State of Health Estimations for Lithium-Ion Batteries Based on MSCNN

Abstract

Lithium-ion batteries, essential components in new energy vehicles and energy storage stations, play a crucial role in health-status investigation and ensuring safe operation. To address challenges such as limited estimation accuracy and a weak generalization ability in conventional battery state of health (SOH) estimation methods, this study presents an integrated approach for SOH estimation that incorporates multiple health indicators and utilizes the multi-scale convolutional neural network (MSCNN) model. Initially, the aging characteristics of the battery are comprehensively analyzed, and then the health indicators are extracted from the charging data, including the temperature, time, current, voltage, etc., and the statistical transformation is performed. Subsequently, Pearson’s method is employed to analyze the correlation between these health indicators and identify those with strong correlations. A regression-prediction model based on the MSCNN model is then developed for estimating battery SOH. Finally, validation using a publicly available lithium-ion battery dataset demonstrates that, under similar operating conditions, the mean absolute error (MAE) for SOH estimation is less than 0.67%, the mean absolute percentage error (MAPE) is less than 0.37%, and the root mean square error (RMSE) is less than 0.74%. The MSCNN has good generalization for datasets with different working conditions.

1. Introduction

The contemporary world is confronted with significant challenges, such as the global energy crisis and environmental pollution. This situation has led major economies to actively pursue energy-saving and emission-reduction policies, achieving broad consensus. Among these strategies, developing electrified transportation and new power systems emerges as an effective means to achieve sustainable development. In 2020, the Chinese government introduced ambitious targets for carbon neutrality and peak carbon emissions, identifying the development of electrified transportation as a crucial pathway towards realizing “dual carbon” goals. This approach not only effectively reduces fuel consumption and carbon emissions but also contributes to improving urban air quality and mitigating noise pollution [1]. To support the advancement of electrified transportation, numerous governments have implemented various incentive policies. For instance, China has rolled out multiple rounds of subsidies for electric vehicle purchases aimed at reducing acquisition costs [2]. The federal and state governments in the United States have allocated funds and established tax incentive programs to bolster both public and private sector infrastructure for electric vehicle charging. At the same time, this new type of power system represents a critical component within modern energy systems that is characterized by cleanliness, low-carbon attributes, safety, efficiency, and rapid clean energy. Electrochemical energy storage has become an important part of the new power system and one of the fastest growing energy-storage technologies due to its fast response time, high energy density, and flexible structure. Lithium-ion batteries are particularly pivotal within this domain, owing to their high energy density, low self-discharge rates, high cycle efficiencies, long service life, and flexibility in configuration, making them indispensable in electrochemical storage technology [3].

Lithium-ion batteries have been widely adopted for use in both the field of electrified transportation and energy storage facilities, serving as the primary energy storage devices for electric vehicles and energy storage facilities, respectively [4]. Both of these represent a critical driver of continued growth in the global sales of lithium-ion batteries. It is forecast that, by 2025, the global installed capacity of lithium-ion batteries will reach 800 GWh, with an estimated market value of approximately $100 billion [5]. Due to their irreversible electrochemical nature, continuous charging-discharging cycles lead to the gradual degradation of their output performance, operational conditions, mechanical stress, and chemical reactions [6]. In order to guarantee the long-term safe and stable operation of the battery-management system and the regular maintenance of essential lithium-ion battery-based energy storage systems (ESSs), the state of health (SOH) serves as a critical indicator of system safety and stability, defined as the ratio of the current maximum capacity to the initial capacity, providing a pivotal metric for assessing battery degradation [7]. When the SOH drops to a certain level, it signifies that the battery no longer meets the requirements for operating new EVs and can consequently be repurposed for utilization in areas such as ES stations [8,9].

Recent years, researchers have made significant strides in studying battery SOH and have proposed various effective methods. Typically, direct measurement is unfeasible; however, the capacity can be derived by integrating the discharge curve from high cut-off voltage to low cut-off voltage. Nevertheless, obtaining a comprehensive charge–discharge curve is challenging in practical applications due to infrequent full charge-discharge cycles, rendering the method suitable primarily for laboratory conditions. Methods estimating health status generally fall into two categories: model-based approaches and data-driven methods [10].

Model-based methodologies employ mathematical formulations to represent the health status of batteries and subsequently predict the battery’s aging trajectory. These model-based approaches primarily encompass electrochemical models, equivalent circuit models (ECM), and empirical models [11]. Electrochemical models are rooted in first-principles modeling and adeptly capture the electrochemical reactions taking place within lithium-ion batteries. The model parameters possess explicit physical significance, establishing a clear coupling relationship between the internal state of the battery and its external characteristics. Wang et al. [12] utilized pseudospectral methods to solve solid-phase diffusion equations while simplifying liquid-phase concentration equations using Galerkin methods. They employed a particle swarm optimization algorithm for identifying 11 parameters within an electrochemical model and estimated the state of charge through a particle filter. In another study, researchers investigated how battery SOH affects ECM parameters [13], observing that, as the SOH diminishes, the ohmic resistance and polarization resistance increase, while the polarization capacitance decreases. An empirical model was proposed to delineate the impacts of the SOH, state of charge (SOC), and temperature on ECM parameters. Furthermore, Zhang et al. [14] proposed a multi-domain parameter-identification method for fractional-order models of lithium-ion batteries which involves determining 25 identification parameters and introduces a multi-domain identification approach based on frequency domain analysis using electrochemical impedance spectroscopy and time domain assessment through dynamic stress test terminal voltage measurements. This multi-domain parameter-identification method yielded more precise results compared to traditional frequency domain or time domain approaches.

