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Shrinking Pareto Fronts to Guide Reservoir Operations by Quantifying Competition Among Multiple Objectives

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Water Resources Research
Authors:
Hong‐Ru Wang
Hong‐Ru Wang
Hong‐Ru Wang
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Fangfang Li at China Agricultural University
Fangfang Li
Guang‐Qian Wang
Guang‐Qian Wang
Guang‐Qian Wang
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Jun Qiu
Jun Qiu
Jun Qiu
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Abstract and Figures

Multiobjective optimization has been widely applied to reservoir operations in order to provide balanced operational schemes considering their multiple functions, including flood control, power generation, and ecological objectives. The Pareto front derived from multiobjective optimization is a set of optimal solutions that cannot quickly provide direct guidance for decision‐makers. In this study, a shrinking method is proposed to reduce the selection range of the optimal solutions on the Pareto front. Based on two proposed indices, that is, competitiveness and the competition efficiency between each pair of dual objectives, the optimal solutions are shrunk twice to accurately focus on the optimal solution region and reduce the difficulty of decision‐making. The proposed methodology is applied to a large‐scale reservoir on the upper reach of the Yellow River, China, simultaneously considering power generation, hydropower output stability, and ecological objectives. The results show that the proposed method could reduce the Pareto front to the solutions performing well in objectives and can be generalized for other multiobjective optimization models.
Schematic diagram of the preferred region of the operational schemes. Gray circles correspond to Pareto solutions, solid circles AŽ1maxZ2A $\mathrm{A}\left({\check{Z}}_{1}^{\mathrm{max}}{Z}_{2}^{A}\right)$ and BZ1BŽ2max $\mathrm{B}\left({Z}_{1}^{B}{\check{Z}}_{2}^{\mathrm{max}}\right)$ are the equivalent maximum solutions of objective Z1 ${Z}_{1}$ and Z2 ${Z}_{2}$; blue‐shaded Ω12 ${{\Omega}}_{12}$ is the preferred region when Z1 ${Z}_{1}$ is more competitive, green‐shaded Ω21 ${{\Omega}}_{21}$ is the preferred region when Z2 ${Z}_{2}$ is more competitive, and the striped shaded area is the preferred region when Z1 ${Z}_{1}$ and Z2 ${Z}_{2}$ have equivalent competitive advantages. The regions are separated by dots D $\mathrm{D}$, D1 ${\hspace*{.5em}\mathrm{D}}_{1}$, and D2 ${\hspace*{.5em}\mathrm{D}}_{2}$ that are located according to the listed equations.
Schematic diagram of the preferred region of the operational schemes. Gray circles correspond to Pareto solutions, solid circles AŽ1maxZ2A $\mathrm{A}\left({\check{Z}}_{1}^{\mathrm{max}}{Z}_{2}^{A}\right)$ and BZ1BŽ2max $\mathrm{B}\left({Z}_{1}^{B}{\check{Z}}_{2}^{\mathrm{max}}\right)$ are the equivalent maximum solutions of objective Z1 ${Z}_{1}$ and Z2 ${Z}_{2}$; blue‐shaded Ω12 ${{\Omega}}_{12}$ is the preferred region when Z1 ${Z}_{1}$ is more competitive, green‐shaded Ω21 ${{\Omega}}_{21}$ is the preferred region when Z2 ${Z}_{2}$ is more competitive, and the striped shaded area is the preferred region when Z1 ${Z}_{1}$ and Z2 ${Z}_{2}$ have equivalent competitive advantages. The regions are separated by dots D $\mathrm{D}$, D1 ${\hspace*{.5em}\mathrm{D}}_{1}$, and D2 ${\hspace*{.5em}\mathrm{D}}_{2}$ that are located according to the listed equations.
… 
Flowchart of the proposed method to shrink the Pareto front.
Flowchart of the proposed method to shrink the Pareto front.
… 
Prior solutions for the wet year of 2009. (a) Pareto optimal surface. (b) PF|z1z2 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{2}}$, that is, Pareto front projection of power generation and stability. (c) PF|z1z3 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power generation and ecological score. (d) PF|z2z3,i.e., $\text{PF}{\vert }_{{\mathrm{z}}_{2}{\mathrm{z}}_{3}},\hspace*{.5em}\mathrm{i}.\mathrm{e}.,$ Pareto front projection of power output stability and ecological score. Hollow diamonds indicate the equivalent extreme points. The color of the solid circles indicates the value of the objective (not shown), with purple representing smaller values. Circles with red edges indicate the prior solutions. The shaded region is the preferred region derived from Equations 19–23, corresponding to Figure 1.
