A two-stage deep learning-based hybrid model for daily energy consumption forecasting with variational mode decomposition and wavelet thresholding denoising

Bowen Hou; Yin Liu; Jiajun Chen; Letian Qi; Jinhao Zhang; Honghao Zhao; Bowen Hou; Yin Liu; Jiajun Chen; Letian Qi; Jinhao Zhang; Honghao Zhao

doi:10.3934/era.2026325

Electronic Research Archive

2026, Volume 34, Issue 10: 7531-7564. doi: 10.3934/era.2026325

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Research article

A two-stage deep learning-based hybrid model for daily energy consumption forecasting with variational mode decomposition and wavelet thresholding denoising

1.
Department of Decision Sciences, School of Business, Macau University of Science and Technology, Macao 999078, China
2.
China Nanguang Group Co., Ltd.: Zhuhai, Guangdong-Macao In-Depth Cooperation Zone, 519000, China
3.
School of Electronic Engineering, Huainan Normal University, Huainan 232038, China
4.
Biomedical Engineering, The Hong Kong Polytechnic University, Hong Kong 999077, China
5.
School of Computer Science and Engineering, Faculty of Innovation Engineering, Macau University of Science and Technology, Macao 999078, China

Received: 07 May 2026 Revised: 16 July 2026 Accepted: 07 August 2026 Published: 01 September 2026

Global energy demand keeps rising amid economic expansion and population growth. Although renewables like wind and solar have grown fast, fossil fuels (especially petroleum) remain central to the global energy mix, underpinning industry, households, policy, and stability. This article presents an improved hybrid forecasting framework, denoted as the VMD-WTD-GRU model with mode-specific denoising, for daily average petroleum consumption, which adopts a two-stage decomposition-targeted denoising strategy. The framework combines the models of variational mode decomposition, wavelet threshold denoising, and the gated recurrent unit networks for petroleum energy forecasting, where different models are designed for time-series data decomposition, noise elimination, sub-series forecasting, and integration, respectively. In general, the proposed hybrid model has achieved excellent predictive performance on multiple evaluation indicators when compared with several candidate models, especially demonstrating that the mode-specific denoising strategy can effectively reduce noise interference while preserving useful trend information in low-frequency components. Such a framework can be easily implemented to forecast petroleum consumption in real application scenarios.
- daily energy consumption prediction,
- deep learning,
- two-stage hybrid model,
- variational mode decomposition,
- wavelet thresholding denoising
Citation: Bowen Hou, Yin Liu, Jiajun Chen, Letian Qi, Jinhao Zhang, Honghao Zhao. A two-stage deep learning-based hybrid model for daily energy consumption forecasting with variational mode decomposition and wavelet thresholding denoising[J]. Electronic Research Archive, 2026, 34(10): 7531-7564. doi: 10.3934/era.2026325

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Abstract

Global energy demand keeps rising amid economic expansion and population growth. Although renewables like wind and solar have grown fast, fossil fuels (especially petroleum) remain central to the global energy mix, underpinning industry, households, policy, and stability. This article presents an improved hybrid forecasting framework, denoted as the VMD-WTD-GRU model with mode-specific denoising, for daily average petroleum consumption, which adopts a two-stage decomposition-targeted denoising strategy. The framework combines the models of variational mode decomposition, wavelet threshold denoising, and the gated recurrent unit networks for petroleum energy forecasting, where different models are designed for time-series data decomposition, noise elimination, sub-series forecasting, and integration, respectively. In general, the proposed hybrid model has achieved excellent predictive performance on multiple evaluation indicators when compared with several candidate models, especially demonstrating that the mode-specific denoising strategy can effectively reduce noise interference while preserving useful trend information in low-frequency components. Such a framework can be easily implemented to forecast petroleum consumption in real application scenarios.

