Tree-Type Irrigation Pipe Network Planning and Design Method Using ICSO-ASV

: Research on tree-type irrigation pipe networks is an important component of agricultural water-saving projects. The optimal design of tree-type irrigation pipe networks is a key aspect regarding the proﬁtability of irrigated agriculture. Meanwhile, swarm intelligence optimization algorithms have good computational ability and can be applied to solve many optimization problems in agricultural engineering. To identify the lowest investment cost for a pipe network, this study deﬁned the concept of an upper water node to ensure the connectivity of tree-type irrigation pipe networks, and therefore, improve the pipenetwork planning model without using preliminary network connection diagrams. In addition, this study proposed an improved chicken swarm optimization algorithm (Improved Chicken Swarm Optimization using Adaptive Search and Variation, ICSO-ASV), which was applied to solve 12 test functions of di ﬀ erent dimensions. The test results show that, compared to the traditional chicken swarm algorithm and other algorithms in the control group, the ICSO-ASV algorithm could e ﬀ ectively improve the global search capability. Finally, the ICSO-ASV algorithm was used to plan and design 15-node and 40-node pipe networks. The calculation results show that the average investment costs of the two pipe networks generated by the ICSO-ASV algorithm were 42.20% and 31.09% lower than those generated by the traditional chicken swarm algorithm, which further veriﬁed the feasibility of applying ICSO-ASV to design tree-type irrigation pipe networks. Thus, the design method proposed in this study can solve the optimal problems of tree-type irrigation pipe networks with varying topologies. The optimal solutions can be generated automatically using the ICSO-ASV algorithm if essential parameters of the pipe network planning model are provided.

To further improve the generality of the design of tree-type irrigation pipe networks, this study proposed a pipe network deployment form based on the upper water node for the traditional tree-type irrigation pipe network model and ICSO-ASV (Improved Chicken Swarm Optimization using Adaptive Search and Variation) for the synchronous design of the pipeline deployment planning and pipe diameter selection; this configuration identifies the planning scheme with the minimum application cost that meets the water supply connectivity requirement of a tree-type pipe network and the specific constraints of a pipe network, therefore obtaining an efficient and practical planning and design method for tree-type irrigation pipe networks. The organization of this paper is as follows. Section 2 describes the improved pipe network planning model based on upper water nodes, which can ensure the connectivity of tree-type irrigation pipe networks. Section 3 proposes an improved chicken swarm optimization algorithm (ICSO-ASV) with a test function experiment, which uses adaptive search and mutation strategies. Section 4 applies the proposed ICSO-ASV algorithm to solve two pipe network test cases by finding the best combination of water-consuming nodes and their corresponding upper water nodes. Finally, conclusions are stated in Section 5.

Traditional Pipe Network Planning Model
In general, a traditional self-pressure pipe network is composed of a water source point, water-consuming nodes, and connecting pipes between the nodes. Given the location of the water-consuming nodes, the design of a pipe network is equivalent to solving a weighted directed graph problem with the pipe lengths as the edges to satisfy the objective conditions. With the goal of obtaining the lowest investment cost for the pipe network, a pipe network planning model was established based on References [10][11][12][13]23]. Here, I c is the investment cost of the pipe network (yuan); N is the number of connecting pipes between the water-consuming nodes; D i and L i are the pipe diameter (mm) and the pipe length (m), respectively, of pipe i; and α, β, and γ are the pipe cost coefficients and index, respectively.
The pressure, pipe velocity, and pipe diameter constraints of the water-consuming nodes in the pipe network are shown in Equations (2)-(4), respectively.
Here, E w is the water surface elevation of the water source point (m); k(i) is the number of connecting pipes between the water source point and pipe i (i = 1, 2, . . . , N); ω is the local head loss coefficient of the pipe network; θ, m, and n represent the pipe head loss coefficients related to the pipe material; Q i is the flow of pipe i (m 3 /h); G i is the ground elevation of the water-consuming node into which pipe i flows (m), where P i is the lowest allowable water pressure of this water-consuming node (m); V max and V min are the maximum and minimum flow velocities (m/s), respectively, allowed by pipe i; and D max and D min are the maximum and minimum pipe diameters (m), respectively, that can be used for pipe i.

