Application of reliability optimization allocation by the goal oriented method

8 Application of reliability optimization allocation by the goal oriented method Chapter Outline 8.1 Introduction ................................................................................................................................. 157 8.2 Case study .................................................................................................................................... 157 8.2.1 Conducting system analysis of the hoisting mechanism................................................ 157 8.2.2 Developing the goal oriented model of the hoisting mechanism................................ 158 8.2.3 Establishing the mathematical model ............................................................................. 160 8.2.4 Solving the optimization allocation mathematical model by improved genetic algorithm............................................................................................................................ 164 8.2.5 Determining the allocation result.................................................................................... 164 8.2.6 Result analysis .................................................................................................................... 167

8.1 Introduction Taking a hoisting mechanism in a nuclear power plant as a case study in this chapter, reliability optimization allocation based on the goal oriented (GO) method is illustrated.

8.2 Case study 8.2.1 Conducting system analysis of the hoisting mechanism 8.2.1.1 To analyze system principle, function, and structure The function of the hoisting mechanism is to achieve 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 raises and lowers the control rods. An electricity schematic brief diagram of the electronic control system and a function diagram of the mechanical executing system are illustrated in Figs. 7 7A and 7 7B, respectively. The hoisting mechanism is a typical multiphase mission system, and has a switch-on phase, startup phase, and operating phase. The function of the switch-on phase is to achieve the power distribution of the direct current supply, PLC, control panel, and so on, the Goal Oriented Methodology and Applications in Nuclear Power Plants. DOI: https://doi.org/10.1016/B978-0-12-816185-2.00008-3 © 2020 Elsevier Inc. All rights reserved.

157

158

Goal Oriented Methodology and Applications in Nuclear Power Plants

function of the startup phase is to breakover control circuit, and the function of the operating phase is to control the mechanical executing system during operation.

8.2.1.2 To determine system characteristics According to system analysis, the time sequence of the three phased missions for a system is, in order, the switch-on phase, startup phase, and operating phase. The switch-on phase provides power distribution for the breakovering control circuit and the operating mechanical executing system, the startup phase provides the control signal for the mechanical executing system. This system is a multiple function system.

8.2.1.3 To define success rule of the system According to the system analysis of the hoisting mechanism, the success rule can be defined as that which can achieve descending and ascending of control rods.

8.2.2 Developing the goal oriented model of the hoisting mechanism 8.2.2.1 To select the goal oriented operator According to the analysis result of the hoisting mechanism and the type description of GO operators, the GO operators used to describe logical relationship and units are selected, as shown in Table 8 1. The Type 1 operator represents a two-state unit with a failure state and Table 8–1 Unit number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Operator type in the goal oriented model of a hoisting mechanism. Operator number

Unit

Type

Property

1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24

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

Virtual 5 1 1 1 1 1 6 6 Virtual 5 1 1 1 1 6 Virtual 5 1 1 1 1 22

Function operator Function operator Function operator Function operator Function operator Function operator Function operator Function operator 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 8 • Application of reliability optimization allocation

Table 8 1

159

(Continued)

Unit number 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Operator number 26 27 28 29 30 31 32 33 34 35 37 38 40 41 42 43 44 47 49 50 51 52 54 55 56 57 58 59 60 62 63 64 65 66 67 68 69 70 71 72 73 74

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 No. (operator) No. (operator) No. (operator)

Unit 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 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 KA6.2 U6.2 BRK6.2 Gearbox Roller Steel rope I Travelling block I Steel rope II Travelling block II 6, 8, 9, 36, 39, 46, 48, 61, 75 45, 53 23, 25

Type

Property

22 1 1 1 1 1 6 1 1 6 6 6 6 6 Virtual 5 1 1 6 5 6 5 6 1 Virtual 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 Logical operator Logical operator Auxiliary operator

160

Goal Oriented Methodology and Applications in Nuclear Power Plants

an operating state, the Type 2 operator represents “OR” logical relation, the Type 5 operator represents the input unit, the Type 6 operator represents the unit with two control signals, the Type 10 represents “AND” logical relation, and the combination of Types 15B and 22 operators represents a multifunction unit. The virtual 5 operator represents the virtual input signal, whose reliability is 1.

8.2.2.2 To establish the goal oriented model According to the above analysis of a hoisting mechanism, the GO model of the system is established, as shown in Fig. 8 1.

8.2.3 Establishing the mathematical model 8.2.3.1 To establish reliability constraint functions Step 1: To determine reliability constraint functions The unit reliability constraint function is given by Ru;lower # Ru # Ru;upper ;

u 5 1; 2; . . .; 57

(8.1)

According to the statistical data of the old type unit and the design expectation of the new type unit, the lower and upper limits of unit reliability are presented in Table 8 2. Step 2: To determine the system reliability constraint function The system reliability constraint function based on the GO method is given by RS 5 PS75 $ RS 5 0:8

(8.2)

where, RS is the predicted reliability of the hoisting mechanism based on the GO method, PS75 is the success probability of signal flow 75 based on the GO method, RS is the target reliability of the hoisting mechanism.

