Reliability of water distribution networks subjected to seismic hazard: Application of an improved entropy function

Reliability Engineering and System Safety 197 (2020) 106828

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Reliability of water distribution networks subjected to seismic hazard: Application of an improved entropy function

T

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Hossein Emamjomeha, Roohollah Ahmady Jazanyb, , Hossein Kayhanic, Iman Hajirasoulihad, Mohammad Reza Bazargan-Larie a

Department of Structural Engineering, IIEES, Tehran, Iran Department of Civil Engineering, East Tehran Branch, Islamic Azad University, Tehran, Iran c Department of Civil Engineering, Pardis Branch, Islamic Azad University, Pardis, Iran d Department of Civil and Structural Engineering, the University of Sheffield, Sheffield, UK e Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Canada b

A R T I C LE I N FO

A B S T R A C T

Keywords: Entropy function Water Distribution Networks (WDNs) Seismic hazard Network failure Serviceability Fragility curve

This study aims at improving an entropy-based measure considering both total and partial failure, for assessing the reliability and redundancy of Water Distribution Networks (WDNs) during seismic events, which was not explicitly addressed in previous literature. In order to capture the effects of connectivity order of the network nodes as well as the seismic hazard of the site in reliability studies, new entropy calculation method is proposed using hazard information of seismic regions and fragility curve of pipelines. Subsequently, the effect of failure of each pipe due to a seismic event, which results in reducing the serviceability and reliability of a WDN, is investigated using the proposed entropy calculation. The results indicate that the proposed entropy function is capable of estimating the reliability trend of WDNs considering different failure patterns caused by different hazard levels. The proposed method enables the engineers and practitioners to find a proper optimum design approach for the water supply network and plays an important role in assessing the performance of WDNs and mitigating the chance of total failure of WDN in hazardous zones.

1. Introduction Water Distribution Network (WDN) is an expensive infrastructure system, which has a major role in providing one of the most essential needs of people in all cities and rural areas with considerable populations. This lifeline system mainly consists of connections, pipes, pumps, tanks and some other hydraulic components. The main objectives of the supplying system besides reserved capacity for firefighting purposes are to provide the required pressure and flow rate for consumers. Since the operational serviceability of WDN considerably affects the social life of the human community, the reliability of this network has received attention during the past few decades [1–5]. The study on the operational serviceability of the pipeline system started in the early 1980s. Shinozuka et al. [6] proposed a method to evaluate the serviceability of the pipeline transmission system affected by the seismic events in Los Angles. Isoyama and Katayama [7] suggested a new method for evaluating the microscopic reliability of the water supply system of Tokyo city by possible flow methods. Javanbarg

and Takada [8,9] proposed a method to simulate the flow rate with respect to two recognized pipeline damages modes including leakage and breakage. They studied the serviceability of Osaka city WDN for ten hospitals by their proposed simulation method and concluded that the flow analyses may lead to estimations with more accuracy for vulnerability assessment of the water supply network. Redundancy, in a WDN, represents the reserved capacity that is provided by alternative supply paths between demand nodes. Higher redundancy decreases the risk of total WDN failure when some of the components fail. Reliability is another concept, which represents the resilience of WDN under adverse conditions and is well correlated with redundancy [10,11]. In other words, a system with more reliability has a higher potential to withstand partial or total failure. During the past decade, different measures were established to evaluate the redundancy of WDN pipelines and different methods for evaluating the reliability of lifelines were developed based on these measures, which resulted in increasing the reliability of lifelines. It was shown that the established reliability measures could be recognized as suitable alternatives to

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Corresponding author. E-mail addresses: [email protected] (H. Emamjomeh), [email protected] (R. Ahmady Jazany), [email protected] (H. Kayhani), i.hajirasouliha@sheffield.ac.uk (I. Hajirasouliha), [email protected] (M.R. Bazargan-Lari). https://doi.org/10.1016/j.ress.2020.106828 Received 19 February 2019; Received in revised form 1 January 2020; Accepted 19 January 2020 Available online 20 January 2020 0951-8320/ © 2020 Elsevier Ltd. All rights reserved.

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examples will be presented to validate the efficiency of the proposed method.

redundancy measures of lifeline networks [12–15]. In other words, increasing the redundancy of WDN can promote WDN reliability. Awumah et al. [10] studied the reliability of the water supply network by including a more explicit statement of the alternate path available in the network. They concluded that the entropy function could very well predict the different levels of reliability for different network layouts. Tanyimboh and Templeman [13] enhanced the suggested method established by Awumah et al. [11] and introduced a more understandable definition of entropy function for WDN by developing an algorithm for computing the reliability of a single source. Hoshiya et al. [14,15] proposed a redundancy index with regard to the entropy of a damaged system with local pipeline failure due to seismic risk. Followed by latter researches, a case study was performed on the Kobe city water supply network to develop a resilience measure against seismic risk [15]. Tanyimboh and Sheahan [16] have performed an investigation to optimize the layout of the water distribution system by using a ratio of minimum-cost/maximum-entropy function. In their study, they showed that there is a possible optimized solution for assigning mechanical properties of WDN by minimizing the entropy of network flows. However, they could not identify the actual role of entropy in their study. Ang and Jowitt [17–19] studied the relationship between total power dissipation inside WDN. These researchers also proposed a simplified method to calculate the entropy of WDN, called Path Entropy Method (PEM), which provides a better insight into the branching-tree network problems. Tanyimboh and Templeman [20] investigated the correlation between entropy and reliability of WDN. They showed that there is a good agreement between maximum flow entropy in the water distribution network and reliability. Bazargan-Lari [21] used a type of entropy for determining the best layout of monitoring stations in a realworld drinking water distribution system threatened by deliberate contamination events. More recently, Hosseini and Emamjomeh [22] and Kashani et al. [23] proposed a modified entropy-based measure that takes into account uncertainties imposed by hydraulic flow and mechanical specifications of WDN pipelines. The results showed that the proposed entropy-based index is an efficient tool for finding an optimum hydraulic flow, designing a new WDN and making a decision to find the best mitigation plan for existing water distribution network subjected to different natural and man-made hazards. Diao et al. [24] proposed a Global Resilience Analysis (GRA) approach to assess the resilience of the WDN system considering an unexpected disaster that results in WDN possible failure mode. The results showed that when calculating GRA, the details of the WDN hydraulic model have great importance in predicting system behaviour for different failure scenarios with acceptable precision. Hosseini et al. [25] reviewed the definitions and measures of complex and large-scale systems. They categorized the previously established methods for evaluating the resilience of the system focusing on qualitative and quantitative aspects of the approaches. In other relevant studies, Cimellaro et al. [26,27] proposed a new resilience index for WDN, which is a combination of three indices including the number of users temporarily without water, the water level in the tank and the water quality. They showed that the proposed index could be used for assessing the functionality of WDN, including the delivering of specific water demand with required pressure and reconstruction process after extreme conditions. In a more recent study, Meng et al. [28] investigated the importance of topological attributes matrices of WDN. They concluded that water source location has a considerable impact on the resilience value of WDN. Based on the above-mentioned research and their limitations, this study proposes a new entropy index considering both uncertainties in hydraulic and mechanical specifications of WDN while Tanyimboh and Templeman [20] and Awumah et al. [10] considered only one of these aspects in their entropy indices. Furthermore, the proposed entropy function of this research can include the effects of seismic hazard of the studied site to assess the reliability of WDN, through considering the fragility curve of pipelines and seismic hazard assessment results for calculating the reliability of water supply network. Then different

