Multi-Source Multi-Sector Sustainable Water Supply Under Multiple Uncertainties: An Inexact Fuzzy-Stochastic Quadratic Programming Approach

This paper presents the development and the first application of a superiority–inferiority-based inexact fuzzy-stochastic quadratic programming (SI-IFSQP) approach for sustainable water supply under multiple uncertainties. SI-IFSQP improves conventional nonlinear programming by tackling multiple uncertainties within an individual parameter; SI-IFSQP is also superior to existing inexact methods due to its reflection of economies of scale and reduction of computational requirements. An interactive solution algorithm with high computational efficiency was also proposed. The application of SI-IFSQP to long-term planning of a multi-source multi-sector water supply system demonstrated its applicability. The close reflection of system complexities, such as multiple uncertainties, scale economies and dynamic parameters, could enhance the robustness of the optimization process as well as the acceptability of obtained results. Corresponding to varied system conditions and decision priorities, the interval solutions from SI-IFSQP could help generate a series of long-term water supply strategies under a number of economic, environmental, ecological, and water-security targets.     

A Fuzzy Robust Nonlinear Programming Model for Stream Water Quality Management

An interval-parameter fuzzy robust nonlinear programming (IFRNP) approach was developed for stream water quality management under uncertainty. The interval and fuzzy robust programming methods were incorporated within a general framework to address uncertainties associated with the nonlinear objective and the left- and right-hand sides of the constraints. A piecewise linearization approach was developed to deal with the nonlinear cost function. IFRNP could explicitly address complexities of various system uncertainties, where parameters were represented as both interval numbers and fuzzy membership functions. Furthermore, the dual uncertain information associated with the lower and upper bounds of each interval parameter could be effectively tackled through the concept of fuzzy boundary interval. The proposed IFRNP method was applied to a case of water quality management in the Guoyang section of the Guo River in Anhui province, China. A number of cost-effective schemes for water quality management were generated, and allowable wastewater discharge amounts were recommended. The results indicated that IFRNP was applicable to water quality management problems, where high nonlinearities and dual uncertainties exist.     

Basin Scale Water Resources Systems Modeling Under Cascading Uncertainties

Global change in climate and consequent large impacts on regional hydrologic systems have, in recent years, motivated significant research efforts in water resources modeling under climate change. In an integrated future hydrologic scenario, it is likely that water availability and demands will change significantly due to modifications in hydro-climatic variables such as rainfall, reservoir inflows, temperature, net radiation, wind speed and humidity. An integrated regional water resources management model should capture the likely impacts of climate change on water demands and water availability along with uncertainties associated with climate change impacts and with management goals and objectives under non-stationary conditions. Uncertainties in an integrated regional water resources management model, accumulating from various stages of decision making include climate model and scenario uncertainty in the hydro-climatic impact assessment, uncertainty due to conflicting interests of the water users and uncertainty due to inherent variability of the reservoir inflows. This paper presents an integrated regional water resources management modeling approach considering uncertainties at various stages of decision making by an integration of a hydro-climatic variable projection model, a water demand quantification model, a water quantity management model and a water quality control model. Modeling tools of canonical correlation analysis, stochastic dynamic programming and fuzzy optimization are used in an integrated framework, in the approach presented here. The proposed modeling approach is demonstrated with the case study of the Bhadra Reservoir system in Karnataka, India.     

A Conditional Value-at-Risk Based Inexact Water Allocation Model

A conditional value-at-risk (CVaR) based inexact two-stage stochastic programming (CITSP) model was developed in this study for supporting water resources allocation problems under uncertainty. A CITSP model was formulated through incorporating a CVaR constraint into an inexact two-stage stochastic programming (ITSP) framework, and could be used to deal with uncertainties expressed as not only probability distributions but also discrete intervals. The measure of risks about the second-stage penalty cost was incorporated into the model, such that the trade-off between system economy and extreme expected loss could be analyzed. The developed model was applied to a water resources allocation problem involving a reservoir and three competing water users. The results indicated that the CITSP model performed better than the ITSP model in its capability of reflecting the economic loss from extreme events. Also, it could generate interval solutions within which the decision alternatives could be selected from a flexible decision space. Overall, the CITSP model was useful for reflecting the decision maker’s attitude toward risk aversion and could help seek cost-effective water resources management strategies under complex uncertainties.     

