·Home ·Table of Contents ·Nuclear Industry  Use of Montecarlo Method as a Tool for Risk Evaluation for Application of Various Plugging Criteria on Water Steam Generators
Dr.sc. Berislav Nadinic dipl.ing.(INETEC Manager for Inspection of Steam Generators)
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1. Introduction
During operation of WWER steam generators various degradation processes can appear on steam generator tubes. The status of degradation can be determined by Eddy Current inspection. Eddy current inspections on WWER steam generators are performing usually by standard bobbin coil technique, but the other kind of probes can be used depending of the inspection objectives (See References 1 and 2.). The Eddy Current inspections are performed in equal time intervals and in scopes which are defined by particular country or/and particular utility. Present situation are different from one country/utility to the other and specific strict international guidelines do not exist yet. It means that in some countries use of Eddy Current method is not recognized by their national Regulatory Body as efficient tool to raise safety of WWER NPP operation. On the other hand in some countries certain basic regulations exist regarding the scope of inspection, used EC technique and interval of inspections, and in some countries very detailed regulations exists regarding inspection of WWER steam generators.
The inspection of WWER steam generator tubes is directly connected to the problem of plugging criteria, because when some data about tube degradation exists than some of the tubes have to be isolated from the service because their condition is dangerous from the safety point of view. So, during operational life of every WWER nuclear power plant which use eddy current inspection of their steam generators, the question of plugging criteria is unavoidable and for now it is one of the most painful topic. Presently the lack of commonly accepted methodology for determination of plugging criteria forced every country/utility to define their own criteria on the basis of none, or some research work. In reality the numerous different plugging criteria based on different determination methodology are using on the sites now. It is interesting that all present plugging criteria is based on depth and they are use in range between 20% and 80%.
For decision making personnel which define policy of steam generator maintenance, the problem of establishing or changing plugging criteria (See Reference 3) is primarily the question of safety and after that an economic question.
To help them, in this article, the Monte Carlo method is demonstrated as a very powerful tool for risk estimation using particular plugging criteria on WWER nuclear power plant. Theoretical approach is discussed and the possibility for comparing risks between different plugging criteria. Also, on example of one WWER nuclear power plant, the developed methodology is demonstrated. The obtained results for various number of plugging criteria and for various inspection intervals showed very clearly, effect of plugging criteria on the safety of operation and also suggest the safe inspection policy.
2. Theoretical approach
The idea of using Monte Carlo method as a method for comparison of risk due to tube leakage (emergency shutdowns) using different plugging criteria is based on the following facts:
 Indications due to degradation processes on WWER steam generator tubes can be reliably detected with EC method using one or more EC technique (bobbin, rotating, array, etc.).
 Regarding plugging criteria different parameters can be base for their determination (for example depth of indication in percentage of tube wall loss, length of indication, volume (voltage) of indications). Today practically all countries with WWER NPP use plugging criteria based on depth of indication, but in future the other parameters are expected to be in use too, because more advanced EC techniques will be used more often (rotating probes of various design)
 Frequency distribution of all found indications regarding plugging parameter can be made on the basis of EC results for every inspection.
 Frequency distribution of indication propagation rate regarding plugging parameter between inspections can be made on the basis of EC results collected during two or more EC inspections.
 Every found indication which will be left in service to the next EC inspection can propagate in unknown extent, but in accordance with frequency distribution determined for the previous interval or intervals. It means that deterministic approach for calculation of future status of the indications (till next inspection) can not be used. Instead deterministic, probabilistic approach has to be used as more appropriate.
 Uncertainty of EC measurement of plugging parameter has to be taken into account in all risk evaluation calculations.
 The risk of emergency shutdown is proportional with the number of indications, which during two inspection intervals reached critical value of plugging parameter regarding tube integrity. For example if indication of 92% depth can sustain normal pressure difference ((7090 bars) for WWERs) and 93% can not, than all indications over 92% depth will be counted for risk evaluation.
 The number of leaking indications over emergency shutdown limit can be calculated if experience with emergency shutdowns exists. It means that rate between leaking indications, which caused shutdown, and number of indications described in previous paragraph is known.
Note: Example of calculation of progression rate of indications during last
inspection interval (A are results of present inspection, B are results of
previous inspection) 
Fig 21:

