predicting pipe failure effects in water distribution networks

14
PREDICTING PIPE FAILURE EFFECTS IN WATER DISTRIBUTION NETWORKS By P. W. JowitP and Chengchao Xu2 ABSTRACT: A practical way of assessing the impact of various pipe failure con- ditions on water distribution networks is described. The method avoids the need to undertake a full network analysis or simulation for each particular component failure. The method assesses the vulnerability of the network to the loss of any particular pipe element, and provides a quantitative estimate of the impact on each nodal demand, and the postfailure utilization of nodal sources and pipe elements. Implementation of the method requires only knowledge of the network configu- ration and a set of typical operating conditions, such as might already be available from a routine network analysis of the intact distribution network. Detailed non- linear network analysis covering all possible failure conditions is thus avoided. In cases of networks supplied by a number of sources of differing quality, the same overall technique can also be used to determine the chemical/biochemicalquality of supply at any point in the network. The results of the method can be combined with pipe failure probabilities to provide measures of network reliability. INTRODUCTION Water project engineers need to assess and monitor the vulnerability of a water distribution network to component failures, couple this with an estimate of the likelihood of these failures, thus providing a measure of the reliability of supply. This need stems from statutory and contractual obli- gations in respect of the level of service offered to consumers/customers. An informed assessment of a network's behavior in both normal and ab- normal circumstances also has an important bearing on capital investment priorities and on the required operational response to local failures within the network. This paper describes a method that allows network vulnerability to pipe element failures to be readily assessed, and the impact on nodal consump- tions, pipe element flows, and the utilization of sources of supply to be predicted. The same general method can be used for networks supplied by a number of sources of differing chemical/biochemical quality in order to assess the quality of supply delivered to any part of the network. This latter problem has been addressed previously by others (Grayman et al. 1988; Liou and Kroon 1987; Males et al. 1985), but the connection with reliability assessment does not appear to have been explored. The present method is based on knowledge of the network configuration together with a known set of network flows (such as might have previously been obtained by network analysis). An overall indication of network vul- nerability is obtained without the need for a detailed network analysis to be undertaken for each of the considered failure conditions. This enables 1Prof.- Cir. Engrg. Systems, Civ. Engrg. Systems Res. Centre, Dept. of Cir. and Offshore Engrg., Heriot-Watt Univ. ; RicCarton, Edinburgh EH14 4AS, U.K. 2Res. Fellow, Dept. of Electrical and Electronic Engrg., Brunel Univ., Uxbridge, Middlesex, UB8 3PH, U.K.; formerly Res. Student Heriot-Watt University. Note. Discussion open until June 1, 1993. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on March 4, 1992. This paper is part of the Journal of Water Resources Planning and Management, Vol. 119, No. 1, January/February, 1993. ISSN 0733-9496/93/0001-0018/$1.00 + $.15 per page. Paper No. 3053, 18 J. Water Resour. Plann. Manage. 1993.119:18-31. Downloaded from ascelibrary.org by Florida Atlantic University on 09/08/13. Copyright ASCE. For personal use only; all rights reserved.

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P R E D I C T I N G P I P E F A I L U R E E F F E C T S IN W A T E R

D I S T R I B U T I O N N E T W O R K S

By P. W. JowitP and Chengchao Xu 2

ABSTRACT: A practical way of assessing the impact of various pipe failure con- ditions on water distribution networks is described. The method avoids the need to undertake a full network analysis or simulation for each particular component failure. The method assesses the vulnerability of the network to the loss of any particular pipe element, and provides a quantitative estimate of the impact on each nodal demand, and the postfailure utilization of nodal sources and pipe elements. Implementation of the method requires only knowledge of the network configu- ration and a set of typical operating conditions, such as might already be available from a routine network analysis of the intact distribution network. Detailed non- linear network analysis covering all possible failure conditions is thus avoided. In cases of networks supplied by a number of sources of differing quality, the same overall technique can also be used to determine the chemical/biochemical quality of supply at any point in the network. The results of the method can be combined with pipe failure probabilities to provide measures of network reliability.