An Integrated Approach of System Dynamics,Orthogonal Experimental Design and Inexact Optimization for Supporting Water Resources Management under Uncertainty

An integrated approach of system dynamics (SD), orthogonal experimental design (OED) and inexact optimization modeling was proposed for water resources management under uncertainty. The developed method adopted a combination of SD and OED to identify key scenarios within multiple factors, through which interval solutions for water demands could be obtained as input data for consequential optimization modeling. Also, optimal schemes could be obtained in the combination of inexact two-stage stochastic programming and credibility constrained programming. The developed method was applied to a real-world case study for supporting allocation of multiple-source water resources to multiple users in Dalian city within a multi-year context. The results indicated that a lower credibility-satisfaction level would generate higher allocation efficiency, a higher system benefit and a lower system violation risk. The developed model could successfully reflect and address the variety of uncertainties through provision of credibility levels, which corresponds to the decision makers’ preference regarding the tradeoffs between system benefits and violation risks.     

IFTSQP: An Inexact Optimization Model for Water Resources Management under Uncertainty

Abstract

An interval-fuzzy two-stage quadratic programming (IFTSQP) method is developed for water resources management under uncertainty. The methodincorporates techniques of interval-parameter programming, two-stage stochastic programming, and fuzzy quadratic programming within a general optimization framework to tackle multiple uncertainties presented as intervals, fuzzy sets and probability distributions. In the model formulation, multiple control variables are adopted to handle independent uncertainties in the model's right-hand sides; fuzzy quadratic terms are used in the objective function to minimize the variation in satisfaction degrees among the constraints. Moreover, the method can support the analysis of policy scenarios that are associated with economic penalties when the promised targets are violated. The developed method is then applied to a case study of water resources management planning. The results indicate that reasonable solutions have been obtained. They can help provide bases for identifying desired water-allocation plans with a maximized system benefit and a minimized constraint-violation risk.     

IB-ICCMSP: An Integrated Irrigation Water Optimal Allocation and Planning Model Based on Inventory Theory under Uncertainty

In this study, an inventory-theory-based inexact chance-constrained multi-stage stochastic programming (IB-ICCMSP) model under multi-uncertainties is developed. IB-ICCMSP integrates inventory theory into an inexact chance-constrained multi-stage stochastic optimization framework. This method can not only effectively address system multiple uncertainties (e.g. discrete intervals and probability density functions) and dynamic features, but also provide water transferring and allocating schemes among multiple stages. The developed model is applied to irrigation water allocation optimization system in Zhangye City, Gansu province, China. Based on the runoff simulation prediction of Yingluo Gorge and water supply–demand balance analysis of the 12 irrigation areas in Zhangye City, different optimal irrigation water measures are generated under different flow levels and different probabilities in the planning year. The obtained results are valuable for supporting the adjustment of the existing irrigation patterns and identifying desired water-allocation plans for irrigation under multi-uncertainties.     

