A novel lithium-ion battery capacity prediction framework based on SVMD-AO-DELM

Accurate and efficient lithium-ion battery capacity prediction plays an important role in improving performance and ensuring safe operation. In this study, a novel lithium-ion battery capacity prediction model combining successive variational mode decomposition (SVMD) and aquila optimized deep extreme learning machine (AO-DELM) is proposed. Firstly, SVMD is used to divide capacity signal and it improves short-term trend prediction, especially for capacity growth that occurs during the degradation process. Secondly, the DELM network outperforms other networks in efficiently extracting time-dependent features, and it is more accurate than other standard ELM-based methods. The AO algorithm is used to find the parameters of the DELM training process for the problem of sensitivity to initial weights. Finally, experiments are conducted to validate the predictive performance of the proposed model based on NASA and CALCE lithium-ion batteries sequences. The MAE (0.0066Ah, 0.0044Ah), RMSE (0.0113Ah, 0.0078Ah), MAPE (0.44%, 0.82%) are effectively reduced, and the R2 (98.94%, 99.87%) is better than the prediction performance of other hybrid models.


Introduction
Electric vehicles commonly employ lithium-ion batteries as their internal power source due to its benefits of high energy and power density [1] [2]. The complexity of the battery degradation process makes it hard to forecast the life. In order to evaluate the state of lithium-ion batteries, the study of the state of health (SOH) and remaining useful life (RUL) prediction of batteries is necessary [3] [4]. This can guarantee a decrease in equipment maintenance costs and an increase in system operational reliability.
The current methods for lithium-ion battery life prediction are mainly divided into model-driven and data-driven methods [5]. The capacity prediction technique based on the model-driven method is based on various electrochemical B Guorong Ding dingguorong2022@163.com 1 Statistics Bureau of Maiji District, Tianshui 741020, Gansu, China 2 College of Science, Wuhan University of Science and Technology, Wuhan 430065, Hubei, China reactions in the battery can extract the corresponding models to estimate the RUL and SOH of the battery [6]. The basic mechanism of the internal reaction of the lithium battery driven by the model analyzes the performance change mode during the operation of the battery, considers the effect of the performance on the internal and external state variables of the battery, and then establishes the cell degradation model [7]. Although substantial progress has been made in modeldriven prediction, major drawbacks remain [8]. In addition, there is no accurate and general battery degradation model that can describe the key parameters associated with battery aging for accurate capacity prediction [9].
In contrast, data-driven methods can effectively avoid the above-mentioned problems inherent to model-based RUL prediction methods, which rely mainly on data collection [8]. The capacity prediction methods for lithium-ion batteries are divided into direct and indirect prediction methods in terms of the selection of characteristic quantities [10]. For indirect forecasting, it has two obvious drawbacks. On the one hand, the target prediction is based on the prediction results of other exogenous variables, which leads to cumulative errors [11]. On the other hand, the multicollinearity presented between the selected variables may inevitably introduce overfitting. In contrast, time series predictions can capture future patterns without the use of any additional exogenous variables. In light of these reasons, direct predictions of capacity series have replaced indirect estimates. Support vector machine (SVM) [12,13], Gaussian regression processes (GPR) [14,15], autoregressive integrated moving average (ARIMA) [16], extreme learning machine (ELM) [17][18][19], long and short-term memory network (LSTM) [9,20] and gated recurrent unit (GRU) [21,22] are commonly used for lithium-ion battery capacity prediction due to their nonlinear mapping and self-learning capabilities. The effectiveness of a single forecast model is limited due to the capacity regeneration problem during degradation. Definite deep learning algorithms may have low prediction accuracy when utilizing poor initial settings, and they are challenging to apply to a variety of time series with unknown features [23].
In response to the capacity regeneration problem during the degradation of lithium-ion batteries, many multiscale decomposition methods have been introduced. Wavelet packet decomposition (WPD) [24], empirical mode decomposition (EMD) [20,25,26] and variational mode decomposition (VMD) have been used to eliminate the capacity regeneration during degradation. However, they all suffer from poor decomposition, too long time or difficult to identify parameters [27][28][29]. Successive variational mode decomposition (SVMD) does not require the exact number of mode components to be set as in VMD when processing a signal, and it uses a continuous approach to identify and extract all components. The SVMD technique helps to increase the speed of convergence and has been used successfully in wind speed prediction and other fields [30,31].
Deep extreme learning machine (DELM) [32,33] is a deep neural network based on ELM, which is currently more effective in time series prediction. In practice, the initial weights of the DELM are usually chosen based on engineering experience. In recent years, many intelligent algorithms have been successfully applied to parameter optimization, such as particle swarm optimization (PSO) [34], grey wolf optimization (GWO) [35], and sparrow search algorithm (SSA) [19,36]. The aquila optimization (AO) [37] with strong find ability can further improve the prediction accuracy and robustness of DELM, so it is used in this study.
Therefore, an SVMD-AO-DELM model for lithium-ion battery capacity prediction is proposed in this paper. Firstly, SVMD is used for decomposing the signal into several intrinsic mode functions (IMFs) representing different frequency. Secondly, the DELM network outperforms the one-way network in extracting time-dependent features efficiently. The AO algorithm is introduced to optimize the parameters of the DELM training process for the problem of sensitivity to initial weights. Finally, the final lithium battery capacity prediction results are obtained by effectively integrating the prediction results of each component.
The remainder of the paper is organized as follows. Section 2 presents some basic methods of SVMD and DELM. In addition, the proposed SVMD-AO-DELM model is presented in Sect. 3. Section 4 discusses the predictive performance of different methods. Finally, Sect. 5 discuss and summarize the main contributions of this study.

