# Optimization of Water and Energy Spatial Patterns in the Cascade Pump Station Irrigation District

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area

^{4}m

^{3}, the annual energy consumption was 18,770 × 10

^{4}kWh. The crop output value was 65,244 × 10

^{4}yuan, and the irrigated area was 21,952 ha. The drip irrigation area comprised 7.56% of the total irrigated area.

## 3. The Mathematical Model and Calculation

#### 3.1. The Multi-Objective Linear Programming Model

#### 3.1.1. Decision Variables

_{ik}is the area of crop k in sub-district i, and d

_{ik}is a constant. In the optimization scheme, x

_{ijk}is taken as the decision variable, which is the area of crops k of types of irrigation technologies j (j = 1 for surface irrigation, j = 2 for drip irrigation) in sub-district i. Thus, in the optimization scheme, the sub-district comprises three parts, the optimized surface irrigation area, the optimized drip irrigation area, and the unchanged present drip irrigation area.

#### 3.1.2. The Water–Energy–Crops–Irrigation Nexus in the CPSID

_{1}of the CPSID is calculated by Equation (1):

_{1}is the annual irrigation water consumption of the CPSID (m

^{3}); i is the sub-district number; I is the total number of sub-districts; j is the number of irrigation technology types, j = 1 is surface irrigation, j = 2 is drip irrigation; J is the total number of irrigation technology types; k is the number of crop types; K is the total number of crop types; q

_{jk}is the net irrigation quota (m

^{3}/ha); n is the number of WCP&C segments; η

_{n}is the water use efficiency of WCP&C segments; and γ

_{ij}is the water use efficiency of the sub-district (which is related to the irrigation technology).

_{2}of the CPSID is calculated according to Equation (2):

_{2}is the annual irrigation water consumption of the CPSID (kWh); p is the pump station number; P is the total number of pump stations; Hp is the pump station head (m); α

_{p}is the pump station efficiency (%); h

_{d}is the pump station head in the drip irrigation system in the sub-district (m; h

_{d}= 50 m), which comprises a head drip irrigation emitter of 10 m and a head loss in the drip irrigation system of 40 m; and ε

_{d}is the efficiency of the pump station of the drip irrigation system (%; ε

_{d}= 82%).

_{3}in the CPSID is calculated according to Equation (3):

_{jk}is the output value of the crops per unit area (yuan/ha), which is related to the type of crop and irrigation technology.

#### 3.1.3. Objective Function

_{1}(X) is the minimum annual irrigation water consumption in the CPSID. The second term in Equation (1) is a constant. Therefore, W

_{1}(X) is calculated according to Equation (4):

_{ijk}}, X is the set of decision variables.

_{2}(X) is the minimum annual energy consumption in the CPSID, and the third term in Equation (2) is a constant, so W

_{2}(X) is expressed as Equation (5):

_{1}(X) is the maximum output value of crops in the CPSID, since the second term in Equation (3) is a constant, Z

_{1}(X) is expressed as Equation (6):

#### 3.1.4. Constraints

- (1)
- Area constraint of each sub-district

- (2)
- Constraints of the annual water amount for a sub-district

_{i}is the upper limit of annual water amount for the sub-district (m

^{3}), as shown in Table 1.

- (3)
- Area constraints of various crops in the CPSID

- (4)
- Constraint conditions of drip irrigation project investment

_{k}is the investment per unit area of the new drip irrigation project (yuan/ha), which depends on the crop type, because different crops are planted in different densities. T is the total investment quota of the drip irrigation projects in the ID (yuan).

- (5)
- Non-negative constraint:

#### 3.2. The Algorithm of the Model Solution

_{ijk}. The multi-objective linear programming model is as follows:

**Step 1**:- Obtain the ideal and the anti-ideal solutions.

**Step 2**:- First stage of the algorithm.

_{ijk}.

**Step 3**:- Second stage of the algorithm.

_{1}, λ

_{2,}λ

_{3,}and xijk.

