{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,26]],"date-time":"2026-09-26T11:50:07Z","timestamp":1790423407406,"version":"4.1.0"},"reference-count":64,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,3,1]],"date-time":"2021-03-01T00:00:00Z","timestamp":1614556800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005416","name":"Norges Forskningsr\u00e5d","doi-asserted-by":"publisher","award":["280617"],"award-info":[{"award-number":["280617"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Contemporary Critical Infrastructures (CIs), such as the power grid, comprise cyber physical systems that are tightly coupled, to form a complex system of interconnected components with interacting dependencies. Modelling methodologies have been suggested as proper tools to provide better insight into the dependencies and behavioural characteristics of these complex systems. In order to facilitate the study of interconnections in and among critical infrastructures, and to provide a clear view of the interdependencies among their cyber and physical components, this paper proposes a novel method, based on a graphical model called Modified Dependency Structure Matrix (MDSM). The MDSM provides a compact perspective of both inter-dependency and intra-dependency between subsystems of one complex system or two distinct systems. Additionally, we propose four parameters that allow the quantitative assessment of the characteristics of dependencies, including multi-order dependencies in large scale CIs. We illustrate the workings of the proposed method by applying it to a micro-distribution network based on the G2ELAB 14-Bus model. The results provide valuable insight into the dependencies among the network components and substantiate the applicability of the proposed method for analyzing large scale cyber physical systems.<\/jats:p>","DOI":"10.3390\/s21051685","type":"journal-article","created":{"date-parts":[[2021,3,1]],"date-time":"2021-03-01T10:25:18Z","timestamp":1614594318000},"page":"1685","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Identifying and Analyzing Dependencies in and among Complex Cyber Physical Systems"],"prefix":"10.3390","volume":"21","author":[{"given":"Aida","family":"Akbarzadeh","sequence":"first","affiliation":[{"name":"Department of Information Security and Communication Technology, Norwegian University of Science and Technology, N-2815 Gj\u00f8vik, Norway"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2966-9683","authenticated-orcid":false,"given":"Sokratis","family":"Katsikas","sequence":"additional","affiliation":[{"name":"Department of Information Security and Communication Technology, Norwegian University of Science and Technology, N-2815 Gj\u00f8vik, Norway"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.cose.2017.04.005","article-title":"Cyber physical systems security: Analysis, challenges and solutions","volume":"68","author":"Ashibani","year":"2017","journal-title":"Comput. Secur."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1109\/37.969131","article-title":"Identifying, understanding, and analyzing critical infrastructure interdependencies","volume":"21","author":"Rinaldi","year":"2001","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Haes Alhelou, H., Hamedani-Golshan, M.E., Njenda, T.C., and Siano, P. (2019). A survey on power system blackout and cascading events: Research motivations and challenges. Energies, 12.","DOI":"10.3390\/en12040682"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.erss.2017.05.010","article-title":"Valuing blackouts and lost leisure: Estimating electricity interruption costs for households across the European Union","volume":"34","author":"Shivakumar","year":"2017","journal-title":"Energy Res. Soc. Sci."},{"key":"ref_5","unstructured":"Lee, R.M., Assante, M.J., and Conway, T. (2016, January 18). Analysis of the cyber attack on the Ukrainian power grid: Defense Use Case. Proceedings of the Electricity Information Sharing and Analysis Center (E-ISAC), Washington, DC, USA."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1016\/j.egypro.2017.11.055","article-title":"Literature review of power system blackouts","volume":"141","author":"Wu","year":"2017","journal-title":"Energy Procedia"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Akbarzadeh, A., Pandey, P., and Katsikas, S. (2019, January 6\u20138). Cyber-Physical Interdependencies in Power Plant Systems: A Review of Cyber Security Risks. Proceedings of the 2019 IEEE Conference on Information and Communication Technology, Allahabad, India.","DOI":"10.1109\/CICT48419.2019.9066188"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Li, X., and Yu, W. (2018, January 8\u201313). A hybrid fuzzy petri nets and neural networks framework for modeling critical infrastructure systems. Proceedings of the 2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Rio de Janeiro, Brazil.","DOI":"10.1109\/FUZZ-IEEE.2018.8491550"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.ijcip.2012.01.003","article-title":"Quantification of dependencies between electrical and information infrastructures","volume":"5","author":"Beccuti","year":"2012","journal-title":"Int. J. Crit. Infrastruct. Prot."