GRS Method for Uncertainty and Sensitivity Evaluation of

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2008, Article ID 798901, 7 pagesdoi:10.1155/2008/798901

Research ArticleGRS Method for Uncertainty and Sensitivity Evaluation ofCode Results and Applications

Horst Glaeser

Gesellschaft fur Anlagen- und Reaktorsicherheit (GRS) mbH, Forschungsinstitute, 85748 Garching, Germany

Correspondence should be addressed to Horst Glaeser, [email protected]

Received 8 May 2007; Accepted 14 February 2008

Recommended by Cesare Frepoli

During the recent years, an increasing interest in computational reactor safety analysis is to replace the conservative evaluationmodel calculations by best estimate calculations supplemented by uncertainty analysis of the code results. The evaluation of themargin to acceptance criteria, for example, the maximum fuel rod clad temperature, should be based on the upper limit of thecalculated uncertainty range. Uncertainty analysis is needed if useful conclusions are to be obtained from “best estimate” thermal-hydraulic code calculations, otherwise single values of unknown accuracy would be presented for comparison with regulatoryacceptance limits. Methods have been developed and presented to quantify the uncertainty of computer code results. The basictechniques proposed by GRS are presented together with applications to a large break loss of coolant accident on a reference reactoras well as on an experiment simulating containment behaviour.

Copyright © 2008 Horst Glaeser. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Best estimate computer codes are used to calculate postulatedloss of coolant accidents and transient in a realistic way andnot in a conservative way. There is an increasing interest incomputational reactor safety analysis to replace the conser-vative evaluation model calculations by best estimate calcu-lations supplemented by a quantitative uncertainty analysis.The USA Code of Federal Regulation (CFR) 10 CFR 50.46[1], for example, allows either to use a best estimate codeplus identification and quantification of uncertainties, or theconservative option using conservative computer code mod-els listed in Appendix K of the CFR, Title 10, Part 50.

Code predictions are uncertain due to several sourcesof uncertainty, like code models as well as uncertainties ofplant and fuel parameters. These uncertainties, for exam-ple, come from scatter of measured values, approximationsof modelling, variation and imprecise knowledge of initialand boundary conditions. Computer code models are devel-oped based on experiments which can simulate the complexbehaviour of a reactor plant under accident conditions in asimplified way only. Most of the experiments are performedin small scale compared to plant size. Uncertainty due toimprecise knowledge of parameter values in calculations is

quantified by ranges and probability distributions. These dis-tributions should be taken into account for input parametersinstead of one discrete value only.

Stochastic variability due to possible component failuresof the reactor plant is not considered in an uncertainty anal-ysis. The single failure criterion is still taken into accountin a deterministic way. This is a superior principle of safetyanalysis and requirements of redundance. The probability ofsystem failures is part of probabilistic safety analyses, not ofdemonstrating the effectiveness of emergency core coolingsystems.

The aim of the uncertainty analysis is at first to iden-tify and quantify all potentially important uncertain param-eters. Their propagation through computer code calculationsprovides probability distributions and ranges for the coderesults. The evaluation of the margin to acceptance crite-ria, for example, the maximum fuel rod clad temperature,should be based on the upper limit of this distribution forthe calculated temperatures, see Figure 1. Uncertainty analy-sis is needed if useful conclusions with regard to predictioncapability, such as maximum cladding temperature, are to beobtained from “best estimate” thermal-hydraulic code calcu-lations, otherwise single values of unknown accuracy wouldbe presented for comparison with limits for acceptance.

mailto:[email protected]

2 Science and Technology of Nuclear Installations

Safety limit

Acceptance criterion (regulatory requirement)

Actual safetymargin

Real value

Margin toacceptance criterion

Upper limit of calculateduncertainty range (tolerance limit)

Calculated uncertainty range

Figure 1: Margin illustration.

Section 2 describes the GRS method, Section 3 presentsexamples of application of the GRS method, and Section 4provides conclusions.

2. DESCRIPTION OF THE GRS METHOD

Among others, GRS method [2] has been developed for thedetermination of uncertainties. The state of knowledge aboutall uncertain parameters is described by ranges and prob-ability distributions, Figure 2. In order to get informationabout the uncertainty of computer code results, a numberof code runs have to be performed. For each of these calcu-lation runs, all identified uncertain parameters are varied si-multaneously. Uncertain parameters are uncertain input val-ues, models, initial and boundary conditions, numerical val-ues like convergence criteria and maximum time step size,and so forth. Model uncertainties are expressed by adding onor multiplying correlations by corrective terms, or by a setof alternative model formulations. Uncertainties in noding,to describe the important phenomena, are to be taken intoaccount in the code validation process. However, alternativenoding schemes can be included in the uncertainty analysis.Code validation results are a fundamental basis to quantifyparameter uncertainties.

