# Solution Selection from a Pareto Optimal Set of Multi-Objective Reservoir Operation via Clustering Operation Processes and Objective Values

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Mei–Wang Fluctuation Similarity Measure

- ${f}_{\alpha}$ is the index used to describe the difference in quantitative variation between ${Z}_{n}$ and ${Z}_{m}$. It is calculated with Equation (2):$${f}_{\alpha}=\sqrt{\frac{1}{N}{\sum}_{i=1}^{N}|{({z}_{m}\left(i\right)-\overline{{z}_{m}})}^{2}-{({z}_{n}\left(i\right)-\overline{{z}_{n}})}^{2}|},$$
- ${f}_{\beta}$ is an index describing the difference between two processes in terms of contour variations. It is calculated with Equations (3)–(5):$${f}_{\beta}=\sqrt{{\sum}_{i=1}^{N}{({\theta}_{m}\left(i\right)-{\theta}_{n}\left(i\right))}^{2}},$$$${\theta}_{m}\left(i\right)=\{\begin{array}{ll}\mathrm{arctan}\left|{k}_{m}\left(i\right)\right|& i=1orN\\ \left|\mathrm{arctan}{k}_{m}\left(i\right)-\mathrm{arctan}{k}_{m}\left(i-1\right)\right|& 2\le i\le N-1,and{k}_{m}\left(i\right)\times {k}_{m}\left(i-1\right)\ge 0\\ \mathrm{arctan}\left|{k}_{m}\left(i\right)\right|+\mathrm{arctan}\left|{k}_{m}\left(i-1\right)\right|& 2\le i\le N-1,and{k}_{m}\left(i\right)\times {k}_{m}\left(i-1\right)0\end{array},$$$${k}_{m}\left(i\right)=\{\begin{array}{ll}\frac{{z}_{m}\left(i+1\right)-{z}_{m}\left(i\right)}{t\left(i+1\right)-t\left(i\right)}& 1\le i\le N-1\\ \frac{{z}_{m}\left(N\right)-{z}_{m}\left(N-1\right)}{t\left(N\right)-t\left(N-1\right)}& i=N\end{array},$$
- The MWFSM between ${Z}_{n}$ and ${Z}_{m}$ is calculated with Equation (6) as follows:$$D{\mathrm{ist}}_{MWF}={f}_{\alpha}\xb7{f}_{\beta},$$

#### 2.2. Clustering by Fast Search and Find of Density Peaks

**.**The detailed process of applying the DPC to $X$ is described as below:

- Calculate distance matrix ${D}^{n\times n}={\left\{{d}_{ij}\right\}}^{n\times n}$ via MWFSM.
- Sort ${d}_{ij}$ in ascending order, assign the top 2% of the data column to ${d}_{c}$.
- Calculate local density $\rho $ according to Equation (7) for each data point in $X$.
- Calculate distance from the nearest larger density point $\delta $ according to Equations (8) and (9) for each data point in $X$.
- Calculate the decision value $\gamma $ according to Equation (10) for each data point in $X$.
- Sort $\gamma $ in ascending order and record the new order.
- Construct the decision value graph where points are represented as $\left(i,{\gamma}_{i}\right)$ with the ascending order in Step 6.
- Select the points of the large γ values as cluster centers according to the decision value graph.
- Allocate the remaining points following the principle of proximity.

#### 2.3. Clustering-Based Solution Selection Method

- The DPC method is applied to set $Z$ and obtains $k$ clusters of decision processes.
- The $k$ clusters generated in step 1 are ranked by size, and the ${i}^{th}$ decision cluster is denoted as $D{C}_{i}$, and the decision cluster with the largest membership is denoted as $D{C}_{1}$.
- An operation pattern set consisting of the solutions corresponding to the centers of the $k$ decision clusters is generated.
- The k-means algorithm is employed to cluster set $F$ and obtain $q$ objective value clusters.
- The $q$ objective value clusters are ranked in descending order, and the ${j}^{th}$ cluster is denoted as $F{C}_{j}$, and the set with the largest membership is denoted as $F{C}_{1}$.
- The intersection of $D{C}_{1}$ and $F{C}_{1}$ is considered. If the intersection set is not empty, it is denoted as ${C}_{aim}$. If the intersection set is empty, the intersection of $D{C}_{1}$ and the next objective cluster $F{C}_{2}$ is determined. This process is repeated until the intersection set is no longer empty, which is then denoted as ${C}_{aim}$.
- The decision process with the minimum accumulative similarity in set ${C}_{aim}$ is identified and recommended.
- The selected solution and the operation pattern set are provided to DMs.

