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Reliability optimization allocation method for nuclear power plants by the goal oriented method
4 Reliability optimization allocation method for nuclear power plants by the goal oriented method Chapter Outline 4.1 Introduction ................................................................................................................................... 83 4.2 Reliability optimization allocation model by the goal oriented method ................................ 84 4.2.1 Constraints function ............................................................................................................ 84 4.2.2 Objective function ............................................................................................................... 87 4.2.3 Reliability optimization allocation mathematics model .................................................. 88 4.3 Intelligent algorithms for solving the optimization allocation model .................................... 88 4.3.1 Solving algorithm I: improved genetic algorithm ............................................................ 89 4.3.2 Solving algorithm II: improved ant colony optimization................................................. 91 4.4 Reliability optimization allocation method by the goal oriented method ............................. 95 4.4.1 Approach of reliability optimization allocation by the goal oriented method............. 95 4.4.2 Process of reliability optimization allocation method by the goal oriented method
97
4.5 Problems......................................................................................................................................... 97 References............................................................................................................................................. 98
4.1 Introduction Reliability optimization allocation is a decision problem in reliability engineering. Its objective is to allocate the system reliability index to design units, so that it can guide the design to ensure the reliability design requirements. In recent years, study on reliability optimization allocation considering different constraints, such as cost constraint, dimension constraint, etc., has been a hot issue of reliability engineering. The majority of researches in this area are devoted to multistate system optimization, multiobjective optimization, active and cold-standby redundancy, fault-tolerance mechanism, multifunction system optimization, and optimization techniques [16]. For nuclear power plants, which have many characteristics, such as multiphase mission, multifunction, and time-sequence, reliability optimization allocation is an essential task in the design of nuclear power plants, as it can provide a more reasonable design in order to save costs and human resources. However, the existing
Goal Oriented Methodology and Applications in Nuclear Power Plants. DOI: https://doi.org/10.1016/B978-0-12-816185-2.00004-6 © 2020 Elsevier Inc. All rights reserved.
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Goal Oriented Methodology and Applications in Nuclear Power Plants
reliability optimization allocation technologies for nuclear power plants systems still have four obvious limitations that can cause great deviations in the reliability allocation results, as follows [714]: • The reliability allocation models are not directly related, accurately describe the system structure and system characteristics, and are affected by subjective experience, therefore the reliability allocation results are difficult to meet the engineering design requirements for complex nuclear power plant systems. • It is not easy to allocate the system reliability index to design units, including redesigned and selected unit versions, quickly, efficiently, and directly. • Production is finalized through multiple design revisions. The models of reliability, such as series, parallel, seriesparallel, etc., are not directly related and exactly describe the product structure, function, and characteristics. Hence, for design changes, it is very difficult to conduct reliability reallocation quickly and efficiently to allocate a reasonable reliability margin for design units. • For the problem of specific optimization allocation, the basic optimization algorithm needs to be improved to obtain satisfactory convergence effects and avoid a local extremum problem. Thus, how to avoid the above limitations of reliability optimization allocation methods for nuclear power plant systems is a research hotspot. Some typical studies have indicated that the reliability optimization allocation method by the goal oriented (GO) method can efficiently solve the above limitations because of its advantages in terms of its ease of creating a model and its representational and analysis power [1518]. Furthermore, the reliability optimization allocation process by the GO method shows not only that the satisfactory optimization allocation result can be obtained, but also shows the advantages, operability, and engineering applicability of this method. In this chapter, the reliability optimization allocation method for complex nuclear power plant systems based on the GO method is illustrated in terms of the reliability optimization allocation model by the GO method, intelligent algorithms for solving optimization allocation model, and the method’s process.
4.2 Reliability optimization allocation model by the goal oriented method The reliability optimization allocation problem is described through the corresponding mathematics model, which contains constraints function and objective function.
4.2.1 Constraints function For reliability optimization allocation of nuclear power plant systems with multiple characteristics, there are three constraints functions—reliability constraint function of unit,
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reliability constraint function of system function, and constraint function of system. It is assumed that a nuclear power plant system consists of m units, and can execute n system functions.
4.2.1.1 Reliability constraint function of the unit The reliability constraint function of the unit consists of the allocated reliability index of the unit, the lower limit and upper limit of the reliability index for the unit, as follows: Ri;lower # Ri # Ri;upper
(4.1)
where, Ri is the allocated reliability of the unit, and Ri;lower and Ri;upper are the lower limit and upper limit of unit reliability, i 5 1; 2; . . .; m, respectively.
4.2.1.2 Reliability constraint function of system function 4.2.1.2.1 Definition For reliability optimization allocation of nuclear power plant systems with multifunction systems, the reliabilities of the system and its functions can be obtained by using the reliability of the design unit and GO algorithm to conduct GO operation according to the GO model. In the GO model, the reliability of the output signal flows for the system and its functions is the reliability of the system and its functions. The reliability of signal flow based on the GO method for reliability optimization allocation is defined as follows: Rx 5 Fx ðRx1 ; Rx2 ; . . .; Rxy Þ
(4.2)
where, Rx is the predicted reliability of xth signal flow, Rxy is the allocated reliability of the yth unit for calculating xth signal flow, Fx ðÞ is the success probability of the xth signal flow obtained using the allocated reliability of the unit to conduct the GO operation according to the GO model and GO algorithm. 4.2.1.2.2 Reliability constraint function of system function The reliability constraint function of system functions indicates that the predicted reliability of system function using allocated reliability of the design unit by the GO method should meet the target reliability of system function, as shown in Eq. (4.3).
RGw 5 Fw ðRw1 ; Rw2 ; . . .; Rwj Þ RGw . RGw
(4.3)
where, RGw is the predicted reliability of wth function, Rwj is the allocated reliability of the jth unit for wth function, Fw ðÞ is the reliability of output signal flow represented by wth function using the allocated reliability of a unit based on the GO method, and RGw is the target reliability of system function, w 5 1; 2; . . .; n, 1 # j # m.
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4.2.1.2.3 Function importance factor constraint function of system function For a nuclear power plant system with multiple functions, the usage frequency of each function and the combination of units to realize different functions are not often the same. In addition, different use demands of each system function are determined by the different environments of the task, and the differentiation of use demand of each system function results in the reliability levels of various functions being different. To deal with the differentiation of functions in reliability optimization allocation, the more reasonable approach is to convert it into a constraint condition and to select an optimal solution in the optimization process. Obviously, the units and functions in a system do not only have a reliability index, but also have a parameter of importance factor. A new index named the importance factor has been defined to measure the importance of a unit or function by relating the reliability of the unit or function. Generally, for the higher importance factor of a unit or function, higher reliability should be allocated to it. Therefore, according to the positive correlation between the importance factor and reliability, an importance factor model is proposed based on the experience in engineering applications, as shown in Eq. (4.4). gðRÞ 5 eðR21Þ
0,g ,1
(4.4)
where, gðÞ is the importance factor of the unit or function, and R is the reliability of the unit or function. The function importance factor constraint indicates the predicted function importance factor by using the allocated reliability of the unit based on the GO method that should meet the allocated function importance factor, as shown in Eq. (4.5).
RGw 5 Fw ðRw1 ; Rw2 ; . . .; Rwj Þ
gðRGw Þ 5 eðRGw 21Þ $ gðRGw Þ 5 eðRGw 21Þ
(4.5)
where, RGw is the predicted reliability of wth function, Rwj is the allocated reliability of the jth unit for wth function, Fw ðÞ is the reliability of output signal flow represented by wth function by using the allocated reliability of the unit based on the GO method, gðRGw Þ is the predicted function importance factor of wth function, w 5 1; 2; . . .; n, 1 # j # m, and gðRGw Þ is the allocated function importance factor by the estimation of function importance factors, which can be obtained by the following steps: • Step 1: The factors which can affect the reliabilities of functions, such as using frequency, the design level, complexity, etc., should be ascertained. Based on these factors, an importance factor hierarchy model of functions which generally consists of a target layer, criterion layer, and object layer can be built. Among these, the target layer corresponds to the system reliability, the criterion layer corresponds to the various factors which can cause a change to the reliability indexes of functions, and the object layer corresponds to the functions of the system, as shown in Fig. 41. • Step 2: Based on expert evaluation obtained according to hierarchy structure, a judgment matrix can be built. Then the weight of each function which represents the influence on
Chapter 4 • Reliability optimization allocation method for nuclear power plants
Target layer
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Target of system reliability
Criterion layer
Influence factor 1
Object layer
Function 1
Influence factor 2
……
Function 2
……
Influence factor M
Function N
FIGURE 4–1 Function importance factor hierarchy model.
the target reliability index of the system under the influence of various factors can be calculated, that is, V 5 fv1 ; v2 ; . . .; vn g. • Step 3: According to V and the target reliability index of the system, the target reliability index of each function can be obtained. • Step 4: The importance factor of each function is easy to determine using Eq. (4.4).
