# Physics-Based SoH Estimation for Li-Ion Cells

^{1}

^{2}

^{*}

## Abstract

**:**

_{x}). The results showed that loss of lithium inventory (LLI) is the main driving factor for cell degradation, followed by loss of cathode active material (LAM

_{C}). SoH estimation was achievable with a mean absolute error lower than 0.75% for SoH values higher than 85% and lower than 3.70% SoH values between 85% and 80% (end of life). The analyses of the results will allow for guidelines to be defined to replicate the presented methodology, characterize new Li-ion cell types, and perform onboard SoH estimation in battery management system (BMS) solutions.

## 1. Introduction

- Defining a suitable EIS-based SoH estimation model which uses degradation indicators directly linked to DMs, allowing principled modelling of the physical phenomena (i.e., physics-based SoH estimation);
- Defining a simple and robust framework that could be exploited for online-SoH estimation by next generation smart BMSs based on (i) an appropriate testing campaign to initialize the SoH estimation model for a new Li-ion chemistry/model and (ii) the capability to run onboard EIS measurement.

## 2. Materials and Methods

#### 2.1. Inputs: Experimental Dataset and DRT Characterization

_{x}) as anode material. The testing campaign has been performed at the CSEM’s Sustainable Energy Center, Neuchâtel (Switzerland) with testing set-up composed of: (i) a cell tester Biologic BCS815 equipped with 32 parallel, 9V−15A channels (±0.01% FSD accuracy on the voltage and ±0.015% FSD accuracy on current, for each available range) with EIS spectroscope multiplexed and able to range from 10 kHz to 10 mHz (Figure 1a) [45]; and (ii) a thermostatic chamber ATT-DM1200T with −45 °C–180 °C temperature range (Figure 1b) [46]. Ten different cycling conditions have been studied and their details together with the corresponding cell’s IDs are given in Table 2. The tests have been performed in the thermostatic chamber at a constant temperature of 20 °C (Figure 1c).

#### 2.2. Methodology

- Peak 2 is attributed to the growth and decomposition of SEI layer and presence of lithium plating on the anode side; it will be used to account for Loss of Lithium Inventory ($LLI$);
- Peak 3 and peak 4 are attributed to cathode degradation (specifically particle cracking for NMC811) and they are therefore used to account for Loss of Cathode Active Material ($LA{M}_{C}$);
- Peak 5 is attributed to graphite degradation and it is hence used to account for Loss of Anode Active Material ($LA{M}_{A}$).

#### 2.2.1. Degradation Indicators

- A set of prior points is selected based on a sliding window of size W and is concatenated with the indicator ${\mathrm{k}}_{i\_calc}$ computed at diagnosis step $i$;
- A linear fitting model is applied to the selected vectors;
- The linear model is used to compute the corrected value of ${\mathrm{k}}_{i}.$

#### 2.2.2. SoH Estimation

_{1}, B

_{1}) are found by fitting the available measurement of SoH. When the cells show “after-knee” behavior instead ($\Delta {\mathrm{R}}_{\Omega ,i}>1\%$), the SoH function includes two additional parameters: the value of SoH ($So{H}_{last}$) and the value of the total degradation indicator ($TD{M}_{last}$) before the knee. As described in Section 2.1, “after-knee” behavior shows an acceleration in capacity fade unrelated to the previous aging behavior. The two additional parameters are introduced to normalize the function because the “knee” does not occur always at the same conditions (i.e., SoH and $TD{M}_{i}$) for each cell. Once these parameters are fixed with the last previous point computed in “pre-knee” condition, the coefficients (A

_{2}, B

_{2}) are found by fitting the available measurements of SoH.

- Training subset: n
_{1}cells are selected and their EIS measurements are used to compute degradation indicators and ohmic indicator. SoH values are fitted to find the parameters of the model described in Equation (6). - Validation subset: n
_{2}cells (i.e., n_{2}= 10 − n_{1}) are used to compute the $TD{M}_{i}$ indicator and the SoH is estimated with the SoH model checking the value of $\Delta {\mathrm{R}}_{\Omega}$. Finally, the estimated value is compared with the one computed by capacity measurement.

## 3. Results

#### 3.1. Degradation Indicators

- $LLI$ (Figure 7a): a monotonic growth is observed until around 400 EqC when the indicator decreases its growth and starts to oscillate;
- $LA{M}_{C}$ (Figure 7c): cell’s cathode is not impacted by degradation until 200 EqC; after this point it shows a constant growth up to 60% at 1000 EqC;
- $LA{M}_{A}$ (Figure 7b): this DM is not affecting cell performances until about 800 EqC. Moreover, its magnitude is one order of magnitude lower than the other two indicators.

