Remaining Useful Life Prediction of Lithium-Ion Batteries Based on Neural Network and Adaptive Unscented Kalman Filter

Abstract

Accurate prediction of remaining useful life (RUL) plays an important role in maintaining the safe and stable operation of Lithium-ion battery management systems. Aiming at the problem of poor prediction stability of a single model, this paper combines the advantages of data-driven and model-based methods and proposes a RUL prediction method combining convolutional neural network (CNN), bi-directional long and short-term memory neural network (Bi-LSTM), SE attention mechanism (AM) and adaptive unscented Kalman filter (AUKF). First, three types of indirect features that are highly correlated with RUL decay are selected as inputs to the model to improve the accuracy of RUL prediction. Second, a CNN-BLSTM-AM network is used to further extract, select and fuse the indirect features to form predictive measurements of the identified degradation metrics. In addition, we introduce the AUKF model to increase the uncertainty representation of the RUL prediction. Finally, the method is validated on the NASA dataset and the CALCE dataset and compared with other methods. The experimental results show that the method is able to achieve an accurate estimation of RUL, a minimum RMSE of up to 0.0030, and a minimum MAE of up to 0.0024, which has high estimation accuracy and robustness.

1. Introduction

Lithium-ion batteries are widely used in communication equipment, grid-level power storage, smartphones, and other fields owing to their advantages, such as high power tolerance, high rated voltage, high storage energy density, and no memory effect [1,2,3]. However, due to the existence of various complex internal aging mechanisms in lithium batteries, irreversible changes such as capacity degradation and an increase in internal resistance will occur during use or storage, their performance will decline, and internal stability will deteriorate. If it is not replaced in time and continues to be used, it may lead to the loss of control of the heat generated by the internal chemical reaction of the Lithium-ion battery, triggering a series of problems such as the thermal runaway effect, which can cause serious safety accidents [4].

In recent years, Lithium-ion batteries have often suffered from problems such as spontaneous combustion and explosions, which have led to safety accidents and caused huge losses to users’ lives and companies’ reputations [5]. On 20 August 2023, a bus in Nanjing, Jiangsu Province, China, burst into flames in the middle of its journey, resulting in a tragic case of two deaths and five scalding injuries, with the cause of the fire being spontaneous combustion of lithium batteries carried by passengers. On 18 February 2024, a battery recycling plant in the south of France caught fire with about 900 tons of lithium batteries, leading to the complete destruction of a 3000-square-meter building. On 18 February 2024, about 900 tons of lithium batteries caught fire in a battery recycling plant in southern France, resulting in a 3000-square-meter building being completely burned down and causing incalculable damage. On 14 April 2024, in a neighborhood in Nanjing, China, lithium batteries for electric vehicles caught fire and started a fire in the early hours of the morning, causing 15 people to be killed and 44 others injured. The frequent occurrence of Lithium-ion battery safety accidents has triggered concerns and worries from all sectors of society about its safety.

In order to reduce the safety hazards caused by aging Lithium-ion batteries, it is usually stipulated that when the Lithium-ion battery capacity decreases to 70% of its rated capacity, the battery’s life is terminated, and it must be replaced for the first time [6,7]. Therefore, it is critical to accurately predict the RUL of Lithium-ion batteries. By accurately predicting the RUL of a battery, it is possible to detect the existence of safety hazards in time, reduce the possibility of accidents, and avoid potential safety problems to a certain extent.

Currently, the RUL prediction methods are mainly classified into three groups [8,9]: model-based methods, data-driven methods, and fusion methods. Model-based methods construct accurate models by analyzing the physicochemical reactions inside the battery, such as the electrochemical model [10], equivalent circuit model [11], Kalman filter [12], etc. Lui et al. [13] proposed an empirical model-based prediction method to estimate the degradation parameters and perform the prediction of the battery RUL by voltage and capacity. However, only the degradation model was simulated, and constant operating conditions were assumed in their prediction. Therefore, their method cannot be used to simulate degradation under any other operating conditions. Gao et al. [14] estimated the battery capacity by analyzing five indirect characteristics measured during battery discharge and based on the Kendall rank correlation coefficient, the predicted capacity was taken as the measured value of PF for RUL prediction. Model-based methods do not require the collection of large amounts of data, but the accuracy of prediction results is highly dependent on the battery model established. In addition, the internal material properties of the battery may change during actual operation, which makes it difficult to establish an accurate model, so model-based methods have been greatly limited in their development.

Data-driven methods do not require consideration of the internal reactions of the battery and the mechanism of capacity degradation but only need to analyze the historical data, from which the degradation law can be learned and predicted [15]. Therefore, data-driven methods are widely used for RUL prediction of batteries, such as support vector machine (SVM) [16], relevance vector machine (RVM) [17], Gaussian process regression (GPR) [18], etc. Wang et al. [19] extracted three health indicators from the charge/discharge curves after analyzing the aging characteristics of the battery from several perspectives, and then proposed a new composite kernel function, and constructed the GPR model by selecting a different pair of mean and kernel functions from several mean and kernel functions, and the result of the study shows that the average absolute error of the estimation was 1.7%, and the root-mean-square error was 2.41%. Wang et al. [20] addressed the difficulty of extracting features, combining local tangent space alignment (LTSA) with an improved LSTM. Feature selection and model parameter settings highly influence the prediction accuracy of data-driven methods. Feature selection has a direct impact on the prediction results, which can effectively reduce the redundancy of data, improve the learning efficiency of neural networks, and improve the accuracy of prediction [21]. Choosing unsuitable model parameters may lead to insufficient generalization ability [22].

