Civil Engineering Reference
In-Depth Information
load bus
i
in pre-earthquake conditions,
R
i
=
|
V
i
,
s
−
V
i
,0
|/
V
i
,0
is the percent
reduction of voltage magnitude, and
w
i
=
R
i
) is a function (
H
being
the Heaviside function) accounting for the small tolerance on voltage
reduction: its value is one for
R
i
H
(0.1
−
10% and zero otherwise. The SSI index
varies between zero, when there is no solution for the power-fl ow analysis,
and 100 when the EPN remains undamaged after the earthquake. In an
analogous manner, the SSI for a WSS can be defi ned as the ratio of the sum
of the delivered water fl ows after an earthquake to that before the earth-
quake
SSI
≤
n
n
∑∑
, ,
. This defi nition is modifi ed from the
metrics proposed by Wang
et
al.
(2010):
SSI
=
100
Q
Q
is
i
0
i
=
1
i
=
1
n
n
∑∑
, where
n
0
(<
n
) is the number of satisfi ed demand nodes after the earthquake and
Q
i
is unchanged before and after the event (demand-driven analysis, while
in the presented model the analysis is head-driven).
0
=
100
Q
Q
i
i
i
=
1
i
=
1
18.6.3 Probabilistic metrics
Metrics that account for the uncertainty affecting the system are normally
defi ned based on other metrics, such as those illustrated before. Typical
examples are the damage consequence index (DCI) and the upgrade benefi t
index (UBI), both defi ned for a WSS (Wang
et
al.
, 2010). The DCI measures
the impact of damage (including breaks and leaks) on each pipe on the
overall system serviceability, in order to identify
critical
links
that signifi -
cantly affect the system seismic performance. It is defi ned at the component
level in terms of the expected value of the SSI, and hence refl ects the func-
tional consequence of damage to all systems' components:
DCI
i
=
(
E
[
SSI
]
−
E
[
SSI
]), where
E
[
SSI
] is the (unconditional) expected
value of
SSI
from a set of simulations in which the
i
th pipe might or might
not be damaged; and
E
[
SSI
|
L
i
] is the conditional expectation of
SSI
from
another set of simulations under the same seismic hazard, but given that
the
i
th pipe is damaged. As damage to the
i
th pipe is certain in the calcula-
tion of
E
[
SSI
|
L
i
], theoretically,
E
[
SSI
|
L
i
] is always smaller than
E
[
SSI
] for
which the pipe might or might not be damaged. Therefore,
DCI
i
is always
positive, and is the percentage reduction of
SSI
given that the
i
th pipe is
damaged.
The UBI measures the impact of an upgrade of an individual pipe on the
overall system serviceability, and refl ects the systemic functional conse-
quence of damage to the whole system(s) at the component level. It is
defi ned as:
UBI
i
E
[
SSI
|
L
i
])
/
(1
−
E
[
SSI
]) in which
E
upgrade
i
[
SSI
]
is the expected value of
SSI
given that the
i
th pipe is 'upgraded'. This means
that the probability of pipe damage given an earthquake is signifi cantly
reduced with respect to its pre-upgrade value.
UBI
i
is the percent increase
of
SSI
given that the
i
th pipe is upgraded, and can be used to identify critical
links in seismic mitigation, as those with relatively large UBI values.
=
(
E
upgrade
i
[
SSI
]
−
E
[
SSI
])/(1
−
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