A Review of Co-simulation Algorithm

doi:10.3901/JME.2024.19.172

Abstract

Abstract: The co-simulation algorithm has been widely used in digital twin, multi-physics domain coupling analysis and parallel computing of complex large-scale dynamic systems.Compared with traditional strongly coupled analysis, it has the advantages of modularity, intelligence, non-intrusive and efficient calculation.With the development of intelligent manufacturing and the deep integration of the industrial Internet, data interfaces and computing platforms based on co-simulation algorithms have developed rapidly.Due to the differences in stiffness, accuracy requirements and numerical characteristics of multi-physics domain models, the numerical stability of co-simulation algorithms has always been a bottleneck restricting their application.First, the basic concepts, classification and coupling models of co-simulation algorithms are introduced.Then, the numerical stability issues introduced by interface variable interpolation are discussed respectively, and the co-simulation stabilization techniques, local error analysis and macro-time step control methods are summarized.Next, the application and algorithm platform of co-simulation algorithms in integrated energy systems, dynamic systems and digital twin technology are summarized.Finally, facing the future of efficient and high-stability co-simulation algorithms, the research and application of co-simulation algorithms are summarized and prospected.

Key words: co-simulation algorithm, numerical properties, weak coupling, platform of the algorithm

CLC Number:

TG156

LI Pu, LU Daixing. A Review of Co-simulation Algorithm[J]. Journal of Mechanical Engineering, 2024, 60(19): 172-186.

References

[1] FAN H，YU Z，XIA S，et al. Review on coordinated planning of source-network-load-storage for integrated energy systems[J]. Frontiers in Energy Research，2021：1-12.
[2] 林俊光，冯彦皓，林小杰，等. 综合能源系统源网荷储动态建模技术进展[J]，热力发电. 2022，51(10)：92-102. LIN Junguang，FENG Yanhao，LIN Xiaojie，et al. Advances in dynamic modelling of source，grid，load and storage in integrated energy systems[J]. Thermal Power Generation，2022，51(10)：92-102.
[3] 夏越，陈颖，杜松怀，等. 综合能源系统多时间尺度动态时域仿真关键技术[J]. 电力系统自动化，2022，46(10)：97-110. XIA Yue，CHEN Ying，DU Songhuai，et al. Key technologies for multi-time-scale dynamic time-domain simulation of integrated energy system[J]. Automation of Electric Power Systems，2022，46(10)：97-110.
[4] BREZINA T，HADAS Z，VETISKA J. Using of co-simulation ADAMS-SIMULINK for development of mechatronic systems[C]//14th International Conference Mechatronika，2011，Trencianske Teplice，Slovakia. IEEE，2011:59-64.
[5] SPIRYAGIN M，SIMSON S，COLE C，et al. Co-simulation of a mechatronic system using Gensys and Simulink[J]. Vehicle system dynamics，2012，50(3)：495-507.
[6] ZHOU X，ZHAO B，GONG G. Control parameters optimization based on co-simulation of a mechatronic system for an UA-based two-axis inertially stabilized platform[J]. Sensors，2015，15(8)：20169-20192.
[7] ADEMUJIMI T，PRABHU V. Digital twin for training bayesian networks for fault diagnostics of manufacturing systems[J]. Sensors，2022，22(4)：1-24.
[8] HAVARD V，JEANNE B，LACOMBLEZ M，et al. Digital twin and virtual reality：a co-simulation environment for design and assessment of industrial workstations[J]. Production & Manufacturing Research，2019，7(1)：472-489.
[9] FITZGERALD J，LARSEN P G，PIERCE K. Multi-modelling and co-simulation in the engineering of cyber-physical systems：towards the digital twin[C]//From Software Engineering to Formal Methods and Tools，and Back，Springer，2019：40-55.
[10] 林小夏，张树有，陈婧，等. 多体动力学与有限元联合仿真的时变载荷历程模型[J]. 浙江大学学报(工学版)，2011，45(09)：1643-1649. LIN Xiaoxia，ZHANG Shuyou，CHEN Jing，et al. Time varying load course model for co-simulation of multibody dynamics and finite element[J]. Journal of Zhejiang University(Engineering Science)，2011，45(09)：1643-1649.