SVMD theory
The SVMD method [38] is based on the VMD model, which continues until all modes are extracted or the reconstruction error is less than the threshold to obtain an approximation of the mode center frequency. The SVMD method does not need to precisely set the number of mode components when processing signals and reduces computation time.
The specific steps of the SVMD method are detailed in the literature [38] and will not be expanded in detail here.

DELM model
DELM is a deep neural network made up of multiple stacks of extreme learning machine auto encoder (ELM-AE), which are trained one by one to unsupervised implement the mapping of samples to deep features [39].
The input weight matrix for ELM-AE is randomly generated and the implied layer matrix is calculated as, The loss function of the network is: The output weight of ELM-AE can be calculated by the following formula: where C is a regularization parameter, which is used to improve the generalization performance of ELM-AE. X is the input sample matrix; f (·) is the activation function; I stands for identity matrix. DELM has reached the global optimization of the network by training ELM-AE one by one. Taking the training process of the (i − 1)th ELM-AE as an example, the output matrix L i−1 calculated by the (i −1)th ELM-AE is used as the input and output of the current ELM-AE. The calculation method of the output matrix L i−1 is as follows: Then calculate the output weight and output matrix L i of the current ELM-AE: where k is the number of stacks ELM-AE; L k is the depth feature finally extracted by DELM. Based on the above training process, DELM realizes the nonlinear mapping of sample data to low-dimensional features. Since the DELM network carries out multi-layer unsupervised learning in the early stage, the number of neurons in each hidden layer is set, so the DELM network has the characteristics of fast learning speed and strong generalization ability. In order to set the weight of DELM network reasonably, this paper adopts the aquila optimization algorithm to find the weight, so as to improve the prediction accuracy of the network and the generalization ability of the model.

AO algorithm
The AO is a highly reliable and consistent swarm intelligence optimization algorithm with powerful optimal solution solving capability, capable of convergence with relatively fast acceleration and strong stability for optimization seeking. The specific steps of the AO algorithm are detailed in the literature [37] and will not be expanded in detail here.

