- Email: [email protected]
Use of multiple objective evolutionary algorithms in optimizing surveillance requirements
ARTICLE IN PRESS
Reliability Engineering and System Safety 91 (2006) 1027–1038 www.elsevier.com/locate/ress
Use of multiple objective evolutionary algorithms in optimizing surveillance requirements S. Martorella,, S. Carlosa, J.F. Villanuevaa, A.I Sanchezb, B. Galvanc, D. Salazarc, M. Cepind a
Department of Chemical and Nuclear Engineering, Polytechnic University of Valencia, Spain Department of Statistics and Operational Research, Polytechnic University of Valencia, Spain c Institute of Intelligent Systems & Numerical Applications in Engineering (IUSIANI), Evolutionary Computation Division (CEANI), Las Palmas de G. C. University, Spain d Reactor Engineering Division, Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia b
Available online 6 January 2006
Abstract This paper presents the development and application of a double-loop Multiple Objective Evolutionary Algorithm that uses a Multiple Objective Genetic Algorithm to perform the simultaneous optimization of periodic Test Intervals (TI) and Test Planning (TP). It takes into account the time-dependent effect of TP performed on stand-by safety-related equipment. TI and TP are part of the Surveillance Requirements within Technical Specifications at Nuclear Power Plants. It addresses the problem of multi-objective optimization in the space of dependable variables, i.e. TI and TP, using a novel flexible structure of the optimization algorithm. Lessons learnt from the cases of application of the methodology to optimize TI and TP for the High-Pressure Injection System are given. The results show that the double-loop Multiple Objective Evolutionary Algorithm is able to find the Pareto set of solutions that represents a surface of nondominated solutions that satisfy all the constraints imposed on the objective functions and decision variables. Decision makers can adopt then the best solution found depending on their particular preference, e.g. minimum cost, minimum unavailability. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction Surveillance Requirements are a part of Nuclear Power Plant (NPP) Technical Specifications, which establish Test Intervals (TI) and often Test Strategy, which is prerequisite for Test Planning (TP). The primary purpose is to assure that safety-related equipment normally in stand-by will be available when it is needed in an accident. Surveillance tests are required to be periodically (e.g. weekly, monthly or quarterly) performed based on the TI.
Abbreviations: EA, Evolutionary Algorithm; GA, Genetic Algorithm; HPIS, High-Pressure Injection System; MOEA, Multiple Objective Evolutionary Algorithm; MOGA, Multiple Objective Genetic Algorithm; MOP, Multiple objectives Optimization Problem; NPP, Nuclear Power Plant; NSGA-II, Nondominated Sorting Genetic Algorithm–II; RAMS+C, Reliability; Availability; Maintainability; Safety and Cost; SOEA, Single Objective Evolutionary Algorithm; SOP, Single Objective Optimization Problem; SPEA2, Strength Pareto Evolutionary Algorithm2; TI, Test Interval; TP, Test Planning/Strategy Corresponding author. E-mail address: [email protected] (S. Martorell). 0951-8320/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2005.11.038
The same TI applies for redundant equipment in different trains of the same safety system. The relative scheduling of the test, i.e. TP/Strategy affects the test-limited risk. By testing stand-by components, failures can be detected and therefore the plant risk can be kept under control. However, excessive testing, i.e. too short TI, could impact adversely the plant safety and in addition may impose unnecessary costs and burden to the plant staff [1]. Furthermore, inappropriate TP can allow simultaneous inoperability of important equipment that may impose high vulnerability as a consequence of the risky configuration of the plant. TI and TP were first established based only on deterministic analysis. However, there is a consensus now on the idea that both TI and TP in Surveillance Requirements have an important potential for risk and cost reduction [2]. One can find in the literature papers devoted to Surveillance Requirements optimization based on Reliability, Availability, Maintainability, Safety and Cost (RAMS+C) criteria [3]. However, most of prior work
ARTICLE IN PRESS 1028
S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
focuses on the optimization of either TI or TP based on single or multiple criteria [2,4–8]. In Ref. [9], it is presented the development and application of an approach to perform the simultaneous optimization of periodic TI and TP accounting for the time-dependent effect of TP performed on safety-related equipment normally in stand-by at NPP. Risk (e.g. system unavailability) and cost are adopted as decision criteria for the multi-objective optimization problem. The application case of the methodology to optimize Surveillance Requirements (TI+TP) for the High-Pressure Injection System (HPIS) shows how a sort of MOEA, referred there as Multiple Objective Genetic Algorithm (MOGA), is able to find a Pareto set of solutions representing a surface of nondominated solutions that satisfy all the constraints imposed on the objective functions and decision variables related with TI+TP. However, the most important lesson learnt from that example was that the simultaneous optimization of TI and TP faces the problem of optimization in the space of dependable variables. In particular, optimization of TP depends on TI. As a result, it was envisaged the chance of arriving at the best possible results by performing the simultaneous optimization of TI and TP in an iterative way, as it would allow more flexibility in introducing constraints to the range of allowed values for TP that depends on TI. Consequently, a more flexible structure of the optimization GA is required to solve the simultaneous optimization of TI and TP in such an iterative way. This paper presents the development and application of a MOEA algorithm with two nested loops to perform such a simultaneous optimization of periodic TI and TP. The advantage of the novel approach is shown step-by-step through several cases of application for the HPIS as compared with the results in the previous work.
2. Problem formulation 2.1. Optimization criteria and decision variables The Multi-objective Optimization Problem (MOP) of Surveillance Requirements is to be formulated like any other, i.e. establishing decision criteria and variables. Now, TI and parameters associated with TP are adopted as the decision variables for the MOP based on availability and costs criteria, the later as a particular case of the RAMS+C informed decision making [3]. In brief, search for solutions to such MOP requires appropriate modeling for simulating how changes on TI+TP variables affect availability and costs attributes. The idea behind optimization is tuning of such variables to arrive to the best compromise between two normally conflicting criteria, such as availability and costs. The problem becomes more complex as a consequence of the constraints imposed in searching for solutions, e.g. goals and range of allowed values for criteria and variables, respectively.
2.2. Models for time-dependent availability and costs Time-dependent probabilistic models for modeling of equipment reliability, availability (or alternatively unavailability) and costs contributions have to be used for simulating how changes on TI+TP variables affect availability and costs attributes. Therefore, such models have to consider explicitly the effects of TI and TP. Herein, a time-dependent unavailability model similar to the one proposed in [2,10] has been used, which could also incorporate explicitly the effect of maintenance by adapting equipment failure rates following the principles introduced in [11]. In addition, a cost model has been derived following [4]. In brief, the time-dependent unavailability model for each component considers three main possible states where the component can be found along its operational life. First, the component is normally in standby; then, the probability of a component failure while in standby increases with the time elapsed since the last test, as a function of its standby failure rate, and it resets to zero after the test. Each component is associated with a time to the first test (connected to TP) and a test interval (i.e. TI). Second, each test introduces a component unavailability contribution that depends directly on the test duration and inversely on TI. Third, repair is assumed once the component has been found failed after a test, which introduces a component unavailability contribution that depends directly on the mean time to repair and the probability of finding the component in the failed state, the later depending in turn on TI. Next, the time-dependent unavailability model for the system, U-model, is derived using the system fault tree and assessing the unavailability of every component over the time. This model takes into account the relationships that apply for the TP and TI of the several components that constitute the system. The system mean unavailability, Um, and maximum time-dependent unavailability, Utm, can be derived for a given period. In addition, the system cost model, C-model, is derived by summing up the test and repair cost contributions over all the system components. Thus, every component is associated with two main cost contributions: cost of performing tests and cost of repairing the failed component, which depend on TI. The cost contributions are normally derived on a year base to obtain an averaged yearly cost for the system, C, in a given period. These three magnitudes, which are derived using the above models that depend explicitly on the TI and TP, participate in the MOP as described in the following section. 2.3. The Multi-objective Optimization Problem The optimization goal can be formulated to minimize the multi-objective vectors function of the form: y ¼ fðxÞ ¼ ðU m ðxÞ; U tm ðxÞ; CðxÞÞ,
(1)
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
where x is the decision vector (vector of decision variables, i.e. TI and TP), y the objectives vector, X is the decision space and Y is called the objective space, that is to say Y ¼ f(X). In addition, the MOP is subjected to the vector of constraints: gðxÞ ¼ ðU m ðxÞ U m ðxi Þ; CðxÞ Cðxi ÞÞ,
U tm ðxÞ U tm ðxi Þ, ð2Þ
where xAX is the {range of values permitted for the decision variables}, yAY the {range of values permitted for the objective vector}, Um(xi) the mean system unavailability for the initial (default) values of the decision vector xi, Utm(xi) the maximum time-dependent unavailability of the system for the initial xi, C(xi) the yearly cost associated with the initial (default) values for the decision vector, xi. A MOP admits multiple optimal solutions, which are always situated in its Pareto optimal (non-dominated) set [5]. A feasible solution belonging to this set, represented by a vector x*, is a Pareto (non-dominated) solution to the MOP if there exist not any feasible vector such that it scores better than the former in all the objective functions considered; in particular for the availability and costs problem regarding Um(x*), Utm(x*) and C(x*). Each of the non-dominated solutions to the MOP is called a Pareto optimal (non-dominated) solution, and all of the feasible non-dominated solutions of the problem constitute the Pareto optimal set. Their corresponding images in the optimization space, i.e. {y* ¼ f(x*)}, constitute the Pareto front. 3. Multiple objective evolutionary algorithm (MOEA) An important advance and proliferation in the use of heuristic techniques for optimization purposes, such as Evolutionary Algorithms (EA), was noticed recently due to their features and efficiency in solving MOP. Genetic Algorithms (GA) is one of the most popular EA. GA can maintain a population of solutions and at the same time search for non-dominated solutions. These attributes meet the requirement of seeking a Pareto optimal set in solving a MOP. There are several GA-based approaches proposed in the literature to solve MOP. The NSGA-II (Nondominated Sorting Genetic Algorithm-II) [12], and the SPEA2 (Strength Pareto Evolutionary Algorithm-2) [13] are examples of last generation MOEA, which are based on the concept of Pareto optimality discussed above to find Pareto optimal sets. However, GA are essentially unconstrained search techniques, so that, it is necessary to adapt the GA to deal with several types of restrictions [14]. The MOEA proposed in [5] is based on the SPEA2, which has been customized to manage the sort of explicit and implicit constraints involved in the MOP described above. The next subsections introduce the fundamentals of the SPEA2-based MOEA and its extension to manage the
1029
problem of optimization in the space of dependable variables that requires a more flexible MOEA. 3.1. SPEA2-based MOEA The MOGA proposed in [5] can be used to search for solutions to the above MOP, in which mean and maximum values associated with the time-dependent system unavailability and the mean yearly cost will act as objective functions. The approximation of the Pareto optimal set involves two, possibly conflicting, objectives. Thus, the distance to the optimal front is to be minimized and the diversity of the generated solutions is to be maximized generation after generation. The first issue is directly related to the question of how to guide the search towards the Pareto optimal front. The second issue addresses the question of which individuals to keep during the evolution process as limited time and storage resources usually apply. In doing so, the SPEA2-based MOEA combines three techniques in a single algorithm: externally stores the Pareto optimal solutions (External Set), uses the concept of Pareto dominance in order to assign scalar fitness values to individuals (Fitness Assignment) and performs clustering to reduce nondominated solutions stored in the external population (Environmental Selection). Fig. 1 gives a schematic view of the stages that constitute this MOEA. This MOEA uses overlapping populations. It starts with an initial population of a given size (popsize). This initial or base population may be provided by the user (initial set) or generated randomly. Each of these individuals is a possible solution to the optimization problem, which is given by the genetic information encoded in its corresponding genome. TI and the time to first test, the later connected to TP, of every component of the safety system will act as decision variables to be encoded into that MOEA. This algorithm generates an auxiliary population, of size nrepl, constituted by the offspring obtained after recombination and mutation (Variation) of certain individuals selected from the base population (Mating Selection). Newly generated offspring is evaluated and then added to the base population. Each individual of the resulting population, composed by popsize+nrepl individuals, is penalized (if necessary) and then scaled to derive a ranking of individuals based on their fitness score. After scaling, the nrepl worst individuals in the ranking are removed in order to return the population to its original size (popsize). Therefore, after replacement, the best individuals remain in the new population constituting the new generation, generically denoted by g þ 1, which descends from the previous one, g. Next step involves Environmental Selection, in which all the non-dominated individuals in the new population (Population Set) are added to the External Set. Next, all the dominated individuals are removed from the extended External Set. If the size of this set (extsize) exceeds a given maximum size the algorithm performs clustering to reduce non-dominated solution from the
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
1030
Learning Machine
Initial Set Best Set
Nondominated Set
Random Set
Population Set (popsize)
External Set (extsize) Initialization
yes Stop
g=0
popsize
Evolution finished ?
