Text

Qualification of Reactor Physics Methods for Application to Monticello
	ML20072M455
Person / Time
Site:	Monticello
Issue date:	08/24/1994
From:	Bonneau C, Dean D, Matis L; NORTHERN STATES POWER CO.
To:	;
Shared Package
ML20072M446	List: ML20072M443; ML20072M455;
References
	NSPNAD-8609, NSPNAD-8609-R02, NSPNAD-8609-R2, NUDOCS 9409020047
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Monticello Nuclear Power Plant Qualification of Reactor Physics Methods for Application to Monticello NSPNAD-8609, Rev 2 August 1994 Northern States Power Company Nuclear Analysis & Design aven remweem+ pn .

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MONTICELLO NUCLEAR GENERATING PLANT QUALIFICATION OF REACTOR PHYSICS METHODS FOR APPLICATION TO MONTICELLO NSPNAD-8609 Revision 2 August 1994 Principal Contributors Anthony Bockelman, NSP Clifford Bonneau, NSP David Dean, NSP Keith Dehnbostel, NSP Thomas Iseman, NSP William Lax, NSP Ryan Maas, NSP Michael Miller, NSP '

Peter Pankratz, NSP Richard Rohrer, NSP Ralph Rye, NSP Richard Streng, NSP Scott Vanevenhoven, NSP Prepared by O ,/

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David W.' Dean, Principa Engineer / /

Reviewed by f r2 e- m t4544 Date t] Y CTi# d A. KnneauyP7essManager [ [

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Approved - Date T Louis P. Matis, Director Fuel esources

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NSPN AD-8609 Rev. 2 Page 1 of 76 j

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i ABSTRACT This document is a Topical Report describing the Northern States Power Company (NSP) qualification of reactor physics methods for application to the Monticello Nuclear Plant.

This document addresses the reactor model description, qualification and quantification of reliability factors and applications to operations and reload safety evaluations of the Monticello plant.

LEGAL NOTICE This report was prepared by or on behalf of Northern States Power Company (NSP) .

It is intended for use by NSP personnel only. Use of any information, apparatus, method, or process disclosed or contained in this report by non-authorized personnel shall be considered unauthorized use, unless said personnel have received prior written permission from NSP to use the contents of this report.

With respect to unauthorized use, neither NSP, nor any person acting on behalf of NSP:

a.Makes any warranty or representation, express or implied, with respect to the accuracy, completeness, usefulness, or use of any information, apparatus, method or process disclosed or contained in this report, or that the use of any such information, apparatus, method, or process may not infringe ' privately owned rights; or

b. Assumes any liabilities with respect to the use of, or for damages resulting f rom the use of, any information, apparatus, method, or process disclosed in the report.

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NSPNAD-8609 Rev. 2 Page 2 of 76

A TABLE OF CONTENTS j PAGE

1.0 INTRODUCTION

. . . . . . . . . . . . . . . . . 6 !

2.0 GENERAL CHARACTERISTICS OF THE NSP CALCULATIONAL MODEL . 6 l

3.0 MODEL VERIFICATION AND RELIABILITY FACTOR DETERMINATION . . . 9 3.1 Control Rod Worth . . . . . . . . . . . . . . . . . . . 11 3.2 Temperature Coefficient . . . . . . . . . . . . . . . . 18 3.3 Void Coefficient . . . . . . . . . . . . . . . . 18 3.4 Doncler Coefficient . . . . . . . . . . . 20 3.5 Isotonics . . . . . . . . . . . . . . . . . . . . . . 20 3.6 Power Distribution Reliability Factor Determination . . . 20 3.6.1 Local Power Distribution . . . . . . . . . . . . . . 20 3.6.2 Inteorated Power Distribution . . . . . . . . . 24 3.6.3 Gamma Scan Comoarisons . . . . . . . . . . . . 25 3.6.4 Standard Power Distribution Comparison . . . . . . 25 3.6.4.1 Axial Power Distribution Comparisons . 25 3.6.4.2 Radial Power Distribution Comparisons . . 26 3.6.4.3 Nodal Power Distributions Comparisons . . 26 3.7 Delayed Neutron Parameters . . . . . . . . . . . . . 26 3.8 Effective Neutron Lifetime . . . . . . . . . . . . 27 4.0 MODEL APPLICATIONS TO REACTOR OPERATIONS . . . . . . . 55 4.1 Predictive Aeolications . . . . . . . . . . . . . 55 4.1.1 Cold Criticals . . . . . . . . . . . . . . . 55 4.1.2 Hot Full Power Criticals . . . . . . . . . . . . 55 4.2 Monitorino Acolications . . . . . . . . . . . . . . . . . 56 4.2.1 Process Computer . . . . . . . . . . . . . 56 4.2.2 Isotonic Inventorv . . . . . . . . . . . . . . . 56 5.0 MODEL APPLICATIONS TO SAFETY EVALUATION CALCULATIONS . . . 60 5.1 Linear Heat Generation Rate (LHGR and A.PLHGR) . . . . 60 5.2 Critical Power Ratio (CPR) . . . . . . . . . . . . . . . 60 5.3 Control Rod Worth . . . . . . . . . . . . 60 5.4 Void Reactivity . . . . . . . . . . . . . . . . 61 5.5 Fuel Temperature (Doooler) Coefficient . . . . . . . . 61 5.6 Delayed Neutrons . . . . . . . . . . . . . . . . . . . . 61 5.7 Prompt Neutron Lifetime . . . . . . . . . . 61