2. Methodology and model formulation In this paper, the concept of entropy function is utilized to calculate the reliability of WDNs. The entropy of a WDN shows the capability of a network for passing a molecule of water from the source node to the demand node. Tanyimboh and Templeman [13, 20] proposed a promising entropy function for WDN, which was based on informational entropy suggested by Shannon [29] as an uncertainty measure. The above-mentioned proposed method employed available information of WDN including the topological layout of various pipeline systems, supply and demand of all WDN nodes, and the flow direction to calculate the entropy function. The most critical values that can considerably affect the optimum entropy value are the distribution of flow and the number of paths between demand and supply nodes. The entropy function developed by Tanyimboh and Templeman [20] is as follows: N

S = K (S0 +

∑ Pn Sn) n=1

(1)

Where K, S0, Pn, Sn, and N are: Boltzman constant, the entropy of external inflows, the ratio of the total inflow of nth node to the total inflow of the network, the outflow entropy of nth node and the total number of WDN nodes respectively. Based on the study by Tanyimboh and Sheahan [16], Boltzman constant is assumed as 1 which is a valid value if all pipelines of WDN are fully serviceable [22]. The entropy of the external inflows (S0) and the outflow entropy of node (Sn), which are hereinafter called as "external entropy" and “internal entropy” respectively, are defined as follows [20]:

S0 = − ∑ Pi,0 LnPi,0 i∈I

∑

Sn = −

i ∈ IEn

(2)

Pnj LnPnj (3)

where I, Pi,0 in Eq. (2) as well as IEn, Pnj in Eq. (3) are the set of all source nodes, probability of selecting of the ith source node, set of outflows from node n and probability of selecting a path between nodes n and j, respectively. The probability of selecting ith source node (Pi,0), probability of selecting a path between nodes n and j (Pnj) and the ratio of the total inflow of nth node to the total inflow of the network (Pn) are quantified using Eqs. (4–6) [20]:

Pi,0 = Pnj =

Pn =

qi,0 T0

(4)

qnj Tn

(5)

Tn T0

(6)

Where qi,0, T0 and Tn are external inflow at source node i, the total supply or demand and the total inflow at node n. The first term of entropy function, as presented in Eq. (1), represents external entropy (S0) and the second term is the internal entropy (Sn) originated from entropy values at each demand node which has a specified contribution to the total WDN entropy based on probability of selecting the nth node (Pn). The probability of selecting a path in a studied WDN by a molecule of water, reaching a specified node n, is considerably affected by several factors including the location of demand nodes, flow configuration and unpredictable events such as potential hazards of the site, and damage state of pipelines due to different failure scenarios. However, the mathematical concept of the portability of selecting a path is based on the equal chance of selecting all paths. Hence, the key issues are the implementation of the practical and actual engineering aspects of 2

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Fig. 1. (a) Schematic view of a simple WDN and the details of corresponding entropy calculations (the flow rate of links is L/s) (b) PEM of Fig 1(a).

Fig 2. Flowchart diagram of the proposed methodology calculation method.

would not be zero otherwise the internal entropy is equal to zero. Path Entropy Method (PEM) [19] was established to identify different possible alternative ways from a source node to all demand nodes to calculate the entropy of WDN. The application of PEM is demonstrated in Fig. 1 using a simple example of gravity WDN with 4 nodes. This simple WDN has one source and three demand nodes. Based on Fig. 1(a), the arrows show the selected flow pattern between source and supply nodes based on the topology of the Nodes. The above-mentioned methodology [16,20] is utilized for estimating the entropy value for the studied WDN. The maximum entropy of this example was obtained with respect to the pipeline flow rates as shown in Fig. 1(a). In this regard, optimum pipeline diameters are obtained using hydraulic analyses considering different hydraulic parameters including topological specifications of the nodes, pipelines lengths, and Hazen–Williams coefficient of pipelines. Fig. 1(b) shows the PEM of the studied sample and the entropy calculation details of studied WDN are presented as follows (See Fig. 1(a)):

Fig. 3. A schematic view of the considered sample WDN for a specific flow pattern (the flow rate unit of links is L/s) (Case A).

uncertainties in the proposed entropy. The entropy of the external inflows (S0) determines the uncertainty associated with the selection of different sources by a molecule of water for reaching a determined supply node. However, the internal entropy (Sn) denotes the uncertainty of selecting a path between a source node by a water molecule to reach the determined supply node. It is evident, if there are two or more paths from one of the sources to the specified supply node to pass a molecule of water, the internal entropy value

( ) + × ln ( ) + × ⎡2 × × ln ( ) + × ln ( ) ⎤ ⎣ ⎦ × ⎡ × ln ( ) + × ln ( ) ⎤ = 1.9073 ⎣ ⎦ 30

5

S = − 30 × ⎡ 30 × ln ⎣ −

17.5 30

−

15 30

5 30

5 30

5 30

17 . 5 30

5 30

5 30

10 30

17 . 5 30

7.5 30

7.5 30

× ln

( ) ⎤⎦ 7.5 30

7.5 30

10 30

(7)

In this research, to include the effect of serviceability of pipelines 3

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Fig. 4. Possible flow patterns for the studied WDN (Case A).

