1

Water Distribution System Reliability under a Fire Flow Condition:a Probabilistic Approach

Julio Gomes1 and Bryan W. Karney2

1 Ph.D Candidate, IPH/UFRGS, CNPq-Brazil Scholarship. Professor, Department of Civil Engineering, Centro Universitário Positivo (UnicenP). Rua Prof. Viriato Parigot de Souza, 5300; Curitiba, 81280-330, Paraná, Brazil. PH (55)(41) 317-3256; FAX (55)(41) 317-3030; email: jgomes@unicenp.br2 Ph.D, Professor, Department of Civil Engineering, University of Toronto. 35 St. George Street; Toronto, M5S 1A4, Ontario, Canada. PH (416) 978-7776; FAX (416) 978-3674; email: karney@civ.utoronto.ca

Abstract

In the technical literature concerned with the design of water distribution systems (WDS), the term ‘fire flow’ is often associated with three distinct challenges: to estimate the actual fire flow needed to fight a fire; to estimate the availability of fire flow in a specific WDS; or, to evaluate the potential impacts of fire flow on WDS reliability. The first two challenges are generally addressed from a deterministic perspective, and, although some probabilistic approaches exist for solving the latter one, they seldom address fire flows explicitly. In general, fire flows are considered as part of a unique nodal flow probability distribution for reliability analysis purposes. Yet one significant question remains consistently unanswered: what is the WDS reliability given that a fire situation occurs? The objective of this study is to present a methodology which can be used to evaluate the WDS reliability under a fire condition. To this end, two different probability distributions are assessed for the nodal flows: one representing the base demands and the other representing the fire flow demands. A Monte Carlo method is used to generate the flows, and steady-state hydraulic simulations are performed to derive both nodal and system reliability indexes for a WDS under a fire condition.

Introduction

Many communities base the design of water distribution mains on providing adequate water for fire protection. Such design practice is based on the fact that fire flow requirements sometimes exceed the normal domestic, industrial, and other demands imposed on the water system (Boulos et al., 1997). The effects of fire demands are difficult to derive precisely since fires occur with random frequency and intensity in different areas, with each area having unique fire protection requirements.

In the technical literature concerned with the design of water distribution systems, the term ‘fire flow’ is often associated with three distinct challenges: to estimate the actual fire flow needed to fight a fire; to estimate the availability of fire flow in a specific water distribution system (WDS); or, to evaluate the potential

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impacts of fire flow on WDS reliability. The first two challenges are generally addressed from a deterministic perspective, and, although some probabilistic approaches exist for dealing with the latter one, they seldom address fire flows explicitly. In general, fire flows are considered as part of a unique nodal flow probability distribution for reliability analysis purposes. Yet, one significant question is consistently unanswered: what is the WDS reliability given that a fire situation occurs? If fire flows are considered more explicitly, the WDS reliability under a fire condition can be evaluated more precisely.

The objective of the study is to present a methodology which can be used to evaluate the WDS reliability under a fire condition. The methodology assumes two different probability distributions for the nodal flows, one representing the base demands and the other one representing the fire flows. Using a Monte Carlo approach, both base and fire demands are generated, and a fire flow condition is assumed at a specific node j. In sequence, a “conditional” reliability index is calculated for each node given the fire condition at that specific node j. The criterion for estimating the “conditional” reliability index is based on whether the estimated nodal pressures are inferior to a specified minimum nodal pressure. The nodal pressures are calculated from steady-state hydraulic simulations using routines from the EPANET programmer’s toolkit. The procedure is repeated by imposing a fire flow condition at each demand node (1 ≤ j ≤ number of demand nodes). Finally, both the nodal and system reliability indexes under a fire condition are evaluated from the nodal “conditional” reliability indexes.

Literature Review

There is currently no universally accepted definition or measure of WDS reliability. In general, reliability is informally understood as the probability that a system performs its mission within specified limits for a given period of time in a specified environment (Lansey et al., 1989); exactly how this is measured varies.

According to Goulter and Bouchart (1990), the reliability of a water distribution network is also defined by the probability of the network failing to perform, either through failure of one or more of its components (e.g., in pipes or pumps), or through the demand on the system being greater than that assumed for design purposes. Therefore, WDS reliability is concerned with two types of failure, mechanical and hydraulic. Mechanical failure encompasses such events as pipe breaks, power outages, pump and control valve failure, etc.; hydraulic failure involves cases when the delivered flow and pressure head are inadequate at one or more demand points (Bao and Mays, 1990).

