Analysis of Structural Reliability of Complex Coefficients Fractional-Order ‎System Using Plane Transformation

Document Type : Research paper


Department of Electrical Engineering, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat, India


The main goal of the paper is to achieve the structural reliability of the failure components in the system that can be modelled as a Transfer Function (TF). The classical reliability of the power system has been a major field for research in the past decades, which has resulted in the reliability of the power grid by integrating the failure rates of the system components. As a result, a gap analysis is carried out by modelling the failure components into TF, and a comparison of structural and classical reliability is explained in the paper. The paper expands on methodology of the mapping technique for transforming a system from one domain to another. By doing so, the transformation of the Complex Coefficients Integer Order (CCIO) and the Complex Coefficients Fractional Order (CCFO) system transfer function becomes the Non-complex Coefficients Integer Order (NCCIO) in nature. Therefore, the root locus plot for the transformed system is observed as the symmetrical structure about the real axis. Therefore, the plane transformation becomes advantageous in the field of structural reliability analysis. The root locus plot for the transformed system into a w-plane becomes reliable as per the symmetrical structure. The reliability index Loss of Load Probability (LOLE) has been evaluated with different forced outage rates of the system components to analysis the classical reliability of the system.


  1.   M. Yang, D. Zhang, and X. Han, “New efficient and robust method for structural reliability analysis and its application in reliability-based design optimization”, Comput. Meth. Appl. Mech. Eng., vol. 366, pp. 113018, 2020.
  2.    M. Tavazoei, and M. Haeri, “A note on the stability of fractional order systems”, Math. Comput. Sim., vol. 79, no. 5, pp. 1566-1576, 2009.
  3.   J. Adams, T. Hartley, and C. Lorenzo, “Fractional-order system identification using complex order-distributions”, IFAC Proc. Vol., vol. 39, no. 11, pp. 200-205, 2006.
  4.   B. Mohammadzadeh, A. Safari, and S. Najafi Ravadanegh, “Reliability and supply security based method for simultaneous placement of sectionalizer switch and DER units”, J. Oper. Autom. Power Eng., vol. 4, no. 2, pp. 165-174, 2016.
  5.   E. Babaei, and N. Ghorbani, “Combined economic dispatch and reliability in power system by using PSO-SIF algorithm”, J. Oper. Autom. Power Eng., vol. 3, no. 1, pp. 23-33, 2015.
  6.   L. Ge, “Application study of complex control algorithm for regenerative furnace temperature”, J. Control Theory Appl., vol. 2, no. 2, pp. 205-207, 2004.
  7.    M. Abolvafaei, and S. Ganjefar, “Maximum power extraction from fractional order doubly fed induction generator based wind turbines using homotopy singular perturbation method”, Int J. Electr. Power Energy Syst., vol. 119, pp. 105889, 2020.
  8.    A. Cortez et al., “Fractional order controllers for irrational systems”, IET Control Theory Appl., vol. 15, no. 7, pp. 965-977, 2021.
  9.    H. Erol, “Delay margin computation in micro grid systems with time delay by using fractional order controller”, Electr. Power Comp. Syst., pp. 1-12, 2021.
  10.  F. Babaei, A. Safari, and J. Salehi, “Evaluation of delays-based stability of LFC systems in the presence of electric vehicles aggregatore”, J. Oper. Autom. Power Eng., vol. 10, no. 2, pp. 165-174, 2022.
  11.  M. Tavazoei, “Passively realizable approximations of non-realizable fractional order impedance functions”, J. Franklin Ins., vol. 357, no. 11, pp. 7037-53, 2020.
  12. A. Bagheri et al., “A practical approach for coordinated transmission switching and OLTC's tap adjustment: DigSILENT-base improved PSO algorithm”, J. Oper. Autom. Power Eng., vol. 9, no. 2, pp. 103-115, 2021.
  13. T. Aziz et al., “Review on optimization methodologies in transmission network reconfiguration of power systems for grid resilience”, Int. Trans. Electr. Energy Syst., vol. 31, no. 3, pp. e12704, 2021.
  14.  A. Khorshidi, T. Niknam, and B. Bahmani, “Synchronization of microgrid considering the dynamics of V2Gs using an optimized fractional order controller based scheme”, J. Oper. Autom. Power Eng., vol. 9, no. 1, pp. 11-22, 2021.
  15.  B. Li et al., “A hybrid approach for transmission grid resilience assessment using reliability metrics and power system local network topology”, Sustain. Resilient Infrastruct., vol. 6, no. 1-2, pp. 26-41, 2021.
  16.  M. Zuo, “System reliability and system resilience”, Frontiers  Eng. Manage., pp. 1-5, 2021.
  17. W. Rui et al., “Reduced-order transfer function model of the droop-controlled inverter via Jordan continued-fraction expansion”, IEEE Trans. Energy Conv., vol. 35, no. 3, pp. 1585-1595, 2020.
  18.  B. Vinagre et al., “Some approximations of fractional order operators used in control theory and applications”, Fractional Calculus Appl. Analysis, vol. 3, no. 3, pp. 231-248, 2000.
  19. F. Yang et al., “Characteristic analysis of the fractional-order hyperchaotic complex system and its image encryption application”, Signal Proc., vol. 169, pp. 107373, 2020.
  20. N. Zendehdel, “Robust agent based distribution system restoration with uncertainty in loads in smart grids”, J. Oper. Autom. Power Eng., vol. 3, no. 1, pp. 1-22, 2015.
  21. Z. Moravej, and S. Bagheri, “Condition monitoring techniques of power transformers: A review”, J. Oper. Autom. Power Eng., vol. 3, no. 1, pp. 71-82, 2015.
  22. P. Salyani, and J. Salehi, “A customer oriented approach for distribution system reliability improvement using optimal distributed generation and switch placement”, J. Oper. Autom. Power Eng., vol. 7, pp. 246-60, 2019.
  23. X. Liu et al., “A resilience assessment approach for power system from perspectives of system and component levels”, Int. J. Electr. Power Energy Syst., vol. 118, pp. 105837, 2020.
  24. L. Chen et al., “Variable coefficient fractional-order PID controller and its application to a SEPIC device”, IET Control Theory Appl., vol. 14, no. 6, pp. 900-908, 2020.
  25. S. Zhao, J. Chang, and R. Hao, “Reliability assessment of the Cayley graph generated by trees”, Discrete Appl. Math., vol. 287, pp. 10-14, 2020.
  26. Z. Lu et al., “Finite-time non-fragile filtering for nonlinear networked control systems via a mixed time/event-triggered transmission mechanism”, Control Theory Tech., vol. 18, no. 2, pp. 168-181, 2020.
  27. M. Gangnet, D. Perny, and P. Coueignoux, “Perspective mapping of planar textures”, Comput. Graph., vol. 8, no. 2, pp. 115-123, 1984.
  28. S. Mei, W. Wei, and F. Liu, “On engineering game theory with its application in power systems”, Control Theory Tech., vol. 15, no. 1, pp. 1-12, 2017.
  29. M. Olson, and M. Hill, “Two-dimensional mapping of in-plane residual stress with slitting”, Experimental Mech., vol. 58, no. 1, pp. 151-166, 2018.
  30. K. Hou et al., “Cooperative control and communication of intelligent swarms: a survey”, Control Theory Tech., vol. 18, no. 2, pp. 114-134, 2020.
  31. S. Abbasi, and H. Abdi, “Return on investment in transmission network expansion planning considering wind generation uncertainties applying non-dominated sorting genetic algorithm”, J. Operation Autom. Power Eng., vol. 6, no. 1, pp. 89-100, 2018.