The selection of parameter values according to their spec-ified probability distributions, their combination, and theevaluation of the calculation results requires a method. Fol-lowing a proposal by GRS, the central part of the method isa set of statistical techniques. The advantage of using thesetechniques is that the number of code calculations neededis independent of the number of uncertain parameters. Ineach code calculation, all uncertain parameters are varied si-multaneously. In order to quantify the effect of these vari-ations on the result, statistical tools are used. Because thenumber of calculations is independent of the number of un-certain parameters, no a priori ranking of input parametersis necessary to reduce their number in order to cut computa-tion cost. The ranking is a result of the analysis as describedlater.

The number of code calculations depends on the re-quested probability content and confidence level of the sta-tistical tolerance limits used in the uncertainty statements of

Parametervalues

System model

Submodels

Modelresults

· · · · · ·

Temperature

Time

PDF

Parametervalue

distributions

System model

Submodels

Modelresult

distributions

Temperature

Time

· · · · · ·Figure 2: Consideration of input parameter value ranges instead ofdiscrete values in the GRS method.

the results. The required minimum number n of these calcu-lation runs is given by Wilks’ formula [3, 4], for example, forone-sided tolerance limits: 1 − an ≥ b, where b × 100 is theconfidence level (%) that the maximum code result will notbe exceeded with the probability a× 100 (%) (percentile) ofthe corresponding output distribution, which is to be com-pared to the acceptance criterion. The confidence level isspecified to account for the possible influence of the sam-pling error due to the fact that the statements are obtainedfrom a random sample of limited size. For two-sided statisti-cal tolerance intervals, the formula is: 1−an−n(1−a)an−1 ≥b. The minimum number of calculations can be found inTable 1.

The probabilistic treatment of parameter uncertaintiesallows quantifying their state of knowledge. This means, inaddition to the uncertainty range, the knowledge is expressedby probability density functions or probability distributions.This interpretation of probability is used for a parameterwith a fixed but unknown or inaccurately known value. Theclassical interpretation of probability as the limit of a relativefrequency, expressing the uncertainty due to stochastic vari-ability, is not applicable here.

The probability distribution can express that some val-ues in the uncertainty range are more likely to be the appro-priate parameter value than others. In the case that no pref-erences can be justified, uniform distribution will be speci-fied, that is, each value between minimum and maximum isequally likely to be the appropriate parameter value. As theconsequence of this specification of probability distributionsof input parameters, the computer code results also show a

Horst Glaeser 3

Table 1: Minimum number of calculations n for one-sided and two-sided statistical tolerance limits.

One-sided statistical tolerance limits Two-sided statistical tolerance limits

b/a 0.90 0.95 0.99 0.90 0.95 0.99

0.90 22 45 230 38 77 388

0.95 29 59 299 46 93 473

0.99 44 90 459 64 130 662

probability distribution, from which uncertainty limits or in-tervals are derived.

A total number of n code runs are performed varying si-multaneously the values of all uncertain input parameters,according to their distribution. The n values of the consid-ered output parameters are ordered: Y(1) < Y(2) · · · <Y(n−1) < Y(n). Therefore, the name-order statistics is usedfor Wilks’ formula. On the basis of this ranking, the 95th per-centile value with a confidence level of 95% is obtained byselecting Y(n) with n = 59 for the one-sided tolerance limit,for example. A 5th percentile value with a confidence level of95% is obtained by selecting Y(1) with n = 59.A (95%/95%)two-sided tolerance limit is obtained by selecting Y(1) andY(n) with n = 93.

Another important feature of the method is that one canevaluate sensitivity measures of the importance of parameteruncertainties for the uncertainties of the results. These mea-sures give a ranking of input parameters. This informationprovides guidance as to where to improve the state of knowl-edge in order to reduce the output uncertainties most effec-tively, or where to improve the modelling of the computercode. Sensitivity measures like standardised rank regressioncoefficients, rank correlation coefficients, and correlation ra-tios permit a ranking of uncertainties in model formulations,input data, and so forth, with respect to their relative contri-bution to code output uncertainty. The difference to otherknown uncertainty methods, for example, [5], is that theranking is a result of the analysis and not of prior estimatesand judgements. This prior setup of a phenomena identifi-cation and ranking table (PIRT) by extensive expert staff-hours in [5] is known to be very costly. Uncertainty state-ments and sensitivity measures are available simultaneouslyfor all single-valued (e.g., peak clad temperature) as well ascontinuous valued (time dependent) output quantities of in-terest. The method relies only on actual code calculationswithout using approximations like fitted response surfaces.Similar methods based on the GRS method, and an alterna-tive uncertainty method is presented in [6].

