# Protection Mechanism in Reliability Evaluation Approach to Multistate System with Common Cause Failure

Authors: Jinlei Qin; Zheng Li; Youchan Zhu
Journal: Complexity
 (2021)

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## Abstract

Except the two types of state, complete failure and perfect functioning, some intermediate states also existed between those two states. This characteristic enables multistate system (MSS) whose components are often of more than two states. Moreover, because of dependence between components, common cause failure (CCF) enhances the failure risk of components during the operating period of MSS. A protection mechanism needs to be adopted because of the importance of certain components. For a MSS with CCF and protection mechanism, its reliability can be evaluated by a proposed approach. Certain components not only can cause failure at one state, but also can cause failure to other components because of the CCF. Under the impact of one type of CCF, those components can constitute a specific group that can overlap with certain components. Universal generating function (UGF) is adopted to incorporate the CCF and protection mechanism of a component. Consequently, reliability indices of a MSS can be evaluated by the proposed approach based on UGF. Examples and comparisons illustrate and validate this method. This method can also help engineers prioritize elements that require protection mechanism under various resource limitations.


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## Body

## 1. Introduction

With the development of industrial technology, product quality is being considered as an important aspect for customers. In addition, the traditional reliability theory cannot be applied in the modernized industrial system and/or component because of its limitation to two states. For example, certain systems (such as those in the manufacturing, production, and oil and gas industry, and energy systems whose total performance can remain at different stages depending on the operating conditions) are referred to as multistate system (MSS) [1]. The analysis of MSS entails the development of new representation, modeling, and quantification technologies. A few researchers had focused on the above-mentioned aspects [2–5]. A new approach based on logical differential calculus is proposed for importance analysis and application of MSS, where MSS states change depending on simultaneous changing of some components’ states [6, 7]. Other technologies including these for assessment and optimization can be applied to MSSs in industries [8–11].Multiple components fail owing to one cause, and this type of failure is named common cause failure (CCF). In real-life industry fields, it often occurred. Unprotected electronic equipment can experience outage in extreme weather conditions such as lightning events. And multiple failures of components often originate from voltage surge because of inappropriate switching in power system. CCFs can be divided into two types according to the origin of the cause: internal CCF (e.g., originating from propagated failures than from one component or multiple failures within system) and external CCF (e.g., abrupt variations in the environment such as those in the temperature and humidity or fire caused by lightning) [12, 13]. From the perspective of maintenance, an implicit two-stage approach was used to analyze the reliability of a nonrepairable MSS incorporating the CCF [14]. Using recursive and exact methods, the authors developed the reliability evaluation procedure for a nonrepairable system, wherein an individual component’s failure could cause CCFs of other components [15]. However, in many real-life scenarios, repair is generally required to improve the reliability and/or availability of a system. An instantaneous availability model aimed at repairable MSSs with public bus performance sharing was suggested [16]. With regard to other aspects, such as the predictive inference of system reliability, a nonparametric approach was developed for a system with CCF [17].CCF reduces the reliability of a technology system because of its interdependent failures (i.e., failures depend on other components). Interdependent failures are also called “interactive failure” and can be categorized into two types based on the consequences of failure of a component in the system: immediate failure dependence and gradual degradation dependence [18]. Life cycle cost and reliability were analyzed considering interdependent degradation [19, 20]. The reliability analysis of MSS with statistics dependent components was studied from the perspective of statistics dependence that exists in the physical and complicated systems [21]. Owing to the inflection of element’s performance distribution from those of other elements or subsystems, the dependency of elements is considered to evaluate MSS reliability based on universal generating function (UGF) technology [22]. Propagated failures are taken into account when a recursive approach was proposed for reliability analysis of MSS with CCF [15, 23]. System reliability can be improved using redundancies, fault detection, and isolation mechanisms. Nevertheless, once the above methods fail, certain components’ failures cannot be detected. This phenomenon is named as imperfect fault cover (IFC). Stochastic multivalue models are put forward to achieve the availability of MSS with the incorporation of IFC [24]. An irrelevancy coverage model is also put forward to cover the irrelevant components, as well as the faulty components [25]. For assessing the risk of large cascading blackout, based on random chemistry, a new approach with high computational efficient is reported [26]. A dynamical stochastic is developed to explore the spread of flowing failure in the communication grid [27]. An MSS and component with CCF were modeled based on the dynamic Bayesian network to perform reliability assessment [28, 29]. A reliability analysis method based on the evidence network was investigated to address the epistemic uncertainty of complex MSS with CCF [30]. Although many other researchers have focused on dependent failure during reliability assessment, inadequate consideration has been paid to the protection of important elements. Risk analysis was applied to reduce the failure risk of industrial systems in related fields [31–33]. Meanwhile, performance optimization and assessment of certain wireless networks and power line engineering were also investigated [34, 35]. Although a rich body of research has been conducted for addressing effects of CCF in reliability analysis of diverse types of systems, protecting mechanism on certain important elements during the reliability assessment of an MSS with CCF is seldom focused on. One contribution is to build protecting mechanisms, which are implemented and extended for improvement of system reliability based on earlier studies. The other contribution is to develop a recursive procedure, which is based on the straightforward and conveniently programmable UGF technology. In this paper, the procedure aims to evaluate the reliability of an MSS with CCF under the protection measures implemented on certain important elements. Here, the protection measures can be considered as a special mechanism. For example, certain components need to be protected and isolated from fire caused by other components. A firewall would be set up to protect the important components.The remainder of this paper is organized as follows. Section2 presents the MSS model. The UGF method for estimating the system performance distribution and reliability indices is presented in Section 3. In addition, a specific method to evaluate an MSS with CCF under protection measures is developed in this section. Section 4 presents several examples to illustrate the validity of the approach. Section 5 concludes this paper.


