An efficient and robust method for lithium-ion battery capacity estimation using constant-voltage charging time

The state-of-health (SoH) estimation based on the constant-voltage (CV) charging data has been an interesting research topic in recent years. However, most of the existing estimation methods based on CV charging data are sensitive to the cut-off condition and/or require a relatively high storage resource as well as computing power, preventing the feasibility in real world applications. To extend the scope of the estimation method based on CV charging data, this paper proposes a quick and robust battery capacity estimation method using a two-layer CV charging time ( T CV )-based model. First, the evolution of T CV -based SoH model with respect to different cut-off currents is investigated, and the detailed mathematical expressions of the model coefficients are derived based on the decoupled dynamic characteristics of the CV charging current. Second, considering the actual sampling periods ( T s s) utilized in the online application, a T s -adaptive moving average filter is proposed to filter the high- frequency measurement noise. Third, experimental results demonstrate that the proposed method can determine SoH with a root-mean-square error of less than 2.05% for two types of tested batteries under different charging protocols. In addition, the comparison study further highlights the superiority of the proposed method in terms of robustness, accuracy, computational cost, and storage consumption.


Introduction
Lithium-ion batteries have become a promising battery technology due to the advantages of high energy density, high power, and relatively long cycle life [1,2]. They have been extensively used, ranging from low power consumer electronics [3,4], to high power traction applications [5,6]. For example, in traction applications like electric vehicles (EVs), lithium-ion batteries are widely used as the energy storage system in battery powered or hybrid electric vehicles to help reduce gas emission and fossil fuel consumption. The battery capacity is one of the core parameters to evaluate the battery performance. The actual capacity degrades as the battery cycles, which influences the driving range of the vehicle, and further increases the "range anxiety" [7]. Therefore, it is essential to monitor the battery state-of-health (SoH) related to the energy capability in real-time for the safe and reliable battery utilization [8,9].

