IAEA Training in level 1 PSA and PSA applications
Basic Level 1. PSA course for analysts
Boolean Algebra and PSA quantification
Boolean Algebra and PSA quantification
Content
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What is a set?
Dealing with abstract sets
System failure states
Boolean Algebra. Operations and Laws
Boolean equation & Cut
-sets
Cut-sets
Example
Slide 2.
Boolean Algebra and PSA quantification
WHAT IS A SET?
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A set is a collection of items that have something in common
The universal set (E or 1) is the set that contains all possible
items
The membership of a set is expressed as x M (hence non
non-membership by x
M)
A set X is a sub
-set of set Y if all the elements of X are also
sub-set
elements of Y: X Y
The null set ( or 0) is the set that contains no elements at
all
The complementary set of A ( ) is the set that contains all
the elements that do not belong to set A
Slide 3.
Boolean Algebra and PSA quantification
DEALING WITH ABSTRACT SETS
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THE IDEA OF A SET IS VERY GENERAL:
One can deal with sets of physical things or with sets of
abstract things
„
„ This lecture will deal with SETS OF SYSTEM STATES
„
„
Slide 4.
Boolean Algebra and PSA quantification
SETS OF SYSTEM STATES
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A
N
K
M VA
XVB
SVC
PA
NVA
PB
NVB
XVC
XVD
XVE
SYS
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Slide 5.
Boolean Algebra and PSA quantification
BOOLEAN ALGEBRA IN PSA
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Boolean Algebra is a simple method of finding the minimal
combinations of failures that cause system failure: MINIMAL
CUT SETS
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This is the first step in the assessment of the system failure
probability
Slide 6.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
THE UNION:
VX
VY
VX
VY
SYSTEM
FINAL STATE
S
S
F
F
S
F
S
F
1 = SUCCESS
2 = FAILURE
3 = FAILURE
4 = FAILURE
Slide 7.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
THE UNION:
E
B
A
3
4
2
1
AUB
E = { system states } = { 1,2,3,4 }
A = { system states that contain
failure of VX } = { 3,4 }
B = { system states that contain
failure of VY } = { 2,4 }
A U B = { system states that
contain all the failures of VX OR
the failures of VY } =
=
{ 2,3,4 }
Slide 8.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
THE UNION:
A U B is normally written as A
+ B and in fault tree notation
it is represented by an OR
gate
SYSTEM
FAILURE
P (TOP) = P(A) + P(B) - P(AB)
FAILURE OF
VX
FAILURE OF
VY
A
B
Using the rare event
approximation: P(AB)
<<< 1
P(TOP) = P(A) + P(B)
Slide 9.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
THE INTERSECTION:
VX
VX
VZ
SYSTEM
FINAL STATE
S
S
1 = SUCCESS
S
F
2 = SUCCESS
F
S
3 = SUCCESS
F
F
4 = FAILURE
VZ
Slide 10.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
THE INTERSECTION:
A
C
3 4
E
2
1
A
C
E = { system states } =
{ 1,2,3,4 }
A = { system states that contain
failure of VX } =
{ 3,4 }
C = { system states that contain
failure of VZ } =
{ 2,4 }
A C = { system states that
contain failure of VX AND
failure of VZ } =
{4}
Slide 11.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
THE INTERSECTION:
SYSTEM
FAILURE
A
C is normally
written as A × C and in
fault tree notation it is
represented by an AND
gate
FAILURE OF
VX
FAILURE OF
VZ
A
C
P (TOP) = P(A) x P(C)
(Assuming that A and C are
independent)
Slide 12.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
COMBINATIONS OF UNIONS AND INTERSECTIONS
COMBINATIONS OF UNIONS AND INTERSECTIONS
VY
VX
VZ
VX
VY
VZ
SYSTEM
FINAL STATE
S
S
S
F
S
F
F
F
S
S
F
S
F
S
F
F
S
F
S
S
F
F
S
F
1 = SUCCESS
2 = SUCCESS
3 = SUCCESS
4 = SUCCESS
5 = FAILURE
6 = FAILURE
7 = SUCCESS
8 = FAILURE
Slide 13.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
COMBINATIONS OF UNIONS AND INTERSECTIONS
A
4
6
7
8
3
E
B
A = { system states that contain
failure of VX } = { 4,6,7,8 }
B = { system states that contain
failure of VY } = { 3,5,7,8 }
5
1
2
C
(A U B)
E = { system states } =
{ 1,2,3,4,5,6,7,8 }
C
C = { system states that contain
failure of VZ } = { 2,5,6,8 }
(A U B)
C = { system failure
states } = { system states that
contain failure of VX OR VY,
AND failure of VZ } = ({ 4,6,7,8 }
U { 3,5,7,8 })
{ 2,5,6,8 } =
{ 3,4,5,6,7,8 }
{ 2,5,6,8 } =
{ 5,6,8 }
Slide 14.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
DISTRIBUTIVE LAW
(A U B)
C = (A
C) U (B
SYSTEM
FAILURE
SYSTEM
FAILURE
FAILURE OF
VY AND VZ
FAILURE OF
VX AND VZ
FAILURE OF
VZ
FAILURE OF
VX OR VY
C)
C
FAILURE OF
VX
A
FAILURE OF
VY
B
FAILURE OF
VX
FAILURE OF
VZ
FAILURE OF
VY
FAILURE OF
VZ
A
C
B
C
Slide 15.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
ABSORPTION LAW
A U (A
A
C
A
C
B
B
B) = A
A
B=C
CUA=A
Slide 16.
