Rapid High-Precision Diagnosis of the Capacity and Internal Resistance of Lithium-Ion Batteries Using Impedance Measurements

We prepared three types of degraded LIBs using the commercial cylindrical LIB NCR18650B manufactured by Panasonic Corporation, with a nominal capacity of 3,350 mAh. The LIB consists of a Li(Ni,Co,Al)O₂ (NCA) cathode and a graphite anode. ²⁰ The first type of LIB was continuously degraded through 80, 200, or 325 repeated charge and discharge cycles between 2.5 V and 4.2 V at 0.5 C and 60 °C (termed the 60 °C-0.5 C cycle). This is because previous studies have shown that LIBs with cyclic aging at high temperatures accelerate SEI growth at the anode. ^15,16 The second type of LIB was degraded by storage at 60 °C for 20 or 215 days, with an SOC of 100% (4.2 V) (termed the 60 °C-aging). This is because LIBs stored at high temperatures and high SOCs have also been reported to accelerate SEI growth at the anode. ⁸ The third type of LIB was degraded via the continuous repetition of 2 C charge and 0.1 C discharge cycles between 2.5 V and 4.2 V at 20 °C (termed the 20 °C-2C cycle). This is because rapid charging at low temperatures has been reported to accelerate Li plating at the anode. ^18,19 Metallic Li was confirmed to be present on the anode of the 20 °C-2C cycle LIBs. Figure S1 in the supplementary material shows a photograph of the anode as a reference (available online at stacks.iop.org/JES/168/090551/mmedia).

The discharge capacity of each LIB was measured under a constant current/constant voltage (CCCV) at 0.1 C and 20 °C using equipment manufactured by Aska Electronic Co., Ltd. As an example, Fig. 2a shows the CCCV discharge curves for each degraded LIB. The capacity was reduced by degrading the LIB in each case. The impedances of the initial LIBs and degraded LIBs were measured under a voltage amplitude of 5 mV from 10^–2 Hz to 10⁵ Hz using a Solartron analytical impedance analyzer CELLTEST-8T. To predict the capacity fade using impedance at any temperature and SOC, AC impedance measurements were conducted between −20 °C and 50 °C at intervals of 10 °C and at SOCs between 0% and 100% at intervals of 10%. The open-circuit voltage (OCV) was also measured as a set. Incidentally, previous studies have also reported that capacity can be estimated using the OCV; ^21–23 however, we used the momentary OCV, not the time dependence of the OCV, as in previous studies. As a result, we prepared a dataset of impedances under 88 measurement conditions (11 SOCs × 8 temperatures) corresponding to the capacity of seven initial LIBs, 21 LIBs degraded through a 20 °C-2C cycle, eight LIBs degraded through 60 °C-aging, and 12 LIBs degraded through a 60 °C-0.5 C cycle (4,220 data points without measurement anomalies).

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Figure 2b shows the Nyquist plots of the impedance spectrum of each LIB at 20 °C and an SOC of 50%. The change in impedance of the LIB degraded through each process differed significantly from that based on other process when the capacity fade was approximately equal in each process. Therefore, we investigated the differences between the three types of degraded LIBs by analyzing the impedance at 20 °C and an SOC of 50% from 10^–2 Hz to 10⁴ Hz using the equivalent circuit model shown in Fig. 3a. Here, CPE (constant phase element) was defined as CPE = 1/((iω)^P T), where P and T are the CPE parameters and ω is the angular frequency. When p equals 1.0, the above equation is typical for the capacitance, where T refers to the capacitance. Figures 3b–3g show the parameters obtained by fitting the impedance spectra. R_S , which is the ohmic resistance, increases with decreasing capacity under each degradation (Fig. 3c). This result is consistent with those reported in previous studies. ^6,8 The R_S of the 60 °C-aged LIBs was higher than that of the other LIBs. This result is slightly different from that reported in previous studies ⁶ owing to differences in the LIB design.

