The Evaluation of Reliability Indices in Water Distribution Networks under Pipe Failure Condition

In this research, reliability indicators of water distribution networks are evaluated under pipe failure conditions. The case studies include two benchmark and one real-life water distribution networks in Iran with more hydraulic constraints. Some important reliability indicators are presented such as resilience index, network resilience, modified resilience index and minimum surplus head index. GANetXL is used to do one-objective and two-objective 20 optimization of the previously mentioned water distribution networks in order to not only minimize the cost, but also maximize the reliability indicators. Moreover, the results of a statistical analysis for each pipe is used to determine the sensitive pipes that are of the most failure probability. GANetXL is an optimization tool in Excel environment and works based on Genetic Algorithm. GANetXL has the capability of being linked to EPANET (Hydraulic simulation software). The results obtained clearly show that network resilience index is of poor performance when compared 25 with the other indexes under pipe failure conditions, especially in real-life networks that include small pipe diameters. It was also showed that if a water distribution network was optimized only in terms of cost, there would be an unacceptable pressure drop at some nodes in case of pipe failure.

et al. 2018) applied a Simple Modified Particle Swarm Optimization (SMPSO) to minimize the cost of water distribution networks.SMPSO then used a novel factor to decrease the inertia weight of the algorithm in proportion with simulation time to facilitate both global and local search.Literature review shows that stochastic models, particularly the GA types, give better results than linear and non-linear optimization models (Pandit and Crittenden 2012).
Objective function is important in optimizing the design of distribution systems.The main negative aspect of the single-objective constrained formulation is that it does not effectively set up a trade-off between cost and reliability/robustness of a design (Todini 2000).Reliability can be considered as the ability of providing an adequate supply under both usual and unusual conditions (Farmani, Savic et al. 2005), including demand uncertainty, pipe failure, etc.One of the most used reliability criteria is the concept of resilience index suggested by (Todini 2000),which is a measure of the ability of the network to handle failures and is related indirectly to system reliability.Several suggestions were made to modify the resilience index introduced by Todini (Prasad and Park 2004, Farmani, Savic et al. 2005, Jayaram and Srinivasan 2008, Reca, Martinez et al. 2008, Raad, Sinske et al. 2010, Baños, Reca et al. 2011, Greco, Di Nardo et al. 2012, Pandit and Crittenden 2012).
Subsequently, a genetic algorithm technique is used in this research as a part of GANetXL (Savić, Bicik et al. 2011) , an add-in to Microsoft Excel. .There are a few applications of GANetXL in water systems, which include the development of a model for optimal management of groundwater contamination (Farmani, Savic et al. 2005, Farmani, Henriksen et al. 2009) and multi-objective optimization of water distribution systems (Piratla and Ariaratnam 2012, Mala-Jetmarova, Barton et al. 2015, Piratla 2016).GANetXL is used to optimize two benchmark networks from literature (Two-loop and Hanoi water networks) in two different conditions including single-objective (cost) and twoobjective (cost and reliability criteria) optimizations.Afterwards, the solutions obtained, as well as the performance of the proposed Resilience Index, Network Resilience, Modified Resilience Index and Minimum Surplus Head Index are discussed.Finally, as the results obtained for the benchmark networks are satisfactory, GANetXL is used to design a real-life water network in Iran in which there are more hydraulic constraints compared with the benchmark networks

Optimization Model for WDN Design
In this paper, WDNs are optimized with pipe diameters as decision variables.Cost is considered as the objective function that must be minimized [Eq.( 1)] and the reliability criteria are modeled in the form of a two-objective function [Eq.( 2)]. (1) Where  1 is network cost,  2 is network reliability,   is cost for unit length of pipe with diameter,   length   and  is pipe numbers in the network.

Constraints
The constraints to the optimization problem are as follows: 1) Explicit system constraints such as conservation of mass of flow, conservation of energy and conservation of mass of constituent, which all are controlled by water network simulator software, EPANET (Rossman 2000, Mala-Jetmarova, Barton et al. 2015).
where   = diameter of pipe

Reliability Indicators
A range of reliability criteria has been introduced to different degrees of complexity.Usually, these criteria give some suggestion of the ability of a WDN to handle changing conditions and are straightforward to analyze so are practical for optimization studies that compare the performance of network design.This section presents the definition of the key criteria and their derivatives as well as the advantages and disadvantages of them.

