# Utilizing the Sobol’ Sensitivity Analysis Method to Address the Multi-Objective Operation Model of Reservoirs

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{3}capacity and a drainage area of 2085 km

^{2}. It is a large-scale hydro project with multiple purposes, including a primary objective of water supply, and a number of secondary objectives, such as flood control, hydro power generation, irrigation, and fish farming. Water supply mainly consists of I&D and agricultural users—the amount is 430 million and 40 million m

^{3}with a required reliability of 95% and 75%, respectively. The dead water level for the reservoir is 47 m and the normal water level is 69 m. After the water diversion project from the Dahuofang reservoir, the ecological objective will be considered to improve the ecological environment of the reservoir lower reaches, with an amount of 160 million m

^{3}. Moreover, the water supply for I&D, agriculture, and ecosystem can be reduced to 90%, 70%, and 50%, respectively, of their normal requirements during droughts. The geographic location of the Biliuhe reservoir is shown in Figure 1.

^{3}/year, the pipeline transmission scale is 900,000 t/d, and the water leakage loss is 4%. Considering the characteristics of the inflow and water consumption of the Biliuhe reservoir, the amount of water diverted from the Dahuofang reservoir should be at full capacity of the pipeline during the year but for four periods, including mid-July, late July, early August, and mid-August, during which the reservoir has adequate inflow. The annual inflow (1958–2021) for each 10 days of the Biliuhe reservoir is shown in Figure 2.

#### 2.2. Sobol’ Method

_{i}and x

_{j}), and so on.

_{i}; ${D}_{{x}_{i},{x}_{j}}$ is the fraction contributed by interaction between parameter x

_{i}and x

_{j}. The Sobol’ indices can be defined as the ratio of the partial variances to the total variance, as follows:

_{i}fixed. In other words, it implies the main effect of parameter x

_{i}and its interactions with other parameters. The corresponding index is defined below:

_{i}means all input parameters except for the parameter x

_{i}. D

_{~xi}is the variance contribution on account of all parameters, except for parameter x

_{i}. The difference between $ST({x}_{i})$ and ${S}_{{x}_{i}}$ is the impact of interactions with other parameters for parameter x

_{i}.

- Each dimension in the uncertainty space is split into N intervals with an equal length, and equally spaced levels corresponding to [0, 1/N), [1/N, 2/N), …, [1–1/N, 1] for each variable;
- A random point in the range is generated in any one interval of each variable. Then, they are combined into multivariate samples that preserve the space-filling property of the marginal distribution [38].
- All of the samples produced by LHS form a N × k matrix (x
_{i,j}), where N represents the number of samples and k denotes the number of variables, and x_{i,j}is the value of the jth variable in the ith sample.

#### 2.3. Ecological Scheduling Method

_{i}and ${\sigma}_{i}{}^{2}$ denote the original mean value and variance of the ith indicator, respectively.

#### 2.4. Scenarios Design

#### 2.4.1. The Objective Functions

- I&D and agricultural objectives: To reflect the situations of I&D and agricultural water supply, the I&D water shortage index (ISI) and agricultural water shortage index (ASI) are used as objective functions. The shortage index (SI) is the indicator to measure the damage degree of water shortage to different users. Therefore, smaller SI values equate to better efficiency for I&D and agricultural water supply. The objective function is as follow:

- where N is the total number of years; T denotes the time periods in a year; ${D}_{i,j}$ and ${W}_{i,j}$ are the water demand and water supply for the ith users (I&D or agriculture) during the jth time period, respectively; and k is an index to reflect the socio-economic impact of water shortage, which in this paper k is 2. Because I&D (or agricultural) the water supply is nonnegative, and it cannot be bigger than the water demand, the possible range of SI is from 0 to 100.
- Water diversion objective: In addition to the ecological objective, as mentioned in Section 2.2 (Formula (7)), the water diversion objective is also considered in this paper. To minimize the long-distance water division cost, the least amount of water should be diverted from the Dahuofang reservoir. The objective function is as follows:

- where D is the annual amount of diverted water; m and n are the number of years and the time periods in one year, respectively; and ${d}_{i,j}$ denotes the amount of diverted water in the ith year for the jth time period. The actual water supply capacity of this project is 288 million m
^{3}/year after deducting the water leakage loss, so the value interval for water diversion objective D is (0, 288 million m^{3}/year). - Constraints:

#### 2.4.2. The Implementation of Sobol’ Method

- Parameters of the model: In the water diversion and water supply combined operation model, the parameters are the water levels on different dispatching lines during the operation time periods. In this paper, the total number of parameters is 160 (agricultural limit line 16, I&D limit line 36, ecological limit line 36, upper water diversion line 36, and lower water diversion line 36).
- Range of parameter values: For the parameters of each dispatching line at different periods, the water level values range from the normal water level (the flood-limit water level in the flood season) to the dead water level.
- Sampling method: The LHS technique is used to compute the Sobol’ sensitivity indices. All of the parameters obey the uniform distribution. A set of 2000 LH samples is used per parameter and a total number of $2000\times \left(160+2\right)=324000$ model simulations are required to compute the Sobol’ indices.
- Sensitivity calculation: The sampled parameters are placed into the reservoir multi-objective operation model to obtain the objective function values for different sampling sequences. Then, the variance-based analysis method is applied to analyze the influence of single and multiple water levels on other objectives at different time periods.

