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Fault Tree Analysis

Fault Tree Analysis. Part 12 – Redundant Structure and Standby Units. Active Redundancy. The redundancy obtained by replacing the important unit with two or more units operating in parallel. Passive Redundancy.

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Fault Tree Analysis

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  1. Fault Tree Analysis Part 12 – Redundant Structure and Standby Units

  2. Active Redundancy The redundancy obtained by replacing the important unit with two or more units operating in parallel.

  3. Passive Redundancy The reserve units can also be kept in standby in such a way that the first of them is activated when the original unit fails, the second is activated when the first reserve unit fails, and so on. If the reserve units carry no load in the waiting period before activation, the redundancy is called passive. In the waiting period, such a unit is said to be in cold standby.

  4. Partly-Loaded Redundancy The standby units carry a weak load.

  5. Cold Standby, Perfect Switching, No Repairs

  6. Life Time of Standby System The mean time to system failure

  7. Exact Distribution of Lifetime If the lifetimes of the n components are independent and exponentially distributed with the same failure rate λ. It can be shown that T is gamma distributed with parameters n and λ. The survivor function is

  8. Approximate Distribution of Lifetime Assume that the lifetimes are independent and identically distributed with mean time to failure μ and standard deviation σ. According to Lindeberg-Levy’s central limit theorem, T will be asymptotically normally distributed with mean nμ and variance nσ^2.

  9. Cold Standby, Imperfect Switching, No Repairs

  10. 2-Unit System • A standby system with an active unit (unit 1) and a unit in cold standby. The active unit is under surveillance by a switch, which activates the standby unit when the active unit fails. • Let be the failure rate of unit 1 and unit 2 respectively; Let (1-p) be the probability that the switching is successful.

  11. Two Disjoint Ways of Survival • Unit 1 does not fail in (0, t], i.e. • Unit 1 fails in the time interval (τ, τ+dτ], where 0<τ<t. The switch is able to activate unit 2. Unit 2 is activated at time τ and does not fail in the time interval (τ,t].

  12. Probabilities of Two Disjoint Events • Event 1: • Event 2: Unit 1 fails Unit 2 working afterwards Switching successful

  13. System Reliability

  14. Mean Time to Failure

  15. Partly-Loaded Redundancy, Imperfect Switching, No Repairs

  16. Two-Unit System Same as before except unit 2 carries a certain load before it is activated. Let denote the failure rate of unit 2 while in partly-loaded standby.

  17. Two Disjoint Ways of Survival • Unit 1 does not fail in (0, t], i.e. • Unit 1 fails in the time interval (τ, τ+dτ], where 0<τ<t. The switch is able to activate unit 2. Unit 2 does not fail in (0, τ], is activated at time τ and does not fail in the time interval (τ,t].

  18. Probabilities of Two Disjoint Events • Event 1: • Event 2: Unit 1 fails at τ Unit 2 still working after τ Switching successful Unit 2 working in (0, τ]

  19. System Reliability

  20. Mean Time to Failure

  21. Cold Standby, Perfect Switching, With Repairs

  22. Possible States of a 2-Unit System with Cold Standby and Perfect Switching

  23. State Space Diagram 4 3 2 0 1

  24. State Equations

  25. Eliminating the Failed State

  26. Laplace Transform • Substitute s=0 • Note that

  27. Solution

  28. Mean Time to Failure

  29. Mean Time to Failure • Take Laplace transform of R(t) • Substitute s=0

  30. Mean Time to Failure

  31. Cold Standby, Perfect Switching, With Repairs,A Main Operating Unit

  32. Possible States

  33. State Space Diagram 4 3 0

  34. State Equations Where

  35. Steady State Probabilities

  36. Availability and Unavailability

  37. Eliminate Failed State from State Equations Where

  38. Treating State 0 as An Absorbing State • Take Laplace transform and let s=0 • Solution

  39. Mean Times to Failure and to Repair • Mean time to failure • Mean time to repair

  40. Cold Standby, Imperfect Switching, With Repairs,A Main Operating Unit

  41. State Space Diagram 4 3 0

  42. Steady State Probabilities

  43. Availability and Unavailability

  44. Mean Time to Failure

  45. Partly-Loaded Standby, Perfect Switching, With Repairs,A Main Operating Unit

  46. Possible States of a 2-Unit System with Partly-Loaded Standby and Perfect Switching

  47. State Space Diagram 4 3 0 1

  48. Steady State Probabilities

  49. L Spares, With Replacements and Repairs

  50. State Space Diagram 0 2 2j 2L 1

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