#### 2.4. Multi-Objective Optimization Model

#### 2.4.1. Objective Functions

#### 2.4.2. Constraints

- The water balance is expressed as$${V}_{i,t+1}={V}_{i,t}+\left({I}_{i,t}-{Q}_{i,t}\right)\xb7\Delta t$$
- The outflow constraint is expressed as$${I}_{i+1,t}={Q}_{i,t}+{q}_{i,t}$$
- The power output constraint is given by$${N}_{i,t}^{min}\le {N}_{i,t}\le {N}_{i,t}^{max}$$
- The storage volume constraint is expressed as$${Z}_{i,t}^{min}\le {Z}_{i,t}\le {Z}_{i,t}^{max}$$
- The boundary condition limit is given by$${Z}_{i,1}={Z}_{i,T+1}={Z}_{i,m}$$

## 3. Case Study

#### 3.1. Study Area

^{3}/s are considered small and medium floods in the TGR. During small and medium floods, the objective of flood risk and ecological influence minimization is adopted.

#### 3.2. Input Data

## 4. Results and Discussion

#### 4.1. NSGA-II Output and Traditional Analysis of Pareto Set

#### 4.2. Clustering of the Trade-Off Frontier

#### 4.3. Clustering of the Reservoir Operation Processes

#### 4.4. Solution Selection

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Determination of the Ecological Flow Based on the Indicators of Hydrologic Alteration (IHA) Method

**Table A1.**Components of the Indicators of Hydrologic Alteration (IHA) results used to develop the artificial flood pulse flow.

Hydrologic Indicators of a High Pulse | Quantile | ||
---|---|---|---|

75% | 50% | 25% | |

Date of rise | 12 | 7 | 2 |

Initial flow of the pulse (m^{3}/s) | 23,500 | 21,150 | 20,250 |

Duration of the high pulse (day) | 8 | 4 | 2 |

No. of high pulses (times) | 2 | 1 | 1 |

Peak flow of the high pulse (m^{3}/s) | 30,800 | 25,450 | 21,200 |

Rise rate (m^{3}/s/d) | 2303 | 1932 | 1558 |

Fall rate (m^{3}/s/d) | −1105 | −1386 | −1766 |

Rise duration (d) | 4 | 2 | 1 |

Fall duration (d) | 4 | 2 | 1 |

^{3}/s. The remainder of the high pulse was assigned the corresponding value of the 50th percentile series. The ecological flow series considering one high pulse is listed in Table A2.

Index Number (Day) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Discharge (m^{3}/s) | 14,800 | 14,900 | 14,800 | 15,500 | 15,100 | 14,700 | 15,100 |

Index Number (Day) | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

Discharge (m^{3}/s) | 15,400 | 15,000 | 18,483 | 21,966 | 25,450 | 23,666 | 21,883 |

Index Number (Day) | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

Discharge (m^{3}/s) | 20,100 | 19,500 | 22,000 | 21,800 | 22,200 | 23,200 | 25,200 |

