- Email: [email protected]
Application of reliability assessment by the goal oriented method
9 Application of reliability assessment by the goal oriented method Chapter Outline 9.1 Introduction ................................................................................................................................. 169 9.2 Case study .................................................................................................................................... 169 9.2.1 Conducting system analysis of electronic control system of the hoisting mechanism ............................................................................................................ 169 9.2.2 Developing the goal oriented model of electronic control system of the hoisting mechanism ......................................................................................................................... 170 9.2.3 Selecting the test unit by qualitative analysis of the goal oriented method .............. 170 9.2.4 Collecting test data ........................................................................................................... 170 9.2.5 Estimating the failure rate of the test unit..................................................................... 170 9.2.6 Evaluating system mean time to failures ........................................................................ 175 9.3 Result analysis ............................................................................................................................. 178
9.1 Introduction Taking the electronic control system of the hoisting mechanism (ECSOHM) in a nuclear power plant as a case study in this chapter, the reliability assessment method based on the goal oriented (GO) method is illustrated. The mean time to failures (MTTF) of ECSOHM is also evaluated in this case.
9.2 Case study 9.2.1 Conducting system analysis of electronic control system of the hoisting mechanism 9.2.1.1 To analyze the system principle, function, and structure The function of the hoisting mechanism is to carry out raising and lowering of control rods. The hoisting mechanism is composed of an electronic control system and a mechanical executing system. The electronic control system mainly achieves power distribution and control function. There are three phased missions, which are the switch-on phase, the startup phase, and the operating phase. The function of the switch-on phase is to achieve the power distribution of the direct current supply, PLC, control panel, etc. The function of the startup phase is to break over the control circuit. The function of the operating phase is to control the Goal Oriented Methodology and Applications in Nuclear Power Plants. DOI: https://doi.org/10.1016/B978-0-12-816185-2.00009-5 © 2020 Elsevier Inc. All rights reserved.
169
170
Goal Oriented Methodology and Applications in Nuclear Power Plants
mechanical executing system. An electricity schematic brief diagram of the electronic control system is shown in Fig. 77A.
9.2.1.2 To define the system success rule According to the analysis of the system principle, function, and structure, we define the system success rule as follows: ECSOHM can achieve power distribution and control the raising and lowering of the control rods, and the correlations in the system are not considered.
9.2.2 Developing the goal oriented model of electronic control system of the hoisting mechanism 9.2.2.1 To select the goal oriented operator According to the system analysis result of ECSOHM, the GO operators for describing function units and logical relationship are determined, as shown in Table 91.
9.2.2.2 To establish the goal oriented model According to the system analysis and GO operators of ECSOHM, the GO model of ECSOHM is established, as shown in Fig. 91. In GO operators of the GO model, the former number is the type of GO operator, and the latter number is the serial number of the GO operator. The number on a signal flow is the serial number of signal flow. The signal flow 47 is the signal flow of the system output.
9.2.3 Selecting the test unit by qualitative analysis of the goal oriented method All minimum cut sets of ECSOHM can be obtained by qualitative analysis of the GO method, as shown in Table 92. In this case, all minimum cut set units are selected as test units except the virtual input units.
9.2.4 Collecting test data 9.2.4.1 To determine the test type of the test unit In this case, a nonsubstitute number tac-tail life test is selected as the test type of all test units.
9.2.4.2 To conduct the unit test and collect the test data According to the test type of the test unit, the unit test is conducted, with each unit having 10 test samples. The test data are collected as shown in Table 93.
