CAPACITY EXPANSION PLANNING MODEL FOR WASTE WATER TREATMENT AND I/I CONTROL1

ABSTRACT: A dynamic programming model is presented for planning the optimal capacity expansion sequence of waste water treatment plants servicing sewerage systems with infiltration/inflow. The model employs a nonlinear deterministic demand function for waste water treatment plant capacity. Model output provides the optimal timing, size, and sequence for treatment plant capacity expansion projects and infiltration/inflow removal projects. An example problem is presented and solved.     

OPTIMAL ALLOCATION OF ARTIFICIAL AERATION ALONG A POLLUTED STREAM USING DYNAMIC PROGRAMMING1

ABSTRACT: When a series of aerators are used to raise the level of dissolved oxygen in a polluted stream through instream artificial aeration augmentation, the system is governed by the basic dissolved oxygen mass balance equation with the existence of artificial aeration as its boundary conditions. A mathematical model is formulated for the optimization of the allocation of aeration capacity to each of the series of aerators subject to a limitation on total available aeration capacity. The objective function is the minimization of the sum of the squares of the aeration costs and the costs incurred by damaging or unnecessarily improving the system. The original constrained allocation problem is simplified by converting it to an unconstrained one via the use of Lagrange multiplier. A discretized dynamic programming algorithm is formulated for finding the optimal allocation policy. A typical optimal aeration capacity allocation policy and its corresponding dissolved oxygen sag profile for the illustrated numerical example is presented, and the relationship between the total available aeration capacity and Lagrange multiplier is also developed treating weighting factors as parameters.     

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.     

DYNAMIC PROGRAMMING FOR OPTIMUM CONJUNCTIVE USE1

The Las Vegas, Nevada area like most semi-arid basins, was developed through exploitation of available ground-water resources. Area growth in this large valley has occurred in a scattered and sporadic manner with development both in incorporated areas and within the County. As a result, today there exist five major water suppliers which are: a water district, three municipalities, and a large corporation, in addition to numerous small water companies and thousands of domestic wells. In the past 20 years the area has grown from a population of less than 50,000 to over 300,000 today. The bulk of the water demand for this growth has been met from the ground-water resource and as a result the basin is being severely mined. Current extractions are over three times the estimated annual replenishment. Rapidly declining water levels are increasing the costs of water and are creating water shortages during periods of peak demand. To meet both the current and anticipated water demands, the Southern Nevada Water Supply Project is being constructed to import additional water from nearby Lake Mead. Agriculture in the area is very limited, and primarily uses reclaimed waste water for irrigation. The chief water demands in the area are thus municipal and industrial, with the former predominating. This study was designed to determine how best the Las Vegas Valley Water District, supplier of 80 percent of the domestic water, might integrate the use of the existing ground water and anticipated imported surface water. Additionally the consequences of application of certain provisions in the Nevada Water Law were examined to determine their effects on the ground-water system and costs of water. To achieve these objectives, a dynamic programming technique was utilized. The problem as formulated consists of a single decision variable, single state variable dynamic programming algorithm evaluated over a fifty-year planning horizon at monthly intervals. Three alternative solutions, with different ground-water law constraints are evaluated. In all solutions certain basic operating rules regarding ground-water pumping distribution and use of surface-water systems are kept constant. The problem is considered as deterministic in all respects. Recharge to ground water is assumed to equal the estimated average annual replenishment evenly distributed over the year and additionally is not considered to be a function of average basin ground water potential. The only surface supply, Lake Mead, is considered to operate at near constant elevation and not be subject to shortage conditions. In light of the size of Lake Mead, the Colorado River flow and the size of Nevada's allotment, 300,000 ac ft, the latter assumption is reasonable. Demand for water is considered as a known function of time. Optimization of conjunctive use for the Water District is based on the objective function of minimizing water production costs. Costs of distributing water are considered to be constant regardless of source, and so are not included. Also, fixed costs of amortizing the pipeline project and well fields are not considered. Results of the study are presented as a series of policy traces under each of the three alternatives considered. These traces describe the ground-water basin response under optimal operating conditions, given an estimate as to the present worth of ground-water pumping rights, and prescribe monthly water-procurement schedules for the operation of the Water District.     

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 CODED ALGORITHM FOR CAPACITY EXPANSION OF A WATER QUALITY MANAGEMENT SYSTEM1

ABSTRACT: This paper describes a mathematical model, an algorithm and a computer program that were specially developed to study the problem of a water quality management system undergoing a rapidly increasing environmental stress. The model output will determine the locations, sizes and the timing of construction of new treatment plants plus an overall treatment plant operating policy so that environmental standards are maintained at a minimum cost. The model, as formulated, is a 0-1 mixed integer programming problem which is solved by decomposing it into a capital budgeting problem (solved by Little's branch and bound algorithm) and an operational policy problem (solved by linear programming). The coded algorithm (in FORTRAN 10) has been tested with a semi-realistic example.     