Termination popsize
g
g+1
no
Environmental Selection
Mating Selection
popsize Replacement
parents Recombination and Mutation
nrepl
popsize + nrepl
Evaluation
Penalization
offsprings
Scaling
Fitness assignment
Variation Evolution
Fig. 1. Main stages of the SPEA2-based MOEA [5].
External Population. After that, the algorithm checks if the termination criterion is achieved. If this criterion is not satisfied, then evolution continues to obtain a new generation as described previously, or the algorithm stops otherwise. The Pareto set of optimal solutions found have to satisfy the constraints imposed on the value of the mean and maximum time-dependent unavailability of the system, and mean yearly costs involved. Other constraints under consideration include limiting the range of allowed values of the decision variables. Constraints are managed in that MOEA following traditional strategies. Thus, the penalization approach is adopted to avoid proliferation of solutions which violate goals on the desired criteria [15]. In parallel, those solutions providing values for decision variables which fall out of range are rejected directly. 3.2. MOEA with flexible evolution As discussed in the introduction section, the simultaneous optimization of TI and TP faces the problem of optimization in the space of dependable variables. In particular, optimization of TP depends on TI. As a result,
it seems advisable performing the simultaneous optimization of TI and TP in an iterative way, as it would allow more flexibility in introducing constraints to the range of allowed values for TP that depends on TI. Consequently, a more flexible structure of the optimization GA, as compared with the previous SPEA2-based MOEA, is required to solve the simultaneous optimization of TI and TP in such an iterative way. The fundamentals of Flexible Evolution Agents are presented in Ref. [16]. Based on the idea in this previous work, a general approach to allow flexibility in driving the optimization process using GA may consist of an iterative process that uses a MOEA (e.g. SPEA2-based MOEA) coupled with a learning machine that is responsible for guiding the search (see Fig. 1). Fig. 2 shows an overview of the basic elements involved in such iterative process. Thus, the learning machine would be fed with the departure point (initial files and constraints) to guide the search directly into the MOP using the MOEA or if necessary adding a priori preferences of the decision-maker to formulate and solve a particular SOP using a Single Objective Evolutionary Algorithm (SOEA). The MOEA or SOEA algorithms would stop after evolving a pre-established number of
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
MOEA - Multiple Objective Evolutionary Algorithm
Preferences ?
Yes
SOEA - Single Objective Evolutionary Algorithm
Single-Objective Optimization Problem (SOP)
Multi-Objective Optimization Problem (MOP)
Learning Machine
1031
Table 1 Main algorithm for the double loop MOEA 1. FOR TI: minTI to maxTI–[External loop-stepTI] 1.1. Set maxTP to TI 1.2. Run MOEA (popsize, extsize, generations, genetic parameters) and get the non dominated solutions contained in External Set–[Internal loop] 1.3. Set FinalArchive ¼ FinalArchive+External Set 1.4. Prune dominated solutions from FinalArchive End FOR 2. Present Final Archive as an output.
Local Search (LOS)
Constraints
Pareto front of nondominated solutions TI+TP (Contour)
Files
Fig. 2. An approach to solve MOP based on an iterative process.
generations. The learning machine would be then responsible for adjusting the necessary constraints (or even adjusting GA parameters) to guide the search in the right direction of the search space using either MOEA or SOEA algorithms, or instead any other algorithm to conduct a Local Search. Then, optimization would be performed in a step-by-step process running and stopping the algorithm according to appropriate rules implemented into the learning machine. This iterative process represents a general framework that has been simplified in this paper to cope with the problem of the simultaneous optimization of TI and TP using a MOEA algorithm with two nested loops. Thus, the learning machine simplifies to be an external loop while the MOEA algorithm is responsible of the internal loop. Next section presents in detail this double-loop MOEA. 3.3. A double-loop MOEA The simultaneous optimization of TI and TP herein can be solved in a new and more direct way following the lessons learnt from Ref. [9] That work evidenced that the optimization of TI allows rough tuning of Surveillance Requirements in Technical Specifications while optimization of TP allows fine tuning of Surveillance Requirements following the well-known principle that suggests staggered testing of redundant trains, which depends on TI. Then, the MOEA-based approach must allow rough and fine tuning of Surveillance Requirements in an iterative way. A double-loop MOEA has been developed to allow such procedure that considers multiple objective search and variability of the constraints. The external loop (i.e. learning machine) is responsible for guiding the selection of TI, while the internal loop (i.e. MOEA) is responsible for guiding the search of the optimal TP once a solution for TI is proposed by the external loop. Table 1 shows the
Unavailability
MOEA with flexible evolution
Opt. TP [TImax] Opt. TP [TIj]
Pareto fronts of non-dominated solutions TP for each TI
Opt. TP {TImin}
Cost Fig. 3. Illustration of the expected performance of the double-loop MOEA.
main algorithm and Fig. 3 illustrates the expected performance of the double-loop MOEA. An exhaustive search procedure over TI is used as external loop since the following application case will consider only a single TI. The external loop starts with minTI (e.g. 720 h) and varies up to maxTI (e.g. 8760 h.) with step stepTI (e.g. 24 h) (see Table 1). The value of stepTI controls the exhaustive enumeration of TI. A short stepTI value increases precision in deriving the Pareto front but also increases the computational cost. However, the external loop may stop before reaching maxTI as the largest values of TI could violate the constraints. The external algorithm controls such a circumstance to reduce the computational effort. The MOEA that has been implemented as internal loop is based on the NSGA-II algorithm [12] as it performs quite fast. This MOEA is responsible for guiding the search of the optimal TP for each TI value proposed by the external loop. In doing so, the MOEA considers that the decision variable is TP ranges in the interval [0,TI] and it obtains a Pareto front ‘‘Opt. TP[TIj]’’ (see Fig. 3) and the associated Pareto set that is stored in External Set. This process is repeated for several TIj values proposed by the external loop that yields to derive the FinalArchive with the final Pareto set found.
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
1032
4. Application example
4.2. Decision variables and criteria
4.1. Problem description
It follows from Table 2 and the several relationships shown above that decision-making process may be based on the following vector of decision variables:
A case of application is provided in this section for the HPIS of a NPP. A simplified HPIS of a NPP with Pressurized Water Reactor is shown in Fig. 4, which has been adapted from the literature [17]. This system is normally in stand-by and consists of three pumps and seven valves organized as shown in Fig. 4. Under accidental conditions the HPIS can be used to remove heat from the reactor in those events in which steam generators are unavailable. For example, in case of a Small-Break Loss-Of-Coolant Accident the HPIS safety function draws water from the Refueling Water Storage Tank (RWST) and must discharge it into the cold legs of the Reactor Cooling System through any of the two injection paths. Normally, pumps discharge into the injection paths A and B through valves 3 and 5, although crossover valves 4, 6 and 7 provide alternative flow paths in case of failure of the normal feed. According to typical requirements for pumps and valves within the HPIS Technical Specifications, TI ¼ 2184 h. In addition, the relevant component unavailability and cost data for this case of application are adopted from [4]. Table 2 shows a possible distribution of TI and TP for the components of the HPIS grouped according to the applicable TI {T1, T2, T3} and the time to first test, i.e. TP {TA, TB, TC, TD, TE, TF}. For example, decision variables associated to pump PA correspond to the set {T1, TA}. According to current policy of TP for a typical HPIS at NPP, the following relationships can also apply concerning TI:
x ¼ fT1; k2; k3; TA; TB; TC; kd; ke; kfg.
For sake of clarity in presenting this case of application, the decision vector can be simplified using typical values {k2 ¼ 3, k3 ¼ 3, kd ¼ 0, ke ¼ 1, kf ¼ 2} according to current practice at NPP, which yields to the following decision vector: x ¼ fT1; TA; TB; TCg.
(4)
Eq. (1) in Section 2.3 is adopted to represent the vector of decision criteria. Thus, the mean unavailability of the HPIS, Um(x), the maximum time-dependent unavailability of the HPIS, Utm(x), and the yearly mean cost, C(x), all three for a given x, constitute the decision criteria. 4.3. Initial conditions First, it is worthy to assess the departure point in order to establish the basis for comparing results after the optimization process. Herein, initial conditions are represented by the particularization of the decision vector in Eq. (4) for the initial set of TI and TP as follows: xi ¼ f2184; 24; 48; 72g.