6.0 REFERENCES

. . . . . . . . . . . . . . . . . . . . . . . . 62 APPENDIX A Statistical Methods for the Determination and Apolication of Uncertainties . . . . . . . . . . . . . . . . . . . . 66 A.1 Application of Normal Distribution Statistics . . . . . . 67 A.2 Apolication of Non-Normal Distribution Statistics . . . . . . 70 APPENDIX B Computer Code Summary Description . . . . . . . . . . . . . 76 NSPNAD-8609 Rev. 2 Page 3 of 76 m

LIST OF TABLES TABLE TITLE PAGE 3.0.1 Reliability Factors for Monticello . . . . . . . . . . . . . 10 3.1.1 Measured to Calculated Rod Worth Comparison . . . . . . 12 3.3.1 EOC Coastdown Statepoints . . . . . . . . . . . . . 19 3.6.1 Full Power Statepoints . . . . . . . . . . . . . . . . . . . 28 3.6.2 Axial Power Distribution Comparison . . . . . . . . . . 30 3.6.3 Radial Power Distribution Comparisons . . . . . . . . . . . 31 3.6.4 Power Distribution Standard Deviations in 20 Axial Planes . 32 4.1.1 Few Rod and In-sequence Cold Criticals . . . . . . . 57 A.1 Single-Sided Tolerance Factors . . . . . . . . . . . 69 NSPNAD-8609 Rev. 2 Page 4 of 76

k LIST OF FIGURES FIGURE DESCRIPTION PAGE 2.0.1 Flow Chart: CASMO-3/ SIMULATE-3 Model . . . . . . . . . . . '8 3.1.1 Control Notch Worth Inventory Versus Exposure Cycle 11 . . 13 3.1.2 Control Notch Worth Inventory Versus Exposure Cycle 12 . 14 3.1.3 Control Notch Worth Inventory Versus Exposure Cycle 13 . . . 15 3.1.4 Control Notch Worth Inventory Versus Exposure Cycle 14 . 16 3.1.5 Control Notch Worth Inventory Versus Exposure Cycle 15 . . . 17 3.6.1 Measured and Calculated Detector Responses BOC Cycle 11 . . 33 3.6.2 Measured and Calculated Detector Responses MOC Cycle 11 . . 34 3.6.3 Measured and Calculated Detector Responses EOC Cycle 11 . . . 35 3.6.4 Measured and Calculated Detector Responses BOC Cycle 12 . . 36 3.6.5 Measured and Calculated Detector Responses MOC Cycle 12 . . 37 3.6.6 Measured and Calculated Detector Responses EOC Cycle 12 . . 38 3.6.7 Measured and Calculated Detector Responses BOC Cycle 13 . 39 ;

