Online State of Health Estimation with Deep Learning Frameworks Based on Short and Random Battery Charging Data Segments

The battery aging test was performed by the NEWARE testing system on a commercial 18650 li-ion battery cell. Figure 2 shows the schematic of the testing system, and the key parameters of the battery are listed in Table I.

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In this work, eight 18650 cells were employed for the test, and they were sorted evenly into two groups, namely 1C-constant and 3C-WLTC cycles. Each basic cycle consists of four steps: charge, rest, discharge, and rest. The cells were charged using the CC-CV protocol. Firstly, a cell is charged at a constant current of 3 A until the voltage reaches an upper limit of 4.2 V. After that, the cell is supplied with a constant voltage of 4.2 V charging until the current drops to 0.15 A. Following this, the cell undergoes a rest step of 30 min. Finally, the cell is discharged until the battery voltage drops to 2.5 V, which is then followed by a rest period of 30 min. The discharging protocols are different for the two test groups, thus resulting in different aging paths. The 1C-constant cells are discharged at a constant current of 3 A, while the 3C-WLTC cells are set to work at the Worldwide harmonized Light vehicles Test Cycle (WLTC) ³⁵ protocol with a maximum value of 9 A, which mimics complex urban and rural driving conditions including low, medium, high and extra high-speed driving modes. Figures 3a and 3b show the current-time curves for the 1C-constant and 3C-WLTC cycling processes, respectively. To minimize the influence of the ambient temperature, the test room was maintained at a constant temperature of 25 °C by a large ventilation system.

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Basic signals such as voltage ($V$) and current ($I$) can be directly measured by the test system. Battery capacity ($Q$) can be obtained by adding the capacity increment (${\rm{\Delta }}Q$) to the initial capacity (${Q}_{0}$) as shown in Eq. 1. Here, ${\rm{\Delta }}Q$ is the integral of current ($I$) with respect to time ($t$) as shown in Eq. 2:

$$\begin{eqnarray}&&Q={Q}_{0}+{\rm{\Delta }}Q,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}Q=\displaystyle \int I{\rm{dt}}\,.\end{eqnarray} \tag{ 2 }$$

Feed-forward neural network (FNN)

FNN is the earliest and simplest neural network model, which includes an input layer, an output layer and several hidden layers (Fig. 4). ³⁶ The neurons in each hidden layer are fully connected with the neurons in adjacent layers, but no loop is formed. In FNN, information has only one flow direction, that is, forward. The structure of the input data determines the size of the input layer, and the parameters in the hidden layer neurons are all learnable and determined through iterative training. The size of the output layer depends on the problem under study.

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Convolutional neural network (CNN)

CNN is a regularized version of FNN. Unlike the fully connected structure of FNN, the convolution operation in CNN realizes local connection and weight sharing between neurons in adjacent layers. ³⁷ Hence, a CNN model has relatively fewer parameters. Generally, a CNN is composed of convolutional layers, pooling layers and fully connected layers ³⁸ (Fig. 5). The convolutional and pooling layers are used for feature extraction and feature map compression, respectively. Global features of input data can be obtained by performing multi-step convolution and pooling operations. This scheme enables an efficient extraction of main features and a significant reduction in computational complexity. Finally, regression or classification can be achieved through the fully connected layers.

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Long short-term memory neural network (LSTM)

LSTM was developed from recurrent neural network (RNN). Compared with FNN (Fig. 6a), RNN contains an additional memory cell as shown in Fig. 6b. The output of FNN is completely determined by the current input, as formulated in Eq. 3:

$$\begin{eqnarray}&&{y}_{F}\left(t\right)={w}_{F}x\left(t\right)+{b}_{F},\end{eqnarray} \tag{ 3 }$$

where $x$ and ${y}_{F}$ represent the input and output of a neuron, respectively, $t$ represents the present time step, ${w}_{F}$ is the weight parameter, and ${b}_{F}$ is the bias parameter. However, for RNN, the memory cell makes its output not only depend on the input of the current time step, but also on the output of the last time step, as shown in Eqs. 4–6:

$$\begin{eqnarray}&&{y}_{R}\left(t\right)={w}_{R}x\left(t\right)+{b}_{R}+c\left(t-1\right),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&c\left(t\right)={y}_{R}\left(t\right),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{y}_{R}\left(t+1\right)={w}_{R}x\left(t+1\right)+{b}_{R}+c\left(t\right),\end{eqnarray} \tag{ 6 }$$

where ${y}_{R}$ represents the output, $t-1$ and $t+1$ represent the previous and next time steps, respectively, ${w}_{R},$ ${b}_{R}$ and $c$ represent the weight, bias, and memory parameters, respectively. This makes RNN more suitable for processing time-series data. However, the gradient vanishing problem may occur when using traditional RNN and dealing with long-time sequences. ³⁹ To solve this problem, LSTM was proposed as an evolutionary RNN with the simple memory cell further evolving into a memory module. This module contains three basic gates (input, output and forget gates), which control the proportion of information stored in the memory cell and output (Fig. 6c). ⁴⁰

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The forward propagation process of a unit in the LSTM neural network is shown in Eqs. 7–12:

$$\begin{eqnarray}&&{\tilde{c}}_{t}=g\left({w}_{c1}{x}_{t}+{w}_{c2}{h}_{t-1}+{b}_{{\rm{c}}}\right),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{i}_{{\rm{t}}}=f\left({w}_{i1}{x}_{t}+{w}_{i2}{h}_{t-1}+{b}_{{\rm{i}}}\right),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{f}_{{\rm{t}}}=f\left({w}_{f1}{x}_{t}+{w}_{f2}{h}_{t-1}+{b}_{{\rm{f}}}\right),\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{o}_{{\rm{t}}}=f\left({w}_{o1}{x}_{t}+{w}_{o2}{h}_{t-1}+{b}_{{\rm{o}}}\right),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{c}_{t}={i}_{{\rm{t}}}{\tilde{c}}_{t}+{f}_{{\rm{t}}}{c}_{t-1},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{h}_{t}={o}_{{\rm{t}}}h\left({c}_{t}\right),\end{eqnarray} \tag{ 12 }$$

where ${x}_{t}$ is the input of a LSTM unit, ${\tilde{c}}_{t}$ is the generated new memory, ${h}_{t}$ and ${h}_{t-1}$ are the hidden outputs of the unit at the present time step and last time step, ${i}_{{\rm{t}}},$ ${f}_{{\rm{t}}}$ and ${o}_{{\rm{t}}}$ represent the input gate, forget gate, and output gate, respectively, ${c}_{t-1}$ is the previous state of the memory cell, ${c}_{t}$ is the present state of the cell, ${w}_{c1},$ ${w}_{c2},$ ${w}_{i1},$ ${w}_{i2},$ ${w}_{f1},$ ${w}_{f2},$ ${w}_{o1}$ and ${w}_{o2}$ are the weight parameters, ${b}_{{\rm{c}}},$ ${b}_{{\rm{i}}},$ ${b}_{{\rm{f}}}$ and ${b}_{{\rm{o}}}$ are the bias parameters, and $g\left(\,.\right),$ $f\left(\,.\right),$ $h\left(\,.\right)$ represent activation functions.

Implementation details

The battery cycle dataset is collected during the aging test (Figs. 2 and 3). Since the traffic conditions and the driving behavior can differ for each EV, the discharging behavior of an EV can be very complex during real driving scenarios. However, battery charging process in EVs typically follows a standardized protocol, namely CC-CV protocol (Fig. 3), which is more stable and controllable. Hence, the charging dataset is selected as the training set and utilized for online SOH estimation. Figure 7 shows the detailed data processing procedure used for the model training and testing. To obtain the model input, linear interpolation is performed on the original experimental V–Q charging curve to obtain the interpolated curve (Fig. 7a). This enables a constant voltage interval (0.01V) between every two adjacent points in each sequential CC charging data segment. The maximum capacity can be read directly from the V–Q charging curve, which quantitively reflects the current battery aging state (Fig. 7a). In this context, the V-Q charging curve can be divided into two parts: the CC part and CV part (Fig. 7b), respectively, as expressed by Eq. 13:

