COST EFFECTIVE OPERATION OF URBAN WATER SUPPLY SYSTEM USING DYNAMIC PROGRAMMING1

ABSTRACT: Efficient operation of a city water supply system is an important goal of all municipalities. Efficient operation should result in minimum operation cost through reduction in total energy use and/ or reduction in on-peak energy consumption. An optimization model was designed for operating the water supply systems of cities using groundwater. The Newton-Raphson pipe network was used for network analysis and a dynamic programming optimization algorithm was used for determining a schedule for pump operation in the pipe network system. The model is most suitable for use in small cities with up to 45,000 in population, but with large-scale disintegration techniques may also be used for larger cities. The savings in operation costs are a function of energy cost and energy use pattern and water use pattern in the area.     

COMPARISON OF RESERVOIR LINEAR OPERATION RULES USING LINEAR AND DYNAMIC PROGRAMMING1

ABSTRACT: Mathematical optimization techniques are used to study the operation and design of a single, multi-purpose reservoir system. Optimal monthly release policies are derived for Hoover Reservoir, located in Central Ohio, using chance-constrained linear programming and dynamic programming-regression methodologies. Important characteristics of the former approach are derived, discussed, and graphically illustrated using Hoover Reservoir as a case example. Simulation procedures are used to examine and compare the overall performance of the optimal monthly reservoir release policies derived under the two approaches. Results indicate that, for the mean detention time and the corresponding safe yield target water supply release under existing design of Hoover Reservoir, the dynamic programming policies produce lower average annual losses (as defined by a two-sided quadratic loss function) while achieving at least as high reliability levels when compared to policies derived under the chance-constrained linear programming method. In making this comparison, the reservoir release policies, although not identical, are assumed to be linear. This restricted form of the release policy is necessary to make the chance-constrained programming method mathematically tractable.     

COMPARISON OF STOCHASTIC AND DETERMINISTIC DYNAMIC PROGRAMMING FOR RESERVOIR OPERATING RULE GENERATION1

ABSTRACT: Two dynamic programming models — one deterministic and one stochastic — that may be used to generate reservoir operating rules are compared. The deterministic model (DPR) consists of an algorithm that cycles through three components: a dynamic program, a regression analysis, and a simulation. In this model, the correlation between the general operating rules, defined by the regression analysis and evaluated in the simulation, and the optimal deterministic operation defined by the dynamic program is increased through an iterative process. The stochastic dynamic program (SDP) describes streamflows with a discrete lag-one Markov process. To test the usefulness of both models in generating reservoir operating rules, real-time reservoir operation simulation models are constructed for three hydrologically different sites. The rules generated by DPR and SDP are then applied in the operation simulation model and their performance is evaluated. For the test cases, the DPR generated rules are more effective in the operation of medium to very large reservoirs and the SDP generated rules are more effective for the operation of small reservoirs.     

STOCHASTIC DYNAMIC PROGRAMMING FOR OPTIMUM RESERVOIR OPERATION1

ABSTRACT. For a multipurpose single reservoir a deterministic optimal operating policy can be readily devised by the dynamic programming method. However, this method can only be applied to sets of deterministic stream flows as might be used repetitively in a Monte Carlo study or possibly in a historical study. This paper reports a study in which an optimal operating policy for a multipurpose reservoir was determined, where the optimal operating policy is stated in terms of the state of the reservoir indicated by the storage volume and the river flow in the preceding month and uses a stochastic dynamic programming approach. Such a policy could be implemented in real time operation on a monthly basis or it could be used in a design study. As contrasted with deterministic dynamic programming, this method avoids the artificiality of using a single set of stream flows. The data for this study are the conditional probabilities of the stream flow in successive months, the physical features of the reservoir in question, and the return functions and constraints under which the system operates.     

A STOCHASTIC DYNAMIC PROGRAMMING BASED APPROACH TO THE OPERATION OF A MULTI-RESERVOIR SYSTEM1

ABSTRACT: An heuristic iterative technique based upon stochastic dynamic programming is presented for the analysis of the operation of a three reservoir ‘Y’ shaped hydroelectric system. The technique is initiated using historical inflow data for the downstream reservoir. At each iteration the optimal policies for the downstream hydroelectric generating unit are used to provide relative weightings or targets for operation of upstream reservoirs. New input inflows to the downstream reservoir are then obtained by running the historical streamflow record through the optimal policies for the upstream reservoirs. These flows are then used to develop a new operating policy for the downstream reservoir and hence new targets for the upstream reservoirs. The process is continued until the operating policies for each reservoir provide the same overall system benefit for two successive iterations. Results obtained from the procedure are compared to the results obtained by historical operation of the system. The procedure is shown to develop operating policies which give benefits which are as close to the historical benefits as can be expected given the choice of the number of storage state variables.     

