}$ | ${\mathbf{\tau}}_{\mathbf{max}}$ | |

$\frac{1}{2\pi {f}_{\mathrm{max}}}$ | $\frac{{10}^{4}}{2\pi {f}_{\mathrm{min}}}$ | $\frac{10}{\pi {f}_{\mathrm{sample}}}$ | ${T}_{\mathrm{relax}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{5}$ |

**Table 2.**Comparison of the RMSE in $\mathrm{mV}$ during the validation cycle of different RC parameter sets.

EIS-Fitting | DRT-Fitting | DRT-Fitting | |||
---|---|---|---|---|---|

Measurement | EIS | EIS | EIS and TDM | ||

Merging Approach | n.a. | n.a. | Separate | Interconnected | Combined |

Regularization Term | n.a. | x | x/$\frac{{\mathit{d}}^{2}\mathit{x}}{\mathit{d}ln(\mathit{\tau}{)}^{2}}$ | x/$\frac{{\mathit{d}}^{2}\mathit{x}}{\mathit{d}ln(\mathit{\tau}{)}^{2}}$ | $\frac{{\mathit{d}}^{2}\mathit{x}}{\mathit{d}ln(\mathit{\tau}{)}^{2}}$ |

Whole Validation Cycle | 72.56 | 73.14 | 45.18 | 41.52 | 43.38 |

A: 1 C Discharge | 42.88 | 46.76 | 28.78 | 26.41 | 26.71 |

B: Relaxation at 0% SOC | 148.12 | 148.17 | 38.86 | 52.44 | 56.87 |

C: 1 C Charge | 49.61 | 56.85 | 95.08 | 60.87 | 59.63 |

D: Relaxation at 100% SOC | 37.35 | 37.35 | 36.33 | 36.47 | 36.74 |

E: DST | 24.14 | 25.63 | 22.65 | 20.59 | 22.16 |

F: Driving Profile | 31.56 | 32.05 | 23.69 | 25.12 | 27.39 |

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**MDPI and ACS Style**

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*, 52.
https://doi.org/10.3390/batteries7030052

**AMA Style**

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.
https://doi.org/10.3390/batteries7030052

**Chicago/Turabian Style**

Wildfeuer, Leo, Philipp Gieler, and Alexander Karger. 2021. "Combining the Distribution of Relaxation Times from EIS and Time-Domain Data for Parameterizing Equivalent Circuit Models of Lithium-Ion Batteries" *Batteries* 7, no. 3: 52.
https://doi.org/10.3390/batteries7030052