Review Reports - Mechanics of Lithium-Ion Batteries: Aging and Diagnostics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript addresses an important and emerging topic: leveraging mechanical deformation for battery diagnostics and aging modeling. The integration of mechanical measurements into SOC estimation and degradation modeling is innovative and relevant for EV applications. However, several aspects require clarification, deeper analysis, and stronger validation to meet the standards of a high-impact journal. 1. More discussion should be provided in the introduction section on SOC and SOH estimation vis different methods/models. A remarkable literature report (https://doi.org/10.1002/eom2.12505) should be read and cited in the revised version. 2. How were laser displacement sensors calibrated? What is the uncertainty in thickness measurements, especially under dynamic conditions? 3. Please quantify thermal deformation and isolate its effect. 4. It is highly recommended to implement adaptive covariance tuning in POLISOC. 5. Pls supplement the revised version with hysteresis modeling explicitly for LFP and NMC. 6. Perform sensitivity and uncertainty analysis for POLIDEMO parameters. 7. The model assumes kinetic-limited SEI growth. How does it account for mechanical stress amplification under high-temperature fast charging? 8. The model uses multiple empirical parameters (e.g., LAM law, SEI growth). How is parameter uniqueness ensured? Was sensitivity analysis performed?

Comments on the Quality of English Language

English should be improved.

Author Response

The authors thank the reviewer for the time and effort spent reviewing this manuscript. The insightful comments helped improve its clarity and overall quality. Responses are provided in red below each reviewer’s comment.

More discussion should be provided in the introduction section on SOC and SOH estimation vis different methods/models. A remarkable literature report (https://doi.org/10.1002/eom2.12505) should be read and cited in the revised version.

In the revised manuscript, the authors have expanded the Introduction to include a more comprehensive discussion of existing models for battery aging predictions and algorithms for battery diagnostics (SOC and SOH estimation). This includes an updated review of traditional and advanced approaches, and the incorporation of additional citations to relevant works in the field. We believe these additions improve the context and background of our study.

How were laser displacement sensors calibrated? What is the uncertainty in thickness measurements, especially under dynamic conditions?

Laser sensors do not need any calibration. They give an output voltage in the range 0-10V proportional to the differential distance measured between the sensor and the target object, placed at a nominal distance of 25mm from the sensor. The measured distance must be in the range of the specific sensor, in our case 10mm. Then, the laser sensors must be mounted on a distance of 25mm from the battery surface and they can read variations of ±5mm with respect to such reference distance. If interested, this is the website of the sensor: https://www.micro-epsilon.it/distance-sensors/laser-sensors/optoncdt-1900/. As you can read from the datasheet, the linearity error (thus the uncertainty) is below ±2μm, corresponding to less than 1% of the measurement, considering the maximum thickness change of about 0.2mm. The sensor have a measuring rate of up to 10kHz, considering the extremely slow dynamic of battery deformation, dynamic is not an issue in this application. This information are included in the manuscript now.

Please quantify thermal deformation and isolate its effect.

Figure 2 d-f quantifying the thickness change contributions and the relative discussion have been added. Thanks for the comment, this discussion was worth to be included.

It is highly recommended to implement adaptive covariance tuning in POLISOC.

This aspect is considered outside the scope of the present work. The results obtained using a fixed covariance value are already sufficiently accurate for the intended application. Implementing an adaptive covariance algorithm, especially in the view of an on-board application, would significantly increase the computational complexity and implementation effort, while providing only a marginal improvement in accuracy, which is not justified given the associated costs (time and complexity).

Pls supplement the revised version with hysteresis modeling explicitly for LFP and NMC.

The modeling framework used to handle voltage and deformation hysteresis within the SOC estimation algorithm for LFP and NMC batteries has already been thoroughly addressed in a previous paper by the authors (Clerici D., 2025, https://doi.org/10.1016/j.apenergy.2025.126740) and is therefore not repeated here in order to keep the present manuscript concise and easy to follow.

The primary objective of this work is to provide a general overview of the role of mechanics in state estimation and battery aging. Providing excessive detail on specific aspects that have already been discussed in dedicated publications would risk obscuring the main narrative and reducing the readability of the paper. For this reason, we deliberately adopted a level of detail that enables a clear understanding of the proposed approach without unnecessarily overburdening the manuscript.

