# Reservoir Operation Policy based on Joint Hedging Rules

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Expression form of Hedging Rules

_{1}, EWA

_{1}and SWA

_{2}, EWA

_{2}are, respectively, SWA, EWA of JHR’s two hedge process, and the reservoir storage can be divided into five sections by them. The five sections are shown in Figure 4, and the process of JHR is shown in Table 1. Section 5 is an abandoned section, in which the storage is larger than normal high-water level or flood limit level. When the supply capacity of the reservoir is in Section 4, the supply capacity of the reservoir is sufficient, and hedge rule can be applied to increase reservoir’s supply and reduce transfer, and in this section, the single-point hedging rule is chosen as the basic form. When the reservoir’s supply capacity is in Section 2, in order to reduce the possibly unacceptably large damage from shortage, another hedge process would reduce the supply of local reservoir and increase transfer. But the supply of local reservoir shouldn’t be too small so as not to cause water shortage due to small water distribution network capacity, water purification capacity, and so on. Similarly, the transferred water shouldn’t be too small too. Thus, compared with traditional HR, JHR’s two hedge rate shouldn’t be too small, so as not to cause water shortage, we made the two smallest hedge rate p

_{e}and q

_{s}.

_{t}is the whole demand needed to be satisfied by the transferred water and reservoir in t period; M

_{t}and L

_{t}are the part which needed to be satisfied by the reservoir and the transferred water in t period (D

_{t}= M

_{t}+ L

_{t}); p and q are hedge rate, their values are, respectively, from 1 to p

_{e}(p

_{e}< 1) and q

_{s}to 1, and p

_{e}, q

_{s}are the smallest hedge rate of the two hedge process. p, q can be acquired by Equations (1)–(3).

_{max}and V

_{p}:

_{e}+ (1 − p

_{e})(V

_{max}− V

_{t})/(V

_{max}− V

_{p}),

_{q}and q

_{s}× V

_{q}:

_{q}− V

_{t})/(V

_{q}− V

_{de}),

_{s}× V

_{q}and V

_{e}:

_{s},

_{max}is the storage of reservoir’s flood limit level in flood season or the storage of reservoir’s normal high-water level in non-flood season; V

_{t}is the supply capacity in t period; V

_{p}is the storage of SWA

_{2}; V

_{q}is the storage of SWA

_{1}; V

_{e}is storage of EWA

_{1}.

#### 2.2. Method Solution and Analysis

_{2}and EWA

_{2}), the lower hedge process would search its optimal decision (SWA

_{1}and EWA

_{1}). In other words, JHR’s two hedge process fit within the framework of leader-follower or Stackelberg game. A bi-level optimization model and an improved particle swarm optimization algorithm [26] can be applied to solve this kind of problem. As a shuffling process is introduced in the improved particle swarm optimization algorithm, it performs better than the particle swarm optimization algorithm in the following aspects: first, it can ensure information sharing between subgroups and the overall group; second, it has less possibility of encountering local optima; third, it can avoid precocious phenomena to a certain extent. This paper proposed a bi-level optimization model, and an improved particle swarm optimization algorithm was applied. The external loops number (T = 100), the loops number of the bi-level optimization model is 50. The flowchart of solving the bi-level optimization model for JHR is shown in Figure 5.

#### 2.3. The Objective Function and Constraints

- Transfer no more than the supply capability or the transfer capacity of the water transfer projectL
_{t}≤ L_{mt}, L_{t}≤ L_{l} - Largest and smallest storage constraints of the reservoirV
_{de}≤ V_{t}≤ V_{max}, - Water balance constraintV
_{t}= V_{t−1}+ x_{t}− M_{t−1}− Q_{It}, - Non-negative constraintsD
_{t}≥ 0; L_{t}≥ 0; x_{t}≥ 0; m ≥ 0; α ≥ 0,

_{t}is the runoff of t period; L

_{mt}is the supply capacity of the transferred water in t period; m is the reservoir’s storage ratio at the end of year; α is a preset parameter between 0 and 1, represent the different importance of transfer and reservoir’s water storage at the end of the year (in the considered case, α is 0.5); L

_{l}is the transfer capacity of the water diversion project.

