article · Mathematics
This research examines the nonlinear chaotic behaviour of electrical power systems through an incommensurate fractional-order mathematical framework. System equilibrium points are established and evaluated using local stability methods tailored for fractional-order dynamics. Through bifurcation analysis, the study traces how adjustments to operational parameters and fractional orders trigger transitions from orderly behaviour into period-doubling cascades and full chaos. The work also explores multistability, revealing scenarios where multiple stable or chaotic attractors coexist simultaneously under identical operating conditions. Computational simulations implemented in numerical software verify these phenomena, using diagnostic tools such as time series, bifurcation diagrams, Lyapunov exponents, and two- and three-dimensional phase portraits to characterise the complex dynamic states of the system.
Electrical power grids must operate reliably to avoid catastrophic supply failures and blackouts. By revealing how complex mathematical conditions can suddenly lead to chaotic behaviour and multiple coexisting operating states, this work helps improve theoretical understanding of electrical instability. Such insights are essential for designing better control mechanisms to prevent unexpected grid disturbances.
This work represents early-stage theoretical and simulation-based research. While it could eventually assist power system software developers and grid operators in designing stability control algorithms, the abstract provides no evidence of experimental testing, physical prototypes, or direct commercial application pathways.
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This paper presents the nonlinear chaotic dynamics of a power system model within an incommensurate fractional-order framework. Equilibrium points are derived and analyzed using Jacobian-based local stability theory adapted to fractional-order systems. Furthermore, bifurcation analysis is employed to examine how variations in system parameters and incommensurate fractional orders influence the emergence of period-doubling cascades and chaotic motion. The study investigates multistability phenomena characterized by the coexistence of multiple attractors under identical system parameters. The simulation is run in MATLAB R2020a, and nonlinear tools such as time series, bifurcation diagrams, Lyapunov exponents, and phase portraits in 2D and 3D projections are used to visualize the findings.
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DOI: 10.3390/math14173164
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