References

[1]	G. E. Halkos, E. C. Gkampoura, Reviewing usage, potentials, and limitations of renewable energy sources, Energies, 13 (2020), 2906. https://doi.org/10.3390/en13112906 doi: 10.3390/en13112906
[2]	A. Yi, M. Yang, Y. Li, Macroeconomic uncertainty and crude oil futures volatility–evidence from China crude oil futures market, Front. Environ. Sci., 9 (2021), 636903. https://doi.org/10.3389/fenvs.2021.636903 doi: 10.3389/fenvs.2021.636903
[3]	J. Zhang, Y. Zhu, X. Zhang, M. Ye, J. Yang, Developing a Long Short-Term Memory (LSTM) based model for predicting water table depth in agricultural areas, J. Hydrol., 561 (2018), 918–929. https://doi.org/10.1016/j.jhydrol.2018.04.065 doi: 10.1016/j.jhydrol.2018.04.065
[4]	Y. Huang, L. Yang, S. Liu, G. Wang, Multi-step wind speed forecasting based on ensemble empirical mode decomposition, long short term memory network and error correction strategy, Energies, 12 (2019), 1822. https://doi.org/10.3390/en12101822 doi: 10.3390/en12101822
[5]	Z. Deng, B. Wang, H. Guo, C. Chai, Y. Wang, Z. Zhu, Unified quantile regression deep neural network with time-cognition for probabilistic residential load forecasting, Complexity, 2020 (2020), 9147545. https://doi.org/10.1155/2020/9147545 doi: 10.1155/2020/9147545
[6]	J. Zhang, X. Xin, Y. Shang, Y. Wang, L. Zhang, Nonstationary significant wave height forecasting with a hybrid VMD-CNN model, Ocean Eng., 285 (2023), 115338. https://doi.org/10.1016/j.oceaneng.2023.115338 doi: 10.1016/j.oceaneng.2023.115338
[7]	S. Sun, Y. Wang, Y. Meng, C. Wang, X. Zhu, Multi-step wind speed forecasting model using a compound forecasting architecture and an improved QPSO-based synchronous optimization, Energy Rep., 8 (2022), 9899–9918. https://doi.org/10.1016/j.egyr.2022.07.164 doi: 10.1016/j.egyr.2022.07.164
[8]	J. Xu, Y. Wei, P. Zeng, VMD-based iterative Boruta feature extraction and CNNA-BiLSTM for short-term load forecasting, Electr. Power Syst. Res., 238 (2025), 111172. https://doi.org/10.1016/j.epsr.2024.111172 doi: 10.1016/j.epsr.2024.111172
[9]	I. Jrhilifa, H. Ouadi, A. Jilbab, N. Mounir, A. Ouaguid, A VMD-deep learning approach for individual load monitoring and forecasting for residential buildings energy management, e-Prime, 8 (2024), 100624. https://doi.org/10.1016/j.prime.2024.100624 doi: 10.1016/j.prime.2024.100624
[10]	L. Liu, J. Zhang, S. Xue, Photovoltaic power forecasting: Using wavelet threshold denoising combined with VMD, Renew. Energy, 249 (2025), 123152. https://doi.org/10.1016/j.renene.2025.123152 doi: 10.1016/j.renene.2025.123152
[11]	X. Guo, Y. Gao, W. Song, Y. Zen, X. Shi, Short-term power load forecasting based on CEEMDAN-WT-VMD joint denoising and BiTCN-BiGRU-Attention, Electronics, 14 (2025), 1871. https://doi.org/10.3390/electronics14091871 doi: 10.3390/electronics14091871
[12]	S. S. Pappas, L. Ekonomou, D. C. Karamousantas, G. Chatzarakis, S. Katsikas, P. Liatsis, Electricity demand loads modeling using AutoRegressive Moving Average (ARMA) models, Energy, 33 (2008), 1353–1360. https://doi.org/10.1016/j.energy.2008.05.008 doi: 10.1016/j.energy.2008.05.008
[13]	A. Azadeh, S. M. Asadzadeh, M. Saberi, A Neuro-fuzzy-stochastic frontier analysis approach for long-term natural gas consumption forecasting and behavior analysis: The cases of Bahrain, Saudi Arabia, Syria, and UAE, Appl. Energy, 88 (2011), 3850–3859. https://doi.org/10.1016/j.apenergy.2011.04.027 doi: 10.1016/j.apenergy.2011.04.027
[14]	A. Azadeh, R. Babazadeh, S. Asadzadeh, Optimum estimation and forecasting of renewable energy consumption by artificial neural networks, Renew. Sustain. Energy Rev., 27 (2013), 605–612. https://doi.org/10.1016/j.rser.2013.07.007 doi: 10.1016/j.rser.2013.07.007
