A Method for Predicting the Life of Lithium-Ion Batteries Based on Successive Variational Mode Decomposition and Optimized Long Short-Term Memory

Abstract

Accurately predicting the remaining lifespan of lithium-ion batteries is critical for the efficient and safe use of these devices. Predicting a lithium-ion battery’s remaining lifespan is challenging due to the non-linear changes in capacity that occur throughout the battery’s life. This study proposes a fused prediction model that employs a multimodal decomposition approach to address the problem of non-linear fluctuations during the degradation process of lithium-ion batteries. Specifically, the capacity attenuation signal is decomposed into multiple mode functions using successive variational mode decomposition (SVMD), which captures capacity fluctuations and a primary attenuation mode function to account for the degradation of lithium-ion batteries. The hyperparameters of the long short-term memory network (LSTM) are optimized using the tuna swarm optimization (TSO) technique. Subsequently, the trained prediction model is used to forecast various mode functions, which are then successfully integrated to obtain the capacity prediction result. The predictions show that the maximum percentage error for the projected results of five unique lithium-ion batteries, each with varying capacities and discharge rates, did not exceed 1%. Additionally, the average relative error remained within 2.1%. The fused lifespan prediction model, which integrates SVMD and the optimized LSTM, exhibited robustness, high predictive accuracy, and a degree of generalizability.

1. Introduction

Lithium-ion batteries are widely used in modern mobile devices, electric vehicles, aerospace, and other fields due to their fast charging and light weight [1]. However, as the number of battery usages and charge–discharge cycles increases, the battery’s performance and safety may decline, potentially causing battery failures or serious accidents [2,3]. Therefore, precise prediction of the remaining lifespan of lithium-ion batteries is essential for battery condition monitoring, fault warning, and battery replacement, which can enhance battery stability and utilization efficiency [4].

At present, researchers mainly use data-driven prediction methods to forecast the remaining lifespan of lithium-ion batteries [1,5]. These methods involve the application of machine learning, deep learning, and other techniques to analyze massive datasets of battery charge–discharge cycles. Input parameters, such as cycle count, current, and voltage, are employed to feed the algorithm. By employing models such as neural networks to learn patterns and rules in the battery data, researchers can bypass some of the complexities and specialized knowledge associated with electrochemical reactions. These methods have demonstrated high adaptability [6]. Machine learning techniques, originating from the field of artificial intelligence, possess powerful predictive capabilities and are extensively utilized in predicting battery lifespan. Commonly used machine learning methods for battery life prediction include supervised learning algorithms such as support vector machine (SVM) [7], relevance vector machine (RVM) [8], extreme learning machine (ELM) [9], ensemble learning [10], and Gaussian process regression (GPR) [11] for constructing battery life prediction models. Fang et al. [12] used RVM to establish the relationship between battery capacity and the extracted health factors. Additionally, they built a health factor prediction model based on ELM. Guo et al. [13] designed a hybrid data-driven prediction model by integrating two stochastic learning algorithms, and demonstrated that the method accurately predicts the SOH and RUL of batteries online. Deep learning-based algorithms are considered highly accurate predictive models [14]. Neural network algorithms are widely used in RUL prediction due to their strong feature learning and nonlinear trend-fitting capabilities. A long short-term memory (LSTM) network is a type of neural network that has a distinctive mechanism for controlling the flow of information over time. Compared to conventional recurrent neural networks (RNN), LSTMs can better preserve important information over long sequences, and prevent the issue of unstable gradients that can occur in RNN [15]. In a comparative study conducted by Reference [16], it was found that the long short-term memory (LSTM) network had the best predictive performance among the four neural networks. Because of its outstanding performance, the LSTM network method has been widely used in fault diagnosis [17], life prediction [18], and other fields. Many scholars have achieved excellent prediction results by using improved methods based on LSTM networks to predict the remaining life of batteries [19,20,21].

