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Applications of reliability modeling and analysis by the goal oriented method
7 Applications of reliability modeling and analysis by the goal oriented method Chapter Outline 7.1 Introduction ................................................................................................................................. 131 7.2 Case study I .................................................................................................................................. 131 7.2.1 Goal oriented modeling method for multilevel standby structures............................. 132 7.2.2 Reliability modeling and analysis..................................................................................... 132 7.2.3 Result analysis .................................................................................................................... 137 7.3 Case study II ................................................................................................................................. 137 7.3.1 Reliability modeling and analysis..................................................................................... 138 7.3.2 Result analysis .................................................................................................................... 144 7.4 Case study III ................................................................................................................................ 144 7.4.1 Reliability modeling and analysis..................................................................................... 145 7.4.2 Result analysis .................................................................................................................... 154
7.1 Introduction This chapter presents three cases to illustrate the method of using reliability modeling and analysis based on the goal oriented (GO) method for nuclear power plant systems with characteristics including an AC power system for a single unit with two 100% capacity divisions, considering the multilevel standby structure, power structure of the control rod drive mechanism (CRDM) with three-state electrical units, and a hoisting mechanism considering multifunction and common cause failure (CCF), respectively.
7.2 Case study I Taking the AC power system for a single unit with two 100% capacity divisions in a nuclear power plant as an example, the conducting a reliability analysis using the GO method is illustrated. In order to verify the feasibility, advantages, and reasonableness of the GO method for reliability analysis of systems considering their characteristics, its analysis results
Goal Oriented Methodology and Applications in Nuclear Power Plants. DOI: https://doi.org/10.1016/B978-0-12-816185-2.00007-1 © 2020 Elsevier Inc. All rights reserved.
131
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Goal Oriented Methodology and Applications in Nuclear Power Plants
are compared with those result of a GO method without considering standby correlations among the multilevel standby structure.
7.2.1 Goal oriented modeling method for multilevel standby structures In power systems of nuclear power plants, the multilevel standby structure often exists to improve the reliability and safety of power systems. Its typical structure is shown in Fig. 71 and the corresponding GO model is shown in Fig. 72. In Fig. 72, the Type 5 and Type 6 operators are used to describe the input unit and switching unit in an input equipment group i, i 5 1; 2; . . .; n. Both primary the equipment group and standby equipment groups are input equipment groups.
7.2.2 Reliability modeling and analysis 7.2.2.1 Analyzing the AC power system 7.2.2.1.1 Step 1: Analyzing system structure and function constituents In a typical nuclear power plant, the alternating current power system includes the power supplies and distribution systems arranged to provide power to the alternating current loads
Input unit Primary equipment group
Input unit
Switching unit Standby equipment group 1 Standby structure 1
Input unit
Switching unit Standby equipment group 2 Standby structure 2
Input unit
Switching unit Standby equipment group n Standby structure n
FIGURE 7–1 Multilevel standby structure.
Chapter 7 • Applications of reliability modeling and analysis
133
Standby structure n Standby structure 2 Standby structure 1
5
18A
18A
…
18A
20
5
6
5
6
5
20
6
20
FIGURE 7–2 Goal oriented model for a multilevel standby structure.
and controls for plant normal operation, startup, and orderly shutdown. These Class 1E electric loads are usually separated into two or more redundant load groups. To assure the reliability of the whole power system, some special protective actions are often adopted when one of the power sources is out of operation, as shown in Fig. 73. As illustrated in Fig. 73, the system input boundaries are normal power supply, unit generator, backup power supply, standby generator A, and standby generator B. The system output boundaries are medium voltage bus A and medium voltage bus B. 7.2.2.1.2 Step 2: Determining multilevel standby structures As we can see from Fig. 73, when normal power supply or the incoming breaker 1M1 (2M1) are unavailable to feed to the medium voltage bus, they may be transferred to another backup AC power supply through the incoming breaker 1M2 (2M2). This transfer may be in the form of a fast bus automatic transfer. When both normal and backup power supply are not available, the medium voltage bus will be powered by redundant onsite standby power sources (diesel generators) through the third breaker 1M3 (2M3) within a time consistent with the requirements of the function under normal and abnormal conditions.
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Normal power supply
Backup power supply
Step-up transformer A
Generator circuit breaker Start-up transformer B
Auxiliary transformer
Unit generator
Interlock Medium voltage bus A
2M1
1M1 1 Bus transfer
1M2
2M2
2 Bus transfer
Interlock Medium voltage bus B
1M3
Standby generator A
2M3
Standby generator B
FIGURE 7–3 Structure of an AC power system for a single unit with two 100% capacity divisions.
7.2.2.1.3 Step 3: Making the system success rule According to the above system analysis, the system success rule is defined as the AC power system supplying power to medium voltage bus A or medium voltage bus B normally.
7.2.2.2 Establishing the goal oriented model of an AC power system 7.2.2.2.1 Step 1: Selecting the goal oriented operator According to the analysis results of an AC power system, the GO operators are selected to describe the unit logical relationship in the system, as presented in Table 71. 7.2.2.2.2 Step 2: Establishing the system goal oriented model According to the above analysis of an AC power system and selected GO operators, the system GO model is established, as shown in Fig. 74. In operators of the GO model, the
Chapter 7 • Applications of reliability modeling and analysis
135
Table 7–1 Goal oriented operator type for the goal oriented model of an AC power system. Operator number
Type
Description
Property
1 2 3 4 5 6 7 8 9 10 12 11 13 15 14 16 17 18 19 21 20 22 23 25 24 26 27
5 1 5 1 2 1 1 5 1 20 18A 6 5 6 20 18A 1 1 20 18A 6 5 20 18A 6 1 2
Normal power supply Start-up transformer A Unit generator Generator circuit breaker OR Auxiliary transformer 1M1 Backup power supply Start-up transformer B Standby
Function operator Function operator Function operator Function operator Logical operator Function operator Function operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Logical operator
1M2 Standby generator A 1M3 Standby Medium voltage bus A 2M1 Standby 2M2 Standby generator B Standby 2M3 Medium voltage bus B OR
former number is the GO operator type, and the latter number is the serial number of the GO operator. The number on a signal flow is the serial number of the signal flow. The signal flow 27 is the signal flow of system output.
7.2.2.3 Processing data According to engineering statistical results, the fault rate and maintenance rate of units are obtained, as presented in Table 72. The availabilities of units are obtained by Eq. (7.1), as presented in Table 72. A5
μ μ1λ
(7.1)
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Goal Oriented Methodology and Applications in Nuclear Power Plants
1 5-1
1-2
2
5
6
2-5 3 5-3
7
1-6
1-7
18A -12
12
16 18A -16
1-17
4
1-4
17 20-10
20-14
27
10 8 5-8
9
1-9
2-27 11
6-11
14
15
System output
26 19 6-20
18 20-19
13 1-18
5-13
6-15
1-26 25
22 5-22
20 18A -21
24 6-24
21
18A -25
23 20-23
FIGURE 7–4 Goal oriented model of an AC power system for a single unit with two 100% capacity divisions.
Table 7–2
Reliability parameters of units for an AC power system.
Operator number
Unit
λ failures/year
μ repairs/year
A
1 2 3 4 6 7 8 9 11 13 15 17 18 20 22 24 26
Normal power supply Start-up transformer A Unit generator Generator circuit breaker Auxiliary transformer 1M1 Backup power supply Start-up transformer B 1M2 Standby generator A 1M3 Medium voltage bus A 2M1 2M2 Standby generator B 2M3 Medium voltage bus B
0.643 0.015 0.032 0.045 0.015 0.003 0.643 0.015 0.003 0.032 0.003 0.0001 0.003 0.003 0.032 0.003 0.0001
43.8 43.8 43.8 40 43.8 175.2 43.8 43.8 175.2 43.8 175.2 4380 175.2 175.2 43.8 175.2 4380
0.9855 0.9997 0.9993 0.9989 0.9997 0.999983 0.9855 0.9997 0.999983 0.9993 0.999983 0.9999 0.999983 0.999983 0.9993 0.999983 0.9999
Chapter 7 • Applications of reliability modeling and analysis
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Table 7–3 Calculating process for the quantitative operation of an AC power systems.
S6
S9
State combination probability of S6 and S9
1 1 2 2
1 2 1 2
4.832448542878858e06 3.217803425733531e04 0.014790817551457 0.984882569657427
Success probability of the corresponding state combination 0.999999484494118 0.999999999998976 0.999999999998976 0.999999999999000
System availability 0.999999999996509
where A is the availability of the unit, and μ and λ are the maintenance rate and fault rate of the unit, respectively.
7.2.2.4 Conducting the quantitative operation The signal flows S6 and S9 are the shared signal, and so the exact algorithm with shared signals should be adopted to conduct a quantitative operation. There are four kinds of state combinations for shared signals, and the calculating process for system availability is presented in Table 73. In Table 73, numbers 1 and 2 represent the fault state and success state of S6 and S9, respectively.