## 2. Modeling of MSS

One of the most important characteristics of an MSS is the multiple states of itsn components, which are generally connected in series-parallel. The performance level s of component i can vary from zero (complete failure) to mi (perfect function). We denote each performance level by his, and all the levels can form the vector hi=hi0,hi1,…,his,…,himi. Furthermore, the performance levels of the overall system would adopt values from the following set, which is obtained as the Cartesian product of all hi:(1)Hn=h1×h2×⋯×hi×⋯×hn.The performance level of componenti at a specified moment will be also assumed to be a discrete random variable Hi∈hi. In addition, Hi would adopt the value his at a certain probability pis such that pis=PrHi=his. All the probabilities pis can constitute a set(2)pi=pi0,pi1,…,pis,…,pimi.Here,∑s=0mipis=1. That is, component i could be in one, and only one, state at any time because all the mi+1 states establish a group that is composed by mutually exclusive events.Here, one assumption is that certain external reasons, such as lightning, which can cause failure or degradation of the system, will not be considered. Based on this situation, the state of an MSS can be ascertained explicitly from the state of its components.The states of an MSS are denoted asj∈0,1,…,M; i.e., the MSS would have M+1 states. Similarly, state 0 denotes complete failure, state M denotes the perfect function, and the others denote the intermediate states. If wj denotes the performance level of an MSS at system state j, all the MSS performance levels can constitute the following set L:(3)L=w0,w1,…,wj,…,wM.Suppose thatW is a discrete random variable to denote certain performance level of MSS and can adopt values from L. Combining (1) and the above modeling of components and MSS, the system function of the MSS can be written as(4)W=ΦH1,H2,…,Hi,…,Hn:Hn⟶L.Essentially, this formula is to map the performance level space of components to the one of system. Likewise, the probability of system at certain performance level can be denoted by(5)qj=PrW=wj,j=0,1,…,M.Several reliability indices can be derived from the above model to quantify the reliability of an MSS. In general, CCFs can be divided into two groups based on the origin of their causes (external and internal causes). CCFs belonging to the latter group are frequently referred to as propagated failures because these can propagate to other components of the same system. For the MSS considered here, component failure because of certain type of internal cause would result in the failure of other components. These failed components may be mutually dependent or even overlay on one or more components. Consequently, the performance level of the MSS would be weakened by an induced decrease in reliability. If the performance level reduces to a value below a boundary valueξ, which is generally named as the system demand, the reliability of the MSS cannot satisfy the demand and is considered unacceptable. In this scenario, the reliability of an MSS with CCF can be determined from the probability that the system satisfies ξ. From (5), the system reliability function can be determined as(6)Rξ=∑j=0MqjΔwj,ξ,where Δwj,ξ=1,wj≥ξ;Δwj,ξ=0,otherwise.Another important index can be defined as the conditional expected performanceΩξ. This indicates the expected performance level of an MSS under the scenario where the MSS is in an acceptable state. This index can be obtained using the system reliability function Rξ:(7)Ωξ=∑j=0MqjwjΔwj,ξRξ.To calculate the two indices, the distribution of the performance level of an MSS should be first obtained according to (5). The UGF technique has been demonstrated to be highly efficient for the MSS’s reliability evaluation of different kinds of system structure. Particularly, for an MSS, whose components are connected by series or parallel, it is highly effective to implement the UGF technique by the recursive approach.


## 3. UGF and Reliability Evaluation

### 3.1. UGF Technique

The high-dimensional combinatorial problem is generally encountered during the process of calculating MSS reliability because of its multistate components. UGF technique can address this problem efficiently because UGF, also called a u-function or universalz-transform [36], is based on the generalization of the ordinary moment-generating function and z-transformation. For a multistate component i at a state gi with a probability mass function (PMF) pigi, its UGF can be defined by(8)uiz=∑gi=0mipigizhigi,i=1,2,…,n.The UGF of a series-parallel MSS can be determined by performing certain algebraic operations on the components’ UGF. To represent the UGF of an MSS with the structure function (4), the composition operator ⊗ϕ needs to be defined as follows by combining the UGF of each components:(9)Uz=⊗ϕu1z,u2z,…,unz=⊗ϕ∑g1=0m1p1g1zh1g1,∑g2=0m2p2g2zh2g2,…,∑gn=0mnpngnzhngn=∑g1=0m1∑g2=0m2⋯∑gn=0mn∏i=0npigizϕh1g1,h2g2,…,hngn.Here, the polynomialUz represents the UGF of the MSS. The items on the right-hand side of the above formula are the feasible mutually exclusive combinations of the UGF of individual independent components. The function ϕh1g1,h2g2,…,hngn needs to be determined according to the task characteristics.The direct derivation ofUz is still challenging for different types of task. As presented in [36], exhibiting the Uz in a recursive method is effective from the perspectives of simplified calculation and derivation clarity. Particularly, for an MSS with complex configuration, the overall system can be represented as the composition of several subsystems consisting of certain components. This characteristic can be expressed as follows:(10)⊗ϕu1z,…,ukz,uk+1z,…,unz=⊗ϕ⊗ϕ1u1z,…,ukz,⊗ϕ2uk+1z,…,unz.Here, the functionsϕ, ϕ1 and ϕ2 represent different operations for different configurations. The configuration of an MSS or its subsystems can generally be considered as the connection style among the subsystems or components. For any series-parallel components in subsystem, the composition operator can be employed recursively to obtain the UGF of the series-parallel components.For continuous energy or material transmission system, such as electric power, oil, and water, its performance level can be defined as productivity. For paralleled components, the total productivity of the system is the sum of productivities of each individual component. The total productivity of two paralleled components (i and j) is equal to the sum of the individual productivities. The function ϕp calculates the sum of the corresponding performance. The performance of the two components in this occasion can be expressed as(11)Uijz=uiz⊗ϕpujz=∑gi=0mi∑gj=0mjpigipjgjzsumhigi,hjgj.Similarly, for series components, the component that is of the minimum productivity would become the limit. Functionϕs calculates the minimum of all the parameters. Consequently, the UGF of two components (i and j) connected in series should adopt the following formula:(12)Uijz=uiz⊗ϕsujz=∑gi=0mi∑gj=0mjpigipjgjzminhigi,hjgj.