Resilience Index (𝐼 𝑟 )
Todini's resilience index is a popular surrogate measure within the WDN research field (Todini 2000, Reca, Martinez et al. 2008, Atkinson, Farmani et al. 2014).It considers surplus hydraulic power as a proportion of available hydraulic power.The resilience index,  , is measured in the continuous range of [0-1] (for feasible solutions of    ≤   ) and is formulated as below (Todini 2000): Where  is the number of supply and demand nodes;  is the set of supply nodes (reservoir/emptying tanks);  denotes the number of pumps;   is the available head at supply node ;    represents the required head at supply node ;   is the demand at node ;   is the supply at input node ;   is representative of head associated with the input node ;   is the power of pump ; and finally  is the specific weight of water.Maximization of the resilience index improves the ability of a pipeline network in encountering failure conditions.

Network
Resilience (  ) Prasad and Park (2004)  Theoretically, the value of network resilience may vary between 0 and 1.However, for real-world systems it never attains a value of 1, since imposing the same diameter to all pipes in a network need not always provide a Paretooptimal solution in Cost-  space, as   is a measure of the combined effect of surplus power and nodal uniformity.

Modified Resilience Index (𝑴𝑹𝑰)
Jayaram and Srinivasan (Jayaram and Srinivasan 2008) proposed a modified resilience index (), which theoretically overcomes the drawback of Todini's resilience index when evaluating networks with multiple sources.
In contrast to Todini's resilience index, the value of the modified resilience index is directly proportional to the total surplus power at the demand nodes.Eq. ( 9) describes , which only considers the solutions with pressures equal to or higher than that required in all nodes.While Todini's   and Prasad's   take values up to a maximum of 1, Jayaram's  can be greater than 1 (Baños, Reca et al. 2011).

Minimum Surplus Head Index (𝑰 𝒎 )
In a WDN, minimum surplus head,   , is defined as the lowest nodal pressure difference between the minimum required and observed pressure, formulated as Maximization of the available surplus head at the most depressed node to some extent improves the reliability of a network (Prasad and Park 2004).

GANetXL
GANetXL is used as the optimization tool in this research.GANetXL has been developed by the Center for Water System of University of Exeter as an add-on in Microsoft Excel (Miri andAfshar 2014, Peirovi, Moghaddam et al. 2020).It is a common optimization tool with spreadsheet-based interface for solving both single-objective and multiobjective optimization problems (Savić, Bicik et al. 2011).The primary advantage of GANetXL is its capability of easy integration with EPANET via Visual Basic.GANetXL incorporates GA for single-objective and NSGA-II for multi-objective optimizations (Deb, Pratap et al. 2002).In addition, it has the capability to apply penalty functions.
In this paper GANetXL is employed in two steps: in the first step for single-objective optimization based on GA and the second step for two-objective optimization based on NSGA-II.GA and NSGA-II parameters such as population size, the number of generations, selection method, crossover and mutation operators, crossover and mutation probability and the type of algorithm were tested and reasonably well-performing parameters selected for final optimization runs.These parameters are presented in Table 1 for three example networks, which are described in the following sections.The crossover and mutation types are described in details in CWS (2011).

Results and Discussion
Three example applications are studied: the Two-loop (Alperovits and Shamir 1977), Hanoi (Fujiwara and Khang 1990) ,which are the benchmark networks, as well as a real-life case study in Iran.

