State of Health Prediction in Electric Vehicle Batteries Using a Deep Learning Model

Abstract

Accurately estimating the state of health (SOH) of lithium-ion batteries plays a significant role in the safe operation of electric vehicles. Deep learning (DL)-based approaches for estimating state of health (SOH) have consistently been the focus of study in recent years. In the current era of electric mobility, the utilization of lithium-ion batteries (LIBs) has evolved into a necessity for energy storage. Ensuring the safe operation of EVs requires a precise assessment of the state-of-health (SOH) of LIBs. To estimate battery SOH accurately, this paper employs a deep learning (DL) algorithm to enhance the estimation accuracy of SOH to obtain accurate SOH measurements. This research introduces the Diffusion Convolutional Recurrent Neural Network (DCRNN) with a Support Vector Machine-Recursive Feature Elimination (SVM-RFE) algorithm (DCRNN + SVM-RFE) for enhancing classification and feature selection performance. The data gathered from the dataset were pre-processed using the min–max normalization method. The Center for Advanced Life Cycle Engineering (CALCE) dataset from the University of Maryland was employed to train and evaluate the model. The SVM-RFE algorithm was used for feature selection of pre-processed data. DCRNN algorithm was used for the classification process to enhance prediction precision. The DCRNN + SVM-RFE model’s performance was calculated using Mean Absolute Percentage Error (MAPE), Mean Squared Error (MAE), Mean Squared Error (MSE), and Root MSE (RMSE) metric values. The proposed model generates accurate results for SOH prediction; all RMSEs are within 0.02%, MAEs are within 0.015%, MSEs were within 0.032%, and MAPEs are within 0.41%. The mean values of RMSE, MSE, MAE, and MAPE were 0.014, 0.026, 0.011, and 0.32, respectively. Experiments confirmed that the DCRNN + SVM-RFE model has the highest accuracy among those that predict SOH.

1. Introduction

The rising instances of vehicles with internal combustion that utilize non-renewable standard fuels have resulted in significant environmental and energy issues. However, the advancement of EVs has been slowed by the high cost of electric automobiles and the quick progress of traditional vehicles. Research on EVs has advanced since the beginning of the 21st century due to concerns over environmental pollution and energy-related difficulties. Through collaboration between government and industry, advancements have been made in infrastructure and EV technologies. In 2016, the worldwide sales of EVs hit 1 million [1]. As of 2018, global sales of light-duty EVs and plug-in hybrid EVs exceeded five million. Over the last 150 years, there has been significant progress and advancement in the field of EV technology.

Figure 1 illustrates the evolution of EVs from their initial form as unchargeable automobiles to their current state with advanced control systems. This development may be categorized into three distinct stages: the early phase in 1832, the intermediate phase in 1960, and the contemporary era. Zero emissions can be achieved if the vehicle is fueled by renewable energy sources [2].

Batteries are extensively utilized as the main source of power for EVs and hybrid EVs (HEVs) because of their notable advantages, like high energy density, minimal environmental impact, and extended cycle life. However, batteries require specific attention when used in EV applications [3]. Improper operations, such as exceeding the suggested voltage, charging/discharging levels, or current, can lead to serious safety concerns for batteries. These actions can significantly speed up the aging process of the batteries and even result in fire or explosion [4]. LIBs are commonly utilized as sources of power in EVs because of their high density of energy, capabilities of fast charging, and minimum rate of self-discharge [5].

Battery Management Systems (BMSs) are crucial for maintaining the safety and optimal functioning of batteries. Important technologies within the BMSs of EVs consist of battery modelling, internal state estimates, and battery charging. A robust battery model is essential for analyzing battery behavior, monitoring the state of battery, designing real-time controllers, managing temperature conditions, and diagnosing faults [4,5]. State estimation is a fundamental and essential function of a BMS, since it serves as the monitor of the power system [6]. The BMS protects the battery packs from short-circuit currents and excessive voltage through the integration of controls, actuators, and sensors [7]. Figure 2 illustrates the fundamental elements of BMSs and applications in EV technology.

1.1. State of Health Estimation

SOH is a metric that quantifies the extent of battery degradation relative to that of a new battery [8]. This information is essential for the vehicle energy management system to fine-tune its controls to maintain the vehicle’s performance and safety within intended limits. Various methods can be used to assess and measure the SOH of an EV’s battery. Recent research have primarily focused on determining either the decrease in capacity (SOHc) or the increase in internal resistance (SOHr) [9]. Deterioration of LIBs is unavoidable during extended periods of cycling or storage. The battery SOH data are essential to the energy management system of EVs to ensure a secure and exceptionally efficient operational state. Examining and analyzing the mechanisms that cause batteries to deteriorate over time, as well as the resulting effects, is crucial for accurately assessing battery health and making reliable forecasts about performance. Analysis of battery aging mechanisms is typically conducted on multiple levels, encompassing elements that influence it, internal side reactions, degradation mechanisms, and external influences [10].

The SOH value of LIBs is an essential measure for quantifying the health of the battery. It functions as an indicator of the battery’s aging status and can be utilized for predicting its lifespan. The SOH of an LIB can be evaluated based on its capacity, which is the ratio of the battery’s total capacity to its nominal capacity, and the expression is described as follows:

$$\mathrm{S}\mathrm{O}\mathrm{H}={\displaystyle \frac{C_{max}^{\prime }}{C_{max}}}\times 100\%$$

(1)

where $C_{max}^{\prime }$ represents the battery’s power after it was charged, and $C_{max}$ represents the nominal capacity of a newer battery [11].

