Inexact Two-stage Stochastic Programming Research Articles

Optimal Allocation Model for Water Resources Coupled with Ecological Value Factors—A Case Study of Dalian, China

The surface water ecosystem has important ecological value and plays an important supporting and guarantee role in the sustainable development of human society. In this study, an inexact two-stage stochastic programming (ITSP) model was developed for supporting water resource allocation for the four main water sectors (industry, municipal, agriculture, and ecological environment). Several scenarios corresponding to different flow patterns, which reflect different probabilities of water resource availability and environmental carrying capacity, were examined. On the basis of traditional water resource allocation, this model adds consideration of ecological value factors, which is conducive to the synergistic efficiency of socio-economic and ecological water consumption. Results revealed that the water resource carrying capacity, ecological value factors, and water environmental capacity are the main factors affecting the optimal allocation of water resources. Furthermore, the optimal allocation scheme for water resources coupled with ecological value factors were determined to realize the coordinated development of social economic benefits and ecological benefits. The current study findings are of great significance for establishing a rational water resource management system for water resource exploitation and utilization. This model can be used to guide various departments in Dalian to formulate an optimal water resources allocation scheme by considering ecological value factors, and provide a basis for realizing the coordinated development of Dalian’s socio-economic development goals, water resource utilization, and environmental quality improvement.

An Optimization Model for Water Management under the Dual Constraints of Water Pollution and Water Scarcity in the Fenhe River Basin, North China

Sustainable watershed development suffers from severe challenges, such as water pollution and water scarcity. Based on an analysis of water quality and water utilization in the Fenhe River Basin, an inexact two-stage stochastic programming model with downside-risk aversion was built for optimal water resource allocations for the four primary water use sectors (industry, domestic use, agriculture, and the environment) in the Fenhe River Basin. The model aims to maximize the comprehensive watershed benefits, including water benefits, water costs, water treatment costs, and downside risks. The constraints are water quality, available water resources, and sectoral demands in different hydrological scenarios. The results show that pollutant emissions decrease as risk-aversion levels increase and show the opposite trend in the midstream and downstream areas. The increase in water resource allocation for agriculture and reduction in ecological water indicate that agriculture suffered the greatest water shortage and risk. Improving water recycling and coordinating the transferred water resources increases the comprehensive benefits and reduces sectoral risks. The model effectively manages rational water allocations under dual constraints and provides support for coordinating socio-economic development and environmental protection in the river basin.

Optimal Allocation of Surface Water Resources at the Provincial Level in the Uzbekistan Region of the Amudarya River Basin

Water users in the Amudarya River Basin in Uzbekistan are suffering severe water use competition and uneven water allocation, which seriously threatens ecosystems, as shown, for example, in the well-known Aral Sea catastrophe. This study explores the optimized water allocation schemes in the study area at the provincial level under different incoming flow levels, based on the current water distribution quotas among riparian nations, which are usually ignored in related research. The optimization model of the inexact two-stage stochastic programming method is used, which is characterized by probability distributions and interval values. Results show that (1) water allocation is redistributed among five different sectors. Livestock, industrial, and municipality have the highest water allocation priority, and water competition mainly exists in the other two sectors of irrigation and ecology; (2) water allocation is redistributed among six different provinces, and allocated water only in Bukhara and Khorezm can satisfy the upper bound of water demand; (3) the ecological sector can receive a guaranteed water allocation of 8.237–12.354 km3; (4) under high incoming flow level, compared with the actual water distribution, the total allocated water of four sectors (except for ecology) is reduced by 3.706 km3 and total economic benefits are increased by USD 3.885B.

