Redundancy improvement using variance of pipe-flow series in water distribution systems considering pressure-deficient modeling approaches

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 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 V_Q = VPF series, N = number of pipes in the network; Q_x = 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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.