[11] STEFAN D，GERHARD H，GUNTER S. Interaction of vehicles and flexible tracks by co-simulation of multibody vehicle systems and finite element track models[J]. Vehicle System Dynamics，2002，37(sup1)：372-384.
[12] MASSAT J，LAURENT C，BIANCHI J，et al. Pantograph catenary dynamic optimisation based on advanced multibody and finite element co-simulation tools[J]. Vehicle System Dynamics，2014，52(sup1)：338-354.
[13] GEAR C，WELLS D. Multirate linear multistep methods[J]. BIT Numerical Mathematics，1984，24(4)：484-502.
[14] KUEBLER R，SCHIEHLEN W. Two methods of simulator coupling[J]. Mathematical and Computer Modelling of Dynamical Systems，2000，6(2)：93-113.
[15] HAFNER I，POPPER N. An overview of the state of the art in co-simulation and related methods.[J]. Simulation Notes Europe，2021，31(4)：185-200.
[16] ARNOLD M. Stability of sequential modular time integration methods for coupled multibody system models[J]. Journal of Computational and Nonlinear Dynamics，2010，5(3)：1-9.
[17] GONZÁLEZ F，NAYA M Á，LUACES A，et al. On the effect of multirate co-simulation techniques in the efficiency and accuracy of multibody system dynamics[J]. Multibody System Dynamics，2011，25(4)：461-483.
[18] LIANG，S，ZHANG H，WANG H. Combinative algorithms for the multidisciplinary collaborative simulation of complex mechatronic products based on major step and convergent integration step[J]. Chinese Journal of Mechanical Engineering，2011，24(3)：355-363.
[19] GLUMAC S，KOVACIC Z. Relative consistency and robust stability measures for sequential co-simulation[C]// 13th International Modelica Conference，2019，Regensburg，Germany. Linkping University Electronic Press，2019：120-157.
[20] GLUMAC S，KOVACIC Z. Calling sequence calculation for sequential co-simulation master[C]//ACM SIGSIM Conference on Principles of Advanced Discrete Simulation，2018，New York，USA. 2018：157-160.
[21] SWEAFFORD T，YOON H. Co-simulation of dynamic systems in parallel and serial model configurations[J]. Journal of Mechanical Science and Technology，2013，27：3579-3587.
[22] KRAFT J，MEYER T，SCHWEIZER B. Reduction of the computation time of large multibody systems with co-simulation methods[C]//IUTAM Symposium on Solver-Coupling and Co-Simulation，2019，Darmstadt，Germany. 2019：131-152.
[23] CHEN W，RAN S，WU C，et al. Explicit parallel co-simulation approach：analysis and improved coupling method based on H-infinity synthesis[J]. Multibody System Dynamics，2021，52，255–279.
[24] WU Q，SUN Y，SPIRYAGIN M，et al. Parallel co-simulation method for railway vehicle-track dynamics[J]. Journal of Computational and Nonlinear Dynamics，2018，13(4)：41004.
[25] BENEDIKT M，WATZENIG D，HOFER A. Modelling and analysis of the non-iterative coupling process for co-simulation[J]. Mathematical and computer modelling of dynamical systems，2013，19(5)：451-470.
[26] BENEDIKT M. Control-oriented aspects of non-iterative co-simulation[J]. Automatisierungstechnik，2014，62(8)：598-606.
[27] STETTINGER G，HORN M，BENEDIKT M，et al. Model-based coupling approach for non-iterative real-time co-simulation[C]//2014 European Control Conference，2014，Strasbourg，France. IEEE，2014：2084-2089.
[28] LI P，YUAN Q. Explicit co-simulation with interface Jacobian approximation[J]. Journal of Mechanical Science and Technology，2022，36(3)：1103-1112.
[29] LI P，YUAN Q，LU D，et al. Improved explicit co-simulation methods incorporating relaxation techniques[J]. Archive of Applied Mechanics，2020，90(1)：17-46.
