“The Optimal Steady-State in Groundwater Management,” by Keith C. Knapp and Eli Feinerman2

Knapp and Feinerman (1985) pose and solve a problem of steady-state allocation of ground water based directly on the underlying dynamic problem. Their dynamic steady-state formulation incorporates both the equations of transient ground. water flow and the discount rate. We wish to discuss two aspects of their analysis. First, we question their assertion that the computational advantages of the dynamic steady.state formulation will justify its substitution for the full transient problem. In fact, the dynamic steady-state problem will often be harder to solve than the properly formulated transient problem. Second, we argue that the dynamic steady-state is a concept that has limited applicability in ground-water management. In cases where the optimal steady state is indeed useful, the dynamic solution is often identical to the static solution.     

OVERPUMPED ARTESIAN WELLS AMONG A WELL GROUP1

ABSTRACT. Overpumped artesian wells are defined and explained. Unified well equations which can be used for artesian, gravity and overpumped artesian wells are derived under both the steady and transient conditions of flow. The location of the boundary between artesian and gravity steady and transient flow systems in artesian aquifers, due to multiple wells among which overpumped artesian wells may exist, is explained.     

A NONLINEAR PROGRAMMING ALGORITHM FOR REAL-TIME HOURLY RESERVOIR OPERATIONS1

ABSTRACT: The optimization of real-time operations for a single reservoir system is studied. The objective is to maximize the sum of hourly power generation over a period of one day subject to constraints of hourly power schedules, daily flow requirement for water supply and other purposes, and the limitations of the facilities. The problem has a nonlinear concave objective function with nonlinear concave and linear constraints. Nonlinear Duality Theorems and Lagrangian Procedures are applied to solve the problem where the minimization of the Lagrangian is carried out by a modified gradient projection technique along with an optimal stepsize determination routine. The dimension of the problem in terms of the number of variables and constraints is reduced by eliminating the 24 continuity equations with a special implicit routine. A numerical example is presented using data provided by the Bureau of Reclamation, Sacramento, California.     

Optimal Location and Management of Waste Disposal Facilities Affecting Ground Water Quality1

ABSTRACT: Numerical simulation of ground water solute transport is combined with linear programming to optimize waste disposal. A discretized form of the equation governing solute transport is included as a set of constraints in a linear program. Two problems are described. First, the management model is used to maximize ground water waste disposal. The model constrains disposal activities so that the quality of local ground water supplies is protected. Parametric programming is shown to be important in evaluating waste disposal tradeoffs at the various facilities. Changes in the velocity field induced by waste water injection cause a nonlinearity in the solute transport equation which is dealt with by employing an iterative procedure. The second problem is aimed at identifying all sites which are suitable for waste disposal in the subsurface. The management model is manipulated so that the optimal value of the dual variables are “unit source impact indicators.” This physical interpretation is valuable in identifying feasible disposal sites. The joint simulation and optimization approach permits the management of complex ground water systems where the aquifer is used simultaneously for waste disposal and water supply.     

POLITICAL RESOLUTION OF ENVIRONMENTAL CONFLICTS1

ABSTRACT Many water resources management problems ultimately require a political decision in order to formulate a course of action to follow. Metagame analysis is a nonquantitative technique that can model the decision-making process and is used in this article to arrive at a politically feasible solution to a typical water pollution problem. The output from the linear programming formulation of the problem is used as input to the “political” metagame analysis. This allows a solution to the pollution conflict that is not only physically and economically feasible according to the linear programming analysis, but politically feasible as well.     

GROUND WATER MANAGEMENT OPTIMIZATION USING GENETIC ALGORITHMS AND SIMULATED ANNEALING: FORMULATION AND COMPARISON1

ABSTRACT: Genetic algorithms (GA) and simulated annealing (SA), two global search techniques, are coupled with MODFLOW, a commonly used groundwater flow simulation code, for optimal management of ground water resources under general conditions. The coupled simulation-optimization models allow for multiple management periods in which optimal pumping rates vary with time to reflect the changing flow conditions. The objective functions of the management models are of a very general nature, incorporating multiple cost terms such as the drilling cost, the installation cost, and the pumping cost. The models are first applied to two-dimensional maximum yield and minimum cost water supply problems with a single management period, and then to a multiple management period problem. The strengths and limitations of the GA and SA based models are evaluated by comparing the results with those obtained using linear programming, nonlinear programming, and differential dynamic programming. For the three example problems examined in this study, the GA and SA based models yield nearly identical or better solutions than the various programming methods. While SA tends to outperform GA in terms of the number of forward simulations needed, it uses more empirical control parameters which have significant impact on solution efficiency but are difficult to determine.     

