High-Precision Battery Temperature Estimation Using the Impedance Method

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Abstract

INTRODUCTION

Recently, frequent fires in secondary lithium batteries have highlighted the importance of battery health diagnostics, particularly thermal runaway prediction. The battery management system (BMS), which is responsible for battery diagnostics, estimates the various states of the battery, including the state of charge (SoC) [1,2], state of health [3,4], and internal battery temperature [5–7], through the voltage, current, and battery surface temperature. Among the diagnostic parameters of BMS, the battery temperature is directly related to safety, and the importance of accurate temperature estimation is increasing significantly as the applications of lithium batteries have recently expanded to transportation systems and large-scale energy storage systems [1–9].

The internal temperature of a battery is estimated using various methods, such as thermistor [10,11], thermocouple [12,13], resistance temperature detector (RTD) [14,15], fiber Bragg grating sensor (FBG) [16,17], Johnson noise thermometry (JNT) [18], thermal imaging camera (TIC) [19], and electrochemical impedance spectroscopy (EIS) [20–22]. Thermistors, thermocouples, RTD, FBG, and JNT measure the battery surface temperature with sensors attached to the battery surface and then estimate the internal temperature of the battery using a physical model [23,24]. Therefore, there is a time delay before the sensor detects rapid temperature changes inside the cell, making the real-time analysis of the internal temperature virtually impossible [25,26] and limiting the accuracy of temperature estimation depending on the sensor location [27,28]. There have been attempts to directly measure the internal temperature by inserting sensors into the battery [26,29]; however, these methods can be difficult to apply depending on the battery structure, and the inserted sensors may distort the internal structure of the battery, affecting the battery performance [30,31]. In addition, all the above methods using sensors incur additional component and process costs [32].

TIC and EIS are noncontact methods for estimating the battery temperature. The TIC method converts infrared energy into electrical signals that are used to visualize the spatial distribution of heat [33]. Although the TIC is favored by several researchers for its ability to easily visualize the temperature distribution on the battery surface, it shares the same limitations as contact sensors in estimating the internal battery temperature because it can only observe the surface temperature [34]. Moreover, the observed surface temperature data can be skewed not only by the object itself, but also by various other factors, including nearby objects and atmospheric radiation [33]. EIS estimates the internal temperature of a battery based on its impedance, which is determined from the output current (or voltage) in response to a sinusoidal input voltage (or current) [35]. Given that the impedance accurately mirrors the real-time conditions within the battery and is relatively unaffected by the surrounding environment, internal temperature estimation using this method is expected to be highly accurate compared with the other aforementioned methods [20,36]. However, the equipment for EIS measurements is expensive, and the process of impedance measurement and analysis can be relatively time-consuming, particularly when applying the typical frequency range (from kilohertz to hertz) used in battery analysis [17,20].

To overcome this limitation of the EIS technique, efforts have been made to use the impedance value at a single frequency for temperature estimation. For example, it is possible to determine a specific reaction step that has a strong temperature dependence but is largely unaffected by other factors (such as SoC) and estimate the temperature by analyzing the impedance at the characteristic frequency at which such a reaction step is activated. A typical reaction step with this characteristic is the combination of the resistance and capacitance of the anode film layer (i.e., solid electrolyte interface (SEI) layer), modeled as an R–C parallel unit, the utilization of which was first reported by Srinivasan [20]. Using a similar approach, several researchers have attempted to estimate the internal temperature of a battery by selecting a characteristic frequency that reflects information from the SEI layer and analyzing the phase angle [20], real impedance [21], imaginary impedance [22], and various derived parameters [26,28] at this frequency. However, despite being one of the most promising methods for the real-time, nondestructive estimation of the internal temperature of batteries known to date, this method suffers from a critical problem: low accuracy in the high-temperature region (>50°C), which is critical for battery safety diagnostics. To the best of our knowledge, the reason for this decrease in the accuracy of temperature estimation at high temperatures has not been identified. Some attempts have been made to improve the measurement accuracy just using empirical models that are solely applicable in the high-temperature region [22]. However, these methods are difficult to use in situations involving different battery systems, charge/discharge protocols, or experimental environments.

