Accurate estimation of the state-of-health (SOH) of lithium-ion batteries is a crucial reference for energy management of battery packs for electric vehicles. It is of great significance in ensuring safe and reliable battery operation while reducing maintenance costs of the battery system. To eliminate the nonlinear effects caused by factors such as capacity regeneration on the SOH sequence of batteries and improve the prediction accuracy and stability of lithium-ion battery SOH, a prediction model based on Variational Modal Decomposition (VMD) and Dung Beetle Optimization -Support Vector Regression (DBO-SVR) is proposed. Firstly, the VMD algorithm is used to decompose the SOH sequence of lithium-ion batteries into a series of stationary mode components. Then, each mode component is treated as a separate subsequence and modeled and predicted directly using SVR. To address the problem of difficult parameter selection for SVR, the DBO algorithm is used to optimize the parameters of the SVR model before training. Finally, the predicted values of each subsequence are added and reconstructed to obtain the final SOH prediction. In order to verify the effectiveness of the proposed method, the VMD-DBO-SVR model was compared with SVR, Empirical Mode Decomposition-Support Vector Regression (EMD-SVR), and VMD-SVR methods for SOH prediction of batteries based on the NASA dataset. Experimental results show that the proposed model has higher prediction accuracy and fitting degree, with prediction errors all within 1% and better robustness.
Keywords: lithium-ion battery; state of health; variational mode decomposition; dung beetle optimization algorithm; support vector regression
With the acceleration of economic globalization and the massive use of fossil fuels, environmental pollution and energy shortage have become increasingly prominent issues. Lithium-ion batteries for energy storage have found extensive applications across various facets of daily life and industrial production, owing to their substantial energy-storage capacity and excellent cycling performance [[
Equivalent circuit models or electrochemical models are the primary approaches utilized in model-based methods. Reference [[
Data-driven methods essentially create a black-box model, where the internal structure of the battery does not need to be explicitly constructed. To construct a prediction model for SOH, it is only necessary to extract and analyze external parameters of the battery that are extremely correlated with SOH, and use them as training data. Reference [[
To address the above issues, we extracted the available capacity of each charge-discharge cycle of the battery and calculated the corresponding SOH data. We used this SOH data as a health indicator and proposed a lithium-ion battery SOH prediction method based on the VMD-DBO-SVR model. Firstly, the VMD method is employed to decompose the original SOH sequence into a series of Intrinsic Mode Function (IMF) components that represent local features at multiple scales. Then, SVR is employed to model and predict each IMF element directly, and to address the difficulty in selecting SVR parameters, the DBO algorithm is utilized to optimize the parameters of the SVR model before model training. Finally, the predicted values of each sub-sequence are combined and reconstructed to derive the ultimate SOH prediction value. The proposed method is evaluated using the NASA dataset.
We compared the accuracy and practicality of our method with other methods using the same dataset in the literature. In reference [[
Battery SOH refers to the current health status of a battery, which is an important indicator of battery performance and service life, as the health status of a battery gradually deteriorates over time. SOH is typically expressed as a percentage and is defined as follows [[
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VMD is a variational method-based technique used to decompose nonlinear and non-stationary signals into multiple Intrinsic Mode Functions (IMF) [[
Step 1: Construct the variational model. The original SOH signal is decomposed into
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Step 2: Introduce a penalty factor
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Step 3: Initialize
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Step 4: Stop the iterative updates until the stopping criteria are satisfied, which are as follows:
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The Dung Beetle Optimization (DBO) algorithm is a population-based intelligent optimization algorithm proposed by Jiankai Xue et al. [[
The DBO algorithm mainly includes four types of behavior: rolling, breeding, foraging, and stealing, corresponding to four types of dung beetles: rolling dung beetle, breeding dung beetle, foraging dung beetle, and thief dung beetle. The algorithm achieves parameter optimization by having each type of dung beetle perform its corresponding operation. The specific four behaviors of the DBO algorithm are as follows:
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The dung beetle rolls a much larger dung ball than itself and usually uses celestial cues such as the sun to navigate in order to maintain the dung ball's motion in a linear path. During the rolling process, the position of the rolling dung beetle is updated according to the following formula:
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where, t is the iteration count,
When a dung beetle confronts an obstruction that blocks its path, it needs to reposition itself by dancing in order to find a new route. To simulate this dance behavior, a tangent function is used to obtain a new rolling direction. After the dung beetle determines a different direction, it continues to roll the ball backward. The position of the dung beetle is updated as follows:
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where,
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In nature, female dung beetles roll their dung balls to a safe place suitable for laying eggs and hide them. Inspired by this behavior, a strategy for selecting boundaries is chosen to mimic the oviposition area of female dung beetles, which is defined as follows:
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Once the oviposition area is determined, the female beetle will select an egg in this area for laying. The boundary range of the oviposition area will dynamically change, which is mainly determined by the value of
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The eggs laid by female beetles will gradually grow. Some matured small beetles will come out of the ground to search for food. The optimal foraging area of small beetles is modeled as follows:
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There are some beetles, called thieves, that steal dung balls from other beetles. From Equation (
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The SVR method finds the best hyperplane that fits the data by continuously reducing the error between predicted and actual values. Its advantage is strong generalization ability and good performance in handling nonlinear problems [[
Assuming a given sample set
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This paper employs the SVR method to establish the SOH prediction model. However, the selection of the penalty factor
The process of DBO for optimizing SVR parameters is illustrated in Figure 1, with the main procedures as follows:
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The flowchart of the combined prediction model based on VMD-DBO-SVR is illustrated in Figure 2.