9.2.5 Estimating the failure rate of the test unit 9.2.5.1 To determine the total time test For each test unit, the total time test is obtained by Tr 5
r X i51
xi 1 ðn 2 rÞxr
(9.1)
Table 9–1 Goal oriented operators of electronic control system of the hoisting mechanism. Operator number 1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 42 43 44 47 49 50 51 52 54 55 56 57 58 59 60 6, 8, 9, 34, 36, 39, 46, 48 45, 53
Description Three-phase power DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 SB2.1 operating KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 SA2.1 operating Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 JS3.1 operating EN8.1 EN8.2 DR6.1 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 operating PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3
Type 0
5 1 1 1 1 1 6 6 50 1 1 1 1 6 50 1 1 1 1 6 6 1 1 1 1 6 1 1 6 1 6 6 6 6 6 50 1 1 6 5 6 5 6 1 50 6 1 1 1 1 10 2
Property
Operator description
Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Logical operator Logical operator
Virtual input signal, whose reliability is 1 Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator AND gate OR gate
172
Goal Oriented Methodology and Applications in Nuclear Power Plants
14 1-14
11
1-15
16
13
6-11
15
1-13
1-16
1
6-17
17 3
1-3
1 5-1
1-2
2
10-6
1-7
4
5
6-23
21
25
6-24
22 1-22
1-21
24
23
19
1-19
1-20
10-9
5-18
5-12
20
9 1-5
12
6-10
18
10-8
1-4
10
7
6
1-26
1-27
26
1-28
1-25
28
27 6-29
29
31 30 1-30
1-31
32 1-33
6-32
33 34 6-35
1034
39 1039
35
40
43
5-42
1-43
38 6-38
1046
6-40
45
36
46 6-47
47 System output
42
41
1036
1-44
2-45
6-41
44 37
56 48
1048
6-37
55 6-56
5-55
57 1-57
58 49 5-49
6-50
59
53 2-53
51 5-51
6-52
1-58
54
50
1-59
1-54
60 52
1-60
FIGURE 9–1 Goal oriented model of electronic control system of the hoisting mechanism.
where Tr is the total time test, r is failure number, xi is the life of the ith test sample, and n is the number of the test sample.
9.2.5.2 To develop the likelihood function According to the sample date of the nonsubstitute number tac-tail life test, the likelihood function is developed as follows:
Table 9–2 All minimum cut sets of electronic control system of the hoisting mechanism. Order
Operator number
Unit
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 42 54 55 56 57 58 59 60 43, 44 49, 51 49, 52 50, 51 50, 52
Three-phase power DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 SB2.1 operating KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 SA2.1 operating Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 JS3.1 operating SA7.1 PRS3.1 operating PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3 EN8.1, EN8.2 LC7.1, LC7.2 LC7.1, AMP7.2 AMP7.1, LC7.2 AMP7.1, AMP7.2
174
Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 9–3
Test data of the selected test unit.
Operator number
Unit
Failure number
Total time (106 h)
2 3 4 5 7 10 11 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 43 44 49 50 51 52 54 56 57 58 59 60
DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 EN8.1 EN8.2 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3
6 3 3 3 5 4 7 7 5 2 7 5 5 5 8 5 3 3 6 6 5 5 5 5 7 7 7 4 4 2 4 3 5 5 5 5 5 5 5 3 6 6 6 6
4.9587 4.7619 4.7619 4.7619 21.7391 5.1282 10.6061 15.5556 21.7391 0.8475 15.5556 15.1515 21.7391 21.7391 38.0952 21.7391 2.2059 2.2059 50.0000 50.0000 21.7391 6.0976 6.0976 21.7391 9.4595 28.0000 43.7500 6.4516 6.4516 3.4483 6.4516 8.5714 38.4615 38.4615 20.0000 20.0000 45.4545 29.4118 8.9286 12.5000 14.2857 14.2857 8.4507 8.4507
Chapter 9 • Application of reliability assessment by the goal oriented method
LðθÞ 5
n! θ2r e2Tr =θ ðn 2 rÞ!
175
(9.2)
where θ is the life of the test unit.
9.2.5.3 To obtain the failure rate of the test unit To use the logarithm derivation for Eq. (9.3), the failure rate of the test unit is obtained by 1 r λ^ 5 5 Tr θ^
(9.3)
where λ^ is the estimated value of the failure rate for the test unit and θ^ is the estimated value of life for the test unit. The failure rate of each test unit is obtained by Eq. (9.4) and Table 93, as shown in Table 94.
9.2.6 Evaluating system mean time to failures 9.2.6.1 To determine the time points In this case, we evaluate the MTTF of ECSOHM after operating for 10,000 hours. These 10,000 hours are divided into 100 time points.
