Aiming at the optimal allocation of irrigation water in a multi-water source project in a water resource shortage area, this study developed a water resource joint scheduling optimization model for the reservoir and the pumping station under deficit irrigation conditions. In the model, the maximum annual yield of the irrigation area was the objective function; the water supply, water spill of the reservoir and replenishment water of the pump station at each stage were the decision variables; and the total annual water supply of the system, the reservoir operation criteria, the water rights of the pumping station, and the water demand of the crop during the entire growth period were the constraint conditions. According to the characteristics of the model, a large system decomposition aggregation dynamic programming (DADP) method is proposed to transform the N + 1 dimensional dynamic programming problem into a N + 1 one-dimensional dynamic programming problem for solution. In addition, this study also uses the real-coded genetic algorithm (RGA) and DADP to compare the algorithms, and discusses the performance of the two algorithms from the optimization of the algorithm and the applicability of the algorithm.

  • An optimization model for the joint operation of the reservoir and the pumping station in the water shortage area is established.

  • A new optimization method for a large-scale system model is proposed.

  • The complex nonlinear problem is transformed into a series of dynamic programming iterative operations.

Graphical Abstract

Graphical Abstract

In water-scarce areas, there has been much interest in research on the optimal allocation of irrigation water resources with the goal of increasing crop yields. Allocation methods involving irrigation water (Zhu et al. 2014; Yang et al. 2021), irrigation quota, irrigation timing, water demand characteristics of crops, growth stage, tillage methods (Anzai et al. 2014; Yamane et al. 2016), competition for water among different crops, and water deficit have all had an effect on optimal water supply decisions concerning irrigation.

Reservoirs, ponds, wells and pumping stations are common water source projects in irrigation areas. The joint operation of multiple water source projects can effectively ensure the irrigation task in dry years. The optimal operation model of the irrigation system with the joint operation of multiple reservoirs and multiple pumping stations is usually based on the minimum operation cost of the irrigation system (mainly the pumping cost of pumping stations) as the objective function. Yu et al. (1994) proposed a nonlinear model with the minimum cost of water replenishment as the objective function; Reca et al. (2014, 2015) proposed a linear model for the joint operation of reservoirs and pumping stations with the goal of minimizing the cost of water replenishment. However, the above studies did not consider the total amount of water replenishment from the perspective of agricultural water rights. With the gradual promotion of water price reform (Molle 2004; Heikkila 2015), regional water rights are strictly divided, and the total amount of water replenishment is limited. Therefore, the existing joint dispatching model needs to take into account the impact of water rights constraints on irrigation decisions.

The optimal operation of reservoir water resources can adopt linear programming (Yoo 2009), nonlinear programming (Cai et al. 2002), and dynamic programming (DP). However, for the optimal scheduling problem of complex irrigation systems characterized by high dimension and a large number of decision variables, the existence of the ‘curse of dimensionality’ (Cheng et al. 2012) is an unavoidable obstacle in classical algorithms. Heuristic algorithms, such as genetic algorithms (Yang et al. 2019) and genetic programming (Fallah-Mehdipour et al. 2013), particle swarm algorithms (Mousavi & Shourian, 2010), and simulated annealing algorithms (Teegavarapu & Simonovic, 2002), have also been widely used to optimize reservoirs. These heuristic algorithms with random search properties have significant advantages over classical algorithms in terms of the number of calculations that can be performed and their calculation time when solving problems that are highly dimensional and nonlinear and have multiple decision variables. However, such methods are difficult to apply to optimization problems with judgmental constraints, such as reservoir operational criteria. In addition, the treatment of equality constraints and the mathematical proof of global optimality using heuristic algorithms remains controversial (Samanipour & Jelovica 2020).

The decomposition coordination method and decomposition aggregation method are both called ‘two-level algorithms’, and the former has been widely used in the optimization of large system models (Li et al. 2014; Gong & Zhu 2022). For the optimal scheduling problem of high-dimensional and complex irrigation systems, the decomposition aggregation method (Litvinchev 1995) can give priority to the optimization of subsystems and replace independent variables with associated variables compared with the decomposition coordination method (Gong & Cheng 2018). In addition, the decomposition aggregation method can also ensure the connection between subsystems in the process of decomposition of the large system, so that the optimal solution of the large system model can be determined by the aggregation model.

In this study, taking the irrigation system consisting of the reservoirs and pumping stations as the research object, considering the water rights constraints, a water resources optimal allocation model for joint operation of water source projects under the condition of insufficient irrigation is established. According to the characteristics of the model, the decomposition aggregation dynamic programming (DADP) method is proposed to solve the problem. The DADP method can decompose the large system model into a series of subsystem models. The subsystem models are solved by using the one-dimensional dynamic programming method to ensure the optimality of each subsystem. Then, the associated variables are used as decision variables, and a series of subsystems is aggregated according to the recursive principle of DP. The aggregation model is also solved by using the DP method. The optimal solution of the aggregation model is used to check the optimal solution of each subsystem, Thus, the global optimal solution of the large scale system model is obtained. At the same time, the real-coded genetic algorithm (RGA) and the DADP method are used to compare and analyze the algorithms.

System generalization

The irrigation system jointly operated by reservoir and pumping station is composed of a single annual regulating irrigation reservoir and a single pump station. The system generalization is shown in Figure 1. In areas where water resources are scarce, when the local main water source projects (such as reservoirs) can not guarantee the agricultural water demand of the irrigation area, under the condition of considering the water right restriction, the pumping station can divert water from the river outside the area to supplement the reservoir, to improve the guaranteed rate of irrigation water.
Figure 1

Schematic diagram of a single-reservoir and a single-pumping-station irrigation system.

Figure 1

Schematic diagram of a single-reservoir and a single-pumping-station irrigation system.

Close modal

In Figure 1, i is the number of crop species in the irrigation area (i = 1,2); j is the crop growth stage number (j = 1,2,…,Mi); Xi,j is the water supply from the reservoir to crop i at growth stage j under deficit irrigation conditions, MCM; Yi,j is the volume of water replenishment from the pumping station at growth stage j of crop i, MCM; LSi,j is the inflow to the reservoir at growth stage j of crop i, MCM; PSi,j is the water spill by the reservoir at growth stage j of crop i, MCM; and EFi,j is the evaporation from the reservoir at growth stage j of crop i, MCM. (Note: MCM is million cubic meters.)