Interval-parameter Two-stage Stochastic Semi-infinite Programming: Application to Water Resources Management under Uncertainty

In this study, an interval-parameter two-stage stochastic semi-infinite programming (ITSSP) method was developed for water resources management under uncertainty. As a new extension of mathematical programming methods, the developed ITSSP approach has advantages in uncertainty reflection and policy analysis. In order to better account for uncertainties, the ITSSP approach is expressed with discrete intervals, functional intervals and probability density functions. The ITSSP method integrates the two-stage stochastic programming (TSP), interval programming (IP) and semi-infinite programming (SIP) within a general optimization framework. The ITSSP has an infinite number of constraints because it uses functional intervals with time (t) being an independent variable. The different t values within the range [0, 90] lead to different constraints. At same time, ITSSP also includes probability distribution information. The ITSSP method can incorporate pre-defined water resource management policies directly into its optimization process to analyze various policy scenarios having different economic penalties when the promised amounts are not delivered. The model is applied to a water resource management system with three users and four periods (corresponding to winter, spring, summer and fall, respectively). Solutions of the ITSSP model provide desired water allocation patterns, which maximize both the system’s benefits and feasibility. The results indicate that reasonable interval solutions were generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of stream flow. The obtained solutions are useful for decision makers to obtain insight regarding the tradeoffs between environmental, economic and system reliability criteria.     

水环境污染系统规划的模糊非线性规划模型

本文将模糊集理论与非线性规划优化方法有机地结合起来，首次提出水环境污染系统规划的模糊非线性规划模型．并将它应用于沈阳市南部污水排放系统的最优化处理规划．在满足水质要求的同时，可以显著地降低污水处理的总费用．     

Dual-Interval Two-Stage Optimization for Flood Management and Risk Analyses

In this study, a dual interval two-stage restricted-recourse programming (DITRP) method is developed for flood-diversion planning under uncertainty. Compared with other conventional methods, DITRP improves upon them by addressing system uncertainties with complex presentations and incorporating subjective information within its optimization framework. Uncertainties in DITRP can be represented as probability distributions and intervals. In addition, the dual-interval concept is presented when the available information is highly uncertain for boundaries of intervals. Moreover, decision makers’ attitudes towards system risk can be reflected using a restricted-resource measure by controlling the variability of the recourse cost. The method has been applied to a case study of flood management. The results indicate that reasonable solutions for planning flood management practice have been generated which are related to decisions of flood-diversion. Several policy scenarios are analyzed, assisting in gaining insight into the tradeoffs between risk and cost.     

Application of first-order and Monte Carlo analysis in watershed water quality models

To achieve effective environmental control, it is important to develop methodologies for dealing with uncertainties in model simulation of pollution behaviour and effects. Several procedures have been proposed to quantify uncertainties in modelling studies. This paper utilizes the two methods that are widely applied, i.e. functional analysis and Monte Carlo Simulation.The first-order part of the functional analysis method provides a measure of uncertainties in dependent variables in terms of uncertainties in independent variables. The procedure is based on first-order terms in the Taylor series expansion of the dependent variable about its mean value with respect to one or more independent variables. The major assumption in this procedure is that all independent and dependent variables are the second moment variables (SMV), which means that the behaviour of any SMV is completely described by its mean and standard deviation. The mathematical simplicity of the procedure allows application by simple input-output models. Consequently, it has been applied to many environmental simulators, e.g. hydrological models, stream water quality models, lake water quality models and ground water pollution models.The Monte Carlo Simulation (MCS) method uses a large number of repeated trials or simulations with the values for stochastic inputs or uncertain variables selected at random from their assumed parent probability distributions to establish an expected range of model uncertainty.     

Regional Water Use Structure Optimization Under Multiple Uncertainties Based on Water Resources Vulnerability Analysis

In this paper, a modeling framework by combining system dynamic (SD) model and optimal allocation model was developed to study water resources vulnerability and optimal water use structure, and the framework was applied in the middle reaches of Heihe River basin, northwest of China. The SD model could describe the dynamical change of water resources vulnerability by integrating water resources with socio-economic effect. The sensitivity analysis of SD model was then conducted to design appropriate scenarios for finding out the optimal development pattern, and based on which, an integrated water-saving scenario with lower water resources vulnerability was identified for optimization modeling. Then, an inexact fuzzy-parameter two-stage programming (IFTSP) model was developed and applied to optimize water use structure among industries under uncertainties. This study addresses the water resources vulnerability analysis in considering both water resources system and socio-economic system. Water resources vulnerability analysis was combined with optimization model to make adaptive water resources management plans. And the optimal allocation schemes under lower water resources vulnerability are more advantageous for regional sustainable development.     