7.2.3 Result analysis The quantitative analysis result of the GO method without considering standby correlations among the multilevel standby structure is about 0.99967, which is less than the quantitative analysis result obtained by the GO method when considering standby correlations among the multilevel standby structure. It is shown that: • If the structure correlation in power systems is not considered, the quantitative analysis result will be biased. Moreover, if there are a large number of multilevel standby structures in the system, these biases will be obvious. • The analysis process of this example shows the advantages of the GO method for reliability analysis of power systems with a multilevel standby structure in terms of establishing the reliability model and conducting quantitative analysis.
7.3 Case study II In order to illustrate this book’s GO method, the power structure of CRDM is taken as an example to conduct reliability analysis using the GO method. This analysis result is then compared with the result obtained using the GO method for a two-state nuclear power electrical system in order to illustrate the advantages of the GO method for reliability analysis of the system with characteristics.
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Goal Oriented Methodology and Applications in Nuclear Power Plants
7.3.1 Reliability modeling and analysis 7.3.1.1 Conducting system analysis 7.3.1.1.1 Step 1: Analyzing the system structure and function constituents CRDM is an electromagnetic lifting device that can raise or lower the drive wire and rod cluster. Three coils used to produce magnetic forces, installed outside the pressure shell, are the components of a CRDM. In the pressure shell, there is an attracted armature. The moving-claw armature and the fixed-claw armature can be used to manipulate the talons, which results in the shaft moving. Moving claws and fixing claws are also used to keep the control rods in a certain position. With the armature of the moving and fixed claws, the control rods can be operated in a step-by-step process. The power structure circuit is used to control the armature, as shown in Fig. 75. Ki is the switch, i 5 1; . . .; 9; Rj is the fuse,
N
A B C K7
K8
K9
R8
R7
V9
N
A B C K4
R9
V8
V7
N
K5
K1
K6
R6
R4
R5
V4
V5
A B C
V6
K2
R1
R2
V1
V2
R3 V3
R10
V10
T3 T4
T1 V11
T5
T6 T7
T8
V12
T9
T2
T10
T11 T12 T13 T14
R27 R28 R29 R30 R31 R32 R33 R34
R35 R36 R37 R38
R21 R22
R19 R20
L5
L6
R23 R24 L9
L10 L11 L12 L13 L14 L15 L16 L17 L18 L19
L7
L8
G6
G7
R42 V15
V16
G9
G11
G12
L1
R25 R26
R39 R40 R41
V13 V14
R11 R12
R13
L2
R15 R16
L3
R14
L4
R17
R18
G3
G4
L20 G5
G10
K3
FIGURE 7–5 Circuit diagram of the power structure of a control rod drive mechanism.
G8
G1
G2
Chapter 7 • Applications of reliability modeling and analysis
139
j 5 1; . . .; 42; Gq is the resistance, q 5 1; . . .; 12; Vw is the turn-off thyristor, w 5 1; . . .; 16; Lm is the inductance, m 5 1; . . .; 20; and Tn is the diode, n 5 1; . . .; 14. 7.3.1.1.2 Step 2: Determining three-state electrical units In the power structure of a CRDM, the resistance, turn-off thyristor, and diode are defined as three-state electrical units, that is, G1, G2, G3, G4, G5, G6, G7, G8, G9, G10, G11, G12, V1, V2, V3, V4, V5, V6, V7, V8, V9, V10, V11, V12, V13, V14, V15, V16, G1, G2, G3, G4, G5, G6, G7, G8, G9, G10, G11, G12. 7.3.1.1.3 Step 3: Making the system success rule According to the above system analysis, the success rule of the power structure for a CRDM is defined as that which can control the armature.
7.3.1.2 Developing the goal oriented model 7.3.1.2.1 Step 1: Selecting the goal oriented operator According to the above system analysis, it needs to select five kinds of GO operators to develop the GO model. The type 5 operator represents the input unit. The type 1 operator represents the two-state unit, whose states are operating and open mode failure. The type 26 operator represents the three-state unit, whose states are operating, open mode failure, and close mode failure. The type 2 operator represents the logical relationship “OR.” The type 10 operator represents the logical relationship “AND.” The GO operators of the power structure for a CRDM are presented in Table 74. Table 7–4
Operator type of units.
Description
Operator number
Type
Property
Power K1 K2 K3 K4 K5 K6 K7 K8 K9 T1 T2 T3 T4 T5 T6
1 2 5 8 14 17 20 24 27 30 12 13 72 75 78 81
5 1 1 1 1 1 1 1 1 1 26 26 26 26 26 26
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 (Continued)
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 74
(Continued)
Description
Operator number
Type
Property
T7 T8 T9 T10 T11 T12 T13 T14 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L18 L19 L20 G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 R1 R2 R3 R4
84 87 90 93 96 99 102 105 36 40 44 48 53 57 61 65 74 77 80 83 86 89 92 95 98 101 104 107 38 42 46 50 55 59 63 67 114 117 120 123 3 6 9 15
26 26 26 26 26 26 26 26 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 26 26 26 26 26 26 26 26 26 26 26 26 1 1 1 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 (Continued)
Chapter 7 • Applications of reliability modeling and analysis
Table 74
(Continued)
Description
Operator number
Type
Property
R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26 R27 R28 R29 R30 R31 R32 R33 R34 R35 R36 R37 R38 R39 R40 R41 R42 V1 V2 V3 V4 V5 V6
18 21 25 28 31 34 35 39 43 47 52 56 60 64 37 41 45 49 54 58 62 66 73 76 79 82 85 88 91 94 97 100 103 106 112 115 118 121 4 7 10 16 19 22
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 26 26 26 26 26 26
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 (Continued)
141
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 74
(Continued)
Description
Operator number
Type
Property
V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 11, 23, 33, 51, 68, 108, 109, 110, 111, 124 9
26 29 32 69 70 71 113 116 119 122 OR
26 26 26 26 26 26 26 26 26 26 2
Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Logical operator
AND
10
Logical operator
7.3.1.2.2 Step 2: Establishing the system goal oriented model The GO model of the power structure of a CRDM is established by using signal flow to connect the GO operator from the No. 1 GO operator to the No. 125 GO operator, as shown in Fig. 76. The system output is the signal flow 125.
7.3.1.3 Processing data In this case, we assume that the switch, fuse, resistance, turn-off thyristor, inductance, and diode are identical units. Their state probabilities are presented in Table 75. In Table 75, P1 is the state probability of operating for the unit, P2 is the state probability of close mode failure for the unit, and P3 is the state probability of open mode failure for the unit.
7.3.1.4 Conducting a quantitative operation Because shared signals exist in Fig. 76, a GO algorithm with shared signals should be used to conduct the GO operation. In this case, the modified algorithm with shared signals is adopted. The signal flow S1 is a shared signal in Fig. 76, and the system reliability of the power structure for a CRDM is obtained by Eq. (7.2). P125 ð1Þ 5 P1 ð1Þ½PG1 P125 ð1jPG1 Þ 1 PG2 P125 ð1jPG2 Þ 5 0:9980 3 0:999999794846927 0:9980
(7.2)
where P125 ð1Þ and P1 ð1Þ are the success probability of signal flow 1 and 125, respectively; PG1 and PG2 are the fault probability and success probability of shared signal, that is, PG1 5 0 and PG2 5 1, respectively; P125 ð1jPG1 Þ is the success probability of the system output at the condition of PG1 , and P125 ð1jPG2 Þ is the success probability of system output at the condition of PG2 .
Chapter 7 • Applications of reliability modeling and analysis
35 2 1-3
1-2
40
7
11
1-40
26-7
2-11
26-12
43
42
44 1-44
45 1-45
26-46
46 51
26-10
47
48 1-48
49 1-49
50 26-50
1 10-125
15 1-15
1-14
26-42
2-51
26-13
1-47 14
1-41
10
9 1-9
1-8
41
13
12
1-43 8
26-38 38
39 1-39
6 1-6
1-5
37 1-37
26-4 4
5
36 1-36
1-35
3
52
26-16
1-53
1-52
16
53
1-54
54
5-1
18 1-18
1-17
23
19 26-19
1-56
2-23
56
1-57
57
1-58
System output
26-55 55
17
1
143
58
26-59
59
26-63
63
68
2-68
20
21 1-21
1-20
22
1-60
26-22
60
1-61
64 24
25 1-25
1-24
27
26-26
28 1-28
1-27
1-30
1-31
66 1-66
67 26-67 124
26-72
29
72
1-73
73
1-74
112
108
74
1-112
2-108
113 26-113
26-114
114
2-33
33
32 26-32
76 1-76
26-75
78
1-77
79 1-79
81 69
77
109 2-109
1-115
115
26-116
116
26-117 117 2-124
1-34 26-78
26-69
62
26
26-29
31
1-62
65 1-65
1-64
75 30
61
34 26-84
26-87
82 1-82
26-81
84
87
1-85
1-88
83
88
1-86
1-89
1-118
2-110
1-83
85
110
80 1-80
111 2-111
118
121 1-121
86
89
70 26-70
90
91 1-91
26-90 93
26-96
26-99
96
99
92 1-92
94 1-94
26-93
1-97
1-100
95 1-95
97
100
1-98
1-101
98
101
71 26-71
103
102 1-103
26-102 105 26-105
104 1-104
106 1-106
26-119
107 1-107
FIGURE 7–6 Goal oriented model of the power structure of a control rod drive mechanism.
