# Multi-Objective Optimal Operations Based on Improved NSGA-II for Hanjiang to Wei River Water Diversion Project, China

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

#### 2.1. Study Area

^{2}[24,25]. The northern part of the Wei River basin is in the Loess Plateau and the southern part is in the Qinling Mountains. Social development has been retarded by water shortages because of rapid economic growth [26]. Water supply to Guanzhong has continually increased from 49.08 billion m

^{3}in 2010 to 54.2 billion m

^{3}in 2015. The ratio of surface water to groundwater is equal in the water supply structure. The development and level of utilization of surface water resources is 44%, higher than the internationally recognized level (40%). Moreover, overexploitation of underground water is frequent, which has damaged the ecological environment of the Wei River basin. Thus, the Project is a necessary engineering measure in the basin to alleviate water shortage, maintain river ecosystem health and promote social and economic sustainability.

^{2}. The Han River basin is in the subtropical monsoon climate zone with an average annual precipitation of 700~1100 mm, and the average surface water resource is 55.5 billion m³. The Danjiangkou reservoir is an important water source in the Han River, and supplies approximately 384 million m

^{3}for the South-to-north Water Diversion Project [28].

#### 2.2. The Hanjiang to Wei River Water Diversion Project

^{3}water per year from Han River to 21 users in Guanzhong area of Shaanxi province, including important cities, counties and industrial parks. The Project will mitigate water shortages, gradually increase ecological water and reduce the mining of groundwater in the Wei River basin.

^{3}/s, and its water loss coefficient is 1.5%. The concrete parameters of the Project are presented in Table 1.

#### 2.3. Data Situation

## 3. Modeling and Methodology

#### 3.1. Simulated Operation Model of Water Quantity

^{3}, which is the total water transfer of the Project (10

^{8}m

^{3}/s); $T$ is the length of operation cycle; $\Delta t$ is the period of reservoir operation with a month as step (s); $M$ is the number of reservoirs, where 1 represents HJX reservoir and 2 represents SHK reservoir; and ${Q}_{s}(m,t)$ is water supplied by HJX and SHK reservoirs in period t (m

^{3}/s).

#### 3.2. Multi-Objective Optimization Operation Model

- (1)
- Water balance$${V}_{}^{m}(t+1)-{V}_{}^{m}(t)=\left[{Q}_{I}^{m}(t)-{Q}_{O}^{m}(t)-{Q}_{S}^{m}(t)\right]\times \Delta t$$
- (2)
- Water level$${Z}_{\mathrm{min}}^{2}\le {Z}_{}^{2}(t)\le {Z}_{\mathrm{max}}^{2}(t)$$
- (3)
- Transferable water quantity$$\sum _{m=1}^{M}{Q}_{S}^{m}(t)\times \Delta t}\le {W}_{\mathrm{max}}^{qty}(t)$$
- (4)
- Maximum overflow$${Q}_{power}^{m}(t)\le {Q}_{\mathrm{max}}^{m}$$$${Q}_{}^{tunnel}(t)\le {Q}_{\mathrm{max}}^{tunnel}$$
- (5)
- Output of power station$${N}_{}^{m}(t)\le {N}_{installed}^{m,\mathrm{max}}$$$${N}_{dry}^{1}(t)\ge {N}_{firm}^{1}$$
- (6)
- Power of pump station$${P}_{}^{m}(t)\le {P}_{installed}^{m,\mathrm{max}}$$
^{8}m^{3}); ${Q}_{I}^{m}(t)$, ${Q}_{O}^{m}(t)$, and ${Q}_{S}^{m}(t)$ represent inflow runoff, outflow runoff and water-transferred flow of reservoir m in period t, respectively (m^{3}/s); ${Z}_{}^{2}(t)$ is water level of SHK reservoir in period t(m); ${Z}_{\mathrm{min}}^{2}$ is dead water level of SHK reservoir (m); and ${Z}_{\mathrm{max}}^{2}(t)$ is the highest water level of SHK reservoir (m), mainly including flood control level in flood season and normal high water level in non-flood season. ${W}_{\mathrm{max}}^{qty}(t)$ is the maximum transferable quantity of water of Han River in period t (10^{8}m^{3}); ${Q}_{}^{m}(t)$ is outflow of hydropower station m in period t; ${Q}_{\mathrm{max}}^{m}$ is the maximum outflow of hydropower station m (m^{3}/s); ${Q}_{}^{tunnel}(t)$ is average transfer flow in Qinling tunnel in period t; ${Q}_{\mathrm{max}}^{tunnel}$ is the maximum water transfer capability of Qinling tunnel (m^{3}/s); ${N}_{}^{m}(t)$ is output of hydropower station m in period t; ${N}_{installed}^{m,\mathrm{max}}$ is installed capacity of the hydropower station m; ${N}_{dry}^{1}(t)$ and ${N}_{firm}^{1}$ represent output in dry season and guaranteed output of the HJX hydropower station, respectively (MW); ${P}_{}^{m}(t)$ is power consumption of the pump station m in period t; and ${P}_{installed}^{m,\mathrm{max}}$ is installed capacity of the pump station m. All variables are non-negative.