Fig. 5. The Path Entropy Method (PEM) diagram of the studied water distribution network for different possible flow patterns (the flow rate of links is L/s) (Case A).

pipeline). The above-mentioned factor also contains both mechanical and hydraulic uncertainties of WDN. Considering that some pipelines are partially to fully unserviceable, the ratio of the total inflow of nth node to the total inflow of the network (Pn) in this study is changed to the probable value of Pn which is hereafter called PP. This implies that PP is the inflow rate of the nth node, which is partially serviceable. By substituting Eqs. (2) and (3) into Eq. (1) and considering partial serviceability, the following equation can be obtained:

due to unpredictable natural disasters and develop an approach to consider the pipeline partial serviceability, a serviceability factor is introduced in the internal entropy term of Eq. (1). This factor considers the effect of connectivity order of nodes, which equals the sum of the possible paths from source to reach a node, and is also attributed to the connectivity conditions and the serviceability probability of pipelines. This factor varies in the range of 0 (for the fully serviceable pipeline with minimum required pressure) to 1 (for unserviceable/disconnected 4

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Fig. 6. The contour map of calculated proposed entropy function for the sample WDN as shown in Fig. 3 subject to failure of one pipeline (Case A). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.) N

S = K (− ∑ Pi, o LnPi, o −

∑ PP ∑ n=1

S = K (− ∑i ∈ I

Pnj LnPnj ) (8)

j ∈ IEn

S = K (− ∑i ∈ I

qi,0 T0 qi,0 T0

q

N

Ln Ti,0 − ∑n = 1 0

q

N

Ln Ti,0 − ∑n = 1 0

Tps T0 Tps Tn

.

Tn Tn

∑j ∈ IEn

.

Tn To

∑j ∈ IEn

qnj T0 qnj T0

qnj

Ln T ) ⇒ o

qnj

Ln T ) o

(11)

where Pp in this research is defined as follows:

Pp =

The ratio of Tps/Tn is considered as serviceability ratio (SRn) of the nth node. Hence Eq. (11) can be rewritten and developed as follows:

Tps (9)

T0

N

S = K (− ∑ Pi, o LnPi, o −

In this equation, Tps is the total supply of the nth node in the proposed method that is connected to the pipe with partial serviceability. By substituting

Eq. (8),

Eqs. (4), (5) and (9) in

n=1

the following expression is obtained:

S = K (− ∑ i∈I

qi,0 T0

Ln

qi,0 T0

N

−

∑ n=1

Tps T0

∑ j ∈ IEn

qnj T0

Ln

qnj To

∑ SRn . Pn ∑ j ∈ IEn

Pnj LnPnj ) (12)

SRn is obtained by statistical calculations based on the serviceability of pipelines (SR) connecting the nth node to the sources using studied flow patterns. In other words, the serviceability of pipelines that make paths between sources to the nth node contributes to calculating SRn. The SR values of pipelines depend on many factors including the age of the WDN, topographic characteristics of the WDN and the mechanical specification of pipelines materials, which in turn affect pipelines' current pressure. However, in this study, SR is a measure of total or

) (10)

Based on the assumed concept of the proposed method to consider the partial serviceability in internal entropy, if the second term of above-mentioned Equation is multiplied by Tn/Tn, the following equation will be obtained: 5

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Fig. 7. The possible flow patterns subjected to different concurrent failure scenarios of two pipelines (Case A).

System can be applied [38]. In this study, to consider the hazard effect on seismic reliability of WDN by entropy function, the SRseismic is defined as follows:

partial failure of pipeline subjected to seismic events in addition to pressure loss of pipeline compared to the minimum required pressure of the pipeline, which in turn affects the severability of WDN. The introduced serviceability ratio or SR is a product of SRseismic and SRNonseismic. If the pipeline current pressure is smaller than the required pressure, SRNonseismic will be equal to the ratio of pipeline current pressure subjected to the current head of source to the minimum required pressure of the pipeline, otherwise SRNonseismic is equal to one. It is worth mentioning that the classic optimization techniques [30,31] can be utilized to find the optimum entropy value of WDN with respect to mechanical specifications of WDN considering that the head of the demand node should be larger than the minimum required head. Based on previous studies [6,8,12,14,32], the velocity of earthquake ground motion results in the serviceability reduction in pipelines or pipeline fracture if a WDN experiences earthquake. Fragility function, which is defined as the probability of overall structural collapse as a function of different ground motion parameters for a specified hazard level [33], was developed by Finite Element Methods (FEM) and statistical tools to find a relationship between the velocity of ground motion and damage of pipeline during the earthquake. In engineering reliability, the fragility curve is defined as a probability of exceeding a determined demand value “D” from defined capacity “C” for structural response excited by the scaled earthquake record considering specified Intensity Measure (IM) value [33]:

PFragility = P [D ≥ C IM = X ]

SRseismic = 1 − DR

Where DR is defined as the Damage Rate of a pipeline, which is a function of the seismic zone PGV. The PGV is obtained based on seismic hazard analysis of the studied area using the Poisson distribution. The PGV is correlated well with Peak Ground Acceleration (PGA) [39,40]. By considering Poisson distribution in seismic hazard analyses, the probability of occurrence of an event during the exposure time (t) is estimated as follows 40]:

P (n > 1, t , τ ) = 1 − et / τ

(16)

where τ is the average recurrence interval of the seismic event (i.e. earthquake) with magnitude (M) greater than a specified value. The probability of exceeding PGV from determined value was developed previously from some studies and some empirical functions for seismic regions were attained [40–42]. As an instance, Heidari et al. [39] proposed an empirical function for obtaining PGV values of Tehran which will be implemented for the reliability assessment of WDN in Section 3 (Case C). Fig. 2 shows the flowchart for calculating the proposed entropy function. Based on this flowchart, SR includes SRsiesmic and SRNon-seismic. It is shown in Fig. 2 that obtaining the fragility curves and hydraulic analysis of WDN are different processes in which the mechanical properties of pipelines are implemented. Based on this flowchart, it is evident that the proposed methodology aims at assessing and improving risk management with respect to the planning phase of WDN.

