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

Abstract

The mathematical model of optimal water quantity allocation for a single main canal in a large-scale irrigation area was constructed that took the minimal sum of the squared deviation of water shortage for water receiving areas controlled by the single main canal in one given irrigation period as the study target, and the total irrigation quantity of the single main canal as a constraint condition. Taking the optimal allocation of water quantity of each branch canal as decision variables, and several branch canals under the irrigation sequence of the main canal as a state variable, this model was solved by the one-dimensional dynamic programming (DP) method, by which the minimal water shortage and corresponding optimal water quantity allocation of each branch canal was calculated. The proposed method could provide a decision-making reference for optimal water resources allocation of single main canal irrigation areas, and also provide the theoretical basis for optimal water quantity allocation of a main canal with rotation irrigation by strips or with segmented rotation irrigation mode in China’s large-scale irrigation areas. Taking Hengliu Main Canal of Zhouqiao Irrigation Area in Jiangsu Province as a study case, optimization results showed that in a medium drought year (p = 75%) and a special drought year (p = 95%), minimal water shortage for water receiving areas controlled by Hengliu Main Canal was respectively 2.57 × 10⁴ m³ and 23.31 × 10⁴ m³ during the ponding period of rice. The corresponding water quantity allocation for each branch canal has reflected a compellent model solution precision and efficiency.

1. Introduction

As an important kind of infrastructure in agricultural water conservancy project, canals are also indispensable components of water transfer modes in large-scale irrigation areas. China’s single large-scale irrigation areas usually cover over 30 × 10⁴ acres of irrigation area, which leads to the characteristics that water conveyance canals usually include multi-stage and long-distance water conveyance with multiple hydraulic structures along canals at all levels. Affected by water rights control policy enforced by the Chinese government that focuses on agricultural water consumption saving in recent years, and combined with still relatively extensive water distribution management, it is necessary to improve agricultural water resources efficiency in irrigation areas by means of scientific water resources allocation methods. According to the current agricultural farming structures of irrigation areas, a system optimization model could be applied to optimize water quantity allocation for each branch canal of a single main canal to improve full utilization of water resources in irrigation areas, which will also promote increased agricultural production and increasing farmers’ income in irrigation areas, and serve China’s rural revitalization strategy.

Aiming at efficient utilization of water resources in irrigation areas, conventional research has mainly covered optimization design for vertical and cross sections of canals [1,2], optimal layout of a canal system [3], water allocation optimization of a canal system [4], and optimal allocation of water resources for irrigation areas including canals and field crops [5,6]. Also there is some relevant research focused on heat preservation for lining canals in alpine areas [7,8]. In recent years, studies have paid more attention to multi-objective optimization, including optimization design for canal sections with considerations of water conveyance efficiency and construction cost demands [1], optimal water delivery with multi-objectives of minimal water loss, maximal net benefit of irrigation and drainage [9], multi-water resources optimal allocation with increasing water net benefit and decreasing water-use amount [10], and optimal allocation of water resources considering minimal water loss of the main canal combined with maximal economic benefit of crops in irrigation areas [11,12]. Considering the uncertainties of agricultural irrigation based on agricultural sustainability, stochastic and multi-objective programming [13], multi-scale multi-objective programming [14], inexact interval programming [15], and interval two-stage stochastic programming [16] were developed to optimize agricultural water resources. Wu et al. [17] optimized the cross-section parameters with Matlab, which has the advantages of calculation stability and simplicity on the optimization design of the cross-sections of canals. Xu et al. [18] studied the optimal scheduling of water distribution in irrigation areas, and the optimal scheduling model of water distribution in dry and branch channels was established with the optimization objective of minimum water loss in the irrigation area, and even with changes of the water level in the main channel, using the NSGA-II algorithm. Wang et al. [19] took the minimum leakage loss in the water allocation process as the goal to establish the water allocation model for a channel system with an improved particle swarm optimization algorithm. Singh [20] presented the formulation and application of a management model for the optimal allocation of available good quality water and land resources to maximize the farm revenue of a canal command area.

Considering the researches carried out at home and abroad, it is clear that the optimal water resources allocation of branch canals for a single main canal with a total given water quantity is an important basis for further research on multi-objective optimization, which requires constructing complex mathematical models and applying modern intelligent algorithms. Taking a single main canal with its controlling branch canals as the research object, a mathematical model of optimal water quantity allocation for a single main canal has been put forward in this paper. Optimization theory of complex systems was applied to solve the constructed model, which is used to obtain the optimal water quantity allocation of each one of the branch canals corresponding to the minimal water shortage for water receiving areas. The study results could provide the theoretical basis for optimal water quantity allocation for main canals with rotation irrigation (either by strips or with segmented rotation irrigation) in large-scale irrigation areas, and also provide a highly informative decision-making reference for managers of relevant irrigation areas.