#### 3.3. Feasible Search Space

^{3}/s); ${Q}_{lower}(t)$ and ${Q}_{upper}(t)$ are the boundaries of the final feasible search space in $t$ (m

^{3}/s); $\Delta q$ is the search step size of the feasible space (m

^{3}/s); and ${Q}_{trf}^{Han}(t)$ ${Q}_{trf}^{1}(t)$ are the maximized adjustable waters of Han River and the HJX reservoir in $t$ (m

^{3}/s), respectively.

#### 3.4. Steps of the Solution and Settings of the Schemes

#### 3.5. Drafting the Operation Chart of Water Supply

- (1)
- The maximum storage capacity: this is determined by design data of the maximum water level in flood season and maximum water level in non-flood season.
- (2)
- The minimum storage capacity: this is decided by design data of dead water level.
- (3)
- The hedging rule curve for abandoned water: this is decided by the outsourcing line composed of the initial water level for each month in the abandoned water year during long time series data.
- (4)
- The hedging rule curve for combined water supply: this is determined by the outsourcing line composed of the initial water level of each month in the year of SHK pump stations operation involved during long time series data.
- (5)
- The hedging rule curve for basic water supply: this is decided by the outsourcing line composed of the initial water level of each month in the year, in this situation, water demand can’t be satisfied and only the SHK reservoir supplies.

## 4. Results and Discussion

#### 4.1. Simulation Model (Scheme 1)

#### 4.2. Improvement in NSGA-II (Schemes 2 and 3)

^{8}m

^{3}, could not satisfy the demand for intake areas. On the contrary, most of the results for scheme 3 satisfies demand. The absolute value of the slope of the Pareto curve of scheme 2 is 0.33, greater than that of 0.27 for scheme 3. This means that the accuracy of the results and speeds of are better with a smaller search space, which is also intuitive. (3) Power generation and energy consumption are positively correlated in the results of the model, in contrast to the mathematical conclusion in (1) above, where they appear to be inversely proportional. This is an odd result: with an increase in the volume of transferred water, both energy consumption and power generation increase. However, according to experience, the greater water quantity is transferred, the smaller power generates.

^{3}can be annually transferred by the Project. Transferring more water and increasing net power generated under the constraint of adjustable water is thus a critical decision for management. Significantly, this appears to obey a preliminary rule, and is explored in greater detail below.

- (1)
- In the design of the Project, if the average annual transferred quantity of water is 1.5 billion, the best water transferred ratio is one where the HJX reservoir transferred approximately 0.8–0.9 billion m
^{3}and the SHK reservoir approximately 0.6–0.7 billion m^{3}. - (2)
- The HJX reservoir should undertake the main task of transferring water in flood season to the intake area and the SHK reservoir, and the pumping flow of the HJX pump station is better controlled at around 50 m
^{3}/s if the adjustable water is sufficient in volume. - (3)
- The SHK pump station is better at reducing operating frequencies and time to save energy. Once the SHK pump station begins operating, it meant the HJX pump station has consumed energy to lift water to the SHK reservoir. The highest volume of water transferred from HJX to SHK is 0.05–0.08 billion m
^{3}. - (4)
- The SHK reservoir should increase water supply to reduce abandoned water in flood season and avoid drastic fluctuations in the water level.