河流水质模型(RWQM1)介绍及实例分析

以河流水质为目标对城市污水系统进行集成管理,是改善河流水质、保护生态环境的有效途径。介绍了国际水协会河流水质1号模型(RWQM1),该模型采用与标准活性污泥模型类似的描述方式,用化学需氧量作为有机物度量描述有机物与营养物,选择表征碳、氢、氧、氮、磷循环的水质成分和模型状态变量,描述河流中好氧与缺氧条件下物质的循环变化过程。该模型可直接与活性污泥模型相连接,用于集成城市污水处理系统的设计规划与监视控制。     

Water Resources Management and Planning under Uncertainty: an Inexact Multistage Joint-Probabilistic Programming Method

In this study, an inexact multistage joint-probabilistic programming (IMJP) method is developed for tackling uncertainties presented as interval values and joint probabilities. IMJP improves upon the existing multistage programming and inexact optimization approaches, which can help examine the risk of violating joint-probabilistic constraints. Moreover, it can facilitate analyses of policy scenarios that are associated with economic penalties when the promised targets are violated within a multistage context. The developed method is applied to a case study of water-resources management within a multi-stream, multi-reservoir and multi-period context, where mixed integer linear programming (MILP) technique is introduced into the IMJP framework to facilitate dynamic analysis for decisions of surplus-flow diversion. The results indicate that reasonable solutions for continuous and binary variables have been generated. They can be used to help water resources managers to identify desired system designs against water shortage and for flood control, and to determine which of these designs can most efficiently accomplish optimizing the system objective under uncertainty.     

城市河流综合整治工程的治理效果预测研究

以湖南省常德市穿紫河综合整治工程为例,利用美国环保局开发的综合水质模型QUAL2E对该工程治理方案进行效果预测。根据综合整治的主要治理措施设计了穿紫河水质模拟的五大类情景,模拟了CODCr、DO、BOD5、NH3 N等4项指标,进一步分析了不同截污程度、引水方案和引水量组合工程措施对提高水质达标率的作用,为穿紫河综合整治工程提供合理化建议。     

Stochastic and Robust Multi-Objective Optimal Management of Pumping from Coastal Aquifers Under Parameter Uncertainty

Combined simulation-optimization approaches have been used as tools to derive optimal groundwater management strategies to maintain or improve water quality in contaminated or other aquifers. Surrogate models based on neural networks, regression models, support vector machies etc., are used as substitutes for the numerical simulation model in order to reduce the computational burden on the simulation-optimization approach. However, the groundwater flow and transport system itself being characterized by uncertain parameters, using a deterministic surrogate model to substitute it is a gross and unrealistic approximation of the system. Till date, few studies have considered stochastic surrogate modeling to develop groundwater management methodologies. In this study, we utilize genetic programming (GP) based ensemble surrogate models to characterize coastal aquifer water quality responses to pumping, under parameter uncertainty. These surrogates are then coupled with multiple realization optimization for the stochastic and robust optimization of groundwater management in coastal aquifers. The key novelty in the proposed approach is the capability to capture the uncertainty in the physical system, to a certain extent, in the ensemble of surrogate models and using it to constrain the optimization search to derive robust optimal solutions. Uncertainties in hydraulic conductivity and the annual aquifer recharge are incorporated in this study. The results obtained indicate that the methodology is capable of developing reliable and robust strategies for groundwater management.     

