Probabilistic Criterion for Evaluation of the Dependability of Critical Electrotechnical Systems
Keywords:
Keywords: dependability, dependability indicators, critical electrotechnical systems, Markov processes, state-based modeling, recovery of operability.
Abstract
Abstract. In the paper, a probabilistic methodology for the quantitative evaluation of the dependability of critical electrotechnical systems operating under destructive impacts is proposed. It is shown that traditional reliability and availability indicators are insufficient for describing system conduct under partial degradation, complete loss of operability, and subsequent recovery processes.
Dependability is interpreted as an integral probabilistic property reflecting a system’s ability to retain and restore critical functions under adverse conditions. A state-based Markov model is applied to describe the evolution of the functional state through a finite set of functional states and probabilistic transitions. Within this framework, dependability is characterized by three complementary probabilistic components: retention of full operability under destructive impact, recovery after partial loss of functionality, and recovery after complete loss of operability.
On this basis, a normalized integral dependability criterion bounded within the interval [0;1] is proposed using weighted aggregation of the probability components. The proposed approach provides a rigorous and flexible tool for comparative analysis and evaluation of critical electrotechnical systems with respect to resilience and recoverability.
References
1. Rinaldi, S.M., Peerenboom, J.P. and Kelly, T.K. (2001). Identifying, Understanding, and Analyzing Critical Infrastructure Interdependencies. IEEE Control Systems, [online] 21(6), pp.11–25. doi:https://doi.org/10.1109/37.969131.
2. Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E. and Havlin, S. (2010). Catastrophic cascade of failures in interdependent networks. Nature, 464(7291), pp.1025-1028. doi:https://doi.org/10.1038/nature08932.
3. Panteli, M.; Mancarella, P. (2015). The Grid: Stronger, bigger, smarter? Presenting a conceptual framework of power system resilience. ІEEE Power and Energy Magazine, 13(3), pp.56–66. doi:https://doi.org/10.1109/MPE.2015.2397334.
4. Laprie, J.C. (1992). Dependability: Basic Concepts and Terminology. Dependability: Basic Concepts and Terminology, Springer, Vienna, 5, pp.3–245. doi:https://doi.org/10.1007/978-3-7091-9170-5_1.
5. Haimes, Y.Y. (2009). On the definition of resilience in systems. Risk Analysis, 29(4), pp.498–501. doi: https://doi.org/10.1111/j.1539-6924.2009.01216.x
6. Hosseini, S., Barker, K. and Ramirez-Marquez, J.E. (2016). A review of definitions and measures of system resilience. Reliability Engineering & System Safety, [online] 145, pp.47–61. doi:https://doi.org/10.1016/j.ress.2015.08.006.
7. Avizienis, A., Laprie, J.-C., Randell, B. and Landwehr, C. (2004). Basic Concepts and Taxonomy of Dependable and Secure Computing. IEEE Transactions on Dependable and Secure Computing, 1(1), pp.11–33. doi:https://doi.org/10.1109/TDSC.2004.2.
8. Panteli, M. and Mancarella, P. (2015). Influence of extreme weather and climate change on the resilience of power systems: Impacts and possible mitigation strategies. Electric Power Systems Research, [online] 127(2), pp.259–270. doi:https://doi.org/10.1016/j.epsr.2015.06.012.
9. Panteli, M., Mancarella, P., Trakas, D.N., Kyriakides, E. and Hatziargyriou, N.D. (2017). Metrics and Quantification of Operational and Infrastructure Resilience in Power Systems. IEEE Transactions on Power Systems, 32(6), pp.4732–4742. doi:https://doi.org/10.1109/TPWRS.2017.2664141.
10. Panteli, M., Pickering, C., Wilkinson, S., Dawson, R. and Mancarella, P. (2017). Power System Resilience to Extreme Weather: Fragility Modeling, Probabilistic Impact Assessment, and Adaptation Measures. IEEE Transactions on Power Systems, 32(5), pp.3747–3757. doi:https://doi.org/10.1109/TPWRS.2016.2641463.