THE OPTIMALITY OF CAPACITY SHARING IN STOCHASTIC DYNAMIC PROGRAMMING PROBLEMS OF SHARED RESERVOIR OPERATION1

ABSTRACT: This paper is concerned with finding an optimal allocation of water entitlements for each of two users of water who share a reservoir. Two instruments of allocation are considered. The first, release sharing, involves sharing the releases from the reservoir; the second, capacity sharing, is concerned with allocating to each user of water a share of inflows, reservoir capacity and leakage and evaporation losses. Stochastic dynamic programming problems of reservoir operation under each type of sharing arrangement are formulated. It is shown that the maximum discounted expected profit from reservoir operation over the life of the storage using capacity sharing is at least as large as that obtained using release sharing and that release sharing is not Pareto efficient.     

COST MINIMIZATION OF NUTRIENT REDUCTION IN WATERSHED MANAGEMENT USING LINEAR PROGRAMMING1

ABSTRACT: Linear programming is applied to identify the least cost strategy for reaching politically specified phosphorus and total suspended solids reduction targets for the Fox-Wolf river basin in Northeast Wisconsin. The programming model uses data collected on annualized unit reduction costs associated with five categories of sources of phosphorus and total suspended solids discharge in each of the 41 sub-watersheds in the basin to determine the least cost management strategy. Results indicate that: (1) cost-effective nutrient reduction requires careful selection of geographic areas and source categories to address throughout the watershed; (2) agricultural sources are the most cost-effective to address in the basin; and (3) care should be exercised in setting nutrient reduction targets, given that there are likely to be significantly increasing marginal costs of nutrient reduction; the model predicts that lowering the most restrictive target by 33 percent would cut reduction expenditures by about 75 percent. Policy implications of the model include support for the investigation and potential development of institutional arrangements that enable cost-effective nutrient reduction activities to occur, such as the creation of an agency with authority over a given watershed, coordinated watershed management activities, or nutrient trading programs.     

IRRIGATION WATER DELIVERY SYSTEM OPERATION VIA AGGREGATE STATE DYNAMIC PROGRAMMING1

ABSTRACT: A modified dynamic programming (DP) approach that is called aggregate state dynamic programming (ASDP) is presented to optimally operate irrigation water delivery systems. ASDP can be applied to multiple reservoir systems without encountering dimensionality problems. In addition, the random nature of water supply and consumptive crop demands can be incorporated into the technique. A case study is presented to display the application of ASDP. Using a sum-of-squared shortages objective, ASDP out performs a traditional separation technique and approaches the theoretical (ideal) optimum. Problem settings that are conducive to the use of ASDP and limitations of the technique are presented.     

SOLID SET IRRIGATION SYSTEM DESIGN USING LINEAR PROGRAMMING1

: Irrigation systems can be formulated to enable linear programming to be used to obtain optimal systems. Linear programming is a readily available technique in most scientific and technical centers which have computer facilities. The objective function towards a minimal cost is based on a suggested classification of the system components, and the constraints mainly refer to required pressures and discharges. A case study in which linear programming is used to determine optimal pipe diameters is given for the solution of a solid set under tree sprinkler system.     

CONSOLIDATION OF A STREAM QUALITY MONITORING NETWORK1

ABSTRACT A method for systematic consolidation of a fixed station water quality monitoring network using dynamic programming is described. The approach utilizes a hierarchical structure; stations are allocated to what are termed primary basins on the basis of a weighted attribute score, and specific station locations within each primary basin are specified using a criterion based on stream order numbers. The method has been applied to the Municipality of Metropolitan Seattle (Metro) stream and river quality monitoring network. The results aided in a 1982 metro decision to reduce the scope of its fixed station monitoring from 81 to 47 stations, at an annual savings of approximately $33,000 per year exclusive of equipment depreciation and indirect costs.     