INTRODUCTION

Water project engineers need to assess and moni tor the vulnerabil i ty of a water distr ibution network to component failures, couple this with an est imate of the l ikel ihood of these failures, thus providing a measure of the reliabil i ty of supply. This need stems from statutory and contractual obli- gations in respect of the level of service offered to consumers/customers. A n informed assessment of a ne twork 's behavior in both normal and ab- normal circumstances also has an impor tant bear ing on capital investment priori t ies and on the required opera t ional response to local failures within the network.

This paper describes a method that allows network vulnerabil i ty to pipe e lement failures to be readi ly assessed, and the impact on nodal consump- tions, p ipe e lement flows, and the util ization of sources of supply to be predicted. The same general method can be used for networks supplied by a number of sources of differing chemical /biochemical quality in order to assess the quality of supply del ivered to any part of the network. This la t ter p roblem has been addressed previously by others (Grayman et al. 1988; Liou and Kroon 1987; Males et al. 1985), but the connect ion with reliabili ty assessment does not appear to have been explored.

The present me thod is based on knowledge of the network configuration together with a known set of network flows (such as might have previously been obta ined by network analysis). A n overall indication of network vul- nerabil i ty is obta ined without the need for a detai led ne twork analysis to be under taken for each of the considered failure conditions. This enables

1Prof.- Cir. Engrg. Systems, Civ. Engrg. Systems Res. Centre, Dept. of Cir. and Offshore Engrg., Heriot-Watt Univ. ; RicCarton, Edinburgh EH14 4AS, U.K.

2Res. Fellow, Dept. of Electrical and Electronic Engrg., Brunel Univ., Uxbridge, Middlesex, UB8 3PH, U.K.; formerly Res. Student Heriot-Watt University.

Note. Discussion open until June 1, 1993. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on March 4, 1992. This paper is part of the Journal of Water Resources Planning and Management, Vol. 119, No. 1, January/February, 1993. �9 ISSN 0733-9496/93/0001-0018/$1.00 + $.15 per page. Paper No. 3053,

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those failure events of concern to be identified quickly, and those of less concern to be screened out.

If appropriate, a full and proper solution of the nonlinear network equa- tions can then be undertaken for the events that are predicted to be of more critical importance. The efficacy of the proposed method is demonstrated for an example network through a comparison of the predicted performance consequent upon a particular element failure, with the actual network be- havior as indicated by a full solution of the nonlinear network equations for the degraded network.

TYPES AND CONSEQUENCES OF FAILURE

A water distribution network comprises elements of transmission (pipes), storage (service/storage reservoirs), sources of delivery power (pumps, ser- vice reservoirs), and control (valves, pumps). These elements, together with the water that flows through them, are all susceptible to failure and thereby a loss of performance for the network (either in whole or in part).

In a branched (no loop) network with a single source of supply and no other service reservoir locations, the loss of any pipe element will disconnect all consumers downstream of the break, resulting in a total and immediate loss of their service. In other circumstances, failure may manifest itself in a less complete way; for example, consumers may be faced with a drop in the delivery head to levels below which full demands cannot be met. This can arise from such diverse causes as the mechanical failure of a booster pump; excessive consumer demands leading to large head losses due to friction in the network; or a revalving of the network, which might have the effect of restricting some of the usual paths of flow from source to consumer. Furthermore, the loss of performance may be progressive. For example, a break in a pipe element will first lead to uncontrolled flow from the network and in general a drop in network heads. Service will be totally lost if any shortfall between inflow (supply) to the network and total outflow (uncontrolled flow plus nodal consumptions) exhausts reserve supplies (e.g. service reservoir storage). Isolation of the burst element will allow the loss of water to be stemmed, network heads to be partially restored, and the network to be operated at a more stable but perhaps unacceptable state. Eventually, repair of the burst main will allow a return to normal operation, with adequate delivery heads and service reservoir levels within normal operating bounds. A detailed study of this type of failure event and a means of assessing the associated reliability and level of service has been described previously by Germanopoulos et al. (1986).