9.2.6.2 To obtain the unit reliability For test units, their reliabilities are obtained by Eq. (9.5) and Table 94. For nontest units, their reliabilities are set at 1. ^
RðtÞ 5 e2λt
(9.4)
where RðtÞ is the reliability of the test unit at time t.
9.2.6.3 To obtain the system reliability The signal flow 2, 3, 4, 5, 13, 15, 26, 27, 28, 31, and 41 are shared signals, so the calculating form of the exact algorithm with a shared signal is adopted to conduct the quantitative analysis. Taking an example of t 5 100 hours, the calculating process of system reliability is presented in Table 95. In the same way, the system reliabilities at different time points are obtained, as presented in Table 96.
9.2.6.4 To evaluate the system mean time to failures The MTTF of ECSOHM is evaluated as follows: MTTF 5
ð 10;000 0
RðtÞdt 5
ð 10;000 0
e20:000002081t dt 9796:68 hours
(9.5)
176
Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 9–4
Failure rate of the selected test unit.
Operator number
Unit
Failure rate (1026/h)
2 3 4 5 7 10 11 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 43 44 49 50 51 52 54 56 57 58 59 60
DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 EN8.1 EN8.2 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3
1.21 0.63 0.63 0.63 0.23 0.78 0.66 0.45 0.23 2.36 0.45 0.33 0.23 0.23 0.21 0.23 1.36 1.36 0.12 0.12 0.23 0.82 0.82 0.23 0.74 0.25 0.16 0.62 0.62 0.58 0.62 0.35 0.13 0.13 0.25 0.25 0.11 0.17 0.56 0.24 0.42 0.42 0.71 0.71
Chapter 9 • Application of reliability assessment by the goal oriented method
Table 9–5
177
Calculating process of system reliability at 100 h.
State of shared signal S2
S3
...
S41
State combination probability
0 0 ^ 1
0 0 ^ 1
... ... ^ 1
0 1 ^ 1
2.807e45 2.399e41 ^ 0.9988
Success probability of system corresponding to state combination of shared signal 0 0 0 0.9991
System reliability (100 h) 0.9979
Table 9–6 Reliability of electronic control system of the hoisting mechanism at different times. Number
Time (h)
Reliability
Number
Time (h)
Reliability
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
0.9979 0.9958 0.9938 0.9917 0.9896 0.9876 0.9855 0.9835 0.9814 0.9794 0.9774 0.9753 0.9733 0.9713 0.9693 0.9673 0.9652 0.9632 0.9612 0.9592 0.9572 0.9552 0.9533 0.9513 0.9493 0.9473 0.9454 0.9434 0.9414 0.9395
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 6900 7000 7100 7200 7300 7400 7500 7600 7700 7800 7900 8000
0.8993 0.8974 0.8956 0.8937 0.8918 0.8900 0.8881 0.8863 0.8845 0.8826 0.8808 0.8789 0.8771 0.8753 0.8735 0.8717 0.8699 0.8680 0.8662 0.8644 0.8626 0.8608 0.8591 0.8573 0.8555 0.8537 0.8519 0.8502 0.8484 0.8466 (Continued)
178
Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 96
(Continued)
Number
Time (h)
Reliability
Number
Time (h)
Reliability
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.9375 0.9356 0.9336 0.9317 0.9298 0.9278 0.9259 0.9240 0.9220 0.9201 0.9182 0.9163 0.9144 0.9125 0.9106 0.9087 0.9068 0.9049 0.9031 0.9012
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
8100 8200 8300 8400 8500 8600 8700 8800 8900 9000 9100 9200 9300 9400 9500 9600 9700 9800 9900 10,000
0.8449 0.8431 0.8414 0.8396 0.8379 0.8361 0.8344 0.8327 0.8309 0.8292 0.8275 0.8257 0.8240 0.8223 0.8206 0.8189 0.8172 0.8155 0.8138 0.8121
9.3 Result analysis In order to verify the advantages and rationality of the new reliability assessment method for complex nuclear power equipment, the results are compared with the results obtained using the Monte Carlo method, which is widely used to evaluate the system reliability index. The main steps of the Monte Carlo method are as follows: (1) the simulation model is developed according to the logical relationship among the system and its units, (2) random numbers of unit reliability at time t are generated, (3) the system reliability at time t is obtained by simulation 1 million times, (4) repeat steps (2) and (3) until the system reliabilities at all time points are obtained, and (5) evaluate system MTTF by Eq. (9.5) using system reliabilities at all time points. The MTTFs of ECSOHM using the GO method and the Monte Carlo method are presented in Table 97. Table 97 shows that: • MTTFs by the GO method are in close proximity to the MTTFs obtained using Monte Carlo simulation. Therefore this indicates that the reliability assessment method based on the GO method is feasible and reasonable. • MTTFs by the Monte Carlo method at different operation numbers fluctuate because of the influence of sampling, but MTTFs obtained using the GO method at different operation numbers are the same. Thus this indicates that the reliability assessment
