Two-phase early prediction method for remaining useful life of lithium-ion batteries based on a neural network and Gaussian process regression

Abstract

Lithium-ion batteries (LIBs) are widely used in transportation, energy storage, and other fields. The prediction of the remaining useful life (RUL) of lithium batteries not only provides a reference for health management but also serves as a basis for assessing the residual value of the battery. In order to improve the prediction accuracy of the RUL of LIBs, a two-phase RUL early prediction method combining neural network and Gaussian process regression (GPR) is proposed. In the initial phase, the features related to the capacity degradation of LIBs are utilized to train the neural network model, which is used to predict the initial cycle lifetime of 124 LIBs. The Pearson coefficient’s two most significant characteristic factors and the predicted normalized lifetime form a 3D space. The Euclidean distance between the test dataset and each cell in the training dataset and validation dataset is calculated, and the shortest distance is considered to have a similar degradation pattern, which is used to determine the initial Dual Exponential Model (DEM). In the second phase, GPR uses the DEM as the initial parameter to predict each test set’s early RUL (ERUL). By testing four batteries under different working conditions, the RMSE of all capacity estimation is less than 1.2%, and the accuracy percentage (AP) of remaining life prediction is more than 98%. Experiments show that the method does not need human intervention and has high prediction accuracy.

Graphical abstract

Keywords

lithium-ion batteries / RUL prediction / double exponential model / neural network / Gaussian process regression (GPR)

1 Introduction

The global transition toward sustainable and recyclable non-fossil energy sources is an essential measure in alleviating the energy crisis and mitigating environmental pressures [1–3]. Harnessing natural energy and converting it into electricity, which can be stored electronically, is instrumental in paving the way for a sustainable future [4–6]. Compared with lead-acid and nickel-metal hydrate batteries, lithium-ion batteries (LIBs) are being increasingly used to replace fossil fuels and vigorously developed for the market of electric vehicles and other transportation sectors because of their elevated specific power, specific energy, long life, and low self-discharge rate [7–9]. As a battery is used, its capacity and remaining useful life (RUL) will decay in different ways under various operating conditions. If the maximum available capacity of LIBs drops to a certain specified threshold, serious consequences, such as rapid performance degradation or even failure, fire, explosion, and other safety incidents, can occur. Therefore, the accurate prediction of battery life is essential for battery maintenance and guarantee of normal battery usage. Moreover, the timely replacement of batteries that will reach their life failure threshold provides a guarantee for the safe operation of electric vehicles [5]. Furthermore, battery life prediction based on early cycle data can significantly save time and cost for experiments, opening up new opportunities for battery production, optimization, and evaluation.

Academics have proposed many approaches to forecast the RUL of batteries in recent years. The currently available research primarily categorizes techniques for estimating the RUL of LIBs into three classes: data-driven methods, model-based methods, and other methods. A mechanistic model-based degradation model was established by analyzing the internal structure of LIBs. This model can be divided into electrochemical models (EMs) [10,11], equivalent circuit models (ECMs) and empirical degradation models [12]. EMs generally employ reaction dynamics and use the porous electrode theory to accurately describe battery states. However, model parameters are susceptible to temperature and other factors, making parameter identification complicated model. ECMs use circuit analysis to build an equivalent mathematical model of a battery, considering the aging mechanism of the storm. However, model construction relies on impedance and other data that are difficult to obtain in practice. ECMs construct exponential and polynomial models based on the degradation trends of LIBs and combine them with adaptive filtering algorithms for RUL prediction [13]. Nevertheless, these models can only approximate battery degradation, whose prediction performance is limited by the combined filtering algorithms [12,14–16]. Although model-based approaches are explained well, they are constructed by approximating the degradation mechanism. Thus, model construction and the solution procedure must be simplified to be implemented in real applications. Moreover, they generate significant RUL errors in early prediction of LIBs.