The different steps of the uncertainty analysis accordingto the GRS method are supported by the software systemfor uncertainty and sensitivity analyses (SUSA) developed byGRS [7]. They provide a choice of statistical tools to be ap-plied during the uncertainty and sensitivity analysis.

3. APPLICATIONS

The GRS method for uncertainty and sensitivity evaluationof code results can be used for different codes to investigatethe combined influence of all potentially important uncer-

tainties on the calculation results. Several applications havebeen performed in GRS to investigate loss of coolant from theprimary and secondary coolant systems of pressurised waterreactors, as well as related experiments. For these analyses,we used the thermal-hydraulic computer code ATHLET. An-other uncertainty and sensitivity analysis was performed cal-culating an experiment simulating containment behaviourusing the computer code COCOSYS.

3.1. Thermal-hydraulic applications using the ATHLETcomputer code

Several uncertainty and sensitivity analyses were performedby GRS using the thermal-hydraulic computer code ATHLETsimulating breaks of the primary and secondary side coolingsystems of pressurised water reactors. These are

(i) separate effects experiment OMEGA heater rod bun-dle Test 9,

(ii) integral experiment LSTF-CL-18, 5% cold leg break,accumulator injection into cold legs,

(iii) PWR 5% cold leg break, accumulator injection intohot legs (Siemens/ KWU reactor),

(iv) integral experiment LOFT L2-5, 2 × 100% cold legbreak, accumulator injection into cold legs,

(v) PWR 2×100% cold leg break, combined ECC injectioninto cold and hot legs,

(vi) PWR 10% steam line break,(vii) PSB-VVER 11% upper plenum break experiment, UP-

11-08 (OECD PSB-VVER Test1).

One out of these applications is described in the followingsection.

3.2. Application to a German PWR reference reactor,2× 100% cold leg break

A double ended cold leg offset shear break design basis ac-cident of a German PWR of 1300 MW electric power is in-vestigated. The fuel rod peak linear heat generation rate is530 W/cm. Loss of off-site power at turbine trip is assumed.ECC injection is into cold and hot legs. The accumulator sys-tem is specified to initiate coolant injection into the primarysystem below a pressure of 2.6 MPa. High- and low-pressureECC injection is available. A single failure is assumed in thebroken loop check valve for ECC injection from accumulator,high- and low- pressure system, and one hot leg accumulatoris unavailable due to preventive maintenance. These assump-tions are considered to be the worst unavailability, agreed be-tween applicants and assessors.


The uncertainty analysis considered 56 uncertain inputparameters. These consist of 37 model parameters, 4 param-eters to select different model correlations for heat trans-fer and friction, 2 for bypass flow cross sections in the re-actor vessel, 1 for temperature of accumulator water, 1 forcore power, 1 for decay heat, 1 for radial power distribu-tion in the core, 1 for hot channel factor, 5 for gap width (5burn-up classes), 1 for fuel thermal conductivity, and 2 forconvergence criteria. The model parameters comprise criti-cal flow, heat transfer, evaporation, condensation, wall andinterfacial shear, form loss, main coolant pump head, andtorque.

A total number of 100 calculations were performed usingthe code ATHLET Mod 1.2, cycle D [8].

3.3. Maximum clad temperature

Figure 3 shows at any point of time, at least 95% of the com-bined influence of all considered uncertainties on the calcu-lated clad temperatures is below the presented uncertaintylimit (one-sided tolerance limit), at a confidence level of atleast 95%. For each instant of time, the desired tolerancelimits were selected from the 100 calculated code results. A“conservative” calculation result is shown for comparison,applying the best estimate code ATHLET with default val-ues of the models and conservative values for the initial andboundary conditions reactor power, decay heat, gap widthof fuel rods between fuel and clad, fuel pellet thermal con-ductivity, and temperature of accumulator water. All theseconservative values were also included in the distributions ofthe input parameters for the uncertainty analysis. The max-imum clad temperature of the conservative calculation doesnot bound the 95%/95% one-sided tolerance limits of theuncertainty analysis over the whole transient time, for exam-ple, after 75 seconds. The regulatory acceptance criterion forpeak clad temperature is 1200◦C.

