Development of stochastic dynamic Nash game model for reservoir operation II. The value of players’ information availability and cooperative behaviors

Stochastic dynamic game models can be applied to derive optimal reservoir operation policies by considering interactions among water users and reservoir operator, their preferences, their levels of information availability and cooperative behaviors. The stochastic dynamic game model with perfect information (PSDNG) has been developed by [Ganji A, Khalili D, Karamouz M. Development of stochastic dynamic Nash game model for reservoir operation. I. The symmetric stochastic model with perfect information. Adv Water Resour, this issue]. This paper develops four additional versions of stochastic dynamic game model of water users interactions based on the cooperative behavior and hydrologic information availability of beneficiary sectors of reservoir systems. It is shown that the proposed models are quite capable of providing appropriate reservoir operating policies when compared with alternative operating models, as indicated by several reservoir performance characteristics. Among the proposed models, the selected model by considering cooperative behavior and additional hydrologic information (about the randomness nature of reservoir operation parameters), as exercised by reservoir operator, provides the highest attained level of performance and efficiency. Furthermore, the selected model is more realistic since it also considers actual behavior of water users and reservoir operator in the analysis.     

Development of stochastic dynamic Nash game model for reservoir operation. I. The symmetric stochastic model with perfect information

Increasing water demands, higher standards of living, depletion of resources of acceptable quality and excessive water pollution due to agricultural and industrial expansions have caused intense social and political predicaments, and conflicting issues among water consumers. The available techniques commonly used in reservoir optimization/operation do not consider interaction, behavior and preferences of water users, reservoir operator and their associated modeling procedures, within the stochastic modeling framework. In this paper, game theory is used to present the associated conflicts among different consumers due to limited water. Considering the game theory fundamentals, the Stochastic Dynamic Nash Game with perfect information (PSDNG) model is developed, which assumes that the decision maker has sufficient (perfect) information regarding the associated randomness of reservoir operation parameters. The simulated annealing approach (SA) is applied as a part of the proposed stochastic framework, which makes the PSDNG solution conceivable. As a case study, the proposed model is applied to the Zayandeh-Rud river basin in Iran with conflicting demands. The results are compared with alternative reservoir operation models, i.e., Bayesian stochastic dynamic programming (BSDP), sequential genetic algorithm (SGA) and classical dynamic programming regression (DPR). Results show that the proposed model has the ability to generate reservoir operating policies, considering interactions of water users, reservoir operator and their preferences.     

A stochastic model for optimal operation of inter-basin water allocation systems: a case study

This paper presents a new methodology for optimal operation of inter-basin water transfer systems by conjunctive use of surface water resources in water donor basin and groundwater resources in water receiving basin. To incorporate the streamflow uncertainty, an integrated stochastic dynamic programming (ISDP) model is developed. In the ISDP, the monthly inflow to the reservoir in the water donor basin, the water storage of the reservoir, and the water storage of the aquifer in the water receiving basin are considered as state variables. A water allocation optimization model is embedded in the main structure of ISDP and a new ensemble streamflow prediction model based on K-nearest-neighbourhood algorithm is also developed and linked to the ISDP. By using a new reoptimization process, the ISDP model provides monthly policies for water allocation to users in water donor and receiving basins. As water users can form a coalition to increase their benefits, several solution concepts in cooperative game theory, namely Nash–Harsanyi, Shapley, Nucleolus, Weak Nucleolus, Proportional Nucleolus, Separable Costs Remaining Benefits (SCRBs) and Minimum Costs Remaining Savings are utilized to determine the profit of each water user. In the last step, stakeholders make negotiation over these solution concepts using the Fallback bargaining theory to reach a unanimous agreement on the final distribution of the total benefit. The methodology is applied to an inter-basin water transfer project and the results show that the Shapley and SCRB solutions concepts can provide better distributions for the total benefit and the total benefit of water users is increased by a factor of 1.6 when they participate in a grand coalition.     

