Fault Tree Analysis Part 3 DigraphBased Fault Tree

Fault Tree Analysis Part 3: Digraph-Based Fault Tree Synthesis Procedure (Tree and NFBL)

LEVELS OF MATHEMATICAL MODELS OF ENGINEERING SYSTEMS Partial Differential Equations Ordinary Differential Equations Algebraic Equations DIGRAPHS V 1 Gain Multi-Valued Logic node Boolean Algebra edge V 2

The value of gain is discretized! ± 1: if a moderate deviation in the input variable causes moderate deviation in the output. ± 10: if the output deviation is very large when compared to the input. 0: if the output deviation is very small compared with the input.

Computation of Output Value

[ Example ] 3 HOT NITRIC ACID 1 2 COOLING WATER 4 WATER LEAKS INTO NITRIC ACID -1 +1 TSURR +1 +1 +1 T 2 +1 +1 -1 -1 U -1 A

T 2(+1) OR M 1(+1) T 1(+1) …… M 4(-1) The Fault-Tree Structure for Tree-Like Digraph

THE MAGNITUDE OF DISTURBANCES Gain Deviation in input from its normal value +10 “Large” Positive Deviations +1 “Normal” Positive Deviations 0 No Change in -1 “Normal” Negative Deviations -10 “Large” Negative Deviation Z is the condition required for the gain to be correct (if implicit in initial conditions it is unstated)

Digraph Model 3 [ EXAMPLE ] 1 2 AIR TO OPEN +1 regular valve P 3 quick opening P 3 M 2 +10 0 failure models valve stuck +1 P 3 -1 M 2 valve reversed M 2

Digraph 3 Models Control Valve (Air to Open) M = Mass Rate P = Pressure T = Temperature 1 2 DEN = Density X = Mass Fraction Output Variable M 2 M 1 (Gain) Input (+1) M 1 , (+1) P 3 , (+1)DEN. 1 (-1) P 3 , if Valve Reversed (+1) Fails Open (-1) Fails Closed (+1) M 2 , (+1) P 3 (-1) P 3 if Valve Reversed ( 0 ) P 3 if Valve Stuck (+1) Fails Open (-1) Fails Close ( 0 ) P 3 if Valve Stuck (-1) Plug (-1) Leak Out (+1) Leak In (-1) Plug (0)M 2 (+1) Leak (-1) Leak if Plug = +10 if Fails closed = +10 Out In

Output P 2 (Gain) Input (+1) P 1 , (+1) P 3 , ( 0 ) P 3 if Valve Stuck , ( 0 ) P 1 if plug = +10 , ( 0 ) P 1 if Fails Closed +10 , (-1) Plug , (-1) Fails Closed , (-1) P 3 if Valve Reversed , (+1) Fails Open , (+1) Leak , (-1) Leak Out P 1 (+1) P 2 , (-1) P 3 , ( 0 ) P 3 if Valve Stuck , (+1) P 3 if Valve Rev , ( 0 ) P 2 if Plug = , ( 0 ) P 2 if Fails Closed = +10 , ( ) Plug , (+1) Fails Closed , -1) Fails Open , ( ) Leak In , (-1) Leak Out T 2 (+1) T 1 , ( 0 ) T 1 if M 2 = -10 , (+1) Leak In T 1 P 3 None (if ) ) (

Output Vapor Fraction 2 Vap. Frac 1 Den 2 Den 1 (Gain) Input

COOPERATIVE CAUSES FOR AN EVENT [ Example ] The simultaneous occurrences of P (+1) and T (-1) Cause brittle fracture in a tank, P +1 (T= -1) fracture -1 (P= +1) T

Glossary • Digraph : nodes connected by edges which have direction. • Edge : the line connecting two nodes. – It indicates a relationship between the two nodes. The number next to the edge is the gain. • Conditional Edge : The relationship between two nodes depends on another event or variable. – For example, the gain between valve position and flow out of the valve is zero if the valve is stuck. The condition is “valve stuck”.

Glossary • Primal node : a node on the system digraph with no inputs. • Input : an edge pointing to the node under consideration. • Local Input : variables or events one nods away from the node being considered. • Gain : change in Output / Change in Input. – Gains may have values of ± 1, ± 10, 0. Zero means no gain.

Glossary Variable and Event Values • These are deviations of the variables and events from their normal value. • ± 10 indicates large or fast deviations which cannot be handled by normal NFBL. • ± 1 is the usual deviation expected in the variable or event. • Zero means no deviation. • Some variables are univariant (can only vary in one direction from their normal value), e. g. a normally open valve cannot be further opened or a fire can only have values of 0, +1, and +10.

Glossary • Feedback Loop (FBL) : A path through the nodes in a digraph which starts and terminates at one node. • Negative Feedback Loop (NFBL) : A feedback loop in which the product of the normal gains around the loop is negative. • Positive Feed Back Loop (PFBL) : The product of the gains around the FBL is positive.

[ Example ] FLOW CONTROL LOOP FAULT TREE The Process is a simple feedback loop for flow control. The flow rate of stream 3 (M 3) is sensed by a flow sensor connected to signal line 4. As the flow increases, the signal in line 4 increases. The flow recorder-controller upon receiving the increased signal from 4 sends a decreased signal to stream 5. This causes the valve to close returning the flow to its desired setting.

