#### Document Type

Report

#### Publication Date

January 1977

#### Abstract

A regional water quality control model is developed by linking a steady-state water quality simulation model with an optimization model. The water quality simulation model can be applied to complex river systems with both point and nonpoint loads using multiple interdependent pollution parameters described by either linear or nonlinear equations. Twelve water quality parameters can be modeled simultaneously: four non conservative constituents (or conservative constituents if the decay rate is set equal to zero); coliform bacteria (MPN); phosphorus; biochemical oxygen demand (BOD); ammonia (NH3); nitrate (NO3); dissolved oxygen (DO); temperature (°C); and algae. The water quality model is used to generate constraint equation for the optimization model. The optimization model is formulated as an integer programming problem in which the integer decision variables are wastewater treatment levels or diffuse source management practices to be determined for each load. The model considers the addition or upgrading of wastewater treatment with structural and nonstructural schemes for both point and diffuse pollution sources. A least cost solution is found subject to water quality standards at surveillance points. Additional constraints can include uniform and zoned uniform treatment. Low flow augmentation and bypass piping can be considered with slight water quality simulation model modification. A simulation model-optimization model iteration procedure is used to find an optimum solution. The regional water quality model is applied to two problems, a hypothetical problem and the Jordan River in Utah. The hypothetical consists of four pollution discharge points, at which seven possible treatment levels are available for six quality constituents: phosphorus; biochemical oxygen demands; ammonia; nitrate; dissolved oxygen; and algae. Water quality standards for three constituents are imposed at five surveillance points along the river. The portion of the Jordan River examined consists of seven pollution discharge points, at which seven treatment levels are available for the same six quality constituents as in the hypothetical problem. Water quality standards for two constituents are imposed at three surveillance points. The cost minimization problem for the Jordan River (1975 flows) required tertiary sand filters at all point loads at an increase from current costs of $1,75,881 per year to meet stream water quality standards. To assist in gaining familiarity with the linked simulation optimization model, several sensitivity studies are performed. The sensitivity of the optimal solution to two model input parameters is investigated. These parameters are the water quality equation coefficients and the water quality stream standards. Substantial reductions in treatment costs were possible by making minor changes in some of the input parameters. In the hypothetical problem, 10 percent increase in the ammonia decay rate or a 15 percent increase in the maximum specific algae growth rate would result in a 16 percent reduction in the minimum system treatment cost. A 10 percent relaxation of the stream standards at all surveillance points would result in a 54 percent reduction in the minimum system treatment costs. The optimal solution’s sensitivity to changes in headwater and point discharge flow is also investigated. The optimal treatment scheme for the projected 1995 flows in the Jordan River was the same as for the 1975 flows. The increase from current costs for the 1995 flows was $2,407,092 per year.

#### Recommended Citation

Finney, Brad A.; Grenney, William J.; Bishop, A. Bruce; and Hughes, Trevor C., "Mixed Integer Programming Models for Water Resources Management" (1977). *Reports.* Paper 492.

https://digitalcommons.usu.edu/water_rep/492