Energy Engineering - RELIABILITY, SAFETY AND RISK ANALYSIS C

Full exam

First name and Last name: Student number: ______________________________________________________________________________________________________________ RELIABILITY SAFETY AND RISK ANALYSIS A+B 22 /07 /2021 Note: 1. Make sure you write your first name, last name, student number and sign every exam sheet; 2. The exam consist s in 2 exercises and 2 open question s; 3. The exam time is 2 hours and 15 minutes . Exercise 1 [7.5 points] Consider a hot standby system of two identical units. The failure rate of the units is � when they are operating and ������� when they are in standby. Both units are healthy at time 0. In case of failure of the operating unit, the maintenance team starts imm ediately the repair, which is characterized by a constant repair rate, �. The failure of the unit in standby is unnoticed until it is asked to operate. In the case in which the operating unit had a failure and the standby unit had previously failed, the m aintenance team starts repairing both units in parallel. This latter repair process, which will result in the repair of both units at the same time, is characterized by a constant repair rate �������< �. E1.1) Draw the Markov diagram of the system, upon proper def inition of the system states. E1.2) Write the transition matrix and the Markov equation in the matrix form. E1.3) Find the Mean Time To Failure of the system. E1.4) Find the expected number of system failures in 1 year , considering steady state probabilities . E1.5) Repeat questio ns E1.1 ) and E1.2 ) assuming that an external event may occur with rate ������� and that it has a probability � of causing the failure of the operating unit and ������� of the standby unit. Exercise 2 [7.5 points] Consider an emergency door with probability � of failure to open on demand. Experts have estimated � based on prior experience with similar emergency doors. They feel that � can take only 3 values: 0.005, 0.010 and 0.025. Their prior estimation of the p robability distribution of � is: ������′(�= 0.005 )= 0.2 ������′(�= 0.010 )= 0.5 ������′(�= 0.025 )= 0.3 E2.1) A reliability test of the emergency door has been performed and 4 failures to open have been observed over 200 demands to open. Find the posterior probability distribution ������′′(�) of �; E2.2) What is the probability that people are not able to escape in case of fire in a room with two different emergency exits, each one equipped with its own emergency door? Use the posterior distribution found in E2.1 ); E2.3) Compute the posterior distribution of � assuming the following prior distribution (different from that used in E2.1 ): • the prior distribution of � is a beta distribution with parameters �= 2.94 and �= 199 .7 • use as experimental evidence the same results of the reliability test used in E2.1 ); Com pare the expected value and variance of � in E2.1 ) and E2.3 ). E2.4) Repeat the estimation of � using the method of maximum likelihood and assuming as only source of information the results of the test in E2.1 ). Question 1 [7.5 points] Q1.1) Define the concept of critical path vector from the point of view of Birnbaum; Q1.2) Define the Criticality importance measure; Q1.3) Find the relationship between Criticality and Birnbaum importance measures ; Q1.4) Compute the Criticality importance measure of component s 1 and 3 in the system below (assuming that �1(������),�2(������) �3(������) are the component 1, 2 and 3 reliabilities, respectively) Q1.5) Illustrate possible decision -making applications of the Criticality importance measure in RAMS analysis. Question 2 [7.5 points] Consider a system of 2 different components (A and B) in parallel. Each component can be in two different states: “working” and “failed” with constant failure ( �� and �������) and repair ( �� and ��) rates. Q2.1) Draw a possible life of the system in the phase space; Q2.2) Introduce the concept of transport kernel for a stochastic transition; Q2.3) Write the conditional probability that the system has a transition in the time interval [������,������+������������ ] given that the prec eding transition occurred at time t’ and that the state thereby entered was: [Component A = “working” and component B = “failed”]; Q2.4) Write the conditional probability that the system enters in state [Component A = “failed” and component B = “failed”] given t hat a transition occurred at time t when the system was in state [Component A = “working” and component B = “failed”]; Q2.5) The Monte Carlo simulation of 3 system lives from time 0, at which all the component are working, to the mission time, ������������ = 100 in arbitr ary units, have been performed. Hereafter, the obtained transition times and type of undertaken transitions are reported: System Life 1: • Component A fails at ������= 10 • Component A is repaired at ������= 21 • Component B fails at ������= 61 • Component A fails at ������= 68 • Component A is repaired at ������= 79 • Component A fails at ������= 88 • Component B is repaired at ������= 94 • No events from time 94 to time 100 System Life 2: • Component A fails at ������= 24 • Component A is repaired at ������= 29 • Component A fails at ������= 65 • Component A is repaired at ������= 78 • No events from time 78 to time 100 System Life 3: • Component A fails at ������= 18 • Component B is fails at ������= 22 • Component B is repaired at ������= 35 • Component B fails at ������= 49 • Component A is repaired at ������= 63 • Component B is repaired at ������= 66 • Component A fails at ������= 91 • Component B fails at ������= 93 • No events from time 93 to time 100 You are required to provide the Monte Carlo estimates and the associated uncertainty of : a) The system reliability at times ������= 25 ,50 ,75 ,100 ; b) The system instantaneous availability at times ������= 25 ,50 ,75 ,100 ; c) The system average availability over its mission time