A Fuzzy Variable Least Core Game for Inter-basin Water Resources Allocation Under Uncertainty

In this paper, a new solution concept, called Fuzzy Variable Least Core (FVLC), is developed for fuzzy cooperative games. The FVLC is able to incorporate fuzzy input variables and result in fuzzy benefit shares of players participating in a coalition. This solution concept is used for water and benefit allocation to water users in inter-basin water transfer systems considering the uncertainties associated with their benefit coefficients. In the proposed water allocation methodology, an Integrated Stochastic Dynamic Programming (ISDP) model is developed to obtain the water rights of players and economic water allocation policies. In the next step, the total net fuzzy benefit of the system is reallocated to water users in an equitable and rational way using a FVLC-based model. In this model, a new algorithm is proposed for converting a multilateral cooperative game with fuzzy variables to some fuzzy bilateral cooperative games, which are solved using the FLVC solution concept. The applicability and efficiency of the proposed methodology is examined by applying it to a large scale inter-basin water transfer project in Iran.     

A Novel Solution for Stochastic Dynamic Game of Water Allocation from a Reservoir Using Collocation Method

In this study, a continuous model of stochastic dynamic game for water allocation from a reservoir system was developed. The continuous random variable of inflow in the state transition function was replaced with a discrete approximant rather than using the mean of the random variable as is done in a continuous model of deterministic dynamic game. As a result, a new solution method was used to solve the stochastic model of game based on collocation method. The collocation method was introduced as an alternative to linear-quadratic (LQ) approximation methods to resolve a dynamic model of game. The collocation method is not limited to the first and second degree approximations, compared to LQ approximation, i.e. Ricatti equations. Furthermore, in spite of LQ related problems, consideration of the stochastic nature of game on the action variables in the collocation method would be possible. The proposed solution method was applied to the real case of reservoir operation, which typically requires considering the effect of uncertainty on decision variables. The results of the solution of the stochastic model of game are compared with the results of a deterministic solution of game, a classical stochastic dynamic programming model (e.g. Bayesian Stochastic Dynamic Programming model, BSDP), and a discrete stochastic dynamic game model (PSDNG). By comparing the results of alternative methods, it is shown that the proposed solution method of stochastic dynamic game is quite capable of providing appropriate reservoir operating policies.     

Optimal Inter-Basin Water Allocation Using Crisp and Fuzzy Shapley Games

In recent years, uneven distribution of available water resources as well as increasing water demands and overexploiting the water resources have brought severe need for transferring water from basins having sufficient water to basins facing water shortages. Therefore, optimal allocation of shared water resources in water transfer projects, considering the utilities of different stakeholders, physical limitations of the system and socioeconomic criteria is an important task. In this paper, a new methodology based on crisp and fuzzy Shapley games is developed for optimal allocation of inter-basin water resources. In the proposed methodology, initial water allocations are obtained using an optimization model considering an equity criterion. In the second step, the stakeholders form crisp coalitions to increase the total net benefit of the system as well as their own benefits and a crisp Shapley Value game is used to reallocate the benefits produced in the crisp coalitions. Lastly, to provide maximum total net benefit, fuzzy coalitions are constituted and the participation rates of water users to fuzzy coalitions are optimized. Then, the total net benefit is reallocated to water users in a rational and equitable way using Fuzzy Shapley Value game. The effectiveness of this method is examined by applying it to a large scale case study of water transfer from the Karoon river basin in southern Iran to the Rafsanjan plain in central Iran.     

Robust Methods for Identifying Optimal Reservoir Operation Strategies Using Deterministic and Stochastic Formulations

Water allocation in a competing environment is a major social and economic challenge especially in water stressed semi-arid regions. In developing countries the end users are represented by the water sectors in most parts and conflict over water is resolved at the agency level. In this paper, two reservoir operation optimization models for water allocation to different users are presented. The objective functions of both models are based on the Nash Bargaining Theory which can incorporate the utility functions of the water users and the stakeholders as well as their relative authorities on the water allocation process. The first model is called GA–KNN (Genetic Algorithm–K Nearest Neighborhood) optimization model. In this model, in order to expedite the convergence process of GA, a KNN scheme for estimating initial solutions is used. Also KNN is utilized to develop the operating rules in each month based on the derived optimization results. The second model is called the Bayesian Stochastic GA (BSGA) optimization model. This model considers the joint probability distribution of inflow and its forecast to the reservoir. In this way, the intrinsic and forecast uncertainties of inflow to the reservoir are incorporated. In order to test the proposed models, they are applied to the Satarkhan reservoir system in the north-western part of Iran. The models have unique features in incorporating uncertainties, facilitating the convergence process of GA, and handling finer state variable discretization and utilizing reliability based utility functions for water user sectors. They are compared with the alternative models. Comparisons show the significant value of the proposed models in reservoir operation and supplying the demands of different water users.     

