Redundancy is an important parameter for the resilient design of water distribution systems (WDSs). In this study, the redundancy of WDSs of a branch and looped water distribution system is increased by adding one-by-one pipes with the objective of minimization of variance of the pipe flows (VPF) and ensuring cost-effective and reliable design of WDNs under pressure-deficient conditions suggested by various pressure dependent analysis (PDA) approaches. Reliability and cost values are compared using different PDA approaches. Three example networks are considered including multi-source network for redundancy evaluations with two PDA approaches. Further, the application on real practical problems of Saoner city, Maharashtra state in India is also considered. Different PDA based approaches are considered and redundancy and cost values are compared based on PDA approaches for all the example networks. The results also demonstrated the advantages of both PDA over demand dependent analysis (DDA).

  • Increases WDSs redundancy by adding pipes to branch networks.

  • Different Pressure deficient analysis is used for improving redundancy of WDS.

  • Minimizes variance of pipe-flow series for cost-effective and reliable designs.

  • Applications are shown on real example of case study network of India.

The objective of water distribution systems (WDSs) is to provide safe and reliable water to urban and rural communities with adequate quantities of good quality water. Normally, demand-dependent analysis (DDA) is performed for analysis of WDSs. However, DDA fails to determine the deficiency of pressure and the flow in any demand node and thus sometimes, at some nodes, negative pressure arises. Therefore, the need of accurate analysis of WDSs using pressure-dependent analysis (PDA) is realized and hence crucial for determining the quantity of water supplied by the network at various nodes to evaluate the short fall in supply. Currently various studies on PDA highlighted the ways to perform PDA analysis using software EPANET 2.0 and 2.2 by incorporating various artificial elements in the existing network or modifying the source code of DDA simulator (Bhave 1978, 1981, 2003; Ang & Jowitt 2006; Babu & Mohan 2012; Gorev & Kodzhespirova 2013; Jun & Guoping 2013; Liu & Yu 2013; Herman et al. 2017; Mahmoud et al. 2017; Paez et al. 2018; Hamed et al. 2022). Some PDA toolkits are also developed and available in an open platform (Eliades et al. 2016). Previous works on WDS redundancy have primarily focused on heuristic approaches and did not consider different pressure-deficient conditions (Abbot 2012). Abbot (2012) considered two parameters: total user benefited (TUB) and vulnerable user benefited (VUB) for improving redundancy of branch WDSs. Level-one redundancy based designed of WDSs are performed by various researchers (Gupta & Bhave 1994; Gupta et al. 2003; Tanyimboh & Templeman 2010; Atkinson et al. 2014; Gupta et al. 2015; Rathi et al. 2018; Rathi & Gupta 2018, 2023; Lu et al. 2022). Rathi et al. (2018) and Rathi & Gupta (2018, 2023) observed that minimizing the variance of pipe-flow (VPF) series improves flow uniformity in the network and increased reliability of the network and those combinations further improved the cost of the network. Some researchers highlighted on how and where to increase the redundancy in branch as well as looped WDNs using a heuristics approach (Abbot 2012). Gupta & Rathi (2017) used a similar concept of addition of a one-by-one pipe in a branch network for joint consideration of layout optimization and water distribution network design by minimizing the VPF series which maximizes reliability for low-cost design. The same concept was used in this study for increasing the reliability for branch and looped WDSs along with consideration of cost constraints. However, Gupta & Rathi (2017) used this method for jointly optimizing layout and pipe network optimizations. Furthermore, Rathi et al. (2018) and Rathi & Gupta (2018) observed that Chiong (1985) and Martínez (2007) suggested the model for VPF series is not appropriate and it is directional dependent. Therefore, Rathi et al. (2018) and Rathi & Gupta (2018) provided the modified and corrected equation of a VPF model (Equation (2)) to maximize the flow uniformity in the network which is then used to evaluate the reliability. The same methodology is used here to optimize the reliability and cost of the network. Also, to date, no study has compared the reliability and cost comparison using different PDA modeling approaches. Also, to date, no study has compared the reliability and cost comparison using different PDA modeling approaches by minimizing the value of the VPF parameter. The PDA of the network is then performed using two methods by Babu & Mohan (2012) and Paez et al. (2018) and the demand satisfaction ratio (DSR) is then calculated. Cost is obtained using LINGO 13.0. Accurate determinations of redundancies in the pipelines are crucial and can be determined using various PDA suggested by different researchers. Various ways for PDA modeling are suggested by various studies such as changes in EPANET source code, by adding various elements or strings in EPANET. Thus, this work mainly focused on increasing reliability of water distribution systems (by minimizing VPF parameters) by adding a one-by-one pipe in the branch and looped network. The VPF is evaluated for various possibilities of additions of pipes in the networks. The possibility which provides minimum variance is considered first for the addition of pipes. Cost is evaluated using LINGO 13.0. The same procedure is repeated. As DDA is not enough to provide accurate analysis, therefore different PDA methods are performed to check the effectiveness of PDA for redundant design of WDSs. Three example networks considered are: branch network and the remaining are two-looped networks. The looped networks which are considered are two kinds of network such as a single-source network and the other is a network consisting of two sources. The selection of those networks demonstrated the easy application on the diverse network. Further, the application of the methodology is shown on a real practical problem of Saoner district in Maharashtra State in India and results are discussed in Section 5. Further, two PDA approaches are considered for redundant-based designed of WDSs (Babu & Mohan 2012; Paez et al. 2018) The results of two PDA modeling approaches are considered for comparison of reliability and cost. The future scope of the study is to compare the results using all the PDA modeling approaches with their advantages and disadvantages. For all the networks, it was observed that Paez et al. (2018) provided higher reliability with slightly higher cost as compared to Babu & Mohan's (2012) method. Further, its applications are shown in the real case study of Saoner District in Maharashtra state in India.

