Optimisation of ISI interval using genetic algorithms for risk informed in-service inspection

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1 Optimisation of ISI interval using genetic algorithms for risk informed in-service inspection Gopika Vinod a, *, H.S. Kushwaha a, A.K. Verma b, A. Srividya b a Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai , India b Indian Institute of Technology, Bombay, Mumbai, India Abstract Risk Informed In-Service Inspection (RI-ISI) aims at prioritising the components for inspection within the permissible risk level thereby avoiding unnecessary inspections. Various constraints such as risk level, radiation exposure to the workers and cost of inspections are encountered, while planning the inspection programme. This problem has been attempted to solve using genetic algorithms, which has already established its suitability in optimizing Surveillance and Maintenance activities in Nuclear Power Plants. The paper describes the application of genetic algorithm in optimizing the ISI of feeders, which are large in number and also fall in the same inspection category. Keywords: Risk informed in-service inspection; Optimization; Markov model; Genetic algorithms 1. Introduction Risk Informed In-Service Inspection (RI-ISI) methodology has emerged as one of the powerful activity in the arena of Risk informed decision-making applications. This methodology has profound application in the nuclear as well as non-nuclear industries. In fact, this technique had its initial successful application in oil and gas industry and in chemical plants. For those applications, consequence analysis has been analysed mostly based on cost and environmental releases (for chemical plants handling hazardous substances). This has eventually led to systematic development of API Publication 581: Base Resource Document Risk-based Inspection, 1st edition May 2000, by American Petroleum Institute. This situation can be extrapolated to Nuclear Power Plants as well. Here our major concern will be the risk to the plant, the radiation exposure to workers, while conducting ISI and the cost involved in inspection activities. Extensive ISI is carried out on Power plants components to provide assurance against plant outages and economic losses from component failures and assure the safety of plant personnel. In case of nuclear power plants, periodic inspection of safety systems and their components is already performed at appropriate intervals to provide assurance that they have adequate availability. Periodic inspection represents yet another form of ISI. The purpose of periodic inspection is to provide an assurance that the likelihood of a failure that could endanger health and safety has not increased significantly since the plant was put into service. During test intervals, component functionality is tested, whereas in ISI interval structural integrity is also tested using various Non-Destructive Testing techniques. Since this is an elaborate procedure, it puts a limitation on duration of inspection, resources required, personnel involved, etc. Mostly random sample of components are selected based on deterministic criteria and ISI is performed accordingly. When ISI is performed on a particular component, its failure frequency is modified. This will have an indirect effect on plant risk level too. It is even more essential to ensure that ISI should be focused on such components so as to keep the risk level within the prescribed limit. The studies on RI-ISI can be defined as complete only if the inspection plan is the optimal solution with respect to risk involved and cost associated. In the case of Nuclear Power Plants, while conducting inspection on radioactive components, the attention should be focused on the man-rem exposure during the ISI. Optimization of test and maintenance

2 308 intervals with respect to system availability and cost have been dealt in detail by engineers and industrialists using different optimization techniques. This study attempts to solve the optimization of ISI on radioactive area so that an optimum inspection plan can be proposed. 