Model construction

Objective function

Considering the contradiction between water supply and water demand in the study area in dry years, as well as the relationship between water and output, the Jensen (1968) model, which is a water production function with high applicability, is adopted as the basic model for research. In this study, the maximum annual output value of two main crops in the irrigation area is taken as the objective function. At the same time, in order to facilitate the joint operation calculation of the reservoir and the pumping station, the ratio of actual evapotranspiration and potential evapotranspiration in the water production function is converted into the ratio of the reservoir water supply and crop water demand (Wardlaw & Barnes 1999), and the objective function is shown in formula (1):
(1)
where: G is the annual output value of the irrigation area, RMB; Mi is the entire growth stages of crop i; (Ym)i is the maximum yield of i crop when the water supply is sufficient, kg/hm2; hi,j are the sensitive indexes of yield response to water shortage at j growth stage of crop i; YSi,j is the maximum water demand in the j growth stage of crop i in irrigation area, MCM; Ai is the planting area of crop i, hm2; Pi is the unit price of crop i, kg/RMB. (Note: RMB is the legal currency of the People's Republic of China.)

Constraint

Water supply constraints

  • (1) Total annual water supply constraint of the system:
    (2)
  • (2) Water supply restriction during the entire growth period of a single crop:
    (3)
    where SK is the total annual water supply of the annual regulation reservoir, MCM; BZ is the water rights of the pumping station, that is, the maximum allowable water replenishment of the pumping station in a year, MCM; IWi is the upper limit of the water supply during the entire growth period of a single crop, MCM.

Scheduling criteria constraints
(4)
where Vi,j is determined according to the water balance equation:
(5)
(6)
In the formula, Vi,j is the reservoir storage capacity at the end of stage j of crop i, MCM; EFi,j is the evaporation from the reservoir during stage j of crop i, MCM; Ei,j is the evaporation from the evaporator, mm, wi,j is the surface evaporation conversion factor for stage j of crop i, and α and β are coefficients.
According to the water balance equation of the system, the operational criteria of the reservoir are as follows:
(7)
Pumping station constraints

  • (1) Constraints on water replenishment capacity of pumping station:
    (8)
  • (2) Water rights constraints:
    (9)

Crop water demand constraints
(10)
where is the average minimum water demand in the j growth stage of crop i, MCM.
Initial and boundary conditions
(11)
where V0 is the initial storage capacity of the reservoir, Vend refers to the storage capacity at the end of reservoir regulation and storage.

Model solution

Decomposition aggregation dynamic programming (DADP)

The DADP method used in this study belongs to a dimension reduction solution method, which can transform a N + 1 dimensional DP problem into a N + 1 one-dimensional DP problem. As shown in Figure 2, the DADP method combines the decomposition aggregation (DA) method and the DP method. The solution process is to decompose high-dimensional large system problems into multiple low-dimensional subsystem problems, and then aggregate the subsystems through the recursive idea of DP. The aggregation model is solved by the DP method. The optimal solution of the aggregation model is used to check the optimal solution of each subsystem, and then the global optimal solution of the large system can be obtained. The solution steps of DADP are as follows:
Figure 2

(a) is the schematic diagram of DP; (b) is the schematic diagram of DADP.

Figure 2

(a) is the schematic diagram of DP; (b) is the schematic diagram of DADP.

Close modal

Step 1: Large system decomposition. For the water-deficient irrigation areas of two main dry crops, the water competition relationship between the two dry crops under limited water resources is considered while studying the impact of water stress on the final yield of dry crops. The optimal allocation model of water resources of the reservoir and the pumping station irrigation system can be divided into two optimal allocation models for irrigation of a single a dry crop, as shown below:

Objective function
(12)
Constraint
(13)
The subsystem model obtained by decomposition still needs to meet the constraints of dispatching criteria, pumping station capacity and crop water demand.
Step 2: Solve the subsystem. The subsystems obtained from the decomposition of large systems are all one-dimensional DP models, which can be solved by classical DP methods. The solution process of subsystems is shown in Figure 3. Where, state variable λi,j is the total water supply of the reservoir to a single crop during the entire growth period, which is discrete in [0, IWi] with a fixed step size d. For each state variable λi,j, set the decision variables Xi,j in their feasible region [0,λi,j] is discrete by step d. The state transfer equation is λi,j−1 = λi,j-Xi,j, the optimal water supply process Xi,j(i = 1,2; j = 1,2,…,Mi) and the objective function value fi (i = 1,2) of the reservoir for each growth stage of dry crops in the subsystem can be obtained. According to the optimal water supply process of the reservoir, the optimal water spill process PSi,j(i = 1,2; j = 1,2,…,Mi) of the reservoir and the optimal water replenishment process Yi,j (i = 1,2; j = 1,2,…,Mi) of the pumping station can be determined.
Figure 3

Flow chart of subsystem DP solution.

Figure 3

Flow chart of subsystem DP solution.

Close modal

The optimization solution process of the subsystem can refer to the relevant research (Wei et al. 2021). But it is worth emphasizing that the DP method adopted by Wei et al. (2021) in the research is only applicable to the solution of one-dimensional DP model. The DADP method proposed in this study is a solution method of the large system model, which can be applied to the dimensionality reduction of high-dimensional DO model.

Step 3: Subsystem aggregation. After the subsystem is solved, a series of corresponding relations of IWifi(IWi) (i = 1,2) can be obtained, and then the subsystem models are aggregated according to the corresponding relations and the recursive idea of DP. The aggregation model is as follows:

Objective function
(14)
Coupling constraint
(15)

Equations (14) and (15) form the aggregation model, which is also a typical one-dimensional mathematical model and can also be solved by DP. Where, state variable ξi is the sum of the total water supply of the large system, and the feasible area is [0, SK + BZ]; The decision variable IWi is the total water supply of the subsystem, and the feasible region is [0,ξi]。 The global optimal G* of the large system model and the optimal water supply IWi*(i = 1,2) corresponding to the optimal solution can be obtained after the solution is completed. Then, the optimal reservoir water supply process Xi,j*(i = 1,2; j = 1,2,…,Mi ), the optimal water replenishment process Yi,j*(i = 1,2; j = 1,2,…,Mi) of the pumping station, and the optimal reservoir water spill process PSi,j*(i = 1,2; j = 1,2,…,Mi) can be obtained by back checking the optimization results of the subsystem.