[30] LI P，LU D，SCHMOLL R，et al. Explicit co-simulation approach with improved numerical stability[C]//IUTAM Symposium on Solver-Coupling and Co-Simulation，2019，Darmstadt，Germany. 2019：131-152.
[31] KRAFT J，KLIMMEK S，MEYER T，et al. Implicit co-simulation and solver-coupling：efficient calculation of interface-Jacobian and coupling sensitivities/gradients[J]. Journal of Computational and Nonlinear Dynamics. 2022，17(4)：41004.
[32] SICKLINGER S，BELSKY V，ENGELMANN B，et al. Interface Jacobian-based co-simulation[J]. International Journal for Numerical Methods in Engineering，2014，98(6)：418-444.
[33] SICKLINGER S，LERCH C，WÜCHNER R，et al. Fully coupled co-simulation of a wind turbine emergency brake maneuver[J]. Journal of Wind Engineering and Industrial Aerodynamics，2015，144：134-145.
[34] VINCENTELLI A. The waveform relaxation method for time-domain analysis of large-scale integrated circuits[J]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems，1982，1(3)：131-145.
[35] MACIEJEWSKI M，GARCIA I C，SCHÖPS S，et al. Application of the waveform relaxation technique to the co-simulation of power converter controller and electrical circuit models[C]//22nd International Conference on Methods and Models in Automation and Robotics，2017，Miedzyzdroje，Poland. IEEE，2017：837-842.
[36] WACK T，SCHLÜTER S，HENNIG T，et al. Application of waveform relaxation in distributed process co-simulation[J]. Chemie Ingenieur Technik，2018，90(10)：1559-1567.
[37] SCHWEIZER B，LU D. Semi-implicit co-simulation approach for solver coupling[J]. Archive of Applied Mechanics，2014，84(12)：1739-1769.
[38] ARNOLD M，CLAUß C，SCHIERZ T. Error analysis and error estimates for co-simulation in FMI for model exchange and co-simulation V2.0[C]//Progress in Differential-Algebraic Equations. Springer，2014：107-125.
[39] GU B，ASADA H. Co-simulation of algebraically coupled dynamic subsystems without disclosure of proprietary subsystem models[J]. Journal of Dynamic Systems, Measurement, and Control，2004，126(1)：1-13.
[40] SCHWEIZER B，LI P，LU D. Implicit co‐simulation methods：stability and convergence analysis for solver coupling approaches with algebraic constraints[J]. Journal of Applied Mathematics and Mechanics，2016，96(8)：986-1012.
[41] 高浩，干锋，戴焕云. 刚柔耦合接触动态粘合算法研究[J]. 振动与冲击，2012，31(23)：123-127. GAO Hao，GAN Feng，DAI Huanyun. A dynamic gluing algorithm for rigid-flexible contact problems[J]. Journal of Vibration and Shock，2012，31(23)：123-127.
[42] WANG J，MA Z，GREGORY M. A gluing algorithm for distributed simulation of multibody systems[J]. Nonlinear Dynamics，2003，34(1-2)：159-188.
[43] TSENG F，HULBERT G. A gluing algorithm for network-distributed multibody dynamics simulation[J]. Multibody System Dynamics，2001，6(4)：377-396.
[44] SCHWEIZER B，LI P，LU D. Explicit and implicit cosimulation methods：stability and convergence analysis for different solver coupling approaches[J]. Journal of Computational and Nonlinear Dynamics，2015，10(5)：1-12.
[45] GOMES C，THULE C，BROMAN D，et al. Co-simulation：a survey[J]. ACM Computing Surveys，2018，51(3)：1-33.
[46] SADJINA S，KYLLINGSTAD L T，SKJONG S，et al. Energy conservation and power bonds in co-simulations：non-iterative adaptive step size control and error estimation[J]. Engineering with Computers，2017，33：607-620.
[47] MOSHAGEN T. On meeting energy balance errors in co-simulations[J]. Mathematical and computer modelling of dynamical systems，2019，25(2)：139-166.
[48] BUSCH M，SCHWEIZER B. Stability of co-simulation methods using hermite and lagrange approximation techniques[C]//ECCOMAS Thematic Conference on Multibody Dynamics，2011：1-10.
