# Optimal Irrigation Water Allocation Using a Genetic Algorithm under Various Weather Conditions

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{6 }m

^{3}is designed to supply water to irrigate 114,500 ha. The areas currently irrigated with this water under modern or traditional irrigation and drainage networks are 60,000 ha in Ramjrd and Marvdasht, 7000 ha from Pol-e-Khan to Band-e-Amir and 47,500 ha from Band-e-Amir to Korbal. The dam also provides drinking water to the cities of Marvdasht and Shiraz. The average inflow of this dam between 1987 and 2009 was 1141 × 10

^{6}m

^{3}, with a maximum of 2509 × 10

^{6}m

^{3}and a minimum of 254 × 10

^{6}m

^{3}[6].

Growth Stage | Parameter | Crop | ||||
---|---|---|---|---|---|---|

Wheat | Corn | Sugar Beet | Barley | Canola* | ||

Establishment | K_{y }**^{,}^{a} | 0.01 | 0.07 | 0.12 | 0 | - |

Early vegetation | 0.2 | 0.4 | 2 | 0.12 | - | |

Late Vegetation | 0.2 | 0.4 | 2 | 0.15 | - | |

Flowering | 0.6 | 1.5 | - | 1.5 | - | |

Yield formation | 0.5 | 0.5 | 0.36 | 0.4 | - | |

Ripening | 0.01 | 0.2 | 0.12 | 0.14 | - | |

Gross margin (10^{6}Rls/ha)^{b} | 17 | 27 | 36 | 15 | 41 | |

Production cost (10^{6}Rls/ha)^{b} | 4 | 12 | 14 | 3 | 7 | |

Planting day ^{b} | 21 October | 22 June | 21 March | 21 October | 12 October | |

Length of growing season(days)^{b} | 220 | 110 | 200 | 210 | 240 |

**is the yield response factor which quantifies the response of yield to water supply. It relates relative yield decrease to relative ET deficit for different growing stages;**

_{y}^{a }[14],

^{b }[15].

^{6 }m

^{3}and the dead capacity 300 × 10

^{6}m

^{3}. Although the proposed optimization model can handle heterogeneous soils, the soil under study was considered to be homogeneous with a field capacity (FC) of 0.3 cm

^{3}/cm

^{3}and a permanent wilting point (PWP) of 0.15 cm

^{3}/cm

^{3}.

#### 2.2. Stochastic Generation of Climatic Data

Weather Condition | Probability Level of Exceedance | ||
---|---|---|---|

Evapotranspiration (%) | Rainfall (%) | Inflow (%) | |

Hot and dry | 20 | 100 (no rain) | 100 |

Dry | 40 | 80 | 80 |

Normal | 50 | 50 | 50 |

Wet | 60 | 20 | 20 |

#### 2.3. Model Formulation

#### 2.3.1. Objective Function

_{i}is the price of crop i (Rls kg

^{−1}); R

_{ij}is the reservoir release for crop i during time interval j; C

_{i}is the production cost of crop i (Rls ha

^{−1}); A

_{i}is the cultivated area of crop i (ha); Y

_{a}is the actual yield (kg ha

^{−1}); Y

_{m}is maximum crop yield under the given management conditions with unlimited water supply (kg ha

^{−1}); ET

_{a}is the actual evapotranspiration (mm); ET

_{m}is the maximum evapotranspiration (mm), which is the product of a crop factor K

_{c }and the reference evapotranspiration; the K

_{c}for three growing stages (initial, middle, end) were extracted from Shabani [14]; λ

_{i,}

_{j}is the sensitivity index of crop i to water stress during time interval j.

_{y}) [16]:

_{y}

^{3}− 0.1768k

_{y}

^{2}+ 0.9464k

_{y}− 0.0177

_{y}for all growing stages of the crops was obtained in Table 1.

_{a }= ET

_{m}, Y

_{a}= Y

_{m}and the relative yield (Y

_{a}/Y

_{m}) is fixed to unity. The Equation (1) may be reduced to [10]:

#### 2.3.2. Constraints

#### Reservoir Constraint

_{j}is the reservoir storage at the beginning of time interval j, (m

^{3}); Q

_{j}is the reservoir inflow during time interval j (m

^{3}); E

_{j}is the reservoir surface area evaporation rate (mm) during time interval j. This is computed from the de Bruin equation [22], further f(S

_{j}) is the reservoir surface area in time interval j (m

^{2}), which is related to the storage of the reservoir and SP

_{j}is the overflow loss from the spillway during time interval j (m

^{3}). Like in [10] the rainfall on the reservoir area is considered as negligible and therefore is not included in the model.