Literature review
Extensive research efforts have been focused on the estimation of battery SoH in recent years, which can be roughly categorized into discharging data-based and charging data-based methods in terms of the working conditions.
The discharging data-based methods can be further divided into indirect estimation and direct identification methods. For the first type, the model parameters and the battery capacity or other aging-sensitive parameters are generally combined into a state vector, and identified through a series of adaptive algorithms, such as least squares estimationbased [10], filter-based [11], and observer-based techniques [12]. It should be noted that the performance of the adaptive algorithm strongly depends on the employed model structure and the algorithm parameters. As for the second type, the vehicle driving data (current, voltage, and temperature) recorded by the battery management system (BMS) are directly used or investigated to extract the feature-of-interests (FoIs) reflecting the battery capacity degradation. Then, considering these variables as input features, some machine learning techniques, such as support vector machine, Gaussian process regression, relevance vector machine, and so on, can be utilized to train an SoH model [13][14][15]. Generally, more computational cost is required with the increasing number of input features. With the development of cloud computing in recent years, this kind of methods has great potential to be implemented in EV applications [16].
Unlike the dynamic and random discharging scenario, the charging scenario is relatively simple, stable, and predictable, thus the charging data have been widely adopted for the battery SoH estimation. One of the commonly used charging data-based methods is differential analysis technique, which mainly includes incremental capacity analysis (ICA) [17,18], differential voltage analysis (DVA) [19,20], and differential thermal voltammetry (DTV) [21,22]. Through the differential operation, the plateaus regions on the measured terminal voltage or the surface temperature curve under the long-term constant-current (CC) charging scenario can be transformed to the identifiable peak/valleys on the differential curve. It has been proved that the height, area, and location of the relevant peak/valleys are effective FoIs, and can be employed to establish the SoH estimation model [16]. However, two main disadvantages exist concerning the differential analysis methods. Firstly, this kind of methods is sensitive to the current rates. For example, to reduce the influence of the polarization effect, ICA/DVA methods are usually implemented at the moderate or even low current rates [16,23]. While for the DTV method, a higher current rate is preferred to guarantee observable heat generation. Secondly, the long-term CC charging process is required to obtain the comprehensive peak/valley information. Nevertheless, in practical applications like EVs, the discharging process is strongly dependent on the driving habits and battery is rarely fully discharged, which subsequently influences the initial charging state. In addition, to reduce the charging time and improve the energy transfer efficiency, multistage CC charging protocol, which contains multiple short-term CC charging processes with different current rates, has been widely used in EVs. Under the above working conditions, FoIs cannot be effectively extracted by the differential analysis method. To overcome these limitations, Ref. [24] proposed a deep-learning approach to estimate the entire CC charging curves only based on the small portions of the charging data, and thus the battery capacity can be extracted under the incomplete charging scenario. In order to reduce the influence of current rate on the ICA, Ref. [25] proposed a robust battery SoH prediction method to correct the peak shift through polarization compensation. With respect to the fast charging scenario, Refs. [26,27]  State-of-health capacity degradation. In addition, Ref. [28] introduced the battery terminal voltage at 80% SoC for about 10 min as the FoI, and proposed a long short-term memory-fully connected network to deal with battery inconsistency caused by fast charging. It has been found that the constant-voltage (CV) charging protocol applied after the CC charging process can effectively reduce the electrode polarization and slow down the battery degradation rate [29,30]. In addition, compared to the CC charging process, the CV charging process is more robust to the preceding discharging process and flexible with the initial charging state. Hence, identifying the battery SoH based on CV charging data has been an interesting topic and has attracted wide attention in recent year. The relevant research work has revealed that the charging time [31][32][33], the capacity [34], the time constant [35,36], and the energy [37,38] extracted based on the CV charging data are effective FoIs to characterize the battery aging state. Among them, the CV charging time (T CV ) is a simple and direct FoI because it can be obtained as soon as the CV charging process is complete, and no sophisticated parameter identification procedure is required. However, T CV is sensitive to the cut-off condition. The random and uncertain factors in practical application may compromise the estimation performance. Besides, due to the gradually decreasing current, the CV charging process is time-consuming, especially when the charging current is below a certain value. Hence, some charging strategies perform the CV charging process for a certain duration [39], where the conventional T CV -based SoH estimation method cannot work. Considering the partial CV charging process, the indirect FoIs are generally extracted to estimate the battery SoH. For example, Ref. [35] employed the current time constant as the input of the established SoH estimation model, and developed a logarithmic function-based prediction model to estimate the reference current time constant. In addition, Ref. [34] selected the CV capacity as the FoI, and iteratively incorporated the Q-V modeling with the open-circuit voltage estimation to reconstruct the complete CV phase. The verification results demonstrated the satisfactory SoH estimation performance even under the partial CV charging scenario. It should be noted that for the indirect FoI-based SoH estimation method, the charging data during the CV process should be recorded to identify the model parameters, at the expense of high storage consumption, especially for the battery system with the long-time CV charging process and/or high sampling frequency. Besides, the parameter identification procedure consumes the computing power of the on-board microcontroller.

Motivations and contributions
It can be concluded from the previous study that the existing CV charging data-based SoH estimation methods using indirect FoIs generally suffer from high storage resource and computing power. While for the direct FoI-based method, two main challenges are summarized as follows: (1) Sensitive to the cut-off condition: the conventional T CV -based SoH model is established under a specific cut-off condition, i.e., the constant cut-off current (I cut ). This limits the utilization of this method in real-world applications with random and uncertain I cut s. (2) Limited to the constant sampling period: in the existing study, the sampling periods in the offline identification and online estimation procedures are generally considered as the same value. However, to save memory space for the on-board BMS, a flexible sampling frequency is recently introduced in real world applications [40]. This may reduce the generalization capability of the established SoH model.
Hence, there is still room to improve the CV charging data-based method, in terms of simultaneously reducing complexity and improving robustness. To bridge the aforementioned research gap, a robust and computationally efficient T CV -based SoH estimation method is proposed in this paper. The main contributions are summarized as: (1) Established a two-layer T CV -based SoH model insensitive to the cut-off condition: the evolution of the correlation between the battery capacity and T CV with respect to different I cut s is investigated. Subsequently, based on a second-order exponential model developed in our previous work [41], the coefficients of the T CV -SoH correlation are mathematically characterized as functions of I cut .
(2) Proposed a sampling period-adaptive filter: the influence of the window length on the performance of a moving average filter (MAF) is discussed in the continuous-time domain. Then, based on the sampling periods utilized in the offline test and the online application, an adjustment method of the reserved buffer length is proposed to realize a sampling period-adaptive MAF. (3) Validated the effectiveness under different charging conditions: based on two battery degradation datasets, the estimation performance is systematically evaluated under different CV charging scenarios, including the constant I cut cut-off condition, the constant T CV cut-off condition, and the different sampling periods. In addition, the conventional T CV -based and some other state-of-art estimation methods are employed to conduct a comparative study.