There is no single definition for the battery SOH. A general description of battery SOH can be given as:

$$\mathrm{S}\mathrm{O}\mathrm{H}\left(t\right)=\mathrm{S}\mathrm{O}\mathrm{H} \left(t_0\right)+{\int }_{\tau =t_0}^t{\delta }_{func} (I, T, SOC, others)d\tau$$

(2)

where $SOH \left(t_0\right)$ represents the initial battery $SOH$, ${\delta }_{func}$ is an aging rate function, which strongly depends on several factors such as current, temperature, SOC, and others that represent some other stress factors such as mechanical vibrations and over-potential [12]. In EV applications, battery aging leads to a reduction in battery capacity and an increase in battery internal resistance. Therefore, the SOH of the battery can be approximated by evaluating its internal resistance or useable capacity, which serves as a predictive measure for changes in the field of computer science. There have been many proposed methods for estimating battery SOH, which can be classified into three groups: model-free, model-based, and data-mining approaches. For the model-free method, given the aged capacity $C_a$ or the increased internal resistance $R_i$, battery SOH can be simply defined as:

$$\left\{\begin{array}{c}\mathrm{SOH}={\displaystyle \frac{C_a}{C_n}}\times 100\%\\ \mathrm{SOH}={\displaystyle \frac{R_i}{R_n}}\times 100\%\end{array}\right.$$

(3)

where $C_n$ and $R_n$ stand for the nominal capacity and internal resistance, respectively, of the new, unused battery [6].

The fundamental operational framework for DL-based SOH prediction of EVs is depicted in Figure 3. The fundamental workflow of the SOH prediction model comprises the following stages: data acquisition, selection of relevant features, classification of features, and evaluation of SOH prediction. The initial step of the model involves gathering real-time data from a specific dataset. Next, the gathered data are subjected to pre-processing, which involves employing data cleaning or data normalization procedures to eliminate undesirable noisy aspects. After pre-processing, the data are then passed on to the feature selection and classification stages. The DL model has the ability to handle complex, non-linear relationships and patterns in time-series data, which are characteristic of battery performance metrics. Unlike traditional ML models, DL, particularly the DCRNN, is better at capturing temporal dependencies and intricate interactions between features, leading to more accurate SOH predictions. This capability is crucial for developing robust BMSs that can operate effectively under diverse conditions and with various battery types [13].

1.2. Research Objectives

The research objectives for this work on SOH prediction using the DCRNN + SVM-RFE model can be outlined as follows:

• To design and implement a novel hybrid model that combines DCRNN with SVM-RFE to enhance the accuracy of SOH predictions in lithium-ion batteries.
• To explore the adaptability of the proposed model to the CALCE dataset, which consists of time-series data capturing various parameters, including voltage, current, temperature, and charge/discharge cycles.
• To pre-process the data using min–max normalization to ensure all features are scaled between 0 and 1, facilitating efficient model training.
• To assess the ability of the DCRNN to capture complex temporal patterns and dependencies in battery performance data, thereby improving SOH prediction accuracy.
• To improve the feature selection process using SVM-RFE, ensuring that the most relevant features are identified and utilized in the SOH prediction model.
• To evaluate the proposed DCRNN + SVM-RFE model using key performance metrics, including MAPE, MSE, MAE, and RMSE.
• To validate the model’s effectiveness with other reviewed approaches and to calculate the efficiency of the research method in enhancing SOH prediction.

This paper includes an extended review of the various prediction methods analyzed in the related works section. The proposed methodology section includes the dataset collection, pre-processing, feature selection, and classification methods used. The experimental section presents the performance evaluation, SOH estimation, and comparison of the research model with current models. Finally, the paper concludes with future directions.

2. Related Works

This section provides an in-depth review of several approaches used in the field of SOH prediction methods for LIBs based on a literature study. The purpose of this survey was to examine the development of strategies for estimating SOH and to identify emerging trends in ML and DL technologies. This survey consolidates information from many studies to offer a comprehensive summary of the present state of SOH prediction algorithms and to offer guidance for future research. An approach to SOH prediction of LIBs was introduced by [14]. A GAN-LSTM-TL network was utilized in this method, where Generative Adversarial Networks (GANs) were utilized to handle the related feature data. A long short-term memory (LSTM) network was employed to mapping the relationship among SOH and LIB characteristics. Transfer learning (TL) was employed to improve the flexibility of the LSTM network and accurately estimate the SOH by addressing the training and testing challenges across different datasets of LIBs. The GAN-LSTM-TL model has the most accurate prediction curve, with the lowest overall error. It achieves a minimal MAE value of 0.0208 and a reduced RMSE value of 0.0275.

In research by [15], the researchers developed an innovative hybrid model named HFCM-LSTM (Hierarchical Feature Coupled Module-LSTM). The purpose of this network was to effectively recover the original data information, resulting in a more precise estimation of the SOH of a battery. The entire approach addresses the issue of inaccurate estimation created by the model because of inadequate data extraction. By validating both datasets, it has been observed that the RMSE of the proposed network remains below 2.3%. This indicates that the method was highly effective. A study by [16] presented a model to enhance the precision of SOH assessment for LIBs that integrated Incremental Capacity Analysis (ICA) with Bidirectional LSTM (Bi-LSTM) neural networks. This model utilizes characteristics of health for predicting the LIB’s SOH. Using the Pearson correlation coefficient, the parameters of health characteristics were subsequently extracted from the charging curve of the LIB. Furthermore, Independent Component Analysis was employed to thoroughly explore the profound connections between the IC curve peaks and SOH. The associated voltages of those peaks were then retrieved and incorporated as supplementary inputs to the model. Ultimately, the verification process relies on the fifth battery specifications from the NASA-LIB data collection. The findings demonstrated that the model effectively decreased the MSE by 55.17%, 49.28%, and 41.47% when compared to individual models.

A hybrid model was created by [17] to forecast the SOH of LIBs. This model was built by combining a Convolutional Neural Network (CNN), an Attention Mechanism (AM), and Bi-LSTM. This model utilized a CNN for feature extraction of the battery time series. Bi-LSTM was employed to capture the connections in both ways. Additionally, it applied the AM to assign weights for accurate evaluation of the SOH of LIBs. Analysis of NASA PCoE LIB data revealed that the hybrid model surpassed other individual models in performance, exhibiting RMSE values for SOH prediction that were consistently below 0.01. A study conducted by [18] developed a method using LSTM to forecast the health condition of LIBs. The accuracy and superiority of the model were validated by two well-known LIB datasets from NASA and CALCE, which offer a potential tool for accurately determining the SOH. The accuracy of predicting battery cell performance was higher when using a 40% cycle as the starting point, compared to using 50% or 60% cycles. This occurs due to the gradual decline in battery capacity data throughout the cycle, leading to reduced forecast errors. However, towards the end of the cycle, the capacity decreases rapidly and fluctuates more, leading to larger prediction errors and an overall increase in average error when using a 60% starting point.