In this paper, we introduce an enhanced method for estimating temperature that employs a single characteristic frequency to precisely ascertain the internal temperature of a cell across an extensive temperature range encompassing high temperatures. First, we analyzed the temperature-dependent impedance variation at characteristic frequencies reported in the literature to understand the problems with existing temperature estimation methods. In particular, we sought to identify the cause of the decrease in the accuracy of temperature estimation at high temperatures. Based on this, the need to change the characteristic frequency for more reliable temperature estimation was presented. Then, another issue that arises as a result was discussed: the effect of inductance. Finally, we showed that, with an upward adjustment of the temperature estimation frequency and a simple mathematical correction of the inductance, a much more accurate temperature estimation can be achieved from the impedance over a wide temperature range, including high temperatures, compared with conventional methods.

EXPERIMENTAL

A lithium-ion battery (model YJ 401215, YJ Power Group Limited, China; cathode active material: LiCoO₂, anode active material: graphite) with a capacity of 38 mAh was used as a test cell for temperature estimation. For temperature control, the cell was placed in a temperature control chamber (S-LTC-120, LAB HOUSE, Korea), and a temperature sensor (DTA-5070, Daesan ENT, Korea) was affixed to the surface of the cell. For the initial characterization of the cells, the chamber was adjusted to 30°C. The cell was then subjected to a constant current (CC) charge at a rate of 0.1 C (3.8 mA) up to 4.2 V, followed by a CC discharge at a rate of 0.1 C down to 3.0 V. The charge and discharge currents were determined based on a rated capacity of 38 mAh.

In the main experiment, the cell was charged to 20% of its full capacity (SoC20) by CC charging at a rate of 0.1 C at 30°C and held at the open circuit voltage (OCV) for 3 h to stabilize it. The chamber temperature was then varied from –10 to 80°C in steps of 5°C. At each temperature, the cell was left in the open circuit state for at least 1 h to stabilize the temperature distribution, and subsequently, the impedance of the cell was measured in the frequency range of 1 MHz to 100 mHz. This process was used in a set of experiments to determine the temperature-dependent impedance of different SoCs (SoC40, 60, and 80).

A simplified module of 1S3P (three cells connected in parallel) was constructed, as a model system for analyzing the temperature estimation in the form of a module with multiple connected cells. Before connecting the three cells, we equalized the state of each cell by performing the following steps: Following the same procedure as the single-cell experiment, each cell was charged under the CC condition at a rate of 0.1 C (3.8 mA) up to 4.2 V in a chamber at 30°C, and then discharged at a rate of 0.1 C down to 3.0 V. Then, each cell was charged under the CC condition at a rate of 0.1 C (3.8 mA) up to SoC50. Now, after modularizing the three cells in the same state by connecting them in parallel, the temperature of the cells in the chamber was maintained within a range of –10 to 80°C. All the subsequent experimental procedures were identical to those performed in the single-cell experiments. All the battery charge/discharge experiments and electrochemical impedance measurements were performed using a Solartron 1470E potentiostat and Solartron 1455 frequency response analyzer (Solartron Analytical, UK).

RESULTS AND DISCUSSION

Fig. 1a shows the Nyquist plot of the cell impedance measured at ambient temperature (20°C) and different SoCs. The measured impedance was observed to have a strong SoC dependence in the low-frequency region, which is consistent with the results reported in the literatures [20–22,38]. In particular, a very high SoC dependence was observed at frequencies below 200 Hz, which indicates that frequencies above 200 Hz may be suitable for temperature estimation because the SoC dependence on the temperature is low. In fact, above this frequency, the anode SEI signal, which is known to have little SoC dependence, appears.

Fig. 1.

(a) Nyquist plot of the cell impedance measured at ambient temperature (20°C) and different SoCs. Inset figure is the magnified plot of the inside the dotted red square, and (b) Bode plot of the variation in the average imaginary impedance with the SoC and temperature above 200 Hz. Inset figure is the magnified plot of the inside the dotted blue square.

Fig. 1b demonstrates a Bode plot of the variation in the average imaginary impedance with the SoC and temperature above 200 Hz. As the imaginary impedance exceeds zero at frequencies of approximately a few tens of kilohertz (inset figure; please note that –Z_Im is used instead of Z_Im for the vertical axis to follow convention), it can be inferred that a relatively large inductance contributes to the high-frequency impedance signal (dotted circle). This high-frequency inductance affects the signal from the anode SEI layer, which appears above approximately 200 Hz (Fig. 1a), and consequently reduces the accuracy of the temperature estimation based on the signal from the SEI layer. Nevertheless, to the best of our knowledge, no studies have adequately considered the effect of high-frequency inductance on temperature estimation. Instead, the existing studies have used the frequency domain of ~500 Hz and below for temperature estimation, where the effect of inductance is expected to be small, that is, where the imaginary impedance is below zero [20–22].

