Maritime University of Szczecin
Akademia Morska w Szczecinie
2012, 32(104) z. 2 pp. 115–122 2012, 32(104) z. 2 s. 115–122
Components validity evaluation in a complex technical
structure
Zbigniew Matuszak
Maritime University of Szczecin, Faculty of Mechanical Engineering, Department of Power Plant Exploitation 70-500 Szczecin, ul. Wały Chrobrego 1–2, e-mail: z.matuszak@am.szczecin.pl
Key words: measurement of validity, flow of streams, energy factors Abstract
The described in literature components validity measures (the order of the smallest cross section measurement of streams, number of occurrences of events in the Fault Tree, Birnbaum, Veseley-Fussell’s measures, the potential of reliability improvement, Lambert’s measure ...) relating mainly to the concepts of reliability and reliability structures is characterized. Concepts of studying functional structures of complex technical systems are presented. Directions and the amount of streams of energy factors flow in the systems will be analyzed. The analysis will be performed for the so-called dynamic functional structures. The validity and importance of the component in the system is assessed on the basis of the number of streams of energy factors which flow to and from the component. The greater the number of those streams, the more responsible for the operation of the system in this state is the component.
Introduction
During the operation of complex technical sys-tems it is often required to identify the component or device that plays the most important part in the system. Until now, special attention was paid to system reliability analysis to select, as a result, the minimum components or sections that are most important for the operation of the system [1]; for obtaining through the system the optimal values of reliability measurement [2, 3]. These issues in the reliability analysis are related to the problem of searching for weak links in the system and are called the validity analysis. From the view point of reliability, the validity of a particular component of the system depends on: the reliability of a particular component, the reliability structure, in which the component is located.
The effect of the first factor is obvious from the view point of the location of the component in the reliability structure; the closer the structure, in which the component is located to the reliability serial structure, the more important the component is. The component validity decreases with the in-crease of the level of reserving. Literature describes the existing qualitative and quantitative validity measurements, which, however, refer mainly to the
analysis of different types of reliability structures and various methods for their analysis [4, 5, 6, 7, 8, 9, 10, 11].
In order to determine accurately the degree of validity of the components in the system, the so- -called components validity measurements have been constructed. There are many of them, and each of them recognizes the problem in a different way. Selection of an appropriate measurement for a specific practical application largely depends on the fact, what aspect of the component validity is in this case most important. Unfortunately, measure-ments of components validity, reported in the litera-ture, as it will be shown hereinafter, refer mainly to the concepts of reliability and reliability structures. In literature, there is lack of methods to determine the component validity in a complex technical structure, taking into account issues other than the information of the components and structures relia-bility.
Order of the smallest cut set
In case of a technical system analysis using fault trees, to determine the component validity one can use the so-called order of the smallest cut set method. In qualitative analysis, the validity of
a particular set usually depends on the number of components in this set. This number is called order of minimal cut set. Very often minimal cut set of the first order is more important (critical) than cut sets of higher orders. If the system has a one- -component cut set, the component failure brings down the system state. This case is related to com-ponents located in serial reliability structure.
Order of the smallest cut set with the i-th com-ponent is given by the IO(i) quantitative measure.
Let C1i, C2i, ..., Cni be describing all cut sets with
the event Ei, then:
)] ( [ min ) ( ,... 2 , 1i i ni k k O i card C I (1)
the value of IO(i) does not depend on the
compo-nents reliability characteristics. For the analysis of systems modeled by means of fault tree, one can use measurement similar to the number of occur-rences of the i-th event in the fault tree [12]. Often the more important component exists in more cut sets.
Another important factor in the qualitative ana-lysis of components validity is the rank of primary events in a given minimal cut set [13]. This is based on the assumption that human errors are more common than failures of active components and active components failures are more common than failures of passive components. Based on the events ranks one can build two- and more-component cut set rankings consisting of various types of events [12].
Birnbaum reliability measurement IB(i|t)
Accepting that r(t)[r1(t),r2(t),...,rn(t)] is the vector of the system components reliability at the moment t, and R[ tr()] is the reliability of the sys-tem, which depends on the reliability of individual components and the system reliability structure, Birnbaum reliability measurement for the i-th com-ponent of the system is defined as [7]:
) ( )] ( [ ) ( )] ( [ ) | ( t f t r F t r t r R t i I i i B (2) where: )] ( [ 1 )] ( [r t Rr t
F – the function of the
sys-tem reliability at the moment t;
fi(t) – probability density function of the time
period before failure for the i-th compo-nent.