#### 3.2. SoH Estimation

## 4. Discussion

_{x}), the defined proxies allowed to quantify the contributions of different DMs during cells’ aging. DMs were distinguished by analyzing the DRT profiles of the cells as explained in [44] on the same cell type investigated in the present work. Indicator filtering (linear and based on a sliding window) was essential to regularize the aging trend and to reduce noise and outliers while preserving information related to steep variations caused by abrupt increase of degradation. The selection of a proper sliding window size was found crucial to encompass cell degradation in a comprehensive way and it depends not only on the type of data but also on the available number of diagnosis points. In this work, the optimal number of prior points to be used for the sliding window was found equal to 7. Looking at SoH estimation, the piecewise model showed good performances for SoH > 85%, while larger errors for SoH < 85%. These larger errors were mainly linked to the small number of available points (i.e., measurements) all belonging to “after-knee” behavior. Therefore, for SoH < 85% only accelerated degradation was available. Two possible solutions can be adopted to mitigate the problem: (i) increase the number of diagnosis points when the cells reach a certain capacity (e.g., SoH < 85%) and (ii) prolong test duration (EoT > 1000 EqC) for those cells that ended the testing campaign with SoH > 85% (i.e., with less severe testing conditions).

- Plan relevant aging tests that cover different aging behaviors and allow to train the SoH model both for “pre-knee” and “after-knee” conditions. The selected protocols should include: (i) reduced DoD condition to appreciate slow degradation (i.e., capacity fade); (ii) nominal conditions, to verify the specifications from the manufacturer; (iii) moderate charging or discharging conditions that accelerates degradation with respect to nominal conditions and that could guarantee both “pre-knee” and “after-knee behavior” (such as cell ID:FC05) and (iv) high charging or discharging rate that guarantee fast degradation and “after-knee” conditions (such as cell ID:FC2);
- Perform diagnosis phase (capacity + EIS measurements) at a fixed number of EqC down to a certain value of SoH (e.g., 85%) and then intensify the number of checks by lowering the number of cycles in each repetition. In this way, more measurements will be available in the region where is mainly occurring the “knee” and accelerated capacity fade, reducing the SoH estimation error;
- Run sensitivity analysis on the ohmic resistance variation parameter $\Delta {\mathrm{R}}_{\Omega ,i}$ to discriminate between “pre-knee” and “after-knee” conditions with a suitable threshold. Validation can be performed graphically on SoH evolution curves as done in Figure 5.

- Run “diagnosis” based on long-EIS measurements (10 kHz–10 mHz). Select an appropriate criterion on when to acquire two consecutives full-EIS measurements. Depending on battery application, this variable could be set based on cycles number, a fixed period of time, or randomly (e.g., exploiting resting periods during application);
- Run “check-up” based on short-EIS measurements only at high frequency (10 kHz–1 kHz) to frequently update ${R}_{\Omega}$, which is crucial to activate additional “diagnosis” measurements whenever the “after-knee” behavior is reached based on the $\Delta {\mathrm{R}}_{\Omega}$ computation (Section 2.2.2). Additional “diagnosis” can also be activated under a certain estimated value of SoH (e.g., 85%);
- Compute the degradation indicators whenever possible to understand if unexpected behaviors are happening inside the cell. This can be done by updating DMs values and by analyzing them over time and/or over cycle number.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Overview of equipment and cells in Sustainable Energy Center laboratory at CSEM: (

**a**) Biologic BCS815 Battery tester; (

**b**) ATT-DM1200T Thermostatic chamber and; (

**c**) NMC 811 cells under investigation inside the thermostatic chamber.

**Figure 3.**Typical DRT profile of the cylindrical commercial cells under investigation. Attribution of physical processes, degradation mechanisms and degradation modes. The readers are addressed to [44] and the literature cited therein for more details, specifically [31,54] for similar methodologies and [9,55,56] for details on the degradation mechanisms.

**Figure 4.**Three-step procedure to correct the degradation indicators applying a linear fitting model.

**Figure 5.**(

**a**) SoH evolution over $TD{M}_{i}[\%]$ values for the cells under investigation. Two regions are depicted, corresponding to “pre-knee” and “after-knee” behaviors; and (

**b**) validation of $\Delta {\mathrm{R}}_{\Omega ,i}$ threshold equal to 1%, identifying all the points with “after-knee” behavior.

**Figure 7.**Example of degradation indicators computed for cell ID:REF (cycling in reference condition, C/3 symmetric with Constant Voltage-phase during charge). The plots show both the not corrected and corrected values of: (

**a**) $LLI$, (

**b**) $LA{M}_{A}$, (

**c**) $LA{M}_{C}$ and (

**d**) $TDM$.

**Figure 8.**(

**a–j**) Evolution of the three DMs indicators ($LLI$, $LA{M}_{A}$ and $LA{M}_{C}$) for the cells under investigations. Details about testing protocols are given in plot titles: (i) DoD: Depth of Discharge; (ii) SoC: SoC interval, (iii) Chg: charging rate (

^{1}including Constant Voltage phase with cut-off current at C/50); (iv) Dsg: discharging rate.

**Figure 9.**(

**a**) SoH functions found with the training cells in Table 3; and (

**b**) fitting coefficients (SoH model described in Equation (6)).

**Figure 10.**SoH model validation applying the model of Figure 9. (

**a**–

**e**) Results of application of SoH model to the validation cells of Table 4. Details about testing protocols are given in plot titles: (i) DoD: Depth of Discharge; (ii) SoC: SoC interval, (iii) Chg: charging rate (

^{1}including Constant Voltage phase with cut-off current at C/50); (iv) Dsg: discharging rate.