The fusion method combines the advantages of multiple algorithms, overcomes the limitations existing in a single model, and achieves better results in terms of prediction accuracy. Fusion methods can be divided into two groups: fusion of multiple data-driven methods and fusion of model-based and data-driven methods. Chen et al. [23] proposed a polynomial empirical model describing nonlinear battery degradation and combined it with an improved GPR fusion into an empirical data hybrid-driven model with high RUL prediction accuracy. Li et al. [24] utilized an algorithm based on a fusion framework of unscented particle filter (UPF) and least squares support vector machines (LSSVM) to solve the problem of difficulty in obtaining observation equations and observations in filtering algorithms.

Owing to various reasons, such as instrument errors and disturbing factors in the measurement, the measured battery capacity data are usually contaminated with various kinds of noise, which may have an incalculable impact on the performance of the model. At the same time, a single Bi-LSTM model does not guarantee accurate prediction, and the complex neural network system is not combined with denoising methods, resulting in the existence of large errors in the prediction results. To solve the above problems, we propose the CNN-Bi-LSTM-AM-AUKF algorithm and extract indirect features from the raw data that are more suitable for battery RUL prediction.

The main contributions of this paper are summarized below:

(1)

In order to select suitable features to improve the prediction accuracy, three indirect features were extracted based on the NASA dataset and the CALCE dataset, and their validity was verified by Pearson’s correlation coefficient.

(2)

To address the instability of the Bi-LSTM method, a CNN-Bi-LSTM-AM prediction model is established. By introducing the SE attention mechanism, the AM model is able to strengthen the LSTM network so that it is able to more effectively screen out the key time series information that has a significant impact on the prediction task during the data processing, which effectively improves the prediction accuracy and stability.

(3)

The uncertainty of the prediction output is characterized using AUKF, and the model’s prediction accuracy is further improved.

The rest of the paper is organized as follows. Section 2 describes the Lithium-ion battery aging test dataset and the extracted indirect features. Section 3 introduces the theory of models and algorithms, including the CNN algorithm, Bi-LSTM algorithm, SE attention mechanism, and AUKF algorithm. Section 4 shows and discusses the experimental results. Section 5 concludes this article.

2. Battery Data Analysis and Feature Extraction

2.1. Definition of Lithium-Ion Battery RUL

Lithium-ion battery’s RUL is the number of charge–discharge cycles experienced when its current capacity reaches 70% of the rated capacity [25]. At this point, the Lithium-ion battery reaches its failure threshold, and its continued use poses a potential safety hazard. The formula for calculating the Lithium-ion battery’s RUL is as follows:

$$RUL=N_{all}-N_c$$

(1)

where $N_{all}$ is the total number of times the battery has been cyclically charged and discharged, and $N_c$ is the number of times the battery has currently experienced cyclic charging and discharging.

2.2. Lithium-Ion Battery Data Set

In this paper, B0005, B0006, B0018, and B0046 from the National Aeronautics and Space Administration (NASA) dataset [26] and CS2-36 and CS2-38 from the Computer Aided Life Cycle Engineering Center (CALCE) dataset [27] are used. For the NASA dataset, the battery model is an 18650-LiCoO2 battery. For charging, the battery was charged in constant current (CC) mode at 1.5 A until the charging voltage reached 4.2 V. Then, it was charged in constant voltage (CV) mode until the charging current was reduced to 20 mA. For discharging, B0005, B0006, and B0018 were discharged in constant current (CC) mode at 2 A, and it was terminated when it reached the cut-off voltage. For the CALCE dataset, the battery model is a columnar lithium cobalt oxide battery. When charging, the battery is charged in CC mode at 1.1 A until the voltage rises to 4.2 V. When discharging, the battery is discharged in CV mode until the current drops to 0.05 A. When RUL reaches 70%, the anomalous data in the CALCE dataset appears frequently and the prediction of the capacity after reaching the RUL is of little practical significance, so it is removed in this paper. The battery capacity curves of the two datasets are shown in Figure 1.

2.3. Data Collection and Pre-Processing

The capacity of a battery is difficult to measure directly during actual operation, so it must be characterized based on extracting features from monitorable data. The complete charge/discharge experiment consists of three stages: the CV charging stage, the CC charging stage, and the CC discharging stage. Taking the B0005 and CS2-38 batteries as an example, the changes in voltage and current in the charging and discharging phases of the batteries under different cycles are shown in Figure 2.

It can be seen that the curves of voltage and current under different cycle times have obvious changes, and the curves are obviously shifted to the left, which indicates that there is a close relationship between the duration of battery charging and discharging and the number of cycles. The figure shows the discharge voltage curve and charge current curve under different cycle times. Figure 3 shows the discharge voltage curve and charge current curve under different cycle times.

