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

: 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 × 10 4 m 3 and 23.31 × 10 4 m 3 during the ponding period of rice. The corresponding water quantity allocation for each branch canal has reﬂected a compellent model solution precision and efﬁciency.


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 × 10 4 acres of irrigation area, which leads to the characteristics that water conveyance canals usually include multistage 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 multiobjective 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], multiscale 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.

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.
Objective function : Water 2022, 14, 3917 3 of 8 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 (m 3 ); YS j = water requirement of the water receiving area controlled by the j-th branch canal (m 3 ); W 0 = water right of single water diversion of the main canal (m 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: (2) According to objective function (1) and state transition Equation ( 3), the merit function of each optimization stage could be obtained as follows: The state variable of the first stage λ 1 is discretized into 0, W 1 , W 2 , . . ., W 0 .For each one of λ 1 , the decision variable X 1 is discretized into 0, X 11 , X 12 , . . ., X 1max , where X 1max is the maximal water quantity allocation corresponding to the state variable of the first stage λ 1 .The inequality of X 1 ≥ λ 1 should be met according to Formula (2).
The state variable of the first stage λ j is also discretized into 0, W 1 , W 2 , . . ., W 0 .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 j ∑ j=1 X j ≥ λ 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: 3 Stage j = m The merit function is as follows: The state variable of the m-th stage λ m is set as W 0 .According to λ m−1 = λ m − X m , Formula (7) could be transformed into the following form: 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).

Basic Information of Hengliu Main Cain System in Zhouqiao Irrigation Area
Zhouqiao Irrigation Area, located in Hongze District of Jiangsu Province, is a largescale 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 m 3 /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, respectively.The basic information of each one of the branch canals in the Hengliu Main Canal system is shown in Taking the ponding period of rice as an example, and considering 110 m 3 /mu as the irrigation quota, and respectively considering 0.9, 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.The agricultural water permits of 2019 for Zhouqiao Irrigation Area was 160 million m 3 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 m 3 and 1.275 million m 3 , 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.

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 m 3 and 0.8039 million m 3 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 m 3 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 m 3 and 0.8039 million m 3 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 m 3 .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. 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, 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. = 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 Tables 4 and 5, Figure 2 indicates that the minimal water shortage of the Hengliu Main Canal system are 2.57 × 10 4 m 3 and 23.31 × 10 4 m 3 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 betterdistributed in a medium drought year (p = 75%, 0.64~0.65 × 10 4 m 3 ) and a special drought year (p = 95%, 5.82~5.83× 10 4 m 3 ) 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.

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 largescale 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.

Figure 2 .
Figure 2. Comparison of the water quantity allocation for each branch canal for the two modes.

Table 2 .
Water 2022, 14, x FOR PEER REVIEW 5 System generalization of the water receiving area controlled by the Hengliu Main Ca

Table 1 .
Basin information of the Hengliu Main Canal.System generalization of the water receiving area controlled by the Hengliu Main Canal.

Table 1 .
Basin information of the Hengliu Main Canal.

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

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.

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.

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.

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.

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.Comparison of the water quantity allocation for each branch canal for the two modes.