A study by [19] proposed an enhanced SOH prediction scheme battery pack using a flexible LSTM model, along with a TL strategy. Subsequently, a refined LSTM network that incorporates TL was suggested to construct the CMM and thoroughly validated by comparing it with widely employed SOH prediction techniques. The validation results obtained from this model demonstrated that the error in cell SOH was less than 3% when 70% of the training data was utilized. Furthermore, the validation findings obtained from several types of LIBs provided additional evidence of the scalability of the technique. A Hybrid Neural Network model for SOH Estimation of LIBs was explained in [20]. The model was called dilated CNN-BiGRU, which combines a dilated CNN and a bidirectional GRU. This approach allows for the extraction of more precise spatial and temporal characteristics from the initial data of the battery, leading to increased efficiency and resilience. The research findings indicate that the model overcomes the data-driven approaches being compared, such as CNN and RNN. Additionally, the MAE and RMSE values of the NASA dataset were limited to a maximum of 1.9% and 3.3%, respectively. The Oxford datasets yielded an MAE value of 0.048% and an RMSE value of 0.08%.

The SOH of LIBs used in EVs was calculated using an Extreme Gradient Boosting (XGBoost) method by [21]. An error analysis was performed on the model to improve the battery’s performance parameter. An accuracy correction model for determining the SOH of LIBs utilizing the XGBoost algorithm was presented. The XGBoost approach was utilized to validate the accuracy of the SOH of LIBs using the ensemble bagging prediction method on a dataset. This model demonstrated excellent predictive performance, with an MAE of 0.0013% and an RMSE of 0.0019%.

A technique was introduced by [22] for evaluating the SOH of LIBs. This method involves generating sample data and using a Temporal Convolutional Neural (TCN) with a Variational Auto-Encoder (VAE). Initially, the charge/discharge curves of the batteries were examined, and specific characteristics were identified that exhibited a strong correlation with deterioration of the SOH. Subsequently, a Variational Autoencoder (VAE) was employed to acquire knowledge about the characteristics and probability distributions of the given dataset to produce data that closely resemble it and augment the sample size. In the end, the non-linear correlation between features and SOH was extracted using a TCN with a collective source and expanded domain to estimate SOH. As demonstrated by the results, the described model’s RMSE was decreased to 0.0073, representing an accuracy enhancement of 64.9% over the benchmark model. An approach was proposed by [23] to calculate the SOC and SOH of LIB. This method utilized DRSN-CW-LSTM networks for estimation. An LSTM network was employed to forecast the SOH of the LIB. Furthermore, the expected SOH value was combined with the SOC values present in the data. By integrating the adaptive noise function into the residual module of the DRSN-CW networks, the detrimental impacts of substandard data of LIBs on SOC estimation can be alleviated, thus improving the precision of SOC estimation. The experimental findings indicate that the MAE and MSE of the DL approach were consistently maintained within 5%.

A study by [24] introduced a precise method for estimating SOH by utilizing temperature prediction and a GRU neural network. The Extreme Learning Machine (ELM) approach was used to estimate complete temperature fluctuation during the process of fixed current charging using random intermittent short-term charging functions. Subsequently, a method named finite difference was utilized to compute unprocessed temperature fluctuations, which was subsequently refined through the implementation of the Kalman filter. The experimental results showed that the mentioned strategy could reliably and robustly forecast SOH. The RMSE, MAE, and max absolute error were limited to 1.2%, 1.02%, and 2.28%, respectively. Additionally, all cells had R-squared values greater than 0.98.

CL-Net was an innovative hybrid architecture proposed in a paper by [25]. It integrated LSTM models with Convolutional LSTM (ConvLSTM) designed specifically for multi-step SOH and forecasting power consumption. The collection of data concerning the SOH and power consumption of batteries was followed by a pre-processing phase aimed at eliminating any discrepancies or errors present in the data. Subsequently, ConvLSTM layers were employed to process the processed data, extract spatiotemporal features, and produce an encoded map. Furthermore, for the last multi-step prediction, LSTM was utilized to decipher encoded attributes and transmit them to linked layers. In the final stage, a comprehensive analysis was conducted on three distinct time series datasets—the NASA, IHEPC, and DEMS datasets—on a variety of combinations of sequential learning models. In comparison to the current leading methodologies, the RMSEs on the NASA and IHEPC datasets were reduced by 0.13 and 0.0052, respectively, using “CL-Net” architecture.

The Bat Algorithm (BA)-ELM model was created by [26] to provide an accurate assessment of the SOH of LIBs. The ELM has been built with swift learning velocity and exceptional generalization capabilities. At the outset, the model’s input variables were ascertained through the application of Spearman and Pearson correlation analyses. The BA-ELM model was constructed by optimizing the link between weights and bias in the ELM model using the BA algorithm. In addition to the radar chart, violin diagram, and scatter plot, five statistical error indicators were utilized to analyze the performance impacts. The results indicate that the model can rapidly integrate and efficiently optimize the ELM network model. The MAE was 0.4326%, and the RMSE was 0.5354%. These were the least significant values among the six models.

The study by [27] introduced a technique for predicting SOH using the XGBoost algorithm, incorporating accuracy correction. The SOH of LIBs was predicted using three battery features: temperature difference, voltage differential, and average voltage. The XGBoost algorithm was initially used to forecast the SOH, and subsequently a Markov (MK) chain was implemented to rectify the anticipated values. The method exhibits a high level of accuracy and possesses robust generalization capabilities. A comparison of XGBoost and MK correction demonstrates that the utilization of MK correction can substantially enhance the estimation efficiency of XGBoost by a range of 10% to 20%.