Integrated Artificial Neural Network (ANN) and Stochastic Dynamic Programming (SDP) Model for Optimal Release Policy

Complexicity in reservoir operation poses serious challenges to water resources planners and managers. These challenges of water reservoir operation are illustrated using a simulation to aid the development of an optimal operation policy for dam and reservoir. To achieve this, a Comprehensive Stochastic Dynamic Programming with Artificial Neural Network (SDP-ANN) model were developed and tested at Sg. Langat Reservoir in Malaysia. The nonlinearity of the natural physical processes was a major problem in determining the simulation of the reservoir parameters (elevation, surface-area, storage). To overcome water shortages resulting from uncertainty, the SDP-ANN model was used to evaluate the input variable and the performance outcome of the Model were compared with the Stochastic Dynamic Programming integrated with auto-regression (SDP-AR) model. The objective function of the models was set to minimize the sum of squared deviation from the desired targeted supply. Comparison result on the performance between SDP-AR model policy with SDP-ANN model found that the SDP-ANN model is a reliable and resilience model with a lesser supply deficit. The study concludes that the SDP-ANN model performs better than the SDP-AR model in deriving an optimal operating policy for the reservoir.     

Common pool water resources management considering a regulator interference: A game theory approach to derive managerial policies for Urmia lake,Iran

An increased need for water, combined with a scarcity of water resources, has resulted in serious challenges regarding water resource management. Those issues have always been a major concern of most research regarding the management and operation of water resources. Various qualitative and quantitative methods, Game Theory (GT) being an example, have been proposed for managing common pool water resources. This study adopts GT to address a case of common pool water resource management, including consideration of the interference of a regulator. Urmia lake in north‐western Iran was chosen as an illustration of a common source of water shared between two provinces acting as littoral stakeholders to the lake. Due to a non‐cooperative behaviour of the stakeholders, the lake is most likely to dry up. To change the game situation, a regulator was inserted into the game as a third player. The game among players is investigated in terms of two static and dynamic game methods. Two strategies are defined for each player. The equilibrium point will be achieved according to the strategies adopted by the players. For static games, a low‐level equilibrium under non‐cooperation conditions is obtained. Dynamic games among players are more likely to occur, rather than static games. The equilibrium of high‐level productivity will be obtained under cooperative conditions among the players.     

Optimizing Multiple-Pollutant Waste Load Allocation in Rivers: An Interval Parameter Game Theoretic Model

In this paper, a new methodology is developed for optimal multiple-pollutant waste load allocation (MPWLA) in rivers considering the main existing uncertainties. An interval optimization method is used to solve the MPWLA problem. Different possible scenarios for treatment of pollution loads are defined and corresponding treatment costs are taken into account in an interval parameter optimization model. A QUAL2Kw-based water quality simulation model is developed and calibrated to estimate the concentration of the water quality variables along the river. Two non-cooperative and cooperative multiple-pollutant scenario-based models are proposed for determining waste load allocation policies in rivers. Finally, a new fuzzy interval solution concept for cooperative games, namely, Fuzzy Boundary Interval Variable Least Core (FIVLC), is developed for reallocating the total fuzzy benefit obtained from discharge permit trading among waste load dischargers. The results of applying the proposed methodology to the Zarjub River in Iran illustrate its effectiveness and applicability in multiple-pollutant waste load allocation in rivers.     

