Gate-Recycling Fractional-Order Grey Forecasting of Water-Sector Energy Consumption

The limited length and structural evolution of time series for water-sector energy consumption constrain reliable multi-step forecasting. To address this problem, this research proposed a gate-recycling fractional-order grey model named GRGM(r,k,1), based on a finite-memory accumulation operator. The parameter r governed geometric attenuation within the active gate, whereas the integer gate length k determined the range of direct historical participation. Their separation decoupled attenuation intensity from memory span. The operator admitted exact recursive inversion. Theoretical analysis established the nonsingularity of the accumulation operator, derived operator-norm and condition-number bounds, and demonstrated finite direct perturbation support and relevant boundary relations. Empirical validation combined fixed-origin, rolling-origin, and external tests against four benchmark models. For Chinese water production and supply energy consumption, GRGM(r,k,1) achieved test MAPE and sMAPE values of 0.99% and 1.00%, respectively. Its test MAPE values for the desalination and wastewater-treatment datasets are 0.68% and 2.60%. The forecasts further indicate sustained consumption growth with a declining annual growth rate. The results suggest that gate recycling provided an interpretable fractional-order mechanism for short-sample grey forecasting by jointly regulating information attenuation and finite memory.