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.     

Waste management with recourse: An inexact dynamic programming model containing fuzzy boundary intervals in objectives and constraints

The existing inexact optimization methods based on interval-parameter linear programming can hardly address problems where coefficients in objective functions are subject to dual uncertainties. In this study, a superiority–inferiority-based inexact fuzzy two-stage mixed-integer linear programming (SI-IFTMILP) model was developed for supporting municipal solid waste management under uncertainty. The developed SI-IFTMILP approach is capable of tackling dual uncertainties presented as fuzzy boundary intervals (FuBIs) in not only constraints, but also objective functions. Uncertainties expressed as a combination of intervals and random variables could also be explicitly reflected. An algorithm with high computational efficiency was provided to solve SI-IFTMILP. SI-IFTMILP was then applied to a long-term waste management case to demonstrate its applicability. Useful interval solutions were obtained. SI-IFTMILP could help generate dynamic facility-expansion and waste-allocation plans, as well as provide corrective actions when anticipated waste management plans are violated. It could also greatly reduce system-violation risk and enhance system robustness through examining two sets of penalties resulting from variations in fuzziness and randomness. Moreover, four possible alternative models were formulated to solve the same problem; solutions from them were then compared with those from SI-IFTMILP. The results indicate that SI-IFTMILP could provide more reliable solutions than the alternatives.     

ON THE MISMATCH BETWEEN DATA AND MODELS OF HYDROLOGIC AND WATER RESOURCE SYSTEMS1

ABSTRACT: Many difficulties exist in the matching of models with data. This paper identifies elements of this problem and discusses considerations involved in model evaluation. The well known multivariate linear regression model is used to illustrate the distinctions between accuracy and precision and between estimation and prediction (because the model is commonly misused.) No amount of additional data will improve the accuracy of a poor model. A high R², while indicative of a good matching between the observed data and model estimates, is a poor criterion for judging adequacy of the model to make good predictions of future events. Model evaluation also includes the problem of introducing secondary data and proxy variables into a model. Secondary data frequently enter, for example, the mass, energy and water budget equations because of difficulties in measuring the primary variables. Proxy variables arise because of a desire to collapse a vector of incomparable values, say, of water quality into a single number. Review of the above issues indicates that model evaluation is a multi-criterion problem, often imbedded in a larger framework where models are intended to meet multiple objectives. The mismatch of models and data has increasing legal and social consequences.     

LINEAR PROGRAMS FOR NONLINEAR HYDROLOGIC ESTIMATION1

ABSTRACT: The minimization of the sum of absolute deviations and the minimization of the absolute maximum deviation (mini-max) were transformed into equivalent linear programs for the estimation of parameters in a transient and linear hydrologic system. It is demonstrated that these two methods yield viable parameter estimates that are globally optimal and reproduce properly the timing and magnitude of hydrologic events and associated variables such as total runoff. The two linear estimation methods compared favorably with the popular least-squares nonlinear estimation method. The generality of the theoretical developments shows that linear program equivalents are adequate competitors of nonlinear methods of hydrologic estimation and parameter calibration.     

DETERMINING SIGNIFICANCE AND PRECISION OF ESTIMATED PARAMETERS FOR RUNOFF FROM SEMIARID WATERSHEDS1

ABSTRACT Significant parameters for predicting thunderstorm runoff from small semiarid watersheds are determined using data from the Walnut Gulch watershed in southern Arizona. Based on these data, thunderstorm rainfall is dominant over watershed parameters for predicting runoff from multiple linear regression equations. In some cases antecedent moisture added significantly to the models. A technique is developed for estimating precision of predicted values from multiple linear regression equations. The technique involves matrix methods in estimating the variance of mean predicted values from a regression equation. The estimated variance of the mean predicted value is then used to estimate the variance of an individual predicted value. A computer program is developed to implement these matrix methods and to form confidence limits on predicted values based on both a normality assumption and the Chebyshev inequality.     