#### 4.3. Feasible Search Space in I-NSGA-II (Schemes 3~9)

^{3}/s, and the Pareto curves from schemes 4 to 9 are above scheme 2 and under scheme 3. The curve of scheme 4 is very close to that of scheme 3, and this results from similar and small search galleries based on close search step size. From results of the quantity of water, only schemes 3 and 4 meet the water demand.

^{3}. We compared schemes 3-1 and 4-1 with scheme 1, and analyzed the relative rate of change (referred hereafter to as “R”) of the model’s objective value, especially the WSI. The values are listed in Table 5.

^{3}/s) is more suitable for demand than scheme 3-1 (Δq = 5 m

^{3}/s). An examination of their corresponding search spaces indicates that restricting their widths would influence the genetic manipulation of individual genes. When the space is too small at time t, this means that the value of the individual gene is stable in the evolutionary iteration, and it adds the difficulties and requirements before time (t − 1). Moreover, the range of the search space fluctuates excessively at time t and (t + 1), which causes the value of the gene to easily become unstable. Therefore, it is suitable to choose 10 m

^{3}/s as search step size to build the search gallery in solving this model from the perspective of water supply.

^{3}/s as search step size to build the search gallery in this model and to meet water demand.

^{3}, and this will promote economic development and increase revenue by RMB 50 billion yuan. The average WSI of the Pareto points in scheme 4-1 is 5.31%, which can ensure water safety in the intake area. In summary, we chose a search step size of 10 m

^{3}/s to build the search gallery in this multi-objective model.

#### 4.4. Operation Chart of the Project

## 5. Conclusions

- (1)
- The simulation results show that the operational framework in this paper is superior to other models designed under the same initial conditions. Therefore, it can be applied to subsequent research.
- (2)
- Setting a reasonable feasible search space with the NSGA-II can help find better optimal solutions. With the same influence of the initial populations of the algorithm and limited computing ability, the qualified rate of the I-NSGA-II is much higher than that of the NSGA-II.
- (3)
- It is determined that 10 m
^{3}/s is the most suitable search step size value for the feasible search space in this case study. Power generation, energy consumption, rate of guaranteed water supply and the WSI of intake area should all be regarded as factors of evaluation to determine the search step size. A very large or too small feasible search space affects the optimization results. - (4)
- It is useful to manage water resources of large-scale IBWT projects in combination with the specific characteristics of the projects, especially under strict constraints of government regulation and the interests of all parties. If the Project operates as shown in Figure 10, it can both meet the demand for water and ensure the optimal conversion of energy in the system according to the needs of the decision makers.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the Hanjiang to Wei River Water Diversion Project, the green arrow indicates the direction of the water flow.

**Figure 5.**Solution steps flowchart of the improved Non-dominated Sorting Genetic Algorithm-II (I-NSGA-II).

**Figure 6.**Water transfer process of Huangjinxia reservoir (HJX) and Sanhekou reservoir (SHK) in scheme 1.

**Figure 7.**The Pareto front solved by NSGA-II and I-NSGA-II, the number mark on the Pareto point represents the amount of water supply.

**Figure 9.**Comparison of water supply, power generation and energy consumption in different schemes (combined with Table 5).

Items | Units | HJX | SHK | ||||
---|---|---|---|---|---|---|---|

Reservoir | Power Station | Pump Station | Reservoir | Power Station | Pump Station | ||

Regulating storage | 10^{8} m^{3} | 0.92 | - | - | 7.10 | - | - |

Inflow | 10^{8} m^{3} | 66.36 | - | - | 8.61 | - | - |

Regulation ability | - | Daily | - | - | Multi-year regulating | - | - |

Normal high water level | m | 450 | - | - | 643 | - | - |

Flood control level | m | 448 | - | - | 642 | - | - |

Dead water level | m | 440 | - | - | 558 | - | - |

Working head | m | - | 36.5 | 104.5 | - | - | 97.7 |

Installed capacity | MW | - | 135 | 126 | - | 60/40 | 20 |

Firm power | MW | - | 0.86 | - | - | - | - |

Maximum outflow | m^{3}/s | - | 435.30 | 70 | 72.71 | - | 18 |

Ecological flow | m^{3}/s | 25 | - | - | 2.71 | - | - |

Scheme | Δq ^{b} | Search Space | Method | Objects |
---|---|---|---|---|

1 | - | All feasible space | Simulated | Water quantity (15) ^{a} |

2 | - | $[0,{Q}_{upper}^{initial}]$ | NSGA-II | Energy consumption Power generation |