Example 1: The Two -loop network
The Two-loop network was originally presented by (Alperovits and Shamir 1977).The network consists of 7 nodes and 8 pipes with two loops, and is fed by gravity from a reservoir with a 210 m fixed head.The network details are available in previous studies (Alperovits and Shamir 1977, Cunha and Sousa 1999, Geem 2009, Savić, Bicik et al. 2011, Pant and Snasel 2021).The minimum pressure head requirement of the other nodes is 30 m above the nodal elevations.
In the first step, as a result of single-objective optimization of the Two-loop network using GA technique in GANetXL, the minimum cost obtained 419000$ with 35000 number of function evaluations (NFEs) which is the same to minimum costs obtained by GA (Savic and Walters 1997), Simulated Annealing (SA) (Cunha and Sousa 1999), Shuffled frog leaping Algorithm (SFLA) (Eusuff and Lansey 2003), Harmony Search (HS) (Geem 2009)and Scatter search (SS) (Lin, Liu et al. 2007) with 250000, 25000, 11323, 5000 and 3215 NFEs, respectively.
As a result, minimum cost is 419000$ for one-objective optimization of this network using GANetXL after 1000 generations that is equal with minimum costs obtained by GA, Simulated Annealing (SA), Shuffled frog leaping Algorithm (SFLA) Harmony Search (HS) and Scatter search (SS) (Savic and Walters 1997, Cunha and Sousa 1999, Geem, Kim et al. 2002, Eusuff and Lansey 2003, Geem 2009).
In the second step, figure 1 (a-d Figure 2 shows the surplus pressure of the minimum pressure head requirement in the nodes of Two-loop network for solutions with maximum reliability criteria and minimum cost.As it is observed, the surplus pressure of the nodes in the solutions with minimum cost is lower than the solutions of maximum reliability criteria (  ,   ,  and   ).Also, the design based on single-objective function (minimum cost), surplus pressure is closer to the minimum allowed pressure in nodes 3, 6, and 7, showing that these nodes are the critical nodes of the network.As a result, if the twoloop network was designed only based on minimum cost, in critical periods such as pipe failures, there would be problems issues at these nodes.
Reliability evaluation should be analyzed under all feasible extreme conditions.Failure of multiple pipes as well as the failure of the reservoir connection line during a firefighting event and/or power or pumping station failures should be evaluated simultaneously.Although an infinite number of failure scenarios are likely, the probability of simultaneous failures in multiple pipes is too low (Tabesh, Tanyimboh et al. 2001).Pipe failures independency can be assumed (Su, Mays et al. 1987) and any likely dependency will be negative.For example, if a pipe failure occurs in the network, the pressure will decrease, and consequently the probability of another pipe failure will decrease as well.However, in case the system is a large-scale WDN, the influence of pressure might not be significant.Other pipe failure reasons, such as damages or traffic loadings, may lead to pipe failures that are completely independent events (Shafiqul Islam, Sadiq et al. 2013).  2 presents the statistical parameters of each pipe in two-loop network under different runs of single-objective optimizations when the objective function is to minimize cost.This table helps the designers to recognize critical and sensitive pipes that have the most probability of failure in the network.For example, maximum and minimum diameters that are allocated to pipe 1 in different runs of GANetXL are 24 and 18 inch, respectively.is why this pipe is not taken into account for failure analysis.Figure 4 shows the performance of solutions with maximum reliability criteria under the failure of pipes 2, 3 and 5.

Example 2: The Hanoi network
The Hanoi network in Vietnam, first presented by Fujiwara and Khang, is a new design as all new pipes are to be selected.The network consists of 32 nodes and 34 pipes organized in three loops.The system is gravity fed by a single reservoir.The network details are given in (Fujiwara and Khang 1990).The minimum required pressure head for all nodes is 30 m and the elevation for all nodes is zero.There are six available pipe diameters to be selected for each new pipe and the pipe cost per meter for the six available pipe diameters are listed in previous studies (Atiquzzaman and Liong 2004, Zecchin, Simpson et al. 2006, Savić, Bicik et al. 2011, Pant and Snasel 2021).
In the first step, as a result of single-objective optimization, GA method in GANetXL obtained a minimum cost of 6.097×10 6 $ with 100000 NFEs for this network while in the previous researches the methods of GA (Savic and Walters 1997), Ant Colony Optimization (ACO) (Zecchin, Simpson et al. 2006), and Shuffled Complex Evolution (SCE) Atiquzzaman and Liong (Atiquzzaman and Liong 2004)reported costs of 6.195, 6.134 and 6.22  nodes No. 13, 30 and 31 is less than 1 m which shows that these nodes are the most critical ones of this network. ,  and criteria have similar performance for all the nodes, but   criterion determinates more surplus pressure for most of the nodes than other criteria in this network unlike the two-loop network.Table 3 shows the statistical parameters for each pipe of Hanoi network due to different runs of single-objective optimizations by GANetXL.According to this table, Pipes No. 4, 5, 6 and 20 that have standard deviation and variation coefficient equal to zero have been chosen for reliability evaluation when there is a failure in the network.

Example 3: The Real-life network
Real-life WDN is located in Iran and it has 37 pipes, 24 nodes and a reservoir with a 962 m fixed head (Fig 7).The design purpose of this network is municipal water supply of city and improving of the existing condition of the WDN (Moghaddam, Alizadeh et al. 2020).For this purpose, a series of pipes which have diameters more than 100 mm are used for future conditions (Rasekh, Afshar et al. 2010).For designing this network, polyethylene pipes (PE-80) with Hazen-Williams coefficient of 130 are used.The nodes and pipes characteristics are presented in (Moghaddam, Alizadeh et al. 2020).In the design of the network, nodes pressure and velocity constraints are between 14-60 m and 0.2-2 m/s, respectively (Department of Technical Affairs 2013).There are more constraints in this example than the other ones.In the second step, the results of figure 8 (a-d (2000).Subsequently, in this study, a random pipe failure has been created using a uniform distribution in the range of [26,37], that is the pipe numbers for existing pipes (Shafiqul Islam, Sadiq et al. 2013) Finally, the failure of the pipes No. 27 and 34 was analyzed in the network.

