{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T12:51:40Z","timestamp":1784119900749,"version":"3.55.0"},"reference-count":45,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,9]],"date-time":"2025-01-09T00:00:00Z","timestamp":1736380800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62273215"],"award-info":[{"award-number":["62273215"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021SFGC1101"],"award-info":[{"award-number":["2021SFGC1101"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Major Scientific and Technological Innovation","award":["62273215"],"award-info":[{"award-number":["62273215"]}]},{"name":"Major Scientific and Technological Innovation","award":["2021SFGC1101"],"award-info":[{"award-number":["2021SFGC1101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In gas-to-methanol processes, optimizing multi-energy systems is a critical challenge toward efficient energy allocation. This paper proposes an entropy-based stochastic optimization method for a multi-energy system in a gas-to-methanol process, aiming to achieve optimal allocation of gas, steam, and electricity to ensure executability under modeling uncertainties. First, mechanistic models are developed for major chemical equipments, including the desulfurization, steam boilers, air separation, and syngas compressors. Structural errors in these models under varying operating conditions result in noticeable model uncertainties. Second, Bayesian estimation theory and the Markov Chain Monte Carlo approach are employed to analyze the differences between historical data and model predictions under varying operating conditions, thereby quantifying modeling uncertainties. Finally, subject to constraints in the model uncertainties, equipment capacities, and energy balance, a multi-objective stochastic optimization model is formulated to minimize gas loss, steam loss, and operating costs. The entropy weight approach is then applied to filter the Pareto front solution set, selecting a final optimal solution with minimal subjectivity and preferences. Case studies using Aspen Hysys-based simulations show that optimization solutions considering model uncertainties outperform the counterparts from a standard deterministic optimization in terms of executability.<\/jats:p>","DOI":"10.3390\/e27010052","type":"journal-article","created":{"date-parts":[[2025,1,9]],"date-time":"2025-01-09T07:00:47Z","timestamp":1736406047000},"page":"52","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Entropy-Based Stochastic Optimization of Multi-Energy Systems in Gas-to-Methanol Processes Subject to Modeling Uncertainties"],"prefix":"10.3390","volume":"27","author":[{"given":"Xueteng","family":"Wang","sequence":"first","affiliation":[{"name":"College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2635-8724","authenticated-orcid":false,"given":"Jiandong","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mengyao","family":"Wei","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Yue","sequence":"additional","affiliation":[{"name":"Shandong Rongxin Group Co., Ltd., Zoucheng 273517, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jcou.2016.05.005","article-title":"Efficient utilization of carbon dioxide in a gas-to-methanol process composed of CO2\/steam\u2013mixed reforming and methanol synthesis","volume":"16","author":"Zhang","year":"2016","journal-title":"J. CO2 Util."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/j.enconman.2016.12.010","article-title":"Coke oven gas to methanol process integrated with CO2 recycle for high energy efficiency, economic benefits and low emissions","volume":"133","author":"Gong","year":"2017","journal-title":"Energy Convers. Manag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1016\/j.energy.2018.08.218","article-title":"Recent developments and trends in optimization of energy systems","volume":"164","author":"Frangopoulos","year":"2018","journal-title":"Energy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"121871","DOI":"10.1016\/j.applthermaleng.2023.121871","article-title":"A review on multi energy systems modelling and optimization","volume":"236","author":"Tesio","year":"2024","journal-title":"Appl. Therm. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"112246","DOI":"10.1016\/j.rser.2022.112246","article-title":"Next frontiers in energy system modelling: A review on challenges and the state of the art","volume":"160","author":"Fodstad","year":"2022","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.esr.2018.06.003","article-title":"A review of approaches to uncertainty assessment in energy system optimization models","volume":"21","author":"Yue","year":"2018","journal-title":"Energy Strategy Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2909","DOI":"10.1007\/s00158-021-03026-7","article-title":"Modeling, analysis, and optimization under uncertainties: A review","volume":"64","author":"Acar","year":"2021","journal-title":"Struct. Multidisc. Optim."