Interval-parameter Two-stage Stochastic Nonlinear Programming for Water Resources Management under Uncertainty

Over the past decades, controversial and conflict-laden water allocation issues among competing interests have raised increasing concerns. In this research, an interval-parameter two-stage stochastic nonlinear programming (ITNP) method is developed for supporting decisions of water-resources allocation within a multi-reservoir system. The ITNP can handle uncertainties expressed as both probability distributions and discrete intervals. It can also be used for analyzing various policy scenarios that are associated with different levels of economic consequences when the promised allocation targets are violated. Moreover, it can deal with nonlinearities in the objective function such that the economies-of-scale effects in the stochastic program can be quantified. The proposed method is applied to a case study of water-resources allocation within a multi-user, multi-region and multi-reservoir context for demonstrating its applicability. The results indicate that reasonable solutions have been generated, which present as combined interval and distributional information. They provide desired water allocation plans with a maximized economic benefit and a minimized system-disruption risk. The results also demonstrate that a proper policy for water allocation can help not only mitigate the penalty due to insufficient supply but also reduce the waste of water resources.     

An Inexact Chance-constrained Quadratic Programming Model for Stream Water Quality Management

Water quality management is complicated with a variety of uncertainties and nonlinearities. This leads to difficulties in formulating and solving the resulting inexact nonlinear optimization problems. In this study, an inexact chance-constrained quadratic programming (ICCQP) model was developed for stream water quality management. A multi-segment stream water quality (MSWQ) simulation model was provided for establishing the relationship between environmental responses and pollution-control actions. The relationship was described by transformation matrices and vectors that could be used directly in a multi-point-source waste reduction (MWR) optimization model as water-quality constraints. The interval quadratic polynomials were employed to reflect the nonlinearities and uncertainties associated with wastewater treatment costs. Uncertainties associated with the water-quality parameters were projected into the transformation matrices and vectors through Monte Carlo simulation. Uncertainties derived from water quality standards were characterized as random variables with normal probability distributions. The proposed ICCQP model was applied to a water quality management problem in the Changsha section of the Xiangjiang River in China. The results demonstrated that the proposed optimization model could effectively communicate uncertainties into the optimization process, and generate inexact solutions containing a spectrum of wastewater treatment options. Decision alternatives could then be obtained by adjusting different combinations of the decision variables within their solution intervals. Solutions from the ICCQP model could be used to analyze tradeoffs between the wastewater treatment cost and system-failure risk due to inherent uncertainties. The results are valuable for supporting decision makers in seeking cost-effective water management strategies.     

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.     

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.     

A Fuzzy Stochastic Dynamic Nash Game Analysis of Policies for Managing Water Allocation in a Reservoir System

In this paper a fuzzy dynamic Nash game model of interactions between water users in a reservoir system is presented. The model represents a fuzzy stochastic non-cooperative game in which water users are grouped into four players, where each player in game chooses its individual policies to maximize expected utility. The model is used to present empirical results about a real case water allocation from a reservoir, considering player (water user) non-cooperative behavior and also same level of information availability for individual players. According to the results an optimal allocation policy for each water user can be developed in addition to the optimal policy of the reservoir system. Also the proposed model is compared with two alternative dynamic models of reservoir optimization, namely Stochastic Dynamic Programming (SDP) and Fuzzy-State Stochastic Dynamic programming (FSDP). The proposed modeling procedures can be applied as an appropriate tool for reservoir operation, considering the interaction among the water users as well as the water users and reservoir operator.     

Stochastic Goal Programming Model for Optimal Blending of Desalinated Water with Groundwater

A stochastic goal programming (GP) model is developed in orderto determine the daily production of desalination plants to meet the requirements of water blending stations (WBS) for major cities in the Eastern Province of the Kingdom of SaudiArabia. The WBS is assumed to be a control point in the systemwhere water is blended to satisfy the desired water quality, downstream of the control point. The desalinated water is blended with brackish groundwater extracted from several groundwater wells. The objective of the model is to minimize the goal deviations from the following priority levels: demand for blended water, control of salinity levels, depletion of groundwater and maximize the use of brackish water, demand forbrackish water at WBS, and production of desalinated water. Anessential element of the model is the input data; unfortunately,available data are not accurate due to the inherent uncertaintyassociated with it. This uncertainty will generate uncertainty in the model output, which affects reliability and confidence associated with the decisions. Thus, reliable planning should consider uncertainties associated with model input parameters.The developed stochastic model shows how Goal Programming (GP)modeling can be used to plan the water resources in the EasternProvince of Saudi Arabia, assuming that both supply and demandare uncertain.     

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 Bayesian Stochastic Optimization Model for a Multi-Reservoir Hydropower System

This paper presents the development of an operating policy model for a multi-reservoir system for hydropower generation by addressing forecast uncertainty along with inflow uncertainty. The stochastic optimization tool adopted is the Bayesian Stochastic Dynamic Programming (BSDP), which incorporates a Bayesian approach within the classical Stochastic Dynamic Programming (SDP) formulation. The BSDP model developed in this study considers, the storages of individual reservoirs at the beginning of period t, aggregate inflow to the system during period t and forecast for aggregate inflow to the system for the next time period t + 1, as state variables. The randomness of the inflow is addressed through a posterior flow transition probability, and the uncertainty in flow forecasts is addressed through both the posterior flow transition probability and the predictive probability of forecasts. The system performance measure used in the BSDP model is the square of the deviation of the total power generated from the total firm power committed and the objective function is to minimize the expected value of the system performance measure. The model application is demonstrated through a case study of the Kalinadi Hydroelectric Project (KHEP) Stage I, in Karnataka state, India.     

A Comparison of Modelling Systems for the Development and Revision of Water Resources Management Plans

In this paper, two modelling systems used for the simulation of water resources management are compared. These modelling systems can be used in the implementation of the European Water Framework Directive or to perform any other kind of integrated assessment with regard to water resources management. In such investigations the use of models is inevitable, as integrated water resources management demands the survey of large areas as well as the inclusion of the different functions of the water cycle and water utilisation processes. Water quantity data provides important input for hydro-chemical, hydro-ecological or hydro-economic models. If no significant water resources management activities are realised in the basin under study, these data can be provided by simple rainfall-runoff models. If significant water resources management activities are realised or planned, the effects of these water resources management activities must be taken into consideration. Then, however, the use of water resources management models becomes necessary. Two such modelling systems, WRAP and WBalMo, are compared. Both have been designed for the development and revision of water resources management plans. Due to different approaches regarding the modelling routines the models lead to different results in the calculation of water quantities. By tracking the simulation algorithms, an understanding of the detected differences becomes possible. By adapting the spatial configuration of the modelled system, equivalent results can be obtained.     

A Review of Multiple Criteria Analysis for Water Resource Planning and Management

Multiple criteria analysis (MCA) is a framework for ranking or scoring the overall performance of decision options against multiple objectives. The approach has widespread and growing application in the field of water resource management. This paper reviews 113 published water management MCA studies from 34 countries. It finds that MCA is being heavily used for water policy evaluation, strategic planning and infrastructure selection. A wide range of MCA methods are being used with the fuzzy set analysis, paired comparison and outranking methods being most common. The paper also examines the motivations for adopting MCA in water management problems and considers future research directions. This study was funded by the eWater CRC ().