A Novel Temperature Compensation Method for Surface Strain of Cylindrical Lithium-ion Batteries

In order to ensure the safe operation of lithium-ion batteries, real-time monitoring of battery status is necessary. The surface strain signal of lithium-ion batteries has the potential to evaluate the battery’s state, but it is significantly affected by temperature. Generally, measuring the battery surface temperature and the thermal expansion coefficient can be performed to quantify and eliminate the influence of temperature on strain, but this increases the cost and complexity of strain measurement. This article proposes a method that eliminates the need to measure the battery temperature and material parameters. By simultaneously measuring the circumferential and axial strains on the battery surface and calculating their difference, the influence of temperature on strain can be minimized. Furthermore, the effectiveness of the proposed method is experimentally tested. Results demonstrate that after applying temperature compensation to commercial lithium-ion batteries, the influence of temperature on strain can be reduced from 16.4 ppm/°C to 1.7 ppm/°C. The strain no longer exhibits sensitivity to current, making it more suitable for evaluating the state of lithium-ion batteries.


Introduction
Lithium-ion batteries are widely used in various fields, such as consumer electronics, electric vehicles, and energy storage, due to their high energy density, high power density, long lifespan, and environmental friendliness.To ensure the safe operation of lithium-ion batteries, real-time monitoring of the internal battery state is necessary for battery management systems [1].Currently, battery management systems typically monitor external parameters such as voltage, current, and temperature to evaluate the battery's state and manage/protect the battery.However, these parameters do not directly provide information about the internal physical and chemical processes of the battery [2].One important characteristic of lithium-ion batteries is that their volume changes during operation.This is mainly due to the concentration dependence of the electrode active materials.As lithium ions are inserted and extracted, the electrode undergoes corresponding expansion or contraction, causing overall deformation of the battery.Therefore, the deformation information of lithium-ion batteries has the potential to reflect the internal state of the battery [3].
The most direct approach to detect electrode deformation is by embedding fiber-optic sensors [4] or thin-film strain gauge sensors [5] on the internal electrode surfaces of the battery.The strain of the electrode has shown a high correlation with its state of charge and can be used to assess the battery's health and charge status.However, embedding sensors inside the battery requires complex processes and incurs higher costs.One-dimensional expansion measurement devices and fiber-optic sensors have been used to measure the expansion and strain signals on the surface of pouch-type lithium-ion batteries [6,7].Relevant studies have shown that the deformation signals on the battery surface can still reflect the internal deformation information.However, the geometrical limitations of these two sensors restrict their application in cylindrical lithium-ion batteries.Resistance strain gauges can be attached to the surface of materials to measure their linear strain.They have a small size and can be attached to the surface of cylindrical lithium-ion batteries to measure strain in various directions.
The surface strain of batteries is influenced by multiple factors, with temperature playing a particularly prominent role [8].On the one hand, the battery itself undergoes thermal expansion, while on the other hand, the strain sensor experiences temperature drift.During operation, the battery generates heat, resulting in an increase in temperature, which is directly proportional to the intensity of the current [9].This poses a challenge when measuring the surface strain of the battery and evaluating its state.One possible compensation approach is to incorporate a temperature sensor to measure the battery's real-time temperature [1,7], calculate the thermal strain of the battery, and subtract it from the actually measured strain to eliminate the effects of temperature-induced strain.However, this approach increases the cost of detection and requires additional calibration of the battery's thermal expansion coefficient.The use of temperature-compensated strain sensors can mitigate temperature drift in the strain sensor [10], but it still fails to eliminate the thermal expansion effect of the battery itself.Hence, there is a need to develop simpler and more practical temperature compensation methods to mitigate or eliminate the influence of temperature on the strain.
By investigating the consistency and differences between circumferential and axial strains on the surface of cylindrical lithium-ion batteries, this study proposes a temperature compensation method for surface strain in lithium-ion batteries.The effectiveness of this method is validated through experimental testing.Compared to existing temperature compensation methods, the proposed approach eliminates the need for additional temperature measurement and determination of the linear expansion coefficient of the battery.Moreover, it achieves temperature compensation for surface strain in cylindrical lithium-ion batteries using a single biaxial strain gauge, making it both economical and practical.
The rest of this paper is organized as follows.Section 2 derived the consistency and differences of surface strain in batteries, elucidating the principles of the proposed method.Section 3 introduced the experimental equipment and testing conditions.In Section 4, the results of the experimental tests were reported and discussed, and the effectiveness of the method has been verified.Finally, Section 5 presents the conclusion.