(13)

In pipeline engineering practice [33–36], the fragility function is presented as Damage Rate (DR) of each pipe of WDN subjected to seismic hazard, and intensity measure would be Peak Ground Velocity (PGV) of the seismic zone where the WDN is located:

DR = P [Dpipe ≥ Cpipe IM = PGV ]

(15)

3. Application to WDNs

(14)

To show the applicability and usefulness of the proposed methodology in assessing the reliability and redundancy of WDN subjected to both total and partial failure during seismic events, in this section, four different examples will be studied. In the first part of this section, the calculation method of the proposed entropy function is addressed for a simple WDN with two loops (Case A). Then, in the second part (Case B), the effect of flow pattern subjected to failure of pipeline on the proposed entropy function is compared to the sample WDN employed for Case A. In the third example, reliability assessment of WDN affected by seismic event for a WDN is studied using fragility curve of the pipeline

where Dpipe and Cpipe are demand and capacity of the pipeline with determined material and geometric specifications due to the specified IM. The calculation method of pipeline fragility curves is based on analytical modeling of pipeline [35–37] considering soil-pipeline interaction modeling [37] and nonlinear time-history analyses using the earthquake ground motion [36] that are consistent with seismic characteristics of the studied site. Regarding the randomness essence of damage ratio, PGV can be represented as an independent variable and damage ratio as a random variable. To describe and examine the dependence of DR and PGV, regression analysis and the Fuzzy Interface 6

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Fig. 8. Contour map of calculated proposed entropy function for the sample WDN shown in Fig. 3 subjected to concurrent failure of two pipes (Case A). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

7

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Now the reliability of sample WDN when a pipeline fails with respect to the assumed flow pattern shown in Fig. 3, will be investigated using the proposed entropy function. Based on the developed methodology, the SR values of the failed pipeline (See SRn in Eq. (12)) are set to zero to omit the serviceability and the entropy function will be calculated with respect to x and y variables that represent flow rates in pipelines of WDN. Fig. 6(a)–(d) shows the entropy values computed based on improved entropy function introduced in Section 2 for the studied sample WDN in this section (see Fig. 3), when one of the pipelines 2, 3, 4 or 5 fails (or is unserviceable), respectively. The green points in Fig. 6 show maximum entropy regarding the predetermined unserviceable pipeline. Based on the assumed flow pattern as shown in Fig. 3, if the pipeline 1 fails, the required demand of node 2 (i.e. 20 liters per second) will not be provided. This indicates that the calculation of the corresponding proposed entropy function was not included in this study. In Fig. 6, x and y variables are flow rates of pipeline 1 and 5 respectively. Regarding Fig. 6(a), when pipeline 2 fails, the proposed entropy function reaches its maximum value if the flow rates of pipelines 1 and 5 are 25 and 30 liters per second, respectively. Based on Fig. 6(d), if the pipeline 3 fails, the water network experiences, generally, smaller entropy values except for the case that x and y as the flow rates of pipelines 1 and 5 are approximately equal to 20 and 5 liters per second, respectively. On the contrary, if pipe 5 fails, the sample WDN experiences the largest entropy when the flow rate of pipeline 1, which corresponds to x variable, be in the range of 17 to 26 liters per second (See Fig. 6(b)). It is shown in Fig. 6(a) and (c) that an increase in the inflow rates in pipelines 1 and 5 (i.e. an increase in x and y variable) leads to an increase in entropy value, when pipeline 2 or 4 fails respectively. However, if pipe 3 fails (Fig. 6(d)) entropy value of WDN does not experience its peak value except for the case that the flow rate in pipe 1(x variable) equals 20 liters per second. To study the effects of concurrent failure of pipelines on the reliability of sample WDN, in this section, different flow patterns when two pipes fail simultaneously are considered using proposed entropy calculations. Similar to the failure of a pipeline addressed in this section, the SR values of the two failed pipelines based on specified flow patterns (See Fig. 3) are set to zero (see Eq. (12)). Different flow scenarios are considered with respect to the concurrent failure of two pipelines (See Fig. 7). Then, the entropy function based on Eq. (12) will be calculated with respect to variables x and y, which represent the flow rates of pipelines 1 and 5 in sample WDN of this section. Contour maps of calculated entropy considering different modes of simultaneous failure of two pipelines are shown in Fig. 8. As indicated, WDN experienced maximum entropy for a specific inflow and outflow rate considering the concurrent total failure of two pipes. For example, if pipelines 1 and 4 or pipelines 2 and 4 are not concurrently serviceable (See Fig. 8(a) and 8(b)), maximum entropy value happens if the flow rate of pipeline 5 exceeds 25 liters per second. Comparing Figs. 8(a) to (h), it is shown that for most failure modes considered, the entropy value reaches the largest value if flow rate in pipeline 1 or 5 (i.e. x and y variable) exceeds 20 or 25 liters per second respectively except for the case that pipelines 1 and 5 are concurrently unserviceable. Flow

Fig. 9. Schematic view of the revised WDN (the flow rate of links is L/s) (Case B).

(Case C); and finally, a real-world example of WDN is presented to show the adequacy of the proposed method of this study to estimate the reliability. 3.1. Case A: reliability of a simple WDN using the proposed methodology In this section, a simple example is considered and the entropy is evaluated by the suggested entropy function (i.e. Eq. (12)) for different patterns of pipeline failure. Fig. 3 shows a schematic view of the considered sample of WDN. The network consists of four nodes and 5 pipelines arranged in two loops. Since the shown sample WDN has more than two connections between node 1 and node 4, it has two degrees of connectivity order. The included source node (Node 1) and demand nodes (i.e. Nodes 2, 3, and 4) provide an ensemble of feasible flow patterns. Increasing the number of possible flow patterns will obviously lead to an increase in uncertainty value related to the calculation of the reliability of a WDN. The topology of the nodes was adjusted so that all possible flow patterns as shown in Fig. 4 would be feasible regarding assumed flow directions of pipeline and satisfy the minimum head of demand nodes. For the sample WDN shown in Fig. 3, if the flow rates of two pipes could be considered as two variables of x and y (for pipes 1 and 5), the flow rates of other pipes will be determined using the principle of mass conservation at each node (see Fig. 3). The possible flow patterns for a WDN are obtained by graph theory; however, they depend on the topology of WDN [13,16,20,22]. Fig. 5 shows PEM [19] for calculating the entropy function of sample WDN shown in Fig. 3. The method of entropy calculation was addressed in the methodology (See Fig. 1 and Eq. (7)). The formulation of PEM [19] is based on the fact that the entropy of the water distribution network increases by increasing the paths for a water molecule to move from a super-source to a super-sink. The diagram of PEM for the studied network and its entropy calculation for different flow patterns as shown in Fig. 4 are presented in Fig. 5. Based on Fig. 5, the entropy value for different flow configurations a, b, c, and d (See Fig. 4) are 1.7155, 1.4452, 1.666, and 1.050 respectively. Larger entropy function values, which are provided by a flow configuration in the studied WDN, represent the greater reliability and resilience of a WDN; thus, flow scenario denoted as “a” (See Fig. 4) is capable of providing the largest reliability among flow scenarios.