In order to ensure reliable delivery of potable water to the user, the components of the water utility must be suited to accommodate a range of expected loading conditions, such as maximum day, maximum hour, average day, average hour, or maximum instantaneous demand. Fire flow demands may also be superimposed on these “normal” demand events (Cullinane et al., 1989).

Goulter et al. (1999) emphasize the importance of looking at emergency loading conditions when they affirm that to ensure delivery of treated water to the user, the WDS must be designed to accommodate a range of expected emergency loading

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conditions. These emergency conditions have been classified by Goulter et al., (1999) as: (1) fire demands; (2) broken links; (3) pump failures; (4) power outages; (5) control valve failures; and (6) insufficient storage capability.

In evaluating a system, it is common to see how the system performs under the following scenarios (Ysusi, 1999): peak-hour or maximum day demand plus fire flow. Evaluating the system at peak-hour demand gives the designer a picture of system-wide performance. On the other hand, placing fire flows at different locations in the system with a background demand equivalent to the maximum day demand will highlight isolated system deficiencies. Obviously, it is theoretically possible for a fire to occur during peak-hour demand, but since this simultaneous occurrence is relatively unlikely, this is not usually considered an appropriate design criterion for the system (Ysusi, 1999).

According to Bao and Mays (1990), fire flows are rare events in which the demand may be one to two orders of magnitude greater than average demand. Such flows do no fit the normal demand distribution well. One option to circumvent the shortcoming of the distribution is to include another random variable for fire flow at each low probability of occurrence (Bao and Mays, 1990).

In short, fire flows are mostly addressed in two different ways for reliability analysis purposes. Firstly, fire flows are used as a check on network design. The procedure is usually to superimpose the fire flows to a specific background flow regime, such as maximum daily demand. In this case, fire flows are not used to derive a reliability index. Alternatively, fire flows are considered as part of a unique flow probability distribution used to derive a reliability index. However, the network behavior under the fire flow condition is not directly addressed since both base and fire demands are represented by a unique probability distribution.

Two issues have been identified as being particularly problematic in reliability assessments. These are what measure is most appropriate for reliability assessment and what is an acceptable level of reliability (Xu and Goulter, 1998). According to Lansey et al. (1989), the real issue of WDS reliability concerns the ability of the system to supply the demands at nodes or demand points within the system at minimum required pressures. From this consideration, a nodal reliability Ri for a node i can be derived from:

)(1 miii HHPR <−=

where )( mii HHP < is the probability that the pressure head at node i (Hi) is less than

a specified pressure head limit ( miH ).

Bao and Mays (1990) offer a broader interpretation, defining the nodal reliability as the probability that a given node receives sufficient water under adequate pressure. Theoretically, the authors recognize that nodal reliability is a joint probability of flow rates and pressure heads being satisfied at given nodes. However, as it is difficult to derive mathematically and compute this joint probability, the usual approach is to compute the conditional probability in terms of pressure heads, provided that the water demand has been satisfied. Gupta and Bhave (1994) present a method termed “node flow analysis” that determines the available nodal flows under

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deficient conditions by considering the nodal demands and pressure heads simultaneously when determining WDS reliability.

Although the nodal reliabilities depict a fairly complete measure of the WDS reliability, it is also convenient to use a single index such as overall system reliability to represent the composite effect of the nodal reliabilities (Bao and Mays, 1990).

Bao and Mays (1990) suggest three heuristic definitions of system reliability (Rs). Firstly, system reliability (Rsm) is defined as the minimum nodal reliability in the system. Secondly, it is defined as the arithmetic mean of all nodal reliabilities (Rsa). Finally, the third approach defines the system reliability (Rsw) as a weighted mean of all nodal reliabilities with the weighting factors represented by the water supply at each node. Xu and Goulter (1998) also follow a heuristic approach to define system reliability (Rs), but they propose a generic expression for calculating Rs:

∑=

=n

iiis RwR

1

where wi = weighting factor defined by the network analysts for node i. Xu and Goulter (1998) advocate adjusting these weighting factors to reflect particular circumstances of each network or the particular interpretation of reliability being used.