The operation of a water distribution network can be regarded as repeated phases of normal operation, uncontrolled failure, and controlled repair. The level of service offered to one or more groups of consumers may become less than satisfactory during the failure and/or repair phases. The overall reliability and level of service offered is a combination of the severity of these failure episodes and their likelihood. Network reliability has been the focus of considerable and diverse research activity in recent years. Much attention has been directed to the related issues of network connectivity, the likelihood of topological node isolation, and the provision of topological redundancy, as described by the work of such as Wagner et al. (1988a, 1988b), Ormsbee and Kessler (1990), Quimpo and Shamsi (1989), Goulter (1988), and Awumah et al. (1991).

The importance of determining network behavior in the failure/postfailure phase has been demonstrated by Germanopoulos et al. (1986) in work

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related to a practical study of the need to augment emergency storage requirements. Their approach combined an extended-period simulation of the network for the hours following a total failure within the network with a statistical model describing the frequency of occurrence of such failures and the random repair times. In this detailed extended-period solution of the nonlinear network equations, the full behavior of the network was tracked as it moved from normal operation, uncontrolled flow from a burst main, isolation of the main, and eventual restoration of full supplies. These detailed simulations can be undertaken for all those failure events that are likely to lead to an overall loss of service. Similar approaches, but using Monte Carlo simulation to generate a distribution of operating scenarios, have been described by Wagner et al. (1988b) and Bao and Mays (1990). Obviously, it would be useful if the critical and not-so-critical failure events could be identified at the outset.

The method outlined in the following offers a simple and practical method of identifying such potentially critical failure events without recourse to a detailed and full solution of the nonlinear network equations for each pos- sible element failure. In particular, it offers a relatively simple way of pin- pointing those failure events of concern that ought perhaps to be the focus of more detailed study. The key idea in what follows is the use of a known set of operational network flows to infer a more detailed description of the flow pattern. This flow pattern is termed the microflow distribution, and it gives the detailed composition in terms of source and destination of the flows in each nodal supply, in each pipe element and each nodal consump- tion.

MICROFLOW DISTRIBUTIONS IN SUPPLY NETWORKS

The microflow distribution is defined in terms of the following variables (see also Fig. 1). Supply node variables:

SC~ = amount of water supplied from node m and consumed at node n

Consumption node variables:

CS~ = amount of water consumed at node n and supplied by node m

(Note that CS~ = SC~.) Pipe element variables:

QSTf = amount of flow in pipe element ij supplied from node m

QC?j = amount of flow in pipe element ij consumed at node n

Sj

Qij Qij

m ~ n QSij QCij J \ Cl

m S i

FIG, 1. Typical Section of Network

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Q~" = amount of flow in element i] supplied form node m

and consumed at node n

The information contained in the microflow distribution gives the depen- dence under the given operating condition of each consumption node on any specified pipe element or source of supply, and can be used to assess the network's behavior in the event of the outage of a particular pipe element or source of supply. Furthermore, a picture of the work network can be gained from a single analysis.

In reality, or course, any outage of a pipe element will result in a rear- rangement of network heads and flows in the remaining pipe elements. But, as is shown later, the proposed method may provide a good prediction of such consequences.

CALCULATION OF MICROFLOW DISTRIBUTIONS

Example Before outlining the approach for the general case, it will be instructive to

develop the method for a small example that can readily be solved by hand. Consider the simple network shown in Fig. 2, comprising four nodes and five pipe elements. Node A acts as both a source of supply and as a nodal con- sumption. Node B is a source node and nodes C and D are demand nodes. Suppose the network properties are such that the direction and magnitude of flows at a particular operating condition are as given in Fig. 2.

By inspection, the flow in pipe elements BA, BC, and BD originate solely from the source at B. Node A receives water directly from a service reservoir at that node, and indirectly from the reservoir at node B by virtue of the flow along element BA. In turn, node A transmits water directly to the nodal consumption at A and indirectly to nodes C and D via element AC.

The contribution by each source (A and B) to the flows exiting from node A are summarized in Table 1, part (a), parameterized by the value q. Clearly, q must lie between zero and 10, but the actual value of q (and thereby the other three values in the example) will be determined by physical factors at the node. The simplest assumption, and in the absence of any contrary evidence, a reasonable one, is that inflows to the node become completely mixed. This corresponds to a value of q = 20/11. Table 1, part (b), gives the full picture.

