# Study on Water Quantity Allocation Optimization for Single Main Canal in Large-Scale Irrigation Area Based on DP Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{4}m

^{3}and 23.31 × 10

^{4}m

^{3}during the ponding period of rice. The corresponding water quantity allocation for each branch canal has reflected a compellent model solution precision and efficiency.

## 1. Introduction

^{4}acres of irrigation area, which leads to the characteristics that water conveyance canals usually include multi-stage and long-distance water conveyance with multiple hydraulic structures along canals at all levels. Affected by water rights control policy enforced by the Chinese government that focuses on agricultural water consumption saving in recent years, and combined with still relatively extensive water distribution management, it is necessary to improve agricultural water resources efficiency in irrigation areas by means of scientific water resources allocation methods. According to the current agricultural farming structures of irrigation areas, a system optimization model could be applied to optimize water quantity allocation for each branch canal of a single main canal to improve full utilization of water resources in irrigation areas, which will also promote increased agricultural production and increasing farmers’ income in irrigation areas, and serve China’s rural revitalization strategy.

## 2. Model Construction of Optimal Water Quantity Allocation for Single Main Canal

_{j}= water quantity allocation of the j-th branch canal (m

^{3}); YS

_{j}= water requirement of the water receiving area controlled by the j-th branch canal (m

^{3}); W

_{0}= water right of single water diversion of the main canal (m

^{3}).

## 3. Model Solution Method

_{j}, this constructed model could be solved by one-dimensional dynamic programming.

- (1)
- Considering the position number of each one of the branch canals as stage variable j, and the total water quantity allocation of former branch canals as state variable λ
_{j}, the state transition equation could be constructed as follows:

- (2)
- According to objective function (1) and state transition Equation (3), the merit function of each optimization stage could be obtained as follows:

_{1}is discretized into 0, W

_{1}, W

_{2}, …, W

_{0}. For each one of λ

_{1}, the decision variable X

_{1}is discretized into 0, X

_{11}, X

_{12}, …, X

_{1max}, where X

_{1max}is the maximal water quantity allocation corresponding to the state variable of the first stage λ

_{1}. The inequality of X

_{1}≥ λ

_{1}should be met according to Formula (2).

_{j}is also discretized into 0, W

_{1}, W

_{2}, …, W

_{0}. For each one of λ

_{j}, the decision variable X

_{j}is also discretized into 0, X

_{j}

_{1}, X

_{j}

_{2}, …, X

_{j}

_{max}, where X

_{j}

_{max}is the maximal water quantity allocation corresponding to the state variable of the j-th stage λ

_{j}. The inequality of $\sum _{j=1}^{j}{X}_{j}}\ge {\lambda}_{j$ should be met, where j = 2, 3, …, m−1.

_{m}is set as W

_{0}. According to ${\lambda}_{m-1}={\lambda}_{m}-{X}_{m}$, Formula (7) could be transformed into the following form:

_{m}is also discretized in its feasible region, by which to obtain the minimal sum of the squared deviation of the water shortage for the water receiving areas F and the corresponding optimal water quantity allocation of the m-th branch canal X

_{m}

^{*}.

_{j}

^{*}(j = 1, 2, 3, …, m).

## 4. Model Application of Study Case

#### 4.1. Basic Information of Hengliu Main Cain System in Zhouqiao Irrigation Area

^{3}/s. There are 8 main canals (sub-trunk canals) with a combined total length of 84.1 km, and 101 branch canals with a combined total length of 267.38 km.

#### 4.2. Optimal Water Quantity Allocation for Each Branch Canal along Hengliu Main Canal

#### 4.2.1. Analysis of Water Requirement during the Ponding Period of Rice Controlled by the Hengliu Main Canal

^{3}/mu as the irrigation quota, and respectively considering 0.9, 0.91, and 0.95 as the water utilization coefficient of branch canals, field canals, and fields, respectively, the irrigation water demand of each piece of the water receiving area corresponding to each branch canal was calculated, and is shown in Table 3.

#### 4.2.2. Analysis of Available Water Supply during the Ponding Period of Rice Controlled by the Hengliu Main Canal

^{3}with an effective irrigation area of 0.3195 million mu. For the 11,635 mu covered by the Hengliu Main Canal and the corresponding planting proportion controlled by each branch canal, considering a medium drought year (p = 75%) and a special drought year (p = 95%), water demand quantity is respectively 27.53% and 21.88% of the irrigation quota with the corresponding available water supply of 1.604 million m

^{3}and 1.275 million m

^{3}, respectively. According to crop planting area of each water receiving area controlled by each branch canal, available water supply for each branch canal under two different year types is obtained which is shown in Table 3.