Methodology

The conceptual model developed here to evaluate the water system reliability under a fire flow condition tries to address the stochastic nature of demand. It assumes that both nodal base and fire flows are uncertain but drawn from a known probability distribution. Unlike previous studies, the uncertainty about base and fire flows is represented by distinct probability distributions. Therefore, two different probability distributions are used to represent those flow categories: one to represent the base demands, and the other for fire demands. At the node under fire condition, the fire flow is added to the base demand and the effect of that condition on the entire system is evaluated.

Among the different nodes, the nodal base demands are correlated while the nodal fire flow demands are assumed to be independent. Moreover, there is no correlation between the base flow and the fire flow at a specific node. Both the nodal base and fire flows follow the same type of probability distribution, and the distribution can be normal or lognormal. Finally, a fire flow condition occurs at only one specific node at each time.

The nodal reliability index is estimated from the concept of nodal “conditional” reliability index and can be understood as the nodal reliability given that a fire flow condition is imposed on a specific node j. Therefore, each node presents more than one reliability index, and it is a function of not only the node itself but also the node where the fire flow condition is assumed. In other words, the “conditional” nodal reliability index has two dimensions: the node itself; and the node where there is a fire flow condition. Figure 1 presents the basic procedure used to

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generate the nodal base flows. The same procedure is used to generate the nodal fire flows, but no correlation is assumed.

IndependentUniformVariables

(U)

IndependentNormalDeviates

(Z)

CorrelatedNormalDeviates

(Y)

CorrelatedNormal or Log-normal

Variables(X)

Figure 1 Generic procedure used to generate flow values.

The routines used to generate the independent uniform distributed variable (U) are available in the Software Compaq Visual Fortran (Standard Edition – version 6.1.0). The algorithm used was the "Prime Modulus Multiplicative Linear Congruential Generator", a modified version of the random number generator by Park and Miller (1988). The sampling procedure adopted was the Latin Hypercube Sampling (LHS) as presented by McKay et al., (1979) and discussed in Gentle (1998).

The second step consists in transforming the independent uniform variables (U) into independent normal deviates (Z). The procedure used herein was the inverse function technique which is simple to summarize. Given UZFZZ ==≤ *)(*]Pr[

then )()(* 1 UFUGZ −== . For the present study, a numerical approximation for the normal distribution inverse function as presented in Zelen and Severo (1964) was used.

The procedure employed herein to transform the independent normal deviates (Z) into correlated normal deviates (Y) is presented in Ang and Tang (1984). Finally, the normal or lognormal variables (X) were calculated from the correlated normal deviates (Y) using the basic definition of a normal deviate. For lognormal variables, the relationships between the statistical parameters of the variables X and ln X as presented in Benjamin and Cornell (1970) were used.

The steady-state hydraulic simulations were performed using some routines provided by the EPANET Programmer's Toolkit, which is a dynamic link library (DLL) of functions that allows developers to customize EPANET's computational engine for specific needs. Furthermore, it is public domain software that can be downloaded from USEPA (2004).

Some discussion about both nodal and system reliability indexes was previously outlined. Basically, those nodal reliability indexes do not distinguish abnormal from normal operation once they are general indexes. In the present study, the main focus is to derive a reliability index that addresses directly the network reliability under an abnormal condition (fire flow). Therefore, the fire is assumed as a certain event.

It seems logical to expect that the nodal reliability index is a function of the node j where the fire flow is assumed. In addition, the nodal reliability index should also include the “relative” frequency of a fire flow condition at node j. One can understand such a “relative” frequency as the probability of a fire flow at a node jgiven that there is a fire flow condition. Considering that nodal pressures below a

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minimum limit represent a failure event, the nodal reliability index (Ri) under a fire flow condition can be defined as follows:

∑=

<−=nodesn

jjNjiii nodefirenodefirePPR

_

1

)(Pr)]./limPr(1[ (1)

where: Ri = reliability index for node i; Pi = pressure at node i; P limi = pressure limit for node i; PRN(fire nodej) = “relative” frequency of a fire condition at node j, estimated by Equation 2.