- All of the water level parameters will be optimized by 500,000 calculations, which is the full search scenario;
- Firstly, the sensitivity parameters are optimized by 5000 calculations to obtain the feasible solutions, then these solutions are subsequently taken as the initial solutions for the global optimization problem of all parameters, and (500,000–5000) calculations are run to search for the final feasible solutions, which is the pre-conditioned full search scenario.

#### 2.5. The Evaluation Indicators

- Generational distance: Generational distance was proposed by Van Veldhuizen and Lamont in 1998 [42]. This indicator is used to measure the gap between the solutions obtained by the algorithm and the real Pareto frontier solutions. It is defined as follows:

- where n is the number of the optimal solutions, d
_{i}is the minimum Euclidean distance between the target space and theoretical Pareto front for the ith individual. The smaller the value for the generation distance, the closer the solutions obtained are to the real Pareto frontier solutions. When all of the solutions obtained by the algorithm are exactly the real Pareto frontier solutions, GD = 0. - Additive indicator: The additive indicator [43] evaluates the minimum distance for which the current solutions can completely dominate the reference solutions. It is defined as follows:

- where Q is the non-inferior solution sets, and ${I}_{\epsilon +}\left(\mathrm{Q}\right)$ is the approaching degree of the non-inferior solution sets and the Pareto front. The smaller the value of the additive indicator, the closer the current solutions are to the reference solutions.
- Hypervolume indicator: The hypervolume indicator [44] is used to evaluate the target space of the dominated solutions. It not only shows the distance between the current solutions and the optimal solutions, but also the distribution of the current solutions in the target space. It is defined as follows:

- where |S| represents the number of non-dominant solution sets and v
_{i}represents the hypervolume formed by the reference point and the ith solution. The bigger the value of the hypervolume indicator, the greater the improvement in the current solutions and the farther away they are from the worst solution.

## 3. Results and Discussion

#### 3.1. The Sensitive Parameters

#### 3.2. The Evaluation Indicators

#### 3.3. The Feasible Solutions Analysis

#### 3.4. The Processing Speed Analysis

#### 3.5. The Implication for Reservoirs Management

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The geographic location of the Biliuhe reservoir. (

**a**) Location of Liaoning Province in China, (

**b**) Location of Biliuhe Reservoir in Liaoning Province, (

**c**) Drainage map for Biliuhe basin.

**Figure 3.**The water diversion and water supply dispatching diagrams for the inter-basin water diversion reservoir.

**Figure 4.**The sensitivity analysis results for different operation objectives. (

**a**) for ISI (

**b**) for ASI (

**c**) for P

_{eco}(

**d**) for water diversion.

**Figure 6.**Comparison of the optimal solutions obtained using the sensitivity analysis pre-processed and unprocessed methods.

**Figure 7.**Comparison of the optimal solutions for each operation objective obtained using the sensitivity analysis pre-processed and unprocessed methods. (

**a**) for ISI objective (

**b**) for ASI objective (

**c**) for ecological objective (

**d**) for water diversion objective.

**Table 1.**The water diversion and water supply dispatching rules for the inter-basin water diversion reservoir.

Combined Dispatching Diagrams | Dispatching Zones | Dispatching Rules |
---|---|---|

water diversion dispatching diagram | water diversion zone I | no water diversion |

water diversion zone II | water diversion on a prorate | |

water diversion zone III | water diversion based on the capacity of pipeline | |

water supply dispatching diagram | water supply zone I | normal water supply for I&D, agriculture, and ecosystem |

water supply zone II | normal water supply for I&D and agriculture, limited water supply for ecosystem | |

water supply zone III | normal water supply for I&D, limited water supply for agriculture and ecosystem | |

water supply zone IV | limited water supply for I&D, agriculture, and ecosystem | |

limiting factors | I&D is 0.9, agriculture is 0.7, ecosystem is 0.5 | |

dispatching time period | 10-days |

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**MDPI and ACS Style**

Wang, H.; Zhao, Y.; Fu, W.
Utilizing the Sobol’ Sensitivity Analysis Method to Address the Multi-Objective Operation Model of Reservoirs. *Water* **2023**, *15*, 3795.
https://doi.org/10.3390/w15213795

**AMA Style**

Wang H, Zhao Y, Fu W.
Utilizing the Sobol’ Sensitivity Analysis Method to Address the Multi-Objective Operation Model of Reservoirs. *Water*. 2023; 15(21):3795.
https://doi.org/10.3390/w15213795

**Chicago/Turabian Style**

Wang, Haixia, Ying Zhao, and Wenyuan Fu.
2023. "Utilizing the Sobol’ Sensitivity Analysis Method to Address the Multi-Objective Operation Model of Reservoirs" *Water* 15, no. 21: 3795.
https://doi.org/10.3390/w15213795