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We then defined a typical internal resistance, R_E , as shown in Fig. 2b, as an indicator of degradation. R_E is related to the sum of the ohmic and charge transfer resistances of the anode and cathode. As R_E largely depends on the temperature and SOC, ^24,25 R_E ^{20 °C,50%} was defined as the R_E at 20 °C and an SOC of 50% to assess the degree of degradation in the used LIBs. The R_E ^{20 °C,50%} for the 60 °C-0.5 C cycle LIBs increased significantly following degradation, where one of the impedance arcs in the Nyquist plot became large. This originates from the increase in R₁ (Fig. 3d). Here, CPE1, parallel to R₁, was an order of magnitude larger T (T₁) than CPE2, as shown in Figs. 3f and 3g. As the cathode generally contains a fine conductive agent, for example, carbon black, the capacity (related to CPE) at the cathode is greater than that at the anode. Furthermore, R₁ was more sensitive to SOC than R₂ . Therefore, we concluded that R₁ is the charge transfer resistance of the cathode and R₂ is the charge transfer resistance of the anode.

The R₁ of the 60 °C-0.5 C cycle LIBs was significantly higher than that of the other LIBs. This was probably caused by cracking and decomposition of the electrode material at the cathode. Although the 60 °C-0.5 C cycle LIBs were presumed to be degraded by SEI growth at the anode, as described in previous reports, ^15,16 remarkable degradation at the cathode was also observed. Similarly, the R_E ^{20 °C,50%} of the 60 °C-aged LIBs (Fig. 2b) also increased as a consequence of degradation; however, this increase mainly originated from an increase in R₂ due to SEI growth during high-temperature storage (Fig. 3e), which agrees with the findings of a previous study. ⁸ Increase in R_S of the 60 °C-aged LIBs is a factor of the increase in the R_E ^{20 °C,50%}. The increase in R_S is probably due to a decrease in electrolyte salt concentration associated with the SEI growth. In contrast, the LIBs subjected to the 20 °C-2C cycle were prepared as LIBs with Li plating at the anode via repeated rapid charge and discharge cycles; therefore, the R_S , R₁, and R₂ values of the 20 °C-2C cycle LIBs changed slightly with reduced capacity. These results clearly show that the change in impedance is not simply correlated with capacity, as various reasons exist for changes in impedance following LIB degradation.

Figure 4 shows a scatter plot of the capacities and R_E ^{20 °C,50%} values of the LIBs degraded through the 20 °C-2C cycle, 60 °C-aging, and 60 °C-0.5 C cycle, which were used for construction of the prediction model. Table S1 in the supplementary material lists the capacities and R_E ^{20 °C,50%} values of each LIB as a reference. The slope of R_E ^{20 °C,50%} with respect to the capacity was steepest for the 60 °C-0.5 C cycle LIBs. Therefore, the power drop is the worst for 60 °C-0.5 C cycle LIBs if the capacity reduction is the same for all three types of degraded LIB. As the data points displaying almost the same capacity fading during identical degradation processes were not dispersed, there were negligible individual differences between the LIBs. Therefore, it is possible but not easy to make predictions of the capacity and R_E ^{20 °C,50%} based on impedance owing to the complex correlations between these parameters.

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We constructed a prediction model for LIB capacity via machine learning using the random forest approach ^26,27 in R statistical computing software. ²⁸ We evaluated the generalization performance of the model using a 10-fold cross-validation (CV). Here, an ntree (the number of trees) of 10,000 and mtry (the number of variable par level) of 8, as hyperparameters, were optimized to achieve the lowest root mean squared error for the predictions with respect to the actual values, RMSE_CV , in the test data. In addition, RMSE_CV indicates the average value for the 10-fold cross-validation, where a low RMSE_CV indicates an accurate prediction of the generic data. The coefficient of determination between the predicted and experimental values, ${{R}_{CV}}^{2},$ was also calculated for reference, where a high ${{R}_{CV}}^{2}$ indicates an accurate prediction. The random forest approach was used to estimate the remaining useful life ²⁹ or impedance of LIBs. ³⁰