This complete mixing assumption is entirely analogous to the gravity model assumption used in transportation engineering for trip estimations in origin-destination surveys (Wilson 1970). Furthermore, it corresponds pre- cisely to the solution that would be obtained if the problem was treated as one of statistical inference and resolved using the principle of maximum statistical entropy (Jayues 1957). Applications of the entropy measure to network flows and related issues of redundancy have also been given by Tanyimboh and Templeman (in press, 1992) and Awumah et al. (1991).

The inferred composition of the consumption at A and the flow in AC can now be used in turn to determine the composition of the consumptions (nodal outflows) at nodes C and D, and the flow in pipe element CD. The resulting composition (in terms of the source) of all pipe and nodal con- sumption flows is shown in Table 2.

Working backward from node D, a similar approach can be applied to detail the composition (in terms of the destination) of all pipe-element and source inflows. The resulting information is shown in Table 3.

And finally, the information contained in Tables 2 and 3 can be combined

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100

)/~ )

/ c / 9O

10 200

40

FIG. 2. Simple Four-Node, Five-Pipe Network

TABLE 1. Composition of Flows from Node A in Terms of Source

From source A From source B Condition (direct) (via pipe BA) Total

(1) (2) (3) (4)

(a) Generalized Solution in Terms of Parameter q

Consumption at node A I CS A = q CSSA = 20 -- q E = 20 m

Flow in pipe AC QSAc = 10 - q QS~c = 80 + q E = 90

[Total] E = 10 E = 100 Y = 110

(b) Particular Solution Under Complete Mixing Assumption T

Consumption at node A | CS A = 20/11 CSSA = 200/11 E = 20 Flow in pipe AC | QS'~c = 90/11 QS~c = 900/11 E = 90

i [Total] J. E = 10 E = 100 E = 110

TABLE 2. Composition of Flows in All Pipe Elements and to All Nodal Con- sumptions in Terms of Source

From source A From source B Total pipeflow/ Element E = 10 E = 100 nodal outflow

(1) (2) (3) (4)

Consumption at node B Flow in pipe BA Flow in pipe BC Flow in pipe BD Consumption at node A Flow in pipe AC Consumption at node C Flow in pipe CD Consumption at node D

c s ' a = o

QSAA = 0

Q & , c = o

QS'ao = o CS fl = 20/11 QSAc = 90/11 CS A = 60/11 QS~D = 30/11 CS A = 30/11

c s g = o

QS~ c = 100 QS~ c = 60 QSg o = 40 CSff~ = 200/11 QS~c = 900/11 CS~ = 1,040/11 QSgD = 520/11 CS~ = 960/11

E = 0 E = 100 E = 60 E = 40 E = 20 Z = 90 E = 100 E = 50 E = 90

to give t h e c o m p l e t e m i c r o d i s t r i b u t i o n o f f l o w s in t he n e t w o r k . T h e s e a re g iven in Fig. 3.

F o r e x a m p l e , as s h o w n in Fig . 3 ( d ) , p i p e e l e m e n t A C ca r r i e s a t o t a l o f 90 un i t s , 90/11 a re f r o m s o u r c e A a n d 900/11 a re f r o m s o u r c e B; 60 u n i t s

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TABLE 3. Composition of Flows in All Pipe Elements and from All Nodal Supplies in Terms of Destination

To node A To node C To node D Total Element Z = 20 E = 100 E = 90 2 = 210

(1) (2) (3) (4) (5)

Flow in pipe CD QC~D = 50 Flow in pipe BD Flow in pipe A C Flow in pipe BC Supply from node A Flow in pipe B A Supply from node B

QC~o = 0 QC'~D = 0 QCAc = 0 QC~c = 0 SC A = 20/11 QC~A = 200/11 SC~ = 200/11

QCCco = 0 QC~o = 0 QCCac = 60 QC~c = 40 SC c = 60/1t Q c L = 600/11 SC~ = 1,040/11

QC~D = 40 QC~c = 30 QC~ = 20 SC~ = 30/11 QC~A = 300/11 SC~ = 960/11

E = 50 E = 40 E = 90 E = 60 E = 10 Z = 100 E = 200

m n V a l u e s of Qi j

for sources A & B

and consumptions

A, C and D

S

u A P P L C Y

D T 0

(a)

