On Fractional Discrete-Time Power Systems: Chaos, Complexity and Control

In this paper, based on the Caputo-like delta fractional difference operator, we will present a fractional discrete model of a 4D Power System. We present an extension of the popular integer-order single-machine infinite-bus formulation to two fractional cases, one with commensurate (equal) fractional orders and another incommensurate (not equal). This extension captures long-memory effects in dynamics and thus offers a consistent mathematical description of the nonlinear behavior of power systems. The orders of the fractional models are analyzed numerically. Using time series evolution, phase-space plots, bifurcation maps, Lyapunov spectra, and the 0–1 chaos test, spectral entropy and C0 complexity metrics, we identify chaotic regimes. Additionally, techniques for controlling chaos are explored to stabilize and regulate the dynamics of the system. Both the fractional formulations exhibit richer dynamical features than their integer counterparts, and for the incommensurate case, the sensitivity to the fractional variations is larger, generating complex nonlinear oscillations. The fractional discrete power system framework provides a new perspective for studying instability, the voltage collapse phenomenon, and chaotic oscillations in power engineering applications.