Common Cause Failures: Extended Alpha Factor method and its
Implementation
Alexandra Sitdikova
Reactor Engineering Division, Jožef Stefan Institute
Jamova 39, SI-1000 Ljubljana, Slovenia
Institute of Physics and Technology, National Research Tomsk Polytechnic University,
30, Lenin Avenue, 634050 Tomsk, Russia
Alexandra.Sitdikova@gmail.com
Marko Čepin
Faculty of Electrical Engineering, University of Ljubljana
Tržaška 25, SI-1000 Ljubljana, Slovenia
marko.cepin@fe.uni-lj.si
ABSTRACT
Contribution of the common cause failures (CCFs) has been recognized as the
dominant, within the results of a safety system reliability analysis and a probabilistic safety
assessment (PSA) when analyzing nuclear safety.
The purpose of this paper is to present a new method for explicit modeling of multiple
components failure event within multiple common cause failure groups simultaneously. The
method is based on Alpha Factor model with few modifications because of the developed
expansion. The assessment for simultaneous assignment of single or multiple failure events to
multiple common cause failure groups is performed. A standard standby safety system was
selected as a case study and a comparison of results with standard Alpha Factor and Beta
Factor methods was made. The results show that consideration of one failure event in several
common cause failure groups gives a larger failure probability of considered systems.
1
INTRODUCTION
Nuclear power plants are designed with redundant safety systems, redundant trains and
redundant equipment for improved reliability and safety. However, very high reliability
theoretically achievable through the use of redundancy is often compromised by single events
that can individually render redundant components unavailable [1]. Such events are known
under the term common cause failure (CCF) events or common cause failures.
CCF events have been recognized as the dominant contributors to the results of the
system reliability analysis and the probabilistic safety assessment (PSA). They are defined as
a subset of dependent failures in which two or more component fault states exist at the same
time, or in short time interval, and thus they represent failures resulting from a shared cause
[2]. For example, environmental CCFs include orbital debris strikes and exposure to
excessively high humidity, temperature or vibration. Neglecting contribution of common
caused failures can result a significant underestimation of risk. [3] CCFs are being
acknowledged as one of the most challenging issues in the PSA, especially within PSA fault
tree (FT) modeling of safety systems within nuclear power plants.
520.1
520.2
This paper presents the method based on Alpha Factor method, but applying for
explicit modeling of single and multiple components failure events simultaneously within
number of several different Common Cause Failure Groups (CCFGs) – sets. Each CCFG is
defined on the basis of specific coupling mechanism. All sets could be sorted by the group
size – k (number of elements in it) and by the number of common elements in each CCFG, – x
(if it is single than x = 1 or multiple x ≥ 1; but in any case x ≥ k). The presented method that
accommodates components failure events to be simultaneously assigned to different CCFGs
given different coupling mechanisms is based on a modification of the well-known Alpha
Factor model. The motivation for this study is the incapability of one of the most widespread
PSA software for fault tree (FT) and event tree (ET) modeling, [5], for simultaneous
assignment of neither one single component failure event, nor multiple components failure
event in more than one CCFG within the fault tree analysis technique. Namely, the software
package provides with a CCF modeling feature based on manual assignment of arbitrary
failure events, i.e. basic events (BEs), to specific CCFG upon selection of proper parametric
CCF model. In the process of this assignment of BE to CCFG, the software does not
accommodate the option for one to assign one BE to several different CCFGs, a scenario quite
probable in practice since given component can experience failure due to different causes,
which if seen as shared causes couple the specific component with other components in
different CCFGs simultaneously.
The method was applied on a selected case study system. The application of the
method enables improved PSA models. The improved models consequently implicate better
results.
2
COMMON CAUSE FAILURE METHODS
Three most known CCF methods include the Beta Factor, the Multi-Greek Letter
(MGL) and the Alpha Factor method. The Beta Factor method is a single parameter method
and assumes that whenever a CCF occurs, all components within a CCFG fail. Although
historical nuclear industry data indicates that common caused events do not always fail all
redundant components, the Beta Factor method does not allow consideration of intermediate
failure criteria (e.g., 2 of 3 failures leading to a specific failure).