An Improved Inexact Two-Stage Stochastic with Downside Risk-Control Programming Model for Water Resource Allocation under the Dual Constraints of Water Pollution and Water Scarcity in Northern China

Water resource allocation aimed at sustainable watershed development suffers from prominent challenges such as water pollution and scarcity, especially in water-deprived regions. Based on analysis of water quality, use, and sectoral demands during the planning period in the Fenhe River Basin, an improved inexact two-stage stochastic programming model with downside risk control was built for optimal resource allocations for the four primary sectors (industry, domestic use, agriculture, and the environment) in the basin. The principal constraints are river water quality and available water resources under the three hydrological scenarios (low, medium, and high). The results show that industrial, domestic, and agricultural water use in the middle and lower reaches were significantly reduced by requiring improved water quality; agriculture suffered the greatest water shortage and risk. As the level of risk control improved, the comprehensive watershed benefits and agricultural risks were gradually reduced. Improving water reuse significantly reduces the risk and increases the benefits. The model can effectively manage rational water allocations under the dual constraints of water quality and quantity, meanwhile alleviating water competition caused by different water benefits to provide support for coordinating the improvement of water quality and socio-economic development in the basin.

Multi-objective optimal allocation of sediment resources under multiple uncertainties

Abstract A model suitable for optimal allocation of sediment resources is utilized to cope with multiple uncertainties in the system and contradictions between allocation units. This approach incorporates inexact two-stage stochastic programming, probabilistic-possibilistic programming and fuzzy random intervals (ITPP-FRI) in an integrated framework. The multiple uncertainties of various parameters in this study are represented by the interval numbers with fuzziness and randomness. The model introduced the concept of possibility and necessity measures to address multiple uncertainties in the system, which are suitable for optimistic and pessimistic decision-makers, respectively. The model makes a tradeoff between multiple objectives including sediment siltation of each allocation unit, and economic benefits of the system. As demonstrated in the Weishan irrigation area, a number of alternative cases can be achieved at different probability levels and under varied sediment diversion scenarios. The result shows that the sediment allocation for the unit of desilting basins was significantly reduced by at least 4.3% compared with the current situation, and the units of transporting sediments to the fields and using them for producing building materials were increased by more than 15% and 5%, respectively. The sediment is allocated to the units in need and the channel siltation is reduced. According to the allocation results, a new pattern of regulating water delivery is put forward for irrigation area in this paper to meet the requirement of using limited water to transport sediment to each unit.

An Inexact Inventory Theory-Based Water Resources Distribution Model for Yuecheng Reservoir, China

In this study, an inexact inventory theory-based water resources distribution (IIWRD) method is advanced and applied for solving the problem of water resources distribution from Yuecheng Reservoir to agricultural activities, in the Zhanghe River Basin, China. In the IIWRD model, the techniques of inventory model, inexact two-stage stochastic programming, and interval-fuzzy mathematics programming are integrated. The water diversion problem of Yuecheng Reservoir is handled under multiple uncertainties. Decision alternatives for water resources allocation under different inflow levels with a maximized system benefit and satisfaction degree are provided for water resources management in Yuecheng Reservoir. The results show that the IIWRD model can afford an effective scheme for solving water distribution problems and facilitate specific water diversion of a reservoir for managers under multiple uncertainties and a series of policy scenarios.

A Comprehensive Forecasting–Optimization Analysis Framework for Environmental-Oriented Power System Management—A Case Study of Harbin City, China

In this study, a comprehensive research framework coupled with electric power demand forecasting, a regional electric system planning model, and post-optimization analysis is proposed for electric power system management. For dealing with multiple forms of uncertainties and dynamics concerning energy utilization, capacity expansions, and environmental protection, the inexact two-stage stochastic robust programming optimization model was developed. The novel programming method, which integrates interval parameter programming (IPP), stochastic robust optimization (SRO), and two-stage stochastic programming (TSP), was applied to electric power system planning and management in Harbin, China. Furthermore, the Gray-Markov approach was employed for effective electricity consumption prediction, and the forecasted results can be described as interval values with corresponding occurrence probability, aiming to produce viable input parameters of the optimization model. Ten scenarios were analyzed with different emissions reduction levels and electricity power structure adjustment modes, and the technique for order of preference by similarity to ideal solution (TOPSIS) was selected to identify the most influential factors of planning decisions by selecting the optimal scheme. The results indicate that a diversified power structure that dominates by thermal power and is mainly supplemented by biomass power should be formed to ensure regional sustainable development and electricity power supply security in Harbin. In addition, power structure adjustment is more effective than the pollutants emission control for electricity power system management. The results are insightful for supporting supply-side energy reform, generating an electricity generation scheme, adjusting energy structures, and formulating energy consumption of local policies.