Boolean Algebra and PSA quantification
OPERATIONS AND LAWS
LAWS OF THE BOOLEAN ALGEBRA
COMMUTATIVE
A + B = B ++ A
A
ASSOCIATIVE
A+B+C
A
A× B × C
DISTRIBUTIVE
A × ( B + C ) = ( A ×× B
B )) ++ (( A
A ×× C
C ))
IDEMPOTENT
A+A = A
A
A× A = A
NULL SET
A+0 = A
A×0 = 0
UNIVERSAL SET
A+1 = 1
A×1 = A
ABSORPTION
A+(A×B) = A
=
=
A
A × B = B ×× A
A
(A+B)+C
( A ×× B
B )) ×× C
C
=
==
A
A ++ (( B
B ++ C
C ))
A
A ×× (( B
B ×× C
C ))
Slide 17.
Boolean Algebra and PSA quantification
BOOLEAN EQUATION & CUT SETS
The combinations of system failures can be obtained from a
FAULT TREE MODEL
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The solution of the fault tree is the BOOLEAN EQUATION
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The Boolean equation is a combinations of UNIONS (+) and
INTERSECTIONS ((×):
×):
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A1 × A2 ×
....× Ai + B1 × B2 ×
....× Bi + ...
…
×....×
×....×
...…
A MINIMAL CUT SET is the minimal combination of failures
that lead to top gate failure
Slide 18.
Boolean Algebra and PSA quantification
EXAMPLE
D
B
C
E
Slide 19.
Boolean Algebra and PSA quantification
EXAMPLE
FAULT TREES
z The following fault trees are intended to
illustrate the concepts presented earlier and
have been simplified as follows:
„ They do not show the specific failure modes of the
components
„ They do not show human failure events
„ They do not show common cause failures
Slide 20.
Boolean Algebra and PSA quantification
EXAMPLE
FAULT TREES
Slide 21.
Boolean Algebra and PSA quantification
EXAMPLE
FAULT TREES
Slide 22.
Boolean Algebra and PSA quantification
EXAMPLE
Minimal Cutsets
D = (M V A + T A N K ) . (X V B + T A N K ) = (M V A . X V B ) + (M V A . T A N K ) + (T A N K .
X V B ) + (T A N K . T A N K ) = (M V A . X V B ) + T A N K
C = S V C + D = S V C + (M V A . X V B ) + T A N K
B = (P A + N V A + C ) . (P B + N V B + C ) = (P A . P B ) + (P A . N V B ) + (P A . C ) +
(N V A . P B ) + (N V A . N V B ) + (N V A . C ) + (C . P B ) + (C . N V B ) + (C . C ) = (P A .
P B ) + (P A . N V B ) + (N V A . P B ) + (N V A . N V B ) + C = (P A . P B ) + (P A . N V B ) +
(N V A . P B ) + (N V A . N V B ) + S V C + (M V A . X V B ) + T A N K
TOP = A1 . A2 . A3
T O P = (X V C + B ) . (X V D + B ) . (X V E + B ) = [(X V C . X V D ) + (X V C . B ) + (B .
X V D ) + (B . B )] . (X V E + B ) = [(X V C . X V D ) + B ] . (X V E + B ) = (X V C . X V D .
X V E ) + (X V C . X V D . B ) + (B . X V E ) + (B . B ) = (X V C . X V D . X V E ) + B =
XVC . XVD . XVE +
PA . PB +
PA . NVB +
NVA . PB +
NVA . NVB +
SVC +
MVA . XVB +
TANK
MINIMAL CUTSETS
Do they make sense????
Slide 23.