Perform sensitivity and uncertainty analysis for POLIDEMO parameters.

A formal global sensitivity analysis was performed for the parameter set estimated for the LCO battery sample, and it is published in a previous paper (https://doi.org/10.1016/j.apenergy.2025.126744). For the same reasons detailed above, the sensitivity analysis of the parameters estimated for the NMC dataset is not reported here.

The model assumes kinetic-limited SEI growth. How does it account for mechanical stress amplification under high-temperature fast charging?

The authors acknowledge that the original sentence may have caused a misunderstanding. The model does not assume a priori a kinetically limited SEI growth law. Instead, the observation of an approximately linear trend in irreversible deformation suggests a linear SEI growth rate, indicating that SEI formation is consistent with a kinetically limited regime under the investigated conditions.

This point has now been clarified in the revised manuscript by explicitly explaining this reasoning in the discussion of the irreversible deformation trend. In addition, in the newly added Model Framework section, we clearly state that the SEI growth law is formulated to account for both kinetic-limited and diffusion-limited regimes, thus allowing the model to remain valid under different operating conditions, including high-temperature fast charging.

The model uses multiple empirical parameters (e.g., LAM law, SEI growth). How is parameter uniqueness ensured? Was sensitivity analysis performed?

A dedicated section has been introduced in the revised manuscript to describe the parameter identification procedure and the strategy adopted to approach parameter uniqueness. In particular, the authors did not used just the capacity as object function for the minimization problem, but also the degradation indicators and the irreversible swelling, as explained in the manuscript. This identification methodology is designed to allow the solution to converge toward a quasi-unique parameter set. As with any optimization-based identification method, strict uniqueness cannot be guaranteed a priori. However, the proposed strategy was specifically conceived to minimize non-uniqueness and improve the robustness of the identified parameters.

A formal global sensitivity analysis was performed for the parameter set estimated for the LCO battery sample, and it is published in a previous paper (https://doi.org/10.1016/j.apenergy.2025.126744). For length limitations, the sensitivity analysis of the parameters estimated for the NMC dataset is not reported here.

Reviewer 2 Report

Comments and Suggestions for Authors Fig.2f Incorrect caption under the drawing Fig.3b The graph should be approximated by a continuous function. Connecting measurement points distorts the graph. Fig.4 The numbering under the drawings is incorrect, (2 x a, 2 x b). C and d are missing. Please provide a description that answers the following questions: 1. How will the proposed algorithm behave in non-laboratory cases, i.e.: a) changes in the charging and discharging current values - in reality, the currents in the battery do not maintain constant values throughout their life cycle, b) lack of equal time periods in charging and discharging cycles, c) should diagnostics be performed from the beginning of the battery's life? Is it possible to diagnose batteries by adding a measurement system to a partially used battery? 2. How to measure mechanical deformation in commercial solutions – laboratory laser measurement methods are generally not applicable here. Do appropriate sensors need to be built into the battery housing? 3. What coefficients should be determined for different battery types? Changing the battery type requires the construction of new calculation models.

Author Response

- Fig.2f Incorrect caption under the drawing.

Corrected, thank you. Fig 2 has been revised including the different contributions to thickness change, the figure the reviewer refers to is fig 2i in the revised manuscript.

- Fig.3b The graph should be approximated by a continuous function. Connecting measurement points distorts the graph.

Figure 3 has been revised, and among other modifications the connection line in fig 3b has been removed as it was unnecessary.

- Fig.4 The numbering under the drawings is incorrect, (2 x a, 2 x b). C and d are missing.

Corrected, thanks.

Please provide a description that answers the following questions:

How will the proposed algorithm behave in non-laboratory cases, i.e.: a) changes in the charging and discharging current values - in reality, the currents in the battery do not maintain constant values throughout their life cycle, b) lack of equal time periods in charging and discharging cycles, c) should diagnostics be performed from the beginning of the battery's life? Is it possible to diagnose batteries by adding a measurement system to a partially used battery?