Out from model-based approaches, data-driven methods extract features from various data sources such as the voltage, current, temperature, and capacity to investigate the interrelationship between battery performance degradation and key features. This approach eliminates the necessity for a detailed understanding of the intricate operational mechanisms and chemical changes within batteries while also not depending on prior knowledge in electrochemistry. By constructing a nonlinear degradation model for battery health state and employing diverse algorithms for data analysis and processing, precise estimations of lithium-ion batteries’ SOH can be achieved with broad applicability. Data-driven methodologies encompass AI techniques, statistical methods, and signal processing [15], with Gaussian process regression (GPR) being a commonly utilized method within artificial intelligence-based approaches [16]. Additionally, the literature has proposed deep Gaussian process regression (DGPR) based on Gaussian processes and deep neural networks for lithium-ion battery SOH assessment [17]. Furthermore, a data-driven prognostics method based on enhanced long short-term memory (LSTM), where the network topology is estimated using the particle swarm optimization (PSO) algorithm, has been introduced [18]. Other notable methodologies include the relevance vector machine [19] and support vector regression methods [20]. In statistical-based approaches, the literature has introduced autoregressive moving average (ARMA) time-series models alongside proposing extended models for SOH estimation. The predictive performance was compared using ARIMA models along with seasonal ARIMA and ARIMAX models [21]. Recent years have witnessed signal processing-based techniques being employed to extract crucial information from raw data for fault diagnosis and estimation. For instance, the variational mode decomposition-replacement entropy (VMD-PE) method was introduced which utilizes a time convolutional neural network (TCN). Furthermore, the discrete wavelet transform (DWT) function was applied to reduce noise during the preprocessing stage [22]. Moreover, a feature-selection algorithm based on maximum correlation-minimal redundancy normalized mutual information (MI) was used to select optimized health indicators; a bidirectional long short-term memory (Bi-LSTM) neural network was leveraged for SOH prognostics [23].

The current study selected a limited set of dimensions related to health indicators with the aim of ensuring a comprehensive exploration and accurate assessment of battery health. However, the complex interdependencies between these input variables may pose a challenge to traditional machine learning techniques. CNNs are more suitable to addressing this challenge, as they are able to model the spatial and temporal complexity of such input data. As a result, CNN-based methods can achieve higher accuracy in SOH estimation, thereby optimizing the performance and lifetime of Li-ion batteries. CNNs are well-suited to managing the spatial and temporal complexity of input data characterizing the state of the battery. Specifically, CNNs can discover voltage and current waveform patterns associated with specific degradation mechanisms such as electrode cracking or capacity decay. On top of CNNs, the MSCNN enhances the model’s ability to process data features at different time scales or different frequencies using multiple convolutional layers with different kernel sizes. And this structural optimization enhances the ability of the model in dealing with multi-scale features. Compared with other neural networks, LSTM mainly focuses on the dynamics of time series, and is weak in local feature extraction in battery data. In contrast, the MSCNN can handle multi-scale features simultaneously and is more effective in capturing both local and global features of the battery state. LSTM is less flexible than the MSCNN in handling multi-scale features. The health state of Li-ion batteries is affected by multiple factors involving different time scales, and the MSCNN is able to capture these features through convolutional kernels at different scales, whereas LSTM mainly relies on the sequential relationship of time-series data, and may not be able to fully utilize the information at all scales. Therefore, the MSCNN is able to have more accurate prediction in estimating the health status of lithium-ion batteries.

In this study, we present a multi-scale convolutional neural network (MSCNN) model for evaluating the health status of lithium-ion batteries across their entire lifecycle using charging data from a publicly available dataset. Initially, we analyze the charging data of lithium-ion batteries to uncover underlying health indicators associated with battery aging. Subsequently, through mathematical calculations, we identify key health indicators and conduct correlation analyses to determine those strongly correlated with battery aging. We then utilize the MSCNN model to forecast the health status of lithium-ion batteries. Finally, we assess our model’s accuracy and generalizability by predicting the battery SOH under various operating conditions and datasets. The experimental results demonstrate that our proposed approach accurately predicts the health status of lithium-ion batteries. The technical framework for our proposed approach is depicted in Figure 1.

In contrast to conventional methods for predicting SOH, this study introduces three novel contributions:

(1)

By statistically analyzing the aging test data of battery voltage, current, and temperature, eight health indicators with strong correlation with battery capacity are mined;

(2)

Compared with CNNs, the MSCNN can effectively extract features from data of different scales, and its estimation accuracy is improved by more than 26%;

(3)

The MSCNN has good generalization for datasets with different working conditions.