The “conservative” calculation is representative for theuse of best estimate computer codes plus conservative ini-tial and boundary conditions. Such an evaluation is possi-ble in the licensing procedure of several countries, but not inthe USA. The uncertainty of code models is not taken intoaccount. The selection of conservative initial and boundaryconditions will bound these model uncertainties. That is ob-viously not the case for the whole transient in the presentexample. An uncertainty analysis quantifies uncertain initialand boundary conditions as well as model uncertainties. Thepeak clad temperatures, however, are bounded due to cumu-lating conservative values of the highly sensitive parametersgap width and pellet thermal conductivity. It is obvious thatthe results are dependent on the extent of conservatism im-plemented in the conservative calculations. Therefore, the USCode of Federal Regulation [1] requires that “uncertaintiesin the analysis method and inputs must be identified and as-sessed so that the uncertainty in the calculated results can beestimated” when a best-estimate computer code is used forthe analysis.

According to the US Code of Federal Regulations, Title10, Section 50.46, the conservative method requires conser-

0

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1000

1200

Max

imu

mcl

adte

mp

erat

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(C)

0 25 50 75 100 125 150

Time (s)

One-sided upper limitReferenceConservative 4

Figure 3: Calculated one-sided 95%/95% uncertainty limit andbest estimate reference calculation compared with a “conservative”calculation of rod clad temperature for a reference reactor during apostulated double ended offset shear cold leg break.

vative models to be applied in conformity with the requiredand acceptable features listed in Appendix K, “ECCS Eval-uation Models” of the Federal Regulations [1]. This is themain reason why, in the USA, an additional margin to licens-ing criteria is available by changing from conservative eval-uation to best estimate calculations plus uncertainty analy-sis.

The confidence level 95% denominates that the 95th per-centile is overestimated conservatively by 95% probabilityproviding a (95%, 95%) statement. This conservatism is thereason why some experts claim that a coverage of a (95%,95%) statement by a conservative calculation is not needed.GRS requires coverage unless other suitable methods forcomparison and quantification of “conservatisms” are pre-sented. This could be achieved by an additional statistical testproving that the conservative calculation bounds the 95thpercentile.

3.3.1. Sensitivity measures

Sensitivity measures indicate the influence of the uncertaintyin input parameters on calculation results. For example, theSpearman rank correlation coefficient is used as sensitivitymeasure. The length of the bars indicates the sensitivity ofthe respective input parameter uncertainty on the first peakclad temperature which occurs during the blowdown phase;see Figure 4. The sensitivity measure gives the variation ofthe result in terms of standard deviations when the input un-certainty varies by one standard deviation (if the input un-certainties are independent). Positive sign means that inputparameter value and result tend to move in the same direc-tion, that is, an increase of uncertain input parameter value

Horst Glaeser 5

−1

−0.8

−0.6

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0.6

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1

Ran

kco

rrel

atio

nco

effici

ent

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57

Index of parameter

Modelfor

criticalheat flux

Minimumfilm boilingtemperature

2-phase

multiplierhorizontal

pipe

Fuel heatconductivity

Reactorpower

Fuel rodgap width

1

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4546

47

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4950

51

52

53

54

55

56

Figure 4: Sensitivity measures of the blowdown PCT with respectto the selected 56 uncertain input parameters (rank correlation co-efficient) for the reference reactor large break.

tends to increase the clad temperature and vice versa. Fornegative sign, the input parameter value and the result tendto move in opposite direction, that is, an increase of the pa-rameter value tends to decrease the clad temperature and viceversa.

The most important parameter uncertainties, out of 56identified potentially important parameters, with respect tothe blowdown peak clad temperature uncertainty are

(i) fuel rod gap width for low burn up (positive sign),(ii) fuel heat conductivity (negative sign),

(iii) minimum film boiling temperature (negative sign),(iv) model for critical heat flux (negative sign: Biasi cor-

relation causes lower clad temperatures due to a laterchange from nucleate to transition boiling comparedto the Hench-Levy correlation),

(v) reactor initial power (positive sign),(vi) 2-phase multiplier in horizontal pipe (negative sign:

higher resistance of water transport to break location⇒higher water content in core due to lower break flow⇒lower clad temperature).

The most important parameters for the peak clad tempera-ture uncertainty during reflood are, according to Figure 5,

(i) fuel heat conductivity (negative sign),(ii) fuel rod gap width for low burn up (positive sign),

(iii) model for 1-phase convection to steam (positive sign,i.e., Mc Eligot correlation tends to cause higher cladtemperatures than Dittus-Boelter II),

(iv) number of droplets (negative sign: number of dropletshigher⇒higher condensation⇒lower PCT),

(v) steam-droplet cooling (negative sign: higher coolingtends to result in lower PCT).