The methodology used in the paper is suggested by Gupta & Rathi (2017) for redundant-based design of the WDN using different PDA approaches. The same model is used here to optimize the reliability and cost of the network (Rathi et al. 2018; Rathi & Gupta 2018, 2023). Two PDA approaches suggested by Babu & Mohan (2012) and Paez et al. (2018) are used to calculate the redundancy of the networks and cost on four example networks. The flow chart of the methodology is given in Figure 1.
Figure 1

Cost and reliability calculation for a level-one redundant system.

Figure 1

Cost and reliability calculation for a level-one redundant system.

Close modal

The various steps used in the methodology are as follows:

  • 1. Convert a network to a branched network: Initially the looped network is converted to a branch network using the minimum spanning tree method.

  • 2. Initial flow distribution: Calculate the flow in each pipe of the branched network considering the continuity equations at each demand node. For the original looped network, use Rathi & Gupta (2018) model for obtaining the flow distribution in the network, which requires solving node-flow continuity and loop flow equations.

  • 3. Calculate VPF: Determine the variance of the pipe-flow series as given by Rathi et al. (2018) and Rathi & Gupta (2018) which are given in Equations (1) and (2):
    (1)
    (2)
    where VQ = VPF series, N = number of pipes in the network; Qx = flow in any pipe x; = mean of pipe-flow series, and q is the demand at node j.
  • 4. Consider pipe additions: various pipe combinations are observed to make branch network to looped network. For each addition of pipes, calculate the new flow distribution using Equations (1) and (2) and obtain the value of VPF.

  • 5. Among the various alternatives, select the pipe which provided minimum value of variance of pipe-flows.

  • 6. Modify the network by adding the selected pipe.

  • 7. Repeat the process: Repeat steps 2–6, adding pipes one at a time until no further reduction in variance is possible or all pipe addition options have been considered.

  • 8. The network consisting of all the possible addition of pipes are then designed by assigning the flow using flow distribution using Equations (1) and (2). The LP model is then formulated using LINGO 13.0. The cost of the network is obtained.