The degradation process of lithium-ion batteries is characterized by complex chemical and physical processes, resulting in capacity fluctuations that are nonlinear and non-stationary during the battery degradation process [22,23,24], which poses challenges in accurately predicting the remaining battery life. To address the nonlinear fluctuations in battery capacity throughout the full life cycle of lithium-ion batteries, a large number of existing studies have employed empirical mode decomposition (EMD) to denoise battery capacity data. These denoised data are then used to predict the RUL using methods such as Gaussian process regression (GPR) [25] or neural networks [26,27]. The EMD algorithm commonly used in the literature suffers from mode-mixing and incomplete decomposition, which can lead to unreliable prediction results. Unlike EMD, the variational mode decomposition (VMD) method is entirely based on a mathematical framework, and exhibits stronger robustness to noise and sampling errors [28]. However, the performance of VMD decomposition depends on the setting of parameters such as the number of modes and penalty factors [29]. Inappropriate parameter selection may have the potential to impede the accurate extraction of genuine signal information. In a related study, the parameter selection for VMD was enhanced utilizing the whale optimization algorithm (WOA), leading to a comprehensive disentanglement of degradation signals in lithium-ion batteries, as demonstrated in [30]. Successive variational mode decomposition (SVMD), based on the VMD algorithm, adaptively determines the number of modes and matches the optimal central frequency by decomposing the signal iteratively. This approach avoids the extraction of redundant modes, reduces the computation time, and improves the convergence speed [31]. The SVMD method has found extensive applications in the realm of temporal data, including wind power forecasting [32], short-term load prediction [33], and fault diagnostics [34]. By using SVMD to decompose battery capacity data into different frequency mode functions and constructing a neural network for prediction, better prediction performance can be achieved.

To address the capacity fluctuation issue in the full life cycle of lithium-ion batteries and its impact on the accuracy of remaining life prediction, this study proposes a lithium-ion battery life prediction method based on SVMD and optimized LSTM. The proposed method accurately captures subtle changes in capacity degradation, and has been validated for its prediction performance using a publicly available dataset.

2. Methods

2.1. Successive Variational Mode Decomposition

Successive variational mode decomposition (SVMD) is an efficient and fast adaptive method based on Variational mode decomposition (VMD). In contrast to the VMD method, SVMD does not require the exact number of modal components to be set when processing signals. Instead, it obtains an approximate value of the modal center frequency and extracts the modal components continuously until all modes have been extracted or the reconstruction error falls below a specified threshold. The mathematical formulations of SVMD are shown below:

$$\left\{\begin{array}{l}\underset{u_n,{\omega }_n,f_r}{\min }\left\{\alpha J_1+J_2+J_3\right\}\\ subject to:u\_n \left(t\right)+f\_r \left(t\right)=f\left(t\right)\end{array}\right.$$

(1)

$$J_1={‖{\partial }_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast u_n\left(t\right)\right]e^{-j{\omega }_{n}t}‖}_2^2$$

(2)

$$J_2={‖{\beta }_n\left(t\right)\ast f_r\left(t\right)‖}_2^2$$

(3)

$$J_3=\sum _{i=1}^{L-1}{‖{\beta }_i\left(t\right)\ast u_n\left(t\right)‖}_2^2$$

(4)

where $\alpha$ is a balancing parameter, $f\left(t\right)$ represents the input signal, and $u_n$ and $f_r$ are the two decomposed signals of $f\left(t\right)$, which, respectively, correspond to the nth mode and the input signal’s residue. ${\omega }_n$ represents the central frequency of the nth mode, $*$ represents the convolution computation, and ${\beta }_i\left(t\right)$ and ${\beta }_n\left(t\right)$ are the impulse responses of the filters.

2.2. Tuna Swarm Optimization

The tuna swarm optimization (TSO) algorithm has few adjustable parameters and is easy to implement [35]. With fewer iterations, it optimizes the convergence rate and computation time. The algorithm starts the optimization process by randomly generating initial individuals or populations in the search space. The starting position of the tuna is as follows:

$${\mathit{Z}}_i^{\mathrm{int}}=\mathit{r}\mathit{a}\mathit{n}\mathit{d}\cdot \left(\mathit{u}\mathit{b}-\mathit{l}\mathit{b}\right)+\mathit{l}\mathit{b}, i=\mathrm{1,2},\dots ,N,$$

(5)

where ${\mathit{Z}}_i^{int}$ is the starting position of the $i$th tuna; $\mathit{u}\mathit{b}$ and $\mathit{l}\mathit{b}$ is the search range of the tuna; $N$ is the number of tuna fish; and $\mathit{r}\mathit{a}\mathit{n}\mathit{d}$ is a random vector uniformly distributed between (0,1).