The variation in the data observed in the figure shows that, as the number of batteries charging and discharging cycles increments, the time period occupied by the CC mode during charging is significantly reduced, while the time required for the battery voltage to reach its lowest point during the discharge phase is also significantly reduced and the rate of voltage reduction is significantly increased, indicating that the performance of the batteries deteriorates with use. The charging and discharging curves and the number of cycles show a certain pattern from which indirect characteristics can be obtained. Therefore, this paper extracts the following three types of indirect features (F1, F2, F3) from the NASA dataset and the CALCE dataset, and their relationships with the time series are shown in Figure 4. Taking B0005 and CS2-38 as an example, their features are specifically described as shown below:

F1: Equal voltage drop time interval. With the aging of the battery, the time it takes for the charging process to decay from a high voltage to a relatively low voltage is gradually reduced, reflecting the aging characteristics of the battery to a certain extent. The high voltage selected in this paper is 3.9 V, and the low voltage is 3.5 V.

F2: The time when the discharge voltage reaches the lowest point. When a Lithium-ion battery is discharged, the time it takes for its voltage to gradually drop from the initial 4.2 V to the cutoff voltage decreases as the battery capacity decreases, so it can be regarded as an indirect characteristic of the RUL of lithium batteries.

F3: CC constant current mode duration. The trend of CC mode duration decreasing is basically the same as the decay trend of Li-ion battery RUL, the shorter the CC mode duration, the more serious the battery aging, which can be regarded as an indirect characteristic.

In order to confirm the effectiveness of the indirect features extracted in this paper in predicting the RUL of lithium batteries, Pearson’s correlation coefficient was used as an evaluation metric for validation. When there are two variables $X(x_1,x_2,\dots ,x_n)$ and $Y(y_1,y_2,\dots ,y_n)$, The formula for the Pearson correlation coefficient is as follows:

$$r=\frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2\sqrt{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}}$$

(2)

where $\overline{x}$ and $\overline{y}$ are the means of the two variables. $n$ is the sample size. The closer the absolute value of the Pearson correlation coefficient is to 1, the stronger the correlation. According to the principle of the Pearson correlation coefficient, these two variables are considered to be extremely strongly correlated when $0.8<\left|r\right|<1$.

Table 1. Pearson correlation coefficient between indirect characteristics and cell capacity.

Feature	Pearson Correlation Coefficient
	B0005	B0007	B0018	B0046	CS2-36	CS2-38
F1	0.9998	0.9997	1.0000	0.8268	1.0000	0.9753
F2	1.0000	1.0000	1.0000	0.8076	1.0000	1.0000
F3	0.9992	0.9989	0.9992	0.9621	0.9999	0.9996

demonstrates the Pearson correlation coefficients between the indirect characteristics and the battery capacity of each group. It can be seen that the Pearson correlation coefficients between these three types of indirect features and the battery capacity are all greater than 0.8, indicating that they are highly correlated with the battery capacity, so these three types of features can be selected as indirect features.

2.4. Normalization

Due to the different scales of the extracted features, the direct use of these features as inputs may affect the convergence speed of the neural network as well as the prediction accuracy, so the normalization of these features is a necessary step. In this paper, Min–max normalization is used to solve the problem of comparability between data. The formula for Min–max normalization is as follows:

$$x^{\prime }=\frac{x-\mathrm{m}\mathrm{i}\mathrm{n} \left(x\right)}{\max \left(x\right)-\mathrm{m}\mathrm{i}\mathrm{n} \left(x\right)}$$

(3)

where $\max \left(x\right)$ is the maximum value of the sample data, $\mathrm{m}\mathrm{i}\mathrm{n} \left(x\right)$ is the maximum minimum value of the sample data, and $x^{\prime }\in \left[0,1\right]$.

3. Hybrid Networks for Battery Capacity Prediction

3.1. Convolutional Neural Network (CNN)

CNN is a feed-forward neural network with convolutional operations and deep structure, which is commonly used in the fields of image recognition, predictive analytics, and mining time series data features. CNN can train models with few features, effectively capturing the features of the data through convolutional operations. At the same time, CNN has a parameter-sharing characteristic, which enables it to use fewer training parameters during training, reducing the difficulty of network training, improving the training speed while ensuring accuracy, and making the network training process more efficient. The basic structure of CNN usually consists of five parts: the input layer, convolutional layer, pooling layer, fully connected layer, and output layer. The convolutional layer uses the convolutional kernel to do convolutional operations to extract local features, and the pooling layer is used to remove redundant information. The basic structure of a CNN is shown in Figure 5.

3.2. Bi-Directional Long Short-Term Memory (Bi-LSTM)

Although CNN performs well in automatically extracting multidimensional spatial features from big data, it appears to be incompetent in dealing with time-dependent time-series data. In contrast, Bi-LSTM is able to effectively address the long-term dependency problem by introducing gating units. Combining CNN and Bi-LSTM can greatly enhance the extraction of time-series features and reduce the time required for computation to a certain extent, thus improving the efficiency and accuracy of the model in processing time-series data. Bi-LSTM is a long short-term memory network combining forward and backward propagation, which can extract information from both front and backward directions at the same time so as to better capture the features of the input sequence. Battery capacity and related features are time series data, and their changes are closely related to time development; in order to make more accurate predictions, the influence of the direction sequence also needs to be taken into account, so Bi-LSTM is very useful for the battery capacity prediction learning task. The Bi-LSTM calculation formula is shown below:

$${\overrightarrow{h}}_t=\left\{\begin{array}{c}{\overrightarrow{z}}_t=\tanh ({\overrightarrow{W}}_z{\overrightarrow{x}}_t+{\overrightarrow{U}}_z{\overrightarrow{h}}_{t-1}+{\overrightarrow{b}}_z)\\ {\overrightarrow{i}}_t=\sigma ({\overrightarrow{W}}_i{\overrightarrow{x}}_t+{\overrightarrow{U}}_i{\overrightarrow{h}}_{t-1}+{\overrightarrow{b}}_i)\\ {\overrightarrow{f}}_t=\sigma ({\overrightarrow{W}}_f{\overrightarrow{x}}_t+{\overrightarrow{U}}_f{\overrightarrow{h}}_{t-1}+{\overrightarrow{b}}_f)\\ {\overrightarrow{o}}_t=\sigma ({\overrightarrow{W}}_o{\overrightarrow{x}}_t+{\overrightarrow{U}}_o{\overrightarrow{h}}_{t-1}+{\overrightarrow{b}}_o)\\ {\overrightarrow{c}}_t={\overrightarrow{z}}_t\odot {\overrightarrow{i}}_t+{\overrightarrow{c}}_{t-1}\odot {\overrightarrow{f}}_t\\ {\overrightarrow{h}}_t={\overrightarrow{o}}_t\odot \tanh \left({\overrightarrow{c}}_t\right)\end{array}\right.$$

(4)

$${\overleftarrow{h}}_t=\left\{\begin{array}{c}{\overleftarrow{z}}_t=\tanh ({\overleftarrow{W}}_z{\overleftarrow{x}}_t+{\overleftarrow{U}}_z{\overleftarrow{h}}_{t+1}+{\overleftarrow{b}}_z)\\ {\overleftarrow{i}}_t=\sigma ({\overleftarrow{W}}_i{\overleftarrow{x}}_t+{\overleftarrow{U}}_i{\overleftarrow{h}}_{t+1}+{\overleftarrow{b}}_i)\\ {\overleftarrow{f}}_t=\sigma ({\overleftarrow{W}}_f{\overleftarrow{x}}_t+{\overleftarrow{U}}_f{\overleftarrow{h}}_{t+1}+{\overleftarrow{b}}_f)\\ {\overleftarrow{o}}_t=\sigma ({\overleftarrow{W}}_o{\overleftarrow{x}}_t+{\overleftarrow{U}}_o{\overleftarrow{h}}_{t+1}+{\overleftarrow{b}}_o)\\ {\overleftarrow{c}}_t={\overleftarrow{z}}_t\odot {\overleftarrow{i}}_t+{\overleftarrow{c}}_{t+1}\odot {\overleftarrow{f}}_t\\ {\overleftarrow{h}}_t={\overleftarrow{o}}_t\odot \tanh \left({\overleftarrow{c}}_t\right)\end{array}\right.$$

(5)

$$h_t={\overrightarrow{h}}_t\oplus {\overleftarrow{h}}_t$$

(6)

$$O_t=g(W{h}_t+b_t)$$

(7)

where ${\overrightarrow{h}}_t$ denotes the forward propagation of the LSTM network; ${\overleftarrow{h}}_t$ denotes the backpropagation of the LSTM network; ${\overrightarrow{W}}_z,{\overrightarrow{W}}_i,{\overrightarrow{W}}_f,{\overrightarrow{W}}_o$ denote the weights of the input layers for forward propagation; ${\overleftarrow{W}}_z,{\overleftarrow{W}}_i,{\overleftarrow{W}}_f,{\overleftarrow{W}}_o$ denote the weights of the input layers of the back propagated; ${\overrightarrow{U}}_z,{\overrightarrow{U}}_i,{\overrightarrow{U}}_f,{\overrightarrow{U}}_o$ denote the forward propagating hidden layer weights; ${\overleftarrow{U}}_z,{\overleftarrow{U}}_i,{\overleftarrow{U}}_f,{\overleftarrow{U}}_o$ denote the hidden layer weights for backpropagation; ${\overrightarrow{b}}_z,{\overrightarrow{b}}_i,{\overrightarrow{b}}_f,{\overrightarrow{b}}_o$ denote the forward propagating offsets; ${\overleftarrow{b}}_z,{\overleftarrow{b}}_i,{\overleftarrow{b}}_f,{\overleftarrow{b}}_o$ denote the back propagated offsets; $O_t$ denotes the final output value; $\sigma$ is the Sigmoid function; $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$ is the hyperbolic tangent function; $\odot$ denotes the element-by-element product; and $\oplus$ denotes the splicing operation.

3.3. SE Attention Mechanism

As a simple attention mechanism with a small number of parameters, a simple structure, and no change in the number of channels and the size of input data, the SE attention mechanism can be embedded as a subnetwork in any neural network. During the training process, Bi-LSTM networks may sometimes fail to pay sufficient attention to the importance of different time nodes and the unique characteristics of time series data. By introducing an attention mechanism into the Bi-LSTM network, we can focus the network on the information that is most critical to the current prediction task. This combination not only preserves the integrity of the temporal information but also reduces the computational burden by reducing the network parameters, thus speeding up the computation. This optimization improves the accuracy and generalization ability of the model prediction, enabling it to better cope with temporal data processing tasks in various scenarios. Compared with other attention mechanisms that improve network performance from the spatial dimension, the SE attention mechanism considers the problem from the relationship between features and channels, thus avoiding the undesirable effects of deepening the number of network layers.