Development of an Optimal Reservoir Operation Scheme Using Extended Evolutionary Computing Algorithms Based on Conflict Resolution Approach: A Case Study

Optimal reservoir operation and water allocation are critical issues in sustainable water resource management due to increasing water demand. Multiplicity of stockholders with different objectives and utilities makes reservoir operation a complicated problem with a variety of constraints and objectives to be considered. In this case, the conflict resolution models can be efficiently used to determine the optimal water allocation scheme considering the utility and relative authority of different stakeholders. In this study, the Nash product is used for formulation of the objective function of a reservoir water allocation model. The Analytic Hierarchy Process (AHP) is used to determine the importance of each stockholder in bargaining for water. The Particle Swarm Optimization algorithm (PSO) and the Imperialism Competitive Algorithm (ICA) are applied to solve the proposed optimization model. System performance indices including reliability, resiliency and vulnerability are used to evaluate the performance of optimization algorithms. The simplest and most often-used reservoir policy (Standard Operating Policy, SOP) is also used in order to evaluate the performance of the proposed models. The proposed model is applied to the Karkheh River-Reservoir system located in south western part of Iran as a case study. Results show the significance of the application of conflict resolution models, such as the Nash theory and proposed optimization algorithms, for water allocation in the regional scale especially in complicated water supply systems.     

Water Resources Allocation Using Solution Concepts of Fuzzy Cooperative Games: Fuzzy Least Core and Fuzzy Weak Least Core

In this paper, two new solution concepts for fuzzy cooperative games, namely Fuzzy Least Core and Fuzzy Weak Least Core are developed. They aim for optimal allocation of available water resources and associated benefits to water users in a river basin. The results of these solution concepts are compared with the results of some traditional fuzzy and crisp games, namely Fuzzy Shapley Value, Crisp Shapley Value, Least Core, Weak Least Core and Normalized Nucleolus. It is shown that the proposed solution concepts are more efficient than the crisp games. Moreover, they do not have the limitation of Fuzzy Shapley Value in satisfying the group rationality criterion. This paper consists of two steps. In the first step, an optimization model is used for initial water allocation to stakeholders. In the second step, fuzzy coalitions are defined and participation rates of water users (players) in the fuzzy coalitions are optimized in order to reach a maximum net benefit. Then, the total net benefit is allocated to the players in a rational and equitable way using Fuzzy Least Core, Fuzzy Weak Least Core and some traditional fuzzy and crisp games. The effectiveness and applicability of the proposed methodology is examined using a numerical example and also applying it to the Karoon river basin in southern Iran.     

A Model for Optimal Allocation of Water to Competing Demands

The present study develops a simple interactive integrated water allocation model (IWAM), which can assist the planners and decision makers in optimal allocation of limited water from a storage reservoir to different user sectors, considering socio-economic, environmental and technical aspects. IWAM comprises three modules—a reservoir operation module (ROM), an economic analysis module (EAM) and a water allocation module (WAM). The model can optimize the water allocation with any of two different objectives or two objectives together. The two individual objectives included in the model are the maximization of satisfaction and the maximization of net economic benefit by the demand sectors. Weighting technique (WT) or simultaneous compromise constraint (SICCON) technique is used to convert the multi-objective decision-making problem into a single objective function. The single objective functions are optimized using linear programming. The model applicability is demonstrated for various cases with a hypothetical example.     

Water Resources Allocation Using a Cooperative Game with Fuzzy Payoffs and Fuzzy Coalitions

In this paper, two fuzzy cooperative games are utilized for modeling equitable and efficient water allocation among water users in both inter-basin and intra-basin water allocation problems. The proposed all-inclusive water allocation approach consists of three main steps, following Sadegh et al. (Water Resour Manage 24(12):2991?C2310, 2010). In the first step, an initial water allocation is carried out using an optimization model taking into account an equity criterion. In the second step, the water users form crisp coalitions with fuzzy characteristic functions to increase the total net benefit of the system and also their own benefits. In the methodology used in this step, the water users do not need to have exact information about their payoffs and they can evaluate their payoffs as fuzzy numbers. In the second step, based on the Hukuhara difference of fuzzy numbers, optimum water allocation strategies are determined using a game with fuzzy characteristic function. In the third step, we applied another methodology which considers a class of fuzzy games with fuzzy characteristic functions and also fuzzy coalitions for water allocation. The methodology of this step is on the basis of the Hukuhara difference and the Choquet integral. The usefulness of the mentioned methodologies is studied by applying them to three defined real life scenarios in a case study of water allocation in Iran. The results showed that the proposed methodologies are professionally appropriate to real-world uncertain problems of equitable and economic inter-basin and intra-basin water resources allocations.     