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

Taking the minimal sum of the squared deviation of the water shortage for the water receiving areas controlled by all of the branch canals of a single main canal in one given irrigation period as the study target, a mathematical model of water quantity allocation optimization for a single main canal in a large-scale irrigation area was constructed that took the optimal allocation of water quantity of each one of the branch canals as decision variables, and the total irrigation quantity of a single main canal as the constraint condition.

$$\mathrm{Objective} \mathrm{function}: F=\min f=\min {\displaystyle \sum _{j=1}^m{\left(X_j-Y{S}_j\right)}^2}$$

(1)

$$\mathrm{Constraint} \mathrm{condition}: {\displaystyle \sum _{j=1}^m{X}_j\le W_0}$$

(2)

where F = minimal sum of the squared deviation of the water shortage for the water receiving areas controlled by all of the branch canals of a single main canal in one given irrigation period; f = sum of the squared deviation of the water shortage for the water receiving areas controlled by all of the branch canals for a single main canal in one given irrigation period; m = quantity of branch canals for a single main canal; j = branch canal number (j = 1, 2, …, m); X_j = water quantity allocation of the j-th branch canal (m³); YS_j = water requirement of the water receiving area controlled by the j-th branch canal (m³); W₀ = water right of single water diversion of the main canal (m³).

3. Model Solution Method

The above constructed model (1)~(2) is a non-linear mathematical model that takes the position number of each one of the branch canals as stage variables, and the optimal allocation of water quantity of each one of the branch canals as decision variables. Considering the total water quantity allocation of former branch canals as state variable λ_j, this constructed model could be solved by one-dimensional dynamic programming.

(1)

Considering the position number of each one of the branch canals as stage variable j, and the total water quantity allocation of former branch canals as state variable λ_j, the state transition equation could be constructed as follows:

$${\lambda }_j=X_j+{\lambda }_{j-1}$$

(3)

(2)

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

① Stage j = 1

$$g_1\left({\lambda }_1\right)=\min {\left(X_1-Y{S}_1\right)}^2$$

(4)

The state variable of the first stage λ₁ is discretized into 0, W₁, W₂, …, W₀. For each one of λ₁, the decision variable X₁ is discretized into 0, X₁₁, X₁₂, …, X_1max, where X_1max is the maximal water quantity allocation corresponding to the state variable of the first stage λ₁. The inequality of X₁ ≥ λ₁ should be met according to Formula (2).

② Stage j = 2, 3, …, m−1

$$g_j\left({\lambda }_j\right)=\min \left[{\left(X_j-Y{S}_j\right)}^2+g_{j-1}\left({\lambda }_{j-1}\right)\right]$$

(5)

The state variable of the first stage λ_j is also discretized into 0, W₁, W₂, …, W₀. For each one of λ_j, the decision variable X_j is also discretized into 0, X_j1, X_j2, …, X_jmax, where X_jmax is the maximal water quantity allocation corresponding to the state variable of the j-th stage λ_j. The inequality of ${\displaystyle \sum _{j=1}^j{X}_j}\ge {\lambda }_j$ should be met, where j = 2, 3, …, m−1.

According to the state transition Equation (3), the merit function of Formula (5) could be transformed into the following form:

$$g_j\left({\lambda }_j\right)=\min \left[{\left(X_j-Y{S}_j\right)}^2+g_{j-1}\left({\lambda }_j-X_i\right)\right]$$

(6)

③ Stage j = m

The merit function is as follows:

$$g_m\left({\lambda }_m\right)=\min \left[{\left(X_m-Y{S}_m\right)}^2+g_{m-1}\left({\lambda }_{m-1}\right)\right]$$

(7)

The state variable of the m-th stage λ_m is set as W₀. According to ${\lambda }_{m-1}={\lambda }_m-X_m$, Formula (7) could be transformed into the following form:

$$g_m\left(W_0\right)=\min \left[{\left(X_m-Y{S}_m\right)}^2+g_{m-1}\left(W_0-X_m\right)\right]$$

(8)

The decision variable X_m is also discretized in its feasible region, by which to obtain the minimal sum of the squared deviation of the water shortage for the water receiving areas F and the corresponding optimal water quantity allocation of the m-th branch canal X_m^*.

Finally, searching of the reverse order is carried out by means of a state transition equation of each stage, which would help determine the minimal sum of the squared deviation of the water shortage for the water receiving areas F and the corresponding optimal water quantity allocation of the each one of the branch canal X_j^* (j = 1, 2, 3, …, m).