2. Battery Dataset

The lithium-ion battery aging data utilized in this study are sourced from the publicly available battery dataset at Xi’an Jiaotong University [24]. These lithium-ion batteries, manufactured by LISHEN (Tianjin, China), have a nominal capacity of 2000 mAh and a nominal voltage of 3.6 V. They have a charging cut-off voltage of 4.2 V and a discharge cut-off voltage of 2.5 V. For analysis, all batteries from batch one and every other odd-numbered battery out of the 15 in batch two are selected. In the original dataset, the eight batteries from batch one are denoted as Group A1 to A8, while the odd-numbered batteries from batch two are denoted as Group B1 to B8. The parameters for the charging and discharging conditions can be found in

Table 1. Charging and discharging conditions for datasets A and B.

Group ID	Charge Condition	Discharge Condition
Group A	CC (4 A)-CV (4.2 V)	CC (2 A) discharge to 2.5 V
Group B	CC (6 A)-CV (4.2 V)	CC (2 A) discharge to 2.5 V

.

Under the pre-set operating conditions, the battery’s aging cycle test, following the charging and discharging strategy, consists of two stages: nominal capacity and aging cycle. The battery charging and discharging conditions are illustrated in Figure 2. During the nominal capacity stage, the initial capacity is first measured, followed by charging at a constant current of 1 A to a cut-off voltage of 4.2 V, then transitioning to constant-voltage charging until the current decreases to 0.04 A. After a 5 min rest period, discharging begins at a constant current of 0.4 A until a lower cut-off voltage of 2.5 V is reached, completing the constant capacity stage. In the aging cycle test stage, charging begins with a constant current of 4 A until it reaches an upper cut-off voltage of 4.2 V, followed by continuous charging at a fixed voltage until the current decreases to 0.1 A. After a resting period of five minutes, discharging starts with a constant current of 2 A until it reaches a lower cut-off voltage of 2.5 V and then remains idle for an additional five minutes before completing one full charge-discharge aging cycle test. For Battery Group B, the constant capacity strategy aligns with that used for Group A; however, during the aging cycle test stage for the Group B batteries, the initial charging occurs at a consistent flow rate of 6 A up to an upper cut-off voltage of 4.2 V before transitioning into continuous charging at fixed voltage until meeting the below-mentioned conditions; subsequently, the test follows similar discharge procedures as described earlier. The aforementioned charge-discharge strategies are employed until the battery capacity diminishes to below 80% (i.e., less than 1600 mAh), signaling the completion or termination point for the aging tests.

The degradation curves of the batteries in Group A and Group B are illustrated in Figure 3. It is evident from the graph that the rate of capacity decay for Group A batteries is comparatively slower than that of Group B, which is attributed to disparities in their charging conditions. The cycle life distribution for the Group A batteries is concentrated around 400 cycles, while Group B exhibits three distinct scenarios, with cycle lives distributed around 150 cycles (two batteries), 230 cycles (one battery), and 300 cycles (five batteries), respectively. The primary reason for the lower cycle life of the Group B batteries can be attributed to their utilization of a fast-charging/fast-discharging operational mode. Additionally, they are subject to external mechanical stress and internal chemical reactions, among other factors. These atypical data complicate SOH estimation for batteries and imposes higher demands on the generalization capability and accuracy of battery SOH estimation models. Further analysis reveals an early-stage phenomenon of capacity regeneration within the battery’s decay curve during cycling. Researchers such as Jia Guo et al., from Aalborg University [25], have investigated this initial-cycle capacity-increase phenomenon observed in lithium-ion batteries. This occurrence originates from the graphite anode. Under significant discharge depth, increased interlayer spacing within the graphite anode promotes lithium-ion diffusion, consequently enhancing the battery capacity.

3. Feature Engineering Approach

In this section, the publicly available lithium-ion battery dataset is initially obtained and a comprehensive analysis of the battery aging data features is performed. Subsequently, mathematical analysis and calculations are utilized to derive relevant health indicators, followed by a Pearson correlation analysis conducted between these indicators and capacity. Ultimately, the health indicators that exhibit stronger correlations with capacity are selected as inputs for the health-state estimation model.

3.1. Extraction of Health Indicators

The quality of the data is a crucial aspect in data-driven methodologies, and the selection of health indicators directly influences the feasibility and accuracy of estimating battery health status. While the direct measurement of battery capacity and internal resistance presents challenges, parameters such as the voltage, current, temperature, and time during the charge-discharge process in lithium-ion batteries are readily detectable. These fundamental physical quantities can be employed in health-state prognostic models to achieve precise prognosis of the SOH for batteries. With an increase in the number of charging cycles, subtle changes occur in the charging voltage–current curves due to a gradual increase in internal resistance [10]. This heightened resistance results in greater heat loss during charge-discharge processes, subsequently impacting the stability of the charging current and voltage [26]. Furthermore, as batteries age, there is a gradual evolution in both physical and chemical properties within the electrode materials which affects the charge transport rates and voltage response [27]. Analysis reveals that critical indicators closely associated with battery aging are embedded within voltage–current cycling curves. In this study, a constant current charging time and constant voltage charging time are initially selected as temporal domain health indicators. Subsequently, a comprehensive analysis is performed on the data obtained from various stages, including the constant current stage and the constant voltage stage. Parameters such as the current flow rate (A), terminal voltages (V), temperatures (°C), etc., are taken into consideration. By segmenting data points at regular intervals for each cycle across voltage–current–temperature profiles, 128 sampling points are acquired. As the cycles progress further, aging-related shifts become apparent; charging voltages exhibit leftward shifts while currents display rightward shifts, respectively, as depicted in Figure 4 and Figure 5.