We carefully selected the impedance measurement frequencies to achieve fast capacity prediction with high precision. The details of the selection are explained later. Figure 5a and 5b shows Bode plots measured from 10^–2 Hz to 10⁵ Hz with 12 points per decade (full impedance), measured at 10^–1, 10⁰, 10¹, 10², and 10³ Hz (selected impedance). Figures 5c and 5d also shows the RMSE_CV of the predicted capacity and measurement time for each condition in the impedance measurements, respectively. The capacity was predicted using the temperature and OCV, as well as the real and imaginary parts of the impedance (Z' and Z'', respectively) as explanatory variables. As the impedance depends on the SOC and temperature, information on the SOC, temperature, and impedance should be added to the explanatory variables for predictions at temperatures from −20 °C to 50 °C and SOCs from 0% to 100%. However, the SOC could not be obtained unless the capacity was measured; thus, we used the OCV as one of the explanatory variables, which can be measured over a short period and has a strong correlation with the SOC.

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The RMSE_CV using the selected impedance was lower than that obtained using the full impedance (Fig. 5c). The measurement time for the selected frequencies was also significantly shorter than that for the full impedance (Fig. 5d). Here, we discuss why the above-mentioned selected impedance values are preferable for the diagnosis. Table I lists the RMSE_CV , ${{R}_{CV}}^{2},$ and measurement time evaluated under each impedance measurement condition. The results indicate that an appropriate frequency region exists for capacity prediction. Impedance at 10⁴–10⁵ Hz had negative effects on the prediction because the individual difference in impedance at 10⁴–10⁵ Hz is larger than the change in impedance with capacity fading. Furthermore, the dispersion of impedances obtained at 10⁵ Hz for the initial LIBs at 20 °C and an SOC of 50% was 2.0 × 10^–3 Ω², which was one order of magnitude larger than the 2.3 × 10^–4 Ω² obtained at 10³ Hz. The accuracy of the prediction did not decrease by partially thinning the impedance measurement points. As a result, the RMSE_CV using impedances measured at 10^–2, 10^–1, 10⁰, 10¹, 10², and 10³ Hz was the lowest among all considered cases. However, measuring the impedance required approximately "1/frequency" seconds per point; therefore, the impedance measurement time at 10^–2 Hz was approximately 100 s. To satisfy the fast measurement requirements and obtain high-precision predictions, the impedance at 10^–2 Hz should be removed from the explanatory variables for the prediction. The measurement time could be reduced to 47 s by skipping the impedance measurements at 10^–2 Hz. The RMSE_CV without impedance at 10^–2 Hz was comparable to that with impedance at 10^–2 Hz. Therefore, using the impedances at 10^–1, 10⁰, 10¹, 10², and 10³ Hz is optimal for fast predictions with high precision. As a side note, the prediction accuracy of capacity may be heightened by including a specific impedance point as well as 10^–2 Hz in the explanatory variables.

Table I.  RMSE_CV , ${{R}_{CV}}^{2},$ and measurement time evaluated under each condition during impedance measurements.

Impedance frequency (Hz)	RMSECV (mAh)	${{R}_{CV}}^{2}$	Measurement time(s)
10−2 to 105 (85 points)	53.67	0.883	1,170
10−2, 10−1, 100, 101, 102, 103, 104, 105	53.24	0.882	199
10−2, 10−1, 100, 101, 102, 103, 104	47.63	0.906	194
10−2, 10−1, 100, 101, 102, 103	43.60	0.919	189
10−2, 10−1, 100, 101, 102	47.96	0.900	184
10−1, 100, 101, 102, 103	44.01	0.916	47
100, 101, 102, 103	44.55	0.915	33
103	58.78	0.848	15