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Tan, R.R., Aviso, K.B., Promentilla, M.A.B., Yu, K.D.S., and Santos, J.R. (2019). Input\u2013Output Models of Infrastructure Systems. Input-Output Models for Sustainable Industrial Systems, Springer.","DOI":"10.1007\/978-981-13-1873-3"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"106898","DOI":"10.1016\/j.ress.2020.106898","article-title":"Modeling and assessing interdependencies between critical infrastructures using Bayesian network: A case study of inland waterway port and surrounding supply chain network","volume":"198","author":"Hossain","year":"2020","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wei, J., and Kundur, D. (2015). Biologically inspired hierarchical cyber-physical multi-agent distributed control framework for sustainable smart grids. Cyber Physical Systems Approach to Smart Electric Power Grid, Springer.","DOI":"10.1007\/978-3-662-45928-7_9"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.ijcip.2014.12.004","article-title":"Critical infrastructure dependencies: A holistic, dynamic and quantitative approach","volume":"8","author":"Hernantes","year":"2015","journal-title":"Int. J. Crit. Infrastruct. Prot."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1080\/17477778.2017.1418640","article-title":"Modelling methodologies for analysing critical infrastructures","volume":"12","author":"Iturriza","year":"2018","journal-title":"J. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1504\/IJCAST.2016.081294","article-title":"A framework for characterising infrastructure interdependencies","volume":"1","author":"Carhart","year":"2016","journal-title":"Int. J. Complex. Appl. Sci. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_17","unstructured":"Blackwell, C. (, January May). A multi-layered security architecture for modelling complex systems. Proceedings of the 4th Annual Workshop on Cyber Security and Information Intelligence Research: Developing Strategies to Meet the Cyber Security and Information Intelligence Challenges Ahead, Oak Ridge, TN, USA."},{"key":"ref_18","unstructured":"Monday, J.L. (2003). Managing disruptions to critical interdependent infrastructures in the context of the 2001 World Trade Center attack. Beyond September 11th: An Account of Post-Disaster Research, Institute of Behavioral Science, Natural Hazards Research and Applications Information Center, University of Colorado."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/j.trb.2010.10.001","article-title":"A generalized modeling framework to analyze interdependencies among infrastructure systems","volume":"45","author":"Zhang","year":"2011","journal-title":"Transp. Res. Part B Methodol."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Dudenhoeffer, D.D., Permann, M.R., and Manic, M. (2006, January 3\u20136). CIMS: A framework for infrastructure interdependency modeling and analysis. Proceedings of the 2006 Winter Simulation Conference, Monterey, CA, USA.","DOI":"10.1109\/WSC.2006.323119"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.ress.2013.06.040","article-title":"Review on modeling and simulation of interdependent critical infrastructure systems","volume":"121","author":"Ouyang","year":"2014","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_22","unstructured":"Nieuwenhuijs, A., Luiijf, E., and Klaver, M. (2008, January 12\u201314). Modeling dependencies in critical infrastructures. Proceedings of the International Conference on Critical Infrastructure Protection, Arlington, VA, USA."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1111\/j.1467-9299.2011.01926.x","article-title":"The state and the threat of cascading failure across critical infrastructures: The implications of empirical evidence from media incident reports","volume":"89","author":"Nieuwenhuijs","year":"2011","journal-title":"Public Adm."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Rausand, M., and Haugen, S. (2020). Risk Assessment: Theory, Methods, and Applications, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781119377351"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.1109\/TSG.2014.2310742","article-title":"Reliability assessment of smart grids considering indirect cyber-power interdependencies","volume":"5","author":"Falahati","year":"2014","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1504\/IJCIS.2013.051606","article-title":"Assessing n-order dependencies between critical infrastructures","volume":"9","author":"Kotzanikolaou","year":"2013","journal-title":"Int. J. Crit. Infrastruct."