Memory-Based Particle Swarm Optimization for Smart Grid Virtual Power Plant Scheduling Using Fractional Calculus

This paper presents an engineering framework for smart grid virtual power plant (VPP) day-ahead scheduling using fractional calculus-enhanced particle swarm optimization, targeting practical deployment in energy management systems. A fractional calculus-enhanced particle swarm optimization algorithm was developed and validated for day-ahead scheduling in virtual power plants using authentic market data and rigorous statistical analysis. The algorithm incorporates Grünwald–Letnikov fractional derivatives with adaptive memory into particle velocity updates, enabling trajectory-aware search that leverages historical exploration patterns. A factorial experiment across 500 independent test cases (50 dates × 10 trials) with controlled random seeds demonstrated that fractional particle swarm optimization increased mean daily profit by $205, representing a 4.1% improvement over standard particle swarm optimization. Wilcoxon signed-rank tests confirmed statistical significance (p < 0.0001, Cohen’s d = 1.08), with superior performance observed in 89.4% of cases. The factorial design identified fractional calculus as the primary performance driver, while advanced scenario generation provided no significant additional benefit. Sensitivity analysis indicated that wind generation variability was the primary predictor of performance variance, with profit difference standard deviations ranging from $34 to $325 depending on meteorological conditions, supporting the use of adaptive computational strategies. Computation required approximately two minutes per optimization on standard hardware. These findings establish fractional calculus as a credible enhancement for operational energy systems and demonstrate that the quality of optimization algorithms outweighs the complexity of forecast uncertainty modeling. The results extend fractional calculus applications from benchmark functions to practical infrastructure scheduling, with projected annual value exceeding $74,000 for a 50-megawatt system. The three-stage optimization architecture is designed for integration with standard energy management systems and SCADA platforms, offering a deployable pathway for smart grid operators.