3.6.8 Measured and Calculated Detector Responses MOC Cycle 13 . . 40 3.6.9 Measured and Calculated Detector Responses EOC Cycle 13 . . 41 3.6.10 Measured and Calculated Detector Responses BOC Cycle 14 . 42 3.6.11 Measured and Calculated Detector Responses MOC Cycle 14 . . . 43 3.6.12 Measured and Calculated Detector Responses EOC Cycle 14 . 44 3.6.13 Measured and Calculated Detector Responses BOC Cycle 15 . . 45 3.6.14 Measured and Calculated Detector Responses MOC Cycle 15 . 46 3.6.15 Measured and Calculated Detector Responses EOC Cycle 15 . . . 47 3.6.16 Observed Differences Density Function Comparison . . . . .. . 48 3.6.17 Cumulative Distribution Function (CDF) Comparison . . . . 49 3.6.18 CDF in the Region of the 95th Percentile Model Comparison 50 3.6.19 Observed Differences Density Function Integrated Reaction Rates Comparison . . . . . . . . . . . . . . . 51 3.6.20 Cumulative Distribution Function (CDF) Integrated Reaction Rates Comparison . . . . . . . . . . . . . . . . 52 3.6.21 CDF in the Region of the 95th Percentile For Integrated Reaction Rates . . . . . . . . . . . . . . 53 3.6.22 Standard Deviation vs Measured Instrument Response . . . 54 4.1.1 Cold Criticals versus Core Average Exposure . . . . . . 58 4.1.2 Hot Criticals . . . . . . . . . . . . . . . . . . . 59 i

NSPN AD-8609 Rev. 2 Page 5 of 76 l

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1.0 INTRODUCTION

This report addresses the reactor model description, qualification and quantification of reliability factors, applications to operations and reload safety evaluations of the Monticello Nuclear Plant (Mnt). This model, based on the Studsvik CMS system of codes, can be used as a substitute for the CASMO/NDH methods previously apprcved for use (Reference 2). Adoption of the methods described here does not preclude the use the earlier CASMO/NDH methods as needed.

A summary description of the computer codes is given in Section 2. This report stresses the aspects of implementation of the NSP model; the individual code descriptions are referenced in Appendix B.

Whenever possible, directly observable parameters (such as reactor critical k,, and measured incore detector fission rates) are utilized. i The Mnt data used in this evaluation span cycles 11 through 15. In i order to be completely objective in the choice of data to be used for -

the comparisons, all Mnt cycles 11 through 15 measurements were reviewed and qualified prior to initiating the comparison calculations.

After the measured data to be used in the benchmark process had been defined, the model calculations were performed and comparisons are presented in this report as part of the quantification of the NSP model calculational uncertainties and reliability factors. A statistical approach was used to derive the uncertainties. These uncertainties are consistent with the model application procedures and methodology.

The uncertainties are evaluated by direct comparison to experimental data.

In order to provide a continuing verification of the conservatism of the i reliability factors determined by Mnt cycles 11 through 15 data, ongoing I comparisons are made each cycle using the statistical methods described in this report. A discussion of the reliability factors is provided in Section 3.

The methods for use of the model and the reliability factors are described relative to reactor operation and reload safety evaluation in Sections 4 and 5, 2.0 GENERAL CHARACTERISTICS OF THE NSP CALCULATIONAL MODEL The Monticello (Mnt) calculational model based on the Studsvik system of codes, is very similar to the calculational model previously approved for use by Yankee Atomic Electric Company for use with Vermont Yankee (see References 4, 5, and 6), and is similar in many respects to the model previously approved for use with Mnt (see Reference 2)The A flow

. code diagram of the Monticello model is shown in Figu'e 2.0.1.

acronyms used in these figures are defined in Appendix B.

In general, the CASMO-3.s7 program is used to generate the 5 lattice physics parameters for input to SIMULATE-3"U. MICBURN-3 is used to model gadolinia containing fuel pins and provides homogenized Gd cross sections for input to CASMO-3. CASMO-3' produces fission product nuclide concentrations, depletion and product chain data, pin power distributions, microscopic and macroscopic cross sections, and other nuclear data input to TABLES-3u. TABLES-3 constructs tables of these nuclear data as functions of local state variables (e.g. water density, fuel temperature etc.) for input to SIMULATE-3.

SIMULATE-3 is a three-dimensional, two-group steady state reactor neutronic and thermal hydraulic simulator. This simulator is used to generate eigenvalues, power distributions, and incore instrument predictions for use in reload safety evaluations, plant support, reload NSPN AD-8609 Rev. 2 Page 6 of 76 J

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design, fuel management, and benchmark comparisons.

ESCORE"A"*" is an EPRI computer code for steady state fuel performance analysis. The Monticello methodology uses ESCORE for fuel temperature predictions to be used as input to MICBURN-3, CASMO-3, and SIMULATE-3 for modeling fuel temperature related effects on the nuclear data (i.e.