#### 2.4. Study Area

## 3. Result

_{t}to M

_{t}. The sum of squares of daily water shortage ratio, transfer, and the storage ratio at the end of the year of SOP, HR, and JHR in each case are shown in Table 2, Table 3 and Table 4.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Tan, Q.-F.; Wang, X.; Wang, H.; Wang, C.; Lei, X.-H.; Xiong, Y.-S.; Zhang, W. Derivation of optimal joint operating rules for multi-purpose multi-reservoir water-supply system. J. Hydrol.
**2017**, 551, 253–264. [Google Scholar] [CrossRef] - Kim, Y. An analysis on the Standard Operating Procedures of School Organization In Terms of Curriculum Management: A Case of Public High School in Equalization Policy Region. J. Curric. Stud.
**2004**, 22, 123–147. [Google Scholar] - Kumar, K.; Kasthurirengan, S. Generalized Linear Two-Point Hedging Rule for Water Supply Reservoir Operation. J. Water Resour. Plann. Manage.
**2018**, 144, 04018051. [Google Scholar] [CrossRef] - Srinivasan, K.; Kumar, K. Multi-Objective Simulation-Optimization Model for Long-term Reservoir Operation using Piecewise Linear Hedging Rule. Water Resour Manage.
**2018**, 32, 1901–1911. [Google Scholar] [CrossRef] - Wan, W.; Zhao, T.; Lund, J.R.; Lei, X.; Wang, H. Optimal Hedging Rule for Reservoir Refill. J. Water Resour. Plann. Manage.
**2016**, 142, 4016051. [Google Scholar] [CrossRef] - Taghian, M.; Rosbjerg, D.; Haghighi, A.; Madsen, H. Optimization of Conventional Rule Curves Coupled with Hedging Rules for Reservoir Operation. J. Water Resour. Plann. Manage.
**2014**, 140, 693–698. [Google Scholar] [CrossRef] - Baes, M.; Bürgisser, M. Hedge algorithm and Dual Averaging schemes. Math. Meth. Oper. Res.
**2012**, 77, 279–289. [Google Scholar] [CrossRef] [Green Version] - Shiau, J.-T. Optimization of Reservoir Hedging Rules Using Multiobjective Genetic Algorithm. J. Water Resour. Plann. Manage.
**2009**, 135, 355–363. [Google Scholar] [CrossRef] - Huang, Q.; Wang, J.; Yu, Z.; Zhang, J.; Xu, B.; Zhong, P.-A. Optimal Hedging Rules for Water Supply Reservoir Operations under Forecast Uncertainty and Conditional Value-at-Risk Criterion. Water
**2017**, 9, 568. [Google Scholar] [Green Version] - Kang, L.; Zhang, S.; Ding, Y.; He, X. Extraction and Preference Ordering of Multireservoir Water Supply Rules in Dry Years. Water
**2016**, 8, 28. [Google Scholar] [CrossRef] - Tayebiyan, A.; Mohammad, T.A.; Al-Ansari, N.; Malakootian, M. Comparison of Optimal Hedging Policies for Hydropower Reservoir System Operation. Water
**2019**, 11, 121. [Google Scholar] [CrossRef] - Ji, Y.; Lei, X.; Cai, S.; Wang, X. Hedging Rules for Water Supply Reservoir Based on the Model of Simulation and Optimization. Water
**2016**, 8, 249. [Google Scholar] [CrossRef] - Wang, J.; Hu, T.; Zeng, X.; Yasir, M. Storage targets optimization embedded with analytical hedging rule for reservoir water supply operation. Water Sci. Technol. Water Supply
**2017**, 18, 622–629. [Google Scholar] [CrossRef] - Sangiorgio, M.; Guariso, G. NN-Based Implicit Stochastic Optimization of Multi-Reservoir Systems Management. Water
**2018**, 10, 303. [Google Scholar] [CrossRef] - Peña-Guzmán, C.; Melgarejo, J.; Prats, D. Forecasting Water Demand in Residential, Commercial, and Industrial Zones in Bogotá, Colombia, Using Least-Squares Support Vector Machines. Math. Probl. Eng.