#### 4.3. Results and Analysis of Optimal Water Quantity Allocation for the Hengliu Main Canal System

^{3}and 0.8039 million m

^{3}in a medium drought year (p = 75%) and a special drought year (p = 95%), respectively. These two water supply quantities are both less than the total water requirement of 103.7 million m

^{3}for all of the branch canals controlled by Hengliu Main Canal, which makes it necessary to carry out optimal water quantity allocation for all branch canals under the certain water rights.

^{3}and 0.8039 million m

^{3}as the total water supply of Hengliu Main Canal for a ponding period of 6.5 days in a medium drought year (p = 75%) and a special drought year (p = 95%), respectively, a one-dimensional dynamic programming can be applied to solve mathematical model (1)~(2) for the total water requirement of 103.7 million m

^{3}. From this, the optimal water quantity allocation of each branch canal can be obtained for a medium drought year (p = 75%) and a special drought year (p = 95%). The optimization results are shown in Table 4. On the other hand, water quantity allocation results by equal proportion for each water receiving area controlled by its branch canal calculated from the total water quantity of Hengliu Main Canal are shown in Table 5, with the same consideration of a medium drought year (p = 75%) and a special drought year (p = 95%). For convenience of discussion, DP-mode was defined as the optimal water quantity allocation with dynamic programming, while EP-mode was defined as water quantity allocation by equal proportion for each water receiving area controlled by its branch canal. The comparison of the water quantity allocation for each branch canal for two above modes is shown in Figure 2.

^{4}m

^{3}and 23.31 × 10

^{4}m

^{3}in the medium and special drought years, respectively, calculated by the two above water quantity allocation modes. Compared with EP-mode, the real water shortage for each branch canal is better-distributed in a medium drought year (p = 75%, 0.64~0.65 × 10