4.2.1.3 Reliability constraint function of the system The reliability constraint function of the system indicates the predicted reliability of the system using the allocated reliability of the design unit by the GO method meets the target reliability of the system, as shown in Eq. (4.6).
RS 5 FS ðR1 ; R2 ; . . .; Rm Þ RS $ RS
(4.6)
where, RS is the predicted reliability of the system, Ri is the allocated reliability of the ith unit, FS ðÞ is the reliability of output signal flow of the system by using the allocated reliability of the unit based on the GO method, and RS is the target reliability of the system, i 5 1; 2; . . .; m.
4.2.2 Objective function The cost is often of concern in engineering, and the reliability optimization allocation methods whose goal is to minimize the cost have been studied by many scholars [19,20]. Thus, the objective function of the reliability optimization allocation problem in this chapter is to minimize the cost, as follows: minCS ðRÞ 5
m X i51
ci ðPi ; Ri ; Ri;min ; Ri;max Þ
(4.7)
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where, CS ðÞ is the cost function of the system; ci ðPi ; Ri ; Ri;min ; Ri;max Þ is the cost function of the design unit, that is, ci ðPi ; Ri ; Ri;min Þ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ , Pi is the basic cost of the ith unit, Ri is the allocated reliability of the ith unit, Ri;min is the lower limit value of reliability of the ith unit, and Ri;max is the upper limit value of reliability of the ith unit.
4.2.3 Reliability optimization allocation mathematics model Combining the above objective function and constraints function, the reliability optimization allocation problem for nuclear power plant systems considering multiple characteristics can be described by three models. • Model I: Reliability optimization allocation model for single-function systems 8 m X > > minC ðRÞ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ > S > > < i51 s:t: Ri;min # Ri # Ri;max > > > > > : gðRGw Þ 5 gðFw ðRw1 ; Rw2 ; . ..; Rwj ÞÞ $ gðRGw Þ RS 5 FS ðR1 ; R2 ; . . .; Rm Þ $ RS
i 5 1; 2; . . .; m w 5 1; 2; . . .; n
(4.8)
• Model II: Reliability optimization allocation model for multiple-function systems without considering differentiation of functions 8 m X > > minC ðRÞ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ > S > > < i51 s:t: Ri;min # Ri # Ri;max i 5 1; 2; . . .; m > > > > > : RGw 5 Fw ðRw1 ; Rw2 ; . . .; RwjÞ $ RGw w 5 1; 2; . . .; n RS 5 FS ðR1 ; R2 ; . . .; Rm Þ $ RS
(4.9)
• Model III: Reliability optimization allocation model for multiple-function systems considering differentiation of functions 8 m X > > minC ðRÞ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ > S > > < i51 s:t: Ri;min # Ri # Ri;max > > > > > : gðRGw Þ 5 gðFw ðRw1 ; Rw2 ; . ..; Rwj ÞÞ $ gðRGw Þ RS 5 FS ðR1 ; R2 ; . . .; Rm Þ $ RS
i 5 1; 2; . . .; m w 5 1; 2; . . .; n
(4.10)
4.3 Intelligent algorithms for solving the optimization allocation model For the complex reliability optimization allocation model of the system, it is difficult to obtain the solution. A large number of researches have proved that applying artificial intelligence methods (such as neural networks, ant colony algorithms, genetic algorithms, hybrid genetic
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algorithms, simulated annealing algorithms, etc.) [2134] to reliability optimization allocation can achieve good results. Nevertheless, intelligent algorithms also have their disadvantages, such as: premature phenomena may occur in the course of use; swinging when approaching the optimal solution; slow convergence and easily falling into local extremum; or the optimization result not being ideal. Therefore it is meaningful to improve the basic intelligent algorithm so that it is applicable for specific problems and can obtain the optimal solution efficiently. In this section, a genetic algorithm (GA) and an improved ant colony optimization (ACO) are illustrated to solve the reliability optimization allocation model.
4.3.1 Solving algorithm I: improved genetic algorithm In order to obtain satisfactory optimization results effectively and quickly, an improved GA is presented to solve the reliability optimization allocation model of a nuclear power plant with multiple characteristics, and it can efficiently overcome the limitations of a basic GA, such as falling into local extremum easily, slower convergence efficiency, etc. The steps to an improved GA are described next.
4.3.1.1 To develop a reliability optimization allocation model according to Eqs. (4.8)(4.10) • Step 1: Determining the quantity of decision variables, that is, the reliability of the design unit. • Step 2: Determining the type of decision variable, that is, the design unit is redesigned or a selected version. • Step 3: Establishing the objective function of the system cost. • Step 4: Establishing the constraints function of the reliability optimization allocation model.
4.3.1.2 To set the parameters of the algorithm • Step 1: Selecting the encoding. In comparison with binary encoding, real encoding has some advantages. First, it does not need the transformation from chromosome to solution before the calculation, which improves the efficiency of the algorithm. Second, it adopts float-type data to reduce the memory requirement. Finally, there is no precision loss for any operator in a GA, therefore the real encoding is applied in the improved GA proposed in this chapter. • Step 2: Setting the number of the population. The determination of this is an important factor to achieve an optimal search. • Step 3: Setting the crossover rate. Its aim is to provide more space to the item of population. • Step 4: Setting the mutation rate. This can improve the ability of the GA to obtain the optimal solution. • Step 5: Setting the stop criterion based on the convergence of the optimal solution.
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4.3.1.3 To operate algorithm • Step 1: Generating the initial population randomly. Because the initial population has an effect on the result and efficiency of the algorithm, it is necessary to generate the population dispersedly in the space of the solution. In general, the initial population is generated randomly or according to a specified method, which results in the nonuniform distribution of the initial population. Then, the efficiency of the algorithm is reduced. In order to provide uniform distribution of the initial population, the generation of the initial population is based on the array that is built on the question in improved GA. In this process, the space of the solution is first divided into S subspaces. Second, the subspace is quantified, and then M chromosomes are chosen. Finally, N chromosomes are chosen from the M 3 S chromosomes to be the initial population. • Step 2: Calculating the fitness of the individual. In the process of searching, GA applies the value of fitness to evaluate the solution and execute the algorithm. In general, the fitness function is variable, and is used to evaluate the individual in the population according to the question. In this chapter, the improved GA applies the objective function to be the fitness function. Therefore the most fit individual has the smallest value of objective function. In the process of searching, the chromosome generated randomly may violate the constraints and exceed the upper bound. In order to guarantee that the constraints are not violated, a penalty term is introduced into the function of fitness, which is used to remove the individual that violates the constraints, and then finishing the calculation of the fitness of that individual. • Step 3: Judging whether the stop criterion is to be met. If the stop criterion is not met, execution of the algorithm should be continued. If it is met, the optimal solution is output. • Step 4: Operating selection. The selection is based on the theory of evolution, which reproduces the best individual. The value of fitness is used to judge whether the individual is the best one. In the selection, an individual that obtains a large value of fitness has more chance of being chosen. In this chapter, the value of fitness is assigned by ranking, and wheel selection is applied. All help the improved GA to find the optimal individual. • Step 5: Operating crossover. This is a key operation in GA. In this chapter, the improved GA applies the discrete recombination to determine the next generation with the same probability of choosing the parents. • Step 6: Operating mutation. After the selection and crossover, operation of the mutation improves the ability of the algorithm to find the optimal solution. When combined with the operation of crossover, it can avoid information loss caused by the operation of copy and crossover, which improves the efficiency and variety of the algorithm. In this chapter, the improved GA applies the self-adaptation mutation to search the individual with a large value of fitness in a small range and searches the individual with a small value of fitness in a large range.