To further explore the aging information concealed within these electrical characteristics, mathematical analyses were conducted on the datasets, encompassing calculations of the mean values, standard deviations, skewness, and kurtosis for the voltages, currents, and temperatures throughout the charging phase [28].

• Mean: In the realm of data processing and statistical analysis, the mean serves as a pivotal metric for denoting the central tendency or position within a dataset.

  $$\overline{x}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}x(i)}$$

  (1)

  where x(i) represents the numerical values of the voltage, current, and temperature at the time of sampling.
• Standard Deviation: This plays a crucial role in data processing and analysis by aiding our understanding of data dispersion or variability. A smaller standard deviation indicates that data points are closer to the mean, while a larger standard deviation suggests a wider distribution of data points. A smaller standard deviation typically signifies more reliable and stable data, as the data points are more concentrated around the mean. This is particularly important for scenarios such as quality control and assessing the reliability of experimental results.

  $$\sigma ={\displaystyle \frac{1}{n}}\sqrt{{\displaystyle \frac{1}{n-1}}{\displaystyle \sum _{i=1}^n{(x(i)-\overline{x})}^2}}$$

  (2)

  where x(i) represents the numerical values of the voltage, current, and temperature at the time of sampling. $\overline{x}$ represents the mean values of the voltage, current, and temperature.
• Skewness: This is a critical statistical measure, especially in skewed distributions where the mean may no longer accurately represent the central tendency due to the influence of skewness.

  $$skewness={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(x(i)-\overline{x})}^3}}{(n-1){\sigma }^3}}$$

  (3)

  where x(i) represents the numerical values of the voltage, current, and temperature at the time of sampling. $\overline{x}$ represents the mean values of the voltage, current, and temperature. $\sigma$ represents the standard deviation values of the voltage, current, and temperature.
• Kurtosis: This statistical measure characterizes the peakedness or flatness of a data distribution.

  $$kurtosis={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(x(i)-\overline{x})}^4}}{(n-1){\sigma }^4}}$$

  (4)

  where x(i) represents the numerical values of the voltage, current, and temperature at the time of sampling. $\overline{x}$ represents the mean values of the voltage, current, and temperature. $\sigma$ represents the standard deviation values of the voltage, current, and temperature.

3.2. Correlation Analysis

Through statistical analysis and calculations of data such as the current, voltage, and temperature, 14 health indicators associated with the condition of a lithium-ion battery’s health can be derived. However, not all of these health indicators demonstrate strong correlations with the battery’s health status. To enhance model accuracy, redundant data are removed to improve the computational efficiency. This study utilizes the Pearson correlation coefficient (r) for assessment, which ranges from −1 to 1. The proximity of its absolute value to 1 indicates the strength of the correlation; approaching 0 suggests a lack of correlation between the health indicator and the battery’s SOH. The calculation formula for Pearson correlation is presented in Equation (5).

$$r={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})}}{\sqrt{{\displaystyle \sum _{i=1}^n{(X_i-\overline{X})}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{(Y_i-\overline{Y})}^2}}}}$$

(5)

The equation involves $X_i$, which represents the value of the health indicator selected at each iteration, and $\overline{X}$, denoting the average value. $Y_i$ stands for the current capacity at each iteration, while $\overline{Y}$ represents the average capacity.

The 12 extracted health indicators are correlated with the health status of lithium-ion batteries, as detailed in

Table 2. The Pearson correlation coefficients between each health indicator and capacity.

Types	Mean	Standard Deviation	Kurtosis	Skewness
Voltage	−0.9692	0.3614	−0.6491	−0.9729
Current	0.9723	0.9731	−0.9726	0.9789
Temperature	0.7811	−0.6912	0.8017	−0.9283

. These correlation coefficients are utilized to select six highly correlated health indicators as inputs for the health-state prognostics model, enabling the precise estimation of lithium-ion battery health status [29]. Additionally, the Pearson correlation coefficients between the constant current charging time and constant voltage charging time are 0.9946 and −0.9547, respectively.

Through comparative analysis, this study selects the constant current charging time, constant voltage charging time, average charging voltage, average charging current, standard deviation of charging voltage, skewness of charging current, skewness of charging voltage, and kurtosis of charging voltage as the model inputs. These eight health indicators demonstrate a correlation coefficient with capacity exceeding 95% and are denoted by HI1–HI8, respectively.

Upon analysis of Figure 6 and Figure 7, it is apparent that the values of four health indicators—namely the constant current charging time, average charging current, standard deviation of charging voltage, and skewness of charging voltage—exhibit an overall declining trend with an increasing cycle number. This pattern closely reflects the relationship between the battery capacity and health indictors, demonstrating a strong positive correlation among these four health factors and a battery’s SOH. In contrast, another set of four health indicators—comprising the constant voltage charging time, average charging voltage, charging current skewness, and charging voltage kurtosis—demonstrates an overall increasing trend with the cycle number. This divergence from the variation in battery capacity with the cycle number indicates a strong negative correlation between the four health factors and capacity.

