Dynamic network flows for generalized Kirchhoff law
A wide range of problems dealing with static and dynamic network flows have been considered in the literature. A discrete time dynamic flow in a static network corresponds to some static flow in a version of the network developed in time. The optimal dynamic flow problem thus reduces to a sufficiently large finite-dimensional mathematical programming problem, which linearly depends on the number of time instants. Continuous-time dynamic flows are reducible to some optimal control problem with a delayed argument. We propose simplified models without restrictions on node capacity. This leads to models with constraints on instantaneous flows. This simplification is introduced by analyzing water flow in irrigation canals. The irrigation canals are separated by partitions into sections, which are regarded as the network arcs; the partitions themselves are the network nodes. The delay time corresponds to the time to traverse a section. The capacity in this case is the capacity of the section upstream from a partition. Instantaneous discharges through a partition have previously been determined ignoring water accumulation (or water losses) in sections. This is attributable to the following reasons: (1) Each section has a sufficiently large reserve capacity, so that changes of water volume in the section on a fixed time interval can be ignored. (2) Water users announce their demand for water (i.e., determine the outflow rate) in advance and typically change it during the irrigation phase. It is therefore meaningless to try to allow with any accuracy for water accumulation in the sections, because periodic recalculations have to be made allowing for changing circumstances. (3) Direct estimates of water accumulation are obtained by periodic measurements of the water level in canal sections, and these measurement results are used to calculate the water discharges at a given instant. There is thus a feedback between the selected discharge and the main system parameters. (4) Methods for forecasting future water levels in canal sections have been developed elsewhere. To a certain extent, these methods can be applied to allow for section capacity constraints. The instantaneous flows thus can be chosen using a comparatively simple model with correction of water volumes in the sections based on observed discharges.
scientific article; zbMATH DE number 1051096