[49] ANDREAS B，MARKUS B，MICHAEL G，et al. Dynamic iteration for coupled problems of electric circuits and distributed devices[J]. SIAM Journal on Scientific Computing，2013，35(2)：315-335.
[50] SCHWEIZER B，LI P，LU D，et al. Stabilized implicit cosimulation method：solver coupling with algebraic constraints for multibody systems[J]. Journal of Computational and Nonlinear Dynamics，2016，11(2)：1-18.
[51] FRANCISCO G，SIAMAK A，ARASH M，et al. Energy-leak monitoring and correction to enhance stability in the co-simulation of mechanical systems[J]. Mechanism and Machine Theory，2019，131：172-188.
[52] JARKKO R，FRANCISCO G，MIGUEL Á N. An automated methodology to select functional co-simulation configurations[J]. Multibody System Dynamics，2020，48(1)：79-103.
[53] SCHWEIZER B，LU D，LI P. Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques[J]. Multibody System Dynamics，2016，36(1)：1-36.
[54] TAMELLIN I，RICHIEDEI D，RODRÍGUEZ B，et al. Eigenstructure assignment and compensation of explicit co-simulation problems[J]. Mechanism and Machine Theory，2022，176：1-26.
[55] GENSER S，BENEDIKT M，WATZENIG D. Approximated beats exact Interface Jacobians at model-based co-simulation - explanation and impact[J]. Mechanism and Machine Theory，2022，176：1-12.
[56] INCI E O，CROES J，BOSMANS J，et al. Offline adaptation of co-simulation time steps by leveraging past co-simulation runs in multi-level mechatronic systems[J]. Mechanism and Machine Theory，2022，171：104740-104763.
[57] ARNOLD M，HANTE S，KUEBIS M. Error analysis for co-simulation with force-displacement coupling[J]. Proceedings in Applied Mathematics and Mechanics，2014，14(1)：43-44.
[58] MEYER T，SCHWEIZER B. Error estimation approach for controlling the communication step size for semi-implicit co-simulation methods[J]. Proceedings in Applied Mathematics and Mechanics，2015，15(1)：63-64.
[59] BUSCH M. Continuous approximation techniques for co-simulation methods：analysis of numerical stability and local error[J]. Journal of Applied Mathematics and Mechanics，2016，96(9)：1061-1081.
[60] MEYER T，KRAFT J，SCHWEIZER B. Co-simulation：error estimation and macro-step size control[J]. Journal of Computational and Nonlinear Dynamics，2021，16(4)：1-26.
[61] SCHIERZ T，ARNOLD M，CLAUß C. Co-simulation with communication step size control in an FMI compatible master algorithm[C]//9th International MODELICA Conference，2012，Munich, Germany. Linkping University Electronic Press，2012：205-214.
[62] BENEDIKT M，STIPPEL H，WATZENIG D. An adaptive coupling methodology for fast time-domain distributed heterogeneous co-simulation[R]. SAE Technical Paper，2010.
[63] BUSCH M. Zur effizienten Kopplung von Simulationsprogrammen[D]. kassel university press GmbH，2012. BUSCH M. For efficient coupling of simulation programs[D]. Kassel University Press GmbH，2012.
[64] DAHMANN J，FUJIMOTO R，WEATHERLY R. The department of defense high level architecture[C]// Winter Simulation Conference，1997，Atlanta, USA. IEEE Computer Society，1997：142-149.
[65] BLOCHWITZ T，OTTER M，J. ÅKESSON，et al. Functional mockup interface 2.0：the standard for tool independent exchange of simulation models[J]，2012：1-13.
[66] 李洋，吴鸣，周海明，等. 基于全能流模型的区域多能源系统若干问题探讨[J]. 电网技术，2015，39(8)：2230-2237. LI Yang，WU Ming，ZHOU Haiming，et al. Study on some key problems related to regional multi energy system based on universal flow model[J]. Power System Technology，2015，39(8)：2230-2237.