Due to the models used in the method of fault trees, it is convenient to express Birnbaum meas-urement for the i-th component at the moment t, as a derivative partial function of the Q0(t) system
unavailability with a respect to operational unavail-ability of the i-th component:
) ( )] ( [ ) ( ) ( ) | ( 0 t q t q Q t q t Q t i I i i B (3) where: )] ( ),..., ( ), ( [ ) (t q1 t q2 t q t
q n – the vector of
una-vailability of the system operational components at the moment t;
)] ( [ ) ( 0 t Q q t
Q – the system unavailability. For numerical analysis the following depend-ence describing the Birnbaum reliability measure-ment can be applied:
) ( ) ( ) | ( 1 t q t Q t i I i m j j B i
(4) where: ) (tQj – the partial unavailability of the j-th minimal cut set containing the i-th com-ponent;
mi(t) – the number of minimal cut sets
contain-ing i-th component;
qi(t) – the operational unavailability of the
i-th component.
If the IB(i|t) measurement has a large value at the
moment t for the i-th component, a small change in the value of its operational unavailability qi(t), leads
to great value changes in the system unavailability Q0(t). In turn, if R[r(t)] is the linear function of r(t),
and when all system components are statistically independent, then the IB(i|t) measurement for the
i-th component is not dependent on the reliability of ri(t), it is dependent only on the reliability of other
components and system reliability structure.
After unfolding the previously quoted structure function into the components due to the state of the i-th component (Xi = 1 if the component is working
properly and Xi = 0 if it is unsuitable), we obtain:
] 0 ), ( [ ]} 0 ), ( [ ] 1 ), ( [ ){ ( ] 0 ), ( ][ ) ( 1 [ ] 1 ), ( [ ) ( )] ( [ i i i i i i i i X t X X t X X t X t X X t X t X X t X t X t X (5)
then assuming components independence after the introduction of the expected structure functions value, the system reliability can be described as follows: ]]} 0 ), ( [ [ ]] 1 ), ( [ [ { ) ( )] ( [ i i i X t X E X t X E t r t r R (6)
where: ]] 1 ), ( [ [ X t Xi
E – the system reliability with
the efficient i-th component R[ri(t)1,r(t)]; ]] 0 ), ( [ [ X t Xi
E – the system reliability with
the i-th component in the state of incapacity )] ( , 0 ) ( [r t r t R i .
Then the Birnbaum measurement can have the form:
r t r t
R t r t r R t r t r R t i I i i i B , 0 , 1 | (7)Similarly, the record using the notion of the system unavailability can be represented:
)] ( , 0 ) ( [ )] ( , 1 ) ( [ ) ( ) ( ) | ( 0 t q t q Q t q t q Q t q t Q t i I i i i B (8)
The equation above is often used to estimate the IB(i|t) measurement in computer programs such as
in CARA Fault-Tree. Referring the above formula to the fault tree one can conclude that the Birnbaum measurement defines the difference between the probability of the peak event, when the i-th compo-nent fails at the moment t, and the probability of occurrence of peak, when the i-th component is fit at the moment t. This difference can be interpreted as the probability that the i-th component is critical for the system at the moment t.
Birnbaum structural measurement B(i) Birnbaum structural validity measurement for the i-th component is defined as the relative number of system states for which the i-th component is critical for the system. This measurement can be described by the following relation [7]:
1 2 ) ( ) (i ni B (9) where:
(i) – represents vectors the total number of critical tracks for the component i. Critical track vector for the i-th component is any vector of the components state for a system that operates if and only if the i-th component is in a fitness state. Birnbaum structural measurement serves designation of the relative number of system states (for all components other than the i-th), which give to the i-th component the status of a critical component.
It can be shown that if all components of the operating system have the operational unavailabil-ity qi = 0.5, then the equality B(i) = IB(i|t0) takes
place.
Vesely-Fussell validity measurement IVF(i|t)
The Vesely-Fussell validity IVF(i|t) measurement
for the i-th event is defined as the conditional prob-ability that there will be at least one minimal cut set containing the i-th component at the moment t, assuming that the system goes into a state of unfit-ness at the moment t [4].