The MGL and Alpha Factor methods are more detailed and allow consideration of
intermediate failure criteria. That is for large number of redundant components, many
possible failure scenarios exist and extensive modeling efforts are required in order to depict
all possible failure combinations. Practicality necessitates some simplification, especially for
large CCFG sizes and components with high degrees of redundancy [2].
2.1
Alpha Factor Method
Several guidelines for modeling CCFs in PSA have been published [1, 2, 3]. Some
give the sources of generic common caused data. This data is presented in the form of Alpha
Factor fractions of the total frequency (αn) in tables for use with CCFG size up to eight. For a
given component group size k, an individual
is the probability that when a CCF occurs, it
involves failure of exactly n of k components. The sum of these fractions for a single CCFG is
equal to one.
The probability of CCF event involving 'n' specific components (1 ≤ n ≤ k) in CCFG
of size 'k' for non-staggered testing scheme , is calculated by using the following equation:
(1)
(2)
Proceedings of the International Conference Nuclear Energy for New Europe, Bovec, Slovenia, Sept. 12-15, 2011
520.3
By equation (2) the normalizing factor
is calculated. Then,
is the probability of
an independent failure of a single component. is the total probability of failure for a given
component. Therefore,
represents a CCF multiplier, often shown as a percentage, intended
to be multiplied by the component failure rate for PSA fault tree CCF basic event data entry.
These CCF multipliers are calculated for each CCF criteria in a CCFG.
The system failure probability is also calculated.
Analyzing a system S made of three components A, B, C, where system failure
probability of two-out-of-three components, with Alpha factor method is given by Eq. (3):
(3)
where P(S) – failure probability system S;
P(A), P(B), P(C) - failure probability of component A, B and C respectively;
P(CAB) - failure probability of two component failures: A and B, common cause;
P(CABC) - failure probability of three component failures: A, B and C, common cause.
The failure probability of a system S depends on failure probability of its components
and on the contribution of common cause failures, which are modeled as one common cause
group consisting of components A, B and C, which may fail due to common cause. Alpha
facto method considers cases where two of three components fail due to common mechanism
or all three components fail due to common mechanism.
It is assumed that failure probabilities of similar components are the same.
(4)
(5)
(6)
The probability of occurrence of any basic event within a given common cause
component group is assumed to depend only on the number and not on the specific
components in that basic event. Using the above notation, the system failure probability can
be written as:
(7)
This method was developed for system with three components considering one CCFG.
For cases of multiple CCFG simultaneously, the modified Alpha Factor method is proposed.
2.2
Modified Alpha Factor Method
The method is based on traditional Alpha Factor Method with few modifications
because of expansion for multiple CCFGs defined for a system. The extension goes by
number of groups in every set and number of common elements. In parallel system of four
trains, the function of one is enough for the system success, so all trains should fail for the
system failure, i.e. the failure criteria is n out of n.
In order to present briefly how the modified Alpha Factor model works, let us assume
an example system with four trains as presented on the Figure 1. The success criteria is 1/4,
the failure criteria is 4/4.
A1
A2
A3
B1
B2
B3
C1
C2
C3
D1
D2
D3
Figure 1: Example system with four trains
Proceedings of the International Conference Nuclear Energy for New Europe, Bovec, Slovenia, Sept. 12-15, 2011
520.4
The fault tree without implementation of CCF is shown on Figure 2 and the one with
CCF implementation on Figure 3.
TOP
A
A1
B
A2
A3
B1
B2
C
B3
C1
D
C2
C3
D1
D2
D3
Figure 2: Fault tree without CCF implementation
TOP
CCF
CCFG 4
elements
Indep.