### 3.2. Expression of Component’s UGF Including CCF

The CCF triggered by componenti can correspond to state 0′ with the performance level hi0′=c. Actually, the performance value can be equal to the component performance value of local failure with zero. If component i is in state 0′, those components involved by this CCF would also be in the failure state with performance c. If a component cannot arouse CCF, the PMF of the corresponding state could be zeroed pi0′=0. The UGF of component i containing the CCF can be expressed by(13)uiz=∑gi=0mipigi∗zhigi+pi0′∗zc.The conditional PMF of componenti that cannot fail owing to the CCF can be denoted by(14)ui′z=∑gi=0mipigi1−pi0′∗zhigi,where pi0′ is the probability of causing the CCF.Let there beb≤n components that can cause the CCF simultaneously and independently. For the component vector β=β1,β2,…,βr,…,βb, r∈1,b, the probability that the CCF originated from one component can be denoted by pβr0′. Because these CCFs can occur independently, the result of combination is 2b.For one combinationδ,0≤δ≤2b−1, the CCF comes from component βr if(15)εr=mod2δ2r−1=1,1≤r≤b.The group of components corresponding to theδ can be achieved after calculating the above formula from δ=0 to δ=2b−1. The probability of the CCF corresponding to each δ is(16)ϕδ=∏r=1bpβr0′εr1−pβr0′1−εr,0≤δ≤2b−1.The group of components causing CCF under the specificδ is(17)νδ=∪r=1bγr,γr=βr,εr=1,∅,εr=0.Assuming that a component fails owing to the CCF, the conditional PMF of its performance level can be denoted byψz=zc. This shows that the component can only be in the failure state with performance level c.When the CCF corresponding to the combinationδ occurs, all the components in νδ would be in CCF mode, and their UGFs would be substituted by ψz. Conversely, the UGFs of components not belonging to νδ must be denoted by (14).


### 3.3. Incorporating Protection into Component’s UGF

Based on the properties of CCF, other failed components caused by componentβr can be expressed by Sβr. For each δ, the group of components affected by the CCF can be obtained by(18)μδ=∪r=1bSβr′,Sβr′=Sβr,εr=1,∅,εr=0.For certain important components, protection measures can be adopted to safeguard the components from CCF. When the CCF corresponding toδ occurs, any component belonging to the group μδ fails when its protection mechanism fails and continues functioning with its typical performance level when its protection mechanism functions well. The protection for component i can fail with probability αi. Based on the whole probability theorem, protection can be incorporated in UGF as follows:(19)u⇀iz=αi∗ψz+1−αi∗ui′z.Because of the protection, the UGF of a component inμδ should be replaced with (19). Here, the performance level is generally equal to zero. That is, ψz=z0.Then, the conditional PMF of system performance would be attained asuδz. Combining (16), system UGF can be computed by(20)Usysz=∑δ=02b−1uδz∗ϕδ.Using the expression of system’s UGF, the relevant indices of reliability can be evaluated by (6) and (7).In addition to the above analysis, the next steps should be adopted to realize the evaluation.(i)
S1. Integrate the CCF into component’s UGF using (13).(ii)
S2. Achieve the conditional UGFui′z of component i according to (14).(iii)
S3. Find out all the combinations of components evoked by the CCF.(iv)
S4. For each combinationδ,0≤δ≤2b−1, fix the corresponding components, which can arouse the CCF based on (15).(v)
S5. Compute the probability of every combinationδ by (16).(vi)
S6. Establish the set of components triggering CCFνδ by (17), and substitute the UGF of all the components belonging to νδ with ψz.(vii)
S7. Determine the group of components influenced by the CCF given a combinationδ according to (18). Furthermore, substitute the UGF of all the components belonging to the group 6 μδ with (19).(viii)
S8. Represent the conditional UGFuδz based on the physical structure of MSS.(ix)
S9. Obtain the UGF of systemUsysz by (20).AfterUsysz is achieved, the reliability indices can be evaluated using (6) and (7).