The SE attention mechanism is mainly realized by two operations:

(1)

Compression. The features with input dimension H × W × C are compressed along the spatial dimension by the global average pooling technique, and the features of dimension 1 × 1 × C are obtained.

(2)

Excitation. The compressed features are input into the fully connected layer to learn, and the results obtained after learning are used as weights to be weighted with the original features through excitation.

The structure of the SE attention mechanism is shown in Figure 6.

3.4. Adaptive Unscented Kalman Filter (AUKF)

The presence of a variety of complex reactions within the battery, coupled with conditions that make it difficult to measure internal variables that degrade the capacity of the battery. In addition, we need a filtering algorithm to solve the problem of noise filtering in complex models. For the above situation, AUKF can provide a better solution. Compared with EKF, AUKF does not need to calculate the Jacobi matrix, has higher estimation accuracy, and can adaptively adjust the process noise and measurement noise covariance, avoiding the linearization error problem caused by the EKF ignoring the second-order and higher-order terms [28,29].

The measurement of voltage, current, time and other data may have errors. Meanwhile, due to the limited number of training samples, the establishment of the CNN-Bi-LSTM-AM model may also have some errors. Therefore, the system noise $Q_k$ and the measurement noise $R$ are added to the state space equations:

$$X_{k+1}=f\left(X_k,i_k\right)+Q_k$$

(8)

$$Y_{k+1}=H\left(X_k,i_k\right)+R_k$$

(9)

The general flow of the AUKF algorithm is as follows:

Step 1: Obtain a set of Sigma point sets using UT transformation

$$X_0={\widehat{X}}_{k-1}$$

(10)

$$X_{k-1}^i={\widehat{X}}_{k-1}+{(\sqrt{\left(n+\lambda \right)P\left(k\right)})}_i,i=\mathrm{1,2},\cdots ,n$$

(11)

$$X_{k-1}^j={\widehat{X}}_{k-1}-{(\sqrt{\left(n+\lambda \right)P\left(k\right)})}_j,j=n+1,n+2,\cdots ,2n$$

(12)

where $k$ is the sampling moment, and $P\left(k\right)$ is the covariance in the state vector at moment $k$.

The weights and variances were calculated as follows:

$$W_0^m=\frac{\lambda }{\lambda +n}$$

(13)

$$W_0^c=\left(1-a^2+\beta \right)+\frac{\lambda }{\lambda +n}$$

(14)

$$W_i^m=W_i^c=\frac{1}{2(\lambda +n)},i=\mathrm{1,2},\cdots ,2n$$

(15)

where $\alpha$ is the state parameter of the control sampling point distribution, $\beta$ is the state distribution parameter. $\lambda$ is obtained from the following equation:

$$\lambda =a^2\times \left(n+k\right)-n$$

(16)

Step 2: One-step prediction of $X$ at moment $k$ by sample points

$$X_{k|k-1}^i=f\left(X_{k-1}^i\right)+w_{k-1}$$

(17)

$${\widehat{X}}_{k|k-1}=\sum _{i=0}^{2n}{W}_i^m{X}_{k|k-1}^i$$

(18)

$$P_{\overline{x},k|k-1}=\sum _{i=0}^{2n}{W}_i^c\left(X_{k|k-1}^i-{\widehat{X}}_{k|k-1}\right){(X_{k|k-1}^i-{\widehat{X}}_{k|k-1})}^T+Q_k$$

(19)

where $w_{k-1}$ is the process noise, ${\widehat{X}}_{k|k-1}$ is the a priori estimate, and $P_{\overline{x},k|k-1}$ is the a priori estimation error covariance of the state quantity.

Step 3: Predicted values of output variables

$$Y_0={\widehat{X}}_{k|k-1}$$

(20)

$$Y_{k-1}^i=Y_0+{(\sqrt{\left(n+\lambda \right)P\left(k\right)})}_i,i=\mathrm{1,2},\cdots ,n$$

(21)

$$Y_{k-1}^i=Y_0-{(\sqrt{\left(n+\lambda \right)P\left(k\right)})}_j,j=n+1,n+2,\cdots ,2n$$

(22)

$$Y_{k|k-1}^j=H\left(Y_{k-1}^i\right)+v_k$$

(23)

where $v_k$ is the observation noise.

Step 4: Find new estimates and covariances of the observations by weighting them

$${\widehat{Y}}_{k|k-1}=\sum _{i=0}^{2n}{W}_i^m{Y}_{k|k-1}^j$$

(24)

$$P_{y,k}=\sum _{i=0}^{2n}{W}_i^c\left(Y_{k|k-1}^i-{\widehat{Y}}_{k|k-1}\right){(Y_{k|k-1}^i-{\widehat{Y}}_{k|k-1})}^T+R_k$$

(25)

Step 5: Observation correction, calculate the joint covariance of state and output variables

$$P_{xy,k}=\sum _{i=0}^{2n}{W}_i^c\left(X_{k|k-1}^i-{\widehat{X}}_{k|k-1}\right){(Y_{k|k-1}^i-{\widehat{Y}}_{k|k-1})}^T$$

(26)

Step 6: Calculate Kalman gain

$$K_k=\frac{P_{xy,k}}{P_{y,k}}$$

(27)