AO-DELM method
In response to the fact that the use of poor initial parameters in deep learning methods may lead to poor prediction accuracy, the AO algorithm is used to optimize the initial weights of the DELM. It can effectively avoid falling into local optima and improve the capture of the best positions. The fitness function is used to calculate and find the best position of the eagle, which will be updated iteratively as the AO algorithm operates until the end condition of the algorithm is met. The process flowchart of AO-DELM is shown in Fig. 1.

SVMD-AO-DELM framework
The proposed SVMD-AO-DELM lithium capacity prediction steps are as follows. Step 1 Obtain a sequence of lithium-ion battery capacities, the SVMD method is used to decompose the sequence to obtain a number of IMFs in order to eliminate the capacity growth phenomenon and other problems arising from battery degradation, and to divide the training and test sets in a certain proportion.
Step 2 Determine the DELM network structure (two layers in this paper), and for each subsequence component, construct the MSE of the training sample as an objective function for the AO algorithm to optimize the initial weights of the DELM.
whereŷ i denotes the predicted obtained data, y i denotes the true value, n denotes the length of the data.
Step 3 Setting the falcon population size N, the maximum number of iterations, the upper and lower bounds of the weights, and updating the parameters to finally obtain the optimal weight parameters.
Step 4 Create the AO-DELM model, then use it to forecast the values of each IMF.
Step 5 Construct multiple evaluation indicators to interpret and evaluate the model.
The flow diagram of SVMD-AO-DELM model is presented in Fig. 2.

Data source
The first experimental dataset is from NASA public dataset of lithium-ion batteries, and four batteries, B0005, B0006, B0007 and B0018, are selected for simulation validation. The rated capacity of each battery is 2Ah.
The second experimental dataset is from the Center for advanced life cycle engineering (CALCE) at the University of Maryland, USA, where six batteries, CS33, CS34, CS35, CS36, CS37, CS38, are selected for model validation. The rated capacity of each battery is 1.2Ah. Figure 3 shows the decay curves of capacity with the number of cycles during the discharge process of NASA and CALCE batteries, respectively. It can be seen that different types of batteries experience capacity growth during discharge, indicating the non-smooth, nonlinear nature of the lithium-ion battery capacity series.

Evaluation indexes
In this study, mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), correlation index (R 2 ), relative accuracy (RA), Theil inequality coefficient (TIC) and the index of agreement (IA) are used to measure the superiority between models. The specific formulae are as follows.
where y i andŷ i are the observed and predicted values, respectively. y is the average value of n observed samples.

