Irrigation pipe distribution network optimization with Jaya Algorithm: a hybrid approach Open Access

CPM

Critical Path Method

EA

Evolutionary Algorithm

FEs

Function evaluations

GA

Genetic Algorithm

HGL

Hydraulic gradient level

HW

Hazen Williams

IPDN

Irrigation pipe distribution networks

JA

Jaya Algorithm

KBC

Kanhan Branch Canal

LP

Linear programming

NLP

Non-linear programming

PSC

Prestressed concrete pipes SA = Sensitivity Analysis

Total cost of the network

and

No. of links and commercial pipes respectively

Unit cost of pipe size b in link a

Hazen-Williams pipe coefficient

Diameter of pipe in mm

Minimum head at node j

Head available at source node O

,

Cost constant

Length of the pipe in meter

Length of pipe size b in link a

Flow through link a in m³/min

and

Two arbitrary numbers in the range of 0–1 for j^th variable during i^th iteration

Friction slope for pipe size b in link a

Modified value of

Value of j^th variable for the best candidate

Value of j^th variable for the worst candidate

Water is a vitally important resource required to be managed efficiently for the sustenance of human beings. In a tropical country like India, growing population and industrialization have aggravated water shortage in the agricultural sector. Irrigation in the country is practiced mainly through open canal networks in which the seepage losses are very high. Though the design efficiencies of most of the projects are to the tune of 41–48%, actually it may reduce to 20–35% because of field difficulties or restrictions, which shows the extent of water loss depriving the tail-enders of the benefits of irrigation practices (Satpute et al. 2012). The maintenance cost of the canal system is high as compared to pipe networks. Also, repetitive maintenance is required for the canals, whereas for pipe networks, better control on the functioning of the system can be achieved. As the pipe system may be buried, freedom of network layout is possible along with the solution to the problem of land acquisition as in the case of canal networks. In light of this fact, concrete steps of replacing the open canals with pipe conveyance systems have become a necessity. This method of conveying water would assure water saving, which otherwise percolates in the ground and becomes unavailable when required (Gajghate & Mirajkar 2020). The available water may cultivate more crops per year, leading to higher benefits for the farmers. The rise in income promotes the economic status of the users, leading to society's overall prosperity. However, the initial investment cost of the IPDN is very high compared to the open canals (Sudan & Gupta 2017). The justification for pressing pipe irrigation stands erect because of the long-term advantages of comparing the land acquisition cost, maintenance cost, water losses, public hygiene, and socio-economic factors. The major part of the initial investment of IPDN is the pipe cost. The fact that the pipe cost directly varies with the pipe diameter makes it very important that optimum pipe sizes be selected for both the performance and total project cost while designing any network. Colebrook-White transition formula was used for circular pipe design, which relates Reynolds number and relative roughness with friction loss (Viccione & Tibullo 2012; Praks & Brkic 2020). With the advancement in computer technology, various optimization techniques have been developed for water network optimization. LP was used for the optimization of the water supply system involving two steps. The first step converted the looped system into a branched network, while in the next step, optimization of the branched network, is carried out (Bhave 1983). LP and NLP are used in two stages, with LP in the first stage to determine the minimum cost design for a given flow. In the second stage, non-linear programming (NLP) is used to find the modified flows, which further reduces the cost of the water distribution network (Qiu et al. 2020). Integer linear programming technique was explored for water distribution system optimization (Alperovits & Shamir 1977). For minimizing the total cost of the irrigation pipeline network, a hybrid model including LP and Genetic Algorithms (GA) was developed (Lapo et al. 2016) to check its effectiveness. Other optimization methods that are used in water networks optimization are differential evolution and mixed integer linear programming (Mansouri et al. 2015), GA (Kadu et al. 2008), simulated annealing (Cunha & Sousa 1999), ant colony (Maier et al. 2003) and shuffled frog leaping algorithm (Ensuff & Lansey 2003).