利益相关者的水资源配置博弈

水资源的准公共产品属性及其在开发利用中存在的外部性使河流自然流态受到严重的扰动,本文基于非合作博弈博弈论,建立主从关系的用户博弈模型分析河流水资源分配,基于个体效益不能达到帕累托最优状态,采用合作博弈方法,建立水资源用户合作博弈模型。合作博弈增加了总效益,但使得上游用户的效益比非合作时减少,有必要对上游用户进行效益补偿:引入微观经济学的无差异曲线与合作联盟形成的条件构建合作效益分配的可行解集,并提出基于边际贡献的效益分配方法在可行解集中挑选唯一的效益分配解。研究结果表明,合作博弈总效益大于非合作博弈的总效益,达到帕累托最优状态;通过合作效益分配,处于先动优势的上游用户的分配效益大于非合作的效益,有利于行动占先者加入合作联盟。     

水市场的博弈分析

本文应用博弈论研究了水资源管理部门如何利用初始水权分配和水资源费对水市场进行有效的宏观调控,以实现水资源的优化配置。分析了市场经济条件下以水权和水市场为基础的水资源配置的过程。根据用水者在水市场上的行为特征,建立了以水资源总效益最大化为目标的两阶段动态博弈模型。给出了求解子博弈精炼纳什均衡条件下市场参与者的行动策略方法,从而得出管理机构相应的最优初始水权分配方案和水资源费率方案,并用算例阐明了建模的思路与方法。     

A Nonlinear Interval Model for Water and Waste Load Allocation in River Basins

In this paper, a new methodology is proposed for simultaneous allocation of water and waste load in river basins. A nonlinear interval number optimization model is used to incorporate the uncertainties of model inputs and parameters. In this methodology, the bounds of the uncertain inputs are only required, not necessarily knowing their probability density or fuzzy membership functions. In the proposed model, the existing uncertainties in water demands and monthly available water are considered in the optimization model. Also the economic and environmental impacts of water allocation to the agricultural water users are taken into account. To have an equitable water and waste load allocation, benefits are reallocated to water users using some solution concepts of the cooperative game theory. Results of applying the methodology to the Dez river system in south-western part of Iran show its effectiveness and applicability for water and waste load allocation in an uncertain environment.     

不确定性多目标模糊规划在水资源优化配置中的应用

在充分考虑水资源系统中供需水不确定性的基础上,基于多目标规划、模糊规划和区间规划原理,以经济效益、社会效益和环境效益最大为目标,以需水量、可供水量和不同子区用水部门间的用水公平性为主要约束,构建一种适用于多水源、多子区、多用户的考虑供需水不确定性的多目标模糊规划模型,以期通过平衡水资源配置系统中用户配置水量、缺水风险与系统收益三者的关系,实现系统综合效益最大。模型以衡水市历年供水、用水及社会经济等数据确定规划年的水资源参数及经济参数,采用区间参数反映系统中的不确定性,通过引入模糊隶属度函数,利用两步交互式算法,将多目标规划转化为单目标规划进行求解,以2025年为衡水市规划水平年,得到规划水平年下的11个子区、4种水源、4个用水行业的最优配水方案。结果表明：本研究制定的水资源优化配置方案可实现“外调水优先利用、地下水控制利用”,同时考虑用水的公平性约束后,在缺水条件下可有效控制水资源向每立方米水效益高的区域和部门流动,有利于多区域、多部门共担缺水风险;该水资源优化配置方案可有效缓解衡水市水资源供需矛盾,实现多水源和多目标之间的协同互补。研究成果可为河北省其他县域的水资源优化配置提供技术参...     

Technical Communication: Two-Phase Stochastic Dynamic Programming Model for Optimal Operation of Irrigation Reservoir

A two-phase stochastic dynamic programming model is developed for optimal operation of irrigation reservoirs under a multicrop environment. Under a multicrop environment, the crops compete for the available water whenever the water available is less than the irrigation demands. The performance of the reservoir depends on how the deficit is allocated among the competing crops. The proposed model integrates reservoir release decisions with water allocation decisions. The water requirements of crops vary from period to period and are determined from the soil moisture balance equation taking into consideration the contribution of soil moisture and rainfall for the water requirements of the crops. The model is demonstrated over an existing reservoir and the performance of the reservoir under the operating policy derived using the model is evaluated through simulation.     