Improved Pipe Network Planning Model
For the pipe network planning problem, current solutions require a preliminary connection diagram of the pipe network corresponding to the actual working conditions of the project and the experience of the design personnel to ensure the single-point water supply principle of a tree-type pipe network [3]. This study introduced the concept of an upper water node into the traditional pipe network planning model and transformed the pipe network deployment planning problem into the selection and combination of water-consuming nodes and their corresponding upper water nodes; this ensured the single-point water supply principle of the tree-type pipe network. Compared to the traditional pipe network deployment method, the improved pipe network planning model does not require the use of a preliminary connection diagram of the pipe network, thereby improving the versatility of the model. The improved pipe network planning model is described below.
The parameters of the water source point N 0 of the irrigation pipe network are denoted as (x 0 , y 0 , E w ), and the parameters of the water-consuming node N i are denoted as (x i , y i , G i ) (i = 1, 2, . . . , N), where x i and y i indicate the ground coordinates of the node and G i indicates the ground elevation of the node. For any water-consuming node N i , its usable set of upper water nodes S i is shown in Equation (5).
Then, for node N i , there must exist one upper water node N h S i such that the pipe i connecting These two nodes have a length of L ih , which can be obtained via Equation (6).
Denoting the diameter of pipe i as D ih , the investment cost of the pipe network can be calculated using Equation (7).
According to the single-point water supply principle of a tree-type irrigation pipe network, to ensure the connectivity of the pipe network, there is one and only one connection pipe i connected to any water-consuming node N i (i = 1, 2, . . . , N) that supplies water to this node. In a self-pressure tree-type irrigation pipe network, the node connected to the other end of pipe i is specified as N h (h i), where the node N h is the upper water node of the node N i , meaning that water flows through pipe i from node N h to node N i . This study introduced the upper water node concept to the improved pipe network planning model to ensure the water supply connectivity of the pipe network deployment. To further improve the robustness of the pipe network, the ground elevation G i of the node N i should not be lower than the ground elevation G h of the node N h . A schematic diagram of the calculation process for each upper water node of the improved pipe network planning model is shown in Figure 1. Note that, if there are multiple water-consuming nodes with the same ground elevation in the pipe network, then the average distance between each water-consuming node and its set of upper water nodes is calculated separately. The water-consuming node with the smallest average distance has priority over the other nodes and becomes the upper water node of the other water-consuming nodes. Water 2020, 12, x FOR PEER REVIEW 5 of 20

Traditional CSO Algorithm
The CSO algorithm simulates the hierarchy of a chicken swarm and the food-searching behavior of different individuals [14]. A chicken swarm consists of several groups, each of which is composed of one rooster, several hens, and several chicks. According to the functional fitness value of the problem-solving function, the chicken swarm is divided into three swarms: the rooster swarm, the hen swarm, and the chick swarm. From all the individuals, the set of individuals with the best fitness values is selected to be the rooster swarm, the set of individuals with the worst fitness values is selected to be the chick swarm, and the remainder of the individuals composes the hen swarm. The mother-child relationship between the chicks and hens is established randomly. The hierarchy, dominance, and mother-child relationships are updated at every generation.
The number of individuals in the chicken swarm is denoted as N. RN, HN, CN, and MN denote the numbers of roosters, hens, chicks, and mothers of chicks, respectively. In each group, the position of the rooster is updated, as shown in Equations (8) and (9).
Here, Randn(0, σ 2 ) represents a random number generated using a Gaussian distribution with a mean of zero and a standard deviation of σ 2 , t indicates the current number of iterations, ε is the smallest constant used to avoid zero-division errors, k is the index of a rooster randomly selected from another group, and fi is the fitness value of the rooster i.
The hens follow the rooster in their group to search for food, and their positions are updated as shown in Equations (10)- (12).