4.3.1.4 To determine the engineering solution In order to avoid the problem of premature convergence in the basic genetic algorithm, population evolution controlled by the abnormal individual in the selection operation, and the
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excellent gene destroyed in the crossover operation, in this chapter, the GA is improved by introducing multipopulation evolution, which enables the subpopulation to evolve in a different path. This will acquire many optimal individuals from different subspaces. When it extents the searching space, it avoids bad convergence. In addition, it promotes the dispersal of an optimal solution by exchanging the optimal individuals in a different subspace, which improves the speed of convergence and accuracy. The different optimal individuals can be obtained by the optimization of different populations, and then the norm can be determined to choose the optimal solution according to the engineering practice. The operation process of improved GA for solving Eqs. (4.8)(4.10) is formulated as shown in Fig. 42.
4.3.2 Solving algorithm II: improved ant colony optimization In order to obtain satisfactory optimization results effectively and quickly, an improved ACO is presented to solve the reliability optimization allocation model of nuclear power plant systems with multiple characteristics, which it can efficiently overcome the limitations of basic ACO, such as falling into local extremum easily and slower convergence efficiency. The steps of an improved ACO are as follows.
4.3.2.1 To develop the reliability optimization allocation model according to Eqs. (4.8)(4.10) • Step 1: Determining the quantity of decision variables, that is, the reliability of the design unit. • Step 2: Determining the type of decision variable, that is, the design unit is redesigned or selected versions are used. • Step 3: Establishing the objective function of system cost. • Step 4: Establishing the constraints function of the reliability optimization allocation model.
4.3.2.2 To set the parameters of the algorithm • Step 1: Setting the population quantity. • Step 2: Setting iteration times designed as stopping criterion. • Step 3: Setting a convergence operator aimed at improving the ability of the ACO to find the optimal solution. • Step 4: Setting the number of nodes with the purpose of establishing an ant colony path diagram.
4.3.2.3 To operate the algorithm • Step 1: Establishing ant colony path diagram. All of the directed paths that allowed the ant individual to walk constitute the ant colony path diagram. Each directed path corresponds to an optimization allocated result, and each node value corresponds to an allocated reliability of the design unit. The ant colony path diagram is shown in Fig. 43.
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Selecting the encoding
Determining the quantity of decision variables
Setting the number of the population Determining the type of decision variable
Setting the crossover rate
Establishing the objective function of system cost
Setting the mutation rate
Setting the stop criterion Establishing the constraints function of the reliability optimization allocation model
Setting the number of the generation
To develop the reliability optimization allocation model
To set the parameters of the algorithm
Generating the initial population
Executing the algorithm N times
Calculating the fitness of the individual Collecting the suboptimal solutions Is the criterion of stop met? N
Obtaining N times results of the decision variables
Operating selection Operating crossover
Selecting the final solution according to the engineering requirements and the used decision variables frequently
Operating mutation Y Output of the optimal solution
To determine the engineering solution
To operate the algorithm
FIGURE 4–2 Operation process of improved genetic algorithm.
In Fig. 43, the node values of each column represent the selectable allocated results of the corresponding design unit. The number of nodes can be obtained by Eq. (4.11). Ni 5
Ri;max 2 Ri;min L
(4.11)
Chapter 4 • Reliability optimization allocation method for nuclear power plants
1
1
1
2
L1
93
2
L2
Lm
FIGURE 4–3 Ant colony path diagram.
where, Ni is the number of nodes of the ith column, and L is the step length of the node interval. The ant colony path diagram can be represented by the cell array, as follows: 8 R1;1 > > < R2;1 R5 ^ > > : RL1;1
R1;2 R2;2 ^ RL2;2
9 ? R1;m > > = ? R2;m ? ^ > > ; ? RLm;m
(4.12)
where, Rj;i is the jth selectable allocated result of the ith unit, i 5 1; 2; ?; m, j 5 1; 2; ?; Li . • Step 2: Initializing the pheromone path diagram. According to the ant colony path diagram, the corresponding pheromone path diagram can be established. The carrier of pheromone is the moving path of an ant individual. And the pheromone concentration of the moving path for the ant individual corresponds with the quality of the objective function value for such a moving path. The pheromone diagram will update with the change in the number of iterations, and the pheromone path diagram can be represented by the cell array, as follows: 8 τ 1;1 > > < τ 2;1 τðLoopÞ 5 > ^ > : τ L1;1
τ 1;2 τ 2;2 ^ τ L2;2
9 ? τ 1;m > > = ? τ 2;m ? ^ > > ; ? τ Lm;m
(4.13)
where, τ j;i is the pheromone element of the jth selectable allocated result of the ith unit; Loop is the iterations times, when Loop 5 1, the pheromone path diagram is the initialization pheromone path diagram, and τ j;i 5 1, i 5 1; 2; . . .; m, j 5 1; 2; . . .; li . • Step 3: Ant colony moving. The process of the formation of a path for each ant is defined as ant colony moving. Each path corresponds to a reliability allocated result, and the reliability allocated result is determined by the pheromone path diagram and the cost of each node. To establish the node probability diagram in order to represent the selected
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probability for an ant individual in the node, the node probability diagram can be represented by the cell array, as follows: 8 P1;1 > > < P2;1 P5 ^ > > : PL1;1
P1;2 P2;2 ^ PL2;2
9 ? P1;m > > = ? P2;m ? ^ > > ; ? PLm;m
(4.14)
where, Pi;j is the selected probability of the jth selectable allocated result of the ith unit, Pi τ i;j ð1=Ci;j ÞÞ, Ci;j is the cost of the jth selectable allocated result Pi;j 5 ðτ i;j ð1=Ci;j ÞÞ=ð Lj51 of the ith unit. Then, the reliability allocated result is obtained using the roulette wheel method to select the node of each column in the ant colony path diagram. • Step 4: Constraint IF and solving the objective function. After the ant colony is moving, the reliability allocated result obtained by each ant individual needs to be judged to see if it meets the constraints based on the GO method. If it meets the constraints, the setting constrain value is 1, that is, constrain 5 1; otherwise, setting constrain 5 0. Then, the ant individual corresponding to the minimum value of the objective function is determined among the ant individuals that satisfied the constraint. • Step 5: Updating the pheromone path diagram. When making the next iteration computation, the pheromone path diagram needs to be updated in order to improve the convergence efficiency and obtain the satisfactory results. The approach of updating the pheromone path diagram is as follows: first, for the ant individual corresponding to the minimum value of objective function in the previous iteration, the formula of updating the pheromone is given by X τ i;j ðLoop 1 1Þ 5 constrain τ i;j ðLoopÞ 1 constrain C
(4.15)
where, X is the convergence operator, and C is the cost of such an ant individual. Then, for other ant individuals, the formula of updating the pheromone is given by τ i;j ðLoop 1 1Þ 5 constrain τ i;j ðLoopÞ
(4.16)
• Step 6: Judging the termination condition. The iteration times is the termination condition of the improved ACO. If it meets the termination condition, the optimization allocation results and system cost will be output. And if it does not meet the termination condition, it will operate the algorithm from step 3: Ant colony moving. Above all, the operation process of the improved ACO for solving Eqs. (4.8)(4.10) is formulated as shown in Fig. 44.
Chapter 4 • Reliability optimization allocation method for nuclear power plants
Determining the quantity of decision variables
Setting the population quantity
95
Establishing the ant colony path diagram
Setting iteration times Initializing the pheromone path diagram Determining the type of decision variable
Setting the convergence operator Ant colony moving Setting the number of nodes
Establishing the objective function of system cost
To set the parameters of the algorithm
Constraint IF and solving the objective function
Selecting the allocation result meeting engineering practice Establishing the constraints function of the reliability optimization allocation model To develop the reliability optimization allocation model
N
Updating the pheromone path diagram
Outputting the optimization allocation results
Judging the termination condition Y
To determine the engineering solution
To operate the algorithm
FIGURE 4–4 Operation process of improved ant colony optimization. GO, Goal oriented.