_{max}) and a lower limit (dead storage, S

_{min}). Thus, it is possible to remove the overflow variable SP

_{j}from Equation (4) and rewrite it as a state Equation (5), where the reservoir storage is the state:

#### Soil Moisture Constraint

_{in})

_{i.j + 1}= (SM

_{in})

_{i.j}+ ERAIN

_{i.j}+ IR

_{i.j}− (ET

_{a})

_{i.j}+ (ASM

_{i.j + 1}− ASM

_{i.j}) − DP

_{ij}

_{in}is the initial soil moisture level (mm); ERAIN

_{ij}is the effective rainfall for crop i in time interval j (mm); which is computed by the procedure described by e.g., [24,25]; IR

_{ij}is irrigation water allocated to crop i in time interval j (mm); ASM

_{i.j}is the available soil moisture (mm) for crop i in time interval j and DP

_{ij}is deep percolation for crop i in time interval j (mm).

_{i.j}in Equation (6)), a sine function was adopted for assessing time patterns of root growth [26]. At the beginning of every time interval, any water added (ERAIN

_{i.j}and IR

_{i.j}) is computed as if it was done instantaneously.

_{ij}≥ IR

_{i.j}(1 − AE)

#### Crop Irrigation Requirements and Reservoir Releases

_{max})

_{i,j}of crop i during a given time interval j depends on the initial soil moisture level, the effective rainfall, the potential evapotranspiration and the remaining moisture. It can therefore be given by [27]:

_{max})

_{i,j}= max {0,(1 − p

_{ij})ASM

_{i.j}+ (ET

_{m})

_{i.j}− (SM

_{ in})

_{i.j}− ERAIN

_{ i.j}}

_{max }is the maximum irrigation requirement (mm); p

_{ij}is the soil moisture depletion factor under no stress condition for crop i in time interval j. This according to Allen et al. [18] depends on the specific evapotranspiration of the crop and the maximum evapotranspiration (ET

_{m}).

_{max}is the maximum release from reservoir to meet the irrigation requirements, ME is the mean irrigation efficiency, including the application efficiency and the conveyance efficiency.

#### ET_{a} Constraint

#### Bounds

_{a}/Y

_{m}) for all crops is therefore set at 0.70.

#### 2.4. Solution Technique

_{i}is the fitness value; F(x) is the objective function value, k is the number of constraints, € is −1 for maximization and +1 for minimization; δ

_{j}is the penalty coefficient; and φ

_{j}is the amount of violation. The GA has been implemented in the MATLAB language [30]. As explained in the Genetic Algorithm and Direct Search users’ guide of the MATLAB software, population size needs to be at least twice the number of variables for enough searches in the variables’ space, so that the individuals in each population span the space being searched. In our case, the number of variables is 145. Therefore, the population size is 300. The mutation probability considered to vary adaptively from 0.2 to a small value when we are near to optimal solutions. It helps us to explore the search space in the beginning of optimization and to prevent missing the near optimized solutions at the end. Different settings for Crossover in genetic algorithm can be used to see which one gives the best results. The implemented coding runs the function GA multi times; varying crossover from zero to one in increments of 0.05. A schematic diagram is given in Figure 2.

## 3. Results and Discussion

#### 3.1. Results from the Stochastic Generation

**Figure 3.**(

**A**) means of rainfall; (

**B**) standard deviation of rainfall; (

**C**) means of evapotranspiration; (

**D**) standard deviation of evapotranspiration; (

**E**) means of inflow; (

**F**) standard deviation of inflow.

**Figure 4.**(

**A**) Ten day rainfall levels; (

**B**) Ten day evapotranspiration levels; (

**C**) Ten day inflow levels.

#### 3.2. Results from the Genetic Algorithm (GA)

#### Performance of the Genetic Algorithm

_{min}) and the minimum desired relative yield (Y

_{a}/Y

_{m}).

**Table 3.**Relative yield (Y

_{a}/Y

_{m}(of crops under study corresponding to various weather conditions.

Crop | Relative Yield (Ya/Ym) | ||
---|---|---|---|

Wet | Normal | Dry | |

Wheat | 1 | 1 | 0.98 |

Barley | 1 | 1 | 1 |

Corn | 1 | 1 | 1 |

Sugar beet | 0.98 | 0.91 | 0.78 |

Canola | 1 | 0.88 | 0.7 |

**Table 4.**Optimum water allocation from reservoir, optimum cultivation area of different crops and total income under various weather conditions.