It is assumed that [X(t)] is the function of the system structure, taking the 0 value when the sys-tem fails, and 1 if it is fit. It is set to X(t) zero-one vector, the components of which take the 0 value, if the component fails, and 1 when the component is working properly. Denoting as mi – the minimal
number of cut sets containing the i-th component; Cij(t) – the j-th minimal cut set containing the i-th
component damaged at the moment t; Di(t) – a
col-lection containing at least one Cij(t) cut set, which is
unsuitable at the moment t, the reasoning can be written in the form:
) ( ... ) ( ) ( ) (t C1 t C2 t C t Di i i imi (10)
The Vesely-Fussell measurement definition is then as follows: } 0 )] ( [ | ) ( { ) | (i t P D t X t IVF i (11)
The Vesely-Fussel measurement can be inter-preted as the probability that the event peak occurs due to the incapacity of the i-th component, assum-ing that the peak event has occurred.
For numerical analysis the reproduced below approximate dependence describing the Vessely- -Fussell measurement is usually accepted:
) ( ) ( ) | ( 0 1 t Q t Q t i I i m j j VF
(12) where: ) (tQj – the partial unavailability of the j-th minimal cut set containing the i-th component,
Q0(t) – the system unavailability.
It should be noted that for a system of n-com-ponents parallel structure (primary events are con-nected with the AND gate directly generating the peak event) all the components are involved in the occurrence of the peak event, and then:
1 ] | ) [( ... ] | ) 2 [( ] | ) 1 [( } 0 )] ( [ { ) ( ... ) ( ) ( 2 1 t n i I t i I t i I t X t D t D t D VF VF VF n (13)
The ICR(i|t) critical measurement
The ICR(i|t) critical measurement is defined as
the probability that the i-th component is critical for the system and fails at the moment t, assuming that the system fails at the moment t [8]. Using the pre-viously adopted symbols of reliability critical measurement that is the conditional probability of a Cr[X(t),Xi 1][Xi(t)0] event, if the system fails at the moment t, it is described by the follow-ing dependence: } 0 )] ( [ | ] 0 ) ( [ ] 1 ), ( [ { t X t X X t X Cr P I i i CR (14) or } 0 )] ( [ { ]} 0 ) ( [ ] 1 ), ( [ { t X P t X X t X Cr P ICR i i (15)
Finally, one can bind the critical measurement with the Birnbaum reliability measurement in the following dependence: ) ( ) ( ) | ( ) | ( 0 t Q t q t i I t i ICR B i (16)
Potential for improving the IIP(i|t) reliability
Potential for improving the IIP(i|t) reliability,
specified for the i-th component is as a measure-ment defined as the system reliability increase, if the i-th component at the moment t will be replaced by an ideal component. If the actual system reliabil-ity is determined as R0(t), and the system reliability
with the i-th component of the reliability of ri = 1
as R0(ri=1)(t), then the potential of improving the
reliability can be presented as [8, 14]: IIP(i|t) = R
0(ri=1)(t) – R0(t) (17)
As mentioned previously, the Birnbaum meas-urement can be interpreted as the probability that the i-th component is critical for the system at the moment t. If Cr[X(t),Xi 1] represents the event, that the system is in a state in which the i-th com-ponent is critical and this event is independent on the component state, then:
) | ( ]} 1 ), ( [ {Cr X t X I i t P B i (18)
The potential reliability increase can be inter-preted as the probability that the i-th component is critical and fails at the moment t, which can be expressed with the formula:
]} 0 ) ( [ ] 1 ), ( [ { ) | (i t PCr X t X X t IIP i i (19)
After substituting the expression (18) to the above equation and after the introduction of operat-ing unavailability qi the dependence between the
Birnbaum reliability measurement and potential for improving reliability is obtained:
) ( ) | ( ) | (i t I i t q t IIP B i (20)
Substituting the expression (4) to the above equation one obtains:
mi j j IP i t Q t I 1 ) ( ) | ( (21) where: ) (tQj – the partial unavailability of the j-th minimal cut set containing the i-th com-ponent;
mi(t) – the total number of minimal cut sets
containing i-th component in the given fault tree.