4.4 Reliability optimization allocation method by the goal oriented method For nuclear power plant systems with multiple characteristics, the reliability optimization allocation based on the GO method, whose goal is to minimize the system cost, is expounded in detail in this section, including its evaluating steps, and formulating the process.
4.4.1 Approach of reliability optimization allocation by the goal oriented method The allocation steps of reliability optimization allocation by the GO method are as follows.
4.4.1.1 Conducting system analysis The system analysis is based on developing a GO model and conducting a GO operation. This has three steps: • Step 1: To conduct the system analysis to determine the principle diagram, engineering drawing, or function flowchart of nuclear power plant systems, the logical relationships among system and system characteristics, such as multiphase mission, multifunction, etc. • Step 2: To define the success rule of the system according to the results of a system analysis. • Step 3: To determine the importance factor of functions if a differentiation of functions exists in the system.
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4.4.1.2 Developing necessary models for describing the reliability optimization allocation problem The GO model is a key element of developing the reliability optimization allocation model. It is used to establish the reliability constraint of system functions and the system. If the nuclear power plant system has a differentiation of functions, the importance factor hierarchy model of the functions should be established. This has two steps: • Step 1: To develop the GO model. GO operators and signal flows are fundamental elements of the GO model. First, select the function operator, logical operator, and auxiliary operator to describe the unit itself, the logical relationship, and the system characteristics according to the system analysis. Then, develop the GO model using the signal flow to connect GO operators. • Step 2: To establish the importance factor hierarchy model of functions, which contain the determining target layer, criterion layer, and object layer.
4.4.1.3 Describing the reliability optimization allocation problem The reliability optimization allocation mathematics model for nuclear power plant systems is the mathematical description for the reliability optimization allocation problem, which consists of reliability constraint functions and the cost-objective function. • Step 1: To establish reliability constraint functions. If the system function is unique, the reliability constraint functions of the unit and system should be established, that is, Eqs. (4.1) and (4.6). If the system has multiple functions without considering their differentiation, the reliability constraint functions of the unit, system functions, and system should be established, that is, Eqs. (4.1), (4.3), and (4.6). If the system has multiple functions with differentiation, the reliability constraint functions of the unit and system, and the function importance factor constraint function should be established, that is, Eqs. (4.1), (4.6), and (4.5). • Step 2: To establish the cost-objective function, that is, Eq. (4.7). • Step 3: To develop the reliability optimization allocation mathematics model combining the above reliability constraint functions and cost-objective function. The reliability optimization allocation mathematics models of single-function systems, multifunction systems without considering differentiation of functions, and multifunction systems considering differentiation of functions are illustrated as Eqs. (4.8), (4.9), and (4.10), respectively.
4.4.1.4 Determining the engineering solution According to engineering practice, the engineering solution can be determined by selecting the reliability optimization allocation results, which are obtained by an intelligent algorithm to solve the reliability optimization allocation mathematics model. The steps are as follows: • Step 1: Solving the reliability optimization allocation mathematic model. Intelligent algorithms have been widely applied to solve the reliability optimization allocation mathematics model, such as GA and ACO presented in this chapter. Setting the
Chapter 4 • Reliability optimization allocation method for nuclear power plants
Analyzing the system principle diagram, engineering drawing, or function flowchart, system characteristics
Selecting the GO operator Develoing the GO model Establishing the GO model
Defining the success rule of the system
Determining the target layer Establishing importance factor hierarchy model of functions
Determining the importance factor of functions Conducting system analysis
Determining the criterion layer Determining the object layer
Developing the necessary models for describing the reliability optimization allocation problem For single-function systems, establishing the reliability constraint functions of the unit and system
Solving the reliability optimization allocation mathematic model Establishing the reliability constraint functions
Setting the parameters of the algorithm
For multi function systems without considering the differentiation of functions, establishing the reliability constraint functions of the unit, system functions, and system
For multi function systems with differentiation of functions, establishing the reliability constraint functions of the unit and system, and the function importance factor constraint function
Operating the algorithm
Selecting the allocation result to meet engineering practice
97
Establishing the objective function
Determining the cost of the unit Determining the cost of the system
Establishing the mathematic model of the reliability optimization allocation problem
Determining the engineering solution
Describing the reliability optimization allocation problem
FIGURE 4–5 Reliability optimization allocation process with the goal of minimizing the system cost based on the goal oriented method.
parameters of the algorithm and operating it are necessary steps. The improved GA and ACO presented in this chapter are described in Sections 4.3.1 and 4.3.2. • Step 2: Selecting the allocation result to meet engineering practice.
4.4.2 Process of reliability optimization allocation method by the goal oriented method The process of the reliability optimization allocation method based on the GO method for nuclear power plant systems is formulated as presented in Fig. 45.
4.5 Problems 1. What are the key elements of the reliability optimization allocation problem? 2. What are the advantages of the reliability optimization allocation method by the GO method?
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3. How many constraint functions are in the reliability optimization allocation model by the GO method? 4. How are the reliability constraint function of unit, reliability constraint function of system function, and constraint function of system constructed, respectively? 5. How is the improved GA operated? 6. How is the improved ACO operated? 7. How is the reliability optimization allocation based on this chapter’s method conducted?
References [1] W. Kuo, W. Rin, Recent advances in optimal reliability allocation, IEEE Trans. Syst. Man Cybern. A: Syst. Hum. 37 (2007) 143156. [2] S. Jiang, S. Yang, An improved multi-objective optimization evolutionary algorithm based on decomposition for complex pareto fronts, IEEE Trans. Cybern. 46 (2) (2016) 421437. [3] A. Zaretalab, V. Hajipour, M. Sharifi, M.E. Shahriari, Knowledge-based archive multi-objective simulated annealing algorithm to optimize series-parallel system with choice of redundancy strategies, Comput. Ind. Eng. 80 (2015) 3344. [4] A. Azadeh, B.M. Shoja, S. Ghanei, M. Sheikhalishahi, Multi-objective optimization problem for multistate series-parallel systems: a two-stage flow-shop manufacturing system, Reliab. Eng. Syst. Saf. 136 (2015) 6274. [5] A. Dolatshahi-Zand, K. Khalili-Damghani, Design of SCADA water resource management control center by a bi-objective redundancy allocation problem and particle swarm optimization, Reliab. Eng. Syst. Saf. 133 (2015) 1121. [6] M.A. Ardakan, A.Z. Hamadani, Reliability optimization of seriesparallel systems with mixed redundancy strategy in subsystems, Reliab. Eng. Syst. Saf. 130 (1) (2014) 132139. [7] V.C. Bueno, Minimal standy redundancy allocation in a k out-of-n: F system of dependent components, Eur. J. Oper. Res. 165 (3) (2005) 786793. [8] G. Levitin, Y. Dai, M. Xie, K.L. Poh, Optimizing survivability of multi-state systems with multi-level protection by multi-processor genetic algorithm, Eur. Reliab. Eng. Syst. Saf. 82 (1) (2003) 93104. [9] G. Levitin, A. Lisnianski, Structure optimization of multistate system with two failure modes, Eur. Reliab. Eng. Syst. Saf. 72 (1) (2001) 7589. [10] F. Altiparmak, B. Dengiz, A.E. Smith, Optimal design of reliable computer networks: a comparison of meta heuristics, Eur. J. Heuristics 9 (6) (2003) 471487. [11] N. Nahas, M. Nourelfath, Ant system for reliability optimization of a series system with multiple-choice and budget constraints, Eur. Reliab. Eng. Syst. Saf. 87 (2005) 112. 2004. [12] X.J. Yi, Y.H. Lai, H.P. Dong, et al., A reliability optimization allocation method considering differentiation of functions, Proc. Int. Conf. Comput. Methods 2 (2015) 515525. [13] X.J. Yi, P. Hou, H.P. Dong, et al., A reliability optimization allocation method for systems with differentiation of functions, in: Proceedings of ASME 2015 International Mechanical Engineering Congress & Exposition, Houston, Texas, November 1319, 2015, 14, IMECE2015-52928, 2015. [14] X.J. Yi, Y.H. Lai, H.P. Dong, et al., A reliability optimization allocation method considering differentiation of functions, Int. J. Comput. Methods 13 (2016). Available from: https://doi.org/10.1142/ S0219876216410206. [15] X.J. Yi, H.N. Mu, P. Hou, Y.H. Lai, A reliability optimization allocation method considering differentiation of functions based on goal oriented method, in: 7th International Conference on Computational Methods, Berkeley, US, August 14, 2016.
