# Irrigation Canal System Delivery Scheduling Based on a Particle Swarm Optimization Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model of Canal System Operation Scheduling Optimization

_{l}is the total amount of water loss (m

^{3}) of the main canal W

_{ml}and the branch canal W

_{bl}in the entire irrigation process; q

_{m}and q

_{j}are the discharges (m

^{3}/s) of the main and branch canals, respectively; A

_{m}and A

_{bj}are water permeability coefficients and m

_{m}and m

_{bj}are indexes of the main and branch canals, respectively; and l

_{m}and l

_{bj}are the valid water delivery lengths (km) and t

_{m}and t

_{bj}are water delivery times (s) of the main and branch canals, respectively.

^{3}/s/km); A and m are the water permeability coefficient and index of main or branch canals, respectively, which depend on the soil of the canal bed, canal lining, depth of groundwater, and other factors; q is the discharge (m

^{3}/s); l is the valid length of canals (km); and t is the duration of water delivery (s).

_{j}) is α times its designed discharge, where α ranges from 0.6 to 1.0.

_{js}) and end time (t

_{je}) of water delivery in the branch canal j must be within the delivery period (T).

_{j}is the duration of water delivery of the branch canal j and T is the whole water delivery time.

_{j}is the water requirement amount of the branch canal j.

_{mi}is the delivery discharge of the main canal at time interval i; i is the time interval number; j is the branch canal number; N is the number of branch canals. x

_{ij}is defined by Equation (13): when the jth branch canal delivers water in the ith time interval, x

_{ij}= 1; and otherwise, x

_{ij}= 0. t

_{i}is the time corresponding to the ith interval.

_{md}is the designed discharge of the main canal.

#### 2.2. Model Solution Algorithm Based on PSO

#### 2.2.1. The PSO Algorithm

_{i}, has two features: position X

_{i}(decision variables) and velocity V

_{i}. Each particle updates itself during the iteration by tracking two extremes: the current optimal position found by the particle (P

_{best}) and the current best position found by the global population (G

_{best}). The conditions for terminating the loop are either reaching the maximum allowed generation or achieving a designated fitness level. The PSO can be updated by:

_{1}and c

_{2}are learning factors; and r

_{1}and r

_{2}are random numbers uniformly distributed in [0, 1].

#### 2.2.2. Coding Design

_{m}, L

_{b}), designed discharge (q

_{md}, q

_{bd}), and leakage loss parameters (A, m) of the main and branch canals have been determined. The constraints include specific water delivery requirements for each branch canal. The problem of optimization of delivery scheduling for irrigation canal water systems is to determine the decision variables, namely start time, water delivery duration, end time, and delivery discharge of each branch canal.

_{j}, the ratio of the water delivery discharge of the branch canal to its designed discharge, and t

_{je}, the end time of water delivery, as decision variables for coding. The duration of water delivery of branch canals j (t

_{j}) can be obtained by dividing its demanded water amount by the water delivery discharge, and then t

_{js}can be deduced from Equation (10). Optimization of α

_{j}can be used to satisfy the constraints of Equations (6) and (7). Optimization of t

_{je}can satisfy the constraints of Equations (9) and (10). Using the aforementioned methods to satisfy the constraints can greatly improve the probability, as well as the efficiency, of finding feasible solutions.

#### 2.2.3. Fitness Function Design

_{fit}is the fitness function; W

_{ml}is the amount of main canal water loss (m

^{3}); W

_{bl}is the amount of branch canal water loss (m

^{3}); W is the total amount of water delivery (m

^{3}) within the whole irrigation time T; and p

_{i}is the penalty function.

_{i}reflects Equation (15), which means the delivery discharge of the main canal must be as close as possible to its designed discharge at any time. To guarantee unified dimensions with p

_{i}, the part ${({q}_{md}-{\displaystyle \sum _{j=1}^{N}{q}_{j}}\cdot {x}_{ij})}^{2}/{q}_{md}$ are designed to reflect Equation (18).

## 3. Applications

#### 3.1. Project Cases

^{3}/s; it delivers water to 24 lateral canals that can be regarded as second-level canals with designed discharge values ranging from 0.03 to 0.18 m

^{3}/s. The soil texture in the canal beds is silt loam; the canals are lined with concrete; and the permeability parameters A and m are 0.86 and 0.4, respectively. The length and discharge values of different canals are listed in Table 1. Data regarding the practical process of water delivery in one irrigation period in the spring of 2015 were collected and are presented in Table 1 and Figure 3(a1,a2).

^{3}/s. It delivers water to 14 branch or lateral canals that can be regarded as second-level canals with designed discharge values ranging from 0.15 to 1.04 m

^{3}/s. The soil texture in the canal beds is sandy loam; the canals are lined with concrete; and the permeability parameters A and m are 1.19 and 0.45, respectively. The length and discharge values of different canals are listed in Table 1. Data regarding the practical process of water delivery in one irrigation period in the spring of 2015 were collected and are presented in Table 1 and Figure 3(a3,a4).