The mathematical formulations of spiral foraging are as follows:

$${\mathit{Z}}_i^{t+1}=\left\{\begin{array}{cc}\begin{array}{c}{\lambda }_1\cdot \left({\mathit{Z}}_{rand}^t+\beta \cdot \left|{\mathit{Z}}_{rand}^t-{\mathit{Z}}_i^t\right|\right)+{\lambda }_2\cdot {\mathit{Z}}_i^t,i=1\\ {\lambda }_1\cdot \left({\mathit{Z}}_{rand}^t+\beta \cdot \left|{\mathit{Z}}_{rand}^t-{\mathit{Z}}_i^t\right|\right)+{\lambda }_2\cdot {\mathit{Z}}_{i-1}^t,i=\mathrm{2,3},\dots ,N\end{array}& \left.\right\}\mathrm{if} \mathit{rand}<\frac{t}{t_{\max }},\\ \begin{array}{c}{\lambda }_1\cdot \left({\mathit{Z}}_{best}^t+\beta \cdot \left|{\mathit{Z}}_{best}^t-{\mathit{Z}}_i^t\right|\right)+{\lambda }_2\cdot {\mathit{Z}}_i^t,i=1\\ {\lambda }_1\cdot \left({\mathit{Z}}_{best}^t+\beta \cdot \left|{\mathit{Z}}_{best}^t-{\mathit{Z}}_i^t\right|\right)+{\lambda }_2\cdot {\mathit{Z}}_{i-1}^t,i=\mathrm{2,3},\dots ,N\end{array}& \left.\right\}\mathrm{if} \mathit{rand}\ge \frac{t}{t_{\max }}.\end{array}\right.$$

(6)

$${\lambda }_1=a+\left(1-a\right)\cdot \frac{t}{t_{\max }},$$

(7)

$${\lambda }_2=\left(1-a\right)-\left(1-a\right)\cdot \frac{t}{t_{\max }},$$

(8)

$$\beta =e^{b\omega }\cdot \mathit{cos}\left(2\pi b\right),$$

(9)

$$\omega =e^{3\mathit{cos}\left(\left(\left(t+1/t\right)-1\right)\pi \right)}$$

(10)

where ${\lambda }_1$ and ${\lambda }_2$ represent the weight control parameters; $t$ represents the current iteration number; $t_{max}$ represents the maximum number of iterations; $a$ is a constant between (0,1); $\beta$ is the spiral equation and $\omega$ is the parameter of the spiral equation; ${\mathit{Z}}_i^t$ represents the position of the $i$th tuna at the $t$th iteration; ${\mathit{Z}}_{i-1}^t$ represents the position of the $i-1st$ tuna at the $t$th iteration; ${\mathit{Z}}_{best}^t$ represents the optimal position of tuna at the $t$th iteration of the population; and ${\mathit{Z}}_{rand}^t$ represents the random position of tuna at the $t$th iteration of the population.

Assuming that spiral search and parabolic cooperative search are conducted simultaneously with equal probability, the specific mathematical model can be described as follows:

$${\mathit{Z}}_i^{t+1}=\left\{\begin{array}{c}{\mathit{Z}}_{best}^t+\mathit{r}\mathit{a}\mathit{n}\mathit{d}\cdot \left({\mathit{Z}}_{best}^t-{\mathit{Z}}_i^t\right)+\delta \cdot p^2\cdot \left({\mathit{Z}}_{best}^t-{\mathit{Z}}_i^t\right), \mathrm{if} \mathit{rand}<0.5,\\ \delta \cdot p^2\cdot {\mathit{Z}}_i^t, \mathrm{if} \mathit{rand}\ge 0.5,\end{array}\right.$$

(11)

$$\delta ={\left(1-\frac{t}{t_{\max }}\right)}^{\left(t/t_{\max }\right)},$$

(12)

where $\delta$ is a random number between (0,1).

During the entire optimization process, TSO updates and calculates all individuals until the final condition is met, and then returns the optimal individual and its corresponding fitness value.

2.3. Long Short-Term Memory Network

A long short-term memory (LSTM) network is a type of recurrent neural network (RNN) that was proposed by Hochreiter to solve the problem of vanishing or exploding gradients in traditional RNNs [36,37]. It is typically used to address long-term dependencies in long time-series data. The details of the LSTM’s structure are presented in Figure 1.