Given the limitations of model-based methods, data-driven and machine learning techniques have become attractive for prediction methods. A data-driven approach [17] does not require an analysis of the internal structure of LIBs [18–20]. By extracting the characteristic parameters from the measured data of external battery variables (e.g., current, voltage, and temperature) [21,22] to describe battery life and obtain the degradation characteristics and trend of a battery to construct RUL prediction models, the effect of the accuracy of a model on prediction accuracy is avoided, and this method is widely used in battery RUL prediction applications [23,24]. In the past years, scholars have developed many data-driven methods for early RUL (ERUL) prediction of LIBs, such as support vector machines (SVMs) [25–27], relevance vector machines (RVMs) [28], autoregressive models, Bayesian models, Gaussian process regression (GPR) [29–31], long short-term memory (LSTM) [32], artificial neural networks (ANNs) [33–35], convolutional neural networks (CNN) [36,37], and recurrent neural networks (RNNs) [38,39]. However, most of the aforementioned methods are single prediction models that suffer from the problems of poor generalization and accuracy. Due to the substantial requirement of historical LIBs capacity data for training, these methods may exhibit suboptimal performance in the early prediction of LIBs RUL.

In addition to the single-model data-driven early prediction methods described above, multi-model hybrid ensemble learning methods have been used to balance accuracy, and their results have been interpreted as ERUL predictions for LIBs. The authors of Refs. [40,41] proposed an unscented Kalman filter (UKF) combined with relevance vector regression (RVR) for predicting the RUL and short-term capacity of batteries. They developed an improved particle swarm optimization (PSO)-support vector regression (SVR) model to estimate RUL at different failure thresholds. The authors of Ref. [42] proposed a data-driven algorithm based on charging data. The optimal feature set was determined by extracting the statistical features of battery charging data and conducting correlation analysis and feature selection. The prediction error caused by local capacity changes was offset using two residual models based on the GPR algorithm. The authors of Ref. [43] proposed early battery aging data to achieve degradation patterns (DPs) recognition and transfer learning (TL), which can effectively enhance the accuracy of state of health (SOH) estimation. The authors of Ref. [44] proposed a fusion neural network model by combining the broad learning system algorithm and the long short-term memory neural network (LSTM NN) to predict the capacity and RUL of LIBs accurately. The authors of Ref. [45] proposed a novel hybrid approach to forecasting battery future capacity and RUL, which is presented by combining the improved variational modal decomposition (VMD), particle filter (PF), and GPR. The authors of Ref. [46] proposed incremental capacity analysis (ICA) analysis and developed improved BLS network-based the SOH estimation technology for LIBs. The authors of Ref. [20] established a linear model and proposed to learn the global mapping function between 124 LIBs, and used the variance of the discharge capacity difference of the 10th and 100th cycles to predict the ERUL of the battery. The authors of Ref. [47] proposed a novel transfer learning strategy in combination with the cycle life prediction technology, using feature extraction and deep learning technology to achieve accurate cycle life prediction. First, the information about the two-stage aging process is learned through an offline basic aging model. Then, the aging trajectory is predicted and the uncertainty is quantified by the Bayesian model transfer technique. The aforementioned ensemble methods require a lot of battery charge−discharge cycles for training, and the network structure is complex and computationally intensive. Accordingly, an ERUL prediction method for LIBs with low computational effort and significant accuracy must be developed.

In this paper, an ERUL prediction method for LIBs is proposed and successfully validated on the LIBs data set used in Refs. [20,48]. This paper is contributive because, in the first phase, feature factors related to capacity degradation are extracted. A neural network is constructed for training, and an initial prediction of each lithium battery cycle life is performed. Moreover, the two feature factors with the most significant Pearson coefficients and the lifetime normalized by the neural network prediction form a 3D space. The closest Euclidean distance between each battery in the test set and the training and validation sets is calculated, and the battery with the shortest distance is considered to have a similar degradation pattern. The capacity degradation trend of the selected LIBs is determined based on the dual exponential model (DEM) [49]. Furthermore, in the second phase, the selected DEM parameters are used as the initial parameter of GPR, and the identified DEM is then trained according to the end-of-period monitoring (EOM) capacity of the test set. The trained GPR is used to predict the RUL of LIBs in a given EOM cycle.

2 Selection of data set and feature factor

2.1 Introduction to data set

The data set contains 124 high-power LiFePO₄ (LFP)/graphite cells with cycle lives ranging from 147 to 2234 cycles (72 different fast-charging strategies with over 90000 cycles) [20], shown in Fig.1. In this paper, 42 of the first 84 LIBs are randomly selected as the training data set (denoted as LIBs-A), 42 are selected as the validation data set (denoted as LIBs-B), and the remaining 40 are selected as the test data set (denoted as LIBs-C).