## 4. Results and Discussion

^{4}yuan, and the value and data presented in Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 were input to the objective functions (4)–(6) and the constraint conditions (7)–(12). The ideal solutions (${Z}_{1}^{*}$, ${W}_{1}^{*}$, ${W}_{2}^{*}$) and the anti-ideal solutions (${Z}_{1}^{-}$,${W}_{1}^{-}$, ${W}_{2}^{-}$) were (765,307,095, 97,249,784, 122,606,014) and (510,911,364, 146,310,129, 192,864,931), respectively.

_{1}= 761,502,174 yuan, W

_{1}= 102,258,392 m

^{3}, and W

_{2}= 129,099,716 kWh, λ = 0.49.

_{1}= 766,902,174 yuan, W

_{1}= 100,353,658 m

^{3}, W

_{2}= 126,016,650 kWh, and λ

_{1}= 0.52, λ

_{2}= 0.81, λ

_{3}= 0.62.

_{ijk}obtained in the second step and the present area of various crops in drip irrigation (Table 5). The optimal scheme of the spatial patterns of crops and irrigation technologies in the Jingdian Phase I Irrigation District were obtained (Table 7).

^{4}m

^{3}and 5515 × 10

^{4}kWh, respectively. Furthermore, it could increase the crop output value by 19,277 × 10

^{4}yuan; that is, the scheme could reduce water and energy consumption by 26.18% and 29.38%, and increase income by 29.55%.

^{4}yuan, 20,000 × 10

^{4}yuan, 30,000 × 10

^{4}yuan, 40,000 × 10

^{4}yuan, and 50,000 × 10

^{4}yuan, respectively. The annual irrigation water consumption, annual energy consumption, and crop output value of different investment optimization schemes can be obtained by using Equations (1)–(3). The annual irrigation water and energy consumption in the ID demonstrated a decreasing trend with an increase in drip irrigation investment (Figure 5a,b). Energy intensity is an important indicator to measure energy conservation in the CPSID [7]. The energy intensity in the ID demonstrated a downward trend with an increase in drip irrigation investment (Figure 5c), indicating that the CPSID can achieve significant energy conservation by saving water. The crop output value demonstrated a rising trend (Figure 5d).

## 5. Conclusions

^{4}yuan of drip irrigation project investment, the optimal scheme of the spatial pattern of crops and irrigation technologies in the ID was obtained. Compared with the present situation, the annual irrigation water consumption was reduced by 3722 × 10

^{4}m

^{3}, and the energy consumption was reduced by 5515 × 10

^{4}kWh, whereas the crop output value was increased by 19,277 × 10

^{4}yuan. This optimization saves water by 26.18%, saves energy by 29.38%, and increases income by 29.55%. The optimal spatial pattern of the ID suggested that crops with low water consumption and high output values should be distributed in the sub-districts with high elevations. Meanwhile, crops with high water consumption and low output values should be distributed in the sub-districts with low elevations. The results showed that the annual irrigation water consumption, energy consumption, and energy intensity decreased with the increasing investment in drip irrigation projects, while the crop output value increased.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Relationship between the drip irrigation project investment and the (

**a**) annual irrigation water consumption, (

**b**) annual energy consumption, (

**c**) energy intensity and (

**d**) crop output value in the CPSID.

Sub-District | Upper Limit of Annual Water Amount (10^{4} m^{3}) | Elevation Range (m) | Water Use Efficiency of Surface Irrigation Unit (%) | Water Use Efficiency of Drip Irrigation Unit (%) | Area (ha) |
---|---|---|---|---|---|

1 | 237 | 1561–1580 | 72 | 94 | 357 |

2 | 1550 | 1580–1596 | 69 | 93 | 2306 |

3 | 4439 | 1596–1620 | 67 | 93 | 6533 |

4 | 1707 | 1620–1638 | 70 | 95 | 2375 |

5 | 600 | 1638–1658 | 69 | 94 | 806 |

6 | 3140 | 1658–1676 | 68 | 92 | 4054 |

7 | 2319 | 1676–1694 | 71 | 93 | 3454 |

8 | 1545 | 1694–1711 | 72 | 94 | 1943 |

9 | 31 | 1711–1738 | 74 | 95 | 35 |

10 | 73 | 1738–1757 | 73 | 95 | 89 |

Pump Station | Pump Station Head (m) | Pump Station Efficiency (%) | Design Discharge of Pump Station (m^{3}/s) |
---|---|---|---|

