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Influence of hydraulic and mechanical
reliability on the overall reliability of water networks
RUDY GARGANO, DOMENICO PIANESE
Department of Hydraulic and Environmental
Engineering "Girolamo Ippolito"
University of Naples "Federico II". Via Claudio n.21. 80125 Naples (ITALY)
Tel.+39 81 7683429 Fax +39 81 5938936
E-mail: gargano@unina.it or pianese@unina.it
ABSTRACT
A problem frequently encountered in studying the overall reliability of
water distribution networks (WDNs) is that of minimising the computational
workload generated by the extremely large number of operating conditions that
normally have to be analysed. An approach is therefore proposed by which it is
possible to fix the maximum approximation error in evaluating a WDN's overall
reliability (MaxErr), and then to determine which of the distribution network's
operating conditions actually need to be investigated, choosing from the
possible configurations that the network might assume following the outage of
one or more components. More specifically, a criterion is suggested which makes
it possible, once the characteristics of the WDN's main electromechanical
components and the stochastic structure of water demand are known, to assess
the incidence of mechanical and hydraulic reliability on overall reliability.
To this end, a hierarchic approach is proposed to estimate the overall
reliability. In the first stage, a large set of daily users' demand is
generated by using a stochastic approach. Starting from these data, for each of
the various network operating conditions taken for reference (e.g.: WDN with
all electromechanical components working; WDN with only one link withdrawn; WDN
with only two links withdrawn, etc.), the nodal heads and flow rate are
evaluated by mathematical modelling. Then, by the Hydraulic Performance Indices
(HPIs), the daily volumes of water supplied to the users are estimated and
compared to actual demand. Subsequently, by statistical analysis of HPIs
reliability is estimated , for each operating condition, by means of suitable
Hydraulic Reliability Indices, HRIs. More specifically, HRI represents the
probability that the index HPI is higher than a predefined threshold value 
. The last stage is to assess the Overall Reliability Indices
(ORIs) by means of a weighted mean of the HRIs. The reliability evaluated by
the ORIs represents the probability that the WDN will provide overall a
distribution service level that is higher than a predefined standard,
regardless of whether the electromechanical components are fully or partially
operative. Finally, some relationships and charts are proposed that are
particularly useful for identifying, for a fixed number of electromechanical
components in the network and for a fixed value of their operation
availability, the maximum number of components simultaneously out that need to
be considered in order to evaluate the overall reliability so that the maximum
error will be at most equal to the predetermined one.
Keywords: water,
network, reliability, probability, availability, stochastic, index.
INTRODUCTION
Over the last ten years or so, research into Water Distribution Networks
(WDNs) has frequently focused on the question of reliability. After the initial
phase which witnessed the use of models developed in other fields of
engineering, researchers subsequently defined more suitable approaches for
studying WDN reliability.
The result of this research effort was essentially to identify two
different types of reliability determined according to the factors that could
result in WDN under-performance: mechanical reliability and hydraulic
reliability.
Mechanical reliability is the WDN's ability to satisfy users even when
one or more electro-mechanical component is withdrawn from service (Su et al.,
1987; Wagner et al., 1988). Hydraulic reliability is a WDN's ability to satisfy
user demand even if the flow demand is randomly and strongly variable in time
and space (Bao and Mays, 1990; Pianese and Villani, 1994).
Although some approaches which attempt to study a WDN's overall (mechanical
+ hydraulic) reliability are available in the technical literature (Cullinane
et al. 1992; Gupta and Bhave, 1994), little or nothing has been done to
investigate the incidence of each of the above aspects on the overall
reliability, which is here defined as the water distribution system's ability
to satisfy user demand in any case, regardless of the factors causing the
restriction on water distribution (i.e. outages of electromechanical components
and/or stochastic variability of demand).
The present paper starts from these considerations to define a criterion
which makes it possible, once the characteristics of the WDN's main
electromechanical components and the stochastic structure of water demand are
known, to assess the incidence of mechanical and hydraulic reliability on
overall reliability.
RELIABILITY INDICES
The approach to studying the reliability of WDNs proposed by Gargano and
Pianese (1998) utilises the hydraulic simulation of numerous operating
conditions in order to take into account both the stochastic variability of
demand and the random variability of the operating conditions in which the WDN
finds itself following the withdrawal from service of electromechanical
components (pipes, joints, pumping stations, etc.).
This furnishes a large sample of data regarding the nodal heads and flow
rates in the various operating conditions, which makes it possible to define
first a series of Hydraulic Performance Indices (HPI) and subsequently a series
of Hydraulic Reliability indices (HRI), so that the overall reliability of a
WDN can be estimated.
