Optimizing regional irrigation water allocation for multi-stage pumping-water irrigation system based on multi-level optimization-coordination model

Journal of Hydrology X 4 (2019) 100038

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Optimizing regional irrigation water allocation for multi-stage pumpingwater irrigation system based on multi-level optimization-coordination model Yao Jianga,d, Lvyang Xiongb, Fuqi Yaoc, Zongxue Xua,d,

T

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a

College of Water Sciences, Beijing Normal University, Beijing 100875, PR China College of Water Resources and Civil Engineering, China Agricultural University, Beijing 100083, PR China c Agricultural Water Conservancy Department, Changjiang River Scientific Research Institute, Wuhan 430015, PR China d Beijing Key Laboratory of Urban Hydrological Cycle and Sponge City Technology, Beijing 100875, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Irrigation water allocation Optimization Multi-stage pumping stations Decomposition-coordination method Multi-objective

The optimal allocation and utilization of irrigation water are particularly significant but complex in multi-stage pumping-water irrigation systems which involve both water distribution processes and machine operation management. This paper proposes a regional optimization-coordination model for optimizing irrigation water allocation in complex multi-stage pumping-water irrigation system, which is developed according to actual water use problems in Guhai pumping-water irrigation system (GPIS) located in Ningxia Hui Autonomous Region, northwest China. The model is a three-level hieratical structure with minimum water deficiency and energy cost as two objectives for better describing the coordination and constraint between multiple levels of irrigation planning and decision making. The three levels deal with the optimal schemes of pumping and allocating water in each single pumping-irrigation subdistrict (PIS) (first level), among multi-stage pumping-irrigation subdistrict of each subsystem (second level) and among different subsystems (third level) respectively. The model is solved by using decomposition-coordination method and each level is connected and iterated through the exchange of coordinating variables. The model is applied and tested in the optimizations of pumping and allocating water for each PIS in each period in GPIS. Results show that the range of water deficiency among different PIS is significantly reduced and pumping-water amount for the lower-stage PIS is particularly increased under different conditions of total water delivery amount. Spatial-temporal contradictions of water supply among multi-stage pumping stations could be effectively alleviated through optimizations. The application indicates that the optimization-coordination model is practical and effective for decision-making of irrigation water management in the complex multi-stage pumping-water irrigation system, and could be a reference for other similar irrigation systems worldwide.

1. Introduction The optimization of irrigation system is an effective measure to realize reasonable, high-efficient and sustainable use of water resources in irrigated areas (Playán and Mateos, 2006). Relevant researches have been widely conducted since 1960s worldwide. With deepening of researches and development of techniques, the connotation of water allocation optimization is increasingly abundant. The scale of researches has been extended from water allocation optimizations on field and crop scale to joint optimizations of multilevel decision-making and different water sources on regional scale (Shang and Mao, 2006; Safavi et al., 2010; Ge et al., 2013; Homayounfar et al., 2014). The research

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contents have also been enlarged from only water quantity allocation concerns to considering water quality and related eco-environmental benefits (Fleifle et al., 2014; Allam et al., 2016; Heydari et al., 2016). As for research methods, the traditional programing methods, like linear programing, non-linear programing and dynamic programing etc. (Singh, 2014), have recently been evolved to intelligent programing methods like genetic algorithm (GA), artificial neural network (ANN), ant colony optimization (ACO), particle swarm optimization (PAO) and simulated annealing algorithm (SA) (Haq et al., 2008; Brown et al., 2010; Noory et al., 2011; Singh, 2015; Nguyen et al., 2016). Optimization models are not just limited to single programing models but developed to coupling models of mathematical models, simulation

Corresponding author. E-mail addresses: [email protected] (Y. Jiang), [email protected] (L. Xiong), [email protected] (Z. Xu).

https://doi.org/10.1016/j.hydroa.2019.100038 Received 8 June 2019; Received in revised form 9 August 2019; Accepted 12 August 2019 Available online 16 August 2019 2589-9155/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Fig. 1. Location of the Guhai pumping-water irrigation system.

objective optimization models have been developed through combining with different theories or methods, such as stochastic multi-objective programming model, fuzzy multi-objective programming model and dynamic multi-objective programming model etc. (Bai et al., 2015; Alizadeh et al., 2017; Li et al., 2019b). It is an important way for the optimization of complex systems to develop multi-objective hierarchical analysis by combining large system theory and multi-objective method. Thus, the optimizations referring to different benefit-based relationships or decision-making levels in complex MPIS could be described using a multi-level multi-objective optimization-coordination model. Therefore, in this paper, a multi-level optimization-coordination model with multi-objectives is developed for optimal irrigation water allocation according to the characteristics of multi-stage pumpingwater irrigation system. Based on the decomposition-coordination method for large system, the model is constructed with a three-level hierarchical structure in order to better describe and coordinate the irrigation planning processes between different decision-making levels in the whole irrigation system. The minimum water requirement and energy cost (shaft power) are taken as two objectives and different periods are considered for detailed schedules for water allocation and operation. The model is then applied and tested to obtain optimal schemes of pumping and allocating water for different stage of pumping station systems and different irrigated areas under conditions of limited water resources. This study aims at dealing with the optimal allocation of irrigation water in complex multi-stage pumping-water irrigation systems with considering different levels of decision-making processes.