(5)
Next, the values of the elements in the objective vector can be derived using the probabilistic models for Um(x), Utm(x) and C(x) [h/year], yielding to the Table 2 TI and TP groups of the HPIS
T1 ¼ 2184 h; T2 ¼ k2 T1; T3 ¼ k3 T1
TI vs. TP
PA
PB
PC
V1
V2
V3
and TP:
T1 T2 T3
TA
TB
TC
TA
TC
TA
TA ¼ 24; TB ¼ 48; TC ¼ 72; TD ¼ TB+kd T1; TE ¼ TB+ke T1; TF ¼ TB+kf T1.
INJECTION PATH A
V4
V5
V6
V7
TE
TF
TC TD
VALVE 3 (V3) PUMP A (PA)
VALVE 6 (V6)
(3)
VALVE 4 (V4)
VALVE 1 (V1) PUMP B (PB)
VALVE 7 (V7)
FROM RWST
VALVE 2 (V2)
VALVE 5 (V5) PUMP C (PC) INJECTION PATH B Fig. 4. HPIS system.
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
following results: yi ¼ fU m ðxi Þ; U tm ðxi Þ; Cðxi Þg ¼ f5:22688E 4; 5:62661E 2; 19752:3g. Fig. 5 represents the evolution of the time-dependent unavailability of the HPIS for the initial case.
cases 1–3 are aimed at deriving the Pareto set of solutions for extreme situations in order to note the importance and role of both TI and TP in the optimization process. On the other hand, running cases 4 and 5 are aimed at deriving the Pareto set using the direct search (case 4) and the novel iterative process (case 5), respectively. 4.6. Problem resolution
4.4. Objective vector and constraints The vector of decision criteria in Eq. (1) acts as objective vector for the MOP subject to a number of constraints. Eq. (2) represents the vector of constraints on the decision criteria, which are limited herein by the inequalities {Um(x)p2.0E4, Utm(x)p3.0E2, C(x)p5.E+4}. In addition, constraints are also imposed to the range of allowed values for the decision variables. Thus, the range of allowed values for TP [TA, TB, TC] is variable as TP depends on T1 for the general case. Therefore, the particular constraint imposed to the range of allowed values will depend on the particular problem (running case) to be solved, as it is discussed in the following section. 4.5. Description of running cases Table 3 summarizes the details of the five running cases that have been considered in this example of application, i.e. problem formulation, objective function and constraints, decision vector, problem resolution, used technique, encoding mechanism, genetic operators and parameters. The idea behind the running cases selected is to demonstrate in a step-by-step approach the principles underlying the algorithm proposed and the advantages and disadvantages of using the novel method. Thus, running
Fig. 6 shows a 3-D plot with five surfaces that summarize the results of the set of analyses. Fig. 7 shows the same results in a 2-D plot with comments highlighting the more relevant points found, which are summarized also in Table 4. All these results represent a single run for each case. In running case 1, the surface ‘‘Opt_TI {T1} {TA ¼ 24, TB ¼ 48, TC ¼ 72}’’ (triangle) is obtained, which represents the Pareto set of solutions to the MOP where only T1 (i.e. TI) acts as decision variable while TA, TB and TC (i.e. TP) remain constant in their initial values. Then, only TI is in scope of this MOP. The relevant points C1A and C1B represent individual solutions belonging to this Pareto set, which minimize Um(x) and C(x), respectively. In running case 2, the surface ‘‘Opt_TP {TA, TB, TC} [T1 ¼ 744]’’ (diamond) is obtained, which represents the Pareto set of solutions to the MOP where only TA, TB and TC (i.e. TP) act as decision variable while T1 (i.e. TI) remains constant. Then, only TP is in scope of this MOP. Now, search for solutions to the MOP problem departs from point {744, 24, 48, 72}, which corresponds to the T1 value (i.e. TI) of the solution with the lowest mean and maximum system unavailability after solving the MOP that considers only T1 (i.e. TI) as decision variable (i.e. point C1A that is solution for Case 1). The relevant point C2 represents an individual solution belonging to this Pareto set, which minimizes C(x).
HPIS
T1=2184 TA=24 TB=48 TC=72 C(x)=19752.3 Um(x)=0.000522688 Utm=0.0562661
Time-dependent Unavailability [U(t)]
0.01
0.001
0.0001
0.00001
0.000001 0
1033
5,000
10,000
15,000
20,000
25,000
30,000
Chronological time [t (hours)] Fig. 5. HPIS Time-dependent unavailability. Initial case.
35,000
40,000
Fitness scaling Termination External Size
Range and step [min, max, step] T1 TA TB TC Resolution Algorithm External loop GA of reference GA Encoding GA parameters Generations Population size Initialization Mating selection Crossover operator Crossover rate Mutation operator Mutation rate Replacement of population (%) Evaluation Constraint handling T1 ¼ 744 [0, 744, 24] [0, 744, 24] [0, 744, 24] SPEA2-based MOEA None SPEA2 Integer {TA,TB,TC} 3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1 Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
SPEA2-based MOEA None SPEA2 Integer {T1}
3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1
Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
{TA, TB, TC}
Case 2
[720,8760,24] TA ¼ 24 TB ¼ 48 TC ¼ 72
{T1}
Case 1
Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1
SPEA2-based MOEA None SPEA2 Integer {TA,TB,TC}
T1 ¼ 1128 [0, 1128, 24] [0, 1128, 24] [0, 1128, 24]
{TA, TB, TC} {Um(x), Utm(x), C(x)} {Um(x)p2.0E4, Utm(x)p3.0E2, C(x)p5.E+4}
Case 3
Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1
SPEA2-based MOEA None SPEA2 Integer {T1,TA,TB,TC}
[720,8760,24] [0, 720, 24] [0, 720, 24] [0, 720, 24]
{T1, TA, TB, TC}
Case 4
No scaling Generations 20 (for each T1)
Pareto dominance NSGA II Constrained domination
30 (for each T1) 20 Randomly Binary tournament One point crossover 0.9 Uniform and triangular 0.3 100
Double loop MOEA T1 NSGA-II Integer {TA,TB,TC}
[720,8760,24] [0, T1, 24] [0, T1, 24] [0, T1, 24]
{T1, TA, TB, TC}
Case 5
1034
Formulation Decision vector Objective vector Constraint vector
Item
Table 3 Description of running cases
ARTICLE IN PRESS
S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038 Case 1.-Opt_TI {TI} [TA=24, TB=48, TC=72] Case 2.-Opt_TP {TA,TB,TC} [TI=744] Case 3.-Opt_TP {TA,TB,TC} [TI=1128] Case4.-Opt_TI+TP {TI,TA,TB,TC}-fixed range [720] Case 5.-Opt_TI+TP {TI,TA,TB,TC}-2-loop
Unavailability [Mean, Max]+ Cost Plot
2.0E-4 1.9E-4 1.8E-4 1.6E-4 1.5E-4 1.4E-4 1.3E-4
1.2E-4 1.1E-4 1.0E-4
(x)
2.8E-2 2.6E-2 2.4E-2 2.2E-2 2.0E-2 1.8E-2 1.6E-2 1.4E-2 1.2E-2
000
Utm
30,
35,0
00
C (x
)
40,0
00
45,
000
Fig. 6. 3-D plot summarizing problem results.
Running case 3 yields to surface ‘‘Opt_TP {TA, TB, TC} [T1 ¼ 1128]’’ (inverted triangle), which represents the Pareto set of solutions to the MOP where only TA, TB and TC (i.e. TP) act as decision variable while T1 (i.e. TI) remains constant. Then, only TP is in scope of the MOP. Now, search for solutions to the MOP problem departs from point {1128, 24, 48, 72}, which corresponds to the T1 value (i.e. TI) of the solution with the lowest Cost after solving the MOP that considers only T1 (i.e. TI) as decision variable (i.e. point C1B for Case 1). The relevant point C3 represents an individual solution belonging to this Pareto set, which minimizes C(x). Running case 4 analysis yields to surface ‘‘Opt_TI+TP {T1, TA, TB, TC}-fixed range [720]’’ (circle), which has been obtained for the optimization case where the whole set of decision variables {T1, TA, TB, TC} participates in the direct search of solutions to the full scope MOP. Then, both TI and TP participate in the MOP. In Case 4, the set of allowed values for TP (i.e. TA, TB, TC) ranges in [0,720]. The relevant points C4A and C4B represent individual solutions belonging to this Pareto set, which minimize Um(x) and C(x), respectively. Comparing solutions obtained for the Case 1 where only T1 (i.e. TI) plays its role in the MOP (triangle) and the
C4B={1392, 120, 720, 696} Unavailability [Mean, Max]+ Cost Plot 2.0E-4
C1B={1128, 24, 48, 72}
1.9E-4 C5B={1392, 1392, 1320, 696}
Case 1.-Opt_TI {TI} [TA=24, TB=48, TC=72]
Case 2.-Opt_TP {TA,TB,TC} [TI=744] Case 3.-Opt_TP {TA,TB,TC} [TI=1128] Case 4.-Opt TI+TP {TI,TA,TB,TC}-fixed range [720] Case 5.-Opt_TI+TP {TI, TA,TB, TC}-2-loop
1.7E-4 1.6E-4 1.5E-4
Um (x)
Um (x)
1.7E-4
1035
1.4E-4 C1A={744, 24, 48, 72} 1.3E-4
C3={1128,600,648,48}
1.2E-4 1.1E-4 C2={744, 744, 432, 384}
1.0E-4 30,000
35,000 C (x) C4A={744, 360, 432, 720}
Fig. 7. 2-D plot summarizing results of the analyses.
40,000
45,000 C5A={720, 720, 408, 360}
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
1036
Table 4 Relevant solutions for TI and TP under the several cases Case 1 Point Solution Decision variable (x) T1 TA TB TC Decision criteria Um(x) Utm(x) C(x) [h/year] CPU time (h) *
Case 2
Case 3
Case 4
Case 5
C1A Min Um(x)
C1B Min C(x)
C2 Min C(x)
C3 Min C(x)
C4A Min Um(x)
C4B Min C(x)
C5A Min Um(x)
C5B Min C(x)
744 24 48 72
1128 24 48 72
744 744 432 384
1128 600 648 48
744 360 432 720
1392 120 720 696
720 720 408 360
1392 1392 1320 696
1.16E4 1.99E2 44854
1.89E4 2.95E2 31912
1.01E4 1.18E2 44438 0.86
1.51E4 1.67E2 31716 0.86
1.01E4 1.18E2 44451
1.99E4 2.31E2 26947
9.79E5 1.12E2 45709
1.97E4 1.99E2 26671
0.86
0.86
4.99
*Note that one evaluation of the objective vector, which involves the simulation of the time-dependent unavailability of the HPIS for 5 years of operational life, takes approximately 1 s of CPU time.
