In this study, a fixed-mix stochastic fractional programming (FSFP) method is developed for balancing the water-allocation conflict between upstream hydropower generation and downstream agricultural irrigation. FSFP has advantages in dealing with ratio-objective problem under uncertainty, reflecting the dynamic and stochastic characteristics over a long-term planning context, as well as analyzing interrelationships between system efficiency and violation risk of water-allocation target. Then, FSFP is firstly applied to Tuyamuyun reservoir in the lower reach of Amu Darya River basin (Central Asia), where multiple scenarios based on different hydropower-generation targets and inflow levels are examined for identifying the complex relationship between hydropower generation and crop irrigation. Major findings and managerial insights can be summarized as: (i) with the reduction of reservoir inflow, water allocation for downstream agricultural irrigation would decrease by 30.4% once the minimum demand is satisfied, and hydropower generation should be higher priority for pursuing higher marginal benefit; (ii) with the shrinking water supply and rising hydropower-generation target, cotton planting should be firstly restrained due to its high water demand and grape planting is encouraged; (iii) under extreme water scarcity (i.e., low and very-low inflow levels), low-level hydropower generation target (i.e., α = 0.45) is desired for meeting the food requirement in the study basin; (iv) for alleviating the water shortage during dry seasons, it is recommended that water storage should be conducted in autumn and winter, and water release for crop irrigation should be implemented during spring and summer. These findings can help managers identify sustainable water-allocation schemes for agricultural irrigation and hydropower generation against water shortage, environmental destruction and energy insecurity in arid regions.

在这项研究中，开发了一种固定混合随机分数规划（FSFP）方法来平衡上游水力发电和下游农业灌溉之间的水资源分配冲突. FSFP在处理不确定条件下的比率目标问题、反映长期规划背景下的动态和随机特征以及分析系统效率与水分配目标违规风险之间的相互关系方面具有优势。然后，FSFP 首次应用于阿姆河流域（中亚）下游的吐亚木云水库，根据不同的水力发电目标和流入水平，研究了多个情景，以识别水力发电与作物灌溉之间的复杂关系。主要发现和管理见解可概括为：（i）随着水库流入量的减少，一旦满足最低需求，下游农业灌溉的水分配将减少 30.4%，水力发电应该是追求更高边际效益的更高优先级；(ii) 在供水量减少、水力发电目标上升的情况下，棉花需水量大，应首先抑制棉花种植，鼓励葡萄种植；(iii) 在极端情况下缺水（即低和极低的流入水位），低水位发电目标（即 α = 0.45）是满足研究流域粮食需求的理想选择；（四）为缓解旱季缺水问题，建议秋冬季蓄水，春夏放水灌溉农作物。这些发现可以帮助管理者确定用于农业灌溉和水力发电的可持续水资源分配计划，以应对干旱地区的水资源短缺、环境破坏和能源不安全。