[15]	J. Bai, J. Wang, J. Ran, X. Li, C. Tu, An improved neural network algorithm for energy consumption forecasting, Sustainability, 16 (2024), 9332. https://doi.org/10.3390/su16219332 doi: 10.3390/su16219332
[16]	L. G. B. Ruiz, R. Rueda, M. P. Cuéllar, M. Pegalajar, Energy consumption forecasting based on Elman neural networks with evolutive optimization, Expert Syst. Appl., 92 (2018), 380–389. https://doi.org/10.1016/j.eswa.2017.09.059 doi: 10.1016/j.eswa.2017.09.059
[17]	G. P. Zhang, Time series forecasting using a hybrid ARIMA and neural network model, Neurocomputing, 50 (2003), 159–175. https://doi.org/10.1016/S0925-2312(01)00702-0 doi: 10.1016/S0925-2312(01)00702-0
[18]	L. Wang, H. Zou, J. Su, L. Li, S. Chaudhry, An ARIMA-ANN hybrid model for time series forecasting, Syst. Res. Behav. Sci., 30 (2013), 244–259. https://doi.org/10.1002/sres.2179 doi: 10.1002/sres.2179
[19]	S. Barak, S. S. Sadegh, Forecasting energy consumption using ensemble ARIMA–ANFIS hybrid algorithm, Int. J. Electr. Power Energy Syst., 82 (2016), 92–104. https://doi.org/10.1016/j.ijepes.2016.03.012 doi: 10.1016/j.ijepes.2016.03.012
[20]	N. Shabbir, L. Kutt, M. Jawad, M. N. Iqbal, P. S. Ghahfaroki, Forecasting of energy consumption and production using recurrent neural networks, Adv. Electr. Electron. Eng., 18 (2020), 190–197.
[21]	K. Ma, X. Nie, J. Yang, L. Zha, G. Li, H. Li, A power load forecasting method in port based on VMD-ICSS-hybrid neural network, Appl. Energy, 377 (2025), 124246. https://doi.org/10.1016/j.apenergy.2024.124246 doi: 10.1016/j.apenergy.2024.124246
[22]	K. Sharma, Y. K. Dwivedi, B. Metri, Incorporating causality in energy consumption forecasting using deep neural networks, Ann. Oper. Res., 339 (2024), 537–572. https://doi.org/10.1007/s10479-022-04857-3 doi: 10.1007/s10479-022-04857-3
[23]	S. Y. Shin, H. G. Woo, Energy consumption forecasting in Korea using machine learning algorithms, Energies, 15 (2022), 4880. https://doi.org/10.3390/en15134880 doi: 10.3390/en15134880
[24]	A. Dedinec, S. Filiposka, A. Dedinec, L. Kocarev, Deep belief network based electricity load forecasting: An analysis of Macedonian case, Energy, 115 (2016), 1688–1700. https://doi.org/10.1016/j.energy.2016.07.090 doi: 10.1016/j.energy.2016.07.090
[25]	Y. Dong, X. Ma, T. Fu, Electrical load forecasting: A deep learning approach based on K-nearest neighbors, Appl. Soft Comput., 99 (2021), 106900. https://doi.org/10.1016/j.asoc.2020.106900 doi: 10.1016/j.asoc.2020.106900
[26]	M. Alipour, J. Aghaei, M. Norouzi, T. Niknam, S. Hashemi, M. Lehtonen, A novel electrical net-load forecasting model based on deep neural networks and wavelet transform integration, Energy, 205 (2020), 118106. https://doi.org/10.1016/j.energy.2020.118106 doi: 10.1016/j.energy.2020.118106
[27]	Y. Wang, N. Zhang, Q. Chen, D. S. Kirschen, P. Li, Q. Xia, Data-driven probabilistic net load forecasting with high penetration of behind-the-meter PV, IEEE Trans. Power Syst., 33 (2018), 3255–3264. https://doi.org/10.1109/TPWRS.2017.2762599 doi: 10.1109/TPWRS.2017.2762599
[28]	D. W. Van der Meer, J. Munkhammar, J. Widén, Probabilistic forecasting of solar power, electricity consumption and net load: Investigating the effect of seasons, aggregation and penetration on prediction intervals, Sol. Energy, 171 (2018), 397–413. https://doi.org/10.1016/j.solener.2018.06.103 doi: 10.1016/j.solener.2018.06.103
[29]	F. Mei, Q. Wu, T. Shi, J. Lu, Y. Pan, J. Zheng, An ultrashort-term net load forecasting model based on phase space reconstruction and deep neural network, Appl. Sci., 9 (2019), 1487. https://doi.org/10.3390/app9071487 doi: 10.3390/app9071487