  • 9. The network is analyzed for one-pipe failure condition to obtain its reliability through the value of DSR. Here, two PDA approaches are used named as Babu & Mohan (2012) and Paez et al. (2018) to obtain the reliability through the DSR.

Demand satisfaction ratio

The DSR (Equation (3)) is used to calculate the reliability of the network. A higher DSR indicates higher reliability and is calculated to identify the network's ability to meet demand under all pipe working conditions as well as one-pipe failure conditions:
(3)

Network design

The network is designed using LINGO 13.0. A trade-off between cost and reliability is obtained.

Example 1: A single-source hypothetical branch network

An example network of a single-source simple branch WDN as shown in Figure 2(a) and 2(b) is considered for evaluation of reliability and cost (Gupta & Rathi 2017). The network shown in Figure 2 consists of a single-source node, nine links (numbered from 1 to 9) and five demand nodes, 1 to 5 and hydraulic gradient level (HGL) at source node (R) is 100 m. The nodal demands, nodal HGL values and pipe details for this network are given in Table 1 (Gupta & Rathi 2017).
Table 1

Nodes and pipe details for branch networks (Gupta & Rathi 2017)

Node IDElevationBase demandLengthDiameter
Sr. no.mCMHLink IDmmm
88 200 500 300 
86 200 350 200 
87 300 400 300 
85 350 300 250 
84 250 300 200 
Resvr R-1 100 – 700 250 
   600 200 
   500 150 
   600 200 
Node IDElevationBase demandLengthDiameter
Sr. no.mCMHLink IDmmm
88 200 500 300 
86 200 350 200 
87 300 400 300 
85 350 300 250 
84 250 300 200 
Resvr R-1 100 – 700 250 
   600 200 
   500 150 
   600 200 
Figure 2

Example network: (a) branched network and (b) looped network.

Figure 2

Example network: (a) branched network and (b) looped network.

Close modal

The inner diameters of the pipes are 100, 150, 200, 250, 300, 350, 400, 500, and 600 mm, whose costs are Rs. 542, 842, 1,150, 1,520, 1,821, 2,283, 2,760, 3,863, and 5,121 per meter, respectively. All pipes are assumed as new pipes with a Hazen–Williams coefficient of 130.

Design of WDN using LINGO and determination of reliability

Pipe diameters and their lengths were evaluated and are shown in Table 2. The network was analyzed for one-pipe failure conditions to obtain its reliability using two PDA methods. The design was achieved using Rathi et al. (2018) and Rathi & Gupta's (2018) model for optimal flow distribution and accordingly a linear programming (LP) model was formulated and solved. The appropriate pipe sizes and lengths are detailed in Table 1. The total construction cost for this network amounted to Rs. 2,506,432.

Table 2

Iteration-wise addition of pipes on priority wise with evaluation of cost and VPF

Iteration numbersPipe additionPipe numbersLength (m)
Diameter (mm)
Babu & Mohan (2012) Paez et al. (2018) 
36.352 77.228 300 
– 463.647 422.771 300 
63.280 83.957 200 
– 286.719 266.042 200 
400 400 300 
300 300 250 
300 300 200 
250 250 250 
     
6 and 7 455.405 22.175 300 
 44.594 477.824 300 
23.285 45.283 200 
 326.714 304.716 200 
400 400 300 
300 300 250 
300 300 200 
250 250 250 
600 600 200 
6, 7, and 8     
422.886 7.022 300 
 77.113 492.977 300 
117.949 258.113 200 
 232.051 91.886 200 
400 400 300 
300 300 250 
300 300 200 
250 300 250 
600 250 200 
577.0715 579.7906 150 
6, 7, 8, and 9 422.886 5.896 300 
 77.113 494.104 300 
117.949 252.243 200 
 232.051 97.756 200 
400 400 300 
300 300 250 
300 300 200 
250 250 250 
600 600 200 
600 600 150 
500 500 200 
Iteration numbersPipe additionPipe numbersLength (m)
Diameter (mm)
Babu & Mohan (2012) Paez et al. (2018) 
36.352 77.228 300 
– 463.647 422.771 300 
63.280 83.957 200 
– 286.719 266.042 200 
400 400 300 
300 300 250 
300 300 200 
250 250 250 
     
