Fractional Stochastic Interval Programming for Optimal Low Impact Development Facility Category Selection under Uncertainty

Uncertainties in nature and human society influence low impact development (LID) facility category selection during LID facility optimization distribution, however the investigation of this area is seldom. There are still two problems with uncertainty which influence LID facility distribution 1) how uncertainty factors affect LID facility selection and 2) in the case of a number of LID facilities of multiple categories are to be set, how to construct the LID facility optimization distribution model for LID facility category selection under uncertainty. To handle the problems, this study develops a fractional stochastic interval programming model to process LID facility category selection under the influence of uncertainty. The model can either process multiple objectives via objective maximization and minimization or process the stochastic uncertainty and interval uncertainty. The study shows that the uncertainties which influence LID facility category selection exist in rainfall, infiltration rate, release coefficient, unit price and budget. and the study reveal that the key constraint of LID facility category selection is the uncertainty parameter characteristic of the LID facility, in which different parameters lead to various LID facility optimization schemes. Results of the model include a series of LID facility optimization distribution schemes in multiple scenarios.Results also provide a series of feasible schemes for decision makers, and the manager can select the most appropriate scheme according to water processing level or budget. The developed model could 1) identifying the uncertainty which impact the LID facility distribution. 2) processing the LID facility category selection under interval uncertainty and stochastic uncertainty during LID facility optimization distribution. The method can also be used to estimate the rationality of the LID facility optimization scheme. Moreover, the proposed method is universal and could be extended to other cases of LID facility category selection under uncertainty.

     

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.     

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.     

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.     

Optimal Long-term Operation of Reservoir-river Systems under Hydrologic Uncertainties: Application of Interval Programming

Long-term basin-wide reservoir-river operation optimization problems are usually complex and nonlinear especially when the water quality issues and hydrologic uncertainties are incorporated. It is due to non-convex functions in water quality modeling and a large number of computational iterations required by most of stochastic programming methods. The computational burden of uncertainty modeling can be reduced by a special combination of uncertainty modeling and interval programming, though the problem solution is still a challenge due to model nonlinearity. In this paper, an integrated water quantity-quality model is developed for optimal water allocation at river-basin scale. It considers water supply and quality targets as well as hydrologic, water quality and water demand uncertainties within the nonlinear interval programming (NIP) framework to minimize the slacks in water supply and quality targets during a long-term planning horizon. A fast iterative linear programming (ILP) method is developed to convert the NIP into a linear interval programming (LIP). The ILP resolves two challenges in NIP, first converting the large non-linear programming (NLP) into a linear programming (LP) with minimum approximation and second reducing the iterations needed in interval programming for NLP into just two iterations for the upper and lower limits of decision variables. This modeling approach is applied to the Zayandehrood river basin in Iran that has serious water supply and pollution problems. The results show that in this river basin at dry conditions when available surface water resources are below 85 % of normal hydrologic state and water demands are 115 % of current water demands, the total dissolved solids (TDS) concentration can be reduced by 50 % at the inlet of the Gavkhuni wetland located downstream of the river basin.     

Agricultural Multi-Water Source Allocation Model Based on Interval Two-Stage Stochastic Robust Programming under Uncertainty

Numerous uncertainties and complexities exist in the agricultural irrigation water allocation system, that must be considered in the optimization of water resources allocation. In this paper, an agricultural multi-water source allocation model, consisting of stochastic robust programming and two-stage random programming and introducing interval numbers and random variables to represent the uncertainties, was proposed for the optimization of irrigation water allocation in Jiamusi City of Heilongjiang Province, China. The model could optimize the water allocaton to different crops of groundwater and surface water. Then, the optimal target value and the optimal water allocation of different water sources distributed to different crops could be obtained. The model optimized the economic benefits and stability of the agricultural irrigation water allocation system via the introduction of a the penalty cost variable measurement to the objective function. The results revealed that the total water shortage changed from [18.6, 32.3]?×?10⁸ m³ to [15.7, 26.2]?×?10⁸ m³ at a risk level ω from zero to five, indicating that the water shortage decreased and the reliability improved in the agricultural irrigation water allocation system. Additionally, the net economic benefits of irrigation changed from [287.21, 357.86]?×?10⁸ yuan to [253.23, 301.32]?×?10⁸ yuan, indicating that the economic benefit difference was reduced. Therefore, the model can be used by decision makers to develop appropriate water distribution schemes based on the rational consideration of the economic benefit, stability and risk of the agricultural irrigation water allocation system.     