COMPUTATIONAL IMPROVEMENT FOR STOCHASTIC DYNAMIC PROGRAMMING MODELS OF URBAN WATER SUPPLY RESERVOIRS1

ABSTRACT: This paper describes two methods that are introduced to improve the computational effort of stochastic dynamic programming (SDP) as applicable to the operation of multiple urban water supply reservoir systems. The stochastic nature of streamflow is incorporated explicitly by considering it in the form of a multivariate probability distribution. The computationally efficient Gaussian Legendre quadrature method is employed to compute the conditional probabilities of streamflow, which accounts for the serial correlation of streamflow into each storage and the cross correlation between the streamflow into various storages. A realistic assumption of cross correlation of streamflow is introduced to eliminate the need to consider the streamflow combinations which are unlikely to occur in the SDP formulation. A “corridor” approach is devised to eliminate the need to consider the infeasible and/or inferior storage volume combinations in the preceding stage in computing the objective function in the recursive relation. These methods are verified in terms of computational efficiency and accuracy by using a hypothetical example of three interconnected urban water supply reservoirs. Therefore, it can be concluded that these methods allow SDP to be more attractive for deriving optimal operating rules for multiple urban water supply reservoir systems.     

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.     

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.     

WETLAND DESIGN METHODS FOR RESIDENTIAL WASTEWATER TREATMENT1

ABSTRACT: Constructed wetlands have recently gained popularity as an alternative method for wastewater treatment. This paper compares two design methodologies currently used for constructed wetlands; Tennessee Valley Authority (TVA) and the Environmental Protection Agency (EPA) methods. A discussion of parameters for both methods is given and a wetland treatment system is designed for an individual residence with typical BOD5 loads and flow rates. Calculation results revealed significant discrepancies in the required constructed wetlands volume, and thus detention time, stemming from inherent differences in the design methodologies. The EPA method relies heavily on plug flow kinetics, and is therefore sensitive to changes in the reaction rate constant and media porosity. Conversely, TVA determines the surface area by sizing in accordance with a recommended hydraulic loading criterion and is affected only by the hydraulic flow rates. This study concluded that a constructed wetland is a viable option under design considerations that are not favorable for traditional on-site wastewater treatment methods. However, it is recommended that conservative values for flow and loading rates be assumed to assure complete treatment for either of the design methods.     

AN ARCHETYPAL LINEAR PROGRAMMING MODEL FOR OPTIMAL INTERTEMPORAL ALLOCATION OF GROUNDWATR TO AGRICULTURAL ACTIVITIES1

ABSTRACT: In the past, researchers have applied a variety of analytical techniques for maximizing the present value of a stock resource - simulation, calculus of variations, stochastic dynamic programming, and optimal control theory. This paper presents a more operational approach - linear programming. A simplified, broadly drawn example from Southwest agriculture is used for demonstrating the model's structure and output. The method is based on a set of state transformation operations that prevent the additivity assumption of linear programming from being violated.     

A STOCHASTIC DYNAMIC PROGRAMMING MODEL FOR THE OPTIMUM OPERATION OF A MULTI-PURPOSE RESERVOIR1

ABSTRACT: The main objective of this paper is to present a stockastic dynamic programming model useful in determining the optimal operating policy of a single multipurpose surface reservoir. It is the unreliability of forecasting the amount of future streamflow which makes the problem of a reservoir operation a stochastic process. In this paper the stochastic nature of the streamflow is taken into account by considering the correlation between the streamflows of each pair of consecutive time intervals. This interdependence is used to calculate the probability of transition from a given state and stage to its succeeding ones. A dynamic programming model with a physical equation and a stochastic recursive equation is developed to find the optimum operational policy. For illustrative purposes, the model is applied to a real surface water reservoir system.     

ON THE CENTRALIZATION OF WASTEWATER TREATMENT FACILITIES1

ABSTRACT. Two factors affecting the centralization of wastewater treatment facilities were investigated; the cost of collection and treatment systems and the performance of treatment plants. Based on computer-generated minimum cost designs, wastewater collection networks were found to be characterized by diseconomies of scale of magnitude similar to the reported economies of scale for wastewater treatment works. The combined costs of collection and treatment are U-shaped functions from which the least cost size of collection and treatment systems were found for particular values of population density. Examination of the day-to-day performance of five metropoitan-area waste-water treatment plants revealed that, for time series shorter than one month, the day-to-day variation in effluent quality was random, although the variation in quantity discharged was distinctly non-random. The performances of all five plants on any given day showed little correlation. This suggests that the decentralization of treatment facilities can produce benefits both through the reduction in quantities of waste discharged at a given point and through in-stream averaging of the varying performances of several treatment plants. Since the cost function of collection and treatment combined is generally flat in the region of the minimum-cost size, little penalty is invoked to gain the potential benefits of treatment plant decentralization.     