11. Raoufi, H. ., Vahidinasab, V. and Mehran, K. (2020). Power Systems Resilience Metrics: A Comprehensive Review of Challenges and Outlook. Sustainability, 12(22), p.9698. doi:https://doi.org/10.3390/su12229698.
12. Jimada-Ojuolape, B. and Teh, J. (2020). Surveys on the reliability impacts of power system cyber–physical layers. Sustainable Cities and Society, 62, pp.102384. doi:https://doi.org/10.1016/j.scs.2020.102384.
13. Wang, Q., Zhang, G. and Wen, F. (2021). A survey on policies, modelling and security of cyber‐physical systems in smart grids. Energy Conversion and Economics, 2(4), pp.197–211. doi:https://doi.org/10.1049/enc2.12051.
14. Zhao, Y., Yang, C., Sun, Y., Ren, H., Cheng, X. and Xie, K. (2021). Reliability Evaluation of Cyber-Physical Power Systems Considering Random Failures in Measurement and Remote Control. Electric Power Components and Systems, 49(4–5), pp. 532–546. doi:https://doi.org/10.1080/15325008.2021.1970288.
15. Aslani, M., Hashemi‐Dezaki, H. and Ketabi, A. (2022). Analytical reliability evaluation method of smart micro-grids considering the cyber failures and information transmission system faults. IET Renewable Power Generation, 16(13), pp. 2816-2839. doi:https://doi.org/10.1049/rpg2.12541.
16. Alvarez-Alvarado, M.S., Donaldson, D.L., Recalde, A.A., Noriega, H.H., Khan, Z.A., Velasquez, W. and Rodriguez-Gallegos, C.D. (2022). Power System Reliability and Maintenance Evolution: A Critical Review and Future Perspectives. IEEE Access, 10, pp.51922–51950. doi:https://doi.org/10.1109/ACCESS.2022.3172697.
17. Sathurshan, M., Saja, A., Thamboo, J., Haraguchi, M. and Navaratnam, S. (2022). Resilience of Critical Infrastructure Systems: A Systematic Literature Review of Measurement Frameworks. Infrastructures, 7(5), p.67. doi:https://doi.org/10.3390/infrastructures7050067.
18. Sujay Kaloti and Chowdhury, B. (2023). Toward Reaching a Consensus on the Concept of Power System Resilience: Definitions, Assessment Frameworks, and Metrics. IEEE Access, 11, pp.81401–81418. doi:https://doi.org/10.1109/ACCESS.2023.3300292.
19. Singh, S., Saket, R.K. and Khan, B. (2023). A comprehensive review of reliability assessment methodologies for grid-connected photovoltaic systems. IET Renewable Power Generation, 17(7), pp. 1859–1880. doi:https://doi.org/10.1049/rpg2.12714.
20. Mohanty, A., Ramasamy, A.K., Renuga Verayiah, Satabdi Bastia, Sarthak Swaroop Dash, Cuce, E., T.M. Yunus Khan and Elahi, M. (2024). Power system resilience and strategies for a sustainable infrastructure: A review. Alexandria Engineering Journal, 105, pp.261–279. doi:https://doi.org/10.1016/j.aej.2024.06.092.
21. Umunnakwe, A., Huang, H., Oikonomou, K. and Davis, K.R. (2021). Quantitative analysis of power systems resilience: Standardization, categorizations, and challenges. Renewable and Sustainable Energy Reviews, 149, p.111252. doi:https://doi.org/10.1016/j.rser.2021.111252.
22. Lu, C., Zheng, S., Dai, X. and Wu, X. (2024). Reliability Assessment of Power Distribution Cyber–Physical Systems Considering the Impact of Wireless Access Networks. Applied Sciences, 14(21), p.9974. doi:https://doi.org/10.3390/app14219974.