119
26-120 120 123
122 26-122
26-123
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 7–5
State probability of the unit.
Unit
P1
P2
P3
Unit type
Power Ki, i 5 1; . . .; 9 Rj, j 5 1; . . .; 42 Lm, m 5 1; . . .; 20 Vw, w 5 1; . . .; 16 Tn, n 5 1; . . .; 14 Gq, q 5 1; . . .; 12
0.9980 0.9991 0.9952 0.9992 0.9615 0.9734 0.9852
0.0311 0.0221 0.0140
0.0020 0.0009 0.0048 0.0008 0.0074 0.0045 0.0008
Two-state Two-state Two-state Two-state Three-state Three-state Three-state
7.3.2 Result analysis In order to verify the feasibility, reasonability, and advantages of the GO method for a reliability analysis system considering the characteristics, its analysis result is compared with the results by a GO method for two-state nuclear power electrical systems. The analysis steps are mainly as follows: conducting the system analysis, developing the GO model, obtaining the probability of two-state units, and operating quantitative analysis. In system analysis, the open mode failure and close mode failure are not considered, so all three-state units in systems are considered as two-state units, whose states are operating state and faulting state. In the GO model, all three-state units are represented by a Type 1 operator, and the system reliability using the existing GO method is 0.99795. Compared analysis results show that: • The system reliability obtained by the GO method considering three-state electrical units is larger than the result using the GO method for two-state nuclear power electrical systems. It meets the engineering practice because the system can be operating when the three-state electronic unit is in close mode failure or open mode failure. Therefore it illustrates that the GO method can obtain a more accurate quantitative analysis result. • The reliability analysis process of the example shows the advantages of the GO method in terms of the GO model and quantitative analysis, as follows: • The GO model is closely related to the system structure, working principle, and function institute, so that different engineer(s) can obtain highly consistent reliability analysis results by the GO method. And it is easy to check. • The quantitative analysis result is obtained by multiple GO operations, which are easy to operate.
7.4 Case study III The hoisting mechanism in a nuclear power plant is taken as an example to conduct reliability analysis by the method considering the multifunctions and CCF in order to illustrate this
Chapter 7 • Applications of reliability modeling and analysis
145
book’s GO method. The analysis result is then compared with the result obtained by fault tree analysis (FTA) and MonteCarlo simulation (MCS), respectively.
7.4.1 Reliability modeling and analysis In order to illustrate the GO method conveniently, we assume that: • The availabilities of interfaces of the system are set 1. • The organizational-level maintenance is adopted, that is, the maintenance work is mainly the replacement of components, and the maintenance time is not more than 2 hours. • To select part of the unit in an electronic control system and mechanical executing system as the units with multiple failure modes.
7.4.1.1 Conducting system analysis 7.4.1.1.1 Step 1: Analyzing the system structure and function constituents The function of the hoisting mechanism is to allow descending and ascending 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, and the mechanical executing system enables the control rods to rise and lower. There are three phased missions, which are the switch-on phase, startup phase, and operating phase. The function of the switch-on phase is to achieve the power distribution of direct current supply, programmable logic controller (PLC), control panel, etc. The function of the startup phase is to breakover the control circuit. And the function of the operating phase is to control the mechanical executing system to operate. A function diagram of the hoisting mechanism is shown in Fig. 77. An electricity schematic brief diagram of an electronic control system is shown in Fig. 77A. A function diagram of a mechanical executing system is shown in Fig. 77B. 7.4.1.1.2 Step 2: Determining the system characteristics According to engineering statistical results, the system units with multiple fault modes are presented in Table 76. Because AMP 7.1, AMP 7.2, EN 8.1, and EN 8.2 are affected by interruption to the environment, which can easily break them down at the same time, they are CCF groups. According to system analysis, the time sequence of the three phased missions for the system is, in order, switch-on phase, startup phase, and operating phase. The switch-on phase provides power distribution for breakovering the control circuit and operating the mechanical executing system, the startup phase provides a control signal for the mechanical executing system. The system is a multiple function system. 7.4.1.1.3 Step 3: Making the system success rule According to the analysis of the hoisting mechanism, the success rule can be defined as it achieving descending and ascending of the control rods.
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Goal Oriented Methodology and Applications in Nuclear Power Plants
(A) DISC1.1
L1
Q1.1
FU1.1
FU1.2
L2
PMR1.1
FU1.3
L3
Q1.3 Q1.2
Q1.7
L
+
KM2.1
G U2.1 24V DC/10A
Q1.8 T1.1
KM2.2
FLT1.1 Q1.4
X3 Slot 2
PLC DO 3.2
X2 Ethernet X8 X17 X16
PLC DO 4.1
DR6.1 X1
DP
I005 I006 I007 I008 I009 I010 I011
PLC DO 3.1
FLT1.2 N P
I001 I002 I003
X2 PN L+ M PLC CPU 8.1
M L+ PS 8.1
I004 DP
AMP7.1
AMP7.2
DP
DP
EN8.2
EN8.1
(B)
Power distribution DR6.1
BRK6.1
Steel rope I
Traveling block I
Control
M6.1
Gearbox
Roller
Control rods
Power distribution BRK6.2
PRS3.1 Control
FIGURE 7–7 Function diagram of the hoisting mechanism.
Steel rope II
Traveling block II
N -
Chapter 7 • Applications of reliability modeling and analysis
147
7.4.1.2 Establishing the goal oriented model of the hoisting mechanism 7.4.1.2.1 Step 1: Selecting the goal oriented operator According to the analysis results of the hoisting mechanism, the operator types corresponding to units and logical operator types corresponding to logical relations are determined, as shown in Table 77. Table 7–6
System units with multiple fault modes.
Unit
Failure mode
Failure mode number
Roller
Shaft key failure Shaft distortion Roller distortion Shaft key failure Circuit failure No output Braking torque is insufficient Shaft key failure Open failure Close failure Open failure Close failure
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
Electric machine M6.1 Gearbox Encoder EN8.1 Encoder EN8.2
Table 7–7
Operator type and reliability parameters of the units.
Operator number
Description
Type
Property
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22
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 Q1.6
5 1 1 1 1 1 6 6 5 1 1 1 1 6 5 1 1 1 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 (Continued)
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 77
(Continued)
Operator number
Description
24 KM2.1 26 KM2.2 27 FLT1.1 28 Q1.4 29 FLT1.2 30 Q1.7 31 U2.1 32 U2.2 33 Q1.8 34 PS8.1 35 PLCDI3.1 37 TAS3.1 38 CPU8.1 40 PLCDI3.2 41 JS3.1 42 JS3.1 operating 43 EN8.1 44 EN8.2 47 PLCDO4.1 49 LC7.1 50 AMP7.1 51 LC7.2 52 AMP7.2 54 SA7.1 55 PRS3.1 operating 56 PRS3.1 57 LS2.1 58 LS3.1 59 LD3.2 60 LD3.3 62 DR6.1 63 KA6.1 64 BRK6.1 65 M6.1 66 KA6.2 67 U6.2 68 BRK6.2 69 Gearbox 70 Roller 71 Steel rope I 72 Traveling block I 73 Steel rope II 74 Traveling block II 6, 8, 9, 36, 39, 46, 48, 61, 75 45, 53 23, 25
Type
Property
22 22 1 1 1 1 1 6 1 1 6 6 6 6 6 5 1 1 6 5 6 5 6 1 5 6 1 1 1 1 6 6 1 1 6 1 1 6 6 1 1 1 1 10 2 15B
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 Auxiliary operator
Chapter 7 • Applications of reliability modeling and analysis
149
7.4.1.2.2 Step 2: Establishing the system goal oriented model According to the above analysis of the hoisting mechanism, the GO model of the system is developed from system input to system output, as shown in Fig. 78. In operators of the GO model, the former number is the type of operator, and the latter number is the serial
6
7
1-7
10-6
10
6-10
6-11
11
1-13
13
1-14
12
5-12
1-16
3
5-18
14
15
1-15
16
17
6-17 18
1-3
5-1
4
1
1-2
10-8
1-4
2
1-5
5
10-9
8
1-19
9
1-20
19
23
15B-23
22-24
22
15B-25
25
24
26 22-26
1-22
20
1-21
21 28
1-29
1-30
30
1-33 36
1036
37
6-63 38
51
43
6-52
1048
50
52
54 53
1-54
1-44 44
6-47
45 42
2-45
41 56
47
1061
6-62
62 1-65
6-66
6-41
34
1-34
63
65
66
1-71
1-64
6-69
1-67
64 69
67
71 1-72 6-70
6-56
1-57 1-58 1-59 1-60
57 58 59 60
FIGURE 7–8 Goal oriented model of the hoisting mechanism.