Real-coded genetic algorithm (RGA)

Wright (1991) proposed an RGA, which is a real-to-individual coding method that can achieve better optimization results than binary-coded genetic algorithms.

For a given optimization problem with m variables, the n initial solution sets are usually generated in a random manner when the RGA is used to solve the problem, where the process of population iteration, selection, crossover and mutation are adopted to ensure the evolution of the population. In this study, a roulette wheel was used for the selection operation, and the random single-point method was used for cross-operation. First, a sequence number t smaller than m was randomly generated, and then the sequence numbers corresponding to a pair of genes t to m were exchanged with each other to complete the crossover operation. For example: a pair of genomes and will become and through the cross-operation; The mutation operation was completed by random single-point mutation.

Considering that the mathematical model established in this study contains many constraints, in the process of solving RGA, the penalty term was constructed to deal with many of these constraints, as follows:
(16)
(17)
(18)
(19)
where, P1, P2, P3 and P4 are the penalty functions of the upper and lower limits of reservoir capacity, the water right constraints of sub-regions and the agricultural total water right constraints of the whole irrigation area respectively, and μ1, μ2, μ3 and μ4 are the penalty factors respectively.
By integrating the objective function of the model with the penalty term, the fitness function of the RGA was constructed as follows:
(20)
where: W is the fitness function value.

Study area

The Yibei(YB) irrigation area is located in the eastern mountainous region of Xinyi City, Jiangsu Province, China. The location of the study area is shown in Figure 4. The annual rainfall is 750 mm in a level year (50% probability) and 571 mm in a dry year (75% probability), and multi-year average evaporation is 968 mm. The area under winter wheat cultivation is 1.26 × 103 hm2, and the area under corn cultivation is 1.47 × 103 hm2. The main irrigation water source in the irrigation area is the Shadun (SD) Reservoir, with a design capacity of 7.5 MCM, a lower limit of 2.0 MCM, and an irrigation water utilization coefficient of 0.48. The test parameters of winter wheat and corn are shown in Tables 1 and 2.
Table 1

Experimental parameters of winter wheat

Growth period
SeedingReturning greenJointingHeadingFillingMature
Sensitive index0.26750.06130.37650.59510.59510.2981
Crop water requirement (mm) 50% probability 97.3 17.3 30.4 92.1 110.7 105 
75% probability 107 19 33.4 101.3 121.8 115.5 
Growth period
SeedingReturning greenJointingHeadingFillingMature
Sensitive index0.26750.06130.37650.59510.59510.2981
Crop water requirement (mm) 50% probability 97.3 17.3 30.4 92.1 110.7 105 
75% probability 107 19 33.4 101.3 121.8 115.5 
Table 2

Experimental parameters of corn

Growth period
SeedingJointingHeading/fillingMature
Sensitive index0.2570.20220.32370.2189
Crop water requirement(mm) 50% probability 81 90 77.4 109.8 
75% probability 92.4 91.8 85.2 115.8 
Growth period
SeedingJointingHeading/fillingMature
Sensitive index0.2570.20220.32370.2189
Crop water requirement(mm) 50% probability 81 90 77.4 109.8 
75% probability 92.4 91.8 85.2 115.8 
Figure 4

Location map of YB Irrigation district.

Figure 4

Location map of YB Irrigation district.

Close modal

The hydrological storage calculation period of traditional reservoirs is usually divided into daily, 10-day, and monthly periods. In this study, the regulation and storage calculation of the irrigation system is divided by crop growth period, which can be applied to the planning solution of the crop water production function for winter wheat and corn in the YB irrigation area. In addition, the incoming water process of the irrigation system was calculated statistically based on the total calendar time of each stage of growth. With reference to hydro-meteorological data from the local weather station, the inflow of the reservoir at each stage is shown in Table 3.

Table 3

Inflow (104 m3)

Level year 32 43 13 12 19 28 37 49 53 42 
Dry year 24 19 18 11 16 18 19 29 26 
Level year 32 43 13 12 19 28 37 49 53 42 
Dry year 24 19 18 11 16 18 19 29 26 

YB irrigation area is located in hilly and mountainous areas. Relying on artesian water diversion alone can not ensure normal agricultural water use in the irrigation area, so water needs to be diverted for recharge while retaining surface water. The existing Haohu (HH) pumping station is used to divert and lift the water from the main canal to supplement the SD reservoir. The characteristic parameters of the pumping station are shown in Table 4.

Table 4

Characteristics of pumping stations

Pumping stationDesign discharge (m3/h)Daily operation duration (h)Water rights (MCM)
HH 3,600 20 
Pumping stationDesign discharge (m3/h)Daily operation duration (h)Water rights (MCM)
HH 3,600 20 

The evaporation losses from irrigation reservoirs were determined based on the evaporation depth for the time period and the average reservoir water area, where the evaporation depth Ej was determined using measured evaporation data from an area near the study area (see Table 5) and corrected using the conversion factor wj. The water area was determined based on the reservoir area and storage volume relationship function provided by the reservoir manager. (With reference to the local water resources planning data, we can obtain that α = 2.117 × 10−3, β = 1.063.)

Table 5

Ej and wj of each period

Period12345678910
Ej 92.8 102 44.2 51.6 107.6 94.6 126.3 134.4 118.6 96.5 
Wj 1.04 1.11 1.03 0.96 0.93 0.92 0.94 0.96 0.97 1.01 
Period12345678910
Ej 92.8 102 44.2 51.6 107.6 94.6 126.3 134.4 118.6 96.5 
Wj 1.04 1.11 1.03 0.96 0.93 0.92 0.94 0.96 0.97 1.01 

Solution results of the DADP method

According to the information provided by the reservoir manager, the initial boundaries V0 of reservoir storage in a level year (50% probability) and in a dry year (75% probability) were 3.18 MCM and 2.86 MCM, respectively, for the SD reservoir. The water supply of winter wheat in the YB irrigation area was discretized in the interval [4.48 MCM, 7.0 MCM] and [4.51 MCM, 5.18 MCM] at 50% probability and 75% probability. The corn water supply at 50% probability and 75% probability is discretized within the interval [0.82 MCM, 3.36 MCM] and [1.27 MCM, 1.94 MCM], and the optimization solution results of the subsystem are shown in Tables 6 and 7:

Table 6

IWi∼ fi(IWi) ∼ (X*,Y*,PS*) correspondence table in a level year (50% probability)

IW1 (104m3)f1(IW1)Xj(j = 1,2,…,6)Yj(j = 1,2,…,6)PSj(j = 1,2,…,6)IW2 (104m3)f2(IW2)Xj(j = 1,2,3,4)Yj(j = 1,2,3,4)PSj(j = 1,2,3,4)
448 0.221 82 19 30 105 116 96 0 0 0 65 113 82 0 0 0 0 0 0 334 1.000 70 66 31 144 52 19 53 216 0 0 0 0 
453 0.231 82 19 35 105 116 96 0 0 0 70 113 82 0 0 0 0 0 0 329 1.000 70 66 31 144 52 19 48 216 0 0 0 0 
458 0.240 82 19 40 105 116 96 0 0 0 75 113 82 0 0 0 0 0 0 324 1.000 70 66 31 144 52 19 43 216 0 0 0 0 
463 0.249 82 19 45 105 116 96 0 0 0 80 113 82 0 0 0 0 0 0 319 1.000 70 66 31 144 52 19 38 216 0 0 0 0 
468 0.257 82 19 50 105 116 96 0 0 0 85 113 82 0 0 0 0 0 0 314 1.000 70 66 31 144 52 19 33 216 0 0 0 0 
473 0.265 82 19 55 105 116 96 0 0 0 90 113 82 0 0 0 0 0 0 309 0.998 70 66 31 142 52 19 28 216 0 0 0 0 
478 0.272 82 19 60 105 116 96 0 0 0 95 113 82 0 0 0 0 0 0 304 0.993 70 66 31 137 52 19 23 216 0 0 0 0 
483 0.279 82 19 65 105 116 96 0 0 0 100 113 82 0 0 0 0 0 0 299 0.988 70 66 31 132 52 19 18 216 0 0 0 0 
… … … … … … … … … … 
540 0.355 82 19 69 144 130 96 0 0 3 140 127 82 0 0 0 0 0 0 242 0.923 70 45 31 96 52 0 0 144 0 0 0 0 
545 0.362 82 19 69 146 133 96 0 0 3 142 130 82 0 0 0 0 0 0 237 0.917 70 42 31 94 52 0 0 139 0 0 0 0 
550 0.368 82 19 69 149 135 96 0 0 3 145 132 82 0 0 0 0 0 0 232 0.911 70 40 31 91 52 0 0 134 0 0 0 0 
555 0.375 82 19 69 152 137 96 0 0 3 147 135 82 0 0 0 0 0 0 227 0.905 70 37 31 89 52 0 0 129 0 0 0 0 
560 0.382 82 19 69 154 140 96 0 0 3 150 137 82 0 0 0 0 0 0 222 0.899 70 35 31 86 52 0 0 124 0 0 0 0 
565 0.388 82 19 69 157 142 96 0 0 3 152 140 82 0 0 0 0 0 0 217 0.893 70 32 31 84 52 0 0 119 0 0 0 0 
570 0.395 82 19 69 159 145 96 0 0 3 155 142 82 0 0 0 0 0 0 212 0.886 70 30 31 81 52 0 0 114 0 0 0 0 
… … … … … … … … … … 
650 0.505 82 19 69 199 185 96 0 0 3 195 182 82 0 0 0 0 0 0 132 0.778 53 0 31 48 6 0 0 32 0 0 0 0 
655 0.512 82 19 69 202 187 96 0 0 3 198 184 82 0 0 0 0 0 0 127 0.771 50 0 31 46 6 0 0 37 0 0 0 0 
660 0.519 82 19 69 204 190 96 0 0 3 200 187 82 0 0 0 0 0 0 122 0.764 47 0 31 44 6 0 0 42 0 0 0 0 
665 0.526 82 19 69 207 192 96 0 0 3 203 189 82 0 0 0 0 0 0 117 0.757 44 0 31 42 6 0 0 47 0 0 0 0 
670 0.533 82 19 69 209 195 96 0 0 3 205 192 82 0 0 0 0 0 0 112 0.750 42 0 31 39 6 0 0 52 0 0 0 0 
675 0.540 82 19 69 212 197 96 0 0 3 208 194 82 0 0 0 0 0 0 107 0.742 39 0 31 37 6 0 0 57 0 0 0 0 
680 0.547 82 20 69 213 200 96 0 0 4 209 197 82 0 0 0 0 0 0 102 0.735 35 0 31 36 6 0 0 62 0 0 0 0 
685 0.554 82 20 69 216 202 96 0 0 4 212 199 82 0 0 0 0 0 0 97 0.727 31 0 30 36 6 0 0 67 0 0 0 0 
690 0.561 82 20 69 219 204 96 0 0 4 215 201 82 0 0 0 0 0 0 92 0.720 29 0 27 36 6 0 0 72 0 0 0 0 
695 0.568 82 20 69 220 208 96 0 0 4 216 205 82 0 0 0 0 0 0 87 0.712 26 0 25 38 6 0 0 77 0 0 0 0 
700 0.575 82 21 69 220 212 96 0 0 5 216 209 82 0 0 0 0 0 0 82 0.704 24 0 22 36 6 0 0 82 0 0 0 0 
