RELIABILITY BLOCK DIAGRAM (RBD) 1. System performance 2. RBD definition 3. Structure function Master ISMP Master ISMP - Castanier 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 »? Master ISMP Master ISMP - Castanier 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) Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier Theorems and 풏 풊ퟏ 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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, Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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. Master ISMP Master ISMP - Castanier 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, Master ISMP Master ISMP - Castanier 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, …). Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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) Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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 Master ISMP Master ISMP - Castanier 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.