6 and 7 455.405 22.175 300 
 44.594 477.824 300 
23.285 45.283 200 
 326.714 304.716 200 
400 400 300 
300 300 250 
300 300 200 
250 250 250 
600 600 200 
6, 7, and 8     
422.886 7.022 300 
 77.113 492.977 300 
117.949 258.113 200 
 232.051 91.886 200 
400 400 300 
300 300 250 
300 300 200 
250 300 250 
600 250 200 
577.0715 579.7906 150 
6, 7, 8, and 9 422.886 5.896 300 
 77.113 494.104 300 
117.949 252.243 200 
 232.051 97.756 200 
400 400 300 
300 300 250 
300 300 200 
250 250 250 
600 600 200 
600 600 150 
500 500 200 

Observations and results

The cost of the network obtained in iteration no. 1 with addition of pipe 6 using Babu & Mohan's (2012) method is calculated as Rs. 1,699,765 and DSR is 0.525, and using Paez et al.’s (2018) method, the cost of the network obtained is calculated as Rs. 1,834,267 and DSR is 0.686. In iteration 2, with combination of pipe 6, pipe 7 is selected based on minimizing the variance. The network is then analyzed for one-pipe failure conditions to obtain its reliability using Babu & Mohan's (2012) method and cost is calculated as 2,063,837 and DSR is 0.575, and using Paez et al.’s (2018) method, the cost of the network obtained is calculated as Rs. 2,187,640 and DSR is 0.7355. Similarly, in iteration no. 3, the addition of pipe no. 8, cost is evaluated as Rs. 2,370,426 and DSR is 0.606, and using Paez et al. (2018), cost is calculated as 2,453,552 and DSR is 0.7699. In iteration 4, using Babu & Mohan (2012), the calculated cost is Rs. 2,641,426 and DSR is 0.606, and using Paez et al. (2018) the evaluated cost is Rs. 2,726,652 and DSR is 0.7711.

From the result it is observed that Paez et al.'s method shows a 62.9% increase in DSR compared to Babu & Mohan (2012), however, the cost is increased by 7.9%. The variance of pipe-flows using Paez et al.’s (2018) method is almost double, indicating greater flow uniformity and higher reliability. Babu & Mohan (2012) provided economical design; however, there is a significant compromise on the reliability. Paez et al. (2018) provided higher reliability with increased cost.

Example 2: single-source two-loop network

The two-looped network (Alperovits & Shamir 1997) shown in Figure 3(a) is considered to obtained reliability and cost using various PDA methods which are Babu & Mohan (2012) and Paez et al. (2018). The loop network is then converted to branch network using minimum spanning tree method as is shown in Figure 3(b).
Figure 3

(a) Two-loop network and (b) conversion to a branch network.

Figure 3

(a) Two-loop network and (b) conversion to a branch network.

Close modal

The network shown in Figure 3 consists of six demand nodes and eight pipes. Node and pipe details are given in Table 3. An additional demand of 30 L/s is considered. The available diameters are 100, 150, 200, 250, 300, 350, 400, 500, and 600 mm, with corresponding costs in Rs. per meter of 542, 842, 1,150, 1,520, 1,821, 2,283, 2,760, 3,863, and 5,121, respectively. All new pipes have a Hazen–William's coefficient of 130.

Table 3

Node and pipe details for two-looped network

NodeElevation (m)Demand (L/S)PipeDia. (mm)PipeDia. (mm)
100 – – – – – 
90 30 400 250 
90 30 350 250 
88 30 350   
88 30 300   
85 30 300   
85 100 250   
NodeElevation (m)Demand (L/S)PipeDia. (mm)PipeDia. (mm)
100 – – – – – 
90 30 400 250 
90 30 350 250 
88 30 350   
88 30 300   
85 30 300   
85 100 250   

Design of WDN using LINGO and determination of reliability

Pipe diameters and their lengths were evaluated and are shown in Table 4. The network is analyzed for one-pipe failure conditions to obtain its reliability using two PDA methods. As discussed in Section 4.1.1, optimal flow distribution is obtained using Rathi & Gupta's (2018) model and design is obtained using the LP problem. The appropriate pipe sizes and lengths are detailed in Table 4. The total network cost of Rs. 73,93,354. The network's reliability, assessed under a no-pipe failure condition, yielded a DSR of 0.7696, as shown in Table 4.

