Max–Min Water Delivery Through Quantised Gates: Exclusion Conditions, Rounding Rules and Accounting Periods at a Canal Node

Gates move in finite steps and ignore changes below a deadband, so a gate opening from closed passes at least one quantum per step. For max–min delivery at a canal node through sealing gates that start the day closed, with exact levels and executed commands, it is proved that direct execution excludes users with plans below 1 − 1/(2k) of a quantum, k the deadband in resolution steps. Minimum deliveries cost the max–min level at most the summed quanta over total demand, and whole steps the summed step volumes over the demand of outlets still above quanta. A θ pulse rule serves every positive plan below the daily pulse capacity within −θ to +1 pulse, and carrying the debt forward bounds the cumulative error by one pulse. For ten recorded seasons at the Dustlik node, Kegeyli canal, Uzbekistan, the gates lowered the max–min level of a static constant-level allocation by under 0.001. In closed-loop Saint–Venant simulation with plan-derived demands, direct execution excluded the smallest regulator on exactly the 77 predicted early-April days of assumed demand, pulse rules served it, and carry-over cut its unweighted mean absolute dekadal error from 6.5% (pulse rule) to 1.0% of demand, shifting deliveries between days.