Doppler coefficient and power defect). ,

The S3 POST" program summarizes SIMULATE-3 results including the measured and predicted incore reaction re.tes. SPM, an NSP developed code, then combines all the'statepoints to calculate overall uncertainties.

The computer code descriptions are summarized in Appendix B.

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i NSPNAD-8609 Rev. 2 Page 7 of 76 I i

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NSPNAD-8609 Rev. 2 Page 8 of 76 t

3.O MODEL VERIFICATION AND RELIABILITY FACTOR DETERMINATION The NSP models have been benchmarked against Mnt measurements made during cycles 11 through 15 for the CASMO-3/ SIMULATE-3 model to quantify the reliability factors to be used in safety related calculations. The resultant reliability factors and biases are summarized in Table 3.0.1.

The remainder of this section is a detailed account of the derivation of these factors.

The term reliability factor (RF) is used to describe the allowances to

be used in safety related calculations to assure conservatism. The uncertainty factor (la) is used to describe the actual model accuracy.

The reliability factor is always larger than the uncertainty factor.

The term bias is used to describe the statistical difference between an observed or measured distribution and the calculated value.

Appendix A describes the statistical methods used in the evaluation of the uncertainties in the following sections.

During each cycle, measured and calculated parameters will be compared in order to verify and uposte the reliability factors determined in this section. Results of the verification and an update for each parameter will be documented in the reload safety evaluation for the reload in which the updated values will be used. The updates to the reliability factors will be in accordance with the methods outlined in this section and in Appendix A.

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NSPNAD-8609 Rev. 2 Page 9 of 76

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TABLE 3.0.1 Reliability Factors for Monticello Reliability Reliability Bias Parameter Factor Factor (expressed as (expressed as applied) %)

RFrpr = .124 12.4 0 APLHGR RFrpv = .124 12.4 0 LHGR 2F,pp = .095 9.5 0 MCPR

.10 10.0 0 Rod Worth RFaons =

RFyoins = .10 10.0 0 _

Void Coefficient RFoop = .10 10.0 0 Doppler Coefficient Delayed Neutron Parameters 4.0 0 A RFi= .04 0 RF, = .04 4.0 Q

NSPNAD-8609 Rev. 2 Page 10 of 76 i l

l 3.1 Control Rod Worth Control rod worth in a BWR cannot be directly measured. Control rod worth can be inferred from various reactor critical conditions. The approach taken is to benchmark the NSP model to these critical l conditions. The data base includes 9 few rod criticals and 24 sequence criticals taken at temperatures ranging from 85 *F to 209 *F. This data represents the actual critical statepoints in cycles 11 through 15. All measured statepoints at temperatures below the boiling point of 212 *F have been included. The results of the comparisons are shown in Table 3.1.1. J The standard deviation of the calculated kg at the critical positions is .0027. This difference includes the measurement uncertainty as well :

as the calculational uncertainty. The typical amount of reactivity being held down by rods is on the order of 10% Ak. Using this value we can calculate an uncertainty in rod worth by dividing the standard deviation by this worth, i.e. .27% Ak / 10% Ak = 2.7%. For conservatism the rod worth reliability factor (RF ) is defined as 10%.

Figures 3.1.1 through 3.1.5 present graphs of control rod notch inventory versus cycle exposure for hot critical conditions for cycles 11 through 15. The best estimate is the predicted control rod notch inventory using CASMO-3/ SIMULATE-3 with the 1%AK reactivity anomaly shown. Measured rod notch inventory is indicated as a dot for each statepoint. All measured values are within the 11%AK bounds. This indicates the well behaved prediction of the model and supports the use ,

of the conservative rod worth reliability factor used above. l l

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NSPNAD-8609 Rev. 2 Page 11 of 76

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Ill Table 3.1.1 Measured to Calculated P.od Worth Comparison Temperature kg Cycle Notches Core Ave.