Component
A Fails
CCFG 2
elements
A_ind
A1_
ind
A2_
ind
A3_
ind
Component
B Fails
CCFG 3
elements
B1_
ind
CCFG 2
elements
B_ind
B2_
ind
B3_
ind
Component
D Fails
Component
C Fails
CCFG 3
elements
CCFG 2
elements
C_ind
C1_
ind
C2_
ind
CCFG 3
elements
C3_
ind
D1_
ind
CCFG 2
elements
D_ind
D2_
ind
CCFG 3
elements
D3_
ind
Figure 3: Fault tree with CCF implementation
One common mechanism is found for components A1, B1, C1 and D1, which is
modeled in one CCFG. The other common mechanism is found for components: A1, B2, C2
and D2, which is modeled in other CCFG. The size k = 4 (Figure 4) means the number of
components in a group.
The probability of a set failure, concerning Boolean logic, could be calculated as a sum
of all combinations of two, three and four element failures and independent failure of element
A1:
12
– A1B1, A1C1, A1D1, B1C1, B1D1, C1D1, A1B2, A1C2, A1D2, B2C2, B2D2,
C2D2;
8
– A1B1C1, A1B1D1, A1C1D1, B1C1D1, A1B2C2, A1B2D2, A1C2D2, B2C2D2;
Proceedings of the International Conference Nuclear Energy for New Europe, Bovec, Slovenia, Sept. 12-15, 2011
520.5
2
– A1B1C1D1, A1B2C2D2;
Summarizing, the probability of failure for given set could be defined as:
, or
(8)
(9)
where
- number of combinations n of k,
.
Pind – independent failure probability of a component;
P2CF - failure probability of two component failures due to common cause;
P3CF - failure probability of three component failures due to common cause;
P4CF - failure probability of four component failures due to common cause.
Figure 4: Example system with two CCFGs: A1B1C1D1, A1B2C2D2
Thus, the probability of failure for m-set Ps with single common component in it is
calculated as follows.
(10)
One more way of applying modified Alpha Factor method is to use the sets with more
than one common element in CCFGs. The probability of multiple failure events in multiple
common cause failure groups is calculated in this case.
The probability of failure of a set with x common components in it (size k > x) could be
described with the following equation:
(11)
where
– the probability of independent failures, sum of probabilities
of each common component (A1, B1, C1, etc.).
The main difference in comparison with usual Alpha factor method is the way of
calculation probabilities of the basic events involving n specific components in a CCFG of
size k (1 ≤ n ≤ k), or
:
(12)
This method is proposed for cases, where the set contains the groups with similar types
of components in one train (i.e. diesel generators, check valves, motor pumps), where all
component failure probabilities are equal. For component groups of different types of
components, where several failure probabilities are used, the mean value of
normalizing
factor and failure frequency of each component
are calculated as follows.
,
,
(13)
(14)
(15)
Proceedings of the International Conference Nuclear Energy for New Europe, Bovec, Slovenia, Sept. 12-15, 2011
520.6
So the probability of common cause failure involving n specific components in CCFG
can now be written as:
(16)
3
CASE STUDY
A simplified model of an auxiliary feedwater system (AFWS) of a light-water PWR
second generation NPP was used for a case study (Figure 5). The AFWS provides a backup
supply of feedwater to the secondary side of the steam generators when the main feedwater
pumps cease to operate that makes the normal main feedwater unavailable [4], [5].
SG 1
MIV 7
CV 7
CV 4
FCV 1
MIV 9
CV 1
MIV 1
MIV 4
MDP 1
CV 9
MIV_CS 1
MIV 12
CST 1
POIV 2
FCV 3
MIV 5
CV 5
TDP
CV 2
MIV 2
FCV 4
POIV 1
MIV_CS 2
CST 2
MIV 11
SG 2
CV 10
FCV 2
MIV 10
MIV 8
CV 8
MIV 6
CV 6
MDP 2
CV 3
MIV 3
Figure 5: Auxiliary Feedwater System (AFWS)
The corresponding FT top event is defined as system failure. AFWS is nonoperational
if either one of the two steam generators does not receive cooling water. The presented
method for assigning single component failure within different CCFGs simultaneously was
applied. Two categories of AFWS components were encompassed by the method, i.e. check
valves (CVs) and isolation valves (MIVs). These components were combined, first, into
groups and then to several sets by size.