3.4. Application to the experiment HDR T31.5simulating containment behaviour

The experiment T31.5 on the HDR containment facility sim-ulates a large break of a main coolant pipe, investigating

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0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57

Index of parameter

Model forone-phaseconvection

to steam

Fuel rodgap width

Steam-droplet

coolingNumber of

droplets Fuel heatconductivity

1

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2324

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41

42

43

4445

4647

48

49

50

51

52

53

54

55

56

Figure 5: Sensitivity measures of the reflood PCT with respect tothe selected 56 uncertain input parameters (rank correlation coeffi-cient) for the reference reactor large break.

steam and gas release into the containment according to thelow pressure scenario of the German risk study. A short termphase was performed with emphasis on pressure buildup inthe containment and the temperature evolution.The hydro-gen distribution was measured during a long term phase over20 hours, when steam and a helium-hydrogen mixture wereinjected.

A total number of 200 calculations were performed us-ing the code COCOSYS V0.2 [9]. At least 95% of the com-bined influence of all considered uncertainties on the cal-culated pressure at a confidence level of at least 95% atany point of time is shown in Figure 6. A total of 79 un-certain parameters were included, consisting of model pa-rameters, of the experimental facility, initial and boundaryconditions.

Sensitivity measures about the influence of the uncer-tainty in input parameters on the pressure in the upperpart of the HDR containment versus time are presented inFigure 7. We see decreasing and increasing high importanceversus time on the maximum pressure. Decreasing influencewith time is due to decreasing energy transport with decreas-ing convection for

(i) free convection, parameter 72, negative sign,(ii) forced convection, parameter 73, negative sign,

(iii) condensation at wall, parameter 74, negative sign.

Increasing with time are the following parameters because ofdecreasing convection:

(i) thickness of liner, parameter 79, negative sign,(ii) surface of liner, parameter 77, negative sign,

(iii) heat capacity of concrete structures, parameter 69,negative sign.

4. CONCLUSIONS

Two applications of the uncertainty method proposed byGRS are presented. A significant advantage of this methodol-ogy is that no a priori reduction in the number of uncertain


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ssu

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Time (s)

Experiment CP409Reference calculationLower limitUpper limit

Figure 6: 95%/95% uncertainty interval, reference calculation andexperimental values for pressure in the upper part of the contain-ment versus time.

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3 1 3 2 3 2

3 3 3 3 3 4 3 4 3 5 3 5

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5 7

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5 9 5 9 6 0 6 0 6 1 6 1 6 2 6 2 6 3 6 3 6 4 6 4 6 5 6 5 6 6 6 6 6 7

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7 0 7 0 7 1 7 1

7 2 7 2

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7 3

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7 5 7 5 7 6

7 6

7 7 7 7 7 8 7 8 7 9

7 9

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5 5 5 5

5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6

6 6 6 6 6 6 6

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7 7 7 7 7 7 7 7 7 7 7 7 7

7 7 7 7 7 7 7 7 7 7 7 8 8 8 8

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1 0 1 0 10 1 0

1 0 1 0 1 0 1 0 1 0 10 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0

1 1 1 1

11 1 1

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1 1 11 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1

1 2 1 2 12

1 2 1 2 1 2

1 2 1 2 1 2 12 1 2 12 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 12 1 2 1 2

1 3 1 3 13 1 3 1 3 1 3 1 3 1 3 1 3 13 1 3 13 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 13 1 3 1 3 1 4 1 4 14 1 4 1 4 1 4 1 4 1 4 1 4 14 1 4 14 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 14 1 4 1 4 1 5 1 5 15 1 5 1 5 1 5 1 5 1 5

1 5 15 1 5 15 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 15 1 5 1 5

1 6 1 6 16 1 6

1 6 1 6 1 6 1 6 1 6 16 1 6 16 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 16 1 6 1 6

1 7 1 7 17

1 7 1 7 1 7 1 7 1 7 1 7 17 1 7 17 1 7

1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 17 1 7 1 7 1 8 1 8 18 1 8 1 8 1 8 1 8 1 8 1 8 18 1 8 18 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 18 1 8 1 8 1 9 1 9 19 1 9 1 9 1 9 1 9 1 9