Chapter 9 • Application of reliability assessment by the goal oriented method
Table 9–7
179
MTTF of ECSOHM using different methods. MTTF hour Operation number
Method
No. 1
No. 2
No. 3
No. 4
No. 5
Operation time (once)
GO MCS
9796.68 9212.24
9796.68 9913.56
9796.68 9545.78
9796.68 9662.10
9796.68 9756.87
12.53 s 72.65 s
GO, goal oriented; MCS, Monte Carlo simulation; MTTF, mean time to failures; ECSOHM, electronic control system of hoisting mechanism.
method based on the GO method can avoid the influence of sampling, so that it can obtain a stable evaluation result. • The operation time of the GO method is much less than the operation time of the Monte Carlo method. Thus it shows that the reliability assessment method based on the GO method has higher efficiency. The process of this chapter’s method shows that it has some obvious advantages, as follows: • Only reliability data of the unit are used to evaluate the system MTTF, therefore this reliability assessment method does not need to conduct a system test to save cost. • The GO model is closely linked to system structure and function, therefore it is easy to check for complex nuclear power equipment, so that it can avoid effectively human errors; the evaluation process of the reliability assessment method based on the GO method is also easy to operate.
9.1 Introduction Taking the electronic control system of the hoisting mechanism (ECSOHM) in a nuclear power plant as a case study in this chapter, the reliability assessment method based on the goal oriented (GO) method is illustrated. The mean time to failures (MTTF) of ECSOHM is also evaluated in this case.
9.2 Case study 9.2.1 Conducting system analysis of electronic control system of the hoisting mechanism 9.2.1.1 To analyze the system principle, function, and structure The function of the hoisting mechanism is to carry out raising and lowering of control rods. The hoisting mechanism is composed of an electronic control system and a mechanical executing system. The electronic control system mainly achieves power distribution and control function. There are three phased missions, which are the switch-on phase, the startup phase, and the operating phase. The function of the switch-on phase is to achieve the power distribution of the direct current supply, PLC, control panel, etc. The function of the startup phase is to break over the control circuit. The function of the operating phase is to control the Goal Oriented Methodology and Applications in Nuclear Power Plants. DOI: https://doi.org/10.1016/B978-0-12-816185-2.00009-5 © 2020 Elsevier Inc. All rights reserved.
169
170
Goal Oriented Methodology and Applications in Nuclear Power Plants
mechanical executing system. An electricity schematic brief diagram of the electronic control system is shown in Fig. 77A.
9.2.1.2 To define the system success rule According to the analysis of the system principle, function, and structure, we define the system success rule as follows: ECSOHM can achieve power distribution and control the raising and lowering of the control rods, and the correlations in the system are not considered.
9.2.2 Developing the goal oriented model of electronic control system of the hoisting mechanism 9.2.2.1 To select the goal oriented operator According to the system analysis result of ECSOHM, the GO operators for describing function units and logical relationship are determined, as shown in Table 91.