$$\begin{eqnarray}&&V=\left\{\begin{array}{c}\left[{V}_{lower},{V}_{lower}+{\rm{\Delta }}V,\,\ldots ,\,{V}_{lower}+M{\rm{\Delta }}V\right],Q=Q\left(V\right)\\ {V}_{upper},Q=\left[{Q}_{lower},{Q}_{lower}+{\rm{\Delta }}Q,\,\ldots ,\,{Q}_{lower}+N{\rm{\Delta }}Q\right]\end{array}\right.,\end{eqnarray} \tag{ 13 }$$

where $Q=Q\left(V\right)$ corresponds to the CC charging process, ${V}_{{\rm{lower}}}$ denotes the lower voltage limit, ${V}_{{\rm{upper}}}$ (${V}_{{\rm{lower}}}+M{\rm{\Delta }}V$) represents the upper voltage limit, and ${\rm{\Delta }}V$ is the sampling interval of the voltage. Here, $V={V}_{{\rm{upper}}}$ corresponds to the CV process, ${Q}_{{\rm{lower}}}$ denotes the lower capacity limit, ${Q}_{{\rm{upper}}}$ (${Q}_{{\rm{lower}}}+N{\rm{\Delta }}Q$) represents the upper capacity limit, and ${\rm{\Delta }}Q$ is the sampling interval of the capacity. Finally, the V-Q curve can be encoded into a $2\times \left(M+N+2\right)$ matrix $\left[{V}_{1},{V}_{2},\,\ldots ,\,{V}_{N+1},{V}_{N+2},\,\ldots ,\,{V}_{M+N+2};\right.$ $\left.{Q}_{1},{Q}_{2},\,\ldots ,\,{Q}_{N+1},{Q}_{N+2},\,\ldots ,\,{Q}_{M+N+2}\right].$ Each element in the matrix represents the first value of the corresponding sequential data segment (Fig. 7b).

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The battery charging process in practical EV scenarios is often incomplete, resulting in incomplete charging data segments collected in real time. Specifically, an available segment can be represented as ${X}_{0i}=\left[{V}_{t},{V}_{t+1},\,\ldots ,\,{V}_{t+w};{Q}_{t},{Q}_{t+1},\,\ldots ,\,{Q}_{t+w}\right],$ $\left(1\leqslant w\leqslant M+N,1\leqslant t\leqslant M+N+2-w\right).$ Figure 7b shows the schematic of a data segment which includes both the voltage and capacity sequences. In this work, each segment comprises 30 points of a V-Q charging curve, resulting in 69 segments within a charging cycle. Considering the data loss risk of EVs as well as the convenience and flexibility for data processing, the capacity sequence was converted to $\left[\left({Q}_{t}-{Q}_{t}\right),\left({Q}_{t+1}-{Q}_{t}\right),\ldots ,\left({Q}_{t+w}-{Q}_{t}\right)\right],$ and the segment becomes ${X}_{i}=\left[{V}_{t},{V}_{t+1},\,\ldots ,\,{V}_{t+w};0,{\rm{\Delta }}{Q}_{t+1},\,\ldots ,\,{\rm{\Delta }}{Q}_{t+w}\right].$ This means the input segment can be obtained directly from an arbitrary EV charging period, which is independent of the historical data in BMS.

To improve the model accuracy and speed up training convergence, the input data is usually normalized before being fed into the neural network. In this work, the min-max normalization was adopted to rescale the segments into the range of [0, 1] without disturbing the data distribution, as shown in Eq. 14:

$$\begin{eqnarray}&&{X}_{normi}=\displaystyle \frac{{X}_{i}-\,{\rm{\min }}\left({X}_{t}\right)}{\max \left({X}_{t}\right)-\,{\rm{\min }}\left({X}_{t}\right)},i\in \left\{1,2,\,\ldots ,\,n\right\},\end{eqnarray} \tag{ 14 }$$

where ${X}_{normi}$ is the normalized input, $n$ is the number of samples in the training or test set, ${X}_{t}$ is the collection of ${X}_{i}$ in the training set for all charging cycles, $\min \left({X}_{t}\right)$ and ${\rm{\max }}\left({X}_{t}\right)$ represent the minimum and maximum values of the training data, respectively.

