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.     

Waste load allocation for water quality management of a heavily polluted river using linear programming.

A waste load allocation model using linear programming has been developed for economic water quality management. A modified Qual2e model was used for water quality calculations and transfer coefficients were derived from the calculated water quality. This allocation model was applied to the heavily polluted Gyungan River, located in South Korea. For water quality management of the river, two scenarios were proposed. Scenario 1 proposed to minimise the total waste load reduction in the river basin. Scenario 2 proposed to minimise waste load reduction considering regional equity. Waste loads, which have to be reduced at each sub-basin and WWTP, were determined to meet the water quality goal of the river. Application results of the allocation model indicate that advanced treatment is required for most of the existing WWTPs in the river basin and construction of new WWTPs and capacity expansion of existing plants are necessary. Distribution characteristics of pollution sources and pollutant loads in the river basin was analysed using Arc/View GIS.     

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.     

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.     

Equitable Waste Load Allocation in Rivers Using Fuzzy Bi-matrix Games

This paper presents a new game theoretic methodology for equitable waste load allocation in rivers utilizing fuzzy bi-matrix games, Non-dominated Sorting Genetic Algorithms II (NSGA-II), cooperative game theory, Bayesian Networks (BNs) and Probabilistic Support Vector Machines (PSVMs). In this methodology, at first, a trade-off curve between objectives, which are average treatment level of dischargers and fuzzy risk of low water quality, is obtained using NSGA-II. Then, the best non-dominated solution is selected using a non-zero-sum bi-matrix game with fuzzy goals. In the next step, to have an equitable waste load allocation, some possible coalitions among dischargers are formed and treatment costs are reallocated to discharges and side payments are calculated. To develop probabilistic rules for real-time waste load allocation, the proposed model is applied considering several scenarios of pollution loads and the results are used for training and testing BNs and PSVMs. The applicability and efficiency of the methodology are examined in a real-world case study of the Zarjub River in the northern part of Iran. The results show that the average relative errors of the proposed rules in estimating the treatment levels of dischargers are less than 5?%.     

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.     

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.     

Waste Load Allocation in Rivers using Fallback Bargaining

In this paper, bargaining process between different stakeholders involved in a waste load allocation problem is simulated using the Fallback Bargaining (FB) concept. The paper considers two main parties in a waste load allocation problem. On the one hand, there are wastewater dischargers intending to minimize their treatment costs and on the other hand, there is an environmental protection agency which monitors the river water quality at a checkpoint downstream of the location of dischargers. In this paper, different alternatives which are combinations of dischargers’ treatment scenarios are defined. A water quality simulation model is utilized to estimate the concentration of the water quality indicator along the river based on a selected alternative. If the concentration of water quality indicator in the selected checkpoint violates the water quality standards, a penalty function is used to calculate the amount of penalty assigned to dischargers. The allocated cost to each discharger is computed considering his treatment scenario as well as the penalty allocated to him. Two kinds of Fallback bargaining procedure termed as Unanimity Fallback Bargaining (UFB) and Fallback bargaining with Impasse (FBI), which both aim at minimizing the maximum dissatisfaction of bargainers in a negotiation problem, are utilized for finding a Compromise Set (CS) of alternatives. In this paper, the best alternative (alternatives) among CS members is (are) selected using a social choice theory namely Condorcet winner. The results of these two approaches are compared and the final alternative is selected which shows the initial Tradable Discharge Permits (TDPs) allocated to dischargers. Finally, in order to decrease the total allocated cost to dischargers, initial allocated TDPs are exchanged between them using the Extended Trading Ratio System (ETRS) developed by Mesbah et al. (Environ Model Software 24:238–246, 2009). The applicability and efficiency of the proposed methodology is investigated by applying it to a case study of the Zarjub River in the northern part of Iran.     

A Coupled Water Quantity–Quality Model for Water Allocation Analysis

As the demand for water continuously increases with population growth and economic development, the gap between water supply and demand in China has become increasingly wide. In recent years worsening water pollution has caused this gap to become much more serious. Conventional allocation pattern, which mainly considers water quantity as the key factor, is no longer satisfying the water allocation need. A coupled water quantity–quality model in a river basin is presented in this paper to provide a tool for water allocation schemes analysis. The pollutants transport and hydrological cycling processes in a river basin are involved in the model. A river network is divided into different reaches. According to the division of river network, the areas out of the river are divided into a series of tanks. In each tank, hydrologic processes, pollutant loading production, water demand of users and water supply are calculated. In river network, hydrodynamics processes and water quality are simulated. Water quantity and quality exchanges between each tank and river are also considered. The time step of water quality calculation is 24 h, the same with that of water quantity calculation. In each time step period, the connections of river reaches and tanks are realized through the exchange of water quantity and quality between rivers and tanks: pollutants discharge from tanks to rivers and water intake from rivers to satisfy water demand in tanks. The water use in each tank includes three types: domestic, industrial and agricultural water use. Water allocation schemes are one of the input conditions of the model. Using the proposed model, in each tank, water demand and deficit of different uses, in both water quantity and quality, can be obtained under different water allocation schemes. According to the water deficit, water allocation schemes are analyzed to make proper allocation schemes. In this aspect, the proposed model can also be thought as a water allocation model. The model is tested and applied to the Jiaojiang River basin, Zhejiang Province, China, to analyze the different water resource allocation schemes.     