IW1 (104m3)f1(IW1)Xj(j = 1,2,…,6)Yj(j = 1,2,…,6)PSj(j = 1,2,…,6)IW2 (104m3)f2(IW2)Xj(j = 1,2,3,4)Yj(j = 1,2,3,4)PSj(j = 1,2,3,4)
448 0.221 82 19 30 105 116 96 0 0 0 65 113 82 0 0 0 0 0 0 334 1.000 70 66 31 144 52 19 53 216 0 0 0 0 
453 0.231 82 19 35 105 116 96 0 0 0 70 113 82 0 0 0 0 0 0 329 1.000 70 66 31 144 52 19 48 216 0 0 0 0 
458 0.240 82 19 40 105 116 96 0 0 0 75 113 82 0 0 0 0 0 0 324 1.000 70 66 31 144 52 19 43 216 0 0 0 0 
463 0.249 82 19 45 105 116 96 0 0 0 80 113 82 0 0 0 0 0 0 319 1.000 70 66 31 144 52 19 38 216 0 0 0 0 
468 0.257 82 19 50 105 116 96 0 0 0 85 113 82 0 0 0 0 0 0 314 1.000 70 66 31 144 52 19 33 216 0 0 0 0 
473 0.265 82 19 55 105 116 96 0 0 0 90 113 82 0 0 0 0 0 0 309 0.998 70 66 31 142 52 19 28 216 0 0 0 0 
478 0.272 82 19 60 105 116 96 0 0 0 95 113 82 0 0 0 0 0 0 304 0.993 70 66 31 137 52 19 23 216 0 0 0 0 
483 0.279 82 19 65 105 116 96 0 0 0 100 113 82 0 0 0 0 0 0 299 0.988 70 66 31 132 52 19 18 216 0 0 0 0 
… … … … … … … … … … 
540 0.355 82 19 69 144 130 96 0 0 3 140 127 82 0 0 0 0 0 0 242 0.923 70 45 31 96 52 0 0 144 0 0 0 0 
545 0.362 82 19 69 146 133 96 0 0 3 142 130 82 0 0 0 0 0 0 237 0.917 70 42 31 94 52 0 0 139 0 0 0 0 
550 0.368 82 19 69 149 135 96 0 0 3 145 132 82 0 0 0 0 0 0 232 0.911 70 40 31 91 52 0 0 134 0 0 0 0 
555 0.375 82 19 69 152 137 96 0 0 3 147 135 82 0 0 0 0 0 0 227 0.905 70 37 31 89 52 0 0 129 0 0 0 0 
560 0.382 82 19 69 154 140 96 0 0 3 150 137 82 0 0 0 0 0 0 222 0.899 70 35 31 86 52 0 0 124 0 0 0 0 
565 0.388 82 19 69 157 142 96 0 0 3 152 140 82 0 0 0 0 0 0 217 0.893 70 32 31 84 52 0 0 119 0 0 0 0 
570 0.395 82 19 69 159 145 96 0 0 3 155 142 82 0 0 0 0 0 0 212 0.886 70 30 31 81 52 0 0 114 0 0 0 0 
… … … … … … … … … … 
650 0.505 82 19 69 199 185 96 0 0 3 195 182 82 0 0 0 0 0 0 132 0.778 53 0 31 48 6 0 0 32 0 0 0 0 
655 0.512 82 19 69 202 187 96 0 0 3 198 184 82 0 0 0 0 0 0 127 0.771 50 0 31 46 6 0 0 37 0 0 0 0 
660 0.519 82 19 69 204 190 96 0 0 3 200 187 82 0 0 0 0 0 0 122 0.764 47 0 31 44 6 0 0 42 0 0 0 0 
665 0.526 82 19 69 207 192 96 0 0 3 203 189 82 0 0 0 0 0 0 117 0.757 44 0 31 42 6 0 0 47 0 0 0 0 
670 0.533 82 19 69 209 195 96 0 0 3 205 192 82 0 0 0 0 0 0 112 0.750 42 0 31 39 6 0 0 52 0 0 0 0 
675 0.540 82 19 69 212 197 96 0 0 3 208 194 82 0 0 0 0 0 0 107 0.742 39 0 31 37 6 0 0 57 0 0 0 0 
680 0.547 82 20 69 213 200 96 0 0 4 209 197 82 0 0 0 0 0 0 102 0.735 35 0 31 36 6 0 0 62 0 0 0 0 
685 0.554 82 20 69 216 202 96 0 0 4 212 199 82 0 0 0 0 0 0 97 0.727 31 0 30 36 6 0 0 67 0 0 0 0 
690 0.561 82 20 69 219 204 96 0 0 4 215 201 82 0 0 0 0 0 0 92 0.720 29 0 27 36 6 0 0 72 0 0 0 0 
695 0.568 82 20 69 220 208 96 0 0 4 216 205 82 0 0 0 0 0 0 87 0.712 26 0 25 38 6 0 0 77 0 0 0 0 
700 0.575 82 21 69 220 212 96 0 0 5 216 209 82 0 0 0 0 0 0 82 0.704 24 0 22 36 6 0 0 82 0 0 0 0 
Table 7