∑=

=nodesn

jjjjN nodefirenodefirenodefire

_

1

)Pr()Pr()(Pr (2)

The nodal “conditional” reliability index, expressed as (1 - Pr[Pi > P limi / fire nodej]) in Eq. (1), represents the probability that the pressure at node i is greater than or equal to a minimum pressure limit given a fire condition is imposed on node j.

As stated, Eq. (2) raises the following question: Given that there is a fire flow condition, what is the probability of that condition happening at node j? The “relative” frequency could be then estimated from observed data and from practitioners’ experience, and it should consider various characteristics such as: land use, population density, construction type, etc. Finally, system reliability (Rs) can be estimated using the same heuristic approaches suggested by Bao and Mays (1990) and discussed previously.

Figure 2 sketches the procedure to derive both the nodal and system reliability indexes from those nodal “conditional” reliability indexes. Such a procedure also permits the derivation of an empirical conditional probability distribution for each node i of the network given a fire condition at node j as well as assigning different pressure limits for each node depending on whether or not a given node experiences a fire flow condition.

Case Study

The network used as a case study was extracted from Xu and Goulter (1998). Figure 3 presents the network topology, and Tables 1 and 2 provide the associated data. Boththe expected nodal base and fire flows were assumed equal to the expected demands presented in Table 1, and the coefficient of variation (CoV), defined as the ratio between standard deviation and mean, was assumed equal to 10% for all nodal flow distributions. The probability distributions were all assumed to be lognormal. Finally, both the correlation between nodal base flows and the correlation between nodal fire flows were assumed to be zero. This assumption of independent demands, while not strictly realistic, is often used in network reliability analysis. A total of 5,000 simulations were performed for a fire condition imposed on a specific node j, and Table 3 presents the resulting nodal “conditional” reliability indexes.

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Start

for j = 1 to thenumber of demand nodes

for i = 1 to thenumber of simulations

Set nodal base flows

Add fire flowat the node j

Performhydraulic simulation

Compare nodal pressures to nodal pressure limits

Calculatethe nodal “conditional”

reliability indexes

Calculate both the nodaland the system reliability

indexes

End

Figure 2 Generic representation of the simulation procedure.

Legend:

Source Demand

J - junction P - pipe

P1

P3 P8 P6

P11

P7

P2

P12

P4

P9

P5

P10R1 J1 J2

J3 J4

J5 J6

J7

J8

R2

J9 J10

J11 J12

J13

P13 P14P15

P16

P17

P18

P19

P20 P21

Figure 3 Network physical representation (adapted from Xu and Goulter, 1998)

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Table 1 Network node information

NodeExpected demand

(m3/s)Ground

level(m)

MinimumPressure head

(m)J1 0.0 27.4 15J2 0.042 23.5 15 J3 0.042 19.0 15J4 0.127 22.0 15J5 0.042 20.5 15J6 0.135 21.4 15J7 0.127 19.6 15J8 0.064 21.4 15J9 0.021 22.6 15

J10 0.0 24.1 15J11 0.021 25.1 15J12 0.021 26.6 15J13 0.0 23.5 15R1 0.0 61.0 0R2 0.0 61.0 0

Source: Xu and Goulter (1998)

Table 2 Network link information.

LinkStartNode

EndNode

Length(m)

Diameter(mm)

HWCoefficient

P1 R1 J1 612 508 120P2 J1 J2 240 457 110P3 J2 J3 1524 406 110P4 J3 J4 1128 305 100P5 J4 J5 1189 381 100P6 J5 R2 640 406 110P7 J6 J4 762 254 100P8 J7 J3 945 254 100P9 J2 J7 1676 381 95

P10 J7 J6 1128 305 100P11 J5 J6 884 305 95P12 J5 J13 1372 381 100P13 J6 J13 762 254 90P14 J8 J7 823 254 90P15 J2 J10 945 305 100P16 J9 J10 579 305 95P17 J10 J11 488 203 100P18 J12 J11 457 152 110P19 J12 J9 503 203 100P20 J8 J9 884 203 100P21 J13 J8 945 305 95

Source: Xu and Goulter (1998).

The results presented in the Table 3 suggest that node 12 is the most problematic in terms of reliability under a fire flow condition and they also suggest that a fire condition at nodes 6 and 7 (highest demand values) seems to be most critical. Overall, the nodal “conditional” reliability indexes give valuable information about the network performance under a fire condition.

