Optimal operation of water-supply systems

Abstract

The traditional water-supply planning problem is characterized by two main steps: (1) project future water requirements based on present rates of economic growth, 'and (2) schedule water development projects to be introduced into the system on time to meet these predicted requirements. If alternative projects are thought to exist, the one thought to cost the least amount is selected. As project costs rise and actual new water availabilities become less, there is a growing awareness that more new water is not necessarily the only answer. Increased efficiency in water use through conservation, reuse, transfer to less consumptive and higher valued applications, and improved management techniques are becoming practical alternatives. These alternatives lead to a need for a restatement of water-supply planning objectives in more precise forms than have heretofore been put forth. The various water- supply planning objective functions including the traditional one are all expressions which maximize the difference between gains and los se s involved with water development. They can be expressed mathematically and differentiated on the basis of how these gains and losses are defined. In the traditional sense, gains derived from meeting projected requirements are assumed to be infinite, and losses are taken to be actual project costs; therefore, maximization of net gains is accomplished by minimizing project costs and gains do not even have to be expressed. Consideration of alternatives, however, requires that gains be expressed quantitatively as benefits to individuals, communities, or regions, i. e. , primary, secondary, or tertiary benefits. The same thing holds for the expression of total costs. An objective function used to express the water-supply problem in the Tucson Basin, Arizona, considers gains as cash revenue to a hypothetical central water-control agency which sells water to the users within the basin. Losses are considered as marginal costs to the agency for producing, treating, and distributing water. The concept of economic demand is used to estimate the amount of water that municipal, agricultural, and industrial users will purchase at different prices. The possible sources of supply considered are groundwater from within the basin, groundwater from the neighboring Avra Valley Basin, reclaimed waste water, and Central Arizona Project water from the Colorado River. Constraints are formulated in order to determine optimal allocations of water under different conditions. The model used is referred to as a pricing model and is optimized by first decomposing the objective function into component parts, each part representing terms involving only one source of water. Then in instances involving inequality constraints, quadratic programming is used. In other instances where equality constraints or unconstrained conditions exist, Lagrangian multipliers and the calculus are used. The se latter conditions arise when it is determined at which point certain constraints become inactive. In the completely general case, this type of decomposition is not possible, but it appears that in many specific uses objective functions of this nature can be profitably decomposed. and optima determined much more conveniently than otherwise possible.