4. Model Application of Study Case

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

Zhouqiao Irrigation Area, located in Hongze District of Jiangsu Province, is a large-scale irrigation area with a designed irrigation area of 0.32 million mu and effective irrigation area of 0.3195 million mu. As the water diversion project, Zhouqiao Sluice has a water diversion capacity of 28 m³/s. There are 8 main canals (sub-trunk canals) with a combined total length of 84.1 km, and 101 branch canals with a combined total length of 267.38 km.

Zhouqiao Irrigation Area has a main canal with a segmented rotation irrigation mode as its operating regime. The Hengliu Main Canal system is chosen from within Zhouqiao Irrigation Area as a study case, which has a length of 4.96 km covering an irrigation area of 11,635 mu. The water receiving area controlled by the Hengliu Main Canal is usually used for rice-wheat crop rotation, whose system generalization and basic information are shown in Figure 1 and

Table 1. Basin information of the Hengliu Main Canal.

Main Canal	Stake Number	Bottom Width b (m)	Slope Ratio m (m)	Water Depth h (m)	Longitudinal Slope i	Roughness n	Design Flow Q (m3/s)
Hengliu Main Canal	0 + 000~2 + 800	4.00	1.50	2.00	0.000125	0.0225	4.14
	2 + 800~4 + 960	4.20	1.50	1.40	0.000125	0.0225	1.86

, respectively. The basic information of each one of the branch canals in the Hengliu Main Canal system is shown in

Table 2. Basic information of each of the branch canals in the Hengliu Main Canal system.

Main Canal	Branch Canal	Length of Branch Canal (km)	Covered Irrigation Area (mu)	Canal Head Structure
Hengliu Main Canal	Beixuyang Branch Canal	2.50	2700	Beixuyang Branch canal head
	Beitouwei Branch Canal	2.52	2750	Beitouwei Branch canal head
	Beiweicheng Branch Canal	2.70	3000	Beiweicheng Branch canal head
	Nanheping Branch Canal	3.99	3185	Nanheping Branch canal head

.

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

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

Taking the ponding period of rice as an example, and considering 110 m³/mu as the irrigation quota, and respectively considering 0.9, 0.91, and 0.95 as the water utilization coefficient of branch canals, field canals, and fields, respectively, the irrigation water demand of each piece of the water receiving area corresponding to each branch canal was calculated, and is shown in

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

Branch Canal	Control Area (mu)	Planting Proportion α	Planting Area (mu)	Crop Water Requirement (104 m3)		Available Water Supply (104 m3)
				Net Water Requirement	Water Requirement Converted to Branch Head	Gross Water Supply		Water Supply Converted to Branch Head
						p = 75%	p = 95%	p = 75%	p = 95%
Beixuyang Branch Canal	2700	0.68	1836	20.20	23.36	25.31	20.12	22.78	18.11
Beitouwei Branch Canal	2750	0.70	1925	21.18	24.49	26.54	21.09	23.88	18.99
Beiweicheng Branch Canal	3000	0.72	2160	23.76	27.48	29.78	23.67	26.80	21.30
Nanheping Branch Canal	3185	0.70	2230	24.53	28.37	30.74	24.44	27.67	21.99
Total	11,635		8151	89.67	103.7	112.37	89.32	101.13	80.39

.

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

The agricultural water permits of 2019 for Zhouqiao Irrigation Area was 160 million m³ with an effective irrigation area of 0.3195 million mu. For the 11,635 mu covered by the Hengliu Main Canal and the corresponding planting proportion controlled by each branch canal, considering a medium drought year (p = 75%) and a special drought year (p = 95%), water demand quantity is respectively 27.53% and 21.88% of the irrigation quota with the corresponding available water supply of 1.604 million m³ and 1.275 million m³, respectively. According to crop planting area of each water receiving area controlled by each branch canal, available water supply for each branch canal under two different year types is obtained which is shown in Table 3.

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

Table 3 shows that the total water supply of all the branch canals controlled by Hengliu Main Canal during the ponding period of rice is 1.0113 million m³ and 0.8039 million m³ in a medium drought year (p = 75%) and a special drought year (p = 95%), respectively. These two water supply quantities are both less than the total water requirement of 103.7 million m³ for all of the branch canals controlled by Hengliu Main Canal, which makes it necessary to carry out optimal water quantity allocation for all branch canals under the certain water rights.