The hidden layer of LTSM is composed of one or multiple memory cells, each containing a forget gate, an input gate, and an output gate. The retention of the previous unit state ${\mathit{S}}_{t-1}$ in the current unit state ${\mathit{S}}_t$ is determined by the forget gate, the calculation of which is presented in Equation (13).

$${\mathit{f}}_t=\sigma \left({\mathit{W}}_f\cdot \left[{\mathit{y}}_{t-1},{\mathit{x}}_t\right]+{\mathit{b}}_f\right)$$

(13)

where ${\mathit{W}}_f$ represents the weight matrix of the forget gate, ${\mathit{x}}_t$ represents the input layer vector at moment $t$, ${\mathit{y}}_{t-1}$ represents the output layer vector at moment $t-1$, ${\mathit{b}}_f$ represents the bias term, ${\mathit{f}}_t$ represents the output matrix of the forget gate, and $\sigma \left(\cdot \right)$ represents the sigmoid function. Upon activation, the unit state ${\mathit{S}}_t$ produces a real-valued vector between 0 and 1, with 1 indicating complete retention and 0 indicating the complete discarding of the state value.

The input gate controls the amount of current network input ${\mathit{x}}_t$ that is preserved in the unit state ${\mathit{S}}_t$. The mathematical formulations are as follows:

$${\mathit{i}}_t=\sigma \left({\mathit{W}}_i\cdot \left[{\mathit{y}}_{t-1},{\mathit{x}}_t\right]+{\mathit{b}}_i\right)$$

(14)

$${\tilde{\mathit{S}}}_t=\tan \mathrm{h}\left({\mathit{W}}_S\cdot \left[{\mathit{y}}_{t-1},{\mathit{x}}_t\right]+{\mathit{b}}_S\right)$$

(15)

$${\mathit{S}}_t={\mathit{f}}_t\times {\mathit{S}}_{t-1}+{\mathit{i}}_t\times {\tilde{\mathit{S}}}_t$$

(16)

where the weight matrix and bias term of the input gate are denoted by ${\mathit{W}}_i$ and ${\mathit{b}}_i$, respectively. A new unit state is constructed by applying the $\tan \mathrm{h}$ activation function, which is subsequently utilized by the output gate to determine the output information. The computation can be expressed as follows:

$${\mathit{O}}_t=\sigma \left({\mathit{W}}_O\cdot \left[{\mathit{y}}_{t-1},{\mathit{x}}_t\right]+{\mathit{b}}_O\right)$$

(17)

$${\mathit{y}}_t={\mathit{O}}_t\tan \mathrm{h}\left({\mathit{S}}_t\right)$$

(18)

where ${\mathit{W}}_O$ represents the weight matrix of the output gate; ${\mathit{b}}_O$ represents the bias term of the output gate.

2.4. Optimization Process of TSO-LSTM

The process of using TSO to find the optimal hyperparameters for LSTM is illustrated in Figure 2. The optimization process can be summarized as follows:

Step 1: Define the objective of optimization. Select the mean squared error (MSE) as the optimization objective.

Step 2: Initialize a group of tuna. Randomly initialize a certain number of tuna, where each tuna represents a set of LSTM network parameters.

Step 3: Evaluate the fitness function. For each tuna, calculate its fitness function value on the training set based on its corresponding LSTM network parameters, i.e., the loss function.

Step 4: Update the position and velocity of each tuna. Update the position and velocity of each tuna according to the position update and velocity update strategies of the tuna algorithm based on its fitness function value.

Step 5: Update the LSTM network parameters. Update the corresponding LSTM network parameters of each tuna based on its position and velocity.

Step 6: Check the termination condition. Check if the termination condition is met, for example, by reaching the maximum number of iterations or a certain fitness value.

Step 7: Output the results. Output the final optimization results, including the optimal LSTM network parameters and their corresponding fitness function values.

2.5. SVMD and TSO-LSTM Prediction Framework

The framework for the fusion prediction model of SVMD and TSO-LSTM is illustrated in Figure 3, and can be roughly divided into the following six steps:

Step 1: Import the batteries’ capacity degradation data into the SVMD algorithm as the prediction parameter, and obtain multiple sets of modal functions $\mathrm{I}\mathrm{M}\mathrm{F}\left(\mathrm{t}\right)$ through signal decomposition.

Step 2: Divide the datasets into training sets and testing sets according to a certain proportion.

Step 3: Initialize the LSTM network parameters, TSO algorithm parameters, and sample data.