The LIBs have a notional rated capacity of 1.1 Ah and an upper and lower cut-off of 3.6 and 2.0 V, respectively. In different charging techniques, the cycle life of individual batteries varies significantly. All batteries are cycled at 30 °C in ambient circumstances until they reach the end of life (EOL), or when their capacity is lower than 80% of nominal.

2.2 Extraction of feature

An in-depth study of the causes of LIBs degradation indicates that the loss of active substances, the loss of lithium-ion storage, and the increase of internal resistance are the three main factors leading to the gradual decline of battery performance. The loss of active material is the capacity loss of the positive and negative active materials of the battery, which is caused by the peeling off of electrode materials, structural changes, or interface reactions. The lithium-ion storage loss is caused by the capacity loss of lithium-ion in the battery, the electrode structure damage in the process of lithium-ion insertion and de-insertion, the reduction reaction of lithium salt in the electrolyte, or the modification of functional substances. The increase of internal resistance is due to the increase of internal resistance of the battery, the change of electrolyte concentration, the accumulation of electrode materials, and the degradation of electrolytes under the action of lithium salt. Increasing internal resistance will lead to energy loss and decline in power output. By studying the degradation causes of LIBs, the features related to battery degradation can be identified, such as the loss of active substances, the loss of lithium-ion storage, and the increase of internal resistance, which provide an important guidance and basis for the prediction of battery RUL and the improvement of battery design.

2.2.1 Features of IC curve

The incremental capacity curve illustrates the correlation between the voltage and capacity of an LIB during discharge (dQ/dV) [50], as shown in Fig.2(b). It reflects the electrochemical reactions within a battery. The relationship between discharge voltage and capacity is given by

(1)V=f( Q),

(2)(f^{-1}{)}^{\prime }=\frac{\text{d}Q}{\text{d}V}=\frac{I\times \text{d}t}{\text{d}V}= I\times \frac{\mathrm{d}t}{\mathrm{d}V},

where Q, I, V, and t are the discharge capacity, current, voltage, and time, respectively.

By observing the gradual changes in the IC peak curve throughout the battery cycle, the aging mechanism of the battery can be understood. The maximum values of the horizontal and vertical coordinates on the IC curve are used as the extracted features of F1 and F2.

2.2.2 Features of discharge voltage curve

Loss of lithium-ion stocks can be reflected by discharge voltage curves. Several features are calculated from the discharge voltage curve to obtain the electrochemical mode of a single cell during cycling. In particular, the evolutionary patterns between cycles of Q(V) and the profile of the discharge voltage are indicated as a function of voltage for a given cycle as shown in Fig.2(a). The voltage range is common to each cycle. Therefore, the features of F3, F4, and F5 are proposed based on comparing the period to capacitance as a function of voltage.

Discharge voltage curves can show the loss of lithium-ion stocks. To determine the electrochemical mode of a single cell during cycling, many properties are identified from the discharge voltage curve. As illustrated in Fig.2(a), the evolutionary patterns between cycles and the profile of the discharge voltage are described as a function of voltage for a specific cycle [20]. Because the voltage range is constant throughout the cycle, the features of F3, F4, and F5 are proposed by comparing the period to capacitance as a function of voltage.

(3)\mathrm{\Delta }Q(V)= Q_{100} (V)-Q_{10}(V), \mathrm{\Delta }Q(V) \in {\mathbb{R}}^p,

(4)\mathrm{\Delta }\overline{Q}(V)=\frac{1}{p}\sum _{i=1}^p\mathrm{\Delta }Q (V),

(5){\mathit{b}}^{\ast }=\underset{b}{\arg min} \frac{1}{d}\Vert q-Xb{\Vert }_2^2 ,

where d is the number of cycles used in the prediction, and q\in R^d is a vector of discharge capacity as a function of the cycle number.

2.2.3 Features of IR curve

The primary contributor to battery aging is the increase in internal resistance. Fig.2(c) displays the IR values of two lithium-iron phosphate batteries with varying lifecycles: 1748 and 426 charge−discharge cycles. The results of this experiment are shown in Fig.2. From Fig.2(c), a significant difference is observed between the first 100 charge/discharge cycles of the two batteries. This discrepancy can be employed to model building and remaining service life prediction, based on which, the features of F7 and F8 are proposed.