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Table 3 Nodal conditional reliability indexes. “Base” case: (CoV = 10% and ρ = 0).Fire at Node

2 3 4 5 6 7 8 9 11 12Node 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Node 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Node 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Node 4 1.0000 1.0000 0.8634 1.0000 0.9982 1.0000 1.0000 1.0000 1.0000 1.0000Node 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Node 6 1.0000 1.0000 0.9930 1.0000 0.5304 0.9876 1.0000 1.0000 1.0000 1.0000Node 7 1.0000 1.0000 1.0000 1.0000 0.9998 0.9910 1.0000 1.0000 1.0000 1.0000Node 8 1.0000 1.0000 0.9998 1.0000 0.9980 0.9876 0.9972 1.0000 1.0000 1.0000Node 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Node 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Node 11 1.0000 1.0000 0.9986 1.0000 0.9944 0.9834 0.9988 1.0000 0.9476 0.9968Node12 0.9998 0.9996 0.9346 0.9998 0.8210 0.6844 0.9304 0.9928 0.8330 0.5030Node 13 1.0000 1.0000 1.0000 1.0000 0.9990 0.9998 1.0000 1.0000 1.0000 1.0000

The next step was to estimate the nodal reliability indexes. It was assumed that the probability of occurrence of a fire condition is the same for all demand nodes, which implies PrN(fire nodej) = 1/10. Table 4 presents the nodal reliability indexes estimated from Eq. (1) as well as the system reliability index estimated from the heuristic approaches previously discussed. It is obvious that a more precise evaluation of the PrN values is required for real cases. This becomes evident when one compares the minimum nodal “conditional” reliability indexes (Table 3) to the “final” nodal reliability indexes (Table 4).

In relation to the system reliability indexes, the index from the minimum value criterion was significantly lower than those obtained from the others. Actually, the smaller the nodal reliability index range, the smaller the difference between the minimum value criterion and the others. In general, as systems become less reliable, there is a tendency for these measures to converge to more similar values.

Table 4 Nodal and System reliability indexes.Nodal Reliability System Reliability

Node Reliability index

Node Reliability index

HeuristicApproach

ReliabilityIndex

1 1.0000 8 0.99832 1.0000 9 1.0000

Minimum value(Equation 2.8)

0.8698

3 1.0000 10 1.00004 0.9862 11 0.9920

Average value(Equation 2.9)

0.9843

5 1.0000 12 0.86986 0.9511 13 0.9999

Flow-weighed value(Equation 2.10)

0.9821

7 0.9991

Sensitivity Analysis

A sensitivity analysis was performed to evaluate the influence of both flow coefficient of variation (CoV) and correlation (ρ) between the nodal base flows on the reliability indexes. Again, the reliability indexes were calculated using a set of 5,000 simulations for a fire condition imposed on a specific node j

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Flow coefficient of variation (CoV)

The flow coefficient of variation (CoV) was sequentially assumed equal to 10%, 20%, 30%, 40% and 50%, and the correlation (ρ) between the nodal base flows was assumed equal to 0. Table 5 shows the influence of the CoV values on the reliabilities indexes. The results showed that both nodal reliability and system reliability decreased as CoV increased. This seems to be an obvious conclusion; however, if the pressure limits are set too high, which means low reliability indexes, the results exhibited the opposite tendency (i.e., the system reliability increases as CoV increases).

Table 5 Sensitivity Analysis. Parameter: flow coefficient of variation (CoV)CoV

10% 20% 30% 40% 50%Node 1 1.0000 1.0000 1.0000 1.0000 1.0000Node 2 1.0000 1.0000 1.0000 1.0000 0.9993Node 3 1.0000 1.0000 0.9994 0.9950 0.9831Node 4 0.9862 0.9603 0.9300 0.8950 0.8618Node 5 1.0000 1.0000 0.9994 0.9957 0.9851Node 6 0.9511 0.9279 0.8927 0.8529 0.8208Node 7 0.9991 0.9834 0.9587 0.9261 0.8940Node 8 0.9983 0.9666 0.9254 0.8811 0.8436Node 9 1.0000 0.9981 0.9812 0.9474 0.9109

