Theoretical and Conceptual Framework - Pattern Recognition for Reliability Assessment of Water Distribution Networks - page 18

as a percentage of the time in which the pressure at the node is above the defined threshold.
Cullinane states it as follows:

r t

m

ki k

∑
=

R

=

2.9

T

k 1

where R i = hydraulic reliability of node i ; r ki = hydraulic reliability of node i during time step
k ; t k = duration of the time step; m = total number of the time steps; and T = length of the
simulation period. r ki = 1 for the nodal pressure p ki equal or above the threshold pressure p min ,
and r ki = 0 in the remaining case of p ki < p min . For equal time intervals, t k = T / m . The
reliability of the entire system consisting of n nodes can be defined as the average of all nodal
reliabilities:

R

n

∑
=

R =

2.10

n

i 1

The above equations assume that the components and sub-components are fully functional,
i.e. 100% available, which is rarely the case. Applying so called expected value of the nodal
reliability includes impacts of the availability on the hydraulic performance. This value can
be determined as follows:

RE = A R + U

R

2.11

ij j ij j

where RE ij = expected value of the nodal reliability while considering pipe j ; A j = availability
of pipe j i.e. the probability that this pipe is operational; U j = unavailability of pipe j i.e. the
probability that it is non-operational; R ij = reliability of node i with pipe j available i.e.
operational; and R i = reliability of node i with pipe j unavailable i.e. non-operational.

Availability A j is determined by Equation 2.2, while U j = 1 - A j . The values for R ij and R i are
calculated by Equation 2.9, running the network simulation once with pipe j operational, and
then again, by excluding it from the layout. With such correction of the nodal reliability, the
overall system reliability can be calculated by Equation 2.11.

Table 2.4 Case network: node properties

Node Elevation Demand

msl l/s

n1
n2
n3
n4
n5
n6
P1
P2

16.8
22.2
17.5
20.2
14.6
14.3
14.0
15.0

3
3
3
3
3
4
-
-

Figure 2.4 Sample case network (Trifunović and Umar, 2003)