4. MSCNN Estimation Model

The CNN was introduced by Turing Award laureate LeCun as a specialized type of neural network designed for processing data with grid-like structures, such as time series and image data. CNNs represent an advancement over traditional neural networks and utilize a hierarchical network structure to establish a mapping from the input to the output, enabling the learning of relationships between the input and output from extensive datasets. The CNN architecture primarily consists of an input layer, convolutional layers, pooling layers, fully connected layers, and regression layers. The input layer is responsible for preprocessing raw data through mean subtraction and normalization. The CNN structure diagram is shown in Figure 8 [30].

Convolution operations serve as feature enhancement techniques that effectively strengthen key features within original input data, thereby improving the model stability and estimation accuracy during task execution while also reducing noise. With an increase in the number of convolutional layers, the network can extract more complex and abstract data features to achieve the comprehensive exploration of high-dimensional characteristics [31]. However, convolution operations alone provide limited feature compression for individual local regions within images. The product of the input x with filter kernel w defines the convolutional layer, subsequently yielding feature mappings post-convolutional layer application, as shown in Equation (6).

$$h_k^l=\varphi (w_k^l\ast x+b_k^l)$$

(6)

where $\ast$ represents the convolution operator, and $h_k^l$, $w_k^l$, and $b_k^l,$ respectively, denote the feature map obtained by the kth layer of the lth layer convolution kernel in the model.

The pooling layer follows the convolution layer and serves to reduce the spatial size of the output from the convolution layer while preserving essential features. This process contributes to reducing the model’s complexity and enhancing its invariance to input data. In regression-estimation tasks, the pooling layer aids in the feature size reduction and extraction of pertinent features, thereby expediting the model’s training and improving its generalization ability [32]. The fully connected layer receives flattened features from either the pooling or convolution layers as its input and is utilized for learning intricate relationships within the input data. In regression-estimation tasks, this layer takes extracted features from the pooling layer as its input and learns weights and biases to establish a nonlinear mapping relationship between the input data and output data [33]. These weight and bias parameters are adjusted based on the training data features to enhance model fitting with the training data for accurate regression estimations.

CNNs primarily use a single convolutional kernel for feature extraction, which is locally perceptive and has a limited global feature-perception ability, affecting the robustness of feature extraction and reducing the accuracy of predicting the health status of lithium-ion batteries. However, the MSCNN, as a deep learning model, uses multiple different scale convolutional kernels in the convolution operation to extract multi-scale features of the health factors of lithium-ion batteries, fusing more abundant local information and fully mining the hidden information within the health factors. It can learn the complex nonlinear relationship between the health statuses of lithium-ion batteries and the selected health factors from a large amount of data, which is difficult for traditional analysis methods or shallow models to capture by capturing the coupling relationship between them. Compared with CNNs, the MSCNN can effectively extract features from data of different scales and has significant advantages in handling complex battery health status estimation problems.

Traditional CNNs are usually designed for image classification tasks and may not be optimized enough for regression tasks; for scale variations in the input data, traditional CNNs may not perform well because a fixed-size convolutional kernel and pooling layer are usually used, and this design makes the network have a fixed sensory field when extracting features. Although traditional CNNs can capture localized features, they have a limited ability to process information at different scales, especially when dealing with regression tasks that require multi-scale information, and may not be able to fully capture features at various scales. For lithium-ion battery health state prediction, data features at different time scales or different frequencies need to be processed. However, traditional CNNs mainly rely on a fixed convolutional kernel size and pooling layer when extracting features, which may not be flexible enough for processing regression tasks with features at different scales. When conducting research on the estimation of a battery’s health status, it is necessary to consider not only capturing these features at different scales but also comprehensively evaluating the battery’s condition. The multi-scale feature-extraction enables the MSCNN to more accurately understand dynamic characteristics, which helps to predict the capacity degradation and determine the battery’s overall state. In this paper, an MSCNN model was constructed using multiple convolutional layers with varying kernel sizes along with max pooling layers which sequentially extract features from input data [34].

The MSCNN structure used in this paper, first, uses a combination of two convolutional pooling layers consecutively. After the first convolutional layer extracts the features, the pooling layer is used for down-sampling to reduce the dimensionality of the input features while retaining important feature information. The second convolutional layer then operates on the pooled features, enabling the further extraction and aggregation of multi-scale information while reducing the dimensionality of the data. This processing enables the network to capture features at different scales, after which the stacking of three convolutional layers is used, enabling the model to extract more complex features and patterns and helping the model to capture advanced features in the input data. And the deep network, by extracting and combining features layer by layer, allows the model to learn richer and more effective feature representations to better understand the intrinsic relationships of the data and better fuse these features into the final prediction. Finally, the pooling layer is used to further reduce the feature dimensions and integrate the input feature information into global information to facilitate subsequent regressor processing. The detailed structure is shown in Figure 9.