23. Swain, K., Cherukuri, M., Indu Sekhar Samanta, Pati, A., Giri, J., Amrutanshu Panigrahi, Qin, H. and Mallik, S. (2024). Fuzzy Markov model for the reliability analysis of hybrid microgrids. Frontiers in Computer Science, 6. doi:https://doi.org/10.3389/fcomp.2024.1406086.
24. Aslani, M., Hashemi-Dezaki, H. and Ketabi, A. (2021). Reliability Evaluation of Smart Microgrids Considering Cyber Failures and Disturbances under Various Cyber Network Topologies and Distributed Generation’s Scenarios. Sustainability, 13(10), p.5695. doi: https://doi.org/10.3390/su13105695.
25. Zhang, Z. Turnbull, B. Z., Kermanshahi, S. K., Pota, H., Damiani, E., Yeun, C. Y. and Hu, J. (2025). A survey on resilient microgrid system from cybersecurity perspective. Applied Soft Computing, [online] 175, p.113088. doi:https://doi.org/10.1016/j.asoc.2025.113088.
26. Riurean, S., Nicolae-Daniel Fîță, Dragoș Păsculescu and Răzvan Slușariuc (2025). Securing Photovoltaic Systems as Critical Infrastructure: A Multi-Layered Assessment of Risk, Safety, and Cybersecurity. Sustainability, [online] 17(10), pp.4397–4397. doi:https://doi.org/10.3390/su17104397.
27. Avizienis, A., Laprie, J.-C., Randell, B. and Landwehr, C. (2004). Basic Concepts and Taxonomy of Dependable and Secure Computing. IEEE Transactions on Dependable and Secure Computing, 1(1), pp.11–33. doi:https://doi.org/10.1109/TDSC.2004.2.
28. Trivedi, K.S. (2002). Probability and Statistics with Reliability, Queuing, and Computer Science Applications, New York: John Wiley & Sons. URL: https://www.amazon.com/Probability-Statistics-Reliability-Queueing-Applications/dp/0471333417.
29. Billinton, R. and Allan, R.N. (1996). Reliability Evaluation of Power Systems, 2nd ed.; Springer: New York. doi:https://doi.org/10.1007/978-1-4899-1860-4.
30. Rinaldi, S.M., Peerenboom, J.P. and Kelly, T.K. (2001) Identifying, Understanding, and Analyzing Critical Infrastructure Interdependencies. IEEE Control Systems Magazine, [online] 21(6), pp.11–25. doi:https://doi.org/10.1109/37.969131.
31. Panteli, M. and Mancarella, P. (2015). The Grid: Stronger, Bigger, Smarter? Presenting a Conceptual Framework of Power System Resilience. IEEE Power and Energy Magazine, 13(3), pp.58–66. doi:https://doi.org/10.1109/MPE.2015.2397334.
32. Li, W. (2005). Risk Assessment of Power Systems: Models, Methods, and Applications. Wiley-IEEE Press: Hoboken. URL: https://ieeexplore.ieee.org/book/5238312.
33. Limnios, N. and Oprişan, G. (2001). Semi-Markov Processes and Reliability. .Boston, MA: Birkhäuser: Boston. doi: https://doi.org/10.1007/978-1-4612-0161-8.
34. Hosseini, S., Barker, K. and Ramirez-Marquez, J.E. (2016). A Review of Definitions and Measures of System Resilience. Reliability Engineering & System Safety, 145, 47–61. doi:https://doi.org/10.1016/j.ress.2015.08.006.
35. Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E. and Havlin, S. (2010). Catastrophic Cascade of Failures in Interdependent Networks. Nature, 464(7291), pp.1025–1028. doi:https://doi.org/10.1038/nature08932.
36. Wang, Y., Chen, C., Wang, J. and Baldick, R. (2016). Research on Resilience of Power Systems Under Natural Disasters—A Review. IEEE Transactions on Power Systems, 31(2), pp.1604–1613. doi: https://doi.org/10.1109/TPWRS.2015.2429656.