Considering the difference between lab test conditions and EV driving conditions, we employed two basic scenarios to evaluate the performance of the deep learning models, i.e., the direct learning scenario and the transfer learning scenario. For direct learning, the model is trained and tested under complex conditions. For transfer learning, the model is trained under simple conditions while tested under complex ones. The training and test sets for these two scenarios are listed in Table III. Here, 3C-WLTC refers to the battery cells discharged according to the WLTC protocol, but with a maximum discharging current of 9 A (3 C). 1C-constant refers to the battery cells discharged with a constant current of 3 A (1 C). #x represents the serial number of a battery cell. For the training set, 75% of the dataset is selected randomly for model training, while the remaining 25% for model validation.

For battery SOH estimation based on FNN, the estimation error is defined as:

$$\begin{eqnarray}&&Error=\left|\widehat{{\rm{SOH}}}-{\rm{SOH}}\right|,\end{eqnarray} \tag{ 15 }$$

where $\widehat{{\rm{SOH}}}$ and ${\rm{SOH}}$ represent the estimated and real SOH values, as calculated in Eqs. 16 and 17, respectively:

$$\begin{eqnarray}&&\widehat{{\rm{SOH}}}=\displaystyle \frac{{\hat{Q}}_{{\rm{\max }}}}{{Q}_{nom}}\times 100 \% ,\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\rm{SOH}}=\displaystyle \frac{{Q}_{{\rm{\max }}}}{{Q}_{nom}}\times 100 \% ,\end{eqnarray} \tag{ 17 }$$

Here, ${Q}_{nom}$ represents the nominal capacity of a battery cell. ${\hat{Q}}_{{\rm{\max }}}$ and ${Q}_{{\rm{\max }}}$ denote the estimated and real maximum capacity values, respectively.

Figure 8 shows the estimation results of the test set based on the FNN model. The starting voltage (horizontal axis) denotes the position of an input sequential data segment on a charging curve, which indicates the corresponding SOC level of a battery cell. For instance, a larger starting voltage corresponds to an input segment extracted at a higher SOC. The observed SOH (vertical axis) indicates the actual aging state of a cell, which decreases as the test proceeds or the mileage increases. In order to comparatively study the performance of the same model in different scenarios, the color bar range is unified as shown in Figs. 8a and 8b. The relative frequency (Fig. 8c), as an index of the quantitative analysis, represents the proportion of input samples with a certain error level, as shown in Eq. 18:

$$\begin{eqnarray}&&Relative\,Frequency=\displaystyle \frac{a}{A}\times 100 \% ,\end{eqnarray} \tag{ 18 }$$

where $a$ represents the number of samples whose error falls within a certain range, and $A$ is the total number of the test samples.

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For the direct learning scenario (Fig. 8a), the estimation error is relatively small (<1%) for most of the region (94.87%, as shown in Fig. 8c). The maximum error is found to be 4.86%, while it is just limited to a small region as marked by the bright yellow color. It can be clearly seen that larger errors (≥2%) are mainly concentrated in the region where the starting voltage is approximately 3.4 V and the observed SOH is above 85%, accounting for 0.81% of all input samples (Fig. 8c). This indicates that the SOH estimation performance for a relatively new battery cell at a low SOC can be slightly weakened. Besides, there are also some small parts with slightly larger errors (1∼2%) for the starting voltage around 3.6 V or 3.9 V, and SOH above 90%. Overall, the large error can be limited to a small region. This means the FNN model can be expected to give a good SOH estimation with most of the input sequences, especially when the battery life fades below 90%.

When it comes to the transfer learning scenario, the maximum error of the FNN model increases to 5.76% (Fig. 8b). Most of larger errors are still concentrated in the region as that of the direct learning scenario (Fig. 8a). However, this larger-error region is found to be slightly expanded. As shown in Fig. 8c, the proportions of the larger error (≥2%) and the slightly larger error (1∼2%) increase to 1.39% and 9.32%, respectively. Hence, compared with the direct learning scenario, the model performance in the transfer learning scenario deteriorates slightly. This is featured with an increase in maximum error and an expansion of the large-error region.

