The alpha factor method, as described here, was adapted from NUREG 4780, Volume I, Section 3.3 [25] and Volume II [26], specifically pages C-4 through C-6 and
current component composed inside of the set 1≤ ≤k m. The symbol, m, is the total number of components for that particular cut set. In the STPNOC case for looking at EDG failure, m would be equal to three as there are three EDGs. Basic events may also be refered briefly as “k-way (common-cause) failure”, except that the case k=1 shall often be termed as an “independent failure”. The probability (for demand-based failure) or frequency (for time-based failure) of a k-way common-cause failure of some specific set of k components, and only those k components, is designated as Qk( )m . To simplify the language, this section will use probabilities, but the discussion applies equally to frequencies, with slight changes of language.
These probabilities of basic events involving failure of a specified set of components, and only those components, are the basic parameters of any system model, in that system (or subsystem) failure that can be expressed in terms of them. For example, the subsystem of interest in this case is an emergency AC power system comprised of three identical Emergency Diesel Generators. Therefore, the minimal cut sets for a 3-out-of-three, all EDGs have to fail; a success logic gate based of a failure of the system can be seen in Equation (5.30).
In Equation (5.30), the symbols A BI, ,I andCIrepresent independent failures (“IndependentFs”) of components (EDGs) one (“A”) through three (“C”), respectively.
The variables cAB, cACandcBC represent a common-cause doublet failure (“2/3CCFs”) for component A and B, A and C, and B and C, respectively. The variable cABC represents a common-cause triplet failure (“3/3CCFs”) for component A, B, and C.
Therefore the probability of each type of failure can then be determined which are shown in Equation (5.31).
(3)
The total probability of the system, ( )P s , and then be determined based on the probabilities and the cut sets which is shown in Equation (5.32)
( ) ( )* ( ) * ( )
Applying Equation (5.31), the system failure probabilities can be altered, resulting in Equation (5.33).
( )
1(3) 33
2(3) 1(3) 3(3)Q
S= Q + Q Q + Q
(5.33)The three terms on the right in Equation (5.33) represent respectively: system failures of each of the three EDG trains independently, failures due to a common-cause doublet failure of one of the (three) pairs of EDGs accompanied by an independent failure of the third, and failures caused by a single common-cause triplet failure.
Other types of failure events are conceivable, but commonly are not considered.
An example would be two common-cause doublet failures. This can give rise to inconsistencies in system models. For our purposes, the system model is Equation (5.33).
The only types of basic failure events considered are those whose probabilities (or frequencies) appear here, and these basic events are considered mutually exclusive. See page C-2 of NUREG 4780 [26] for further discussion on this matter.
For reference, the need is now to define several additional quantities and express
once the required component has been selected. The probability of k-way failure that includes some specific component is shown in Equation (5.34).
1 ( )
Equation (5.34) uses the combination rule of statistics. This rule is shown as the binomial coefficient, which is displayed as Equation (5.35).
!
Similarly, the probability of some k-way basic event is shown in Equation (5.36).
( )m
From Equation (5.34), the quantity defined in Equation (5.37) is the probability of some k-way basic event that includes failure of some specified component.
( )
Using the previous Equations (5.36) and (5.37), one can determine an “alpha factor” shown in (5.38). This alpha factor is defined as the probability of some k-way basic event occurring over the probability of some basic event occurring; i.e., the probability of a k-way basic event occurs, given that a basic event occurred.
( )
Equation (5.38) uses Equation (5.36) and the fact that every basic event involves some positive number of component failures. Therefore, it can be noted that Equation (5.38) also represents the probability of some k-way basic event given that some basic event occurs.
Equation (3.39) represents an expected number of component failures given that some basic event occurs designated byαt.
( )
An algebraic manipulation of Equations (5.39) and (5.37) can be performed to get a better interpretation of αt. To start this approach, Equation (5.39) can be written in the form of Equation (5.40).
( ) ( ) 1 ( )
To understand Equation (5.40) is important to solve for αt as shown in Equation (5.41). This represents the size of common-cause component group multiplied by the probability of some k-way basic event that includes failure of some specified component over the probability of some basic event occurring.
( ) ( )
( ) ( ) ( )
Equation (5.42) can then be simplified into Equation (5.43) using the definition shown in Equation (5.39).
( )
To better understand, the Equation (5.43) is split into two parts as demonstrated in Equation (5.44) using the definition shown in Equation (5.38).
( )
The right side of Equation (5.44) can be defined as the probability of some k-way basic event. The left side of Equation (5.44) is then defined as the probability of some k-way basic event over the probability of some basic event; multiplied by, the probability of some k-way basic event that includes failure of some specified component; divided by, the probability of some k-way basic event that includes failure of some specified component over the probability of some basic event.
Equation (5.44) can then be solved for the probability Qk( )m . The result is displayed in Equation (5.45).
( )
In any application of PRA it is the Qk( )m that are ultimately required, for the insertion into some system failure equation, such as Equation (5.33). In the alpha factor method it is the αk( )m and Qt that are supplied to the analyst from industrial averages [19,20,23]. These data values then permit the computation of Qk( )m as seen in Equation (5.45). The result is Equation (C.23) from NUREG-4780 Volume II [26].
Equation (5.45) can be written out in a more simple form which is shown Equation (5.46) as it appears identical to this particular equation shown in Table C-1 of the NUREG 4780 [26]
Therefore, the example of three EDGs, the specific failure probabilities for independent, common-cause doublet, and common-cause triplet based off the alpha factors, are shown in Equations (5.47) through (5.48), respectively.
(3) 1 1
1 2 2 3 3
α αt = + α + α (5.50)
Plugging in these values found in Equations (5.47)-(5.50) into Equation (5.33) will reveal the formula shown in (5.51).
3
3 2 3
1 3( 2 1 ) 3
s t t t
t t t t
Q α Q α α Q α Q
α α α α
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞
=⎜ ⎟ + ⎜ ⎟⎜ ⎟ + ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ (5.51)
Equation (5.51), like in Equation (5.33), has the three terms on the right-hand side. These terms still represent respectively system failures of each of the three EDG trains, failures due to a common-cause doublet failure of one of the (three) pairs of EDGs accompanied by an independent failure of the third, and failures caused by a single common-cause triplet failure.