## References

- Yu, Y.; Wang, C.; Wang, P.; Hou, J.; Qian, J. Assessment of multi-objective reservoir operation in the middle and lower Yangtze River based on a flow regime influenced by the Three Gorges Project. Ecol. Inform.
**2017**, 38, 115–125. [Google Scholar] [CrossRef] - Labadie, J.W. Optimal operation of multireservoir systems: State-of-the-art review. J. Water Resour. Plan. Manag.
**2004**, 130, 93–111. [Google Scholar] [CrossRef] - Reddy, M.J.; Kumar, D.N. Optimal reservoir operation using multi-objective evolutionary algorithm. Water Resour. Manag.
**2006**, 20, 861–878. [Google Scholar] [CrossRef] [Green Version] - Chou, F.N.F.; Wu, C.W. Stage-wise optimizing operating rules for flood control in a multi-purpose reservoir. J. Hydrol.
**2015**, 521, 245–260. [Google Scholar] [CrossRef] - Dai, L.; Zhang, P.; Wang, Y.; Jiang, D.; Dai, H.; Mao, J.; Wang, M. Multi-objective optimization of cascade reservoirs using NSGA-II: A case study of the Three Gorges-Gezhouba cascade reservoirs in the middle Yangtze River, China. Hum. Ecol. Risk Assess.
**2017**, 23, 814–835. [Google Scholar] [CrossRef] - Coello, C.A.C.; Lamont, G.B.; Van Veldhuizen, D.A.; Goldberg, D.E.; Koza, J.R. Evolutionary Algorithms for Solving Multi-Objective Problems; Springer: New York, NY, USA, 2007; ISBN 9780387310299. [Google Scholar]
- Van Veldhuizen, D.A.; Lamont, G.B. Multiobjective evolutionary algorithms: Analyzing the state-of-the-art. Evol. Comput.
**2000**, 8, 125–147. [Google Scholar] [CrossRef] - Zhu, D.; Mei, Y.; Xu, X.; Chen, J.; Ben, Y. Optimal operation of complex flood control system composed of cascade reservoirs, navigation-power junctions, and flood storage areas. Water
**2020**, 12, 1883. [Google Scholar] [CrossRef] - Zhou, A.; Qu, B.Y.; Li, H.; Zhao, S.Z.; Suganthan, P.N.; Zhangd, Q. Multiobjective evolutionary algorithms: A survey of the state of the art. Swarm Evol. Comput.
**2011**, 1, 32–49. [Google Scholar] [CrossRef] - Deb, K. Multi-objective Optimisation Using Evolutionary Algorithms: An Introduction. In Multi-Objective Evolutionary Optimisation for Product Design and Manufacturing; Springer: London, UK, 2011. [Google Scholar]
- Reed, P.M.; Hadka, D.; Herman, J.D.; Kasprzyk, J.R.; Kollat, J.B. Evolutionary multiobjective optimization in water resources: The past, present, and future. Adv. Water Resour.
**2013**, 51, 438–456. [Google Scholar] [CrossRef] [Green Version] - Adeyemo, J.A. Reservoir operation using multi-objective evolutionary algorithms—A review. Asian J. Sci. Res.
**2011**, 4, 16–27. [Google Scholar] [CrossRef] - Chang, L.C.; Chang, F.J. Multi-objective evolutionary algorithm for operating parallel reservoir system. J. Hydrol.
**2009**, 377, 12–20. [Google Scholar] [CrossRef] - Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] [Green Version] - Qin, H.; Zhou, J.; Lu, Y.; Li, Y.; Zhang, Y. Multi-objective Cultured Differential Evolution for Generating Optimal Trade-offs in Reservoir Flood Control Operation. Water Resour. Manag.
**2010**, 24, 2611–2632. [Google Scholar] [CrossRef] - Fotovatikhah, F.; Herrera, M.; Shamshirband, S.; Chau, K.W.; Ardabili, S.F.; Piran, M.J. Survey of computational intelligence as basis to big flood management: Challenges, research directions and future work. Eng. Appl. Comput. Fluid Mech.
**2018**, 12, 411–437. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Luo, J.; Sun, X.; Xie, J. Optimal reservoir flood operation using a decomposition-based multi-objective evolutionary algorithm. Eng. Optim.
**2019**, 51, 42–62. [Google Scholar] [CrossRef] - Liu, D.; Huang, Q.; Yang, Y.; Liu, D.; Wei, X. Bi-objective algorithm based on NSGA-II framework to optimize reservoirs operation. J. Hydrol.
**2020**, 585, 124830. [Google Scholar] [CrossRef] - Malekmohammadi, B.; Zahraie, B.; Kerachian, R. Ranking solutions of multi-objective reservoir operation optimization models using multi-criteria decision analysis. Expert Syst. Appl.
**2011**, 38, 7851–7863. [Google Scholar] [CrossRef] - Tzeng, G.-H.; Huang, J.-J. Multiple Attribute Decision Making: Methods and Applications a State of the Art Survey; CRC Press: Boca Raton, FL, USA, 2011; ISBN 9781439861578. [Google Scholar]
- Kao, C. Weight determination for consistently ranking alternatives in multiple criteria decision analysis. Appl. Math. Model.
**2010**, 34, 1779–1787. [Google Scholar] [CrossRef] - Jaini, N.; Utyuzhnikov, S. Trade-off ranking method for multi-criteria decision analysis. J. Multi-Criteria Decis. Anal.
**2017**, 24, 121–132. [Google Scholar] [CrossRef] - Provost, F.; Fawcett, T. Data Science and its Relationship to Big Data and Data-Driven Decision Making. Big Data
**2013**, 1, 51–59. [Google Scholar] [CrossRef] - Taboada, H.A.; Coit, D.W. Data mining techniques to facilitate the analysis of the pareto-optimal set for multiple objective problems. In Proceedings of the 2006 IIE Annual Conference and Exposition, Orlando, FL, USA, 20–24 May 2006. [Google Scholar]
- Suwal, N.; Huang, X.; Kuriqi, A.; Chen, Y.; Pandey, K.P.; Bhattarai, K.P. Optimisation of cascade reservoir operation considering environmental flows for different environmental management classes. Renew. Energy
