RELIABILITY BLOCK DIAGRAM (RBD)
1. System performance 2. RBD definition 3. Structure function atrIM Castanier - ISMP Master 4. Reliability function of a non repairable complex system 5. Reliability metrics for a repairable complex system
59 System performance
Problem: Individual definition of the failure modes and components What about the global performance of the « system »?
atrIM Castanier - ISMP Master
60 Reliability Block Diagram definition
Definition A Reliability Block Diagram is a success-oriented graph which illustrates from a logical perspective how the different functional blocks ensure the global mission/function of the system. The structure of the reliability block diagram is mathematically described through structure functions which allow to assess some reliability measures of the (complex) atrIM Castanier - ISMP Master system. Function model
EThe valve can be closed S and stop the flow SDV1
61 Reliability Block Diagram definition
Decomposition of a complex system in functional blocks
Component
1 8 9 4 4 2 7 8 10 atrIM Castanier - ISMP Master 5 6 3 9 10
Functional block 1
62 Reliability Block Diagram definition
The classical architectures « Serie » structure A system which is operating if and only of all of its components are operating is called a serie structure/system.
« Parallel » structure A system which is operating if at least one of its components is atrIM Castanier - ISMP Master operating is called a parallel structure/system.
« oo » structure A system which is operating if at least of its component are operating is called a over structure/system
63 Reliability Block Diagram definition
Example Let consider 2 independent safety-related valves V1 and V2 which are physically installed in series.
In normal condition, the valves are opened and automatically close for shuting down the process in atrIM Castanier - ISMP Master emergency case. Questions : Identify the functions of the system Draw the reliability block diagram
64 Structure function
Definition Let consider a system with n independent components and let define its state vector with:
The binary function defined by atrIM Castanier - ISMP Master
Is called system structure function or simply structure.
65 Structure function
Write the structure function: For a serie system For a parallel system For a 2oo3 system atrIM Castanier - ISMP Master
66 Structure function
Definition A component i is said non-relevant if for all where is the vector for which the state of the component i is 1 or 0 A system is coherent if all of its components are relevant and its structure function is non-decreasing in its parameters atrIM Castanier - ISMP Master Theorems and 풏 풊 ퟏ 풊
67 Reliability function of a non repairable complex system
The state of the component at time is a random variable denoted Let define:
The reliability of component : ) The reliability of system S
atrIM Castanier - ISMP Master if independence on the components
68 Reliability function of a non repairable complex system
Serie system C1 C2 C3
Reliability assessment for a system with independent components
∑ ∫ System failure rate atrIM Castanier - ISMP Master function Let remark: Application: What is the reliability of a serie system with independent components when the failure rates are constant? What is the associated MTTF? 69 Reliability function of a non repairable complex system C1
Parallel system C2
Reliability assessment for a system with independent components C3
atrIM Castanier - ISMP Master If the component failure rates are constant:
Application: What is the reliability function, the MTTF and the failure rate of a parallel system with 2 independent components when ? 70 Reliability function of a non repairable complex system
C1 2oo3 system C2 2/3 The system structure function C3 휙 푿(푡) = 푋 푡 푋 푡 + 푋 푡 푋 푡 + 푋 푡 푋 푡 −2푋 (푡)푋 (푡)푋 (푡)
The associated reliability function:
푅 (푡)= 푅 푡 푅 푡 + 푅 푡 푅 푡 + 푅 푡 푅 푡 −2푅 (푡)푅 (푡)푅 (푡)
If 푧 푡 = 휆, atrIM Castanier - ISMP Master 푅 푡 =3푒 −2푒
6휆 푒 − 푒 → 푧 푡 = 2휆 3푒 −2푒
5 1 푀푇푇퐹 = 푅 푡 푑푡 = 6 휆
71 Reliability function of a non repairable complex system
Structure Fonction de fiabilité MTTF 1oo1
1oo2
2oo3
Reliability functions 1 atrIM Castanier - ISMP Master 0,8
0,6 1oo1
R(t) 0,4 1oo2 1oo3 0,2 0 0 0,5 1 1,5 2 2,5 372 72 Reliability function of a non repairable complex system
Redundancies The reliabilitlty improvement for a critical functionn is: The search in the improvement in the intrinsic reliability of the function (choice of very high reliable components) The introduction of active or passive redundancies. It the components are working in parallel (in the same time), we call active redundancy. The components share the function load as far as one of the components fails. atrIM Castanier - ISMP Master If one component is in standby for ensuring the function as soon as the other component fails, we call passive redundancy. It the stanby component is not involved in the another function and that we can assume no failure of this component in this interval, we talk about cold standby. If the standby component ensures a reduced working load, we said partly loaded redundancy.