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[16] X.J. Yi, B.S. Dhillon, J. Shi, H.N. Mu, P. Hou, Reliability optimization allocation method for multifunction systems with multistate units based on goal-oriented methodology, ASCE-ASME J. Risk Uncertainty Eng. Syst. B: Mech. Eng. 3 (4) (2017). Available from: https://doi.org/10.1115/1.4037123. [17] X.J. Yi, Y.C. Xie, J. Shi, H.N. Mu, P. Hou, A new reliability optimization allocation method for nuclear power systems based on goal oriented methodology, in: ASME International Conference on Nuclear Engineering 25, Shang Hai, China, July 26, 2017. [18] J.E. Ramirez-Marquez, D.W. Coit, A heuristic for solving the redundancy allocation problem for multistate series-parallel system, Eur. Reliab. Eng. Syst. Saf. 83 (3) (2004) 341349. [19] G. Levitin, A. Lisnianski, Optimizing survivability of vulnerable series-parallel multi-state systems, Eur. Reliab. Eng. Syst. Saf. 79 (3) (2003) 319331. [20] G. Levitin, A. Lisnianski, Optimal multilevel protection in series-parallel systems, Eur. Reliab. Eng. Syst. Saf. 81 (1) (2003) 93102. [21] R.S. Parpinelli, H.S. Lopes, A.A. Freitas, et al., Data mining with an ant colony optimization algorithm, IEEE Trans. Evol. Comput. 6 (4) (2002) 321332. [22] C. Blum, Ant colony optimization: introduction and recent trends, Phys. Life Rev. 2 (4) (2005) 353373. [23] M. Middendorf, F. Reischle, H. Schmeck, et al., Multi colony ant algorithms, J. Heuristics 8 (3) (2002) 305320. [24] M. Dorigo, G.A. Caro, L.M. Gambardella, et al., Ant algorithms for discrete optimization, Artif. Life 5 (2) (1999) 137172. [25] M. Middendorf, F. Reischle, H. Schmeck, et al., Multi colony ant algorithms, J. Heuristics 8 (3) (2002) 305320. [26] M. Dorigo, L.M. Gambardella, Ant colony system: a cooperative learning approach to the traveling salesman problem, IEEE Trans. Evol. Comput. 1 (1) (1997) 5366. [27] H. Kawamura, M. Yamamoto, K. Suzuki, et al., Multiple ant colonies algorithm based on colony level interactions, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 83 (2) (2000) 371379. [28] G.R. Harik, F.G. Lobo, D.E. Goldberg, The compact genetic algorithm, IEEE Trans. Evol. Comput. 3 (4) (1999) 287297. [29] D. Whitley, A genetic algorithm tutorial, Stat. Comput. 4 (2) (1994) 6588. [30] M. Srinivas, L.M. Patnaik, Genetic algorithms: a survey, IEEE Comput. 27 (6) (1994) 1726. [31] D.E. Goldberg, J.H. Holland, Genetic algorithms and machine learning, Mach. Learn. 3 (2) (1988) 9599. [32] K. Deb, Evolutionary multi-criterion optimization, in: Genetic and Evolutionary Computation Conference, 2010, pp. 25772602. [33] P.G. Espejo, S. Ventura, F. Herrera, et al., A survey on the application of genetic programming to classification, IEEE Trans. Syst. Man Cybern. C: Appl. Rev. 40 (2) (2010) 121144. [34] X. Yu, Introduction to evolutionary algorithms, Ann. Conf. Comput. 9 (4) (2010) 348349.
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4.1 Introduction Reliability optimization allocation is a decision problem in reliability engineering. Its objective is to allocate the system reliability index to design units, so that it can guide the design to ensure the reliability design requirements. In recent years, study on reliability optimization allocation considering different constraints, such as cost constraint, dimension constraint, etc., has been a hot issue of reliability engineering. The majority of researches in this area are devoted to multistate system optimization, multiobjective optimization, active and cold-standby redundancy, fault-tolerance mechanism, multifunction system optimization, and optimization techniques [16]. For nuclear power plants, which have many characteristics, such as multiphase mission, multifunction, and time-sequence, reliability optimization allocation is an essential task in the design of nuclear power plants, as it can provide a more reasonable design in order to save costs and human resources. However, the existing
Goal Oriented Methodology and Applications in Nuclear Power Plants. DOI: https://doi.org/10.1016/B978-0-12-816185-2.00004-6 © 2020 Elsevier Inc. All rights reserved.
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reliability optimization allocation technologies for nuclear power plants systems still have four obvious limitations that can cause great deviations in the reliability allocation results, as follows [714]: • The reliability allocation models are not directly related, accurately describe the system structure and system characteristics, and are affected by subjective experience, therefore the reliability allocation results are difficult to meet the engineering design requirements for complex nuclear power plant systems. • It is not easy to allocate the system reliability index to design units, including redesigned and selected unit versions, quickly, efficiently, and directly. • Production is finalized through multiple design revisions. The models of reliability, such as series, parallel, seriesparallel, etc., are not directly related and exactly describe the product structure, function, and characteristics. Hence, for design changes, it is very difficult to conduct reliability reallocation quickly and efficiently to allocate a reasonable reliability margin for design units. • For the problem of specific optimization allocation, the basic optimization algorithm needs to be improved to obtain satisfactory convergence effects and avoid a local extremum problem. Thus, how to avoid the above limitations of reliability optimization allocation methods for nuclear power plant systems is a research hotspot. Some typical studies have indicated that the reliability optimization allocation method by the goal oriented (GO) method can efficiently solve the above limitations because of its advantages in terms of its ease of creating a model and its representational and analysis power [1518]. Furthermore, the reliability optimization allocation process by the GO method shows not only that the satisfactory optimization allocation result can be obtained, but also shows the advantages, operability, and engineering applicability of this method. In this chapter, the reliability optimization allocation method for complex nuclear power plant systems based on the GO method is illustrated in terms of the reliability optimization allocation model by the GO method, intelligent algorithms for solving optimization allocation model, and the method’s process.
4.2 Reliability optimization allocation model by the goal oriented method The reliability optimization allocation problem is described through the corresponding mathematics model, which contains constraints function and objective function.
4.2.1 Constraints function For reliability optimization allocation of nuclear power plant systems with multiple characteristics, there are three constraints functions—reliability constraint function of unit,
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reliability constraint function of system function, and constraint function of system. It is assumed that a nuclear power plant system consists of m units, and can execute n system functions.
4.2.1.1 Reliability constraint function of the unit The reliability constraint function of the unit consists of the allocated reliability index of the unit, the lower limit and upper limit of the reliability index for the unit, as follows: Ri;lower # Ri # Ri;upper
(4.1)
where, Ri is the allocated reliability of the unit, and Ri;lower and Ri;upper are the lower limit and upper limit of unit reliability, i 5 1; 2; . . .; m, respectively.
4.2.1.2 Reliability constraint function of system function 4.2.1.2.1 Definition For reliability optimization allocation of nuclear power plant systems with multifunction systems, the reliabilities of the system and its functions can be obtained by using the reliability of the design unit and GO algorithm to conduct GO operation according to the GO model. In the GO model, the reliability of the output signal flows for the system and its functions is the reliability of the system and its functions. The reliability of signal flow based on the GO method for reliability optimization allocation is defined as follows: Rx 5 Fx ðRx1 ; Rx2 ; . . .; Rxy Þ
(4.2)
where, Rx is the predicted reliability of xth signal flow, Rxy is the allocated reliability of the yth unit for calculating xth signal flow, Fx ðÞ is the success probability of the xth signal flow obtained using the allocated reliability of the unit to conduct the GO operation according to the GO model and GO algorithm. 4.2.1.2.2 Reliability constraint function of system function The reliability constraint function of system functions indicates that the predicted reliability of system function using allocated reliability of the design unit by the GO method should meet the target reliability of system function, as shown in Eq. (4.3).