techniques and vector optimization theory etc. (Singh and Panda, 2013; Makaremi et al., 2017; Naghibi et al., 2017). Mathematical descriptions of optimization questions have begun to consider the trade-off of different aspects by using multiple objectives (e.g. economic benefits and environmental implication) rather than just single objective (Singh, 2012; Marques et al., 2015; Li et al., 2018). Meanwhile several uncertainty theories have been widely introduced into optimization models for considering the uncertainties in actual irrigation optimization problems in recent years (Li et al., 2017; Zhang et al., 2018). In these researches, the optimization in multi-stage pumping-water irrigation systems (MPIS) is a more complicated problem, which not only involve optimizations of water distribution system but also the regulations of pumping station system. In recent years, several attempts have been done to develop different optimization models to assist the operation of complex pumping-water irrigation system (Makaremi et al., 2017; Ming et al., 2017; Zhang et al., 2019). Most of researches are focused on the optimal design and operation of pumping stations related to the energy costs such as pumping head optimization, operation scheduling of pumping stations and operation efficiency improved (García et al., 2014; Moradi-Jalal et al., 2003; Napolitano et al., 2016). While water allocation optimization in the MPIS refers to both water distribution management and pumping station operation, which is usually characterized by multi-objective and multi-level structure on a regional scale. However, the mathematical sophistication of optimization models, characteristics of multi-dimension or oversimplification of irrigation network etc. restricted the model applications and acceptances (Boulos et al., 2001). As a result, it is meaningful to develop reasonable, effective and practical optimization models for regional optimization of irrigation water use in MPIS rather than overly complicate and sensitive models. The decomposition-coordination method for large system has an advantage of decomposing complex problems into the combination of relatively simple optimization problems, which reduce the dimensions of optimizations on a regional scale. For example, Shangguan et al. (2002) developed a three-level optimization model to solve optimal water allocation from canals to crops; Jiang et al. (2016) developed a regional economic optimization model with two-level structures for regional optimal allocation of irrigation water. Lu et al. (2019) developed a multi-level non-linear optimization model for groundwater remediation management with considering the leader-follower relationships among multiple levels. Meanwhile, multi-objective programing is also widely applied to deal with harmonization management between multiple aspect of economy, environment, ecology and society (Hu et al., 2016; Davijani et al., 2016; Li et al., 2019). Various multi-

2. Irrigation system description The Guhai pumping-water irrigation system (GPIS) (36°11′-37°18′N, 105°35′-106°12′E) is located in the middle arid area of the Ningxia Hui Autonomous Region, northwest China. The GPIS is 100 km long from north to south and 10 km wide from east to west, covering an acreage of approximately 738 km2 (Fig. 1). While the elevation varies from about 1205 m in the north to about 1621 m in the south. The study area has a continental arid climate characterized by average annual precipitation of only 286 mm, 70% of which occurring during July to September. While the mean annual reference evapotranspiration (ETo) reaches to about 996 mm. The growing season is generally from March to September, and the major irrigated crops include corn, spring wheat and some economic crops (e.g. Lycium barbarum) etc. While the local agriculture is mostly dependent on irrigation due to low rainfall. Two different irrigation methods are mainly applied in GPIS, which are 2

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Table 1 Detailed information of the Guhai pumping-water irrigation system. Irrigated area (hm2) Total

Total pumping head (m)

Net pumping head (m)

Designed pumping flow (m3 s−1)

Designed operation mode (actual + spare units)

47 0 0 0 357 170 553 0 355 62 724 2117

115 0 0 0 2514 2180 2545 1933 3666 795 2031 3900

47.5 56.3 20.2 33.8 48.2 40.8 31.0 38.6 61.2 30.0 36.3 35.8

44.5 48.4 16.6 30.2 43.5 36.0 27.4 33.6 35.7 27.1 33.2 31.2

12.7 12.7 12.7 12.4 12.3 11.9 10.6 10.0 8.5 8.4 5.9 3.5

5+3 5+3 5+3 5+3 5+3 5+3 4+3 4+3 1+3 4+3 3+2 4+2

130 1687 343 11,476 4207 2155 8914 4359

159 602 263 1237 812 38 1349 153

289 2289 607 12,713 5019 2192 10,262 4513

30.7 58.2 34.1 37.5 25.8 50.2 50.6 19.7

27.6 54.9 29.3 33.6 20.4 46.1 44.8 15.5

20.0 19.4 18.7 11.2 9.4 6.4 5.4 3.5

6+1 4+1 8+4 9+2 6+1 6+1 7+1 6+1

7855 7855 5311 1681

0 2657 4718 2516

0 399 701 1804

0 3056 5419 4321

49.0 43.0 43.0 43.9

43.7 34.1 34.2 39.0

8.5 6.5 4.0 1.8

6+1 8+2 6+1 4+1

40,827

58,457

11,901

70,358

Pumpingirrigation subdistrict

Approved pumping-water amount (104 m3)

Actual pumping-water amount in 2017 (104 m3)

Traditional irrigation

Highefficient irrigation

Subsystem1

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 1-10 1-11 1-12

13,251

12,316 12,250 12,250 12,250 12,250 10,251 8322 6462 5357 3143 2653 1914

68 0 0 0 2157 2010 1992 1933 3311 733 1307 1783

Subsystem2

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8

21,500

20,656 18,223 16,631 16,242 8852 6173 5200 1872

Subsystem3

3-1 3-2 3-3 3-4

10,000

44,751

Subsystem

Total

traditional irrigation (e.g. flood or border irrigation) and high-efficient irrigation (e.g. drip irrigation) respectively. The GPIS is characterized by its machine-driven irrigation due to larger elevation difference of approximately 420 m from north to south. There are 25 major pumping stations and more than 200 pumping units as well as several secondary pumping-water systems in total at present. The source of irrigation water comes from the Yellow River in the north, and is transported from north to south through multi-stage pumping stations (Fig. 1). The total designed pumping flow is 41.2 m3 s−1, and the designed capacity of delivery water is 487 million m3. The GPIS mainly consists of three subsystems, each of which is controlled by several pumping stations and main channels (Fig. 1). Subsystem 1 pumps water through 12 major pumping stations with total pumping head of 479.7 m. Subsystem 2 pumps water mainly through 8 major pumping stations with total pumping head of 382.5 m. Subsystem 3 contains 4 stages of pumping stations with total pumping head of 253.1 m. The details of each subsystem are listed in Table 1. Water delivery and allocation are complicated processes in GPIS, which not only includes the joint operation of multi-stage pumping stations, but also the allocation of water amount among different subsystems and pumping stations as well as different irrigated areas. Water allocation in the whole system depends on the available water delivery amount, crop water requirement, the operation of pumping stations and the regulation of canal systems. Therefore, the GPIS can be regarded as a large system with multilevel decision-making processes and complex structures, which could be described and solved by using decomposition-coordination method for large system.