Step 4: Calculate the fitness by using the mean squared error (MSE) obtained from training the sample data with the LSTM as the initial fitness value, and optimize it using the TSO algorithm. The optimal fitness value is obtained by iterating and updating the parameters until the termination condition is met.

Step 5: Import the optimal parameters obtained in Step 4 into the LSTM network for training, and output the training results.

Step 6: Based on the test results obtained in Step 5, effectively integrate the prediction results to obtain the final capacity prediction signal $\tilde{C}\left(t\right)$.

3. Results

3.1. Battery Datasets Description

The battery public datasets used in this study were obtained from Idaho National Laboratory, which is home to the National Aeronautics and Space Administration (NASA) PCoE Research Center [38], and the Center for Advanced Life Cycle Engineering (CALCE) at the University of Maryland in the United States [39]. Five batteries with various types, capacities, and discharge rates were selected from the datasets. The specifications of these batteries are presented in

Table 1. Battery-related parameters.

Battery	Type	Size/mm	Capacity/(A·h)	Discharge Rate/C
CS2-34	Prismatic	5.4 × 33.6 × 50.6	1.1	0.5
CS2-36	Prismatic	5.4 × 33.6 × 50.6	1.1	1
CS2-37	Prismatic	5.4 × 33.6 × 50.6	1.1	1
CX2-37	Prismatic	6.6 × 33.8 × 50	1.35	0.5
B0007	Cylindrical	18 × 65	2	1

.

The B0007 battery dataset was obtained by using charging, discharging, and impedance measurements under an ambient temperature of 24 °C. The charging process was conducted at a constant current of 1.5 A until the voltage reached 4.2 V, followed by a constant-voltage charging mode until the charging current decreased to 20 mA. The discharge process was conducted at a constant current of 2 A until the battery voltage dropped to 2.5 V. Then, an electrochemical impedance spectroscopy was conducted at a frequency range of 0.1 Hz to 5 kHz to measure the impedance of the battery.

For the CS2 and CX2 series, all batteries were charged and discharged at a room temperature of 24 °C. During charging, a constant current of 0.5 C was applied until the battery voltage reached 4.2 V, followed by a constant voltage charging mode until the charging current dropped to the cutoff current (50 mA). During discharging, a constant current was applied until the battery voltage dropped to 2.7 V. CX2-37 and CS2-34 were discharged at a rate of 0.5 C, while CS2-36 and CS2-37 were discharged at a rate of 1 C.

Figure 4 displays the capacity degradation curves of the five batteries. The three CS2 batteries exhibit similar degradation trends, characterized by slow degradation in the initial stage and accelerated degradation in the later stage. B0007 and CX2-37 exhibit relatively stable degradation, while all five batteries exhibit notable capacity fluctuations in their degradation curves.

3.2. Battery Capacity Sequence Decomposition Results

The CS2-34 battery capacity degradation data are used as an example in this study to decompose the capacity degradation signals into appropriate modal functions IMF(t) based on their frequency. Figure 5 shows the decomposition results.

3.3. Evaluation Indicators

To comprehensively evaluate the effectiveness of the selected method, this study selects three indicators to assess the model’s performance: root mean square error (RMSE), mean absolute percentage error (MAPE), and absolute error (MAE). The calculation formulas for these indicators are as follows:

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\sum _{t=1}^n\left(C_t-{\tilde{C}}_t\right)}^2}$$

(19)

$$\mathrm{MAPE}=\frac{1}{n}\sum _{t-1}^n\frac{\left|C_t-{\tilde{C}}_t\right|}{C_t}$$

(20)

$$\mathrm{MAE}=\frac{1}{n}\sum _{t=1}^n\left|C_t-{\tilde{C}}_t\right|$$

(21)

where $C_t$ and ${\tilde{C}}_t$ represent the actual and predicted capacity values, respectively, and $n$ represents the number of samples.

3.4. Battery Life Prediction Results

To demonstrate the effectiveness of the SVMD-TSO-LSTM prediction model, we compared it with two other combined models: VMD-TSO-LSTM and EMD-TSO-LSTM. The combined EMD-TSO-LSTM model is denoted M1, the combined VMD-TSO-LSTM model is denoted M2, and the combined SVMD-TSO-LSTM model is denoted M3.