Table 4

Analysis of a two-loop network using methodology

Iteration numbersPipe additionPipe numberLength (m)
Diameter (mm)
Babu & Mohan (2012) Paez et al. (2018) 
579.6133 260.7428 300 
– 420.3867 739.2572 300 
354.8061 279.2632 300 
– 645.1939 720.7368 300 
1,000 1,000 250 
1,000 1,000 250 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
Variance using Babu & Mohan (2012) is calculated as 4.89, cost = Rs. 6,113,940, DSR = 0.668 
Variance using Paez et al. (2018) is calculated as 6.0506, cost = Rs. 6,289,209, DSR = 0.762 
    
772.8683 226.0616 300 
– 227.1317 773.9384 300 
659.4200 268.1241 300 
– 340.5800 731.8759 250 
1,000 1,000 250 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
Variance using Babu & Mohan (2012) is 6.1142, cost = Rs. 6,995,949, DSR = 0.74915 
Variance using Paez et al. (2018) is calculated as 7.207, cost = 7,393,354, DSR = 0.7696 
Iteration numbersPipe additionPipe numberLength (m)
Diameter (mm)
Babu & Mohan (2012) Paez et al. (2018) 
579.6133 260.7428 300 
– 420.3867 739.2572 300 
354.8061 279.2632 300 
– 645.1939 720.7368 300 
1,000 1,000 250 
1,000 1,000 250 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
Variance using Babu & Mohan (2012) is calculated as 4.89, cost = Rs. 6,113,940, DSR = 0.668 
Variance using Paez et al. (2018) is calculated as 6.0506, cost = Rs. 6,289,209, DSR = 0.762 
    
772.8683 226.0616 300 
– 227.1317 773.9384 300 
659.4200 268.1241 300 
– 340.5800 731.8759 250 
1,000 1,000 250 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
1,000 1,000 200 
Variance using Babu & Mohan (2012) is 6.1142, cost = Rs. 6,995,949, DSR = 0.74915 
Variance using Paez et al. (2018) is calculated as 7.207, cost = 7,393,354, DSR = 0.7696 

Observations and results

The results are shown in Table 4. The values of variance cost and DSR using Babu & Mohan (2012) and Paez et al.’s (2018) methods is evaluated. The following observations were obtained:

  • 1. Paez et al. (2018) provided higher DSR; however, cost increases by 2.8% as compared to Babu & Mohan's (2012) method. The increase in variance of pipe-flows using Paez et al.’s (2018) method improved reliability though the cost increase is relatively marginal compared to the gain in reliability.

  • 2. Trade-off: Paez et al.'s (2018) method offers an optimized balance between reliability and cost for this configuration. The modest increase in cost justifies the reliability boost, particularly in systems where pressure reliability is crucial.

  • 3. Babu & Mohan (2012): The lower DSR indicates vulnerability to failures, making it less suitable for critical systems requiring higher levels of service reliability. Paez et al. (2018): Despite better performance, the cost increase may not be acceptable in scenarios where budget constraints are a priority. Additionally, the variance could indicate a potential overdesign, particularly in areas with lower demand.

