Mathematical methods for use in planning regional wastewater treatment systems

A mathematical method presented here deals with regionalization of wastewater systems, a complex public sector planning problem. The method proposed focuses on generating alternative physical plans efficiently and systematically so that planning issues other than economic efficiency may be meaningfully integrated into the process of comparing alternative plans. Such a method, although simple in concept, can aid analysts in developing insights. Two types of alternative plans can be generated by the method, single-time period plans and simplified multiperiod plans. In generating alternative plans, the method takes advantage of the structure of a branch-and-bound algorithm. A branch-and-bound tree may be transformed into a matrix called the imputed value incidence matrix which displays the incidence relationship between each of the alternative plans and the state of variables (regional facilities) associated with it. Once the matrix is constructed the imputed value of a given variable or a given set of variables can be obtained from the matrix. An application of the method to a realistic example problem is presented and the interpretation of imputed values is discussed.U.S. Department of the InteriorU.S. Geological SurveyOpe

Japanese regional wastewater treatment systems

The Japanese government is undertaking a major program to provide additional sewage collection and treatment because of the severe water pollution that has resulted from rapid urbanization and industrialization. The program is characterized by strong regional planning and by the utilization of central treatment plants serving many communities and, in some cases, industries. Through a study carried out in Japan many important planning issues were identified, including: (1) cultural and historical factors; (2) cost effectiveness, including economies of scale, treatment effectiveness, and short-run and long-run flexibility; (3) potential interrelationships with drainage, water supply, and land use; (4) interactions between planning agencies and local citizens; and (5) institutional arrangements between governmental bodies. The Japanese experience is a significant one in the history of planning wastewater treatment systems and provides some guidance as similar programs are initiated in other developed and developing countries throughout the world. In general, planning such systems represents a complex public-sector problem and calls for an interdisciplinary approach.U.S. Department of the InteriorU.S. Geological SurveyOpe

Mathematical models for use in planning regional water resources and energy systems

Existing and projected energy facilities will, in the near future, place major demands on the country's water resources. These demands compete with many other uses of the resources, including municipal and industrial uses, navigation, irrigation, and water quality maintenance. The possible development of coal conversion facilities presents another potential water demand. Complex public sector problems such as: 1) the extent and development of coal conversion capacity, 2) interbasin transfer of water, 3) cooling technologies for large energy facilities, 4) diversion of Lake Michigan water, and 5) allowable withdrawal and consumptive uses of river water, all arise from the interlocking nature of the water resources-energy system. Although mathematical models cannot solve these problems directly, they can be useful in gaining insight into major issues associated with policy alternatives. With the aid of such models, quantitative trends such as costs and water development patterns associated with each decision alternative can be more readily identified. In this report, mathematical models are presented for use in planning a regional allocation of water for energy facilities as well as for other water uses. These models include components for the interrelated water and energy subsystems. The use of these models in conjunction with other existing models in order to provide a better picture of the overall system is discussed. Since the models use widely available computer codes, they are practical and easy to utilize. Example applications are presented, with a discussion of computational results.U.S. Geological SurveyU.S. Department of the InteriorOpe

Nonlinear programming model of a wastewater treatment system: Sensitivity analysis and a robustness constraint

A method for sensitivity analysis in nonlinear programming is described and then illustrated using a least-cost model of a secondary wastewater treatment system. A sensitivity equation approach is used to calculate normalized sensitivity coefficients, which approximate the percent changes in model variables and objective function due to a small parameter variation. Design changes predicted by the sensitivity coefficients are confirmed by a perturbation analysis of the optimal solution. Sensitivity concepts are used to develop a robustness measure which is incorporated into the constraint set of the nonlinear model. Robustness is narrowly defined as the ability of a model solution to maintain a level of performance that meets the system design criteria even if the actual values of model parameters are not exactly the same as the values assumed for design. A gradient optimization procedure is used to examine the tradeoff between total cost and the robustness measure. A preliminary analysis shows that the trends in robust wastewater treatment plant design are in direct conflict with the optimal decisions obtained when minimizing cost without a constraint on robustness but are in agreement with those designs observed to work in practice. The robustness constraint method presented should be applicable to other optimization models of water resources systems.U.S. Department of the InteriorU.S. Geological SurveyOpe

Methods for generating alternative solutions to water resources planning problems

An optimization model is generally not a perfect representation of a complex water resources planning problem because not all important issues can be captured in a model. Optimization models can be used in a planning process to generate planning alternatives that are good and different so that the analyst and the decision maker can examine a wide range of alternatives t.0 gain insight and understanding. This approach is called modeling to generate alternatives (MGA). Several new MGA methods, a random method, a branch and bound/screening (BBS) method and a Fuzzy BSJ method, are described. This work also provides an assessment of the potential use of these methods as well as the HSJ method for generating good and different alternative solutions; the methods are illustrated using a wastewater treatment system planning problem, which is formulated as a mixed integer programming (MIP) model.U.S. Geological SurveyU.S. Department of the InteriorOpe

Incorporating a rule-based model of judgment into a wastewater treatment plant design optimization model