Step 7: Update the state and error variance

$${\widehat{X}}_k={\widehat{X}}_{k|k-1}+K_k(Y_k-{\widehat{Y}}_{k|k-1})$$

(28)

$$P_k=P_{x,k|k-1}-K_k{P}_{y,k}{K}_k^T$$

(29)

Step 8: Update observation noise covariance and process noise covariance

$${\mu }_k=Y_k-H({\widehat{X}}_{k-1},i_k)$$

(30)

$$c_k=\frac{{\sum }_{i=k-L+1}^k{\mu }_k{u}_k^T}{L}$$

(31)

$$R_k=c_k+\sum _{i=0}^{2n+1}{W_i^c(Y_{k|k-1}^i-Y_k+c_k)(Y_{k|k-1}^i-Y_k+c_k)}^T$$

(32)

$$Q_k=K_k{c}_k{K}_k^T$$

(33)

where $L$ is the slide window size.

In summary, this paper proposes a hybrid method (CNN-Bi-LSTM-AM-AUKF). CNN is used to capture and extract critical degraded feature information hidden in indirect features as a way to enhance the performance of the overall prediction model. Bi-LSTM learns bi-directional sequential features from the output feature information of these CNN layers, and it is able to efficiently utilize the long-term dependency property in the sample data for training. In addition, the application of the SE attention mechanism further strengthens the effect of feature extraction, enabling the model to focus more on the information that is crucial to the prediction task while reducing the attention to useless information, thus improving the accuracy and reliability of the prediction. The predicted measurements from the CNN-Bi-LSTM-AM network obtained with uncertainty in the prediction measurements will be properly represented by AUKF in a transparent manner. The method consists of the following main steps:

(1)

Three types of indirect features were extracted from the NASA dataset and the CALCE dataset and normalized as inputs to the neural network.

(2)

Build a prediction model based on CNN-Bi-LSTM-AM. The extracted indirect features are input into the neural network and battery capacity as an output of the neural network.

(3)

The output values of the CNN-Bi-LSTM-AM network are used as inputs to the AUKF, which are fed into the AUKF framework in order to update the parameters of the observation equations and predict the Lithium-ion battery’s RUL.

The algorithmic framework used in this study is shown in Figure 7.

4. Results and Discussion

The proposed approach in this article is implemented on an NVIDIA Geforce RTX 3060 GPU, and Matlab-2022b was used to code the model. The main parameter settings of the experimental setup are shown in

Table 2. The main parameter.

Model	Parameter Name	Parameter Value
CNN	Kernel number	32/128
	Kernel size	1
	Stride	1
	Activation function	ReLU
Bi-LSTM	Optimizer	Adam
	Batch-Size	128/512
	Learn Rate	0.01/0.04
	Epoch	1000/1500
	Layer	1
	Neurons	128/256
	Dropout	0.3
AUKF	control sampling point distribution	0.1
	state distribution	2
	slide window	16

.

In order to evaluate the prediction performance of different algorithms, the root mean square error (RMSE) and the mean absolute error (MAE) are chosen as the evaluation targets in this paper. Their detailed definitions are as follows:

$$RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{(y_i^{\prime }-y_i)}^2}$$

(34)

$$MAE=\frac{1}{n}\sum _{i=1}^n|y_i^{\prime }-y_i|$$

(35)

where $y_i^{\prime }$ is the predicted value, $y_i$ is the true value, and $n$ is the number of samples.

Algorithms such as ELM [30], KELM [31], and SVR [32] are commonly used for time series-related data prediction, while algorithms such as DBO [33], INFO [34], WOA [25] are commonly used to optimize the parameters of neural networks in order to improve the prediction accuracy. In order to prove the superiority of the CNN-Bi-LSTM-AM-AUKF algorithm, this paper compares the CNN-Bi-LSTM-AM-AUKF with the three algorithms. The three algorithms are described as follows:

(1)

DBO-SVR. Using the parameter-seeking superiority of the DBO algorithm, the kernel parameters of the SVR method have been optimized to solve the SVR parameter selection problem.

(2)

INFO-KELM. In order to enhance the performance of KELM, the INFO algorithm is used, this algorithm with its powerful optimization ability and fast convergence speed, the regularization coefficients and kernel function parameters of KELM are precisely adjusted and optimized.

(3)

WOA-ELM. The WOA algorithm has the advantages of strong optimization ability and global search, and the ELM model is faster than traditional learning algorithms while guaranteeing learning accuracy.

We set different starting points for prediction (40% and 60% of the total amount of data) and compared the prediction results of the different algorithms with six groups of cells to verify the robustness and adaptability of the algorithms. Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show the comparison of the capacity prediction results of the different prediction methods in six groups of batteries with different prediction starting points.