EVALUTING STREAM STANDARD VIOLATIONS USING A WATER QUALITY DATA BASE1

ABSTRACT: Cumulative density functions (c.d.f.'s) for water quality random variables may be estimated using data from a routine grab sampling program. The c.d.f. may then be used to estimate the probability that a single grab sample will violate a given stream standard and to determine the anticipated number of violations in a given number of samples. Confidence limits about a particular point on the c.d.f. may be used to reflect the accuracy with which the sample estimate represents the true c.d.f. Methods are presented here for calculating such confidence limits using both a normal model and a nonparametric model. Examples are presented to illustrate the usefulness of an estimated c.d.f. and associated confidence limits in assessing whether an observed number of standard violations is the result of natural variability or represents real degradation in water quality.     

PRACTICAL IMPLICATIONS IN THE USE OF STOCHASTIC DYNAMIC PROGRAMMING FOR RESERVOIR OPERATION1

ABSTRACT: A stochastic dynamic programming model is applied to a small hydroelectric system. The variation in number of stage iterations and the computer time required to reach steady state conditions with changes in the number of storage states is investigated. The increase in computer time required to develop the storage probability distributions with increase in the number of storage states is reviewed. It is found that for an average of seven inflow states, the largest number of storage states for which it is computationally feasible to develop the storage probability distributions is nine. It is shown that use of the dynamic program results based on a small number of storage states results in unrealistically skewed storage probability distributions. These skewed distributions are attributed to “trapping” states at the low end of the storage range.     

THE EFFECTS OF ARTIFICIAL CIRCULATION ON A SMALL NORTHEASTERN IMPOUNDMENT1

ABSTRACT: Artificial circulation was applied at East Sidney Lake, a small, eutrophic impoundment in New York, to improve the water quality of the reservoir and tailwater. Treatment was successful at both reducing the stability of the reservoir and maintaining higher mean oxygen concentrations in the bottom waters. Discharge waters had lower metals and phosphorus concentrations during treatment years. However, vertical temperature differences, although minimal, were still sufficient to permit chemical stratification and some phosphorus release from the sediments. Frequent mixing events during periods of low stability, and runoff from storm events, also appeared to increase transport of phosphorus into the epilimnion. Overall, treatment did not result in decreases in algal populations or improvements in water clarity.     

AN EVALUATION OF EIGHT WASTELOAD ALLOCATION METHODS1

Evidence suggests that there is no superior wasteload allocation method. Eight allocation strategies have been evaluated based on: total cost of implementation, equity in distributing costs and loads among dischargers, effectiveness in use of available waste assimilative capacity, and sensitivity to changes in water-quality-related variables. One method, which allocated equal percentages of the maximum allowable dissolved oxygen deficit, was eliminated as a feasible strategy because it led to excessive costs and overly conservative load estimates. The other seven methods remained viable alternatives. Two methods proved to be insensitive to changes in the water-quality-related variables tested, which may be advantageous in certain applications. This report presents seven workable alternatives that may be used in wasteload allocation and demonstrates a procedure to determine the practicability of other methods.     

THE EFFECTS OF ARTIFICIAL CIRCULATION ON A HYPEREUTROPHIC LAKE1

ABSTRACT: Crystal Lake, a small urban lake in Robbinsdale, MN, had been artificially circulated for 12 years before a detailed water quality evaluation was undertaken. In 1986, the circulation system was shut off for a two-year assessment. Although the lake remained hypereutrophic, the use of the lake, which included shoreline fishing and feeding ducks and geese, did not appear to be seriously impaired by the absence of artificial circulation. The circulation system was returned to service in October 1987 (there are 16 diffusers in this 0.31 km^?2 lake). Continuous limnological data from October 1987 through October 1988, plus several sampling dates in 1989, compares to the two non-circulation years (1986 and 1987) as follows: there was a two- to three-fold increase in the lake's concentration of total phosphorus, total Kjeldahl nitrogen, and chlorophyll and a similar decrease in Secchi disk transparency. The surface oxygen concentration was reduced and the deep waters were nearly anoxic. In fact, following a wind storm in 1988, the entire lake became anoxic due to the mixing of high BOD throughout the water column, and a summertime fish-kill resulted. All of these occurrences are related to the artificial circulation of the lake.     

