{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:37:01Z","timestamp":1759970221979,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,11]],"date-time":"2025-01-11T00:00:00Z","timestamp":1736553600000},"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":["52207083"],"award-info":[{"award-number":["52207083"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The optimization of the symmetry of MTDC systems after a contingency is crucial for the stable and economic operation of the MTDC systems. In this paper, a multi-objective optimal control method for the power flow symmetry of MTDC systems for the large-scale integration of offshore wind farms is proposed. A mirror relationship between the available headroom of DC lines and VSCs and their actual power flow distribution performance is established. A corresponding symmetry index is established for the MTDC network, and the multi-objective optimization problem is converted into a series of single-objective problems by the normal boundary intersection method, and solved by the original dyadic interior point method, so as to obtain the Pareto optimal solution with uniform distribution. The compromise optimal solution is decided according to the entropy weight double-basis point method, which provides decision-making guidance for the operators. The simulation results show that the normal boundary intersection method can solve the multi-objective dynamic optimal control problem of the VSC-HVDC system quickly and efficiently, and improve the symmetry of the power flow in an MTDC network.<\/jats:p>","DOI":"10.3390\/sym17010105","type":"journal-article","created":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T04:01:52Z","timestamp":1736740912000},"page":"105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Adaptive Voltage Reference-Based Multi-Objective Optimal Control Method for the Power Flow Symmetry of Multi-Terminal DC Systems with the Large-Scale Integration of Offshore Wind Farms"],"prefix":"10.3390","volume":"17","author":[{"given":"Yuanshi","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiwen","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tongxin","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yilei","family":"Li","sequence":"additional","affiliation":[{"name":"State Grid Nanjing Power Supply Company, Nanjing 210008, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinye","family":"Du","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chaoyang","family":"Yuan","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongrui","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wu, M., Ma, D., Xiong, K., and Yuan, L. (2024). Deep Reinforcement Learning for Load Frequency Control in Isolated Microgrids: A Knowledge Aggregation Approach with Emphasis on Power Symmetry and Balance. Symmetry, 16.","DOI":"10.3390\/sym16030322"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"83","DOI":"10.3390\/sym16010083","article-title":"Microgrid Operation Optimization Method Considering Power-to-Gas Equipment: An Improved Gazelle Optimization Algorithm","volume":"1","author":"Wu","year":"2024","journal-title":"Symmetry"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5665","DOI":"10.1109\/TSG.2024.3420805","article-title":"POMDP-Based Dispatch Scheme for Residential Distributed Energy Resources Under Customer Fatigue Consideration","volume":"15","author":"Chen","year":"2024","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"124697","DOI":"10.1016\/j.apenergy.2024.124697","article-title":"Mitigating power grid impact from proactive data center workload shifts: A coordinated scheduling strategy integrating synergistic traffic-data-power networks","volume":"377","author":"Zhang","year":"2025","journal-title":"Appl. Energy"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3839","DOI":"10.1049\/rpg2.13064","article-title":"Low-carbon Economic Schedule of the H2 DRI-EAF Steel Plant Integrated with a Power-to-hydrogen System Driven by Blue Hydrogen and Green Hydrogen","volume":"18","author":"Zou","year":"2024","journal-title":"IET Renew. Power Gen."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1109\/TSG.2024.3417294","article-title":"A Tri-Level Demand Response Framework for EVCS Flexibility Enhancement in Coupled Power and Transportation Networks","volume":"16","author":"Qian","year":"2024","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"121703","DOI":"10.1016\/j.apenergy.2023.121703","article-title":"A novel non-intrusive load monitoring method based on ResNet-seq2seq networks for energy disaggregation of distributed energy resources integrated with residential houses","volume":"349","author":"Zhang","year":"2023","journal-title":"Appl. Energy"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1109\/TSTE.2021.3064375","article-title":"Optimal Operation of Energy Hubs with Large-Scale Distributed Energy Resources for Distribution Network Congestion Management","volume":"12","author":"Hu","year":"2021","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"122460","DOI":"10.1016\/j.apenergy.2023.122460","article-title":"Windfall profit-aware stochastic scheduling strategy for industrial virtual power plant with integrated risk-seeking\/averse preferences","volume":"357","author":"Xiao","year":"2024","journal-title":"Appl. Energy"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"124476","DOI":"10.1016\/j.apenergy.2024.124476","article-title":"Unsupervised learning for efficiently distributing EVs charging loads and traffic flows in coupled power and transportation systems","volume":"377","author":"Qian","year":"2025","journal-title":"Appl. Energy"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1109\/TSTE.2012.2191986","article-title":"Coordinated