Regarding the key hyperparameters of the model settings, they are mainly related to the number of convolutional and pooling layers, and the size of the convolutional kernel when performing sense-field computation. The specific settings can be found in

Table 3. Detailed structural parameters of the MSCNN model.

Layer Name	Specific Parameters
Convolution layer 1	Convolution kernel 11 × 1, channel 16
Pooling layer 1	Convolution kernel 2 × 1, step size 2
Convolution layer 2	Convolution kernel 5 × 1, channel 32
Pooling layer 2	Convolution kernel 2 × 1, step size 2
Convolution layer 3	Convolution kernel 3 × 1, channel 64
Convolution layer 4	Convolution kernel 3 × 1, channel 128
Convolution layer 5	Convolution kernel 3 × 1, channel 64
Pooling layer 3	Convolution kernel 2 × 1, step size 2

, and the settings of these parameters are based on several debugging sessions and references from the literature.

To evaluate the predictive performance of the proposed method for assessing the health status of lithium-ion batteries, three statistical measures—the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE)—are utilized to assess the model’s output [35]. The calculation formulas (7)–(9) are as follows:

$$MAPE={\displaystyle \frac{1}{\omega }}{\displaystyle \sum _{i=1}^{\omega }\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}\times 100\%$$

(7)

$$MAE={\displaystyle \frac{1}{\omega }}{\displaystyle \sum _{i=1}^{\omega }\left|y_i-{\widehat{y}}_i\right|}\times 100\%$$

(8)

$$MAPE=\sqrt{{\displaystyle \frac{1}{\omega }}{\displaystyle \sum _{i=1}^{\omega }{(y_i-{\widehat{y}}_i)}^2}}$$

(9)

where $y_i$ represents the actual capacity value of the battery, ${\widehat{y}}_i$ denotes the predicted capacity value, and $\omega$ stands for the number of cycles. These three metrics have a range of values from [0, +∞). Lower values of these indicators indicate an enhanced predictive performance of the model algorithm, leading to a closer alignment between the predicted and observed values.

5. Results

The battery datasets utilized in this study, designated as Group A and Group B, are obtained from the same batch and subjected to identical charging and discharging protocols. As a result, batteries within the same batch exhibited similar processes of health degradation. In deviation from traditional practices, individual battery data are not segregated into distinct training and test sets; instead, specific battery data are allocated for training purposes while data from other batteries served as the test set. To evaluate the accuracy and generalization capability of the proposed method, three distinct experimental methodologies were employed for validation.

5.1. Comparative Analysis of SOH Estimation Using Diverse Methodologies

In this section, each battery within the same group is sequentially designated as the test set, while the remaining seven batteries serve as the training set, as depicted in Figure 10.

To validate the accuracy of selecting the MSCNN model structure, this section compares the results with those obtained using both the basic CNN architecture and the LSTM model. The architectural details of the basic CNN model are delineated in

Table 4. The architecture of basic CNN model.

Layer Name	Specific Parameters
Convolutional Layer	Convolution Kernel 3 × 1, Channel 16
Pooling Layer	Convolution Kernel 2 × 1, Stride 2

.

• The SOH estimation results of the three different models for the battery cells in Group A

This section validates the efficacy of three distinct models in predicting the health state of the eight battery cells within Group A. Taking the third battery cell in Group A as an example, Figure 11 illustrates the health state estimation results generated by these models. The black line denotes the actual SOH values of the batteries, while the green and blue lines represent predictions from the LSTM and CNN models, respectively. Figure 12 illustrates the health state estimation errors generated by the three models. The red and blue scattered points represent estimations from the CNN and LSTM models, respectively. Additionally, the black scattered points depict estimations from our proposed MSCNN model. Comparative analysis reveals that, in contrast to both the CNN and LSTM models, estimations from our MSCNN model align more closely with the actual SOH values, underscoring its superior accuracy in SOH estimation.

To provide a more intuitive comparison of the estimation accuracy of the three models,

Table 5. SOH estimation error of battery condition in Group A for three different models.

Battery ID	Index	MSCNN (%)	CNN (%)	LSTM (%)
Group A 1	MAE	0.2976	0.4282	0.5543
	MAPE	0.1646	0.2350	0.3007
	RMSE	0.5228	0.7325	0.8438
Group A 2	MAE	0.2913	0.5698	0.4171
	MAPE	0.1565	0.3210	0.2266
	RMSE	0.5828	0.8895	0.7846
Group A 3	MAE	0.2434	0.7367	0.3975
	MAPE	0.1319	0.4187	0.2145
	RMSE	0.3694	1.0651	0.6680
Group A 4	MAE	0.2296	0.6098	0.2878
	MAPE	0.1223	0.3392	0.1538
	RMSE	0.4016	0.8367	0.5815
Group A 5	MAE	0.3053	0.4051	0.3250
	MAPE	0.1622	0.2184	0.1739
	RMSE	0.4024	0.6162	0.5239
Group A 6	MAE	0.3359	0.4989	0.5454
	MAPE	0.1775	0.2613	0.2939
	RMSE	0.4591	0.6745	0.6377
Group A 7	MAE	0.2969	0.3881	0.6456
	MAPE	0.1604	0.2113	0.3481
	RMSE	0.4046	0.5598	0.7694
Group A 8	MAE	0.7507	1.2399	0.8265
	MAPE	0.4222	0.6922	0.4849
	RMSE	1.0465	1.5744	1.1571