Methodology
Given that the ratio of the outer diameter to the thickness of the cylindrical lithium-ion battery casing is significantly greater than 20, we can consider the casing as a thin-walled cylinder.During the operation of the battery, the electrode that undergoes lithium intercalation expands, resulting in pressure exerted on the inner surface of the casing.When the thin-walled cylinder is subjected to uniform internal pressure, the surface stress can be expressed by the following equation: where r V , c V , and a V represent the radial stress, circumferential stress, and axial stress, respectively, p is the internal pressure, D represents the outer diameter of the battery casing, and t represents the thickness of the casing.The ratio between D and t is significantly greater than 20.The surface strain of the cylinder can be separated into circumferential strain and axial strain, which can be calculated using the following equation: where c H and a H represent the circumferential strain and axial strain, respectively, E is Young's modulus of the battery casing, and P represents the Poisson's ratio of the casing.
By substituting Equations ( 1) ~ (3) into Equations ( 4) ~ ( 5) and considering D/t >> 20, we have: The material commonly used for the casing of cylindrical lithium-ion batteries is stainless steel, which typically has a Poisson's ratio of 0.3.According to Equation ( 8), the ratio between c H and a H is approximately 4.3.This implies that the circumferential strain on the battery surface is significantly greater than the axial strain.Consequently, it is customary to solely measure the circumferential strain on the surface of cylindrical lithium-ion batteries to reflect the deformation of the electrode.However, the notable difference between circumferential strain and axial strain indicates that the result of subtracting axial strain from circumferential strain can still reflect the deformation of the electrode.
When there is a change in temperature, the battery casing undergoes thermal expansion and resulting in thermal strain.Due to the isotropic nature of the casing material, the circumferential thermal strain is equal to the axial thermal strain.The strain gauges also experience temperature drift due to their thermal expansion and thermal sensitivity of resistance.By subtracting the axial strain from the circumferential strain, the resulting difference will exclude the strain changes caused by the temperature variations mentioned earlier.
In conclusion, by subtracting the axial strain from the circumferential strain, the resulting difference not only retains the ability to reflect the electrode deformation but also eliminates the thermal expansion strain of the battery casing and the temperature drift of the strain gauges.The proposed temperature compensation method in this study involves two steps: x The circumferential strain and axial strain on the surface of the cylindrical lithium-ion battery are simultaneously measured.This can be achieved by attaching a dual-axis strain gauge to the measurement point, which features two mutually perpendicular sensitive grids.x The difference between the circumferential strain and axial strain is considered.By connecting the two sensitive grids of the dual-axis strain gauge to the adjacent branches of the measurement bridge, the output of the bridge corresponds to the difference between the circumferential strain and axial strain.

Experiments
In this work, the commercial batteries (TERRAE INR-18650-30E, 2900 mAh) with NMC cathode and graphite anode are used.The charging and discharging of the batteries are controlled through a comprehensive testing system (NEWARE CT-4008-20V10A-NA).Real-time measurement of strain signal on the battery surface is achieved by using dual-axis strain gauges (SIGRMA BSF120-2BB-T) and a strain acquisition device (SIGRMA ASMB4-8).Temperature signals on the battery surface are measured using K-type thermocouples and a temperature acquisition device (ANBAI AT4508).All experiments are conducted in a thermostatic chamber (NEWARE MHW-25), where the environmental temperature surrounding the batteries is controlled.
The first experiment was conducted to verify that the difference between circumferential and axial strain can reflect electrode deformation.The battery current rate was set to 0.1 C, and the strain signals were collected during the discharge period.Due to the sufficiently low current rate and the battery being placed in a thermostatic chamber, it can be assumed that there is no temperature change during the testing process.Therefore, the measured strain is solely caused by electrode deformation.
The second experiment was conducted to validate that taking the difference between a circumferential strain and an axial strain can eliminate the influence of temperature on the strain.The battery was placed in the thermostatic chamber.The initial temperature of the chamber was set to 25°C, and then it was gradually increased in increments of 1°C from 25°C to 40°C.After each temperature change, the battery was allowed to settle for half an hour to reach thermal equilibrium.During this period, the strain on the battery surface was monitored.
The third experiment was conducted to study the effect of temperature variations on strain during the battery operation process and to validate the effectiveness of the proposed method.The experiment involved measuring the strain and temperature of the battery during discharge at various current rates, namely 0.2 C, 0.6 C, and 1.0 C. Prior to each discharge, the battery was charged at a rate of 0.3 C until reaching a cut-off voltage of 4.2 V, followed by a three-hour rest period to ensure the battery was in the same initial state.