1 9 19

1 9 19 1 9

1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 19 1 9 1 9 2 0 2 0 20 2 0 2 0 2 0 2 0 2 0 2 0 20 2 0 20 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 20 2 0 2 0 2 1 2 1

21 2 1 2 1 2 1 2 1 2 1 2 1 21 2 1 21 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1

2 2 2 2

22 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 22

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2

2 3 2 3 23 2 3 2 3 2 3

2 3 2 3 2 3 23

2 3 23 2 3

2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 23

2 3 2 3

2 4 2 4 24 2 4 2 4 2 4 2 4 2 4 2 4 24 2 4 24 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 24 2 4 2 4

2 5 2 5 25

2 5 2 5

2 5 2 5 2 5

2 5 25

2 5 25 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 25 2 5 2 5 2 6 2 6 26 2 6 2 6 2 6 2 6 2 6 2 6 26 2 6 26 2 6 2 6 2 6 2 6 2 6 2 6 2 6 2 6 2 6 26 2 6 2 6 2 7 2 7 27 2 7 2 7 2 7 2 7 2 7 2 7 27 2 7 27 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 27 2 7 2 7 2 8 2 8 28

2 8 2 8 2 8 2 8 2 8 2 8 28 2 8 28 2 8 2 8 2 8 2 8 2 8 2 8 2 8 2 8 2 8 28 2 8 2 8

2 9 2 9

29 2 9 2 9 2 9 2 9 2 9 2 9 29 2 9 29

2 9 2 9 2 9 2 9 2 9 2 9 2 9 2 9 2 9 29 2 9 2 9

3 0 3 0 30

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3 0 3 0 3 0 3 0 3 0 3 0 30 3 0 3 0 3 1 3 1 31 3 1 3 1 3 1 3 1 3 1 3 1

31 3 1 31 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 31 3 1 3 1 3 2

3 2 32

3 2 3 2 3 2 3 2 3 2 3 2 32 3 2 32 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 32 3 2 3 2

3 3 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 3 3 3 4 3 4

34 3 4 3 4 3 4

3 4 3 4 3 4 34 3 4 34 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 34 3 4 3 4 3 5 3 5 35 3 5 3 5 3 5 3 5 3 5

3 5 35 3 5 35 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 35 3 5 3 5

3 6 3 6 36 3 6 3 6 3 6 3 6 3 6 3 6 36 3 6 36 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 36 3 6 3 6 3 7 3 7 37 3 7

3 7 3 7 3 7 3 7 3 7 37 3 7 37 3 7 3 7 3 7 3 7 3 7 3 7 3 7 3 7 3 7 37 3 7 3 7

3 8 3 8 38 3 8 3 8 3 8 3 8 3 8 3 8 38 3 8 38 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 38 3 8 3 8 3 9 3 9 39 3 9 3 9 3 9 3 9 3 9 3 9 39 3 9 39 3 9 3 9 3 9 3 9 3 9 3 9 3 9 3 9 3 9 39 3 9 3 9 4 0 4 0 40 4 0 4 0 4 0 4 0 4 0 4 0 40 4 0 40 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 40 4 0 4 0

4 1 4 1 41 4 1 4 1 4 1 4 1 4 1 4 1 41 4 1 41 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 41 4 1 4 1 4 2 4 2 42

4 2 4 2 4 2

4 2 4 2 4 2

42 4 2 42

4 2 4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2 42 4 2 4 2

4 3 4 3 43

4 3 4 3 4 3 4 3 4 3 4 3 43 4 3 43 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 43 4 3 4 3

4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4

4 5 4 5 45 4 5 4 5 4 5 4 5 4 5 4 5 45 4 5 45 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 45 4 5 4 5

4 6 4 6

46 4 6 4 6 4 6

4 6 4 6 4 6 46 4 6 46 4 6

4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 46 4 6 4 6 4 7 4 7

47 4 7

4 7 4 7 4 7 4 7

4 7 47

4 7 47 4 7

4 7 4 7 4 7 4 7 4 7 4 7 4 7 4 7 47 4 7 4 7 4 8 4 8 48 4 8 4 8 4 8 4 8 4 8 4 8 48 4 8 48 4 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 48 4 8 4 8

4 9 4 9 49 4 9 4 9 4 9 4 9 4 9 4 9 49 4 9 49 4 9 4 9 4 9 4 9 4 9 4 9 4 9 4 9 4 9 49 4 9 4 9 5 0 5 0 50 5 0 5 0 5 0 5 0 5 0 5 0 50 5 0 50 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 50 5 0 5 0 5 1 5 1