2. Problem formulation This study originated with an aim to optimize the ISI interval with respect to constraints such as risk, cost and radiation exposure for RI-ISI studies. While employing the Risk matrix suggested by EPRI, it has been found that several components may fall in the same category. If we take different matrix for risk consequence, environmental as well as cost of inspection, it is not necessary to obtain a similar solution. In Pressurized Heavy Water Reactors (PHWRs), there are 306 inlet and 306 outlet feeders. The failure of feeder results in Small Loss of Coolant Accident (SLOCA), which can pose a threat to reactor core. The feeder failures are prevented by conducting ISI. The pipe segments feeders can have different failure rates. These feeders have different costs of inspection, depending on the number of welds present. During testing operations, the technicians are subjected to radiation exposure with limit-dose prescribed by regulatory body. Assuming a constant exposure rate, the minimization of the dose is equivalent to that of the exposure time. Since the feeders are placed in radioactive environment, the radiation exposures to the workers are of concern, and so we impose a restriction on the exposure time allowed to the workers, along with the cost constraint. These feeders are required to be inspected in a gap of five years, with individual inspection planned during the planned shutdown [1]. While applying the RI-ISI methodology in feeders, it has been found that they are all falling in the same risk category. This category specifies that they should be inspected between 6th and 10th service life of plant, which leaves a restriction on the choice of feeders to be inspected every year. Usually ISI activity is taken up during the plant planned shutdown period, which leaves a restriction on the choice of feeders to be inspected every year. In order to find the feeders that should be inspected with respect to constraints, such as risk, exposure time and cost, an optimization problem has been framed. Fig. 1 presents the schematic of feeders present in a typical PHWR, which clearly depicts the complexity of the job involved due to the positioning of feeders. The problem has been formulated to optimize the inspection interval of the feeders, by suggesting the suitable year of its inspection during this time gap based on cost and radiation exposure constraints Estimation of feeder failure frequency Plant operating experience from PHWR has shown that wall thinning has been observed in feeders due to the presence of Erosion Corrosion degradation mechanism. Hence, at any time point of plant life, the feeder can be found to be in one of the three states: Success state, Flaw state or Degraded state. The degraded state can be the state of leak or rupture. However, when an ISI is carried out in flaw state, the pipe element could be brought back to success state as well. This condition allows the application of Markov model for representing the pipe element incorporating the effects of ISI taken up on it. Fig. 2 presents the above described Markov model. Fig. 1. Schematic of feeder layout in PHWR.

3 309 Piping system state S ¼ success state F ¼ flaw state D ¼ degraded (leak or rupture) State transitions Fig. 2. Markov model for single failure state. w ¼ occurrence of flaw v ¼ inspect and repair flaw l 0 ¼ occurrence of degraded state ¼ l L þ l C l L ¼ frequency of leakage l C ¼ frequency of rupture This Markov model would be applied to a pipe element such as a weld or small section of pipe that is uniquely defined in terms of the presence or absence of degradation mechanisms, loading conditions, and status in the inspection program. The model in Fig. 2 has been suggested by Fleming et al. [2], to examine the singular role of the ISI program which can influence the total failure rate of pipe segments but has little if any impact on the conditional probability that a failure will be a rupture. The limitations of this model are: (1) it does not distinguish between leaks and ruptures, cannot model leak before break, and cannot be used to examine the role of leak detection as a means to reduce pipe rupture frequencies and (2) leaks and ruptures are only permitted once the system is in the flaw state. Since all feeders are inspected to be free of detectable flaws at the beginning of commercial operation, the appropriate boundary conditions are: S{t ¼ 0} ¼ 1 D{t ¼ 0} ¼ F{t ¼ 0} ¼ 0 The time dependent solutions for the state probabilities are given by 1 D{t} ¼ 1 2 ðr 1 2 r 2 Þ ðr 1 e r2t 2 r 2 e r1t Þ ð2:1þ where the terms A; r 1 ; and r 2 are defined according to: A ¼ f þ l 0 þ v ð2:2þ r 1 ¼ 2A þ p ffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 2 4fl 0 ð2:3þ 2 r 2 ¼ 2A þ p ffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 2 4fl 0 2 ð2:4þ The transition frequency from success state to flaw state, f; can be estimated either from service experience or limit state formulation. However, in the case study described in Section 4, operating experience value has been adopted. For estimating the transition frequency from flaw state to degraded state, Thomas model [3] could be used. Thomas model estimates the leak frequency and rupture frequency required for computing the degraded state transition frequency, l 0 : The effect