Abstract

The optimal design of water distribution networks (WDNs) has always attracted the attention of researchers due to the complexities involved. The general cost optimization problem is categorized as the NLP-Hard problem. Several evolutionary algorithms (EAs) that can explore the entire search from multiple starting points have been developed in the last three decades. The EAs have more chances of reaching the global optimal solution. However, whether these algorithms converge to a global optimal solution or not is always doubtful, as these algorithms do not guarantee the same solution in the two different runs. A two-source benchmark network is considered for which several better solutions have been reported by different researchers in the last 15 years. The available solutions of this network are analysed based on certain parameters, usually observed in the global optimal solution. The main objective of the study is to explore the possibility of achieving a better than the current best-solution, as the global optimality of the current best is not confirmed. Further, the proposed methodology is applied to test the current best solutions of two additional networks.

1 Introduction

A water distribution network (WDN) is the costliest component of any water supply scheme. Hence, its optimal design is desirable. A network consists of interconnected pipes, which carry water from one or more sources to consumer points scattered all over the city. The water may be supplied by gravity, pumping, or by a combination of both, depending on the topographical features. An existing network may require strengthening and expansion to meet the increased demand. Further, valves of various types are required in the network to facilitate control of flow and pressure, isolation of part of the network, removal of water during flushing, etc. The simplest optimal design problem consists of the design of a single source gravity network with a branched layout in which pipe sizes are to be obtained to minimize the capital cost of the network subject to the constraints of meeting the minimum residual pressure requirements at each delivery point (called nodes) while satisfying the required demands.

The complexities in the network design increase with the presence of loops, size of the network, supply from multiple sources, provision of pumps, existing pipes, time-varying demand and multiple demand patterns, requirement of pressure management, hydraulic and mechanical uncertainties, etc. The design of a looped network requires additional constraints to be imposed to satisfy the conditions of flow balancing at the nodes and energy balancing in the loops. Several evolutionary algorithms (EAs) have been developed in the last three decades. Any new EA, or modified EA, is usually tested with benchmark networks of different complexities, and the results are compared with the current best solution to show the effectiveness of the proposed EA. Sometimes new networks are also introduced to show the application of the proposed EA.

The main challenge in the application of EA is of the non-reproducibility of the same solutions in different runs due to randomness in search behavior. Therefore, several runs are performed and the results of the best run are considered as the optimal. In the case of the benchmark network, a designer repeats the run until a better, or nearer to the current best, is obtained. However, for a new network, no previous solution for comparison is available and a designer stops running the model after making some trials. Then the question is: “Is there any possibility of checking the final solution for its global optimality using certain parameters?”

This paper is aimed at the study of a few benchmark problems from the literature. A typical benchmark problem is initially considered, which is used by many researchers to test their algorithms by either providing a better solution or a similar solution with reduced execution time/effort. The few most feasible solutions for the selected benchmark network are analysed using characteristics usually observed in a global optimal solution (Sonak and Bhave 1993). The result of this analysis is used to explore the possibility of the availability of a better solution than the current best-solution, thereby confirming the global optimality of the current best-solution. The methodology is also applied to two more networks from the literature for which the global optimal solution is not yet confirmed.

2 Typical benchmark problems

Some of the typical benchmark problems with their characteristics are presented in Table 1. Full details of the network are available in the reference paper of researchers (Column 1) who introduced the network. These networks are commonly referred to by the name of the author who introduced it, by the name of the city, or by its typical characteristics as shown in Column 2. Typical characteristics of the network, optimal cost obtained when first introduced, its best-known current solution, % difference between the best-cost solution and the first best solution, the reference paper reporting current best for the first time, and some remarks are given in Columns 3 – 8, respectively, in Table 1. The following can be observed from Table 1:

The optimal solutions are obtained by some of the researchers by considering pipe sizes as continuous variables. Some researchers considered pipe sizes as discrete variables, however, and provided a split pipe size solution in which some of the pipes consist of two different sizes. Herein, while selecting network solutions for further study, only those solutions are considered in which pipe diameters are discrete variables with single pipe size for the entire length.

The global optimal solution for some of the benchmark networks is available, as many researchers reported the same solution through different EAs. Therefore, these networks are not considered in further study.

The difference between the current best-cost solution, and that obtained by the first-used optimization algorithm, varies from 0.38% to 50.52%, thereby indicating the improvement in optimization techniques to reach to global optimal solution.

The networks are classified as small, medium, and large networks depending upon the size of the search space. The search space for a network with X pipes and Y number of available pipe sizes is given by Y ^X. The network is considered as small if the size of the search space is less than 10¹⁰, medium for the search space size in between 10¹⁰ to 10¹⁰⁰, and large if the search space exceeds 10¹⁰⁰. The two-loop network is a small network, Belerama and Ramnagar networks are large networks, and the others considered in Table 1 are medium networks.

Even medium-sized networks are posing challenges to researchers to obtain the global optimal solution, and better solutions than the previous best solutions are being obtained from time to time. The Kadu network, GoYang network, and Ramnagar network are considered to check the global optimality of their current best solution.

The Kadu network is initially considered for further analysis for which the better than previously published best solution is reported at a number of occasions in the last 15 years.

Table 1 Typical benchmark problems.

First Used by	Commonly referred as	Network Characteristic	Cost of Solution when introduced	Cost of Best-Known Solution*	% difference in cost	Obtained by	Remarks
Alperovits and Shamir (1977)	Two-loop Network	Single source, 2 loops, 8 pipes, 14 pipe sizes	479,525 units**	419,000 units	12.62	Many	Small network
Fujiwara and Khang (1990)	Hanoi Network	Single source, 3 loops, 34 pipes, 6 pipe sizes	$6.319 million	$6.081 million	3.77	Many	Medium network
Schaake and Lai (1969)	New York City Tunnel Network	Single source, 3 loops, 21 tunnels for duplication, 11 pipe sizes	$78.09 million	$38.64 million	50.52	Geem (2009)	Medium network
Kadu et al. (2008)	Two-Source Network or Kadu network	Two-source, 9 loops, 34 pipes, 14 pipe sizes	INR 131,678,935	INR 125,019,790	5.06	Gangwani et al. (2024a)	Medium network
Ormsbee and Kessler (1990)	FOWM system	Single source 6 loops, 22 pipes, multiple loading, strengthening and expansion, 7 pipe sizes	$5,339,886 for Maximum day loading & $9,283,700 for peak-hour loading	$4,459,430*** for Maximum day loading & $7,102,781*** for peak-hour loading	20.23     23,49	Rathi et al. (2019)	Medium network
Reca and Martinez (2006)	Balerma	Four sources, 80 loops, 454 pipes, 10 pipe sizes	€2.002 million	€1.94 million	3.10	Tolson et al. (2009)	Large network
Moosavian and Lence (2019)	Farhadgerd	One source, 10 loops, 68 pipes,  9 pipe sizes	$17.80 million	$17.498 million	1.70	Palod et al. (2021)	Medium network
Kim et al. (1994)	GoYang Network	One source, 9 loops, 30 pipes, 8 pipe sizes	179,428,600 Won	177,009,557 Won	1.35	Batmaz and Ibrahim (2024)	Medium network
Gangwani et al. (2024b)	Ramnagar Network	One source, 83 loops, 375 pipes, 16 pipe sizes	INR 37,837,223 (FSS) INR 34,289,277 (DSSR)	INR 34,166,160	9.38 0.36	Gangwani et al. (2024b)	Large-network

* Each pipe of a single commercial size, **Pipes with continuous diameters, ***Split pipe solutions

3 Kadu network (Kadu et al. 2008)

3.1 Network description

The Kadu network was introduced in Kadu et al. 2008. It has two reservoirs with 26 nodes, 34 pipes, and 9 loops. Supply from both reservoirs is unrestricted. The layout of the network is shown in Figure 1. The minimum supply levels of reservoirs 1 and 2 are 100.00 m and 95.00 m, respectively. The node numbers and link numbers are shown in the Figure 1. The nodal demands are also shown in Figure 1. Other details can be obtained from Kadu et al. (2008). The Hazen–Williams pipe friction head loss formula is as shown in Equation 1.

