Appoximate Sytem Reliability Evaluation Up MTTF Down 0 MTBF MTTR () Time Fo many engineeing ytem component, MTTF MTBF i.e. failue ate, failue fequency, f Fequency, Duation and Pobability Indice: failue ate o fequency aveage epai o down time U unavailability (expected annual outage time) U =. U = + µ = /m /m+ / = m+ = T = f.
Appoximate Equation fo Seie Sytem µ µ Fo a eie ytem with n component, Sytem Failue Fequency, = Σ i Sytem Unavailability, U = Σ i i Sytem Aveage Down Time, = U
Deivation of Appoximate Seie Equation: Up Up µ Dn Up Up Up µ Dn Up µ µ µ 3 Up Dn µ 4 Dn Dn 3 Up Dn -Component Repaiable Sytem P up = P = ( µ µ + µ )( + µ ) µ µ equivalent component µ P up = ( µ µ + µ )( + µ ) = P up = ( µ + µ ) = + + U + fo << and
Fo Component Seie Sytem: U = + = + = U (ytem failue ate) Fo a eie ytem with n component, = Σ i U = Σ i i = U
Appoximate Equation fo Paallel Sytem µ µ Fo a -component paallel ytem, =. ( + ) fo i. i << =. / ( + ) U =. Fo a 3-component paallel ytem, =.. 3 (. +. 3 +. 3 ) = (.. 3 + + 3 3 ) µ = Σ µ i U =.
Deivation of Appoximate Paallel Equation: Up Up µ Dn Up 3 Up Dn µ µ µ 4 Dn Dn P dn = P 4 = ( + µ )( + µ ) -Component Repaiable Sytem equivalent component, µ, µ, µ P down = ( + µ )( + µ ) = ( + µ ) = + ( + ) + ). ( + ) fo i. i <<
Fo Component Paallel Sytem: =. ( + ) µ = µ + µ (depatue ate fom State 4) i.e. =. / ( + ) U =. Anothe fom, = ( ) + ( ) Sytem failue occu if, fail followed by failue of duing epai of OR fail followed by failue of duing epai of
3 Component Paallel Sytem Sytem failue occu if, A fail, followed by failue of B duing epai of A, followed by failue of C duing ovelapping epai of A and B OR A fail, followed by failue of C duing epai of A, followed by failue of B duing ovelapping epai of A and C OR B fail, followed by failue of A duing epai of B, followed by failue of C duing ovelapping epai of A and B OR 3 moe tatement fo failue equence BCA, CAB, CBA AB AC = A ( B A )( C ) + A ( C A )( B ) A + B A + C AB BC + B ( A B )( C ) + B ( C B )( A ) A + B B + C AC BC + C ( A C )( B ) + C ( B C )( A ) A + C B + C = A B C ( A + B + C ) µ = µ i U =. A B C
Seie Paallel Sytem Find the fequency of failue, the aveage duation of failue, and the unavailability of the ytem below. All the component ae identical with an aveage epai time of 0 hou and failue ate of 0.05 failue pe yea. 3 4 5 - Netwok Reduction - Minimal Cut-et Appoach
Minimal Cut Set Appoach Appoximate Equation Reliability netwok conit of cut et (in eie) Seie Equation Cut et conit of component (in paallel) Paallel Equation identical component = 0 h = 0.05 f/y 5 3 4 Cut Set (f/y) (h) U (h/y) & 5 5 ( + 5 ) 5 /( + 5 ) =.4E-5 = 0 =.4E-4 = U = U
Appoximate Equation Maintenance µ µ i = failue ate, f/y i = aveage epai time, h i = maint outage ate, o/y i = aveage maint time, h Sytem failue occu if, Event A: OR Event B: out fo maintenance followed by failue of duing the maintenance time of out fo maintenance followed by failue of duing the maintenance time of Event p p U p A ( ) " ( )( B ( ) "+ " "+ ( ) ( " "+ " "+ ) ) Failue event duing maintenance, p = ( ) + ( ) U p = ( )( p = U p / p " "+ ) + ( ) ( " "+ )
Maintenance (continued) Thi technique can be extended to include 3 o moe paallel component. Sytem policy mut be conideed to decide if maintenance will take place when anothe maintenance o failue outage aleady exit. Oveall ytem failue i contibuted fom - component failue when no maintenance - component failue duing maintenance ince both contibution caue ytem failue, pinciple of eie ytem can be ued T = + U T = + = U + U T = U T T
Common Mode (Caue) Failue imultaneou outage of o moe component caued by a ingle extenal event -component paallel ytem U U D U µ µ µ µ µ U D 4 D D =. ( + ) + = p + = + + U =.
Example: µ = 693.5 epai/yea µ = 839.5 epai/yea µ = 36.5 epai/yea
Common Mode (Caue) Failue U U D U µ µ 5 4 D D D D µ µ µ U D =. ( + ) + = p + U = + = U/
Example.5: A econd ode minimal cutet of a ytem contain identical component with = 0. f/y, = h and a common mode failue ate of 0.0 f/y. Evaluate the value of ytem, and U if a) only independent failue ae conideed. b) common mode failue ae conideed uing the 4-tate model with µ = 36.5 epai/yea. c) common mode failue ae conideed uing the 5-tate model with = 8 h. Cae (a) Cae (b) Cae (c) (f/y). ( + ) = 0.000096. ( + ) + = 0.00 U/ (h) + + + = 7.94 = 6 = 5.85 U (h/y). = 0.0006575. = 0.8 + = 0.36 - although i only 0% of the independent failue ate, the ytem i dominated by the common mode failue - The ytem i the ame fo both the 4-tate and the 5- tate common mode failue model - In the 4-tate model, ytem ovelapping outage time aociated with the independent failue - In the 5-tate model, ytem