According to the results in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, the prediction performance of the proposed method is obviously better than other methods, and the predicted values and predicted trends are the closest to the real situation of the batteries. In other words, the CNN-Bi-LSTM-AM-AUKF algorithm has excellent prediction accuracy and good robustness. The DBO algorithm performs well in terms of convergence speed and solution accuracy, and greatly avoids the problem of falling into local optimality, so the prediction accuracy is more desirable; in the INFO-KELM algorithm, INFO searches for optimization through different weighted average rules of vectors, and the global searching ability and convergence have been improved to a certain extent, and the prediction accuracy of KELM is improved by the introduction of the kernel function, but its prediction accuracy still cannot escape from the problem of local optimization, so its prediction accuracy is still unsatisfactory; in the WOA-ELM algorithm, the WOA algorithm is used to optimize the input weights of the ELM algorithm and the thresholds of the hidden layer neurons because of its strong global search ability, which solves the problem that the ELM algorithm is susceptible to the poor performance of the network caused by the random initialization of the weights and thresholds and effectively improves the prediction accuracy of the ELM algorithm. However, in Figure 9a, the prediction effect of the WOA-ELM algorithm is not ideal, which is due to the fact that the training samples are too small, and the WOA algorithm is not effectively trained to misjudge the best individuals. In addition, during the prediction process of the AUKF algorithm, the predicted values of CNN-Bi-LSTM-AM are used as the a posteriori capacity of the particle filtering algorithm to guide the update of the AUKF model. The proposed method can capture battery degradation trends relatively effectively, and it provides the ability to track the capacity regeneration that occurs during battery aging.

In order to visually demonstrate the advantages of the method in this paper,

Table 3. Comparison of RUL prediction results by different methods (40% of test set).

Algorithm	RMSE						MAE
	B0005	B0007	B0018	B0046	CS2-36	CS2-38	B0005	B0007	B0018	B0046	CS2-36	CS2-38
DBO-SVR	0.0115	0.0096	0.0137	0.0193	0.0176	0.0104	0.0097	0.0075	0.0112	0.0168	0.0128	0.0078
INFO-KELM	0.0382	0.0121	0.0318	0.0286	0.0222	0.0161	0.0329	0.0091	0.0269	0.0251	0.0156	0.0111
WOA-ELM	0.0246	0.0098	0.0150	0.0520	0.0209	0.0196	0.0211	0.0086	0.0135	0.0458	0.0142	0.0142
CNN-Bi-LSTM-AM-AUKF	0.0067	0.0065	0.0082	0.0114	0.0059	0.0094	0.0038	0.0034	0.0056	0.0092	0.0051	0.0073

and

Table 4. Comparison of RUL prediction results by different methods (60% of test set).

Algorithm	RMSE						MAE
	B0005	B0007	B0018	B0046	CS2-36	CS2-38	B0005	B0007	B0018	B0046	CS2-36	CS2-38
DBO-SVR	0.0089	0.0074	0.0109	0.0120	0.0358	0.0034	0.0081	0.0065	0.0088	0.0110	0.0342	0.0026
INFO-KELM	0.0165	0.0057	0.0175	0.0246	0.0158	0.0060	0.0155	0.0050	0.0147	0.0224	0.0127	0.0045
WOA-ELM	0.0423	0.0127	0.0129	0.0179	0.0133	0.0059	0.0384	0.0105	0.0112	0.0152	0.0103	0.0046
CNN-Bi-LSTM-AM-AUKF	0.0033	0.0032	0.0079	0.0093	0.0114	0.0030	0.0022	0.0026	0.0049	0.0074	0.0098	0.0024

list the comparison between the CNN-Bi-LSTM-AM-AUKF algorithm and some of the currently available RUL prediction algorithms.

As can be seen from Table 2 and Table 3, under the same prediction conditions, the RMSE and MAE of the RUL of the four groups of batteries predicted by this method are smaller than those of the other methods. In addition, the accuracy of the prediction changes due to the difference in the starting point of the prediction. This is because different prediction starting points lead to changes in the number of training sets, and the later the prediction starting point, the more training samples the model will obtain, resulting in effective model performance and improved prediction accuracy.

In order to see whether the improved algorithms in this paper are effective and to further prove the accuracy and generality of the proposed method, this paper also compares with Bi-LSTM and CNN-Bi-LSTM-AM. Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 demonstrate the prediction results of the three algorithms under different prediction starting points.

From Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19, the CNN-Bi-LSTM-AM-AUKF proposed in this paper predicts the capacity most accurately and with the smallest error. The single Bi-LSTM model has a weak generalization ability, which leads to a large error between the predicted value and the true value. The CNN-Bi-LSTM-AM model effectively improves the prediction accuracy; however, the results are not very satisfactory. Compared with the traditional model-based and data-driven approaches, the proposed CNN-Bi-LSTM-AM-AUKF has better RUL prediction performance due to the fact that the model integrates the advantages of both data-driven and model-based approaches. The CNN-Bi-LSTM-AM model effectively improves the prediction accuracy; however, the results are not very satisfactory. Compared with the traditional model-based and data-driven approaches, the proposed CNN-Bi-LSTM-AM-AUKF has better RUL pre-diction performance due to the fact that the model integrates the advantages of both data-driven and model-based approaches; CNN extracts the degraded feature information hidden in the indirect features and improves the performance of the overall prediction model. Bi-LSTM learns bi-directional sequential features from the feature information extracted from the CNN layer, making full use of the long-term dependency features of the sample data for learning. The AM mechanism strengthens the feature extraction effect and allows the model to focus more attention on the important information and suppress the useless information to achieve better accuracy. Compared with the traditional data-driven RUL prediction methods, the proposed CNN-Bi-LSTM-AM-AUKF algorithm can characterize the uncertainty of RUL during battery degradation. Compared with the traditional data-driven RUL prediction methods, the proposed CNN-Bi-LSTM-AM-AUKF algorithm can characterize the uncertainty of RUL during battery degradation.