8.2.3.2 To establish the cost objective function The basic cost of the unit is presented in Table 8 3. The cost objective function is given by minCS 5

57 X

Cu

(8.3)

u51

where, CS and Cu are the system cost and the unit cost, respectively.

8.2.3.3 To develop the mathematical model According to Eqs. (8.1) (8.3), the reliability optimization allocation mathematical model of the hoisting mechanism is given by

Chapter 8 • Application of reliability optimization allocation

6

7

10

1-7

10-6

11

6-10

6-11

13

14

1-13

1-14

161

15

1-15

12

5-12

16 1-16 17 3

6-17

5-18 18

1-3 4

8

1 5-1

19

10-8

1-4

1-2

23

15B-23

1-19

22

2 1-5

5

10-9

9

1-20

26

25 22-24

22-26

15B-25

27

1-27

24 1-22

20

1-21

21

31

1-31 28 1-28

1-29

1-30

30 32

6-32 1 33 1-33 36

1036

6-35

37 6-37

63

35 6-63 38 6-38

39

1-43

6-50

1048

50 2-53

45 42

2-45

1-44 44

6-52

52

62

71

64 69

65 6-69

67 1-67

1-72 6-70

1075

1-74 74

68 1-68

55 6-56

1-57 57 1-58 58 1-59 59 1-60

60

FIGURE 8–1 Goal oriented model of a hoisting mechanism.

72

70

73 1-73

51 5-51

6-62

66

6-41

56 1-54

61

6-66

41

54 53

1061

1-65

43

40

47

5-55

49

6-47

1-71 1-64

5-42

5-49

46

1046

1039

6-40

34

1-34

75

System output

162

Goal Oriented Methodology and Applications in Nuclear Power Plants

Table 8–2

Reliability range of units.

Unit number

Unit

Ru,lower

Ru,upper

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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

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 Q1.6 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 EN8.1 EN8.2 PLCDO4.1 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3 DR6.1 KA6.1

0.990420774 0.991335728 0.995164937 0.994606708 0.996506464 0.997181778 0.990795425 0.996193234 0.995509935 0.994320573 0.994739220 0.996477798 0.991207085 0.995737612 0.996382659 0.998274276 0.993713936 0.992419114 0.999941052 0.991391005 0.993030838 0.996909464 0.990971469 0.993695224 0.993588089 0.992416126 0.991934722 0.995098816 0.994952623 0.997016679 0.991359407 0.999292344 0.990559997 0.992524242 0.996169655 0.995309475 0.998442791 0.999760835 0.992424063 0.999057185 0.999356180 0.991297393 0.994203596 0.997619981 0.995492702 0.836052592

0.999998716 0.999964801 0.999962322 0.999962539 0.999904760 0.999929675 0.999971782 0.999993812 0.999988060 0.999914129 0.999920881 0.999923421 0.999941782 0.999951130 0.999969225 0.999918363 0.999986501 0.999920480 0.999954418 0.999992874 0.999928070 0.999927449 0.999977760 0.999906544 0.999913073 0.999978215 0.999958470 0.999907165 0.999973539 0.999918758 0.999904576 0.999932252 0.999938502 0.999920112 0.999918298 0.999924440 0.999927297 0.999975710 0.999916491 0.999913440 0.999904525 0.999938579 0.999955210 0.999953944 0.999931260 0.999983803 (Continued)

Chapter 8 • Application of reliability optimization allocation

Table 8 2

163

(Continued)

Unit number

Unit

Ru,lower

Ru,upper

47 48 49 50 51 52 53 54 55 56 57

BRK6.1 M6.1 KA6.2 U6.2 BRK6.2 Gearbox Roller Steel rope I Traveling block I Steel rope II Traveling block II

0.809500295 0.897892198 0.843934938 0.843500939 0.879587878 0.876691086 0.860363406 0.829581600 0.888205086 0.836052592 0.809500295

0.999975772 0.999952517 0.999902142 0.999961980 0.999047600 0.999856580 0.999832420 0.999274200 0.999481620 0.999268590 0.999368050

Table 8–3

Basic cost of a unit.

Unit number

Unit

Pi

Unit number

Unit

Pi

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

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 Q1.6 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

4.441 8.093 5.235 9.606 5.661 2.188 0.758 6.708 4.411 3.324 2.447 7.802 11.091 0.635 7.768 5.295 2.014 7.997 6.038 5.602 2.251 11.82 5.788 4.305 2.790 6.623 9.255 0.633 8.486

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

PLCDI3.2 JS3.1 EN8.1 EN8.2 PLCDO4.1 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3 DR6.1 KA6.1 BRK6.1 M6.1 KA6.2 U6.2 BRK6.2 Gearbox Roller Steel rope I Traveling block I Steel rope II Traveling block II

8.296 12.698 4.768 5.667 5.842 10.842 9.537 5.818 11.418 6.052 0.039 10.012 0.624 4.443 0.842 7.48 11.417 4.944 3.845 11.367 9.697 1.508 4.299 11.306 10.122 3.991 7.559 8.717

164

Goal Oriented Methodology and Applications in Nuclear Power Plants

Table 8–4 Parameters of improved genetic algorithm for solving Eq. (8.4). Parameter

Value

Coding scheme Crossover rate Population quantity Iterative times Mutation rate Operation times

Real coding 0.8 200 500 0.15 20

8 57 X > > > minCS 5 Cu > < u51 s:t: > >  > > RS 5 PS75 $ RS 5 0:80 : Ru;lower # Ru # Ru;upper ;

(8.4) u 5 1; 2; . . .; 57

8.2.4 Solving the optimization allocation mathematical model by improved genetic algorithm 8.2.4.1 To set the parameters of genetic algorithm The parameters of improved genetic algorithm (GA) are presented in Table 8 4.