## 3.1. UGF Technique

The high-dimensional combinatorial problem is generally encountered during the process of calculating MSS reliability because of its multistate components. UGF technique can address this problem efficiently because UGF, also called a u-function or universalz-transform [36], is based on the generalization of the ordinary moment-generating function and z-transformation. For a multistate component i at a state gi with a probability mass function (PMF) pigi, its UGF can be defined by(8)uiz=∑gi=0mipigizhigi,i=1,2,…,n.The UGF of a series-parallel MSS can be determined by performing certain algebraic operations on the components’ UGF. To represent the UGF of an MSS with the structure function (4), the composition operator ⊗ϕ needs to be defined as follows by combining the UGF of each components:(9)Uz=⊗ϕu1z,u2z,…,unz=⊗ϕ∑g1=0m1p1g1zh1g1,∑g2=0m2p2g2zh2g2,…,∑gn=0mnpngnzhngn=∑g1=0m1∑g2=0m2⋯∑gn=0mn∏i=0npigizϕh1g1,h2g2,…,hngn.Here, the polynomialUz represents the UGF of the MSS. The items on the right-hand side of the above formula are the feasible mutually exclusive combinations of the UGF of individual independent components. The function ϕh1g1,h2g2,…,hngn needs to be determined according to the task characteristics.The direct derivation ofUz is still challenging for different types of task. As presented in [36], exhibiting the Uz in a recursive method is effective from the perspectives of simplified calculation and derivation clarity. Particularly, for an MSS with complex configuration, the overall system can be represented as the composition of several subsystems consisting of certain components. This characteristic can be expressed as follows:(10)⊗ϕu1z,…,ukz,uk+1z,…,unz=⊗ϕ⊗ϕ1u1z,…,ukz,⊗ϕ2uk+1z,…,unz.Here, the functionsϕ, ϕ1 and ϕ2 represent different operations for different configurations. The configuration of an MSS or its subsystems can generally be considered as the connection style among the subsystems or components. For any series-parallel components in subsystem, the composition operator can be employed recursively to obtain the UGF of the series-parallel components.For continuous energy or material transmission system, such as electric power, oil, and water, its performance level can be defined as productivity. For paralleled components, the total productivity of the system is the sum of productivities of each individual component. The total productivity of two paralleled components (i and j) is equal to the sum of the individual productivities. The function ϕp calculates the sum of the corresponding performance. The performance of the two components in this occasion can be expressed as(11)Uijz=uiz⊗ϕpujz=∑gi=0mi∑gj=0mjpigipjgjzsumhigi,hjgj.Similarly, for series components, the component that is of the minimum productivity would become the limit. Functionϕs calculates the minimum of all the parameters. Consequently, the UGF of two components (i and j) connected in series should adopt the following formula:(12)Uijz=uiz⊗ϕsujz=∑gi=0mi∑gj=0mjpigipjgjzminhigi,hjgj.


## 3.2. Expression of Component’s UGF Including CCF

The CCF triggered by componenti can correspond to state 0′ with the performance level hi0′=c. Actually, the performance value can be equal to the component performance value of local failure with zero. If component i is in state 0′, those components involved by this CCF would also be in the failure state with performance c. If a component cannot arouse CCF, the PMF of the corresponding state could be zeroed pi0′=0. The UGF of component i containing the CCF can be expressed by(13)uiz=∑gi=0mipigi∗zhigi+pi0′∗zc.The conditional PMF of componenti that cannot fail owing to the CCF can be denoted by(14)ui′z=∑gi=0mipigi1−pi0′∗zhigi,where pi0′ is the probability of causing the CCF.Let there beb≤n components that can cause the CCF simultaneously and independently. For the component vector β=β1,β2,…,βr,…,βb, r∈1,b, the probability that the CCF originated from one component can be denoted by pβr0′. Because these CCFs can occur independently, the result of combination is 2b.For one combinationδ,0≤δ≤2b−1, the CCF comes from component βr if(15)εr=mod2δ2r−1=1,1≤r≤b.The group of components corresponding to theδ can be achieved after calculating the above formula from δ=0 to δ=2b−1. The probability of the CCF corresponding to each δ is(16)ϕδ=∏r=1bpβr0′εr1−pβr0′1−εr,0≤δ≤2b−1.The group of components causing CCF under the specificδ is(17)νδ=∪r=1bγr,γr=βr,εr=1,∅,εr=0.Assuming that a component fails owing to the CCF, the conditional PMF of its performance level can be denoted byψz=zc. This shows that the component can only be in the failure state with performance level c.When the CCF corresponding to the combinationδ occurs, all the components in νδ would be in CCF mode, and their UGFs would be substituted by ψz. Conversely, the UGFs of components not belonging to νδ must be denoted by (14).


## 3.3. Incorporating Protection into Component’s UGF

Based on the properties of CCF, other failed components caused by componentβr can be expressed by Sβr. For each δ, the group of components affected by the CCF can be obtained by(18)μδ=∪r=1bSβr′,Sβr′=Sβr,εr=1,∅,εr=0.For certain important components, protection measures can be adopted to safeguard the components from CCF. When the CCF corresponding toδ occurs, any component belonging to the group μδ fails when its protection mechanism fails and continues functioning with its typical performance level when its protection mechanism functions well. The protection for component i can fail with probability αi. Based on the whole probability theorem, protection can be incorporated in UGF as follows:(19)u⇀iz=αi∗ψz+1−αi∗ui′z.Because of the protection, the UGF of a component inμδ should be replaced with (19). Here, the performance level is generally equal to zero. That is, ψz=z0.Then, the conditional PMF of system performance would be attained asuδz. Combining (16), system UGF can be computed by(20)Usysz=∑δ=02b−1uδz∗ϕδ.Using the expression of system’s UGF, the relevant indices of reliability can be evaluated by (6) and (7).In addition to the above analysis, the next steps should be adopted to realize the evaluation.(i)
S1. Integrate the CCF into component’s UGF using (13).(ii)
S2. Achieve the conditional UGFui′z of component i according to (14).(iii)
S3. Find out all the combinations of components evoked by the CCF.(iv)
S4. For each combinationδ,0≤δ≤2b−1, fix the corresponding components, which can arouse the CCF based on (15).(v)
S5. Compute the probability of every combinationδ by (16).(vi)
S6. Establish the set of components triggering CCFνδ by (17), and substitute the UGF of all the components belonging to νδ with ψz.(vii)
S7. Determine the group of components influenced by the CCF given a combinationδ according to (18). Furthermore, substitute the UGF of all the components belonging to the group 6 μδ with (19).(viii)
S8. Represent the conditional UGFuδz based on the physical structure of MSS.(ix)
S9. Obtain the UGF of systemUsysz by (20).AfterUsysz is achieved, the reliability indices can be evaluated using (6) and (7).