For SOH estimation based on CNN, the maximum error is 4.06% in the direct learning scenario (Fig. 9a), which is lower than that of the FNN model (Fig. 8a). The estimation error is also relatively small (<1%) for most of the region (94.93%, Fig. 9c). When the starting voltage is approximately 3.4 V, 3.7 V and 4.0 V, corresponding to the low SOC, middle SOC and high SOC, respectively, the CNN estimation performance experiences a striking degradation. Fortunately, these errors are just limited to much small regions, which can be possibly avoided for practical applications.

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As for the CNN performance in the transfer learning scenario, the maximum error increases to 4.25% (Fig. 9b), while it shows a similar pattern to that of the direct learning scenario (Fig. 9a). Meanwhile, compared with the direct learning scenario in terms of the small-error (<1%) region, the transfer learning scenario shrinks to 93.10% (Fig. 9c). This further indicates that the transfer learning can induce a slight increase in error level as well as a certain degradation in estimation capability.

Figure 10 shows the battery SOH estimation results based on LSTM. For the direct learning scenario, the maximum error is found to be 1.53% (Fig. 10a), which is much smaller compared with that of FNN and CNN (Figs. 8a and 9a). The overall estimation error is relatively small (<1%) for almost the whole region (99.23%, Fig. 10c). Specifically, it can be found that the LSTM model performs better for SOH ranging from 82.5% to 95%. This implies that LSTM shows an excellent performance with respect to the direct learning capability.

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In the transfer learning scenario, the maximum error of the LSTM model just increases to 2.19%, and the extent of the larger-error region also varies little (Fig. 10b). Based on the quantitative analysis, the proportion of the samples in the small-error region (< 1%) decreases by 2%, while the proportion for a slightly larger error (1∼2%) increases by 1.98% (Fig. 10c). This indicates that, compared with the direct learning scenario, the transfer learning scenario just induces a slight decay in estimation performance for the LSTM model.

Our above analysis shows that the SOH estimation capability can be closely related to the type of deep learning models. The degradation in estimation capability from direct learning to transfer learning differs for the present three deep learning models. However, a similar estimation error pattern can be found between the two scenarios for each model. The present results show that the LSTM model seems to have the best performance, which is followed by CNN and FNN. This is probably due to the memory cell structure of the LSTM framework which can be beneficial for time-series data analysis. ⁴²

To give a quantitative and direct comparison of the present models, three typical result-based parameters are comparatively studied, i.e., the maximum error, the average error and the relative frequency. The relative frequency is presented by Eq. 18. The maximum error and the average error are calculated by Eqs. 19 and 20:

$$\begin{eqnarray}&&Maximum\,Error=\,{\rm{\max }}\left(Erro{r}_{1},Erro{r}_{2},\ldots ,Erro{r}_{n}\right),\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&Average\,Error=\displaystyle \frac{\displaystyle {\sum }_{i=1}^{n}Erro{r}_{i}}{n},\end{eqnarray} \tag{ 20 }$$

where $Erro{r}_{i}$ represents an estimation error for a sample in the test set, which is calculated by Eq. 15, and $n$ is the number of samples in the test set. Here, the maximum error was used to highlight the worst estimation result of a deep learning model. In the direct learning scenario, the maximum error of the LSTM model is 1.53%, which shows a 2.53% and 3.33% decline as compared to that of CNN and FNN, respectively (Fig. 11a). Meanwhile, in the transfer learning scenario, the maximum error of the LSTM model is 2.19%, which still exhibits a 2.06% and 3.57% decline as compared to that of CNN and FNN, respectively (Fig. 11a). From this perspective, the LSTM model has the lowest maximum error level for both direct and transfer learning scenarios, while the CNN model shows the smallest difference in maximum error between these two scenarios.

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On the other hand, we also used the average error to highlight the general SOH estimation capability of the present models (Fig. 11b). Similar to the maximum error distribution, the LSTM model also shows the lowest average error level among the three deep learning models. For the direct learning scenario, the LSTM model shows the lowest average error of 0.28%, which is followed by the CNN model (0.38%) and FNN model (0.39%). For the transfer learning scenario, the LSTM model shows the lowest average error of 0.30%, which is followed by the CNN model (0.43%) and FNN model (0.48%). This further indicates that the LSTM framework enables more accurate and robust estimation performance in complex operating conditions for batteries.