For calculation the probability of sets failure were used Modified Alpha Factor
method, Alpha Factor method and Beta Factor method.
The following Table 1 and Table 2 present the value of probabilities of a CCF
involving n components in CCFG size k. Notation CV# means failure of check valve #.
Notation MIV# means failure of isolation valve #.
Table 1: Value of PnCF for Modified Alpha Factor method
Proceedings of the International Conference Nuclear Energy for New Europe, Bovec, Slovenia, Sept. 12-15, 2011
520.7
CVs Sets
CV1CV6, CV1CV8
CV1CV4CV7,CV1CV2CV3
CV1CV5CV9CV10
MIVs Sets
MIV4, MIV10;
MIV4, MIV9
MIV4, MIV5, MIV6;
MIV4, MIV1, MIV7;
MIV4, MIV11, MIV12
PIND
8,79E-05
8,40E-05
8,13E-05
PIND
P2CF
1,21E-05
2,33E-05
2,18E-05
P2CF
P3CF
P4CF
PS
1,00E-04
5,51E-06
1,13E-04
5,39E-06 1,85E-06 1,09E-04
P3CF
PSUM
9,55E-05 2,27E-06
9,77E-05
9,33E-05 1,51E-05 3,29E-06
1,12E-04
Table 2: Value of PnCF for Alpha Factor method
CVs Sets
CV1CV6, CV1CV8
CV1CV4CV7, CV1CV2CV3
CV1CV5CV9CV10
MIVs Sets
MIV4, MIV10; MIV4, MIV9
MIV4, MIV5, MIV6;
MIV4, MIV1, MIV7;
MIV4, MIV11, MIV12
PIND
1,76E-04
2,52E-04
3,25E-04
PIND
1,91E-04
P2CF
2,42E-05
7,76E-06
2,43E-06
P2CF
9,07E-06
P3CF
P4CF
PS
2,00E-04
1,65E-05
2,76E-04
1,35E-06 7,40E-06 3,36E-04
P3CF
PSUM
2,00E-04
2,80E-04 5,05E-06 9,86E-06
2,95E-04
Table 3: Results comparison for applied CCF methods
PCV1
PMIV4
PTOP2
Modified Alpha factor Alpha factor Method
Method
3,21E-04
8,12E-04
2,09E-04
4,95E-04
2,45E-04
2,69E-04
Beta factor Method
1,00E-04
1,00E-04
2,88E-04
Table 3 comprises the comparison of results for applied methods of calculation CCF,
considering TOP event and two categories of AFWS components check valves and isolation
valves. Due to the fact that Alpha Factor modeling techniques more failure combinations, the
meaning of failure probability could be two to three orders higher than for Beta Factor
method. The modified Alpha Factor method gives good results, what can let use it for CCF
analysis.
4
CONCLUSION
The modified Alpha Factor method for CCF analysis and prevention has been
examined to show that it is useful for assessment the potential CCF at nuclear power plants.
This method will help calculate probabilities of single and multiple events components failure
events simultaneously within number of several different CCFGs, combined in set.
Proposed method has several differences from standard Alpha Factor method. It can
be seen as its upgrade. The main advantage of the discussed model is fact that implementation
of such CCF approach could be especially useful for CCFG within several numbers of
Proceedings of the International Conference Nuclear Energy for New Europe, Bovec, Slovenia, Sept. 12-15, 2011
520.8
different types of components. Examples include number of system trains with components
placed in one room or on the same floor, the number of similar components of the same
producer with the same physical and technological characteristics.
The proposed method presents the explicit modeling of CCF. The disadvantage is
associated with the fact that it requires more specific calculations for several parameters
which is connected with more efforts and more data, which may lead to a higher uncertainty.
ACKNOWLEDGMENTS
The Slovenian Research Agency supported this research (project J2-2182).
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Proceedings of the International Conference Nuclear Energy for New Europe, Bovec, Slovenia, Sept. 12-15, 2011