73 Reliability function of a non repairable complex system 1 Performance analysis of the passive S 2 redundancies 3 Passive redundancy, perfect replacement, no repair Cold passive redundancy, imperfect replacement, atrIM Castanier - ISMP Master no repair Partly load passive redundancy, imperfect replacement, no repair
74 Reliability function of a non repairable complex system
Reliability optimization The objective in the product design can be to maximize the reliability on a given time horizon. This maximization could be done with a budget constrainst or at the lower cost. Other criteria could be associated (minimization of the weight) or some extra constraints (volume, design, …). atrIM Castanier - ISMP Master This optimization consists in finding the best compromise between the reliability improvement and the other criterion. Multi-objective optimization technics (Operations Research) can be used to solve such a problem
75 Reliability function of a non repairable complex system Problem 1 : We search the optimal structure of a serie system which has to perform k sub-functions. The problem is to maximize the reliability on a fixed interval with a max investisment budget . Assumption: Each of the subfunctions can be performed by a specific component at a unitary cost Let consider the failure rates be atrIM Castanier - ISMP Master constant . This is equivalent to find the vector such as
with
76 Problem 2 : Find an optimal structure of a serie system for performing a set of sub- functions. We search to maximize the reliability on a fixed horizon at a minimal cost. Assumption:
Every function is performed by a specific component with a unit cost 푐 and an operating cost per unit of time 푔 . Let assume that the number of components is fixed and respective constant failure rates 휆 . Heuristic for the reliability maximization and cost minization: Step 0 : Elaborate the objective functions for the initial system structure: ∑ . 푅 풏 , 푡 = 푒 , 퐸 퐶 풏 , 푡 = 푐 + 푔 퐸 min( 푇 , 푡) atrIM Castanier - ISMP Master Step v Add a component at the least reliable block 푎 ; 푛 = 푛 +1 and 푛 = 푛 , 푖 ≠ 푎
. Evaluate the reliability function of the block 푎, 푅 푛 , 푡 =1−(1− 푒 ) Evaluate the system reliability 푅 풏 , 푡 = ∏ 푅 (푛 , 푡) 푅 (푛 , 푡) Estimate the mean time of failure before : 퐸 min( 푇 , 푡) = ∫ 푅 풏 , 푢 푑푢
Estimate the average cost 퐸 퐶 풏 , 푡 = ∑ 푛 푐 + 푔 퐸 min( 푇 , 푡) Dessiner 푅 풏 , 푡 , 퐸 퐶 풏 , 푡 77
atrIM Castanier - ISMP Master Pareto frontier
1
78 Analyzis of repairable systems
A repairable system is a system subject to degradation and be in a better state after a repair. Example : For a simple system C1
X(t)
1 atrIM Castanier - ISMP Master t 0 What are the different questions in case of repairable systems? What is the probability that the system is operating at a given date? What is the average operation time on a given time horizon? What is the time percentage the system is unavailable?