不确定条件下的多水源联合供水调度模型

本文针对城市供水调度系统中存在的不确定性与复杂性,运用区间两阶段随机规划的方法,建立了多水源联合供水调度的优化模型。该模型以供水调度系统成本最小为目标函数,引入概率分布及区间数表示不确定性,模拟了地表水源、地下水源、外来水源等多种水源联合供水过程,并对多种水源的调水目标进行优化。以区间形式给出优化结果,为决策者提供宽裕的决策空间。利用该方法,可充分考虑系统中不确定因素对系统成本的影响,更真实的反映多水源联合供水系统的实际情况。     

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.     

Optimization of Industrial Structure Considering the Uncertainty of Water Resources

This paper developed a stochastic linear fractional programming model for industry optimization allocation base on the uncertainty of water resources incorporating chance constrained programming and fractional programming. In this paper, the stochastic linear fractional programming is used in the real word. The development SLFP has the following advantages: (1) The model can compare the two aspects of the targets; (2) The model can reflect the system efficiency intuitively; (3) The model can deal with uncertain issues with probability distribution; (4) The model can give different optimal plans under different risk conditions. The model has a significant value for the industry optimization allocation under uncertainty in local and areas to achieve the maximum economic benefits and the full use of the water resources.     

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.     

A Hybrid Dynamic Dual Interval Programming for Irrigation Water Allocation under Uncertainty

Along with the economic development in Canada, the shortage of irrigation water has become a serious concern (Bouwer 1993; Hennessy 1993). In this study, a model of Dynamic Dual Interval Programming (DDIP) is developed and applied to the irrigation water allocation systems with uncertainty. DDIP method improves the existing dynamics interval programming by explicitly addressing the system uncertainties with a dual interval that had higher system reliability. The solution of DDIP is computationally effective, and its decision variables are incorporated into the solutions for final decision. In order to obtain the optimal allocation schemes in a dynamic process, the developed DDIP was applied to an irrigation water system. The results from this case study revealed that optimal solution can be obtained through the DDIP approach from the agriculture water management activities for feasible decisions. These decisions reflect the high uncertainty of the information in the boundaries of dual intervals. The solution presents a maximum benefit under limited yearly uncertain natural resources. Furthermore, the information obtained though this model may help the authority to make optimal decisions and to reduce the risk for uncertain situations.     

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.     

Modeling of Water Resources Allocation and Water Quality Management for Supporting Regional Sustainability under Uncertainty in an Arid Region

In this study, a scenario-based interval-stochastic fraticle optimization with Laplace criterion (SISFL) method is developed for sustainable water resources allocation and water quality management (WAQM) under multiple uncertainties. SISFL can tackle uncertainties presented as interval parameters and probability distributions; meanwhile, it can also quantify artificial fuzziness such as risk-averse attitude in a decision-making issue. Besides, it can reflect random scenario occurrence under the supposition of no data available. The developed method is applied to a real case of water resources allocation and water quality management in the Kaidu-kongque River Basin, where encounter serve water deficit and water quality degradation simultaneously in Northwest China. Results of water allocation pattern, pollution mitigation scheme, and system benefit under various scenarios are analyzed. The tradeoff between economic activity and water-environment protection with interval necessity levels and Laplace criterions can support policymakers generating an effective and robust manner associated with risk control for WAQM under multiple uncertainties. These discoveries avail local policymakers gain insight into the capacity planning of water-environment to satisfy the basin’s integrity of socio-economic development and eco-environmental sustainability.     