summarizes the RMSE, MAE, and MAPE of the three models. Through analysis and comparison, it can be seen that the MSCNN estimation model selected in this paper has an MAE of less than 0.75% and a MAPE of less than 0.42% for the SOH estimation of the battery cells in Group A. The RMSE is also less than 1.04%. Compared with the traditional CNN model, the error of the MSCNN model selected in this paper is on average lowered by 41.11%, 41.81%, and 38.53% in terms of the MAE, MAPE, and RMSE, respectively. In comparison with the LSTM model, the MAE error value was reduced by an average of 30.38%, the MAPE error value was reduced by an average of 31.04%, and the RSME error value was reduced by an average of 30.94%.

2.

The SOH estimation results of the three different models for the battery condition of Group B

In Figure 13, the SOH estimation results of Cell No. 8 of Battery Group B for the three different models are depicted. The black scattered points represent the actual SOH values, while the green, blue, and red scattered points represent the estimation values of the CNN model, LSTM model, and MSCNN model proposed in this paper, respectively. It is observed that, compared with the CNN and LSTM models, the red scattered points of the MSCNN estimation model closely align with the actual SOH values, indicating higher accuracy. Figure 14 illustrates the health state estimation error generated by the three models. The red and blue scattered points represent estimations from the CNN and LSTM models, respectively. The comparison indicates that the estimation accuracy of the other two models is comparatively lower than that of the MSCNN model.

Table 6. The estimated error of SOH for batteries in Group B.

Battery ID	Index	MSCNN (%)	CNN (%)	LSTM (%)
Group B 1	MAE	0.2635	0.4348	0.5970
	MAPE	0.1457	0.2363	0.3231
	RMSE	0.4869	0.6453	0.8126
Group B 2	MAE	0.7729	1.1465	0.8390
	MAPE	0.1435	0.6408	0.4679
	RMSE	1.1771	1.6760	1.3057
Group B 3	MAE	0.2851	0.4305	0.3710
	MAPE	0.1563	0.2337	0.2021
	RMSE	0.4544	0.6113	0.5676
Group B 4	MAE	0.5046	0.8654	0.6557
	MAPE	0.1497	0.4666	0.3603
	RMSE	0.7007	1.0555	0.9456
Group B 5	MAE	0.6681	0.7798	0.8586
	MAPE	0.3612	0.4296	0.4696
	RMSE	0.7381	0.9371	0.9678
Group B 6	MAE	0.4601	0.8777	0.7036
	MAPE	0.2535	0.4770	0.3872
	RMSE	0.6699	1.0263	0.9372
Group B 7	MAE	0.3696	0.4745	0.3343
	MAPE	0.2007	0.2643	0.1831
	RMSE	0.5655	0.8973	0.6969
Group B 8	MAE	0.2399	0.8799	0.6196
	MAPE	0.1338	0.4837	0.3409
	RMSE	0.4724	1.0091	0.8038

displays the estimated errors of SOH estimation for batteries in Group B. The analysis reveals a slightly higher estimated error for Group B compared to Group A. Additionally, it is noted that abnormality exists, as evidenced by a two-fold increase in error for one battery within this group. However, apart from one instance (the third battery), most batteries exhibit an MAE of less than 0.67%, a MAPE of less than 0.37%, and an RMSE of less than 0.74%. Notably, accurate estimations were obtained even for batteries with significantly lower cycle counts (e.g., Batteries No. 5 and 8) when compared to others within their group under similar working conditions.

This section validates the battery dataset and estimation results of the three models under two identical working conditions. The findings indicate that the MSCNN health estimation model proposed in this study demonstrates superior predictive accuracy compared to both the CNN and LSTM models. In comparison to the baseline CNN structure, the model exhibits an average reduction of 38.02% in the MAE, 47.02% in the MAPE, and 32.46% in the RMSE values. Furthermore, when compared to the LSTM model, it shows an average reduction of 27.18% in the MAE, 39.26% in the MAPE, and 26.01% in the RMSE values.

5.2. Estimated Health Status Results of Various Battery Cells

In this section, Battery No. 4 in Group A does not show abnormal capacity changes and is consistent with daily battery capacity degradation. And the precision of the SOH estimation for battery No. 4 in Section 5.1 is also the highest, while the remaining cells are utilized for training. Specifically, Battery Cell No. 4 is designated as the test set, and the number of batteries in the training set is incrementally increased, as illustrated in Figure 15.

As depicted in Figure 16, the SOH estimation values of the MSCNN model gradually converge with the actual values as the number of batteries in the training set increases. However, once the training set comprises three batteries (approximately 75% of the total), further increasing its size does not significantly enhance the model accuracy. Figure 17 illustrates that the overall estimation error remains within 1.5%. Specifically, the initial 200 cycles exhibit predominantly sub-0.5% error distribution. Subsequently, as the number of test batteries increases, there is a noticeable escalation in the error magnitude and volatility compared to the first 200 cycles.