Fig. 10. Possible flow pattern for the revised WDN shown in Fig. 9 (Case B). 8

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Fig. 11. The Path Entropy Method (PEM) diagram of the revised water distribution network for different flow patterns (the flow rate of links is L/s) (Case B).

3.2. Case B: effects of flow pattern on reliability

patterns are affected by many parameters such as pipeline diameters and topographic characteristics of nodes in real-world WDNs that are explicitly considered in the case study D of this research (see Section 3.4).

To consider the effect of network configuration on studied entropy function, the pipes configuration of the studied WDN (See Fig. 3) was revised by changing the position of pipeline 5 to connect nodes 2 and 3 (see Fig. 9). As it is shown in Fig. 9, the revised WDN still consists of 9

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Fig. 12. Contour map of the proposed entropy function for the sample WDN (Case b), as shown in Fig. 9, assuming that (a) Pipeline 1, (b) pipeline 2, (c) pipeline 3 and (d) pipeline 4 are unserviceable, respectively (Case B). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

largest value when the flow direction of pipeline 5 is changed compared to the assumed flow pattern as shown in Fig. 9 and its flow rate goes beyond 13 liters per second. Based on Fig. 12(b), if pipeline 2 is unserviceable, the flow rate of pipeline 5 should surpass 3 liters per second so that the entropy value of revised WDN reaches its maximum value. Comparing Fig. 12(a) and (d), it is indicated that the failure of pipelines 3 and 4 leads to, on average, larger entropy value considering the studied range of x and y compared to the range of entropy value which corresponds to the failure of pipelines 1 and 2. Considering Fig. 12(a)–(d), it is found that if pipeline 1 is unserviceable, the flow direction of pipeline 5 should be changed compared to assumed direction shown in Fig. 9 to obtain the largest entropy value; whereas, if pipeline 2 is unserviceable, the maximum entropy happens considering the assumed flow direction of pipeline 5 shown in Fig. 9. However, both the flow directions in pipeline 5 can lead to the largest entropy values if pipeline 3 or 4 is unserviceable. Fig. 13 also shows the entropy values of WDN when one of the pipelines is 50% serviceable. Comparing Figs. 12 and 13, shows that the entropy value increases when some pipes are partially serviceable compared to the corresponding value for the case that those pipelines are unserviceable. Similar to unserviceable conditions of pipelines (See

two loops including one inflow and three outflows while the diagonal pipeline is perpendicular to inflow and outflow directions. Different possible flow patterns for the revised WDN are shown in Fig. 10. A similar topology of WDN nodes as case A is considered so that all possible flow patterns as shown in Fig 10 would be possible and feasible practically (Fig. 10). The PEM diagram and entropy value for the revised WDN are summarized in Fig. 11. According to this figure, the maximum entropy value corresponds to the pattern “a”, whereas flow configuration “d” provides minimum entropy in this ensemble. By comparing Figs. 5 and 11, it can be realized that changing a path between supply nodes of WDN changes the entropy values. In order to evaluate the failure probability of the studied revised WDN, in this section, the proposed entropy function is examined by different flow patterns when some pipelines completely fail (SR=0). Fig. 12 shows the calculated entropy values for the cases that a pipeline fails in the revised WDN. Each contour plot line of Fig. 12 represents an entropy value for the specified flow pattern (See Fig. 9) and also four areas (See Fig. 12(a) and (d)) that are painted dark blue represent the hydraulic behavior of each water distribution sub-network. Based on this figure, x and y are the flow rates of pipelines 1 and 5 respectively. In the case of failure of pipeline 1 (See Fig. 12(a)), the entropy has the 10

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Fig. 13. Contour map of the proposed entropy function for the sample WDN, as shown in Fig. 9, assuming that (a) Pipeline 1, (b) pipeline 2, (c) pipeline 3 and (d) pipeline 4 are 50% serviceable, respectively (Case B). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 12), when the pipeline 1 is 50% serviceable, the flow direction of pipeline 5 should be changed compared to the assumed flow direction of pipeline 5 as shown in Fig. 9 so that the proposed entropy function reaches its peak value. While the maximum entropy value of revised WDN is provided for both opposite flow directions of pipeline 5 when one of the pipelines 2, 3 or 4 is unserviceable.

and demand nodes for the studied network [10]. In this section, a new WDN having 17 pipes and 12 demand nodes has been taken into account (see Fig. 14). In order to include the effects of different flow patterns on the calculated reliability values, six different flow configurations are considered using different pipeline diameters; moreover, it is assumed that some pipes have failed due to improper maintenance of pipeline before earthquake event (See Fig. 15 and Table 1). Fig. 15 shows possible different flow patterns for the aforementioned WDN in which the pipes 4, 5, 6 and 10 have failed. Moreover, Table 1 summarizes the utilized pipe diameter and flow rates for considered flow patterns. The diameter of pipes in this WDN varies in the range of 100 and 600 mm to provide actual conditions in practice. All nodes are assumed to be in the same height level, while the lengths of all pipelines are assumed to be 100 cm. Moreover, the minimum demand of 30 liters per second is considered for all nodes. Very smooth pipes are used here, so the Hazen–Williams coefficient is set to 130. It is assumed that the studied WDN is located in Tehran, Iran, where is classified as one of the most high seismically active regions in the Middle East. Heidari et al. [39] proposed the following empirical equation to predict PGV with respect to vertical displacement amplitude (Pd) of the first three seconds