Figure 6a shows a scatter plot of the predicted values and measured capacities for predictions using the temperature, OCV, and impedances at 10^–1, 10⁰, 10¹, 10², and 10³ Hz as the explanatory variables. Figure 6a also shows the borderlines within which 90% of the data were contained. We then constructed a prediction model in which 90% of the data were contained within an error of ± 29 mAh for a capacity of 2,987–3,432 mAh. The RMSE_CV and ${{R}_{CV}}^{2}$ were 44.01 mAh and 0.916, respectively. Knowledge of the degradation of internal resistance is important when assessing whether to reuse or recycle LIBs. Therefore, we predicted R_E ^{20 °C,50%} as an index of internal resistance degradation, using the same explanatory variable as that for capacity prediction. Figure 6b shows a scatter plot of the predicted values and the measured R_E ^{20 °C,50%}. Ninety percent of the data were contained in the prediction model within an error of ± 0.0016 Ω for R_E ^{20 °C,50%} at 0.0569–0.113 Ω. The RMSE_CV and ${{R}_{CV}}^{2}$ were 0.0033 Ω and 0.961, respectively. It is thought that R_E ^{20 °C,50%} is easy to predict because R_E ^{20 °C,50%} is obtained by impedance. However, the temperature and SOC for the diagnosis are not always 20 °C and 50%, respectively; thus, predicting R_E ^{20 °C,50%} is not simple. Consequently, the capacity and internal resistance can be simultaneously predicted with high precision within 1 min Hereafter, we used the temperature, OCV, and impedances at 10^–1, 10⁰, 10¹, 10², and 10³ Hz as explanatory variables for prediction of the capacity and R_E ^{20 °C,50%}.

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To validate the generalization performance of the prediction model, we prepared additional degraded LIBs using different deterioration test conditions. LIBs were tested via repeated charges of 1 C or 2.5 C, followed by discharges of 0.1 C between 2.5 V and 4.2 V at 20 °C (20 °C-1C cycle or 20 °C-2.5 C cycle). We also prepared LIBs degraded via 60 °C-0.5C-450 cycles to validate the prediction accuracy of the constructed prediction model in the extrapolation area. Table S2 in the supplementary material lists the capacities and R_E ^{20 °C,50%} values of the additional degraded LIBs. Furthermore, we prepared complex-degraded LIBs to validate the prediction accuracy of the prediction model, for example, one LIB was degraded through a 20 °C-2C cycle, followed by 60 °C-aging or a 60 °C-0.5 C cycle. Table II lists the capacities and R_E ^{20 °C,50%} values of the complex-degraded LIBs. Figure 7 shows a scatter plot of the capacities and R_E ^{20 °C,50%} of the additional degraded LIBs and complex-degraded LIBs.

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The data for LIBs degraded via the 20 °C cycle were plotted on the curve shown in Fig. 7, regardless of the C-rate from 1 C to 2.5 C. Therefore, we confirmed the accuracy of the prediction model using two LIBs degraded through a 20 °C-1C cycle and two LIBs degraded through a 20 °C-2.5 C cycle, for which the actual capacity was 3,170–3,217 mAh. Figure 8a shows a contour plot of the average capacity prediction error with respect to the measured capacity values of the four LIBs at each temperature and SOC, ranging from −20 °C to 50 °C and 0%–100%, respectively. Most of the capacity prediction error for the four LIBs in the full range of temperatures and SOCs (Fig. 8a) was comparable to an RMSE_CV of 44.01 mAh for the constructed prediction model. In particular, high-precision prediction occurred at approximately 0 °C. The reason for this is not clear but is currently under consideration. The difference in the C-rate slightly affected the capacity prediction error under the degraded condition, that is, a C-rate from 1 C to 2.5 C at 20 °C, because the prediction error for the 20 °C-2.5 C cycle LIBs was similar to that of the 20 °C-1C cycle LIBs. Figure 8b shows a contour plot of the average R_E ^{20 °C,50%} prediction error with respect to the measured values of the four LIBs at each temperature and SOC, ranging from −20 °C to 50 °C and 0%–100%, respectively. The overall prediction error for R_E ^{20 °C,50%} was small. Moreover, high-precision prediction was achieved at approximately 0 °C. The difference in the C-rate from 1 C to 2.5 C slightly affected the prediction error for R_E ^{20 °C,50%}, which was identical to the results of the capacity prediction.