(1)

Where:

hl	=	head loss,
ω, α, and β	=	constants,
L	=	pipe length,
CHW	=	Hazen-Williams pipe roughness coefficient, and
D	=	pipe diameter.

Kadu et al. (2008) considered the values of ω, α, and β as 2.234 ×10¹², 1.85, and 4.87, respectively, for the discharge in m³/min, diameter in mm, and C_HW for all pipes as 130. Kadu et al. (2008) developed the computer program “GANET” in C-language for the design of WDNs using the above values of ω, α, and β.

Figure 1 A two-source network (Kadu et al. 2008).

3.2 Network’s design solutions

The optimal solutions obtained by different researchers for the Kadu network using their respective algorithms are presented in Table 2. However, most of the other researchers used EPANET (Rossman 2000) for hydraulic simulation, which has the default values of ω, α, β, as 10.667, 1.852, 4.871, respectively, for the flow in m³/s, and diameter in m. Gangwani et al. (2024a) checked the feasibility of all the solutions using the current version of EPANET 2.0 (Build 2.00.12.01) in which ω, α, and β are 10.667, 1.852, and 4.871, respectively, and observed that some of the solutions are infeasible. All feasible solutions are ranked herein based on their cost as shown in Column 6 of Table 2.

Table 2 Optimal solutions for Kadu two-source network by different methods.

Reference Papers	Method used	Optimal Cost INR (Indian Rupees)	Total Number of Function Evaluations	Remark	Rank
(1)	(2)	(3)	(4)	(5)	(6)
Kadu et al. (2008)	GA - head deficit-based penalty	131,678,935a	120,000	FSS	R19
	GA - head deficit-based penalty	126,368,865a,b	25,200	RSS	-
Haghighi et al. (2011)	GA-ILP	131,312,815b	4,440	FSS	-
Ezzeldin et al. (2014)	Integer discrete PSO with NR	125,843,995c	-	FSS	R11
		125,501,130	-	FSS	R8
Siew et al.  (2014)	PF-MOEA	125,460,980	436,000	FSS	R7
	PF-MOEA	125,826,425	82,400	RSS	R10
Barlow and Tanyimboh (2014)	GA-based Memetic Algorithm	124,690,000b	142,000	FSS	-
Mohammadi-Aghdam et al. (2015)	AM-PSO	130,666,000	22,000	FSS	R18
Jabbary et al. (2016)	CFOnet	126,535,915	259,476	FSS	R13
Abdy Sayyed et al. (2019)	GA - head deficit-based penalty (DDA)	128,381,245	8,300	RSS	R17
	GA - head and flow deficit-based penalty	126,365,955	7,000	RSS	R12
		125,754,310	7,600	RSS-Altered	R9
		127,891,665 (only q reqd)	-	RSS	R16
Cassiolato et al. (2021)	MINLP model solved using a deterministic Mathematical Programming Approach	124,986,030b,c	-	FSS	-
		125,136,870b	-	FSS	-
Palod et al. (2021)	Rao I Algorithm	126,825,885	24,950	FSS	R14
	Rao II Algorithm	125,434,170	19,700	FSS	R6
Tanyimboh et al. (2021)	GA with redundant binary code	127,368,355	339,000	FSS	R15
Gangwani et al. (2023)	GA - head and flow deficit-based penalty	125,209,860	157,760	FSS	R3
Gangwani et al. (2024a)	GA - head and flow deficit-based penalty	125,019,790	11,25,300	DSSR	R1
		125,076,190	267,000	DSSR	R2
		125,266,260	160,500	DSSR	R4
		125,300,560	231,000	DSSR	R5

GA – Genetic Algorithm, ILP – Integer Linear Programming, PF-MOEA- Penalty Free Multi-Objective Evolutionary Algorithm, AM-PSO – Accelerated Momentum Particle Swarm Optimization, NR-Newton Raphson, CFOnet- Central Force Optimization, FSS – Full Search Space, RSS – Reduced Search Space, DSSR – Dynamic Search Space Reduction, Mixed Integer Non-Linear Programming;

 ^a α, β, ω as 1.85, 4.87, 10.68;   ^b infeasible solution with current EPANET parameters (α, β, ω) = (1.852, 4.871, 10.667). ^c ω = 10.6744

It can be further observed from Column 2 of Table 2 that Ezzeldin et al. (2014) were the first to get improved solutions with two different values, as ω, α, and β (10.6744, 1.852, 4.8704 and 10.667, 1.852, 4.871). At nearly the same time, Siew et al. (2014) also presented a better solution than that obtained by Ezzeldin et al. (2014). Palod et al. (2021) were the next to obtain an improved cost solution. Recently, Gangwani et al. (2023, 2024a) obtained nine better solutions than those reported by Palod et al. (2021). Five of them are shown in Table 2.

3.3 Details of feasible design solutions

The best 15 solutions are considered herein for further analysis. These solutions are varying in cost from 125.019 to 128.381 million (INR). The variation in cost is about 2.68 percent. However, the function evaluations required to identify these solutions varied from a few thousand to a few million. Details of pipe diameters, nodal heads, and pipe flows for these solutions are presented in Tables S1, S2, and S3 in the supplementary material, respectively. These solutions are arranged by rank, from lowest to highest cost solutions.

3.4 Analysis of feasible solutions

Analysis of these solutions is carried out using parameters described below, to explore the possibility of the existence of a better solution, or considering the current best solution as the global optimal solution. Also, it suggests the methodology that a designer can use to check the final solution for its global optimality.

Number of pipes with minimum sizes

Any looped network gets converted to one of its branching configurations during optimization when no constraints on the minimum size of pipes are imposed (Sonak and Bhave 1993, Bhave 2003). This is due to the concave nature of the cost-discharge relationship which allows flows to get concentrated in a few pipes (called primary pipes) and eliminates some pipes (called secondary pipes). However, as a looped nature must be maintained, the optimal design solution provides loop-forming links of minimum size. The prior selection of primary and loop-forming pipes is a challenging task, and therefore, the optimization algorithm is allowed to determine primary and loop-forming secondary pipes on its own.

The two-source network with 9 loops requires the removal of 10 pipes (one pipe from each loop, and one to completely isolate two branched networks, one from each source). As a looped nature is to be maintained, there must be at least 10 pipes of minimum sizes. Out of the 15 solutions, 12 of them have 10 pipes of minimum sizes. Thus, the researchers who stopped at the other 3 solutions could have tried more to get a better solution. Further analysis of 12 solutions shows that two distinct branching configurations are observed on the removal of minimum pipe sizes, as shown in Figure 2. The configuration shown in Figure 2(a) is observed in 9 cases (R1 to R7, R11, and R15) and Figure 2(b) in 3 cases (R10, R13, and R14). The only difference between the two is that instead of pipe 19, pipe 23 is of minimum size in the configuration shown in Figure 2(b). Thus, solution R1 meets the criteria of the minimum number of pipes of minimum sizes and not more than 10 pipes of minimum size are present in any other solution.

Figure 2 Identified branching configurations showing primary and secondary pipes.

The comparison of pipe sizes of the remaining pipes in alternative solutions shows that solutions R2 and R3 have one pair of pipes different as in R1; while the other solutions from R4 to R15 have more than two primary pipes having different sizes as in R1. In R1, pipe 18 is 400 mm and pipe 27 is 250 mm, while the same are 350 mm and 300 mm, respectively, in solution R2. The pair of pipes that are different in R3 as compared to R1 are pipes 13 and 14. While both pipes are 500 mm in R1, pipe 13 is 600 mm, and pipe 14 is 450 mm.

