Abstract
Water resources systems are often characterized by multiple objectives. Typically, there is no single optimal solution which can simultaneously satisfy all the objectives but rather a set of technologically efficient non-inferior or Pareto optimal solutions exists. Another point regarding multi-objective optimization is that interdependence and contradictions are common among one or more objectives. Therefore, understanding the competition mechanism of the multiple objectives plays a significant role in achieving an optimal solution. This study examines cascade reservoirs in the Heihe River Basin of China, with a focus on exploring the multi-objective competition mechanism among irrigation water shortage, ecological water shortage and the power generation of cascade hydropower stations. Our results can be summarized as follows: (1) the three-dimensional and two-dimensional spatial distributions of a Pareto set reveal that these three objectives, that is, irrigation water shortage, ecological water shortage and power generation of cascade hydropower stations cannot reach the theoretical optimal solution at the same time, implying the existence of mutual restrictions; (2) to avoid subjectivity in choosing limited representative solutions from the Pareto set, the long series of non-inferior solutions are adopted to study the competition mechanism. The premise of sufficient optimization suggests a macro-rule of ‘one falls and another rises,’ that is, when one objective value is inferior, the other two objectives show stronger and superior correlation; (3) the joint copula function of two variables is firstly employed to explore the multi-objective competition mechanism in this study. It is found that the competition between power generation and the other objectives is minimal. Furthermore, the recommended annual average water shortage are 1492 × 104 m3 for irrigation and 4951 × 104 m3 for ecological, respectively. This study is expected to provide a foundation for selective preference of a Pareto set and insights for other multi-objective research.
Highlights
Real-world systems always refer to multiple objective optimization in their operations.Multi-objective optimization problems (MOPs) usually require the simultaneous optimization of some incommensurable and competitive objectives
The primary goals of this study are: (1) to analyze the three-dimensional and two-dimensional spatial distributions of the Pareto set obtained on account of reservoir dispatching obtained in the preliminary work; (2) to establish a formula for quantitatively describing the relationship between two objectives and to inform the law of water use; and (3) to explore the multi-objective competition mechanism for the overall impact of one objective on the other two objectives with the copula function constructing the joint sequence of two targets
The range changes to [0.7968,1] and the normalized value of Obj-2 decreases by 1.7388 for each increase of Obj-1. Converting to their own respective target values of the scheme, when the independent variable is between 925~2078 × 104 m3, the average annual ecological water shortage will increase by 9.0553 × 104 m3 for each reduction of 1 × 104 m3 of the average annual irrigation water
Summary
Real-world systems always refer to multiple objective optimization in their operations. Copula function is chosen to construct joint sequence values of two targets This is the first time that copula is applied to explore multi-objective competition mechanism whereby the overall impact of one objective on the other two objectives can be evaluated. The primary goals of this study are: (1) to analyze the three-dimensional and two-dimensional spatial distributions of the Pareto set obtained on account of reservoir dispatching obtained in the preliminary work; (2) to establish a formula for quantitatively describing the relationship between two objectives and to inform the law of water use; and (3) to explore the multi-objective competition mechanism for the overall impact of one objective on the other two objectives with the copula function constructing the joint sequence of two targets. It provides guidance for the unified allocation of water resources and provides insights for other multi-objective problems
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