## 4. Illustrative Examples

### 4.1. Analytic Example

A type of series-parallel MSS shown in Figure1 can be illustrated to validate the above method for the reliability evaluation of MSSs with CCF and protection. Flowing materials such as water, oil, or steam can be transported from A to B by the multistate components. The structure of system in this example is presented as follows. The MSS is configured by two subsystems: Sub1 and Sub2. Each subsystem is connected in series. Within each subsystem, these components are connected in parallel. Because of the CCF, the components C12 and C21 constitute the common group CG1. The component C13 and the second subsystem form CG2. For example, a feeding water system consisting of pumps in a power plant can be divided into two subsystems. C21 and C22 are vital and need to be protected from the fire originating from C12 and C13.Figure 1
Series-parallel structure of MSS.Performance level of MSS in this example will depend on components’ states and system configuration. The state number of each component and probabilities of these components at each state are listed in Table1. For each component, the events of component in working, local failure, and failure to cause CCF constitute the total spaces. Thus, the sum of the probabilities in those states is one. However, the probability of protection failure is not dependent on the component and should be handled independently.Table 1
Parameters of components.
ComponentWorking stateLocal failureFailure that causes CCFFailure of protectionProb.Tons/hProb.Tons/hProb.Prob.Prob.C110.6830.2550.07--C120.6450.2580.060.05-C130.6530.1470.150.06-C210.660.2100.2-0.3C220.520.3560.15-0.4Based on the above analysis, the UGF of each component can be represented as(21)uC11z=0.68∗z3+0.25∗z5+0.07∗z0,uC12z=0.64∗z5+0.25∗z8+0.06∗z0+0.05∗zc,uC13z=0.65∗z3+0.14∗z7+0.15∗z0+0.06∗zc,uC21z=0.6∗z6+0.2∗z10+0.2∗z0,uC22z=0.5∗z2+0.35∗z6+0.15∗z0.Their conditional UGF can be obtained by (14) as follows:(22)uC11′z=uC11z,uC12′z=0.6737∗z5+0.2632∗z8+0.0631∗z0,uC13′z=0.6915∗z3+0.1489∗z7+0.1596∗z0,uC21′z=uC21z,uC22′z=uC22z.The components causing CCF areC12 and C13, such that β1=C12,β2=C13. Therefore, the components affected by these are Sβ1=C21, Sβ2=C21,C22. The parameters corresponding to different combinations are listed in Table 2. These values of ϕδ will be used to calculate the UGF of system according to(20).Table 2
Parameter values of different combinationsδ.
δεrϕδνδμδr=1r=2000=1−0.05×1−0.06=0.893∅∅110=0.05×1−0.06=0.047C12C21201=1−0.05×0.06=0.057C13C21,C22311=0.05×0.06=0.003C12,C13C21,C22Because of the protection measures, according to (19), the UGF of C21 and C22 can be expressed as follows:(23)u⇀C21z=0.3∗z0+1−0.3∗uC21′z=0.42∗z6+0.14∗z10+0.44∗z0,u⇀C22z=0.4∗z0+1−0.4∗uC22′z=0.3∗z2+0.21∗z6+0.49∗z0.According to the physical structure of the example, its conditional UGF can be obtained by(24)uδz=usub1z⊗ϕsusub2z=uC11′z⊗ϕpuC12′z⊗ϕpuC13′z⊗ϕsuC21′z⊗ϕpuC22′z.Forδ=0, there is no element in the sets νδ and μδ. Therefore, in (24), no element’s UGF needs to be substituted. There are 14 items in the UGF expression in the calculation result. The performance values and corresponding probabilities are displayed in Figure 2(a).Figure 2
UGF results of four combinations.
(a)(b)(c)(d)Forδ=1, νδ=C12 and μδ=C21. Therefore, uC12′z and uC21′z in (24) should be substituted with ψz and u⇀C21z, respectively. Similarly, the 9 items in the result are displayed in Figure 2(b).Forδ=2, νδ=C13 and μδ=C21,C22. Therefore, uC13′z, uC21′z and uC22′z in (24) should be substituted by ψz, u⇀C21z and u⇀C22z, respectively. Similarly, there are 10 items in the UGF expression in the result as shown in Figure 2(c).Forδ=3, νδ=C12,C13 and μδ=C21,C22. Therefore, in (24), uC12′z and uC13′z need to be substituted by ψz. uC21′zand uC22′z should be substituted by u⇀C21z and u⇀C22z, respectively. In the result, the UGF expression in the result has 4 items, as shown in Figure 2(d).Finally, UGF of the whole system can be achieved by (20) like this:(25)Usysz=∑δ=03uδz∗ϕδ.The UGF expression of the whole system has 14 items and is shown in cyan color in Figure3.Figure 3
Comparison of UGF expression of the whole system with and without protection.The reliability function of the entire system with protection is depicted in cyan color in Figure4. From point A, we obtain R8=0.6269.Figure 4
Reliability function of system with and without protection.Similarly, the curve of the conditional expected performance of this system with protection is plotted in cyan color in Figure5. Figure 5(a) is the entire graph of conditional expected performance, and Figure 5(b) shows details at ξ=8. From Point A, we obtain Ω8=9.7674.Figure 5
Conditional expected performance of system with and without protection. (a) The whole landscape. (b) The local landscape.
(a)(b)Although there are only five components in the system, the proposed approach is also suitable for system with a large number of components. This is because the entire procedure can be programmed conveniently using an ordinary programming language. For example, the total time required for the calculation using MATLAB in the computing environment applied by the authors is 1.9536 s. The present main configuration of the computer is as follows: main frequency = 2.4 GHz and memory = 4 GB.