37. Kishor Shridharbhai Trivedi and Bobbio, A. (2017). Reliability and Availability Engineering: Modeling, Analysis, and Applications. Cambridge, United Kingdom ; New York, Ny: Cambridge University Press. URL: https://www.amazon.com/Reliability-Availability-Engineering-Modeling-Applications/dp/1107099501.
38. IEEE Guide for Electric Power Distribution Reliability Indices. (2012). doi:https://doi.org/10.1109/ieeestd.2012.6209381.
39. Ouyang, M., Dueñas-Osorio, L. and Min, X. (2012). Three-Stage Resilience Analysis Framework for Urban Infrastructure Systems. Structural Safety, 36–37, pp.23–31. doi:https://doi.org/10.1016/j.strusafe.2011.12.004.
2. Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E. and Havlin, S. (2010). Catastrophic cascade of failures in interdependent networks. Nature, 464(7291), pp.1025-1028. doi:https://doi.org/10.1038/nature08932.
3. Panteli, M.; Mancarella, P. (2015). The Grid: Stronger, bigger, smarter? Presenting a conceptual framework of power system resilience. ІEEE Power and Energy Magazine, 13(3), pp.56–66. doi:https://doi.org/10.1109/MPE.2015.2397334.
4. Laprie, J.C. (1992). Dependability: Basic Concepts and Terminology. Dependability: Basic Concepts and Terminology, Springer, Vienna, 5, pp.3–245. doi:https://doi.org/10.1007/978-3-7091-9170-5_1.
5. Haimes, Y.Y. (2009). On the definition of resilience in systems. Risk Analysis, 29(4), pp.498–501. doi: https://doi.org/10.1111/j.1539-6924.2009.01216.x
6. Hosseini, S., Barker, K. and Ramirez-Marquez, J.E. (2016). A review of definitions and measures of system resilience. Reliability Engineering & System Safety, [online] 145, pp.47–61. doi:https://doi.org/10.1016/j.ress.2015.08.006.
7. Avizienis, A., Laprie, J.-C., Randell, B. and Landwehr, C. (2004). Basic Concepts and Taxonomy of Dependable and Secure Computing. IEEE Transactions on Dependable and Secure Computing, 1(1), pp.11–33. doi:https://doi.org/10.1109/TDSC.2004.2.
8. Panteli, M. and Mancarella, P. (2015). Influence of extreme weather and climate change on the resilience of power systems: Impacts and possible mitigation strategies. Electric Power Systems Research, [online] 127(2), pp.259–270. doi:https://doi.org/10.1016/j.epsr.2015.06.012.
9. Panteli, M., Mancarella, P., Trakas, D.N., Kyriakides, E. and Hatziargyriou, N.D. (2017). Metrics and Quantification of Operational and Infrastructure Resilience in Power Systems. IEEE Transactions on Power Systems, 32(6), pp.4732–4742. doi:https://doi.org/10.1109/TPWRS.2017.2664141.
10. Panteli, M., Pickering, C., Wilkinson, S., Dawson, R. and Mancarella, P. (2017). Power System Resilience to Extreme Weather: Fragility Modeling, Probabilistic Impact Assessment, and Adaptation Measures. IEEE Transactions on Power Systems, 32(5), pp.3747–3757. doi:https://doi.org/10.1109/TPWRS.2016.2641463.
11. Raoufi, H. ., Vahidinasab, V. and Mehran, K. (2020). Power Systems Resilience Metrics: A Comprehensive Review of Challenges and Outlook. Sustainability, 12(22), p.9698. doi:https://doi.org/10.3390/su12229698.
12. Jimada-Ojuolape, B. and Teh, J. (2020). Surveys on the reliability impacts of power system cyber–physical layers. Sustainable Cities and Society, 62, pp.102384. doi:https://doi.org/10.1016/j.scs.2020.102384.
13. Wang, Q., Zhang, G. and Wen, F. (2021). A survey on policies, modelling and security of cyber‐physical systems in smart grids. Energy Conversion and Economics, 2(4), pp.197–211. doi:https://doi.org/10.1049/enc2.12051.