2.2.4 Features of temperature profile

Changes in temperature exert an effect on the mobility of ions and the conductivity of electrolytes within a cell, which are key factors for degrading the performance of cells from the electrochemical perspective. The overall temperature fluctuates with increasing number of cycles, as illustrated in Fig.2(d), and the feature of F9 is proposed for this reason.

2.3 Summary of feature factors

In this paper, nine feature factors are extracted as data-driven input variables for the initial prediction of life span in the first phase. The feature factors are selected as indicated in Tab.1 and distributed as shown in Fig.3.

3 Methods

This section details the DEM, neural network structure, and GPR algorithm for two-phase ERUL predictions. DEM is the best model for predicting battery physical degradation, and neural networks are used to handle the intricate mapping relationship between characteristic factors and battery cycle life. Preliminary predictions of cycle life and RUL were achieved. GPR is a nonparametric Bayesian regression method that predicts the predicted results and estimates the associated uncertainties. It does this by constructing a probability model using historical data. A series of performance evaluation indicators were proposed to evaluate cycle life and predict performance, including root mean square error (RMSE) and mean absolute percentage error (MAPE). These indicators can be used to compare the accuracy and stability of different algorithms or methods in predicting the RUL of LIBs, assessing their feasibility and practicality.

3.1 DEM algorithm

It is shown that DEM is the best model for LIBs physics-based degradation model [49,51]. DEM is based on the observation of the gradual degradation of the battery in the process of use, assuming that the capacity degradation of the battery can be described by two exponential functions. Using this model, the degradation trend of the battery can be predicted according to the current state of the battery, i.e., the number of cycles or capacity degradation. Therefore, this paper uses it to characterize the capacity degradation tendency of LIBs, whose equation could be described as

(6)y_k=a \times \exp (b\times k)+c\times \exp (d\times k),k =1,2, \dots ,n,

where k represents the kth cycle, and y_k represents the capacity of that cycle.

3.2 Structure of neural network

The neural network includes two phases of forward and backward propagation. The introduced nonlinear activation function provides the network with the capability to fit nonlinear data into the structure of the neural network as illustrated in Fig.4.

The network structure has input layer neurons that represent the feature factors extracted in Section 2.3 and output layer neurons that represent RUL. The hidden layer neurons are set by the conditions, and the hidden layer output is calculated by the weights and deviations between the hidden and previous layers.

(7)\begin{array}{rl}& v(x_1,x_2,\dots ,x_n)=\varphi (\sum _{j=1}^l{u}_i(x_i)+b)=\varphi (\sum _{j=1}^l(w_{ij}\times x_i)+b_i),\\ & j=1,2,\dots ,l,\end{array}

where l is the number of neurons in the hidden layer, and \varphi is the hidden layer activation function. The final output of the multilayer perceptron is described as

(8)y(x_1, x_2,\dots , x_n)=\varphi (\sum _l^n(w_{jk}\times x_i)+b_j),k =1,2, \dots ,m,

where m is the number of neurons in the output layer , and w_jk and b_j are the connection weights and deviations between the hidden and output layers, respectively.

3.3 GPR algorithm

GPR is a set of random variables, which is regarded as an efficient method for regressing a Gaussian process. The coordinates of the ith point in space is x⁽ⁱ⁾, the data output from that point is y⁽ⁱ⁾, and the expression through the GPR model is described as

(9)y^{(i)}= f(x^{(i)})+{\epsilon }^{(i)}, f(x^{(i)}\text{) =}{x}^{\text{T}}w,i= 1,2,\dots ,n,

where {\epsilon }^{(i)} denotes the noise variable that follows N\left(0,{\sigma }^2\right). In GPR, a finite number of random variables f\left(x^{(i)}\right) follows the joint Gaussian distribution described as

(10)f(x)\sim N\left(\mu (x),k\left(x,x^{\mathrm{\prime }}\right)\right),

where x=\left[x_i,x_j,x_m,\cdots \right] is the dimension extracted from the Gaussian process, x_i represents the ith dimension; \mu (x): R^n\to R^n stands for mean function, which returns the mean value of each dimension; k\left(x,x^{\mathrm{\prime }}\right): R^n\times R^n\to R^n is the covariance function (also called kernel function n), which returns the covariance matrix between the dimensions of two vectors [45].