#### 3.2. Results from the PSO Algorithm and Comparison

_{1}and c

_{2}were both 2.0. The approximate operation time for one iteration was 20 s on an Intel

^{®}core i5 CPU. To reflect actual water delivery needs, the water delivery interval was set to 12 h for the Fengjiashan irrigation system and was set to 6 h for the Shitouhe irrigation system. The optimization results and the actual delivery processes are displayed in Table 2 and Figure 3 (for brevity, Figure 3 reveals the actual and typical optimization and delivery schedule).

_{max}) was reduced from 33 to 22 in the first system and from 38 to 24 in the second system; the water delivery time was reduced from 396 to 264 h and from 228 to 144 h; and the PSO-planned leakage losses relative to the canal water requirements were close to 5.40% and 7.46% for the two systems. By contrast, the conventional plan had losses of 7.29% and 8.97% of actual delivered water (see row “R

_{1}” in Table 2). This proved that the optimal delivery schedule can improve the water efficiency of canal irrigation systems.

_{3}) were smaller than 1. Ratios in individual delivery intervals (n

_{ex}) were seldom larger than 1; the largest ratio value was 1.079, which is acceptable for the main canal capacity and safe for canal water delivery [43]. The mean ratio values of the main canal delivery discharge to the designed discharge at all irrigation times (R

_{2}) ranged from 0.833 to 0.952, which means the delivery discharge was close to capacity.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Comparisons of water delivery through irrigation canal systems for traditional and optimized schedules. Note: (

**a1**,

**a2**,

**b1**,

**b2**) represents the irrigation scheduling of branch canal No. 11 of the north canal of the Fengjiashan irrigation district; (

**a3**,

**a4**,

**b3**,

**b4**) represents the irrigation scheduling of the west main canal in the Shitouhe river irrigation district. In (

**a2**,

**b2**), from bottom to top, each rectangle successively represents the water delivery discharge and duration from the second-level canal numbered 1 to 24; (

**a4**,

**b4**) are numbered 1 to 14; the height of each rectangle represents the value of discharge and the length represents the duration of water delivery; the scale of the vertical axis is 0.2 m

^{3}/s in (

**a2**,

**b2**) and 0.4 m

^{3}/s in (

**a4**,

**b4**).

j | q_{b} | L_{m} | L_{j} | W_{j} | j | q_{b} | L_{m} | L_{j} | W_{j} |
---|---|---|---|---|---|---|---|---|---|

(m^{3}/s) | (km) | (km) | 10^{3} (m^{3}) | (m^{3}/s) | (km) | (km) | 10^{3} (m^{3}) | ||

The system of branch canal No. 11 of north canal in Fengjiashan irrigation district | |||||||||

1 | 0.06 | 0.17 | 0.68 | 28.64 | 13 | 0.05 | 3.24 | 0.54 | 28.60 |

2 | 0.03 | 0.23 | 0.65 | 9.38 | 14 | 0.16 | 3.96 | 2.75 | 115.55 |

3 | 0.18 | 0.75 | 0.71 | 57.53 | 15 | 0.03 | 3.96 | 0.61 | 12.09 |

4 | 0.15 | 0.75 | 0.87 | 33.68 | 16 | 0.16 | 4.62 | 2.53 | 126.91 |

5 | 0.05 | 1.38 | 0.40 | 17.78 | 17 | 0.3 | 5.37 | 3.20 | 174.80 |

6 | 0.15 | 1.76 | 1.13 | 53.92 | 18 | 0.12 | 5.91 | 1.61 | 40.79 |

7 | 0.08 | 1.76 | 0.64 | 15.80 | 19 | 0.17 | 6.63 | 2.63 | 66.41 |

8 | 0.12 | 2.33 | 1.19 | 52.24 | 20 | 0.05 | 7.34 | 1.09 | 26.96 |

9 | 0.04 | 2.33 | 1.11 | 30.76 | 21 | 0.03 | 7.37 | 0.26 | 3.46 |

10 | 0.08 | 2.84 | 0.96 | 17.28 | 22 | 0.03 | 2.01 | 0.15 | 13.33 |

11 | 0.05 | 2.84 | 0.60 | 17.18 | 23 | 0.08 | 5.91 | 0.43 | 16.89 |

12 | 0.12 | 3.24 | 2.39 | 73.58 | 24 | 0.03 | 8.25 | 0.18 | 3.16 |

The system of west main canal in Shitouhe river irrigation district | |||||||||

1 | 0.20 | 0.224 | 1.219 | 28.64 | 8 | 1.04 | 3.541 | 1.978 | 28.60 |

2 | 0.15 | 0.224 | 0.684 | 9.38 | 9 | 0.26 | 4.455 | 1.263 | 115.55 |

3 | 0.35 | 0.903 | 1.351 | 57.53 | 10 | 0.28 | 4.455 | 1.542 | 12.09 |

4 | 0.30 | 0.903 | 0.567 | 33.68 | 11 | 0.25 | 5.213 | 1.478 | 126.91 |

5 | 0.40 | 1.548 | 1.781 | 17.78 | 12 | 0.22 | 5.213 | 1.335 | 174.80 |

6 | 0.20 | 2.405 | 1.768 | 53.92 | 13 | 0.85 | 5.861 | 3.662 | 40.79 |

7 | 0.20 | 2.92 | 1.936 | 73.58 | 14 | 1.00 | 5.861 | 6.831 | 3.16 |

_{b}is the designed discharge; L

_{m}is the distance between the second-level canal j and the headwork; L

_{j}is the length of the second-level canal j; W

_{j}is the amount of required water.