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Although predicting a value in an extrapolation area is usually difficult, we were able to confirm the prediction accuracy of the constructed prediction model in the extrapolation area. We prepared the LIBs degraded by the 60°C-0.5C-450 cycle, which lay in the extrapolation area of the data set (80–325 cycles) used for the prediction model. Figure 9a shows a contour plot of the average capacity prediction error with respect to the measured value of the two LIBs degraded through the 60 °C-0.5C-450 cycle (Table SII). Under conditions of −20 °C to 50 °C and SOCs from 0% to 100%, the predicted capacities of the LIBs degraded through the 60 °C-0.5C-450 cycle were ≥90 mAh higher than the actual values of approximately 2,910 mAh, although the prediction model was constructed with an RMSE_CV of 44.01 mAh. This yielded a lower limit for the predicted capacity, which was restricted by actual values of approximately 3,000 mAh for LIBs degraded through the 60 °C-0.5C-325 cycle.

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Therefore, we reconstructed the capacity prediction model by adding data extrapolated from the measured impedances to the dataset above. The extrapolated curve was defined as y = ax² + bx + c (y: Z' or Z''; x: capacity), with the addition of extrapolated data at 2,500, 2,600, 2,700, 2,800, and 2,900 mAh. Figures 9b and 9c shows the capacity dependence of Z' and Z'' at 20 °C and an SOC of 50% for the 60 °C-0.5 C cycle LIBs at each frequency. Incidentally, the capacities from our data could not be fitted perfectly using a linear regression of the impedance, as was the case in a previous study. ⁶ Data denoted by the red squares in Figs. 9b and 9c indicate the added data at a temperature and SOC of 20 °C and 50%, respectively. Figure 9d shows a scatter plot of the predicted values and measured capacities when extrapolated data were added to the dataset to construct the prediction model. Figure 9d also shows the borderlines, which contain 90% of the data. The prediction accuracy for capacity was slightly affected by additional extrapolated data. Figure 9e shows a contour plot of the average prediction error for capacity with respect to the measured value of two LIBs degraded through the 60 °C-0.5C-450 cycle with the addition of the extrapolated data. The prediction accuracy improved significantly from Figs. 9a to 9e by using the reconstructed model with the addition of extrapolated data. An extrapolated area always exists unless degraded LIBs with a capacity of 0 mAh are prepared. Therefore, to the greatest extent possible, extrapolated data should be added to the dataset during construction of the prediction model to improve the generalization performance. However, we note that the choice of an extrapolated curve is important for improving the prediction, although y = ax² + bx + c was used for the extrapolated curve in this study.

As LIBs that have been degraded through multiple processes are generally more common in LIB markets than those degraded through only one process, we validated the prediction accuracy of the prediction model using data from complex-degraded LIBs. The arrows for the complex-degraded LIBs in Fig. 7 indicate the changes that occurred between the first and the second degradation process. The change in capacity and R_E ^{20 °C,50%} of batteries B-1 or E-1 can be explained by a simple combination of the 20 °C-2C cycle and 60 °C-0.5 C cycle. The change in capacity and R_E ^{20 °C,50%} of battery A-1 can also be explained by a simple combination of 60 °C-aging and the 20 °C-2C cycle. In contrast, the change in capacity and R_E ^{20 °C,50%} of batteries C-1 and D-1 cannot be explained by a simple combination of the first and second processes. In batteries A-2 and B-2, the capacity did not decrease as a result of the second degradation because the total quantity of the 2 C charge was low owing to the high resistance of the LIBs after the first degradation. As mentioned above, a variety of degraded LIBs can be prepared to validate the generalization performance of the constructed prediction model.