14. Zhao, Y., Yang, C., Sun, Y., Ren, H., Cheng, X. and Xie, K. (2021). Reliability Evaluation of Cyber-Physical Power Systems Considering Random Failures in Measurement and Remote Control. Electric Power Components and Systems, 49(4–5), pp. 532–546. doi:https://doi.org/10.1080/15325008.2021.1970288.
15. Aslani, M., Hashemi‐Dezaki, H. and Ketabi, A. (2022). Analytical reliability evaluation method of smart micro-grids considering the cyber failures and information transmission system faults. IET Renewable Power Generation, 16(13), pp. 2816-2839. doi:https://doi.org/10.1049/rpg2.12541.
16. Alvarez-Alvarado, M.S., Donaldson, D.L., Recalde, A.A., Noriega, H.H., Khan, Z.A., Velasquez, W. and Rodriguez-Gallegos, C.D. (2022). Power System Reliability and Maintenance Evolution: A Critical Review and Future Perspectives. IEEE Access, 10, pp.51922–51950. doi:https://doi.org/10.1109/ACCESS.2022.3172697.
17. Sathurshan, M., Saja, A., Thamboo, J., Haraguchi, M. and Navaratnam, S. (2022). Resilience of Critical Infrastructure Systems: A Systematic Literature Review of Measurement Frameworks. Infrastructures, 7(5), p.67. doi:https://doi.org/10.3390/infrastructures7050067.
18. Sujay Kaloti and Chowdhury, B. (2023). Toward Reaching a Consensus on the Concept of Power System Resilience: Definitions, Assessment Frameworks, and Metrics. IEEE Access, 11, pp.81401–81418. doi:https://doi.org/10.1109/ACCESS.2023.3300292.
19. Singh, S., Saket, R.K. and Khan, B. (2023). A comprehensive review of reliability assessment methodologies for grid-connected photovoltaic systems. IET Renewable Power Generation, 17(7), pp. 1859–1880. doi:https://doi.org/10.1049/rpg2.12714.
20. Mohanty, A., Ramasamy, A.K., Renuga Verayiah, Satabdi Bastia, Sarthak Swaroop Dash, Cuce, E., T.M. Yunus Khan and Elahi, M. (2024). Power system resilience and strategies for a sustainable infrastructure: A review. Alexandria Engineering Journal, 105, pp.261–279. doi:https://doi.org/10.1016/j.aej.2024.06.092.
21. Umunnakwe, A., Huang, H., Oikonomou, K. and Davis, K.R. (2021). Quantitative analysis of power systems resilience: Standardization, categorizations, and challenges. Renewable and Sustainable Energy Reviews, 149, p.111252. doi:https://doi.org/10.1016/j.rser.2021.111252.
22. Lu, C., Zheng, S., Dai, X. and Wu, X. (2024). Reliability Assessment of Power Distribution Cyber–Physical Systems Considering the Impact of Wireless Access Networks. Applied Sciences, 14(21), p.9974. doi:https://doi.org/10.3390/app14219974.
23. Swain, K., Cherukuri, M., Indu Sekhar Samanta, Pati, A., Giri, J., Amrutanshu Panigrahi, Qin, H. and Mallik, S. (2024). Fuzzy Markov model for the reliability analysis of hybrid microgrids. Frontiers in Computer Science, 6. doi:https://doi.org/10.3389/fcomp.2024.1406086.
24. Aslani, M., Hashemi-Dezaki, H. and Ketabi, A. (2021). Reliability Evaluation of Smart Microgrids Considering Cyber Failures and Disturbances under Various Cyber Network Topologies and Distributed Generation’s Scenarios. Sustainability, 13(10), p.5695. doi: https://doi.org/10.3390/su13105695.
25. Zhang, Z. Turnbull, B. Z., Kermanshahi, S. K., Pota, H., Damiani, E., Yeun, C. Y. and Hu, J. (2025). A survey on resilient microgrid system from cybersecurity perspective. Applied Soft Computing, [online] 175, p.113088. doi:https://doi.org/10.1016/j.asoc.2025.113088.