IWi∼ fi(IWi) ∼ (X*,Y*,PS*) correspondence table in a dry year (75% probability)

IW1 (104m3)f1(IW1)Xj(j = 1,2,…,6)Yj(j = 1,2,…,6)PSj(j = 1,2,…,6)IW2 (104m3)f2(IW2)Xj(j = 1,2,3,4)Yj(j = 1,2,3,4)PSj(j = 1,2,3,4)
451 0.219 85 8 21 106 126 105 0 0 5 105 131 103 0 0 0 0 0 0 194 0.607 48 28 60 58 49 29 49 129 0 0 0 0 
454 0.224 85 8 24 106 126 105 0 0 8 105 131 103 0 0 0 0 0 0 191 0.603 47 28 58 58 48 29 47 129 0 0 0 0 
457 0.23 85 8 27 106 126 105 0 0 11 105 131 103 0 0 0 0 0 0 188 0.599 45 28 57 58 47 29 45 129 0 0 0 0 
460 0.235 85 8 30 106 126 105 0 0 14 105 131 103 0 0 0 0 0 0 185 0.594 44 28 55 58 45 29 44 129 0 0 0 0 
463 0.24 85 8 33 106 126 105 0 0 17 105 131 103 0 0 0 0 0 0 182 0.59 43 28 53 58 44 29 42 129 0 0 0 0 
466 0.244 85 8 36 106 126 105 0 0 20 105 131 103 0 0 0 0 0 0 179 0.585 41 28 52 58 42 29 41 129 0 0 0 0 
469 0.249 85 8 39 106 126 105 0 0 23 105 131 103 0 0 0 0 0 0 176 0.581 40 28 50 58 41 29 39 129 0 0 0 0 
472 0.253 85 8 42 106 126 105 0 0 26 105 131 103 0 0 0 0 0 0 173 0.576 39 28 48 58 40 29 37 129 0 0 0 0 
475 0.258 85 8 45 106 126 105 0 0 29 105 131 103 0 0 0 0 0 0 170 0.572 37 28 47 58 38 29 36 129 0 0 0 0 
478 0.262 85 8 48 106 126 105 0 0 32 105 131 103 0 0 0 0 0 0 167 0.567 36 28 45 58 37 29 34 129 0 0 0 0 
481 0.266 85 8 51 106 126 105 0 0 35 105 131 103 0 0 0 0 0 0 164 0.562 35 28 43 58 36 29 32 129 0 0 0 0 
484 0.27 85 8 54 106 126 105 0 0 38 105 131 103 0 0 0 0 0 0 161 0.558 33 28 42 58 34 29 31 129 0 0 0 0 
487 0.274 85 8 57 106 126 105 0 0 41 105 131 103 0 0 0 0 0 0 158 0.553 33 28 39 58 34 29 28 129 0 0 0 0 
490 0.278 85 8 60 106 126 105 0 0 44 105 131 103 0 0 0 0 0 0 155 0.548 33 28 36 58 34 29 25 129 0 0 0 0 
493 0.282 85 8 62 107 126 105 0 0 46 106 131 103 0 0 0 0 0 0 152 0.544 33 28 33 58 34 29 22 129 0 0 0 0 
496 0.285 85 8 63 109 126 105 0 0 47 108 131 103 0 0 0 0 0 0 149 0.539 33 28 30 58 34 29 19 129 0 0 0 0 
499 0.289 85 8 64 111 126 105 0 0 48 110 131 103 0 0 0 0 0 0 146 0.534 33 28 27 58 34 29 16 129 0 0 0 0 
502 0.293 85 8 65 113 126 105 0 0 49 112 131 103 0 0 0 0 0 0 143 0.528 33 28 24 58 34 29 13 129 0 0 0 0 
505 0.297 85 8 65 116 126 105 0 0 49 115 131 103 0 0 0 0 0 0 140 0.523 33 28 21 58 34 29 10 129 0 0 0 0 
508 0.3 85 8 65 119 126 105 0 0 49 118 131 103 0 0 0 0 0 0 137 0.518 33 28 18 58 34 29 7 129 0 0 0 0 
511 0.304 85 8 65 122 126 105 0 0 49 121 131 103 0 0 0 0 0 0 134 0.512 33 28 15 58 34 29 4 129 0 0 0 0 
514 0.306 85 8 65 125 126 105 0 0 49 124 131 103 0 0 0 0 0 0 131 0.507 33 28 12 58 34 29 1 129 0 0 0 0 
517 0.311 85 8 65 128 126 105 0 0 49 127 131 103 0 0 0 0 0 0 128 0.511 33 28 9 58 34 29 0 127 0 0 0 0 
IW1 (104m3)f1(IW1)Xj(j = 1,2,…,6)Yj(j = 1,2,…,6)PSj(j = 1,2,…,6)IW2 (104m3)f2(IW2)Xj(j = 1,2,3,4)Yj(j = 1,2,3,4)PSj(j = 1,2,3,4)
451 0.219 85 8 21 106 126 105 0 0 5 105 131 103 0 0 0 0 0 0 194 0.607 48 28 60 58 49 29 49 129 0 0 0 0 
454 0.224 85 8 24 106 126 105 0 0 8 105 131 103 0 0 0 0 0 0 191 0.603 47 28 58 58 48 29 47 129 0 0 0 0 
457 0.23 85 8 27 106 126 105 0 0 11 105 131 103 0 0 0 0 0 0 188 0.599 45 28 57 58 47 29 45 129 0 0 0 0 
460 0.235 85 8 30 106 126 105 0 0 14 105 131 103 0 0 0 0 0 0 185 0.594 44 28 55 58 45 29 44 129 0 0 0 0 
463 0.24 85 8 33 106 126 105 0 0 17 105 131 103 0 0 0 0 0 0 182 0.59 43 28 53 58 44 29 42 129 0 0 0 0 
466 0.244 85 8 36 106 126 105 0 0 20 105 131 103 0 0 0 0 0 0 179 0.585 41 28 52 58 42 29 41 129 0 0 0 0 
469 0.249 85 8 39 106 126 105 0 0 23 105 131 103 0 0 0 0 0 0 176 0.581 40 28 50 58 41 29 39 129 0 0 0 0 
472 0.253 85 8 42 106 126 105 0 0 26 105 131 103 0 0 0 0 0 0 173 0.576 39 28 48 58 40 29 37 129 0 0 0 0 
475 0.258 85 8 45 106 126 105 0 0 29 105 131 103 0 0 0 0 0 0 170 0.572 37 28 47 58 38 29 36 129 0 0 0 0 
478 0.262 85 8 48 106 126 105 0 0 32 105 131 103 0 0 0 0 0 0 167 0.567 36 28 45 58 37 29 34 129 0 0 0 0 
481 0.266 85 8 51 106 126 105 0 0 35 105 131 103 0 0 0 0 0 0 164 0.562 35 28 43 58 36 29 32 129 0 0 0 0 
484 0.27 85 8 54 106 126 105 0 0 38 105 131 103 0 0 0 0 0 0 161 0.558 33 28 42 58 34 29 31 129 0 0 0 0 
487 0.274 85 8 57 106 126 105 0 0 41 105 131 103 0 0 0 0 0 0 158 0.553 33 28 39 58 34 29 28 129 0 0 0 0 
490 0.278 85 8 60 106 126 105 0 0 44 105 131 103 0 0 0 0 0 0 155 0.548 33 28 36 58 34 29 25 129 0 0 0 0 
493 0.282 85 8 62 107 126 105 0 0 46 106 131 103 0 0 0 0 0 0 152 0.544 33 28 33 58 34 29 22 129 0 0 0 0 
496 0.285 85 8 63 109 126 105 0 0 47 108 131 103 0 0 0 0 0 0 149 0.539 33 28 30 58 34 29 19 129 0 0 0 0 
499 0.289 85 8 64 111 126 105 0 0 48 110 131 103 0 0 0 0 0 0 146 0.534 33 28 27 58 34 29 16 129 0 0 0 0 
502 0.293 85 8 65 113 126 105 0 0 49 112 131 103 0 0 0 0 0 0 143 0.528 33 28 24 58 34 29 13 129 0 0 0 0 
505 0.297 85 8 65 116 126 105 0 0 49 115 131 103 0 0 0 0 0 0 140 0.523 33 28 21 58 34 29 10 129 0 0 0 0 
508 0.3 85 8 65 119 126 105 0 0 49 118 131 103 0 0 0 0 0 0 137 0.518 33 28 18 58 34 29 7 129 0 0 0 0 
511 0.304 85 8 65 122 126 105 0 0 49 121 131 103 0 0 0 0 0 0 134 0.512 33 28 15 58 34 29 4 129 0 0 0 0 
514 0.306 85 8 65 125 126 105 0 0 49 124 131 103 0 0 0 0 0 0 131 0.507 33 28 12 58 34 29 1 129 0 0 0 0 
517 0.311 85 8 65 128 126 105 0 0 49 127 131 103 0 0 0 0 0 0 128 0.511 33 28 9 58 34 29 0 127 0 0 0 0 