In addition, we also employed the relative frequency to give a direct comparison with respect to the overall estimation performance as shown in Fig. 12. Obviously, more small-error (<1%) samples can be observed for LSTM compared to CNN and FNN. This results in fewer samples falling within the relatively large error (1∼2% and ≥2%) regions. The detailed information on the relative frsequency is listed in Table IV.

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To further illustrate the detailed pattern of the SOH estimation results, the estimation values are depicted as a function of the reference values in Fig. 13, represented by discrete points. Here, the green line denotes the ideal estimation reference without any errors. The grey region represents the ±1% error band, and the pink region denotes the ±2% error band. For the direct learning of FNN (Fig. 13a), most of the estimation errors can be controlled within the ±2% error band for SOH below 90%. However, a certain part of the results shows an obvious underestimation as SOH further surpasses 90%. As for the transfer learning of FNN (Fig. 13b), the estimation distribution is similar to that of the direct learning. However, the underestimation becomes more striking as SOH exceeds 85%. This means the FNN model tends to cause larger SOH estimation error or uncertainty for new batteries. For CNN (Figs. 13c and 13d), the estimation performance seems more stable compared with FNN (Figs. 13a and 13b). The underestimation is not obvious except when the SOH exceeds 95%. Moreover, the transfer learning of CNN just exhibits slight weakening compared to the direct learning. As for LSTM (Figs. 13e and 13f), it shows an excellent SOH estimation performance during the whole battery life cycle for both the direct and transfer learning scenarios. Most of the estimation errors fall within the ±1% error band, with minor underestimation observed only at the initial and final stages of the battery life cycle. Similarly, the direct learning still shows a minor advantage over the transfer learning especially at the end stage of the battery life cycle.

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Figure 14 presents the relationship between the battery SOH and the number of cycles. The experimental SOH is represented by the black line and points, while the scope of the estimation results for each SOH value are depicted by blue error bars. Similar to Fig. 13, the grey region corresponds to the ±1% error band, and the pink region represents the ±2% error band. Overall, the tested battery cell undergoes 385 cycles throughout its entire lifespan, and the SOH exhibits a degradation trend with fluctuations as the cycle number increases. For the FNN model (Figs. 14a and 14b), the SOH is generally underestimated until approximately 220 cycles, and the maximum estimation error exceeds 2% prior to around 110 cycles. However, once after 220 cycles, the estimation results predominantly fall within the ±1% error band. For CNN (Figs. 14c and 14d), the estimated SOH results exhibit a relatively consistent pattern throughout the battery's lifespan. However, the SOH is generally underestimated for most of the cycles, particularly in the transfer learning scenario. In the case of LSTM (Figs. 14e and 14f), the SOH estimation results are concentrated within the ±1% error band, while the worst estimation results in the transfer learning scenario display an error close to −2% after 280 cycles. In summary, the LSTM model outperforms the FNN and CNN models, and the CNN model shows a more uniform distribution with respect to the estimation error compared to the FNN model. Additionally, all three models experience a slight decline in performance under the transfer learning scenario compared to the direct learning scenario.

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Overall, according to the above comparative study, the LSTM-based deep learning model exhibits outstanding SOH estimation performance in both the simple direct learning and complex transfer learning scenarios. Moreover, it should be noted that, the input segment of the present models is just limited to a small voltage span (0.29 V) during the CC charging process. This means it just requires a short charging data segment to realize the real-time and reasonable SOH estimation. Specifically, the time required for most of the charging data segments will not exceed 20 min (Fig. 15a), especially in the early charging stage which just needs about 10 min to realize the estimation with a high accuracy (Figs. 13f and 15a). Accordingly, the required charging capacity (ΔSOC) for most of the data segments is below 30% (Fig. 15b), especially in the early charging stage which just needs around 13% and 19% ΔSOC to realize the estimation for aged and new batteries, respectively.

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