Table 7. SOH estimation errors for individual cells within Battery No. 4 from Group A under different training set sizes.

ID	MAE (%)	MAPE (%)	RMSE (%)
Experiment 1	0.5757	0.3216	0.8127
Experiment 2	0.5061	0.2833	0.7032
Experiment 3	0.4453	0.2497	0.6353
Experiment 4	0.3443	0.1853	0.4499
Experiment 5	0.2723	0.1486	0.4276
Experiment 6	0.2799	0.1541	0.4269

illustrates the numerical values of the SOH estimation errors for Battery Cell No. 4 in Group A, with varying numbers of training sets. In Experiment 4, where the number of training sets is increased to four, the error values are as follows: MAE = 0.3443%, MAPE = 0.1853%, and RMSE = 0.4499%. Similarly, in Experiment 6, with an increase in the number of training sets to six, the error values are observed as: MAE = 0.2799%, MAPE = 0.1541%, and RMSE = 0.4269%.

This section focuses on predicting the SOH for select batteries within similar operating conditions by sequentially increasing the number of batteries used in the model training set from one to several units, starting with a random selection from Group A (Battery No. 4). The results show that the proposed method can achieve relatively accurate estimations even when using a smaller training set.

5.3. Integration of Data for Cross-Operating Condition SOH Estimation

To assess the generalization capability and predictive accuracy of the proposed SOH estimation model, this study explores health status estimation under diverse operational conditions. Specifically, the first, second, and third individual cells from Group A as well as the first, second, and third individual cells from Group B are chosen as training sets for sequential SOH estimation of the remaining cells in Group A, as shown in Figure 18.

The experimental findings indicate that estimations across different operating conditions yield lower precision compared to estimations within identical operating conditions. The inclusion of data from diverse operating conditions in the training set results in slightly larger errors than those observed when predicting within a uniform operating condition. Notably, Experiment No. 1 exhibits significant error; specifically, concerning Battery No. 4 in Group A, its MAE increases by 72.06%, its MAPE by 36.75%, and its RMSE by 66.74% compared to estimations made within similar operating conditions. Conversely, Experiment No. 4 demonstrates minimal error; however, it still shows an increase in the MAE by 13.44%, MAPE by 16.19%, and RMSE by 15.96% relative to estimations made within similar operating conditions.

Figure 19 demonstrates the SOH estimate versus the true value for cell No. 8, having the largest estimation error in group A across operating conditions. Figure 20 shows that the estimation error can be divided into two “V” shapes, the first of which has an error of less than 1%, while the second, which is more volatile than the first, has an error of no more than 3.5%.

The detailed estimate error outcomes for various experiments are summarized in

Table 8. Accuracy of SOH estimation across operating conditions for Group A batteries.

ID	MAE (%)	MAPE (%)	RMSE (%)
Experiment 1	0.6316	0.3485	0.8362
Experiment 2	0.5583	0.3106	0.7965
Experiment 3	0.5026	0.2796	0.7191
Experiment 4	0.4993	0.2737	0.6332
Experiment 5	0.9886	0.5507	1.3888

.

In this section, we conduct the estimation of batteries’ SOH across different operating conditions. The training set consists of a total of six batteries, including three from Group A and three from Group B. Subsequently, we predict the SOH of all remaining batteries. Our experimental results indicate that the accuracy of cross-operating condition SOH estimation is lower than that within the same operating condition. Notably, experiment four demonstrates the highest precision, with recorded MAE, MAPE, and RMSE values of 0.4993%, 0.2737%, and 0.6332%, respectively.

6. Conclusions and Outlook

In response to the challenges of low estimation accuracy and weak generalization ability in traditional SOH estimation methods, this study introduces a novel approach for SOH estimation that integrates multiple health indicators with an MSCNN model. Utilizing the publicly available lithium-ion battery dataset from Xi’an Jiaotong University, we extract health indicators from charging data and conduct a correlation analysis to select eight highly correlated health indicators as inputs for the predictive model. Subsequently, a regression-based predictive model is constructed using the MSCNN framework to forecast the health status of lithium-ion batteries. We examine SOH estimations for two sets of battery data using three distinct models. The findings suggest that the accuracy of cross-condition SOH estimation is lower than that of within-condition estimations. Notably, our developed MSCNN model exhibits significantly enhanced precision compared to basic CNN models, consistently yielding lower error metrics than LSTM models. Moreover, it is noted that, when the training ratio reaches 75%, our model achieves optimal accuracy in estimating the SOH among batteries operating under similar conditions. However, the MSCNN model requires a large amount of high-quality data for training. If the data are insufficient or of poor quality (e.g., a small number of training sets in Experiment 2 and the presence of training sets that do not belong to the same operating conditions in Experiment 3), the accuracy of the model in predicting the health state of the batteries is degraded. And, although MSCNN is capable of automatically extracting features, choosing the appropriate feature scale and convolutional layer for a given task may still be challenging. It is necessary to explore more effective data enhancement and preprocessing methods in future research to improve the robustness of the model to different data qualities and defects, as well as performing model fusion combining the MSCNN with other models in order to enhance the processing of time-series data and improve the estimation accuracy of battery health status.