SUPPLY FROH

A B

0 2 0 0 / 1 1

0 6 0 0 / 1 1

0 3 0 0 / 1 1

pipe BA

S

u A P P L C Y

D T 0

(b)

SUPPLY FROM

A B

0 0

0 40

0 2 0

p ipe BC

S

u A P P L C Y

D T 0

SUPPLY FRON

A B

0 0

0 4O

0 0

pipe BD

SUPPLY FRON

A 8

A 0 0

C 60/11 600/11

D 30/111300/11

p i p e AC

SUPPLY FROH

A B

u A 0 0 P P L C 520 /11 Y - - - : - T = T D

0 pipe CD

(c) (d) (e)

FIG. 3. Microflow Distribution for All Pipe Elements Giving Source and Desti- nation of each Flow Component

are delivered to C, with 60/11 from A and 600/11 from B; 30 units are delivered to D, with 30/11 from A and 300/11 from B.

This microdistribution of flows can be used to assess the performance of the network should a particular pipe element be lost from the operational network. These aspects will be demonstrated with respect to a more detailed example, but first, the general procedure is outlined. As shown earlier, for a small network, the method can be undertaken by hand, but in general it is best suited to software implementation.

General Case Consider a generalized node, labeled node i, with a set N[ of nodes

associated with incident pipe elements ki; k E N[, and a set )ViE of nodes associated with emergent pipe elements ij; j E Nff (see Fig. 1) The node

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has a consumption outflow of C~ and a source inflow of &. (A node can act as both a supply and consumption node). The incident flows are of mag- nitude Q~ delivered along pipe elements ki. The emergent flows have mag- nitudes Q~i, delivered along elements i]. Now denote by QSg that part of the flow Qij that originates from the source at node m, and denote by QC;I that part of flow Q~j that supplies the consumption flow at node n. Also, denote by CS m that portion of the consumption at i met by the source node m and SQ' that portion of the supply at node i destined to meet the consumption at node n.

With these definitions and with the perfect mixing assumption introduced previously, the following relationships can be shown to hold for each node of interest i, source node m, and destination node n. Composit ion by source: QS[~, CS'~; for i = m

c s m = Sm Cm �9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) Q~,,,+S,,,

k~Ntm

Srrt

QS,~j = ~. Qk~ +Sm ~cN~

For i ~ rn

Z QS~ k~N[

CST' - k~N{ Qki + Si

Z QS~ k~N[

Qs~ Qki + Si

k~N {

Q,~j; j c N~ . . . . . . . . . . . . . . . . . . . . . . . . (2)

ci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Qo; J ~ NF . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

Composit ion by destination: QCf, i, SC?; for i = n

SC,~ - C~ & = C~ & . . . . . . . . . . . . . . . . . . . . (5)

j~N E k~N 1

c. QC~ =

Z Qkn+Sn kENI

For i 4= n

Z o G ]~N E SC?

Qk, + S~ k~N~

Z O_C~j JENiE

Oct , - Qki + Si

kcN[

Qk.; k r N~ . . . . . . . . . . . . . . . . . . . . . . . . (6)

- - s, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

Oki; k @ Nf . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

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Note also that

CS'2 = S C ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

With the further definition of Qg= to represent that portion of the flow in pipe element ij that originates from the source of supply at node m and is delivered for consumption at node n, then

Qg= = QS~QC'~} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10) Q,j

QUALITY CONSIDERATIONS

It might also be noted that the foregoing expressions for Q S g ; QCF/, and Qpn can be used to assess the water quality offered to a consumer in cir- q

cumstances where the sources of supply are of differing chemical/biochem- ical quality.

For example, if the concentration of some conservation determinant is x m at nodal source m, then the concentration xij in any pipe element ij will be

Q ~ ' x m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11) xij = ~m Qij

Extension to nonconservative is straightforward, given the appropriate decay rate and travel times in the pipe elements, and is consistent with the approach described by others (Grayman et al. 1988; Liou and Kroon 1987; Males et al. 1985).