of ISI activity is accounted by the reverse transition, v: Eq. (2.5) presents the formula to be used for estimating v v ¼ P f1 P FD =ðt I þ T R Þ where ð2:5þ P f1 ¼ probability that piping element with a flaw will be inspected per inspection interval. The value will be 1 if it is in the inspection program or else it will be 0. P FD ¼ probability that a flaw will be detected given this element is inspected. This is the reliability of inspection program and equivalent to Probability of Detection. T I ¼ mean time between inspections for flaw, its typically 10 years for nuclear power plants. T R ¼ mean time to repair once detected, is in order of days, 200 h. These transition frequencies are applied to Markov model shown in Fig. 2, which computes the time dependent degraded state probabilities. Failure of feeder leads to SLOCA, which is a critical initiating event considered in Probabilistic Safety Assessment. The sum of the steady state failure frequencies of all feeders form the initiating event frequency of SLOCA, which is defined as the objective function of the optimization problem. Design life of 40 years is taken as time under consideration Determining the objective, cost and radiation exposure function The frequency of Small LOCA from feeder pipes is considered as objective function. The objective is to minimize the risk associated with the failure of the feeders, with respect to the year of ISI as T: The leakage frequencies and rupture frequencies were evaluated for all 306 feeders depending on their length, diameter, thickness and weld numbers using Thomas model [3]. The implicit constraints of this problem are: The max cost associated with the maintenance of the pipes, should not exceed $4500, in any year. The max exposure time to radiation, in any year, should not exceed 3000 min. Cost and exposure time has been assumed for each feeder. The problem has been formulated as SLOCA feeders ¼ X306 l feederi i¼1 ð2:6þ

4 310 subject to constraints such as C j ¼ X306 c i y i # 4500 i¼1 ð2:7þ y i ¼ 1 for j ¼ x i ; where x i is the year of allocation of the feederelse, y i ¼ 0 E j ¼ X306 e i y i # 3000 i¼1 ð2:8þ y i ¼ 1 for j ¼ x i ; where x i is the year of allocation of the feederelse, y i ¼ 0 6 # j # 10 where C j and E j are constraints for cost and exposure time, respectively. The explicit constraints for this problem are: The year of maintenance should range between 6 and 10. The year of maintenance should be an integer. Genetic algorithm (GA) has been chosen as an appropriate optimisation method, since it has established as an efficient method in optimising the surveillance test interval for safety systems in Nuclear Power Plants. The GA is a stochastic global search method that mimics the metaphor of natural biological evolution. Owing to its simplicity, flexibility, easy operation, minimal requirements and global perspective, GA has been successfully used in a wide variety of problems in the field of engineering [4 6]. The GA uses the objective (fitness) function itself, not derivative or other auxiliary quantities as preferred in classical optimization methods. The GA does not guarantee global optima. But GA can accommodate any problem, for which classical methods do not work well or take too much computational time. Hence, the above described optimisation problem has been tackled using GA approach. 3. Steady-state GA as optimisation method GA operates on a population of potential solutions applying the principle of survival of the fittest to produce better approximations to a solution. At each generation, a new set of approximations is created by the process of selecting pipe segments according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics. This process leads to the evolution of populations of pipe segments that are better suited to their environment than the pipe segments that they were created from, just as in natural adaptation. GA differs substantially from traditional search and optimization methods. The four significant differences are: GAs search a population of points in parallel, not a single point. GAs do not require derivative information or other auxiliary knowledge; only the objective function and corresponding fitness levels influence the directions of search. GAs use probabilistic transition rules, not deterministic ones. GAs work on an encoding of the parameter set rather than the parameter set itself (except in where real-valued pipe segments are used). The main feature of the SSGA [4] is the utilization of overlapping populations, as it can be observed in Fig. 3. The algorithm starts with an initial population of a given size. The number of pipe segments that constitute this base Fig. 3. Steady-state GA scheme.