The optimal irrigation water volume of winter wheat and corn in the entire growth period obtained by the large system can be checked back in Tables 6 and 7, and the optimal water supply and spill processes of the reservoirs, and the optimal water replenishment of the pumping stations in different periods of the irrigation system can be obtained.

Algorithm optimality

In order to compare the performance of the algorithms of DADP and RGA, the optimization of the algorithm is analyzed from four aspects: the optimization of the objective function value, the stability of the algorithm, the convergence of the algorithm and the solving speed of the algorithm.

To compare the optimality of the objective function, the sensitivity analysis of the two algorithms is required first. As shown in Table 8, for DADP, in a level year (50% probability), the optimal value of the objective function is (1.86E + 07) RMB, and in the dry year (75% probability), the optimal value of the objective function is (1.38E + 07) RMB. There is no parameter demand calibration in the solution process.

Table 8

Sensitivity analysis of DADP and RGA

Level year (50% probability) DADP Objective function      
1.86E + 07      
RGA Population size Objective function Crossover pm Objective function Mutation pe Objective function 
20 1.68E + 07 0.3 1.57E + 07 0.1 1.66E + 07 
40 1.71E + 07 0.4 1.63E + 07 0.2 1.82E + 07 
60 1.55E + 07 0.5 1.68E + 07 0.3 1.69E + 07 
80 1.63E + 07 0.6 1.82E + 07 0.4 1.74E + 07 
100 1.82E + 07 0.7 1.76E + 07 0.5 1.80E + 07 
Dry year (75% probability) DADP Objective function      
1.38E + 07      
RGA Population size Objective function Crossover pm Objective function Mutation pe Objective function 
20 1.25E + 07 0.3 1.31E + 07 0.1 1.31E + 07 
40 1.33E + 07 0.4 1.22E + 07 0.2 1.29E + 07 
60 1.29E + 07 0.5 1.37E + 07 0.3 1.36E + 07 
80 1.37E + 07 0.6 1.34E + 07 0.4 1.37E + 07 
100 1.35E + 07 0.7 1.24E + 07 0.5 1.35E + 07 
Level year (50% probability) DADP Objective function      
1.86E + 07      
RGA Population size Objective function Crossover pm Objective function Mutation pe Objective function 
20 1.68E + 07 0.3 1.57E + 07 0.1 1.66E + 07 
40 1.71E + 07 0.4 1.63E + 07 0.2 1.82E + 07 
60 1.55E + 07 0.5 1.68E + 07 0.3 1.69E + 07 
80 1.63E + 07 0.6 1.82E + 07 0.4 1.74E + 07 
100 1.82E + 07 0.7 1.76E + 07 0.5 1.80E + 07 
Dry year (75% probability) DADP Objective function      
1.38E + 07      
RGA Population size Objective function Crossover pm Objective function Mutation pe Objective function 
20 1.25E + 07 0.3 1.31E + 07 0.1 1.31E + 07 
40 1.33E + 07 0.4 1.22E + 07 0.2 1.29E + 07 
60 1.29E + 07 0.5 1.37E + 07 0.3 1.36E + 07 
80 1.37E + 07 0.6 1.34E + 07 0.4 1.37E + 07 
100 1.35E + 07 0.7 1.24E + 07 0.5 1.35E + 07 

For the RGA, in a level year (50% probability), the optimal value of the objective function is (1.82E + 07) RMB, when the population size is 100, the crossover rate pm is 0.6, and the variation rate pe is 0.2; In a dry year (75% probability), the optimal value of the objective function is (1.37E + 07) RMB. At this time, the population size is 80, the crossover rate pm is 0.5, and the variation rate pe is 0.4. Although the objective function values of the above two algorithms are close, the DADP proposed in this study is still slightly better than RGA. At the same time, DADP does not need calibration parameters in the solution process, while RGA has three parameters that need sensitivity analysis. The essence of the DADP algorithm is to decompose the high-dimensional DP model into multiple one-dimensional DP models. The solution results are not affected by additional parameters and have better algorithm operability.

According to the parameters specified in Table 8, DADP and RGA were run 10 times to verify the stability of each algorithm. As shown in Table 9, the objective function value obtained by DADP is the best in the 10 times of operation results. Compared with the best solution, the worst solution, the average and the standard deviation, DADP is still better than RGA. Therefore, the above results show that the proposed DADP has better algorithm stability.

