A Novel Sigmoid Fractional Binary Swarm Intelligent Solution for Optimal Reactive Power Dispatch

Reliable, economical, and secure operation of a power system depends heavily on how well its reactive power is managed. Optimal reactive power dispatch (ORPD) coordinates generator voltage magnitudes, transformer tap settings, and switched shunt compensation so that bus voltages stay within acceptable limits and transmission losses are kept low. Because these controls interact through nonlinear power-flow relations, and because tap positions and shunt banks are physically discrete while generator voltages are continuous, the problem is non-convex, multimodal, and mixed-integer. Continuous optimizers must round their solutions and therefore never search the discrete space directly, while conventional single-lag velocity updates discard the information of earlier iterations and lose swarm diversity. To address both limitations simultaneously, a sigmoid fractional binary particle swarm optimization–gravitational search algorithm (FBPSOGSA) is proposed. The velocity is updated through a truncated Grünwald–Letnikov fractional-order expansion of depth four, which retains a weighted memory of the four most recent velocities, and is mapped to a bit-switching probability through a sigmoid (S-shaped) transfer function that drives the discrete controls directly in binary space. The method is validated on the IEEE 30-bus test system for two objectives. For active power loss minimization, the FBPSOGSA attains 5.4877 MW at a fractional order of 0.2, the lowest value among all compared methods, corresponding to improvements of 19.57% over FAHLCPSO, 13.44% over OGSA, 0.96% over KHA, and 0.89% over GSA. For voltage deviation minimization, it attains 0.1059 p.u. at a fractional order of 0.1, again the lowest among all compared methods, with improvements of 15.82% over PSOGSA, 9.80% over FPSOGSA, 8.07% over FHPSO, and 64.26% over KHA. All reported control settings satisfy the system operating constraints, and the consistency of the solutions is confirmed over 25 independent runs.