A RAPID METHOD OF ESTIMATING MEAN AREAL RAINFALL1

ABSTRACT. A special case of generalized trend surface analysis is examined. This includes a linear surface. It is shown that for most hydrologic problems this case determines mean areal rainfall sufficiently accurately. Based on this conclusion, equations for rapid computation of mean areal rainfall are derived for this linear case. Results of the linear case are compared with other traditional methods of estimating mean areal rainfall.     

CONFIDENCE INTERVALS FOR FLOOD EVENTS UNDER A PEARSON 3 OR LOG PEARSON 3 DISTRIBUTION1

ABSTRACT: The Pearson type 3 (P3) and log Pearson type 3 (LP3) distributions are very frequently used in flood frequency analysis. Existing methods for constructing confidence intervals for quantiles (X_p) of these two distributions are very crude. Most of these methods are based on the idea of adjusting confidence intervals for quantiles Y_p of the normal distribution to obtain approximate confidence inervals for quantiles X_p of the P3/LP3 distribution. Since there is no theoretical reason why this “base” distribution, Y, should be taken to be normal, we search in the present study for the best possible base distribution for producing confidence intervals for P3/LP3 quantiles. We consider a group of base distributions such as the normal, log normal, Weibull, Gumbel, and exponential. We first assume that the skew coefficient, γ of X, to be known, and develop a method for adjusting confidence intervals for Y_p to produce approximate confidence intervals for X_p. We then compare this method (Method A) with another method (Method B) introduced by Stedinger. Simulation shows that the performance of each of these two methods depends on the base distribution Y that is being used, but as a whole, the normal distribution appears to be the best-fit distribution for producing confidence intervals for P3/LP3 quantiles when γ is assumed to be known. We then extend our method (Method A) to the more important case of unknown coefficient of skewness. It is shown that by taking Y to be Weibull, fairly accurate confidence intervals for P3/LP3 quantiles can be obtained for quite a wide range of sample sizes and coefficients of skewness commonly found in hydrology. The case of the P3 distribution with negative skewness needs further research.     

An inexact two-stage mixed integer linear programming method for solid waste management in the City of Regina

In this study, an interval-parameter two-stage mixed integer linear programming (ITMILP) model is developed for supporting long-term planning of waste management activities in the City of Regina. In the ITMILP, both two-stage stochastic programming and interval linear programming are introduced into a general mixed integer linear programming framework. Uncertainties expressed as not only probability density functions but also discrete intervals can be reflected. The model can help tackle the dynamic, interactive and uncertain characteristics of the solid waste management system in the City, and can address issues concerning plans for cost-effective waste diversion and landfill prolongation. Three scenarios are considered based on different waste management policies. The results indicate that reasonable solutions have been generated. They are valuable for supporting the adjustment or justification of the existing waste flow allocation patterns, the long-term capacity planning of the City's waste management system, and the formulation of local policies and regulations regarding waste generation and management.     

IMPROVED RIVER BASIN UTILIZATION THROUGH SYSTEMS ANALYSIS1

ABSTRACT. Theoretical and practical results are summarized for a study to determine optimal water resource allocation in a proposed water conservancy district. The area of this district, which covers several river basins, contains a large number of existing and proposed facilities such as reservoirs and diversions. The operation of all of these facilities was to be determined along with the sizing of the proposed facilities in order to optimize given objective functions. Related efforts in optimal river basin utilization were surveyed, and linear programming was selected as an expedient optimization technique. The problem is formulated by identifying time stages which together constitute a repetitive cycle such as a year. With these stages, it is possible to associate operational and capacity variables with network components, which are branches and nodes. Objective functions are assembled for the component variables. Constraint equations are written in terms of the variables to reflect network nodal continuity, capacity restrictions, and adjudications such as water rights. A numerical example is considered in which the existing and proposed facilities are aggregated to produce a small, tractable number of facilities. This paper examines the example results and suggests future improvements for models of this type.     

