Tie Set Method
Tie Set:
minimal path of the system (input to output path without traversing a node twice)
A
C
A tie set fails if any component in it fails
– tie set components in series
All the tie sets must fail for the system to fail
- tie sets in parallel
E
B
D
Tie Sets:
AC, BD - 2nd order
AED, BEC – 3rd order
RS = P(T1 ∪ T2 ∪ T3 ∪ T4)
A
C
T1
B
D
T2
A
B
E
D
T3
B
B
E
C
T4
= P(T1) + P(T2) + P(T3) + P(T4)
- P(T1 ∩ T2) - P(T1 ∩ T3) - P(T1 ∩ T4)
- P(T2 ∩ T3) - P(T2 ∩ T4) - P(T3 ∩ T4)
+ P(T1 ∩ T2 ∩ T3) + P(T1 ∩ T2 ∩ T4)
+ P(T1 ∩ T3 ∩ T4) + P(T2 ∩ T3 ∩ T4)
- P(T1 ∩ T2 ∩ T3 ∩ T4)
Tie Set Method contd..
P(T1) = RA.RC
P(T2) = RB.RD
P(T3) = RA.RD.RE
P(T4) = RB.RC.RE
P(T1 ∩ T2) = P(T1).P(T2) = RA.RB RC.RD
P(T1 ∩ T3) = P(T1).P(T3) = RA.RC RD.RE
P(T1 ∩ T4) = P(T1).P(T4) = RA.RB RC.RE
..
P(T3 ∩ T4)
= P(T1 ∩ T2 ∩ T3)
= P(T1 ∩ T2 ∩ T4) = P(T1 ∩ T3 ∩ T4)
= P(T2 ∩ T3 ∩ T4) = P(T1 ∩ T2 ∩ T3 ∩ T4)
= RA.RB RC.RD .RE
RS = RA.RB + RC.RD + RA.RD.RE + RB.RC.RE
- RA.RB RC.RD - RA.RB RD.RE - RA.RB RC.RE
- RA.RC RD.RE - RB.RC RD.RE + 2RA.RB RC.RD .RE
If RA = RB = RC = RD = RE = R, then
RS = 2R2 +2R3 – 5R4 + 2R5
If R = 0.99,
RS = 0.99979805
QS = 0.00020195
1
Cut-Set Method for Large Complex Systems
In simple systems, the minimal cutsets can be determined
from visual inspection
1
2
Min. Cuts: 1,3 and 2,3
3
For large complex systems, a computer program can be developed
using Matrix Methods
1.
2.
3.
4.
5.
6.
Obtain Reliability Network Model
Deduce Connection Matrix
Determine Minimal Paths (i.e. Tie Sets) from Connection Matrix multiplication
Build Incidence Matrix
Determine Minimal Cutsets from Incidence Matrix column operations
Use approximations and obtain Reliability indices
Connection Matrix Technique
Obtain the reliability network of the system
Label the nodes (order of matrix = # of nodes)
Show the direction of flow
(uni-directional or bi-directional)
A
2
D
3
1
to nodes
2
3
4
1
0
0
0
A
1
E
0
0
C =M
D
1
B
E
1
0
4
E
1
B
1
2
3
4
C
Connection Matrix:
Rows – from nodes
Columns – to nodes
Elements – connection between nodes
0: no connection between nodes
1: connection between a node and itself
Multiply the matrix by itself repeatedly, until the resulting matrix remains unchanged
(Boolean Algebra)
1
2
3
4
M2
1 A+BE B+AE AC+BD
1
E
C+DE
= 0
0
E
1
EC+D
0
0
0
1
1
M3 = 2
3
4
1 A+BE B+AE AC+BD+BEC+AED
0
1
E
C+DE
0
E
1
EC+D
0
0
0
1
= M4
2
Incidence Matrix Technique
A
2
C
4
E
1
B
Step 1 (Find 1st Order Min Cuts):
If all elements in a column are ‘1’,
that component forms a 1st order min cut.
D
3
Minimal Paths (tie sets):
1. AC 2. BD 3. AED 4. BEC
Incidence Matrix:
shows components in each row of
minimal path
1
2
3
4
A
components
B
C
D
E
1
0
1
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
0
Step 2 (Find 2nd Order Min Cuts):
a. Combine 2 columns at a time
b. All ‘1’ columns forms 2nd order cutsets
c. Eliminate cutsets containing 1st order cutsets
AB, CD
Step 3 (Find 3rd Order Min Cuts):
a. Combine 3 columns at a time
b. All ‘1’ columns forms 3rd order cutsets
c. Eliminate cutsets containing 2nd order cutsets
ABC, ABD, ABE, ACD,ADE, BCD, BCE, CDE
Find Higher Order Min Cuts:
Repeat above method
Event Trees
Event Trees can be applied to:
Continuously operated systems (mainly independent events)
Standby & sequential logic systems (dependent events)
e.g. safety and mission oriented systems
event trees are widely used for these systems
Pictorial representation of all the events that can occur in a system.
Events:
success – represented by vertical line upwards
failure – vertical line downwards
partial failure can also be considered
Probability of encountering a state =
product of the probabilities leading to that state
P(1, 2) = P(1) . P(2)
P(1, 2) = P(1) . P(2)
1
1
2
2
2
2
1, 2
1, 2
1, 2
1, 2
2-component system
3
Event Trees: Continuously operated systems
- mainly independent events
- components can be taken in any order while creating the event tree
- starting point is usually the normal operating condition of the system
Example:
A
C
E
B
D
Event Tree
A
C
E
B
D
4
Reduced Event Tree
A
C
E
B
D
Reduced Event Tree
Reduced Data
A
C
E
B
D
5
Event Trees:
Standby & Sequential Logic Systems
- dependent events
- events must be taken in sequential order while creating the event tree
- starting point is the initiating event
Example: Cooling system of a continuous process plant
Sequential order of events when the normal cooling system fails
(e.g. by a pipe break)
1.
2.
3.
4.
Electric power (EP) supplies emergency (back-up) cooling system
Flow detector (D) detects normal cooling system failure and triggers
operation of electric pumps P1 and P2
Emergency cooling system maintains the system
Both pumps - system success, one pump - partial success
Initiating event – Normal coolant failure (pipe break)
P1
Pipe break – EP – D
P2
6
7