**Figure 11.**Box plot representation of the (

**a**) error; and (

**b**) the absolute error for the different SoH ranges introduced in Table 5 with 5 cells in the training set.

**Figure 12.**Analysis of all the possible combinations of training sets with a number of cells between 2 and 8: (

**a**) MBE and MAE results over the whole SoH range; and (

**b**) box plot results in the SoH range 90–85%.

**Table 1.**Main characteristics of the cylindrical commercial cell under investigation. Current-rate (C-rate) values are calculated with respect to the nominal capacity.

Cell Name | INR21700 M50 |
---|---|

Manufacturer | LG Chem |

Cathode chemistry | NMC811 |

Anode chemistry | Graphite-SiO_{x} |

Nominal capacity [mAh] | 5010 |

Nominal voltage [V] | 3.63 |

Standard charge current [mA] | 1455 (C-rate: C/3) |

Standard discharge current [mA] | 970 (C-rate: C/5) |

Standard cycling current [mA] | 1455 (C-rate: C/3) |

Maximum voltage [V] | 4.2 |

Minimum voltage [V] | 2.5 |

Current cut-off [mA] | 50 (C-rate: C/100) |

Weight [g] | 68.0 |

**Table 2.**Overview of cycling aging testing protocols applied to commercial Li-ion cells. The cycling test type is compared to the reference case (1st row).

Cell ID | Cycling Test Type | DoD [%] | SoC Interval [%] | Charging Rate | Discharging Rate |
---|---|---|---|---|---|

REF | Reference case (by datasheet) | 100 | 0–100 | C/3 ^{1} | C/3 |

REF_w/oCV | Reference without CV phase | 100 | 0–100 | C/3 | C/3 |

DOD20 | Reduced DoD | 20 | 80–100 | C/3 | C/3 |

DOD60 | Reduced DoD | 60 | 20–80 | C/3 | C/3 |

FC05 | Faster charging rate | 100 | 0–100 | C/2 ^{1} | C/3 |

FC1 | Faster charging rate | 100 | 0–100 | 1C ^{1} | C/3 |

FC2 | Faster charging rate | 100 | 0–100 | 2C ^{1} | C/3 |

FD05 | Faster discharging rate | 100 | 0–100 | C/3 ^{1} | C/2 |

FD1 | Faster discharging rate | 100 | 0–100 | C/3 ^{1} | 1C |

FD2 | Faster discharging rate | 100 | 0–100 | C/3 ^{1} | 2C |

^{1}Including Constant Voltage phase with cut-off current at C/50.

**Table 3.**Training subset for SoH estimation. The presence of “pre-knee” and “after-knee” SoH behavior is highlighted in the last two columns. Details about the testing protocols can be found in Table 2.

Cell ID | SoH [%] at EoL/EoT | EqC at EoL/EoT | Pre-Knee SoH Behavior | After-Knee SoH Behavior |
---|---|---|---|---|

DOD20 | 92.7% | 1000 | ✓ | ✕ |

FC05 | 79.5% | 892 | ✓ | ✓ |

FC1 | 63% | 92 | ✕ | ✓ |

FD05 | 86.6% | 1000 | ✓ | ✕ |

FD1 | 84.6% | 1000 | ✓ | ✕ |

**Table 4.**Validation subset with results of MBE and MAE. Details about the testing protocols in Table 2.

Cell ID | Mean Biased Error [%] | Mean Absolute Error [%] |
---|---|---|

DOD60 | 0.69% | 0.73% |

REF | −0.11% | 0.38% |

REF_w/oCV | 1.27% | 1.28% |

FD2 | −1.54% | 1.56% |

FC2 | −7.46% | 7.46% |

**Table 5.**MBE and MAE in different SoH ranges for all the combinations of 5 training cells and 5 validation cells.

# | SoH Range | Mean Biased Error [%] | Mean Absolute Error [%] |
---|---|---|---|

1 | 100% > SoH ≥ 95% | −0.05% | 0.38% |

2 | 95% > SoH ≥ 90% | 0.01% | 0.40% |

3 | 90% > SoH ≥ 85% | 0.43% | 0.71% |

4 | 85% > SoH ≥ 80% | 3.57% | 3.65% |

5 | SoH < 80% | 0.65% | 6.78% |

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**MDPI and ACS Style**

Iurilli, P.; Brivio, C.; Carrillo, R.E.; Wood, V.
Physics-Based SoH Estimation for Li-Ion Cells. *Batteries* **2022**, *8*, 204.
https://doi.org/10.3390/batteries8110204

**AMA Style**

Iurilli P, Brivio C, Carrillo RE, Wood V.
Physics-Based SoH Estimation for Li-Ion Cells. *Batteries*. 2022; 8(11):204.
https://doi.org/10.3390/batteries8110204

**Chicago/Turabian Style**

Iurilli, Pietro, Claudio Brivio, Rafael E. Carrillo, and Vanessa Wood.
2022. "Physics-Based SoH Estimation for Li-Ion Cells" *Batteries* 8, no. 11: 204.
https://doi.org/10.3390/batteries8110204