Prior solutions for the wet year of 2009. (a) Pareto optimal surface. (b) PF|z1z2 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{2}}$, that is, Pareto front projection of power generation and stability. (c) PF|z1z3 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power generation and ecological score. (d) PF|z2z3,i.e., $\text{PF}{\vert }_{{\mathrm{z}}_{2}{\mathrm{z}}_{3}},\hspace*{.5em}\mathrm{i}.\mathrm{e}.,$ Pareto front projection of power output stability and ecological score. Hollow diamonds indicate the equivalent extreme points. The color of the solid circles indicates the value of the objective (not shown), with purple representing smaller values. Circles with red edges indicate the prior solutions. The shaded region is the preferred region derived from Equations 19–23, corresponding to Figure 1.
… 
Preferred solutions for the normal year of 2010. (a) Pareto optimal surface. (b) PF|z1z2 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{2}}$, that is, Pareto front projection of power generation and stability. (c) PF|z1z3 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power generation and the ecological score. (d) PF|z2z3 $\text{PF}{\vert }_{{\mathrm{z}}_{2}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power output stability and the ecological score. Hollow diamonds indicate the equivalent extreme points. The color of the solid circles indicates the value of the objective (not shown), with purple representing smaller values. Circles with red edges indicate the preferred solutions. The shaded region is the preferred region derived from Equations 19–23 corresponding to Figure 1.
Preferred solutions for the normal year of 2010. (a) Pareto optimal surface. (b) PF|z1z2 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{2}}$, that is, Pareto front projection of power generation and stability. (c) PF|z1z3 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power generation and the ecological score. (d) PF|z2z3 $\text{PF}{\vert }_{{\mathrm{z}}_{2}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power output stability and the ecological score. Hollow diamonds indicate the equivalent extreme points. The color of the solid circles indicates the value of the objective (not shown), with purple representing smaller values. Circles with red edges indicate the preferred solutions. The shaded region is the preferred region derived from Equations 19–23 corresponding to Figure 1.
… 
Preferred solutions for the dry year of 2015. (a) Pareto optimal surface. (b) PF|z1z2 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{2}}$, that is, Pareto front projection of power generation and stability. (c) PF|z1z3 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power generation and ecological score. (d) PF|z2z3 $\text{PF}{\vert }_{{\mathrm{z}}_{2}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power output stability and ecological score. Hollow diamonds indicate the equivalent extreme points. The color of the solid circles indicates the value of the objective (not shown), with purple indicating smaller values. Circles with red edges indicate the preferred solutions. The shaded region is the preferred region derived from Equations 19–23, corresponding to Figure 1.
+1
Preferred solutions for the dry year of 2015. (a) Pareto optimal surface. (b) PF|z1z2 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{2}}$, that is, Pareto front projection of power generation and stability. (c) PF|z1z3 $\text{PF}{\vert }_{{\mathrm{z}}_{1}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power generation and ecological score. (d) PF|z2z3 $\text{PF}{\vert }_{{\mathrm{z}}_{2}{\mathrm{z}}_{3}}$, that is, Pareto front projection of power output stability and ecological score. Hollow diamonds indicate the equivalent extreme points. The color of the solid circles indicates the value of the objective (not shown), with purple indicating smaller values. Circles with red edges indicate the preferred solutions. The shaded region is the preferred region derived from Equations 19–23, corresponding to Figure 1.
… 
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1. Introduction
Reservoirs store and regulate natural water resources, playing a vital role in water supply, power generation,
flood control, and shipping (Biemans etal.,2011; Chen etal.,2019; Kibler & Tullos,2013). Consequently, the
construction and operation of reservoirs affect the ecosystem, mainly manifesting as changes in hydrological
conditions downstream. In most cases, the human-controlled discharge process is more stable, altering the flow
regime formed by long-term natural evolution that is required by downstream aquatic ecosystems, thus endanger-
ing them (Jiang etal.,2019).
Fish have a high trophic level in aquatic ecosystems, and their population size and species richness reflect the
changes in the aquatic ecosystems (Mišetić etal.,2003; Wu etal.,2014; Zhao etal.,2015). In some systems, fish
spawning coincides with the onset of flooding, and they gather at the spawning grounds to lay eggs when the
flood increases and leave after it finishes (Tao etal.,2017). The key ecological requirements for the hydro-envi-
ronment in the upper reach of the Yellow River in China during the spawning season include the flow regime, as
the development of the fish gonads requires stimulation of the flow rising process; the water depth, as the hatch-
ing of benthic eggs requires a gentle water flow with little amplitude variation; and water temperature, with the
appropriate temperature for fish spawning and egg incubation being approximately 14°C–15°C (Li etal.,2020).
Table 2 in Bejarano etal.(2017) describes the changes in the environmental and biological responses caused by
short-term flow regime changes, including the destruction of the habitats of fish and other species, impact of mi-
gration activities, and reduction of species diversity. With increased attention paid to the ecological environment,
improving the current reservoir operation schemes and restoring the river’s ecological flow processes, especially
during critical physiological processes, such as fish spawning and migration, have become key objectives in
reservoir operations.