In this study, the temperature was first estimated according to the conventional method in the high-frequency range to analyze the validity of the conventional temperature estimation method in the presence of an inductance. The minimum value in the test frequency range was 200 Hz, as shown in Fig. 1a. The maximum value of the test frequency range was considered as the frequency below the intercept, where the imaginary impedance became zero; thus, the effect of the inductance might have been significant. In fact, the intercept frequency decreased gradually with increasing temperature (red arrow in the inset of Fig. 1b) [25]. We arbitrarily chose 20 kHz as the maximum value of the test frequency, which is higher than the intercept frequency at the ambient temperature (20°C), to focus on the high-temperature region. Thus, the frequency range used for temperature estimation in this work was 20 kHz–200 Hz.

The methodology used for the temperature estimation in this study is described below, with an example of the temperature analysis at a test frequency of 200 Hz. First, determining the imaginary impedance at various temperatures over a wide SoC range showed that the impedance values did not differ significantly depending on the SoC (Fig. 2a). This is consistent with the results shown in Fig. 1, where the SoC dependence of the impedance at 200 Hz is negligible. Fig. 2a also shows that the absolute value of the imaginary impedance decreases with increasing temperature and that it follows a typical Arrhenius relationship (inset figure in Fig. 2a). It is noteworthy that the Arrhenius relationship is satisfied in the typical cell operating range below about 50°C, but deviates at temperatures above that. The Arrhenius relationship below 50°C is written as:

Fig. 2.

(a) Imaginary impedances at different SoCs and temperatures at the frequency of 200 Hz. Inset figure represents the Arrhenius relationship between imaginary impedance and temperature, and (b) correlation between the estimated temperature (T_fit) calculated from Eq. 2 and the actual temperature (T_real) measured by the temperature sensor.

Here, the parameters A, B, and C were determined as 5.25×10^–5 Ω, 2598 K, and –4.14×10^–2 Ω, respectively, via least squares fitting. By substituting the derived parameters into Eq. 1 and rewriting the expression for the temperature, the estimated temperature (T_fit) can be expressed as:

Fig. 2b illustrates the correlation between the estimated temperature (T_fit) calculated using Eq. 2 and the actual temperature (T_real) measured by the temperature sensor. The two temperatures were in almost exact agreement below about 50°C. However, above this temperature, T_fit increasingly deviated from T_real as the temperature increased (T_fit<T_real). Accordingly, the absolute value of the difference between the two temperatures (|T_real-T_fit| : this value will henceforth be referred to as the temperature estimation deviation, TED) also showed a relatively small deviation regardless of the temperature below 50°C. However, above 50°C, the deviation tended to diverge with increasing temperature. These results are attributed to the tendency to deviate from the Arrhenius relationship above 50°C, as described above, and are consistent with the results reported by Spinner et al. [22].

We analyzed the TED over the entire range of test frequencies using the method described above. Fig. 3a shows the difference between T_fit and T_real (i.e., T_fit-T_real) over the entire test frequency range. Below 50°C, T_fit-T_real was relatively small regardless of the frequency. However, above this temperature, the difference was highly dependent on frequency: As the frequency increased from 200 Hz at a certain temperature, T_fit-T_real became progressively smaller; however, as the frequency increased beyond 16 kHz, the difference suddenly increased drastically. To determine the suitability of the frequencies used for temperature estimation over a wide range of temperatures, we statistically processed T_fit-T_real at all temperatures for each test frequency to obtain a box-whisker plot of the TED as a function of the test frequency (Fig. 3b). Two main features can be observed in Fig. 3b. First, both the box size and out-of-box point value gradually decreased in size and value as the test frequency increased (Note: The out-of-box points in the figure are those determined at the highest temperature (80°C)). Subsequently, the box size and point value reached a minimum at approximately 12.5 kHz, and then both values increased rapidly with increasing test frequency. This inaccuracy in temperature estimation at high frequencies is most likely due to the inductance, whose effect increases with increasing frequency (Fig. 1b).

Fig. 3.

(a) Difference between the estimated temperature (T_fit) and the actual temperature (T_real) over the entire test frequency range, and (b) box-whisker plot of the temperature estimation deviation (TED) as a function of the test frequency, reproduced from Fig. 3a.

