{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T16:53:47Z","timestamp":1787504027488,"version":"build-2736575974"},"reference-count":34,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,4,17]],"date-time":"2025-04-17T00:00:00Z","timestamp":1744848000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2022YFF0610601"],"award-info":[{"award-number":["2022YFF0610601"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The traditional consensus-based distributed economic dispatch strategy may lose system convergency and cause imbalanced power when facing an information attack on the individual power generation unit; thus, it is unable to achieve the dispatching goal. Taking into consideration several kinds of attack behaviors that may exist in a distributed control system, this paper develops models of node attacks from the two aspects of action mode and deployment scope, and analyzes the influence of attack behaviors on the distributed economic dispatch system. Based on the idea of the W-MSR algorithm that deletes the information received from nodes that may be attacked, a distributed security consensus-based economic dispatch strategy is synthetized with the incremental cost of the power generation unit as the consensus variable. Based on the graph robustness index, this paper gives its conditions along with its proof that the communication network topology of the system should be satisfied when adopting the W-MSR algorithm. The simulation results of the IEEE-39 bus distribution network show that the strategy proposed in this paper can effectively counter various information attacks, enhancing both the security and economic efficiency of the distributed economic dispatch system. In addition, the (F + 1, F + 1)-robust graph is a necessary and sufficient condition to achieve the consensus of the dispatch strategy.<\/jats:p>","DOI":"10.3390\/s25082551","type":"journal-article","created":{"date-parts":[[2025,4,17]],"date-time":"2025-04-17T20:05:56Z","timestamp":1744920356000},"page":"2551","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Distributed Secure Economic Dispatch Strategy Based on Robust Graph Theory and W-MSR Algorithm"],"prefix":"10.3390","volume":"25","author":[{"given":"Jian","family":"Le","sequence":"first","affiliation":[{"name":"School of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China"},{"name":"Institute of Next Generation Power Systems and International Standards, Wuhan University, Wuhan 430072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jing","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongke","family":"Lang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weihao","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,17]]},"reference":[{"key":"ref_1","unstructured":"Li, H., Zhang, C., Wu, L., and Tong, Z. (2024, January 4\u20136). Economic Optimal Dispatching of Microgrid Based on Improved Finite Time Consistency. Proceedings of the CSU-EPSA, Cologne, Germany."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"123690","DOI":"10.1016\/j.apenergy.2024.123690","article-title":"Two-Stage Robust Operation of Electricity-Gas-Heat Integrated Multi-Energy Microgrids Considering Heterogeneous Uncertainties","volume":"371","author":"Zhang","year":"2024","journal-title":"Appl. Energy"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6960","DOI":"10.1109\/TPWRS.2024.3369664","article-title":"Distributionally Robust Multistage Dispatch with Discrete Recourse of Energy Storage Systems","volume":"39","author":"Zheng","year":"2024","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"109709","DOI":"10.1016\/j.engappai.2024.109709","article-title":"An Improved Reinforcement Learning-Based Differential Evolution Algorithm for Combined Economic and Emission Dispatch Problems","volume":"140","author":"Wang","year":"2025","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.egyr.2022.10.088","article-title":"Weighted Matrix Based Distributed Optimization Method for Economic Dispatch of Microgrids via Multi-Step Gradient Descent","volume":"8","author":"Bai","year":"2022","journal-title":"Energy Rep."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3892","DOI":"10.1109\/TII.2017.2785366","article-title":"An ADMM-Based Distributed Algorithm for Economic Dispatch in Islanded Microgrids","volume":"14","author":"Chen","year":"2018","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4500","DOI":"10.1109\/TIA.2021.3091663","article-title":"Distributed Robust Dynamic Economic Dispatch of Integrated Transmission and Distribution Systems","volume":"57","author":"Chen","year":"2021","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_8","first-page":"199","article-title":"Incremental Cost Based Hierarchical Distributed Control Strategy for AC\/DC Hybrid Microgrid","volume":"48","author":"Yang","year":"2024","journal-title":"Autom. Electr. Power Syst."},{"key":"ref_9","first-page":"109","article-title":"Distributed Harmonic Mitigation Strategy of Distribution Network Based on Consensus Algorithm","volume":"46","author":"Lin","year":"2022","journal-title":"Autom. Electr. Power Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"119873","DOI":"10.1016\/j.jenvman.2023.119873","article-title":"A Hybrid Forecasting Approach for China\u2019s National Carbon Emission Allowance Prices with Balanced Accuracy and Interpretability","volume":"351","author":"Mao","year":"2024","journal-title":"J. Environ. Manag."},{"key":"ref_11","first-page":"103510","article-title":"A Consensus-Based Control for Precise Reactive Power-Sharing in AC Microgrids","volume":"60","author":"Hennane","year":"2023","journal-title":"Sustain. Energy Technol. Assess."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3630","DOI":"10.1109\/TSG.2018.2833108","article-title":"Distributed Consensus Based Algorithm for Economic Dispatch in a Microgrid","volume":"10","author":"Wang","year":"2019","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5541","DOI":"10.1049\/iet-gtd.2018.5219","article-title":"Distributed Hierarchical Consensus Algorithm for Economic Dispatch in Smart Grid","volume":"13","author":"Wu","year":"2019","journal-title":"IET Gener. Transm. Distrib."