1 | 81 | 84 | 13.17 |

2 | 80 | 83 | 13.14 |

3 | 80 | 83 | 13.13 |

4 | 30 | 84 | 13.12 |

5 | 27 | 84 | 12.96 |

6 | 37 | 83 | 11.57 |

7 | 29 | 75 | 7.70 |

8 | 32 | 75 | 6.27 |

9 | 27 | 75 | 5.79 |

10 | 29 | 75 | 3.36 |

11 | 27 | 70 | 1.45 |

12 | 43 | 60 | 0.20 |

13 | 30 | 60 | 0.10 |

Water Conveyance Pipe and Channel Segment | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Water Use Efficiency | 0.97 | 0.98 | 0.97 | 0.97 | 0.98 | 0.99 | 0.95 | 0.96 | 0.96 | 0.99 | 0.99 | 0.99 | 0.99 |

Irrigation Technologies | Surface Irrigation | Drip Irrigation | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Crop | W | M | F1 | C | F2 | V | P | O | C | F2 | V | P | O |

Net Irrigation Quota (m^{3}/ha) | 4080 | 4135 | 3416 | 5200 | 3084 | 4540 | 3426 | 3750 | 3120 | 1696 | 2497 | 1884 | 2070 |

Output Value per Unit Area (yuan/ha) | 18,240 | 22,260 | 29,700 | 24,000 | 44,625 | 54,000 | 25,200 | 25,200 | 28,800 | 53,550 | 64,800 | 30,240 | 30,240 |

Investment per Unit Area of Drip Irrigation (yuan/ha) | - | - | - | - | - | - | - | - | 46,500 | 45,000 | 60,000 | 57,000 | 52,500 |

**Table 5.**The present situation of the spatial patterns of crops and irrigation technologies in the Jingdian Phase I Irrigation District (in 2020, ha).

Sub-District | Surface Irrigation | Drip Irrigation | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

W | M | F1 | C | F2 | V | P | O | C | F2 | V | P | O | |

1 | 64 | 112 | 20 | 12 | 42 | 27 | 36 | 18 | 0 | 26 | 0 | 0 | 0 |

2 | 415 | 726 | 127 | 39 | 310 | 174 | 233 | 116 | 0 | 166 | 0 | 0 | 0 |

3 | 1176 | 2058 | 359 | 60 | 878 | 494 | 659 | 329 | 50 | 0 | 200 | 270 | 0 |

4 | 428 | 748 | 131 | 40 | 319 | 179 | 239 | 120 | 0 | 0 | 0 | 0 | 171 |

5 | 145 | 253 | 44 | 14 | 109 | 61 | 81 | 41 | 0 | 58 | 0 | 0 | 0 |

6 | 730 | 1277 | 223 | 38 | 545 | 306 | 409 | 204 | 30 | 182 | 110 | 0 | 0 |

7 | 622 | 1088 | 190 | 58 | 464 | 261 | 348 | 174 | 0 | 100 | 0 | 149 | 0 |

8 | 372 | 503 | 114 | 35 | 278 | 171 | 218 | 104 | 0 | 0 | 148 | 0 | 0 |

9 | 22 | 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 45 | 0 | 44 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Crop | Crop Area Constraint | Scheme Comparison | ||
---|---|---|---|---|

Upper Limit | Lower Limit | Present | Optimal | |

Wheat | 6029 | 2010 | 4019 | 2010 |

Maize | 10,167 | 3389 | 6778 | 3389 |

Flax | 1878 | 626 | 1252 | 1878 |

Chinese wolfberry | 564 | 188 | 376 | 268 |

Fruit tree | 5748 | 1739 | 3477 | 5748 |

Vegetable | 3654 | 1066 | 2131 | 3654 |

Potato | 4196 | 1321 | 2642 | 4196 |

Oil sunflower | 1916 | 639 | 1277 | 809 |

**Table 7.**The optimal scheme of spatial patterns of crops and irrigation technologies in the Jingdian Phase Irrigation District (ha).

