Abstract
This paper presents a comprehensive study about the dynamic behavior of a fractional ordered three-machine infinite bus (TMIB) power system model using Grunwald–Letnikov’s method. The study investigates nonlinear behaviors including chaos, coexisting behavior and multistability behaviors, using nonlinear tools such as phase portraits, bifurcation analysis, Lyapunov exponents and Lyapunov dimensions. The results demonstrate that the TMIB system exhibits chaos behavior, which is resulting instability in rotor angle through multiscroll chaotic attractors. Furthermore, it is found that the presence of coexisting attractors and multistability leads to undesired state switching and pose a potential threat to the stability of the TMIB power system. These findings provide valuable insights into the nonlinear behavior of TMIB power system via varying fractional order range and can be used to develop effective countermeasures to address potential stability issues arise in TMIB and similar modern power systems. The simulation is conducted in MATLAB, and the obtained results illustrate the efficacy of the work.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ugalde-Loo CE, Acha E, Licéaga-Castro E (2013) Multi-machine power system state-space modelling for small-signal stability assessments. Appl Math Model 37:10141–10161. https://doi.org/10.1016/j.apm.2013.05.047
Gupta PC, Singh PP (2022) Multistability, multiscroll chaotic attractors and angle instability in multi-machine swing dynamics. IFAC-PapersOnLine 55:572–578. https://doi.org/10.1016/j.ifacol.2022.04.094
Gupta PC, Banerjee A, Singh PP (2018) Analysis of global bifurcation and chaotic oscillation in distributed generation integrated novel renewable energy system. In: 2018 15th IEEE India council international conference (INDICON). IEEE, Coimbatore, India, pp 1–5
Gupta PC, Banerjee A, Singh PP (2019) Analysis and control of chaotic oscillation in FOSMIB power system using AISMC technique. In: 2019 IEEE students conference on engineering and systems (SCES). IEEE, Allahabad, India, pp 1–6
Das P, Gupta PC, Singh PP (2021) Bifurcation, chaos and PID sliding mode control of 3-bus power system. In: 2020 3rd international conference on energy, power and environment: towards clean energy technologies. IEEE, Shillong, Meghalaya, India, pp 1–6
Gupta PC, Singh PP (2023) Multistability, coexisting behaviours and control of fractional order dissipative small scale grid with disturbances and noise. Eur Phys J Spec Top. https://doi.org/10.1140/epjs/s11734-023-00927-0
Nayfeh MA, Hamdan AMA, Nayfeh AH (1990) Chaos and instability in a power system - Primary resonant case. Nonlinear Dyn 1:313–339. https://doi.org/10.1007/BF01865278
Chen H-K, Lin T-N, Chen J-H (2005) Dynamic analysis, controlling chaos and chaotification of a SMIB power system. Chaos, Solitons Fractals 24:1307–1315. https://doi.org/10.1016/j.chaos.2004.09.081
Wang X, Chen Y, Han G, Song C (2015) Nonlinear dynamic analysis of a single-machine infinite-bus power system. Appl Math Model 39:2951–2961. https://doi.org/10.1016/j.apm.2014.11.018
Wei DQ, Luo XS (2009) Noise-induced chaos in single-machine infinite-bus power systems. EPL (Europhysics Letters) 86:50008. https://doi.org/10.1209/0295-5075/86/50008
Wei DQ, Zhang B, Qiu DY, Luo XS (2010) Effect of noise on erosion of safe basin in power system. Nonlinear Dyn 61:477–482. https://doi.org/10.1007/s11071-010-9663-0
Kopell N, Washburn R (1982) Chaotic motions in the two-degree-of-freedom swing equations. IEEE Trans Circuits Syst 29:738–746. https://doi.org/10.1109/TCS.1982.1085094
Wang X, Lu Z, Song C (2019) Chaotic threshold for a class of power system model. Shock Vibr 2019:1–7. https://doi.org/10.1155/2019/3479239
Chang S-C (2020) Stability, chaos detection, and quenching chaos in the swing equation system. Math Prob Eng 2020:1–12. https://doi.org/10.1155/2020/6677084
Zhusubaliyev ZT, Mosekilde E, Churilov AN, Medvedev A (2015) Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay. Eur Phys J Spec Top 224:1519–1539. https://doi.org/10.1140/epjst/e2015-02477-8
Zhusubaliyev ZT, Mosekilde E (2015) Multistability and hidden attractors in a multilevel DC/DC converter. Math Comput Simul 109:32–45. https://doi.org/10.1016/j.matcom.2014.08.001
Gupta PC, Singh PP (2022) Chaos, multistability and coexisting behaviors in small-scale grid: Impact of electromagnetic power, random wind energy, periodic load and additive white Gaussian noise. Pramana 97:3. https://doi.org/10.1007/s12043-022-02478-w
Hilfer R (2000) Applications of fractional calculus in physics. WORLD SCIENTIFIC
Baleanu D (ed) (2017) Fractional calculus: models and numerical methods, 2nd edn. World Scientific, New Jersey
Debbouche N, Ouannas A, Momani S, Cafagna D, Pham V-T (2022) Fractional-order biological system: chaos, multistability and coexisting attractors. Eur Phys J Spec Top 231:1061–1070. https://doi.org/10.1140/epjs/s11734-021-00308-5
Ma C, Jun M, Cao Y, Liu T, Wang J (2020) Multistability analysis of a conformable fractional-order chaotic system. Physica Scripta 95:075204. https://doi.org/10.1088/1402-4896/ab8d54
Tolba MF, AbdelAty AM, Soliman NS, Said LA, Madian AH, Azar AT, Radwan AG (2017) FPGA implementation of two fractional order chaotic systems. AEU—Int J Electron Commun 78:162–172. https://doi.org/10.1016/j.aeue.2017.04.028
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Gupta, P.C., Singh, P.P. (2024). Chaos and Multistability in Fractional Order Power System: Dynamic Analysis and Implications. In: Shaw, R.N., Siano, P., Makhilef, S., Ghosh, A., Shimi, S.L. (eds) Innovations in Electrical and Electronic Engineering. ICEEE 2023. Lecture Notes in Electrical Engineering, vol 1109. Springer, Singapore. https://doi.org/10.1007/978-981-99-8289-9_4
Download citation
DOI: https://doi.org/10.1007/978-981-99-8289-9_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-99-8288-2
Online ISBN: 978-981-99-8289-9
eBook Packages: EnergyEnergy (R0)