**2020**, 158, 453–464. [Google Scholar] [CrossRef] - Dumedah, G.; Berg, A.A.; Wineberg, M.; Collier, R. Selecting Model Parameter Sets from a Trade-off Surface Generated from the Non-Dominated Sorting Genetic Algorithm-II. Water Resour. Manag.
**2010**, 24, 4469–4489. [Google Scholar] [CrossRef] - Sato, Y.; Izui, K.; Yamada, T.; Nishiwaki, S. Data mining based on clustering and association rule analysis for knowledge discovery in multiobjective topology optimization. Expert Syst. Appl.
**2019**, 119, 247–261. [Google Scholar] [CrossRef] - Liao, T.W. Clustering of time series data—A survey. Pattern Recognit.
**2005**, 38, 1857–1874. [Google Scholar] [CrossRef] - Xu, D.; Tian, Y. A Comprehensive Survey of Clustering Algorithms. Ann. Data Sci.
**2015**, 2, 165–193. [Google Scholar] [CrossRef] [Green Version] - Wang, X.; Mei, Y.; Cai, H.; Cong, X. A new fluctuation index: Characteristics and application to hydro-wind systems. Energies
**2016**, 9, 114. [Google Scholar] [CrossRef] [Green Version] - Rodriguez, A.; Laio, A. Clustering by fast search and find of density peaks. Science
**2014**, 344, 1492–1496. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Liu, R.; Wang, H.; Yu, X. Shared-nearest-neighbor-based clustering by fast search and find of density peaks. Inf. Sci.
**2018**, 450, 200–226. [Google Scholar] [CrossRef] - Zhang, Y.; Zhou, J.; Li, C.; Chen, F. Integrated utilization of the Three Gorges Cascade for navigation and power generation in flood season. Shuili Xuebao
**2017**, 48, 31–40. [Google Scholar] [CrossRef] - Ban, X.; Diplas, P.; Shih, W.R.; Pan, B.; Xiao, F.; Yun, D. Impact of Three Gorges Dam operation on the spawning success of four major Chinese carps. Ecol. Eng.
**2019**. [Google Scholar] [CrossRef] - Tsai, W.P.; Chang, F.J.; Chang, L.C.; Herricks, E.E. AI techniques for optimizing multi-objective reservoir operation upon human and riverine ecosystem demands. J. Hydrol.
**2015**, 530, 634–644. [Google Scholar] [CrossRef] - Ameur, M.; Kharbouch, Y.; Mimet, A. Optimization of passive design features for a naturally ventilated residential building according to the bioclimatic architecture concept and considering the northern Morocco climate. Build. Simul.
**2020**, 13, 677–689. [Google Scholar] [CrossRef] - Marler, R.T.; Arora, J.S. Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim.
**2004**, 26, 369–395. [Google Scholar] [CrossRef] - Jain, A.K. Data clustering: 50 years beyond K-means. Pattern Recognit. Lett.
**2010**, 31, 651–666. [Google Scholar] [CrossRef] - Gentle, J.E.; Kaufman, L.; Rousseuw, P.J. Finding Groups in Data: An Introduction to Cluster Analysis. Biometrics
**1991**, 47, 788. [Google Scholar] [CrossRef] - Caliński, T.; Harabasz, J. A dendrite method for cluster analysis. Commun. Stat.
**1974**, 3, 1–27. [Google Scholar] - Taboada, H.A.; Baheranwala, F.; Coit, D.W.; Wattanapongsakorn, N. Practical solutions for multi-objective optimization: An application to system reliability design problems. Reliab. Eng. Syst. Saf.
**2007**, 92, 314–322. [Google Scholar] [CrossRef] - Satopää, V.; Albrecht, J.; Irwin, D.; Raghavan, B. Finding a “kneedle” in a haystack: Detecting knee points in system behavior. Proc. Int. Conf. Distrib. Comput. Syst.
**2011**, 166–171. [Google Scholar] [CrossRef] [Green Version] - Berndt, D.; Clifford, J. Using dynamic time warping to find patterns in time series. Knowl. Discov. Databases Workshop
**1994**, 398, 359–370. [Google Scholar] - Yuan, G.; Sun, P.; Zhao, J.; Li, D.; Wang, C. A review of moving object trajectory clustering algorithms. Artif. Intell. Rev.
**2017**, 47, 123–144. [Google Scholar] [CrossRef] - Rousseeuw, P.J. Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math.
**1987**, 20, 53–65. [Google Scholar] [CrossRef] [Green Version] - Arbelaitz, O.; Gurrutxaga, I.; Muguerza, J.; Pérez, J.M.; Perona, I. An extensive comparative study of cluster validity indices. Pattern Recognit.
**2013**, 46, 243–256. [Google Scholar] [CrossRef] - Aghabozorgi, S.; Shirkhorshidi, A.S.; Ying Wah, T. Time-series clustering—A decade review. Inf. Syst.
**2015**, 53, 16–38. [Google Scholar] [CrossRef] - Ruiz, L.G.B.; Pegalajar, M.C.; Arcucci, R.; Molina-Solana, M. A time-series clustering methodology for knowledge extraction in energy consumption data. Expert Syst. Appl.
**2020**, 160, 113731. [Google Scholar] [CrossRef] - Wang, H.; Brill, E.D.; Ranjithan, R.S.; Sankarasubramanian, A. A framework for incorporating ecological releases in single reservoir operation. Adv. Water Resour.
**2015**, 78, 9–21. [Google Scholar] [CrossRef] - Mathews, R.; Richter, B.D. Application of the indicators of hydrologic alteration software in environmental flow setting. J. Am. Water Resour. Assoc.
**2007**, 43, 1400–1413. [Google Scholar] [CrossRef]