The use of a rule-based modeling technique for the formal consideration of poorly modeled issues in a water quality management problem is illustrated in the context of wastewater treatment plant design. Sludge bulking is a poorly understood problem in activated sludge wastewater treatment plants. An engineer must use judgement gained from experience when he designs an activated sludge plant to prevent bulking from causing the plant to fail. An attempt was made to use fuzzy logic in order to model that judgement. Results from research were taken from the literature and used independently as constraints to an activated sludge wastewater plant design optimization model to see their effect on the optimal design. Some of the research results were then formulated as rules in a rule-based system which relates design variable values to the likelihood of a design experiencing bulking problems. The weights of association of those rules to the conclusion that a given design would experience bulking problems and the logical interaction of those rules were calibrated using an experienced engineer's evaluation of a set of 15 plant designs. The consistency of the engineer's and the judgement model's evaluations were then checked with a second set of 15 designs. The model of judgement could be used to evaluate the bulking potential of any design. In the particular example developed, the judgement model was incorporated into a wastewater treatment plant design optimization model so that the cost effectiveness of constraint combinations could be examined. The tradeoff between cost and the likelihood of experiencing bulking problems was examined for a typical plant design problem.U.S. Department of the InteriorU.S. Geological SurveyOpe

Generating alternative solutions for dynamic programming models of water resources problems

A technique is presented to generate alternatives that are different from each other, but good with respect to modeled objectives, for problems that can be modeled by dynamic programming. The technique is compared to other possible approaches, and relevant concepts of difference among alternatives are discussed. Application to a floodplain management model shows that the technique can produce sets of different alternatives for water resources problems.U.S. Geological SurveyU.S. Department of the InteriorOpe

Reliability of reservoir operation under hydrologic uncertainty

Sensitivity analyses were performed to examine typical stochastic programming (SP) modeling issues for a hypothetical single reservoir system. The elements considered include the partitions of inflow and storage states, the hydrologic characteristics of inflows, the types of system performance functions, and the tradeoffs between conflicting objectives. Simulation studies were conducted to verify the modeling outcomes and to provide insights for possible improvements of the system performance. Results from these analyses show that (1) both the numbers and the discrete increment values of the inflow and storage states affect an SP model's accuracy; (2) the uncertainty associated with the coefficients of variation of the inflows consistently has a greater impact on the system performance than the influence of the serial correlations; (3) in a sample study with flood control being the only objective, the use of either a convex function or a concave function alone for flood damages will not lead to an optimal operation policy which always prevents excessive flood release when there exists some unused storage space in the reservoir; (4) the preferences between the conflicting objectives have been shown to affect both the expected system performance and the individual operation decisions; and (5) modification of the discrete optimal solution, using a simple interpolation scheme, may improve the reservoir performance without resorting to a more complex model. A case study of Lake Shelbyville, Illinois was conducted based on the findings of sensitivity analyses for the hypothetical reservoir system using SP. An -- ad hoc approach was used to estimate accurately the agricultural and property damages in the optimization procedure. The optimal pool levels of Lake Shelbyville in the summer months were found to be roughly 2 to 5 ft lower than the current target level which is 599.7 ft. When the summer pool was forced to reach this target level using a penalty function approach in the SP model, the annual expected damages would increase by 9%. Generally, it would take more than one month for Lake Shelbyville to resume the summer pool from the winter drawdown level. Therefore, a transition period longer than one month between the winter drawdown and the summer recovery of lake levels is recommended for consideration if future modification is made in the rule curve.U.S. Geological SurveyU.S. Department of the InteriorOpe

Flood plain management through allocation of land uses–a dynamic programming model

Despite heroic structural measures, flood damages continue to rise. This research develops a means for identifying more nearly optimal patterns of land use with particular reference to timing, depth, and duration of flooding. The major premise is that flood plain management is best viewed as a problem of allocating land uses to land parcels. A dynamic programming model is developed to determine what combination of downstream uses, which require flood protection, and upstream uses, which may increase runoff or provide protection through longer water retention, should be encouraged. The dynamic programming model and an associated simplified routing technique are demonstrated on a real watershed. Desirable extensions of the model are identified. One major result of the project is the realization of a need to classify watersheds by the degree of effective interdependence among land use decisions so as to determine the most appropriate types of analytical model s and public sector interventions for particular cases. Thinking about flood management as a problem of land use allocation is shown to be a fruitful conceptualization for exploring the issues, for developing models, and for identifying appropriate public sector interventions.U.S. Geological SurveyU.S. Department of the InteriorOpe

Fate of Kaluza-Klein Bubble

We numerically study classical time evolutions of Kaluza-Klein bubble space-time which has negative energy after a decay of vacuum. As the zero energy Witten's bubble space-time, where the bubble expands infinitely, the subsequent evolutions of Brill and Horowitz's momentarily static initial data show that the bubble will expand in terms of the area. At first glance, this result may support Corley and Jacobson's conjecture that the bubble will expand forever as well as the Witten's bubble. The irregular signatures, however, can be seen in the behavior of the lapse function in the maximal slicing gauge and the divergence of the Kretchman invariant. Since there is no appearance of the apparent horizon, we suspect an appearance of a naked singularity as the final fate of this space-time.Comment: 13 pages including 10 figures, RevTeX, epsf.sty. CGPG-99/12-8, RESCEU-6/00 and DAMTP-2000-30. To appear in Phys. Rev.