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mathew, J., Ma, L., Tan, A., Weijnen, M., and Lee, J. (2011, January 3\u20135). Engineering asset management and infrastructure sustainability. Proceedings of the 5th World Congress on Engineering Asset Management (WCEAM 2010), Cincinnati, OH, USA.","DOI":"10.1007\/978-0-85729-493-7"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Luiijf, E., Nieuwenhuijs, A., Klaver, M., van Eeten, M., and Cruz, E. (2008, January 13\u201315). Empirical findings on critical infrastructure dependencies in Europe. Proceedings of the International Workshop on Critical Information Infrastructures Security, Rome, Italy.","DOI":"10.1007\/978-3-642-03552-4_28"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Macaulay, T. (2008). Critical Infrastructure: Understanding Its Component Parts, Vulnerabilities, Operating Risks, and Interdependencies, CRC Press.","DOI":"10.1201\/9781420068368"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.futures.2016.09.003","article-title":"A collaborative dynamic scenario model for the interaction of critical infrastructures","volume":"84","author":"Turoff","year":"2016","journal-title":"Futures"},{"key":"ref_31","unstructured":"Faramehr, S. (2020). Investigating Dependencies between Railway System and Other Infrastructure Systems: Using a Scenario-Based Case Study Approach. [Ph.D. Thesis, UCL (Univeristy College London)]."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1007\/s11518-016-5295-3","article-title":"Modeling and simulation of the vulnerability of interdependent power-water infrastructure networks to cascading failures","volume":"25","author":"Zhang","year":"2016","journal-title":"J. Syst. Sci. Syst. Eng."},{"key":"ref_33","first-page":"4637","article-title":"Modeling of interconnected critical infrastructure systems using complex network theory","volume":"9","author":"Zhu","year":"2017","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1002\/tee.22798","article-title":"A new model approach of electrical cyber physical systems considering cyber security","volume":"14","author":"Wang","year":"2019","journal-title":"IEEJ Trans. Electr. Electron. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.aci.2017.01.001","article-title":"Interdependencies and reliability in the combined ICT and power system: An overview of current research","volume":"14","author":"Foros","year":"2018","journal-title":"Appl. Comput. Inform."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Wirtz, N., and Monti, A. (October, January 28). A Flexible Framework to Investigate Cascading in Interdependent Networks of Power Systems. Proceedings of the 2020 6th IEEE International Energy Conference (ENERGYCon).","DOI":"10.1109\/ENERGYCon48941.2020.9236542"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Sen, A., and Basu, K. (2019, January 9\u201313). On connectivity of interdependent networks. Proceedings of the 2019 IEEE Global Communications Conference (GLOBECOM), Waikoloa, HI, USA.","DOI":"10.1109\/GLOBECOM38437.2019.9014143"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Satumtira, G., and Due\u00f1as-Osorio, L. (2010). Synthesis of modeling and simulation methods on critical infrastructure interdependencies research. Sustainable and Resilient Critical Infrastructure Systems, Springer.","DOI":"10.1007\/978-3-642-11405-2_1"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1061\/(ASCE)1076-0342(2001)7:1(1)","article-title":"Leontief-based model of risk in complex interconnected infrastructures","volume":"7","author":"Haimes","year":"2001","journal-title":"J. Infrastruct. Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Setola, R., and Theocharidou, M. (2016). Modelling dependencies between critical infrastructures. Managing the Complexity of Critical Infrastructures, Springer.","DOI":"10.1007\/978-3-319-51043-9_2"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1016\/j.ress.2010.06.010","article-title":"An approach for modelling interdependent infrastructures in the context of vulnerability analysis","volume":"95","author":"Johansson","year":"2010","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_42","unstructured":"Torres, J.L.S. (2013). Vulnerability, Interdependencies and Risk Analysis of Coupled Infrastructures: Power Distribution Network and ICT. [Ph.D. Thesis, Universit\u00e9 de Grenoble]."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Hadjsaid, N., Tranchita, C., Rozel, B., Viziteu, M., and Caire, R. (2009, January 15\u201318). Modeling cyber and physical interdependencies-Application in ICT and power grids. Proceedings of the 2009 IEEE\/PES Power Systems Conference and Exposition, Seattle, WA, USA.","DOI":"10.1109\/PSCE.2009.4840183"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1504\/IJCIS.2008.016092","article-title":"Modelling interdependent infrastructures using interacting dynamical models","volume":"4","author":"Rosato","year":"2008","journal-title":"Int. J. Crit. Infrastruct."