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.     

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.     

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.     

Surrogate-Based Stochastic Multiobjective Optimization for Coastal Aquifer Management under Parameter Uncertainty

Linked simulation-optimization (S/O) approaches have been extensively used as tools in coastal aquifer management. However, parameter uncertainties in seawater intrusion (SI) simulation models often undermine the reliability of the derived solutions. In this study, a stochastic S/O framework is presented and applied to a real-world case of the Longkou coastal aquifer in China. The three conflicting objectives of maximizing the total pumping rate, minimizing the total injection rate, and minimizing the solute mass increase are considered in the optimization model. The uncertain parameters are contained in both the constraints and the objective functions. A multiple realization approach is utilized to address the uncertainty in the model parameters, and a new multiobjective evolutionary algorithm (EN-NSGA2) is proposed to solve the optimization model. EN-NSGA2 overcomes some inherent limitations in the traditional nondominated sorting genetic algorithm-II (NSGA-II) by introducing information entropy theory. The comparison results indicate that EN-NSGA2 can effectively ameliorate the diversity in Pareto-optimal solutions. For the computational challenge in the stochastic S/O process, a surrogate model based on the multigene genetic programming (MGGP) method is developed to substitute for the numerical simulation model. The results show that the MGGP surrogate model can tremendously reduce the computational burden while ensuring an acceptable level of accuracy.

     

基于改进引力搜索算法的风电功率短期预测优化调度研究

考虑到风电功率本身的随机波动特性,在制定日前市场经济预调度方案过程中,需要将风电功率不确定性因素考虑进来。因此,利用机会约束规划方法来建立包含风电场在内的日前经济调度随机优化模型,并选用改进的引力搜索算法来求解所建立的模型。最后,选用具体算例,结合IEEE-30节点网络调试系统进行模型仿真,并利用仿真结果探讨了负荷水平对系统所容纳的风电容量的影响、风电功率预测偏差对预调度惩罚成本的影响、风电预测功率下不同成本系数对日前市场的影响。研究结果表明,利用改进引力搜索算法进行优化仿真的收敛速度比粒子群算法和遗传算法的收敛速度快,可靠性高。研究所得成果为日前市场经济下风电功率短期预测研究提供参考。     

灰绿耦合雨洪系统多目标优化建模与应用

以中山市某开发区为例,以灰绿基础设施总成本、径流系数和节点总溢流量为目标函数,以子汇水区内涝风险等级作为罚函数构建灰绿耦合排水系统多目标优化模型,基于NSGA-Ⅱ算法,编写SWMM接口程序,调用SWMM进行雨洪模型计算及方案寻优,对24h设计暴雨条件下的灰色调蓄设施和绿色LID设施的规模进行多目标优化设计及综合评估。结果表明:绿色LID设施径流系数和节点总溢流量削减量均随其成本的增加呈近似线性下降的趋势,并随重现期增加而加快;灰色调蓄设施容积与节点总溢流量曲线存在明显拐点,拐点后规模增大的边际效益趋0;提出的灰绿耦合基础设施优化模型及求解程序对初始种群设置技术要求低,且能稳定收敛,可为城市雨洪管理方案的制定提供支持和依据。     

漳卫南灌区农业水资源优化配置研究

考虑到不确定条件下漳卫南灌区农业水资源管理的复杂性,为了解决当灌区水资源用户供水目标不能满足需求时的水资源优化配置问题,结合LFP模型与TSP模型的优点,开发了一种分式两阶段随机规划模型(FTSP)。选择漳卫南灌区最大控制性工程岳城水库的两个大型供水灌区作为验证实例,模型应用结果表明,不同决策情景所对应的经济效益和缺水风险不同,最优决策实现了经济效益和缺水风险之间的平衡;不同径流水平下,各用户的正常灌溉面积会发生相应变化,高径流水平时所有用户均能得到正常灌溉。