The behavior of the proposed algorithm under non-laboratory conditions is addressed as follows:

a) Variations in charging and discharging current values.
In general, diagnostic algorithms are applied during charging phases only, in order to operate under controlled and approximately constant current conditions. This approach is commonly adopted in real-world applications, where charging profiles can be designed to ensure sufficient repeatability. The proposed differential expansion–based algorithm follows the same principle. This aspect has now been explicitly specified in the manuscript for clarity.

b) Unequal charging and discharging durations.
Unequal time periods for charging and discharging do not affect the proposed diagnostic approach, as the algorithm relies exclusively on data collected during the charging process. This has been clarified in the revised text.

c) Beginning-of-life reference and application to partially used batteries.
The proposed algorithm requires a differential expansion curve at the beginning of life as a reference for diagnostic purposes. This condition is typically satisfied in battery pack manufacturing, where fresh cells are routinely tested before integration. Moreover, the beginning-of-life reference can be reasonably assumed to be identical for cells of the same model. Therefore, it is not necessary to establish an individual reference for each battery in a pack, and the algorithm can also be applied to partially used batteries, provided that an appropriate beginning-of-life reference curve is available for the same cell type.

How to measure mechanical deformation in commercial solutions – laboratory laser measurement methods are generally not applicable here. Do appropriate sensors need to be built into the battery housing?

The authors would like to note that the spin-off of the research group filed a patent about the implementation of deformation sensing in battery pack, as cited in the manuscript (Device and method to measure and estimation of state of charge and state of health of a battery, WO2022269538A1, 2022), and they are currently developing this measurement technology with the goal of providing a low-cost solution that can be easily integrated into battery packs at an industrial level. For these reasons, the details of the measurement methodology in commercial solutions are not disclosed in this article.

What coefficients should be determined for different battery types? Changing the battery type requires the construction of new calculation models.

The authors are not entirely sure which coefficients the reviewer is referring to.
If the question concerns the state-of-charge estimation method (POLISOC), only the SOC–deformation characteristic of the battery needs to be identified. This requirement is analogous to voltage-based methods, which rely on the SOC–OCV relationship. Moreover, compared to voltage-based approaches, the proposed method has the advantage that no additional parameters need to be estimated, such as the resistances and capacitances of an equivalent circuit model typically required in voltage-based SOC estimation.

If instead the question refers to the aging model (POLIDEMO), it should be noted that, as with all battery aging models, a certain level of parameterization is required depending on the specific battery model. The power of a physics-based model is that the parameter set derived from a single aging experiment allows to define a model that is expected to retain validity under any aging conditions. For the proposed model, two aging tests at different current amplitudes are sufficient to identify the parameters of the LAM law (Eq. 10). Additionally, a calendar aging test can be performed to independently estimate the parameters of the SEI growth law, thereby improving the robustness of the model. If a calendar aging test is not available, the SEI parameters can be estimated jointly with the other aging parameters. This concept was made clearer in the revised manuscript with a dedicated section to parameters identification (4.2).

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript explains and demonstrates very nicely the use of mechanical swelling data to model and estimate key battery indicators. Such development and demonstrations are very important as alternative directions that the battery management system (BMS) can undertake. I find the physics-based model sound; the demonstration result to data is also neat. I would recommend its publication. I have some minor comments for the authors to consider, mostly to help improve the clarity of the manuscript and to provide future readers with a quicker path to identify relevant information.