3 | Δq = 5 | $[{Q}_{lower},{Q}_{upper}]$ reference Equations (14) and (15) | I-NSGA-II | |

4 | Δq = 10 | |||

5 | Δq = 20 | |||

6 | Δq = 30 | |||

7 | Δq = 40 | |||

8 | Δq = 50 | |||

9 | Δq = 60 |

^{a}—10

^{8}m

^{3};

^{b}—m

^{3}/s.

Parameters | NSGA-II/I-NSGA-II |
---|---|

Number of decision variables | 672 |

Population size | 500 |

Generation | 1000 |

Number of objective functions | 3 |

Mutation probability | 0.2 |

Crossover probability | 0.4 |

Index | Designed | Simulated | ||||||
---|---|---|---|---|---|---|---|---|

W^{a} | E_{power}^{c} | E_{pump}^{c} | W^{a} | E_{powe}^{c} | E_{pump}^{c} | |||

to Intake Area | to SHK | to Intake Area | to SHK | |||||

HJX | 9.19 | 0.50 | 3.87 | 3.84 | 9.32 | 0.47 | 3.70 | 3.88 |

SHK | 5.31 | 1.32 | 0.20 | 5.21 | 1.46 | 0.19 | ||

Total | 15.00 | 5.19 | 4.04 | 15.00 | 5.16 | 4.07 |

^{a}—10

^{8}m

^{3};

^{c}—10

^{8}kWh.

Objectives | Model | Simulated | Multi Objective Optimized | Relative Change Rate | ||
---|---|---|---|---|---|---|

Scheme | 1 | 3-1 | 4-1 | 3-1 | 4-1 | |

Water Supplying ^{a} | HJX to intake area | 9.32 | 7.91 | 8.52 | −15.13% | −8.58% |

HJX to SHK | 0.47 | 0.51 | 0.58 | +8.51% | +23.40% | |

SHK to intake area | 5.21 | 6.58 | 5.90 | +26.3% | +13.24% | |

Total | 15.00 | 15.00 | 15.00 | 0 | 0 | |

WSI | 10.21% | 6.92% | 3.25% | - | - | |

Power generation ^{c} | HJX power station | 3.70 | 3.90 | 3.78 | +5.41% | +2.16% |

SHK power station | 1.46 | 1.53 | 1.55 | +4.79% | +6.16% | |

Total | 5.16 | 5.43 | 5.33 | +5.23% | +3.29% | |

Energy consumption ^{c} | HJX pump station | 3.88 | 3.50 | 3.78 | −9.79% | −2.58% |

SHK pump station | 0.19 | 0.20 | 0.23 | +5.26% | +21.05% | |

Total | 4.07 | 3.70 | 4.01 | −9.79% | −2.58% |

^{a}—10

^{8}m

^{3};

^{c}—10

^{8}kWh.

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**MDPI and ACS Style**

Wu, L.; Bai, T.; Huang, Q.; Wei, J.; Liu, X.
Multi-Objective Optimal Operations Based on Improved NSGA-II for Hanjiang to Wei River Water Diversion Project, China. *Water* **2019**, *11*, 1159.
https://doi.org/10.3390/w11061159

**AMA Style**

Wu L, Bai T, Huang Q, Wei J, Liu X.
Multi-Objective Optimal Operations Based on Improved NSGA-II for Hanjiang to Wei River Water Diversion Project, China. *Water*. 2019; 11(6):1159.
https://doi.org/10.3390/w11061159

**Chicago/Turabian Style**

Wu, Lianzhou, Tao Bai, Qiang Huang, Jian Wei, and Xia Liu.
2019. "Multi-Objective Optimal Operations Based on Improved NSGA-II for Hanjiang to Wei River Water Diversion Project, China" *Water* 11, no. 6: 1159.
https://doi.org/10.3390/w11061159