An enhanced Random Forest (RF) model for predicting the SOH of LIB was presented by [28]. To begin with, a comprehensive set of LIB SOH prediction data was obtained by conducting cycles of charging and discharging experiments on 18,650 LiFePO4 batteries, thereby accumulating numerous LIB experimental data points. SOH prediction analyses for multiple LIBs were subsequently performed utilizing four conventional ML models. The findings suggest that the Class RF model produces comparatively more accurate predictions, thereby validating the viability of applying the RF method to the estimation of LIB SOH in the investigation. Thirdly, the PSO algorithm was utilized to further optimize the RF model, which was then followed by an examination of the SOH estimation and the comparative analysis is shown in

Table 1. Comparative analysis of reviewed existing works.

Ref.	Approach Used	Dataset Used	Advantages	Disadvantages
[14]	GAN-LSTM-TL	NASA, CALCE	TL improves flexibility across datasets and provides the most accurate prediction, with the lowest MAE and RMSE values	High complexity in combining GAN, LSTM, and TL
[15]	HFCM-LSTM	NASA, Oxford	Data recovery was effective and RMSE value was consistently below 2.3%	Limited information on handling varied battery conditions, and extensive tuning was required
[16]	ICA-Bi-LSTM	NASA-LIB	Detailed health characteristics were utilized, and the MSE was decreased significantly compared to individual models	Complex data extraction
[17]	CNN-Attention-Bi-LSTM	NASA PCoE LIB	Combines CNN for feature extraction and AM for weight assignment. High accuracy, with RMSE values consistently below 0.01	Model complexity and integration of CNN, AM, and Bi-LSTM were challenging
[18]	LSTM	NASA, CALCE	Higher accuracy with 40% cycle starting point	Larger prediction errors towards end of cycle, and performance varies with starting point
[19]	LSTM-TL	NCM, LCO (CALCE) LFP battery dataset (MIT)	Adaptability across voltage ranges and battery types. Error less than 3% with 70% data	Transfer learning adds complexity
[20]	Dilated CNN-BiGRU	Oxford, NASA	Improved spatial and temporal feature extraction and high efficiency with low RMSE	High complexity
[21]	XGBoost	Battery 6	High accuracy and excellent performance, with an MAE of 0.0013% and an RMSE of 0.0019%	Requires careful hyperparameter tuning. Less effective with smaller datasets
[22]	TCN-VAE	Cobaltite, lithium ternary batteries	Enhanced accuracy, with an RMSE of 0.0073	Complexity in combining TCN with VAE. Requires detailed feature extraction
[23]	DRSN-CW-LSTM	Various LIB datasets	Effective joint estimation of SOC and SOH. High noise resistance	Complex integration of noise function
[24]	GRU-ELM	OXFORD	Robust SOH prediction, with an RMSE of 1.2% and an MAE of 1.02%. High R-squared values	Requires extensive temperature data
[25]	CL-Net (ConvLSTM)	NASA, IHEPC, DEMS	Reduces RMSE on NASA and IHEPC datasets. Effective for multi-step SOH forecasting	Complexity in ConvLSTM and LSTM integration
[26]	BA-ELM	NASA	Optimal weight and bias tuning. Lowest MAE (0.4326%) and RMSE (0.5354%)	Requires Bat algorithm for optimization
[27]	XGBoost-MK	Battery 6, Battery 7	High prediction accuracy with XGBoost	Performance depends on feature selection
[28]	RF-PSO	LiFePO4 batteries	Enhanced accuracy with PSO optimization. Effective for large datasets	Requires large training dataset

.

Research Gap Analysis

The studies present a wide range of methodologies, including deep learning architectures like LSTM, ConvLSTM, CNN, and GAN and hybrid models combining different neural network architectures. These approaches leverage the temporal and spatial characteristics of battery data to accurately estimate SOH. Several studies introduce novel model architectures, such as combining GANs with LSTM for feature processing or incorporating attention mechanisms to improve feature weighting. These innovations enhance the creation of more accurate and robust SOH estimation models. Performance metrics such as RMSE, MAE, and MAPE are commonly used to evaluate the accuracy of SOH estimation models. The provided metrics showcase the efficacy of the suggested methods in accurately forecasting SOH. In the future, establishing benchmark datasets and evaluation protocols would help in assessing the relative performance of various SOH estimation models more accurately. Investigating the synergies between different model architectures and feature selection methods could potentially lead to more robust and accurate SOH estimation models. Addressing these research gaps would contribute to the advancement of LIB SOH estimation methods, ultimately supporting the development of more efficient and reliable BMSs for various applications, including electric vehicles. A comprehensive summary of previous studies in the literature review is presented, which also lays the foundation for the proposed method of enhancing SOH estimation capabilities.

3. Proposed Methodology of the Model

This research proposes a model to improve battery SOH prediction in EVs by combining a Diffusion Convolutional Recurrent Neural Network (DCRNN) with the SVM-RFE algorithm DCRNN + SVM-RFE for enhancing classification and feature selection performance. By incorporating these techniques, the model aims to resolve the challenges of battery prediction in EV environments and enhance detection accuracy. This research developed a novel DCRNN + SVM-RFE algorithm for predicting the SOH performance of LIBs in EVs. The model that has been presented consists of several stages: accumulation of datasets, pre-processing, feature extraction, and classification. In the pre-processing stage, the normalization technique is incorporated by using min–max normalization to improve model performance. After pre-processing, SVM-RFE is used for feature selection from the processed data and classification to minimize the complexity of the model. The DCRNN algorithm is used for classification to produce effective output. The effectiveness of this DCRNN + SVM-RFE model is assessed with various battery samples using the CALCE dataset and is subsequently compared to pre-existing models. The DCRNN + SVM-RFE algorithm’s overall process is depicted in Figure 4. It consists of the following stages: data collection, pre-processing, feature selection, classification, training, testing, and performance measures. The following sections explain and discuss the implementation of these stages.