1075
1-74 74
68 1-68
72
70 73 1-73
55
5-55
6-50
2-53
5-51
1-43
1039
46
1046
61
5-42
5-49
49
39
40
6-40
33
6-37
35
6-38
32
6-32
29
6-35
31
1-31 1-28
27
1-27
75
System output
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Goal Oriented Methodology and Applications in Nuclear Power Plants
number. The number on a signal flow is the serial number of the signal flow. The signal flow 75 is the signal flow of system output.
7.4.1.3 Data processing According to the engineering statistical results, the failure rate and maintenance rate of units are obtained, as shown in Table 78. And the reliability parameters of units are obtained by Eq. (7.3), as shown in Table 78. Table 7–8
Reliability parameters of the units.
Operator number
Unit
1026 failures/hour
Repairs/hour
Availability
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 27 28 29 30 31 32 33 34 35 37 38 40 41 42 43
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 Q1.6 KM2.1 KM2.2 FLT1.1 Q1.4 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 PLCDI3.1 TAS3.1 CPU8.1 PLCDI3.2 JS3.1 JS3.1 operating EN8.1
12 1 1.1 1.1 1.1 2.3 4.2 4.8 15 1.7 2.3 7.6 1.7 4.8 25 2.3 2.3 7.5 2.3 1.8 1.8 3.6 2.3 3.6 2.3 9.6 9.6 2.3 7.7 1 5 3.1 1 5.6 30 1.1 2.1
0.5 0.6 0.7 0.7 0.7 0.8 1 0.6 2 1 0.8 1.2 1 0.6 2 0.8 0.8 0.9 0.8 1 1 0.5 0.8 0.5 0.8 1.3 1.3 0.8 0.5 0.5 1.5 0.5 0.5 0.8 0.7 0.6 0.6
0.999976000575986 0.999998333336111 0.999998428573898 0.999998428573898 0.999998428573898 0.999997125008266 0.999995800017640 0.999992000064000 0.999992500056250 0.999998300002890 0.999997125008266 0.999993666706778 0.999998300002890 0.999992000064000 0.999987500156248 0.999997125008266 0.999997125008266 0.999991666736111 0.999997125008266 0.999998200003240 0.999998200003240 0.999992800051840 0.999997125008266 0.999992800051840 0.999997125008266 0.999992615439148 0.999992615439148 0.999997125008266 0.999984600237156 0.999998000004000 0.999996666677778 0.999993800038440 0.999998000004000 0.999993000049000 0.999957144693799 0.999994666695111
(Continued)
Chapter 7 • Applications of reliability modeling and analysis
Table 78
151
(Continued)
Operator number
Unit
1026 failures/hour
Repairs/hour
Availability
44
EN8.2 PLCDO4.1 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 operating PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3 DR6.1 KA6.1 BRK6.1 M6.1
66 67 68 69
KA6.2 U6.2 BRK6.2 Gearbox
70
Roller
71 72 73 74
Steel rope I Travelling block I Steel rope II Travelling block II
0.6 0.6 0.5 1.2 2 1.2 2 0.9 1.4 1.6 1 1 0.7 0.7 1.1 1 0.7 0.8 0.8 0.7 1 1.3 0.7 1 0.9 0.7 0.7 0.7 1.3 0.6 1.3 0.6
0.999994666695111
47 49 50 51 52 54 55 56 57 58 59 60 62 63 64 65
1.1 2.1 1 4.2 8.9 4.2 8.9 7.5 12 2.7 1.7 1.7 5.3 5.3 7.3 1.2 38 25 11 23 1.2 9.6 38 29 15 9 11 23 10 36 10 36
8 !21 n > X > > λi > AC 5 11 > μi > > > i51 > > n > X > < λC 5 λi i51 > > > λC > > μC 5 n > > X λi > > > > > : μ i51 i
0.999998000004000 0.999996500012250 0.999995550019802 0.999996500012250 0.999995550019802 0.999991666736111 0.999991428644897 0.999998312502848 0.999998300002890 0.999998300002890 0.999992428628755 0.999992428628755 0.999993363680405 0.999998800001440 0.999945717232493 0.999922148918406
0.999998800001440 0.999992615439148 0.999945717232493 0.999954335418683 0.999938575201809
0.999992307751479 0.999940003599784 0.999992307751479 0.999940003599784
(7.3)
where AC , λC , and μC are the availability, failure rate, and maintenance rate of the unit, respectively; λi and μi are the failure rate and maintenance rate of the failure mode i of the unit, respectively, for i (see Table 76).
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This case adopts the β CCF model, and the parameters of each CCF group are estimated based on the β model and the impact vector assessment method, as presented in Table 79.
7.4.1.4 Operating quantitative analysis based on the goal oriented method The signal flows 2, 3, 4, 5, 13, 15, 21, 26, 29, 30, 31, 34, 62, and 70 are shared signals, therefore the calculating form of the exact algorithm with a shared signal is adopted to conduct the GO operation. The calculating process of system availability is presented in Table 710.
7.4.1.5 Operating qualitative analysis based on the goal oriented method The qualitative analysis results are obtained as shown in Table 711. The analysis procedures are as follows. 7.4.1.5.1 Step 1: Obtaining unit-level minimum cut sets of the system The unit-level minimum cut sets of the system based on the GO model are obtained, as presented in column 1 of Table 711. 7.4.1.5.2 Step 2: Obtaining fault-mode-level minimum cut sets of the system All fault-mode-level minimum cut sets of the system are obtained, as presented in columns 2 and 3 of Table 711. Table 7–9
Parameters of the common cause failure groups.
CCF group
RI
Cm
β
AMP 7.1 and AMP 7.2 EN 8.1 and EN 8.2
0.9999964 0.9999963
4.45e07 8.00e07
0.2 0.3
Table 7–10
Calculating process of system availability.
State of shared signal S2
S3
...
S70
State combination probability of the shared signal
0 0 ^ 1
0 0 ^ 1
... ... ^ 1
0 1 ^ 1
3.8909e70 1.2827e66 ^ 0.9993735
RI item Non-RI item System availability
Success probability of the system corresponding to the state combination of the shared signal 0 0 0 0.999865 0.999238584577500 1.2441e06 0.999238267113163
Chapter 7 • Applications of reliability modeling and analysis
Table 7–11 Qualitative analysis results by the goal oriented method. Order
Unit-level minimum cut sets
Fault-mode-level minimum cut sets
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 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 27 28 29 30 31 32 33 34 35 37 38 40 41 42 47 54 55 56 57 58 59
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 27 28 29 30 31 32 33 34 35 37 38 40 41 42 47 54 55 56 57 58 59 (Continued)
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 711
(Continued)
Order
Unit-level minimum cut sets
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
60 62 63 64 65
66 67 68 69 70
71 72 73 74 43,44
49, 51 49, 52 50, 51 50, 52
Fault-mode-level minimum cut sets F4 F5 F6 F7 F8 F1 F2 F3 F9, F11 F9, F12 F10, F11 F10, F12
60 62 63 64 F4 F5 F6 66 67 68 F7 F8 F1 F2 F3 71 72 73 74 F9, F11 F9, F12 F10, F11 F10, F12 49, 51 49, 52 50, 51 50, 52
7.4.2 Result analysis The qualitative analysis result of this chapter’s GO method can be verified by FTA, whose main steps are: (1) system analysis, (2) development of the system fault tree, and (3) all minimum cut sets of the system are obtained using the FussellVesely method. The FTA model and analysis process of the hoisting mechanism are very complex, and are difficult to describe in this chapter, thus, only the quantitative results are show, as obtained according to all minimum cut sets of the system. The quantitative analysis results by this chapter’s GO method can be verified by MCS, whose main steps are: (1) random numbers of success probability of operators in the GO model that are generated subject to their availabilities, (2) the simulation model is set up based on the logical relationship between the system and its units, and (3) the success probability of the system is obtained by simulation for 10,000, 100,000, and 1 million times, respectively.
Chapter 7 • Applications of reliability modeling and analysis
Table 7–12
155
Analysis results from the different methods.
Method
System availability
Operation time (seconds)
GO (quantitative analysis) GO (qualitative analysis) MSC (100,000) MSC (1 million) FTA
0.9992383 0.99929366 0.99931 0.99926 0.99929366
153.6 8.9 232.538 987.125
The analysis results obtained from the different methods are presented in Table 712. Table 712 shows that: • The qualitative analysis result obtained using the GO method is consistent with the result obtained by FTA. This therefore indicates that the qualitative analysis method of the GO method is reasonable. Moreover, compared with FTA: • The GO model is developed from a system functional diagram and system structure diagram, and so it is able to avoid the subjective engineer experience for a reliability model, and is more concise and easier to check. • The qualitative analysis result can be obtained by multiple GO operations, therefore its analysis procedure is easier to operate. • The quantitative analysis result obtained by the GO method is in close proximity to the result obtained by MCS at the condition of 1 million times simulation. Therefore this indicates that the quantitative analysis method of the GO method is feasible and reasonable. Moreover, compared with MCS: • The GO operation is not affected by the simulation time. • The GO operation has higher operational efficiency. • The qualitative analysis result obtained by GO method is larger than its quantitative analysis result, which shows that the CCF cannot be ignored. • The analysis process of the example using the GO method shows that both the qualitative and accurate quantitative analysis results of the system, considering CCF and multifunction, are obtained by the GO operation quickly. Compared with FTA and MCS, this illustrates that the GO method is easier and more efficient to operate.