Results and discussion
Figure 1 depicts the measured strains obtained in the first experiment.The strain prior to discharge is set to zero.During the discharge process, the axial strain shows minimal variation and is significantly smaller than the circumferential strain.The circumferential strain gradually decreases and can be divided into three main stages, divided by the yellow dashed lines in the graph.This phenomenon bears a resemblance to the characteristic strain curve of graphite anode [1], reflecting the distinct phase transitions occurring within the graphite anode.The difference strain curve obtained by subtracting the axial strain from the circumferential strain preserves the three-stage characteristic of the graphite electrode deformation.Therefore, the difference strain can also be used to reflect the electrode deformation information.Figure 2 illustrates the surface strains of the battery at various temperatures in the second experiment.The symbols represent the measured values, while the solid lines represent the least squares fitting lines of the measured values, with their slopes indicating the average coefficient of linear expansion, which represents the rate of influence of temperature on the strain.The strain at 25°C is set as zero.It can be observed that as the temperature increases, both the circumferential and axial strains exhibit a similar rate of increase, indicating that temperature affects the circumferential and axial strains approximately equally.The average coefficient of linear expansion for the circumferential strain is slightly larger than that of the axial strain.This is due to the larger circumferential strain and smaller axial strain on the surface caused by the thermal expansion of the jelly-roll structure inside the battery, as expressed by Equation (8).By subtracting the circumferential strain from the axial strain, the difference strain, shown as the solid green line in Figure 3, is obtained.The rate of change of the different strains with temperature (1.7 ppm/°C) is significantly smaller than that of the circumferential strain (16.4 ppm/°C) and axial strain (14.7 ppm/°C).This indicates that subtracting the circumferential and axial strains can effectively reduce the influence of temperature.Figure 3 presents the measurement results from the third experiment.The strain curves during discharge exhibit a sensitivity to the applied current.With the same discharge capacity, higher currents result in larger strain values.The significant differences in strain for different currents under the same initial state and discharge capacity are unrelated to the total amount of lithium concentration within the electrode.From Figure 3(c), it can be observed that the temperature curves of the battery at different currents show significant differences.Under the same discharge capacity, higher currents lead to higher temperatures.This is because the Joule heating power is positively correlated with the current.Furthermore, it was observed that both circumferential and axial strain curves exhibited a sudden increase toward the end of the discharge.This phenomenon can also be observed in the temperature curve, where the temperature values rapidly rise during the late stage of discharge, possibly due to significant polarization effects.Consequently, the differences in strain for different currents and the abrupt increase observed during the late stage of discharge may be attributed to the thermal strain caused by temperature.
By subtracting the circumferential strain from the axial strain, the difference strain is obtained, as the strain after temperature compensation is shown in Figure 3(d).The difference strain curves at different currents exhibit a high level of consistency.Throughout the discharge period, the strain exhibits a monotonically decreasing trend.This not only confirms the above hypothesis but also validates the effectiveness of the proposed method.With temperature compensation, the strain's sensitivity to the applied current disappears, making it more suitable for monitoring the state of lithium-ion batteries under dynamic operating conditions.

Conclusion
This paper investigates the consistency and differences between circumferential and axial strains on the surface of cylindrical lithium-ion batteries.The circumferential strain caused by lithium insertion and extraction is significantly larger than the axial strain.In contrast, the thermal expansion of the battery casing due to temperature variations and the temperature drift of the strain gauge result in equal circumferential and axial thermal strains.A temperature compensation method for surface strain in cylindrical lithium-ion batteries is proposed, which is achieved by measuring the differential signal between circumferential and axial strains on the battery surface.
The effectiveness of the proposed method was validated through experimental testing.The method was applied to compensate for the surface strain of commercial lithium-ion batteries and compared to the circumferential strain before temperature compensation.The influence of temperature on strain was reduced from 16.4 ppm/°C to 1.7 ppm/°C.Furthermore, after temperature compensation, the surface strain of the battery no longer exhibited sensitivity to current variations, allowing the strain signal on the battery surface to be more suitable for assessing the battery's state.

Figure 1 .
Figure 1.The strain curves during 0.1 C multiplier current discharge.