stations, and took other secondary pumping stations as parts of water distribution. Therefore, each pumping station had only one upper-stage station and one lower-stage station. Each pumping station and its water supply subdistrict were defined as a pumping-irrigation subdistrict (PIS), and each subsystem (numbered 1 to 3) could thus be regarded as a cascade system of multi-stage pumping-irrigation subdistricts. The GPIS could be thus taken as a large system that consists of three cascade pumping-irrigation subsystems and an initial pumping station (Quanyan station). The water was delivered from the Yellow River through the initial pumping station, and then was transported to each subsystem and each PIS through multi-stage pumping and allocating, and finally to different irrigated areas. Meanwhile, two types of irrigated areas were identified in the model so as to differentiate these two different irrigation methods. The water allocation for each PIS refers to the total water amount allocated to its water supply subdistrict including traditional and high-efficient irrigated areas. The simplified network of water supply in GPIS is shown in Fig. 2. Based on the decomposition-coordination method for large system, an optimization-coordination model with a three-level hierarchical structure was developed for GPIS so as to better describe its multi-level and interrelated characteristics. Each level developed a simple optimization model with two objectives for considering the water limitation and energy cost in the processes of water allocation. The first level dealt with the optimization of water allocation within each single PIS, and thus contained 24 independent models in total. The second level dealt with the optimization of water allocation among multi-stage pumping stations, which contained three independent models corresponding to three subsystems. The third level performed the optimization as a whole for water allocation among three subsystems. These three levels were solved individually and coordinated through the information exchanges of pumping flow, water requirement and energy cost of each PIS and subsystem. The whole model was operated iteratively among three levels until the ending criterion was reached, while the optimal solutions of each level were constrained by the updated coordinated variables in each iteration. The detailed

3. Mathematic model 3.1. Model generalization For simplification, the optimization model only considered the processes of pumping and allocating water among the major pumping 3

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Fig. 2. Simplified network of water supply relationships in the Guhai pumping-water irrigation system (P refers to the pumping station and C refers to its water supply subdistrict).

Fig. 3. Structure of the three-level optimization-coordination model.

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Table 2 Definitions of symbols in the optimization-coordination model. Symbol

Unit

n j m i Z (n, j ) F (n, j ) R (n, j, i) K (n, j, i) X (n, j, i) Y (n, j, i) s (n, j, i, m)

kW – 104 m3 104 m3 104 m3 104 m3 –

q (n, j, m) H (n, j ) η (n, j, m) Rmin (n, j ) /Rmax (n, j ) Kmin (n, j ) /Kmax (n, j ) S (n, j, i) Q (n, j, i) qmax (n, j, m)

m3 s−1 m – 104 m3 104 m3 104 m3 m3 s−1 m3 s−1

mmax (n, j ) Z (n, j, m) Zmax (n, j, i) Z (n) F (n) C (n, j ) D (n, j, i) Dmin (n, j ) ω (n, j ) η (n, j ) ΔTi Qmax (n, j ) Qmin (n, j )

– kW kW kW – kW/m3 s−1 104 m3 104 m3 – – s m3 s−1 m3 s−1

Qs (n, j, i) Qsmax (n, j ) S (n, i) Q (n, i) F Z C (n) D (n, i) Qmax (n, 1) Qmax η (n) S W (n, j ) W (n) W Fmin (n, j ) /Fmax (n, j ) Zmin (n, j ) /Zmax (n, j ) Fmin (n) /Fmax (n) Zmin (n) /Zmax (n) Fmin /Fmax Zmin /Zmax

m3 s−1 m3 s−1 104 m3 m3 s−1 – kW kW/m3 s−1 104 m3 m3 s−1 m3 s−1 – m3 – – – – kW – kW – kW

Definition Number of subsystem (1, 2…N) Stage of pumping-irrigation subdistrict (1, 2…J) Number of pumping unit (1, 2… M) Irrigation period (1, 2 … I) Shaft power of pumping station in pumping-irrigation subdistrict j of subsystem n Quadratic sum of water deficiency in pumping-irrigation subdistrict j of subsystem n Total water requirement of traditional irrigated crops in pumping-irrigation subdistrict j of subsystem n in period i Total water requirement of high-efficient irrigated crops in pumping-irrigation subdistrict j of subsystem n in period i Water allocation for traditional irrigated crops in pumping-irrigation subdistrict j of subsystem n in period i Water allocation for high-efficient irrigated crops in pumping-irrigation subdistrict j of subsystem n in period i State of pumping units at pumping station in pumping-irrigation subdistrict j of subsystem in period i n, which is 0-1 variable, and 0 at startup and 1 at shutdown Pumping flow of unit m at pumping station in pumping-irrigation subdistrict j in subsystem n Net pumping head of pumping station in pumping-irrigation subdistrict j in subsystem n Operating efficiency of unit m in the operating situation at pumping station in pumping-irrigation subdistrict j of subsystem n Minimum/maximum total water requirement of traditional irrigated crops in pumping-irrigation subdistrict j of subsystem n Minimum/maximum total water requirement of high-efficient irrigated crops in pumping-irrigation subdistrict j of subsystem n Total water allocation in pumping-irrigation subdistrict j of subsystem n in period i Total pumping flow of pumping station in pumping-irrigation subdistrict j of subsystem n in period i Allowed maximum pumping flow of unit m at pumping station in pumping-irrigation subdistrict j of subsystem n Maximum number of units at pumping station in pumping-irrigation subdistrict j of subsystem n Shaft power of unit m at pumping station in pumping-irrigation subdistrict j of subsystem n Maximum shaft power of unit m at pumping station in pumping-irrigation subdistrict j of subsystem n Total energy cost of subsystem n, i.e. the sum of shaft power of all pumping stations in subsystem n Quadratic sum of total water deficiency in subsystem n Energy cost per unit of pumping flow at pumping station in pumping-irrigation subdistrict j of subsystem n Total water requirement in pumping-irrigation subdistrict j of subsystem n in period i Minimum water requirement in pumping-irrigation subdistrict j of subsystem n, which is the sum of Rmin (n, j) and Kmin (n, j) Water conveyance efficiency of main canals in pumping-irrigation subdistrict j of subsystem n Water transport efficiency of pumping station in pumping-irrigation subdistrict j of subsystem n Duration of period i Maximum pumping flow of pumping station in pumping-irrigation subdistrict j of subsystem n Minimum pumping flow of pumping station in pumping-irrigation subdistrict j of subsystem n, which is determined by minimum water requirement Flow of water delivery within pumping-irrigation subdistrict j during period i Maximum flow of water delivery within pumping-irrigation subdistrict j, which is determined by canal capacity Total water allocation for subsystem n in period i Total pumping flow of subsystem n in period i, which is equal to the pumping flow of pumping station in subdistrict 1 of subsystem n Quadratic sum of water deficiency amount in the whole system Total energy cost of the whole system Energy cost per unit of pumping flow in subsystem n Total water requirement of subsystem n in period i Maximum pumping flow of pumping station in subdistrict 1 of subsystem n Maximum pumping flow of the initial pumping station in the whole system Operating efficiency of pumping stations in subsystem n Approved total water diversion amount in the whole system Objective of the first-level model, after being transformed to a single objective Objective of the second-level model, after being transformed to single objective Objective of the third-level model, after being transformed to single objective Minimum/allowable maximum value of the objective function for minimizing water deficiency in the first level Minimum/allowable maximum value of the objective function for minimizing energy cost in the first level Minimum/allowable maximum value of the objective function for minimizing water deficiency in the second level Minimum/allowable maximum value of the objective function for minimizing energy cost in the second level Minimum/allowable maximum value of the objective function for minimizing water deficiency in the third level Minimum/allowable maximum value of the objective function for minimizing energy cost in the third level