### 4.2. Comparison with the Cases without Protection

A comparison with the case without protection on the MSS can be performed to validate the effectiveness of the protection mechanism. The difference between the system with and without protection lies in the incorporation of the S6 and S7 in subsection C of Section3. If an MSS has no protection mechanism, the elements in the set νδ∪μδ should be substituted by ψz while calculating uδz. Based on the example in Section A, the procedure for calculating uδz is as follows:Forδ=0 , νδ∪μδ is empty. Therefore, the result does not vary with the result in the above example.Forδ=1, νδ∪μδ=C12,C21. Thus, the UGF of C12 and C21, uC12′z and uC21′z in (24) should be substituted by ψz. Its UGF expression in the result has 5 items.Forδ=2, νδ∪μδ=C13,C21,C22. Therefore uC13′z, uC21′z, and uC22′z in (24) should be substituted by ψz. The result includes one item in the UGF as follows:(26)u2z=1∗z0.Forδ=3, νδ=C12,C13 and μδ=C21,C22, and therefore νδ∪μδ=C12,C13,C21,C22. In (24), uC12′z, uC13′z, uC21′z and uC22′z need to be substituted by ψz. The result for this case is as follows:(27)u3z=1∗z0.Lastly, the UGF of whole system without protection will be obtained according to (20). There are also 14 items in its UGF, as shown in magenta in Figure 3.In this case, its reliability function is plotted in magenta in Figure4. Point B displays a lower value R8=0.6017 than that of Point A in Figure 4. Similarly, the curve of the conditional expected performance of the system at different performance levels is shown in magenta in Figure 5. A comparison of these curves reveals that these values are extremely close. We obtain Ω8=9.8110 from Point B.From the comparison results, it can be found that the reliability results of system with protection have higher values than the ones of the system without protection. This validates that the protection mechanism is effective and justified. Similarly, the curve of conditional expected performance of system without protection is higher than the one of the system with protection because it is inverse proportional to the reliability function. This also provides a proof to the effectiveness of protection.


### 4.3. Numerical Example

Here is another numerical example (with more number of components and CCFs) to prove the efficiency of the suggested approach. The MSS topology structure in this example is shown in Figure6. This system consists of three subsystems: Sub1, Sub2, and Sub3. Sub1 and Sub2 are connected in series, and their combination is parallel to Sub3. There are three common groups: CG1, CG2, and CG3. The components in each common group are marked using dotted lines. In CG1, component C11 causes CCF, and C21 is affected by the CCF. In addition, C21 is also protected with a certain probability. In CG2, component C12 triggers the CCF, and C23 would fail and would also be protected. In CG3, component C31 triggers the CCF and causes the failure of C32 and C33. In CG3, component C33 would be protected with a certain probability.Figure 6
System topology of numerical example.Performance level of MSS will also vary with the states variation of components. All the states and probabilities of these components at each state in the system are recorded in Table3.Table 3
Parameters of all the components in the numerical example.
ComponentWorking stateLocal failureFailure to cause CCFFailure of protectionProb.Tons/hProb.Tons/hProb.Prob.Prob.C110.6240.2550.070.06—C120.6450.2590.060.05—C210.7150.1470.15—0.2C220.670.2100.2——C230.6340.2260.15—0.3C310.6540.1470.150.06—C320.770.2100.1——C330.5530.360.15——The UGF of this system in different cases can be procured conveniently based on the proposed method. A comparison of these 26 items in the UGFs is depicted in Figure7.Figure 7
Comparison of items in the UGF under system with and without protection.The reliability function of the system with and without protection can be compared. The results are displayed in Figure8. Evidence shows that system with protection has a higher reliability level than that without protection. Point A has a higher value R15=0.6212 than that for Point B R15=0.5336. It also validates the effectiveness of protection against system safety.Figure 8
Comparison of reliability of system with and without protection.In addition, there is no significant difference in conditional expected performance between the systems with and without protection. The whole graphs are shown in Figure9(a). As shown in the figure, the values at Point A Ω15=18.0709 and Point B Ω15=18.0633 are similar. Further details are shown in the local graphs in Figure 9(b).Figure 9
Conditional expected performance comparison. (a) The whole graphs. (b) The local graphs.
(a)(b)The above analysis reveals that the recommended method is also suitable for a more complicated system with large-scale components. Furthermore, less time-consumption is another advantage over other similar methods. The time-consumption for this example under a configuration similar to that in Subsection A of Section4 is 2.1484 s. The time cost is generally reasonable for engineers. Based on the time-consumption results from the above two examples, this procedure implementing protection on important component is also acceptable in realistic engineering system.In addition, the data or information of system coming from the computing result is crucial for reliability engineers. When making decision on which components need to be protected, those reliability indices provide scientific evidence. In particular, when there are many components in a large system, engineers can choose the most valuable components among the system and implement the protection on them based on the reliability evaluation indices.