Case 4 surface (circle), it could be concluded that optimizing TI means a rough tuning of Surveillance Requirements that yields to very good scores of both Um(x) (point C1A) and C(x) (C1B). In addition, fine tuning of Surveillance Requirements is possible adjusting TP, as it is shown for the two Optimal-P sets derived after solving two MOP keeping constant T1, T1 ¼ 744 (Case 2) and T1 ¼ 1128 (Case 3), respectively, while TA, TB and TC (i.e. TP) act as decision variables (diamond and inverted triangle). After this study, one may conclude that the simultaneous optimization of TI+TP (rough and fine tuning) is necessary due to two main reasons:
Fine tuning represents the well-known effect that staggered test (i.e. a more or less uniform distribution of TP along TI) yields to better scores in minimizing the mean and maximum point estimate of the system Unavailability [1]. Thus, it is possible to evolve from the rough surface (triangle) to the base surface (circle) keeping constant T1 (e.g. 744 or 1128) and optimizing the remaining decision variables (diamond and inverted triangle). However, for example, a value of T1 ¼ 1392 is not an allowed solution for the MOP that considers only T1 as decision variable keeping TP (i.e. TA, TB, TC) at their initial values, since it will violate the constraints imposed on both mean and maximum unavailability. Therefore, following the principles of the preceding paragraph, one can expect that fine tuning of parameters TP simultaneously with rough tuning of parameter TI will allow reducing scores for mean and maximum unavailability and, in this way, bringing the opportunity to go beyond 1128 for T1.
Taking into account the lesson learnt from above running Cases 1 to 4, the next running Case 5 is developed that yields to surface ‘‘Opt_TI+TP {T1, TA, TB, TC}-2-
loop’’ (square), which has been obtained for the optimization case where the whole set of decision variables {T1, TA, TB, TC} participates in the iterative search of solutions to the full scope MOP using the double-loop MOEA. Then, both TI and TP participate in the MOP. However now, for Case 5 the set of allowed values for TP (i.e. TA, TB, TC) is variable and range in the interval [0, T1]. The relevant points C5A and C5B represent individual solutions belonging to this Pareto set, which minimize Um(x) and C(x), respectively. One can realize from Fig. 7 (see also Table 4) that a wider Pareto front is found in running Case 5 as compared to Case 4. In addition, the double-loop MOEA algorithm arrives at a Pareto set of solutions (i.e. Case 5) that dominates the Pareto set of solutions found after the direct search (i.e. Case 4), as they score better in all three criteria. This effect is more evident as larger the value for T1. This comparison between Case 4 and 5 demonstrates the advantage of a variable range to allow full staggering of tests within T1, which yields to better results concerning the scores of all three decision criteria. The main disadvantage of the double-loop approach is the computational cost (see Table 4). Thus, CPU time for running Case 5 is more than five times the CPU time for running Case 4. However, the CPU time may be reduced by using a larger stepTI (e.g. 48 h), which on the contrary would yield to a worse shape of the P-front. Thus, a stepTI ¼ 48 h would reduce the CPU time by approximately 2.5 h keeping a Pfront of good quality. The evaluation of the objective vector (1 s of CPU time per evaluation) constitutes the major part of the computational cost of the optimization process. This means the computational time is almost proportional to the number of visited and evaluated points in the solution space. Based on it, the proportion of the number of visited points related to the total number of possible solutions has been estimated to be approximately 0.11% for Case 4 and 0.66% for Case 5. Exhaustive enumeration of the total
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
number of solutions would involve the evaluation of approximately 2.8 millions of feasible solutions. 5. Discussion This paper introduces the problem of the simultaneous optimization of Surveillance Requirements of safetyrelated equipment based on reliability, availability, maintainability, safety and costs criteria using a double-loop MOEA. The viability of the methodology is shown based on the use of time-dependent probabilistic models that allow simulating how changes of TI and TP/strategy affect availability and costs attributes. The models are used to assess objective functions of the MOP being solved in the cases of application using two MOEA-based approaches: (1) the SPEA2-based MOEA that performs the direct search of solution to the full MOP and (2) the double-loop NSGA-II-based MOEA that performs the novel iterative process. The idea behind the five running cases of application is to demonstrate in a step-by-step approach the principles underlying the performance of the double-loop MOEA algorithm proposed and the advantages and disadvantages of using this novel method. Starting from the results obtained for the cases of application aimed at the optimization of TI+TP of the HPIS based on availability and costs criteria, the conclusions are as follows:
Optimization of TI (i.e. T1 for the case of application) allows rough tuning of Surveillance Requirements in Technical Specifications. Optimization of TP (i.e. TA, TB and TC for the case of application) allows fine tuning of Surveillance Requirements in Technical Specifications following the wellknown principle that suggest staggered testing of redundant trains. Fine tuning could follow rough tuning in an iterative way yielding to good results. This idea gives credit to prior work where TI and TP are optimized independently to each other. The example of application shows that the best results are derived when considering the simultaneous optimization of TI and TP as a consequence of the constraints imposed on decision variables and objective functions. The best results are obtained in performing the simultaneous optimization of TI and TP in an iterative way, as it allows more flexibility in introducing constraints to the range of values for TP in the interval [0, T1]. By doing so, it is possible to find a larger value for T1, which, in turn, allows a wider range of allowed values for TA, TB and TC. Then, the Pareto set of solutions found improve that derived in the previous work.
1037
Comparing Case 4 and 5 one can realize the advantage of a variable range (double-loop MOEA) to allow full staggering of tests within T1, which yields to the best results concerning the scores of all three decision criteria. The double-loop NSGA-II-based MOEA (Case 5) is able to find the Pareto set of solutions, which represents a surface of non-dominated solutions that satisfy all the constraints imposed on the objective functions and decision variables. Decision makers can adopt then the best solution found depending on the particular preference, e.g. minimum cost, minimum unavailability, and so on. In spite of having used the NSGA-II, it must be noted that the SPEA2 Genetic Algorithm, belonging also to the last generation of Evolutionary Multi-objective Optimization algorithms, may have been used as the MOEA that supports the iterative process of the double-loop MOEA. The main disadvantage of the double-loop approach is the computational cost. Thus, CPU time for running Case 5 is more than five times the CPU time for running Case 4. However, the CPU time may be reduced by using a larger stepTI, which on the contrary would yield to a worse shape of the Pareto front. Therefore, it is necessary the appropriate balance of CPU time and precision in deriving the Pareto front. It may seem that the case of application of the HPIS is a small problem (with four decision variables) that is easy to solve by exhaustive search or exhaustive enumeration of variables. However, it must be noted that one evaluation of the objective vector using the time-dependent probabilistic models takes approximately 1 s of CPU time. Thus, the CPU time required for the exhaustive search has been estimated to be approximately 150 times the CPU time of running Case 5.
References [1] Samanta P, Kim IS, Mankamo T, Vesely WE. Handbook of methods for risk-based analysis of technical specifications. NUREG/CR-6141, 1995. [2] Cepin M. Optimization of safety equipment outages improves safety. Reliab Eng Syst Saf 2002;77:71–80. [3] Martorell S, Villanueva JF, Carlos S, Nebot Y, Sa´nchez A, Pitarch JL, Serradell V. RAMS+C informed decision-making with application to multi-objective optimization of technical specifications and maintenance using genetic algorithms. Reliab Eng Syst Saf 2005;87/ 1:65–75. [4] Martorell S, et al. Simultaneous and multi-criteria optimization of TS requirements and maintenance at NPP. Ann Nucl Energy 2002;29: 147–68. [5] Martorell S, Sanchez A, Carlos S, Serradell V. Alternatives and challenges in optimizing industrial safety using genetic algorithms. Reliab Eng Syst Saf 2004;86:25–38. [6] Marseguerra M, Zio E. Optimizing maintenance and repair policies via a combination of genetic algorithms and Monte Carlo simulation. Reliab Eng Syst Saf 2000;68(1):69–83. [7] Busacca PG, Marseguerra M, Zio E. Multi-objective optimization by genetic algorithms: application to safety systems. Reliab Eng Syst Saf 2001;72(1):59–74. [8] Lapa C, et al. Surveillance test policy optimization through genetic algorithms using non-periodic intervention frequencies
ARTICLE IN PRESS 1038
[9]
[10]
[11]
[12]
S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038 and considering seasonal constraints. Reliab Eng Syst Saf 2003; 81:103–9. Martorell S, Carlos S, Villanueva JF, Sanchez A, Cepin M. Optimization of surveillance requirements at NPP considering time-dependent scheduling of test and preventive maintenance. In: Proceedings of the European safety and reliability 2005 (ESREL’05) conference. Tri City, Poland; A.A. Balkema Publishers; 2005b. p. 1363–1370 ISBN:0-415-38340-4. Cepin M, Mavko B. Probabilistic safety assessment improves surveillance requirements in technical specifications. Reliab Eng Syst Saf 1997;56:69–77. Martorell S, Sanchez A, Serradell V. Age-dependent reliability model considering effects of maintenance and working conditions. Reliab Eng Syst Saf 1999;64:19. Deb K, Pratap A, Agarwal S, Meyarivan T. A fast and elitist multiobjective genetic algorithm. NSGA-II. IEEE Trans Evol Comput 2002;6(2):182–97.
[13] Zitzler E, Laumanns M, Thiele L. SPEA2: improving the strength pareto evolutionary algorithm for multiobjective optimization. Evolutionary methods for design, optimisation, and control. Barcelona, Spain: CIMNE; 2002. p. 95–100. [14] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading, MA: EEUU: Addison-Wesley; 1989. [15] Martorell S, Carlos S, Sanchez A, Serradell V. Constrained optimization of test intervals using a steady-state genetic algorithm. Reliab Eng Syst Saf 2000;67:215–32. [16] Winter G, Galva´n B, Alonso S, Gonza´lez B. Evolving from genetic algorithms to flexible evolution agents. Late breaking papers of the genetic and evolutionary computation conference, GECCO 2002, In: Erick CP, editor. San Mateo, California; Morgan Kaufmann Publishers; 2002. p. 466-473 ISBN: 1-55860-878-8. [17] Harunuzzaman M, Aldemir T. Optimization of standby safety system maintenance schedules in nuclear power plants. Nucl Technol 1996; 113:354.