Traditional CSO Algorithm
The CSO algorithm simulates the hierarchy of a chicken swarm and the food-searching behavior of different individuals [14]. A chicken swarm consists of several groups, each of which is composed of one rooster, several hens, and several chicks. According to the functional fitness value of the problem-solving function, the chicken swarm is divided into three swarms: the rooster swarm, the hen swarm, and the chick swarm. From all the individuals, the set of individuals with the best fitness values is selected to be the rooster swarm, the set of individuals with the worst fitness values is selected to be the chick swarm, and the remainder of the individuals composes the hen swarm. The mother-child relationship between the chicks and hens is established randomly. The hierarchy, dominance, and mother-child relationships are updated at every generation.
The number of individuals in the chicken swarm is denoted as N. RN, HN, CN, and MN denote the numbers of roosters, hens, chicks, and mothers of chicks, respectively. In each group, the position of the rooster is updated, as shown in Equations (8) and (9).
Here, Randn(0, σ 2 ) represents a random number generated using a Gaussian distribution with a mean of zero and a standard deviation of σ 2 , t indicates the current number of iterations, ε is the smallest constant used to avoid zero-division errors, k is the index of a rooster randomly selected from another group, and f i is the fitness value of the rooster i.
The hens follow the rooster in their group to search for food, and their positions are updated as shown in Equations (10)- (12).
Here, Rand represents a uniform random number within the range of [0,1], r 1 is the index of the rooster in the same group as hen i, r 2 is the index of a randomly selected rooster or hen superior to hen i, and r 1 r 2 .
Chicks follow their mothers in their group to search for food, and their positions are updated as shown in Equation (13).
x t+1 Here, x t m,j indicates the position of the mother of chick i and FL [0,2], which represents the foraging and following coefficient of a chick following its mother.

Improved Control Coefficient Pair of the Hen Swarm
In the traditional CSO algorithm, the process of updating the position of a hen is primarily affected by factors such as the rooster in the hen's group, another randomly selected rooster or hen, and the control coefficients S 1 and S 2 of the two previously mentioned individuals, as shown in Equation (10). In Equation (10), S 1 and S 2 represent the degree of closeness of the corresponding hen individual to the rooster individual in that hen's group and the degree of competition with other individuals, respectively. In each iteration, the values of S 1 and S 2 are directly related to the fitness values of the rooster individual in the hen's group and the other randomly selected individual, without including a comprehensive consideration of the impact of the current overall status of the swarm on the hen individual, as shown in Equations (11) and (12). However, the calculation processes of S 1 and S 2 are independent of each other, making it difficult to directly coordinate the impact of the rooster in the hen's group and the other individual on the hen. Based on the above analysis, this study proposed an improved control coefficient pair C 1 and C 2 for the hen swarm, as shown in Equations (14) and (15).
Here, F avg is the current average functional fitness value of the chicken swarm; F best is the functional fitness value of the current optimal chicken swarm individual; |F avg − F best | is the difference between the average value of the population fitness value and the optimal individual value, indicating the proximity of individuals in the population to the optimal individual; b = t/T max is the proportion coefficient of the algorithm iteration process, where T max is the maximum number of iterations of the algorithm and b (0, 1); and the coefficient C 1 is primarily affected by the fitness value of the objective function and the iterative process of the algorithm. In the equations, the weight coefficients are the constants α and β, where α,β [1,1.5]. We denote the sum of C 1 and C 2 as the constant R; accordingly, As the iterative process of the algorithm changes, the value of the coefficient C 1 gradually decreases from α + β to zero and the value of the coefficient C 2 gradually increases from R − (α + β) to R. Considering that C 1 and C 2 characterize the impact of the rooster in the hen's group and the competing individual on the hen individual, respectively, in the early stage of the algorithm's execution, the difference between the average value of the chicken swarm and the optimal individual is relatively large and a larger C 1 can increase the proportion of learning of the hen individual from the rooster in the hen's group and reduce the competition with other individuals; accordingly, the hen individual can quickly approach the rooster individual in its group in terms of its fitness value, thereby accelerating the convergence rate of the algorithm. In the later stage of the algorithm's execution, the difference between the average value of the chicken swarm and the optimal individual is relatively small and a larger C 2 can increase the degree of competition between the hen individual and other individuals and reduce the proportion of learning from the rooster individual in the hen's group; this allows the hen swarm to maintain good population diversity and avoids the premature convergence of the algorithm. Therefore, the improved control coefficient pair proposed in this study can adapt to the changes in the iterative process of the algorithm because it fully takes into consideration the impact of the current overall state of the chicken swarm on the hen individual and adaptively adjusts the impact of the rooster in the hen's group and other individuals on the hen individual via mutually restrained cooperation.

