Energy Engineering - RELIABILITY, SAFETY AND RISK ANALYSIS C

Exam simulation 2021

Other

Name:___________________ ID:___________________ RELIABILITY, SAFETY AND RISK ANALYSIS : Exam Simulation (17 /0 5/20 21 ) Note: 1. Make sure to write your name and ID number 2. The exam consists of 2 numerical problem s and 2 question s on the course topics 3. Exam time is 2 hour s and 15 minut es Problem 1 (7.5 points) Consid er the following system (reliability block diagram) : A. Draw a fault tree for the failure of the system , i.e. “no connection between nodes “1” and “2 ” and write the corresponding structure function B. Obtain the reduced for m of the structure function and f ind the minimal cut sets. C. Find the probability of the top event assuming that the component’s individual unreliabilities are: qA = 10 -3 qB = 2x10 -3 qC = 10 -2 qD = 9x10 -3 D. Consider a case in which A and B unreliabilities ar e due to internal component failures (A’ and B’) and to a failure of their electric power generator (G) , w ith unreliabilit ies qA’ = 0.9 10 -3, qB’ = 1.9 10 -3 and qG = 10 -4. Assume that the same electric power generator is shared between A and B. C ompute th e top ev ent probability in this case. Problem 2 (7.5 points) Consider a 2 -out -of-4 parallel structure of identical components. sharing a common load. In normal operation, four components equally share the load; they may fail with failure rate 4 (expone ntial process). When one component fails, the remaining three have to carry the whole load and the failure rate immediately increases to 3 (exponential process); similarly, if a nother component fails, the remaining two carry the entire load with a failure rate increasing to 2 (exponential process). The repair process of the fa iled components do es not start un til at least two components are failed and it is characterized by a constant repair rate s. T he repairmen team restores the functionality of two components at the same time . When only two components are working, an additional failure will trigger the shut -down of the plant to prevent severe damages, since one component alone is not enough to provide the required load. In case of shut d own t he repair process is such that the repairmen team restores the functionality of two components at the same time and simultaneously activate the component in shut down . A. Draw the Markov diagram of the system, upon proper definition of the system states. B C B D A A 1 2 Name:___________________ ID:___________________ B. Write the transition matrix and the Kolmogorov equation in matrix form. C. Find the MTTF. Try to make some comments on this result. D. Mo dify the Markov diagram to account for an external event hitting the system with rate c and failing the working components with a probability p. Question 1 Monte Carlo simulation for system reliability (7.5 points): Consider a plant made by 2 identical components in parallel (A and B). Both components can be in three states: • 0: healthy • 1: partially degraded (still able to perform its function) • 2: faulty a) list the possible states of the plant, indicate those corresponding to a system failure and draw a possible plant life in the phase space; b) Assume that: • the transition rates between the component states are constant and given by: • 0→ 1: • 0→ 2: • 1→ 0: • 1→ 2: • 2→ 0: • 2→ 1: • both c omponents start to work in state 0 at time t=0: • what is the probability that the first system transition will occur between time and ? • What is the probability that the first transition will be made by component A? • What i s the pro bability that component A will enter in state 1 as a result of the first transition? Part A+B (10 Credit course ): Question 2 (7.5 points): 2.a) Define the Birnbaum importance measure. 2.b) Show why the Birnbaum importance measure of a generic component j may, under proper assumptions, be interpreted as the probability of component j being critical. Part C (8 Credit Course ): Question 2 (7.5 points): Consider a component of a safety system under periodic maintenance. Let be the time interval between two successive maintenance interventions and the maintenance intervention duration. Consider the following causes of failure: • Random failure of the component. The random failure time, , is characterized by the probabilit y density function • Switching failure on demand, with probability of occurrence • Maintenance disabling the component, with probability You are required to compute the average system unavailability over the componen t time horizon .