EFFECTS OF TIME STEP SIZE IN IMPLICIT DYNAMIC ROUTING1

ABSTRACT The effects of the size of the Δt time step used in the integration of the implicit difference equations of unsteady open-channel flow are determined for numerous typical hydrographs with durations in the order of days or even weeks. Truncation errors related to the size of the Δt time step cause a numerical distortion (dispersion and attenuation) of the computed transient. The magnitude of the distortion is related directly to the size of the time step, the length of channel reach, and the channel resistance and inversely to the time of rise of the hydrograph. The type of finite difference expression which replaces spatial derivatives and non-derivative terms in the partial differential equations of unsteady flow has an important influence on the magnitude of the numerical distortion, as well as the numerical stability of the implicit difference equations. Time step sizes in the range of 3 to 6 hrs generally tend to minimize the combination of required computation time and numerical distortion of transients having a time of rise of the order of several days.     

A SEPARABLE LINEAR ALGORITHM FOR HYDROPOWER OPTIMIZATION1

ABSTRACT: A deterministic, separable, linear algorithm is presented for maximizing aggregate hydropower production. The method is iterative and amenable to solution using standard LP software. The utility of the technique is demonstrated using several test applications involving a hypothetical single-purpose hydropower reservoir and a monthly increment 20-year flow record from the Gunpowder River in Maryland. The separable linearized forms solved quickly using MPSX on a variety of IBM hardware: 3090-400 VF, 3084 QX, dual processor 4381-3, and an AT/370 personal computer. For comparison purposes, the original nonlinear nonseparable version of the model was also solved using MINOS. This yielded a value of aggregate hydropower marginally higher than that using MPSX. The separable, linearized methodology proved to be a useful and an efficient means of generating good starting points for MINOS. The use of these warm starts effected substantial reductions in MINOS execution times.     

IDENTIFICATION OF AN OPTIMAL GROUNDWATER MANAGEMENT STRATEGY IN A CONTAMINATED AQUIFER1

A groundwater hydraulic management model is used to identify the optimal strategy for allocating limited fresh-water supplies and containing wastes in a hypothetical aquifer affected by brine contamination from surface disposal ponds. The present cost of pumping from a network of potential supply and interception wells is minimized over a five-year planning period, subject to a set of hydraulic, institutional, and legal constraints. Hydraulic constraints are formulated using linear systems theory to describe drawdown and velocity variables as linear functions of supply and interception well discharge decision variables. Successful validation of the optimal management strategy suggests that the model formulation can feasibly be applied to define management options for locally contaminated aquifer systems which are used to fulfill fresh-water demands.     

Thermodynamic-dynamic analysis of gamma type free-piston stirling engine charged with hydrogen gas as working fluid

ABSTRACT

In this study, the combined thermodynamic and dynamic model of a new concept of gamma type free-piston Stirling engine is conducted. The engine consists of two identical displacer cylinders, a power cylinder, a linear alternator, and three-cushion pistons. Two displacer cylinders are symmetrically positioned on each side of the power cylinder for minimizing the rotational vibrations. Hydrogen is used as the working gas and the effect of gas temperature on the specific heat capacity is considered. The analysis carried out in this study involves the prediction of the thermodynamic-dynamic performance characteristics of the engine. In the thermodynamic section of the analysis, the working space of the engine is divided into 31 nodal volumes and the gas pressures in nodal volumes are assumed to be equal to each other. The conservation of mass and energy equations is obtained for each nodal volume. Instantaneous gas temperatures of nodal volumes are calculated by the first law of thermodynamics given for the unsteady open systems. The dynamic section of the analysis involves the motion equations of displacer, power and cushion pistons. The motion equations are derived using the Newton method. In the calculations done for variable specific heat capacity, it has been determined that there is 1% cyclic work reduction compared to the constant heat capacity. It is estimated that the maximum effective power that can be produced by the linear alternator will be around 1.6 kW. The working frequency range of the proposed engine is found to be suitable to generate electrical power.