Model evolution
Based on the existing research [32,42], due to the increased resistance and the decreased diffusion at the electrode/electrolyte interface, T CV generally increases with the degrading battery capacity (Cap) for a certain I cut , as schematically shown in Fig. 1(a), where the measured points are extracted from the test data of the employed lithium iron phosphate (LFP) battery. Considering the tendency of the related data, the linear function expressed as (1) is employed to describe the relationship between Cap and T CV , where Cap est denotes the estimated battery capacity, K and B denote the slope and the intercept of the linear function, respectively.
As shown in Fig. 1(a), K can be further expressed as where ΔCap and ΔT CV denote the change of Cap and T CV respectively, Cap new and Cap aged denote the battery capacity in the new and aged states, respectively, T CV,new and T CV,aged denote T CV corresponding to the new-state and aged-state batteries, respectively. It is evident from (2) that for a certain ΔCap, the value of K depends on ΔT CV . The battery CV charging current curves at different aging states are schematically presented in Fig. 1(b). It can be observed that in a certain aging state, there exists a specific one-to-one mapping correlation between the CV charging current and the time instant, thus ΔT CV can be expressed as a variable with respect to I cut , i.e., where f TCV,new (I cut ), f T CV,aged (I cut ), and f ΔTCV (I cut ) denote the functions mapping I cut to T CV,new , T CV,aged , and ΔT CV , respectively. It can be concluded from (1) to (3) that the coefficients of the correlation between Cap and T CV demonstrate diverse values with respect to different I cut s, as exemplarily illustrated in Fig. 2.
Therefore, Eq. (1) can be further expressed as where f K (I cut ) and f B (I cut ) denote the functions mapping I cut to K and B in (1), respectively. Hence, in order to extend the scope of the T CV -based SoH estimation method and improve the robustness of this method under different CV charging conditions, the correlations between Cap and T CV corresponding to different I cut s are required to be established in advance.

Mathematical expression of model
According to the aforementioned analysis, the evolution of K with respect to I cut is closely related to the change of ΔT CV with I cut . Hence, it is critical to establish the correlation between ΔT CV and I cut . Based on our previous work, the charging current under CV scenario can be quantitatively expressed as the sum of two current components with different scales of time constants [41], i.e., where t denotes the CV charging time, and t = 0 denotes the start of the CV charging process, I 1 (t) and I 2 (t) denote the fast-dynamic and the slow-dynamic currents, respectively, τ 1 and τ 2 denote the time constants corresponding to I 1 (t) and I 2 (t), respectively, and τ 1 < τ 2 .
The evolution of I(t), I 1 (t) and I 2 (t) are schematically shown in Fig. 3. It can be seen from Fig. 3 that at the end of the CV charging process, I 1 (t) has converged to zero due to the fast-dynamic characteristic, and I (t) at this stage mainly contains I 2 (t), as Therefore, T CV is mainly determined by the slow-dynamic current component. Based on (6), the expression of T CV with respect to I cut is Specifically, τ 2 and I 2 (0) can be obtained based on the nonlinear least squares method, and the identified parameters are closely related to the predefined I cut , which is equivalent to the size of the dataset. The evolution of identified τ 2 and I 2 (0) corresponding to different I cut s is exemplarily shown in Fig. 4. The fitted linear regression functions are also presented in the figure for a quantitative comparison.
It can be observed from Fig. 4 that both τ 2 and I 2 (0) demonstrate strong dependencies with I cut . In addition, the change magnitude of I 2 (0) is far more less than the change magnitude of τ 2 , thus the value of I 2 (0) can be approximately regarded as constant. Therefore, τ 2 and I 2 (0) can be further expressed as τ 2 = a 1 I cut + a 2 and I 2 (0) ≈ a 3 , respectively.
Based on the aforementioned analysis, the detailed expression of ΔT CV with respect to I cut is  ΔT CV = T CV,aged − T CV,new − τ 2,aged ln where τ 2,new and τ 2,aged denote τ 2 for the new-state and aged-state batteries, respectively, I 2,new (0) and I 2,aged (0) denote I 2 (0) for the new-state and aged-state batteries, respectively, k 1 , k 2 , k 3 , and k 4 denote the function coefficients required to be identified. The above parameters satisfy In addition, for a certain Cap, B in (1) can be further expressed as a function of I cut , i.e., where b 1 , b 2 , b 3 , b 4 , and b 5 are function coefficients required to be identified, and satisfy It can be concluded that with the knowledge of I cut , the correlation between Cap and T CV can be obtained online by using the established function expressed as (8) and (9). Subsequently, the actual battery capacity can be identified by substituting T CV into the obtained correlation.