The lithium-ion battery prediction model in this paper consists of four steps:
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The dataset used in this experiment is from the Prognostics Center of Excellence (PCoE) at NASA. The dataset includes aging test data for three 18,650 lithium-ion batteries with a rated capacity of 2 Ah each, labeled B5, B6, and B7. All data were collected at a temperature of 24 °C using the CC-CV cycle test method for aging testing. Firstly, the battery is charged with a steady current of 1.5 A. When the battery voltage hits 4.2 V, the steady voltage mode is applied to continue charging the battery until the charging current drops below 20 mA. Then, the battery is discharged at a steady current of 2 A until the voltages of B5, B6, and B7 drop to 2.7 V, 2.5 V, and 2.2 V, respectively. B5–B7 batteries have undergone 168 charge-discharge cycles, and the data includes measured voltage, current, temperature, and the available capacity for each cycle. Therefore, this dataset is mainly used for predicting the SOH and RUL, and estimating the State of Charge (SOC) of lithium-ion batteries. We extract the available capacity from each charge-discharge cycle and process it as SOH degradation data according to the SOH definition, enabling us to estimate the SOH of lithium-ion batteries. The detailed parameters of the 3 batteries are shown in Table 1, and the SOH degradation curves are shown in Figure 3.
To more comprehensively demonstrate the adaptability of the VMD-DBO-SVR prediction method, in this experiment, 50% and 60% of the SOH data were, respectively, selected as the training set, and the remaining 50% and 40% of the data were applied as the test set to test the performance of the model.
When using the VMD algorithm for signal decomposition, it is necessary to select the mode number K beforehand. If the chosen number of modes is insufficient, certain significant information in the initial signal may be lost. On the other hand, if the selected mode number is too large, it may lead to frequency aliasing. Therefore, in this study, the mode number K is determined by examining the arrangement of center frequencies under various decomposition mode numbers. Taking the B5 battery as an example, the center frequencies under different K values are shown in Table 2.
As shown in Table 2, when K = 6, the central frequencies of the third and fourth mode components are close, indicating an over-decomposition phenomenon. Therefore, K = 5 is determined. The time-domain and corresponding frequency-domain plots of the SOH signal of battery B5 after VMD decomposition are shown in Figure 4 and Figure 5, respectively.
In Figure 4, IMF
The parameter settings used in this paper for optimizing with the DBO algorithm are as follows: the number of dung beetle populations is pop = 30; the proportion of roller dung beetles, breeding dung beetles, foraging dung beetles, and thief dung beetles in the dung beetle population is 0.2, 0.2, 0.2, and 0.4, respectively; the dimension of variable parameters is dim = 2; the upper limit for the number of iterations is
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The performance of the proposed model is evaluated using the above indicators in this paper. A smaller value of MAPE and RMSE indicates a more accurate prediction result, while a value closer to 1 for RA demonstrates a better prediction performance of the model.
When 60% of the data from three lithium-ion batteries were taken as the training set (B5, B6, and B7 with 100 cycles each), the fitting of the SOH prediction results based on the VMD-DBO-SVR model and the actual test results of the lithium-ion batteries are shown in Figure 6. The corresponding prediction errors are shown in Table 3, where ST represents the starting point of the prediction.
As shown in Figure 6, the predicted SOH values for the three batteries are very close to the true values, indicating that the proposed method can effectively predict the trend of battery SOH and has good prediction accuracy. As shown in Table 3, the best prediction performance among the three batteries is B7, with an RA of 99.74% and RMSE and MAPE of 0.5019 and 0.2594%, respectively. The worst prediction performance is for B6, but its RA value is also as high as 99.41%, and RMSE and MAPE are only 0.2765 and 0.5863%, respectively. These results demonstrate that the VMD-DBO-SVR model proposed in this paper has good applicability to different lithium-ion batteries and can maintain high prediction accuracy.