[67] 王伟亮，王丹，贾宏杰，等. 能源互联网背景下的典型区域综合能源系统稳态分析研究综述[J]. 中国电机工程学报，2016，36(12)：3292-3306. WANG Weiliang，WANG Dan，JIA Hongjie，et al. Review of steady -state analysis of typical regional integrated energy system under the background of energy internet[J]. Proceedings of the CSEE. 2016，36(12)：3292-3306.
[68] 马凯琪，吴迪，郑灏. 综合能源系统混合仿真技术路线探讨[J]. 供用电，2018，35(7)：28-33. MA Kaiqi，WU Di，ZHENG Hao. A look at hybrid simulation technology for integrated energy system[J]. Distribution & Utilization，2018，35(7)：28-33.
[69] GUSAIN D，CVETKOVIĆ M，PALENSKY P. Simplifying multi-energy system co-simulations using energysim[J]. SoftwareX，2022，18：1-8.
[70] CHAGAS I，TOMIM M. Co-simulation applied to power systems with high penetration of distributed energy resources[J]. Electric Power Systems Research，2022，212：1-10.
[71] 孙浩，陈永华，吴维宁，等. 区域综合能源仿真分析系统的开发与研制[J]. 电器与能效管理技术，2022(09)：18-25. SUN Hao，CHEN Yonghua，WU Weining，et al. Research on simulation system of integrated community energy[J]. Low Voltage Apparatus，2022(09)：18-25.
[72] LIU S，JIA Y L，SONG Y H. The electrical-thermal co-simulation of integrated engineering system based on modular modeling method[C]// 11th World Congress on Intelligent Control and Automation，2014，Shenyang，China. 2014：6067-6074.
[73] XU X，JIA H，CHIANG H，et al. Dynamic modeling and interaction of hybrid natural gas and electricity supply system in microgrid[J]. IEEE Transactions on Power Systems：A Publication of the Power Engineering Society，2015，30(3)：1212-1221.
[74] AZEROUAL M，BOUJOUDAR Y，ALJARBOUH A，et al. Advanced energy management and frequency control of distributed microgrid using multi-agent systems[J]. International Journal of Emerging Electric Power Systems，2021，23(5)：755-766.
[75] CHRISTINA A，ANASTASIOS I D. Co-simulation of fuzzy control in buildings and the HVAC system using BCVTB[J]. Advances in Building Energy Research，2018，12(2)：195-216.
[76] BASTIAN W，WOLFRAM R，DANIEL S，et al. Co-simulation of geothermal applications and HVAC systems[J]. Energy Procedia，2017，125：345-352.
[77] MARIJA T，JAN L M H，MICHAEL W. Co-simulation of innovative integrated HVAC systems in buildings[J]. Journal of Building Performance Simulation，2009，2(3)：209-230.
[78] 孙加亮，田强，胡海岩. 多柔体系统动力学建模与优化研究进展[J]. 力学学报，2019，51(06)：1565-1586. SUN Jialiang，TIAN Qiang，HU Haiyan. Advances in dynamic modeling and optimization of flexible multibody systems[J]. Chinese Journal of Theoretical and Applied Mechanics，2019，51(06)：1565-1586.
[79] LARS D，TOBIAS L，ANNIKA S. Modelling and simulation of catenary-pantograph interaction[J]. Vehicle System Dynamics，2019，33：490-501.
[80] JORGE A，JOAO P，MANUEL P，et al. A computational procedure for the dynamic analysis of the catenary-pantograph interaction in high-speed trains [J]. Journal of Theoretical and Applied Mechanic，2012，50(3)：681-699.
[81] AMBRÓSIO J，POMBO J，RAUTER F，et al. A memory-based communication in the co-simulation of multibody and finite element codes for pantograph-catenary interaction simulation[J]. Computational Methods in Applied Sciences，2009，12：231-252.
[82] KUEBLER R，SCHIEHLEN W，SCHOLZ C. Simulator coupling for multibody systems[J]. Proceedings in Applied Mathematics and Mechanics，2002，1(1)：33-36.
[83] KUEBLER R，SCHIEHLEN W. Modular Simulation in Multibody System Dynamics[J]. Multibody System Dynamics，2000，4(2-3)：107-127.