Sub-District | Surface Irrigation | Drip Irrigation | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

W | M | F1 | C | F2 | V | P | O | C | F2 | V | P | O | |

1 | 0 | 0 | 0 | 131 | 0 | 143 | 0 | 0 | 57 | 26 | 0 | 0 | 0 |

2 | 0 | 1964 | 0 | 0 | 0 | 176 | 0 | 0 | 0 | 166 | 0 | 0 | 0 |

3 | 2009 | 1425 | 0 | 0 | 0 | 0 | 1940 | 638 | 50 | 0 | 200 | 270 | 0 |

4 | 0 | 0 | 367 | 0 | 0 | 0 | 1837 | 0 | 0 | 0 | 0 | 0 | 171 |

5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 58 | 748 | 0 | 0 |

6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 30 | 1785 | 2239 | 0 | 0 |

7 | 0 | 0 | 1511 | 0 | 0 | 0 | 0 | 0 | 0 | 1794 | 0 | 149 | 0 |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1795 | 148 | 0 | 0 |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 | 0 | 0 | 0 |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 89 | 0 | 0 | 0 |

Sub-District | Drip Irrigation Area (ha) | Annual Irrigation Water Consumption (10^{4} m^{3}) | Annual Energy Consumption (10 ^{4} kWh) | Crop Output (10^{4} Yuan) | ||||
---|---|---|---|---|---|---|---|---|

Present | Optimal | Present | Optimal | Present | Optimal | Present | Optimal | |

1 | 26 | 83 | 216 | 242 | 190 | 209 | 1063 | 1390 |

2 | 166 | 166 | 1409 | 1510 | 1372 | 1470 | 6935 | 6212 |

3 | 520 | 520 | 4035 | 4009 | 4423 | 4394 | 19,268 | 15,592 |

4 | 171 | 171 | 1552 | 1373 | 1863 | 1650 | 6743 | 6236 |

5 | 58 | 806 | 546 | 270 | 719 | 390 | 2426 | 5158 |

6 | 322 | 4054 | 2854 | 1246 | 4047 | 1919 | 12,331 | 24,154 |

7 | 249 | 1943 | 2109 | 1317 | 3749 | 2299 | 10,041 | 14,544 |

8 | 148 | 1943 | 1405 | 498 | 2257 | 860 | 6155 | 10,571 |

9 | 0 | 35 | 28 | 9 | 45 | 17 | 69 | 187 |

10 | 0 | 89 | 66 | 22 | 106 | 46 | 213 | 477 |

Irrigation district indicators | 1660 | 9810 | 14,219 | 10,497 | 18,770 | 13,255 | 65,244 | 84,521 |

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**MDPI and ACS Style**

Bai, C.; Yao, L.; Wang, C.; Zhao, Y.; Peng, W.
Optimization of Water and Energy Spatial Patterns in the Cascade Pump Station Irrigation District. *Sustainability* **2022**, *14*, 4943.
https://doi.org/10.3390/su14094943

**AMA Style**

Bai C, Yao L, Wang C, Zhao Y, Peng W.
Optimization of Water and Energy Spatial Patterns in the Cascade Pump Station Irrigation District. *Sustainability*. 2022; 14(9):4943.
https://doi.org/10.3390/su14094943

**Chicago/Turabian Style**

Bai, Chen, Lixiao Yao, Cheng Wang, Yongxuan Zhao, and Weien Peng.
2022. "Optimization of Water and Energy Spatial Patterns in the Cascade Pump Station Irrigation District" *Sustainability* 14, no. 9: 4943.
https://doi.org/10.3390/su14094943