5 311 population, is selected by the user. Each of these pipe segments is a possible solution to the optimization problem, which is given by the genetic information encoded in its corresponding genome. This algorithm generates an auxiliary population, constituted by the offspring obtained after the reproduction of certain pipe segments selected from the base population. Newly generated offspring is evaluated and then added to the base population. Each pipe segments of the resulting population, composed by initial and auxiliary population pipe segments, is penalized (if necessary) and then scaled to derive a ranking of pipe segments based on their fitness score. After scaling, the worst pipe segments, equal to the number of auxiliary population, in the ranking are removed in order to return the population to its original size. Therefore, after replacement, the best pipe segments remain in the new population constituting the new generation. The number of pipe segments to be replaced is fixed by the user and determines the amount of overlapping between two consecutive generations. One should realize the new offspring may or may not be present in the new generation, depending on their fitness score. Once the new population has been generated, 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 Encoding Real encoding has been used [4], since there are large numbers of decision variables. Real codification is performed by organizing the parameters to be optimized inside an array of real and independent variables x: It is in the definition of this array where the explicit constraints that apply to each real variable, x; can be established. Explicit constraints involve different aspects concerning, for example, with the type of variable, either discrete or continuous, and the range of valid values for each variable. In GA terminology, each solution identifies an pipe segments of the population of candidate solutions to the optimization problem, which is characterized by the particular genetic information stored in x: GAs work in parallel with a population constituted by a number of different pipe segments. Each array stores a feasible solution to the optimization problem depending on the values adopted by the pipe segment variables that encode the solution parameters Selection After initialization, the algorithm undertakes the evolutionary cycle according to the schematics of the SSGA shown in Fig. 3. Evolution continues in following generations ðg. 0Þ with the selection of some pipe segments from the base population to be parents in the reproduction stage. The roulette wheel method, which is a stochastic sampling method that picks the pipe segments by simulating the roulette wheel, is used in selecting pipe segments for next generation. Each pipe segments is assigned a probability of being selected based on its fitness score relative to the rest of the population. Thus, the better score the more likely an pipe segments will be selected. The probability associated with an pipe segments, i; is given by pðiþ ¼ wðiþ popsize X i¼1 wðiþ i ¼ 1; ; popsize ð3:1þ where wðiþ is the fitness score for that pipe segments. The roulette wheel selector is used to select a group of two, or more, pipe segments from the base population and then picks the one with the best score in this group. This process is repeated until the required number of selected pipe segments is reached Crossover The two-point crossover scheme was used for this problem, since the length of the chromosome is very large (306), and so the one-point crossover would not be very effective Mutation A probability is assigned for selecting any particular gene for mutation. If a particular gene is selected for mutation, then we interchange it by a random number generated between 6 and 10 (6 and 10 being the limits of the ISI interval) Penalisation The dynamic penalization method is used for penalizing those pipe segments that violate implicit constraints. Penalty functions artificially create an unconstrained optimization problem, which cope with infeasible solutions by weighting a penalty based on judgment [4]. A typical penalty functions, based on traditional calculus-based methods, is used, which is formulated as follows pða; b; iþ ¼Kðg; aþsvcðb; iþ ð3:2þ where the parameters a and b are defined by the user and Kðg; aþ is an a-function that establishes the pressure on infeasible pipe segments taking into account the number of generations evolved, g: Finally, SVCðb; iþ is a b-function that represents the sum of violated constraints for a given pipe segments i; which is formulated as follows: SVCðb; iþ ¼ X D b j ðiþ ð3:3þ j The sum in Eq. (3.3) extends to the total number of constraints to which each pipe segments, i; is subjected,