Reliability Engineering and System Safety 91 (2006) 1027–1038 www.elsevier.com/locate/ress
Use of multiple objective evolutionary algorithms in optimizing surveillance requirements S. Martorella,, S. Carlosa, J.F. Villanuevaa, A.I Sanchezb, B. Galvanc, D. Salazarc, M. Cepind a
Department of Chemical and Nuclear Engineering, Polytechnic University of Valencia, Spain Department of Statistics and Operational Research, Polytechnic University of Valencia, Spain c Institute of Intelligent Systems & Numerical Applications in Engineering (IUSIANI), Evolutionary Computation Division (CEANI), Las Palmas de G. C. University, Spain d Reactor Engineering Division, Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia b
Available online 6 January 2006
Abstract This paper presents the development and application of a double-loop Multiple Objective Evolutionary Algorithm that uses a Multiple Objective Genetic Algorithm to perform the simultaneous optimization of periodic Test Intervals (TI) and Test Planning (TP). It takes into account the time-dependent effect of TP performed on stand-by safety-related equipment. TI and TP are part of the Surveillance Requirements within Technical Specifications at Nuclear Power Plants. It addresses the problem of multi-objective optimization in the space of dependable variables, i.e. TI and TP, using a novel flexible structure of the optimization algorithm. Lessons learnt from the cases of application of the methodology to optimize TI and TP for the High-Pressure Injection System are given. The results show that the double-loop Multiple Objective Evolutionary Algorithm is able to find the Pareto set of solutions that represents a surface of nondominated solutions that satisfy all the constraints imposed on the objective functions and decision variables. Decision makers can adopt then the best solution found depending on their particular preference, e.g. minimum cost, minimum unavailability. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction Surveillance Requirements are a part of Nuclear Power Plant (NPP) Technical Specifications, which establish Test Intervals (TI) and often Test Strategy, which is prerequisite for Test Planning (TP). The primary purpose is to assure that safety-related equipment normally in stand-by will be available when it is needed in an accident. Surveillance tests are required to be periodically (e.g. weekly, monthly or quarterly) performed based on the TI.
Abbreviations: EA, Evolutionary Algorithm; GA, Genetic Algorithm; HPIS, High-Pressure Injection System; MOEA, Multiple Objective Evolutionary Algorithm; MOGA, Multiple Objective Genetic Algorithm; MOP, Multiple objectives Optimization Problem; NPP, Nuclear Power Plant; NSGA-II, Nondominated Sorting Genetic Algorithm–II; RAMS+C, Reliability; Availability; Maintainability; Safety and Cost; SOEA, Single Objective Evolutionary Algorithm; SOP, Single Objective Optimization Problem; SPEA2, Strength Pareto Evolutionary Algorithm2; TI, Test Interval; TP, Test Planning/Strategy Corresponding author. E-mail address: [email protected] (S. Martorell). 0951-8320/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2005.11.038
The same TI applies for redundant equipment in different trains of the same safety system. The relative scheduling of the test, i.e. TP/Strategy affects the test-limited risk. By testing stand-by components, failures can be detected and therefore the plant risk can be kept under control. However, excessive testing, i.e. too short TI, could impact adversely the plant safety and in addition may impose unnecessary costs and burden to the plant staff [1]. Furthermore, inappropriate TP can allow simultaneous inoperability of important equipment that may impose high vulnerability as a consequence of the risky configuration of the plant. TI and TP were first established based only on deterministic analysis. However, there is a consensus now on the idea that both TI and TP in Surveillance Requirements have an important potential for risk and cost reduction [2]. One can find in the literature papers devoted to Surveillance Requirements optimization based on Reliability, Availability, Maintainability, Safety and Cost (RAMS+C) criteria [3]. However, most of prior work
ARTICLE IN PRESS 1028
S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
focuses on the optimization of either TI or TP based on single or multiple criteria [2,4–8]. In Ref. [9], it is presented the development and application of an approach to perform the simultaneous optimization of periodic TI and TP accounting for the time-dependent effect of TP performed on safety-related equipment normally in stand-by at NPP. Risk (e.g. system unavailability) and cost are adopted as decision criteria for the multi-objective optimization problem. The application case of the methodology to optimize Surveillance Requirements (TI+TP) for the High-Pressure Injection System (HPIS) shows how a sort of MOEA, referred there as Multiple Objective Genetic Algorithm (MOGA), is able to find a Pareto set of solutions representing a surface of nondominated solutions that satisfy all the constraints imposed on the objective functions and decision variables related with TI+TP. However, the most important lesson learnt from that example was that the simultaneous optimization of TI and TP faces the problem of optimization in the space of dependable variables. In particular, optimization of TP depends on TI. As a result, it was envisaged the chance of arriving at the best possible results by performing the simultaneous optimization of TI and TP in an iterative way, as it would allow more flexibility in introducing constraints to the range of allowed values for TP that depends on TI. Consequently, a more flexible structure of the optimization GA is required to solve the simultaneous optimization of TI and TP in such an iterative way. This paper presents the development and application of a MOEA algorithm with two nested loops to perform such a simultaneous optimization of periodic TI and TP. The advantage of the novel approach is shown step-by-step through several cases of application for the HPIS as compared with the results in the previous work.
2. Problem formulation 2.1. Optimization criteria and decision variables The Multi-objective Optimization Problem (MOP) of Surveillance Requirements is to be formulated like any other, i.e. establishing decision criteria and variables. Now, TI and parameters associated with TP are adopted as the decision variables for the MOP based on availability and costs criteria, the later as a particular case of the RAMS+C informed decision making [3]. In brief, search for solutions to such MOP requires appropriate modeling for simulating how changes on TI+TP variables affect availability and costs attributes. The idea behind optimization is tuning of such variables to arrive to the best compromise between two normally conflicting criteria, such as availability and costs. The problem becomes more complex as a consequence of the constraints imposed in searching for solutions, e.g. goals and range of allowed values for criteria and variables, respectively.
2.2. Models for time-dependent availability and costs Time-dependent probabilistic models for modeling of equipment reliability, availability (or alternatively unavailability) and costs contributions have to be used for simulating how changes on TI+TP variables affect availability and costs attributes. Therefore, such models have to consider explicitly the effects of TI and TP. Herein, a time-dependent unavailability model similar to the one proposed in [2,10] has been used, which could also incorporate explicitly the effect of maintenance by adapting equipment failure rates following the principles introduced in [11]. In addition, a cost model has been derived following [4]. In brief, the time-dependent unavailability model for each component considers three main possible states where the component can be found along its operational life. First, the component is normally in standby; then, the probability of a component failure while in standby increases with the time elapsed since the last test, as a function of its standby failure rate, and it resets to zero after the test. Each component is associated with a time to the first test (connected to TP) and a test interval (i.e. TI). Second, each test introduces a component unavailability contribution that depends directly on the test duration and inversely on TI. Third, repair is assumed once the component has been found failed after a test, which introduces a component unavailability contribution that depends directly on the mean time to repair and the probability of finding the component in the failed state, the later depending in turn on TI. Next, the time-dependent unavailability model for the system, U-model, is derived using the system fault tree and assessing the unavailability of every component over the time. This model takes into account the relationships that apply for the TP and TI of the several components that constitute the system. The system mean unavailability, Um, and maximum time-dependent unavailability, Utm, can be derived for a given period. In addition, the system cost model, C-model, is derived by summing up the test and repair cost contributions over all the system components. Thus, every component is associated with two main cost contributions: cost of performing tests and cost of repairing the failed component, which depend on TI. The cost contributions are normally derived on a year base to obtain an averaged yearly cost for the system, C, in a given period. These three magnitudes, which are derived using the above models that depend explicitly on the TI and TP, participate in the MOP as described in the following section. 2.3. The Multi-objective Optimization Problem The optimization goal can be formulated to minimize the multi-objective vectors function of the form: y ¼ fðxÞ ¼ ðU m ðxÞ; U tm ðxÞ; CðxÞÞ,
(1)
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
where x is the decision vector (vector of decision variables, i.e. TI and TP), y the objectives vector, X is the decision space and Y is called the objective space, that is to say Y ¼ f(X). In addition, the MOP is subjected to the vector of constraints: gðxÞ ¼ ðU m ðxÞ U m ðxi Þ; CðxÞ Cðxi ÞÞ,
U tm ðxÞ U tm ðxi Þ, ð2Þ
where xAX is the {range of values permitted for the decision variables}, yAY the {range of values permitted for the objective vector}, Um(xi) the mean system unavailability for the initial (default) values of the decision vector xi, Utm(xi) the maximum time-dependent unavailability of the system for the initial xi, C(xi) the yearly cost associated with the initial (default) values for the decision vector, xi. A MOP admits multiple optimal solutions, which are always situated in its Pareto optimal (non-dominated) set [5]. A feasible solution belonging to this set, represented by a vector x*, is a Pareto (non-dominated) solution to the MOP if there exist not any feasible vector such that it scores better than the former in all the objective functions considered; in particular for the availability and costs problem regarding Um(x*), Utm(x*) and C(x*). Each of the non-dominated solutions to the MOP is called a Pareto optimal (non-dominated) solution, and all of the feasible non-dominated solutions of the problem constitute the Pareto optimal set. Their corresponding images in the optimization space, i.e. {y* ¼ f(x*)}, constitute the Pareto front. 3. Multiple objective evolutionary algorithm (MOEA) An important advance and proliferation in the use of heuristic techniques for optimization purposes, such as Evolutionary Algorithms (EA), was noticed recently due to their features and efficiency in solving MOP. Genetic Algorithms (GA) is one of the most popular EA. GA can maintain a population of solutions and at the same time search for non-dominated solutions. These attributes meet the requirement of seeking a Pareto optimal set in solving a MOP. There are several GA-based approaches proposed in the literature to solve MOP. The NSGA-II (Nondominated Sorting Genetic Algorithm-II) [12], and the SPEA2 (Strength Pareto Evolutionary Algorithm-2) [13] are examples of last generation MOEA, which are based on the concept of Pareto optimality discussed above to find Pareto optimal sets. However, GA are essentially unconstrained search techniques, so that, it is necessary to adapt the GA to deal with several types of restrictions [14]. The MOEA proposed in [5] is based on the SPEA2, which has been customized to manage the sort of explicit and implicit constraints involved in the MOP described above. The next subsections introduce the fundamentals of the SPEA2-based MOEA and its extension to manage the
1029
problem of optimization in the space of dependable variables that requires a more flexible MOEA. 3.1. SPEA2-based MOEA The MOGA proposed in [5] can be used to search for solutions to the above MOP, in which mean and maximum values associated with the time-dependent system unavailability and the mean yearly cost will act as objective functions. The approximation of the Pareto optimal set involves two, possibly conflicting, objectives. Thus, the distance to the optimal front is to be minimized and the diversity of the generated solutions is to be maximized generation after generation. The first issue is directly related to the question of how to guide the search towards the Pareto optimal front. The second issue addresses the question of which individuals to keep during the evolution process as limited time and storage resources usually apply. In doing so, the SPEA2-based MOEA combines three techniques in a single algorithm: externally stores the Pareto optimal solutions (External Set), uses the concept of Pareto dominance in order to assign scalar fitness values to individuals (Fitness Assignment) and performs clustering to reduce nondominated solutions stored in the external population (Environmental Selection). Fig. 1 gives a schematic view of the stages that constitute this MOEA. This MOEA uses overlapping populations. It starts with an initial population of a given size (popsize). This initial or base population may be provided by the user (initial set) or generated randomly. Each of these individuals is a possible solution to the optimization problem, which is given by the genetic information encoded in its corresponding genome. TI and the time to first test, the later connected to TP, of every component of the safety system will act as decision variables to be encoded into that MOEA. This algorithm generates an auxiliary population, of size nrepl, constituted by the offspring obtained after recombination and mutation (Variation) of certain individuals selected from the base population (Mating Selection). Newly generated offspring is evaluated and then added to the base population. Each individual of the resulting population, composed by popsize+nrepl individuals, is penalized (if necessary) and then scaled to derive a ranking of individuals based on their fitness score. After scaling, the nrepl worst individuals in the ranking are removed in order to return the population to its original size (popsize). Therefore, after replacement, the best individuals remain in the new population constituting the new generation, generically denoted by g þ 1, which descends from the previous one, g. Next step involves Environmental Selection, in which all the non-dominated individuals in the new population (Population Set) are added to the External Set. Next, all the dominated individuals are removed from the extended External Set. If the size of this set (extsize) exceeds a given maximum size the algorithm performs clustering to reduce non-dominated solution from the
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
1030
Learning Machine
Initial Set Best Set
Nondominated Set
Random Set
Population Set (popsize)
External Set (extsize) Initialization
yes Stop
g=0
popsize
Evolution finished ?