USE OF MICROFLOW DISTRIBUTIONS IN PERFORMANCE ASSESSMENT UNDER PIPELINE FAILURE CONDITIONS

The foregoing results are now used to consider the effect of pipe-element outage on the network shown in Fig. 4. The corresponding network infor- mation is given in Tables 4 and 5. The network comprises seven nodes, one of which is a constant-head reservoir source. The values of the ultimate demands (i.e. the demand that would be delivered given sufficient head at the node) at the remaining eight nodes are given in Table 4.

In simulating network performance for a failed condition of the network, it is important that the full effects of the failure incident are modeled prop- erly. Typical of the loss of service arising in a real network from the isolation of a failed pipe is a fall in one or more nodal heads and the consequent inability to deliver the required demands. It is important, therefore, that the simulation is head driven rather than demand driven. In the present paper, the particular steady-state simulation program used to model the performance (under both normal and abnormal conditions) accommodates this pressure dependency of the demands on nodal heads, and as such it is particularly useful in studies of network performance under abnormal/failed conditions. Details of this simulation algorithm have been described else- where by Germanopoulos (1985). In the examples presented as follows, the delivered demand is related to the ultimate demand through the relationship:

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FIG. 4.

P

/

/

-i

- - ~ ( ,,

/ B

Simple Seven-Node, Five-Pipe Network

TABLE 4. Nodal Data for Network Shown in Figure 4

Ground level Demand Node (m AOD) I/s

(1) (2) (3)

A a

B C D E F G

74.5 20 20 20 20 20 20

20 55 30 70 15 15

~Reservoir node.

TABLE 5. Pipe Data for Network Shown in Figure 4

Length Diameter Hazen Williams Pipe (m) (m) coefficient (1) (2) (3) (4)

AB AC BC BD CE DE EG DF FG

2000 3000 3600 1200 2100 3100 1500 1700 1300

0.45 0.30 0.30 0.30 0.30 0.30 0.30 0.25 0.25

100 100 100 100 100 100 100 100 100

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Ci = C* [ 1 - aiexp ( - b i p~ l . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

where SP~ = service pressure at node i; ai, bi, and P[ = constants for each particular node i; and C[ = nominal (ultimate) demand at node i.

As can be seen, delivered demands C approach the ultimate demand C* as the service pressure SPi increases beyond P*. In the present example, the numerical values used for all nodes are a = 10; b = 5; and P* = 35 m. These chosen values are illustrative and imply almost full provision of demand at heads in excess of 35 m, with some falloff in provision below that level.

The network behavior is first simulated under normal conditions and then under a series of pipe failure modes, corresponding to the failure and iso- lation of each of the nine pipes, in turn. In this sense, each pipe failure relates to that situation obtaining after the isolation of the failed pipe.

The microflow distribution under normal operation for each pipe element is shown in Table 6. The corresponding nodal heads and the corresponding demands actually delivered to each node are as shown in the 1st row of Table 7. Thus it can be seen that the normal operating heads in the network are generally sufficient to deliver something approaching the nominal or ultimate demand values specified in Table 5. For example, node D has a nominal demand of 30 l/s, and actually receives 29.1 1/s at a head of 40.5 m. The remaining rows of Table 7 show the same operational parameters corresponding to each pipe element failure condition. Thus the 5th row shows the operational performance of the network in terms of nodal heads and consumptions corresponding to the removal of pipe BD. It can be seen that failure of this element causes serious shortfalls in the demands delivered at nodes D, E, F, and G. The importance of the head-driven nature of the simulations should be clear.

Adopting the simple criterion that unsatisfactory performance at a node is characterized by a service head less than the nominal minimum acceptable value (i.e. SPi < P~), the problem nodes arising from each pipe failure are readily identified from the exhaustive set of simulations given in Table 7. The nodes so affected are given in Table 8, together with the system deficit in delivered demand as calculated from the numbers given in Table 7.