Risk-based agricultural water allocation under multiple uncertainties

A risk-based interval-stochastic optimization modelling approach is developed for agricultural water allocation in response to the complexity arising from uncertainties and risk in agricultural water management systems. The approach comprises conditional-value-at-risk (CVaR) model, inexact two-stage stochastic programming (ITSP) model with imprecise probabilities (IPs), and random-boundary intervals (RBIs) within a general framework. The approach can simultaneously balance expected benefits, penalties, and risks from agricultural water allocation, and can address uncertainties of agricultural water supply and demand in the form of probability distributions and intervals with random boundaries. As demonstrated in Hulan River irrigation area, northeast China, the objective of the approach is to allocate limited agricultural water resources to make a trade-off between various subareas under different risk-aversion levels and possible runoff discharges. Most inputs to the approach are expressed as interval numbers that are generated by statistical simulation, based on which various agricultural water allocation schemes are obtained. Irrigation water performance based on the optimal results are also analyzed. Results validate the applicability of the approach incorporating multiple uncertainties and risk-aversion measures in optimization models, and generating agricultural water allocation schemes in the form of interval numbers.

Planning of Electric Power Systems Considering Virtual Power Plants with Dispatchable Loads Included: An Inexact Two-Stage Stochastic Linear Programming Model

In this study, an inexact two-stage stochastic linear programming (ITSLP) method is proposed for supporting sustainable management of electric power system under uncertainties. Methods of interval-parameter programming and two-stage stochastic programming were incorporated to tackle uncertainties expressed as interval values and probability distributions. The dispatchable loads are integrated into the framework of the virtual power plants, and the support vector regression technique is applied to the prediction of electricity demand. For demonstrating the effectiveness of the developed approach, ITSLP is applied to a case study of a typical planning problem of power system considering virtual power plants. The results indicate that reasonable solutions for virtual power plant management practice have been generated, which can provide strategies in mitigating pollutant emissions, reducing system costs, and improving the reliability of power supply. ITSLP is more reliable for the risk-aversive planners in handling high-variability conditions by considering peak-electricity demand and the associated recourse costs attributed to the stochastic event. The solutions will help decision makers generate alternatives in the event of the insufficient power supply and offer insight into the tradeoffs between economic and environmental objectives.

Inexact stochastic optimization model for industrial water resources allocation under considering pollution charges and revenue-risk control

In this study, an inexact two-stage stochastic downside risk-aversion programming is developed for regional industrial water resources allocation under considering system return-risk and various environment control strategies. In the model, interval-parameter programming, two-stage stochastic programming, and downside risk measure are introduced into an integrated framework for reflecting the complexity and uncertainty of industrial system, and avoiding the expected revenue risk. The method could not only reflect industrial water resources allocation characteristic among multiple users and suppliers, but also provide an effective linkage between economic cost and the system stability. The model is applied to a real case of industrial water resources allocation management in Chongqing city, China, where regional industrial system has faced with lots of difficulties and complexities in water resources utilization and water environmental protection. The impact of pollutants emission reduction and risk-aversion attitude on water resources allocation for different industry sectors, system benefits, and pollutants emissions were analyzed. The results indicated that the total pollutants emission amount control and the expected revenue risk can be used as effective measures for regional industry structure adjustment from terminal environmental and macro-economic perspective. The model has a significant value for regional industrial water optimization allocation under uncertainty to achieve the maximum economic benefits and the effective utilization of multiple water resources.

Effects of Important Factors on Effluent Trading: A Bayesian Estimation-Based Inexact Two-Stage Stochastic Programming Method

We proposed a Bayesian estimation-based inexact two-stage stochastic programming (BITSP) for identifying factors’ effect on effluent trading. BITSP incorporates nutrient fate modeling with Bayesian estimation and inexact two-stage stochastic programming (ITSP). Based on the water quality protocols, Bayesian estimation is used to analyze parameter uncertainty of nutrient modeling as well as provide the random inputs for the optimization process. ITSP can then be used for dealing with multiple uncertainties associated with randomness and intervals. A case study for water management in the Xiangxihe watershed is conducted. Results reveal that strict environmental limits increase the desire for permit trading program. The results also reveal that treatment rate have an obvious effect on effluent trading through changing the buying and selling behavior of point sources