Measures of validity independent of time
Historically, the first measurements of validity were proposed by Birnbaum. It should be noted that the measurement of the component ci validity at the
moment t can be defined as follows [8, 14]:
t h
l R
t h Rt
Ii i i B , 0, (22) where:
l Rt
h
R
t R
t R R
t
h i, 1 ,..., i1 ,, i1,..., n (23) It should be noted that h
li,R
t
is the system reliability, when the component ci is fit, h
0i,R
t
is the system reliability, if ci is unfit (at the moment
t). Thus, the formula (22) determines the loss of system reliability due to the failure of the compo-nent ci. It can also be interpreted as the probability
that the failure of the component ci at the moment t
will lead to the system failure or as the probability that at the moment t the system is in a state, for which the component ci is critical.
An equivalent definition of the Birnbaum meas-urement is as follows:
t R R i i B j j R R h t I (24)From the above formulas it follows that the Birnbaum measurement of the component ci
validi-ty does not depend on the reliabilivalidi-ty of this compo-nent, but only on the system reliability structure, time and reliability of the other components.
There are several validity measurements inde-pendent on time. Most of them are weighted Birn-baum measurements. The most famous of these is the measurement bearing the name of Barlow- -Proschan [5, 6]: f
t I
t t I i B i i P B d 0
(25) where:
i' i t Ff is the failure density of the compo-nent ci, 1 0 i P B I (26)
n i i P B I 1 1 (27)The Barlow and Proschan measurement is equal to the probability that the reason of system failure is the component ci failure. This measure can also be
treated as the Birnbaum averaged measurement due to Fi(t). Also popular is a type of measurement
described by the formula (25), different in the fact that the integration takes place in the interval [0, t]. This means that this measurement makes the probability that the system fails by the moment t and that the cause of system damage is the compo-nent ci.
Another measure of the system components validity was proposed by Natvig. According to the proposal, the component validity measurement is the loss of the remaining operation time of the sys-tem up time due to the failure of the component in question. If the system components fail inde-pendently of each other, the Natviga measurement has the form [10, 11]:
k
R
t
R
t
I
t t I i B i i N i N ln d (28)kn factor is necessary to ensure the I sum to N i
unity.
Another measurement of validity, independent of time, is the Bergman measurement:
0 d t t I t tf k I i B i E i E (29)whose usefulness is comparable to the Barlow- -Proschan measurement and the Natvig measure-ment.
A more general value has the i P I measurement defined as:
0 dF t t I t k I i i B p p i p (30)The Barlow-Proschan measurement and the Bergman measurement are special cases of meas-urement defined in formula (27) as:
i P B p i p I I 0 (31) i E p i p I I 0 (32)
On the basis of the concept of a so-called critical component validity, validity measurement of an component in a system was introduced by Lambert.
The component ci is critical if:
a) the system is working properly when the com-ponent ci is fit;
b) the system is damaged, if the component ci is
unfit.
In the system with serial structure all the components are critical. In case of other structures an component becomes critical, if all the other components of the cut sets, to which the component belongs, fail.
The probability that at the moment t the system is in such a state that the component ci has failed by
the moment t, is equal to:
hli,Rt h0i,Rt
Fi t (33) Considering the above-defined events, provided that the system has failed by the moment t, one obtains the Lambert validity measurement in the form:
Rt h t F t R h t R l h Ii i i i L 1 , 0 , (34) ConclusionAll existing validity measurements are based on the knowledge of components reliability and relia-bility structures. There are no methods determining the validity of an component in a complex technical structure taking into account other parameters than the information on the components and structures reliability.
In the papers [15, 16, 17, 18, 19, 20, 21, 22] the author proposes using a method based on the evalu-ation of the amount of streams of energy factors flow through individual components (devices) in the functional structure of the technical system. In figure 1 a fragment of the functional structure of the system consisting of nine components in state 1 (system operation allows changing the functional structure) is presented. Other states of this system operations are shown in figures 2 and 3.