Withdrawn Exposure (*F)

(GWD/MTU) 60 12.802 85 0.9921 11 0.9936 64 12.802 106 12.802 106 0.9948 644 0.9936 394 12.802 113 13.666 129 0.9964 12 152 0.9928 1436 13.666 128 13.666 128 0.9938 728 0.9903 734 16.922 141 19.926 206 0.9896 1498 66 15.025 91 0.9905 - - -

13 0.9904 106 15.025 91 15.025 91 0.9897 978 0.9908 678 15.025 91 23.878 200 0.9876 2040 0.9851 1518 24.789 164 16.683 109 0.9907 14 1 J8 0.9913 1076 16.683 111 16.683 118 0.9936 734 0.9924 108 21.252 122 21.252 123 0.9919 864 0.9895 738 22.494 152

23.330 209 0.9905 892 0.9923 1502 25.193 154 25.193 142 0.9920 1542 16.217 108 0.9933 15 118 0.9963 114 16.217 108 16.217 113 0.9939 984 0.9963 774 16.217 107 16.310 200 0.9979 2560 0.9962 1516 16.310 147 17.833 181 0.9939 1632 0.9922 762 20.368 137 22.419 129 0.9928 702 Mean kg = 0.9923 a= .0027 NSPN AD-8609 Rev. 2 Page 12 0f 76

Figure 3.1.1 Control Notch Worth inventory Versus Exposure Cycle 11 1000 , , , , , , , , , i 900 - --

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3.2 Temperature Coefficient The range of values of moderator temperature coefficients encountered in current BWR lattices does not include any that are significant from the safety point of view. The small magnitude of this coefficient, relative to that associated with steam voids and combined with the long time-constant associated with transfer of heat from the fuel to the coolant, makes the reactivity contribution of moderator temperature change insignificant during rapid transients. ,

For the reasonr ;cated above, current core design criteria do not impose limits on the value of the temperature coefficient, and effects of minor design changes on the coefficient usually are not calculated.

3.3 Void Coefficient The void coefficient in a BWR cannot be directly measured, i.e., there ,

are always present the effects of other parameters such as control rods, Doppler coefficient, xenon etc. The magnitude of the uncertainty in the void coefficient can be inferred, however, from comparisons of predicted versus measured critical statepoints where the effect of the other parameters is minimized. Table 3.3.1 gives calculated values for the measured critical statepoints from EOC coastdown for cycles 11 through

15. The standard deviation of the calculated km's is .0020 Ak for the coastdown cases. The total core reactivity held down by voids for l the average void fraction (35%) at full power is on the order of 5% ak.

An average %Ak/%AV can be calculated from Table 3.3.1 which represents the error in the predicted and measured value. % sk/% 6V = .0083. ,

Multiplying by the average percent void gives the error in terms of ak. .

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% Ak = .0083

35% = 0.29%. Therefore the uncertainty in void can be calculated by dividing by the total void worth at 35% which gives 0.29% !

/ 5% = 5.8% uncertainty. This uncertainty includes components of error from exposure, xenon and Doppler. Therefore, a reliability factor of 10% in void coefficient is deemed appropriately conservative for safety '

related calculation.

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NSPNAD-8609 Rev. 2 Page 18 of 76

Table 3.3.1 EOC Coastdown Statepoints Cycle Cycle Power Void kg Exposure (t) (t)

(GWD/MTU) 11 5.624 100 34.5 1.0009 6.352 99 36.9 1.0017 6.756 92 34.3 1.0016 7.256 84 31.0 1.0015 7.764 74 27.6 1.0014 8.159 66 24.7 1.0016 12 5.478 100 37.8 1.0002 6.830 96 33.2 1.0004 7.148 91 32.7 0.9999 13 7.373 100 36.7 0.9975 8.229 87 31.4 0.9970 8.724 78 28 3 0.9968 9.103 71 25.7 0.9969 9.729 59 21.4 0.9969 10.165 51 18.4 0.9968 14 7.454 100 35.8 0.9992 8 237 93 33.7 0.9990 8.882 83 29.8 0.9988 15 9.332 100 32.1 0.9982 10.101 91 29.8 0.9972 11.197 73 25.0 0.9960 Mean Km = .9990 a =.0020 NSPNAD-8609 Rev. 2 Page 19 of 76

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3.4 Doppler Coefficient Measurements can be made in a power reactor which are directed at determining the Doppler coefficient at various power levels. In a BWR the uncertainty associated with such measurements (e.g. rod repositioning, void feedback) are such that results are not reliable for direct validation of the calculational model. Consequently, an indirect approach is taken.

The primary variable in the calculation of Doppler effects using the CASMO-3/ SIMULATE-3 model i. the fuel temperature. A change in fuel l temperature associated with a power change results in a reactivity change due to the change in the resonance absorption.