The rated capacity of a lithium-ion battery is considered to be reached when its capacity degradation reaches 70% to 80% of the initial capacity, which is referred to as the ending of life (EOL) of the battery [2,3]. To ensure the safe and reliable use of lithium-ion batteries, we selected the life thresholds for the batteries based on the characteristics of their original capacity degradation data. Specifically, we chose 75% of the nominal capacity as the life threshold for the battery B0007, and 80% of the nominal capacity for batteries CS2-34, CS2-36, CS2-37, and CX2-37. We divided the full life cycle data into two parts, with the first 50% and 30% used to train the predictive models, and the remaining 50% and 30% used to test the models. The final results are presented in Figure 6.

Using 50% of the initial cycling data to train the three models, the effective prediction of the capacity signal for five different batteries with varying nominal capacities, types, and discharge rates is demonstrated. The prediction curves of the three models follow the fluctuation trend of the capacity signal at different positions, which confirms the feasibility of the signal extraction method employed in this study for predicting a battery’s remaining life while considering the relaxation phenomenon of lithium-ion batteries. Among the five batteries, the SVMD method yields better overall prediction results than the EMD method. Although the prediction curve of the M1 model deviates after 600 cycles in CS34, it still maintains high accuracy in predicting the remaining life of a battery before it reaches its end-of-life. The prediction performance of the M2 model is generally between that of the M1 and M3 models. This is due to the influence of the number of channels on the decomposition effect of the VMD method, which can be improved by setting an appropriate number of channels. Compared to the EMD method, the SVMD method requires fewer parameters and exhibits better robustness towards the prediction target. Using only 30% of the cycling data for training the three models results in a slight decrease in prediction performance due to the reduced amount of training data.

Figure 7 shows the estimation error curves for the five lithium-ion batteries, providing a better comparison of the predictive performance of the three models. The M3 model exhibits a smaller fluctuation amplitude in the prediction error curve, with the relative error fluctuation of most batteries (except for B0007 and CS2-37, which have limited sample data) remaining within $\pm 0.02$ (A·h). The model can still effectively track the fluctuation trend in the capacity attenuation signal, with the M3 model’s prediction curve closer to the original data curve. In particular, the M3 model outperforms the other two prediction models in the CX2-37 battery life prediction.

4. Discussion

4.1. Analysis of Battery Capacity Sequence Decomposition Results

The original battery capacity data are noisy, but after denoising using SVMD, the IMF1 component becomes smooth and free from interference caused by capacity fluctuation signals. This improves the accuracy of predicting the lifespan of lithium-ion batteries. Although the remaining modal functions contain noise, they may still contain valuable information. The seven remaining modal functions are categorized into a low-frequency group (IMF2, IMF3, IMF4) and a high-frequency group (IMF5, IMF6, IMF7, IMF8) based on their frequencies. Figure 8 illustrates the comparison between the accumulated modal functions of the two groups and the original signal. The fluctuations observed in the low-frequency modal functions are consistent with the areas of significant amplitude changes in the original signal, whereas the high-frequency modal functions capture the subtle variations in the capacity degradation signal with high accuracy. As a result, more accurate predictions can be obtained by separately forecasting the various modal functions with distinct frequencies, and then integrating them effectively.

4.2. Analysis of Battery Life Prediction Results

To assess the predictive accuracy of the models in estimating the remaining available cycle times of lithium-ion batteries before reaching their end-of-life threshold, we utilized parameters $RU{L}_{error}$ and $D_{error}$ for evaluation. The specific formula for this evaluation is provided below:

$$RU{L}_{error}=\left|RU{L}_{ture}-RU{L}_{pre}\right|$$

(22)

$$D_{error}=\frac{\left|RU{L}_{ture}-RU{L}_{pre}\right|}{RU{L}_{ture}}$$

(23)

The parameter $RU{L}_{ture}$ represents the actual remaining cycle life of the lithium-ion battery, $RU{L}_{pre}$ represents the predicted value of the remaining cycle life, $RU{L}_{error}$ represents the error between the predicted and actual remaining cycle life, and $D_{error}$ represents the relative error in the predicted cycle life. The remaining cycle life prediction results using 50% and 30% of the training data are shown in Figure 9, with specific information provided in

Table 2. Lithium-ion batteries’ RUL prediction error using the 50% training set.