RGw 5 Fw ðRw1 ; Rw2 ; . . .; Rwj Þ RGw . RGw
(4.3)
where, RGw is the predicted reliability of wth function, Rwj is the allocated reliability of the jth unit for wth function, Fw ðÞ is the reliability of output signal flow represented by wth function using the allocated reliability of a unit based on the GO method, and RGw is the target reliability of system function, w 5 1; 2; . . .; n, 1 # j # m.
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4.2.1.2.3 Function importance factor constraint function of system function For a nuclear power plant system with multiple functions, the usage frequency of each function and the combination of units to realize different functions are not often the same. In addition, different use demands of each system function are determined by the different environments of the task, and the differentiation of use demand of each system function results in the reliability levels of various functions being different. To deal with the differentiation of functions in reliability optimization allocation, the more reasonable approach is to convert it into a constraint condition and to select an optimal solution in the optimization process. Obviously, the units and functions in a system do not only have a reliability index, but also have a parameter of importance factor. A new index named the importance factor has been defined to measure the importance of a unit or function by relating the reliability of the unit or function. Generally, for the higher importance factor of a unit or function, higher reliability should be allocated to it. Therefore, according to the positive correlation between the importance factor and reliability, an importance factor model is proposed based on the experience in engineering applications, as shown in Eq. (4.4). gðRÞ 5 eðR21Þ
0,g ,1
(4.4)
where, gðÞ is the importance factor of the unit or function, and R is the reliability of the unit or function. The function importance factor constraint indicates the predicted function importance factor by using the allocated reliability of the unit based on the GO method that should meet the allocated function importance factor, as shown in Eq. (4.5).
RGw 5 Fw ðRw1 ; Rw2 ; . . .; Rwj Þ
gðRGw Þ 5 eðRGw 21Þ $ gðRGw Þ 5 eðRGw 21Þ
(4.5)
where, RGw is the predicted reliability of wth function, Rwj is the allocated reliability of the jth unit for wth function, Fw ðÞ is the reliability of output signal flow represented by wth function by using the allocated reliability of the unit based on the GO method, gðRGw Þ is the predicted function importance factor of wth function, w 5 1; 2; . . .; n, 1 # j # m, and gðRGw Þ is the allocated function importance factor by the estimation of function importance factors, which can be obtained by the following steps: • Step 1: The factors which can affect the reliabilities of functions, such as using frequency, the design level, complexity, etc., should be ascertained. Based on these factors, an importance factor hierarchy model of functions which generally consists of a target layer, criterion layer, and object layer can be built. Among these, the target layer corresponds to the system reliability, the criterion layer corresponds to the various factors which can cause a change to the reliability indexes of functions, and the object layer corresponds to the functions of the system, as shown in Fig. 41. • Step 2: Based on expert evaluation obtained according to hierarchy structure, a judgment matrix can be built. Then the weight of each function which represents the influence on
Chapter 4 • Reliability optimization allocation method for nuclear power plants
Target layer
87
Target of system reliability
Criterion layer
Influence factor 1
Object layer
Function 1
Influence factor 2
……
Function 2
……
Influence factor M
Function N
FIGURE 4–1 Function importance factor hierarchy model.
the target reliability index of the system under the influence of various factors can be calculated, that is, V 5 fv1 ; v2 ; . . .; vn g. • Step 3: According to V and the target reliability index of the system, the target reliability index of each function can be obtained. • Step 4: The importance factor of each function is easy to determine using Eq. (4.4).
4.2.1.3 Reliability constraint function of the system The reliability constraint function of the system indicates the predicted reliability of the system using the allocated reliability of the design unit by the GO method meets the target reliability of the system, as shown in Eq. (4.6).
RS 5 FS ðR1 ; R2 ; . . .; Rm Þ RS $ RS
(4.6)
where, RS is the predicted reliability of the system, Ri is the allocated reliability of the ith unit, FS ðÞ is the reliability of output signal flow of the system by using the allocated reliability of the unit based on the GO method, and RS is the target reliability of the system, i 5 1; 2; . . .; m.
4.2.2 Objective function The cost is often of concern in engineering, and the reliability optimization allocation methods whose goal is to minimize the cost have been studied by many scholars [19,20]. Thus, the objective function of the reliability optimization allocation problem in this chapter is to minimize the cost, as follows: minCS ðRÞ 5
m X i51
ci ðPi ; Ri ; Ri;min ; Ri;max Þ
(4.7)
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where, CS ðÞ is the cost function of the system; ci ðPi ; Ri ; Ri;min ; Ri;max Þ is the cost function of the design unit, that is, ci ðPi ; Ri ; Ri;min Þ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ , Pi is the basic cost of the ith unit, Ri is the allocated reliability of the ith unit, Ri;min is the lower limit value of reliability of the ith unit, and Ri;max is the upper limit value of reliability of the ith unit.
4.2.3 Reliability optimization allocation mathematics model Combining the above objective function and constraints function, the reliability optimization allocation problem for nuclear power plant systems considering multiple characteristics can be described by three models. • Model I: Reliability optimization allocation model for single-function systems 8 m X > > minC ðRÞ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ > S > > < i51 s:t: Ri;min # Ri # Ri;max > > > > > : gðRGw Þ 5 gðFw ðRw1 ; Rw2 ; . ..; Rwj ÞÞ $ gðRGw Þ RS 5 FS ðR1 ; R2 ; . . .; Rm Þ $ RS
i 5 1; 2; . . .; m w 5 1; 2; . . .; n
(4.8)
• Model II: Reliability optimization allocation model for multiple-function systems without considering differentiation of functions 8 m X > > minC ðRÞ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ > S > > < i51 s:t: Ri;min # Ri # Ri;max i 5 1; 2; . . .; m > > > > > : RGw 5 Fw ðRw1 ; Rw2 ; . . .; RwjÞ $ RGw w 5 1; 2; . . .; n RS 5 FS ðR1 ; R2 ; . . .; Rm Þ $ RS
(4.9)
• Model III: Reliability optimization allocation model for multiple-function systems considering differentiation of functions 8 m X > > minC ðRÞ 5 Pi eððRi 2Ri;max Þ=ðRi;min 2Ri;max ÞÞ > S > > < i51 s:t: Ri;min # Ri # Ri;max > > > > > : gðRGw Þ 5 gðFw ðRw1 ; Rw2 ; . ..; Rwj ÞÞ $ gðRGw Þ RS 5 FS ðR1 ; R2 ; . . .; Rm Þ $ RS
i 5 1; 2; . . .; m w 5 1; 2; . . .; n
(4.10)
4.3 Intelligent algorithms for solving the optimization allocation model For the complex reliability optimization allocation model of the system, it is difficult to obtain the solution. A large number of researches have proved that applying artificial intelligence methods (such as neural networks, ant colony algorithms, genetic algorithms, hybrid genetic
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algorithms, simulated annealing algorithms, etc.) [2134] to reliability optimization allocation can achieve good results. Nevertheless, intelligent algorithms also have their disadvantages, such as: premature phenomena may occur in the course of use; swinging when approaching the optimal solution; slow convergence and easily falling into local extremum; or the optimization result not being ideal. Therefore it is meaningful to improve the basic intelligent algorithm so that it is applicable for specific problems and can obtain the optimal solution efficiently. In this section, a genetic algorithm (GA) and an improved ant colony optimization (ACO) are illustrated to solve the reliability optimization allocation model.
4.3.1 Solving algorithm I: improved genetic algorithm In order to obtain satisfactory optimization results effectively and quickly, an improved GA is presented to solve the reliability optimization allocation model of a nuclear power plant with multiple characteristics, and it can efficiently overcome the limitations of a basic GA, such as falling into local extremum easily, slower convergence efficiency, etc. The steps to an improved GA are described next.
4.3.1.1 To develop a reliability optimization allocation model according to Eqs. (4.8)(4.10) • Step 1: Determining the quantity of decision variables, that is, the reliability of the design unit. • Step 2: Determining the type of decision variable, that is, the design unit is redesigned or a selected version. • Step 3: Establishing the objective function of the system cost. • Step 4: Establishing the constraints function of the reliability optimization allocation model.