The hydraulic analysis of R1 is carried out by reducing the size of primary pipes one by one, to the next available lower size, to check for a possible reduction in cost. In all cases, the network solution becomes infeasible. The above two observations indicate that no reduction in a single pipe size is possible, however, a change in the diameters of a pair of pipes, or a group of pipes, may result in a reduction in cost while maintaining feasibility. However, identifying that pair through an informal search is not possible, as a huge number of feasible and infeasible solutions exist in the vicinity of the R1. Thus, the possibility of the existence of a better solution cannot be ruled out with only this indicator.

Critical nodes

In the optimization of a gravity-fed network, the available head difference between the source node and any demand node is supposed to be utilized completely to reduce the network cost to the maximum possible extent. However, complete utilization of available heads at all the nodes is not possible, as sufficient pressure is necessary to maintain the flow on the downstream side of the network. Critical nodes in a network are those where the available head is fully utilized. Therefore, it is desirable to have as many as possible critical nodes. The comparison of hydraulic analysis results of the 15 best solutions shows that node 26 is critical in only R13, having available HGL equal to minimum HGL. In the remaining 14 solutions, no node becomes critical. This is mainly due to the imposed condition of a single uniform discrete size for each pipe. Therefore, herein, a node with a minimum excessive head above the required HGL is considered as the most critical node in the remaining solutions. Node 6 in R5 and R14, node 13 in R11, node 12 in R1 and R2, node 16 in R3, node 20 in R10, node 24 in R4, R6 and R8, node 25 in R15, and node 26 in R7, R9, and R12, are observed to be the most critical. It is worth noting that most critical nodes in different solutions are on the boundaries of branching trees. Node 26 is observed to be most critical in the largest number of solutions, i.e., 4 numbers. The two best solutions, R1 and R2, have node 12 as the most critical node. The excess head at node 12 is 0.15 m in R1, indicating a possibility of a better solution than R1.

Sink nodes

Sink nodes are the nodes where all the connected pipes are the inflow pipes. There could be one or many sink nodes in the network depending upon the topography and other conditions, like flow direction in loop-forming links. As there is no further distribution of flow from sink nodes, these nodes are expected to have no or minimum excessive pressure. Further, nodes that have minimum HGL requirements are most likely to be the sink nodes. In the Kadu network, four nodes, nodes 22, 24, 25, and 26, have the same minimum HGL requirement of 80 m. These nodes are interconnected with pipes 32, 33, and 34, which are of minimum pipe sizes in both the identified branching configurations as shown in Figures 2(a) and 2(b). However, all four nodes can rarely become sink nodes. The flow direction in pipes 32, 33, and 34 decides whether one or more of them are sink nodes. Sink nodes observed in different solutions are provided in Table 3. Further, one or more nodes from nodes 6, 7, 12, 13, 16, and 22, on the boundaries of two branching configurations, are likely to be sink nodes.

Table 3 Sink nodes and available residual heads in different solutions of the Kadu network.

Network Solution Number	Sink Node 1	Residual head (m)	Sink Node 2	Residual head (m)	Sink Node 3	Residual head (m)	Sink Node 4	Residual head (m)	Sink Node 5	Residual head (m)
R1	24	0.17	16	0.17	26	0.22	7	3.21
R2	16	0.18	26	0.36	7	3.21
R3	24	0.16	16	0.18	26	0.22	7	3.21
R4	16	0.19	26	0.37	7	3.21
R5	25	0.32	22	1.11	16	0.69	7	2.44
R6	24	0.04	26	0.05	7	0.53
R7	26	0.24	24	0.50	16	0.79	6	1.26
R8	24	0.37	16	0.68	26	0.69	7	3.57
R9	24	0.45	26	0.67	16	0.67	7	3.62
R10	26	0.59	7	1.13	16	1.19
R11	13	0.07	26	0.22	7	1.39	24	1.7
R12	26	0.03	16	0.04	12	0.05	24	0.12	7	2.08
R13	26	0	16	0.06	23	0.71	6	1.18
R14	6	0.07	16	0.09	25	0.17
R15	25	0.24	16	0.53	22	1.10	7	2.65

It can be observed from Table 3 that the number of sink nodes varied from 3 to 5. Solutions R2, R4, R6, R10, and R14 have only three sink nodes. Nodes 7, 16, and 26 in R2, R4, and R10, nodes 7, 24, and 26 in R6, nodes 6, 16, and 25 in R14. Herein, at least one better solution with 4 sink nodes exists, therefore solutions with 4 sink nodes are better than solutions with 3 sink nodes for this network. Node 16 is observed as a sink node in the maximum number of solutions (13 out of 15). The exceptions are R6 and R11. Solution R6 has only 3 sink nodes, while node 13, instead of 16, becomes a sink node in R11. Node 7 is seen as a sink node in 12 out of 15 cases. In three cases, R7, R13, and R14, node 6 becomes a sink node instead of 7. Similarly, node 26 is seen as a sink node in 12 out of 15 cases. In three cases R5, R14, and R15, node 25 instead of node 26 becomes a sink node. Node 24 is observed as sink node in 8 cases, and all the time with node 26. Node 22 in R5, and node 12 in R12, are observed as sink nodes. Thus, the likely four sink nodes are 6 or 7, 13 or 16, 22 or 24, and 25 or 26. Solution R1 has the same four sink nodes: 7, 16, 24 and 26, and node 12 is the most critical. Now, let us compare R1 with R12, which has 5 sink nodes. This also includes node 12 as a sink node. For node 12, to become a sink node, the flow direction in pipe 15 should be from node 13 to node 12. The minimum HGL requirement at node 12 is 85 m, and at node 13 is 82 m. This requires the available HGL at node 13 to be more than 85 m, which increases the cost as observed in solution R12. This indicates that the best solution for this network is with only four sink nodes.

Considering the excessive head of 0.15 at the most critical node 12, and also excessive heads at sink nodes, there lies a possibility for a better cost solution than R1. Considering this possibility, the Rao-II algorithm (Palod et al. 2021) with a higher number of function evaluations (NFEs) is used. A better cost solution of INR 124,963,610 in 42,150 NFEs is obtained by Gangwani et al. 2024b. The solution contains 10 pipes of minimum size with a branching configuration, as shown in Figure 2(a) (same as in R1). The critical node is observed as node 6 with no excessive pressure, instead of node 12 in R1. The sink nodes are observed as nodes 7, 24, 16, and 26 (same as seen in R1). The excessive pressure head is reduced to 0.02 m, 0.15 m, 0.22 m, and 0.2 5 m. As no excessive pressure is available at least at one of the nodes, there are more chances of the present solution being the global optimum solution.

The analysis of best solutions indicates that parameters like the number of minimum pipe sizes in the network, the number of critical nodes and the number of sink nodes as critical nodes, excessive pressure at critical and sink nodes can be used to explore the possibility of a better solution than the current best solution. The application of the methodology is now extended to two more networks.

4 GoYang network (Kim et al. 1994)

4.1 Network description

The GoYang network was introduced by Kim et al. (1994). It includes 22 demand nodes, 30 pipes, 9 loops, and a constant head pump of 4.52 kW linking to a reservoir with a head of 71 m, as shown in Figure 3. The Hazen-Williams roughness coefficient for each new pipe is 100. The minimum required pressure head above the ground elevation at each node is 15 m. A set of 14 commercial pipe diameters was used in this design problem. The node and pipe data are available in Geem (2006).

Figure 3 GoYang network (Kim et al. 1994).

4.2 Network’s design solutions

The selected design solutions from different researchers are presented in Table 4. It can be observed from the table that only four better solutions are provided after the original design. The best cost solution was obtained by Eryiğit (2015). Six other EAs, or their modifications, were tested by other researchers (Jain and Khare 2021; Palod et al. 2021; Gangwani et al. 2024a), and reported the second best solution. Recently, Batmaz and Ibrahim (2024) obtained the best cost solution, as reported by Eryiğit (2015). All feasible solutions are ranked herein, based on the cost as shown in Column 6 of Table 4.