1 2 4 water power air 3 9 5 8 7 6 water water water water water water power water power water air air air air
Fig. 1. State of operation 1 of the technical system in question
1 2 9 5 7 3 4 8 6 water water water water water water water power power water power water air air air air air air power air
Fig. 2. State of operation 2 of the technical system in question 1 2 9 5 air power 3 4 8 7 6 water water water water water water water power air water power water air air air air air power air power
The basis of research is to analyze the functional structures of complex technical systems. Directions and the amount of streams of energy factors flow in the systems will be analyzed. The analysis will be performed for the so-called dynamic functional structures. These are structures, in which during operation directions of streams of energy factors flow, the amount of streams of energy factors and the number of working components of the system are changed. Based on these changes one can dis-tinguish different states of the system. For individ-ual states of work tables of streams of energy fac-tors flow in the functional system structure are worked out (Tab. 1–3).
The validity and importance of the component in the system is evaluated based on the amount of
streams of energy factors that flow to and from the component. The greater the number of these streams, the more responsible is the component for the operation of the system in this state. The validi-ty of an component in the structure is described by the total sum of streams of energy factors IKM(i) (the
stream measurement), which flows to and from the component so. The dependence can be described as:
) ( ) ( ) (i s i s i IKM i o (35)
or for the representation of that measurement by a number from the interval 0,1:
)] ( ) ( [ ) ( ) (i k I i k s i s i I KM KM i o KM M (36)
Table 1. Streams of energy factors flow in the functional structure for state of operations 1
to Component number from 1 2 3 4 5 6 7 8 9 The sum of streams from
1 – 1p – – – – – – 1w 2 2 – – – – – – – – – – 3 1p, 1w 1p – 1a, 1w – – – – – 5 4 – – – – – – – 1w 1a 2 5 – – – – – 1w 1a 1w – 3 6 – – – – – – 1w – – 1 7 – – 1a – – – – – – 1 8 – – – – – – – – – – 9 – 1w – – 1a, 1w – – – – 3
The sum of streams to 2 3 1 2 2 1 2 2 2
a – air stream; p – power stream; w – water stream
Table 2. Streams of energy factors flow in the functional structure for state of operations 2
to Component number from 1 2 3 4 5 6 7 8 9 The sum of streams from
1 – 1p – – – – – – 1w 2 2 – – – – – – – – – – 3 1p, 1w 1p – 1a, 1w – – 1p – – 6 4 – – – – – – – 1a, 1w 1a 3 5 – – – – – 1w 1a 1w – 3 6 – – – – – – 1w – – 1 7 – – 1a – – – – – – 1 8 – – – – – – – – – – 9 1a 1w – – 1a, 1w – – – – 4
The sum of streams to 3 2 1 2 2 1 3 2 2
a – air stream; p – power stream; w – water stream
Table 3. Streams of energy factors flow in the functional structure for state of operations 3
to Component number from 1 2 3 4 5 6 7 8 9 The sum of streams from
1 1p – – – – – – – 1p, 1w 3 2 – – – – – – – – – 3 1p, 1w 1a – 1a, 1w – – – – – 5 4 – – – – – – – 1a, 1w 1a 3 5 – – – – – 1a, 1w 1a, 1w 1w – 5 6 – – – – – – 1w – – 1 7 – – 1a – – – – – – 1 8 – – – – – – – – – 9 – 1w – – 1a, 1p, 1w – – – 4
The sum of streams to 3 2 1 2 3 2 3 3 2
where:
n i KM KM I i k 1 1 )] ([ – the factor to ensure summation to unity; n – the number of components in the system.
The assumed research procedure is as follows: – the choice of complex technical systems with
variable functional structure to be analyzed; – preparation of functional structures for different
operating states;
– drafting tables of streams of energy factors flow in the functional structures of the system for dif-ferent operating states of the system;
– mathematical description of the components validity on the basis of tables of streams of energy factors flow;
– developing validity measurements of the system components based on the streams of energy factors flow;
– estimation of selected validity measurements based on methods represented in the previous chapter;
– comparing the results obtained (the important components) by means of the proposed method of streams of energy factors flow based on com-ponents reliability and reliability structures of the system;
– an attempt to develop the method of components validity measurement in a complex technical system, binding the method of streams of energy factors flow with the method based on compo-nents reliability and reliability structures evalua-tion.
The objects of research will be complex auto-nomous energy systems, particularly systems of marine engine rooms. Preliminary results of the research of the described validity measurement, presented in [15, 16, 17, 18, 19, 20, 21, 22] give an optimistic outlook on the further development of this method.
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