3.3. Case C: correlation assessment between the proposed entropy function and reliability value In order to examine the validity of the proposed method to be used as a reliability index, a more realistic case of a pipeline network is presented and evaluated by the proposed entropy function. Moreover, the outputs are compared to the entropy index and reliability index defined by Tanyimboh and Templeman [5] and Tanyimboh and Sheahan [16]. Based on Awumah et al. [10], large varieties of flow configurations can be considered for assessing the reliability of studied WDN in this section (See Fig. 14), but six configurations were selected regarding logical numbers of loops and limited number of specified sources as well as predetermined assumptions about the source nodes 11

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the studied WDN is not a real-world case study and is just designed to show the applicability and efficiency of the proposed methodology, the proposed entropy method was calculated for different PGV values including 0, 25, 60 and 80 cm/s. These values are for sure in the abovementioned suggested range for Tehran. The developed fragility curves of utilized pipelines (see Fig. 16), which were presented by Iranian Water Resources Management Company [45], were employed to obtain the DR which is required in calculations of the proposed entropy function (See Eqs. (12) and 15). Since a limited number of pipeline specifications are presented in Fig. 15, DR values at intermediate diameters were estimated by linear interpolation. DR of studied WDN is then used for calculating the SRseismic via Eq. (15). Finally, Eq. (12) is utilized to calculate the proposed entropy values. One of the most accurate equations for estimating the reliability of the water network was proposed by Tanyimboh and Templeman [5] as follows:

R=

+

1 T

⎞ ⎛ M M P (0) T (0) + ∑m = 1 P (m) T (m) + ∑m = 1 P (m , n) T (m , n) + ⋯⎟ ⎜ ∀ n ≠ m ⎠ ⎝

1⎛ 1 2⎜

⎞ M M − P (0) − ∑m = 1 P (m) − ∑m = 1 P (m , n) + ⋯⎟ ∀ n≠m ⎝ ⎠ (18)

where R, P(0),P(m), P(m,n), T(0)، T(m),T(m,n), m and T are reliability of water distribution network, probability of inaccessibility of all pipelines, probability of inaccessibility of pipeline “m”, probability of inaccessibility of both pipelines “m” and “n”, number of pipelines of studied WDN and sum of demand nodes of studied WDN respectively. The reliability of the studied sample WDN of this section is calculated based on Eq. (18). Table 2 summarizes the reliability calculated by Eq. (18) and entropy values proposed by this study for the third studied sample WDN and different possible flow patterns (See Fig. 14) using the fragility curves of pipelines as shown in Fig. 16. The third row of Table 2, presents the proposed entropy value in the case that all pipes

Fig. 14. Schematic view of sample WDN (the flow rate of links is liter/second) (Case C).

of “P” wave of earthquake for Tehran city [39]:

PGV = 0.937 × LogPd + 1.638

(17)

where PGV value varies in the range of 0 to 100 cm/s [39]. Similarly, some studies focus on presenting a promising relation between PGV and various magnitude of earthquake [40-44] for some seismic zones. Since

Fig. 15. Studied flow patterns for the WDN as shown in Fig. 13 (Case C). 12

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Table 1 Pipeline diameter and flow rate at different flow rates and different flow patterns as shown in Fig. 15. Pipe number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Pipe diameter (mm) a b

c

d

e

f

Flow rate (liter/sec) a b

c

d

e

f

302 192 226 175 179 178 119 135 361 228 138 275 239 182 169 184 162

325 227 232 0.1 0.1 194 139 147 337 231 190 234 293 149 185 178 149

251 232 302 182 0.1 0.1 163 148 390 0.1 193 244 223 233 199 163 146

326 265 186 0.1 161 0.1 130 180 336 185 237 270 221 212 177 213 100

284 268 225 0.1 184 0.1 143 176 368 0.1 240 286 188 215 184 200 105

175.6 61.6 87.6 42.9 44.2 37.5 12.0 10.8 268.9 86.3 19.9 139.5 89.3 35.0 39.8 40.5 17.0

210.4 90.6 92.1 0.0 0.0 48.4 17.0 13.7 234.1 92.0 48.9 100.3 142.4 21.1 44.8 38.5 14.1

117.5 89.7 174.4 49.2 0.0 0.0 27.6 13.9 327.0 0.0 48.0 110.9 83.5 69.4 55.4 28.0 13.9

212.9 129.4 56.3 0.0 35.2 0.0 15.0 23.4 231.6 55.7 87.7 133.5 70.4 59.9 42.8 50.1 4.4

158.0 130.2 90.3 0.0 51.0 0.0 20.2 22.6 286.5 0.0 88.5 154.4 48.6 60.7 48.0 44.1 5.2

310 206 226 209 157 0.1 123 157 354 227 160 265 209 231 172 200 139

186.2 71.6 87.5 65.3 32.8 0.0 12.9 15.9 258.3 86.9 29.9 129.0 67.4 67.4 40.7 44.6 11.9

proposed entropy function agrees well with the direct reliability method (Eq. (18)) and it can be utilized in the design and evaluation of WDNs in hazardous zones. It is worth noting that the proposed entropy function considers the fragility curves variations, which depend on seismic characteristics of the studied site and geotechnical properties of soil as well as geometric and mechanical specifications of pipelines. In other words, the fragility curve can considerably affect the entropy results due to changes in pipeline diameters, material properties of pipelines, soil type, and PGV as important seismic design parameters. Furthermore, the hydraulic parameters (See Table 1), which are computed using hydraulic analyses, are also utilized for calculation of the proposed entropy function (See Eq. (12)). 3.4. Case D: evaluating the serviceability of Kobe city WDN using the proposed entropy function

Fig. 16. Fragility curve of the utilized pipeline (Iran Water Resources Management Company (2015)).