Most of the optimization techniques involve a number of input parameters and it is very important to achieve reasonable results. More parameters require extensive trials and tuning, which is time consuming, to evaluate the optimum results. The computation complexity is very high for more extensive networks (Shende & Chau 2019). This fact motivated the authors to carry out the present study of the application of JA for the optimization of IPDN. JA is a recent, simple and powerful global optimization algorithm developed by Rao in 2016. The prime advantage of JA is that it requires only a few control parameters reducing the extensive trials and thus minimizing the computation complexity of the problem. It makes the wide application of JA simple and popular in various fields of engineering and sciences (Pandey 2016; Kumar & Yadav 2018; Rao et al. 2018; Varade & Patel 2018; Paliwal et al. 2020).

The objective of the present study is to design the IPDN diameters using CPM. The layout of the irrigation networks and water requirement (demand) is predefined, and the optimization is oriented to find the minimum cost of IPDN. The factors affecting the layout of the irrigation networks are the geographic proximity, spatial distribution of the customers, water demand and available head. Networks considered for the study are two adjoining distributaries of the Pench Irrigation Project. Bakhari distributary is a 33-pipe network, while Nandgaon distributary is a small five-pipe network. Both the networks are designed for steady-state conditions. As the study area belongs to a semi-arid region and the network will be underground, temperature stresses due to weather conditions changes will not be that significant as far as pipe material is concerned. For the present study, the prestressed concrete (PSC) pipes are used for the design using Hazen-Williams (HW) formula as it develops desired positive pressures. The networks are designed using CPM to obtain the initial design diameters. The optimization of the initially obtained diameters is carried out using LP. A computer model is developed for JA using MATLAB (R2015a) to optimize the obtained designed diameters for IPDN. Finally, a cost comparison of the networks is carried out. Also, sensitivity analysis is carried out to determine the effect of JA's control parameter on the best value of the objective function (cost). Hybrid approach of obtaining diameters using CPM and thereby optimizing the cost using the novel JA is the main aim of the present study. The comparison results indicate that JA can be promisingly applied to real networks for IPDN optimization, thus reducing the network's total cost with lesser efforts.

The Left Bank Canal (LBC) of the Pench irrigation project, India, consists of four branch canals, viz. Kanhan, Mouda, Ramtek, and Kandri. The average annual rainfall for the study area is 1,150 mm, which occurs from June to September. Minimum or no rain is received during the remaining period. The mean minimum and maximum temperatures are around 12 and 45 °C. The soil found is medium to deep clayey, black cotton soil. The rocky layer is encountered at a depth of 7–8 m, therefore does not hinder the pipe network. The schematic sketch shown in Figure 1 gives an overview of the Pench project and also shows the area selected for the study.

Two distributaries of different sizes of the Kanhan Branch Canal (KBC), having different command areas, are considered distinctly for the analysis viz. Bakhari (820 Ha) and Nandgaon distributaries (183 Ha). The irrigation system existing for the study area is the canal system. This canal system is proposed to be replaced with the piped irrigation system. The design and optimization of the developed IPDNs are then carried out. Figure 2 represents the proposed replaced canal system with the pipe network for Bakhari and Nandgaon distributaries, not the scale. Bakhari network is a 33-pipe system with a single source, as shown in Figure 2(a). The off-taking point from KBC is at 990 m. R1 (source) location coordinates are 21.34°N and 79.22°E with a head of 308.98 m. Figure 2(b) represents the Nandgaon distributary, which is proposed to be replaced by a five-pipe network. It is adjoining the Bakhari distributary and takes off from the KBC at 3,750 m. The coordinates of the source, R2, are 21.33°N and 79.22°E, and the head at R2 is 307 m.

For the present work, the initial diameters are obtained for the Bakhari and Nandgaon networks using CPM, as shown in Tables 1 and 2. The Hazen-Williams pipe coefficient used is 130. The discharge in each pipe as shown in Tables 1 and 2 are used to calculate the diameters. These initial diameters are used as inputs for the model formulation of LP and JA for optimization.

