Safety Instrumented Systems Verification: Practical Probabilistic Calculations

1oo1 Architecture

A single PEC (Figure F-2) represents a minimum system. No fault tolerance is provided by this system. No failure mode protection is provided. The electronic circuits can fail safely (outputs de-energized, open circuit) or dangerously (outputs frozen or energized, short circuit). Since the effects of on-line diagnostics should be modeled, four failure categories are included:

  • DD - dangerous detected,

  • DU - dangerous undetected,

  • SD - safe detected, and

  • SU - safe undetected.


Figure F-2: 1oo1 Architecture

PFD Fault Tree for 1oo1

Figure F-3 shows the fault tree for dangerous failures. The system will fail dangerously if the unit fails dangerous detected (DD) or dangerous undetected (DU).


Figure F-3: PFD Fault Tree for the 1001 Architecture

Using rough, first order approximation techniques, a simple formula can be generated from the fault tree for the probability of dangerous failure (probability of failure on demand), PFD (assuming perfect periodic test and repair)


where

  • RT = average repair time

  • TI = test interval for a periodic inspection

It should be pointed out that the approximation techniques are only valid for very small system failure rates.

Since many safety evaluations are done using average probability of failure on demand (PFDavg), the equation for PFDavg should be derived. The average approximation is given by:


substituting t = TI and integrating


integrating t ?


when evaluated gives


for the 1oo1 architecture assuming perfect test and repair.

When imperfect periodic test and repair is considered, the fault tree and the equation get more complicated. The equivalent...

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