7.1 Introduction This chapter presents three cases to illustrate the method of using reliability modeling and analysis based on the goal oriented (GO) method for nuclear power plant systems with characteristics including an AC power system for a single unit with two 100% capacity divisions, considering the multilevel standby structure, power structure of the control rod drive mechanism (CRDM) with three-state electrical units, and a hoisting mechanism considering multifunction and common cause failure (CCF), respectively.
7.2 Case study I Taking the AC power system for a single unit with two 100% capacity divisions in a nuclear power plant as an example, the conducting a reliability analysis using the GO method is illustrated. In order to verify the feasibility, advantages, and reasonableness of the GO method for reliability analysis of systems considering their characteristics, its analysis results
Goal Oriented Methodology and Applications in Nuclear Power Plants. DOI: https://doi.org/10.1016/B978-0-12-816185-2.00007-1 © 2020 Elsevier Inc. All rights reserved.
131
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are compared with those result of a GO method without considering standby correlations among the multilevel standby structure.
7.2.1 Goal oriented modeling method for multilevel standby structures In power systems of nuclear power plants, the multilevel standby structure often exists to improve the reliability and safety of power systems. Its typical structure is shown in Fig. 71 and the corresponding GO model is shown in Fig. 72. In Fig. 72, the Type 5 and Type 6 operators are used to describe the input unit and switching unit in an input equipment group i, i 5 1; 2; . . .; n. Both primary the equipment group and standby equipment groups are input equipment groups.
7.2.2 Reliability modeling and analysis 7.2.2.1 Analyzing the AC power system 7.2.2.1.1 Step 1: Analyzing system structure and function constituents In a typical nuclear power plant, the alternating current power system includes the power supplies and distribution systems arranged to provide power to the alternating current loads
Input unit Primary equipment group
Input unit
Switching unit Standby equipment group 1 Standby structure 1
Input unit
Switching unit Standby equipment group 2 Standby structure 2
Input unit
Switching unit Standby equipment group n Standby structure n
FIGURE 7–1 Multilevel standby structure.
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133
Standby structure n Standby structure 2 Standby structure 1
5
18A
18A
…
18A
20
5
6
5
6
5
20
6
20
FIGURE 7–2 Goal oriented model for a multilevel standby structure.
and controls for plant normal operation, startup, and orderly shutdown. These Class 1E electric loads are usually separated into two or more redundant load groups. To assure the reliability of the whole power system, some special protective actions are often adopted when one of the power sources is out of operation, as shown in Fig. 73. As illustrated in Fig. 73, the system input boundaries are normal power supply, unit generator, backup power supply, standby generator A, and standby generator B. The system output boundaries are medium voltage bus A and medium voltage bus B. 7.2.2.1.2 Step 2: Determining multilevel standby structures As we can see from Fig. 73, when normal power supply or the incoming breaker 1M1 (2M1) are unavailable to feed to the medium voltage bus, they may be transferred to another backup AC power supply through the incoming breaker 1M2 (2M2). This transfer may be in the form of a fast bus automatic transfer. When both normal and backup power supply are not available, the medium voltage bus will be powered by redundant onsite standby power sources (diesel generators) through the third breaker 1M3 (2M3) within a time consistent with the requirements of the function under normal and abnormal conditions.
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Normal power supply
Backup power supply
Step-up transformer A
Generator circuit breaker Start-up transformer B
Auxiliary transformer
Unit generator
Interlock Medium voltage bus A
2M1
1M1 1 Bus transfer
1M2
2M2
2 Bus transfer
Interlock Medium voltage bus B
1M3
Standby generator A
2M3
Standby generator B
FIGURE 7–3 Structure of an AC power system for a single unit with two 100% capacity divisions.
7.2.2.1.3 Step 3: Making the system success rule According to the above system analysis, the system success rule is defined as the AC power system supplying power to medium voltage bus A or medium voltage bus B normally.
7.2.2.2 Establishing the goal oriented model of an AC power system 7.2.2.2.1 Step 1: Selecting the goal oriented operator According to the analysis results of an AC power system, the GO operators are selected to describe the unit logical relationship in the system, as presented in Table 71. 7.2.2.2.2 Step 2: Establishing the system goal oriented model According to the above analysis of an AC power system and selected GO operators, the system GO model is established, as shown in Fig. 74. In operators of the GO model, the
Chapter 7 • Applications of reliability modeling and analysis
135
Table 7–1 Goal oriented operator type for the goal oriented model of an AC power system. Operator number
Type
Description
Property
1 2 3 4 5 6 7 8 9 10 12 11 13 15 14 16 17 18 19 21 20 22 23 25 24 26 27
5 1 5 1 2 1 1 5 1 20 18A 6 5 6 20 18A 1 1 20 18A 6 5 20 18A 6 1 2
Normal power supply Start-up transformer A Unit generator Generator circuit breaker OR Auxiliary transformer 1M1 Backup power supply Start-up transformer B Standby
Function operator Function operator Function operator Function operator Logical operator Function operator Function operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Auxiliary operator Logical operator Function operator Function operator Logical operator
1M2 Standby generator A 1M3 Standby Medium voltage bus A 2M1 Standby 2M2 Standby generator B Standby 2M3 Medium voltage bus B OR
former number is the GO operator type, and the latter number is the serial number of the GO operator. The number on a signal flow is the serial number of the signal flow. The signal flow 27 is the signal flow of system output.
7.2.2.3 Processing data According to engineering statistical results, the fault rate and maintenance rate of units are obtained, as presented in Table 72. The availabilities of units are obtained by Eq. (7.1), as presented in Table 72. A5
μ μ1λ
(7.1)
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Goal Oriented Methodology and Applications in Nuclear Power Plants
1 5-1
1-2
2
5
6
2-5 3 5-3
7
1-6
1-7
18A -12
12
16 18A -16
1-17
4
1-4
17 20-10
20-14
27
10 8 5-8
9
1-9
2-27 11
6-11
14
15
System output
26 19 6-20
18 20-19
13 1-18
5-13
6-15
1-26 25
22 5-22
20 18A -21
24 6-24
21
18A -25
23 20-23
FIGURE 7–4 Goal oriented model of an AC power system for a single unit with two 100% capacity divisions.
Table 7–2
Reliability parameters of units for an AC power system.
Operator number
Unit
λ failures/year
μ repairs/year
A
1 2 3 4 6 7 8 9 11 13 15 17 18 20 22 24 26
Normal power supply Start-up transformer A Unit generator Generator circuit breaker Auxiliary transformer 1M1 Backup power supply Start-up transformer B 1M2 Standby generator A 1M3 Medium voltage bus A 2M1 2M2 Standby generator B 2M3 Medium voltage bus B
0.643 0.015 0.032 0.045 0.015 0.003 0.643 0.015 0.003 0.032 0.003 0.0001 0.003 0.003 0.032 0.003 0.0001
43.8 43.8 43.8 40 43.8 175.2 43.8 43.8 175.2 43.8 175.2 4380 175.2 175.2 43.8 175.2 4380
0.9855 0.9997 0.9993 0.9989 0.9997 0.999983 0.9855 0.9997 0.999983 0.9993 0.999983 0.9999 0.999983 0.999983 0.9993 0.999983 0.9999
Chapter 7 • Applications of reliability modeling and analysis
137
Table 7–3 Calculating process for the quantitative operation of an AC power systems.
S6
S9
State combination probability of S6 and S9
1 1 2 2
1 2 1 2
4.832448542878858e06 3.217803425733531e04 0.014790817551457 0.984882569657427
Success probability of the corresponding state combination 0.999999484494118 0.999999999998976 0.999999999998976 0.999999999999000
System availability 0.999999999996509
where A is the availability of the unit, and μ and λ are the maintenance rate and fault rate of the unit, respectively.
7.2.2.4 Conducting the quantitative operation The signal flows S6 and S9 are the shared signal, and so the exact algorithm with shared signals should be adopted to conduct a quantitative operation. There are four kinds of state combinations for shared signals, and the calculating process for system availability is presented in Table 73. In Table 73, numbers 1 and 2 represent the fault state and success state of S6 and S9, respectively.
7.2.3 Result analysis The quantitative analysis result of the GO method without considering standby correlations among the multilevel standby structure is about 0.99967, which is less than the quantitative analysis result obtained by the GO method when considering standby correlations among the multilevel standby structure. It is shown that: • If the structure correlation in power systems is not considered, the quantitative analysis result will be biased. Moreover, if there are a large number of multilevel standby structures in the system, these biases will be obvious. • The analysis process of this example shows the advantages of the GO method for reliability analysis of power systems with a multilevel standby structure in terms of establishing the reliability model and conducting quantitative analysis.