Table 9

Results of ten runs of DADP and RGA

RunLevel year (50% probability)
Dry year (75% probability)
DADPRGADADPRGA
1.86E + 07 1.66E + 07 1.38E + 07 1.31E + 07 
1.86E + 07 1.82E + 07 1.38E + 07 1.22E + 07 
1.86E + 07 1.69E + 07 1.38E + 07 1.37E + 07 
1.86E + 07 1.74E + 07 1.38E + 07 1.34E + 07 
1.86E + 07 1.80E + 07 1.38E + 07 1.24E + 07 
1.86E + 07 1.68E + 07 1.38E + 07 1.31E + 07 
1.86E + 07 1.71E + 07 1.38E + 07 1.29E + 07 
1.86E + 07 1.55E + 07 1.38E + 07 1.36E + 07 
1.86E + 07 1.63E + 07 1.38E + 07 1.37E + 07 
10 1.86E + 07 1.82E + 07 1.38E + 07 1.35E + 07 
Best 1.86E + 07 1.82E + 07 1.38E + 07 1.37E + 07 
Worst 1.86E + 07 1.55E + 07 1.38E + 07 1.22E + 07 
Average 1.86E + 07 1.71E + 07 1.38E + 07 1.32E + 07 
Standard deviation 8.31E + 05 5.04E + 05 
RunLevel year (50% probability)
Dry year (75% probability)
DADPRGADADPRGA
1.86E + 07 1.66E + 07 1.38E + 07 1.31E + 07 
1.86E + 07 1.82E + 07 1.38E + 07 1.22E + 07 
1.86E + 07 1.69E + 07 1.38E + 07 1.37E + 07 
1.86E + 07 1.74E + 07 1.38E + 07 1.34E + 07 
1.86E + 07 1.80E + 07 1.38E + 07 1.24E + 07 
1.86E + 07 1.68E + 07 1.38E + 07 1.31E + 07 
1.86E + 07 1.71E + 07 1.38E + 07 1.29E + 07 
1.86E + 07 1.55E + 07 1.38E + 07 1.36E + 07 
1.86E + 07 1.63E + 07 1.38E + 07 1.37E + 07 
10 1.86E + 07 1.82E + 07 1.38E + 07 1.35E + 07 
Best 1.86E + 07 1.82E + 07 1.38E + 07 1.37E + 07 
Worst 1.86E + 07 1.55E + 07 1.38E + 07 1.22E + 07 
Average 1.86E + 07 1.71E + 07 1.38E + 07 1.32E + 07 
Standard deviation 8.31E + 05 5.04E + 05 

Figure 5(a) shows the convergence trend of DADP and RGA. It is known that the running solution of the DADP method is independent of the number of iterations of the algorithm, and the optimal objective function value can be obtained by running it once. Therefore, the convergence rate of DADP is much earlier than that of RGA. Figure 5(b) shows the running time of the algorithm. Obviously, although RGA can be solved quickly when the number of iterations is small, as the number of iterations increases to 500 generations, the running time of the algorithm will exceed that of the DADP method.
Figure 5

(a) is the convergence trend of the algorithm; (b) is the solution time of the algorithm.

Figure 5

(a) is the convergence trend of the algorithm; (b) is the solution time of the algorithm.

Close modal

In general, compared with RGA, DADP has outstanding performance in optimization of the objective function, algorithm stability, algorithm convergence and algorithm solving time, and has better algorithm optimization.

Algorithm applicability

The applicability of DADP and RGA is compared and analyzed. This study considers the scheduling process and results of the irrigation system. For the water source project, the most common dispatching method is the standard operation policy (SOP) (Faber & Stedinger 2001; Ngo et al. 2007). Therefore, in order to analyze the dispatching effect of the optimization algorithm, the dispatching process and results of SOP, DADP and RGA are comprehensively compared and analyzed.

As shown in Figure 6(a) and 6(b), the water supply trends of DADP and RGA are basically consistent in a level year (50% probability) and a dry year (75% probability), with slight differences only in some periods. SOP is characterized by more water supply in the early stage, less water supply in the later stage and concentrated water shortage. In the case of insufficient water resources in the irrigation area, the water deficit at different growth stages of dry crops has a significant impact on the final yield. Obviously, SOP does not fully consider the relationship between water supply and the yield of crops. Therefore, compared with SOP, both DADP and RGA can formulate reasonable scheduling strategies for the scheduling process of the irrigation system.
Figure 6

(a) Reservoir water supply process under 50% probability; (b) reservoir water supply process under 75% probability.

Figure 6

(a) Reservoir water supply process under 50% probability; (b) reservoir water supply process under 75% probability.

Close modal

It can be seen from Table 10 that in a level year (50% probability), the total output value of the irrigation area obtained by DADP and RGA is 8.4% and 5.8% higher than that of SOP respectively; In a dry year (75% probability), it increased by 4.5% and 3.8% respectively. In terms of irrigation water volume, compared with SOP and RGA, DADP increases the water supply to 0.15 MCM and 0.3 MCM respectively in a level year (50% probability) and 0.22 MCM and 0.43 MCM, respectively, in a dry year (75% probability). The operation results of the above irrigation system show that the scheduling results of the irrigation system can be optimized using DADP and RGA. In addition, compared with RGA, DADP can also increase the total water supply of the irrigation system, reduce water shortage, effectively improve the utilization rate of water resources in the water shortage irrigation area, and alleviate the water conflict in the irrigation area on the basis of improving the output value of the irrigation area.

Table 10

Irrigation system operation results (MCM)

MethodProbabilitySupplyReplenishmentSpillEvaporationShortageTotal value (107 RMB)
SOP 50% 767 600 161 585 1.72 
75% 623 600 166 991 1.32 
DADP 50% 782 600 146 570 1.86 
75% 645 600 144 969 1.38 
RGA 50% 752 600 146 600 1.82 
75% 602 600 143 1,012 1.37 
MethodProbabilitySupplyReplenishmentSpillEvaporationShortageTotal value (107 RMB)
SOP 50% 767 600 161 585 1.72 
75% 623 600 166 991 1.32 
DADP 50% 782 600 146 570 1.86 
75% 645 600 144 969 1.38 
RGA 50% 752 600 146 600 1.82 
75% 602 600 143 1,012 1.37 

In general, the analysis of the scheduling process and results of the irrigation system show that DADP has better algorithm applicability than RGA, and is more suitable for planning and solving the optimal allocation of water resources in such a joint scheduling model.

On the basis of considering regional water rights restrictions, this study establishes a water resources optimal operation model for joint irrigation of reservoirs and pumping stations. The model is solved using the DADP method. Aiming at the more difficult judgment constraints such as the operation criteria, it is coupled with the recursive process of DP to simultaneously solve the optimal water supply, water spill process of the irrigation system and the optimal water replenishment process of the pumping station, and formulate an effective joint operation strategy of reservoir and pumping station. The analysis results of the optimization and applicability of the algorithm show that the optimization model and solution method established in this paper can not only realize the optimal allocation of water resources in complex irrigation systems, but also improve the agricultural output value of irrigation areas, and provide a theoretical basis for the formulation of similar irrigation system operation schemes, while ensuring the sustainable development of a regional economy. In addition, the insufficient irrigation model and the solution method of large systems established in this paper have certain reference significance for future research on the optimal operation of the ‘Reservoir (group) and Pumping station (group)’ joint irrigation with more complex topology.

This work was supported by the National Natural Science Foundation of China (NSFC) [grant number 52079119].

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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