Node 10 1.0000 0.9996 0.9923 0.9711 0.9413Node 11 0.9920 0.9319 0.8635 0.8109 0.7749Node 12 0.8698 0.7811 0.7210 0.6838 0.6602

NodalReliability

Node 13 0.9999 0.9899 0.9624 0.9273 0.8936Minimum 0.8698 0.7811 0.7210 0.6838 0.6602Average 0.9843 0.9645 0.9405 0.9143 0.8899System

Reliability Weighted by nodal

flows

0.9821 0.9609 0.9337 0.9030 0.8749

Nodal base flow correlation (ρ)

The correlation (ρ) between the nodal base flows was assumed equal to: 0 (independent), 0.25, 0.50, 0.75, and 0.999 (highly correlated), and the CoV was assumed equal to 10%. Table 6 reveals the influence of the ρ-values on the reliabilities indexes. The results indicated that both nodal and system reliability decreased as ρ increased. This can be explained by the fact that a higher ρ value implies that nodes having flows of the same magnitude are more likely.

From the ρ-value sensitivity analysis, one can see that the usual hypothesis of independence between nodal base flows (ρ = 0) is not a conservative one. It is important to acknowledge that changes in ρ-values produced smaller reliability index variations than changes in CoV-values.

A worst-case scenario was performed by adopting CoV = 50% and ρ = 0.999. The results in terms of system reliability indexes were: 0.6443 (minimum value);

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0.8004 (average); and, 0.7839 (weighted by nodal flows). Therefore, the evaluation of both CoV and ρ values can be critical to reliability analysis depending on the network properties.

Table 6 Sensitivity Analysis. Parameter: Correlation between nodal base flows (ρ)ρρρρ

ρρρρ = 0 ρρρρ = 0.25 ρρρρ = 0.50 ρρρρ = 0.75 ρρρρ = 0.999Node 1 1.0000 1.0000 1.0000 1.0000 1.0000Node 2 1.0000 1.0000 1.0000 1.0000 1.0000Node 3 1.0000 1.0000 1.0000 1.0000 1.0000Node 4 0.9862 0.9800 0.9738 0.9680 0.9624Node 5 1.0000 1.0000 1.0000 1.0000 1.0000Node 6 0.9511 0.9452 0.9384 0.9323 0.9264Node 7 0.9991 0.9965 0.9929 0.9888 0.9834Node 8 0.9983 0.9893 0.9790 0.9683 0.9593Node 9 1.0000 0.9997 0.9975 0.9936 0.9877

Node 10 1.0000 1.0000 0.9994 0.9978 0.9957Node 11 0.9920 0.9615 0.9331 0.9077 0.8860Node 12 0.8698 0.8154 0.7833 0.7597 0.7416

NodalReliability

Node 13 0.9999 0.9986 0.9955 0.9910 0.9854Minimum 0.8698 0.8154 0.7833 0.7597 0.7416Average 0.9843 0.9759 0.9687 0.9621 0.9560System

Reliability Weighed by nodal flows

0.9821 0.9754 0.9690 0.9630 0.9572

Conclusions

The results from the case study illustrate that the “relative” frequencies of fire events as well as the relative nodal position in the network are important for the evaluation of system reliability. Moreover, the differences between the system reliability indexes estimated from the different criteria were due mainly to both the nodal reliability index and nodal flow ranges. Small ranges of both variables resulted in small differences among the system reliability indexes.

The sensitivity analysis revealed that the reliability indexes decreased as either CoV or ρ increased. Therefore, the usual hypothesis of independence among the nodal base flows (ρ = 0) is not a conservative one. In comparison, the reliability indexes were more sensitive to changes in CoV than in ρ.

Further research could be directed towards a better representation of the physical phenomena. The present study relies on steady-state simulations to derive the nodal pressure distributions. Demand variations during the fire duration were not addressed nor were transient effects caused by a sudden change in nodal demands due to hydrant operation. A better understanding and representation of the physical phenomena could lead to more realistic nodal pressure distributions and better criteria to define the pressure limits. Despite these limitations, the proposed methodology is conceptually simple and it can be used to derive empirical pressure distributions for each node given a fire condition at a specific node j as well as to analyze the relationship between the pressure limit criteria and the reliability indexes.

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