3.1. Dataset Details

The research incorporates experimental data obtained from the LIB datasets provided by the University of Maryland Center for Advanced Life Cycle Engineering (CALCE). The CALCE battery research group utilizes advanced technology to perform cell-to-cell cycle testing, disassembly, and cell assembly, as well as abuse testing (such as overcharging, overheating, and short-circuiting) and material analysis. The purpose of these activities is to guarantee the dependable and secure advancement and functioning of battery parts and systems in real-life scenarios [13]. The CS2 dataset has six sub-datasets. The aging conditions for the cycle are constant voltage charge and constant current discharge.

This study utilizes data obtained from four different series of LiCoO2 batteries with a capacity of 1.1 Ah. The batteries were subjected to testing using an Arbin battery analyzer at ambient temperature. A conventional constant current/constant voltage was utilized during the testing procedure, with 0.5 C of a constant current. Subsequently, the voltage was sustained at 4.2 V till the charging current diminished to 0.05 A, all the while guaranteeing that it remained below the 1 C constant flow electric cutoff voltage of 2.7 V. After experiencing numerous cycles, the battery dropped to 80% of its rated capacity of 0.88 Ah. The battery dataset had a significant number of data anomalies, which were regarded as noise within the dataset. Nevertheless, noise elimination techniques were not implemented in this study. As a result, the derived battery capacity data might not be correctly reflected in the battery capacity data 13. As a result, an algorithm for prediction that exhibits enhanced generalization is necessary. By implementing the suggested model to the dynamic data patterns of the CALCE and utilizing real-world data, it is expected that accurate and significant SOH predictions will be generated, thereby making a valuable contribution to the progress of estimating battery state in EVs. The dataset was split into training and testing sets. Additionally, the dataset utilized in this work is available with open access at: https://calce.umd.edu/battery-data (accessed on 7 May 2024).

3.2. Data Pre-Processing

Data pre-processing is a fundamental step before applying any analysis or modelling techniques to the data. It involves transforming raw data into a clean and usable format to ensure the quality and effectiveness of the analysis. Data cleansing and normalization are components of data pre-processing for LIB time series data used to forecast SOH. The unprocessed data, based on time series, is converted into SOH-based data after the process of data cleaning. Data cleaning, also referred to as data cleansing, is the foundational step in data pre-processing. It involves meticulously examining and correcting errors or inconsistencies present within raw data to enhance their quality. To improve the accuracy and performance of DL models, it is essential to effectively manage the experimental data. Initially, the data are cleansed by eliminating missing data and outliers, which are assessed using intermediate data or moving averages, resulting in the battery data exhibiting periodic degradation traits. Periodically averaging the data to eliminate short-term fluctuations and pre-processing the data ensures that no incorrect values exist that could introduce confounding factors into the model [29]. Data normalization is a technique applied to datasets to improve their quality and prepare them for further analysis or modelling. The dataset based on SOH did not have the same scale. Data normalization is a frequently used technique in in-depth modelling algorithms to enhance the model’s convergence and the accuracy of the forecast [30]. The process of normalization will be carried out using the min–max method, which involves scaling the data to a range of 0 to 1. Therefore, the normalization of the dataset based on SOH using min–max normalization is expressed by the following equation:

$$x_n={\displaystyle \frac{x-x_{min}}{x_{max}-x_{min}}}$$

(4)

where $x_{min}$ indicates the minimum value of the real data, $x_n$ denotes the processed data; the original data are represented by $x$, and $x_{max}$ means the maximum value. Consequently, this methodology facilitates quick and simple data normalization, all the while sticking to a specified range.

3.3. SVM-RFE Model for Feature Selection

Feature selection is a strategy utilized to reduce the number of features by selecting a small subset of relevant ones and removing irrelevant, redundant, or noisy characteristics. Feature selection typically results in improved learning performance, namely higher accuracy in learning, reduced computing expense, and enhanced model interpretability. It is alternatively referred to as variable subset selection, variable selection, or attribute selection.

This technique enhances the efficiency of data mining algorithms, enhances the accuracy of predictions, and improves the understandability of the results. Irrelevant characteristics are characterized by their lack of meaningful information, whereas redundant features do not contribute any further information beyond what is already captured by the currently selected features. Feature selection is now essential for identifying the optimal subset of features and is applied across many domains such as biology, health, finance, image processing, production, and manufacturing. The suggested model utilizes RFE for the process of selecting features. RFE is a strategy used for choosing the most suitable feature subset by considering the acquired model and the accuracy of classification. The traditional RFE method removes features one by one based on their impact on the classification efficiency after constructing a model. A novel RFE methodology was suggested, wherein the evaluation of “feature importance” is prioritized over “classification accuracy” using an SVM model. Consequently, the least significant features are selected for elimination [31].

Figure 5 illustrates the detailed process of feature selection utilizing a feature-importance-based RFE approach. When a model is trained using a training data set, it is possible to obtain feature weights that indicate the significance of each feature. After assigning weights to all features, the feature with the smallest weight value is removed. Subsequently, the classifier undergoes retraining by utilizing the remaining characteristics until all available features have been utilized for training. The feature-importance-based RFE approach allows for comprehensive rating of the features. This methodology is an integrated method of feature selection that has demonstrated robust performance, overcoming the limitations of both filter and wrapper techniques. N denotes the number of features.

Utilizing the SVM-based RFE algorithm, the significance of every feature is determined. The SVM seeks to identify the region of a hyperplane that maximally separates the different classes. The utilization of kernel approaches in the SVM guarantees effective separation of high-dimensional data. The feature weight vectors are derived by utilizing a linear kernel, and those weights are utilized to assess the significance of the features. The quantity of classes is represented by c, and the decision function of the hyperplane by $B\left(y\right)$. If the data contain many classes (i.e., exceeding two), the value of p, which represents the total number of hyperplanes, can be determined using the equation p = c (c − 1)/2. Datasets consisting of multiple classes are defined by Equation (6), whereas Equation (5) concerns the decision functions under the binary classification scenario.

$$B\left(y\right)=sign(x\ast \mu )$$

(5)

$$B\left(y\right)=sign\left(x\ast {\mu }_i\right),i=\mathrm{1,2},3,\dots .,p$$

(6)

In the context of a linear choice function, the vector $y$ represents the components of a certain spectrum, while the vector μ is perpendicular to the hyperplane that defines the function called linear decision. The decision boundary is an essential concept of the SVM, which distinguishes between classes and is determined by certain observations known as support vectors. In the decision-making process, a weighted vector measures the significance of multiple variables. The weighted vector is positioned at the location of a decision limit with the high margin. If the weighted vectors have a high value for a specific feature, this signifies that the mentioned features can effectively distinguish between the classes. The weight values employed in SVM-RFE to determine the significance of variables are derived from Equation (7).

$$V_S={\displaystyle \frac{1}{p}}{\sum }_{j=1}^p{V}_j$$

(7)

SVM-RFE is an effective wrapper and scalable approach that is commonly used in bio-informatics, in comparison to other feature selection methods [31].