Example 3: multiple source network

The network shown in Figure 4(a) consists of two source nodes, nine demand nodes, and 14 pipes. Each demand node has a demand of 50 m3 per hour (CMH), and each pipe is 750 m in length, with a Hazen–Williams coefficient of 130, as detailed in Table 5. The network is converted into branch network as shown in Figure 4(b).
Table 5

Nodal and pipe details for multiple source networks

ElevationBase demandLengthDiameterRoughness
Node IDmCMHLink IDmmm
90 50 P-1 750 300 130 
90 50 P-2 750 300 130 
90 50 P-3 750 300 130 
88 50 P-4 750 300 130 
88 50 P-5 750 300 130 
88 50 P-6 750 300 130 
85 100 P-7 750 300 130 
85 100 P-8 750 250 130 
85 500 P-9 750 250 130 
R-1 100 P-10 750 250 130 
R-2 100 P-11 750 250 130 
   P-12 750 250 130 
   P-13 750 200 130 
   P-14 750 200 130 
ElevationBase demandLengthDiameterRoughness
Node IDmCMHLink IDmmm
90 50 P-1 750 300 130 
90 50 P-2 750 300 130 
90 50 P-3 750 300 130 
88 50 P-4 750 300 130 
88 50 P-5 750 300 130 
88 50 P-6 750 300 130 
85 100 P-7 750 300 130 
85 100 P-8 750 250 130 
85 500 P-9 750 250 130 
R-1 100 P-10 750 250 130 
R-2 100 P-11 750 250 130 
   P-12 750 250 130 
   P-13 750 200 130 
   P-14 750 200 130 
Figure 4

(a) Two-source network and (b) conversion to a branch network.

Figure 4

(a) Two-source network and (b) conversion to a branch network.

Close modal

The diameters of the pipe available for this network are 100, 150, 200, 250, 300, 350, 400, 500, and 600, for which corresponding costs per unit length are Rs. 542, 842, 1,150, 1,520, 1,821, 2,283, 2,760, 3,863, and 5,121 respectively. A Hazen–Williams coefficient is assumed as 130.

Design of WDNs using LINGO and determination of reliability

Pipe diameters and their lengths were evaluated and are shown in Table 6. The network was analyzed for one-pipe failure condition to obtain its reliability using two PDA methods. The design was achieved using the same Rathi et al. (2018) and Rathi & Gupta (2018) models for optimal flow distribution and accordingly an LP model was formulated and solved. The appropriate pipe sizes and lengths are detailed in Table 6. The total network cost was Rs. 7,554,025. The reliability assessment under a no-pipe failure condition revealed a DSR of 0.849 as given in Table 6.