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Le, J., Zhou, Q., Zhao, L., and Wang, Y. (2019). Research on the Real-Time Distributed Economic Dispatch Strategy for Microgrids. Energies, 12.","DOI":"10.3390\/en12204007"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"7020","DOI":"10.1109\/ACCESS.2019.2959600","article-title":"Multi-Agent Based Fully Distributed Economic Dispatch in Microgrid Using Exact Diffusion Strategy","volume":"8","author":"He","year":"2020","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1258","DOI":"10.1109\/TSG.2023.3302309","article-title":"A Transfer Learning-Based Method for Cyber-Attack Tolerance in Distributed Control of Microgrids","volume":"15","author":"Xia","year":"2024","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_17","first-page":"44","article-title":"Data Protection of Distributed Economic Dispatch Based on Blockchain","volume":"20","author":"Liu","year":"2022","journal-title":"Electr. Power Inf. Commun. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2750","DOI":"10.1049\/iet-rpg.2019.1355","article-title":"Real-Time Distributed Economic Dispatch Scheme of Grid-Connected Microgrid Considering Cyberattacks","volume":"14","author":"Huang","year":"2020","journal-title":"IET Renew. Power Gener."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5333","DOI":"10.1016\/j.jfranklin.2023.03.043","article-title":"Attack Resilient Strategy for Event-Triggered Distributed Control Scheme of Multi-Energy Systems","volume":"360","author":"Liu","year":"2023","journal-title":"J. Frankl. Inst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4308","DOI":"10.1109\/TAC.2019.2954363","article-title":"Resilient Distributed Optimization Algorithm Against Adversarial Attacks","volume":"65","author":"Zhao","year":"2020","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.sysconle.2015.02.005","article-title":"Consensus of Second-Order Multi-Agent Systems in the Presence of Locally Bounded Faults","volume":"79","author":"Dibaji","year":"2015","journal-title":"Syst. Control Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1109\/JSAC.2013.130413","article-title":"Resilient Asymptotic Consensus in Robust Networks","volume":"31","author":"LeBlanc","year":"2013","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1109\/TCNS.2019.2924235","article-title":"Resilient Consensus Through Event-Based Communication","volume":"7","author":"Wang","year":"2020","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"ref_24","unstructured":"Tan, S., Yang, S., and Xu, J.-X. (2013, January 10\u201313). Consensus Based Approach for Economic Dispatch Problem in a Smart Grid. Proceedings of the IECON 2013\u201439th Annual Conference of the IEEE Industrial Electronics Society, Vienna, Austria."},{"key":"ref_25","first-page":"2288","article-title":"Convergence Speed Analysis and Optimization for Distributed Control of Virtual Power Plant","volume":"40","author":"Zhao","year":"2016","journal-title":"Power Syst. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.1109\/TPWRS.2012.2188912","article-title":"Convergence Analysis of the Incremental Cost Consensus Algorithm Under Different Communication Network Topologies in a Smart Grid","volume":"27","author":"Zhang","year":"2012","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1495","DOI":"10.1109\/TAC.2010.2088690","article-title":"Distributed Function Calculation via Linear Iterative Strategies in the Presence of Malicious Agents","volume":"56","author":"Sundaram","year":"2011","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.automatica.2019.04.029","article-title":"Resilient Consensus for Multi-Agent Systems Subject to Differential Privacy Requirements","volume":"106","author":"Fiore","year":"2019","journal-title":"Automatica"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"LeBlanc, H.J., and Koutsoukos, X.D. (2011, January 12\u201314). Consensus in Networked Multi-Agent Systems with Adversaries. Proceedings of the 14th International Conference on Hybrid Systems: Computation and Control, Chicago, IL, USA.","DOI":"10.1145\/1967701.1967742"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1109\/TII.2012.2219061","article-title":"An Overview of Recent Progress in the Study of Distributed Multi-Agent Coordination","volume":"9","author":"Cao","year":"2013","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1109\/TAC.2011.2158130","article-title":"Consensus Computation in Unreliable Networks: A System Theoretic Approach","volume":"57","author":"Pasqualetti","year":"2012","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1109\/MCS.2007.338264","article-title":"Information Consensus in Multivehicle Cooperative Control","volume":"27","author":"Ren","year":"2007","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_33","first-page":"112","article-title":"Distributed Economic Dispatch Based on Consensus Algorithm of Multi Agent System for Power System","volume":"36","author":"Xie","year":"2016","journal-title":"Electr. Power Autom. Equip."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1109\/TPWRS.2014.2299436","article-title":"Distributed Consensus-Based Economic Dispatch with Transmission Losses","volume":"29","author":"Binetti","year":"2014","journal-title":"IEEE Trans. Power Syst."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/25\/8\/2551\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:16:41Z","timestamp":1760030201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/25\/8\/2551"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,17]]},"references-count":34,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["s25082551"],"URL":"https:\/\/doi.org\/10.3390\/s25082551","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,17]]}}}