6 312 where for a given constraint, j; the corresponding is defined by the expression 8 9 L 0 if g j ðiþ, gj >< >= D b j ðiþ ¼ g j ðiþ 2 g L ð3:4þ j >: g L ; if g j ðiþ $ g L j >; j which represents the degree of violation associated with restriction j; where g j ðiþ represents the value achieved by the pipe segments i with regard to constraint j; whereas g L j is the maximum value allowed for this constraint j: There is another family of penalty functions, where their origin comes from simulated annealing, that also provides good results. This family is defined as follows mða; b; iþ ¼exp½2Pða; b; iþš ð3:5þ where Pða; b; iþ is derived using Eq. (3.2). This function introduces penalization relative to the worst score encountered in the population after the evaluation stage for the current generation, which also depends on the number of generations evolved, g: Thus, the penalized objective function for a given pipe segments, i; is as follows fðiþ ¼f ðiþ ^ ½1 2 mða; b; iþšf ðwþ ð3:6þ where f ðiþ represents the raw score of the initial objective function for the pipe segments i; and f ðwþ is the corresponding score for the worst pipe segments found in the population at current generation g: The sign of Eq. (3.6) differs whether the objective function is minimized or maximized. In the former case, the penalization is added to the initial objective function score, while it is subtracted otherwise. In both cases, penalization alters the initial objective function score evaluated for each pipe segments making it worse from the optimization point of view as long as it violates any constraint. The dynamic penalization feature is provided in this solution using an appropriate function Kðg; aþ: Fig. 4 shows how a self-adaptive penalization is achieved, by the family of penalty functions mða; b; iþ derived using Eqs. (3.1) (3.5). It is assumed that the exponential term reaches saturation at a given value, named m s ; which is very close to zero. Then, at the first generation, g ¼ 1; an initial tolerance value d 1 is admitted, which is reduced as the algorithm evolves until a certain generation, g ¼ Gs; where a smaller tolerance value d Gs is allowed, which is less indulgent with the pipe segments that violate constraints. In Fig. 5, a family of exponential functions is generated, mða; b; iþ; ranging between the curves with exponents Kð1; aþd b and KðGs; aþd b : Therefore, Kðg; aþ determines the pressure on unfeasible pipe segments depending on the current generation, g; that is being evolved. The pressure on unfeasible pipe segments has to rise as the number of generations evolved increases. Different expressions for Kðg; aþ have been proposed. In this solution, the function selected is given by Kðg; aþ ¼K 1 ½ðg 2 1Þp þ 1Š a ð3:7þ where g corresponds to the current generation number, and K 1 and p are the two parameters to be determined according to the desired values of d 1 and d Gs ; and the number of generations, Gs, for a given m s : Using Eqs. (3.2), (3.5) and (3.7) we can determine the expressions for K 1 and p: Thus, considering only one implicit constraint and using Eqs. (3.2) and (3.5), we have mða; bþ ¼exp½2Kða; bþd b Š ð3:8þ Fig. 4. Penalty function saturation.

7 313 Fig. 5. Evolution of a a-set family of penalty with generation functions g: and taking into account Eq. (3.6) for g ¼ 1ðd ¼ d 1 Þ and g ¼ Gsðd ¼ d Gs Þ we obtain: ln 1 m K 1 ¼ s ð3:9þ p ¼ d b 1 d b=a 1 21! d Gs Gs 2 1 ð3:10þ Suitable values for d 1 ; d Gs ; Gs and ms are 0.01, 0.001, 1000 and 0.01, respectively. In addition, common values for the couple ða; bþ include (0.5, 2), (1, 2) and (2, 2). However, a sensitivity study is required in order to select the appropriate parameters depending on the particular optimization problem Scaling Once the score of each pipe segments of the population has been evaluated using the appropriate objective function and penalized, if necessary, the evolution continues towards the scaling stage. Scaling of scores is performed to avoid two problems that can appear in the evolution of GAs. The first one comprises the appearance, at early generations, of few extra-ordinary pipe segments, named super-pipe segments, among other mediocre ones. This could lead after a few generations, if not avoided, to a situation in which the extraordinary pipe segments take over a significant portion of the population, leading the GA to premature convergence. The second problem may occur late in a run when the average fitness score of the whole population is close to the fitness score of the best members in the population. If this situation is left alone, average members will probably get nearly the same number of copies in future generations and survival of the fittest necessary for improvement becomes a random walk among the mediocre pipe segments. Using an appropriate scaling method it is possible to control the effects of selective pressure, therefore, softening them at the beginning of the run and hardening them at the end of the evolution. The linear scaling function has been chosen here from the different methods available which calculates the scaled fitness score for a given pipe segments i; wðiþ; using the penalized objective function value in Eq. (3.6), fðiþ; as follows wðiþ ¼afðiÞþb ð3:11þ where coefficients a and b are calculated for each generation Replacement After scaling all the pipe segments in the population, the algorithm undergoes replacement as shown in Fig. 3. Several replacement methods are possible depending on the type of GA being used. For a SSGA, it only consists of performing a ranking of the total number of popsize þ nrep pipe segments according to their fitness score given by Eq. (3.11) and, then, removing the nrepl worst fitted in order to restore the population to its initial size (popsize). Therefore, the new population includes the best pipe segments encountered after the algorithm has evolved from the previous generation, g; to the new generation, g þ 1: 3.8. Termination According to the schematics of the SSGA in Fig. 6, the evolutionary cycle is repeated through generations until the termination criterion is satisfied. The termination criterion specifies the conditions that must be satisfied to consider that the best solution has been found, or otherwise, it is