An alternative approach to the operation of multinational reservoir systems: Application to the Amistad &; Falcon system (Lower Rio Grande/Río Bravo)

An optimization approach for the operation of international multi-reservoir systems is presented. The approach uses Stochastic Dynamic Programming (SDP) algorithms – both steady-state and real-time – to develop two models. In the first model, the reservoirs and flows of the system are aggregated to yield an equivalent reservoir, and the obtained operating policies are disaggregated using a non-linear optimization procedure for each reservoir and for each nation's water balance. In the second model a multi-reservoir approach is applied, disaggregating the releases for each country's water share in each reservoir. The non-linear disaggregation algorithm uses SDP-derived operating policies as boundary conditions for a local time-step optimization. Finally, the performance of the different approaches and methods is compared. These models are applied to the Amistad-Falcon International Reservoir System as part of a binational dynamic modeling effort to develop a decision support system tool for a better management of the water resources in the Lower Rio Grande Basin, currently enduring a severe drought.     

Water Costs Allocation in Complex Systems Using a Cooperative Game Theory Approach

The management of complex water resource systems that address water service recovery costs and consider adequate contributions and priorities require methods that integrate technical, economic, environmental, social and legal aspects into a comprehensive framework. In Europe, the Water Framework Directive (WFD) 2000/60/EC recommends that the pricing politics in a river basin take into account the cost recovery and the economic sustainability of the water use. However, the current cost allocation methods do not consider the user’s willingness to pay and often do not permit a total cost recovery. Thus, a new approach is required that includes these requirements when defining water rates. This article presents a methodology to allocate water service costs in a water resource system among different users that attempts to fulfil the WFD requirements. The methodology is based on Cooperative Game Theory (CGT) techniques and on the definition of the related characteristic function using a mathematical optimisation approach. The CGT provides the instruments that are necessary to analyse situations that require a cost-sharing rule. The CGT approach can define efficient and fair solutions that provide the appropriate incentives among the parties involved. Therefore, the water system cost allocation has been valued as a game in which it is necessary to determine the right payoff for each player that is, in this case, a water user. To apply the CGT principles in a water resources system, the characteristic function needs to be defined and evaluated using an adequate modelling approach; in this study, it is evaluated using the optimisation model WARGI. (Sechi and Zuddas 2000). The so-called “core” represents the game-solution set. It represents the area of the admissible cost allocation values from which the boundaries on the cost values for each player can be supplied. Within the core lie all of the allocations that satisfy the principles of equity, fairness, justice, efficiency and that guarantee cost recovery. The core of a cooperative game can represent a useful instrument to define the water cost rates. Furthermore, it can be used as a valid support in water resource management to achieve the economic analysis required by the WFD. The methodology was applied to a multi-reservoir and multi-demand water system in Sardinia, Italy.     

Incorporating Economic and Political Considerations in Inter-Basin Water Allocations: A Case Study

When an inter-basin water transfer is expected among basins with some level of unfriendliness or hostility, ignoring political considerations, which are generally not integrated in economic investigations, can impede an integrated and efficient management. In this paper, a new economic-political methodology is proposed for the optimal and efficient allocation of water resources among water users in inter-basin water transfer systems. The proposed framework quantifies both the economic payoffs using an “n-person real fuzzy cooperative game”, and the political formation prospect of any coalition, using a Modified Political Accounting System (MPAS). The proposed economic-political methodology is applied to a large scale inter-basin water allocation problem including water donor and receiving basins struggling with water scarcity. The results show how including political considerations in the study may provide a more satisfactory solution compared to the just cost-effective water allocations.     

水库汛限水位实时动态控制方案优选研究

水库汛限水位动态控制方案优选属多阶段多目标群决策过程,在水库调度中具有重要作用。针对水库汛限水位动态控制多目标的矛盾性,本文将基于协商对策的多目标群决策模型,应用于水库汛限水位实时动态控制方案优选,选取洪水资源利用率、保证率可靠度、调洪最高水位和最大下泄流量作为优选指标,并采用基于博弈论的组合赋权法将经验权重与数学权重融合,确定各指标权重。以大伙房水库为例,分析其汛限水位实时动态控制时,决策者不同偏好时的最优值。结果表明,采用该模型优选出的方案可以为决策者提供较为直观和合理的决策依据。