On the basis of the previous facts the global scheme of using Monte Carlo method can be established as given in Figure 21. The data used in Monte Carlo simulation consists of the following populations represented with their distribution functions:
 Population of indication which will be left in service till next inservice inspection
 Population of eddy current measurement tolerance (this population is established on artificial flaws and on the basis of comparison of EC data and destructive examination of indications found on pulled tubes from operating steam generators.)
 Population of progression rates of indications during chosen past period or periods. (For this period the last inspection interval is usually chosen but the options of choosing last two, or more intervals, together, also exists, depending of the nature and behavior of degradation processes.)
Fitting of population frequency distribution with most suitable distribution function can be performed with the aid of different statistic tests as Chi^{2}, Smirnov Kolmogorov, p significant test, etc.
After establishing frequency distributions the Monte Carlo method can be applied using some of commercial statistic computer programs (for example Statistica, Mathematica or similar).
In Monte Carlo method, the computer uses random number simulation techniques to mimic various statistical populations numerous numbers of times. Basically is the same as performing same experiment numerous numbers of times. The number of cases (one case is equivalent to one inspection interval) which will be generate by computer program can vary depending on the speed of available computer and quality of available software. The usual number of generated cases is 1000 till 1000000. In next text this number of cases we will call X.
After generating X cases the number of indications which reach critical parameter per each case has to be counted. After that average value can be determined for all X cases, as well as, minimum and maximum value.
Comparison of average numbers of indications that reach critical parameter obtained by different plugging criteria is proportional with the risk of their application.
If the rate between number of indications with leakage over allowed limit and indications that reach critical parameter is known, based on the experience of emergency shutdowns, the risk of emergency shutdown (number of leaking tubes) during next inspection interval can be calculated.
3. Example
3.1 Used data
To demonstrate described method one WWER 440 unit with active degradation mechanisms on steam generator tubes is taken as an example. The data about taken example is the following:
 The chosen WWER 440 unit is performing ISI in the scope of 100% (all tubes, whole length) every four years. The latest two inspections were performed during 1995 and 1999.
 At the end of last fuel cycle two tubes with two indications had leakage which caused emergency shutdown. The rate of indications with leakage, which caused emergency shutdowns, and number of indications that reach critical parameter was 0.02.
 The plant use plugging criteria based on depth of indication (60%).
 Before NPP startup management wanted to know the following:
 How different plugging criteria can affect on risk of the emergency shutdown in the next 4year interval up to the next eddy current in service inspection.
 How reducing of time interval on one year can affect on the risk of the emergency shutdown using different plugging criteria.
 Plugging criteria taken in account for analysis are 30%, 40%, 50%, 60%, 70% and 80%.
3.2 Results obtained by Monte Carlo method
Results obtained by Monte Carlo method on the basis of 1000 cases (time intervals between inservice inspections) are the following:
Plugging criteria
 Average number of 100% indication per 6 steam generator (1000 intervals per 4 years)
 Maximum number of indications found in one 4 year interval for 6 SG
 Minimum number of indications found in one 4 year interval for 6 SG

30%
 50.595
 72
 28

40%
 168.358
 210
 119

50%
 341.1433
 405
 297

60%
 569.924
 641
 514

70%
 811.158
 881
 745

80%
 1067.236
 1158
 986

After 4 year interval:

Plugging criteria
 Average number of 100% indication per 6 steam generator (1000 intervals per 1 year)
 Maximum number of indications found in one year interval for 6 SG
 Minimum number of indications found in one year interval for 6 SG