In sum, the accuracy of temperature estimation in the frequency range proposed in the literature is very low at high temperatures and gradually improves with increasing frequency. However, when the frequency is above a certain value, the accuracy of temperature estimation decreases owing to the effect of the inductance. Given these results, it is clear how to improve the accuracy of high temperature estimation by impedance analysis: increasing the characteristic frequency for temperature estimation while removing the effects of high-frequency inductance. Before experimentally validating our proposed method, we systematically analyze why existing methods fail to estimate high temperatures and why increasing the frequency improves the accuracy of high temperature estimation. This enables the setting of new test frequencies based on an understanding of the mechanism.

First, we made an in-depth analysis on why the TED increased at high temperatures when using traditional methods. Fig. 4a shows the impedance spectra measured at different temperatures at a medium-charge level (SoC60), displayed as a Nyquist plot. The impedance spectra appear as three depressed semicircles (see the inset). These three semicircles are often attributed to three resistor–capacitor (or constant phase element (CPE)) parallel combinations, which typically correspond to the anode SEI resistance and capacitance (R₁-CPE₁), anode charge transfer resistance and double-layer capacitance (R₂-CPE₂), and cathode charge transfer resistance and double-layer capacitance (R₃-CPE₃). The impedance spectra were analyzed using a simplified equivalent circuit, as shown in the inset of Fig. 4a. The three capacitance components were represented using the CPE to account for the nonideality of the signal [20,40–42]. When comparing the values estimated through the complex nonlinear least square fitting (represented by solid lines) with the actual values, the chi-square test showed a good fit with a χ² value of approximately 10^–4 or less.

Fig. 4.

(a) Impedance spectra measured at different temperatures at a medium-charge level (SoC60). Inset figure is the impedance spectrum at 25°C with the equivalent circuit used to fit the data. Open symbols and solid lines represent the experimental data and the fitted curves, respectively, and (b) variations of the minimum frequencies f_m,1, f_m,2, and f_m,3 with test frequency, reproduced from Fig. 4a, together with the boundary frequency f_b, located between f_m,1 and f_m,2.

To further analyze each reaction signal, the time constant and reaction frequency of each reaction step were determined as follows [37,39]: The impedance Z of the parallel R-CPE is generally expressed as:

Here, both Q and n are frequency-independent variables, where Q is the prefactor of the CPE, and n is an exponential term of the CPE, mainly owing to the surface heterogeneity of the system [37,48]. By integrating Eq. 4 into Eq. 3, we obtain

In addition, the general impedance relation for a time constant distribution is expressed as

By comparing Eq. 5 and 6, we can derive the time constant for each reaction step i (τ_i) as follows [37,39,48]:

Here, f_m refers to the frequency at which the minimum imaginary impedance of each of the three depressed semicircles appears (for instance, we marked its position on the 30°C Nyquist plot in the inset of Fig. 4a with a red arrow).

Fig. 4b illustrates the minimum frequencies f_m,1, f_m,2, f_m,3 corresponding to the three reaction combinations of R₁-CPE₁, R₂-CPE₂, and R₃-CPE₃ across all the test temperatures reproduced from Fig. 4a. f_b, located between f_m,1 and f_m,2 in the figure, signifies the boundary frequency of the two semicircles originating from the R₁-CPE₁ and R₂-CPE₂ combinations. The boundary frequency (i.e., the frequency with the localized maximum imaginary impedance) was calculated using the following procedure [48].

Eq. 5 can be decomposed into real and imaginary components.

Hence, the imaginary impedances corresponding to the combinations R₁-CPE₁ and R₂-CPE₂ are expressed as

After estimating the resistive components from Fig. 4a, a second derivative of log(–Z_im) by log(f) gives three real solutions. The largest and smallest of the three real solutions are the values at the inflection points, and the other corresponds to the boundary frequency f_b.

The boundary frequency information at each temperature provides guidelines on which characteristic frequencies should be used for temperature estimation. For example, f_b = 500 Hz at 25°C. This means that at frequencies above 500 Hz, the influence of R₁-CPE₁ is dominant, while below that, the influence of R₂-CPE₂ is dominant. As mentioned earlier, previous researchers assumed that the frequency region below 500 Hz fully reflects the response signal of R₁-CPE₁ (the signal from the anode SEI layer) and estimated the battery temperature from the imaginary impedance near this frequency. However, as shown in Fig. 4b, as the temperature increases above 25°C, f_b increases beyond 500 Hz, and conversely, as the temperature decreases below 25°C, f_b decreases below 500 Hz. This indicates that, at a temperature exceeding 25°C, the signal from R₂-CPE₂, which is a combination of anode charge transfer resistance and double-layer capacitance, becomes dominant, contrary to what was previously believed. The signal attributed to R₂-CPE₂ is highly SoC dependent; therefore, if it becomes dominant, the TED can be large. This is believed to be the crucial reason for the low accuracy of high-temperature estimations in previous studies.

