# Extraction and Preference Ordering of Multireservoir Water Supply Rules in Dry Years

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Optimization of Multireservoir Water Supply System

#### 2.1.1. Hedging Policy for Single Reservoir Operations

#### 2.1.2. Allocation Policy of Common Water Demand between Parallel Reservoirs

_{1,t}and R

_{2,t}are releases from Reservoirs 1 and 2, respectively. Assume that the percentage of target demand allocated for each reservoir are PE

_{1}and PE

_{2}, which are determined according to the reservoir storage capacities and respective inflow conditions. For example, PE

_{1}= 0.4 and PE

_{2}= 0.6. Then the water supply target for Reservoir 1 is D

_{1,t}= PE

_{1}× D

_{t}and D

_{2,t}= PE

_{2}× D

_{t}for Reservoir 2 shown as Figure 2b. For a parallel configuration with more than two reservoirs, the allocation policy of D

_{t}is also in proportion.

#### 2.1.3. Objective Functions

_{i}

_{,t}for each parallel reservoir i by the coefficient PE

_{i}and planned common water demand. Then reservoirs could operate in order according to their respective time-varying hedging rules, represented by a parameter vector (SWA

_{i}

_{,t}, EWA

_{i}

_{,t}, HF

_{i}

_{,t}). So, in optimization of a multireservoir water supply system, our purpose is to find the optimal hedging policy that makes the system perform best. Two objective functions are concerned: (1) minimizing the total deficit ratio (TDR) of all demands of the entire system in operation horizon and (2) minimizing the maximum deficit ratio (MDR) of water supply in a single period.

#### 2.1.4. Solution Technique to Extract Design Alternatives

**Step 1**: Create initial parent population ${P}_{0}$ of size $\mathrm{N}$ and set the generation number $g=0$; the $j\text{th}$ individual ${\text{ind}}^{j}$ represents a scheme of hedging policy, i.e., variables $SW{A}_{i,t}^{j}$, $EW{A}_{i,t}^{j}$, $H{F}_{i,t}^{j}$ with $t=1,\mathrm{...},\mathrm{T}$ and $i=1,\mathrm{...},\text{NM}$.

**Step 2**: For different ${\text{ind}}^{j}$ in ${P}_{0}$, implement the water supply operations of reservoirs one by one, then calculate objectives ${\text{TDR}}_{j}$ and ${\text{MDR}}_{j}$.

**Step 3**: Sort ${P}_{0}$ based on non-domination and assign a rank to each individual equal to its non-domination level (1 is the best level, 2 is the next-best level, and so on).

**Step 4**: Generate an offspring population ${O}_{g}$ of size $\mathrm{N}$ using binary tournament selection, crossover, and mutation operators.

**Step 5**: Evaluate offspring population ${O}_{g}$; the same as step 2.

**Step 6**: Combine the parent and offspring population to form a mating pool ${M}_{g}$ of size $2\mathrm{N}$, ${M}_{g}={P}_{g}\cup {O}_{g}$.

**Step 7**: Sort ${M}_{g}$ by the fast non-dominated sorting algorithm to identify all non-dominated fronts ${F}_{1}$,${F}_{2}$,…

**Step 8**: Estimate the crowding distance of each individual in different non-dominated fronts (crowding-distance-assignment).

**Step 9**: Perform the crowded comparison operator on ${M}_{g}$ to generate a new parent population, ${P}_{g+1}$.

**Step 10**: Set $g=g+1$, and go to Step 4. Repeat steps 4–9 until the stopping criterion is satisfied ($g=\mathrm{max}\text{gen}$).

#### 2.2. Preference Ordering according to Quantitative Criteria

#### 2.2.1. Evaluation Criteria

#### Reservoir Performance Indices

#### Shortage Indices

_{t}= shortage in period t; TD

_{t}= demand in period t; and T = number of periods.

_{i}= number of days in the ith year (365 or 366); and DPDa

_{i}= sum of all DPDs in the ith year.

#### 2.2.2. The SEABODE Approach

_{min}; (2) among the result set, only retain the alternatives that are efficient of order (k

_{min}− 1) with highest degree p

_{max}. Definitions of the two domination theorems used in the SEABODE approach are given as follows.

#### Efficiency of Order k

#### Efficiency of Order k with Degree p

#### 2.2.3. Numerical Illustration

_{min}= 3, p

_{max}= 3, a7 is the most preferred choice.