In this section, the reliability of WDN of Kobe city regarding the specific serviceability is considered. Kobe city WDN is a gravity flow water distribution system, which is extended geographically in the east-west direction. Fig. 17(a) and (b) shows the WDN system of Kobe city and the studied distribution area of the WDN system respectively (See distribution area A in Fig. 17(a)). Based on Hoshyia et al. [14,15], a new transmission pipeline was added to enhance the serviceability of the studied area (see dashed line in Fig. 17(b)). Fig. 18(a) and (b) shows the simplified flow pattern model of studied water distribution area in WDN of Kobe city with and without the new transmission pipeline respectively. In this figure, nodes and links were illustrated by circles and lines, respectively, along with their respective nodes and link numbers. Table 3 presents the mechanical properties of the initial pipelines (i.e. before the addition of the new transmission pipelines) including pipeline materials, links and their connected node numbers, pipeline diameters, link lengths, flow rates (liter/Second), DR and the required demand node. The DRs listed in this table were considered as well as the corresponding value obtained by Hoshyia et al. [14,15]. In addition, SRNonsiemic of pipelines obtained considering the minimum required

are 100% serviceable because PGV=0 cm/s and SR=100 except for those pipelines which failed before seismic hazard and they are unserviceable with respect to different flow patterns as shown in Fig 15. Similarly, 2nd, 3rd and 4th rows of Table 2 present the proposed entropy value considering that all pipes are partially serviceable due to earthquake events having PGV values of 25, 60 and 80 cm/s for different flow patterns as shown in Fig 15. Based on this table, it is shown that the trend of increasing and decreasing the reliability value of studied WDN based on Eq. (18) with respect to different flow patterns (See Fig. 15) agrees well with entropy values calculated based on the proposed method of this study. In other words, by comparing the entropy values (see rows 2 to 5 in Table 2) and reliability values, for each of the two flow configurations considered (e.g. columns “a” and “b” of Table 2), increase or decrease in the reliability of two different patterns is consistent with increase or decrease of entropy values for any serviceability levels (e.g. See (PGV=60 cm/s) in the row 5th). Hence, it is concluded that the proposed method for calculating the reliability by

Table 2 Computed reliability value and proposed entropy for different flow patterns as shown in Fig. 15. Flow configuration

a

b

c

d

w

f

R (Eq. (18)) S(PGV=0 cm/s) S(PGV=25 cm/s) S(PGV=60 cm/s) s(PGV=80 cm/s)

0.999027 5.7562 5.5330 5.4049 4.1468

0.998933 5.7161 5.4930 5.3803 4.1067

0.998872 5.6587 5.4356 5.3662 3.9677

0.998777 5.5772 5.3540 5.2972 3.9617

0.998555 5.5326 5.3095 5.2903 3.9607

0.998485 5.4749 5.2517 5.1249 3.9555

13

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Fig. 17. (a) WDN system of Kobe city (b)studied distribution area A of Kobe city WDN [14].

N7 and d1 to d5 respectively, were defined to simulate Kobe WDN perfectly. The total lengths of dummy links are equal to the length of the new transmission pipeline and the dummy nodes simulate the location of the interconnection of transmission pipeline and WDN. Table 4 lists the mechanical properties of the pipelines after addition of the new transmission pipelines. Based on Javanbarg and Takada [8], since the dummy lines had flexible joints, DRs of these dummy lines are set to zero (see Table 4). In addition, flow rates of dummy lines were considered to be equal to the flow rates of the new transmission pipelines. It should be noted that SR of this study includes the effects of hydraulic characteristics and seismicity of the site. Based on the above-mentioned description, the reliability values of

head of 24 m for each node located in the residential area [46]. Flow rates are calculated by hydraulic analyses using EPANET software program [47]. In this study [14], the DR values of links 30 and 3 are set to zero since the seismic flexible joints, which are capable of accommodating the large displacement for the pipeline, have been utilized for these links [14,15]. The material types of pipelines were Copper and ductile Steel. Based on Fig. 18(b), to connect the new transmission pipeline to links of studied WDN regarding engineering data of Kobe WDN and to simplify the hydraulic model of the studied distribution area, two sources were established which are designated as S1 and S2. Furthermore, some dummy links and nodes, which were designated as N1 to 14

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Fig. 18. The simplified hydraulic model of Kobe city WDN (a) before (b) after addition of new transmission pipeline (posterior network model) [14].

Table 3 Mechanical specification of studied WDN of Kobe city before addition of transition pipeline [14]. Link specification

1 9 11 10 12 4 6 5 8 7 14 13 19 2 15 17 16 18 3 20

C1 C C C S2 S S S S S S S S S S S S S S(flexible joint) S(flexible joint)

End node

Link Length (km)

Diameter (mm)

DR

Flow rate (L/s)

2 3 3 4 4 5 5 6 7 8 9 10 10 11 12 12 13 13 14 15

1.434 0.268 0.573 0.445 0.548 1.632 0.445 0.997 0.613 0.608 1.551 0.531 2.074 0.987 0.435 1.348 1.265 0.627 2.226 0.461

900 900 300 900 700 300 300 300 500 500 400 500 500 700 300 700 300 700 900 900

0.717 0.422 0.62 0.329 0.08 0.537 0.046 0.549 0.062 0.023 0.222 0.238 0.242 0.219 0.064 0.162 0.158 0.354 0 0

235.3 131.5 19.4 131.4 22.3 78 6.2 32.4 70.9 38.8 109.8 114.9 59.6 97.4 34.1 52.9 18 80.3 243.4 117.8

Node Speciation Node No Demand node (m3/day) 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2237 1678 3356 4475 2797 2237 3356 3356 5593 2237 5593 3915 3915 5034

4. Conclusions

Kobe city before and after the addition of the new transmission pipeline were calculated based on the proposed entropy function of the present study. Furthermore, the redundancy and hydraulic reliability values of studied WDN based on previous studies [8,14] are presented for comparison purposes in Table 5. Table 5 summarizes the comparison of the proposed method and the previously developed methods, i.e. redundancy [14,15] and reliability [8]. Based on this table, adding the transmission pipeline results in an increase of proposed entropy value by up to 9.4%. Comparing improvement of redundancy value by Hoshiya et al. [14,15], Hydraulic reliability value by Javanbarg and Takada [8] and the corresponding value of the proposed method, it is shown that the proposed method satisfactorily captures the improvement of reliability after adding the transmission pipeline. While improvement of entropy values in the above-mentioned methods are compared satisfactorily; the improvement of redundancy values proposed by Javanbarg and Takada [8] with respect to the addition of new transition pipeline was considerably different from the others (See Table 5). This may be due to considering only the mechanical properties of pipelines of WDN in the redundancy method [8] with greater importance level compared to the rest of the above-mentioned methods.