Table 4 Optimal solutions for GoYang network by different methods.

Reference Papers	Method used	Optimal Cost (INR)	Total NFE	Remark	Rank
Kim et al. (1994)	(Original Design)	179,428,600	-	FSS	R5
Kim et al. (1994)	NLP	179,142,700	-	FSS	R4
Geem (2006)	HS	177,135,800	10,000	FSS	R3
Jain and Khare (2021)	RAO-II	177,010,355	1,400	FSS	R2
Eryiğit (2015)	AIS	176,958,824b	-	FSS	-
		176,994,561b	-	FSS	-
		177,009,557	-	FSS	R1

NFE - Number of Function Evaluations, NLP – Non-Linear Programming, Harmony Search, AIS- Artificial Immune System;   ^bInfeasible solution with current EPANET parameters (α, β, ω) = (1.852, 4.871, 10.667).

4.3 Details of feasible design solutions

The best cost solution is analyzed as per the methodology tested above on the two source network by Kadu et al. (2008). Details of nodal heads and pipe flows for these solutions are presented in Tables S4 and S5 in the supplementary material, respectively. These solutions are arranged by rank from lowest to highest cost solutions.

4.4 Analysis of best-cost solution

Analysis of the best-cost solution (R1) is carried out based on the parameters as seen earlier to explore the possibility of the existence of a better solution for the network.

Number of pipes with minimum sizes

The number of pipes with the minimum size in the best-cost solution is 23, which far exceeds the minimum required number of 9 pipes to convert the loop into a branch network. This indicates that some of the primary pipes are also of the minimum size. The minimum diameter pipes as observed in different solutions ranked from 1 to 5 are shown in Figure S1 with a red Column. It can be seen that in every loop at least one pipe has red columnour.

Critical nodes

The hydraulic analysis of R1 shows that no node is critical with zero excessive pressure. The most critical node is 14, having a residual head 0.02 m above the minimum required, which is marginally excessive.  

Sink nodes

Sink nodes observed in different solutions are provided in Table S6 (supplementary material). Nodes 15 and 22 are observed as sink nodes in most of the solutions. The other sink nodes are 18 and 20. However, the solutions with nodes 18 and 20 as sink nodes are costlier. In R1, two sink nodes are 15 and 22, with residual heads of 0.16 m and 0.83 m, respectively. The excess residual heads at the sink nodes, coupled with a small excess head at the critical node 14, indicate the possibility of a better cost solution for this network. However, chances are very low, as two more nodes (9 and 10) also have minimal excessive heads. An attempt was made to use the Rao-II algorithm with an increased number of generations. However, better than the current best could not be obtained.

Further, as seen earlier, a change in the diameters of a pair of pipes or a group of pipes may result in a reduction in cost while maintaining feasibility. However, identifying that pair through informal search is not possible. As herein, 23 out of 30 pipes are of minimum size in the best-cost solution, a partial enumeration is also considered. In a partial enumeration, a minimum size pipe can be kept as it is or can be increased to the next higher size. The remaining 7 pipes can have 3 alternatives, one size up, the same size, or one size down. So, the total number of combinations to be analyzed to check the feasibility and cost will be 3⁷ × 2²³ (= 8,390,795 INR). In comparison to the NFE required for obtaining the optimal solution, the NFE required to check the global optimality is too high and can not be recommended. 

The analysis of R1 thus indicates that it is most likely to be a global optimal solution. The possibility of the existence of a better solution cannot be ruled out. However, if a better solution exists, it will be very close to R1 in cost.

A peculiar observation from Figure S1 showing the R1 solution is that the pipe 27, which is downstream of pipes 24 and 25, is of a larger size (100 mm) in comparison to the size of pipes 24 and 25, which are 80 mm. Pipes 24 and 25 can be observed feeding to pipe 27, and therefore pipe 27 carries more flows as compared to the flows in pipes 24 and 25.

5 Ramnagar network (Gangwani et al. 2024a)

5.1 Network description

The Ramnagar network, as shown in Figure 4, was first used by Gangwani et al. (2024a) to illustrate their design methodology. The network consists of 375 pipes, 292 junctions, 83 loops, and a Ground Service Reservoir with a lowest supply level of 327.205 m. The design diameters for the network must be selected from 16 commercially available diameters, which leads to a total search space of 16³⁷⁵. The details of the network can be referred from Gangwani et al. (2024a).

Figure 4 Ramnagar network.

5.2 Network’s design solutions

Ramnagar network is recently used by Gangwani et al. (2024a, 2024b). The design solutions proposed are presented in Table 5. The network has not been used by other researchers to test their design algorithms until now.

Table 5 Optimal solutions for Ramnagar network by different methods.

Reference Papers	Method used	Optimal Cost (INR)	Total Number of Function Evaluations	Remark	Rank
Gangwani et al. (2024a)	GA	37,837,223	8,81,600	FSS	R5
		34,289,227	3,82,500	DSSR	R3
Gangwani et al. (2024b)	Modified Rao-II HDB-SA-DDA	34,236,624	16,15,000	FSS	R2
	Modified Rao-II HDB-SA-PDA	34,429,738	1,499,500	FSS	R4
	Modified Rao-II HFDB-SA-PDA	34,166,161	1,591,500	FSS	R1

GA – Genetic Algorithm, FSS – Full Search Space, RSS – Reduced Search Space, DSSR – Dynamic Search Space Reduction, HDB – Head Deficit Based; HFDB – Head Flow Deficit Based, SA-Self Adaptive, DDA-Demand Driven Analysis, PDA- Pressure Driven Analysis.

5.3 Details of feasible design solutions

These solutions are varying in cost from 3.417 to 3.783 million INR. The variation in cost is about 16.82 percent. However, the function evaluations required to identify these solutions varied from a few Lakhs to a few million. These solutions are arranged by rank from lowest to highest cost solutions.

5.4 Analysis of feasible solutions

Analysis of the best solution is carried out using the parameters discussed earlier to explore the possibility of the existence of a better solution or to consider the current best solution as the global optimal solution.

Number of pipes with minimum sizes

A single source network with 83 loops requires the removal of 83 pipes (one from each loop) to convert it to the branched network. In the best solution for the Ramnagar network, 307 out of 375 pipes are of minimum size. There is at least one pipe of minimum size in each loop, as can be observed in the best solution, as well as in other solutions, see Figure S2. Thus, this criterion is satisfied by R1.

Critical nodes

Node 53 in R1 has the lowest excess head at 0.01 m, and nodes 59 and 248 also have very little excess head. Critical nodes for all the solutions are shown in Table S7. It can be observed that in R4, nodes 53 and 156 are critical with no excess residual pressure. Also, nodes 59 and 248 in R2 have lower excess pressure values as compared to those in R1. This indicates a high possibility of the existence of a better than the current best solution.

Sink nodes

The total no. of sink nodes are either 63 or 64 in different solutions for the Ramnagar network. Most of these nodes are the end nodes of branching trees that originated from loops. The details of sink nodes and available excess residual heads are given in Table S8. The excess residual heads at the sink nodes further emphasize the possibility of the existence of a better cost solution for this network.

A formal search using the EA with an increased number of functional evaluations, or an informal search can be carried out to identify better solutions than in R1. In addition, informal search is used to verify the observations made using the proposed indicative parameters. For the informal search, the solution R1 is shown with column codes in Figure S2. The red column is used to show 80 mm diameter pipes; yellow for 100, 150, and 200 mm; green for 250, 300, 350, and 400 mm; dark blue for 450, 500, and 600 mm; and light blue for greater than 600 mm size. Similarly, nodes are coded based on available residual nodal pressure heads: red for residual pressure heads below 8.4 m; yellow for residual pressure heads between 8.4 and 9 m; green for residual pressure between 9 and 12 m; dark blue for residual pressure heads between 12 and 15; and light blue for greater than 15 m. These column codes provided easy visualization of pressure heads on the downstream nodes of any pipe; and shows a comparison of the size of any pipe with the sizes of its upstream and downstream pipes.