3.4. DCRNN Model for Classification

After the features are processed and selected, the classification step using DCRNN is performed on the resulting feature vector. This section provides a brief review of background concepts like CNN and RNN to understand the DCRNN approach.

A CNN is an algorithm for ML that is trained to discover the parameters of a collection of fixed-length filters. Multiple CNN layers can be stacked together to construct CNN networks. In general, these architectural designs are characterized by the presence of a Dropout layer superimposed on every CNN. While CNNs are typically utilized for image identification in the 2D domain, they are also occasionally employed for forecasting time series in the 1D domain. The kernel support adjusts the extent of temporal variability that the filters can capture. Specifically, filters with a greater duration can capture more extended temporal patterns compared to small filters.

RNNs are specifically developed to address the constraints of feedforward neural networks when it comes to simulating sequences. RNNs consist of units, sometimes known as neurons, that can retain information about previous observations. RNNs have the capability to analyze input data by considering both current and previously observed information. Out of all the various suggested topologies for RNNs, LSTM has shown significant effectiveness in accurately representing long-term relationships between input data points over time. In more precise terms, the LSTM unit executes the subsequent operations recursively when provided with the current observation $\left(x\left(t+1\right)\right)$ and the input vector $x\left(t\right)$:

$$j\left(t\right)=\sigma (V_j\left[x\left(t\right),g\left(t-1\right)\right]+d_j$$

(8)

$$\tilde{a}=\tanh (V_a\left[x\left(t\right),g\left(t-1\right)\right]+d_a)$$

(9)

$$e\left(t\right)=\sigma (V_e\left[x\left(t\right), g\left(t-1\right)\right]+d_e$$

(10)

$$a\left(t\right)=e\left(t\right) ʘ a\left(t-1\right)+j\left(t\right) ʘ \tilde{a}\left(t\right)$$

(11)

$$p\left(t\right)=\sigma (V_p\left[x\left(t\right),g\left(t-1\right)\right]+d_p$$

(12)

$$g\left(t\right)=p\left(t\right) ʘ \tanh \left(a\left(t\right)\right)$$

(13)

Element-wise matrix multiplication, denoted by ʘ, is performed on the matrices $V_j, V_a, V_e, {\mathrm{a}\mathrm{n}\mathrm{d} V}_p$, and the learnable kernels and biases $d_j, d_a, d_e, \mathrm{a}\mathrm{n}\mathrm{d} d_p$, respectively. The input, output gates, forget, input modulation, and cell, represented by $j, \tilde{a}, e, a,\mathrm{a}\mathrm{n}\mathrm{d} p$, respectively, collectively execute operations to enable the LSTM to selectively retain or discard information from the input data. Input data consist of g and x in a sequential fashion, and the hidden state g contains the information that the LSTM unit recalls from prior observations. There have been other variations of LSTM units suggested in academic publications, and the previous explanation only pertains to its most prevalent implementation. An instance of the GRU in the RNN is commonly employed as a simplified alternative to the LSTM. An example of the application of this GRU is the DCRNN, which was recently introduced.

ML algorithms were originally developed to construct models from data whose domains are defined by Euclidean principles. However, the application of these algorithms to different data structures, such as graphs, is a complex challenge. To be more specific, a graph is defined as an ordered pair $C=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ represents the set of nodes and $\mathcal{E}$ represents the set of edges. Assuming the graph contains attributes, including qualities of nodes and edges, $C$ is defined as $\mathsf{Y}\mathsf{є}{R}^{N X P}, W \mathsf{є}{R}^{N X N}$, where the number of nodes is represented by N and P represents the quantity of attributes associated with the nodes. The feature matrix $\mathsf{Y}$ represents the features of the nodes, whereas the weighted matrix $W$ represents the relationships between the nodes, such as the relationship matrix of the network. Conventional ML models like LSTM or CNN are capable of efficiently handling $\mathsf{Y}$, but they cannot incorporate the knowledge that is represented by $W$. The inclusion of relational information in the feature matrix is an effort to improve it, as suggested in recent academic literature. This section presents a brief overview of a significant solution predicated on the idea that the relationship between two nodes can be represented as a diffusion process. The probability of a random walk from node one to node two can be determined by utilizing the state transition matrix $B_0^{-1}·W$. Here, $B_O$ refers to the graph’s out-degree diagonal matrix.

The diffusion process provides valuable insights into the impact that each node has on all the others. The implementation of appropriate convolutional techniques can be utilized to enhance the feature space ($\mathsf{Y}$) by utilizing contextual knowledge and performing filtering. When a signal of graph $\mathsf{Y}\mathsf{є}{R}^{N X P}$ and the filter $f_{\mathsf{\theta }}$ undergo K-step Diffusion Convolution (DC), the resulting product is denoted as $\ast C$. The DC of K-steps between a graph signal $\mathsf{Y}\mathsf{є}{R}^{N X P}$ and $f_{\mathsf{\theta }}$ is referred to as $\ast C$ and is mentioned as:

$${\mathsf{Y}}_{\ast C} f_{\theta }={\sum }_{k=0}^{K-1}{{ (\theta }_{k,1}({B_O}^{-1} W) }^k+{{ \theta }_{k,2}({B_O}^{-1} W^T) }^k ) . \mathsf{Y}$$