In this research, some modifications to the entropy method were proposed to consider the failure uncertainties of the water distribution network (WDN) using damage index. Implementing the proposed damage index in the entropy function enables the inclusion of seismic hazard effects using pipeline fragility curves and Peak Ground Velocity (PGV), and provides a novel approach to evaluate the reliability of WDN considering the site seismicity. 1 The obtained entropy function values suggest that the nodes having larger demand should be connected to sources using more paths. Based on Section 3 (case A) and considering similar WDN configuration, decreasing the connecting paths of the largest demand node by up to 0.33% results in a reduction of the entropy value by up to 38%. However, using the proposed methodology (e.g. See Figs. 7 and 8), the above-mentioned concept is valid when all the paths have similar probabilities of failure. 2 Based on the results of Section 3 (Case A and B), it is clear that the paths with lower probabilities of failure generally have larger probabilities of passing the water molecule with respect to their 15

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Based on the presented example addressed in Section 3 (See Figs. 14 and 15) and similar PGV value (See Table 2), configuration “a” which has the largest number of serviceable pipelines, provides both the largest value of entropy and reliability. 4 Based on the evidence of case C (See Table 2), it is indicated that increasing the PGV values up to 80 cm/s leads to a decrease of entropy values, on average, up to 28% for all studied flow patterns (See patterns a to f in Fig. 15). This shows that fragility curves, which are calculated based on seismic characteristics of the studied site and geotechnical properties of soil as well as geometric and mechanical specifications of pipelines, can considerably affect the proposed entropy function. 5 Considering Case D (See Figs. 17 to 18), it is illustrated that for WDN of Kobe city adding a new pipeline between some nodes of WDN to provide backup facility for the downtown area of the city leads to increase in redundancy index and proposed entropy value up to 6.2% and 9.4% respectively. This confirms that the increasing trend of entropy value proposed by the present study is satisfactorily compared with the other corresponding values. 6 Based on the above conclusions, the most important advantage of the proposed method of this study is the simple calculation of the reliability by entropy function considering the effect of Damage Rate (DR) with respect to PGV, which correlates well with probabilistic hazard assessment for a seismic zone.

Table 4 Mechanical specification of studied WDN of Kobe city after addition of transmission pipeline [14]. Link specification Link No Material type

1-d1 N2 1-d2 N3 1-d3 N4 1 9 11 10 12 4 6 5 N1 8 7 N5 14 13 19 2 15 17 16 18 3 20 10-d4 N6 N7 3-d5

C1 S (seismic joint) S2 S(flexible joint) S(flexible joint) S(flexible joint) C C C C S S S S S(flexible joint) S S S(flexible joint) C C C S S S S S S (flexible joint) S (flexible joint) C S (flexible joint) S (flexible joint) C

End node

Link Length (km)

Diameter (mm)

DR

Flow rate (L/ s)

d1 d1

1.234 0.625

900 500

0.663 0

235.3 235.3

d2 d2

0.537 0.26

700 500

0.125 0

97.4 97.4

d3

0.47

900

0

243.4

d3

0.155

500

0

243.4

2 3 3 4 4 5 5 6 6

0.2 0.268 0.573 0.22 0.548 1.632 0.445 0.997 0.13

900 900 300 900 700 300 300 300 300

0.161 0.422 0.62 0.179 0.08 0.537 0.046 0.549 0

235.3 131.5 19.4 131.4 22.3 78 6.2 32.4 32.4

7 8 8

0.613 0.608 0.604

500 500 500

0.023 0.023 0

70.9 38.8 38.8

9 10 10 11 12 12 13 13 14

0.671 0.531 2.074 0.45 0.435 1.348 1.265 0.627 1.756

400 500 500 700 300 700 300 700 900

0.103 0.231 0.242 0.106 0.064 0.162 0.158 0.354 0

109.8 114.9 59.6 54.3 34.1 52.9 18 80.3 132

15

0.461

900

0

117.8

d4 d4

0.88 0.732

400 500

0.133 0

109.8 109.8

d5

1.167

700

0

131.4

d5

0.225

900

0.182

131.4

Some recommendations for the future investigations of the current study include: (1) Fuzzy method can be used to consider the possibility of pipeline failure in the methodology of the present research; (2) A methodology can be developed for optimizing the reliability of WDN based on the proposed Entropy function; (3) The proposed Entropy function can be implemented in risk management with respect to quantitative terms of operating practice of WDN with inclusion of internal, external, and environmental factors of water supply functioning and other hazardous conditions like pressure fluctuations, pressure loss [48] due to improper pump selection and earlier pipe reconstruction cables; (4) The proposed methodology can be improved by considering the practical aspects of WDN design procedure which were not included in the present research such as pipeline aging, and connection type, and environmental factors of water supply functioning; (5) The proposed entropy function can be used for decision-making in developing WDN along with cost and benefit analysis considering urban development results in reconstruction or renovation of WDN [48].

demand nodes. This concept will be valid if the required minimum head of each demand node is satisfied. Based on examples presented in Section 3 (See Figs. 7,8,12,13), depending on which pipeline is unserviceable, the optimum flow rate of each pattern would be different from the corresponding values of previous entropy function [5]. In other words, the configuration of the water distributions network strongly affects entropy function value considering the probability of failure. 3 By comparing the improved entropy function of this research and reliability measure suggested by Tanyimboh and Templeman [5] (See Case C), it is shown that the proposed entropy can accurately estimate both the trend of reliability changes and the flow patterns.

CRediT authorship contribution statement Hossein Emamjomeh: Methodology, Software, Resources. Roohollah Ahmady Jazany: Methodology, Writing - original draft. Hossein Kayhani: Methodology, Software. Iman Hajirasouliha: Methodology, Writing - review & editing, Supervision. Mohammad Reza Bazargan-Lari: Supervision.

Declaration of Competing Interest The authors of this research paper claim no conflict of interest.

Table 5 Comparison of redundancy, reliability proposed by previous studied [8,14] and the proposed entropy. Studied Methods

Redundancy (Hoshyia et al. [14]) Redundancy (Javanbarg and Takada [8]) ) Hydraulic reliability (Javanbarg and Takada [8]) Proposed entropy function

New transmission pipe Before addition

After addition

0.563 0.569 0.848 5.292

0.598 0.965 0.98 5.799

16

Improvement (%)

6.2% 70% 15.5% 9.4%

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Supplementary materials [24]

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ress.2020.106828.

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