Adaptive Mutation Factors
In this paper, the adaptive mutation factors V 1 and V 2 are introduced into the position update process for the rooster and chick swarms, as shown in Equation (16).
Here, C 1 and C 2 are the improved control coefficient pair and γ is a constant used to ensure that the value ranges of V 1 and V 2 do not exceed 1. Because C 1 and C 2 can change adaptively according to the functional fitness value and the algorithm iteration process, the mutation factors V 1 and V 2 can also change adaptively to dynamically adjust the search operations of the rooster and chick swarms as the algorithm optimization process progresses.
After the position update of a rooster individual is completed, a random number β 1 [0,1] is generated. If β 1 < V 1 and f i > P i are satisfied, the rooster individual performs an adaptive mutation operation, as shown in Equation (17). After the position update of a chick individual is completed, a random number β 2 [0,1] is generated. If β 2 < V 2 is satisfied, then the chick individual is randomly reset.
x t+1 Here, P i represents the optimal value of the rooster individual i, x max − x min represents the boundary distance of the feasible solution domain of the function, and F is the coefficient of mutation such that the rooster individual uses the pre-mutation position as a center to randomly generate a new feasible solution within a small range.
In the early stage of the algorithm, the control coefficients satisfy C 1 > C 2 and the mutation factors satisfy V 2 > V 1 , meaning that the probability of the rooster swarm performing a mutation operation is relatively small, while the probability of the chick swarm performing a mutation operation is relatively large; this is conducive to expanding the global search range of the chicken swarm to avoid falling into a local optimum in the early stage of the algorithm while maintaining the normal optimization of the rooster swarm. In the later stage of the algorithm, the control coefficients satisfy C 2 > C 1 and the mutation factors satisfy V 1 > V 2 , suggesting that the probability of the rooster swarm performing a mutation operation is relatively large, while the probability of the chick swarm performing a mutation operation is relatively small; this is conducive to leading the chicken swarm to search around the optimal solution of the function, thereby improving the accuracy of the algorithm.
The population size of the ICSO-ASV algorithm is denoted as N, the number of roosters is denoted as RN, the number of chicks is denoted as CN, and the probability of the mutation operation for a rooster individual and a chick individual are denoted as V 1 and V 2 , respectively. Therefore, the time complexity of the calculation of a single iteration is O(N + V 1 × RN + V 2 × CN), where 0 < V 1 < 1 and 0 < V 2 < 1. If the maximum number of iterations of the algorithm is denoted by T max , then the time complexity of the ICSO-ASV algorithm is O( The flow of the ICSO-ASV algorithm is described as follows.
Step 1: Initialize the chicken swarm. Set the population size N, the number of roosters RN, the number of hens HN, the number of chicks CN, the number of chick mothers MN, the update period G, and the maximum number of iterations T max . Calculate the fitness values of all the individuals in the chicken swarm, initialize the individual optimal value and the global optimal value, and set the number of iterations to t = 1.
Step 3: If t % G = 1, reorder all the individuals in the flock according to their fitness values, establish the corresponding hierarchical order, and divide the subgroups.
Step 4: Update the rooster individual according to Equations (8) and (9). If β 1 < V 1 and f i > P i are satisfied, then perform the adaptive mutation operation on the rooster individual according to Equation (17); update the hen individuals according to Equation (10); and update the chick individuals according to Equation (13). If β 2 < V 2 is satisfied, then randomly reset the chick individual.
Step 5: Update and save the individual and global optimal values of the chicken swarm.
Step 6: If the algorithm satisfies the iteration stop condition, then stop the iteration and output the optimal feasible solution; otherwise, go back to step 2.

ICSO-ASV Algorithm Performance Analysis
In this study, the performance of the ICSO-ASV algorithm was analyzed using 12 benchmark test functions, the results of which were then compared to those of the following six algorithms: PSO (Particle Swarm Optimization) [24], BA (Bat Algorithm) [25], CSO [14], ADLCSO [18], ICSO-a [19], and ICSO-b [20]. The 12 benchmark test functions are shown in Table 1

Test Functions Equation Scope
Shifted sphere Shifted rotated elliptic 5.12,5.12] Ackley N.4   Table 2. The population size of all algorithms was set to 100, the number of independent executions was 50, the maximum number of iterations was 500, and all other general parameters were kept consistent. Table 2. Algorithm parameter settings.