Sampling period-adaptive moving average filter (MAF)
In order to reduce the influence of the high-frequency measurement noise on the current measurement, the moving average filter (MAF) is utilized in this study due to the advantages of simple realization and low computational cost [43].

Characteristic analysis of MAF
The MAF calculates the output by averaging a series of input data Fig. 4. Evolution of (a) τ 2 and (b) I 2 (0) with respect to I cut (exemplarily with the test data from the adopted LFP battery).
[ − (a 1 I cut + a 2 )ln(I cut ) + (a 1 I cut + a 2 )ln(a 3 )] within a certain window length (T w ), and the relevant expression in continuous-time domain is [44].
where I mea and I avg denote the measured and the filtered current, respectively. Based on (10), Eq. (11) can be obtained after the Laplace transformation.
Substituting s = jω into (11), the transfer function of MAF (H MAF ) is where ω and f denote the radian frequency and the frequency of the input signal, respectively, and ω = 2πf.
The magnitude and phase responses of MAF with different T w s are shown in Fig. 5, and the corresponding cut-off frequencies (f cut s) are marked in Fig. 5(a). As can be seen, the high-frequency components are overall attenuated by the MAF, which is similar to the characteristic of an ideal low-pass filter. Besides, f cut monotonically reduces as T w increases, which will lead to more amplitude attenuation and larger phase lag for the filter output. Hence, the characteristic of the MAF output is closely related to the value of T w .

Discrete-time realization
Based on (10), the discrete-time expression of MAF is [44].
where T s denotes the sampling period, N b denotes the length of the buffer to store I mea , and T w = N b T s . Specifically, the "first-in, first-out" method is utilized to manipulate the current measurements, and the filter algorithm is only conducted at the end of the CV charging process. According to the aforementioned analysis, the performance of the MAF is mainly influenced by T w , which is dependent on N b and T s . In the practical BMS, considering the storage and computation capabilities of the employed microcontroller, T s may be different from that utilized in the offline test [26,40]. If N b in (13) is not adjusted according to the actual T s , the trajectory of the filtered current will demonstrate different dynamic characteristics even in the same aging state, which will deteriorate the battery capacity estimation accuracy. Therefore, to ensure the generality of the established correlation, the actual buffer length in the practical application (N b,on ) should yield to where N b,off is the buffer length utilized in the offline correlation establishment procedure, T w,off and T w,on denote the window lengths used in the offline and online procedures, respectively, T s,off and T s,on denote the offline and online sampling period, respectively.

Framework of the proposed method
The scheme of the proposed battery capacity estimation method is shown in Fig. 6. It mainly includes two parts, i.e., offline identification and online estimation.
The offline identification process is conducted based on the test data of the selected reference battery, and it mainly consists of two layers. In the first layer, the CV charging current is smoothed using the MAF, and the employed N b,off as well as T s,off are recorded for the online estimation. Then, for a specific current range [I low , I up ], the correlations between Cap and T CV corresponding to different I cut s are fitted at a certain current interval ΔI, i.e., K and B in (1) corresponding to I low + kΔI [k = 0, 1, 2, …, (I up -I low )/ΔI] are obtained. In the second layer, k 1 -k 4 in (8) are identified by fitting the relationship between 1/K and I cut , then b 1 -b 5 in (9) are further determined by fitting the relationship between B and I cut . Lastly, the obtained k 1 -k 4 and b 1 -b 5 are stored in the on-board The online estimation process can be summarized as four steps. In step 1, the time instants at the beginning (t 0 ) and the end (t end ) of the CV charging process are recorded. Besides, according to T s,on , N b,on is adjusted based on (14), and the measured current is stored in a buffer during the CV charging process. In step 2, the recorded data is preprocessed when the CV charging process is finished. T CV is derived by subtracting t 0 from t end , i.e., T CV = t end -t 0 . Meanwhile, the filtered I cut is obtained by applying the MAF on the data stored in the buffer. In step 3, according to the filtered I cut , the quantitative correlation between Cap and T CV can be established by using (8) and (9), which have been determined in the offline identification process. In step 4, the actual battery capacity can be obtained by substituting T CV into the correlation established in step 3.

