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Copula-Based Research on the Multi-Objective Competition Mechanism in Cascade Reservoirs Optimal Operation

State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area, Xi’an University of Technology, Xi’an 710048, China
State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China
Environmental Change Institute, University of Oxford, Oxford OX1 3QY, UK
Author to whom correspondence should be addressed.
Water 2019, 11(5), 995;
Received: 10 April 2019 / Revised: 2 May 2019 / Accepted: 9 May 2019 / Published: 12 May 2019
(This article belongs to the Special Issue Managing Water Resources in Large River Basins)
PDF [4414 KB, uploaded 27 May 2019]


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. View Full-Text
Keywords: multi-objective competition mechanism; cascade reservoirs operation; copula function; Pareto set multi-objective competition mechanism; cascade reservoirs operation; copula function; Pareto set

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Zhao, M.; Huang, S.; Huang, Q.; Wang, H.; Leng, G.; Liu, S.; Wang, L. Copula-Based Research on the Multi-Objective Competition Mechanism in Cascade Reservoirs Optimal Operation. Water 2019, 11, 995.

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