3.2.1. First level: optimization of water allocation within each single pumping-irrigation subdistrict in each period The purpose of this level is to optimize water allocation between traditional irrigated areas and high-efficient irrigated areas as well as the regulation of pumping units in order to minimize the water deficiency amount and energy cost within each single PIS. The detailed mathematical model is as follows: Objective function: To minimize the total water deficiency amount and total shaft power of the pumping station in each PIS, i.e.

structure of the optimization-coordination model is presented in Fig. 3.

3.2. Model formulation The detailed mathematical description of the optimization-coordination model is presented in this section. To present the model, a list of notations is provided in Table 2. It should be noted that the water requirement defined for each level is gross water requirement with considering the water use efficiency of irrigation system.

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minF (n, j ) =

(1)

i=1 I

minZ (n, j ) =

Qs (n, j, i)ΔTi η (n, j ) = S (n, j, i)

∑ [(R (n, j, i) − X (n, j, i))2 + (K (n, j, i) − Y (n, j, i))2]

(2) Water amount constraints: The actual water allocation should be larger than the minimum water requirement, and less than the maximum water allocation that is determined by canal capacity, i.e.

M

∑ ∑ 9.81s(n, j, m, i) q (n, j, m) H (n, j)/η (n, j, m) i=1 m=1

(15)

(2)

I

Constraints: (1) Minimum and maximum irrigation water constraints:

∑ S (n, j, i) ≥ Dmin (n, j) i= 1

(16)

S (n, j, i) ≤ ΔTi Qs max (n, j )

(17)

I

∑ X (n, j, i) ≤ Rmax (n, j)

Rmin (n, j ) ≤

(3)

i=1

The sum of water allocation for each PIS in subsystem n should be less than the water amount allocated to this subsystem n, i.e.

I

Kmin (n, j ) ≤

∑ Y (n, j, i) ≤ Kmax (n, j)

(4)

i=1

J

∑ S (n, j, i) ≤ S (n, i)

(2) Available total water constraints: (5)

(3) Pumping flow constraints: The pumping flow of each station should be larger than the minimum water flow determined by minimum water requirement, and lower than the maximum water flow determined by canal capacity, i.e.

(3) Pumping flow constraints: The total pumping flow of all operating units should not be lower than the actual pumping flow of this station, i.e.

Qmin (n, j ) ≤ Q (n, j, i) ≤ Qmax (n, j )

M

∑ s (n, j, m, i) q (n, j, m) ≥ Q (n, j, i)

The pumping flow of each operating unit should not be larger than its maximum capacity of pumping water, i.e.

q (n, j, m) ≤ qmax (n, j, m)

Q (n, 1, i) = Q (n, i)

3.2.3. Third level: optimization of water allocation among subsystems in each period This level is to optimize water allocation among different subsystems with limited water diversion amount. The objective is to minimize water deficiency amount and energy cost (total shaft power) of the whole system. The objective functions are as follows:

M

∑ s (n, j, m, i) ≤ mmax (n, j)

(8)

m=1

N

(5) Power constraints: The actual shaft power of pumping station should not be larger than its allowed maximum shaft power, i.e.

minF =

Z (n, j, m) ≤ Zmax (n, j, m)

minZ =

3.2.2. Second level: optimization of water allocation among multi-stage pumping-irrigation subdistricts in each subsystem in each period The model of this level was formulated for optimizing water allocation among different stages of PIS in each subsystem. The objective is to minimize the total water deficiency amount and total shaft power of each subsystem. The objective functions are formulated as follows:

j=1

J

minZ (n) =

J

n= 1

i))2

(24)

(2) Water amount constraints: The water allocation for subsystem n should be larger than the minimum water requirement of this subsystem n, i.e.

(12)

∑ S (n, i) ≥ ∑ Dmin (n, j)

I

J

i= 1

j=1

(25)

The sum of water allocation of all subsystem should be less than the approved total water diversion amount in the whole system, i.e.

(13)

Constraints: (1) Water balance constraints: The pumping flow for a pumping station is equal to the sum of the diverting flow to its water supply subdistrict and the pumping flow of its following station, i.e.

ω (n, j ) Q (n, j, i) − Qs (n, j, i) = Q (n, j + 1, i)

(23)

(11)

where

D (n, j, i) = R (n, j, i) + K (n, j, i)

(22)

N

∑ Q (n, i) ≤ Qmax

I

j=1 i=1

n=1 i=1

The total pumping flow of all subsystems should be lower than the maximum pumping flow of the initial pumping station, i.e.