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The results of the investigations in figure 10 shows that only the failure in Pipe No. 18 can influence the pressure nodes.Consequently, this pipe is one of the most sensitive pipes in this network.However, reliability performance in the failure conditions is similar to no failure conditions in figure 11.Finally, for this network that includes low diameter in existing pipes,   has not a suitable performance because of making the uniformity in pipes connected to a node leads to the decrease of the diameter of new pipes.Thus, the capability of the surplus pressure decreases due to the 380 increase in head-loss in the pipes.

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criteria obtained using GA and NSGA-II in GANetXL.As it is observed, when the cost is the basis for the design and optimization of Real-life network, velocity variation are so high in the pipes.This can lead to some issues in the network.But in the presented solutions with maximum reliability criteria (  , and   ), velocity variations are not only low but almost uniform.

Fig 2 .
Fig 2. Surplus pressure of nodes in two-loop network for solutions of maximum reliability criteria and minimum cost In this paper, to evaluate reliability of the candidate solutions of maximum   ,   ,  and   criteria, the nodal pressures have been investigated under pipe failure conditions.Table 2 presents the statistical parameters of each pipe

Fig. 3
Fig. 3 Surplus pressure of nodes in two-loop network for solution with maximum reliability criteria under failure of pipes No. (a) 2, (b) 3 and (c) 5 Figure 3 shows that, nodeNo.6 encounters with a serious pressure loss with failure in pipe No. 2, 3 and 5 in represented solutions by   criterion.In represented solutions based on   ,  and for all the pipes of the network the diameter was 609.6 mm while in the obtained solution with maximum   , the diameter of pipes No. 4 and 6 was 25.4 mm and other pipes were 609.6 mm.Consequently,   criterion is of lower performance than any other criterion under pipe failure condition.
Fig 4. Pareto front of two-objective function optimization of the Hanoi network, (a) Cost-  , (b) Cost-  , (c) Cost-, (d) Cost-  Figure 5 shows the surplus pressure in comparison with minimum allowed pressure in the nodes of the Hanoi network for solutions of maximum reliability criteria and minimum cost.In the cost-based optimization, surplus pressure in

Fig 5 .
Fig 5. Nodal surplus pressure of Hanoi network for solutions of maximum reliability criteria and minimum cost

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The results of figure6.(a) and (b) shows that by failure in pipes No. 4 and 5 the surplus pressure in most of the nodes for solutions of maximum   criterion is more than solutions with maximum   ,   and .In effect of pipes No. 4 and 5 failures, nodes reactions to pressure changes are similar because these two pipes are along.However, due to failure in pipe No. 6, none of the nodes of the network meet lack of pressure and the figure 6.(c) shows that the solutions with maximum   and criteria has more capability to supply pressure in most of the networks.In figure 315 6.(d) there is no significant difference in represented solutions with reliability criteria values.The nodes with no values in the graph are those that have negative pressures.
Fig 6.Surplus pressure of Hanoi network nodes for solutions of maximum reliability criteria when pipes No. (a) 4, (b) 5, (c) 6 and (d) 20 are lost due to failure

Fig 7 .
Fig 7. Layout of Real-life network

Fig 8 .
Fig 8. Pareto front of two-objective function optimization of the Real-life network, (a) Cost-  , (b) Cost-  , (c) Cost-, (d) Cost- The results shown in figure9demonstrate that in the cost-based optimization, surplus pressure in the nodes number 13 and 23 is less than 1m that explains these nodes are the most critical ones in the network. and criteria have similar and more successful performance compared to   in terms of the surplus pressure for all the nodes in the network.  has less capability than other criteria to create surplus pressure in the network.
Fig 10.Surplus pressure of nodes in Real-life network for solutions of maximum reliability criteria under failure of ;   = kth commercially available pipe size;  = number of available pipe sizes;   = hydraulic-head available at node ;    = minimum hydraulic-head required at node ;    =maximum hydraulic- head at node ;  = number of demand nodes;    = minimum velocity required at pipe  and    =maximum velocity at pipe ;  = number of pipes.
introduced another reliability measure called network resilience (  ), which incorporates the effects of both surplus power and reliable loops.Reliable loops can be ensured if the pipes connected to the same node do not vary greatly in diameter.If  1 ,  2 , ...,   (where  1 ≥  2 ≥ ⋯ ≥   ) are the diameters of the pipes connected to node j, then uniformity of that node is given by Eq. 7,

Table 1 .
Optimum GA and NSGA-II values for the three case studies in this paper

Table 2 .
Statistical parameters for diameters obtained for each pipe of two-loop network

Table 3 .
Statistical parameters for diameters obtained for each pipe of Hanoi network

Table 4 .
Statistical parameters for diameters obtained for each new pipe of Real-life network