51 5 1 5 1 5 1 5 1 5 1 5 1 51 5 1 51 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 51 5 1 5 1 5 2 5 2 52 5 2 5 2 5 2 5 2 5 2 5 2 52 5 2 52 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 52 5 2 5 2

5 3 5 3 53 5 3 5 3 5 3 5 3 5 3 5 3 53 5 3 53 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 53 5 3 5 3

5 4 5 4 54 5 4 5 4 5 4 5 4 5 4 5 4 54 5 4 54 5 4

5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 54 5 4 5 4

5 5 5 5 55 5 5 5 5 5 5 5 5 5 5 5 5 55 5 5 55 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 55 5 5 5 5

5 6 5 6 56 5 6

5 6 5 6 5 6 5 6 5 6 56 5 6 56 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 56 5 6 5 6

5 7 5 7 57

5 7 5 7 5 7 5 7 5 7 5 7 57 5 7 57 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 57 5 7 5 7 5 8 5 8 58 5 8 5 8 5 8 5 8 5 8 5 8 58 5 8 58 5 8

5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 8 58 5 8 5 8

5 9 5 9 59 5 9 5 9 5 9 5 9 5 9 5 9 59 5 9 59 5 9 5 9 5 9 5 9 5 9 5 9 5 9 5 9 5 9 59 5 9 5 9 6 0 6 0 60

6 0 6 0 6 0 6 0 6 0 6 0 60 6 0 60 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 60 6 0 6 0 6 1 6 1 61 6 1 6 1 6 1 6 1 6 1 6 1 61 6 1 61 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 61 6 1 6 1 6 2 6 2 62 6 2 6 2 6 2 6 2 6 2 6 2

62 6 2 62 6 2 6 2 6 2 6 2 6 2 6 2 6 2 6 2 6 2 62 6 2 6 2

6 3 6 3 63

6 3 6 3 6 3 6 3 6 3 6 3 63 6 3 63 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 63 6 3 6 3 6 4 6 4

64 6 4 6 4 6 4 6 4 6 4

6 4 64 6 4 64 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 64 6 4 6 4 6 5 6 5 65 6 5 6 5 6 5 6 5 6 5 6 5 65

6 5 65 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 65 6 5 6 5 6 6

6 6 66

6 6 6 6 6 6

6 6 6 6 6 6

66 6 6 66

6 6 6 6 6 6

6 6 6 6 6 6 6 6 6 6 6 6 66 6 6 6 6

6 7 6 7 67 6 7 6 7 6 7 6 7 6 7 6 7 67

6 7 67 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 67 6 7 6 7 6 8 6 8

68 6 8

6 8 6 8 6 8 6 8

6 8 68

6 8 68 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 68 6 8 6 8

6 9 6 9

69 6 9

6 9 6 9 6 9 6 9 6 9

69 6 9 69 6 9 6 9 6 9 6 9 6 9

6 9 6 9 6 9 6 9 69 6 9 6 9

7 0 7 0

70 7 0

7 0 7 0 7 0 7 0

7 0 70

7 0 70 7 0

7 0 7 0 7 0 7 0 7 0 7 0 7 0 7 0 70 7 0 7 0

7 1 7 1 71

7 1 7 1 7 1

7 1 7 1 7 1

71 7 1

71 7 1

7 1 7 1 7 1 7 1

7 1 7 1 7 1 7 1 71 7 1 7 1

7 2 7 2 72 7 2 7 2 7 2 7 2 7 2 7 2 72 7 2 72 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 72 7 2 7 2

7 3 7 3

73 7 3 7 3 7 3 7 3 7 3 7 3 73 7 3 73 7 3 7 3 7 3 7 3 7 3 7 3 7 3 7 3 7 3 73 7 3 7 3

7 4 7 4 74 7 4 7 4 7 4 7 4 7 4 7 4 74 7 4 74 7 4 7 4 7 4 7 4 7 4 7 4 7 4 7 4 7 4 74 7 4 7 4

7 5 7 5 75 7 5 7 5 7 5 7 5 7 5 7 5 75 7 5 75 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 75 7 5 7 5

7 6 7 6 76

7 6 7 6 7 6 7 6 7 6 7 6 76 7 6 76 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 76 7 6 7 6