Algorithm
Parameter Settings A comparative analysis was conducted for the ICSO-ASV algorithm, three standard algorithms (PSO, BA, and CSO), and three ICSOAs (ADLCSO, ICSO-a, and ICSO-b) based on the 30D and 50D benchmark test functions. The results of each algorithm based on the 30D benchmark test functions are shown in Table 3. Table 3 indicates that, of the three standard algorithms, the BA algorithm had certain advantages in solving some of the test functions, whereas the CSO algorithm could find the global optimal value of the function f 6 . Both of These algorithms were superior to the PSO algorithm; however, the results of the three standard algorithms for most of the test functions were within the same or similar order of magnitude. Of the three ICSOAs, the ADLCSO algorithm obtained the best single-peak function solution results and the ICSO-a algorithm obtained the best multi-peak function solution results. Overall, These two algorithms performed better than the ICSO-b algorithm. Table 3 indicates that the results of the ICSO-ASV algorithm were better than those of the control algorithms overall. For the shifted functions f 1 and f 3 and the multi-peak functions f 6 -f 8 and f 10 -f 12 , the ICSO-ASV algorithm could not only find the global optimal value but also had a standard deviation close to 0, indicating that it had a better solution accuracy and stability than the control algorithms.
For the function f 5 , the average value generated by the ICSO-ASV algorithm was slightly worse than those generated by the ICSO-a and ICSO-b algorithms, but the global optimal value could still be found. For the other benchmark functions, the solution results of the ICSO-ASV algorithm had obvious advantages. The accuracies of the solutions were improved by more than five orders of magnitude compared to those of the standard CSO algorithm for half of the test functions, especially for functions f 3 , f 11 , and f 12 , and the solution accuracy of the ICSO-ASV algorithm was higher than that of the optimal control algorithm by more than three orders of magnitude.
The iterative curves of the mean fitness values of each algorithm based on the 50D test functions are shown in Figure 2, where some results are logarithmic. Figure 2 indicates that, of the three standard algorithms, the CSO algorithm had certain advantages in terms of the convergence speed, but the BA algorithm could obtain better optimization results for multiple functions. When comparing the CSO algorithm to the ICSOAs, the ICSO-a algorithm had certain advantages in terms of the convergence speed and could obtain better optimization results for multiple functions. The ICSO-ASV algorithm had relatively obvious advantages in terms of the solution accuracy and avoiding local optima in comparison to the control algorithms. The optimization curves of the functions f 4 , f 7 , f 11 , and f 12 show that, as the number of iterations increases, the ICSO-ASV algorithm did not fall into a local optimum and its solution accuracy could be further improved.

Analysis of the Improved Pipe Network Planning Model
The decision variables of the improved pipe network planning model include the length Lih and the diameter Dih of the connecting pipe between the water-consuming node Ni (i = 1, 2, …, N) and its upper water node Nh (h = 1, 2, …, N, h ≠ i), meaning that the solution space of the planning model is {Lih, Dih}. If there are N water-consuming nodes in the pipe network, then the dimension of the solution space is 2N. Considering that the pipe length and the pipe diameter that meet market standards are both discrete variables, the solution space of the ICSO-ASV algorithm also needs to be discretized. Note that, for the water-consuming node Ni, there exists a continuous feasible solution {xl, xd} [0,1] in the solution space of the ICSO-ASV algorithm corresponding to the solution space of the planning model {Lih, Dih}. For the connection pipe length Lih, the upper water node Nh needs to be selected from the set of upper water nodes Si belonging to the water-consuming node Ni; the

Analysis of the Improved Pipe Network Planning Model
The decision variables of the improved pipe network planning model include the length L ih and the diameter D ih of the connecting pipe between the water-consuming node N i (i = 1, 2, . . . , N) and its upper water node N h (h = 1, 2, . . . , N, h i), meaning that the solution space of the planning model is {L ih , D ih }. If there are N water-consuming nodes in the pipe network, then the dimension of the solution space is 2N. Considering that the pipe length and the pipe diameter that meet market standards are both discrete variables, the solution space of the ICSO-ASV algorithm also needs to be discretized. Note that, for the water-consuming node N i , there exists a continuous feasible solution {x l , x d } [0,1] in the solution space of the ICSO-ASV algorithm corresponding to the solution space of the planning model {L ih , D ih }. For the connection pipe length L ih , the upper water node N h needs to be selected from the set of upper water nodes S i belonging to the water-consuming node N i ; the corresponding relationship between x l and N h indexed as T L in the set S i is shown in Equation (18). For the pipe diameter D ih , the index of T D in the pipe diameter set R can be determined according to Equation (18).
Here, |S i | represents the number of elements in the set S i and |R| represents the number of elements in the set R.
Because all water-consuming nodes in the pipe network need to meet the node pressure constraints, a penalty function is used to convert the pipe network planning model and the constraints into an unconstrained objective function, as shown in Equation (19). Here, where t indicates the current number of iterations, and K is the penalty function factor constructed according to the node pressure constraints, where the parameter settings are the same as those in Robinson and Rahmat-Samii [26]. A flow chart of a pipe network design based on the ICSO-ASV algorithm is shown in Figure 3.