**Figure 3.**The location of the Three Gorges Reservoir–Gezhouba (TGR–GZB) cascade reservoirs in China.

**Figure 5.**Output of non-dominated sorting genetic algorithm II (NSGA-II) in the objective and decision spaces: (

**a**) the trade-offs between the eco-friendly objective and flood control target in objective space; (

**b**) the corresponding reservoir operation processes in decision space.

**Figure 6.**Three representative solutions selected through a traditional analysis approach. (

**a**) The objective values of the representative solutions; (

**b**) the corresponding reservoir operation processes of the representative solutions.

**Figure 8.**Clustering results on reservoir operation processes via dynamic time warping (DTW). Clusters are drawn in each subfigure separately. The cluster centers are shown in the last subfigure.

**Figure 9.**Clustering results on reservoir operation processes via Euclidean distance (ED). Clusters are drawn in each subfigure separately. The cluster centers are shown in the last subfigure.

**Figure 10.**Clustering results on the reservoir operation processes via Mei–Wang fluctuation similarity measure (MWFSM). Clusters are drawn in each subfigure separately. The cluster centers are shown in the last subfigure.

**Figure 11.**Results of the clustering-based method for solution selection (CMSS) considering the MWFSM-DPC method. (

**a**) Reservoir operation processes in decision space; (

**b**) objective values in objective space.

Description | DTW-DPC | ED-DPC | MWFSM-DPC | |
---|---|---|---|---|

No. of clusters in decision space | 5 | 5 | 7 | |

Sil | 0.503 | 0.429 | 0.346 | |

No. of solutions in the cluster | cluster 1 | 53 | 40 | 14 |

cluster 2 | 46 | 97 | 9 | |

cluster 3 | 36 | 28 | 16 | |

cluster 4 | 40 | 15 | 36 | |

cluster 5 | 25 | 20 | 71 | |

cluster 6 | 20 | |||

cluster 7 | 34 |

**Table 2.**Number of solutions in the intersection between cluster 5 of MWFSM-DPC and the Pareto clusters.

Pareto Cluster | Cluster 1 | Cluster 2 | Cluster 3 | Cluster 4 | Cluster 5 |

No. of solutions in the intersection | 0 | 9 | 0 | 19 | 0 |

Pareto Cluster | Cluster 6 | Cluster 7 | Cluster 8 | Cluster 9 | Cluster 10 |

No. of solutions in the intersection | 2 | 22 | 0 | 0 | 19 |

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**MDPI and ACS Style**

Kong, Y.; Mei, Y.; Wang, X.; Ben, Y.
Solution Selection from a Pareto Optimal Set of Multi-Objective Reservoir Operation via Clustering Operation Processes and Objective Values. *Water* **2021**, *13*, 1046.
https://doi.org/10.3390/w13081046

**AMA Style**

Kong Y, Mei Y, Wang X, Ben Y.
Solution Selection from a Pareto Optimal Set of Multi-Objective Reservoir Operation via Clustering Operation Processes and Objective Values. *Water*. 2021; 13(8):1046.
https://doi.org/10.3390/w13081046

**Chicago/Turabian Style**

Kong, Yanjun, Yadong Mei, Xianxun Wang, and Yue Ben.
2021. "Solution Selection from a Pareto Optimal Set of Multi-Objective Reservoir Operation via Clustering Operation Processes and Objective Values" *Water* 13, no. 8: 1046.
https://doi.org/10.3390/w13081046