^{4}m

^{3}) and a special drought year (p = 95%, 5.82~5.83 × 10

^{4}m

^{3}) when calculated using DP-mode, which contributes to alleviate water utilization contradictions of water consumers. By means of optimal water quantity allocation using DP-mode, the decreasing range of the minimal sum of the squared deviation of the water shortage for the water receiving areas is 0.76% and 0.64% corresponding to a medium drought year (p = 75%) and a special drought year (p = 95%), respectively, compared with water quantity allocation calculated using EP-mode.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Liu, D.; Hu, Y.X.; Fu, Q.; Li, T.X. Optimization and parameter analysis for channel cross section with concrete lining in northern irrigation district. Trans. Chin. Soc. Agric. Eng.
**2015**, 31, 107–114. [Google Scholar] - Shang, G.L.; Liu, D.; Hu, Y.X. Optimization of irrigation channel section based on cat swarm algorithm and analysis on design parameters. J. Drain. Irrig. Mach. Eng.
**2016**, 34, 128–132. [Google Scholar] - Xu, Z.C. Optimal model of channel layout based on minimum spanning trees. Trans. Chin. Soc. Agric. Eng.
**2017**, 33, 124–130. [Google Scholar] - Akram, A.A.; Mendelsohn, R. Agricultural water allocation efficiency in a developing country canal irrigation system. Environ. Dev. Econ.
**2017**, 22, 571–593. [Google Scholar] [CrossRef][Green Version] - Kanooni, A.; Monem, M.J. Integrated stepwise approach for optimal water allocation in irrigation canals. Irrig. Drain.
**2014**, 63, 12–21. [Google Scholar] [CrossRef] - Khandelwal, S.S.; Dhiman, S.D. Optimal allocation of land and water resources in a canal command area in the deterministic and stochastic regimes. Water Resour. Manag.
**2018**, 32, 1569–1584. [Google Scholar] [CrossRef] - Hao, J.C.; Lou, Z.K.; Gao, F. Numerical simulation and slope coefficient optimization of frost heave of trapezoidal channel. J. Irrig. Drain.
**2017**, 36, 81–86. [Google Scholar] - Wang, Y.; Liu, J.C.; Liu, Q.H.; Wang, Z.Z. Shape optimization of a trapezoidal canal structure for coupled temperature-water-soil conditions in cold regions. J. Tsinghua Univ.
**2019**, 59, 645–654. [Google Scholar] [CrossRef] - Xu, S.Q.; Wang, Y.J.; Qiao, H.Q.; Guo, X.T.; Li, Z.Y.; Xu, E.D. Optimal operation of irrigation and drainage channels based on multiobjective genetic algorithm. J. Northeast. Agric. Univ.
**2020**, 51, 80–86. [Google Scholar] - Fu, Q.; Xiao, Y.Y.; Cui, S.; Liu, D.; Li, T.X. Multi-water resources optimal allocation of irrigation district based on fuzzy multi-objective programming. Trans. Chin. Soc. Agric. Mach.
**2017**, 48, 222–227. [Google Scholar] - Li, R.H.; Chang, Y.L.; Wang, Z.C. Study of optimal allocation of water resources in Dujiangyan irrigation district of China based on an improved genetic algorithm. Water Sci. Technol.
**2021**, 21, 2989–2999. [Google Scholar] [CrossRef] - Pan, Q.; Guo, P.; Zhang, F.; Luo, B.; Zhang, X.X. Study on multi-objective optimal allocation of agricultural water resources in Huangyang Irrigated Area considering canal leakage. J. Water Resour. Water Eng.
**2020**, 31, 166–173. [Google Scholar] - Xu, Y.W.; Wang, Y.H.; Liang, D.L.; Fu, Q.; Zhou, Y.; Chen, X.H. Irrigation water resources allocation in Jinxi irrigation district based on agricultural sustainability. Trans. Chin. Soc. Agric. Mach.
**2020**, 51, 299–309. [Google Scholar] - Li, M.; Sun, H.; Liu, D.; Singh, V.P.; Fu, Q. Multi-scale modeling for irrigation water and cropland resources allocation considering uncertainties in water supply and demand. Agric. Water Manag.
**2021**, 246, 106687. [Google Scholar] [CrossRef] - Gong, X.H.; Zhang, H.B.; Ren, C.F.; Sun, D.Y.; Yang, J.T. Optimization allocation of irrigation water resources based on crop water requirement under considering effective precipitation and uncertainty. Agric. Water Manag.
**2020**, 239, 106264. [Google Scholar] [CrossRef] - Cheng, K.; Wei, S.; Ren, Y.T.; Fu, Q. Optimal allocation of agricultural water resources under the background of China’s agricultural water price reform-a case study of Heilongjiang province. Appl. Math. Model.
**2021**, 97, 636–649. [Google Scholar] [CrossRef] - Wu, F.; Ma, D.; Zai, S.M.; Wu, Y.B.; Feng, X.F. Optimization design of channel cross section based on MATLAB. J. North. China Univ. Water Resour. Electr. Power
**2018**, 39, 86–90. [Google Scholar] - Xu, S.Q.; Gao, K.R.; Yue, J.; Wang, Y.C.; Qiao, H.Q.; Wang, Y.J. Based on the NSGA-Ⅱ irrigation channels of two-stage water delivery and distribution optimization scheduling. J. Northeast. Agric. Univ.
**2020**, 51, 71–78. [Google Scholar] - Wang, Q.J.; Yue, C.F.; Li, Y.Z.; Liu, X.F. Optimal allocation of water resources with two-level channel based on improved particle swarm optimization algorithm. Agric. Res. Arid. Areas
**2019**, 37, 26–33. [Google Scholar] - Singh, A. Optimal allocation of water and land resources for maximizing the farm income and minimizing the irrigation-induced environmental problems. Stoch. Environ. Res. Risk Assess.
**2017**, 31, 1147–1154. [Google Scholar] [CrossRef]

Main Canal | Stake Number | Bottom Width b (m) | Slope Ratio m (m) | Water Depth h (m) | Longitudinal Slope i | Roughness n | Design Flow Q (m^{3}/s) |
---|---|---|---|---|---|---|---|

Hengliu Main Canal | 0 + 000~2 + 800 | 4.00 | 1.50 | 2.00 | 0.000125 | 0.0225 | 4.14 |

2 + 800~4 + 960 | 4.20 | 1.50 | 1.40 | 0.000125 | 0.0225 | 1.86 |

Main Canal | Branch Canal | Length of Branch Canal (km) | Covered Irrigation Area (mu) | Canal Head Structure |
---|---|---|---|---|

Hengliu Main Canal | Beixuyang Branch Canal | 2.50 | 2700 | Beixuyang Branch canal head |

Beitouwei Branch Canal | 2.52 | 2750 | Beitouwei Branch canal head | |

Beiweicheng Branch Canal | 2.70 | 3000 | Beiweicheng Branch canal head | |

Nanheping Branch Canal | 3.99 | 3185 | Nanheping Branch canal head |

**Table 3.**Analysis of water supply and demand during the ponding period of rice for each piece of the water receiving area corresponding to each branch canal in the Hengliu Main Canal system in different year types.