Control of Cascaded Current-source Converter Based Offshore Wind Farm","volume":"3","author":"Popat","year":"2012","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1186\/s41601-023-00329-3","article-title":"Integrated Risk Measurement and Control for Stochastic Energy Trading of a Wind Storage System in Electricity Markets","volume":"8","author":"Xiao","year":"2023","journal-title":"Prot. Control Mod. Power Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1676","DOI":"10.1109\/TSTE.2023.3243163","article-title":"A Multi-Port DC Power Flow Controller Integrated with MMC Stations for Offshore Meshed Multi-Terminal HVDC Grids","volume":"14","author":"Zhang","year":"2023","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2116","DOI":"10.1109\/JESTPE.2018.2881768","article-title":"Hybrid Topology of a Diode-Rectifier-Based HVDC System for Offshore Wind Farms","volume":"7","author":"Chang","year":"2019","journal-title":"IEEE J. Emerg. Sel. Top. Power Electron."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3869","DOI":"10.1049\/rpg2.13070","article-title":"Research on the Optimal Operation of a Prosumer Micro Energy Network Centred on Data Centres","volume":"18","author":"Zhang","year":"2024","journal-title":"IET Renew. Power Gen."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2889","DOI":"10.1109\/TSTE.2020.2980970","article-title":"On the dynamic modeling of marine VSC-HVDC power grids including offshore wind farms","volume":"11","author":"Castro","year":"2020","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1297","DOI":"10.1109\/TEC.2018.2814604","article-title":"Frequency Support From a DC-Grid Offshore Wind Farm Connected Through an HVDC Link: A Communication-Free Approach","volume":"33","author":"Kou","year":"2018","journal-title":"IEEE Trans. Energy Convers."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1682","DOI":"10.1109\/TPWRD.2021.3095529","article-title":"Control of the Parallel Operation of DR-HVDC and VSC-HVDC for Offshore Wind Power Transmission","volume":"37","author":"Nami","year":"2022","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2392","DOI":"10.1109\/TIE.2012.2227904","article-title":"Stability Analysis of Four PMSG-Based Offshore Wind Farms Fed to an SG-Based Power System Through an LCC-HVDC Link","volume":"60","author":"Wang","year":"2013","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"121256","DOI":"10.1016\/j.apenergy.2023.121256","article-title":"A Dynamics-Constrained Method for Distributed Frequency Regulation in Low-Inertia Power Systems","volume":"344","author":"Li","year":"2023","journal-title":"Appl. Energy"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1109\/TPWRS.2023.3264591","article-title":"Interaction Between Grid-Forming Converters with AC Grids and Damping Improvement Based on Loop Shaping","volume":"39","author":"Wang","year":"2023","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1109\/TSTE.2016.2614223","article-title":"Power Sharing Control Strategy of Multiterminal VSC-HVDC Transmission Systems Utilizing Adaptive Voltage Droop","volume":"8","author":"Abdelwahed","year":"2016","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"564","DOI":"10.35833\/MPCE.2018.000878","article-title":"Small-Signal Analysis of DC Microgrid and Multi-Objective Optimization Segmented Droop Control Suitable for Economic Dispatch","volume":"8","author":"Liu","year":"2020","journal-title":"J. Mod. Power Syst. Clean. Energy"},{"key":"ref_24","first-page":"169","article-title":"Adaptive Voltage Droop Method of Multiterminal VSC-HVDC Systems for DC Voltage Deviation and Power Sharing","volume":"34","author":"Wang","year":"2018","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1109\/TPWRS.2017.2719002","article-title":"Adaptive droop control of VSC-MTDC system for frequency support and power sharing","volume":"33","author":"Wang","year":"2017","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3541","DOI":"10.1109\/TPWRS.2020.2976817","article-title":"Stability Criterion for Inertial and Primary Frequency Droop Control in MTDC Grids with Implications on Ratio-based Frequency Support","volume":"35","author":"Vennelaganti","year":"2020","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3550","DOI":"10.1109\/TPWRD.2020.3044978","article-title":"Autonomous DC Line Power Flow Regulation Using Adaptive Droop Control in HVDC Grid","volume":"36","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1441","DOI":"10.1109\/TPWRS.2012.2186988","article-title":"Impact of DC Line Voltage Drops on Power Flow of MTDC Using Droop Control","volume":"27","author":"Haileselassie","year":"2012","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1109\/TSG.2023.3270112","article-title":"Data-Driven Affinely Adjustable Robust Volt\/VAr Control","volume":"15","author":"Shi","year":"2024","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"381","DOI":"10.35833\/MPCE.2021.000307","article-title":"Adaptive Reference Power Based Voltage Droop Control for VSC-MTDC Systems","volume":"11","author":"Wang","year":"2023","journal-title":"J. Mod. Power Syst. Clean. Energy"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4373","DOI":"10.1109\/TPEL.2017.2715039","article-title":"Robust Droop and DC-Bus Voltage Control for Effective Stabilization and Power Sharing in VSC Multiterminal DC Grids","volume":"33","author":"Davari","year":"2018","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1109\/JPETS.2018.2842041","article-title":"Droop Control for a Multi-Line Current Flow Controller in Meshed Multi-Terminal HVDC Grid Under Large DC Disturbances","volume":"5","author":"Wang","year":"2018","journal-title":"IEEE Power Energy Technol. Syst. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"8948","DOI":"10.1109\/TIE.2016.2631136","article-title":"Optimized Power Redistribution of Offshore Wind Farms Integrated VSC-MTDC Transmissions After Onshore Converter Outage","volume":"64","author":"Dong","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4745","DOI":"10.1109\/TPWRS.2021.3067199","article-title":"VSC-HVDC Interties for Urban Power Grid Enhancement","volume":"36","author":"Sun","year":"2021","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2442","DOI":"10.1109\/TPWRD.2016.2632860","article-title":"Fast Frequency Response from Offshore Multiterminal VSC\u2013HVDC Schemes","volume":"32","author":"Adeuyi","year":"2017","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1109\/TSTE.2015.2497340","article-title":"Coordinated Operation and Control of VSC Based Multiterminal High Voltage DC Transmission Systems","volume":"7","author":"Raza","year":"2016","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1109\/TPWRS.2020.2999443","article-title":"Adaptive Droop Control of Multi-Terminal HVDC Network for Frequency Regulation and Power Sharing","volume":"36","author":"Ambia","year":"2021","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3213","DOI":"10.1109\/TPWRD.2021.3125254","article-title":"A Combined Hierarchical and Autonomous DC Grid Control for Proportional Power Sharing with Minimized Voltage Variation and Transmission Loss","volume":"37","author":"Zhang","year":"2022","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1109\/JESTPE.2014.2338738","article-title":"DC Voltage Control and Power Sharing in Multiterminal DC Grids Based on Optimal DC Power Flow and Voltage-Droop Strategy","volume":"2","author":"Rouzbehi","year":"2014","journal-title":"IEEE J. Emerg. Sel. Top. Power Electron."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2083","DOI":"10.1109\/TPWRS.2016.2601104","article-title":"Improved Analytical Model for the Study of Steady State Performance of Droop-Controlled VSC-MTDC Systems","volume":"32","author":"Xiao","year":"2017","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1109\/TPWRS.2020.3020039","article-title":"Minimization of AC-DC Grid Transmission Loss and DC Voltage Deviation Using Adaptive Droop Control and Improved AC-DC Power Flow Algorithm","volume":"36","author":"Zhang","year":"2021","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1812","DOI":"10.1109\/TPWRS.2019.2953044","article-title":"Distributed Control of VSC-MTDC Systems Considering Tradeoff Between Voltage Regulation and Power Sharing","volume":"35","author":"Wang","year":"2020","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3739","DOI":"10.1109\/TPWRS.2020.2986168","article-title":"Distributed Voltage Regulation and Automatic Power Sharing in Multi-Terminal HVDC Grids","volume":"35","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2196","DOI":"10.1109\/TPWRD.2019.2963447","article-title":"A Novel Method to Determine Droop Coefficients of DC Voltage Control for VSC-MTDC System","volume":"35","author":"Li","year":"2020","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"12143","DOI":"10.1109\/TPEL.2024.3409537","article-title":"General Linearized Model of Voltage Source Converter with Fixed Nodal Admittance Matrix","volume":"39","author":"Zhang","year":"2024","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5008","DOI":"10.1109\/TPWRS.2023.3323660","article-title":"Robust optimization of scale and revenue for integrated power-to-hydrogen systems within energy, ancillary services, and hydrogen markets","volume":"39","author":"Gu","year":"2023","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.eng.2024.05.014","article-title":"Feasibility of Scaling up the Cost-Competitive and Clean Electrolytic Hydrogen Supply in China","volume":"39","author":"Pan","year":"2024","journal-title":"Engineering"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1109\/TPWRD.2017.2759302","article-title":"DC Voltage Regulation and Frequency Support in Pilot Voltage Droop-Controlled Multiterminal HVdc Systems","volume":"33","author":"Kirakosyan","year":"2018","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/OAJPE.2024.3354079","article-title":"Adaptive Voltage Reference Based Controls of Converter Power Sharing and Pilot Voltage in HVDC System for Large-Scale Offshore Wind Integration","volume":"11","author":"Zhang","year":"2024","journal-title":"IEEE Open J. Power Energy"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"5288","DOI":"10.1109\/TSG.2020.3000726","article-title":"Multi-Objective Adaptive Robust Voltage\/VAR Control for High-PV Penetrated Distribution Networks","volume":"11","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"4404","DOI":"10.1109\/TPWRS.2022.3210106","article-title":"Modeling the Reserve Capacity of Wind Power and the Inherent Decision-Dependent Uncertainty in the Power System Economic Dispatch","volume":"38","author":"Pan","year":"2023","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1109\/TPWRS.2006.873010","article-title":"Evenly Distributed Pareto Points in Multi-Objective Optimal Power Flow","volume":"21","author":"Roman","year":"2006","journal-title":"IEEE Trans. Power Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/105\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:27:05Z","timestamp":1759919225000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/105"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,11]]},"references-count":52,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010105"],"URL":"https:\/\/doi.org\/10.3390\/sym17010105","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,1,11]]}}}