(14)

where $\mathsf{\theta } є R^{K X 2}$ are the filter parameters, ${B_O}^{-1}W$ is the diffusion process’s state transition matrix, and ${B_O}^{-1} W^T$ is a transpose of it. The DC operator can serve as a fundamental component of a DC layer in a neural network. The parameter θ can be learned using standard training methods, such as backpropagation. More precisely, in the following manner, this layer may be trained to transform the feature matrix $\mathsf{Y}\mathsf{є}{R}^{N X P}$ to an output $H є R^{N X Q}$:

$${\mathrm{H}}_{:,q}=\sigma \left({\sum }_{p=1}^P{\mathsf{Y}}_{:,p\ast C}{f\Theta }_{q,p,:,:}\right)$$

(15)

where $\Theta є R^{QXPXKX2}$ is the trainable parameter tensor. The RNN unit can be transformed into the DCGRU by substituting the matrix multiplications mentioned with the DC operation. To ensure accuracy, the authors introduce an altered edition of the RNN GRU that was discussed. This modified version is technically defined by the following equations:

$$r\left(t\right)=\sigma ({\Theta }_r\ast C \left[\mathsf{Y}\left(t\right), H \left(t-1\right)\right]+{\mathrm{b}}_r)$$

(16)

$$C\left(t\right)=\tanh ({\Theta }_c\ast C \left[\mathsf{Y}\left(t\right), (r\left(t\right)ʘ H \left(t-1)\right)\right]+{\mathrm{b}}_c)$$

(17)

$$u\left(t\right)=\sigma ({\Theta }_u\ast C \left[\mathsf{Y}\left(t\right), H \left(t-1\right)\right]+{\mathrm{b}}_u)$$

(18)

$$\mathrm{H}\left(\mathrm{t}\right)=u\left(t\right)ʘH \left(t-1\right)+(1-u\left(t\right)ʘ C\left(t\right)$$

(19)

where $ʘ$ is the tensor multiplication at element-wise. The reset, update, and cell gates are denoted by the symbols r, u, and C, respectively and execute analogous functions to the gates outlined in Section 3.4. The kernel parameters learned during training are denoted as ${\Theta }_r$, ${\Theta }_u$, ${\mathrm{a}\mathrm{n}\mathrm{d} \Theta }_c$, together with their respective biases ${\mathrm{b}}_r$, ${\mathrm{b}}_u$, and ${\mathrm{b}}_c$. The variable $\mathsf{Y}\left(t\right)$ represents the input of the model at time t, while H(t) represents the output of the model at the same time [31,32,33].

This paper employs a two-step approach for SOH prediction, specifically utilizing SVM-RFE and DCRNN. Firstly, SVM-RFE is used for selecting an essential subset. By eliminating irrelevant or redundant data, the model becomes more efficient and can focus on the crucial elements that affect SOH. Once the SVM-RFE algorithm has identified the most important characteristics, they are fed into a DCRNN model. DCRNN combines the advantages of both CNNs and RNNs. CNNs can detect patterns in the characteristics of data, whereas RNNs are capable of processing input that is arranged in a sequential manner. The DCRNN model is probably trained using CALCE battery data that include known SOH values. This training enables the model to acquire knowledge about the connections between the chosen features and the relevant SOH. After undergoing training, the model can make predictions about the SOH of a battery using new and previously unseen feature data.

4. Experimental Analysis

4.1. Experimental Setup

This section presents the results of experiments carried out utilizing the DCRNN + SVM-RFE research model. Performance evaluation of this research was carried out using the CALCE dataset. The DCRNN + SVM-RFE model was developed using the MATLAB 2020A Simulink tool. The computer environment consisted of a Core i7-620M central processor unit, 16 gigabytes of RAM, and a 64-bit version of the Windows 10 operating system. The number of samples and specifications of each train and test sets of the CALCE dataset are presented in

Table 2. Summary of the CALCE dataset.

Testing Batteries	Training Batteries	Training and Validation Samples	Testing Samples
#35	#36, #37, #38	2901	882
#36	#35, #37, #38	2847	936
#37	#36, #36, #38	2814	969
#38	#35, #36, #37	2787	996

.

4.2. Performance Metrics

To evaluate the performance of the DL DCRNN + SVM-RFE model developed in this work, metrics like MAPE, MAE, MSE, and RMSE were utilized. The following expressions serve as metric values for evaluating the differences between the model’s actual value and predicted value [33]:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(\widehat{Y_i}{-Y}_i\right)}^2}$$

(20)

$$MAE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left|\widehat{Y_i}{-Y}_i\right|$$

(21)

$$MSE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(\widehat{Y_i}{-Y}_i\right)}^2$$

(22)

$$MAPE={\displaystyle \frac{100\%}{N}}{\sum }_{i=1}^N\left|{\displaystyle \frac{\widehat{Y_i}{-Y}_i}{Y_i}}\right|$$

(23)

4.3. Performance Evaluation

This section provides an assessment of the comparative analysis conducted on the proposed predictive model. The data collected from the dataset were given as input to the proposed model. The data from the dataset were divided into test and training sets. According to these sets, the performance evaluation was performed with all the models. The performance of the DCRNN + SVM-RFE method in battery SOH prediction was measured based on various parameters like MAPE, MSE, MAE, and RMSE.

As shown in

Table 3. Proposed model’s performances of SOH estimation using various samples.

Battery Samples	RMSE	MAE	MSE	MAPE
#35	0.012	0.009	0.025	0.30
#36	0.014	0.010	0.023	0.28
#37	0.018	0.012	0.030	0.38
#38	0.013	0.014	0.026	0.32
AVERAGE	0.014	0.011	0.026	0.32

, the proposed DCRNN + SVM-RFE model produces excellent SOH estimation results across various battery samples. The results indicate that all RMSEs are within 0.02%, MAEs are within 0.015%, MSEs are within 0.032%, and MAPEs are within 0.40%. The average values of RMSE, MAPE, MSE, and MAE are 0.014, 0.011, 0.026, and 0.32. As a result, the proposed network demonstrates the ability to accurately estimate SOH values in the presence of various battery samples.