Crop | Wet | Normal | Dry | ||||
---|---|---|---|---|---|---|---|

Water (mm) | Area (ha) | Water (mm) | Area (ha) | Water (mm) | Area (ha) | ||

Wheat | 399 | 73,916 | 381 | 74,776 | 412 | 56,308 | |

Corn | 626 | 19,090 | 696 | 15,466 | 721 | 875 | |

Sugar beet | 890 | 2,992 | 951 | 2477 | 858 | 98 | |

Barley | 447 | 15,412 | 346 | 15,584 | 567 | 5,004 | |

Canola | 544 | 2,183 | 462 | 1,731 | 620 | 642 | |

TOTAL | Deficit irrigation | 2,906 | 113,594 | 2,836 | 110,034 | 3,178 | 62,927 |

Full irrigation | - | 111,952 | - | 107,157 | - | - | |

Total income (10^{6}Rls) | Deficit irrigation | 1,621,097 | 1,541,267 | 847,417 | |||

Full irrigation | 1,584,404 | 1,522,755 | - |

^{6}Rls) under wet weather condition to 847,417(10

^{6 }Rls) under dry weather condition (Table 4).

^{6 }Rls) and total cropped areas increased from 111,952 to 113,594 (ha). The total farm income thus increased by extending the total cropped area, and allowing deficit irrigation for some crops.

## 4. Conclusions

## Appendix

_{DD}= conditional probability that a dry day is followed by a dry day; P

_{DW}= conditional probability that a wet day is followed by a dry day; P

_{WD}= conditional probability that a dry day is followed by a wet day; P

_{WW}= conditional probability that a wet day is followed by a wet day. However, by definition, P

_{WD}= 1 − P

_{DD}and P

_{DW}= 1 − P

_{WW}and thus, only two probabilities need be calculated from historical data, the other two being calculated from these:

_{WD(i) }= the probability that if day i is dry, then day i + 1 will be wet. These probabilities, in combination with a random number generator from uniform distribution, are then used to generate series of wet and dry days. Given the state of the preceding day (W or D), a random number of uniform distribution is generated and compared with the appropriate probability (P

_{WD}if preceding day D and P

_{WW}if preceding day W). If the number generated is greater than the probability, then the day is recorded as wet, otherwise, it is dry. The process is continued until the end of the year, the last day of one year becoming the preceding day for the start of the next. Given that a day is wet, rainfall amounts are calculated by sampling from the gamma distribution [10].

## A1. Monte Carlo Simulation

## A2. ARMAX Model

_{t}= ω ( B ) x

_{t}+ θ ( B ) e

_{t}

_{t}= inflow time series; x

_{t}= precipitation time series; e

_{t}= normally independently distributed white noise residual with mean zero and variance σ

^{2}; φ(B) = 1 + φ

_{1}B − φ

_{2}B

^{2}− .... − φ

_{p}B

^{p}non seasonal autoregressive (AR) operator of order p; θ(B) = 1 − θ

_{1}B − θ

_{2}B

^{2}− .... − θ

_{q}B

^{P}non seasonal moving average (MA) operator of order q and ω(B) = ω

_{0}− ω

_{1}B − ω

_{2}B

^{2}− ... − ω

_{r}B

^{r}operator of order r in the numerator of the transfer r function; B = backward shift operator defined by By

_{t}= y

_{t}

_{−1}.

## Conflicts of Interest

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**MDPI and ACS Style**

Sadati, S.K.; Speelman, S.; Sabouhi, M.; Gitizadeh, M.; Ghahraman, B.
Optimal Irrigation Water Allocation Using a Genetic Algorithm under Various Weather Conditions. *Water* **2014**, *6*, 3068-3084.
https://doi.org/10.3390/w6103068

**AMA Style**

Sadati SK, Speelman S, Sabouhi M, Gitizadeh M, Ghahraman B.
Optimal Irrigation Water Allocation Using a Genetic Algorithm under Various Weather Conditions. *Water*. 2014; 6(10):3068-3084.
https://doi.org/10.3390/w6103068

**Chicago/Turabian Style**

Sadati, Somayeh Khanjari, Stijn Speelman, Mahmood Sabouhi, Mohsen Gitizadeh, and Bijan Ghahraman.
2014. "Optimal Irrigation Water Allocation Using a Genetic Algorithm under Various Weather Conditions" *Water* 6, no. 10: 3068-3084.
https://doi.org/10.3390/w6103068