[30]	I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, et al., Generative adversarial nets, in Advances in Neural Information Processing Systems 27, Curran Associates, Inc., (2014), 2672–2680.
[31]	Z. Wang, T. Hong, Generating realistic building electrical load profiles through the Generative Adversarial Network (GAN), Energy Build., 224 (2020), 110299. https://doi.org/10.1016/j.enbuild.2020.110299 doi: 10.1016/j.enbuild.2020.110299
[32]	M. N. Fekri, A. M. Ghosh, K. Grolinger, Generating energy data for machine learning with recurrent generative adversarial networks, Energies, 13 (2020), 130. https://doi.org/10.3390/en13010130 doi: 10.3390/en13010130
[33]	M. Ranjbar, M. Rahimzadeh, Advancing gasoline consumption forecasting: A novel hybrid model integrating transformers, lstm, and cnn, preprint, arXiv: 2410.16336.
[34]	J. He, Y. Li, X. Xu, D. Wu, Energy consumption forecasting for oil and coal in China based on hybrid deep learning, PLOS One, 20 (2025), e0313856. https://doi.org/10.1371/journal.pone.0313856 doi: 10.1371/journal.pone.0313856
[35]	C. Zhai, X. He, Z. Cao, M. Abdou-Tankari, Y. Wang, M. Zhang, Photovoltaic power forecasting based on VMD-SSA-Transformer: Multidimensional analysis of dataset length, weather mutation and forecast accuracy, Energy, 324 (2025), 135971. https://doi.org/10.1016/j.energy.2025.135971 doi: 10.1016/j.energy.2025.135971
[36]	D. Venkateswaran, R. Venkateswaran, C. Shin, W. Choi, Y. Cho, Novel VMD-Informer-LSTM architecture for solar radiation forecasting with signal decomposition and sparse attention for smart energy applications, Expert Syst. Appl., 305 (2026), 130612. https://doi.org/10.1016/j.eswa.2025.130612 doi: 10.1016/j.eswa.2025.130612
[37]	J. Liu, Y. Zhang, K. Wen, Y. Ding, Analysis of different neural network models based on variational mode decomposition and dung beetle optimizer algorithm for the prediction of air-conditioning energy consumption in multifunctional complex large public buildings, Energy Build., 334 (2025), 115518. https://doi.org/10.1016/j.enbuild.2025.115518 doi: 10.1016/j.enbuild.2025.115518
[38]	Y. Ruan, G. Wang, H. Meng, F. Qian, A hybrid model for power consumption forecasting using VMD-based the long short-term memory neural network, Front. Energy Res., 9 (2022), 772508. https://doi.org/10.3389/fenrg.2021.772508 doi: 10.3389/fenrg.2021.772508
[39]	Y. Yang, W. Li, D. Liu, Monthly runoff prediction for Xijiang River via gated recurrent unit, discrete wavelet transform, and variational modal decomposition, Water, 16 (2024), 1552. https://doi.org/10.3390/w16111552 doi: 10.3390/w16111552
[40]	Y. Yu, Q. Shang, T. Xie, A hybrid model for short-term traffic flow prediction based on variational mode decomposition, wavelet threshold denoising, and long short-term memory neural network, Complexity, 2021 (2021), 7756299. https://doi.org/10.1155/2021/7756299 doi: 10.1155/2021/7756299
[41]	L. Wang, M. An, W. Zhu, W. Xiang, R. He, Forecasting of short-term traffic flow using LSTM ensembled with variational mode decomposition and wavelet denoising, Int. J. Intell. Syst. Technol. Appl., 24 (2026), 1–27. https://doi.org/10.1504/IJISTA.2026.152033 doi: 10.1504/IJISTA.2026.152033
[42]	N. Fang, Z. Liu, S. Fan. Short-term wind power prediction method based on CEEMDAN-VMD-GRU hybrid model, Energies, 18 (2025), 1465. https://doi.org/10.3390/en18061465 doi: 10.3390/en18061465
[43]	W. Jiang, Q. Lu, Y. Xu, Z. Chen, T. Gong, Short-term wind speed prediction based on denoising algorithm of enhanced successive variational mode decomposition and integrated parallel prediction model, Energy, 338 (2025), 138857. https://doi.org/10.1016/j.energy.2025.138857 doi: 10.1016/j.energy.2025.138857