30%
 0.00
 0
 0

40%
 0.00
 0
 0

50%
 0.001
 1
 0

60%
 0.397
 17
 0

70%
 19.17
 37
 8

80%
 68.21
 98
 45

After 1 year interval:

Fig 3.1: Number of indications with 100% depth after interval of 4 years obtained by Monte Carlo method (1000 cases or 4000 reactor years) 
Fig 3.2: Number of indications with 100% depth after interval of 1 year obtained by Monte Carlo method (1000 cases or 1000 reactor years) 
Fig 3.3: Number of leaking tubes after 4 and 1 year interval between EC inspections obtained by Monte Carlo method

4. Conclusions
On the basis of the previous the following general conclusions can be drawn:
 Presented method is easy for application because it is based only on eddy current data and eventual operating experience with leaking tubes. It means that every WWER nuclear power plant, which has EC inspection databases, can use it successfully.
 The quality of results is directly proportional with the quality of EC equipment (testers, probes, software, etc.) and the skills of EC data analysts.
 The quality of results is directly proportional with the size of EC databases. It is obvious that the database which contain results of inspections of all tubes in all steam generators will generate very confident results in comparison with the data base which contain only results from the tube sample from every or some steam generators.
 Describe method is not valid for determination of tube with indication behavior during postulated accidents such as LOCA, MSLB etc. It is valid only for evaluation of normal operation.
 Monte Carlo is useful method for evaluation of operational risk for steam generator tube behavior up to next EC inspections using different plugging criteria
 Decision making process about EC inspection interval, value of plugging criteria, as well, as scope of inspection can be significantly improved by using results of Monte Carlo method. All previous can have extremely positive influence on operational safety and economy.
 Monte Carlo method can be used with great success on evaluation of plugging criteria based on different parameters (for example, not only between several depth based plugging criteria, but for example between depth based and length based plugging criteria, or any combination)
 Crucial prerequisite for use of Monte Carlo method is existence of confident and consistent data bases based on EC inspections. The quality of EC data has to be checked very carefully prior to use with Monte Carlo method. The EC procedures, guidelines for EC data analysis and collection, qualification of personnel together with their experience, comparison of EC results and results obtained by destructive methods on pulled tubes and characteristic of used equipment have to be evaluate before use of Monte Carlo method. In some cases (see Ref. 4) the quality of EC data is so poor because of various reasons (analysts training, multiple location errors in collection of data) that it is without any sense to take such EC data bases for any other further use.
Regarding used example the following conclusions can be drawn:
 The most significant contributor to risk of tube leakage is interval between EC inspections. Fouryear interval between inspections leads in high risk of tube leakage.
 For oneyear inspection interval the acceptable plugging criteria values are all up to 60% with practically no risk for appearance of leaking tubes.
 Using shorter inspection interval of 1 year in combination with very high plugging criteria (for example 70%) will be much safer than combination of long inspection interval (4 years) with low plugging criteria (for example 30%). For our example the rate between risk for tube leakage for plugging criteria of 30% associated with 4 year interval and risk of plugging criteria of 70% associated with 1 year interval is 2.63:1.
5. References
 B. Nadinic: "Inspection Strategy For Wwer Steam Generators", Regional Workshop Steam Generator Degradation and Inspection 14  18 June 1999, SaintDenis, France
 B. Nadinic: "Experience In Eddy Current Testing Of Wwer Steam Generators", Regional Workshop Steam Generator Degradation and Inspection 14  18 June 1999, SaintDenis, France
 B. Nadinic: "Tube integrity evaluation on the basis of eddy current examination results", Regional Workshop Steam Generator Degradation and Inspection 14  18 June 1999, SaintDenis, France
 N. Davidenko and others: "On the Integrity of the Elements of VVER Steam Generators of Concern Rosenergoatom", IAEA Regional Workshop on Steam Generator  Inspection and Degradation Saint Denis 14 June18June 1999.