79 Availability of a single unit
C1
At time t :
: probability of being in the good state ( ) : probability of being in the failed state ( ) To ensure the operation at time ), it needs: The system is operating at t and there is no failure in The system is on repair at t and be operational at Then atrIM Castanier - ISMP Master By dividing by and if tends to 0, First-order differential equation with the following solution
80 So, The (instantaneous) availability at time t: 1 퐴 푡 = 휇+휆푒 휆 + 휇 The asymptotic availability 푡 →∞ or « mean » or « average »: 휇 푀푇푇퐹 퐴 = = 휆 + 휇 푀푇푇퐹 + 푀푇푇푅
The average availabilty on an interval 푡 , 푡 1 퐴 푡 , 푡 = 퐴 푡 푑푡 푡 − 푡 The average reliability from the fiLa disponibilité moyenne depuis la mise en service 1 퐴 휏 = 퐴 푡 푑푡 휏 La disponibilité moyenne asymptotique atrIM Castanier - ISMP Master 1 퐴 = lim 퐴 휏 = lim 퐴 푡 푑푡 → → 휏 Moreover, . The asymptotic unavailability is 퐴̅ =1− 퐴 = ≈ 휆. 푀푇푇푅 . The expected number of failures on an interval : 푊(푡)≈ The operational unavailability: 퐴̅ =1− 퐴 =
81 Availability of a serie system C1 C2
휆 1 Markov graph휇 Assumption: 휆 = 휆 = 휆 et 휇 = 휇 = 휇 0 휆
휇 2 −2휆 휆 휆 One-step transition matrix 푀 = 휇 −휇 0 to solve 푡 = 푃 푡 푀 by Laplace transform. 휇 0 −휇 The Laplace transformed is the solution of: 푝 +2휆 −휆 −휆 ℒ (푝),ℒ (푝),ℒ (푝) = 1,0,0 푝퐼 − 푀 = 1,0,0 −휇 푝 + 휇 0 −휇 0 푝 + 휇
atrIM Castanier - ISMP Master The availability is given by the sum of the operating states given the initial condition 푝 +2휆 −휆 −휆 1 푀 ℒ퐴 푝 = 1,0,0 −휇 푝 + 휇 0 0 = ∆ −휇 0 푝 + 휇 0 Nota: This is equivalent to a 2-state transition graph and we directly have
82 Availability of a serie system
In the general case, we can prove: So, the system repair rate is
atrIM Castanier - ISMP Master And, because the system failure rate is and the asymptotic availability , we have
Nota: If 푨풊 is closed to one, by the structure function approach, we can approximate: 퐴 푡 = 퐸 휙 푿(푡) ≅ ∏ 퐴 푡 = ∏ 휇 +휆 푒 and 퐴 = ∏
83 Availability of a parallel system C1 C2
Only if C3 The availability is independance
And so, the asymptotic availibility can be approximated atrIM Castanier - ISMP Master
84 84 Résolution du cas à 2 redondances actives identiques et un réparateur Markov transition graph where Ei is the event « i components are operating »
−2휆 2휆 0 One-step transition matrix 푀 = 휇 − 휆 + 휇 휆 to solve 푡 = 푃 푡 푀 with Laplace. 0 휇 −휇 The Laplace transformed is the solution of 푝 +2휆 −2휆 0 ℒ (푝),ℒ (푝),ℒ (푝) = 1,0,0 푝퐼 − 푀 = 1,0,0 휇 푝 + 휆 + 휇 −휆 0 −휇 푝 + 휇 And because the availability is the sum of the operating state 푝 +2휆 −2휆 0 1 푀 + 푀 ℒ퐴 푝 = 1,0,0 휇 푝 + 휆 + 휇 −휆 1 = ∆ 0 −휇 푝 + 휇 0 With the matrix determinant, we have atrIM Castanier - ISMP Master ∆ = 푝 푝 + 푝 2휇 +3휆 + 휇 +2휆 +2휆휇 = 푝 푝 − 푆 푝 − 푆 푝 + 푝 2휇 + 3휆 + 휇 +2휆휇 퐴 퐵 퐶 ℒ퐴 푝 = = + + 푝 푝 − 푆 푝 − 푆 푝 푝 − 푆 푝 − 푆
퐴푆 푆 = 휇 +2휆휇 With 푆 =− (3휆 + 2휇 + 휆 + 4휆휇) ; 푆 =− (3휆 + 2휇 − 휆 + 4휆휇) 푎푛푑 퐴 + 퐵 + 퐶 =1 − 퐴 + 퐵 푆 − 퐴 + 퐶 푆 =2휇 + 3휆 And finally we can prove 휇 + 2휆휇 2휆 푆 푒 − 푆 푒 퐴 푡 = − 휇 + 2휆휇 + 2휆 푆 푆 푆 − 푆 85 Optimizing a structure
The structure optimization is in this context to define: The nature of the redundancy The number of redundant components/functions The number of repairmen The problem is a complex discrete optimization problem especially because of the dependencies due to the repairmen availabilities. atrIM Castanier - ISMP Master It the structure is complex, an optimization method based on stochastic simulation (MCMC or Generalized SPNN) could be used. We can also used the previous approx if the approximation quality is relevant (low repair time comparing to operating time)
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