7-PDF457-459_System Reliability Theory Models and Statistical

Pressure sensors
fail to detect
Pressure sensors
fail independently
Random failure of
pressure sensor 1
Pressure sensors
Random failure of
pressuresensor 2
Explicitmodeling of common cause failureof asystemwith twopressuresensors.
(Adapted from Summers and Raney 1999.)
Fig. 10.17
is the percentage of common causeDU failuresamong all DU failuresof acomponent.
Similarly, the spurious trip rate AST may be written as
where A is the rate of independent ST failures that only affects one component, and
AKA is the rate of common cause ST failures that will cause failure of all the system
components at the same time. The common cause factor
is the percentage of common cause ST failures among all ST failures of a component.
Since there may be different failure mechanisms leading to DU and ST failures, BDU
and BST need not be equal.
Diagnostic Self-Testingand Common Cause failures
may be classified in two main types:
Common cause failures
1. Multiple failures that occur at the same time due to a common cause
2. Multiple failures that occur due to a common cause, but not necessarily at the
same time
As an example of type 2, consider a redundant structure of electronic components
that are exposed to a common cause: increased temperature. The components will
fail due to the common cause, but usually not at the same time. If we have an SIS
with an adequate diagnostic coverage with respect to this type of failure, we may be
able to detect the first common cause failure and take action before the system fails.
A system failure due to the common cause may therefore be avoided.
Remark: If the common cause, increased temperature,is due to a cooling fan failure,
this should be explicitly modeled as illustrated in Example 10.12. Monitoring the
condition of the cooling fan would in this case give an earlier warning than diagnostic testing of the electronic components, and a higher probability of successful
shutdown before a system common cause failure occurs. A similar example is discussed in IEC61508-6withoutmentioningany explicitmodelingof the cooling fan.0
When we have identified the causes of potential common cause failures ( e g , by
applying a checklist), we should carefully split the potential common cause failures
in the two types (1 and 2) above. For each cause leading to failures of type 2 we
should evaluate the ability of the diagnostic self-testing to reveal the failure (or the
failure cause), the time required to take action, and the probability that this action
will prevent a system failure.
It seems obvious that the common cause factor ,!?for an SIS good diagnostic
coverage should be lower than for a system with no, or a poor, diagnostic coverage.
Weshouldthereforebe carefuland not useestimatesfor from old-fashionedsystems
when analyzing a modern SIS with good diagnostic coverage.
Example 10.13
Parallel System
Reconsider the parallel system of two firedetectorsin Example 10.4,and assume that
DU failures occur with a common cause factor BDU.The PFD of the parallel system
is from (10.10) and (10.13) approximately
With respect to spurious trips, the system is a series system, and the trip rate is
The rate of spurious trips will therefore decrease when BSTincreases.
By using the same data as in Example 10.4, A.DU
= 0.21 . lop6 hours-'
t = 2190 hours, and BDU = BST = 0.10,we get form (10.27)
PFD(BDu) % 5.71 .
+ 2.30. lo-'
2.31 . lo-'
We observe that with realistic estimates of ADU and t,PFDDu is dominated by the
common cause term in (10.27). We may therefore use the approximation
when h D u T is small.
Example 10.14 2-out-of-3 System
The probability of failure on demand for a 2-out-of-3 system is from (10.12) and
With a local alarm on the logic solver we may avoid almost all independent spurious
trips. All common cause failures will, on the other hand, result in a system spurious
trip, and we therefore have
k e 3 (BST) = B S T ~ S T
With the same data as in Example 10.13 we get from (10.29)
PFD(@Du)% 1.71 .
+ 2.30. lop5
2.32. lop5
As in Example 10.13 we observe that with realistic estimates of ADU and t,P F D D ~
is dominated by the common cause term in (10.29). We may therefore use the approximation
when h ~ uist
In Example 10.13and Example 10.14we saw that the PFDDu (BDu) was dominated
by the common cause term of the expressions (10.27) and (10.29), respectively when
k ~ uistsmall. It is straightforward to show that the same applies to all koon systems,
where n 3 2, and k 5 n . We will therefore have that
when k ~ uist small. When @DU > 0, we will therefore get approximately the same
result for all types of koon configurations, and the result is nearly independent of the
number n of components, as long as n 3 2. This may not be a realistic feature of the
@-factormodel. A more realistic alternative to the B-factor model has been proposed
as part of the PDS approach that is described in Section 10.7.
IEC61508 recommends using the @-factormodel with a single "plant specific" @
that is determined by using a checklist for all voting configurations (see IEC61508-6,
appendix D). This makes a comparison between different voting logics rather meaningless. Corneliussen and Hokstad (2003) have criticized the @-factormodel and
introduced a multiple @-factor(MBF) model, that is a generalization of the @-factor
Some reliability data sources (see Chapter 14) present the total failure rates,
while other data sources only present the independent failure rates. The data in
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