8 314 Fig. 6. Termination criterion scheme. accepted that the GA has evolved for a maximum number of generations. One can find several termination criteria in the literature. The simplest criterion, and also the most frequently used, consists of limiting the number of generations the GA has to evolve, named G; without checking whether a suitable solution has been found or not. It is assumed that after having evolved for G generations the algorithm should have found a good solution. Other termination criteria are based on the search for population convergence. It has to be distinguished between two types of convergence: (1) fitness score convergence towards the same score; and (2) genetic convergence, or diversity, towards the same pipe segments (the same genome x). Both criteria may be necessary to admit that convergence is achieved. The termination criterion implemented in the problem determines whether evolution must finish by checking all three aspects: 1. Maximum number of generations (G). 2. Fitness score convergence (pconv). 3. Genetic convergence, or diversity (popdiv). Firstly, the current generation, g; is compared with the maximum number of generations allowed, G; defined by the user. If the current generation equals G the algorithm stops. Otherwise, the termination criterion proceeds with checking convergence. Convergence is first checked based on the fitness score of population pipe segments in the current generation. In this solution optimization means minimizing either risk or cost functions, or consequently, the fitness score convergence criterion is formulated as follows Pc ¼ 1 2 lwðmþ 2 kwðiþll kwðiþl $ pconv ð3:12þ where wðmþ is the minimum fitness score encountered, kwðiþl is the averaged fitness score of the population in the current generation, and pconv is the convergence level required by the user. If the fitness score convergence criterion is not achieved, i.e. the condition pc $ pconv is not satisfied, the SSGA continues evolution. Otherwise, the algorithm proceeds with checking genetic convergence, or population diversity, analyzing whether the genes of all the population pipe segments in the current generation have achieved convergence. To decide when genetic convergence has been achieved, couples of pipe segments must be compared regarding the genetic information stored in their genes. This process is repeated for all the pipe segments (popsize) in the current generation in order to evaluate the population diversity as follows Div ¼ 2 popsizeðpopsize 2 1Þ popsize X i¼1 X L d ik;jk Xi21 k¼1 # popdiv ð3:13þ where popdiv is the maximum diversity allowed by the user, i k and j k are the k nth genes of the genomes i and j; respectively, L is the length of the genome and, in addition ( d ik;jk ¼ 0; if li k 2 j k l, 1 ð3:14þ 1; otherwise in which 1 establishes the accuracy allowed when comparing genes. If the corresponding genes of two different pipe segments, i.e. genes in the same position, are identical, the value of d ik;jk ; from Eq. (3.14) equals to zero. As shown in Fig. 6, if the genetic convergence criterion is not achieved, i.e. the condition Div, popdiv is not j¼1 L

9 315 Table 1 Representative input parameters for some feeders Feeder name Leak frequency (per year) Rupture frequency (per year) Cost ($) Exposure time (min) A A A B B B B B satisfied, the SSGA continues evolution, or otherwise the algorithm stops. Evaluation of the population diversity is a very important topic, since it is possible to reach a situation with many different pipe segments in the population with similar fitness scores. This population would probably satisfy the fitness score convergence criterion, but with a so high diversity that it would not be possible to assure that the optimum solution has been reached. In fact, it could be expected that if evolution continues one of the pipe segments might lead to a better or optimum value. Controlling population diversity is of particular interest for unimodal functions, since at the end of the optimization process all the pipe segments of the population should equal the global optimum, and therefore, showing a population diversity equal to zero. One advantage of the customized termination criterion is that the maximum number of generations allowed can be extremely high without necessarily supposing an excessive computational cost, since the algorithm will stop if fitness scores and diversity convergence is achieved. However, it has to be taken into account that computing diversity is always a time consuming process, especially if the genome length is high. Even more, diversity convergence may not be achieved for the case of multi-modal functions that present local optima, making possible that different pipe segments, that is different genetic information, provide the same best fitness score (the best solution). Fortunately, by establishing a maximum number of generations allowed, G; one can guarantee that the SSGA will always