4.3.1.2 To set the parameters of the algorithm • Step 1: Selecting the encoding. In comparison with binary encoding, real encoding has some advantages. First, it does not need the transformation from chromosome to solution before the calculation, which improves the efficiency of the algorithm. Second, it adopts float-type data to reduce the memory requirement. Finally, there is no precision loss for any operator in a GA, therefore the real encoding is applied in the improved GA proposed in this chapter. • Step 2: Setting the number of the population. The determination of this is an important factor to achieve an optimal search. • Step 3: Setting the crossover rate. Its aim is to provide more space to the item of population. • Step 4: Setting the mutation rate. This can improve the ability of the GA to obtain the optimal solution. • Step 5: Setting the stop criterion based on the convergence of the optimal solution.
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4.3.1.3 To operate algorithm • Step 1: Generating the initial population randomly. Because the initial population has an effect on the result and efficiency of the algorithm, it is necessary to generate the population dispersedly in the space of the solution. In general, the initial population is generated randomly or according to a specified method, which results in the nonuniform distribution of the initial population. Then, the efficiency of the algorithm is reduced. In order to provide uniform distribution of the initial population, the generation of the initial population is based on the array that is built on the question in improved GA. In this process, the space of the solution is first divided into S subspaces. Second, the subspace is quantified, and then M chromosomes are chosen. Finally, N chromosomes are chosen from the M 3 S chromosomes to be the initial population. • Step 2: Calculating the fitness of the individual. In the process of searching, GA applies the value of fitness to evaluate the solution and execute the algorithm. In general, the fitness function is variable, and is used to evaluate the individual in the population according to the question. In this chapter, the improved GA applies the objective function to be the fitness function. Therefore the most fit individual has the smallest value of objective function. In the process of searching, the chromosome generated randomly may violate the constraints and exceed the upper bound. In order to guarantee that the constraints are not violated, a penalty term is introduced into the function of fitness, which is used to remove the individual that violates the constraints, and then finishing the calculation of the fitness of that individual. • Step 3: Judging whether the stop criterion is to be met. If the stop criterion is not met, execution of the algorithm should be continued. If it is met, the optimal solution is output. • Step 4: Operating selection. The selection is based on the theory of evolution, which reproduces the best individual. The value of fitness is used to judge whether the individual is the best one. In the selection, an individual that obtains a large value of fitness has more chance of being chosen. In this chapter, the value of fitness is assigned by ranking, and wheel selection is applied. All help the improved GA to find the optimal individual. • Step 5: Operating crossover. This is a key operation in GA. In this chapter, the improved GA applies the discrete recombination to determine the next generation with the same probability of choosing the parents. • Step 6: Operating mutation. After the selection and crossover, operation of the mutation improves the ability of the algorithm to find the optimal solution. When combined with the operation of crossover, it can avoid information loss caused by the operation of copy and crossover, which improves the efficiency and variety of the algorithm. In this chapter, the improved GA applies the self-adaptation mutation to search the individual with a large value of fitness in a small range and searches the individual with a small value of fitness in a large range.
4.3.1.4 To determine the engineering solution In order to avoid the problem of premature convergence in the basic genetic algorithm, population evolution controlled by the abnormal individual in the selection operation, and the
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excellent gene destroyed in the crossover operation, in this chapter, the GA is improved by introducing multipopulation evolution, which enables the subpopulation to evolve in a different path. This will acquire many optimal individuals from different subspaces. When it extents the searching space, it avoids bad convergence. In addition, it promotes the dispersal of an optimal solution by exchanging the optimal individuals in a different subspace, which improves the speed of convergence and accuracy. The different optimal individuals can be obtained by the optimization of different populations, and then the norm can be determined to choose the optimal solution according to the engineering practice. The operation process of improved GA for solving Eqs. (4.8)(4.10) is formulated as shown in Fig. 42.
4.3.2 Solving algorithm II: improved ant colony optimization In order to obtain satisfactory optimization results effectively and quickly, an improved ACO is presented to solve the reliability optimization allocation model of nuclear power plant systems with multiple characteristics, which it can efficiently overcome the limitations of basic ACO, such as falling into local extremum easily and slower convergence efficiency. The steps of an improved ACO are as follows.
4.3.2.1 To develop the reliability optimization allocation model according to Eqs. (4.8)(4.10) • Step 1: Determining the quantity of decision variables, that is, the reliability of the design unit. • Step 2: Determining the type of decision variable, that is, the design unit is redesigned or selected versions are used. • Step 3: Establishing the objective function of system cost. • Step 4: Establishing the constraints function of the reliability optimization allocation model.
4.3.2.2 To set the parameters of the algorithm • Step 1: Setting the population quantity. • Step 2: Setting iteration times designed as stopping criterion. • Step 3: Setting a convergence operator aimed at improving the ability of the ACO to find the optimal solution. • Step 4: Setting the number of nodes with the purpose of establishing an ant colony path diagram.
4.3.2.3 To operate the algorithm • Step 1: Establishing ant colony path diagram. All of the directed paths that allowed the ant individual to walk constitute the ant colony path diagram. Each directed path corresponds to an optimization allocated result, and each node value corresponds to an allocated reliability of the design unit. The ant colony path diagram is shown in Fig. 43.
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Selecting the encoding
Determining the quantity of decision variables
Setting the number of the population Determining the type of decision variable
Setting the crossover rate
Establishing the objective function of system cost
Setting the mutation rate
Setting the stop criterion Establishing the constraints function of the reliability optimization allocation model
Setting the number of the generation
To develop the reliability optimization allocation model
To set the parameters of the algorithm
Generating the initial population
Executing the algorithm N times
Calculating the fitness of the individual Collecting the suboptimal solutions Is the criterion of stop met? N
Obtaining N times results of the decision variables
Operating selection Operating crossover
Selecting the final solution according to the engineering requirements and the used decision variables frequently
Operating mutation Y Output of the optimal solution
To determine the engineering solution
To operate the algorithm
FIGURE 4–2 Operation process of improved genetic algorithm.
In Fig. 43, the node values of each column represent the selectable allocated results of the corresponding design unit. The number of nodes can be obtained by Eq. (4.11). Ni 5
Ri;max 2 Ri;min L
(4.11)
Chapter 4 • Reliability optimization allocation method for nuclear power plants
1
1
1
2
L1
93
2
L2
Lm
FIGURE 4–3 Ant colony path diagram.