LINEAR PROGRAMMING USE FOR EVALUATING WATER RESOURCES AND COST AND BENEFIT ALLOCATION1

ABSTRACT A linear programming model for a river basin was developed to include almost all water-related economic activity both for consumers and producers. The model was so designated that the entire basin or basin sub-division could be analyzed. The model included seven sectors, nine objective function criteria, and three river-flow levels. Economic basis for conflicts among sectors over incidence of cost allocation and level of economic activity can be traced to some chosen objective. The disposal of untreated household waste water, particularly from the rural household, directly into the river was consistent with maximizing net benefits and minimizing costs. The optimum resource allocation, water-treatment plants, farms and industry activities would change with flow level. For each of the three industries analyzed separately, paper, wool and tanning, public treatment of industrial waste water was the optimal treatment process in one or more of the solutions. Lake shoreline was the dominant feature determining lake-resource valuation. Implied capital value varied from $126 per shoreline foot to over $250 depending on discount rate. Implied prices on lake surface ranged from $42 to $147 per acre. Strong economic forces encouraged small lot sizes for vacation cottages.     

INDUCED OXIDATION OF STREAMS AND WATER QUALITY CONTROL INSTITUTIONS1

The proof is rapidly mounting that instream aeration is a technically and economically feasible supplement to secondary treatment of wastes, an alternative to tertiary waste treatment. Some phenomena remain unexplained; and competing technologies, especially those employing molecular oxygen, are being explored vigorously. However, there are convincing arguments in favor of aeration on theoretical grounds, proved technology is available; and the first installations have already been made. While remaining questions are being further investigated, it is time to consider in more detail the broader aspects, as to the institutional, legal, and political hurdles to be overcome before advantage can be taken of the new technology. This paper first summarizes the state-of-the-art as far as technology is concerned; and then outlines the institutional problem.     

MATHEMATICAL ANALYSIS OF ARTIFICIAL RECHARGE FROM BASINS1

ABSTRACT: One of the most common methods of artificial recharge to the ground water is from basins. In this paper, seven analytical solutions that describe artificial recharge from basins are presented. Most of these solutions are derived by directly solving the general partial differential equation for ground water flow. The solutions differ in that they use different boundary conditions, basin shapes, and consider the nonlinearity of the artificial recharge problem differently. Use of each analytical solution is demonstrated in this paper by application to an example problem. A comparison of each analytical solution presented in this paper was made to give suggestions on their use, their ease of implementation, and their relative agreement. Although no attempt is made in the paper to conclude which analytical solution is best for all problems, some general conclusions can be stated on the applicability of the various analytical solutions. Of the analytical solutions presented in this paper, Glovers and Hantush's solutions for rectangular recharge basins are highly recommended. Baumanns solution for a circular basin also gave fairly reliable results and is very easy to evaluate numerically.     

REGIONAL CHARACTERISTICS OF THE TOTAL DISSOLVED MINERALS IN STREAM WATERS OF ILLINOIS1

ABSTRACT Using water analyses from 67 gaging stations, discharge-frequency-weighted mean concentrations (QFC) and average annual yield (AAY) per unit area were determined for the total dissolved mineral content of Illinois streams. The resultant data indicates that total dissolved mineral contents are controlled by regional rather than local factors. In most cases plausible explanation for the magnitudes can be found in regional patterns of natural and demographic conditions. Although the data suggest that total dissolved minerals are increasing, the relative contributions of natural versus anthropogenic phenomena are difficult to delimit at this level of investigation.     

AN APPLICATION OF MATHEMATICAL PROGRAMMING IN PLANNING SURFACE WATER STORAGE1

Designing a surface reservoir involves the concept of reservoir yield. This concept embodies three basic information items: hydrologic regime, active storage volume, and reservoir release policy. In the actual case presented below, the magnitude of the active storage was prescribed by a legal procedure, so that the planning issue became that of determining the reservoir yield given the hydrological information. A stochastic dynamic programming model was formulated to derive a schedule of seasonal optimal reservoir releases and their respective probabilities of occurrence. This schedule is the reservoir yield. The yearly cycle was divided into three seasons representing the actual climatic conditions, and conditional probabilities linking streamflows in consecutive seasons were estimated. An operating policy was postulated, based on the same set of legal decisions that prescribed the active storage volume, and target reservoir releases were assumed. Similarly, target storages at the end of each season were set up. The optimizing/ minimizing criterion in the dynamic programming formulation was the sum of squares of deviations of actual releases and final storage volumes from their respective targets.