The objectives of reservoir operation are often incommensurable or even conflicting. Multiobjective optimization
models, which simultaneously consider different objectives of reservoirs, such as power generation and ecologi-
cal protection, are required to address these problems (Bai etal.,2019; W. He etal.,2020). Multiobjective opti-
mization models are solved to determine the quantitative relationship among objectives and formulate a scheme
balancing them (Tang etal.,2019).
The existing multiobjective optimization models of reservoir operations, which consider resource exploitation
and ecology, mostly use the weighting of each objective to reduce dimensionality or obtain a set of Pareto optimal
Abstract Multiobjective optimization has been widely applied to reservoir operations in order to provide
balanced operational schemes considering their multiple functions, including flood control, power generation,
and ecological objectives. The Pareto front derived from multiobjective optimization is a set of optimal
solutions that cannot quickly provide direct guidance for decision-makers. In this study, a shrinking method
is proposed to reduce the selection range of the optimal solutions on the Pareto front. Based on two proposed
indices, that is, competitiveness and the competition efficiency between each pair of dual objectives, the
optimal solutions are shrunk twice to accurately focus on the optimal solution region and reduce the difficulty
of decision-making. The proposed methodology is applied to a large-scale reservoir on the upper reach of the
Yellow River, China, simultaneously considering power generation, hydropower output stability, and ecological
objectives. The results show that the proposed method could reduce the Pareto front to the solutions performing
well in objectives and can be generalized for other multiobjective optimization models.
WANG ET AL.
© 2022. American Geophysical Union.
All Rights Reserved.
Shrinking Pareto Fronts to Guide Reservoir Operations by
Quantifying Competition Among Multiple Objectives
Hong-Ru Wang1 , Fang-Fang Li1 , Guang-Qian Wang2,3, and Jun Qiu2,3
1College of Water Resources & Civil Engineering, China Agricultural University, Beijing, China, 2State Key Laboratory
of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing, China, 3State Key
Laboratory of Plateau Ecology and Agriculture, Qinghai University, Xining, China
Key Points:
• A method of evaluating
competitiveness among
multiobjectives is proposed
• Competition efficiency indices
between dual objectives determine the
decision directions
• The bilevel Pareto front shrinking
approach narrows down the Pareto
solutions by nearly 1 order
Correspondence to:
J. Qiu,
aeroengine@tsinghua.edu.cn
Citation:
Wang, H.-R., Li, F.-F., Wang, G.-Q.,
& Qiu, J. (2022). Shrinking Pareto
fronts to guide reservoir operations by
quantifying competition among multiple
objectives. Water Resources Research,
58, e2021WR029702. https://doi.
org/10.1029/2021WR029702
Received 26 JAN 2021
Accepted 28 JAN 2022
Author Contributions:
Conceptualization: Fang-Fang Li, Jun
Qiu
Data curation: Hong-Ru Wang
Formal analysis: Hong-Ru Wang, Jun
Qiu
Funding acquisition: Fang-Fang Li,
Jun Qiu
Investigation: Hong-Ru Wang, Jun Qiu
Methodology: Fang-Fang Li, Jun Qiu
Resources: Fang-Fang Li
Software: Hong-Ru Wang
Supervision: Fang-Fang Li, Guang-Qian
Wang, Jun Qiu
Validation: Hong-Ru Wang, Fang-Fang
Li, Guang-Qian Wang, Jun Qiu
Visualization: Hong-Ru Wang, Jun Qiu
Writing – original draft: Hong-Ru
Wang
Writing – review & editing: Fang-Fang
Li, Guang-Qian Wang
10.1029/2021WR029702
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  • Jul 2019
  • J HYDROL
Frequent occurrence of flood and ice disasters caused by river channel sedimentation in the Upper Yellow River of China has seriously threatened the lives and property of downstream residents, which has become the most challenging issues in harnessing the Yellow River. In this paper, we aim at optimizing water-sediment regulation that considers two cascade reservoirs (Longyangxia and Liujiaxia) as the regulatory bodies using one multi-objective model (sediment transport and hydropower generation) and two single-objective models (sediment transport only, and hydropower generation only). We propose an innovative approach (FSS-MOPSO) that hybrids the Feasible Search Space (FSS) with the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm to search the optimal solutions for the multi-objective joint operation of cascade reservoirs. We analyze the transformation rules of water-sediment regulation among five objectives (hydropower generation, sediment transport, water supply, flood control and ice control) under various optimization schemes. The results indicate that the conflict between the hydropower generation objective and the sediment transport objective is prominent. An extreme case indicates that an increase in hydropower output by 2.31 billion kW·h (17.6% increase) would greatly reduce the amount of sediment transport (73.5% decrease) while only makes little effects on the other three objectives. The results demonstrate that the optimal water-sediment regulation not only can ensure to meet water demands in the future (2030) but can provide an important guideline to safely operate cascade reservoirs during ice and flood periods. The research findings contribute to the identification of the relationship among objectives and strategy recommendations on water-sediment regulation for efficient cascade reservoirs operation in the Upper Yellow River.
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