Termination popsize
g
g+1
no
Environmental Selection
Mating Selection
popsize Replacement
parents Recombination and Mutation
nrepl
popsize + nrepl
Evaluation
Penalization
offsprings
Scaling
Fitness assignment
Variation Evolution
Fig. 1. Main stages of the SPEA2-based MOEA [5].
External Population. After that, the algorithm checks if the termination criterion is achieved. If this criterion is not satisfied, then evolution continues to obtain a new generation as described previously, or the algorithm stops otherwise. The Pareto set of optimal solutions found have to satisfy the constraints imposed on the value of the mean and maximum time-dependent unavailability of the system, and mean yearly costs involved. Other constraints under consideration include limiting the range of allowed values of the decision variables. Constraints are managed in that MOEA following traditional strategies. Thus, the penalization approach is adopted to avoid proliferation of solutions which violate goals on the desired criteria [15]. In parallel, those solutions providing values for decision variables which fall out of range are rejected directly. 3.2. MOEA with flexible evolution As discussed in the introduction section, the simultaneous optimization of TI and TP faces the problem of optimization in the space of dependable variables. In particular, optimization of TP depends on TI. As a result,
it seems advisable performing the simultaneous optimization of TI and TP in an iterative way, as it would allow more flexibility in introducing constraints to the range of allowed values for TP that depends on TI. Consequently, a more flexible structure of the optimization GA, as compared with the previous SPEA2-based MOEA, is required to solve the simultaneous optimization of TI and TP in such an iterative way. The fundamentals of Flexible Evolution Agents are presented in Ref. [16]. Based on the idea in this previous work, a general approach to allow flexibility in driving the optimization process using GA may consist of an iterative process that uses a MOEA (e.g. SPEA2-based MOEA) coupled with a learning machine that is responsible for guiding the search (see Fig. 1). Fig. 2 shows an overview of the basic elements involved in such iterative process. Thus, the learning machine would be fed with the departure point (initial files and constraints) to guide the search directly into the MOP using the MOEA or if necessary adding a priori preferences of the decision-maker to formulate and solve a particular SOP using a Single Objective Evolutionary Algorithm (SOEA). The MOEA or SOEA algorithms would stop after evolving a pre-established number of
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
MOEA - Multiple Objective Evolutionary Algorithm
Preferences ?
Yes
SOEA - Single Objective Evolutionary Algorithm
Single-Objective Optimization Problem (SOP)
Multi-Objective Optimization Problem (MOP)
Learning Machine
1031
Table 1 Main algorithm for the double loop MOEA 1. FOR TI: minTI to maxTI–[External loop-stepTI] 1.1. Set maxTP to TI 1.2. Run MOEA (popsize, extsize, generations, genetic parameters) and get the non dominated solutions contained in External Set–[Internal loop] 1.3. Set FinalArchive ¼ FinalArchive+External Set 1.4. Prune dominated solutions from FinalArchive End FOR 2. Present Final Archive as an output.
Local Search (LOS)
Constraints
Pareto front of nondominated solutions TI+TP (Contour)
Files
Fig. 2. An approach to solve MOP based on an iterative process.
generations. The learning machine would be then responsible for adjusting the necessary constraints (or even adjusting GA parameters) to guide the search in the right direction of the search space using either MOEA or SOEA algorithms, or instead any other algorithm to conduct a Local Search. Then, optimization would be performed in a step-by-step process running and stopping the algorithm according to appropriate rules implemented into the learning machine. This iterative process represents a general framework that has been simplified in this paper to cope with the problem of the simultaneous optimization of TI and TP using a MOEA algorithm with two nested loops. Thus, the learning machine simplifies to be an external loop while the MOEA algorithm is responsible of the internal loop. Next section presents in detail this double-loop MOEA. 3.3. A double-loop MOEA The simultaneous optimization of TI and TP herein can be solved in a new and more direct way following the lessons learnt from Ref. [9] That work evidenced that the optimization of TI allows rough tuning of Surveillance Requirements in Technical Specifications while optimization of TP allows fine tuning of Surveillance Requirements following the well-known principle that suggests staggered testing of redundant trains, which depends on TI. Then, the MOEA-based approach must allow rough and fine tuning of Surveillance Requirements in an iterative way. A double-loop MOEA has been developed to allow such procedure that considers multiple objective search and variability of the constraints. The external loop (i.e. learning machine) is responsible for guiding the selection of TI, while the internal loop (i.e. MOEA) is responsible for guiding the search of the optimal TP once a solution for TI is proposed by the external loop. Table 1 shows the
Unavailability
MOEA with flexible evolution
Opt. TP [TImax] Opt. TP [TIj]
Pareto fronts of non-dominated solutions TP for each TI
Opt. TP {TImin}
Cost Fig. 3. Illustration of the expected performance of the double-loop MOEA.
main algorithm and Fig. 3 illustrates the expected performance of the double-loop MOEA. An exhaustive search procedure over TI is used as external loop since the following application case will consider only a single TI. The external loop starts with minTI (e.g. 720 h) and varies up to maxTI (e.g. 8760 h.) with step stepTI (e.g. 24 h) (see Table 1). The value of stepTI controls the exhaustive enumeration of TI. A short stepTI value increases precision in deriving the Pareto front but also increases the computational cost. However, the external loop may stop before reaching maxTI as the largest values of TI could violate the constraints. The external algorithm controls such a circumstance to reduce the computational effort. The MOEA that has been implemented as internal loop is based on the NSGA-II algorithm [12] as it performs quite fast. This MOEA is responsible for guiding the search of the optimal TP for each TI value proposed by the external loop. In doing so, the MOEA considers that the decision variable is TP ranges in the interval [0,TI] and it obtains a Pareto front ‘‘Opt. TP[TIj]’’ (see Fig. 3) and the associated Pareto set that is stored in External Set. This process is repeated for several TIj values proposed by the external loop that yields to derive the FinalArchive with the final Pareto set found.
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
1032
4. Application example
4.2. Decision variables and criteria
4.1. Problem description
It follows from Table 2 and the several relationships shown above that decision-making process may be based on the following vector of decision variables:
A case of application is provided in this section for the HPIS of a NPP. A simplified HPIS of a NPP with Pressurized Water Reactor is shown in Fig. 4, which has been adapted from the literature [17]. This system is normally in stand-by and consists of three pumps and seven valves organized as shown in Fig. 4. Under accidental conditions the HPIS can be used to remove heat from the reactor in those events in which steam generators are unavailable. For example, in case of a Small-Break Loss-Of-Coolant Accident the HPIS safety function draws water from the Refueling Water Storage Tank (RWST) and must discharge it into the cold legs of the Reactor Cooling System through any of the two injection paths. Normally, pumps discharge into the injection paths A and B through valves 3 and 5, although crossover valves 4, 6 and 7 provide alternative flow paths in case of failure of the normal feed. According to typical requirements for pumps and valves within the HPIS Technical Specifications, TI ¼ 2184 h. In addition, the relevant component unavailability and cost data for this case of application are adopted from [4]. Table 2 shows a possible distribution of TI and TP for the components of the HPIS grouped according to the applicable TI {T1, T2, T3} and the time to first test, i.e. TP {TA, TB, TC, TD, TE, TF}. For example, decision variables associated to pump PA correspond to the set {T1, TA}. According to current policy of TP for a typical HPIS at NPP, the following relationships can also apply concerning TI:
x ¼ fT1; k2; k3; TA; TB; TC; kd; ke; kfg.
For sake of clarity in presenting this case of application, the decision vector can be simplified using typical values {k2 ¼ 3, k3 ¼ 3, kd ¼ 0, ke ¼ 1, kf ¼ 2} according to current practice at NPP, which yields to the following decision vector: x ¼ fT1; TA; TB; TCg.
(4)
Eq. (1) in Section 2.3 is adopted to represent the vector of decision criteria. Thus, the mean unavailability of the HPIS, Um(x), the maximum time-dependent unavailability of the HPIS, Utm(x), and the yearly mean cost, C(x), all three for a given x, constitute the decision criteria. 4.3. Initial conditions First, it is worthy to assess the departure point in order to establish the basis for comparing results after the optimization process. Herein, initial conditions are represented by the particularization of the decision vector in Eq. (4) for the initial set of TI and TP as follows: xi ¼ f2184; 24; 48; 72g.