Looking at the column-coded R1 solution, changes in the diameter of the pipe or a pair of pipes is carried out to reduce the cost. The feasibility of changes is assured through hydraulic analysis by observing the available pressure values. The informal search process is used as below:

There are a few pipes observed in the yellow column (size 100–200 mm), with immediate upstream and downstream pipes in the red column (80 mm). The sizes of these pipes were reduced to 80 mm, and the feasibility of change is developed. Also, one pipe is observed in the green column (size 250–400 mm) with immediate upstream and downstream pipes in the red column (80 mm) or yellow column (size 100–200 mm). The size of this pipe was reduced to 200 mm and feasibility of change is determined. These pipes are coloured blue in Figure S2. If determined to be feasible, pipe size is reduced from 100 to 80 mm, or 250 to 200 mm.

There are groups of pipes encircled red in Figure S2. Therein, the smaller length pipe is 100 mm, and the larger length pipe is 80 mm. These diameters are swapped to reduce cost, if observed feasible.

Also, by trial and error, the possibility of changing the larger diameter to a smaller diameter can be checked to reduce the cost. Some changes are observed as feasible and effected. 

In this process, few new solutions in the vicinity of R1 are observed. The best solution (R1*) has a cost of INR 34,089,252.21, which is 0.225% less than R1. The number of pipes with the minimum diameter increased to 313 from 307. The minimum residual head of 0.01 m is observed at node J-53. The excess residual head at critical nodes are shown in Table S7. This informal search for better solution verified the existence of better solutions than the current best available in the literature and the usefulness of proposed indicative parameters. The pipe diameters that are changed from the R1 solution are given in Table S9.

6 Summary and conclusions

Evolutionary techniques have gained importance over deterministic search techniques because of their capability to search the entire dataset through multiple starting points and with a higher probability of reaching the global optimal solution. However, these techniques require high computational efforts and also suffer from the drawback of non-reproducibility of the same solution in two different runs. Therefore, when these techniques are applied to a new network, some efforts are needed to fine-tune the associated parameters of EAs through repeated execution of the program, and then a few runs are required for comparison of best solutions in different runs. As seen from the solutions of the benchmark Kadu network obtained using GA, PSO, NSGA, CFO, Rao-I, and Rao-II, various researchers stopped at different solutions. Sometimes better than the available best solution at that time is obtained. Three parameters (number of pipes with minimum sizes, number of critical nodes, and number of sink nodes becoming critical) are proposed for checking the final solution for its global optimality and exploring the existence of a better solution. These parameters are applied to the current best solutions of the Kadu, GoYang, and Ramnagar networks. The current best solution for the Kadu network showed the possibility of the existence of a better solution, and when the Rao II algorithm with increased population size and number of generations was used, a better solution was obtained. A remote possibility of the existence of a better solution is observed when proposed parameters are used on the current best solution of the GoYang network. However, no better solution could be obtained through a formal or informal search. The analysis of the current best solution for the Ramanagar network with proposed parameters indicated a high possibility of the existence of better solutions. An informal search on the Ramnagar network using the current best solution indicated several possible changes to reduce the cost. Therefore, the proposed indicative parameters are found to be useful in assessing the optimal solution. When the existence of a better solution is a possibility, further EA runs can be performed to reach near the global optimal solution. A better solution for the Kadu network and a few better solutions for the Ramnagar network could be noticed with the proposed methodology. The current best solution for the GoYang network is shown to have a remote possibility of the existence of a better solution.

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Supplementary Materials

Table S1 Diameters of different solutions for Kadu two-source network.

Pipe	Length (m)	Diameter in mm
		R1	R2	R3	R4	R5	R6	R7	R8	R9	R10	R11	R12	R13	R14	R15
1	300	900	900	900	900	900	1,000	900	900	900	900	900	900	900	1,000	900
2	820	900	900	900	900	900	900	900	900	900	900	900	900	900	900	900
3	940	350	350	350	350	400	350	350	400	350	400	350	350	350	350	400
4	730	300	300	300	300	250	300	300	250	300	250	300	300	300	250	250
5	1620	150	150	150	150	150	150	150	150	150	150	150	150	150	150	150
6	600	250	250	250	250	200	200	250	200	250	200	250	200	300	250	200
7	800	800	800	800	800	800	800	800	800	800	800	800	800	800	800	800
8	1400	150	150	150	150	150	150	150	150	150	150	150	150	150	150	150
9	1175	450	450	450	450	450	450	450	400	450	600	450	600	600	600	400
10	750	500	500	500	500	500	500	500	500	500	600	500	700	600	600	600
11	210	750	750	750	750	800	800	900	900	800	900	800	800	900	900	800
12	700	700	700	700	700	700	700	700	700	750	700	700	700	700	750	750
13	310	500	500	600	600	500	500	500	600	500	500	600	500	500	600	600
14	500	500	500	450	450	500	450	500	450	450	500	450	450	500	400	450
15	1960	150	150	150	150	150	150	150	150	150	150	150	150	150	150	150
16	900	500	500	500	500	500	500	500	500	500	500	500	500	500	450	500
17	850	350	350	350	350	350	350	350	350	350	350	350	350	350	400	350
18	650	400	350	400	350	350	300	400	350	350	350	400	300	400	350	450
19	760	150	150	150	150	150	150	150	200	150	450	150	450	500	500	150
20	1100	150	150	150	150	150	150	150	150	150	150	150	200	150	150	150
21	660	700	700	700	700	700	750	700	700	700	600	700	700	600	600	700
22	1170	150	150	150	150	150	150	150	150	150	150	150	150	150	150	150
23	980	450	450	450	450	450	450	450	500	450	150	450	150	150	150	450
24	670	350	350	350	350	350	350	350	350	350	350	350	350	450	450	350
25	1080	700	700	700	700	700	700	700	700	700	600	700	500	500	500	700
26	750	250	250	250	250	250	250	250	250	250	250	250	250	200	250	250
27	900	250	300	250	300	300	300	250	300	250	300	250	300	350	300	250
28	650	300	300	300	300	250	300	300	300	300	300	300	300	250	250	250
29	1540	200	200	200	200	250	200	200	200	200	200	200	200	250	250	250
30	730	300	300	300	300	250	250	250	250	300	300	300	300	250	300	250
31	1170	150	150	150	150	150	150	150	150	200	150	150	200	150	150	150
32	1650	150	150	150	150	150	150	150	150	150	150	150	150	150	150	150
33	1320	150	150	150	150	150	150	150	150	150	150	150	150	150	150	150
34	3250	150	150	150	150	150	150	150	150	150	150	150	150	150	150	150

Table S2 Available nodal heads for different solutions of Kadu two-source network.