Different kernel functions provide varying properties of Gaussian processes. In this paper, the Matern 5/2 kernel function is chosen, which has a stronger generalization capability. The kernel function is described as

(11)k\left(x_i,x_j\right)= K_{(M,5/2)}= {\delta }_f^2\left(1+\frac{\sqrt{5}d}{l}+\frac{5d^2}{3l}\right)\exp \left(-\frac{\sqrt{5}d}{l}\right),

where d=\sqrt{{\left(x-x^{\mathrm{\prime }}\right)}^{\text{T}}(x-x^{\mathrm{\prime }})} is the Euclidean distance, and {\sigma }_f and l are hyperparameters.

The prior distribution of the function space could be described as

(12)f(x)\sim N\left(\mu (x),k\left(x,x^{\mathrm{\prime }}\right)\right).

Set this process describe as

(13)f(x)-N\left({\mu }_f,K_{ff}\right).

Assuming the observed data, i.e., the given discrete data are (x*, y*), and assuming that y* and f(x) conform to the joint Gaussian distribution, then the joint probability could be described as

(14)\left[\begin{array}{c}f(x)\\ y^{\ast }\end{array}\right] \sim N\left(\left[\begin{array}{l}{\mu }_f\\ {\mu }_y\end{array}\right]\cdot \left[\begin{array}{ll}{K}_{ff}& K_{fy}\\ K_{fy}^{\text{T}}& K_{yy}\end{array}\right]\right),

where K_{ff}=k( x,x) ,K_{fy}=k(x, x^{\ast }),K_{yy} =k(x^{\ast },x^{\ast }), X is the independent variable to be predicted, and X is the known observation independent variable. The Bayesian probability expression can be used to deduce

(15)f(x)\sim N\left(K_{fy}^{\text{T}}+{\mu }_f,K_{yy}-K_{fy}^{\text{T}}{K}_{ff}^{-1}{K}_{fy}\right).

The equation for the predicted average could be described as

(16)y_{\text{mean}}=K_{fy}^{\text{T}}{K}_{ff}^{-1}{y}^{\ast }.

The predicted error matrix equation could be described as

(17)y_{\sigma }=K_{yy} -K_{fy}^{\text{T}}{K}_{ff}^{-1}{K}_{fy}.

Solving hyper parameters using maximum likelihood estimation, the equation could be described as

(18)\begin{array}{rl}\log p(y\mid \delta ,l)& =\log N\left(0,K_y(\delta ,l)\right)\\ & =-\frac{1}{2}{y}^{\text{T}}{K}_{yy}^{-1}y-\frac{1}{2}\log \left|K_{yy}\right|-\frac{N}{2}\log (2\pi ).\end{array}

3.4 Proposed integrated forecasting model

The steps of the two-phase forecasting are as follows.

3.4.1 Neural network training and preliminary prediction

1) The data set was divided in the same way as in the Ref. [20], with the training data set marked as A, the validation data set marked as B, and the test data set marked as C.

2) Feature factors were extracted as the input data set X_{\text{train}}, and the number of LIBs charge/discharge cycles was used as the label set Y_{\text{test}} to train the neural network test data set for the initial prediction.

3) The DEM for each cell in the test data set was determined based on the initial predicted cycle life.

3.4.2 Prediction of RUL

1) The identified DEM parameters were used as the initial parameter of GPR.

2) The EOM of LIBs was tested by using the trained GPR.

3) EOM was utilized to make predictions on the capacity of LIBs and the corresponding confidence interval for future cycles was estimated.

4) When the capacity predicted by the EOM model reached the fault threshold, the RUL of the test set and its corresponding confidence interval were calculated.

The flowchart of the method proposed in this paper is shown in Fig.5.

3.5 Evaluation indicators for cycle life and RUL

In the initial prediction phase, the MAPE and RMSE were used as assessment metrics for cycle life prediction [20]. The equations are

(19)\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=\frac{1}{n}\sum _{i=1}^n\frac{\left|N_i-{\hat{N}}_i\right|}{N_i}\times 100\mathrm{\%},

(20)\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(N_i-{\hat{N}}_i\right)}^2},

where n is the number of tested LIBs, and N_i and {\hat{N}}_i represent the ith actual and predicted cycle life.

