A fixed-mix stochastic fractional programming method for optimizing agricultural irrigation and hydropower generation in Central Asia
Introduction
Water resources are pivotal for ensuring food security and stable energy supply in a reservoir system. Rapid population growth and economic development have brought about soaring water demands and intensified the water-use conflicts between agricultural irrigation and energy generation (Lee and Jung, 2018; Li et al., 2020; Babamiri et al., 2021). For example, in a river basin, the upstream region often increases reservoir water storage for power generation in winter and reduces the downstream water supply, which affects the downstream agricultural production in summer. The release of water from the upstream region in the non-growing season often leads to flooding in the downstream countries, which would result in serious water and food crisis (Thushara De Silva and Hornberger, 2019). According to FAO (2020), 3.2 billion people have been suffering water scarcity and unbalanced water allocation has resulted in an annual loss of 23.8 billion US$ (Veldkamp et al., 2017; Tan et al., 2020). Optimizing water allocation of crop irrigation and hydropower generation is desired for guaranteeing sustainable development of the region. However, some parameters of the reservoir system may be uncertain due to the variations of natural and socioeconomic conditions. For example, randomness of precipitation may lead to random inflow of reservoir, which can increase the complexity of decision-making process in reservoir system (Martinsen et al., 2019; Nie et al., 2021). Therefore, it is necessary to develop an effective method to generate sustainable reservoir management schemes for agriculture irrigation and hydropower generation under uncertainty.
Previously, many stochastic programming methods were developed for tackling resources management problems with random uncertainty (Marchant et al., 2018; Kashanian et al., 2020; Xiao et al., 2022). For example, Yao et al. (2019) proposed a multi-stage programming method for optimizing regional water resources, where the uncertainty of available water was expressed as discrete variables. Şahin et al. (2020) developed a multi-stage stochastic technique for making the demand response optimization under uncertainty, where upper and lower bounds of the relevant scenarios were obtained through a scenario group sub-problem approach. Li et al. (2021a) proposed a multi-stage stochastic programming method for planning water-resource management, where the uncertainty of economic and technical data expressed as distribution probabilities. In general, multistage stochastic programming (MSP) can effectively conduct analysis of multiple policy scenarios with uncertain data and adequately reflect the dynamic variations of system conditions (Li et al., 2008). However, MSP may encounter difficulties in tackling large-scale water management problems with persistent hydrological records. Fixed-mix stochastic programming (FSP), has advantage in simplifying the solution process by using a fixed probability level, is advanced on the base of MSP. Therefore, FSP model can be better applied to deal with large-scale practical problems over a long-term planning context (Liu et al., 2014). However, traditional FSP methods often consider a maximum crop yield or a minimum water allocation as a single-objective function, which neglects the conflicting objectives within the reservoir system.
In arid regions, managers usually prefer to achieve maximum system output with minimum water allocation (i.e., marginal benefit) due to limited water resources (Dai et al., 2021; Li et al., 2021b). Fractional programming (FP) can optimize system efficiency (i.e., marginal benefit) through ratio functions (cost/volume, output/input, or cost/time) and has been commonly used in water management problems (Tan and Zhang, 2018; Arya and Singh, 2019). For instance, Mani et al. (2016) developed a linear fractional programming method for dealing with the conjunctive-use problem of reservoir water and groundwater, where the objective is maximizing the ratio of total groundwater allocation to the total deficit of reservoir storages. Niu et al. (2019) developed a multi-objective linear fractional programming model for optimizing crop planting and water resource management, where the coordination of agricultural production, ecological protection and water saving target was achieved with an optimal system efficiency. Although FP is effective in handling ratio-objective problems, it cannot tackle the uncertainty within system (Wang et al., 2018). However, no studies were reported on coupling FSP and FP methods for planning water resources in a reservoir system.
Therefore, this study aims to develop a novel fixed-mix stochastic fractional programming (FSFP) method for planning water resources in reservoir system under uncertainty. Compared with traditional multi-stage stochastic method, the FSFP method can address bi-objective problems and reflect dynamic variations of large-scale practical problems over a long-term planning context. In comparison with conventional fractional programming method, the FSFP method can effectively tackle the dynamic, interactive and uncertain characteristics of water resources management system. The major novelty and contribution of this study can be listed as: (i) this is the first attempt for integrating techniques of FP and FSP into a general framework for simultaneously dealing with conflicting objectives and stochastic uncertainty over a multistage context; (ii) FSFP is applied to the lower reach of Amu Darya River basin for optimizing water resources allocation, where multiple scenarios associated with different hydropower generation targets are analyzed; (iii) optimal alternatives can be obtained for balancing the water conflict between upstream hydropower generation and downstream agricultural irrigation in different seasons, which can provide desired decision support for sustainable development in Central Asia.
Section snippets
Methodology
Based on the simple rebalancing rules (i.e., permitting to modify the decisions at each time stage by introducing the economic recourse), fixed-mix stochastic programming (FSP) method can deal with the random uncertainty expressed as probability distributions. A FSP method can be formulated as follows:subject to:withsubject to:
where f is the marginal benefit; x is decision variable decided before the random variable ξ is revealed; D(ξ), h(ξ) and T(ξ)
Study area
Amu Darya River is the largest river in Central Asia with the length of 2540 km and annual average flow of 65.6 km3. Its runoff mainly originates from glacier and snow melting during wet seasons (from April to August) (Jalilov et al., 2016). The Tuyamuyun reservoir is the largest reservoir in the lower reach of Amu Darya River basin located on the border between Turkmenistan and Uzbekistan, which regulates the annual runoff of the Amu Darya downstream with a storage capacity of 7.8 km3 (Olsson
Result and discussion
Fig. 2 shows the marginal benefit and total water allocation under all policy scenarios (α levels). Result shows marginal benefit would reduce with deceased α level. The highest marginal benefit would occur under α = 1 (i.e., 81.84 × 10−3 US$/m3); the lowest one would occur when α = 0.3 (i.e., 73.12 × 10−3 US$/m3). High α level indicates higher hydropower generation target (i.e. a preference of water allocation for upstream hydropower generation) and lower marginal benefit; low α level
Conclusions
In this study, a fixed-mix stochastic fractional programming (FSFP) method has been developed through integrating fractional programming (FP) and fixed-mix stochastic programming (FSP) into a general framework. FSFP can deal with bi-objective problem associated with system benefit maximization and water use minimization, and reflect the dynamics of decision making process over a multistage context under uncertainty. Moreover, it can help analyze the interrelationship between system efficiency
CRediT authorship contribution statement
Y.X. Zhou: Writing – original draft, Writing – review & editing, Software, Methodology, Conceptualization. Y.P. Li: Conceptualization, Funding acquisition, Validation, Supervision, Writing – review & editing. G.H. Huang: Writing – review & editing. Y.F. Zhang: Writing – review & editing, Data curation, Formal analysis. Y. Ma: Writing – review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (XDA20060302), the Natural Science Foundation of China (51779008), and the National Foreign Expert Project (G2021111016L). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.
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