**2016**, 1–10. [Google Scholar] [CrossRef] - Peng, Y.; Chu, J.; Peng, A.; Zhou, H. Optimization Operation Model Coupled with Improving Water-Transfer Rules and Hedging Rules for Inter-Basin Water Transfer-Supply Systems. Water Resour. Manage.
**2015**, 29, 3787–3806. [Google Scholar] [CrossRef] - Tatano, H.; Okada, N.; Kawai, H. Optimal operation model of a single reservoir with drought duration explicitly concerned. Stoch Environ. Res. Risk Assess.
**1992**, 6, 123–134. [Google Scholar] [CrossRef] - Neelakantan, T.R.; Pundarikanthan, N.V. Hedging Rule Optimisation for Water Supply Reservoirs System. Water Resour. Manage.
**1999**, 13, 409–426. [Google Scholar] [CrossRef] - Shiau, J.-T.; Lee, H.C. Derivation of Optimal Hedging Rules for a Water-supply Reservoir through Compromise Programming. Water Resour. Manage.
**2005**, 19, 111–132. [Google Scholar] [CrossRef] - You, J.-Y.; You, J.; Cai, X. Hedging rule for reservoir operations: 2. A numerical model. Water Resour. Res.
**2008**, 44, 44. [Google Scholar] [CrossRef] - Srinivasan, K.; Philipose, M.C. Effect of Hedging on Over-Year Reservoir Performance. Water Resour. Manage.
**1998**, 12, 95–120. [Google Scholar] [CrossRef] - Draper, A.J.; Lund, J.R. Optimal Hedging and Carryover Storage Value. J. Water Resour. Plann. Manage.
**2004**, 130, 83–87. [Google Scholar] [CrossRef] [Green Version] - Hu, T.; Zhang, X.-Z.; Zeng, X.; Wang, J. A Two-Step Approach for Analytical Optimal Hedging with Two Triggers. Water
**2016**, 8, 52. [Google Scholar] [CrossRef] - Guo, X.; Hu, T.; Zhang, T.; Lv, Y. Bilevel model for multi-reservoir operating policy in inter-basin water transfer-supply project. J. Hydrol.
**2012**, 424, 252–263. [Google Scholar] [CrossRef] - Hui, R.; Lund, J.; Zhao, T. Optimal Pre-storm Flood Hedging Releases for a Single Reservoir. Water Resour. Manage.
**2016**, 30, 5113–5129. [Google Scholar] [CrossRef] - Jiang, Y.; Hu, T.; Huang, C.; Wu, X. An improved particle swarm optimization algorithm. Appl. Math. Comput.
**2007**, 193, 231–239. [Google Scholar] [CrossRef] - Li, Y.; Xiong, W.; Zhang, W.; Wang, C.; Wang, P. Life cycle assessment of water supply alternatives in water-receiving areas of the South-to-North Water Diversion Project in China. Water Res.
**2016**, 89, 9–19. [Google Scholar] [CrossRef] [PubMed] - Lee, S. Reservoir Operations Applying Discrete Hedging Rule Curves Depending on Current Storage to Cope with Droughts. J. Korean Soc. Hazard. Mitigation
**2017**, 17, 107–115. [Google Scholar] - Maiolo, M.; Pantusa, D. An optimization procedure for the sustainable management of water resources. Water Sci. Technol. Water Supply
**2015**, 16, 61–69. [Google Scholar] [CrossRef] - Ding, W.; Zhang, C.; Cai, X.; Li, Y.; Zhou, H. Multiobjective hedging rules for flood water conservation. Water Resour. Res.
**2017**, 53, 1963–1981. [Google Scholar] [CrossRef]