Probabilistic reservoir operation using Bayesian stochastic model and support vector machine

Available water resources are often not sufficient or too polluted to satisfy the needs of all water users. Therefore, allocating water to meet water demands with better quality is a major challenge in reservoir operation. In this paper, a methodology to develop operating strategies for water release from a reservoir with acceptable quality and quantity is presented. The proposed model includes a genetic algorithm (GA)-based optimization model linked with a reservoir water quality simulation model. The objective function of the optimization model is based on the Nash bargaining theory to maximize the reliability of supplying the downstream demands with acceptable quality, maintaining a high reservoir storage level, and preventing quality degradation of the reservoir. In order to reduce the run time of the GA-based optimization model, the main optimization model is divided into a stochastic and a deterministic optimization model for reservoir operation considering water quality issues.The operating policies resulted from the reservoir operation model with the water quantity objective are used to determine the released water ranges (permissible lower and upper bounds of release policies) during the planning horizon. Then, certain values of release and the optimal releases from each reservoir outlet are determined utilizing the optimization model with water quality objectives. The support vector machine (SVM) model is used to generate the operating rules for the selective withdrawal from the reservoir for real-time operation. The results show that the SVM model can be effectively used in determining water release from the reservoir. Finally, the copula function was used to estimate the joint probability of supplying the water demand with desirable quality as an evaluation index of the system reliability. The proposed method was applied to the Satarkhan reservoir in the north-western part of Iran. The results of the proposed models are compared with the alternative models. The results show that the proposed models could be used as effective tools in reservoir operation.     

A fuzzy-Markov-chain-based analysis method for reservoir operation

In this study, a fuzzy-Markov-chain-based stochastic dynamic programming (FM-SDP) method is developed for tackling uncertainties expressed as fuzzy sets and distributions with fuzzy probability (DFPs) in reservoir operation. The concept of DFPs used in Markov chain is presented as an extended form for expressing uncertainties including both stochastic and fuzzy characteristics. A fuzzy dominance index analysis approach is proposed for solving multiple fuzzy sets and DPFs in the proposed FM-SDP model. Solutions under a set of α-cut levels and fuzzy dominance indices can be generated by solving a series of deterministic submodels. The developed method is applied to a case study of a reservoir operation system. Solutions from FM-SDP provide a range of desired water-release policies under various system conditions for reservoir operation decision makers, reflecting dynamic and dual uncertain features of water availability simultaneously. The results indicate that the FM-SDP method could be applicable to practical problems for decision makers to obtain insight regarding the tradeoffs between economic and system reliability criteria. Willingness to obtain a lower benefit may guarantee meeting system-constraint demands; conversely, a desire to acquire a higher benefit could run into a higher risk of violating system constraints.     

An interval-parameter multi-stage stochastic programming model for water resources management under uncertainty

In this study, an interval-parameter multi-stage stochastic linear programming (IMSLP) method has been developed for water resources decision making under uncertainty. The IMSLP is a hybrid methodology of inexact optimization and multi-stage stochastic programming. It has three major advantages in comparison to the other optimization techniques. Firstly, it extends upon the existing multi-stage stochastic programming method by allowing uncertainties expressed as probability density functions and discrete intervals to be effectively incorporated within the optimization framework. Secondly, penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. Thirdly, it cannot only handle uncertainties through constructing a set of scenarios that is representative for the universe of possible outcomes, but also reflect dynamic features of the system conditions through transactions at discrete points in time over the planning horizon. The developed IMSLP method is applied to a hypothetical case study of water resources management. The results are helpful for water resources managers in not only making decisions of water allocation but also gaining insight into the tradeoffs between environmental and economic objectives.     

A stochastic conflict resolution model for water quality management in reservoir–river systems

In this paper, optimal operating rules for water quality management in reservoir–river systems are developed using a methodology combining a water quality simulation model and a stochastic GA-based conflict resolution technique. As different decision-makers and stakeholders are involved in the water quality management in reservoir–river systems, a new stochastic form of the Nash bargaining theory is used to resolve the existing conflict of interests related to water supply to different demands, allocated water quality and waste load allocation in downstream river. The expected value of the Nash product is considered as the objective function of the model which can incorporate the inherent uncertainty of reservoir inflow. A water quality simulation model is also developed to simulate the thermal stratification cycle in the reservoir, the quality of releases from different outlets as well as the temporal and spatial variation of the pollutants in the downstream river. In this study, a Varying Chromosome Length Genetic Algorithm (VLGA), which has computational advantages comparing to other alternative models, is used. VLGA provides a good initial solution for Simple Genetic Algorithms and comparing to Stochastic Dynamic Programming (SDP) reduces the number of state transitions checked in each stage. The proposed model, which is called Stochastic Varying Chromosome Length Genetic Algorithm with water Quality constraints (SVLGAQ), is applied to the Ghomrud Reservoir–River system in the central part of Iran. The results show, the proposed model for reservoir operation and waste load allocation can reduce the salinity of the allocated water demands as well as the salinity build-up in the reservoir.     