An Optimization Model for Water Management Based on Water Resources and Environmental Carrying Capacities: A Case Study of the Yinma River Basin, Northeast China

In this study, an inexact two-stage stochastic programming (ITSP) model was developed for supporting water resources allocation for the four main water use sectors (industry, municipal, environmental, and agriculture) and total amount control of the pollutant emissions. The Yinma River Basin in northeast China was selected for a case study. A number of scenarios corresponding to different flow levels were examined. The flow levels reflect different probabilities of water resource availability and environmental carrying capacity. The results revealed that the optimal allocation strategies for each sector depend on water resource carrying capacity, wastewater treatment capacity, the total amount of regional control, and the water environment carrying capacity. Water ecology projects were identified that are needed to treat contaminated water and to address the insufficient carrying capacity for pollutant emissions generated in water-using processes. The results will be helpful for establishing sensible water management systems that integrate the development and utilization of water resources and protect the environment, and for providing a basis for water pollution prevention plans, the model can be used to guide management interventions to improve the water environment by regional pollutant emission control and the improvement of carrying capacity in the Yinma River Basin.

Robust cost-risk tradeoff for day-ahead schedule optimization in residential microgrid system under worst-case conditional value-at-risk consideration

With the deregulation of electricity market and the penetration of renewable energy, microgrid system operators may encounter more difficulties in operation management when facing complex economic, technological, and political uncertainties. In this paper, a robust cost-risk tradeoff model is developed for day-ahead schedule optimization in residential microgrid system under uncertainties. This method is an integration of inexact two-stage stochastic programming and worst-case conditional value-at-risk theory, and could handle uncertainties with inexact or partly known probability distribution information. Besides, by introducing the financial risk measurement, it could also hedge against the worst-case scenario caused by multiple independent uncertainties. The proposed model was applied to a hypothetical residential microgrid system with combined heat and power generation for obtaining optimal day-ahead schedule strategies under variable conditions with respect to renewable energy generation, power demand, and electricity market price. The obtained solutions demonstrate that the proposed model could reflect better tradeoff information between economic operation and stable performance according to different risk-aversion attitudes. In general, more conservative risk attitude would be coupled with higher system cost, which implies higher system stability is at the expense of the economic costs. The developed robust cost-risk tradeoff method would be expected to have a potential for wide applications.

Inexact Two-Stage Stochastic Programming for Water Resources Allocation under Considering Demand Uncertainties and Response—A Case Study of Tianjin, China

In this paper, an inexact two-stage stochastic programming model was developed for supporting regional water resource allocation management under uncertainties. The proposed model is an integrated framework of interval parameter programming and two-stage stochastic programming, which can tackle uncertain parameters expressed as interval values with probability distribution information. The proposed model was successfully applied to a typical heavy industrial city suffering water shortage in the east of China, Tianjin. The uncertainties of future water demand were taken into account through generating a set of representative scenarios. The results indicated that different scenarios would affect the water distribution patterns, water shortages, total benefits and system cost. The results revealed that the transferred water from Luan River and Changjiang River would still be the main water resource for each water user. Besides, nearly all water demand in planning horizon would be guaranteed through the reasonable dispatch except under high demand level scenario, in which a small proportion of water requirement in agricultural, municipal and environmental sectors would not be satisfied. The developed method could be used by environmental managers to identify the optimal water supply plan from multiple sources to different end-user sectors under system uncertainties.

An inexact optimization model for regional electric system steady operation management considering integrated renewable resources

In this study, an inexact two-stage stochastic fuzzy programming (ITSFP) is developed for regional power generation planning with considering the intermittency and fuzziness of renewable energy power output. ITSFP incorporates interval-parameter programming (IPP), two-stage stochastic programming (TSP), and fuzzy credibility constrained programming (FCCP) within a general optimization framework which can tackle uncertainties expressed as intervals, probability distributions, and fuzzy sets. The developed method is applied to a regional electric power system over a one-day optimization horizon coupled with air pollution control. The power generation schemes, imported electricity, and system cost under various environmental goals and risk preferences are analyzed. The obtained results indicate that the model can provide a linkage between predefined electric power generation schedule and the relevant economic implications, as well as more reasonable decision alternatives for decision makers by loosening system constraints at specified confidence level. Besides, the fuzziness of forecast error corresponding to the variability of renewable energy resources could be effectively reflected. Moreover, the results are useful for addressing the trade-off between system economy and system risk.