Experimental setup and test procedure
Two groups of lithium-ion batteries, including four 2.5 Ah LFP batteries (numbered from #1 to #4) and three 4.8Ah nickel cobalt aluminum oxide (NCA) batteries (numbered from #5 to #7), are adopted for the test. Specifically, the batteries in each group are selected with the similar characteristics, thus the influence of the battery inconsistency on the estimation result is not considered in this study The tests for the LFP batteries are performed by an 8-channel Arbin BT2000 cycle-based tester, and all three NCA batteries are charged/ discharged by a 16-channel NBT5V20AC16-T battery cycler. All of the tests are conducted at the temperature around 25 • C, and the test data are recorded with the predefined T s of 1 s. The test procedures are presented in Table 1.In this study, all single battery results are based on the test data of batteries #1 and #5 for the LFP and the NCA batteries, respectively.

Analysis of offline identification performance
A. Determination of I cut range It can be concluded from Section 4 that in the offline identification process, the correlation between Cap and T CV should be established in a certain range of I cut , that is, [I low , I up ]. Hence, it is critical to determine an appropriate range of I cut according to the dynamic characteristics of the CV charging current at different aging states. The evolution of the CV charging current throughout the aging process is shown in Fig. 7. As can be seen, for LFP battery, the variation rate of the current curve overall decreases as cycle number increases, which means that T CV corresponding to a certain I cut possesses an increasing trend with respect to the degrading capacity. By contrast, for NCA battery, the time for the   current to decline to approximately 0.74 A are almost identical at different aging states, as shown the point A in Fig. 7(b), indicating that the corresponding T CV cannot be used to reflect the battery capacity loss. The correlation coefficient (r) is utilized to further determine the appropriate range of I cut , and r is expressed as [36].
where x and y represent T CV and the battery capacity, respectively, x‾ and y‾ denote the mean values of x and y, respectively, N d denotes the total number of data points for four LFP or three NCA batteries. The evolution of the absolute values of the correlation coefficient (|r|) versus I cut is shown in Fig. 8. As can be seen, |r| for the tested LFP battery is generally larger than 0.95, and the values progressively approaches 1 with the increasing I cut , that is, Cap demonstrates a stronger linear dependency on T CV with higher I cut . For the employed NCA battery, there is a sudden drop of |r| at around 0.74 A, which is consistent with the trend of the current evolution shown in Fig. 7(b). Besides, although the correlation is significantly improved when I cut is larger than 0.74 A, the corresponding |r|s are still   less than the values in the lower I cut range. This is because within this I cut range, there is no obvious one-to-one mapping correlation between Cap and T CV when Cap degrades less than 4 Ah, as shown in the inset of Fig. 8  , the relationships between I cut and 1/K as well as B for two types of batteries are depicted in Fig. 9, where the solid lines represent the fitted f 1/K (I cut ) and f B (I cut ). The fitted function coefficients are listed in Table 2, and the values of R-square (R 2 ) are also listed in the table to measure the fitting accuracy. It can be concluded from Fig. 9 and Table 2 that the adopted fitting functions can accurately describe the variation trends of 1/K and B.

