Spatiotemporal Decomposition for Distributed Multi-Time-Scale Supervisory Scheduling of Cascade Pumping Stations

Economical operation of cascade pumping stations requires temporal feedback and spatial coordination, whereas a single day-ahead schedule cannot correct deviations arising during execution. This study develops a distributed multi-time-scale supervisory scheduling framework based on spatiotemporal decomposition. A convex parent model represents interval-average discharges, lumped canal-storage balances, delivery targets, tariff-based pumping cost, and station-level operating limits; within-reach hydraulic dynamics and unit-level electromechanical behavior are not resolved. Hourly day-ahead, 15 min intraday, and 5 min real-time models are linked through reference mapping, measured-state feedback, and remaining-delivery correction. The cascade is partitioned into station–canal subsystems, and the alternating direction method of multipliers coordinates local copies of each boundary flow without imposing equal adjacent-station discharges. In a deterministic five-station representative-day case, the proposed method reduced the absolute delivery error, flow-variation index, terminal canal-state deviation, and specific operating cost by 90.01%, 16.42%, 95.12%, and 3.42%, respectively, relative to decentralized local feedback without consensus exchange. Its delivered volume, specific energy, and specific operating cost differed from centralized multi-time-scale scheduling by only 0.006%, 0.006%, and 0.026%, respectively. All 313 distributed rolling problems converged, with mean intraday and real-time solution times of 6.583 s and 0.461 s. In constructed day-ahead cases containing 5–40 stations, the objective gap remained below 0.012% and the maximum interface mismatch below 1.0 × 10−4 m3 s−1. These results support the numerical applicability of the coordination method to larger serial cascades sharing the same aggregate convex structure, but do not establish dynamic-hydraulic, unit-level, or field-scale validity.