## 4.1. Analytic Example

A type of series-parallel MSS shown in Figure1 can be illustrated to validate the above method for the reliability evaluation of MSSs with CCF and protection. Flowing materials such as water, oil, or steam can be transported from A to B by the multistate components. The structure of system in this example is presented as follows. The MSS is configured by two subsystems: Sub1 and Sub2. Each subsystem is connected in series. Within each subsystem, these components are connected in parallel. Because of the CCF, the components C12 and C21 constitute the common group CG1. The component C13 and the second subsystem form CG2. For example, a feeding water system consisting of pumps in a power plant can be divided into two subsystems. C21 and C22 are vital and need to be protected from the fire originating from C12 and C13.Figure 1
Series-parallel structure of MSS.Performance level of MSS in this example will depend on components’ states and system configuration. The state number of each component and probabilities of these components at each state are listed in Table1. For each component, the events of component in working, local failure, and failure to cause CCF constitute the total spaces. Thus, the sum of the probabilities in those states is one. However, the probability of protection failure is not dependent on the component and should be handled independently.Table 1
Parameters of components.
ComponentWorking stateLocal failureFailure that causes CCFFailure of protectionProb.Tons/hProb.Tons/hProb.Prob.Prob.C110.6830.2550.07--C120.6450.2580.060.05-C130.6530.1470.150.06-C210.660.2100.2-0.3C220.520.3560.15-0.4Based on the above analysis, the UGF of each component can be represented as(21)uC11z=0.68∗z3+0.25∗z5+0.07∗z0,uC12z=0.64∗z5+0.25∗z8+0.06∗z0+0.05∗zc,uC13z=0.65∗z3+0.14∗z7+0.15∗z0+0.06∗zc,uC21z=0.6∗z6+0.2∗z10+0.2∗z0,uC22z=0.5∗z2+0.35∗z6+0.15∗z0.Their conditional UGF can be obtained by (14) as follows:(22)uC11′z=uC11z,uC12′z=0.6737∗z5+0.2632∗z8+0.0631∗z0,uC13′z=0.6915∗z3+0.1489∗z7+0.1596∗z0,uC21′z=uC21z,uC22′z=uC22z.The components causing CCF areC12 and C13, such that β1=C12,β2=C13. Therefore, the components affected by these are Sβ1=C21, Sβ2=C21,C22. The parameters corresponding to different combinations are listed in Table 2. These values of ϕδ will be used to calculate the UGF of system according to(20).Table 2
Parameter values of different combinationsδ.
δεrϕδνδμδr=1r=2000=1−0.05×1−0.06=0.893∅∅110=0.05×1−0.06=0.047C12C21201=1−0.05×0.06=0.057C13C21,C22311=0.05×0.06=0.003C12,C13C21,C22Because of the protection measures, according to (19), the UGF of C21 and C22 can be expressed as follows:(23)u⇀C21z=0.3∗z0+1−0.3∗uC21′z=0.42∗z6+0.14∗z10+0.44∗z0,u⇀C22z=0.4∗z0+1−0.4∗uC22′z=0.3∗z2+0.21∗z6+0.49∗z0.According to the physical structure of the example, its conditional UGF can be obtained by(24)uδz=usub1z⊗ϕsusub2z=uC11′z⊗ϕpuC12′z⊗ϕpuC13′z⊗ϕsuC21′z⊗ϕpuC22′z.Forδ=0, there is no element in the sets νδ and μδ. Therefore, in (24), no element’s UGF needs to be substituted. There are 14 items in the UGF expression in the calculation result. The performance values and corresponding probabilities are displayed in Figure 2(a).Figure 2
UGF results of four combinations.
(a)(b)(c)(d)Forδ=1, νδ=C12 and μδ=C21. Therefore, uC12′z and uC21′z in (24) should be substituted with ψz and u⇀C21z, respectively. Similarly, the 9 items in the result are displayed in Figure 2(b).Forδ=2, νδ=C13 and μδ=C21,C22. Therefore, uC13′z, uC21′z and uC22′z in (24) should be substituted by ψz, u⇀C21z and u⇀C22z, respectively. Similarly, there are 10 items in the UGF expression in the result as shown in Figure 2(c).Forδ=3, νδ=C12,C13 and μδ=C21,C22. Therefore, in (24), uC12′z and uC13′z need to be substituted by ψz. uC21′zand uC22′z should be substituted by u⇀C21z and u⇀C22z, respectively. In the result, the UGF expression in the result has 4 items, as shown in Figure 2(d).Finally, UGF of the whole system can be achieved by (20) like this:(25)Usysz=∑δ=03uδz∗ϕδ.The UGF expression of the whole system has 14 items and is shown in cyan color in Figure3.Figure 3
Comparison of UGF expression of the whole system with and without protection.The reliability function of the entire system with protection is depicted in cyan color in Figure4. From point A, we obtain R8=0.6269.Figure 4
Reliability function of system with and without protection.Similarly, the curve of the conditional expected performance of this system with protection is plotted in cyan color in Figure5. Figure 5(a) is the entire graph of conditional expected performance, and Figure 5(b) shows details at ξ=8. From Point A, we obtain Ω8=9.7674.Figure 5
Conditional expected performance of system with and without protection. (a) The whole landscape. (b) The local landscape.
(a)(b)Although there are only five components in the system, the proposed approach is also suitable for system with a large number of components. This is because the entire procedure can be programmed conveniently using an ordinary programming language. For example, the total time required for the calculation using MATLAB in the computing environment applied by the authors is 1.9536 s. The present main configuration of the computer is as follows: main frequency = 2.4 GHz and memory = 4 GB.


