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.     

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.     

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.     

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.     

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.     

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.     

Reservoir operation using grey fuzzy stochastic dynamic programming

This paper presents an optimal regulation programme, grey fuzzy stochastic dynamic programming (GFSDP), for reservoir operation. It is composed of a grey system, fuzzy theory and dynamic programming. The grey system represents data by covering the whole range without loss of generality, and the fuzzy arithmetic takes charge of the rules of reservoir operation. The GFSDP deals with the multipurpose decision‐making problem by fuzzy optimization theorem. The practicability and effectiveness of the proposed approach is tested on the operation of the Shiman reservoir in Taiwan. The current M5 operating rule curves of this reservoir also are evaluated. The simulation results demonstrate that this new approach, in comparison with the M5 rule curves, has superior performance with regard to the total water deficit and number of monthly deficits. Copyright © 2002 John Wiley & Sons, Ltd.     

A dual two-stage stochastic model for flood management with inexact-integer analysis under multiple uncertainties

This study introduces a hybrid optimization approach for flood management under multiple uncertainties. An inexact two-stage integer programming (ITIP) model and its dual formation are developed by integrating the concepts of mixed-integer and interval-parameter programming techniques into a general framework of two-stage stochastic programming. The proposed approach provides a linkage to pre-defined management policies, deals with capacity-expansion planning issues, and reflects various uncertainties expressed as probability distributions and discrete intervals for a flood management system. Penalties are imposed when the policies are violated. The marginal costs are determined based on dual formulation of the ITIP model, and their effects on the optimal solutions are investigated. The developed model is applied to a case study of flood management. The solutions of binary variables represent the decisions of flood-diversion–capacity expansion within a multi-region, multi-flow-level, and multi-option context. The solutions of continuous variables are related to decisions of flood diversion toward different regions. The solutions of dual variables indicate the decisions of marginal costs associated with the resources of regions’ capacity, water availability, and allowable diversions. The results show that the proposed approach could obtain reliable solutions and adequately support decision making in flood management.     

Development of a fuzzy-queue-based stochastic quadratic program for water resources planning

In this study, a fuzzy-queue (FQ)-based inexact stochastic quadratic programming (SQP) method is developed through coupling FQ technique with inexact SQP. FQ-SQP improves upon the existing stochastic programming methods by considering the effects of queuing phenomenon during the water resources allocation process. FQ-SQP cannot only handle uncertainties expressed as interval values, random variables, and fuzzy sets, but also tackle nonlinearity in the objective function; more importantly, it can reflect the effects of FQ on water resources allocation and system benefit. The FQ-SQP model is applied to a case study of planning water resources management, where FM/FM/1 (fuzzy exponential interarrival time, fuzzy exponential service time, and one server) queue is incorporated within the SQP modeling framework. Based on α-cut analysis technique, interval solutions with fuzzy arrival and service rates have been generated, which result in different water resources allocation patterns as well as changed waiting water amounts and system benefits. Results indicate that consideration of queuing problem impacts on water resources allocation can provide more useful information for decision makers and gain in-depth insights into the effects of queuing problems for water resources allocation.     

Factorial two-stage stochastic programming for water resources management

This study presents a factorial two-stage stochastic programming (FTSP) approach for supporting water resource management under uncertainty. FTSP is developed through the integration of factorial analysis and two-stage stochastic programming (TSP) methods into a general modeling framework. It can handle uncertainties expressed as probability distributions and interval numbers. This approach has two advantages in comparison to conventional inexact TSP methods. Firstly, FTSP inherits merits of conventional inexact two-stage optimization approaches. Secondly, it can provide detailed effects of uncertain parameters and their interactions on the system performance. The developed FTSP method is applied to a hypothetical case study of water resources systems analysis. The results indicate that significant factors and their interactions can be identified. They can be further analyzed for generating water allocation decision alternatives in municipal, industrial and agricultural sectors. Reasonable water allocation schemes can thus be formulated based on the resulting information of detailed effects from various impact factors and their interactions. Consequently, maximized net system benefit can be achieved.     