**Table 2.**Water delivery scheduling results of 15 repeated optimizations compared with the traditional schedules.

No. of Computation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Average Value | Actual Process |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

The system of branch canal No. 11 of north canal in Fengjiashan irrigation district | |||||||||||||||||

T_{max} | 21 | 21 | 22 | 23 | 22 | 23 | 24 | 22 | 21 | 22 | 24 | 21 | 21 | 21 | 24 | 22 | 33 |

R_{1} (%) | 5.41 | 5.19 | 5.12 | 5.54 | 5.12 | 5.10 | 5.76 | 5.53 | 5.00 | 5.59 | 5.07 | 5.72 | 5.65 | 5.45 | 5.76 | 5.40 | 7.29 |

α | 0.930 | 0.948 | 0.957 | 0.900 | 0.952 | 0.969 | 0.908 | 0.959 | 0.978 | 1.000 | 0.953 | 0.908 | 0.992 | 0.878 | 0.932 | 0.944 | 0.595 |

R_{2} | 0.909 | 0.952 | 0.869 | 0.869 | 0.909 | 0.909 | 0.869 | 0.909 | 0.909 | 0.869 | 0.869 | 0.909 | 0.909 | 0.909 | 0.833 | 0.893 | 0.618 |

n_{ex} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | -- | 0 |

R_{3} | 0.993 | 0.991 | 0.978 | 0.965 | 0.983 | 0.966 | 0.944 | 0.978 | 1.008 | 0.977 | 0.927 | 1.079 | 0.988 | 0.999 | 0.938 | 0.981 | 0.997 |

The system of west main canal in Shitouhe river irrigation district | |||||||||||||||||

T_{max} | 25 | 24 | 24 | 25 | 25 | 25 | 24 | 24 | 24 | 24 | 24 | 25 | 24 | 23 | 24 | 24 | 38 |

R_{1} (%) | 7.34 | 7.18 | 7.34 | 7.21 | 7.13 | 7.56 | 7.20 | 7.59 | 7.22 | 7.64 | 8.06 | 7.84 | 8.11 | 7.17 | 7.25 | 7.46 | 8.97 |

α | 0.789 | 0.814 | 0.827 | 0.926 | 0.902 | 0.747 | 0.879 | 0.703 | 0.793 | 0.755 | 0.647 | 0.709 | 0.630 | 0.948 | 0.824 | 0.793 | 0.445 |

R_{2} | 0.843 | 0.878 | 0.878 | 0.843 | 0.843 | 0.843 | 0.878 | 0.878 | 0.878 | 0.878 | 0.878 | 0.843 | 0.878 | 0.916 | 0.878 | 0.869 | 0.555 |

n_{ex} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | -- | 0 |

R_{3} | 0.924 | 0.978 | 0.963 | 0.984 | 0.982 | 0.956 | 0.945 | 0.950 | 0.925 | 1.036 | 0.957 | 0.921 | 0.956 | 1.032 | 0.955 | 0.964 | 0.934 |

_{max}is the maximum water delivery interval; R

_{1}is the ratio of leakage loss to the water requirement in the whole canal system (%); α is the percent of the average value of the second-level canal delivery discharge to its designed discharge; R

_{2}is the ratio of the first-level canal delivery discharge to its designed discharge in all irrigation periods; n

_{ex}is the number of time intervals when the delivery discharge of the first-level canal exceeds its designed discharge in all irrigation periods; R

_{3}is the ratio of the maximum delivery discharge of the first-level canal to its designed discharge in all irrigation periods; Considering the actual water delivery needs, 12 h was considered as the water delivery interval for the Fengjiashan irrigation system and 6 h for the Shitouhe irrigation system.

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**MDPI and ACS Style**

Liu, Y.; Yang, T.; Zhao, R.-H.; Li, Y.-B.; Zhao, W.-J.; Ma, X.-Y.
Irrigation Canal System Delivery Scheduling Based on a Particle Swarm Optimization Algorithm. *Water* **2018**, *10*, 1281.
https://doi.org/10.3390/w10091281

**AMA Style**

Liu Y, Yang T, Zhao R-H, Li Y-B, Zhao W-J, Ma X-Y.
Irrigation Canal System Delivery Scheduling Based on a Particle Swarm Optimization Algorithm. *Water*. 2018; 10(9):1281.
https://doi.org/10.3390/w10091281

**Chicago/Turabian Style**

Liu, Ye, Ting Yang, Rong-Heng Zhao, Yi-Bo Li, Wen-Ju Zhao, and Xiao-Yi Ma.
2018. "Irrigation Canal System Delivery Scheduling Based on a Particle Swarm Optimization Algorithm" *Water* 10, no. 9: 1281.
https://doi.org/10.3390/w10091281