Node	Min HGL   (m)	Available head (m)
		R1	R2	R3	R4	R5	R6	R7	R8	R9	R10	R11	R12	R13	R14	R15
Res.1	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100
Res.2	95	95	95	95	95	95	95	95	95	95	95	95	95	95	95	95
3	85	98.28	98.28	98.28	98.28	98.27	98.98	98.29	98.29	98.29	98.28	98.28	98.32	98.29	98.98	98.26
4	85	95.04	95.03	95.04	95.03	95.00	95.79	95.08	95.06	95.06	95.03	95.04	95.17	95.06	95.75	94.97
5	85	87.33	87.33	87.33	87.32	90.73	87.66	87.61	87.45	87.51	90.79	90.81	87.28	87.83	88.74	90.84
6	85	85.35	85.35	85.35	85.35	85.08	85.38	85.79	85.55	85.64	85.24	85.31	85.18	86.18	85.07	85.67
7	82	85.21	85.21	85.21	85.21	82.44	82.53	85.96	85.57	85.62	83.13	83.39	84.08	87.78	85.99	84.65
8	82	87.81	87.82	87.82	87.82	88.51	88.52	88.89	88.32	88.32	89.42	89.77	90.91	89.17	89.20	91.53
9	85	91.13	91.12	91.12	91.12	91.10	91.99	91.2	91.16	91.17	91.14	91.17	91.41	91.13	91.82	91.01
10	85	88.31	88.28	88.30	88.28	88.27	89.29	88.41	88.36	89.19	88.34	88.37	88.81	88.19	89.77	88.96
11	85	86.44	86.41	87.53	87.51	86.41	87.43	86.55	87.59	87.33	86.51	87.60	87.03	86.36	89.01	88.18
12	85	85.15	85.13	85.38	85.36	85.13	85.30	85.28	85.44	85.19	85.27	85.44	85.05	85.12	85.31	86.02
13	82	82.38	82.39	82.39	82.40	83.05	83.12	83.39	82.87	82.85	84.98	82.07	85.68	84.65	84.69	82.53
14	82	92.95	92.95	92.95	92.95	93.51	93.48	94.15	93.49	93.49	94.16	94.15	93.46	94.15	94.15	93.52
15	85	88.01	88.03	88.01	88.02	88.00	88.84	88.09	88.05	87.93	88.05	88.1	88.15	88.29	86.71	87.91
16	82	82.17	82.18	82.18	82.19	82.69	83.16	83.26	82.68	82.67	83.19	82.43	82.04	82.06	82.09	82.53
17	82	89.56	89.56	89.56	89.57	90.12	90.95	90.74	90.09	90.09	90.10	90.51	91.67	90.24	90.20	90.12
18	85	85.45	85.53	85.45	85.53	85.50	86.22	85.55	85.50	85.11	85.55	85.65	85.25	86.33	85.44	85.39
19	82	86.11	83.97	86.11	83.96	83.98	82.28	86.28	86.19	85.39	83.99	84.38	82.07	85.36	85.12	87.61
20	82	82.33	82.35	82.34	82.36	83.04	82.92	83.3	82.81	82.77	82.23	83.85	82.40	82.95	83.00	83.06
21	82	86.35	86.36	86.35	86.36	86.82	87.52	87.46	86.87	86.86	87.10	86.91	85.15	83.18	83.22	86.80
22	80	85.09	85.11	85.09	85.11	81.11	81.01	80.88	85.62	85.62	85.62	80.71	87.07	80.46	85.71	81.1
23	82	83.01	83.23	83.01	83.22	83.19	83.64	83.15	83.08	82.06	83.24	83.43	82.03	82.71	83.29	83.03
24	80	80.17	81.05	80.16	81.05	80.97	80.04	80.5	80.37	80.45	81.06	81.7	80.12	83.24	81.79	80.59
25	80	80.52	80.77	80.53	80.77	80.32	80.93	81.28	80.97	80.95	80.73	81.88	80.54	80.05	80.17	80.24
26	80	80.22	80.36	80.22	80.37	81.52	80.15	80.24	80.69	80.67	80.59	80.22	80.03	80.00	80.41	81.48

Table S3 Available pipe flows for different solutions of Kadu two-source network.

Pipe	Direction in Input file		Flow in m3/sec
	From Node	To Node	R1	R2	R3	R4	R5	R6	R7	R8	R9	R10	R11	R12	R13	R14	R15
1	1	3	6086.17	6088.74	6087.06	6089.62	6105.54	6044.75	6059.65	6074.2	6071.41	6091.84	6083.34	6008.85	6074.59	6070.21	6123.12
2	3	4	4982.17	4984.74	4983.06	4985.62	5001.55	4940.75	4955.65	4970.2	4967.41	4987.84	4979.34	4904.85	4970.59	4966.21	5019.12
3	4	5	615.88	615.88	615.85	615.85	636.47	633.81	605.8	611.41	609.18	633.89	632.86	623.62	594.86	585.29	624.88
4	5	6	225.88	225.88	225.85	225.85	246.47	243.81	215.8	221.41	219.18	243.89	242.86	233.62	204.86	195.29	234.88
5	6	7	5.77	5.66	5.71	5.60	27.71	28.89	-6.31	0.42	2.39	24.57	23.34	17.25	-21.22	-15.75	16.62
6	7	8	-180.23	-180.34	-180.29	-180.4	-158.29	-157.11	-192.31	-185.58	-183.61	-161.43	-162.66	-168.75	-207.22	-201.75	-169.38
7	4	9	4096.29	4098.86	4097.21	4099.77	4095.08	4036.94	4079.85	4088.79	4088.24	4083.95	4076.48	4011.23	4105.73	4110.92	4124.25
8	10	6	31.89	31.78	31.86	31.75	33.24	37.08	29.89	31.01	35.21	32.68	32.48	35.62	25.93	40.96	33.74
9	13	8	-875.59	-875.15	-875.01	-874.59	-877.91	-872.39	-881.2	-876.74	-878.59	-1674.1	-775.74	-1828.6	-1690	-1687.8	-843.18
10	8	14	-1427.8	-1427.5	-1427.3	-1427	-1408.2	-1401.5	-1445.5	-1434.32	-1434.2	-2207.5	-1310.4	-2369.3	-2269.2	-2261.6	-1384.6
11	14	2	-5025.8	-5023.3	-5024.9	-5022.4	-5006.5	-5067.3	-5052.4	-5037.8	-5040.6	-5020.2	-5028.66	-5103.2	-5037.4	-5041.8	-4988.9
12	9	10	2597.88	2605.13	2598.81	2606.04	2601.03	2533.87	2582.76	2591.22	2570.2	2590.15	2586.16	2488.01	2655.58	2622.65	2628.57
13	10	11	1333.68	1333.52	1334.74	1334.59	1330.03	1330.53	1328.87	1332.67	1331.44	1315.77	1336.53	1296.54	1317.92	1319.52	1337.06
14	11	12	841.68	841.52	842.74	842.59	838.03	838.53	836.87	840.67	839.44	823.77	844.53	804.54	825.92	827.52	845.06
15	12	13	25.68	25.52	26.74	26.59	22.03	22.53	20.87	24.67	23.44	7.77	28.53	-11.46	9.92	11.52	29.06
16	9	15	988.41	983.73	988.4	983.73	984.05	993.07	987.09	987.57	1008.04	983.8	980.32	1013.22	940.15	978.28	985.68
17	15	18	358.41	353.73	358.4	353.73	354.05	363.07	357.09	357.57	378.04	353.8	350.32	383.22	310.15	348.28	355.68
18	10	19	542.31	549.83	542.21	549.7	547.76	476.26	534.01	537.54	513.54	551.7	527.15	465.84	621.73	572.16	567.77
19	13	20	4.63	4.08	4.97	4.45	0.31	10.42	7.11	5.30	6.05	766.17	-71.98	842.54	778.13	777.54	-17.63
20	13	16	8.64	8.59	8.78	8.73	11.63	-3.50	6.96	8.11	7.99	27.69	-11.75	86.58	33.79	33.82	1.87
21	14	17	2962.01	2959.76	2961.64	2959.39	2962.25	3029.75	2970.83	2967.48	2970.38	2176.64	3082.26	2097.81	2132.2	2144.19	2968.31
22	19	20	40.10	25.33	40.05	25.27	18.9	-15.4	35.3	37.76	32.86	26.52	13.8	-10.74	31.46	29.34	44.32
23	20	21	-820.39	-819.64	-820.17	-819.42	-793.67	-882.24	-836.48	-824.78	-827.51	-50.59	-934.55	-37.17	-9.68	-9.51	-789.54
24	21	16	531.36	531.41	531.22	531.27	528.37	543.50	533.04	531.89	532.01	512.31	551.75	453.42	506.21	506.18	538.13
25	21	17	-2203.50	-2201.6	-2203.1	-2201.3	-2237.2	-2287.4	-2230	-2208.78	-2211.7	-1417.7	-2342.61	-1333.5	-1392.8	-1384.5	-2243.1
26	18	23	154.41	149.73	154.4	149.73	150.05	159.07	153.09	153.57	174.04	149.8	146.32	179.22	106.15	144.28	151.68
27	19	24	226.21	248.50	226.16	248.43	252.86	215.65	222.71	223.77	204.68	249.18	237.35	200.59	314.27	266.82	247.46
28	20	25	229.12	213.05	229.19	213.13	176.88	241.27	242.89	231.84	230.42	207.28	240.37	232.97	183.26	180.4	180.23
29	21	26	95.73	94.57	95.72	94.57	159.16	105.64	104.49	96.11	96.13	98.77	100.31	86.89	120.86	112.78	159.39
30	17	22	350.53	350.14	350.53	350.14	317.05	334.37	332.82	350.7	350.73	350.97	331.65	356.33	331.45	351.72	317.24
31	23	24	34.41	29.73	34.40	29.73	30.05	39.07	33.09	33.57	54.04	29.8	26.32	59.22	-13.85	24.28	31.68
32	24	25	-9.38	8.24	-9.44	8.16	12.91	-15.27	-14.20	-12.65	-11.28	8.98	-6.33	-10.19	30.42	21.10	9.13
33	25	26	9.74	11.29	9.75	11.30	-20.21	15.99	18.68	9.19	9.14	6.26	24.04	12.78	3.69	-8.50	-20.63
34	26	22	-26.53	-26.14	-26.53	-26.14	6.95	-10.37	-8.82	-26.7	-26.73	-26.97	-7.65	-32.33	-7.45	-27.72	6.76