where, Ni is the number of nodes of the ith column, and L is the step length of the node interval. The ant colony path diagram can be represented by the cell array, as follows: 8 R1;1 > > < R2;1 R5 ^ > > : RL1;1
R1;2 R2;2 ^ RL2;2
9 ? R1;m > > = ? R2;m ? ^ > > ; ? RLm;m
(4.12)
where, Rj;i is the jth selectable allocated result of the ith unit, i 5 1; 2; ?; m, j 5 1; 2; ?; Li . • Step 2: Initializing the pheromone path diagram. According to the ant colony path diagram, the corresponding pheromone path diagram can be established. The carrier of pheromone is the moving path of an ant individual. And the pheromone concentration of the moving path for the ant individual corresponds with the quality of the objective function value for such a moving path. The pheromone diagram will update with the change in the number of iterations, and the pheromone path diagram can be represented by the cell array, as follows: 8 τ 1;1 > > < τ 2;1 τðLoopÞ 5 > ^ > : τ L1;1
τ 1;2 τ 2;2 ^ τ L2;2
9 ? τ 1;m > > = ? τ 2;m ? ^ > > ; ? τ Lm;m
(4.13)
where, τ j;i is the pheromone element of the jth selectable allocated result of the ith unit; Loop is the iterations times, when Loop 5 1, the pheromone path diagram is the initialization pheromone path diagram, and τ j;i 5 1, i 5 1; 2; . . .; m, j 5 1; 2; . . .; li . • Step 3: Ant colony moving. The process of the formation of a path for each ant is defined as ant colony moving. Each path corresponds to a reliability allocated result, and the reliability allocated result is determined by the pheromone path diagram and the cost of each node. To establish the node probability diagram in order to represent the selected
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Goal Oriented Methodology and Applications in Nuclear Power Plants
probability for an ant individual in the node, the node probability diagram can be represented by the cell array, as follows: 8 P1;1 > > < P2;1 P5 ^ > > : PL1;1
P1;2 P2;2 ^ PL2;2
9 ? P1;m > > = ? P2;m ? ^ > > ; ? PLm;m
(4.14)
where, Pi;j is the selected probability of the jth selectable allocated result of the ith unit, Pi τ i;j ð1=Ci;j ÞÞ, Ci;j is the cost of the jth selectable allocated result Pi;j 5 ðτ i;j ð1=Ci;j ÞÞ=ð Lj51 of the ith unit. Then, the reliability allocated result is obtained using the roulette wheel method to select the node of each column in the ant colony path diagram. • Step 4: Constraint IF and solving the objective function. After the ant colony is moving, the reliability allocated result obtained by each ant individual needs to be judged to see if it meets the constraints based on the GO method. If it meets the constraints, the setting constrain value is 1, that is, constrain 5 1; otherwise, setting constrain 5 0. Then, the ant individual corresponding to the minimum value of the objective function is determined among the ant individuals that satisfied the constraint. • Step 5: Updating the pheromone path diagram. When making the next iteration computation, the pheromone path diagram needs to be updated in order to improve the convergence efficiency and obtain the satisfactory results. The approach of updating the pheromone path diagram is as follows: first, for the ant individual corresponding to the minimum value of objective function in the previous iteration, the formula of updating the pheromone is given by X τ i;j ðLoop 1 1Þ 5 constrain τ i;j ðLoopÞ 1 constrain C
(4.15)
where, X is the convergence operator, and C is the cost of such an ant individual. Then, for other ant individuals, the formula of updating the pheromone is given by τ i;j ðLoop 1 1Þ 5 constrain τ i;j ðLoopÞ
(4.16)
• Step 6: Judging the termination condition. The iteration times is the termination condition of the improved ACO. If it meets the termination condition, the optimization allocation results and system cost will be output. And if it does not meet the termination condition, it will operate the algorithm from step 3: Ant colony moving. Above all, the operation process of the improved ACO for solving Eqs. (4.8)(4.10) is formulated as shown in Fig. 44.
Chapter 4 • Reliability optimization allocation method for nuclear power plants
Determining the quantity of decision variables
Setting the population quantity
95
Establishing the ant colony path diagram
Setting iteration times Initializing the pheromone path diagram Determining the type of decision variable
Setting the convergence operator Ant colony moving Setting the number of nodes
Establishing the objective function of system cost
To set the parameters of the algorithm
Constraint IF and solving the objective function
Selecting the allocation result meeting engineering practice Establishing the constraints function of the reliability optimization allocation model To develop the reliability optimization allocation model
N
Updating the pheromone path diagram
Outputting the optimization allocation results
Judging the termination condition Y
To determine the engineering solution
To operate the algorithm
FIGURE 4–4 Operation process of improved ant colony optimization. GO, Goal oriented.
4.4 Reliability optimization allocation method by the goal oriented method For nuclear power plant systems with multiple characteristics, the reliability optimization allocation based on the GO method, whose goal is to minimize the system cost, is expounded in detail in this section, including its evaluating steps, and formulating the process.
4.4.1 Approach of reliability optimization allocation by the goal oriented method The allocation steps of reliability optimization allocation by the GO method are as follows.
4.4.1.1 Conducting system analysis The system analysis is based on developing a GO model and conducting a GO operation. This has three steps: • Step 1: To conduct the system analysis to determine the principle diagram, engineering drawing, or function flowchart of nuclear power plant systems, the logical relationships among system and system characteristics, such as multiphase mission, multifunction, etc. • Step 2: To define the success rule of the system according to the results of a system analysis. • Step 3: To determine the importance factor of functions if a differentiation of functions exists in the system.
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Goal Oriented Methodology and Applications in Nuclear Power Plants
4.4.1.2 Developing necessary models for describing the reliability optimization allocation problem The GO model is a key element of developing the reliability optimization allocation model. It is used to establish the reliability constraint of system functions and the system. If the nuclear power plant system has a differentiation of functions, the importance factor hierarchy model of the functions should be established. This has two steps: • Step 1: To develop the GO model. GO operators and signal flows are fundamental elements of the GO model. First, select the function operator, logical operator, and auxiliary operator to describe the unit itself, the logical relationship, and the system characteristics according to the system analysis. Then, develop the GO model using the signal flow to connect GO operators. • Step 2: To establish the importance factor hierarchy model of functions, which contain the determining target layer, criterion layer, and object layer.
4.4.1.3 Describing the reliability optimization allocation problem The reliability optimization allocation mathematics model for nuclear power plant systems is the mathematical description for the reliability optimization allocation problem, which consists of reliability constraint functions and the cost-objective function. • Step 1: To establish reliability constraint functions. If the system function is unique, the reliability constraint functions of the unit and system should be established, that is, Eqs. (4.1) and (4.6). If the system has multiple functions without considering their differentiation, the reliability constraint functions of the unit, system functions, and system should be established, that is, Eqs. (4.1), (4.3), and (4.6). If the system has multiple functions with differentiation, the reliability constraint functions of the unit and system, and the function importance factor constraint function should be established, that is, Eqs. (4.1), (4.6), and (4.5). • Step 2: To establish the cost-objective function, that is, Eq. (4.7). • Step 3: To develop the reliability optimization allocation mathematics model combining the above reliability constraint functions and cost-objective function. The reliability optimization allocation mathematics models of single-function systems, multifunction systems without considering differentiation of functions, and multifunction systems considering differentiation of functions are illustrated as Eqs. (4.8), (4.9), and (4.10), respectively.
4.4.1.4 Determining the engineering solution According to engineering practice, the engineering solution can be determined by selecting the reliability optimization allocation results, which are obtained by an intelligent algorithm to solve the reliability optimization allocation mathematics model. The steps are as follows: • Step 1: Solving the reliability optimization allocation mathematic model. Intelligent algorithms have been widely applied to solve the reliability optimization allocation mathematics model, such as GA and ACO presented in this chapter. Setting the
Chapter 4 • Reliability optimization allocation method for nuclear power plants
Analyzing the system principle diagram, engineering drawing, or function flowchart, system characteristics
Selecting the GO operator Develoing the GO model Establishing the GO model
Defining the success rule of the system
Determining the target layer Establishing importance factor hierarchy model of functions
Determining the importance factor of functions Conducting system analysis
Determining the criterion layer Determining the object layer
Developing the necessary models for describing the reliability optimization allocation problem For single-function systems, establishing the reliability constraint functions of the unit and system
Solving the reliability optimization allocation mathematic model Establishing the reliability constraint functions
Setting the parameters of the algorithm
For multi function systems without considering the differentiation of functions, establishing the reliability constraint functions of the unit, system functions, and system
For multi function systems with differentiation of functions, establishing the reliability constraint functions of the unit and system, and the function importance factor constraint function
Operating the algorithm
Selecting the allocation result to meet engineering practice
97
Establishing the objective function
Determining the cost of the unit Determining the cost of the system
Establishing the mathematic model of the reliability optimization allocation problem
Determining the engineering solution
Describing the reliability optimization allocation problem
FIGURE 4–5 Reliability optimization allocation process with the goal of minimizing the system cost based on the goal oriented method.
parameters of the algorithm and operating it are necessary steps. The improved GA and ACO presented in this chapter are described in Sections 4.3.1 and 4.3.2. • Step 2: Selecting the allocation result to meet engineering practice.
4.4.2 Process of reliability optimization allocation method by the goal oriented method The process of the reliability optimization allocation method based on the GO method for nuclear power plant systems is formulated as presented in Fig. 45.
4.5 Problems 1. What are the key elements of the reliability optimization allocation problem? 2. What are the advantages of the reliability optimization allocation method by the GO method?
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Goal Oriented Methodology and Applications in Nuclear Power Plants
3. How many constraint functions are in the reliability optimization allocation model by the GO method? 4. How are the reliability constraint function of unit, reliability constraint function of system function, and constraint function of system constructed, respectively? 5. How is the improved GA operated? 6. How is the improved ACO operated? 7. How is the reliability optimization allocation based on this chapter’s method conducted?
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