Analysis of online estimation performance
In order to validate the effectiveness and robustness of the proposed method, three cases, that is, constant I cut cut-off condition, constant T CV cut-off condition, and different T s,on s, as illustrated in Fig. 10, are considered in this section.
A. Case 1: constant I cut cut-off condition In order to evaluate the estimation performance, the scatter plots of Cap est versus Cap with different I cut s are demonstrated in Fig. 11, where the reference line, that is, the solid line, represents the ideal estimation results, that is, Cap est = Cap. The closer the points approach the reference line, the more accurate the estimation results. Specifically, Cap est s are calculated based on the reference regression functions, which are identified based on the test data of batteries #1 (LFP) and #5 (NCA) in this study. As can be seen, the points in Fig. 11 can overall track the  reference line, indicating a satisfactory estimation accuracy for the employed two types of batteries.
In addition, the root-mean-square error (RMSE) between the normalized Cap and Cap est , calculated as (16), is utilized to quantitatively measure the estimation accuracy, where Cap nom denotes the nominal battery capacity, The estimation RMSE with respect to different I cut s for the tested batteries are calculated and plotted in Fig. 12. It can be observed that the calculated RMSEs corresponding to the two tested batteries demonstrate the opposite variation trends with respect to I cut , which can be attributed to the distinct evolution of |r|, as shown in Fig. 8. For example, for the LFP battery, the RMSE overall reduces with an increasing I cut . This is mostly due to the fact that T CV at higher I cut demonstrates a stronger correlation with Cap, resulting in a better estimation performance. Even with the varied values, the overall RMSEs for the respective two batteries are less than 1.35% and 2.05% within a certain range of I cut .
B. Case 2: constant T CV cut-off condition The estimation results under different T CV cut-off conditions for the two tested batteries are exemplarily demonstrated in Fig. 13. As can be seen, for the LFP battery, the estimation results under a lower T CV cut-off condition are much closer to the reference line. While for the NCA battery, the increased cut-off T CV corresponds to a better estimation performance.
The detailed evolution of estimation RMSEs with respect to T CV is  demonstrated in Fig. 14 to further evaluate the estimation performance. It is clear from Fig. 14 that the RMSEs for the LFP and NCA batteries are generally less than 1.45% and 2.05%, respectively, indicating the feasibility of the proposed method under the constant T CV cut-off condition. In addition, as can be seen, the estimation RMSEs for the LFP and NCA batteries demonstrate the increasing and decreasing variation trends versus T CV , respectively, which are consistent with the estimation results shown in Fig. 13. This is because a larger T CV at a certain aging state corresponding to a lower I cut , suggesting a weaker correlation for the LFP battery and a stronger correlation for the NCA battery, as illustrated in Fig. 8. Therefore, in this case, the estimation accuracy is closely related to the predefined T CV , and the estimation performance may significantly deteriorate when the corresponding I cut is outside the appropriate range. C. Case 3: different T s,on s To verify the necessity of the T s -adaptive MAF, the proposed SoH estimation method is implemented based on the test data when T s,on s are set as 1, 5, and 10 s. The length of the filter window is set to 30 in this study. The estimation results with constant and adaptive buffer lengths are graphically represented in Fig. 15, where I cut s are set as 0.30 and 0.35 A for the LFP and NCA batteries, respectively. As can be seen, the influence of T s,on on the estimation results is negligible for the proposed method. By contrast, for the method with the constant buffer length, when T s,on is larger than 1 s, the estimated capacities are generally lower than the actual values, as shown in Fig. 15(a) and (c). In addition, the difference between the reference and the actual values is enlarged with the increasing T s,on , especially for the employed LFP battery.
The evolution of the estimation RMSE versus I cut is illustrated in Fig. 16 to make a comprehensive investigation.
For the LFP battery, only the results in the range of [0.15 A 0.45 A] are presented for a clear comparison. It can be seen from Fig. 16 that based on the T s -adaptive MAF, the produced RMSE curves with different T s,on s almost overlap. While for the estimation results with the constant buffer length, the estimation error corresponding to larger T s,on significantly rises as I cut increases. Especially for the LFP battery, the RMSE corresponding to T s,on = 10 s is larger than 10% when I cut is set as 0.45 A. In order to illustrate the above phenomena, the measured current and the trajectory of the smoothed current based on different smoothing parameters are exemplarily presented in Fig. 17(a), where T s,off and T s,on are set as 1 and 5 s, respectively. Specifically, curves #1 and #2 represent the smoothed current trajectories based on the adaptive and constant buffer length, that is, N b,on,1 = 6 and N on,2 = 30. It can be observed from Fig. 17(a) that curve #1 can track the measured trajectory. By comparison, curve #2 overall lags curve #1, which is consistent with the analysis in Section 3. This means that for a specific I cut , T CV extracted from curve #2 is larger than that extracted from curve #1, leading to the battery capacity estimation lower than the actual value, as illustrated in Fig. 17(b). It should be noted that the difference between T CV,1 and T CV,2 is enlarged in the high I cut region. This causes the rising RMSE with the increasing I cut under the constant buffer length scenario, as shown in Fig. 16(a) and (c).