A prediction of the same effects can now be made independently from the information arising from the microflow distribution, and which is the

TABLE 6. Microflow Distribution Showing Destination of Flow in each Pipe

Flow Node Pipe (I/s) B C D E F G (1) (2) (3) (4) (5) (6) (7) (8) AB 139.5 19.8 21.2 29.1 41.0 14.4 14.0 AC 58.7 0 32.6 0 26.0 0 0.13 BC 38.2 0 21.2 0 16.9 0 0.09 BD 81.5 0 0 29.1 24.1 14.4 13.9 CE 43.1 0 0 0 42.9 0 0.22 DE 24.2 0 0 0 24.1 0 0.12 EG 0.35 0 0 0 0 0 0.35 DF 28.2 0 0 0 0 14.4 13.8 FG 13.8 0 0 0 0 0 13.8

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result of just a single steady-state network solution for the normal (non- failed) condition. Inspection of the values QC~, representing the contri- bution made by the pipe ij to the consumption at node n, gives a quick indication of which nodal demands are likely to be affected, and a measure of the shortfall.

Suppose the loss of any pipe results in the loss of all its flow contributions to other pipes and to all nodes. Suppose further that the network is assumed to be a directed, capacitated network at these reduced flow levels. Adopting the simple criterion that any nodal shortfall denotes a problem, then it is straightforward to identify the suspect nodes from Table 6, together with an estimate of the shortfall in delivered demand. The results are given in Table 8, where a comparison can be made with the actual problem nodes and their shortfalls as revealed by full, pressure-dependent simulation. Table

TABLE 8. Predicted and Actual Effects on Demand Nodes for each Pipe Failure State (Head-Based Definition of Nodal Failure)

Failed pipe I/s (1) (2)

AB 139.5 AC 58.7 BC 38.2 BD 81.5 CE 43.1 DE 24.2 EG 0.3 DF 28.2 FG 13.8

FAILED NODES SYSTEM DEFICIT

Simulation (3)

B, C, D, E, F, G C, D, E, F, G

E, G D, E , F , G D , E , F , G

G

F , G G

Prediction (4)

B, C, D, E, F, G C, E, G C, E, G

D, E , F , G E, G E, G

G

Simulation

I/s (%) (5) (6)

99.1 50 18.4 9

5.3 3 46.8 24 14.8 7

1.7 1 0 0 3.3 2 0.7 0

Prediction

I/s (%) (7) (8)

139.5 70 58.7 30 38.2 19 81.5 41 43.1 22 24.2 12

0.4 0 28.2 14 13.8 7

Notes: Failed nodes simulation =- nodal head < 35 m; and failed nodes prediction -= any shortfall in delivered demand. And total demand with no failed pipes is 198.2 1/s.

TABLE 9. Predicted and Actual Effects on Demand Nodes for each Pipe Failure State (Demand-Based Definition of Nodal Failure)

Failed Failed Nodes

pipe Simulation Prediction (t) (2) (3)

AB AC BC BD CE DE EG DF FG

B, C, D, E, F, G C, E, G

D , E , F , G E, G

B, C, D, E, F, G C, E, C, E,

D , E , F , G E E

F, G G

Note: Failed nodes simulation ~ shortfall in delivered demand > 10%; and failed nodes prediction =- shortfall in delivered demand >10%.

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9 gives the corresponding data when a failed node is defined as having a demand shortfall of 10% or more. In this case, the correspondence between actual (simulated) and predicted nodal failures is less good, particularly for those nodes remote from the supply and which have multiple paths from the source.

As expected, in the full simulation some redistribution of flow occurs and the network displays some resilience to the isolation of one of its pipe elements. Consequently, the overall system deficits are less than those pre- dicted by the simple microflow model. Nevertheless, the simple model does a fair job in identifying problem nodes and critical pipe failures.

RELIABILITY ESTIMATES USING MICROFLOW DISTRIBUTION

The foregoing has sought to estimate the hydraulic impact of particular pipe element failures from the information given in a single hydraulic anal- ysis of the complete network. If individual link reliabilities are introduced, then it is possible to combine levels of deficit and their likelihood, and so provide a variety of reliability measures. For example, suppose that pipe failures are assumed statistically independent, with Pi~ representing the fail- ure probability of pipe ij. Further assume that simultaneous pipe failures are statistically insignificant. Under such circumstances, the probability R of no pipe failures is the series product of terms (1 - pij) , or approximately

R = 1 - ~ p i j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13) /]

The corresponding estimate of the expected shortfall at some node n arising from the failure of one of the total number of pipe elements is estimated from the microflow distribution as

E D n = ~ Q C } ' p i j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

The use of these and related estimates will be described in detail in another publication.