Table S4 Available nodal heads for different solutions of GoYang network.

Node Number	Ground Level (m)	Minimum HGL (m)	Available HGL (m)
			R1	R2	R3	R4	R5
Res.	71.0		71.00	71.00	71.00	71.00	71.00
1	71.0	86.0	86.62	86.62	86.62	86.62	86.62
2	56.4	71.4	81.36	85.33	81.36	85.33	85.33
3	53.8	68.8	80.07	82.53	80.18	84.97	85.00
4	54.9	69.9	78.93	81.48	79.08	84.03	84.46
5	56.0	71.0	78.39	80.20	78.86	83.51	84.19
6	57.0	72.0	77.58	78.51	77.76	82.49	83.94
7	53.9	68.9	79.06	81.62	79.32	84.67	84.39
8	54.5	69.5	78.80	81.2	78.99	84.01	84.33
9	57.9	72.9	77.81	79.11	77.99	82.79	83.99
10	62.1	77.1	77.19	78.27	77.62	82.32	83.64
11	62.8	77.8	77.85	78.83	77.95	82.64	83.76
12	58.6	73,6	76.41	76.76	76.86	81.61	82.99
13	59.3	74.3	76.47	76.76	76.78	81.44	82.88
14	59.8	74.8	74.82	75.13	75.19	80.72	81.30
15	59.2	74.2	74.36	74.68	74.75	80.06	80.86
16	53.6	68,6	79.25	81.91	79.55	84.28	84.69
17	54.8	69.8	78.60	81.55	79.17	83.81	83.89
18	55.1	70.1	78.61	81.54	79.16	84.01	83.90
19	54.2	69.2	78.50	81.56	79.17	83.39	83.73
20	54.5	69.5	77.87	81.18	79.01	82.51	83.34
21	62.9	77.9	78.95	82.64	79.86	83.12	83.99
22	61.8	76.8	77.63	81.16	79.09	81.93	83.32

^a the minimum required nodal head is 15 m above the ground level

Table S5 Pipe flows for different solutions of GoYang network.

Pipe	Direction in Input file		Flow in LPS in solution
	From Node	To Node	R1	R2	R3	R4	R5
1	1	2	29.51	29.51	29.51	29.51	29.51
2	2	3	16.13	15.16	15.35	17.08	16.36
3	3	4	9.57	9.16	9.39	8.66	10.35
4	4	5	7.87	6.95	7.87	7.65	8.67
5	5	6	4.13	3.42	4.87	4.70	6.38
6	6	12	2.80	3.47	4.36	4.30	4.50
7	12	15	3.05	3.07	3.10	2.62	3.11
8	2	22	5.23	5.55	7.20	4.97	6.74
9	2	21	3.65	3.86	2.82	3.48	2.65
10	21	22	3.28	3.50	2.46	3.11	2.29
11	22	20	-0.74	-0.20	0.4	-1.17	-0.23
12	19	20	2.18	1.65	1.04	2.61	1.67
13	2	19	2.73	3.17	2.37	2.21	2.00
14	19	17	-0.82	0.15	-0.04	-1.77	-1.05
15	3	16	2.67	2.29	2.32	2.44	2.85
16	16	17	1.42	1.04	1.07	1.19	1.60
17	17	18	-0.32	0.27	0.11	-1.5	-0.37
18	7	18	0.96	0.37	0.53	2.14	1.01
19	3	7	3.07	2.9	2.83	5.16	2.34
20	7	8	1.38	1.79	1.56	2.29	0.6
21	4	8	1.02	1.53	0.85	0.33	1.00
22	8	9	1.85	2.77	1.85	2.07	1.05
23	5	11	2.88	2.66	2.13	2.09	1.43
24	10	11	-2.39	-2.17	-1.64	-1.60	-0.94
25	6	10	1.91	1.45	1.10	1.20	1.66
26	6	9	-1.36	-2.28	-1.37	-1.58	-0.56
27	10	13	3.95	3.28	2.39	2.45	2.25
28	12	13	-0.68	-0.03	0.83	1.25	0.96
29	13	14	2.84	2.82	2.79	3.27	2.78
30	15	14	-2.11	-2.09	-2.06	-2.54	-2.05
Pump 70			29.51	29.51	29.51	29.51	29.51

Table S6 Sink nodes and available excess residual heads in different solutions of GoYang network.

Network Solution Number	Sink Node 1	Residual head (m)	Sink Node 2	Residual head (m)	Sink Node 3	Residual head (m)
R1	15	0.16	22	0.83
R2	15	0.48	22	4.36	18	11.44
R3	15	0.55	18	9.06	20	9.51
R4	22	5.13	15	5.86
R5	22	6.52	15	6.66

Table S7 Critical nodes and available residual heads in different solutions of Ramnagar network.

Network Solution Number		Critical Node	1	2	3		4	5	6	7	8	9	10
R1	Node Id		J-53	J-248		J-59	J-192	J-27	J-57	J-48	J-193	J-191	J-190
	Residual Head (m)		0.01	0.03		0.04	0.17	0.23	0.24	0.25	0.27	0.3	0.34
R2	Node Id		J-59	J-237		J-248	J-53	J-192	J-57	J-27	J-193	J-48	J-191
	Residual Head (m)		0.01	0.01		0.01	0.07	0.17	0.21	0.25	0.27	0.3	0.3
R3	Node Id		J-237	J-59		J-53	J-57	J-192	J-246	J-247	J-193	J-191	J-48
	Residual Head (m)		0.07	0.08		0.11	0.13	0.14	0.21	0.21	0.24	0.27	0.28
R4	Node Id		J-53	J-156		J-237	J-248	J-59	J-57	J-48	J-85	J-84	J-192
	Residual Head (m)		0.000	0.000		0.010	0.020	0.070	0.180	0.210	0.210	0.280	0.360
R5	Node Id		J-248	J-15		J-237	J-16	J-27	J-250	J-252	J-251	J-10	J-28
	Residual Head (m)		0.04	0.11		0.35	0.41	0.41	0.44	0.46	0.61	0.73	0.75
R1*	Node Id		J-53	J-59		J-248	J-192	J-237	J-193	J-27	J-57	J-191	J-48
	Residual Head (m)		0.01	0.03		0.05	0.10	0.13	0.20	0.23	0.23	0.23	0.25

Table S8 Sink nodes and available excess residual heads in different solutions of Ramnagar network.