7.3 Case study II In order to illustrate this book’s GO method, the power structure of CRDM is taken as an example to conduct reliability analysis using the GO method. This analysis result is then compared with the result obtained using the GO method for a two-state nuclear power electrical system in order to illustrate the advantages of the GO method for reliability analysis of the system with characteristics.
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Goal Oriented Methodology and Applications in Nuclear Power Plants
7.3.1 Reliability modeling and analysis 7.3.1.1 Conducting system analysis 7.3.1.1.1 Step 1: Analyzing the system structure and function constituents CRDM is an electromagnetic lifting device that can raise or lower the drive wire and rod cluster. Three coils used to produce magnetic forces, installed outside the pressure shell, are the components of a CRDM. In the pressure shell, there is an attracted armature. The moving-claw armature and the fixed-claw armature can be used to manipulate the talons, which results in the shaft moving. Moving claws and fixing claws are also used to keep the control rods in a certain position. With the armature of the moving and fixed claws, the control rods can be operated in a step-by-step process. The power structure circuit is used to control the armature, as shown in Fig. 75. Ki is the switch, i 5 1; . . .; 9; Rj is the fuse,
N
A B C K7
K8
K9
R8
R7
V9
N
A B C K4
R9
V8
V7
N
K5
K1
K6
R6
R4
R5
V4
V5
A B C
V6
K2
R1
R2
V1
V2
R3 V3
R10
V10
T3 T4
T1 V11
T5
T6 T7
T8
V12
T9
T2
T10
T11 T12 T13 T14
R27 R28 R29 R30 R31 R32 R33 R34
R35 R36 R37 R38
R21 R22
R19 R20
L5
L6
R23 R24 L9
L10 L11 L12 L13 L14 L15 L16 L17 L18 L19
L7
L8
G6
G7
R42 V15
V16
G9
G11
G12
L1
R25 R26
R39 R40 R41
V13 V14
R11 R12
R13
L2
R15 R16
L3
R14
L4
R17
R18
G3
G4
L20 G5
G10
K3
FIGURE 7–5 Circuit diagram of the power structure of a control rod drive mechanism.
G8
G1
G2
Chapter 7 • Applications of reliability modeling and analysis
139
j 5 1; . . .; 42; Gq is the resistance, q 5 1; . . .; 12; Vw is the turn-off thyristor, w 5 1; . . .; 16; Lm is the inductance, m 5 1; . . .; 20; and Tn is the diode, n 5 1; . . .; 14. 7.3.1.1.2 Step 2: Determining three-state electrical units In the power structure of a CRDM, the resistance, turn-off thyristor, and diode are defined as three-state electrical units, that is, G1, G2, G3, G4, G5, G6, G7, G8, G9, G10, G11, G12, V1, V2, V3, V4, V5, V6, V7, V8, V9, V10, V11, V12, V13, V14, V15, V16, G1, G2, G3, G4, G5, G6, G7, G8, G9, G10, G11, G12. 7.3.1.1.3 Step 3: Making the system success rule According to the above system analysis, the success rule of the power structure for a CRDM is defined as that which can control the armature.
7.3.1.2 Developing the goal oriented model 7.3.1.2.1 Step 1: Selecting the goal oriented operator According to the above system analysis, it needs to select five kinds of GO operators to develop the GO model. The type 5 operator represents the input unit. The type 1 operator represents the two-state unit, whose states are operating and open mode failure. The type 26 operator represents the three-state unit, whose states are operating, open mode failure, and close mode failure. The type 2 operator represents the logical relationship “OR.” The type 10 operator represents the logical relationship “AND.” The GO operators of the power structure for a CRDM are presented in Table 74. Table 7–4
Operator type of units.
Description
Operator number
Type
Property
Power K1 K2 K3 K4 K5 K6 K7 K8 K9 T1 T2 T3 T4 T5 T6
1 2 5 8 14 17 20 24 27 30 12 13 72 75 78 81
5 1 1 1 1 1 1 1 1 1 26 26 26 26 26 26
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 (Continued)
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 74
(Continued)
Description
Operator number
Type
Property
T7 T8 T9 T10 T11 T12 T13 T14 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L18 L19 L20 G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 R1 R2 R3 R4
84 87 90 93 96 99 102 105 36 40 44 48 53 57 61 65 74 77 80 83 86 89 92 95 98 101 104 107 38 42 46 50 55 59 63 67 114 117 120 123 3 6 9 15
26 26 26 26 26 26 26 26 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 26 26 26 26 26 26 26 26 26 26 26 26 1 1 1 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 (Continued)
Chapter 7 • Applications of reliability modeling and analysis
Table 74
(Continued)
Description
Operator number
Type
Property
R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26 R27 R28 R29 R30 R31 R32 R33 R34 R35 R36 R37 R38 R39 R40 R41 R42 V1 V2 V3 V4 V5 V6
18 21 25 28 31 34 35 39 43 47 52 56 60 64 37 41 45 49 54 58 62 66 73 76 79 82 85 88 91 94 97 100 103 106 112 115 118 121 4 7 10 16 19 22
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 26 26 26 26 26 26
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 (Continued)
141
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 74
(Continued)
Description
Operator number
Type
Property
V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 11, 23, 33, 51, 68, 108, 109, 110, 111, 124 9
26 29 32 69 70 71 113 116 119 122 OR
26 26 26 26 26 26 26 26 26 26 2
Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator Logical operator
AND
10
Logical operator
7.3.1.2.2 Step 2: Establishing the system goal oriented model The GO model of the power structure of a CRDM is established by using signal flow to connect the GO operator from the No. 1 GO operator to the No. 125 GO operator, as shown in Fig. 76. The system output is the signal flow 125.
7.3.1.3 Processing data In this case, we assume that the switch, fuse, resistance, turn-off thyristor, inductance, and diode are identical units. Their state probabilities are presented in Table 75. In Table 75, P1 is the state probability of operating for the unit, P2 is the state probability of close mode failure for the unit, and P3 is the state probability of open mode failure for the unit.
7.3.1.4 Conducting a quantitative operation Because shared signals exist in Fig. 76, a GO algorithm with shared signals should be used to conduct the GO operation. In this case, the modified algorithm with shared signals is adopted. The signal flow S1 is a shared signal in Fig. 76, and the system reliability of the power structure for a CRDM is obtained by Eq. (7.2). P125 ð1Þ 5 P1 ð1Þ½PG1 P125 ð1jPG1 Þ 1 PG2 P125 ð1jPG2 Þ 5 0:9980 3 0:999999794846927 0:9980
(7.2)
where P125 ð1Þ and P1 ð1Þ are the success probability of signal flow 1 and 125, respectively; PG1 and PG2 are the fault probability and success probability of shared signal, that is, PG1 5 0 and PG2 5 1, respectively; P125 ð1jPG1 Þ is the success probability of the system output at the condition of PG1 , and P125 ð1jPG2 Þ is the success probability of system output at the condition of PG2 .
Chapter 7 • Applications of reliability modeling and analysis
35 2 1-3
1-2
40
7
11
1-40
26-7
2-11
26-12
43
42
44 1-44
45 1-45
26-46
46 51
26-10
47
48 1-48
49 1-49
50 26-50
1 10-125
15 1-15
1-14
26-42
2-51
26-13
1-47 14
1-41
10
9 1-9
1-8
41
13
12
1-43 8
26-38 38
39 1-39
6 1-6
1-5
37 1-37
26-4 4
5
36 1-36
1-35
3
52
26-16
1-53
1-52
16
53
1-54
54
5-1
18 1-18
1-17
23
19 26-19
1-56
2-23
56
1-57
57
1-58
System output
26-55 55
17
1
143
58
26-59
59
26-63
63
68
2-68
20
21 1-21
1-20
22
1-60
26-22
60
1-61
64 24
25 1-25
1-24
27
26-26
28 1-28
1-27
1-30
1-31
66 1-66
67 26-67 124
26-72
29
72
1-73
73
1-74
112
108
74
1-112
2-108
113 26-113
26-114
114
2-33
33
32 26-32
76 1-76
26-75
78
1-77
79 1-79
81 69
77
109 2-109
1-115
115
26-116
116
26-117 117 2-124
1-34 26-78
26-69
62
26
26-29
31
1-62
65 1-65
1-64
75 30
61
34 26-84
26-87
82 1-82
26-81
84
87
1-85
1-88
83
88
1-86
1-89
1-118
2-110
1-83
85
110
80 1-80
111 2-111
118
121 1-121
86
89
70 26-70
90
91 1-91
26-90 93
26-96
26-99
96
99
92 1-92
94 1-94
26-93
1-97
1-100
95 1-95
97
100
1-98
1-101
98
101
71 26-71
103
102 1-103
26-102 105 26-105
104 1-104
106 1-106
26-119
107 1-107
FIGURE 7–6 Goal oriented model of the power structure of a control rod drive mechanism.
119
26-120 120 123
122 26-122
26-123
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 7–5
State probability of the unit.