Taking 1.0113 million m³ and 0.8039 million m³ as the total water supply of Hengliu Main Canal for a ponding period of 6.5 days in a medium drought year (p = 75%) and a special drought year (p = 95%), respectively, a one-dimensional dynamic programming can be applied to solve mathematical model (1)~(2) for the total water requirement of 103.7 million m³. From this, the optimal water quantity allocation of each branch canal can be obtained for a medium drought year (p = 75%) and a special drought year (p = 95%). The optimization results are shown in

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

Branch Canal	Water Requirement Converted to Branch Head (104 m3)	Optimal Water Quantity Allocation for Each Branch Canal (104 m3)		Minimal Water Shortage (104 m3)		Satisfaction Degree (%)		Objective Value
		p = 75%	p = 95%	p = 75%	p = 95%	p = 75%	p = 95%	p = 75%	p = 95%
Beixuyang Branch Canal	23.36	22.72	17.54	0.64	5.82	97.26	75.09	0.41	33.87
Beitouwei Branch Canal	24.49	23.85	18.66	0.64	5.83	97.39	76.19	0.41	33.99
Beiweicheng Branch Canal	27.48	26.84	21.65	0.64	5.83	97.67	78.78	0.41	33.99
Nanheping Branch Canal	28.37	27.72	22.54	0.65	5.83	97.71	79.45	0.42	33.99
Total	103.7	101.13	80.39	2.57	23.31	97.52	77.52	1.65	135.84

. On the other hand, water quantity allocation results by equal proportion for each water receiving area controlled by its branch canal calculated from the total water quantity of Hengliu Main Canal are shown in

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

Branch Canal	Water Requirement Converted to Branch Head (104 m3)	Optimal Water Quantity Allocation for Each Branch Canal (104 m3)		Minimal Water Shortage (104 m3)		Satisfaction Degree (%)		Objective Value
		p = 75%	p = 95%	p = 75%	p = 95%	p = 75%	p = 95%	p = 75%	p = 95%
Beixuyang Branch Canal	23.36	22.78	18.11	0.58	5.25	97.52	77.52	0.34	27.58
Beitouwei Branch Canal	24.49	23.88	18.98	0.61	5.51	97.52	77.52	0.37	30.31
Beiweicheng Branch Canal	27.48	26.80	21.30	0.68	6.18	97.52	77.52	0.46	38.16
Nanheping Branch Canal	28.37	27.67	21.99	0.70	6.38	97.52	77.52	0.50	40.67
Total	103.7	101.13	80.39	2.57	23.31	97.52	77.52	1.66	136.72

, with the same consideration of a medium drought year (p = 75%) and a special drought year (p = 95%). For convenience of discussion, DP-mode was defined as the optimal water quantity allocation with dynamic programming, while EP-mode was defined as water quantity allocation by equal proportion for each water receiving area controlled by its branch canal. The comparison of the water quantity allocation for each branch canal for two above modes is shown in Figure 2.

Combined with Table 4 and Table 5, Figure 2 indicates that the minimal water shortage of the Hengliu Main Canal system are 2.57 × 10⁴ m³ and 23.31 × 10⁴ m³ in the medium and special drought years, respectively, calculated by the two above water quantity allocation modes. Compared with EP-mode, the real water shortage for each branch canal is better-distributed in a medium drought year (p = 75%, 0.64~0.65 × 10⁴ m³) and a special drought year (p = 95%, 5.82~5.83 × 10⁴ m³) when calculated using DP-mode, which contributes to alleviate water utilization contradictions of water consumers. By means of optimal water quantity allocation using DP-mode, the decreasing range of the minimal sum of the squared deviation of the water shortage for the water receiving areas is 0.76% and 0.64% corresponding to a medium drought year (p = 75%) and a special drought year (p = 95%), respectively, compared with water quantity allocation calculated using EP-mode.

5. Conclusions

Aiming at water supply uniformity for water receiving areas controlled by their respective branch canals in single main canal system, the mathematical model for optimal water quantity allocation of a single main canal was constructed, and a one-dimensional dynamic programming model solution was proposed, by which the optimal water quantity allocation of each branch canal controlled by a single main canal with a single total water supply right was obtained.

According to the optimization and analysis of the case study in different year types, the proposed model and its solution has been proved feasible (with high solution efficiency) for the management of water resources in single main canal systems in an irrigation area. On the basis of the optimization result above, it would be effective to carry out studies on optimal operation of sluice groups along a main canal, which can effectively improve optimal water resources allocation by long-distance automatic controlling of sluice groups. More importantly, the research achievements in this paper provides an important theoretical basis for further studies on the optimization of water resources allocation for main canals with rotation irrigation modes (strips or segmented rotation irrigation) in China’s large-scale irrigation areas. For multiple canal systems with complicated operation regimes, large-scale decomposition-dynamic programming aggregation could be effectively applied to realize optimal water resources allocation by taking a single main canal as a subsystem.

A limitation of this study is that short-time rainfall forecasting has not been considered, and this may reduce the optimization benefit of this constructed model, which is something that should be focused on in future studies.