Non-Uniform Flow Effect on Optimal Waste Load Allocation in Rivers

In this study, the effects of non-uniform flow due to: (i) inflow from tributaries and (ii) the presence of a downstream control structure (such as a weir or a barrage), on the optimal waste load allocation decision and the resulting cost-equity trade-off relationships, have been investigated. These effects are illustrated with in the framework of a typical cost-equity multi-objective optimization model for optimal waste load allocation in rivers. This framework consists of an embedded river water quality simulator with gradually varied flow and transport (BOD-DO) modules and a cost-equity multi-objective optimization model. A multi-objective evolutionary algorithm known as Non-dominated Sorting Genetic Algorithm-II is used for solving the optimization problem. The optimal fraction removal levels, the treatment cost and the system inequity measure are under predicted in certain reaches of the river, if the uniform flow assumption is made, while actually non-uniform flow conditions exist. This effect is quite pronounced when the flow non-uniformity results from a downstream control structure such as a weir.     

An Interval Parameter Model for Cooperative Inter-Basin Water Resources Allocation Considering the Water Quality Issues

In this paper, a new methodology based on interval optimization and game theory is developed for optimal operation of an inter-basin water transfer system considering efficiency, equity and sustainability criteria. A linear version of the agricultural water production function is proposed and used for incorporating deficit irrigation. The interval programming and cooperative game theory are utilized for equitable reallocation of benefits to water users in both water donor and receiving basins. To assess the sustainability of water allocation policies, water quality and environmental flow in the donor basin and groundwater table drawdown in the receiving basin are taken into consideration. The effectiveness of this methodology is examined by applying it to a large scale case study of water transfer from the Karoon river basin in south-west to the Rafsanjan plain in the central part of Iran.     

信息熵在水污染物总量区域公平分配中的应用

在综合已有公平分配准则的基础上，通过综合考虑待分配区域的经济、自然等客观因素，筛选出反映公平准则的相关指标。借鉴信息熵的概念，计算单位指标负荷污染物量信息熵，应用熵值法得到各指标的权重，加权求和得到总的信息熵。构建以加权信息熵最大化为目标的规划模型，设定合理的运算规则和约束条件，制定基于公平性的水污染物总量的区域初始分配方案。将该方法实际应用于天津市各区县的水污染物分配，采用信息熵权衡公平准则指标得到方案一到四，结果表明：东丽区、汉沽区和中心城区是重点削减区域，其次是塘沽区和宁河县。该方法可以定量化地反映多种公平准则，得到公平可行的分配方案。     

A Multi-objective Linear Programming Model with Interval Parameters for Water Resources Allocation in Dalian City

Water resources management has been of concern for many researchers since the contradiction between increased water demand and decreased water supply has become obvious. In the real world, water resources systems usually have complexities among social, economic, natural resources and environmental aspects, which leads to multi-objective problems with significant uncertainties in system parameters, objectives, and their interactions. In this paper, a multi-objective linear programming model with interval parameters has been developed wherein an interactive compromising algorithm has been introduced. Through interactive compromising conflicts among multi-objectives, a feasible solution vector can be obtained. The developed model is then applied to allocation of multi-source water resources with different water qualities to multiple users with different water quality requirements for the Dalian city for 2010, 2015 and 2020 planning years. The model pursues the maximum synthesis benefits of economy, society and the environment. The results indicate that the proportion of reused water to the total water amount is gradually increasing, and the proportion of agricultural water consumption to the total water consumption is gradually decreasing. The allocation of multi-source water resources to multiple users is improved due to incorporation of uncertain factors into the model that provide useful decision support to water management authorities.     

A Fuzzy-Stochastic Modeling Approach for Multiple Criteria Decision Analysis of Coupled Groundwater-Agricultural Systems

The complexity of water resources management increases when decisions about environmental and social issues are considered in addition to economic efficiency. Such complexities are further compounded by multiple uncertainties about the consequences of potential management decisions. In this paper, a new fuzzy-stochastic multiple criteria decision-making approach is proposed for water resources management in which a variety of criteria in terms of economic, environmental and social dimensions are identified and taken into account. The goal is to evaluate multiple conflicting criteria under uncertainties and to rank a set of management alternatives. The methodology uses a simulation-optimization water management model of a strongly interacting groundwater-agriculture system to enumerate criteria based on these bio-physical process interactions. Fuzzy and/or qualitative information regarding the decision-making process for which quantitative data is not available are evaluated in linguistic terms. Afterwards, Monte Carlo simulation is applied to combine these information and to generate a probabilistic decision matrix of management alternatives versus criteria in an uncertain environment. Based on this outcome, total performance values of the management alternatives are calculated using ordered weighted averaging. The proposed approach is applied to a real world example, where excessive groundwater withdrawal from the coastal aquifer for irrigated agriculture has resulted in saltwater intrusion, threatening the economical basis of farmers and associated societal impacts. The analysis has provided potential decision alternatives which can serve as a platform for negotiation and further exploration. Furthermore, sensitivity of different inputs to resulting rankings is investigated. It is found that decision makers’ risk aversion and risk taking attitude may yield different rankings. The presented approach suits to systematically quantify both probabilistic and fuzzy uncertainties associated with complex hydrosystems management.     