An explicit finite element method for dy—namic analysis in three—medium coupling system and its application

In this paper,an explicit finite element method to analyze the dynamic responses of three-medium coupled systems with any terrain is developed on the basis of the numerical simulation of the continuous conditions on the bounda-ries among fluid saturated porous medium,elastic single-phase medium and ideal fluid medium.This method is a very effective one with the characteristic of high calculating speed and small memory needed because the formulae for this explicit finite element method have the characteristic of decoupling,and which does not need to solve sys-tem of linear equations.The method is applied to analyze the dynamic response of a reservoir with considering the dynamic interactions among water,dam,sediment and basement rock.The vertical displacement at the top point of the dam is calculated and some conclusions are given.     

三种介质耦联系统动力反应分析的显式有限元法及其应用

在笔者已建立的流体饱和多孔介质动力分析的显式有限元法的基础上，提出了可分析任意形状的流体饱和多孔介质-单相弹性固体介质-理想流体介质耦联的复杂系统的动力响应的显式有限元方法．该方法建立的有限元方程列式具有解耦特征，不需求解联立方程组，因而极大地提高了计算效率．将这一方法用于分析了考虑库水、坝、淤泥层和基岩这一复杂系统动力相互作用问题的斜坝面的地震响应，并给出了一些计算结果．      

Orthogonal collocation and alternating-direction procedures for unsaturated flow problems

The alternating-direction collocation method has recently been developed for general parabolic equations. In order to test the applicability of the procedure to highly nonlinear problems, an alternating-direction collocation algorithm is developed to simulate two-dimensional flow in unsaturated porous media. The algorithm employs an alternating-direction solution procedure within the framework of a modified Picard iteration scheme. Numerical behaviour of the new procedure is compared to the behaviour of a standard two-dimensional collocation formulation. The new method is also tested on several infiltration problems of practical interest, including a layered and sloping soil. Results demonstrate the method to be accurate and highly mass conservative. The algorithm also produces significant savings in both execution time and storage.     

Advance first order second moment (AFOSM) method for single reservoir operation reliability analysis: a case study

Reservoir system reliability is the ability of reservoir to perform its required functions under stated conditions for a specified period of time. In classical method of reservoir system reliability analysis, the operation policy is used in a simple simulation model, considering the historical/synthetic inflow series and a number of physical bounds on a reservoir system. This type of reliability analysis assumes a reservoir system as fully failed or functioning, called binary state assumption. A number of researchers from various research backgrounds have shown that the binary state assumption in the traditional reliability theory is not extensively acceptable. Our approach to tackle the present problem space is to implement the algorithm of advance first order second moment (AFOSM) method. In this new method, the inflow and reservoir storage are considered as uncertain variables. The mean, variance and covariance of uncertain variables are determined using moment values of reservoir state variables. For this purpose, a stochastic optimization model developed based on the constraint state formulation is applied. The proposed model of reliability analysis is used to a real case study in Iran. As a result, monthly probabilities of water allocation were computed from AFOSM method, and the outputs were compared with those from Monte Carlo method. The comparison shows that the outputs from AFOSM method are similar to those from the Monte Carlo method. In term of practical use of this study, the proposed method is appropriate to determine the monthly probability of failure in water allocation without the aid of simulation.     

Intelligent reservoir operation system based on evolving artificial neural networks

We propose a novel intelligent reservoir operation system based on an evolving artificial neural network (ANN). Evolving means the parameters of the ANN model are identified by the GA evolutionary optimization technique. Accordingly, the ANN model should represent the operational strategies of reservoir operation. The main advantages of the Evolving ANN Intelligent System (ENNIS) are as follows: (i) only a small number of parameters to be optimized even for long optimization horizons, (ii) easy to handle multiple decision variables, and (iii) the straightforward combination of the operation model with other prediction models. The developed intelligent system was applied to the operation of the Shihmen Reservoir in North Taiwan, to investigate its applicability and practicability. The proposed method is first built to a simple formulation for the operation of the Shihmen Reservoir, with single objective and single decision. Its results were compared to those obtained by dynamic programming. The constructed network proved to be a good operational strategy. The method was then built and applied to the reservoir with multiple (five) decision variables. The results demonstrated that the developed evolving neural networks improved the operation performance of the reservoir when compared to its current operational strategy. The system was capable of successfully simultaneously handling various decision variables and provided reasonable and suitable decisions.     