## 4.2. Comparison with the Cases without Protection

A comparison with the case without protection on the MSS can be performed to validate the effectiveness of the protection mechanism. The difference between the system with and without protection lies in the incorporation of the S6 and S7 in subsection C of Section3. If an MSS has no protection mechanism, the elements in the set νδ∪μδ should be substituted by ψz while calculating uδz. Based on the example in Section A, the procedure for calculating uδz is as follows:Forδ=0 , νδ∪μδ is empty. Therefore, the result does not vary with the result in the above example.Forδ=1, νδ∪μδ=C12,C21. Thus, the UGF of C12 and C21, uC12′z and uC21′z in (24) should be substituted by ψz. Its UGF expression in the result has 5 items.Forδ=2, νδ∪μδ=C13,C21,C22. Therefore uC13′z, uC21′z, and uC22′z in (24) should be substituted by ψz. The result includes one item in the UGF as follows:(26)u2z=1∗z0.Forδ=3, νδ=C12,C13 and μδ=C21,C22, and therefore νδ∪μδ=C12,C13,C21,C22. In (24), uC12′z, uC13′z, uC21′z and uC22′z need to be substituted by ψz. The result for this case is as follows:(27)u3z=1∗z0.Lastly, the UGF of whole system without protection will be obtained according to (20). There are also 14 items in its UGF, as shown in magenta in Figure 3.In this case, its reliability function is plotted in magenta in Figure4. Point B displays a lower value R8=0.6017 than that of Point A in Figure 4. Similarly, the curve of the conditional expected performance of the system at different performance levels is shown in magenta in Figure 5. A comparison of these curves reveals that these values are extremely close. We obtain Ω8=9.8110 from Point B.From the comparison results, it can be found that the reliability results of system with protection have higher values than the ones of the system without protection. This validates that the protection mechanism is effective and justified. Similarly, the curve of conditional expected performance of system without protection is higher than the one of the system with protection because it is inverse proportional to the reliability function. This also provides a proof to the effectiveness of protection.


## 4.3. Numerical Example

Here is another numerical example (with more number of components and CCFs) to prove the efficiency of the suggested approach. The MSS topology structure in this example is shown in Figure6. This system consists of three subsystems: Sub1, Sub2, and Sub3. Sub1 and Sub2 are connected in series, and their combination is parallel to Sub3. There are three common groups: CG1, CG2, and CG3. The components in each common group are marked using dotted lines. In CG1, component C11 causes CCF, and C21 is affected by the CCF. In addition, C21 is also protected with a certain probability. In CG2, component C12 triggers the CCF, and C23 would fail and would also be protected. In CG3, component C31 triggers the CCF and causes the failure of C32 and C33. In CG3, component C33 would be protected with a certain probability.Figure 6
System topology of numerical example.Performance level of MSS will also vary with the states variation of components. All the states and probabilities of these components at each state in the system are recorded in Table3.Table 3
Parameters of all the components in the numerical example.
ComponentWorking stateLocal failureFailure to cause CCFFailure of protectionProb.Tons/hProb.Tons/hProb.Prob.Prob.C110.6240.2550.070.06—C120.6450.2590.060.05—C210.7150.1470.15—0.2C220.670.2100.2——C230.6340.2260.15—0.3C310.6540.1470.150.06—C320.770.2100.1——C330.5530.360.15——The UGF of this system in different cases can be procured conveniently based on the proposed method. A comparison of these 26 items in the UGFs is depicted in Figure7.Figure 7
Comparison of items in the UGF under system with and without protection.The reliability function of the system with and without protection can be compared. The results are displayed in Figure8. Evidence shows that system with protection has a higher reliability level than that without protection. Point A has a higher value R15=0.6212 than that for Point B R15=0.5336. It also validates the effectiveness of protection against system safety.Figure 8
Comparison of reliability of system with and without protection.In addition, there is no significant difference in conditional expected performance between the systems with and without protection. The whole graphs are shown in Figure9(a). As shown in the figure, the values at Point A Ω15=18.0709 and Point B Ω15=18.0633 are similar. Further details are shown in the local graphs in Figure 9(b).Figure 9
Conditional expected performance comparison. (a) The whole graphs. (b) The local graphs.
(a)(b)The above analysis reveals that the recommended method is also suitable for a more complicated system with large-scale components. Furthermore, less time-consumption is another advantage over other similar methods. The time-consumption for this example under a configuration similar to that in Subsection A of Section4 is 2.1484 s. The time cost is generally reasonable for engineers. Based on the time-consumption results from the above two examples, this procedure implementing protection on important component is also acceptable in realistic engineering system.In addition, the data or information of system coming from the computing result is crucial for reliability engineers. When making decision on which components need to be protected, those reliability indices provide scientific evidence. In particular, when there are many components in a large system, engineers can choose the most valuable components among the system and implement the protection on them based on the reliability evaluation indices.


## 5. Conclusion

To evaluate the reliability of series-parallel MSSs with CCF and protection, a reliability evaluation approach is recommended in this paper. Components that can evoke the CCF may trigger failures of different subgroups of system components. These subgroups can also overlay with certain components. Because a few components are important for the system, the protection mechanisms are implemented on these ones. The factors of CCF and protection can be melted in the UGF based on the UGF technology. Because it is repeated for each combination, this approach can be programmed suitably by repetition. Some indices such as reliability and the expected conditional performance level can be calculated. To prove the validity of protection mechanism, a comparison with components without protection can be performed. This approach would also help in deciding which component is the most valuable and therefore, needs to be protected by reliability engineers. According to the assumptions in this approach, the restriction of system structure lies in the series-parallel MSS now. Probabilistic and fuzzy UGF would be considered to extend this approach in complex MSS such as bridge andG(k/n) system in the future.



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*Source: 6691362-2021-03-27.xml*