A Model for Solving the Optimal Water Allocation Problem in River Basins with Network Flow Programming When Introducing Non-Linearities

The allocation of water resources between different users is a traditional problem in many river basins. The objective is to obtain the optimal resource distribution and the associated circulating flows through the system. Network flow programming is a common technique for solving this problem. This optimisation procedure has been used many times for developing applications for concrete water systems, as well as for developing complete decision support systems. As long as many aspects of a river basin are not purely linear, the study of non-linearities will also be of great importance in water resources systems optimisation. This paper presents a generalised model for solving the optimal allocation of water resources in schemes where the objectives are minimising the demand deficits, complying with the required flows in the river and storing water in reservoirs. Evaporation from reservoirs and returns from demands are considered, and an iterative methodology is followed to solve these two non-network constraints. The model was applied to the Duero River basin (Spain). Three different network flow algorithms (Out-of-Kilter, RELAX-IV and NETFLO) were used to solve the allocation problem. Certain convergence issues were detected during the iterative process. There is a need to relate the data from the studied systems with the convergence criterion to be able to find the convergence criterion which yields the best results possible without requiring a long calculation time.     

信息熵在水污染物总量区域公平分配中的应用

在考虑排污权的公平准则的基础上,通过综合考虑区域的经济、自然等客观因素,构建了以加权信息熵最大化为目标的规划模型。该模型应用熵值法得到上述各因素的指标的权重,并通过计算单位化指标负荷污染物量的信息熵,进而得到总信息熵。通过设定合理的运算规则和约束条件,制定出基于公平性的水污染物总量的区域初始分配方案。将模型应用于天津市的水污染物分配计算,结果表明：天津市东丽区、汉沽区和中心城区是水污染物重点削减区域,其次是塘沽区和宁河县。应用结果表明,本文方法可以定量化地反映多种公平准则,进而得到公平可行的分配方案。     

基于梯形模糊数的不确定性河道生态需水模型及其应用

针对目前各类河道内生态需水模型缺少不确定性方面的研究，本文将模糊数引入到河道内生态需水估算方法中，建立了模糊生态流速和模糊生态水力半径的概念。以抛物线型过水断面河道为例，在扩展原理的基础上，建立了流量与模糊生态流速、模糊生态水力半径的具体函数关系，提出了基于梯形模糊数的不确定性河道内生态需水模型，设计了算法流程，以杜柯河壤塘站为例进行了实例研究，计算并给出了不确定性河道内生态需水量的表达结果，还利用Tennant法对所建模型的实用性进行了验证，最后对计算结果的表达形式做了分析和解释。     

Runoff modelling using radar data and flow measurements in a stochastic state space approach.

In urban drainage the estimation of runoff with the help of models is a complex task. This is in part due to the fact that rainfall, the most important input to urban drainage modelling, is highly uncertain. Added to the uncertainty of rainfall is the complexity of performing accurate flow measurements. In terms of deterministic modelling techniques these are needed for calibration and evaluation of the applied model. Therefore, the uncertainties of rainfall and flow measurements have a severe impact on the model parameters and results. To overcome these problems a new methodology has been developed which is based on simple rain plane and runoff models that are incorporated into a stochastic state space model approach. The state estimation is done by using the extended Kalman filter in combination with a maximum likelihood criterion and an off-line optimization routine. This paper presents the results of this new methodology with respect to the combined consideration of uncertainties in distributed rainfall derived from radar data and uncertainties in measured flows in an urban catchment within the Emscher river basin, Germany.     

Identification and Quantification of Uncertainties in a Hydrodynamic River Model Using Expert Opinions

Hydrodynamic river models are applied to design and evaluate measures for purposes such as safety against flooding. The modelling of river processes involves numerous uncertainties, resulting in uncertain model outcomes. Knowledge of the type and magnitude of uncertainties is crucial for a meaningful interpretation of the model results and the usefulness of results in decision making processes. The aim of this study is to identify the sources of uncertainty that contribute most to the uncertainties in the model outcomes and quantify their contribution to the uncertainty in the model outcomes. Experts have been selected based on an objective Pedigree analysis. The selected experts are asked to quantify the most important uncertainties for two situations: (1) the computation of design water levels and (2) the computation of the hydraulic effect of a change in the river bed. For the computation of the design water level, the uncertainties are dominated by the sources that do not change between the calibration and the prediction. The experts state that the upstream discharge and the empirical roughness equation for the main channel have the largest influence on the uncertainty in the modeled water levels. For effect studies, the floodplain bathymetry, weir formulation and discretization of floodplain topography contribute most to the uncertainties in model outcomes. Finally, the contribution of the uncertainties to the model outcomes show that the uncertainties have a significant effect on the predicted water levels, especially under design conditions.