The algorithm in SIMULATE-3 that determines the model change in reactivity due to the fuel temperature change uses data calculated by CASMO-3. The approach is to determine the accuracy of CASMO-3 in calculating the change in the resonance integral (RI) due to a known fuel temperature increase, then use engineering judgement to bound this uncertainty to assure conservatism.

Comparisons of CASMO-3 calculations to critical experiments (references 4, 23, 24, 25, 26, 27, 28, and 33) have determined that the uncertainty of CASMO-3 is well within the measurement uncertainty. In view of this, a 10% reliability factor placed on the Doppler coefficient is judged adequate to assure a conservative value.

3.5 Isotocics The benchmarking of CASMO-3 to Yankee Rowe and Zion data is thoroughly discussed in references 4 and 36.

3.6 Power Distribution Reliability Factor Determination The purpose of this section is to discuss the methods used to determine the power distribution reliability factors. Reliability factors have been determined for the local fuel pin power in a node and for the total

! fuel bundle power. These factors can then be applied to the calculation of the linear heat generation rate (LHGR), the average planar linear heat generation rate (APLHGR) and the critical power ratio (CPR) respectively.

The statistics presented in Sections 3.6.1 and 3.6.2 follow those presented in the Prairie Island Topical, see reference 1.

3.6.1 Local Power Distribution The model reliability factor for calculating power distributions is based on comparisons of measured and predicted traversing incore probe (TIP) flux detector signals for normal operating core conditions.

The signals from the detectors are corrected by the on-site process computer to account for such things as detector sensitivity, drift, and background. It is these corrected signals, or reaction rates, which have been compared to simulated reaction rates calculated with the NSP models in order to derive model reliability factors.

The reliability factor, RF, is defined as a single value of ATPF/TPF, such that TPFi (I, J, K) times 1 + ATPF/TPF, has a 95%

probability at a 95% confidence level of being conservative with .

respect to TPF, (I,J,K). The subscripts c and m denote calculated and measured values. TPF (I,J,K) is the total pin peaking factor for all I,J,K locations in the core. This value cannot be measured directly. What is measured by the detector system is the NSPNAD-8609 Rev. 2 Page 20 of 76

reaction rate in the instrument thimble. This measured reaction rate is a local value. RR, = (Er (measured).

These measurements are collapsed dewn to 24 axial nodal values in each thimble consistent with the nodalization of SIMULATE-3. The CASMO-3/ SIMULATE-3 model has been used to calculate the reaction l rates in the instrument thimbles: RR c = $Er(calculated) , The observed difference distribution (ODD) has then been calculated by simply taking the relative difference of these two values:

CDD = (RR, - RR ) / RR, for all measured locations in the core.

It is important to note that the ODD is not the difference between nodal powers but rather is the difference between local fission .

rate values. It is assumed that the ODD is equal to ATPF/TPF,. (

This is a valid assumption since the calculated and measured I reaction rates are local fission rate values as is the TPF, the l only difference is the location.

The observed difference distribution determined above includes the uncertainties in the calculational model as well as the uncertainties in the measurement instrumentation. The calculational model uncertainty includes uncertainty in the calculation of the nodal power and in the conversion factors from t

nodal power to the pin powar which is taken to be the same as the total uncertainty in the calculated reaction rates. Therefore, the total uncertainty in the local pin power can be written as follows:

RFrpp = o ppm r where o rnm is determined from the ODD determined above.

The simulated detector signals are calculated in a manner which is consistent with the calculation of local power peaking factors for the purpose of safety evaluations; see Section 5.1. The first step is to compute the power distribution under consideration.

The resolution used is 24 axial levels per fuel assembly.

The predicted detector signals are obtained directly from SIMULATE-3 calculated two group fluxes and fission cross sections in the instrument locations.

A total of 68 core statepoints, or TIP traces, were chosen for the purpose of comparing measured and simulated in-core reaction rates for the CASMO-3/ SIMULATE-3 model. These statepoints span operating cycles 11 through 15 of Monticello. The specific core conditions for each of the statepoints are given in Table 3.6.1.

Typical examples of the comparisons of measured and predicted reaction rates are provided in Figures 3.6.1 through 3.6.15. The data is presented in sets of three figures, one set for each cycle, three TIP trace maps per cycle (BOC, MOC, EOC). Each figure in each set presents the differences between the measured and predicted axial reaction rates for all instrumented locations in the core and the core average axial reaction rates (lower right hand corner),

The measurements are represented as squares at the 24 axial levels. The predicted reaction rates are shown as lines.