Pipe Network Design Case I
This study used several algorithms, including GA (Genetic Algorithm) [27], BA [25], SABA (Self-Adaptive Bat Algorithm) [28], CSO [14], ADLCSO [18], ICSO-a [19], and ICSO-ASV to design pipe network cases. Pipe network design case I included one water source point and 14 water-consuming nodes. Note that the improved pipe network planning model proposed in this paper can ensure the water supply connectivity of the pipe network by selecting the upper water node of the water-consuming node and uses the node parameters to automatically calculate the lengths of all pipe connections; therefore, there is no need to use a preliminary connection diagram of the pipe network. The node parameters of the pipe network are shown in Table 4, and the unit prices of the pipes are shown in Table 5. In the objective function, Equation (19), the pipe network cost parameters were calculated by fitting the data in Table 5, where the pipe network cost coefficients were α = 1.5 and β = 5.37 × 10 −4 and the pipe network cost index was γ = 1.92 [3,9]. In the pressure-constrained equation, Equation (2), of the water-consuming nodes in the pipe network, the local head loss coefficient was ω = 1.1. Taking the rigid plastic pipe as the standard, the pipe head loss coefficients were θ = 9.48 × 10 4 , m = 1.77, and n = 4.77; the lowest water pressure of each node was P = 10 m; and in the pipe velocity constraint equation, Equation (3) Initialize node parameters of treetype irrigation pipe network Construct the upper water node set of each water node using Equation (5) Initialize the ICSO-ASV Calculate the solution of ICSO-ASV using Equations (8)-(10) and (13)-(16) Whether rooster and chick meet the mutation conditions?
Discretize the feasible solution space of ICSO-ASV using Equation (18) Operate the mutation Y N Calculate the investment cost of the pipe network using Equation (19) and update the optimal global solution t ≥ tmax?
Output the ICSO-ASV results End N Y Figure 3. Flowchart of an irrigation pipe network design based on the ICSO-ASV algorithm. Figure 3. Flowchart of an irrigation pipe network design based on the ICSO-ASV algorithm.  8  2259  925  216  20  1  2681  377  220  20  9  1785  1098  216  20  2  2339  1652  220  20  10  1329  769  215  20  3  1121  1250  219  20  11  2018  421  214  20  4  1435  1359  219  25  12  562  2213  214  20  5  2071  1143  218  25  13  898  1212  212  20  6  2408  851  217  25  14  1314  2020  210  20  7 964 1637 216 20 In this case, the population size of each algorithm was set to 40, the maximum number of iterations was 500, and the number of independent executions was 50. The design results of each algorithm based on the objective function, Equation (19), are shown in Table 6, which details the optimal solution, mean value, and worst solution for the investment cost of the pipe network. The results show that, of the three standard algorithms, the design results based on the GA algorithm were superior overall compared to those of the traditional CSO algorithm in terms of the solution accuracy and stability, indicating that the traditional CSO algorithm was subject to falling into a local optimum when solving low-dimensional discrete combinatorial optimization problems. Compared to the traditional BA algorithm, the SABA algorithm showed a better performance. In terms of the optimal design results, all three ICSOAs obtained satisfactory design results. The design results based on the ICSO-ASV algorithm were superior overall compared to those of the other six algorithms in terms of the solution accuracy and stability. The ICSO-ASV algorithm yielded an optimal result of 27,611 yuan for the pipe network investment cost, and its mean design result was 42.20% lower than that of the traditional CSO algorithm. The design results indicate that the ICSO-ASV algorithm effectively reduced the probability of the algorithm falling into a local optimum, resulting in excellent algorithm performance. Therefore, the pipe network design method based on the ICSO-ASV algorithm proposed in this paper had relatively good accuracy. The optimal design result of the pipe network based on the ICSO-ASV algorithm is shown in Figure 4.