∑ Z (n, j) = ∑ ∑ C (n, j) Q (n, j, i) H (n, j) j=1

I

Q (n, i) ≤ Qmax (n, 1)

I

j=1 i=1

(21)

Constraints: (1) Pumping flow constraints: The pumping flow of subsystem n should be lower than the allowed maximum pumping flow of first-stage station in this subsystem n, i.e.

(10)

where 9.81 is the acceleration of gravity (m s−2).

∑ F (n, j) = ∑ ∑ (D (n, j, i) − S (n, j,

N

∑ Z (n) = ∑ ∑ C (n) × Q (n, i) n=1

Z (n, j, m) = 9.81q (n, j, m) H (n, j )/ η (n, j, m)

I

n=1 i=1

N

(9)

J

N

∑ F (n) = ∑ ∑ (D (n, i) − S (n, i))2 n=1

where Z (n, j, m) is formulated as follow:

minF (n) =

(20)

(7)

(4) Operating units number constraints: The number of operating units should not be larger than the maximum number of this pumping station, i.e.

J

(19)

The pumping flow of the first-stage station of subsystem n should be equal to the total pumping flow of this subsystem n, i.e.

(6)

m=1

0<

(18)

j=1

X (n, j, i) + Y (n, j, i) ≤ S (n, j, i)

N

I

∑ ∑ S (n, i)/η (n) ≤ S n=1 i=1

(26)

where Qmax is related to canal capacity and river water level, and can be determined based on the designed flow of main canals in the third-level model in this paper.

(14) 6

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3.3. Model solution The three-level optimization-coordination model was solved based on the decomposition-coordination method for large system. The multiobjective model in each level was transformed to a single-objective function using the minimum deviation method, which is a modified method based on global criterion method (Proos et al., 2001; Marler and Arora, 2004) and the idea of relative deviation. The mathematical formulations of multiple objectives in the three levels could thus be respectively described as follows:

The first level: W (n, j ) F (n, j ) − Fmin (n, j ) Z (n, j ) − Zmin (n, j ) + = Fmax (n, j ) − Fmin (n, j ) Zmax (n, j ) − Zmin (n, j )

The second level:

W (n) =

(27)

F (n) − Fmin (n) Z (n) − Zmin (n) + Fmax (n) − Fmin (n) Zmax (n) − Zmin (n) (28)

The third level:

W=

F − Fmin Z − Zmin + Fmax − Fmin Zmax − Zmin

(29)

The execution started from the first level through initially giving a set of pumping flow (Q(n,j,i)) and water allocation (S(n,j,i)). Taking water allocation and pumping flow of each level model as the coordinated variables, the coupling constraints should be satisfied for water allocation using Eqs. (5) and (6) and Eqs. (18) and (20). The steps for solving are presented in a flowchart (Fig. 4), which is described as follows: (1) the initial values for pumping flow and water allocation were given to the first-level model based on the historical data. (2) under the given pumping flow and water allocation of each pumping station, the first-level model was operated for optimal allocation of pumping flow among units (i.e. q(n,j,m)) and water amount between traditional and high-efficient irrigated areas (i.e. X (n,j,i) and Y(n,j,i)). The optimal schemes of operation and water allocation were obtained for each PIS in each subsystem, as well as the energy cost per unit of pumping water (C(n,j)) and water requirement (D(n,j,i)) within each PIS. (3) the energy cost C(n,j) and water requirement D(n,j,i) were then input into the second level. Then under the initially given Q(n,i) and S(n,i) of each subsystem, the second level was optimized based on the imported C(n,j) and D(n,j,i) to obtain the optimal allocation of pumping flow and water allocation among multi-stage pumping stations (Q(n,i,j) and S(n,i,j)). Meanwhile, the energy cost per unit of pumping water (C(n)) and water requirement (D(n,i)) for each subsystem were also obtained and imported to the third level as initial conditions for optimization. (4) with the approved total water delivery amount (S), the third-level model was operated based on the C(n) and D(n,i) to obtain the optimal allocation of pumping flow and water supply among different subsystems (Q(n,i) and S(n,i)), as well as total energy cost (Z) and water requirement (D) of the whole system. (5) the optimized results (Q and S) from the second- and third-level models were transferred back to the first- and second-level models as new coordinated variables, respectively. (6) with the feedback of updated coordinated variables (Q and S), the steps of (2)–(5) were conducted repeatedly. (7) this calculation was repeatedly iterated until the objective value of the third level (W) reached the criterion as follow:

|W p − W p - 1| ≤ε Wp -1

Fig. 4. Flow chart for solving steps of the optimization-coordination model.

4. Model application 4.1. Model setup considering different scenarios The basic data required by the model were collected from the Guhai Irrigation District Management Department, including total amount of approved water delivery, crop pattern, irrigation schedules and parameters of each pumping station (e.g. total/net pumping head, number and parameters of pumping units, designed operation mode, designed/ max pumping flow, irrigated areas of each PIS etc.), as well as the actual pumping water and water allocation schemes for each PIS in recent years. The water requirements in different levels of the model were calculated according to several terms, i.e. crop pattern, irrigated area, irrigation schedules and water use efficiency of canal systems. According to field surveys, high-efficient irrigation methods are mainly applied to local economic crops, while other crops still used

(30)

where p is the number of iterations, ε is the maximum threshold of the difference between two iterations, and was set to 0.005 here.