In order to visualize whether the improved algorithm in this paper is effective or not,

Table 5. Comparison of RUL prediction results by different methods (40% of test set).

Algorithm	RMSE						MAE
	B0005	B0007	B0018	B0046	CS2-36	CS2-38	B0005	B0007	B0018	B0046	CS2-36	CS2-38
Bi-LSTM	0.0165	0.0106	0.0402	0.0385	0.0373	0.0203	0.0119	0.0080	0.0351	0.0349	0.0274	0.0147
CNN-Bi-LSTM-AM	0.0105	0.0070	0.0121	0.0296	0.0314	0.0180	0.0083	0.0056	0.0103	0.0263	0.0254	0.0144
CNN-Bi-LSTM-AM-EKF	0.0092	0.0069	0.0095	0.0232	0.0241	0.0156	0.0086	0.0055	0.0085	0.0207	0.0206	0.0128
CNN-Bi-LSTM-AM-AUKF	0.0067	0.0065	0.0082	0.0114	0.0059	0.0094	0.0038	0.0034	0.0056	0.0092	0.0051	0.0073

and

Table 6. Comparison of RUL prediction results by different methods (60% of test set).

Algorithm	RMSE						MAE
	B0005	B0007	B0018	B0046	CS2-36	CS2-38	B0005	B0007	B0018	B0046	CS2-36	CS2-38
Bi-LSTM	0.0169	0.0176	0.0250	0.0266	0.0268	0.0129	0.0149	0.0160	0.0230	0.0257	0.0219	0.0103
CNN-Bi-LSTM-AM	0.0094	0.0065	0.0122	0.0197	0.0182	0.0095	0.0087	0.0060	0.0102	0.0178	0. 0141	0.0079
CNN-Bi-LSTM-AM-EKF	0.0041	0.0044	0.0087	0.0116	0.0171	0.0071	0.0031	0.0040	0.0072	0.0105	0.0152	0.0051
CNN-Bi-LSTM-AM-AUKF	0.0033	0.0032	0.0079	0.0093	0.0114	0.0030	0.0022	0.0026	0.0049	0.0074	0.0098	0.0024

demonstrate the comparison of the prediction results of this paper’s algorithm and Bi-LSTM and CNN-Bi-LSTM-AM.

According to Table 4 and Table 5, it can be seen that the improved algorithm in this paper demonstrates significant progress and improvement in prediction accuracy. Compared with the traditional Bi-LSTM algorithm, the RMSE of RUL prediction is reduced by 0.0338 on average, and the MAE is reduced by 0.0304 on average. Compared with the traditional CNN-Bi-LSTM-AM algorithm, the RMSE of RUL prediction is reduced by 0.0167 on average, and the MAE is reduced by 0.0154 on average. Compared with the traditional CNN-Bi-LSTM-AM-EKF algorithm, the RMSE of RUL prediction is reduced by 0.0092 on average, and the MAE is reduced by 0.0097 on average. The method obtains the minimum RMSE up to 0.0030, the minimum MAE up to 0.0024.

The improved algorithm in this paper has better prediction results. CNN is used to extract degraded feature information hidden in indirect features to improve the performance of the overall prediction model. Bi-LSTM neural network learns bi-directional sequential features from feature information extracted by the CNN layer, utilizing the long-term dependence features of the sample data. The SE attention mechanism allows the model to focus more attention on the important information, suppressing the useless information, reinforcing the feature extraction effect, and improving the prediction performance of the model. The predictive output obtained from the CNN-Bi-LSTM-AM network with uncertainty is characterized by AUKF.

5. Conclusions

Accurately predicting the RUL of Lithium-ion batteries is critical to ensuring the safety and stability of their operation. For improving the RUL prediction accuracy and solving the problem of insufficient generalization ability of a single model, this paper combines the advantages of data-driven and model-based methods and proposes the CNN-Bi-LSTM-AM-AUKF algorithm, which extracts three types of indirect features from the NASA dataset and the CALCE dataset as inputs to the model to perform RUL prediction for batteries under different operating conditions. The features of the proposed algorithm are summarized as follows:

(1)

CNN is used to extract the degraded feature information hidden in the indirect features, Bi-LSTM is used to obtain the long and short-term dependencies in the features in a bidirectional way, and the SE attention mechanism is used to strengthen the feature extraction effect to achieve better accuracy.

(2)

The uncertainty of the prediction output is characterized using AUKF. Compared with EKF, AUKF improves the prediction accuracy.

(3)

Experimental results based on the NASA dataset and CALCE dataset show that the CNN-Bi-LSTM-AM-AUKF algorithm has better prediction results compared to the traditional algorithm. The minimum RMSE obtained by this method for RUL prediction of Li-electronic batteries can be up to 0.0030, and the minimum MAE can be up to 0.0024, which has high estimation accuracy and stability.

The algorithm proposed in this paper can accurately predict the RUL of batteries, ensuring that aging batteries can be replaced in time and prevent safety accidents from occurring during the use or storage of the batteries, which provides some assistance in the use and management of batteries, the extension of battery life, and the enhancement of the reliability of the battery management system. In practical applications, batteries often exist in the form of battery packs. Therefore, we will investigate the inconsistency in the RUL of individual cells in a battery pack in the future.