Assessment of data and parameter uncertainties in integrated water-quality model

In integrated urban drainage water quality models, due to the fact that integrated approaches are basically a cascade of sub-models (simulating sewer system, wastewater treatment plant and receiving water body), uncertainty produced in one sub-model propagates to the following ones depending on the model structure, the estimation of parameters and the availability and uncertainty of measurements in the different parts of the system. Uncertainty basically propagates throughout a chain of models in which simulation output from upstream models is transferred to the downstream ones as input. The overall uncertainty can differ from the simple sum of uncertainties generated in each sub-model, depending on well-known uncertainty accumulation problems. The present paper aims to study the uncertainty propagation throughout an integrated urban water-quality model. At this scope, a parsimonious bespoke integrated model has been used allowing analysis of the combinative effect between different sub-models. Particularly, the data and parameter uncertainty have been assessed and compared by means of the variance decomposition concept. The integrated model and the methodology for the uncertainty decomposition have been applied to a complex integrated catchment: the Nocella basin (Italy). The results show that uncertainty contribution due to the model structure is higher with respect to the other sources of uncertainty.     

An Inexact Two-stage Fuzzy-stochastic Programming Model for Water Resources Management

An inexact two-stage fuzzy-stochastic programming (ITFSP) method is developed for water resources management under uncertainty. Fuzzy sets theory is introduced to represent various punishment policies under different water availability conditions. As an extension of conventional two-stage stochastic programming (TSP) method, two special characteristics of the proposed approach make it unique compared with existing approaches. One is it could handle flexible penalty rates, which are much reasonable for both of the authorities and users, and have seldom been considered in the TSP framework. The other is uncertain information expressed as discrete intervals and probability distribution functions can be effectively reflected in the optimization processes and solutions. After formulating the model, a hypothetical case is employed for demonstrating its applicability under two scenarios, where the inflow is divided into four and eight intervals, respectively. The results indicate that reasonable solutions have been obtained. They provide desired allocation patterns with maximized system benefit under two feasibility levels. The solutions present as stable intervals with different risk levels in violating the water demands, and can be used for generating decision alternatives. Comparisons of the solution from the ITFSP with that from the ITSP (inexact two-stage stochastic programming) and TSP approach are also undertaken. It shows that the ITFSP could produce more system benefit than existing methods and deal with flexible penalty policies for better water management and utilization.     

Water quality management of a stretch of river Yamuna: An interactive fuzzy multi-objective approach

This paper presents an Interactive Fuzzy Multi-Objective Linear Programming (IFMOLP) model for water quality management in a river basin. The IFMOLP model formulated will first evaluate dissolved oxygen (DO) concentrations or DO deficits at a point in different reaches depending on the overall Biochemical Oxygen Demand (BOD) concentration present in the respective drain. Subsequently, the model incorporates the aspirations and conflicting objectives of the decision maker (DM) by taking into consideration the aspects relevant for pollution control boards as well as dischargers responsible for generating wastewater. The uncertainty associated with specifying the water quality criteria (based on DO concentration or DO deficit) and treatment cost to remove pollution level is incorporated by interacting the decision maker. In this process DM is asked to specify the reference aspiration levels of achievement for the values of all membership functions generated with respect to each objective. This provides flexibility for the pollution control authorities and dischargers to specify their aspirations. IFMOLP model developed herein is then used in a case study for the evaluation of optimal BOD removal in different drains located across the river Yamuna at New Delhi, India. The presented model will simulate the allocation of waste load efficiencies with satisfactory results which will indicate usefulness of the model in managing more complex river basins along with better flexible policies of water management.     

人工湿地污水处理系统优化设计模型

针对污水处理厂进水水质和水量的不确定性,考虑进水水量、TSS、BOD5、TN、TP的区间变化,以出水时TSS、BOD5、TN、TP的质量浓度标准为约束条件,单元面积为决策目标,以费用最小为目标函数,建立基于一级动力学污染物去除模型的人工湿地污水处理系统区间数优化设计模型。该模型包含初沉池、人工湿地床(表面流和潜流式)两个单元,考虑了设计中的不确定性因素。实例计算表明,模型适用性较好。