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4.2. Results and discussion

Table 3 Different scenarios of total water diversion amount. Scenario

Total pumping-water amount (104 m3)

Changes compared to the present value

Approved maximum water amount Present Scenario 1 Scenario 2 Scenario 2 Minimum water requirement

44,751

–

40,826 42,868 38,784 36,258 34,436

– Increased by 5% Decreased by 5% Decreased by 10% –

4.2.1. Optimal water allocation among multi-stage pumping stations Under the level of present water diversion amount, the comparison of optimized results of water allocation with present values for GPIS is shown in Table 4. In the present situation, the pumping stations at all stages pumped and allocated water at the same time, which easily caused larger water shortage in the whole system especially in the lower-stage PIS since the limited total water diversion and unreasonable planning of water allocation. While through optimizations, the differences between water allocation and requirement (hereafter referred to as water deficiency) in each PIS was decreased from the present values of −5.10–6.33 million m3 to the optimized values of −0.03–0.66 million m3, with a significant reduction of variation range (Table 4) (positive value represented water surplus). It indicated that the optimized water allocation for each PIS was much closer to the water requirement of each PIS, and the optimal scheme was more reasonable. In the present situation, water deficiency in subsystem 1 was the most significant since it has maximum pumping-water stages (12 stages) and highest pumping head (428.7 m), and thus greater difficulties in the coordination between different stages of pumping station. As for subsystem 3, the pumping-water amount was larger than its total requirement due to its only 4-stage pumping stations and small irrigated areas. After optimizations, the total amount of water allocation was increased in subsystem 1, while were slightly decreased in subsystem 2 and 3 (Table 4). Meanwhile the pumping and allocating water amounts of the lower-stage PIS were increased in all subsystems (Table 4), which significantly alleviated the contradiction of water allocation between upper-stage and lower-stage PIS at present. In addition, despite the shaft power was increased for many pumping stations

traditional irrigation methods. Thus, the irrigated areas and irrigation schedules of economic crops and other crops were used to calculate water requirement (i.e. D(n,j,i)) for the two types of irrigated areas respectively. The water diversion in GPIS lasts from April to August, meanwhile there is a winter irrigation in November. Thus, in this study, the irrigation period was subdivided into 6 of time intervals, and every time interval was set to 30 days equally. The parameters of each pumping station and units were determined according to the collected actual parameters. The maximum number of pumping units within each station was identified based on the total number of units actually installed in this pumping station, but not considered spare units. The present situation adopted the actual total water diversion amount and water allocation scheme in GPIS in 2017. Three scenarios of total water diversion amount in the whole system were defined according to the approved maximum water diversion and minimum total water requirement (Table 3), where the total water diversion was gradually increased/reduced by 5% until it approached the maximum/minimum limitation.

Table 4 Present values and optimized results of water allocation among multi-stage pumping stations in the Guhai pumping-water irrigation system. Subsystem

Pumping-irrigation subdistrict

Pumping-water amount (104 m3)

Water allocation (104 m3)

Present

Optimized

Water requirement

Present

Water deficiency (104m3)

Shaft power(kW)

Optimized

Present

Optimized

Present

Optimized

Subsystem 1

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 1-10 1-11 1-12 Total

12,316 12,250 12,250 12,250 12,250 10,251 8322 6462 5357 3143 2653 1914 12,316

12,031 11,945 11,945 11,945 11,945 10,356 9011 7377 6201 3863 3330 2114 12,031

52.63 0.00 0.00 0.00 1111.58 937.85 1143.67 818.24 1643.17 362.36 847.23 1485.10 8401.83

46.99 0.00 0.00 0.00 1235.36 1185.18 1137.02 784.42 1388.16 347.26 525.06 1174.87 7824.31

65.76 0.00 0.00 0.00 1136.93 965.90 1168.46 838.27 1665.30 386.43 863.91 1506.24 8597.20

−5.64 0.00 0.00 0.00 123.78 247.33 −6.65 −833.83 −255.01 −15.10 −322.17 −310.23 −577.52

13.12 0.00 0.00 0.00 25.35 28.04 24.79 20.02 22.14 24.07 16.68 21.15 195.37

21,835 23,621 8101 14,738 21,229 14,702 9085 8650 7619 3393 3510 2379 138,861

21,329 22,983 7882 14,340 20,656 13,367 8853 8887 7937 3754 3964 2365 136,316

Subsystem 2

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 Total

20,656 18,223 16,631 16,242 8852 6173 5200 1872 20,656

20,580 18,297 17,100 16,712 10,371 8133 7162 2024 20,580

187.86 998.67 324.30 5286.63 1866.45 811.81 4284.00 1690.06 15449.77

206.31 1126.66 323.80 5458.35 2232.56 810.49 3773.49 1560.21 15491.87

187.51 997.70 322.99 5284.93 1864.52 809.39 4281.12 1687.06 15435.22

18.45 128.00 −0.50 171.72 366.11 −1.32 −510.50 −129.85 42.10

−0.35 −0.97 −1.31 −1.70 −1.92 −2.43 −2.88 −3.00 −14.54

22,745 44,293 21,599 24,172 7998 12,589 10,313 1281 144,990

22,662 40,027 19,988 22,385 8432 14,927 12,782 1247 142,450

Subsystem 3

3-1 3-2 3-3 3-4 Total

7855 7855 5311 1681 7855

7123 6411 4812 2027 7123

0.00 1710.24 3029.07 2187.50 6926.81

0.00 2343.98 3629.79 1880.76 7854.53

0.00 1776.46 3094.38 2252.09 7122.94

0.00 633.74 600.73 −306.74 927.72

0.00 66.22 65.32 64.60 196.13

13,671 10,664 7236 2614 34,185

13,776 9671 7285 3503 34,234

40,827

39,734

30778.40

31170.71

31155.36

392.30

376.96

318,036

313,000

Total

Note: Water deficiency was defined as the difference between water allocation and requirement, and the positive values represented water surplus while negative values represented water deficit. 8

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Fig. 5. Present values and optimized results of water allocation among multi-stage pumping-irrigation subdistricts in each period.

after optimizations due to its increased pumping flow. The total shaft power of the whole system changed insignificantly as compared with the present values (Table 4). Under the level of present water diversion amount, the optimal schemes of pumping and allocating water for each PIS in different periods are shown as Figs. 5 and 6. The optimized water allocation basically met the water requirement of each PIS in different periods (Fig. 5). Since the water requirements are generally larger in the key periods of crop growth (June to August), the pumping and allocating water amounts were relatively larger during these months than others for each PIS in the optimal schemes (Figs. 5 and 6). Meanwhile, the pumping and allocating water amounts during different periods were all increased for the lower-stage PIS as compared with the present schemes, while they were decreased in some periods for the upper-stage PIS due to the constraints of the total water amount (Figs. 5 and 6). In

general, the spatiotemporal contradiction of water allocation in the multi-stage pumping-water system was relieved through the optimizations of water supply regulation. Both water requirements in key periods of crop growth and pumping-water amount for lower-stage pumping stations were effectively satisfied.