To further verify the SOH prediction accuracy of the proposed model with insufficient training data, 50% of the data from three batteries were selected as the training set (B5, B6, and B7 with 84 cycles each), and the remaining 50% were used as the test set for SOH prediction. When 50% of the data was used for training, the fitting of the lithium-ion battery SOH prediction results based on the VMD-DBO-SVR model to the true test results is shown in Figure 7, and the corresponding prediction errors are shown in Table 4.
When using fewer data as the training set, the prediction accuracy will decrease as shown in Figure 7, indicating that the less effective information provided during modeling from the early stage of prediction, the greater the error in the prediction results. However, the proposed model in this paper still has a good predictive effect. As shown in Table 4, even for battery B6 with the worst prediction accuracy, its RA value still reaches 99.21%, and the RMSE and MAPE values are only 0.8227 and 0.7892%, respectively. Compared with 60% training data, the corresponding RA value only decreases by 0.2%, while the RMSE and MAPE values increase by only 0.3227 and 0.2029%, respectively, indicating that the proposed prediction model has good generalization ability.
To validate the effectiveness and superiority of the VMD-DBO-SVR model proposed in this paper for predicting the SOH, three batteries were selected with 50% of their data employed as the training set, and compared with three other prediction models: SVR, EMD-SVR, and VMD-SVR. Figure 8 shows the comparison of the prediction results under different algorithms, and the corresponding prediction errors are shown in Table 5.
Figure 8 reveals that the single model SVR has the worst prediction performance, and its estimation error gradually increases in the later stage, indicating poor prediction accuracy. Although the EMD-SVR model reduces the error, the prediction accuracy is still poor and has not stabilized. The VMD-SVR model overcomes the end effect and modal aliasing phenomenon of EMD decomposition, and compared with the EMD-SVR model, the prediction performance is significantly improved, but the prediction accuracy is still not high enough. The VMD-DBO-SVR model proposed in this paper preprocesses the initial SOH data using the VMD decomposition method to reduce noise in the original data. After optimizing the SVR model parameters using the DBO algorithm, the prediction model is trained to accurately predict the overall degradation trend of the battery and has good tracking ability for capacity regeneration, resulting in the best prediction performance.
From Table 5, it can be seen that the SVR model has a shorter prediction time, but its error is too large, which may result in unsatisfactory prediction results in practical applications. Compared with the EMD-SVR and VMD-SVR methods, the proposed model has a significant improvement in prediction accuracy with a relatively small increase in prediction time. Taking the B5 battery as an example, the RMSE of SVR, EMD-SVR, and VMD-SVR models are 1.5486, 0.9775, and 0.5797, and their prediction time are 0.6875 s, 3.2114 s, and 3.4732 s, respectively, while the RMSE and prediction time of the proposed model are 0.4771 and 3.7297 s, proving that the proposed model has higher accuracy than the other three models with only a small increase in prediction time.
To further demonstrate the superiority of the proposed VMD-DBO-SVR prediction method, still using battery B5 as an example and under the same initial conditions with the same training set, we compared the prediction results of our method with those of recently published models in related literature. The comparison results are shown in Table 6.
From Table 6, it can be seen that although Reference [[
In this section, we first conducted simulations with 60% of the training data to validate the precision of the proposed method on three batteries. Then, we reduced the training data to 50% to demonstrate that the prediction accuracy decreases with the decrease of training data, but the proposed model still has good prediction performance. Furthermore, we compared the proposed model with SVR, EMD-SVR, and VMD-SVR models from prediction accuracy and running time, the experimental results showed that the proposed model had significantly improved prediction accuracy with a small increase in computation time. Finally, we compared the proposed model with relevant methods in recent literature to confirm its superior predictive performance.
Accurately predicting the SOH of lithium-ion batteries can improve their safety during operation and prevent accidents. We propose a SOH prediction model based on Variational Mode Decomposition (VMD) and Dung Beetle Optimization-Support Vector Regression (DBO-SVR). Through verification and analysis, the primary conclusions of this paper are as follows:
- (
1 ) The VMD algorithm can decompose the battery SOH sequence into multiple stationary mode components, which can effectively reduce noise interference, such as capacity regeneration and testing errors, and minimize prediction errors. - (
2 ) The selection of kernel parameters in the SVR method directly affects the accuracy of SOH prediction. To address this issue, we proposed a DBO optimization algorithm to provide the optimal parameters for the SVR method. The combination of the two methods can improve the prediction accuracy and stability of SOH. - (
3 ) NASA battery dataset was employed to validate the prediction performance of the proposed VMD-DBO-SVR model. The results showed that the VMD-DBO-SVR model had good prediction accuracy and stability, and the prediction error was maintained within 1%.