These results suggest that, to reduce the TED and improve the accuracy, imaginary impedance values in the higher frequency region where the signal of R₁-CPE₁ is fully represented, that is, the impedance values above the boundary frequency f_b, need to be used. However, as shown in Fig. 3b, increasing the test frequency above a certain level increases the TED owing to the influence of the inductance. Therefore, the next thing we studied was whether the corrected imaginary impedance with the inductance removed could achieve a high estimation accuracy in the high frequency region dominated by the R₁-CPE₁ signal.

Fig. 5 shows a Bode plot of the imaginary impedance as a function of the frequency in the high-frequency region (100–630 kHz), where the influence of the stray inductance is dominant. As the relationship between the frequency and the imaginary impedance is linear regardless of the experimental conditions (i.e., SoC and temperature), it can be represented by a typical inductive reactance relationship as follows:

Fig. 5.

Imaginary impedance as a function of the frequency in the high-frequency range (100 to 630 kHz). Inset figure is the box-whisker plot showing inductance value.

where L is the stray inductance determined from the slope of the figure to be approximately 1.36 μH (inset figure). The L value is largely independent of SoC and temperature because the inductance is just sensitive to the intrinsic cell characteristics, such as electrode porosity [43], electrode geometry [44], and complex electron transport pathways through terminals/connectors/electrodes [45–47], rather than the cell state or operating conditions. Eq. 12 is used to derive the imaginary impedance (Z^'_Im) with the measured inductance effect removed.

When the impedance (Z^'_Im) corrected for the effect of L at 20 kHz is applied to Eq. 1 to re-parameterize the variables, A, B, and C are determined to be 1.65×10^–4 Ω, 1993 K, and 3.24×10^–2 Ω, respectively, and the corrected estimated temperature T_fit,corr becomes

Fig. 6a presents the differences between the corrected estimated temperature (T_fit,corr) and the actual temperature (T_real) over the entire range of test frequencies and temperatures. The corresponding TEDs are plotted as a box-whisker plot in Fig. 6b. Compared with the TED before correction, |T_fit-T_real| (Fig. 3b), the TED after correction, |T_fit,corr-T_real|, is significantly reduced in the high-frequency range, particularly at high temperatures, indicative of significant improvement of temperature estimation accuracy thanks to the inductance elimination. Notably, the frequency at which the box size and out-of-box point value reach a minimum has been changed from 12.5 kHz before correction (Fig. 3b) to 8 kHz after correction. That is, for the test conditions used in this study, 8 kHz was the frequency that provided the most accurate estimates of high temperature.

Fig. 6.

(a) Difference between the corrected estimated temperature (T_fit,corr) and the actual temperature (T_real) over the entire range of test frequencies, (b) box-whisker plot of the temperature estimation deviation (TED) as a function of the test frequency, reproduced from Fig. 6a, (c) the averages of the maximum TED values for all test frequencies before and after inductance correction, and (d) the maximum TED values for the limited frequency range (8–20 kHz) before averaging.

Now, we analyzed in more detail the TEDs before and after correction in Fig. 6c. The averages of the maximum TED values for all test frequencies before and after inductance correction were estimated to be 9.63°C and 6.99°C, respectively. When the frequency range was limited to the high-frequency region, 8–20 kHz, the corresponding values before and after correction became 12.42°C and 1.92°C, respectively. This is the dramatic improvement in the accuracy of temperature estimation (also see Fig. 6d, which shows the data before averaging). Note that the temperature at which the boundary frequency between R₁-CPE₁ and R₂-CPE₂ becomes 20 kHz is 109°C, simply by math. This suggests that the method used in this study can be used to theoretically estimate extremely high cell temperatures with high accuracy. It would be interesting to consider its application in high-temperature all-solid-state batteries in future studies [49–51].