MULTIOBJECTIVE PROGRAMMING MODEL FOR SEDIMENT SLUICING AND POWER GENERATION IN RESERVOIR OPERATION

I. INTRODUCTIONReservoir operation study in planning stage is an important task for the water resourcesdevelopment design. In the past. much attention has been paid to making the most ot' floodcontol. power generation, navigation. and water supply, but less to the unfavorable effectsdue to reservoir sedimentation. According to the survey of 425 reservoirs in Japan with a total storage capacity of 13.2 billion mJ a volume of 825 million m3 of sediment was depositedby the end of 1979. This…     

Adaptive volume penalization for ocean modeling

The development of various volume penalization techniques for use in modeling topographical features in the ocean is the focus of this paper. Due to the complicated geometry inherent in ocean boundaries, the stair-step representation used in the majority of current global ocean circulation models causes accuracy and numerical stability problems. Brinkman penalization is the basis for the methods developed here and is a numerical technique used to enforce no-slip boundary conditions through the addition of a term to the governing equations. The second aspect to this proposed approach is that all governing equations are solved on a nonuniform, adaptive grid through the use of the adaptive wavelet collocation method. This method solves the governing equations on temporally and spatially varying meshes, which allows higher effective resolution to be obtained with less computational cost. When penalization methods are coupled with the adaptive wavelet collocation method, the flow near the boundary can be well-resolved. It is especially useful for simulations of boundary currents and tsunamis, where flow near the boundary is important. This paper will give a thorough analysis of these methods applied to the shallow water equations, as well as some preliminary work applying these methods to volume penalization for bathymetry representation for use in either the nonhydrostatic or hydrostatic primitive equations.     

Reservoir performance under uncertainty in hydrologic impacts of climate change

Relatively few studies have addressed water management and adaptation measures in the face of changing water balances due to climate change. The current work studies climate change impact on a multipurpose reservoir performance and derives adaptive policies for possible future scenarios. The method developed in this work is illustrated with a case study of Hirakud reservoir on the Mahanadi river in Orissa, India, which is a multipurpose reservoir serving flood control, irrigation and power generation. Climate change effects on annual hydropower generation and four performance indices (reliability with respect to three reservoir functions, viz. hydropower, irrigation and flood control, resiliency, vulnerability and deficit ratio with respect to hydropower) are studied. Outputs from three general circulation models (GCMs) for three scenarios each are downscaled to monsoon streamflow in the Mahanadi river for two future time slices, 2045–65 and 2075–95. Increased irrigation demands, rule curves dictated by increased need for flood storage and downscaled projections of streamflow from the ensemble of GCMs and scenarios are used for projecting future hydrologic scenarios. It is seen that hydropower generation and reliability with respect to hydropower and irrigation are likely to show a decrease in future in most scenarios, whereas the deficit ratio and vulnerability are likely to increase as a result of climate change if the standard operating policy (SOP) using current rule curves for flood protection is employed. An optimal monthly operating policy is then derived using stochastic dynamic programming (SDP) as an adaptive policy for mitigating impacts of climate change on reservoir operation. The objective of this policy is to maximize reliabilities with respect to multiple reservoir functions of hydropower, irrigation and flood control. In variations to this adaptive policy, increasingly more weightage is given to the purpose of maximizing reliability with respect to hydropower for two extreme scenarios. It is seen that by marginally sacrificing reliability with respect to irrigation and flood control, hydropower reliability and generation can be increased for future scenarios. This suggests that reservoir rules for flood control may have to be revised in basins where climate change projects an increasing probability of droughts. However, it is also seen that power generation is unable to be restored to current levels, due in part to the large projected increases in irrigation demand. This suggests that future water balance deficits may limit the success of adaptive policy options.     

THEORETICAL APPROACH TO THE SOLUTION OF THE INFILTRATION PROBLEM

ABSTRACT

The one-dimensional transient downward entry of water in unsaturated soils is investigated theoretically. The mathematical equation describing the infiltration process is derived by combining Darcy's dynamic equation of motion with the continuity and thermodynamic state equations adjusted for the unsaturated flow conditions. The resulting equation together with the corresponding initial and boundary conditions constitues a mathematical initial boundary value problem requiring the solution of a nonlinear partial differential equation of the parabolic type. The volumetric water content is taken as the dependent variable and the time and the position along the vertical direction are taken as the independent variables. The governing equation is of such nature that a solution exists for t > 0 and is uniquely determined if two relationships are defined, together with the specified state of the system, at the initial time t = 0 and at the two boundaries. The two required relations are those of pressure versus permeability and pressure versus volumetric water content.