[84] PEIRET A，GONZÁLEZ F，KÖVECSES J，et al. Multibody system dynamics interface modelling for stable multirate co-simulation of multiphysics systems[J]. Mechanism and Machine Theory，2018，127：52-72.
[85] ROCCATELLO A，MANCÒ S，NERVEGNA N. Modelling a variable displacement axial piston pump in a multibody simulation environment[J]. Journal of Dynamic Systems，Measurement，and Control，2007，129(4)：819-829.
[86] JARKKO R，FRANCISCO G，MIGUEL Á N，et al. On the cosimulation of multibody systems and hydraulic dynamics[J]. Multibody System Dynamics，2020，50(2)：143-167.
[87] LIAO Y，DU I. Cosimulation of multi-body-based vehicle dynamics and an electric power steering control system[J]. Proceedings of the Institution of Mechanical Engineers，Part K：Journal of Multi-body Dynamics，2001，215(3)：141-151.
[88] SILVA S，SUNDARAM P，'AMBROSIO J. Co-simulation platform for diagnostic development of a controlled chassis system[J]. SAE Transactions，2006，115：1-12.
[89] KUMAR A，VIJAYAKUMAR R，SUBRAMANIAN V A. Numerical fluid-structure interaction analysis for a flexible marine propeller using co-simulation method[J]. International Journal of Maritime Engineering，2021，163(A2)：81-89.
[90] 李田，张继业，张卫华. 横风下高速列车流固耦合动力学联合仿真[J]. 振动工程学报，2012，25(2)：138-145. LI Tian，ZHANG Jiye，ZHANG Weihua. Co-simulation of high-speed train fluid-structure interaction dynamics in crosswinds[J]. Journal of Vibration Engineering，2012，25(2)：138-145.
[91] MARTIN B，BERNHARD S. Coupled simulation of multibody and finite element systems：an efficient and robust semi-implicit coupling approach[J]. Archive of Applied Mechanics，2012，82(6)：723-741.
[92] ZEISE P，SCHWEIZER B. Dynamics，stability and bifurcation analysis of rotors in air ring bearings[J]. Journal of Sound and Vibration，2022，521：1-35.
[93] 陶飞，张辰源，刘蔚然，等. 数字工程及十个领域应用展望[J]. 机械工程学报，2023，59(13)：193-215. TAO Fei，ZHANG Chenyuan，LIU Weiran，et al. Digital engineering and its ten application outlooks[J]. Journal of Mechanical Engineering，2023，59(13)：193-215.
[94] 刘大同，郭凯，王本宽，等. 数字孪生技术综述与展望[J]. 仪器仪表学报，2018，39(11)：1-10. LIU Datong，GUO Kai，WANG Benkuan，et al. Summary and perspective survey on digital twin technology[J]. Chinese Journal of Scientific Instrument，2018，39(11)：1-10.
[95] 黄海松，陈启鹏，李宜汀，等. 数字孪生技术在智能制造中的发展与应用研究综述[J]. 贵州大学学报(自然科学版)，2020，37(5)：1-8. HUANG Haisong，CHEN Qipeng，LI Yiting，et al. Research overview on the development and application of digital twin technology in intelligent manufacturing[J]. Journal of Guizhou University，2020，37(5)：1-8.
[96] 陈根. 数字孪生[M]. 北京：电子工业出版社，2020. CHEN Gen. Digital twin[M]. Beijing: Publishing House of Electronics Industry，2020.
[97] 李鹏，潘凯，刘小川. 美国空军机体数字孪生计划的回顾与启示[J]. 航空科学技术，2020，31(9)：1-10. LI Peng，PAN Kai，LIU Xiaochuan. Retrospect and enlightenment of the AFRL airframe digital twin program[J]. Aeronautical Science & Technology，2020，31(9)：1-10.
[98] 陶飞，刘蔚然，张萌等. 数字孪生五维模型及十大领域应用[J]. 计算机集成制造系统，2019，25(1)：1-18. TAO Fei，LIU Weiran，ZHANG Meng，et al. Five-dimension digital twin model and its ten applications[J]. Computer Integrated Manufacturing Systems，2019，25(1)：1-18.