Branch Canal | Control Area (mu) | Planting Proportion α | Planting Area (mu) | Crop Water Requirement (10^{4} m^{3}) | Available Water Supply (10^{4} m^{3}) | ||||
---|---|---|---|---|---|---|---|---|---|

Net Water Requirement | Water Requirement Converted to Branch Head | Gross Water Supply | Water Supply Converted to Branch Head | ||||||

p = 75% | p = 95% | p = 75% | p = 95% | ||||||

Beixuyang Branch Canal | 2700 | 0.68 | 1836 | 20.20 | 23.36 | 25.31 | 20.12 | 22.78 | 18.11 |

Beitouwei Branch Canal | 2750 | 0.70 | 1925 | 21.18 | 24.49 | 26.54 | 21.09 | 23.88 | 18.99 |

Beiweicheng Branch Canal | 3000 | 0.72 | 2160 | 23.76 | 27.48 | 29.78 | 23.67 | 26.80 | 21.30 |

Nanheping Branch Canal | 3185 | 0.70 | 2230 | 24.53 | 28.37 | 30.74 | 24.44 | 27.67 | 21.99 |

Total | 11,635 | 8151 | 89.67 | 103.7 | 112.37 | 89.32 | 101.13 | 80.39 |

**Table 4.**Optimal water quantity allocation of each branch canal in the Hengliu Main Canal system during the ponding period of rice in different year types using DP-mode.

Branch Canal | Water Requirement Converted to Branch Head (10^{4} m^{3}) | Optimal Water Quantity Allocation for Each Branch Canal (10^{4} m^{3}) | Minimal Water Shortage (10^{4} m^{3}) | Satisfaction Degree (%) | Objective Value | ||||
---|---|---|---|---|---|---|---|---|---|

p = 75% | p = 95% | p = 75% | p = 95% | p = 75% | p = 95% | p = 75% | p = 95% | ||

Beixuyang Branch Canal | 23.36 | 22.72 | 17.54 | 0.64 | 5.82 | 97.26 | 75.09 | 0.41 | 33.87 |

Beitouwei Branch Canal | 24.49 | 23.85 | 18.66 | 0.64 | 5.83 | 97.39 | 76.19 | 0.41 | 33.99 |

Beiweicheng Branch Canal | 27.48 | 26.84 | 21.65 | 0.64 | 5.83 | 97.67 | 78.78 | 0.41 | 33.99 |

Nanheping Branch Canal | 28.37 | 27.72 | 22.54 | 0.65 | 5.83 | 97.71 | 79.45 | 0.42 | 33.99 |

Total | 103.7 | 101.13 | 80.39 | 2.57 | 23.31 | 97.52 | 77.52 | 1.65 | 135.84 |

**Table 5.**Water quantity allocation of each branch canal in the Hengliu Main Canal system during the ponding period of rice in different year types using EP-mode.

Branch Canal | Water Requirement Converted to Branch Head (10^{4} m^{3}) | Optimal Water Quantity Allocation for Each Branch Canal (10^{4} m^{3}) | Minimal Water Shortage (10^{4} m^{3}) | Satisfaction Degree (%) | Objective Value | ||||
---|---|---|---|---|---|---|---|---|---|

p = 75% | p = 95% | p = 75% | p = 95% | p = 75% | p = 95% | p = 75% | p = 95% | ||

Beixuyang Branch Canal | 23.36 | 22.78 | 18.11 | 0.58 | 5.25 | 97.52 | 77.52 | 0.34 | 27.58 |

Beitouwei Branch Canal | 24.49 | 23.88 | 18.98 | 0.61 | 5.51 | 97.52 | 77.52 | 0.37 | 30.31 |

Beiweicheng Branch Canal | 27.48 | 26.80 | 21.30 | 0.68 | 6.18 | 97.52 | 77.52 | 0.46 | 38.16 |

Nanheping Branch Canal | 28.37 | 27.67 | 21.99 | 0.70 | 6.38 | 97.52 | 77.52 | 0.50 | 40.67 |

Total | 103.7 | 101.13 | 80.39 | 2.57 | 23.31 | 97.52 | 77.52 | 1.66 | 136.72 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gong, Y.; Zou, W.; Yuan, X.; Yang, X.; Chen, Y. Study on Water Quantity Allocation Optimization for Single Main Canal in Large-Scale Irrigation Area Based on DP Method. *Water* **2022**, *14*, 3917.
https://doi.org/10.3390/w14233917

**AMA Style**

Gong Y, Zou W, Yuan X, Yang X, Chen Y. Study on Water Quantity Allocation Optimization for Single Main Canal in Large-Scale Irrigation Area Based on DP Method. *Water*. 2022; 14(23):3917.
https://doi.org/10.3390/w14233917

**Chicago/Turabian Style**

Gong, Yi, Wenhao Zou, Xiuwei Yuan, Xiaoling Yang, and Yongfeng Chen. 2022. "Study on Water Quantity Allocation Optimization for Single Main Canal in Large-Scale Irrigation Area Based on DP Method" *Water* 14, no. 23: 3917.
https://doi.org/10.3390/w14233917