Pipe Network Design Case II
With the expansion of the scale of the pipe network, that is, with an increase in the number of pipe network nodes, the solution space of the improved pipe network planning model increased exponentially and the difficulty of searching for the optimal solution of the model correspondingly increased. To further verify the scalability of the proposed design method, the second case investigated in this paper, pipe network design case II, included one water source point and 39 water-consuming nodes; the parameters of each node are shown in Table 7.

Pipe Network Design Case II
With the expansion of the scale of the pipe network, that is, with an increase in the number of pipe network nodes, the solution space of the improved pipe network planning model increased exponentially and the difficulty of searching for the optimal solution of the model correspondingly increased. To further verify the scalability of the proposed design method, the second case investigated in this paper, pipe network design case II, included one water source point and 39 water-consuming nodes; the parameters of each node are shown in Table 7.  20  2184  521  224  25  1  1810  1470  238  30  21  1904  816  223  25  2  2586  1510  237  30  22  575  734  223  20  3  1535  431  236  25  23  2420  1150  221  25  4  2210  1165  235  30  24  611  1335  221  25  5  1570  689  235  25  25  1363  1511  221  25  6  2681  1570  235  30  26  2681  377  220  20  7  1727  920  234  20  27  2339  1652  220  20  8  427  1765  234  25  28  1121  1250  219  20  9  1738  1526  233  30  29  1435  1359  219  25  10  1649  1389  232  30  30  2071  1143  218  25  11  1906  899  231  25  31  2408  851  217  25  12  373  1209  231  25  32 The results of the pipe network design based on each algorithm are shown in Table 8. According to the result analysis, the design result based on the traditional CSO algorithm was superior to that based on the GA algorithm, indicating that the CSO algorithm had certain advantages over the GA algorithm when solving high-dimensional optimization problems. Compared to the traditional BA algorithm, the SABA algorithm could still obtain better design results. Compared to the control algorithms, the pipe network design based on the ICSO-ASV algorithm could still obtain relatively more stable optimal results. The ICSO-ASV algorithm yielded an optimal solution of 127,410 yuan as the pipe network investment cost, and its mean design result was 31.09% lower than that of the traditional CSO algorithm. Therefore, the pipe network design method based on the ICSO-ASV algorithm proposed in this paper had good scalability. To further improve the practicality of this pipe network design method, we designed a software system to enable the intelligent design of tree-type irrigation pipe networks (See details in Supplementary Materials). After entering the node parameters of the pipe network and setting the relevant algorithm parameters, the system automatically completes the pipe network design process and outputs the results. The 2D and 3D results that were given by the ICSO-ASV algorithm for pipe network design case II are shown in Figures 5 and 6, respectively.

Conclusions
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Conclusions
This paper proposed an improved planning model for tree-type irrigation pipe networks and verified the effectiveness of the pipe network design method based on the ICSO-ASV algorithm using two pipe network cases with different topologies. The results indicated that the improved pipe network planning model could ensure the pipe connectivity of tree-type irrigation pipe networks via different combinations of water-consuming nodes and upper water nodes. The use of a preliminary connection diagram of the pipe network was not required, and the developed model had relatively good versatility and scalability.
A test function experiment demonstrated that the ICSO-ASV algorithm based on the improved control coefficient pair and adaptive mutation factors had a better global search ability and optimal solution accuracy than the control algorithms. The two pipe network test cases demonstrated that the pipe network design method based on the ICSO-ASV algorithm could effectively reduce the investment cost of a pipe network, and therefore has better practicability than the control algorithms.
The planning and design problem of tree-type irrigation pipe networks is relatively complex, making it difficult to describe the variability of a network with a single model. In future studies, the instructive research on the optimal design of irrigation pipe networks is as follows: (1) In the design process of irrigation pipe networks, the minimization of investment cost should not be the only issue that designers need to consider; the other issues that also need to be considered, include terrain conditions, operation management, and reliability. (2) To enhance the computational efficiency and accuracy, other new swarm intelligence optimization algorithms will be explored in the field of irrigation pipe network optimization.