R1		R2		R3		R4		R5		R1		R2		R3		R4		R5
Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)	Sink Node Id	Residual Head (m)
J-192	0.17	J-192	0.17	J-192	0.14	J-85	0.21	J-251	0.61	J-76	3.42	J-232	3.16	J-26	3.33	J-127	2.37	J-157	4.13
J-193	0.27	J-193	0.27	J-193	0.24	J-192	0.36	J-249	0.75	J-153	3.44	J-208	3.27	J-232	3.34	J-139	2.42	J-219	4.15
J-194	0.35	J-194	0.35	J-194	0.32	J-193	0.46	J-192	1.11	J-154	3.45	J-95	3.28	J-95	3.4	J-26	2.71	J-95	4.17
J-195	0.44	J-195	0.44	J-195	0.41	J-46	0.48	J-254	1.12	J-150	3.46	J-207	3.29	J-276	3.4	J-44	2.73	J-276	4.23
J-46	0.48	J-186	0.5	J-186	0.47	J-194	0.54	J-193	1.22	J-232	3.47	J-22	3.41	J-277	3.46	J-22	3.05	J-277	4.29
J-186	0.49	J-251	0.58	J-185	0.75	J-251	0.59	J-194	1.29	J-85	3.62	J-157	3.56	J-22	3.48	J-164	3.09	J-7	4.39
J-251	0.6	J-5	0.66	J-46	0.92	J-195	0.63	J-195	1.39	J-205	3.73	J-205	3.64	J-44	3.55	J-232	3.25	J-46	4.47
J-249	0.74	J-249	0.72	J-251	1.15	J-186	0.69	J-256	1.41	J-276	3.9	J-88	3.83	J-199	3.61	J-168	3.37	J-103	4.6
J-185	0.77	J-185	0.78	J-182	1.18	J-249	0.73	J-186	1.45	J-277	3.96	J-93	3.94	J-96	3.62	J-103	3.39	J-96	4.65
J-5	0.81	J-46	0.8	J-260	1.22	J-153	0.75	J-260	1.65	J-284	4	J-213	3.97	J-284	3.65	J-173	3.43	J-208	4.76
J-254	1.12	J-220	0.82	J-249	1.3	J-154	0.76	J-185	1.72	J-95	4.25	J-276	4.11	J-103	3.79	J-170	3.45	J-207	4.78
J-220	1.15	J-254	1.1	J-153	1.41	J-150	0.9	J-258	2.02	J-103	4.33	J-228	4.12	J-208	3.92	J-166	3.46	J-284	4.81
J-182	1.21	J-219	1.2	J-154	1.42	J-185	0.97	J-243	2.05	J-157	4.33	J-103	4.13	J-207	3.94	J-174	3.84	J-199	4.99
J-9	1.35	J-182	1.21	J-150	1.47	J-5	1.05	J-9	2.07	J-213	4.35	J-277	4.17	J-287	4.09	J-76	3.94	J-275	5.09
J-256	1.4	J-260	1.32	J-5	1.48	J-254	1.11	J-85	2.11	J-228	4.43	J-284	4.25	J-173	4.14	J-208	4.07	J-173	5.11
J-260	1.5	J-9	1.33	J-179	1.56	J-88	1.15	J-182	2.16	J-287	4.44	J-212	4.27	J-76	4.25	J-207	4.09	J-205	5.13
J-219	1.52	J-256	1.38	J-254	1.67	J-256	1.39	J-88	2.3	J-127	4.61	J-173	4.48	J-275	4.26	J-228	4.22	J-287	5.25
J-179	1.59	J-40	1.44	J-243	1.72	J-182	1.4	J-22	2.45	J-17	4.64	J-127	4.5	J-205	4.29	J-113	4.34	J-127	5.43
J-40	1.67	J-243	1.56	J-73	1.8	J-220	1.43	J-26	2.49	J-212	4.66	J-139	4.52	J-281	4.39	J-276	4.35	J-139	5.45
J-61	1.74	J-179	1.59	J-256	1.96	J-157	1.54	J-179	2.53	J-88	4.72	J-17	4.56	J-164	4.44	J-277	4.41	J-164	5.55
J-243	1.88	J-7	1.76	J-9	2.14	J-260	1.55	J-235	2.56	J-281	4.75	J-287	4.69	J-17	4.66	J-213	4.43	J-281	5.55
J-7	1.95	J-73	1.98	J-130	2.14	J-243	1.63	J-37	2.58	J-139	4.76	J-164	4.94	J-113	4.71	J-205	4.44	J-113	5.6
J-258	2.01	J-258	1.99	J-220	2.21	J-40	1.68	J-5	2.69	J-275	4.78	J-113	4.95	J-168	4.73	J-160	4.54	J-168	5.8
J-73	2.05	J-235	2.09	J-235	2.24	J-9	1.77	J-17	2.71	J-113	5.35	J-275	4.97	J-166	4.79	J-212	4.73	J-170	5.85
J-63	2.13	J-199	2.26	J-85	2.28	J-179	1.78	J-33	3.01	J-173	5.75	J-281	4.99	J-170	4.83	J-284	4.81	J-166	5.93
J-235	2.42	J-44	2.59	J-40	2.55	J-219	1.81	J-40	3.15	J-164	5.94	J-85	5.05	J-88	4.94	J-17	5.08	J-174	6.23
J-26	2.69	J-26	2.88	J-7	2.56	J-95	1.92	J-153	3.55	J-114	6.15	J-168	5.18	J-174	5.26	J-114	5.15	J-114	6.4
J-199	2.7	J-66	2.91	J-258	2.57	J-258	2.01	J-154	3.56	J-168	6.28	J-170	5.23	J-213	5.33	J-275	5.21	J-160	6.67
J-44	2.71	J-76	2.92	J-219	2.59	J-73	2.03	J-232	3.69	J-166	6.33	J-166	5.32	J-114	5.52	J-287	5.25	J-213	6.8
J-22	3.12	J-153	2.95	J-157	2.71	J-235	2.14	J-220	3.77	J-170	6.4	J-174	5.61	J-212	5.64	J-199	5.27	J-212	7.11
J-208	3.36	J-154	2.96	J-127	2.79	J-7	2.19	J-150	3.84	J-160	6.76	J-114	5.76	J-160	6.34	J-281	5.56	J-73	9.09
J-207	3.38	J-150	3.07	J-139	2.86	J-96	2.32	J-44	3.85	J-174	6.88	J-160	6.02					J-76	10.72

Table S9 No. of pipes with the diameters for R1* solution of Ramnagar network.

Sr. No.	Diameter	No. of pipes	Pipe id
1	400	1	1
2	350	1	P-104
3	300	2	7, P-110
4	250	5	P-136, P-157, P-158, P-160, P-161
5	200	14	3, 5, 12,13, 40, P-4, P-6, P-112, P-143, P-144, P-145, P-156, P-163, P-165a
6	150	20	11, 16, 19, 54, 55, 75, P-8, P-10, P-17, P-116, P-118, P-259a, P-259b, P-260, P-272, P-284, P-338, P-339, P-340, P-341
7	100	19	8, 9, 17, 39, 49, 58, 63, 73, P-15, P-16, P-30, P-38, P-101, P-107, P-108, P-168, P-232, P-343, P-345
8	80	313	Remaining pipes

 Figure S1 Identified branching configurations showing primary and secondary pipes for R1, R2, R3, R4, and R5 solutions of GoYang network.

Figure S2 Identified branching configurations showing primary and secondary pipes for R1, R2, R3, R4, and R5 solutions for the Ramnagar network.