stop. 4. Results and discussions The leakage and rupture frequencies are computed using Thomas approach. These frequencies are applied to the Markov model. According to the ISI interval assigned to each feeder during each iteration, its steady state failure frequency is computed. These failure frequencies of all 306 feeders are added together to obtain the Small LOCA frequency, which forms the objective or fitness function of the optimisation problem. (i) (ii) Assumptions All the feeders are assumed to be series. All the feeders are assumed to be uncorrelated. So Small LOCA frequency is taken as the sum of the individual feeder failure frequency (iii) The cost and exposure time considered in this case study are only representative figures, which are computed based on the length of the feeder and number of welds present on the feeder. (iv) The mutation and crossover probability for genetic probability assumed based on the best fit values. Table 1 presents some typical values of frequency and constraint values for some feeders used in the problem. The objective function and constraints for the problem has been described in Section 2. With these constraints and the steps involved in GAs, the optimum total feeder frequency was found to be /year, which is obtained by substituting in Eq. (2.6). Feeder failure frequency for each feeder is obtained by applying the optimum ISI interval obtained from the solution in the Markov model. Since ISI activity on feeders are taken up in a span of 5 years, the yearly allocation of number of feeders for inspection, cost incurred and radiation exposure time needed are computed. The overall results are presented in Table 2. Fitness score or genetic convergence and population divergence are computed as per Eqs. (3.12) and (3.13), respectively. Iterations are terminated when genetic convergence and population divergence criteria are satisfied or when the maximum number of generations are reached. The problem is terminated after 5000 generations, which is termed as maximum generations, G: The computer time Table 2 Cost and exposure time factors per year 6th year 7th year 8th year 9th year 10th year Number of feeders Cost($)/year Exposure time (in minutes)

10 Conclusions Fig. 7. Evaluation of convergence during optimisation. taken for obtaining the solution is 600 s on Pentium III processor. Fig. 7 presents the fitness score and genetic convergence while optimizing the risk function. It has been found that all the feeders could be allocated the optimum ISI interval with respect to the cost and exposure time allowed. This analysis would be even more realistic, if exact values of exposure time and cost could be utilized. Mainly cost here is focused on the cost of inspection resources and workman. But these can still be increased if repair activity is also undertaken along with inspection. Exposure time is a very important parameter to be taken into consideration, which is very difficult to estimate. Usually, after shutdown, when the radiation levels falls below a certain acceptable limit in vault area, the inspection activities will be started. This radiation dose for each feeder is dependent on its location in vault area, background radiation present and radiation from feeder if any, during the inspection of welds in the feeder. However, a representative figure has been taken in our analysis, since the scope of the subject is to substantiate the applicability of genetic algorithms in RI-ISI applications. It is important to note that the GA provides a number of potential solutions to a given problem and the choice of final solution is left to the user. In cases where a particular problem does not have one individual solution, for example a family of Pareto-optimal solutions, as is the case in multiobjective optimization and scheduling problems, then the GA is potentially useful for identifying these alternative solutions simultaneously. When nuclear power plant goes for a major shutdown, it calls for optimum feeder inspection since there will be restriction on man-rem exposure as well as on available resources. With GA, the disadvantage of linear search has been avoided largely. Nevertheless, the solution obtained could not be termed as the global solution, but could necessarily be the closest solution to global optima. Since our requirement is a more close solution rather than a precise solution, the application of GA to this type of problem could be justified. References [1] CAN/CSA-N-285.4, Periodic Inspection of CANDU Nuclear Power Plant components. [2] Fleming KN, Gosselin S, Mitman J. Application of Markov models and service data to evaluate the influence of inspection on pipe rupture frequencies. Proceedings of the ASME Pressure Vessels and Piping Conference, Boston; August [3] Lydell BOY. Pipe failure probability the Thomas paper revisited. Reliab Engng Syst Safety 2000;68: [4] Martorell S, Carlos S, Sanchez A, Serradell V. Constrained optimization of test intervals using a steady-state genetic algorithm. Reliab Engng Syst Safety 2000;67: [5] Giuggioli Busacca P, Marseguerra M, Zio E. Multiobjective optimization by genetic algorithms: application to safety systems. Reliab Engng Syst Safety 2001;72: [6] Goldberg DE. Genetic algorithms in search, optimisation, and machine learning. Reading, MA: Addison-Wesley; 1989.