Since the partial differential equation has strong non-linear terms, a discrete solution is obtained by approximating the derivatives with finite-differences at discrete mesh points in the solution domain and integrated for the corresponding initial and boundary conditions. The use of an implicit difference scheme is employed in order to generate a system of simultaneous non-linear equations that has to be solved for each time increment. For n mesh points the two boundary conditions provide two equations and the repetition of the recurrence formula provides n—2 equations, the total being n equations for each time increment. The solution of the system is obtained by matrix inversion and particularly with a back-substitution technique. The FORTRAN statements used for obtaining the solution with an electronic digital computer (IBM 704) are presented together with the input data.

Analysis of the errors involved in the numerical solution is made and the stability and convergence of the solution of the approximate difference equation to that of the differential equation is investigated. The method applied is that of making a Fourier series expansion of a whole line of errors and then following the progress of the general term of the series expansion and also the behavior of each constituent harmonic. The errors (forming a continuous function of points in an abstract Banach space) are represented by vectors with the Fourier coefficients constituting a second Banach space. The amplification factor of the difference equation is shown to be always less than unity which guarantees the stability of the employed implicit recurrence scheme.

Experiments conducted on a vertical column packed uniformly with very fine sand, show a satisfactory agreement between the theoretically and experimentally obtained values. Many experimental results are shown in an attempt to explain the infiltration phenomenon with emphasis on the shape and movement of the wet front, and the effects of the degree of compaction, initial water content and deaired water on the infiltration rate.     

Optimal operation of reservoirs for downstream water quality control using linked simulation optimization

A methodology is developed for optimal operation of reservoirs to control water quality requirements at downstream locations. The physicochemical processes involved are incorporated using a numerical simulation model. This simulation model is then linked externally with an optimization algorithm. This linked simulation–optimization‐based methodology is used to obtain optimal reservoir operation policy. An elitist genetic algorithm is used as the optimization algorithm. This elitist‐genetic‐algorithm‐based linked simulation–optimization model is capable of evolving short‐term optimal operation strategies for controlling water quality downstream of a reservoir. The performance of the methodology developed is evaluated for an illustrative example problem. Different plausible scenarios of management are considered. The operation policies obtained are tested by simulating the resulting pollutant concentrations downstream of the reservoir. These performance evaluations consider various scenarios of inflow, permissible concentration limits, and a number of management periods. These evaluations establish the potential applicability of the developed methodology for optimal control of water quality downstream of a reservoir. Copyright © 2007 John Wiley & Sons, Ltd.     

Evaluation of stochastic reservoir operation optimization models

This paper investigates the performance of seven stochastic models used to define optimal reservoir operating policies. The models are based on implicit (ISO) and explicit stochastic optimization (ESO) as well as on the parameterization–simulation–optimization (PSO) approach. The ISO models include multiple regression, two-dimensional surface modeling and a neuro-fuzzy strategy. The ESO model is the well-known and widely used stochastic dynamic programming (SDP) technique. The PSO models comprise a variant of the standard operating policy (SOP), reservoir zoning, and a two-dimensional hedging rule. The models are applied to the operation of a single reservoir damming an intermittent river in northeastern Brazil. The standard operating policy is also included in the comparison and operational results provided by deterministic optimization based on perfect forecasts are used as a benchmark. In general, the ISO and PSO models performed better than SDP and the SOP. In addition, the proposed ISO-based surface modeling procedure and the PSO-based two-dimensional hedging rule showed superior overall performance as compared with the neuro-fuzzy approach.     

Multi-objective and multi-scheme research on water and sediment regulation potential of reservoirs in the upper Yellow River

The problems of the deteriorating relation between water and sediment, and the escalating conflict between water supply and demand in the upper Yellow River(YR), need to be addressed. Reservoirs of Longyangxia, Liujiaxia, and Heishanxia(Long-Liu-Hei) in the upper YR are taken as the research object, and a multi-objective, water-sediment optimal operation model for cascade reservoirs Long-Liu-Hei has been developed. This operation model considers the comprehensive requirements of water supply, po...