An inexact stochastic optimization model for agricultural irrigation management with a case study in China

In this study, an inexact stochastic optimization model (ITSOM) is developed for agricultural irrigation management with a case study in China. Functional intervals are introduced into the modeling framework to much accurately address the spatial and temporal variation of system components. According to the results of case study, the developed model shows effectiveness in dealing with functional information of system parameters, and brings no difficulty in obtaining optimal water allocation patterns. It is indicated that the surface water resource (i.e. Heshui River) has better be used as the major source, and proper exploration of groundwater can curtail the related expense and further increase the system net benefit. Among eight farms, hybrid rice farm is going to obtain the greatest amount of water than the others, while watermelon farm has the priority to get water due to its highest benefit and penalty rate. In comparison, water allocations to rapeseed and tea farms are to be minimal within the respective fluctuation ranges. Scenario analysis is also conducted to clarify the differences between ITSOM and a conventional interval two-stage stochastic programming (ITSP) model. A total of 60 scenarios are initiated respectively linking to 60 monthly ITSP models for the entire planning horizon. The results show that the optimal objective function values of all ITSP models always fall into the range of that obtained from ITSOM. As each ITSP solution can only correspond to the system condition under a certain time point, it is highly vulnerable to system variation.     

Single multipurpose reservoir design: a modified optimal control problem by chance-constrained programming

This paper presents a chance-constrained programming model for optimal control of a multipurpose reservoir and its modification to a model for single reservoir design. An algorithm is developed for solving complex stochastic problems of multipurpose reservoir planning and design. The complexity of the problem is resolved by a two-step algorithm: (1) transformation of chance constraints on the state and control variables is performed at the first step; and (2) the choice of optimum control or optimal reservoir storage is carried out in the second step. The method of iterative convolution is chosen for the first step, while linear programming is selected for the second step. The algorithm allows the use of random inflows and random demands together with other deterministic demands. The reservoir design problem is presented as a modified optimal control problem. The procedure is illustrated with an example of a hypothetical reservoir design problem with three different types of downstream releases (hydropower production, municipal water supply, and irrigation).     

Two solution methods for dynamic game in reservoir operation

During the last decade, a number of models have been developed to consider the conflict in dynamic reservoir operation. Most of these models are discrete dynamic models which are developed based on game theory. In this study, a continuous model of dynamic game and its corresponding solutions are developed for reservoir operation. Two solution methods are used to solve the model of continuous dynamic game, namely the Ricatti equations and collocation methods. The Ricatti equations method is a closed form solution, requiring less computational efforts compared with discrete models. The collocation solution method applies Newton's method or a quasi-Newton method to find the problem solution. These approaches are able to generate operating policies for dynamic reservoir operation. The Zayandeh-Rud river basin in central Iran is used as a case study and the results are compared with alternative water allocation models. The results show that the proposed solution methods are quite capable of providing appropriate reservoir operating policies, while requiring rather short computational times due to continuous formulation of state and decision variables. Reliability indices are used to compare the overall performance of the proposed models. Based on the results from this study, the collocation method leads to improved values of the reliability indices for total reservoir system and utility satisfaction of water users, compared to the Ricatti equations method. This is attributed to the flexible structure of the collocation model. When compared to alternative water allocation models, lower values of reliability indices are achieved by the collocation method.     

Performance evaluation of implicit stochastic reservoir operation optimization supported by long-term mean inflow forecast

This study applies implicit stochastic optimization (ISO) to develop monthly operating rules for a reservoir located in Northeast Brazil. The proposed model differs from typical ISO applications as it uses the forecast of the mean inflow for a future horizon instead of the current-month inflow. Initially, a hundred different 100-year monthly inflow scenarios are synthetically generated and employed as input to a deterministic operation optimization model in order to build a database of optimal operating data. Later, such database is used to fit monthly reservoir rule curves by means of nonlinear regression analysis. Finally, the established rule curves are validated by operating the system under 100 new inflow ensembles. The performance of the proposed technique is compared with those provided by the standard reservoir operating policy (SOP), stochastic dynamic programming (SDP) and perfect-forecast deterministic optimization (PFDO). Different forecasting horizons are tested. For all of them, the results indicate the feasibility of using ISO in view of its lower vulnerability in contrast to the SOP as well as the proximity of its operations with those by PFDO. The results also reveal that there is an optimal choice for the forecasting horizon. The comparison between ISO and SDP shows small differences between both, justifying the adoption of ISO for its simplified mathematics as opposed to SDP.     