Table II lists the prediction error for the measured capacity (ΔQ) and R_E ^{20 °C,50%} (ΔR_E ^{20 °C,50%}) of each complex-degraded LIB using impedances at 20 °C and an SOC of 50%. Most of the error in the complex-degraded LIBs was comparable to an RMSE_CV of 44.01 mAh for the constructed prediction model. The prediction accuracy for each complex-degraded LIB in Table II was comparable to that at any condition from −20 °C to 50 °C and SOCs from 0% to 100%. Comparatively, the error for battery C-2 was slightly large due to its location in the extrapolation area; however, the error could be reduced to 19 mAh from 81 mAh using the above-mentioned reconstructed prediction model with the addition of extrapolated data. The errors for batteries A-1 and E-2 were significantly larger than those for the other complex-degraded LIBs. One of the reasons is that these batteries were also located in the extrapolation area. The error for batteries A-1 and E-2 could be reduced to 172 mAh and 141 mAh, respectively, when the capacity prediction model was reconstructed with additional data extrapolated from the measured impedances of 20 °C-2C cycle LIBs, 60 °C-aged LIBs, and 60 °C-0.5 C cycle LIBs (the extrapolated curve was defined as y = ax² + bx + c). However, as the errors remained large even when extrapolated data were added to the dataset, the complex degradation effect cannot be ignored in severely degraded LIBs. To accurately predict the resistances of complex-degraded LIBs, the impedance data of these LIBs should be added to the data set for the prediction model in advance. Otherwise, impedance data for complex-degraded LIBs are created as a simple combination of the degradation processes used for model construction. Furthermore, regarding the R_E ^{20 °C,50%} of the complex-degraded LIBs, with the exception of B-2 and C-2, which lay within the extrapolation area, all values were comparable to an RMSE_CV of 0.0033 Ω for the constructed prediction model.

We show the simultaneous prediction of the capacity and typical resistance of NCA/graphite LIBs with high accuracy in a short period of time. This diagnostic method using selected impedances is probably applicable to LIBs with a graphite anode and a Li(Ni,Mn,Co)O₂ (NMC) cathode ^6,8,9,17 because main degradation causes of the LIBs are the same as those of NCA/graphite LIBs (SEI growth at the anode, Li plating at the anode, electrode material cracking and decomposition at the cathode). However, the correlation between capacity and impedance of LiCoO₂ (LCO)/graphite ⁷ or LiFePO₄ (LFP)/graphite ^2,16,18 LIBs may be a little bit different from that of NCA/graphite LIBs because the contribution of the transition metal dissolution to capacity fade cannot be ignored in LCO/graphite or LFP/graphite LIBs. Note that a dataset for prediction should be prepared in each battery type as the capacities and impedances depends on battery types. The diagnosis for the LIBs as mentioned above will be validated in future work and the diagnostic method using selected impedances will be applied to various batteries such as lithium-sulfur batteries ^20,31 and sodium-sulfur batteries. ³²

To develop a nondestructive and rapid method of diagnosing degradation in used LIBs with high precision at any temperature or SOC, we attempted to simultaneously predict the capacity and internal resistance via machine learning using the impedance, temperature, and OCV. First, we prepared three types of degraded LIBs using 18650-type cylindrical LIBs, then developed a dataset by measuring the discharge capacity at 0.1 C as well as the impedance and OCV at each temperature and SOC, which ranged from −20 °C to 50 °C and 0%–100%, respectively. To evaluate degradation of the internal resistance, a typical internal resistance (R_E ^{20 °C,50%}) was defined using the impedance at 20 °C and an SOC of 50%. Second, we trained the capacity or R_E ^{20 °C,50%} using impedance, temperature, and OCV via machine learning. Using the OCV, temperature, and impedances at 10⁻¹, 10⁰, 10¹, 10², and 10³ Hz as explanatory variables, we constructed a prediction model in which 90% of the data lay within an error of ±25 mAh for a discharge capacity of 2,987–3,432 mAh for all three types of degraded LIBs. The prediction model was able to estimate the capacity after less than 1 min of impedance measurement. Furthermore, the R_E ^{20 °C,50%} could be predicted with high precision using the same explanatory variables. Third, we prepared different types of degraded LIBs and validated the generalization performance of the prediction model. This model can be applied in most cases, but is inappropriate for LIB data in the extrapolation area. To solve this problem, extrapolated curves were created using the obtained data; the resulting data can be added to the dataset to reconstruct the prediction model. The practical diagnostic method proposed in this study promotes the reuse of used LIBs and greatly contributes to constructing a battery circulation economy.

We would like to thank Dr. Sasaki at Toyota Central R&D Laboratories for valuable discussions, and Dr. Nozaki at Toyota Central R&D Laboratories for sample preparation support. We would also like to thank Editage (www.editage.com) for English language editing. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.