[99] HATLEDAL L I，SKULSTAD R，LI G，et al. Co-simulation as a fundamental technology for twin ships[J]. Modeling，Identification and Control (MIC)，2020，41(4)：297-311.
[100] 陶飞，张贺，戚庆林，等. 数字孪生模型构建理论及应用[J]. 计算机集成制造系统，2021，27(1)：1-15. TAO Fei，ZHANG He，QI Qinglin，et al. Theory of digital twin modeling and its application[J]. Computer Integrated Manufacturing Systems，2021，27(1)：1-15.
[101] SCHLEICH B，ANWER N，MATHIEU L，et al. Shaping the digital twin for design and production engineering[J]. CIRP Annals，2017，66(1)：141-144.
[102] 周林，毛志杰，陈英梅，等. 基于多领域联合建模的卫星通信装备数字孪生构建方法[C]//2020中国系统仿真与虚拟现实技术高层论坛，2020. 北京国信融合信息技术研究院，2020：20-24. ZHOU Lin，MAO Zhijie，CHEN Yingmei, et al. Construction method of digital twin of satellite communication equipment based on multi-domain joint modeling[C]//2020 China System Simulation and Virtual Reality Technology High-Level Forum, 2020. Beijing Research Institute of Information Technology，2020：20-24.
[103] REZA S，JAHANSHAHI M，AHMAD P，et al. FMI real-time co-simulation-based machine deep learning control of HVAC systems in smart buildings：Digital-twins technology[J]. Transactions of the Institute of Measurement and Control，2023，45(4).
[104] DUGAN R，MCDERMOTT T. An open-source platform for collaborating on smart grid research[C]//2011 IEEE Power and Energy Society General Meeting，2011，Detroit，USA. IEEE Computer Society，2011：1-7.
[105] LIN H，VEDA S，SHUKLA S，et al. GECO：Global event-driven co-simulation framework for interconnected power system and communication network[J]. IEEE Transactions on Smart Grid，2012，3(3)：1444-1456.
[106] KRAMMER M，MARKO N，BENEDIKT M. Interfacing real-time systems for advanced co-simulation-The ACOSAR approach[C]//STAF Doctoral Symposium，2016，Vienna，Austria. 2016：1-8.
[107] JOPPICH W，KURSCHNER M. MpCCI - A tool for the simulation of coupled applications[J]. Concurrency and computation：practice and experience. 2006，18(2)：183-192.
[108] THULE C，LAUSDAHL K，GOMES C，et al. Maestro：The INTO-CPS co-simulation framework[J]. Simulation Modelling Practice and Theory，2019，92：45-61.
[109] LACOURSIÈRE C. FMI Go! A runtime environment for FMI based simulation[C]//12th International Modelica Conference，2017，Prague，Czech Republic. 2017：653-662.
[110] ANDERSSON C，ÅKESSON J，FÜHRER C. PyFMI：A python package for simulation of coupled dynamic models with the functional mock-up interface[R]. Centre for Mathematical Sciences，Lund University Lund，Sweden，2016.
[111] SCHUTTE S，SCHERFKE S，TROSCHEL M. Mosaik：A framework for modular simulation of active components in Smart Grids[C]//2011 IEEE First International Workshop on Smart Grid Modeling and Simulation，2011，Brussels，Belgium. 2011：55-60.
[112] PALMINTERI B，KRISHNAMURTHY D，TOP P，et al. Design of the HELICS high-performance transmission distribution communication market co-simulation framework[C]//Workshop on Modeling & Simulation of Cyber-physical Energy Systems，2017，Pittsburgh，USA. IEEE，2017：1-6.
[113] SCHERFKE S. AIOMAS Documentation [EB/OL]. https://aiomas.readthedocs.io/en/latest/.
[114] HATLEDAL L，SKULSTAD R，LI G，et al. Co-simulation as a Fundamental Technology for Twin Ships[J]. Modeling，Identification and Control (MIC)，2020，41(4):297-311.