LP is an old but effective method for optimization. A globally optimal solution can be reached for the branched water networks using LP. It can solve problems that are simple as well as complex in nature. For the present work, the optimization of the total cost of the two networks (Bakhari and Nandgaon) is carried out using LP with the constraints of minimum head loss and length of each pipe as variable.

The optimization problem is formulated to minimize the total cost of the network without violating the head constraints. A single pipe is divided into two parts to utilize two different diameters for the same pipe. The reference diameters used for these two diameters (one on the higher side and the other on the lower side) are the diameters obtained using the CPM.

The constraints are as follows:

where,

• – Head available at source node O;

• – Minimum head at node j;

• – Friction slope for the pipe size b in link ⁠;

• = 2.215 × 10¹²;

• P – 1.85; and

• r – 4.87.

In the present study, cost of pipe diameter for each pipe and optimized length of each pipe is considered to calculate the total cost of the network. Tables 3 and 4 show each pipe length's division in two parts to achieve the optimized pipe sizes to minimize the total cost for the Bakhari and Nandgaon networks.

The penalty is additive for the minimization problem, whereas for the maximization problem, it is subtractive. Thus, a modified function value is obtained. Further processing of the function is carried out using the best and the worst values of the reformed function. Rejection of the solution containing the violation is achieved as the penalty function's formulation is done accordingly. The maximum number of generations is already set initially as the termination criteria. As soon as the termination criteria are reached, the model stops processing, and results are provided.

Equation (8) gives the upgraded value of the variables, which contains two arbitrary numbers, and ⁠. These two numbers play a vital role in the up-grading of the variables, improving the fitness function. The arbitrary number is linked to a term representing the gap between the present value and the best value of the variable for a particular iteration. Thus, the term shows the inclination to move closer to the best solution. Similarly, is linked to a term representing the gap between the present value and the worst value of the variable for a particular iteration. Hence, the term shows the inclination of the solution to depart from the worst solution. It shows that the algorithm tends to get nearer to success (i.e., approaching the best solution) and attempts to avoid failure (i.e., departing from the worst solution). Hence, it is named Jaya (meaning victory in Sanskrit). The value of is adopted only if it gives a better function value. The function value is compared with the modified value, and if found to be better than the previous value, then the modified value of replaces the previous value, and the process is continued till the final termination criteria are satisfied. The arbitrary numbers and facilitates an adequate search space, while (⁠⁠) term enhances the algorithm's exploration ability. The flow chart depicting the working of JA can be referred from (Rao 2016). In the present study, the single pipe length for two different diameters is the design variable (m), and the number of candidate solutions (population size) considered is 50 for maximum function evaluations (FE) of 1000. Each set of population is evaluated 10 times.

The effect of the input parameters on the output values is studied using sensitivity analysis (SA) (Renault 2000). SA provides a clear idea for the selection of parameters to obtain the desired output. In optimization formulation, the input parameters are very important to achieve reasonable conclusions. Various algorithms have been used to optimize the parameters and to enhance the performance and descent ability (Deng et al. 2020a, 2020b; Song et al. 2020). In the present study, a newly developed JA is used for the optimization of IPDN. The input parameters required are the population size and FEs to obtain the best value of the objective function. FEs are the product of population size and generations and signify the actual number of times for which the objective function value is evaluated. Population size defines the extent of search space; hence is a crucial parameter for Evolutionary Algorithm (EA) performance. The number of possible combinations of the variables increases with the larger population size. More combinations lead to increased computational time without any assurance of significant improvement in the optimal value. Therefore, the selection of particular population size has to be done with proper analysis. Hence, for the current study, SA is carried out for Bakhari network, in which three population sizes are selected, and its effect is analyzed on the best value of the objective function, which is the cost in this case. The three population sizes selected are 10, 50, and 100 for the maximum FEs of 1000, which is kept constant for all three population sizes. For each population size, 10 runs are carried out before reporting the final optimal values. The variation of the best value of the objective function (cost of the network) with the population size is shown in Figure 3.