8.2.4.2 To perform the genetic algorithm The system cost and the allocated reliabilities of units at different operation times are shown in Table 8 5 and Fig. 8 2, respectively.

8.2.5 Determining the allocation result 8.2.5.1 To verify the allocation result of each operating time The predicted reliabilities of the hoisting mechanism using allocated reliabilities of units based on the GO method at different operation times are presented in Table 8 5, which also shows that they meet the reliability constraints.

8.2.5.2 To determine the reliability allocation results Because the system cost of 14th operation time is the minimum system cost in 20 operation times, the allocation result of the 14th operation time is selected as the final allocation result, as presented in Table 8 6.

Chapter 8 • Application of reliability optimization allocation

Table 8–5

165

Cost and predicted reliability of the system at different operation times.

No. (operation time)

CS

Predicted reliabilities of the system

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 10 20

281.80007912 281.80007912 281.80007911 281.80007921 281.80007911 281.80007914 281.80007911 281.80007912 281.80007911 281.80007778 281.80007912 281.80007060 281.80007911 281.80007054 281.80007913 281.80007899 281.80007904 281.80007912 281.80007916 281.80007908

0.80000013 0.80000016 0.80000017 0.80000011 0.80000013 0.80000011 0.80000015 0.80000011 0.80000012 0.80000013 0.80000013 0.80000019 0.80000011 0.80000010 0.80000000 0.80000013 0.80000012 0.80000013 0.80000002 0.80000009

0.999 0.998

Reliability

0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.99 5 10 15

Operation times

20

5

10

15

20

25

30

Unit number

FIGURE 8–2 Allocated reliabilities of units at different operation times.

35

40

45

50

55

166

Goal Oriented Methodology and Applications in Nuclear Power Plants

Table 8–6

Allocated reliability of units at the 14th operation time.

Unit number

Unit

Ru

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 31 32 33 34 35 36 37 38 39 40 41 42 43 44

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 Q1.6 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 EN8.1 EN8.2 PLCDO4.1 LC7.1 AMP7.1 LC7.2 AMP7.2 SA7.1 PRS3.1 LS2.1 LS3.1 LD3.2 LD3.3

0.999998716 0.999964801 0.999962322 0.999962539 0.999904760 0.999929675 0.999971782 0.999993812 0.999988060 0.999914129 0.999920881 0.996556398 0.992948187 0.999951130 0.999969225 0.998274276 0.999986501 0.999920480 0.999941052 0.999992874 0.999928070 0.999927449 0.999977760 0.999906544 0.999913073 0.999978215 0.997522627 0.999907165 0.996535942 0.997016679 0.999904576 0.999292344 0.990559997 0.999920112 0.996169655 0.995309475 0.998442791 0.999760835 0.999916491 0.999913440 0.999356180 0.999938579 0.999955210 0.999953944 (Continued)

Chapter 8 • Application of reliability optimization allocation

Table 8 6

167

(Continued)

Unit number

Unit

Ru

45 46 47 48 49 50 51 52 53 54 55 56 57

DR6.1 KA6.1 BRK6.1 M6.1 KA6.2 U6.2 BRK6.2 Gearbox Roller Steel rope I Traveling block I Steel rope II Traveling block II

0.996903771 0.999983803 0.999975772 0.999952517 0.999902142 0.999961980 0.99904760 0.99985658 0.99983242 0.99927420 0.99948162 0.99926859 0.99936805

8.2.6 Result analysis • According to Fig. 8 2, the allocated reliabilities of each unit at different operation times are very stable. This indicates that the new reliability optimization allocation method proposed in this chapter is feasible, and the improved GA has good performance. • According to Table 8 5, the system cost at different operation times is about 281.8. This shows that the satisfactory convergence solution can be obtained by the new reliability optimization allocation method proposed in this chapter. • According to Table 8 6, the predicted system reliabilities using the allocated reliabilities of the units at different operation times are all larger than the target reliability of the system. This indicates that the new reliability optimization allocation method proposed in this chapter is effective and reasonable. • According to the reliability allocation process of example, it shows that the new reliability optimization allocation method proposed in this paper has obvious engineering applicability, advantage and operability, as follows: • The allocation model based on GO model can relate the system structure and system characteristics so that it can easy to conduct reliability re-allocation at the situation of design changes quickly and efficiently, • The system reliability index can easy to allocate to design unit, moreover the reliability allocation results are the satisfactory convergent solutions.