“Aging” is used throughout this manuscript by the authors. The term “aging” for batteries is more commonly used to refer ‘calendar aging’, but the relevant process in this manuscript is ‘cycle aging’. To avoid the confusion, I suggest the authors to replace “Aging” in the title to “Cycle Aging”, as well as indicate early in the manuscript that all ‘aging’ discussed in this paper is referring to cycle aging.
Several acronyms are not explicitly defined, and they are better to be specified at the first appearance. The more important ones are ‘DOD’, ‘DST’, and the other less important ones include ‘rmse’, ‘LCO’, ‘LFP’, ‘NMC’, etc.
I find it confusing about which data is original and which is not. Is all data, except for Michigan aging dataset from ref. 6, original from the authors and first shown in this manuscript? (For example, the data shown in Fig. 1b?) If the data came from another published source, the origin needs to be properly cited.
The specific chemistry of the NMC needs to be specified. Are all relevant NMC in this manuscript NMC111? It should be clarified at the beginning of section 2.
Figure 2 shows the thickness change as a direct number. It will be best to specify the overall size of the three batteries (two prismatic and one pouch) in section 2.1 so the readers can have a better sense of how large the relative thickness change is.
For section 2.2, it is never described in what comprises an aging cycle. Cycle aging depends strongly on the cycle rate, which should be provided, at a bare minimum.
In Fig 2f, the axis is labeled as “Discharge capacity”, but I believe it should be “Charge capacity”.
In Fig 3a,b, there is no explicit explanation on what the top x-axis is. Description should be added to explain that these are the date of cycling and are in dd/mm format.
Please indicate explicitly, either in caption or in text, that Fig. 3 is showing the “LCO battery” results.
In Fig 3b, the y-axis is labeled as “Resistance”, and the text in line 124 says “Ohmic resistance”. It will be better to label Fig 3b clearly whether this is “Ohmic resistance”, “diffusion resistance”, or “total (Ohmic + diffusion) resistance”.
I am confused on what the “reference” curves are, and how they are determined, in Fig. 4 and Fig. 5. Are the “References” determined by Coulomb counting?
Please indicate explicitly, either in caption or in text, what battery chemistry is for Fig. 6.
For Fig. 6, what is the direction of the capacity in the plot, is it showing charging or discharging?
Line 265-276 seem to have many grammar issues that these two paragraphs are not easy to read and comprehend.
In line 303, “high current charging cycle (C/2, Figure 6a)”, I believe the correct figure here is “Figure 6d”?
Please give a short description of the terms in equation 10. Most terms in this equation are not explained.
For equation 11, for right-most side, is the equation missing a term of R_int,0?
Some small typos I caught:
1. Line 377, “to be estimate” → “to be estimated”
2. Line 408, “cracks surface” → “crack surfaces”
3. There should be more, but I didn’t carefully look for typos and grammar mistakes.

Author Response

“Aging” is used throughout this manuscript by the authors. The term “aging” for batteries is more commonly used to refer ‘calendar aging’, but the relevant process in this manuscript is ‘cycle aging’. To avoid the confusion, I suggest the authors to replace “Aging” in the title to “Cycle Aging”, as well as indicate early in the manuscript that all ‘aging’ discussed in this paper is referring to cycle aging.

The aging model POLIDEMO described in this work accounts for both cycle aging and calendar aging. In particular, when a zero current is applied, only calendar aging mechanisms—such as SEI growth—are active over a time interval equivalent to the imposed cycle duration.

Similarly, the differential expansion–based algorithm detects battery aging by monitoring phase transformations in the electrodes, regardless of whether aging originates from cycling or calendar conditions. Therefore, the term “aging” is intentionally used in a broader sense throughout the manuscript.

Several acronyms are not explicitly defined, and they are better to be specified at the first appearance. The more important ones are ‘DOD’, ‘DST’, and the other less important ones include ‘rmse’, ‘LCO’, ‘LFP’, ‘NMC’, etc.

In the revised manuscript all the acronyms have been specified at the first appearance.

I find it confusing about which data is original and which is not. Is all data, except for Michigan aging dataset from ref. 6, original from the authors and first shown in this manuscript? (For example, the data shown in Fig. 1b?) If the data came from another published source, the origin needs to be properly cited.

The aging data on the LCO cell, generated by the authors, were previously published in another article, which is now explicitly cited in the caption of Figure 3. The mechanical characterization data of LFP and LCO cells shown in Figure 2 were also published in another previous works by the authors and are properly referenced in the caption of figure 2. In contrast, the mechanical characterization data of the NMC cell, also reported in Figure 2, are presented here for the first time. The Michigan aging dataset from Ref. 6 is the only dataset not generated by the authors. We have revised the manuscript to clearly state the origin of each dataset and to ensure that all previously published data are appropriately cited.

The specific chemistry of the NMC needs to be specified. Are all relevant NMC in this manuscript NMC111? It should be clarified at the beginning of section 2.