The number of trials with a different number of the population reveals that more population size takes away from the best solution while lower population size approaches towards the desired solution that is the minimum cost. On analysis, as shown in Figure 3, it is observed that the best value of objective function lies in a particular band ranging from 78.50 to 82.00 Million Rupees (1.06–1.11 Million $). Since the best value of the objective function is found out at the population size of 50, there is little possibility of lying of the optimal solution in the population size more than 100. Hence, the analysis is stopped at the population size of 100. Thus, the optimal solution is reported as the value obtained with the population size of 50 in the sixth run having the maximum FEs as 1000.

The design of the Bakhari and Nandgaon networks is carried out using CPM. The initial diameters resulted from CPM are optimized using LP and JA. The optimization of both the networks is carried out to optimize the diameters and minimize the total pipe cost. LINGO 17.0 solver is used to carry out the LP optimization, whereas the code is developed in MATLAB (R2015a) for JA optimization. The costs are calculated for optimized pipe diameters from both approaches. In the model development of LP, two diameters are considered, one on either side of the obtained designed diameter from CPM to reach the most optimized values. The number of pipes in the Bakhari network is 33. Each pipe is divided into two parts; thus, two variable diameters are obtained for two different lengths. For example, the total length of pipe 2 for the Bakhari network is 1,200 m. It is divided into two parts, viz. L21 and L22. Thus, two variables are generated for a single pipe. Therefore for 33 pipes of the Bakhari network, 66 variables are generated, and similarly, for the Nandgaon network, 10 variables are generated. The first part of the pipe represents the diameter on the higher side of the designed diameter available commercially. The second part represents the diameter on the lower side of the designed diameter. The number of constraints is 135 for Bakhari network, whereas, for the Nandgaon network, it is 21. The results for the optimized diameters and total costs for Bakhari and Nandgaon networks using LP and JA are explained in the following subsections. For the diameter optimization using JA, the code is developed in MATLAB (R2015a). The code is developed to find out the first part of the pipe only; hence the number of variables for the Bakhari distributary is 33 and that for Nandgaon is 5, thus, simplifying the code compared to LP. The second part of the pipe can be calculated, as the pipe's total length is already known.

The optimized diameters and the total cost for the Bakhari network using LP and JA are shown in Table 5. The estimation of total network cost is done using the rates (Chief Engineer 2012) according to the pipe diameters obtained from CPM and the obtained lengths from LP and JA in the networks. In Table 5, column 1 and column 5 represent the two parts (Part I and Part II) of a single link. Column 2 and column 6 represent the pipe diameters adopted for the I and II parts of the pipe length. Column 3 and column 4 represent the optimized lengths obtained for the I part of pipe length using LP and JA, respectively. Similarly, column 7 and column 8 show the optimized lengths obtained for the II part of pipe length using LP and JA, respectively. Column 9 and column 10 show the total cost of the pipes (Part I and Part II) obtained using LP and JA, respectively. To calculate the total cost, rates of the different sizes of the commercially available pipes (Chief Engineer 2012) as shown in Table A of Appendix I are used. The PSC pipes available in the markets are used as per their prices to calculate the total cost. Table 5 shows that using LP optimization, for eight pipes (L11, L111, L171, L191, L241, L291, L301, L311), the first part of the length is completely eliminated, thereby eliminating diameter leads to cutting down the cost of the network. For 10 pipes (L32, L42, L52, L72, L82, L92, L152, L222, L262, L282), the second part is zero depicts that the diameter on the higher side is required for the entire length of these 10 pipes. For the rest of the 15 pipes, the cost is shared by both the sizes of pipes.

Also, it is clear from Table 5 that, using JA, there are two pipes (L272, L332), for which the second part of the length of pipe is eliminated, whereas, for three pipes (L41, L311, L321), the cost of pipe is shared by two diameters on either side of the designed diameters obtained from CPM. For the rest of the 28 pipes, the first part of the pipe is wholly eliminated; thereby, the entire length is served by the diameter on the lower side with reference to the diameter obtained in CPM, thus reducing the network's total cost.