(5)
Next, the values of the elements in the objective vector can be derived using the probabilistic models for Um(x), Utm(x) and C(x) [h/year], yielding to the Table 2 TI and TP groups of the HPIS
T1 ¼ 2184 h; T2 ¼ k2 T1; T3 ¼ k3 T1
TI vs. TP
PA
PB
PC
V1
V2
V3
and TP:
T1 T2 T3
TA
TB
TC
TA
TC
TA
TA ¼ 24; TB ¼ 48; TC ¼ 72; TD ¼ TB+kd T1; TE ¼ TB+ke T1; TF ¼ TB+kf T1.
INJECTION PATH A
V4
V5
V6
V7
TE
TF
TC TD
VALVE 3 (V3) PUMP A (PA)
VALVE 6 (V6)
(3)
VALVE 4 (V4)
VALVE 1 (V1) PUMP B (PB)
VALVE 7 (V7)
FROM RWST
VALVE 2 (V2)
VALVE 5 (V5) PUMP C (PC) INJECTION PATH B Fig. 4. HPIS system.
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
following results: yi ¼ fU m ðxi Þ; U tm ðxi Þ; Cðxi Þg ¼ f5:22688E 4; 5:62661E 2; 19752:3g. Fig. 5 represents the evolution of the time-dependent unavailability of the HPIS for the initial case.
cases 1–3 are aimed at deriving the Pareto set of solutions for extreme situations in order to note the importance and role of both TI and TP in the optimization process. On the other hand, running cases 4 and 5 are aimed at deriving the Pareto set using the direct search (case 4) and the novel iterative process (case 5), respectively. 4.6. Problem resolution
4.4. Objective vector and constraints The vector of decision criteria in Eq. (1) acts as objective vector for the MOP subject to a number of constraints. Eq. (2) represents the vector of constraints on the decision criteria, which are limited herein by the inequalities {Um(x)p2.0E4, Utm(x)p3.0E2, C(x)p5.E+4}. In addition, constraints are also imposed to the range of allowed values for the decision variables. Thus, the range of allowed values for TP [TA, TB, TC] is variable as TP depends on T1 for the general case. Therefore, the particular constraint imposed to the range of allowed values will depend on the particular problem (running case) to be solved, as it is discussed in the following section. 4.5. Description of running cases Table 3 summarizes the details of the five running cases that have been considered in this example of application, i.e. problem formulation, objective function and constraints, decision vector, problem resolution, used technique, encoding mechanism, genetic operators and parameters. The idea behind the running cases selected is to demonstrate in a step-by-step approach the principles underlying the algorithm proposed and the advantages and disadvantages of using the novel method. Thus, running
Fig. 6 shows a 3-D plot with five surfaces that summarize the results of the set of analyses. Fig. 7 shows the same results in a 2-D plot with comments highlighting the more relevant points found, which are summarized also in Table 4. All these results represent a single run for each case. In running case 1, the surface ‘‘Opt_TI {T1} {TA ¼ 24, TB ¼ 48, TC ¼ 72}’’ (triangle) is obtained, which represents the Pareto set of solutions to the MOP where only T1 (i.e. TI) acts as decision variable while TA, TB and TC (i.e. TP) remain constant in their initial values. Then, only TI is in scope of this MOP. The relevant points C1A and C1B represent individual solutions belonging to this Pareto set, which minimize Um(x) and C(x), respectively. In running case 2, the surface ‘‘Opt_TP {TA, TB, TC} [T1 ¼ 744]’’ (diamond) is obtained, which represents the Pareto set of solutions to the MOP where only TA, TB and TC (i.e. TP) act as decision variable while T1 (i.e. TI) remains constant. Then, only TP is in scope of this MOP. Now, search for solutions to the MOP problem departs from point {744, 24, 48, 72}, which corresponds to the T1 value (i.e. TI) of the solution with the lowest mean and maximum system unavailability after solving the MOP that considers only T1 (i.e. TI) as decision variable (i.e. point C1A that is solution for Case 1). The relevant point C2 represents an individual solution belonging to this Pareto set, which minimizes C(x).
HPIS
T1=2184 TA=24 TB=48 TC=72 C(x)=19752.3 Um(x)=0.000522688 Utm=0.0562661
Time-dependent Unavailability [U(t)]
0.01
0.001
0.0001
0.00001
0.000001 0
1033
5,000
10,000
15,000
20,000
25,000
30,000
Chronological time [t (hours)] Fig. 5. HPIS Time-dependent unavailability. Initial case.
35,000
40,000
Fitness scaling Termination External Size
Range and step [min, max, step] T1 TA TB TC Resolution Algorithm External loop GA of reference GA Encoding GA parameters Generations Population size Initialization Mating selection Crossover operator Crossover rate Mutation operator Mutation rate Replacement of population (%) Evaluation Constraint handling T1 ¼ 744 [0, 744, 24] [0, 744, 24] [0, 744, 24] SPEA2-based MOEA None SPEA2 Integer {TA,TB,TC} 3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1 Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
SPEA2-based MOEA None SPEA2 Integer {T1}
3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1
Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
{TA, TB, TC}
Case 2
[720,8760,24] TA ¼ 24 TB ¼ 48 TC ¼ 72
{T1}
Case 1
Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1
SPEA2-based MOEA None SPEA2 Integer {TA,TB,TC}
T1 ¼ 1128 [0, 1128, 24] [0, 1128, 24] [0, 1128, 24]
{TA, TB, TC} {Um(x), Utm(x), C(x)} {Um(x)p2.0E4, Utm(x)p3.0E2, C(x)p5.E+4}
Case 3
Pareto dominance SPEA2 Dynamic penalization and rejection Linear scaling Generations 100
3000 100 Randomly Tournament selector One point crossover 0.9 Flip mutation 0.3 1
SPEA2-based MOEA None SPEA2 Integer {T1,TA,TB,TC}
[720,8760,24] [0, 720, 24] [0, 720, 24] [0, 720, 24]
{T1, TA, TB, TC}
Case 4
No scaling Generations 20 (for each T1)
Pareto dominance NSGA II Constrained domination
30 (for each T1) 20 Randomly Binary tournament One point crossover 0.9 Uniform and triangular 0.3 100
Double loop MOEA T1 NSGA-II Integer {TA,TB,TC}
[720,8760,24] [0, T1, 24] [0, T1, 24] [0, T1, 24]
{T1, TA, TB, TC}
Case 5
1034
Formulation Decision vector Objective vector Constraint vector
Item
Table 3 Description of running cases
ARTICLE IN PRESS
S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038 Case 1.-Opt_TI {TI} [TA=24, TB=48, TC=72] Case 2.-Opt_TP {TA,TB,TC} [TI=744] Case 3.-Opt_TP {TA,TB,TC} [TI=1128] Case4.-Opt_TI+TP {TI,TA,TB,TC}-fixed range [720] Case 5.-Opt_TI+TP {TI,TA,TB,TC}-2-loop
Unavailability [Mean, Max]+ Cost Plot
2.0E-4 1.9E-4 1.8E-4 1.6E-4 1.5E-4 1.4E-4 1.3E-4
1.2E-4 1.1E-4 1.0E-4
(x)
2.8E-2 2.6E-2 2.4E-2 2.2E-2 2.0E-2 1.8E-2 1.6E-2 1.4E-2 1.2E-2
000
Utm
30,
35,0
00
C (x
)
40,0
00
45,
000
Fig. 6. 3-D plot summarizing problem results.
Running case 3 yields to surface ‘‘Opt_TP {TA, TB, TC} [T1 ¼ 1128]’’ (inverted triangle), which represents the Pareto set of solutions to the MOP where only TA, TB and TC (i.e. TP) act as decision variable while T1 (i.e. TI) remains constant. Then, only TP is in scope of the MOP. Now, search for solutions to the MOP problem departs from point {1128, 24, 48, 72}, which corresponds to the T1 value (i.e. TI) of the solution with the lowest Cost after solving the MOP that considers only T1 (i.e. TI) as decision variable (i.e. point C1B for Case 1). The relevant point C3 represents an individual solution belonging to this Pareto set, which minimizes C(x). Running case 4 analysis yields to surface ‘‘Opt_TI+TP {T1, TA, TB, TC}-fixed range [720]’’ (circle), which has been obtained for the optimization case where the whole set of decision variables {T1, TA, TB, TC} participates in the direct search of solutions to the full scope MOP. Then, both TI and TP participate in the MOP. In Case 4, the set of allowed values for TP (i.e. TA, TB, TC) ranges in [0,720]. The relevant points C4A and C4B represent individual solutions belonging to this Pareto set, which minimize Um(x) and C(x), respectively. Comparing solutions obtained for the Case 1 where only T1 (i.e. TI) plays its role in the MOP (triangle) and the
C4B={1392, 120, 720, 696} Unavailability [Mean, Max]+ Cost Plot 2.0E-4
C1B={1128, 24, 48, 72}
1.9E-4 C5B={1392, 1392, 1320, 696}
Case 1.-Opt_TI {TI} [TA=24, TB=48, TC=72]
Case 2.-Opt_TP {TA,TB,TC} [TI=744] Case 3.-Opt_TP {TA,TB,TC} [TI=1128] Case 4.-Opt TI+TP {TI,TA,TB,TC}-fixed range [720] Case 5.-Opt_TI+TP {TI, TA,TB, TC}-2-loop
1.7E-4 1.6E-4 1.5E-4
Um (x)
Um (x)
1.7E-4
1035
1.4E-4 C1A={744, 24, 48, 72} 1.3E-4
C3={1128,600,648,48}
1.2E-4 1.1E-4 C2={744, 744, 432, 384}
1.0E-4 30,000
35,000 C (x) C4A={744, 360, 432, 720}
Fig. 7. 2-D plot summarizing results of the analyses.
40,000
45,000 C5A={720, 720, 408, 360}
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
1036
Table 4 Relevant solutions for TI and TP under the several cases Case 1 Point Solution Decision variable (x) T1 TA TB TC Decision criteria Um(x) Utm(x) C(x) [h/year] CPU time (h) *
Case 2
Case 3
Case 4
Case 5
C1A Min Um(x)
C1B Min C(x)
C2 Min C(x)
C3 Min C(x)
C4A Min Um(x)
C4B Min C(x)
C5A Min Um(x)
C5B Min C(x)
744 24 48 72
1128 24 48 72
744 744 432 384
1128 600 648 48
744 360 432 720
1392 120 720 696
720 720 408 360
1392 1392 1320 696
1.16E4 1.99E2 44854
1.89E4 2.95E2 31912
1.01E4 1.18E2 44438 0.86
1.51E4 1.67E2 31716 0.86
1.01E4 1.18E2 44451
1.99E4 2.31E2 26947
9.79E5 1.12E2 45709
1.97E4 1.99E2 26671
0.86
0.86
4.99
*Note that one evaluation of the objective vector, which involves the simulation of the time-dependent unavailability of the HPIS for 5 years of operational life, takes approximately 1 s of CPU time.