4.2.2. Optimal schemes of water allocation and operation schedules within single pumping-irrigation subdistrict In the optimizations, detailed water allocation schemes in each period within each single PIS were obtained and presented as allocation ratio in Fig. 7. The allocation ratio was defined as the ratio of water allocation between different irrigated areas during each period to the total water allocation of this pumping-irrigation subdistrict. Since the water allocation between traditional irrigated and high-efficient irrigated areas was based on its water requirement in different periods, the 9

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Fig. 6. Present values and optimized results of pumping-water amount among multi-stage pumping stations in each period (taking subsystem 2 and 3 as an example).

flow referred to the total flow of all operating units in the pumping station, and was regulated mainly by changing the number of operating units and its pumping flow in each period. The pumping flow of each pumping station was the largest in the peak periods of water requirement (July to August), thus the number of operating units also reached to maximum (Table 5). In contrast, the number of operating units was relatively less in the period of lower water requirement (April, May and November) in order to guarantee appropriate pumping flow and efficiency of each unit, which could improve the total operational efficiency and reduce the energy cost in each pumping station.

water allocation schemes within each single PIS were affected by crop water requirements and also closely related to the irrigated area and crop pattern. With the increases of water requirement in key growing periods (June to August), water allocations in these months were all increased for both traditional and high-efficient irrigated areas, which accounted for more than 20% of the total water allocation of most PIS (Fig. 7). While the ratio of water allocation between these two types of irrigated areas was mainly determined by the planting ratio of economic crops and grain crops. The PIS with higher planting ratio of economic crops generally had a larger ratio of water allocation for highefficient irrigated areas. For example, the water allocation ratio of highefficient irrigated areas was approximately 80% in the PIS of 2-1 and 23 in different periods due to more than 50% of the planting ratio of economic crops (Fig. 7 and Table 1). The water allocation for highefficient irrigation usually needs sedimentation in ponds firstly for removing sediment from the Yellow River, thus the results were helpful for local irrigation managers to make decisions in advance. While the 12 to 1-4 and 3-1 pumping stations were only employed to lift water for lower-stage PIS, there were nearly no irrigated areas in these PIS and thus no water allocation in the optimizations. Meanwhile the operating state of all units and its pumping flow within each pumping stations were optimized in this study, and the optimized operation schedules are presented as Table 5. The pumping

4.2.3. Optimized results under different scenarios of water supply The effects of water diversion amount changes on the optimizations were discussed further. When the total water diversion amount was adequate (e.g. scenario 1), the pumping-water amount in each PIS was restricted by both energy cost and water requirement. Hence, the water allocation for each PIS was not quite larger than its water requirements (Fig. 8). There were also minor differences between the pumping-water amount in scenario 1 and that in present optimal schemes (Fig. 9). While with relatively less water diversion amount (e.g. scenario 3), the optimizations were mainly constrained by water requirements. Hence, pumping-water amount would be maximized as far as possible in order to reduce the water deficiency. As shown in Figs. 8 and 9, the 10

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Fig. 7. Allocation ratio of irrigation water for different irrigated areas during each period relative to the total water allocation in this pumping-irrigation subdistrict (taking subsystem 2 and 3 as an example). Table 5 Optimized results of pumping-water amount (PWA) (m3 s−1) and corresponding operating units at each pumping station in each period (take subsystem 1 as an example). Pumping station

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 1-10 1-11 1-12

April

May

June

July

August

November

PWA

Operating units

PWA

Operating units

PWA

Operating units

PWA

Operating units

PWA

Operating units

PWA

Operating units

7.64 7.56 7.56 7.56 7.56 5.95 5.17 4.13 3.54 2.31 1.97 1.37

#1-#3 #1#3#4 #1#3#4 #1#3#4 #1#3-#5 #1#3#4 #1#2#4 #1#4#6 #3#4 #1#5 #2 #1#6

7.64 7.56 7.56 7.56 7.56 5.95 5.17 4.13 3.54 2.31 1.97 1.37

#1-#3 #1#3#4 #1#3#4 #1#3#4 #1#3-#5 #3#4 #1#2#4 #1#4#6 #3#4 #1#5 #2 #1#6

10.50 10.42 10.42 10.42 10.42 8.21 7.20 5.94 4.89 3.02 2.68 1.57

#1-#4 #1-#5 #1-#4 #1-#4 #1-#4 #1-#4 #1-#6 #3-#6 #2-#4 #4#6 #1#5 #1#2

11.55 11.47 11.47 11.47 11.47 8.84 7.65 6.31 5.29 3.16 2.69 1.48

#1-#6 #1-#6 #1-#6 #1-#6 #1-#4#6 #1-#4 #1-#6 #1-#4 #2-#4 #4#5 #1#5 #1#2

8.29 8.21 8.21 8.21 8.21 6.13 5.11 3.89 3.10 1.77 1.50 0.89

#1-#4 #1-#4 #1-#4 #1-#4 #1-#4 #1-#4 #1#2#4 #2#3 #3#4 #2 #5#6 #3

5.96 5.88 5.88 5.88 5.88 4.87 4.46 4.08 3.56 2.33 2.05 1.47

#1#4 #1#4 #1#4 #1#4 #1#4-6 #3#4 #1#2#4 #3#4 #3#4 #1#5 #3#4 #1#2

Note: # represented the serial number of pumping unit in each pumping station.