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Fioriti, V., D\u2019Agostino, G., and Bologna, S. (2010, January 22\u201324). On modeling and measuring inter-dependencies among critical infrastructures. Proceedings of the 2010 Complexity in Engineering, Rome, Italy.","DOI":"10.1109\/COMPENG.2010.14"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Gueye, A., Mbaye, B., Fall, D., Diop, A., and Kashihara, S. (2020, January 8\u20139). A Matrix Model to Analyze Cascading Failure in Critical Infrastructures. Proceedings of the International Conference on Innovations and Interdisciplinary Solutions for Underserved Areas, Nairobi, Kenya.","DOI":"10.1007\/978-3-030-51051-0_15"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Chen, L., and Whyte, J. (2020, January 5\u20137). Analysing interdependencies of complex engineering systems using a digital twin-driven design structure matrix. Proceedings of the Construction Research Congress 2020: Computer Applications. American Society of Civil Engineers Reston, Reston, VA, USA.","DOI":"10.1061\/9780784482865.045"},{"key":"ref_48","unstructured":"Peterson, T. (2016, January 18\u201321). Exploring System Space with Graph Theory. Proceedings of the INCOSE International Symposium, Scotland, UK."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/014461900370807","article-title":"Analytical design planning technique (ADePT): A dependency structure matrix tool to schedule the building design process","volume":"18","author":"Austin","year":"2000","journal-title":"Constr. Manag. Econ."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Eppinger, S.D., and Browning, T.R. (2012). Design Structure Matrix Methods and Applications, MIT Press.","DOI":"10.7551\/mitpress\/8896.001.0001"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Thulasiraman, K., Arumugam, S., Nishizeki, T., and Brandst\u00e4dt, A. (2016). Handbook of Graph Theory, Combinatorial Optimization, and Algorithms, Chapman and Hall\/CRC.","DOI":"10.1201\/b19163"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1038\/scientificamerican0503-60","article-title":"Scale-free networks","volume":"288","author":"Bonabeau","year":"2003","journal-title":"Sci. Am."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1038\/nature10011","article-title":"Controllability of complex networks","volume":"473","author":"Liu","year":"2011","journal-title":"Nature"},{"key":"ref_54","unstructured":"Gen, M., Cheng, R., and Lin, L. (2008). Network Models and Optimization: Multiobjective Genetic Algorithm Approach, Springer."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1016\/0031-3203(91)90097-O","article-title":"A simulated annealing algorithm for the clustering problem","volume":"24","author":"Selim","year":"1991","journal-title":"Pattern Recognit."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1109\/4235.585892","article-title":"Ant colony system: A cooperative learning approach to the traveling salesman problem","volume":"1","author":"Dorigo","year":"1997","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Atashpaz-Gargari, E., and Lucas, C. (2007, January 25\u201328). Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition. Proceedings of the 2007 IEEE Congress on Evolutionary Computation, Singapore.","DOI":"10.1109\/CEC.2007.4425083"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1016\/j.asoc.2014.08.024","article-title":"A survey on the imperialist competitive algorithm metaheuristic: Implementation in engineering domain and directions for future research","volume":"24","author":"Hosseini","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s00163-016-0229-0","article-title":"Supporting design via the system operational dependency analysis methodology","volume":"28","author":"Guariniello","year":"2017","journal-title":"Res. Eng. Des."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1038\/nature08932","article-title":"Catastrophic cascade of failures in interdependent networks","volume":"464","author":"Buldyrev","year":"2010","journal-title":"Nature"},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Bang-Jensen, J., and Gutin, G.Z. (2008). Digraphs: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-1-84800-998-1"},{"key":"ref_62","unstructured":"Sanchez, J., Caire, R., and Hadjsaid, N. (2013, January 5\u20136). Ict and electric power systems interdependencies modeling. Proceedings of the International ETG-Congress 2013, Symposium 1: Security in Critical Infrastructures Today, Berlin, Germany."},{"key":"ref_63","unstructured":"Akbarzadeh, A., and Katsikas, S. (July, January 27). Identifying Critical Components in Large Scale Cyber Physical Systems. Proceedings of the IEEE\/ACM 42nd International Conference on Software Engineering Workshops, Seoul, Korea."},{"key":"ref_64","first-page":"21","article-title":"Infrastructure systems interdependencies and risk informed decision making (RIDM): Impact scenario analysis of infrastructure risks induced by natural, technological and intentional hazards","volume":"6","author":"Stapelberg","year":"2008","journal-title":"J. Syst. Cybern. Inform."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/5\/1685\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:30:54Z","timestamp":1760160654000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/5\/1685"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,1]]},"references-count":64,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["s21051685"],"URL":"https:\/\/doi.org\/10.3390\/s21051685","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,1]]}}}