More specifically, the daily volumes of water supplied to the users are
estimated and compared to actual demand through an expression already utilised
in the technical literature by several authors (Wagner et al., 1988; Gupta and
Bhave, 1994). For the j-th node and the d-th day, the local Hydraulic
Performance Index, 
, is introduced. This is defined as:
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n = number of time intervals Dt into which the day has been subdivided
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(1) |
where the pressure availability coefficient 
is defined as:
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(2) |
For the whole network, the global Hydraulic Performance Index (
) evaluated with reference to the d-th day is given by the
weighted mean of the indices 
evaluated in each
distribution point in the WDN. The weighting factor is given by the ratio
between users demand in node j and the overall water demand in the WDN.
For a given operating condition (e.g.: WDN with all electromechanical
components working; with one link withdrawn; with two links withdrawn, etc.) we
can determine a series of 
and 
values that can then
be subjected to statistical analysis.
This analysis makes it possible to estimate, for each operating
condition, reliability by means of suitable Hydraulic Reliability Indices,
HRIs. More specifically, after defining a threshold value 
, HRI is defined as its probability of being exceeded:
(3)
Eq.(3) can be used to obtain the 
relative to the nodes
j, or 
for the WDN.
Finally, once the 
have been estimated
for each operating condition taken into consideration (the superscript l
represents one of the L operating condition examined), the next step is to
assess the network's overall reliability by means of a weighted mean of the 
. The weighting factor (Pl) is given by the probability
that the network will have to work in the way hypothesised when the relative 
was estimated. For
instance, with reference to the generic node j, the local Overall Reliability
Index (ORIj) is given by:
(4)
(
, where R is the overall number of electromechanical elements
in the WDN).
Substituting the HRIj in Eq.(4) with HRInet, we
likewise get the Overall Reliability Index for the whole WDN, ORInet.
It can be shown that according to the total probability theorem (Mood et
al.,1963), the ORIs yield the probability that altogether (i.e. regardless of
the specific operating conditions of the WDN) the index HPI will be greater
than the threshold value 
. Therefore, the reliability evaluated using the ORIs
represents the probability that the WDN will provide overall a distribution
service level that is higher than a predefined standard, regardless of whether
the electromechanical components are fully or partially operative.
If we use Ai (Ui=1- Ai) to indicate the
availability (unavailability) of the i-th element, i.e. the probability that
the element in question will (not) be available for operation when needed
(Dhillon, 1988), Eq. (4) assumes the following more explicit expression:
(5)
Similarly, the ORInet for the whole network is given by:
(6)
The superscripts of the HRIs in Eqs.(5)-(6) identify the various
operating conditions that can arise in a WDN. Hence: "0" indicates the
condition in which no element is withdrawn from service; "w" one of the R
conditions with only one element withdrawn from service while all the others
are working properly; "wy" one of the R×(R-1)/2 operating conditions with only 2
elements out at the same time while all the others are working properly, etc.
APPROXIMATIONS INTRODUCED IN EVALUATING OVERALL
RELIABILITY
The need to reduce the computational workload involved in evaluating the
overall reliability of a distribution network, whether simple or complex, can
be successfully met by truncating Eqs.(5)-(6) from a certain addend on.
Continuing in this way will make it possible to omit those terms of Eqs.(5)-(6)
which besides depending on several network operating conditions also have a
lower probability of occurrence.
As we have 
for the
electromechanical components (and this is also true for the more vulnerable
elements), the probability of the WDN operating with a given number of
components withdrawn from service at the same time falls considerably as this
number increases.
The problem that arises is therefore to determine the addend from which
to effect the truncation in Eqs.(5)-(6) or, more precisely, the maximum number
of simultaneous outages that needs to be considered in order to assess the
network's overall reliability, given that the approximation error will not
exceed a predetermined value.
To this end, it is worthwhile identifying the MaxErr value that
maximises, for a given WDN, the approximation introduced by omitting a series
of addends of the sum after a certain term in Eqs.(5)-(6).
Eqs.(5)-(6) point out that the maximum truncation error is given by the
probability that any one of the operating conditions neglected following the
truncation of the above relations will arise. Similarly, the MaxErr can be
calculated as the complement to 1 of the probabilities of there arising any one
of the operating conditions taken into consideration for the evaluation of the
overall reliability. Hence:
(7)
In Eq.(7), 
represents those
operating conditions, each of which characterised by m components
simultaneously withdrawn from service, which are considered in the estimate of
the ORIs, while M indicates the maximum number of elements simultaneously out
that are considered in the evaluation of the overall reliability.
As the working status of the individual components are mutually
incompatible events, likewise the operating conditions indicated with 
are also are mutually
incompatible events. Therefore, according to Eq.(7), the MaxErr is given by:
(8)
The fact that MaxErr entails an increase in the approximation error in
the estimate of the ORIs can be seen in Eqs.(5)-(6). The omitted addends in
Eqs.(5)-(6) are made up of the sum of the products of the probabilities that
the WDN will operate in certain working conditions, for the corresponding HRI
where, by definition, 
.
SIMPLIFICATION OF THE STUDY OF OVERALL
RELIABILITY BY MaxErr
Once we know the characteristics of the electromechanical components in
the WDN and, in particular, the availability Ai of the single
elements, the MaxErr induced by the truncation of Eqs.(5)-(6) can be evaluated
from Eq.(8).