26. Riurean, S., Nicolae-Daniel Fîță, Dragoș Păsculescu and Răzvan Slușariuc (2025). Securing Photovoltaic Systems as Critical Infrastructure: A Multi-Layered Assessment of Risk, Safety, and Cybersecurity. Sustainability, [online] 17(10), pp.4397–4397. doi:https://doi.org/10.3390/su17104397.
27. Avizienis, A., Laprie, J.-C., Randell, B. and Landwehr, C. (2004). Basic Concepts and Taxonomy of Dependable and Secure Computing. IEEE Transactions on Dependable and Secure Computing, 1(1), pp.11–33. doi:https://doi.org/10.1109/TDSC.2004.2.
28. Trivedi, K.S. (2002). Probability and Statistics with Reliability, Queuing, and Computer Science Applications, New York: John Wiley & Sons. URL: https://www.amazon.com/Probability-Statistics-Reliability-Queueing-Applications/dp/0471333417.
29. Billinton, R. and Allan, R.N. (1996). Reliability Evaluation of Power Systems, 2nd ed.; Springer: New York. doi:https://doi.org/10.1007/978-1-4899-1860-4.
30. Rinaldi, S.M., Peerenboom, J.P. and Kelly, T.K. (2001) Identifying, Understanding, and Analyzing Critical Infrastructure Interdependencies. IEEE Control Systems Magazine, [online] 21(6), pp.11–25. doi:https://doi.org/10.1109/37.969131.
31. Panteli, M. and Mancarella, P. (2015). The Grid: Stronger, Bigger, Smarter? Presenting a Conceptual Framework of Power System Resilience. IEEE Power and Energy Magazine, 13(3), pp.58–66. doi:https://doi.org/10.1109/MPE.2015.2397334.
32. Li, W. (2005). Risk Assessment of Power Systems: Models, Methods, and Applications. Wiley-IEEE Press: Hoboken. URL: https://ieeexplore.ieee.org/book/5238312.
33. Limnios, N. and Oprişan, G. (2001). Semi-Markov Processes and Reliability. .Boston, MA: Birkhäuser: Boston. doi: https://doi.org/10.1007/978-1-4612-0161-8.
34. Hosseini, S., Barker, K. and Ramirez-Marquez, J.E. (2016). A Review of Definitions and Measures of System Resilience. Reliability Engineering & System Safety, 145, 47–61. doi:https://doi.org/10.1016/j.ress.2015.08.006.
35. Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E. and Havlin, S. (2010). Catastrophic Cascade of Failures in Interdependent Networks. Nature, 464(7291), pp.1025–1028. doi:https://doi.org/10.1038/nature08932.
36. Wang, Y., Chen, C., Wang, J. and Baldick, R. (2016). Research on Resilience of Power Systems Under Natural Disasters—A Review. IEEE Transactions on Power Systems, 31(2), pp.1604–1613. doi: https://doi.org/10.1109/TPWRS.2015.2429656.
37. Kishor Shridharbhai Trivedi and Bobbio, A. (2017). Reliability and Availability Engineering: Modeling, Analysis, and Applications. Cambridge, United Kingdom ; New York, Ny: Cambridge University Press. URL: https://www.amazon.com/Reliability-Availability-Engineering-Modeling-Applications/dp/1107099501.
38. IEEE Guide for Electric Power Distribution Reliability Indices. (2012). doi:https://doi.org/10.1109/ieeestd.2012.6209381.
39. Ouyang, M., Dueñas-Osorio, L. and Min, X. (2012). Three-Stage Resilience Analysis Framework for Urban Infrastructure Systems. Structural Safety, 36–37, pp.23–31. doi:https://doi.org/10.1016/j.strusafe.2011.12.004.
Published
2026-04-23
Section
Electric Power Generation, Transmission and Distribution Systems
This work is licensed under a Creative Commons Attribution 4.0 International License.