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Gao, W., and Lin, Y. (2023). Energy dispatch for CCHP system in summer based on deep reinforcement learning. Entropy, 25.","DOI":"10.3390\/e25030544"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Xu, R., Lin, F., Shao, W., Wang, H., Meng, F., and Li, J. (2024). Multi-time-scale optimal scheduling strategy for marine renewable energy based on deep reinforcement learning algorithm. Entropy, 26.","DOI":"10.3390\/e26040331"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.enconman.2017.12.008","article-title":"Optimal integrated sizing and operation of a CHP system with Monte Carlo risk analysis for long-term uncertainty in energy demands","volume":"157","author":"Urbanucci","year":"2018","journal-title":"Energy Convers. Manag."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"107374","DOI":"10.1016\/j.compchemeng.2021.107374","article-title":"Sustainable retrofit of petrochemical energy systems under multiple uncertainties using the stochastic optimization method","volume":"151","author":"Qian","year":"2021","journal-title":"Comput. Chem. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Dong, J., Song, Z., Zheng, Y., Luo, J., Zhang, M., Yang, X., and Ma, H. (2024). Robust optimization research of cyber\u2013physical power system considering wind power uncertainty and coupled relationship. Entropy, 26.","DOI":"10.3390\/e26090795"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/j.envsoft.2015.11.019","article-title":"Modelling to generate alternatives with an energy system optimization model","volume":"79","author":"DeCarolis","year":"2016","journal-title":"Environ. Model. Softw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.energy.2019.02.021","article-title":"Probabilistic optimization in operation of energy hub with participation of renewable energy resources and demand response","volume":"173","author":"Rakipour","year":"2019","journal-title":"Energy"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.enconman.2017.04.074","article-title":"Stochastic optimization of energy hub operation with consideration of thermal energy market and demand response","volume":"145","author":"Nojavan","year":"2017","journal-title":"Energy Convers. Manag."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"140393","DOI":"10.1016\/j.jclepro.2023.140393","article-title":"Optimal scheduling of zero-carbon integrated energy system considering long- and short-term energy storages, demand response, and uncertainty","volume":"435","author":"Song","year":"2024","journal-title":"J. Clean. Prod."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"109169","DOI":"10.1016\/j.epsr.2023.109169","article-title":"Optimal scheduling of regional integrated energy system considering multiple uncertainties and integrated demand response","volume":"217","author":"Xiao","year":"2023","journal-title":"Electr. Power Syst. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2640","DOI":"10.1109\/TIA.2021.3106573","article-title":"Optimal scheduling of integrated demand response-enabled community-integrated energy systems in uncertain environments","volume":"58","author":"Li","year":"2022","journal-title":"IEEE Trans. Ind. Applicat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/j.apenergy.2012.04.017","article-title":"An efficient scenario-based stochastic programming framework for multi-objective optimal micro-grid operation","volume":"99","author":"Niknam","year":"2012","journal-title":"Appl. Energy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1049\/rpg2.12292","article-title":"Daily operation of multi-energy systems based on stochastic optimization considering prediction of renewable energy generation","volume":"16","author":"Azizi","year":"2022","journal-title":"IET Renew. Power Gener."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"119629","DOI":"10.1016\/j.energy.2020.119629","article-title":"Stochastic optimal operation model for a distributed integrated energy system based on multiple-scenario simulations","volume":"219","author":"Mei","year":"2021","journal-title":"Energy"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"112886","DOI":"10.1016\/j.rser.2022.112886","article-title":"Stochastic simulation-optimization framework for the design and assessment of renewable energy systems under uncertainty","volume":"168","author":"Sakki","year":"2022","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"115395","DOI":"10.1016\/j.apenergy.2020.115395","article-title":"Multi-objective stochastic expansion planning based on multi-dimensional correlation scenario generation method for regional integrated energy system integrated renewable energy","volume":"276","author":"Lei","year":"2020","journal-title":"Appl. Energy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"144187","DOI":"10.1016\/j.jclepro.2024.144187","article-title":"Metal-organic frameworks for atmospheric water extraction: Kinetic analysis and stochastic programming under climate variability","volume":"482","author":"Kim","year":"2024","journal-title":"J. Clean. Prod."