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6.33 million m3). The scenario analysis further indicated that the optimization-coordination model could better coordinate water allocation between different stages of PIS under different conditions of water diversion. 5. Conclusion A regional multi-level multi-objective optimization-coordination model was formulated for optimizing water allocation in complex multi-stage pumping-water irrigation system, which taking the Guhai pumping-water irrigation system (GPIS) of Ningxia Hui Autonomous Region northwest China as a case study. The model had a three-level hieratical structure and was solved based on the decomposition-coordination method for large system. The first level dealt with the optimal water allocation among different irrigated areas within each single pumping-irrigation subdistrict (PIS). The second level sought out the optimal schemes of pumping and allocating water among multistage pumping-irrigation subdistrict in each subsystem. The third level concerns the optimal allocation of water supply among different subsystems. In each level, minimizing water requirement and energy cost (shaft power) were taken as two objectives, and different irrigation periods were considered to obtain detailed schedules for water allocation and operation. The three levels could be solved independently, and meanwhile was connected through the exchange of irrigation water allocation and energy cost of each level. This model could better describe the coordination and constraints of multi-level planning and decision-making processes, as well as the effects of multiple factors including the operating rules of pumping stations, reasonable allocation between different pumping stations and units, and water limitations. Optimal schemes of water allocation for each PIS of different subsystem were proposed, including schemes of pumping and allocating water among each PIS, and water allocation and operation within each PIS. The optimized results showed that the range of water deficiency among different PIS was reduced from the present values of −5.10–6.33 million m3 to the optimized values of −0.03–0.66 million m3 under the level of present water delivery amount. Particularly, both the pumping-water amount and water allocation were increased for the lower-stage PIS in each subsystem. Meanwhile the water requirement of each PIS in each period could be better satisfied to certain extend through optimizations, particularly in the peak period of irrigation (e.g. June-August). Scenarios analysis for three water delivery levels showed that the ranges of water deficiency were smaller than the present range under all levels of water delivery amount. Model application indicated

Fig. 8. Relative difference between water allocation and corresponding water requirement in each pumping-irrigation subdistrict under different water delivery scenarios.

pumping-water amount in each PIS was decreased as the decrease of the total water diversion amount, and its water deficiency was increased accordingly. However, the variation range of water deficiency in the whole irrigation system was still lower than the present values, despite the total water deficiency was larger than the present (Fig. 8). The pumping-water amount in each PIS was decreased by 3–15% in scenario 3 as compared to the optimized results at present, while the pumping-water amount in lower-stage PIS was still increased as compared with the present (Fig. 9). Particularly, water allocation was more uniformly distributed among different PIS and similar to its water requirement without significant deficits (< 15%) in all scenarios (Fig. 8). For example, when the total water delivery amount was reduced by 10% (scenario 3), the water deficiency of each PIS ranged from −0.07 to −1.91 million m3, which was less than the present range (−5.10 to

Fig. 9. Relative difference between optimal pumping-water amount and its present values among multi-stage pumping stations under different water delivery scenarios. 12

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that the model could better balance the spatial-temporal contradictions of water supply among different subsystems and multi-stage pumping stations under different conditions of water delivery. In summary, this paper presents a practical and efficient method for optimizing water allocation in complex multi-stage pumping-water systems, which can also be applied for references in other similar irrigation systems. Additionally, further studies can be conducted for better decisionmaking support such as considering the uncertainty of crop water requirement and predicting water requirement in advance.

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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research was supported by the 13th Five-year National Key Research and Development Program of the Chinese Ministry of Science and Technology (grant numbers: 2016YFC0400206-04) and the National Natural Science Foundation of China (grant numbers: 91647202). We are especially grateful to the Guhai Irrigation District Management Department for providing basic data. References Allam, A., Tawfik, A., Yoshimura, C., Fleifle, A., 2016. Simulation-based optimization framework for reuse of agricultural drainage water in irrigation. J. Environ. Manage. 172, 82–96. Alizadeh, M.R., Nikoo, M.R., Rakhshandehroo, G.R., 2017. Hydro-environmental management of groundwater resources: a fuzzy-based multi-objective compromise approach. J. Hydrol. 551, 540–554. Boulos, P.F., Wu, Z., Orr, C.H., Moore, M., Hsiung, P., Thomas, D., 2001. Optimal Pump Operation of Water Distribution Systems Using Genetic Algorithms. AWWA Distribution System Symposium. American Water Works Association, Denver. Brown, P.D., Cochrane, T.A., Krom, T.D., 2010. Optimal on-farm irrigation scheduling with a seasonal water limit using simulated annealing. Agric. Water Manage. 97 (6), 892–900. Bai, T., Chang, J.X., Chang, F.J., Huang, Q., Wang, Y.M., Chen, G.S., 2015. Synergistic gains from the multi-objective optimal operation of cascade reservoirs in the Upper Yellow River basin. J. Hydrol. 523, 758–767. Davijani, M.H., Banihabib, M.E., Anvar, A.N., Hashemi, S.R., 2016. Optimization model for the allocation of water resources based on the maximization of employment in the agriculture and industry sectors. J. Hydrol. 533, 430–438. Fleifle, A., Saavedra, O., Yoshimura, C., Elzeir, M., Tawfik, A., 2014. Optimization of integrated water quality management for agricultural efficiency and environmental conservation. Environ. Sci. Pollut. R. 21 (13), 8095–8111. García, I.F., Moreno, M.A., Díaz, J.R., 2014. Optimum pumping station management for irrigation networks sectoring: case of Bembezar MI (Spain). Agric. Water Manage. 144, 150–158. Ge, Y.C., Li, X., Huang, C.L., Nan, Z.T., 2013. A Decision Support System for irrigation water allocation along the middle reaches of the Heihe River Basin, Northwest China. Environ. Modell. Softw. 47, 182–192. Haq, Z.U., Anwar, A.A., Clarke, D., 2008. Evaluation of a genetic algorithm for the irrigation scheduling problem. J. Irrig. Drain. Eng. 134 (6), 737–744. Heydari, F., Saghafian, B., Delavar, M., 2016. Coupled quantity-quality simulation-optimization model for conjunctive surface-groundwater use. Water Resour. Manage. 30 (12), 4381–4397. Homayounfar, M., Lai, S.H., Zomorodian, M., Sepaskhah, A.R., Ganji, A., 2014. Optimal crop water allocation in case of drought occurrence, imposing deficit irrigation with proportional cutback constraint. Water Resour. Manage. 28 (10), 3207–3225. Hu, Z., Wei, C., Yao, L., Li, L., Li, C., 2016. A multi-objective optimization model with

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