Energy Engineering - RELIABILITY, SAFETY AND RISK ANALYSIS C

Full exam

First name and Last name: Student number: ______________________________________________________________________________________________________________ RELIABILITY SAFETY AND RISK ANALYSIS A+ B 11 /06 /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 (1 hour and 45 minutes for the students attending the exam remotely ). Exercise 1 [ 7.5 points] A fire emergency system is made by a liquid Storage Tank (ST), and three pumps in parallel. Pump P1 is powered by two Electric Generators (EG1, EG2), which work in parallel. Pump P1 stops when both EGs fail. P umps P2 and P3 are Electric Pumps (EP) that depend on the functioning of a common standby diesel generator G. The failure of the system occurs if the storage tank is empty or if all the three pumps fail to function. The system is represented in the figure below and the failure probabilities of each component are reported in the table below: Description Failure Probability ������������ Probability that the storage tank ST is empty pST = 10 -5 Probability that the pump P1 fails pP1 = 10 -3 Probability that the pump P2 fails pP2 = 10 -3 Probability that the pump P3 fails pP3 = 10 -3 Probability that the electric generator EG1 fails pEG1 = 0.5∙10 -3 Probability that the electric generator EG2 fails pEG2 = 0.5∙10 -3 Probability that the generator G fails pG = 0.5∙10 -3 E1.1) Draw the fault tree of the top event “failure of the system” ; E1.2) Write the structure function ; E1.3) F ind the minimal cut sets ; E1.4) Find the probability of the top event ; E1.5) Compute the Birnbaum Importance Measure of the Tank and the three pumps. Comment the results. E1.6) Consi der now a common cause failure for pumps P2 and P3. Using the beta factor model, ( ������ = 0.05 ), repeat E1.3 and E1.4. Exercise 2 [ 7.5 points] Consider an electrical network equipped with a protection system from an external threat (such as lightning, extreme weather, etc). The protection system can fail with constant failure rate ������� and the failure remains undetected until the electrical netwo rk is subjected to an external threat, and, in this situation, the electric network fails. In addition, the electrical network can have an internal detectable failure with constant failure rate �. After the internal failure, the electrical network is repa ired by a maintenance team. Repair time is exponentially distributed with constant repair rate �. Whenever the electrical network is repaired, its protection system from external hazard is also checked and, if necessary, repaired. Let ������ denote the rate o f occurrence of the external threat. You are required to: E2.1) Draw the Markov diagram of the system, upon proper definition of the system states; E2.2) Write the transition matrix and the M arkov equation in matrix form; E2.3) Find the MTTF; E2.4) Find the steady -state availability of the electric network; E2.5) Find the expected number of failure of the protection system in a period of time T, and the expected number of maintenance interventions. Assume that the system has reached an asymptotic behaviour. Question 1 [7.5 points] Q1.1) Illustrate the inverse transform method for sampling random numbers from a generic probability distribution ������������(������). Q1.2) Show that the samples obtained according to the procedure in Q1.1) are distributed according to the desired probability distri bution. Q1.3) A component of a network system has probability p to be failed. Show the application of the method of the inverse transform to simulate the state of the component. Q1.4) Consider the network system of the Figure below: You are required to provide the Monte Carlo estimation of the failure probability of the network, i.e., the probability of no connection between nodes S and T, simulating the system state M=5 times using the attached random numbers. Arc Number ������ Failure Probability ������������ 1 0.65 2 0. 25 3 0.15 4 0. 1 5 0.03 6 0.4 1) 0.5508 2) 0.7081 3) 0.2909 4) 0.5108 5) 0.8929 6) 0.8963 7) 0.1256 8) 0.2072 9) 0.0515 10) 0.4408 11) 0.0299 12) 0.4568 13) 0.6491 14) 0.2785 15) 0.6763 16) 0.5909 17) 0.0240 18) 0.5589 19) 0.0493 20) 0.4151 21) 0.2835 22) 0.6931 23) 0.4405 24) 0.1569 25) 0.5446 26) 0.7803 27) 0.3064 28) 0.2220 29) 0.3880 30) 0.9364 31) 0.9760 32) 0.6724 33) 0.9028 34) 0.8458 35) 0.3780 36) 0.0922 37) 0.6534 38) 0.5578 39) 0.3616 40) 0.2251 41) 0.4065 42) 0.4689 43) 0.2692 44) 0.2918 45) 0.4577 46) 0.8605 47) 0.5863 48) 0.2835 49) 0.2780 50) 0.4546 51) 0.2054 52) 0.2014 53) 0.5140 54) 0.0872 55) 0.4836 56) 0.3622 57) 0.7077 58) 0.7467 59) 0.6911 60) 0.6892 61) 0.3736 62) 0.6681 63) 0.3398 64) 0.5728 65) 0.3258 66) 0.4451 67) 0.0615 68) 0.2427 69) 0.9716 70) 0.2306 71) 0.6915 72) 0.6505 73) 0.7239 74) 0.4751 75) 0.5967 76) 0.0670 77) 0.0726 78) 0.1990 79) 0.1519 80) 0.1001 81) 0.1293 82) 0.5533 83) 0.1878 84) 0.9521 85) 0.6816 86) 0.5410 87) 0.7072 88) 0.2639 89) 0.9267 90) 0.8392 Question 2 [7.5 points] Q2.1) Let A and B be the failure events of two different components of the same plant. When are A and B dependent from a mathematical point of view? Q2.2) Illustrate the Beta -factor model for dependent failure analysis and show an example of its applicatio n to an industrial system. Q2.3) Assume that a two -train redundant standby safety system is periodically tested every 10 days. During each test, the two trains are tested at the same time. The Table below reports the result of the tests (S = success, F = f ailure). Estimate the parameter ������ of the Beta -factor model. Day 0 Day 10 Day 20 Day 30 Day 40 Day 50 Day 60 Day 70 Day 80 Day 90 Comp. 1 S S F F S S S S F F Comp. 2 S F S F S S S F F S