9.2.2.2 To establish the goal oriented model According to the system analysis and GO operators of ECSOHM, the GO model of ECSOHM is established, as shown in Fig. 91. In GO operators of the GO model, the former number is the type of GO operator, and the latter number is the serial number of the GO operator. The number on a signal flow is the serial number of signal flow. The signal flow 47 is the signal flow of the system output.
9.2.3 Selecting the test unit by qualitative analysis of the goal oriented method All minimum cut sets of ECSOHM can be obtained by qualitative analysis of the GO method, as shown in Table 92. In this case, all minimum cut set units are selected as test units except the virtual input units.
9.2.4 Collecting test data 9.2.4.1 To determine the test type of the test unit In this case, a nonsubstitute number tac-tail life test is selected as the test type of all test units.
9.2.4.2 To conduct the unit test and collect the test data According to the test type of the test unit, the unit test is conducted, with each unit having 10 test samples. The test data are collected as shown in Table 93.
9.2.5 Estimating the failure rate of the test unit 9.2.5.1 To determine the total time test For each test unit, the total time test is obtained by Tr 5
r X i51
xi 1 ðn 2 rÞxr
(9.1)
Table 9–1 Goal oriented operators of electronic control system of the hoisting mechanism. Operator number 1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 42 43 44 47 49 50 51 52 54 55 56 57 58 59 60 6, 8, 9, 34, 36, 39, 46, 48 45, 53
Description Three-phase power DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 SB2.1 operating KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 SA2.1 operating Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 JS3.1 operating EN8.1 EN8.2 DR6.1 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 operating PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3
Type 0
5 1 1 1 1 1 6 6 50 1 1 1 1 6 50 1 1 1 1 6 6 1 1 1 1 6 1 1 6 1 6 6 6 6 6 50 1 1 6 5 6 5 6 1 50 6 1 1 1 1 10 2
Property
Operator description
Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Logical operator Logical operator
Virtual input signal, whose reliability is 1 Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator AND gate OR gate
172
Goal Oriented Methodology and Applications in Nuclear Power Plants
14 1-14
11
1-15
16
13
6-11
15
1-13
1-16
1
6-17
17 3
1-3
1 5-1
1-2
2
10-6
1-7
4
5
6-23
21
25
6-24
22 1-22
1-21
24
23
19
1-19
1-20
10-9
5-18
5-12
20
9 1-5
12
6-10
18
10-8
1-4
10
7
6
1-26
1-27
26
1-28
1-25
28
27 6-29
29
31 30 1-30
1-31
32 1-33
6-32
33 34 6-35
1034
39 1039
35
40
43
5-42
1-43
38 6-38
1046
6-40
45
36
46 6-47
47 System output
42
41
1036
1-44
2-45
6-41
44 37
56 48
1048
6-37
55 6-56
5-55
57 1-57
58 49 5-49
6-50
59
53 2-53
51 5-51
6-52
1-58
54
50
1-59
1-54
60 52
1-60
FIGURE 9–1 Goal oriented model of electronic control system of the hoisting mechanism.
where Tr is the total time test, r is failure number, xi is the life of the ith test sample, and n is the number of the test sample.
9.2.5.2 To develop the likelihood function According to the sample date of the nonsubstitute number tac-tail life test, the likelihood function is developed as follows:
Table 9–2 All minimum cut sets of electronic control system of the hoisting mechanism. Order
Operator number
Unit
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 42 54 55 56 57 58 59 60 43, 44 49, 51 49, 52 50, 51 50, 52
Three-phase power DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 SB2.1 operating KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 SA2.1 operating Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 JS3.1 operating SA7.1 PRS3.1 operating PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3 EN8.1, EN8.2 LC7.1, LC7.2 LC7.1, AMP7.2 AMP7.1, LC7.2 AMP7.1, AMP7.2
174
Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 9–3
Test data of the selected test unit.