Unit
P1
P2
P3
Unit type
Power Ki, i 5 1; . . .; 9 Rj, j 5 1; . . .; 42 Lm, m 5 1; . . .; 20 Vw, w 5 1; . . .; 16 Tn, n 5 1; . . .; 14 Gq, q 5 1; . . .; 12
0.9980 0.9991 0.9952 0.9992 0.9615 0.9734 0.9852
0.0311 0.0221 0.0140
0.0020 0.0009 0.0048 0.0008 0.0074 0.0045 0.0008
Two-state Two-state Two-state Two-state Three-state Three-state Three-state
7.3.2 Result analysis In order to verify the feasibility, reasonability, and advantages of the GO method for a reliability analysis system considering the characteristics, its analysis result is compared with the results by a GO method for two-state nuclear power electrical systems. The analysis steps are mainly as follows: conducting the system analysis, developing the GO model, obtaining the probability of two-state units, and operating quantitative analysis. In system analysis, the open mode failure and close mode failure are not considered, so all three-state units in systems are considered as two-state units, whose states are operating state and faulting state. In the GO model, all three-state units are represented by a Type 1 operator, and the system reliability using the existing GO method is 0.99795. Compared analysis results show that: • The system reliability obtained by the GO method considering three-state electrical units is larger than the result using the GO method for two-state nuclear power electrical systems. It meets the engineering practice because the system can be operating when the three-state electronic unit is in close mode failure or open mode failure. Therefore it illustrates that the GO method can obtain a more accurate quantitative analysis result. • The reliability analysis process of the example shows the advantages of the GO method in terms of the GO model and quantitative analysis, as follows: • The GO model is closely related to the system structure, working principle, and function institute, so that different engineer(s) can obtain highly consistent reliability analysis results by the GO method. And it is easy to check. • The quantitative analysis result is obtained by multiple GO operations, which are easy to operate.
7.4 Case study III The hoisting mechanism in a nuclear power plant is taken as an example to conduct reliability analysis by the method considering the multifunctions and CCF in order to illustrate this
Chapter 7 • Applications of reliability modeling and analysis
145
book’s GO method. The analysis result is then compared with the result obtained by fault tree analysis (FTA) and MonteCarlo simulation (MCS), respectively.
7.4.1 Reliability modeling and analysis In order to illustrate the GO method conveniently, we assume that: • The availabilities of interfaces of the system are set 1. • The organizational-level maintenance is adopted, that is, the maintenance work is mainly the replacement of components, and the maintenance time is not more than 2 hours. • To select part of the unit in an electronic control system and mechanical executing system as the units with multiple failure modes.
7.4.1.1 Conducting system analysis 7.4.1.1.1 Step 1: Analyzing the system structure and function constituents The function of the hoisting mechanism is to allow descending and ascending 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, and the mechanical executing system enables the control rods to rise and lower. There are three phased missions, which are the switch-on phase, startup phase, and operating phase. The function of the switch-on phase is to achieve the power distribution of direct current supply, programmable logic controller (PLC), control panel, etc. The function of the startup phase is to breakover the control circuit. And the function of the operating phase is to control the mechanical executing system to operate. A function diagram of the hoisting mechanism is shown in Fig. 77. An electricity schematic brief diagram of an electronic control system is shown in Fig. 77A. A function diagram of a mechanical executing system is shown in Fig. 77B. 7.4.1.1.2 Step 2: Determining the system characteristics According to engineering statistical results, the system units with multiple fault modes are presented in Table 76. Because AMP 7.1, AMP 7.2, EN 8.1, and EN 8.2 are affected by interruption to the environment, which can easily break them down at the same time, they are CCF groups. According to system analysis, the time sequence of the three phased missions for the system is, in order, switch-on phase, startup phase, and operating phase. The switch-on phase provides power distribution for breakovering the control circuit and operating the mechanical executing system, the startup phase provides a control signal for the mechanical executing system. The system is a multiple function system. 7.4.1.1.3 Step 3: Making the system success rule According to the analysis of the hoisting mechanism, the success rule can be defined as it achieving descending and ascending of the control rods.
146
Goal Oriented Methodology and Applications in Nuclear Power Plants
(A) DISC1.1
L1
Q1.1
FU1.1
FU1.2
L2
PMR1.1
FU1.3
L3
Q1.3 Q1.2
Q1.7
L
+
KM2.1
G U2.1 24V DC/10A
Q1.8 T1.1
KM2.2
FLT1.1 Q1.4
X3 Slot 2
PLC DO 3.2
X2 Ethernet X8 X17 X16
PLC DO 4.1
DR6.1 X1
DP
I005 I006 I007 I008 I009 I010 I011
PLC DO 3.1
FLT1.2 N P
I001 I002 I003
X2 PN L+ M PLC CPU 8.1
M L+ PS 8.1
I004 DP
AMP7.1
AMP7.2
DP
DP
EN8.2
EN8.1
(B)
Power distribution DR6.1
BRK6.1
Steel rope I
Traveling block I
Control
M6.1
Gearbox
Roller
Control rods
Power distribution BRK6.2
PRS3.1 Control
FIGURE 7–7 Function diagram of the hoisting mechanism.
Steel rope II
Traveling block II
N -
Chapter 7 • Applications of reliability modeling and analysis
147
7.4.1.2 Establishing the goal oriented model of the hoisting mechanism 7.4.1.2.1 Step 1: Selecting the goal oriented operator According to the analysis results of the hoisting mechanism, the operator types corresponding to units and logical operator types corresponding to logical relations are determined, as shown in Table 77. Table 7–6
System units with multiple fault modes.
Unit
Failure mode
Failure mode number
Roller
Shaft key failure Shaft distortion Roller distortion Shaft key failure Circuit failure No output Braking torque is insufficient Shaft key failure Open failure Close failure Open failure Close failure
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
Electric machine M6.1 Gearbox Encoder EN8.1 Encoder EN8.2
Table 7–7
Operator type and reliability parameters of the units.
Operator number
Description
Type
Property
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22
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 Q1.6
5 1 1 1 1 1 6 6 5 1 1 1 1 6 5 1 1 1 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 (Continued)
148
Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 77
(Continued)
Operator number
Description
24 KM2.1 26 KM2.2 27 FLT1.1 28 Q1.4 29 FLT1.2 30 Q1.7 31 U2.1 32 U2.2 33 Q1.8 34 PS8.1 35 PLCDI3.1 37 TAS3.1 38 CPU8.1 40 PLCDI3.2 41 JS3.1 42 JS3.1 operating 43 EN8.1 44 EN8.2 47 PLCDO4.1 49 LC7.1 50 AMP7.1 51 LC7.2 52 AMP7.2 54 SA7.1 55 PRS3.1 operating 56 PRS3.1 57 LS2.1 58 LS3.1 59 LD3.2 60 LD3.3 62 DR6.1 63 KA6.1 64 BRK6.1 65 M6.1 66 KA6.2 67 U6.2 68 BRK6.2 69 Gearbox 70 Roller 71 Steel rope I 72 Traveling block I 73 Steel rope II 74 Traveling block II 6, 8, 9, 36, 39, 46, 48, 61, 75 45, 53 23, 25
Type
Property
22 22 1 1 1 1 1 6 1 1 6 6 6 6 6 5 1 1 6 5 6 5 6 1 5 6 1 1 1 1 6 6 1 1 6 1 1 6 6 1 1 1 1 10 2 15B
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 Auxiliary operator
Chapter 7 • Applications of reliability modeling and analysis
149
7.4.1.2.2 Step 2: Establishing the system goal oriented model According to the above analysis of the hoisting mechanism, the GO model of the system is developed from system input to system output, as shown in Fig. 78. In operators of the GO model, the former number is the type of operator, and the latter number is the serial
6
7
1-7
10-6
10
6-10
6-11
11
1-13
13
1-14
12
5-12
1-16
3
5-18
14
15
1-15
16
17
6-17 18
1-3
5-1
4
1
1-2
10-8
1-4
2
1-5
5
10-9
8
1-19
9
1-20
19
23
15B-23
22-24
22
15B-25
25
24
26 22-26
1-22
20
1-21
21 28
1-29
1-30
30
1-33 36
1036
37
6-63 38
51
43
6-52
1048
50
52
54 53
1-54
1-44 44
6-47
45 42
2-45
41 56
47
1061
6-62
62 1-65
6-66
6-41
34
1-34
63
65
66
1-71
1-64
6-69
1-67
64 69
67
71 1-72 6-70
6-56
1-57 1-58 1-59 1-60
57 58 59 60
FIGURE 7–8 Goal oriented model of the hoisting mechanism.
1075
1-74 74
68 1-68
72
70 73 1-73
55
5-55
6-50
2-53
5-51
1-43
1039
46
1046
61
5-42
5-49
49
39
40
6-40
33
6-37
35
6-38
32
6-32
29
6-35
31
1-31 1-28
27
1-27
75
System output
150
Goal Oriented Methodology and Applications in Nuclear Power Plants
number. The number on a signal flow is the serial number of the signal flow. The signal flow 75 is the signal flow of system output.
7.4.1.3 Data processing According to the engineering statistical results, the failure rate and maintenance rate of units are obtained, as shown in Table 78. And the reliability parameters of units are obtained by Eq. (7.3), as shown in Table 78. Table 7–8
Reliability parameters of the units.