[44]	J. Xiao, Y. Li, L. Xie, D. Liu, J. Huang, A hybrid model based on selective ensemble for energy consumption forecasting in China, Energy, 159 (2018), 534–546. https://doi.org/10.1016/j.energy.2018.06.161 doi: 10.1016/j.energy.2018.06.161
[45]	T. Shojaei, A. Mokhtar, Forecasting energy consumption with a novel ensemble deep learning framework, J. Build. Eng., 96 (2024), 110452. https://doi.org/10.1016/j.jobe.2024.110452 doi: 10.1016/j.jobe.2024.110452
[46]	K. Dragomiretskiy, D. Zosso, Variational Mode Decomposition, IEEE Trans. Signal Process., 62 (2014), 531–544. https://doi.org/10.1109/TSP.2013.2288675 doi: 10.1109/TSP.2013.2288675
[47]	Y. Hou, X. Deng, Y. Xia, Precipitation prediction based on variational mode decomposition combined with the crested porcupine optimization algorithm for long short-term memory model, AIP Adv., 14 (2024), 065315. https://doi.org/10.1063/5.0204644 doi: 10.1063/5.0204644
[48]	D. P. Bertsekas, Multiplier methods: A survey, Automatica, 12 (1976), 133–145. https://doi.org/10.1016/0005-1098(76)90077-7 doi: 10.1016/0005-1098(76)90077-7
[49]	M. Han, Y. Liu, J. Xi, W. Guo, Noise smoothing for nonlinear time series using wavelet soft threshold, IEEE Signal Process. Lett., 14 (2007), 62–65. https://doi.org/10.1109/LSP.2006.881518 doi: 10.1109/LSP.2006.881518
[50]	Y. M. Liu, Application of improved VMD and wavelet threshold denoising method in bolt detection, in 2020 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR), IEEE, (2020), 14–20. https://doi.org/10.1109/ICWAPR51924.2020.9494619
[51]	D. Valencia, D. Orejuela, J. Salazar, J. Valencia, Comparison analysis between rigrsure, sqtwolog, heursure and minimaxi techniques using hard and soft thresholding methods, in 2016 XXI Symposium on Signal Processing, Images and Artificial Vision (STSIVA), IEEE, (2016), 1–5. https://doi.org/10.1109/STSIVA.2016.7743309
[52]	F. X. Diebold, R. S. Mariano, Comparing predictive accuracy, J. Bus. Econ. Stat., 13 (1995), 253–263. https://doi.org/10.1080/07350015.1995.10524599 doi: 10.1080/07350015.1995.10524599
[53]	Y. Yang, Y. Gao, H. Zhou, J. Wu, S. Gao, Y. G. Wang, Multi-Granularity Autoformer for long-term deterministic and probabilistic power load forecasting, Neural Netw., 188 (2025), 107493. https://doi.org/10.1016/j.neunet.2025.107493 doi: 10.1016/j.neunet.2025.107493
[54]	Z. Qu, Y. Meng, X. Hou, R. Chi, Y. Ai, Z. Wu, Integrated energy short-term multivariate load forecasting based on PatchTST secondary decoupling reconstruction for progressive layered extraction multi-task learning network, Expert Syst. Appl., 269 (2025), 126446. https://doi.org/10.1016/j.eswa.2025.126446 doi: 10.1016/j.eswa.2025.126446
[55]	W. Zhao, J. Zhao, X. Shen, J. Liu, S. Fan, A two-stage uncertainty forecasting method for offshore wind power based on FEDformer and SDCDM, Energy, 348 (2026), 140452. https://doi.org/10.1016/j.energy.2026.140452 doi: 10.1016/j.energy.2026.140452
[56]	Z. Yang, J. Li, H. Wang, C. Liu. An Informer model for very short-term power load forecasting, Energies, 18 (2025), 1150. https://doi.org/10.3390/en18051150 doi: 10.3390/en18051150
[57]	H. Wang, B. R. Sarker, J. Li, J. Li, Adaptive scheduling for assembly job shop with uncertain assembly times based on dual Q-learning, Int. J. Prod. Res., 59 (2021), 5867–5883. https://doi.org/10.1080/00207543.2020.1794075 doi: 10.1080/00207543.2020.1794075
[58]	F. Zhao, Z. Fu, L. Wang, H. Sang, A heterogeneous graph reinforcement learning framework with question-aware neighborhood aggregation and interoption prompt attention for dynamic flexible job shop scheduling problem, IEEE Trans. Ind. Inform., 22 (2026), 2863–2874. https://doi.org/10.1109/TII.2025.3646962 doi: 10.1109/TII.2025.3646962

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