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.

An Inexact Two-Stage Stochastic Programming Model for Sustainable Utilization of Water Resource in Dalian City

An Inexact Two-Stage Stochastic Programming Model for Sustainable Utilization of Water Resource in Dalian City

An inexact risk management model for agricultural land-use planning under water shortage

Water resources availability has a significant impact on agricultural land-use planning, especially in a water shortage area such as North China. The random nature of available water resources and other uncertainties in an agricultural system present risk for land-use planning and may lead to undesirable decisions or potential economic loss. In this study, an inexact risk management model (IRM) was developed for supporting agricultural land-use planning and risk analysis under water shortage. The IRM model was formulated through incorporating a conditional value-at-risk (CVaR) constraint into an inexact two-stage stochastic programming (ITSP) framework, and could be used to control uncertainties expressed as not only probability distributions but also as discrete intervals. The measure of risk about the second-stage penalty cost was incorporated into the model so that the trade-off between system benefit and extreme expected loss could be analyzed. The developed model was applied to a case study in the Zhangweinan River Basin, a typical agricultural region facing serious water shortage in North China. Solutions of the IRM model showed that the obtained first-stage land-use target values could be used to reflect decision-makers’ opinions on the long-term development plan. The confidence level α and maximum acceptable risk loss β could be used to reflect decisionmakers’ preference towards system benefit and risk control. The results indicated that the IRM model was useful for reflecting the decision-makers’ attitudes toward risk aversion and could help seek cost-effective agricultural land-use planning strategies under complex uncertainties.

Optimizing Water Allocation under Uncertain System Conditions in Alfeios River Basin (Greece), Part A: Two-Stage Stochastic Programming Model with Deterministic Boundary Intervals

The enactment of the Water Framework Directive, constituting the basis of the European water policy, introduced various challenges and complexities for water resources management. River basins are exposed to a plethora of environmental stresses, resulting in degradation of their quantitative and qualitative status. This led to the reduction of clean available water, increasing competition among water users and imposing the need for optimal water allocation for each river unit. In most countries (including those in the Mediterranean), water resources management is characterized by lack of effective operational strategies combined with the absence of permanent measuring systems and low financial means, hampering the implementation of efficient river monitoring. Therefore, water resources management is indicated by high uncertainty and by imprecise and limited data, which may be easily approximated through estimates of intervals. In the present work, optimal water allocation under uncertain system conditions is undertaken for the Alfeios River Basin (Greece) based on an inexact two-stage stochastic programming methodology developed by Huang and Loucks (2000). It combines ordinary two-stage stochastic programming with uncertainties expressed as deterministic boundary intervals. Stable intervals for optimized water allocation targets and probabilistic water allocation and shortages are estimated under a baseline scenario and four water and agricultural policy future scenarios.

A Novel Inexact Two-Stage Stochastic Robust-Compensation Model for Electric Supply Environmental Management Under Uncertainty

In this study, a novel inexact two-stage stochastic robust-compensation programming (ITSP-RC) model is developed for CO2 emission reduction management under uncertainties. This model is attempted to integrate ITSP and stochastic RC programming into a general framework and apply the ITSP-RC for power management and CO2 emission reduction management, such that the developed model can tackle uncertainties described in terms of interval values and probability distributions over a two-stage context. Moreover, it can reflect dynamic and randomness of the energy systems during the planning horizon. The developed method has been applied to a case to solve CO2 emission management problem in electric supply environmental management. A number of scenarios corresponding to different adoption rate levels of carbon capture, utilization, and storage technology are examined. With the RC programming, regional energy systems would have a stable financial budget. The result suggests that the methodology is applicable for reflecting complexities of large-scale energy management systems and addressing CO2 emissions reduction issue with the planning period.