Operator number
Unit
1026 failures/hour
Repairs/hour
Availability
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 27 28 29 30 31 32 33 34 35 37 38 40 41 42 43
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 Q1.6 KM2.1 KM2.2 FLT1.1 Q1.4 FLT1.2 Q1.7 U2.1 U2.2 Q1.8 PS8.1 PLCDI3.1 TAS3.1 CPU8.1 PLCDI3.2 JS3.1 JS3.1 operating EN8.1
12 1 1.1 1.1 1.1 2.3 4.2 4.8 15 1.7 2.3 7.6 1.7 4.8 25 2.3 2.3 7.5 2.3 1.8 1.8 3.6 2.3 3.6 2.3 9.6 9.6 2.3 7.7 1 5 3.1 1 5.6 30 1.1 2.1
0.5 0.6 0.7 0.7 0.7 0.8 1 0.6 2 1 0.8 1.2 1 0.6 2 0.8 0.8 0.9 0.8 1 1 0.5 0.8 0.5 0.8 1.3 1.3 0.8 0.5 0.5 1.5 0.5 0.5 0.8 0.7 0.6 0.6
0.999976000575986 0.999998333336111 0.999998428573898 0.999998428573898 0.999998428573898 0.999997125008266 0.999995800017640 0.999992000064000 0.999992500056250 0.999998300002890 0.999997125008266 0.999993666706778 0.999998300002890 0.999992000064000 0.999987500156248 0.999997125008266 0.999997125008266 0.999991666736111 0.999997125008266 0.999998200003240 0.999998200003240 0.999992800051840 0.999997125008266 0.999992800051840 0.999997125008266 0.999992615439148 0.999992615439148 0.999997125008266 0.999984600237156 0.999998000004000 0.999996666677778 0.999993800038440 0.999998000004000 0.999993000049000 0.999957144693799 0.999994666695111
(Continued)
Chapter 7 • Applications of reliability modeling and analysis
Table 78
151
(Continued)
Operator number
Unit
1026 failures/hour
Repairs/hour
Availability
44
EN8.2 PLCDO4.1 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 operating PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3 DR6.1 KA6.1 BRK6.1 M6.1
66 67 68 69
KA6.2 U6.2 BRK6.2 Gearbox
70
Roller
71 72 73 74
Steel rope I Travelling block I Steel rope II Travelling block II
0.6 0.6 0.5 1.2 2 1.2 2 0.9 1.4 1.6 1 1 0.7 0.7 1.1 1 0.7 0.8 0.8 0.7 1 1.3 0.7 1 0.9 0.7 0.7 0.7 1.3 0.6 1.3 0.6
0.999994666695111
47 49 50 51 52 54 55 56 57 58 59 60 62 63 64 65
1.1 2.1 1 4.2 8.9 4.2 8.9 7.5 12 2.7 1.7 1.7 5.3 5.3 7.3 1.2 38 25 11 23 1.2 9.6 38 29 15 9 11 23 10 36 10 36
8 !21 n > X > > λi > AC 5 11 > μi > > > i51 > > n > X > < λC 5 λi i51 > > > λC > > μC 5 n > > X λi > > > > > : μ i51 i
0.999998000004000 0.999996500012250 0.999995550019802 0.999996500012250 0.999995550019802 0.999991666736111 0.999991428644897 0.999998312502848 0.999998300002890 0.999998300002890 0.999992428628755 0.999992428628755 0.999993363680405 0.999998800001440 0.999945717232493 0.999922148918406
0.999998800001440 0.999992615439148 0.999945717232493 0.999954335418683 0.999938575201809
0.999992307751479 0.999940003599784 0.999992307751479 0.999940003599784
(7.3)
where AC , λC , and μC are the availability, failure rate, and maintenance rate of the unit, respectively; λi and μi are the failure rate and maintenance rate of the failure mode i of the unit, respectively, for i (see Table 76).
152
Goal Oriented Methodology and Applications in Nuclear Power Plants
This case adopts the β CCF model, and the parameters of each CCF group are estimated based on the β model and the impact vector assessment method, as presented in Table 79.
7.4.1.4 Operating quantitative analysis based on the goal oriented method The signal flows 2, 3, 4, 5, 13, 15, 21, 26, 29, 30, 31, 34, 62, and 70 are shared signals, therefore the calculating form of the exact algorithm with a shared signal is adopted to conduct the GO operation. The calculating process of system availability is presented in Table 710.
7.4.1.5 Operating qualitative analysis based on the goal oriented method The qualitative analysis results are obtained as shown in Table 711. The analysis procedures are as follows. 7.4.1.5.1 Step 1: Obtaining unit-level minimum cut sets of the system The unit-level minimum cut sets of the system based on the GO model are obtained, as presented in column 1 of Table 711. 7.4.1.5.2 Step 2: Obtaining fault-mode-level minimum cut sets of the system All fault-mode-level minimum cut sets of the system are obtained, as presented in columns 2 and 3 of Table 711. Table 7–9
Parameters of the common cause failure groups.
CCF group
RI
Cm
β
AMP 7.1 and AMP 7.2 EN 8.1 and EN 8.2
0.9999964 0.9999963
4.45e07 8.00e07
0.2 0.3
Table 7–10
Calculating process of system availability.
State of shared signal S2
S3
...
S70
State combination probability of the shared signal
0 0 ^ 1
0 0 ^ 1
... ... ^ 1
0 1 ^ 1
3.8909e70 1.2827e66 ^ 0.9993735
RI item Non-RI item System availability
Success probability of the system corresponding to the state combination of the shared signal 0 0 0 0.999865 0.999238584577500 1.2441e06 0.999238267113163
Chapter 7 • Applications of reliability modeling and analysis
Table 7–11 Qualitative analysis results by the goal oriented method. Order
Unit-level minimum cut sets
Fault-mode-level minimum cut sets
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 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 27 28 29 30 31 32 33 34 35 37 38 40 41 42 47 54 55 56 57 58 59
1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 27 28 29 30 31 32 33 34 35 37 38 40 41 42 47 54 55 56 57 58 59 (Continued)
153
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Goal Oriented Methodology and Applications in Nuclear Power Plants
Table 711
(Continued)
Order
Unit-level minimum cut sets
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
60 62 63 64 65
66 67 68 69 70
71 72 73 74 43,44
49, 51 49, 52 50, 51 50, 52
Fault-mode-level minimum cut sets F4 F5 F6 F7 F8 F1 F2 F3 F9, F11 F9, F12 F10, F11 F10, F12
60 62 63 64 F4 F5 F6 66 67 68 F7 F8 F1 F2 F3 71 72 73 74 F9, F11 F9, F12 F10, F11 F10, F12 49, 51 49, 52 50, 51 50, 52
7.4.2 Result analysis The qualitative analysis result of this chapter’s GO method can be verified by FTA, whose main steps are: (1) system analysis, (2) development of the system fault tree, and (3) all minimum cut sets of the system are obtained using the FussellVesely method. The FTA model and analysis process of the hoisting mechanism are very complex, and are difficult to describe in this chapter, thus, only the quantitative results are show, as obtained according to all minimum cut sets of the system. The quantitative analysis results by this chapter’s GO method can be verified by MCS, whose main steps are: (1) random numbers of success probability of operators in the GO model that are generated subject to their availabilities, (2) the simulation model is set up based on the logical relationship between the system and its units, and (3) the success probability of the system is obtained by simulation for 10,000, 100,000, and 1 million times, respectively.
Chapter 7 • Applications of reliability modeling and analysis
Table 7–12
155
Analysis results from the different methods.
Method
System availability
Operation time (seconds)
GO (quantitative analysis) GO (qualitative analysis) MSC (100,000) MSC (1 million) FTA
0.9992383 0.99929366 0.99931 0.99926 0.99929366
153.6 8.9 232.538 987.125
The analysis results obtained from the different methods are presented in Table 712. Table 712 shows that: • The qualitative analysis result obtained using the GO method is consistent with the result obtained by FTA. This therefore indicates that the qualitative analysis method of the GO method is reasonable. Moreover, compared with FTA: • The GO model is developed from a system functional diagram and system structure diagram, and so it is able to avoid the subjective engineer experience for a reliability model, and is more concise and easier to check. • The qualitative analysis result can be obtained by multiple GO operations, therefore its analysis procedure is easier to operate. • The quantitative analysis result obtained by the GO method is in close proximity to the result obtained by MCS at the condition of 1 million times simulation. Therefore this indicates that the quantitative analysis method of the GO method is feasible and reasonable. Moreover, compared with MCS: • The GO operation is not affected by the simulation time. • The GO operation has higher operational efficiency. • The qualitative analysis result obtained by GO method is larger than its quantitative analysis result, which shows that the CCF cannot be ignored. • The analysis process of the example using the GO method shows that both the qualitative and accurate quantitative analysis results of the system, considering CCF and multifunction, are obtained by the GO operation quickly. Compared with FTA and MCS, this illustrates that the GO method is easier and more efficient to operate.