CONCLUSIONS

The foregoing described a simple method for estimating the impact of component failures on overall network and individual nodal performance. The basis of the method was the calculation of the microflow distribution, which was derived from a normal solution of the network flow equations. This microflow distribution described the role of each link flow in providing supply to each demand node and in distributing the supply available from each source of supply. The performance of the method compared favorably with the network performance was determined from a simulation of the network incorporating full pressure dependency of consumer demands on nodal heads. Of course, the extent of the quantitative agreement between the failed network behavior, as represented by the pressure-dependent sim- ulation, and the prediction based on the microflow distribution depends on the values of the pressure dependency parameters a, b, and P* and the other network characteristics. And for practical applications of the tech- nique, some calibration of the pressure dependency of nodal demands is required. Nevertheless, the present paper has sought to show how such pressure dependency in postbreakdown behavior can be simulated and its effects anticipated through a detailed representat ion-- termed the micro-

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flow dis t r ibut ion--of the network's nonfailed behavior. It might also be noted that the consequences of failure in a real network will depend on the state of the system and the demands on the system just prior to failure. The technique presented can easily be used to take the output from an extended period simulation to give a time profile of reliability and so forth. The robustness of the technique applied to larger and more complex networks is under investigation.

The application of the microflow distribution in the determination of reliability measures and the variation of water quality in the network was also indicated.

APPENDIX. REFERENCES

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Bao, Y., and Mays, L. W. (1990). "Model for water distribution system reliability." J. Hydr. Engrg., ASCE, 116(9), 1119-1137.

Germanopoulos, G. (1985). "A technical note on the inclusion of pressure dependent demand and leakage terms in water supply network models." Civ. Eng. Syst., 2(3), 171-179.

Germanopoulos, G., Jowitt, P. W., and Lumbers, J. P. (1986). "Assessing the reliability of supply and level of service for water distribution systems." Proc., ICE, Part 1, 80(Apr.), 413-428.

Goulter, I. C. (1988). "Measures of inherent redundancy in water distribution net- work layouts." J. Inf. and Optimisation Sci., 9(3), 363-390.

Grayman, W. G., Clark, R. M., and Males, R. M. (1988). "Modeling distribution- system water quality: Dynamic approach." J. Water Resour. Plng. and Mgmt., 114(3), 295-311.

Jaynes, E. T. (1957a). "Information theory and statistical mechanics." The Physical Rev., 106, 620-630.

Jaynes, E. T. (1957b). "Information theory and statistical mechanics." The Physical Rev., 108, 71-190.

Liou, C. P., and Kroon, J. R. (1987). "Modeling the propagation of waterborne substances in distribution networks." J. Am. Water Works Assoc., 79(11), 54-58.

Males, R. M., Clark, R. M., Wehrman, P. J., and Gates, W. E. (1985). "Algorithm for mixing problems in water systems." J. Hydr. Engrg., ASCE, 111(2), 206-219.

Ormsbee, L., and Kessler, A. (1990). "Optimal upgrading of hydraulic-network reliability." J. Water Resour. Plng. and Mgmt., ASCE, 116(6), 784-802.

Quimpo, R. G., and Sbamsi, U. M. (1989). "Network reliability for water distribution system management." Proc. 5th IAHR Int. Syrup. on Stochastic Hydraulics. Univ. of Birmingham, Birmingham, U.K.

Wagner, J. M., Shamir, U., and Marks, D. H. (1988a). "Water distribution relia- bility: Analytical methods." J. Water Res. Plng. and Mgmt., ASCE, 114(3), 253- 275.

Wagner, J. M., Shamir, U., and Marks, D. H. (1988b). "Water distribution relia- bility: Simulation methods." J. Water Res. Ping. and Mgmt., ASCE, 114(3), 276- 294.

Wilson, A. G. (1970). Entropy in urban and regional traffic modelling. Pion Press Ltd., London, U.K.

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