Model	Battery	${\mathit{RUL}}_{\mathit{ture}}$	${\mathit{RUL}}_{\mathit{pre}}$	${\mathit{RUL}}_{\mathit{error}}$	${\mathit{D}}_{\mathit{error}}$
M1	B0007	28	31	3	0.1071
	CS2-34	156	158	2	0.0128
	CS2-36	121	113	8	0.0661
	CS2-37	196	211	15	0.0765
	CX2-37	138	146	8	0.0580
M2	B0007	28	36	8	0.2857
	CS2-34	156	150	6	0.0385
	CS2-36	121	114	3	0.0248
	CS2-37	196	203	7	0.0357
	CX2-37	138	150	12	0.0870
M3	B0007	28	26	2	0.0714
	CS2-34	156	155	1	0.0064
	CS2-36	121	122	1	0.0083
	CS2-37	196	200	4	0.0204
	CX2-37	138	134	4	0.0290

and

Table 3. Lithium-ion batteries’ RUL prediction error using the 30% training set.

Model	Battery	${\mathit{RUL}}_{\mathit{ture}}$	${\mathit{RUL}}_{\mathit{pre}}$	${\mathit{RUL}}_{\mathit{error}}$	${\mathit{D}}_{\mathit{error}}$
M1	B0007	68	74	6	0.0882
	CS2-34	296	293	3	0.0101
	CS2-36	254	259	5	0.0197
	CS2-37	336	352	16	0.0476
	CX2-37	338	353	15	0.0444
M2	B0007	68	72	4	0.0589
	CS2-34	296	299	3	0.0101
	CS2-36	254	263	9	0.0354
	CS2-37	336	353	17	0.0506
	CX2-37	338	359	21	0.0621
M3	B0007	68	71	3	0.0441
	CS2-34	296	294	2	0.0068
	CS2-36	254	255	1	0.0039
	CS2-37	336	345	9	0.0268
	CX2-37	338	345	7	0.0207

.

The predictive accuracy of the remaining useful life (RUL) for battery B0007 is affected by the limited number of available data samples. Among the three models, M3 demonstrated superior prediction stability and robustness. It achieved lower actual errors in predicting the RUL of battery B0007 compared to M1 and M2, and exhibited relative errors of less than 3% for the other four batteries. With the reduction of training data to the first 30% of the dataset, the prediction errors of all three models increased. However, M3 demonstrated better performance, maintaining an average relative error of only 2.04% and outperforming the other two models, which had a maximum error of less than 10. The SVMD and TSO-LSTM models demonstrated high prediction accuracy, robustness, and stability, surpassing the other two models.

Figure 10 summarizes the performance indicators of the three models when using the first 30% and 50% of cycle data as the training set for prediction, where smaller values of RMSE and MAPE indicate the higher prediction accuracy of the model. When using the first 30% cycle data, the overall MAPE of the M3 model is below 1%, with the RMSE of B0007 and CS2-37 batteries exceeding 1%; however, both are controlled within 1.5%, which is superior to the M1 and M2 models, and the RMSEs of the remaining three groups of batteries are all below 1%. When using the first 50% cycle data, M3′s performance indicators are superior to the other two models, with the MAPE and RMSE of CX2-37 battery below 0.26%; the maximum MAPE (except for CS2-36) is 1.08% for the other four batteries, which are within 0.7%, and the maximum RMSE does not exceed 1%. The performance of M3 in predicting the remaining useful life of five batteries with different types, capacities, and discharge rates surpasses that of the other two models.

5. Conclusions

This study proposes an SVMD and TSO-LSTM-based model for accurately predicting the remaining life of lithium-ion batteries. The main conclusions are as follows:

• The SVMD method is suitable for decomposing nonlinear and non-stationary signals. By applying this method, the capacity attenuation data of lithium-ion batteries are decomposed into modal functions with different scales, which reduces the impact of capacity fluctuations on the prediction accuracy during the actual use of lithium-ion batteries. This approach effectively accounts for the nonlinear and non-stationary characteristics of battery capacity data, leading to a more accurate prediction of the remaining life of the battery.
• The model for predicting the remaining life of lithium-ion batteries, based on SVMD and TSO-LSTM, demonstrates high prediction accuracy and stability. Compared to other models proposed in this study, the maximum MAPE for batteries of different types, discharge rates, and capacities using the SVMD and TSO-LSTM prediction model does not exceed 1%, and the average relative error is within 2.1%. This model exhibits good generalization performance and stable robustness.