For instance, for M=[0,1,2], Eq.(8) assumes the following expressions:
(9)
Even if Eq.(8) and the more explicit Eqs.(9) do not give cause for
difficulties in calculation, even for fairly complex networks, it is
nevertheless worthwhile considering the particular case in which all components
have an equal availability A.
In this case, whatever the value of M, Eq.(8) assumes the form:
(10)
and the Eqs.(9) assume the form:
(11)
Eq. (10) and the relations derived from it make it possible to define
the MaxErr according to just three variables: the overall number R of
components, the maximum number M of elements considered to be simultaneously
withdrawn from service, and the availability A. In actual fact, in engineering
terms it is more profitable to make the MaxErr, in Eq.(10) equal to a value
representing the threshold for approximation acceptability, and with reference
to the various values that M can assume, to define the functional relationship 
between the
availability and the overall number R of components in a distribution network.
In the case in question, Fig.1 reports an example of some MaxErr curves
deduced from Eq.(10) and, more specifically, from the first of Eqs.(11). These
curves make it possible to study, for pre-defined values of MaxErr
(respectively for: 2.5%, 4%, and 5%), the effect on the overall reliability
evaluation involved in the hypothesis considering, at most, the network
operating conditions with just a single component withdrawn from service. This
hypothesis has been assumed by many authors (Su et al., 1987; Cullinane et al.
1992) in order to simplify the study of mechanical reliability.
With reference to Fig.1, if the point representing the WDN lies above
the curve MaxErr=2.5%, or on the same curve, the approximation on the
evaluation of system overall reliability obtained by omitting all the operating
conditions in the network with more than one component simultaneously out is,
at most, equal to 2.5% of the overall reliability calculated. Obviously,
if the point lies below this curve, the
approximation error may turn out to be greater.
Even more interesting, because of its practical implications, is the chart
in Fig.2. This assumes that MaxErr=2.5% and reports the curves obtained from
Eqs.(11) for certain values of 
. If the availability A of the components and the overall
number R of components making up the WDN are known, this chart makes it
possible to define the maximum number of outages that need to be considered in
order to evaluate system overall reliability for MaxErr=2.5%.
CONCLUSIONS
A WDN's ability to satisfy user demand depends both on the correct
operation of its electromechanical components (pipes, joints, pumping stations,
etc.) and on the amount and variability of the user flow demand.
The present paper has addressed the issue of evaluating a WDN's overall
(hydraulic + mechanical) reliability and has made reference to specific
dimensionless indices defined as the overall reliability indices.
Evaluating these indices exactly would normally require a considerable
calculation workload even for small WDNs. Starting from these considerations,
in the paper a methodology has been provided for evaluating the maximum error
that may be committed by evaluating the above indices in an approximated way.
The approximation is obtained by neglecting the less probable failure
events, e.g. those regarding the failure of several electromechanical
components in the network, that is to say by truncating at a given addend the
sums that make it possible to determine the exact value of the overall
reliability indices proposed in the paper (relations (5) and (6)).
Finally, the paper has proposed some charts that are particularly useful
for identifying, for a fixed overall number of electromechanical components in
the network and for a fixed value of their mean operation availability, the
maximum number of components simultaneously out that need to be considered in
order to evaluate the overall reliability so that the maximum error will be at
most equal to the predetermined one.
In short, these charts make it possible to define, a priori, the
computational workload to be carried out in order to give a sufficiently
reliable evaluation of the network's overall reliability (hydraulic +
mechanical).
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Fig.1 Curves of the
function A=A(R, MaxErr, M) for M=1 and MaxErr=2.5%, 4%, 5% |
Fig.2 Curves of the
MaxErr=2.5% for M=0,1,2,3 components simultaneously out |
ACKNOWLEDGEMENTS
The paper has been developed as part of the research financed by the
C.N.R. (contract No. 96.02870.PF42) and by the M.U.R.S.T. (Innovative
methodologies for water systems planning and protection: hydraulic, mechanic,
and quality aspects).
REFERENCES
[1]
Bao, Y. &
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[2]
Cullinane, M. J.,
Lansey, K. E. & Mays, L.W.(1992). "Optimization-availability-based design
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[3]
Dhillon B.S.
(1988). "Mechanical reliability: theory, models, and applications." AIAA
Education Series, Washinton, D.C.
[4]
Gargano, R. &
Pianese, D. (1998). "Reliability as a tool for hydraulic network planning."
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[5]
Gupta, R. &
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[6]
Mood, A.M.,
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[7]
Pianese, D. &
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[8]
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[9] Wagner, J.M., Shamir, U. & Marks, D.H.
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= flow demand in the j-th node during the k-th 
= pressure availability coefficient of the node j during
the k-th 
= minimum piezometric head needed to fully meet demand
= elevation of the lowest user served in the j-th node 
= piezometric head on the j-th node during the k-th 