The RUL prediction segment uses three commonly used metrics, i.e., AE (absolute error), AP (accuracy percentage), and TS (trend stability), to assess the predictive accuracy and stability of the RUL prediction results for test LIBs. The equations of the three indicators are [20]

(21)\text{AE =}\left|N-\hat{N}\right|,

(22)\mathrm{A}\mathrm{P}=(1-\frac{\left|N-\hat{N}\right|}{N})\times 100\mathrm{\%},

(23)\text{TS}=\sqrt{\frac{1}{N}\sum _{i=t_{\text{EOM}}+1}^{t_{\text{EOM}}+N}{(y_i-\hat{y})}^2},

(24)\text{RMSE =} \sqrt{\frac{1}{N}\sum _{i=1}^N{(y_i-\hat{y})}^2}\times 100\mathrm{\%},

(25)\text{MAPE =} \frac{1}{n}\sqrt{\sum _{i=1}^n\left|\frac{y-\hat{y}}{y}\right|}\times 100\mathrm{\%},

where N and {\hat{N}}_i represent the actual RUL and predicted RUL of the test data set, y_i and {\hat{y}}_i represent the actual capacity and predicted capacity of the cycle, and t_EOM represents a certain number of charge/discharge cycles for predicting the RUL of LIBs.

4 Prediction results of remaining life and discussion

This section presents experimental methodologies and findings of RUL prediction, with a focus on the early cycle life prediction phase and the RUL prediction phase.

4.1 Phase 1: Prediction of initial cycle life

By preprocessing the data set and extracting the characteristic factors, the lifespan of LIBs was preliminarily predicted using a neural network, and the prediction results were verified. Fig.4 shows the neural network structure used, and Fig.3 shows the distribution of characteristic factors. The neural network utilized two fully connected layers, i.e., FC1 and FC2. The rectified linear unit (ReLU) was used as the activation function after the fully connected layer with 350 neurons. This choice of activation function allowed for a high degree of nonlinear fitting in the model. The maximum training epoch was set to 2500, and the training process was optimized by minimizing the validation error. The computer used in this paper was equipped with a Gen Intel® Core™ I5-12400F 2.50 GHz CPU and a 2GB GPU for memory. The entire training time was only 6 s.

4.1.1 Prediction results of LIBs cycle

The experimental results for predicting the LIB cycle life are shown in Fig.6(a). The RMSE and MAPE results are shown in Tab.2. Fig.6(b) shows the coordinates of the eigenvalues of the data set in 3D space.

4.1.2 Selection of capacity degradation model

The two characteristic factors with the two largest Pearson coefficients and the lifetime normalized by the neural network prediction were combined to form a 3D space, as shown in Fig.6(b). The Euclidean distance between them was calculated, and the shortest distance was considered to have a similar degradation trend, which was used as a reference for battery capacity decay in the test set to determine the initial DEM. The capacity degradation curve of the test data set is shown in Fig.7. The blue dashed line represents the capacity degradation curve of the actual test set, while the red solid line represents the personalized prediction curve that aligns with it.

In this paper, the battery sample row algorithm was selected to verify the test data set under four different working conditions. C1 working condition: 5C (67%) −4C, C9 working condition: 5.3C (54%) −4C, C25 working condition: 5.6C (19%) −4.6C, C30 working condition: 5.9C (60%) −3.1C. The DEM parameters were determined based on neural network prediction. Tab.3 shows the DEM parameters (a, b, c, and d) identified in Phase 1. Fig.8 shows the curve fitting between the test and matched batteries.

4.2 Phase 2: Prediction of RUL

In this section, the prediction of RUL using GRP algorithms and the verification of RUL predictions with 5 batteries at different working conditions were illustrated.

4.2.1 Experimental setup

In the RUL prediction experiment, the number of cycles was used as the input data for GPR, and the capacity of the test set was used as the output label. In this experiment, three different values of EOM were set: 200, 300, and 400. The goal of this phase is to predict the capacity of LIBs in a certain period based on the corresponding cycle count (more significant than the EOM value). Consequently, when the expected capacity of LIBs reached the EOL threshold of 0.88 Ah, the RUL of LIBs was calculated.