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Albert, C.G., Callies, U., and Von Toussaint, U. (2022). A Bayesian approach to the estimation of parameters and their interdependencies in environmental modeling. Entropy, 24.","DOI":"10.3390\/e24020231"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Xing, X., Wang, J., and Sun, S. (2024). Nonlinear dynamic models with uncertainties measured by fuzzy sets for radiator-heated buildings. IEEE Trans. Fuzzy Syst., 1\u20137.","DOI":"10.1109\/TFUZZ.2024.3493201"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.energy.2017.08.058","article-title":"The integrated coke-oven gas and pulverized coke gasification for methanol production with highly efficient hydrogen utilization","volume":"140","author":"Xiang","year":"2017","journal-title":"Energy"},{"key":"ref_28","unstructured":"Yue, Y., Du, H., and Wu, H. (2020). Process Operation Manual for Main Equipment of Methanol Production Lines, Shandong Rongxin Group Co. Ltd."},{"key":"ref_29","unstructured":"Incropera, F.P., DeWitt, D.P., Bergman, T.L., and Lavine, A.S. (1996). Fundamentals of Heat and Mass Transfer, Wiley."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1194","DOI":"10.1016\/j.energy.2019.03.160","article-title":"Modeling of a steam boiler operation using the boiler nonlinear mathematical model","volume":"175","author":"Trojan","year":"2019","journal-title":"Energy"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"432","DOI":"10.3182\/20140824-6-ZA-1003.01510","article-title":"Dynamic modeling and simulation of compressor trains for an air separation unit","volume":"47","author":"Dominic","year":"2014","journal-title":"IFAC Proc. Vol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1321","DOI":"10.1016\/j.procir.2018.03.271","article-title":"Model-based method for condition monitoring and diagnosis of compressors","volume":"72","author":"Engelberth","year":"2018","journal-title":"Procedia CIRP"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jcat.1996.0156","article-title":"A steady-state kinetic model for methanol synthesis and the water gas shift reaction on a commercial Cu\/ZnO\/Al2O3 Catalyst","volume":"161","author":"Bussche","year":"1996","journal-title":"J. Catal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2883","DOI":"10.1016\/0009-2509(86)80019-7","article-title":"Chemical equilibria in methanol synthesis","volume":"41","author":"Graaf","year":"1986","journal-title":"Chem. Eng. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1063\/1.555927","article-title":"Estimation of the thermodynamic properties of C-H-N-O-S-halogen compounds at 298.15 K","volume":"22","author":"Domalski","year":"1993","journal-title":"J. Phys. Chem. Ref. Data"},{"key":"ref_36","unstructured":"Lennart, L. (1999). System Identification: Theory for the User, Prentice-Hall. [2nd ed.]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.jhydrol.2010.07.043","article-title":"Bayesian approach for uncertainty quantification in water quality modelling: The influence of prior distribution","volume":"392","author":"Freni","year":"2010","journal-title":"J. Hydrol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1016\/j.ress.2010.02.015","article-title":"A Bayesian approach for quantification of model uncertainty","volume":"95","author":"Park","year":"2010","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s43586-020-00001-2","article-title":"Bayesian statistics and modelling","volume":"1","author":"Depaoli","year":"2021","journal-title":"Nat. Rev. Methods Prim."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"00037","DOI":"10.1051\/itmconf\/20182300037","article-title":"Kernel density estimation and its application","volume":"23","author":"Weglarczyk","year":"2018","journal-title":"ITM Web Conf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3542","DOI":"10.1016\/j.camwa.2012.09.003","article-title":"MOMCMC: An efficient Monte Carlo method for multi-objective sampling over real parameter space","volume":"64","author":"Li","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Shen, Y., Li, D., and Wang, W. (2024). Multi-energy load prediction method for integrated energy system based on fennec fox optimization algorithm and hybrid kernel extreme learning machine. Entropy, 26.","DOI":"10.3390\/e26080699"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1137\/10079731X","article-title":"Direct multisearch for multiobjective optimization","volume":"21","author":"Madeira","year":"2011","journal-title":"SIAM J. Optim."},{"key":"ref_44","first-page":"56","article-title":"Monte Carlo sampling-based methods for stochastic optimization","volume":"19","author":"Bayraksan","year":"2014","journal-title":"Surv. Oper. Res. Manag. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"114186","DOI":"10.1016\/j.eswa.2020.114186","article-title":"Effects of the entropy weight on TOPSIS","volume":"168","author":"Chen","year":"2021","journal-title":"Expert Syst. Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:25:45Z","timestamp":1759919145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,9]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["e27010052"],"URL":"https:\/\/doi.org\/10.3390\/e27010052","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,9]]}}}