Operator number
Unit
Failure number
Total time (106 h)
2 3 4 5 7 10 11 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 43 44 49 50 51 52 54 56 57 58 59 60
DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 EN8.1 EN8.2 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3
6 3 3 3 5 4 7 7 5 2 7 5 5 5 8 5 3 3 6 6 5 5 5 5 7 7 7 4 4 2 4 3 5 5 5 5 5 5 5 3 6 6 6 6
4.9587 4.7619 4.7619 4.7619 21.7391 5.1282 10.6061 15.5556 21.7391 0.8475 15.5556 15.1515 21.7391 21.7391 38.0952 21.7391 2.2059 2.2059 50.0000 50.0000 21.7391 6.0976 6.0976 21.7391 9.4595 28.0000 43.7500 6.4516 6.4516 3.4483 6.4516 8.5714 38.4615 38.4615 20.0000 20.0000 45.4545 29.4118 8.9286 12.5000 14.2857 14.2857 8.4507 8.4507
Chapter 9 • Application of reliability assessment by the goal oriented method
LðθÞ 5
n! θ2r e2Tr =θ ðn 2 rÞ!
175
(9.2)
where θ is the life of the test unit.
9.2.5.3 To obtain the failure rate of the test unit To use the logarithm derivation for Eq. (9.3), the failure rate of the test unit is obtained by 1 r λ^ 5 5 Tr θ^
(9.3)
where λ^ is the estimated value of the failure rate for the test unit and θ^ is the estimated value of life for the test unit. The failure rate of each test unit is obtained by Eq. (9.4) and Table 93, as shown in Table 94.
9.2.6 Evaluating system mean time to failures 9.2.6.1 To determine the time points In this case, we evaluate the MTTF of ECSOHM after operating for 10,000 hours. These 10,000 hours are divided into 100 time points.
9.2.6.2 To obtain the unit reliability For test units, their reliabilities are obtained by Eq. (9.5) and Table 94. For nontest units, their reliabilities are set at 1. ^
RðtÞ 5 e2λt
(9.4)
where RðtÞ is the reliability of the test unit at time t.
9.2.6.3 To obtain the system reliability The signal flow 2, 3, 4, 5, 13, 15, 26, 27, 28, 31, and 41 are shared signals, so the calculating form of the exact algorithm with a shared signal is adopted to conduct the quantitative analysis. Taking an example of t 5 100 hours, the calculating process of system reliability is presented in Table 95. In the same way, the system reliabilities at different time points are obtained, as presented in Table 96.
9.2.6.4 To evaluate the system mean time to failures The MTTF of ECSOHM is evaluated as follows: MTTF 5
ð 10;000 0
RðtÞdt 5
ð 10;000 0
e20:000002081t dt 9796:68 hours
(9.5)
176
Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 9–4
Failure rate of the selected test unit.
Operator number
Unit
Failure rate (1026/h)
2 3 4 5 7 10 11 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 38 40 41 43 44 49 50 51 52 54 56 57 58 59 60
DISC1.1 FU1.1 FU1.2 FU1.3 Q1.1 PMR1.1 SB2.1 KA2.1 Q2.1 MSR2.1 KA2.2 SA2.1 Q1.2 Q1.3 T1.1 Q1.4 KM2.1 KM2.2 FLT1.1 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 TAS3.1 NODE PLCDI3.1 PLCDI3.2 CPU8.1 PLCDO4.1 JS3.1 EN8.1 EN8.2 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3
1.21 0.63 0.63 0.63 0.23 0.78 0.66 0.45 0.23 2.36 0.45 0.33 0.23 0.23 0.21 0.23 1.36 1.36 0.12 0.12 0.23 0.82 0.82 0.23 0.74 0.25 0.16 0.62 0.62 0.58 0.62 0.35 0.13 0.13 0.25 0.25 0.11 0.17 0.56 0.24 0.42 0.42 0.71 0.71
Chapter 9 • Application of reliability assessment by the goal oriented method
Table 9–5
177
Calculating process of system reliability at 100 h.
State of shared signal S2
S3
...
