Chance-Constrained Water-Quality Management
Water-quality management is the practice of protecting the physical, chemical, and biologic characteristics of various water resources. Historically, such efforts have been guided toward the goal of assessing and controlling the impacts of human activities on the quality of water. To implement water-quality management measures in a conscientious manner, one must acknowledge both the activities of the society and the inherently random nature of the stream environment itself (Ward and Loftis, 1983). In particular, the environments in which decisions are to be made concerning in-stream water-quality management are inherently subject to many uncertainties. The stream system itself, through nature, is an environment abundant with ever-changing and complex processes, both physically and biologically.
Public Law 92-500 (PL 92-500) in the United States provided impetus for three essential tasks, one of which is to regulate waste-water discharge from point sources from industrial plants, municipal sewage treatment facilities, and livestock feedlots. It also requires treatment levels based on the best available technology. However, if a stream segment is water-quality-limited, in which the waste assimilative capacity is below the total waste discharge authorized by PL 92-500, more stringent controls may be required.
For streams under water-quality-limited conditions or where effluent standards are not implemented, the waste-load-allocation (WLA) problem is concerned with how to effectively allocate the existing assimilative capacity of the receiving water body among several waste dischargers without detrimental effects to the aquatic environment. As an integral part of water-quality management, WLA is an important but complex decision-making task. The results of WLA have profound implications on regional environmental protection. A successful WLA decision requires sound understanding of the physical, biologic, and chemical processes of the aquatic environment and good appreciation for legal, social, economical, and environmental impacts of such a decision.
Much of the research in developing predictive water-quality models has been based on a deterministic evaluation of the stream environment. Attempts to manage such an environment deterministically imply that the compliance with water-quality standards at all control points in the stream system can be ensured with absolute certainty. This, of course, is unrealistic. The existence of the uncertainties associated with stream environments should not be ignored. Thus it is more appropriate in such an environment to examine the performance of the constraints of a mathematical programming model in a probabilistic context. The random nature of the stream environment has been recognized in the WLA process. Representative WLA using a chance-constrained formulation can be found elsewhere (Lohani and Thanh, 1979; Yaron, 1979; Burn and McBean, 1985; Fujiwara et al., 1986, 1987; Ellis, 1987; Tung and Hathhorn, 1990).
In the context of stochastic management, the left-hand-side (LHS) coefficients of the water-quality constraints in a WLA model are functions of various random water-quality parameters. As a result, these LHS coefficients are random
variables as well. Furthermore, correlation exists among these LHS coefficients because (1) they are functions of the same water-quality parameters and (2) some water-quality parameters are correlated with each other. Moreover, the water-quality parameters along a stream are spatially correlated. Therefore, to reflect the reality of a stream system, a stochastic WLA model should account for the randomness of the water-quality parameters, including spatial and cross-correlations of each parameter.
The main objective of this section is to present methodologies to solve a stochastic WLA problem in a chance-constrained framework. The randomness of the water-quality parameters and their spatial and cross-correlations also are taken into account. A six-reach example is used to demonstrate these methodologies. Factors affecting the model solution to be examined are (1) the distribution of the LHS coefficients in water-quality constraints and (2) the spatial correlation of water-quality parameters.