WATER RECLAMATION BY PHYSICOCHEMICAL TREATMENT OF WASTEWATER1

ABSTRACT. Treatment of a high strength acidic industrial chemical waste, which had a COD of about 3000 mg/1 and a pH of 3, was attempted by activated carbon adsorption to evaluate the feasibility of yielding effluents of reusable qualities. The experimental methods which were employed in this investigation included batch and column studies. The former was used to evaluate the rate and equilibrium of carbon adsorption while the latter was used to determine treatment efficiencies and performance characteristics. Parallel operations of fixed-bed and expanded-bed adsorbers were contrasted in the column study. Results of this investication indicate that activated carbon was very efficient in removing the COD of the chemical waste. Initial pH adjustments for the wastewater to 7.0 or 11.4 did not increase the COD removal efficiency. Turbidity and nitrogen (total Kjeldahl- and ammonia-N) were removed to some extents by the carbon bed adsorbers while phosphorus (total-and ortho-P) was totally unaffected by the carbon treatment.     

NITROGEN ACCUMULATION IN A CONSTRUCTED WETLAND FOR DAIRY WASTEWATER TREATMENT1

ABSTRACT: A free water surface (FWS) constructed wetland was installed at a dairy in Glendale, Arizona, to study the potential of such a wetland to remove nitrogen (N) from wastewater. The study objectives were: (1) to determine N removal from the wastewater, and (2) to evaluate N accumulation in soil and plant tissues. The system consisted of eight cells (70 × 9 × 1.5 m) planted with Typha domingensis, Scirpus validus, and Phragmites australis. The four cells in series were lined with plastic, and the four cells in a parallel series were lined with clay. Cells received approximately 180 m³/d of partially treated dairy effluent. Plant tissues and soil samples were collected above and below ground from 24 locations during one year. Total N removal from wastewater was about 17 percent. Clay‐lined cells accumulated more N in the soil and less N in plant biomass compared with plastic lined cells. Plant biomass accounted for approximately 60 percent of total N accumulated in cells with dense plant communities. Ninety percent of accumulated soil N was organic. Total N accumulated in soil reached a maximum (1,100 mg/kg) eight months after the introduction of wastewater.     

GOALS IN WASTEWATER TREATMENT1

Within a few years all domestic wastewater effluents in the United States will be subjected to a minimum of a properly operated primary and secondary treatment process. This implies a very high degree of removal of the more readily biologically degradable material as measured by the BOD test. This practice will to a large degree negate the value of the BOD test as a pollution parameter. Organic carbon appears to be a more suitable means for determining the strength of a wastewater or for controlling the operation of physical and chemical treatment processes. Studies were conducted to determine the effect of time of passage on the ratio of organic carbon to BOD, COD and carbohydrate. The ratio was found to vary with both time and the state of the sample. The carbon content was reduced to a lesser degree than the other parameters. Organic carbon content appears to correlate better with COD than with BOD. The efficiency of a chemical precipitation process can be determined on the basis of organic carbon removal.     

OPTIMAL CAPITAL INVESTMENT IN THE EXPANSION OF AN EXISTING WATER RESOURCES SYSTEMS1

The study of the optimal expansion of existing water resources systems is of continuing importance because of the rising demand and limited supply of water in many areas of the world, particularly in the southwestern part of the United States of America. This study is concerned with the investigation of the optimal expansion of a realistic water resources system to meet an increasing demand for municipal and industrial use, irrigation, energy, and recreation over a planning horizon of T years. A number of possible dam sites are available for the further regulation of river (canal) flows in the basin and/or the regulation of imported waters into the basin. To maximize, over the set of alternative projects, the sum of discounted present value of net earnings subject to the demands and various institutional, physical and budgetary limits, an optimization problem (Problem I) was formed as a 0-1 mixed integer programming problem and was decomposed into the set of all feasible combinations (Problem II). The economic return was determined for each combination (Problem III). Problem II was solved by a branch and bound procedure which selected each feasible combination of dams while the optimal return for each such combination (Problem III) was found by a network analysis code.     

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.     

DESIGN OF DESALTING PLANTS FOR CONJUNCTIVE OPERATION1

ABSTRACT The selection of an evaporator design must reflect the balancing of captial cost (primarily in heat exchanger surface and vessels) against operating cost (primarily steam cost) to achieve minimum cost. In a conjunctive plant the tendency is to select a low-capital cost, high-operating cost plant. In addition, it is advantageous to use a high-capacity plant which needs to be operated much less of the time than a plant which is sized just at the needed rate. For example, in the study of a possible system to satisfy a future increase of 450 MGD in water supply to New York Qty, a plant of 750 MGD capacity was selected as optimum. This plant, of the advanced VTE-MSF process type, would have a performance ratio of 9 lb product/1000 Btu as compared to 10-13 normally used for base-loaded plants. Steam would be supplied by a multi-unit dual-purpose nuclear power plant. The most economical type of energy supply would be “interruptible”; the steam would be used by a low pressure turbine to generate electricity during periods of peak electrical demand but would be available to the desalting plant at other times. The low pressure turbine would be available as spinning reserve during desalting plant operation. It is estimated that the desalting plant would have a load factor of 27 per cent over its life.     

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.