Case 4 surface (circle), it could be concluded that optimizing TI means a rough tuning of Surveillance Requirements that yields to very good scores of both Um(x) (point C1A) and C(x) (C1B). In addition, fine tuning of Surveillance Requirements is possible adjusting TP, as it is shown for the two Optimal-P sets derived after solving two MOP keeping constant T1, T1 ¼ 744 (Case 2) and T1 ¼ 1128 (Case 3), respectively, while TA, TB and TC (i.e. TP) act as decision variables (diamond and inverted triangle). After this study, one may conclude that the simultaneous optimization of TI+TP (rough and fine tuning) is necessary due to two main reasons:
Fine tuning represents the well-known effect that staggered test (i.e. a more or less uniform distribution of TP along TI) yields to better scores in minimizing the mean and maximum point estimate of the system Unavailability [1]. Thus, it is possible to evolve from the rough surface (triangle) to the base surface (circle) keeping constant T1 (e.g. 744 or 1128) and optimizing the remaining decision variables (diamond and inverted triangle). However, for example, a value of T1 ¼ 1392 is not an allowed solution for the MOP that considers only T1 as decision variable keeping TP (i.e. TA, TB, TC) at their initial values, since it will violate the constraints imposed on both mean and maximum unavailability. Therefore, following the principles of the preceding paragraph, one can expect that fine tuning of parameters TP simultaneously with rough tuning of parameter TI will allow reducing scores for mean and maximum unavailability and, in this way, bringing the opportunity to go beyond 1128 for T1.
Taking into account the lesson learnt from above running Cases 1 to 4, the next running Case 5 is developed that yields to surface ‘‘Opt_TI+TP {T1, TA, TB, TC}-2-
loop’’ (square), which has been obtained for the optimization case where the whole set of decision variables {T1, TA, TB, TC} participates in the iterative search of solutions to the full scope MOP using the double-loop MOEA. Then, both TI and TP participate in the MOP. However now, for Case 5 the set of allowed values for TP (i.e. TA, TB, TC) is variable and range in the interval [0, T1]. The relevant points C5A and C5B represent individual solutions belonging to this Pareto set, which minimize Um(x) and C(x), respectively. One can realize from Fig. 7 (see also Table 4) that a wider Pareto front is found in running Case 5 as compared to Case 4. In addition, the double-loop MOEA algorithm arrives at a Pareto set of solutions (i.e. Case 5) that dominates the Pareto set of solutions found after the direct search (i.e. Case 4), as they score better in all three criteria. This effect is more evident as larger the value for T1. This comparison between Case 4 and 5 demonstrates the advantage of a variable range to allow full staggering of tests within T1, which yields to better results concerning the scores of all three decision criteria. The main disadvantage of the double-loop approach is the computational cost (see Table 4). Thus, CPU time for running Case 5 is more than five times the CPU time for running Case 4. However, the CPU time may be reduced by using a larger stepTI (e.g. 48 h), which on the contrary would yield to a worse shape of the P-front. Thus, a stepTI ¼ 48 h would reduce the CPU time by approximately 2.5 h keeping a Pfront of good quality. The evaluation of the objective vector (1 s of CPU time per evaluation) constitutes the major part of the computational cost of the optimization process. This means the computational time is almost proportional to the number of visited and evaluated points in the solution space. Based on it, the proportion of the number of visited points related to the total number of possible solutions has been estimated to be approximately 0.11% for Case 4 and 0.66% for Case 5. Exhaustive enumeration of the total
ARTICLE IN PRESS S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038
number of solutions would involve the evaluation of approximately 2.8 millions of feasible solutions. 5. Discussion This paper introduces the problem of the simultaneous optimization of Surveillance Requirements of safetyrelated equipment based on reliability, availability, maintainability, safety and costs criteria using a double-loop MOEA. The viability of the methodology is shown based on the use of time-dependent probabilistic models that allow simulating how changes of TI and TP/strategy affect availability and costs attributes. The models are used to assess objective functions of the MOP being solved in the cases of application using two MOEA-based approaches: (1) the SPEA2-based MOEA that performs the direct search of solution to the full MOP and (2) the double-loop NSGA-II-based MOEA that performs the novel iterative process. The idea behind the five running cases of application is to demonstrate in a step-by-step approach the principles underlying the performance of the double-loop MOEA algorithm proposed and the advantages and disadvantages of using this novel method. Starting from the results obtained for the cases of application aimed at the optimization of TI+TP of the HPIS based on availability and costs criteria, the conclusions are as follows:
Optimization of TI (i.e. T1 for the case of application) allows rough tuning of Surveillance Requirements in Technical Specifications. Optimization of TP (i.e. TA, TB and TC for the case of application) allows fine tuning of Surveillance Requirements in Technical Specifications following the wellknown principle that suggest staggered testing of redundant trains. Fine tuning could follow rough tuning in an iterative way yielding to good results. This idea gives credit to prior work where TI and TP are optimized independently to each other. The example of application shows that the best results are derived when considering the simultaneous optimization of TI and TP as a consequence of the constraints imposed on decision variables and objective functions. The best results are obtained in performing the simultaneous optimization of TI and TP in an iterative way, as it allows more flexibility in introducing constraints to the range of values for TP in the interval [0, T1]. By doing so, it is possible to find a larger value for T1, which, in turn, allows a wider range of allowed values for TA, TB and TC. Then, the Pareto set of solutions found improve that derived in the previous work.
1037
Comparing Case 4 and 5 one can realize the advantage of a variable range (double-loop MOEA) to allow full staggering of tests within T1, which yields to the best results concerning the scores of all three decision criteria. The double-loop NSGA-II-based MOEA (Case 5) is able to find the Pareto set of solutions, which represents a surface of non-dominated solutions that satisfy all the constraints imposed on the objective functions and decision variables. Decision makers can adopt then the best solution found depending on the particular preference, e.g. minimum cost, minimum unavailability, and so on. In spite of having used the NSGA-II, it must be noted that the SPEA2 Genetic Algorithm, belonging also to the last generation of Evolutionary Multi-objective Optimization algorithms, may have been used as the MOEA that supports the iterative process of the double-loop MOEA. The main disadvantage of the double-loop approach is the computational cost. Thus, CPU time for running Case 5 is more than five times the CPU time for running Case 4. However, the CPU time may be reduced by using a larger stepTI, which on the contrary would yield to a worse shape of the Pareto front. Therefore, it is necessary the appropriate balance of CPU time and precision in deriving the Pareto front. It may seem that the case of application of the HPIS is a small problem (with four decision variables) that is easy to solve by exhaustive search or exhaustive enumeration of variables. However, it must be noted that one evaluation of the objective vector using the time-dependent probabilistic models takes approximately 1 s of CPU time. Thus, the CPU time required for the exhaustive search has been estimated to be approximately 150 times the CPU time of running Case 5.
References [1] Samanta P, Kim IS, Mankamo T, Vesely WE. Handbook of methods for risk-based analysis of technical specifications. NUREG/CR-6141, 1995. [2] Cepin M. Optimization of safety equipment outages improves safety. Reliab Eng Syst Saf 2002;77:71–80. [3] Martorell S, Villanueva JF, Carlos S, Nebot Y, Sa´nchez A, Pitarch JL, Serradell V. RAMS+C informed decision-making with application to multi-objective optimization of technical specifications and maintenance using genetic algorithms. Reliab Eng Syst Saf 2005;87/ 1:65–75. [4] Martorell S, et al. Simultaneous and multi-criteria optimization of TS requirements and maintenance at NPP. Ann Nucl Energy 2002;29: 147–68. [5] Martorell S, Sanchez A, Carlos S, Serradell V. Alternatives and challenges in optimizing industrial safety using genetic algorithms. Reliab Eng Syst Saf 2004;86:25–38. [6] Marseguerra M, Zio E. Optimizing maintenance and repair policies via a combination of genetic algorithms and Monte Carlo simulation. Reliab Eng Syst Saf 2000;68(1):69–83. [7] Busacca PG, Marseguerra M, Zio E. Multi-objective optimization by genetic algorithms: application to safety systems. Reliab Eng Syst Saf 2001;72(1):59–74. [8] Lapa C, et al. Surveillance test policy optimization through genetic algorithms using non-periodic intervention frequencies
ARTICLE IN PRESS 1038
[9]
[10]
[11]
[12]
S. Martorell et al. / Reliability Engineering and System Safety 91 (2006) 1027–1038 and considering seasonal constraints. Reliab Eng Syst Saf 2003; 81:103–9. Martorell S, Carlos S, Villanueva JF, Sanchez A, Cepin M. Optimization of surveillance requirements at NPP considering time-dependent scheduling of test and preventive maintenance. In: Proceedings of the European safety and reliability 2005 (ESREL’05) conference. Tri City, Poland; A.A. Balkema Publishers; 2005b. p. 1363–1370 ISBN:0-415-38340-4. Cepin M, Mavko B. Probabilistic safety assessment improves surveillance requirements in technical specifications. Reliab Eng Syst Saf 1997;56:69–77. Martorell S, Sanchez A, Serradell V. Age-dependent reliability model considering effects of maintenance and working conditions. Reliab Eng Syst Saf 1999;64:19. Deb K, Pratap A, Agarwal S, Meyarivan T. A fast and elitist multiobjective genetic algorithm. NSGA-II. IEEE Trans Evol Comput 2002;6(2):182–97.
[13] Zitzler E, Laumanns M, Thiele L. SPEA2: improving the strength pareto evolutionary algorithm for multiobjective optimization. Evolutionary methods for design, optimisation, and control. Barcelona, Spain: CIMNE; 2002. p. 95–100. [14] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading, MA: EEUU: Addison-Wesley; 1989. [15] Martorell S, Carlos S, Sanchez A, Serradell V. Constrained optimization of test intervals using a steady-state genetic algorithm. Reliab Eng Syst Saf 2000;67:215–32. [16] Winter G, Galva´n B, Alonso S, Gonza´lez B. Evolving from genetic algorithms to flexible evolution agents. Late breaking papers of the genetic and evolutionary computation conference, GECCO 2002, In: Erick CP, editor. San Mateo, California; Morgan Kaufmann Publishers; 2002. p. 466-473 ISBN: 1-55860-878-8. [17] Harunuzzaman M, Aldemir T. Optimization of standby safety system maintenance schedules in nuclear power plants. Nucl Technol 1996; 113:354.