The above conclusion indicates that the model solves various noise interference problems through the VMD algorithm, and solves the problem of difficult SVR parameter selection through the DBO optimization algorithm, thereby improving prediction accuracy and achieving the preliminary research objectives of this article.
In actual SOH prediction for lithium-ion batteries, it is sometimes difficult to directly measure the available capacity of the battery, making the approach proposed in this paper unsuitable. Therefore, the next research direction of this paper is to use easily measurable feature factors that characterize the degradation pattern of SOH.
Graph: Figure 1 Optimization process of SVR parameters using DBO algorithm.
Graph: Figure 2 VMD-DBO-SVR forecasting model process.
Graph: Figure 3 SOH degradation curves of NASA dataset battery B5, B6, and B7.
DIAGRAM: Figure 4 Time-domain diagram of VMD decomposition for B5 battery.
DIAGRAM: Figure 5 Spectrum diagram of B5 battery after VMD decomposition.
Graph: Figure 6 SOH prediction results with 60% data set of three batteries as training set. (a) B5; (b) B6; (c) B7.
Graph: Figure 7 SOH prediction results of three batteries with 50% dataset as training set. (a) B5; (b) B6; (c) B7.
Graph: Figure 8 Comparison of SOH prediction results under different models for three batteries. (a) B5; (b) B6; (c) B7.
Table 1 Detailed parameters of the experimental dataset.
Number Temperature/°C Discharge Current Capacity/Ah Shutdown Voltage/V B5 24 2A/CC 2 2.7 B6 24 2A/CC 2 2.5 B7 24 2A/CC 2 2.2
Table 2 Center frequency of B5 battery under different K values.
K Center Frequency/Hz 2 1.97 × 10−5 0.233 - - - - 3 1.97 × 10−5 0.166 0.328 - - - 4 1.96 × 10−5 0.095 0.233 0.357 - - 5 1.96 × 10−5 0.093 0.167 0.292 0.401 - 6 1.96 × 10−5 0.066 0.1664 0.224 0.330 0.402
Table 3 Prediction results of the VMD-DBO-SVR model based on the training dataset of 60%.
Battery MAPE/% RMSE RA B5 0.3511 0.3488 0.9964 B6 0.5863 0.5019 0.9941 B7 0.2594 0.2765 0.9974
Table 4 Prediction results of the VMD-DBO-SVR model based on a 50% dataset as training set.
Battery MAPE/% RMSE RA B5 0.3906 0.4771 0.9961 B6 0.7892 0.8227 0.9921 B7 0.3318 0.4828 0.9966
Table 5 SOH Prediction Errors of Different Models for three Batteries.
Battery Model MAPE/% RMSE RA Prediction Time/s B5 SVR 1.7833 1.5486 0.9821 0.6875 EMD-SVR 1.1607 0.9775 0.9883 3.2114 VMD-SVR 0.6467 0.5797 0.9935 3.4732 VMD-DBO-SVR 0.3906 0.4771 0.9961 3.7297 B6 SVR 1.9822 1.6458 0.9801 0.7751 EMD-SVR 1.3974 1.3233 0.9860 3.2046 VMD-SVR 1.0090 0.9082 0.9899 3.6104 VMD-DBO-SVR 0.7892 0.8148 0.9921 3.9867 B7 SVR 1.4489 1.4325 0.9855 0.6658 EMD-SVR 1.1933 1.0556 0.9880 3.1699 VMD-SVR 0.7364 0.6431 0.9926 3.1105 VMD-DBO-SVR 0.3318 0.4828 0.9966 3.4405
Table 6 Comparison of prediction results between proposed method and other method in literature.
Battery Model MAPE/% RMSE B5 IALO-SVR [ 0.7400 0.6841 ABMS-CEEMDAN-LSTM [ 1.3145 1.0948 VMD-DBO-SVR 0.3906 0.4771
Conceptualization, C.W. and J.F.; methodology, C.W. and J.F.; software, X.H. and X.X.; validation, J.M. and C.W.; formal analysis, J.M.; investigation, J.F.; resources, C.W. and X.H.; data curation, X.X.; writing—original draft preparation, C.W. and J.F.; writing—review and editing, C.W.; visualization, X.X.; supervision, J.F.; project administration, C.W.; funding acquisition, C.W. and X.H. All authors have read and agreed to the published version of the manuscript.
The data supporting the results of this study was obtained from the NASA Prognostics Center of Excellence (PCoE). Available online: https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository (accessed on 15 July 2022).
The authors declare no conflict of interest.
By Chunling Wu; Juncheng Fu; Xinrong Huang; Xianfeng Xu and Jinhao Meng
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