Prediction results
Considering previous statements in the literature that the predictive effectiveness of a single model is always limited, no further single prediction algorithm is listed in this paper. Accurate prediction of lithium-ion battery capacity is a prerequisite for better prediction performance, and to verify the effectiveness and practical relevance of the proposed method, the VMD combined with ELM, KELM and a single DELM model is compared based on battery ageing data from NASA and CALCE. However, the accuracy of models constructed based on the data-driven algorithm usually depends on the size of the training set and the suitability of the algorithm. Therefore, to better represent the SVMD-AO-DELM prediction framework established in this paper, 70%, 50% and 30% of the full-life cycle data of the dataset were used to train the model during training, and then the remaining 30%, 50% and 70% of the full-life cycle data were used to test the model to obtain the final prediction results. To save the length of the article, only the results predicted using 50% of the dataset are shown here.
Several models were first trained using the 50% dataset, as shown in Fig. 4. It can be seen that when using SVMD-ELM, the ELM random weight assignment resulted in very unstable predictions with relatively large fluctuations in the predictions. In particular, the degraded data for batteries CS33 and CS34 have significant fluctuations with large errors in prediction, with RMSE of 0.4274 and 0.4086 for CS33 and 0.1434 for CS34.
To further demonstrate the effectiveness of the proposed method, only 30% of the measured capacity data was used to train the capacity prediction model for Li-ion batteries, with the latter 70% of the data used to test. The battery capacity prediction results are shown in Fig. 4a-j. It is much more difficult to train an accurate prediction model using only 30% of the measurement data compared to half of the ageing data. The results show an increase in MAE, RMSE and MAPE compared to the 70% and 50% training samples, but the overall prediction accuracy is still above 99%.
For model training with small samples, it can be seen from the figure that the prediction accuracy of both SVMD-ELM and SVMD-KELM is significantly reduced compared to each other. The prediction accuracy of SVMD-DELM has significantly increased as a result of the special nature of the DELM deep network structure, which can capture the degradation trend of the capacity with a small amount of data. The fitness curve is shown in Fig. 4k. After applying AO to find the weights of DELM, the prediction accuracy is further improved, with an RMSE of 0.0129 for CS33 and 0.0040 for CS34, representing 97.21% improvement in prediction accuracy. The prediction results based on four batteries from NASA and six batteries from CALCE show that the proposed method can capture the degradation trend relatively accurately using the actual capacity of the previous cycle as input, and that overall, the method has good prediction accuracy. It can be seen that the proposed model in this paper is computationally less expensive and has superior time series processing capability to better predict the future capacity of lithium-ion batteries. Similar results can be derived from Table 1. The relationship between the observed and expected values is determined by the fitted curve. As observed in Fig. 4l, even for diverse data sets, the SVMD-AO-DELM prediction curves are rather near to the actual curves. Additionally, even at the location where the capacity had been generated, the match between the expected and actual numbers is pretty strong, and the prediction accuracy is unaffected. Hence, the performance improves as the scatter point gets nearer to this line. The scatter distribution roughly follows the shape of the regression line. This indicates that the suggested model has a special ability to fit data with significant volatility.
To further illustrate the effectiveness of the proposed model, seven evaluation metrics were used to evaluate the model, where the mean value of each metric refers to the arithmetic mean of the prediction metrics obtained using 30%, 50% and 70% training samples. It can be seen that the SVMD-AO-DELM model proposed in this paper has a relatively good prediction effect on the capacity prediction of lithium-ion batteries, and the prediction accuracy has also been improved compared to other models. The RMSE for different batteries is shown in Fig. 5.
The boxplot of the eight models' absolute prediction errors is shown in Fig. 6. It should be emphasized that the relative error percentage is the proportion of the absolute error value to the real value, whereas the absolute value of the prediction error refers to the absolute value of the true value and the projected value. Among them, it can be seen that the SVMD-AO-DELM model distribution of absolute error is comparatively nearer to zero, has a narrow range of change, and has less outliers than other comparison models.
The capacity prediction model presented in this research is contrasted with a number of alternative approaches using the same sample data. As seen in Table 2, LSTM, GRU, and Elman neural network prediction accuracy declines with a reduction in training data. The model presented in this research still has a high level of prediction accuracy, lessens the heavy reliance on data, and requires less training time.

Conclusion
Accurate and efficient lithium-ion battery capacity prediction plays an important role in improving performance and ensuring safe operation. This paper proposes a semi-supervised model, SVMD-AO-DELM, to predict lithium-ion battery capacity. The SVMD technique can greatly improve the performance of short-term trend prediction, especially for the capacity growth phenomenon. For the high frequency and irregular variation of time series data, the prediction capability of the DELM method is improved by introducing unsupervised feature learning. The SVMD-AO-DELM framework provides high accuracy and stability for lithium-ion battery capacity prediction at different time scales. The experimental results show that the framework can reduce prediction uncertainty and outperform other machine learning models in terms of prediction accuracy. However, this study only makes use of a single capacity data and does not consider the current-voltage situation of lithium-ion batteries during charging and discharging. Some indirect features will be extracted to predict capacity in the subsequent paper, and other neural network methods will be combined to further improve the prediction accuracy.
Author contribution Guorong Ding contributed to the conceptualization, data curation, methodology, supervision, validation, software, writing-review and editing. Hongxia Chen was involved in writing-review and editing.
Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Data availability Data will be made available on request.

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Ethical approval This article does not involve ethical issues.