Table 6 shows that the Nandgaon network has only five pipes out of which, for two pipes (L421, L431), the first part of the length is eliminated. For L402, the second part is eliminated, and for only one pipe, L411, the cost is shared by two pipe sizes. The column description for Table 6 is similar to that for the Bakhari network (Table 5). Table 6 shows the optimized diameters for the Nandgaon network using JA, which indicates that out of five pipes, only one pipe, L432, bears the cost of the higher diameter for the entire length. For the rest of the four pipes (L401, L411, L421, L441), the first part of the pipe is completely eliminated; therefore, the pipe diameters on the lower side are utilized to reduce the total cost of the network.

The comparison of the total cost for the Bakhari and Nandgaon networks using LP and JA is presented in Table 7. For the present study, the PSC pipes are used for the design using Hazen-Williams (HW) formula as it develops desired positive pressures (Chief Engineer 2012). It is clear from Table 7 that JA gives more cost-effective results for small and relatively larger networks. The total cost of Bakhari using LP is estimated as 84.99 Million Rupees ($1.15 Million), whereas it comes out to be 78.74 Million Rupees ($1.11 Million) using JA, which shows a significant percentage cost reduction of 7.36. On the other hand, for the Nandgaon network, a small five pipe system network, the total cost estimated using JA is lesser than that using LP, but the percentage reduction is comparatively less. The total length of the Bakhari network is 18,355 m, and the total cost estimated for the complete network is 78.74 Million Rupees using JA. Therefore, the network's per meter cost is found out to be 4,289.89 Rupees ($58.11).

Similarly, the total length of the Nandgaon network is 4,020 m, and its total cost is 45.31 Million Rupees. The per meter cost obtained for the Nandgaon network is 11,270.26 Rupees ($152.65). The Nandgaon network, being a small five-pipe system with a total network length of 4,020 m, has little room for optimization against the Bakhari network with 33 pipes, having a total network length of 18,355 m. Hence the per meter cost obtained is higher for Nandgaon relative to the Bakhari network. The per meter cost comparison using both the optimization techniques shows that JA outstands the results.

The overall efficiency of the canal system considered for the study is 45% (Pench Irrigation Project Report, 2001). With the consideration of leakage losses to the maximum of 30% and eliminating seepage and evaporation losses entirely, the efficiency achieved is 70%. With the saved water, the area irrigated for the Bakhari distributary will increase to 1471. Ha, which was 820 Ha with the existing canal system. For Nandgaon, it will increase to 328 Ha, which was 183 with the canal system. It indicates a 44% increase in the irrigation area, which is considerable. This study may further be extended to obtain the entire construction cost of the systems.

The optimal design of IPDN is presented using the CPM. The diameters thus obtained in CPM are optimized using LP and JA. JA being a parameterless algorithm-specific algorithm, a sensitivity analysis is carried out to select the population size. For the population size of 50 in the sensitivity analysis with FES count of 1000, the model lead to the optimum solution. The cost per meter of the Bakhari network is 4,630.54 Rupees ($62.72) using LP, and that using JA, it comes out to be 4,289.89 Rupees ($58.11). Similarly, the cost per meter for the Nandgaon, which is a smaller network, is 11,493.02 Rupees ($155.67) using LP whereas, it is 11,270.26 Rupees ($152) using JA. The Nandgaon network, being a small five-pipe system with a total network length of 4,020 m, has little room for optimization against the Bakhari network with 33 pipes, having a total network length of 18,355 m. Hence the per meter cost obtained is higher for Nandgaon relative to the Bakhari network. The results indicate that JA's cost is lesser than that obtained using LP for both networks. The percentage decrease in cost for the Bakhari network is 7.36 while for the Nandgaon network is 1.94, respectively. It can be concluded that JA has reached a better-optimized solution than LP for the IPDN. Thus, JA can be promisingly implemented for the optimization of real existing IPDN. The study can be extended to the entire command area network for further reduction in the cost. Also, it can be explored for the layout optimization as well.

The cooperation extended by the Pench Irrigation Division, Nagpur Maharashtra, India, under Vidarbha Irrigation Development Corporation, is gratefully acknowledged for providing the necessary data to carry out the present study.

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