Comparison with conventional T CV -based method
In this section, the conventional T CV -based method is conducted for a comparative study. Since the current is recorded by the high-precision sensor in the laboratory environment, the additional random noise with a standard deviation of 10 mA is added on the measured current to simulate the measurement noise. The capacity estimation results by the conventional and the proposed T CV -based methods are plotted in Fig. 18, and the corresponding RMSEs are listed in Table 3 to make a quantitative comparison.
It can be observed from Fig. 18 that, compared with the proposed method, the conventional method generally produces the results higher than the actual values, leading to larger estimation errors, as shown in Table 3. This is because is affected by the noise disturbance, and the actual CV charging current cannot reach the predefined cutoff value, which resulting in a lower T CV and a higher capacity estimation, in comparison to the actual value. By contrast, the proposed method can update the coefficients of the reference regression function according to the actual I cut . Hence, the produced estimation results are close to the actual values.

Comparison with different FoIs and offline modeling methods
To further evaluate the performance of the proposed method, three other FoIs proposed in the recent literature and the feedforward neural network (FNN) [45,46] as one of the popular offline modeling methods are employed to make a comprehensive comparison. The employed FoIs include the time constant of the decoupled CV charging current (τ CV ) [36], the energy of the CV charging process (E CV ) [38], and the peak area (PA) under the IC curve, i.e., the interval capacity [18,47], during the CC charging process. Specifically, τ CV s are obtained using the dynamic-decoupled parameter identification method based on CV charging data, as reported in our previous work [36]. E CV during each cycle is calculated as [38].
where V t is the battery terminal voltage. For PA, it can be directly obtained from the voltage charging curve under the CC charging scenario, i.e. [18], where I CC denotes the current under the CC charging scenario, V low and V up denote the lower and upper bounds, respectively. The evolutions of the IC curves of the tested LFP and NCA batteries are plotted in Fig. 19 Tables 4 and 5, respectively, where the required data represent the dataset used for the FoI identification, t model and t FoI represent the computational cost of the offline model construction and the FoI extraction processes, respectively, and both of them are obtained by averaging from 10 runs. The algorithms in this study are implemented on a Lenovo ThinkCentre computer with Intel Core i5-7400 (3-GHz) CPU and 16-GB RAM.
It can be seen from Tables 4 and 5 that all methods can yield the satisfactory estimation results, especially the PA-based method for the LFP battery and the τ CV -based method for the NCA battery. With respect to the computational cost, due to the two-layer offline modeling process, t model of the proposed T CV -based method is generally longer compared with the τ CV -based method. However, the fitting method consumes less time than the FNN method to establish the offline model. In addition, it can be observed from Tables 4 and 5 that t FoI of T CV is approximately    zero, since it is obtained as soon as the CV charging process is finished.
By comparison, the τ CV -based method consumes the most time, which is supposed to be caused by the relatively complex parameter identification process. It is worth noting that although E CV can be immediately acquired when the charge is finished, the integral computation is continuously conducted throughout the CV charging process, as expressed in (17), which means that the entire CV charging data are required. For the PA-based methods, the charging data within the predetermined voltage interval are required, which are 15.43% and 25.14% of the complete CC charging data for the LFP and NCA batteries, respectively. By contrast, the proposed method only requires the data with the length of N b at the end of the CV charging process, indicating the lowest storage consumption. In summary, the proposed method shows an overall satisfactory performance comprehensively considering the estimation accuracy, computational cost, and storage consumption.

Conclusion
In this paper, we develop an online battery SoH estimation method adaptive to the flexible CV charging profile. The coefficients of the conventional T CV -based SoH model are further expressed as the functions of the actual I cut , and a T s -adaptive MAF is proposed to acquire the accurate I cut in real time. Verification results demonstrate that within the appropriate I cut range, the proposed method guarantees satisfactory capacity estimations with the RMSE within 1.45% and 2.05% for the respective LFP and NCA batteries under different CV charging profiles. Benefited from the adaptive buffer length, the estimation accuracy is almost maintained in the case of different T s,on s. In addition, compared with the conventional T CV -based method, the RMSEs of the proposed method are reduced to within 2% for both types of the tested batteries at a certain I cut , considering the noise disturbance, and the comparison among different FoIs and modeling methods further validate the overall superiority of the proposed method in terms of the estimation accuracy, computational cost, as well as storage consumption.
In this paper, the ambient temperature, the charging current in the previous CC process, i.e., the initial value of the CV charging current, and the CV charging voltage are considered as constant throughout the aging process. However, these variables are generally flexible and uncertain in practical applications, such as the fast-charging scenario in real-world EVs, and this will compromise the performance of the established SoH model. Hence, in our future work, the impacts of the aforementioned stress factors will be considered to further improve the robustness of the proposed method.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
Data will be made available on request.