Table 6

Analysis of a multi-source network using methodology

Iteration numberPipe additionPipe numberLength (m)Diameter (mm)
Babu & Mohan (2012) Paez et al. (2018) 
1.873408 656.4233 300 
 748.1266 93.57669 300 
29.23173 42.61083 300 
 720.7683 707.3892 300 
750 750 300 
750 750 300 
750 750 300 
7 750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
Variance using Babu & Mohan (2012) is calculated as 3.427, Cost = Rs. 5,748,258, DSR = 0.725 
Variance using Paez et al. (2018) is calculated as 3.548, cost = Rs. 6,179,591, DSR = 0.733 
4 and 7     
319.3097 314.5131 300 
 430.6903 435.4869 300 
110.4536 111.7446 300 
 639.5464 638.2554 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
Variance using Babu & Mohan (2012) is calculated as 3.0967, cost = Rs. 6,667,141, DSR = 0.771 
Variance using Paez et al. (2018) is calculated as 3.092, cost = Rs. 6,668,196, DSR = 0.7725 
4,7,14     
197.1803 192.3184 300 
 552.8197 557.6816 300 
26.88431 26.73961 300 
 723.1157 723.2604 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
14 750 750 200 
Variance using Babu & Mohan (2012) is calculated as 3.542, cost = Rs. 7,135,557, DSR = 0.836 
Variance using Paez et al. (2018) is calculated as 3.529, cost = Rs. 7,137,064, DSR = 0.8378 
4, 7, 14, 13     
169.9677 166.866 300 
– 580.0323 583.1331 300 
18.28134 17.43478 300 
– 731.7187 732.5652 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
13 750 750 200 
14 750 750 200 
Variance using Babu & Mohan (2012) is calculated as 4.048, cost = Rs. 7,552,837, DSR = 0.848 
Variance using Paez et al. (2012) is calculated as 4.036, cost = 7,554,025, DSR = 0.849 
Iteration numberPipe additionPipe numberLength (m)Diameter (mm)
Babu & Mohan (2012) Paez et al. (2018) 
1.873408 656.4233 300 
 748.1266 93.57669 300 
29.23173 42.61083 300 
 720.7683 707.3892 300 
750 750 300 
750 750 300 
750 750 300 
7 750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
Variance using Babu & Mohan (2012) is calculated as 3.427, Cost = Rs. 5,748,258, DSR = 0.725 
Variance using Paez et al. (2018) is calculated as 3.548, cost = Rs. 6,179,591, DSR = 0.733 
4 and 7     
319.3097 314.5131 300 
 430.6903 435.4869 300 
110.4536 111.7446 300 
 639.5464 638.2554 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
Variance using Babu & Mohan (2012) is calculated as 3.0967, cost = Rs. 6,667,141, DSR = 0.771 
Variance using Paez et al. (2018) is calculated as 3.092, cost = Rs. 6,668,196, DSR = 0.7725 
4,7,14     
197.1803 192.3184 300 
 552.8197 557.6816 300 
26.88431 26.73961 300 
 723.1157 723.2604 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
14 750 750 200 
Variance using Babu & Mohan (2012) is calculated as 3.542, cost = Rs. 7,135,557, DSR = 0.836 
Variance using Paez et al. (2018) is calculated as 3.529, cost = Rs. 7,137,064, DSR = 0.8378 
4, 7, 14, 13     
169.9677 166.866 300 
– 580.0323 583.1331 300 
18.28134 17.43478 300 
– 731.7187 732.5652 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 300 
750 750 250 
750 750 250 
10 750 750 250 
11 750 750 250 
12 750 750 250 
13 750 750 200 
14 750 750 200 
Variance using Babu & Mohan (2012) is calculated as 4.048, cost = Rs. 7,552,837, DSR = 0.848 
Variance using Paez et al. (2012) is calculated as 4.036, cost = 7,554,025, DSR = 0.849 

Observations and results

The results are shown in Table 6. The values of variance cost and DSR using Babu & Mohan (2012) and Paez et al.’s (2018) methods is evaluated. The following observations were obtained:

  • 1. Paez et al. (2018) yields a slight improvement in DSR over Babu & Mohan (2012) with an increase in cost by 7.5%. The marginal difference in variance between the two methods suggests that both approaches are similarly effective in this configuration, with neither offering a significant advantage in terms of flow uniformity.

  • 2. Babu & Mohan (2012): Although more cost-effective, the slightly lower DSR may pose a problem in high-demand scenarios where failure prevention is critical.

Example 4: A case study of a WDN of Saoner in Maharashtra State in India

The network consists of 71 nodes and 72 pipes and one source node as shown in Figure 5 and is also considered for application of reliability and cost calculation using Babu & Mohan (2012) and Paez et al.’s (2018) methods. The PDA applications are shown in Figure 6. The pipe and node details are not provided due to confidentiality and space problem. The cost and reliability calculations are shown in Table 7.
Table 7