Real-time multipurpose reservoir operation: a case study

Abstract

A real-time operational methodology has been developed for multipurpose reservoir operation for irrigation and hydropower generation with application to the Bhadra reservoir system in the state of Karnataka, India. The methodology consists of three phases of computer modelling. In the first phase, the optimal release policy for a given initial storage and inflow is determined using a stochastic dynamic programming (SDP) model. Streamflow forecasting using an adaptive AutoRegressive Integrated Moving Average (ARIMA) model constitutes the second phase. A real-time simulation model is developed in the third phase using the forecast inflows of phase 2 and the operating policy of phase 1. A comparison of the optimal monthly real-time operation with the historical operation demonstrates the relevance, applicability and the relative advantage of the proposed methodology.     

A stochastic model of water-table elevation

A second order stochastic differential equation is used for modeling of water-table elevation. The data were sampled at the Borden Aquifer as a part of a tracer experiment. The purpose of the water-table data collection was to determine presence of a water flow. We argue that the water-table surface is a simple plane oscillating up and down in time according to an equation for a stochastic oscillator. We derive the model, estimate its parameters and provide arguments for goodness-of-fit of the model.     

A stochastic model of water-table elevation

A second order stochastic differential equation is used for modeling of water-table elevation. The data were sampled at the Borden Aquifer as a part of a tracer experiment. The purpose of the water-table data collection was to determine presence of a water flow. We argue that the water-table surface is a simple plane oscillating up and down in time according to an equation for a stochastic oscillator. We derive the model, estimate its parameters and provide arguments for goodness-of-fit of the model.     

Risk assessment of agricultural irrigation water under interval functions

In recent years, water shortages and unreliable water supplies have been considered as major barriers to agricultural irrigation water management in China, which are threatening human health, impairing prospects for agriculture and jeopardizing survival of ecosystems. Therefore, effective and efficient risk assessment of agricultural irrigation water management is desired. In this study, an inexact full-infinite two-stage stochastic programming (IFTSP) method is developed. It incorporates the concepts of interval-parameter programming and full-infinite programming within a two-stage stochastic programming framework. IFTSP can explicitly address uncertainties presented as crisp intervals, probability distributions and functional intervals. The developed model is then applied to Zhangweinan river basin for demonstrating its applicability. Results from the case study indicate that compromise solutions have been obtained. They provide the desired agricultural irrigation water-supply schemes, which are related to a variety of tradeoffs between conflicting economic benefits and associated penalties attributed to the violation of predefined policies. The solutions can be used for generating decision alternatives and thus help decision makers to identify desired agricultural irrigation targets with maximized system benefit and minimized system-failure risk. Decision makers can adjust the existing agricultural irrigation patterns, and coordinate the conflict interactions among economic benefit, system efficiency, and agricultural irrigation under uncertainty.     

Development of an inexact two-stage stochastic model with downside risk control for water quality management and decision analysis under uncertainty

Water quality management along rivers involves making water-allocation plans, establishing water quality goals, and controlling pollutant discharges, which is complicated itself but further challenged by existence of uncertainties. In this study, an inexact two-stage stochastic downside risk-aversion programming (ITSDP) model is developed for supporting regional water resources allocation and water quality management problems under uncertainties. The ITSDP method is a hybrid of interval-parameter programming, two-stage stochastic programming, and downside risk measure to tackle uncertainties described in terms of interval values and probability distributions. A water quality simulation model was provided for reflecting the relationship between the water resources allocation, wastewater discharge, and environmental responses. The proposed approach was applied to a hypothetical case for a shared stream water quality management with one municipal, three industrial and two agricultural sectors. A number of scenarios corresponding to different river inflows and risk levels were examined. The results demonstrated that the model could effectively communicate the interval-format and random uncertainties, and risk-aversion into optimization process, and generate a trade-off between the system economy and stability. They could be helpful for seeking cost-effective management strategies under uncertainties, and gaining an in-depth insight into the water quality management system characteristics, and make cost-effective decisions.