**Figure 9.**The comparative results based on JHR, HR, and SOP (

**a**,

**b**show the sums of the squares of daily shortage ratio and the amount of transferred water, respectively).

Scope | Transfer | Reservoir | Hedge Rate | |
---|---|---|---|---|

Section 1 | V_{de} < V_{t} ≤ EWA_{1} | D_{t} − L_{t} × q_{s} | supply capacity | |

Section 2 | EWA_{1} < V_{t} ≤ SWA_{1} | D_{t} − q × L_{t} | q × L_{t} | q ∈ (q_{s}, 1) |

Section 3 | SWA_{1} < V_{t} ≤ SWA_{2} | M_{t} | L_{t} | |

Section 4 | SWA_{2} < V_{t} ≤ V_{max} | M_{t} × p | D_{t} − M_{t} × p | p ∈ (p_{e}, 1) |

Section 5 | V_{t} > V_{max} | M_{t} × p_{e} | D_{t} − M_{t} × p_{e} |

**Table 2.**Sums of squares of daily shortage ratio, transfer, and the storage ratio at the end of the year of JHR.

Cases | xd, xd | d, xd | m, xd | w, xd | xd, d | d, d | m, d | w, d |
---|---|---|---|---|---|---|---|---|

Sums of squares of daily shortage ratio | 49.07 | 30.14 | 12.36 | 21.38 | 0.00 | 0.00 | 0.00 | 0.00 |

Transfer (0.1 billion) | 5.86 | 5.86 | 5.86 | 5.86 | 10.42 | 9.17 | 8.87 | 8.95 |

Sum of Yuqiao supply (0.1 billion) | 3.9 | 5.11 | 6.75 | 5.86 | 3.54 | 4.8 | 5.09 | 5.02 |

Storage ratio | 0.22 | 0.30 | 0.32 | 0.32 | 0.33 | 0.38 | 0.85 | 0.61 |

**Table 3.**Sums of squares of daily shortage ratio, transfer, and the storage ratio at the end of the year of HR.

Cases | xd, xd | d, xd | m, xd | w, xd | xd, d | d, d | m, d | w, d |
---|---|---|---|---|---|---|---|---|

Sums of squares of daily shortage ratio | 49.11 | 30.15 | 11.57 | 20.82 | 0.22 | 0.00 | 0.00 | 0.00 |

Transfer (0.1 billion) | 5.86 | 5.86 | 5.86 | 5.86 | 10.38 | 10.11 | 10.04 | 10.04 |

Sum of Yuqiao supply (0.1 billion) | 3.88 | 5.13 | 6.81 | 5.96 | 3.52 | 3.86 | 3.92 | 3.92 |

Storage ratio | 0.22 | 0.30 | 0.30 | 0.32 | 0.34 | 0.42 | 0.90 | 0.63 |

**Table 4.**Sums of squares of daily shortage ratio, transfer, and the storage ratio at the end of the year of SOP.

Cases | xd, xd | d, xd | m, xd | w, xd | xd, d | d, d | m, d | w, d |
---|---|---|---|---|---|---|---|---|

Sums of squares of daily shortage ratio | 51.46 | 31.19 | 9.12 | 19.12 | 0.00 | 0.00 | 0.00 | 0.00 |

Transfer (0.1 billion) | 5.86 | 5.86 | 5.86 | 5.86 | 10.04 | 10.04 | 10.04 | 10.04 |

Sum of Yuqiao supply (0.1 billion) | 4.19 | 5.65 | 7.34 | 6.55 | 3.92 | 3.92 | 3.92 | 3.92 |

Storage ratio | 0.12 | 0.12 | 0.12 | 0.12 | 0.21 | 0.40 | 0.90 | 0.63 |

Shortage Ratio | 0%–20% | 20%–40% | 40%–60% | 60%–80% | >80% |
---|---|---|---|---|---|

SOP | 1 | 21 | 64 | 0 | 0 |

HR | 6 | 88 | 58 | 0 | 0 |

JHR | 5 | 88 | 56 | 0 | 0 |

Shortage Ratio | 0%–20% | 20%–40% | 40%–60% | 60%–80% | >80% |
---|---|---|---|---|---|

SOP | 1 | 0 | 42 | 0 | 0 |

HR | 9 | 88 | 8 | 0 | 0 |

JHR | 11 | 88 | 15 | 0 | 0 |

Shortage Ratio | 0%–20% | 20%–40% | 40%–60% | 60%–80% | >80% |
---|---|---|---|---|---|

SOP | 1 | 28 | 110 | 17 | 0 |

HR | 10 | 88 | 107 | 0 | 0 |

JHR | 5 | 95 | 106 | 0 | 0 |

Shortage Ratio | 0%–20% | 20%–40% | 40%–60% | 60%–80% | >80% |
---|---|---|---|---|---|

SOP | 0 | 28 | 163 | 21 | 0 |

HR | 1 | 57 | 190 | 0 | 0 |

JHR | 1 | 53 | 192 | 0 | 0 |

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**MDPI and ACS Style**

Men, B.; Wu, Z.; Li, Y.; Liu, H.
Reservoir Operation Policy based on Joint Hedging Rules. *Water* **2019**, *11*, 419.
https://doi.org/10.3390/w11030419

**AMA Style**

Men B, Wu Z, Li Y, Liu H.
Reservoir Operation Policy based on Joint Hedging Rules. *Water*. 2019; 11(3):419.
https://doi.org/10.3390/w11030419

**Chicago/Turabian Style**

Men, Baohui, Zhijian Wu, Yangsong Li, and Huanlong Liu.
2019. "Reservoir Operation Policy based on Joint Hedging Rules" *Water* 11, no. 3: 419.
https://doi.org/10.3390/w11030419