Alternatives | Criteria (m = 3) | 3-Pareto-Optimal | [k, p]-Pareto-Optimal | ||||
---|---|---|---|---|---|---|---|

c1 | c2 | c3 | [2,1] | [2,2] | [2,3] | ||

a1 | 6.33 | 2.45 | 51.31 | √ | √ | √ | |

a2 | 13.91 | 3.68 | 36.54 | √ | √ | ||

a3 | 4.12 | 6.01 | 58.15 | √ | √ | ||

a4 | 8.62 | 7.57 | 46.22 | √ | |||

a5 | 12.35 | 9.74 | 32.13 | √ | √ | ||

a6 | 10.11 | 11.96 | 23.15 | √ | √ | ||

a7 | 1.05 | 15.51 | 15.20 | √ | √ | √ | √ (preferred) |

a8 | 5.71 | 26.53 | 5.22 | √ | √ | √ | |

a9 | 2.43 | 31.26 | 13.84 | √ | √ | ||

a10 | 3.57 | 43.22 | 9.01 | √ | √ |

**Figure 4.**Two-dimensional Pareto plots of the three criteria: (

**a**) c1 versus c2; (

**b**) c1 versus c3; (

**c**) c3 versus c2.

#### 2.3. Framework of the Proposed Methodology in Application

## 3. Case Study

#### 3.1. System Description

Reservoir | Active Storage Capacity K (10^{2} Million m^{3}) | PE (Predefined) | |
---|---|---|---|

Normal Season (September–May) | Flood Season (June–August) | ||

Baozhusi | 13.40 | 10.58 | - |

Tingzikou | 17.32 | 4.72 | 0.7 |

Shengzhong | 6.72 | 6.72 | 0.3 |

#### 3.2. Dry Years and Water Demand Data

**Table 3.**Average natural inflows of the demonstration dry years (Y = 20) and corresponding target demands in 2020 level year (m

^{3}/s).

Reservoir | Months | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | June | July | August | September | October | November | December | ||

Average Inflows | Baozhusi | 202.3 | 180.3 | 185.8 | 289.7 | 472.8 | 532.7 | 754.9 | 684.6 | 681.6 | 512.4 | 318.6 | 222.9 |

Shengzhong | 106.1 | 92.5 | 101.3 | 184.7 | 301.2 | 345.3 | 502.7 | 537.8 | 508.7 | 350.1 | 200.1 | 132.2 | |

Tingzikou | 257.7 | 197.1 | 230.3 | 415.9 | 670.6 | 686.1 | 1691.3 | 1266.4 | 1342.9 | 653.0 | 467.8 | 317.5 | |

Demands (2020) | Baozhusi | 39.5 | 51.2 | 48.4 | 52.9 | 54.9 | 67.6 | 54.2 | 49.7 | 33.9 | 39.6 | 41.0 | 39.8 |

Shengzhong | 106.4 | 111.3 | 114.4 | 98.3 | 291.9 | 237.7 | 179.1 | 169.1 | 78.6 | 72.4 | 86.1 | 127.0 | |

Tingzikou | 159.6 | 167.0 | 171.6 | 147.4 | 437.9 | 356.5 | 268.7 | 253.6 | 117.9 | 108.6 | 129.1 | 190.5 |

#### 3.3. Parameter Settings and Search Ranges of Decision Variables

_{c}= 0.9, mutation probability p

_{m}= 1/108, crossover and mutation distribution indexes η

_{c}= 20 and η

_{m}= 20 [20,22,23]. Moreover, a genetic algorithm (GA) with real-coded pattern and elite strategy is applied to find the extreme optimal MDR and TDR separately. The parameter setting for GA is: population size = 200, maximum number of iterations = 1000, crossover and mutation probabilities p

_{c}= 0.9, p

_{m}= 0.05 [11], elites to be reserved are the top 5% of individuals in each generation.

## 4. Results and Discussion

#### 4.1. Decision Space Obtained by the Optimization Model

**Figure 8.**(

**a**) The decision space A derived by NSGA-II; (

**b**) results of single-objective optimizations obtained by GA.

#### 4.2. Preferred Alternatives Determined by the SEABODE

Reservoir | Four-Dimensional Criterion Space C | ||||
---|---|---|---|---|---|

1-α | 2-γ | 3-υ | 4-GSI | ||

Baozhusi | Range | = 1 | = 1 | = 0 | = 0 |

Std. | – | – | – | – | |

Shengzhong | Range | [0.658, 0.825] | [0.297, 0.637] | [0.345, 0.520] | [0.203, 0.541] |

Std. | 0.054 | 0.079 | 0.061 | 0.112 | |

Tingzikou | Range | [0.592, 0.842] | [0.342, 0.665] | [0.346, 0.635] | [0.434, 1.487] |

Std. | 0.092 | 0.097 | 0.089 | 0.336 |

Reservoir | C | Three-dimensional subspaces | |||
---|---|---|---|---|---|

(1-2-3-4) | (1-2-3) | (1-2-4) | (1-3-4) | (2-3-4) | |

Shengzhong | 29 | 15 | 7 | 28 | 29 |

Tingzikou | 69 | 18 | 2 | 63 | 63 |

Reservoir | 4-Pareto | [3,1] | [3,2] | [3,3] | [3,4] | [2,1] | [2,2] | [2,3] | [2,4] | [2,5] | [2,6] |
---|---|---|---|---|---|---|---|---|---|---|---|