S41
State combination probability
0 0 ^ 1
0 0 ^ 1
... ... ^ 1
0 1 ^ 1
2.807e45 2.399e41 ^ 0.9988
Success probability of system corresponding to state combination of shared signal 0 0 0 0.9991
System reliability (100 h) 0.9979
Table 9–6 Reliability of electronic control system of the hoisting mechanism at different times. Number
Time (h)
Reliability
Number
Time (h)
Reliability
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
0.9979 0.9958 0.9938 0.9917 0.9896 0.9876 0.9855 0.9835 0.9814 0.9794 0.9774 0.9753 0.9733 0.9713 0.9693 0.9673 0.9652 0.9632 0.9612 0.9592 0.9572 0.9552 0.9533 0.9513 0.9493 0.9473 0.9454 0.9434 0.9414 0.9395
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 6900 7000 7100 7200 7300 7400 7500 7600 7700 7800 7900 8000
0.8993 0.8974 0.8956 0.8937 0.8918 0.8900 0.8881 0.8863 0.8845 0.8826 0.8808 0.8789 0.8771 0.8753 0.8735 0.8717 0.8699 0.8680 0.8662 0.8644 0.8626 0.8608 0.8591 0.8573 0.8555 0.8537 0.8519 0.8502 0.8484 0.8466 (Continued)
178
Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 96
(Continued)
Number
Time (h)
Reliability
Number
Time (h)
Reliability
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.9375 0.9356 0.9336 0.9317 0.9298 0.9278 0.9259 0.9240 0.9220 0.9201 0.9182 0.9163 0.9144 0.9125 0.9106 0.9087 0.9068 0.9049 0.9031 0.9012
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
8100 8200 8300 8400 8500 8600 8700 8800 8900 9000 9100 9200 9300 9400 9500 9600 9700 9800 9900 10,000
0.8449 0.8431 0.8414 0.8396 0.8379 0.8361 0.8344 0.8327 0.8309 0.8292 0.8275 0.8257 0.8240 0.8223 0.8206 0.8189 0.8172 0.8155 0.8138 0.8121
9.3 Result analysis In order to verify the advantages and rationality of the new reliability assessment method for complex nuclear power equipment, the results are compared with the results obtained using the Monte Carlo method, which is widely used to evaluate the system reliability index. The main steps of the Monte Carlo method are as follows: (1) the simulation model is developed according to the logical relationship among the system and its units, (2) random numbers of unit reliability at time t are generated, (3) the system reliability at time t is obtained by simulation 1 million times, (4) repeat steps (2) and (3) until the system reliabilities at all time points are obtained, and (5) evaluate system MTTF by Eq. (9.5) using system reliabilities at all time points. The MTTFs of ECSOHM using the GO method and the Monte Carlo method are presented in Table 97. Table 97 shows that: • MTTFs by the GO method are in close proximity to the MTTFs obtained using Monte Carlo simulation. Therefore this indicates that the reliability assessment method based on the GO method is feasible and reasonable. • MTTFs by the Monte Carlo method at different operation numbers fluctuate because of the influence of sampling, but MTTFs obtained using the GO method at different operation numbers are the same. Thus this indicates that the reliability assessment
Chapter 9 • Application of reliability assessment by the goal oriented method
Table 9–7
179
MTTF of ECSOHM using different methods. MTTF hour Operation number
Method
No. 1
No. 2
No. 3
No. 4
No. 5
Operation time (once)
GO MCS
9796.68 9212.24
9796.68 9913.56
9796.68 9545.78
9796.68 9662.10
9796.68 9756.87
12.53 s 72.65 s
GO, goal oriented; MCS, Monte Carlo simulation; MTTF, mean time to failures; ECSOHM, electronic control system of hoisting mechanism.
method based on the GO method can avoid the influence of sampling, so that it can obtain a stable evaluation result. • The operation time of the GO method is much less than the operation time of the Monte Carlo method. Thus it shows that the reliability assessment method based on the GO method has higher efficiency. The process of this chapter’s method shows that it has some obvious advantages, as follows: • Only reliability data of the unit are used to evaluate the system MTTF, therefore this reliability assessment method does not need to conduct a system test to save cost. • The GO model is closely linked to system structure and function, therefore it is easy to check for complex nuclear power equipment, so that it can avoid effectively human errors; the evaluation process of the reliability assessment method based on the GO method is also easy to operate.