4.2.2 Model training and results for estimation

In this section, GPR was utilized to predict the remaining life of the battery in the second phase, taking the identified DEM parameters as its initial average function, using the more flexible Matern 5/2 kernel function, and introducing a constant term as the offset to enhance the flexibility of the algorithm. GPR was trained with a test library capacity of 200, 300, and 400 cycles before EOM.

Fig.9 shows the predicted capacity of four individual cells and their corresponding 99% confidence intervals. For each test LIB, 200, 300, and 400 EOM cycles were used to estimate prediction performance. When the predicted capacity reached EOL, the RULs were calculated. The performance of the proposed method in predicting RUL was evaluated using AE, AP, and TS.

Taking the EOM (estimated capacity consumption) of LIB-cell1 equal to 200 as an example, Fig.9(a) shows the prediction results and their confidence intervals. The AE of the proposed method is 9 ± 3 cycles, the AP is 99.1071% ± 0.03968%, and the TS is 0.0084 Ah. Tab.4 lists the accuracy results of four LIBs. This research method shows minor errors in RUL prediction and confidence interval prediction, which shows that using DEM as the initial parameter of GPR is helpful in improving the accuracy of capacity prediction.

As indicated in Tab.4, the method proposed in this paper has achieved good results on the data sets above. There are small errors in the prediction of RUL and confidence intervals, but the prediction accuracy can reach more than 98%. A feature factor extraction method related to capacity degradation was proposed in the first phase. The neural network structure can accurately predict the cycle life of the battery. The experimental results also demonstrate the effectiveness of the neural network, making it suitable for prediction and diagnostic data model. In addition, when the training data set is small, the GPR exhibits an excellent accuracy, as observed from the RUL prediction results.

To further prove the superiority of the algorithm, this paper takes Cell1 EOM = 400 as an example. The general method adopted the first extraction of feature factors for training and prediction, and the calculation time was the time used for prediction. A comparison of the results is shown in Fig.10 and Tab.5.

In addition, similar results are achieved by experiments conducted with different EOM cycle settings, which further verifies the effectiveness of this method in predicting ERUL. In addition, as the computing power of vehicle chips improves, it becomes possible to execute more complex and accurate prediction models in real-time or near real-time environments. Predicting RUL for LIBs typically requires consideration of numerous factors, including battery capacity degradation and internal resistance growth. More complex models can accurately capture these factors and provide more precise RUL prediction results.

5 Conclusions

In this paper, a method was proposed for predicting the EOL usable capacity of LIBs. In the first phase, the critical feature factors were extracted from the discharge curves of the LIBs in both the training and validation sets. These characteristic factors included battery discharge curves of voltage, current, temperature, and other parameters. The neural network was utilized as the prediction model. The appropriate weight parameters were obtained by training the data in the training set. The weight parameters were verified using the validation data set. Finally, the cycle life of the LIBs of the test device was preliminarily predicted, and the estimated value of ERUL was obtained. Then, the degradation library that closely matched the expected battery capacity degradation trajectory was selected from the training and validation data set as the reference degradation curve for early prediction of RUL in the test set. This method eliminated the need for manual selection, making it widely applicable for selecting large quantities of batteries.

In the second phase, the identified DEM parameters were utilized as the initial mean function for the GPR algorithm. Then the GPR for ERUL prediction was trained using the LIBs capacity data from the test set, which contained only a small number of charge and discharge cycles. The experimental results show that the method has a high prediction accuracy, confirming the feasibility and effectiveness of the algorithm.

In addition, the method proposed can also be extended. For example, various characteristic factors and degradation models can be explored to enhance the accuracy and adaptability of predictions. In addition, advanced optimization algorithms can be used to optimize the GPR parameters, and data-driven models can be used to predict the cycle life of LIBs. The data-driven model can reduce the complexity of battery model and obtain accurate battery models without consuming significant time and resources in simulating practical applications.

The method proposed provides a reliable and effective approach for predicting the early life of LIBs and offers valuable insights for future research in related fields. At the same time, it is also significant to determine the future method for health management of LIBs, as it will play a crucial role in numerous other engineering applications.

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