Figure 6 shows the graphical plot of the proposed model RMSE parameter for SOH prediction. The RMSE values vary from a maximum of 0.018 to a minimum of 0.012. In the #35 sample, the RMSE is 0.012, indicating a relatively low error. Lower RMSE values indicate better model performance.

The graphical plot of MAE values for different battery samples is shown in Figure 7. The MAE values vary from a maximum of 0.014 to a minimum of 0.009. An average value of 0.011 was obtained by the DCRNN + SVM-RFE model. The MAE values vary across samples but generally remain relatively low. For #35, the MAE is the lowest, at 0.009, indicating better accuracy in predictions. Like RMSE, lower MAE values indicate better performance.

Figure 8 illustrates the MSE values of the proposed model. The MSE values vary from a maximum of 0.030 to a minimum of 0.023, with an average of 0.026. The MSE is calculated as the mean of the squares of the errors. Lower MSE values indicate better model performance. The MSE achieved its minimum value of 0.023 for battery sample #36.

Figure 9 illustrates the MAPE values of the proposed model. The MAPE values vary from a maximum of 0.38 to a minimum of 0.28. Low MAPE values were consistently attained by the model, suggesting minimal error in the estimation of SOH. The MAPE with the highest value is observed was #37 (at 0.38), indicating relatively higher error compared to the other samples.

Table 4. Comparison of performance analysis.

Model	RMSE	MAE	MAPE	MSE
LSTM [18]	0.08	-	9.98	-
LSTM-TL [19]	0.59	2.38	-	-
CNN-BiGRU [20]	1.83	3.21	-	-
TCN [22]	0.0170	0.013	-	-
DRSN-CW- LSTM [23]	-	4.73	-	0.79
CL-Net [25]	0.205	0.151	-	0.042
BA-ELM [26]	0.5354	0.4326	0.44	-
RF improved [28]	0.840	-	0.968	-
GAN-LSTM-TL [32]	0.0275	0.0208	-	-
PROPOSED MODEL	0.014	0.011	0.32	0.026

compares the proposed model’s testing performance with existing models included in the review. The performance analysis was compared with models like GAN-LSTM-TL, LSTM, LSTM-TL, CNN-BiGRU, TCN, DRSN-CW-LSTM, CL-Net, BA-ELM, and RF improved. The proposed model produces better results compared to other models.

Figure 10 shows a graphical plot of the comparison of RMSE values with various models. Based on the test performance comparison made with the existing models, it is clear that the DCRNN + SVM-RFE model has a lower error rate than the other models. After comparing the models, it was determined that the TCN model has the smallest RMSE, at 0.0170. Models such as CNN-BiGRU and RF improved exhibit relatively higher RMSE values, indicating lower accuracy compared to the proposed model. The research model has an RMSE of 0.014, which is lower than the other models. Figure 11 shows a graphical plot of the comparison of MAE values. The MAE value of the research model was 0.011, which is lower than the compared models. The TCN model also demonstrates a low MAE of 0.013, followed by the GAN-LSTM-TL model (0.0208). Greater MAE values are observed in models such as DRSN-CW-LSTM and CNN-BiGRU, which signifies more substantial differences between the predicted and observed SOH values.

Figure 12 shows a graphical plot of the comparison of MSE values. The proposed model stands out with an MSE of 0.026, showcasing its superior accuracy compared to the DRSN-CW-LSTM & CL-Net models. Figure 13 shows a graphical plot of the comparison of MAPE values. The proposed model achieves a MAPE of 0.32%, indicating a high level of precision in its predictions. The LSTM model exhibits a relatively higher MAPE value (9.98), indicating lower accuracy compared to the proposed model. As a result, according to this comparison, the DCRNN + SVM-RFE model obtained better results than the compared models in this research.

The proposed DCRNN + SVM-RFE model demonstrates excellent performance in SOH prediction, with minimal error values across various evaluation metrics. This indicates its potential for accurately estimating the SOH of LIBs. This paper conducted a comparative analysis with various existing models, demonstrating the superiority of the research model in SOH prediction across different battery samples. This thorough evaluation enhances the credibility of the proposed approach. However, there are few limitations present in the work. The effectiveness of the proposed work is highly influenced by the quality and representativeness of the training data. Any biases or limitations in the dataset could affect the generalizability and reliability of the model’s predictions.

5. Conclusions

In this research, a combined DCRNN + SVM-RFE model for the SOH estimation of LIBs was proposed. The research model has a series of workflows including dataset collection, pre-processing of data, selection of features, and classification. Data collected from the dataset are pre-processed by min–max normalization. After pre-processing the data, feature selection is performed by the SVM-RFE algorithm. To improve the accuracy of the system, the Diffusion Convolutional Recurrent Neural Network (DCRNN) algorithm is used for the classification of features. The dataset was partitioned for training and evaluating the proposed model. DCRNN + SVM-RFE model performance was assessed by SOH parameters such as RMSE, MAE, MSE, and MAPE. The proposed model generates accurate results, with all RMSEs being within 0.02%, MAEs being within 0.015%, MSEs being within 0.032%, and MAPEs being within 0.41%. The mean values of RMSE, MAPE, MSE, and MAE were 0.014, 0.011, 0.026, and 0.32. The proposed results were compared with the various models like GAN-LSTM-TL, LSTM, LSTM-TL, CNN-BiGRU, TCN, DRSN-CW-LSTM, CL-Net, BA-ELM, and RF improved. The comparative analysis showed that the proposed DCRNN + SVM-RFE model was the best method for SOH prediction for different battery samples, with minimum error values of RMSE, MAE, MSE, and MAPE. In conclusion, the proposed SOH estimation method for LIBs, which is based on DCRNN + SVM-RFE, has been rigorously validated and has yielded excellent results. Future works could explore the accuracy of the proposed model by testing it under a wider range of operating conditions, including variations in temperature, humidity, and cycling patterns. This would offer a more extensive comprehension of its performance in various real-life situations.