Shengzhong | 29 | 29 | 28 | 14 | 5 | 5 | 5 | 5 | 5 | 2 | 1 |

Tingzikou | 69 | 68 | 60 | 17 | 0 | 17 | 15 | 9 | 1 | 0 | 0 |

GA-MDR | GA-TDR | NSGA-II + SEABODE | |||
---|---|---|---|---|---|

Shengzhong (a86) | Tingzikou (a54) | ||||

Objectives | MDR (%) | 34.41 | 68.76 | 56.13 | 60.30 |

TDR (%) | 19.16 | 8.30 | 8.51 | 8.42 | |

Shengzhong | α | 0.379 | 0.825 | 0.825 | 0.821 |

γ | 0.159 | 0.529 | 0.529 | 0.504 | |

υ | 0.320 | 0.494 | 0.494 | 0.520 | |

GSI | 3.371 | 0.200 | 0.204 | 0.204 | |

Tingzikou | α | 0.296 | 0.842 | 0.804 | 0.842 |

γ | 0.164 | 0.648 | 0.610 | 0.665 | |

υ | 0.344 | 0.688 | 0.561 | 0.603 | |

GSI | 4.316 | 0.482 | 0.597 | 0.440 |

#### 4.3. Areas for Future Research

_{i}

_{,t}(i is the reservoir index of parallel reservoirs, t is the time index) as decision variables. However, we noted that the variation ranges of the decision variables SWA

_{i}

_{,t}(0 ≤ SWA

_{i}

_{,t}≤ D

_{i}

_{,t}) and EWA

_{i}

_{,t}(D

_{i}

_{,t}≤ EWA

_{i}

_{,t}≤ D

_{i}

_{,t}+ K

_{i}) are associated to PE

_{i}

_{,t}because D

_{i}

_{,t}= PE

_{i}

_{,t}× D

_{t}(D

_{t}is the water demand in period t and its value is known beforehand). That way, the boundaries on SWA

_{i}

_{,t}and EWA

_{i}

_{,t}vary with the value of PE

_{i}

_{,t}during iterative computations of NSGA-II; even the initial solutions are produced in different search spaces depending on the randomly generated PE

_{i}

_{,t}. There are correlations between some decision variables, which makes the optimization problem very complicated and varying (violating the consistency requirement for optimization problems). This type of problem is seldom seen in multiobjective optimizations. For most applications of multiobjective evolutionary algorithms (MOEAs), the search spaces of decision variables are fixed. Thereupon, fixed PE

_{i}

_{,t}values were used in our study, which were determined according to the reservoir storage capacities and average reservoir inflow conditions. We and maybe other researches are expecting to see PE

_{i}

_{,t}as decision variables and explore new techniques to deal with constraints of this type that are dynamically changing. This is another interesting area for future research.

## 5. Summary and Conclusions

- A simple multireservoir operation policy includes the hedging policy for water supply operation of a single reservoir and the proportional allocation policy of downstream common water demand between two parallel reservoirs is suggested by the authors.
- A new methodology, which aims at deriving a certain number of noninferior multireservoir water supply rules in dry years or drought conditions and making a preference ordering among the extracted alternatives according to several specified evaluation criteria, is proposed. The proposed methodology is implemented as the following two main steps (1) an optimization model of multireservoir operation using the NSGA-II algorithm to obtain a number of noninferior hedging rules in terms of objectives TDR and MDR, which forms a decision space containing enough design alternatives for decision makers to choose between; and (2) a multicriteria decision-making procedure to further eliminate the alternatives according to the SEABODE approach.
- In order to illustrate the application and effect of the proposed methodology, the three-reservoir water supply system in Jialing River is employed as a case study. Results show that we can always find the preferred alternative among a sizeable number of noninferior alternatives for each reservoir with the help of the SEABODE approach. The proposed methodology was able to sieve through the numerous noninferior alternatives and short-list a small number of preferred alternatives to present to decision makers for further consideration.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Kang, L.; Zhang, S.; Ding, Y.; He, X. Extraction and Preference Ordering of Multireservoir Water Supply Rules in Dry Years. *Water* **2016**, *8*, 28.
https://doi.org/10.3390/w8010028

**AMA Style**

Kang L, Zhang S, Ding Y, He X. Extraction and Preference Ordering of Multireservoir Water Supply Rules in Dry Years. *Water*. 2016; 8(1):28.
https://doi.org/10.3390/w8010028

**Chicago/Turabian Style**

Kang, Ling, Song Zhang, Yi Ding, and Xiaocong He. 2016. "Extraction and Preference Ordering of Multireservoir Water Supply Rules in Dry Years" *Water* 8, no. 1: 28.
https://doi.org/10.3390/w8010028