NMC batteries referred in the manuscript are NMC111, both the one tested in our lab both the ones of the dataset [6]. It has been clearly specified in the revised manuscript.

Figure 2 shows the thickness change as a direct number. It will be best to specify the overall size of the three batteries (two prismatic and one pouch) in section 2.1 so the readers can have a better sense of how large the relative thickness change is.

Tabel 1 has been included in the revised manuscript showing the specifications of three battery samples (including their thickness).

For section 2.2, it is never described in what comprises an aging cycle. Cycle aging depends strongly on the cycle rate, which should be provided, at a bare minimum.

The authors agree with the reviewer and apologize for the lack of clarity in the original manuscript. In the revised version, the aging cycle protocol is now explicitly described, including the current profile, current intensity, depth of discharge, and operating temperature, thereby providing all the necessary information to properly interpret the cycle aging conditions.

In Fig 2f, the axis is labeled as “Discharge capacity”, but I believe it should be “Charge capacity”.

Corrected, thank you.

In Fig 3a,b, there is no explicit explanation on what the top x-axis is. Description should be added to explain that these are the date of cycling and are in dd/mm format.

Thank you for the comment, an upper label is included in the revised manuscript stating “Experimental timeline [day/month].

Please indicate explicitly, either in caption or in text, that Fig. 3 is showing the “LCO battery” results.

The caption of figure 3 was made clearer specifying that it displays the results of the aging test on LCO battery. It is also specified in the text of the revised manuscript.

In Fig 3b, the y-axis is labeled as “Resistance”, and the text in line 124 says “Ohmic resistance”. It will be better to label Fig 3b clearly whether this is “Ohmic resistance”, “diffusion resistance”, or “total (Ohmic + diffusion) resistance”.

Figure 3b refers to the Ohmic resistance, coherent with the text. The label has been updated in the revised manuscript.

I am confused on what the “reference” curves are, and how they are determined, in Fig. 4 and Fig. 5. Are the “References” determined by Coulomb counting?

Yes, the reference curves are obtained using Coulomb counting, which is a common practice in this type of analysis. The estimated state of charge provided by the proposed algorithm is compared against a reference SOC computed exclusively through Coulomb counting. This approach is appropriate in the experimental context considered here, since Coulomb counting is performed using the high-accuracy current sensors integrated in the battery cycler, and the tests are initialized from a well-defined SOC condition (100%).

It is acknowledged that such conditions are not representative of an industrial on-board environment, where Coulomb counting alone is generally unreliable due to sensor drift and unknown initial SOC. This limitation motivates the use of more advanced SOC estimation algorithms in practical applications. This clarification has now been explicitly included in the revised manuscript.

Please indicate explicitly, either in caption or in text, what battery chemistry is for Fig. 6.

Figure 6 refers to the LCO battery chemistry. Specifically, Fig. 6 shows the differential analysis of the LCO battery that underwent the aging tests conducted in our laboratory, whose results are presented in Fig. 3. This information has now been explicitly stated in the revised manuscript, either in the figure caption or in the main text.

For Fig. 6, what is the direction of the capacity in the plot, is it showing charging or discharging?

The capacity direction in Figure 6 corresponds to charging. Differential analyses are generally performed during charging, as this is the condition in which the current can be controlled in real-world applications. It is clearly written now in the revised manuscript.

Line 265-276 seem to have many grammar issues that these two paragraphs are not easy to read and comprehend.

The authors have carefully reformulated the mentioned passages to improve clarity and readability, and we hope that these revisions enhance the understanding of the described method.

In line 303, “high current charging cycle (C/2, Figure 6a)”, I believe the correct figure here is “Figure 6d”?

Yes, corrected.

Please give a short description of the terms in equation 10. Most terms in this equation are not explained.

All the terms in Equation 10 have been explained in the line following the equation.

For equation 11, for right-most side, is the equation missing a term of R_int,0?

Yes, corrected the equation in the revised manuscript.

Some small typos I caught:

Line 377, “to be estimate” → “to be estimated”
Line 408, “cracks surface” → “crack surfaces”
There should be more, but I didn’t carefully look for typos and grammar mistakes.