5 SET PT. FRC 1 FLOW CONTROL 4 2 3 LOOP FLOW AIR TO OPEN -10 +1 -1 STUCK -1 FLOW SENSOR REVERSED +1 0 FLOW SENSOR VA LVE REV ERS ED +1 C FR C -10 +10 CK AL U ST FR C 0 +1 -10 D SE R VE RE 0 +10 -10 +1 -1 +10 0 V ALV E S TUC K FR ON M U AN

Discussions with the designer and operator indicate the following events are known to occur in this process. Sensor : Fails (High , Low , Stuck), Reversed. Controller : Fails (High , Low , Stuck) , On Manual, Loss of Air (Causes Signal 5 to go down ), Reversed. Valve : Fails (Open , Closed , Stuck ), Reversed. The system is normally operating with flow in lines 1, 2, and 3. The event that could be a hazard is “Flow in stream 3 too high (M 3 (+1)). ”

M 3 (+1) OR M 2 (+1) OR M 1 (+1) P 5 (+1) If the fault tree is constructed by treating the digraph as a tree, then ……. .

Development of Fault Tree “ What could cause this ? ” or “ Which nodes are inputs to the node representing the current event ? ” + “ Nothing else happens which will cancel the original effect. ” ( ON A NFBL or NFFL )

THE GENERAL FAULT – TREE STRUCTURES OF NFBL (1) M 2 ( +1 ) OR AND M 1 ( +1 ) process disturbance AND NOT ( P 5 (-1) ) NO control loop correction P 5 (+1) control loop disturbance NOT ( M 1 (-1) ) NO process disturbance to cancel P 5 (+1)

THE GENERAL FAULT – TREE STRUCTURES OF NFBL (1) M 2 ( +1 ) OR AND M 1 ( +1 ) process disturbance AND NOT ( P 5 (-1) ) NO control loop correction P 5 (+1) control loop disturbance M 2 ( +1 ) (2) NOT ( M 1 (-1) ) NO process disturbance to cancel P 5 (+1) OR AND M 1 ( +1 ) AND OR P 5(0) not always true P 5 ( +1 ) OR M 5 ( +1 ) M 1(0) nearly always true

THE GENERAL FAULT – TREE STRUCTURES OF NFBL (1) M 2 ( +1 ) OR AND M 1 ( +1 ) process disturbance AND NOT ( P 5 (-1) ) NO control loop correction P 5 (+1) control loop disturbance M 2 ( +1 ) (2) NOT ( M 1 (-1) ) NO process disturbance to cancel P 5 (+1) OR AND M 1 ( +1 ) OR P 5(0) not always true Nearly always true AND P 5 ( +1 ) OR M 5 ( +1 ) M 1(0) nearly always true

THE GENERAL (3) FAULT – TREE STRUCTURES OF M 2 ( +1 ) OR OR AND M 1( +1 ) P 5 ( +1 ) AND P 5(0) M 1 ( +1 ) P 5 ( +1 ) NFBL

THE GENERAL FAULT – TREE (3) STRUCTURES OF M 2 ( +1 ) OR OR P 5 ( +1 ) AND M 1( +1 ) AND P 5(0) M 1 ( +1 ) (4) P 5 ( +1 ) M 2 ( +1 ) OR AND M 1 ( +1 ) P 5(0) NFBL

A disturbance propagates through a control loop if 1. An external disturbance enters the system and the control loop is inactive; 2. The disturbance is caused by the control loop itself; or 3. The disturbance is extremely large in magnitude.

DISTURBANCES THROUGH A NEGATIVE FEEDBACK LOOP + + _ + + VARIABLE DEVIATION Generally, ( +10 ) defined as that value of which causes a +1 deviation. ( NFBL cannot completely cancel disturbance. ) to have at least

THE GENERAL FAULT – TREES STRUCTURES OF NFBL M 2 ( +1 ) (5) OR Very Nearly true M 1 ( +10 ) And P 5 (-1 ) AND M 1 (+1) P 5 ( +1 ) P 5 ( 0) (6) E OR AND OR component failure (primary or secondary) external disturbance enters loop large disturbance enters loop Loop variable causes disturbance loop variable fails to cancel disturbance

OUTPUT ( Value ) OR UNCONTROLLABLE INPUTS PASS THROUGH THE NFBL OR (1) INPUT (Value to give large or fast disturbance ) NOT ON NFBL (2) PRIMARY FAILURE (3) SECONDARY FAILURE CAUSING EVENT (4) SET POINT CHANGE CONTROL LOOP CAUSES THE DEVIATION CONTROLLABLE DIST RBANCES PASS THROUGH THE NFBL OR AND EOR LOCAL EDGE CONDITIONS WHICH CAUSES REVERSE GAIN ON NFBL INPUT (Value to give desired output value) ON NFBL LOOP INACTIVE OR INPUT (value for controllable disturbance into the NFBL) NOT ON NFBL LOCAL EDGE CONDITIONS WHICH GIVES A ZERO GAIN ON THE NFBL INPUT (value = 0) ON THE NFBL

GENERAL STRUCTURE FOR OUTPUT VARIABLES ON A NFBL OUTPUT ( value = 0 ) OR LOCAL EDGE CONDITIONS INPUT ( value= 0 ) WHICH GIVE ZERO ON THE NFBL GAIN ON THE NFBL

[ EXAMPLE ] M 3 (+1) FLOW CONTROL LOOP OR M 2 (+1) OR OR M 1 (+10) AND Valve Mech. Fails Open (+1) M 1(+1) EOR OR Valve Reversed P 5 (+1) Page 2 Valve Stuck P 5 (0) OR FRC On Manual FRC Stuck Flow Sensor Stuck P 4 (0) OR M 2 (inconsistent)

P 5 (+1) OR OR Set Pt. (+1) EOR FRC Fails High AND (no +1 disturbance) OR Flow Sensor Fails Low FRC Reversed (+1) P 4 (-1) OR AND Line 4 Ruptures (no +1 disturbance) EOR Flow Sensor Reversed M 2 (-1) (inconsistent)