7 7 7 7

77 7 7

7 7 7 7

7 7 7 7

7 7 77

7 7 77

7 7 7 7 7 7

7 7 7 7 7 7

7 7 7 7 7 7 77

7 7 7 7

7 8 7 8 78 7 8 7 8 7 8 7 8 7 8 7 8 78 7 8 78

7 8 7 8 7 8 7 8 7 8

7 8 7 8 7 8 7 8 78 7 8 7 8

7 9 7 9 79

7 9 7 9

7 9 7 9

7 9 7 9

79 7 9

79 7 9

7 9 7 9 7 9 7 9

7 9 7 9 7 9 7 9

79 7 9 7 9

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ran

kco

rrel

atio

nco

effici

ent

0 200 400 600 800 1000 1200

Time (s)

Figure 7: Sensitivity measures for pressure in the upper part of thecontainment versus time.

input parameters by expert judgement or screening calcula-tions is necessary to limit the calculation effort. All poten-tially important parameters may be included in the uncer-tainty analysis. The method accounts for the combined influ-ence of all identified input uncertainties on the results. Thiswould be difficult or even impossible to achieve by a prioriexpert judgement of loss of coolant accidents or transients.

The number of calculations needed is independent of thenumber of uncertain parameters accounted for in the analy-sis. It does, however, depend on the requested tolerance lim-its, that is, the requested probability coverage (percentile) ofthe combined effect of the quantified uncertainties, and onthe requested confidence level of the code results. The tol-erance limits can be used for quantitative statements aboutmargins to acceptance criteria.

Another important feature of the method is that it pro-vides sensitivity measures of the influence of the identifiedinput parameter uncertainties on the results. The measurespermit an uncertainty importance ranking. This informationprovides guidance as to where to improve the state of knowl-edge in order to reduce the output uncertainties most effec-tively, or where to improve the modelling of the computercode. Different to other known uncertainty methods, theranking is a result of the analysis and its inputs and not of ana priori expert judgement. Uncertainty statements and sen-sitivity measures are available simultaneously for all single-valued (e.g., peak cladding temperature) as well as contin-uous valued (time dependent) output quantities of interest.The method relies only on actual code calculations withoutthe use of approximations like fitted response surfaces. Themethod proposed by GRS has been used in different applica-tions by various international institutions including licens-ing.

A challenge in performing uncertainty analyses is thespecification of ranges and probability distributions of in-put parameters. Investigations are underway to transformdata measured in experiments and post test calculationsinto thermal-hydraulic model parameters with uncertainties.Care must be taken to select suitable experimental and ana-lytical information to specify uncertainty distributions. Thisis a general experience gained in applying different uncer-tainty methods.

ACKNOWLEDGMENTS

This work performed by GRS was funded by the GermanFederal Ministry for Economy under Contract no. RS 1142.The significant contributions of my colleagues H. Bartalsky,A. Hora, B. Krzykacz-Hausmann, and T. Skorek to this paperare gratefully acknowledged.

REFERENCES

[1] 10 CFR 50.46, “Acceptance criteria for emergency core coolingsystems for light water nuclear power reactors,” Appendix K,“ECCS Evaluation Models”, to 10 CFR Part 50, Code of FederalRegulations, 1996.

[2] E. Hofer, “Probabilistische Unsicherheitsanalyse von Ergeb-nissen umfangreicher Rechenmodelle,” GRS-A-2002, January1993.

[3] S. S. Wilks, “Determination of sample sizes for setting tolerancelimits,” Annals of Mathematical Statistics, vol. 12, no. 1, pp. 91–96, 1941.

[4] S. S. Wilks, “Statistical prediction with special reference to theproblem of tolerance limits,” Annals of Mathematical Statistics,vol. 13, no. 4, pp. 400–409, 1942.

[5] B. E. Boyack, I. Catton, R. B. Duffey, et al., “Quantifying reactorsafety margins,” Nuclear Engineering and Design, vol. 119, no. 1,pp. 1–15, 1990.

[6] Bemuse Phase III Report, “Uncertainty and sensitivity analy-sis of the LOFT L2-5 test,” Tech. Rep. NEA/CSNI/R(2007)4,Nuclear Energy Agency, Issy-les-Moulineaux, France, October2007.

[7] M. Kloos and E. Hofer, SUSA - PC, a personal computer ver-sion of the program system for uncertainty and sensitivity analy-sis of results from computer models, version 3.2, user’s guide and

Horst Glaeser 7

tutorial, Gesellschaft fur Anlagen- und Reaktorsicherheit,Garching, Germany, August 1999.

[8] G. Lerchl and H. Austregesilo, ATHLET Mod 1.2 User’s Manual.GRS-P-1 / Vol.1, Rev. 2a, November 2000.

[9] W. Klein-Heßling, COCOSYS V0.2, User Manual. GRS-P-3/1.

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