Analysis of example network and case study using methodology

ParametersBabu & Mohan (2012) Paez et al. (2018) 
Example 1: Hypothetical branch network 
Total cost (INR) Rs. 2,641,426 Rs. 2,726,652 
Reliability (DSR) 0.606 0.771 
Performance variance 5.163 10.661 
Trade-off Lower cost and slightly less reliability Higher cost but more reliability 
Example 2: Single-source two-loop network 
Total cost (INR) Rs. 6,995,949 Rs. 7,393,354 
DSR 0.749 0.769 
Variance 6.114 7.207 
Trade-off Slightly lower cost but slightly lower reliability Slightly higher cost but slightly higher reliability 
Example 3: Multiple source network 
Variance 4.048 4.036 
Total cost (Rs.) 7,552,837 7,554,025 
(DSR) 0.848 0.849 
Trade-off Lower due to higher variance Higher due to better DSR 
Example 4: Case study of Saoner in Maharashtra State in India 
Variance 0.04947 0.065 
Total cost (Rs.) 473,763.90 494,763.90 
(DSR) 0.31 0.762 
ParametersBabu & Mohan (2012) Paez et al. (2018) 
Example 1: Hypothetical branch network 
Total cost (INR) Rs. 2,641,426 Rs. 2,726,652 
Reliability (DSR) 0.606 0.771 
Performance variance 5.163 10.661 
Trade-off Lower cost and slightly less reliability Higher cost but more reliability 
Example 2: Single-source two-loop network 
Total cost (INR) Rs. 6,995,949 Rs. 7,393,354 
DSR 0.749 0.769 
Variance 6.114 7.207 
Trade-off Slightly lower cost but slightly lower reliability Slightly higher cost but slightly higher reliability 
Example 3: Multiple source network 
Variance 4.048 4.036 
Total cost (Rs.) 7,552,837 7,554,025 
(DSR) 0.848 0.849 
Trade-off Lower due to higher variance Higher due to better DSR 
Example 4: Case study of Saoner in Maharashtra State in India 
Variance 0.04947 0.065 
Total cost (Rs.) 473,763.90 494,763.90 
(DSR) 0.31 0.762 
Figure 5

Real case study of a WDN of Saoner in Maharashtra State, India (Singh & Rathi 2025).

Figure 5

Real case study of a WDN of Saoner in Maharashtra State, India (Singh & Rathi 2025).

Close modal
Figure 6

PDA modeling using the EPANET (a) Babu & Mohan's (2012) method and (b) Paez et al.’s (2018) method (Singh & Rathi 2025).

Figure 6

PDA modeling using the EPANET (a) Babu & Mohan's (2012) method and (b) Paez et al.’s (2018) method (Singh & Rathi 2025).

Close modal

A trade-off between cost and reliability is obtained using Paez et al.(2018) and Babu & Mohan's (2012) methods for all three networks and it is observed that Paez et al.’s (2018) methodology provided more reliability with higher cost. Detailed analysis is further added after each example network. Furthermore, the real case study of Saoner of Maharashtra State in India is also considered and the results are also discussed.

  • Redundancy is a crucial parameter for the resilient design of WDSs. Few works are focused on where and how to add the pipes one by one in water networks to increase the redundancy in the systems and which would be especially helpful in case of branch networks where there is a budget constraints to the water authorities.

  • Considering this aspect, this study mainly focused on the calculation of reliability and cost in branch and looped WDSs for one-pipe failure conditions, i.e. level-one redundant systems.

  • Gupta & Rathi (2017) used the concept of addition of one-by-one pipe in branch networks for joint consideration of layout optimization and water distribution network design by minimizing the VPF series which maximizes reliability for low-cost design. The same concept is used in this study for increasing the reliability for branch and looped WDSs along with consideration of cost constraints. Further, the flow distribution is evaluated using the modified model provided by Rathi & Gupta (2018) and Rathi et al. (2018).

  • Two PDA models used are Babu & Mohan (2012) and Paez et al. (2018) and DSR and cost is obtained.

  • Applications are shown on three example networks consisting of branch and looped networks including two source networks. The application of the methodology is also shown on a real-life network of Saoner district in Maharashtra State in India.

  • Further, for the entire networks, it is observed that Paez et al. (2018) provided higher reliability with slightly higher cost as compared to Babu & Mohan's (2012) method.

  • A trade-off between cost and reliability is obtained using Paez et al. (2018) and Babu & Mohan's (2012) method for all the three networks and it is observed that Paez et al.’s (2018) methodology provided more reliability with higher cost.

  • Further, the real case study of Saoner of Maharashtra State in India is also considered and the results are also discussed. The future scope of the study is to compare the results all the PDA modeling approaches with their advantages and disadvantages.

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

The authors declare there is no conflict.

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