Now, we estimated the temperature of a simplified module connected by 1S3P using the same method as described above, to verify that the improvement in the temperature estimation accuracy by removing the inductance effect is valid even when multiple cells are connected. Fig. 7a depicts the relationship between the estimated temperature before correction (T_fit,mod) and the actual temperature (T_real,mod) of the 1S3P module over the entire range of test frequencies. Fig. 7b presents the difference between the two temperatures in a box-whisker plot. Similar to the previous single-cell results, very large TED values were observed regardless of the frequency. Out-of-box points with very large values were also observed, which were attributed to inaccurate temperature estimation in the high-temperature region. Note that the frequency at which the box size and out-of-box point value reach a minimum was approximately 12.5 kHz for the single cell (Fig. 3b) but decreased to 6.3 kHz for the module (Fig. 7b). This is due to the fact that the high-frequency inductance of the module (1.71 μH, inset in Fig. 7a) is approximately 1.32 times larger than the inductance of a single cell (1.36 μH, inset in Fig. 5).

Fig. 7.

(a) Difference between the estimated temperature (T_fit,mod) and the actual temperature (T_real,mod) of the 1S3P module over the entire test frequency range, and (b) box-whisker plot of the temperature estimation deviation (TED) as a function of the test frequency, reproduced from Fig. 7a.

Fig. 8a presents the differences between the corrected estimated temperature (T_fit,mod,corr) and the actual temperature (T_real,mod) for the module over the entire range of test frequencies. The corresponding TEDs is plotted as a box-whisker plot in Fig. 8b. Compared with the TED before correction |T_fit,mod – T_real,mod| in the high-frequency range (Fig. 7b), the corresponding TED after correction |T_fit,mod – T_{real,mod,corr}| was significantly reduced, particularly at high temperatures. Notably, the frequency at which the box size and out-of-box point value reach a minimum for the module was changed to 8 kHz after correction. This is the same frequency at which the minimum appeared in a single cell, suggesting that the impedance obtained after removing the high-frequency inductance contains only pure cell characteristics. As a result, the highest temperature estimation accuracy is obtained at the same frequency (8 kHz in this study), regardless of whether it is a single cell or a module.

Fig. 8.

(a) Difference between the corrected estimated temperature (T_fit,corr,mod) and the actual temperature (T_real,mod) for the module over the entire range of test frequencies, (b) box-whisker plot of the temperature estimation deviation (TED) as a function of the test frequency, reproduced from Fig. 8a, (c) the averages of the maximum TED values for all test frequencies before and after inductance correction, and (d) the maximum TED values for the limited frequency range (5–12.5 kHz) before averaging.

The averages of the maximum TED values over the entire test frequency before and after correction were 15.81°C and 10.85°C, respectively, for the module (Fig. 8c). When the frequency range was reduced to 5 to 12.5 kHz, where the averages of the maximum TED values were minimized, the corresponding TED values before and after correction became 19.12°C and 3.29°C, respectively (also see Fig. 8d, which shows the data before averaging). In other words, even in situations where the high-frequency inductance increased owing to the connection of multiple cells, the methodology presented in this study could estimate the temperature with a relatively high accuracy.

CONCLUSIONS

In this study, we aimed to improve the accuracy of battery temperature estimation at high temperatures using impedance technique, by identifying and figuring out the problems in conventional temperature estimation. The experimental results are summarized as follows:

(1) Estimating battery temperature over a wide temperature range (–10 to 80°C) from the imaginary impedance at the characteristic frequencies used in previous studies showed that the temperature estimation deviation became very large, especially at high temperatures (> 50°C). Equivalent circuit analysis demonstrated that when the temperature exceeded 25°C, the signal due to the combination of anode charge transfer resistance and double-layer capacitance became dominant at that characteristic frequency. The strong SoC dependence of this signal is believed to be the main reason for the low accuracy of high temperature estimates in previous studies.

(2) When increasing the characteristic frequency to improve the accuracy of the high temperature estimation by allowing only the influence of the anode film layer to exist, the accuracy was actually reduced due to the influence of high-frequency inductance. To address this, the inductance effect was removed over the entire frequency range with the inductance determined from the linear relationship between frequency and imaginary impedance. This resulted in a significant improvement in the estimation accuracy, particularly at high temperatures (average maximum deviation of temperature estimation: 12.42°C before correction vs. 1.92°C after correction (over a narrow frequency range: 8 to 20 kHz)).

(3) An improvement in the accuracy of temperature estimation by increasing the characteristic frequency and eliminating the high-frequency inductance was also observed in the battery module (1S3P). In particular, despite the fact that the inductance, which was about 1.32 times that of a single cell, disturbed the high-frequency signal, the temperature estimation deviation improved significantly after inductance correction (average maximum deviation of temperature estimation: 19.12°C before correction vs. 3.29°C after correction (over a narrow frequency range: 5 to 12.5 kHz)).

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