The distribution of observed differences between measured and calculated instrument signals for all 68 core statepoints was determined. For each trace, 2 of the 24 axial values were excluded NSPNAD-8609 Rev. 2 Page 2I of 76

1 from consideration. These excluded values correspond to the top and bottom nodes. These locations are areas of steep flux gradients, and small errors in instrument position result in large differences in measured to calculated values. Since the reaction rates in these areas are always smaller (i.e., the high power point will never occur in the top or bottom nodes) these values were excluded from the determination of the observed differences density function. The reliability factors developed here include the measurement uncertainty as well as the calculational uncertainty. However, known problems with the TIP measurement system such as TIP tube mislocation and channel bowing make the measurement uncertainty very large relative to the calculational uncertainty. A 95%/95% confidence level was determined from the observed difference density function determined above.

The method of normalizing the calculated and measured reaction rates was used to adjust the average of all 24 detectors at the remaining 22 axial locations to 1.0. This normalization technique was used to put the measured and predicted values on a common basis which is consistent with the definition of the local peaking factors. The measurement uncertainty in core thermal power is accounted for in the transient and LOCA analysis.

All data was retained in the data base. The total number of nodal' observations used was 35,904. The total number of observations eliminated was 3,264.

All subsequent statistical analysis has been performed using the methods described in Appendix A.2. To ensure a conservative reliability factor at all power levels, the sample was divided into subsamples as a function of power (see Figure 3.6.22) . A standard deviation was calculated for each subsample using the methods described in Appendix A.2. Figure 3.6.22 shows a distinct power dependence for the absolute difference. Therefore, to assure conservatism in the application, the reliability factor will be applied as a relative rather than an absolute value.

The distribution of observed differences is shown in Figure 3.6.16. The following statistics therefore represent the total data base as described above using relative differences.

The first step using this method is to determine the mean relative difference of the measured to calculated values (pme) and the standard deviation (emc) :

D

{0 "2

1 pmc = = 0.002 n ;

a

{ (ej pme)2

"' = 0. 071 ome = h n-1 where: e,- ith observed difference n = total number of observations The second step is to transform the ei to standard measure using NSPN AD-8609 Rev. 2 Page 22 of 76

the following formula:

Z, = amc Y

and the resulting variates Z were then sorted into ascending order (see Figure 3.6.17) . A value of Z was chosen as an estimate of the 95th percentile of the distribution, i = 34,109. This gives the 95th percentile of Z to be Zwn = Qn = 1. 6 8 9 which implies that 95% of the errors are likely to be less than 1.689 standard deviations from the mean. It remains then to calculate a 95% confidence interval on On using the following formula VarO,, = ales =

nf i f where: q= the quantile (.95) n= number of independent observations in sample fi= ordinate of the density function of the distribution function at the abscissa q Due to the dependence of the observed differences with axial height, the total number of observations was reduced by a factor of 5 to determine the total number of independent observations.

The factor of 5 was chosen to conservative bound based on the Prairie Island topical, Reference 1, value of 3.0 which is applicable to 48 axial data points rather than 24.

It is necessary to obtain an estimate of f (.95), and this was done by applying a linear regression analysis on a short interval of the cumulative distribution function (CDF) of Z in the region of the 95th percentile (see Figure 3.6.18) . The estimated slope of the CDF (estimated from the straight line in Figure 3.6.18) is an estimate of the ordinate density function. The slope is 1

calculated as 0.143.

This gives:

VarO,5 = 0.95(1-0.95) = 0.00032

'35904' O.143 2 5

and op, = gVarO,5 = 0. 018 The estimate of the upper limit on Q 8 95 K, 0,3 = 1. 64 5 0. 018 = 0. 029 thus:

NSPNAD-8609 Rev. 2 Page 23 of 76 l

1 l

0,3 s 1.6 89 +0. 029 .

i

)

The upper limit is then 1.689 + .029 = 1.718 which gives the I following as the 95% confidence level that the calculated reaction rate (IU() will be conservative with respect to the measured ;

reaction rate (RR ) .

RR, = RR, (1 + pmc + (Q95 + IQa 95) o amc)

RR, = RR, (1 + .124)

Therefore ante = .124 with the bias absorbed into the reliability factor. Note that this value includes measurement error which adds conservatism to the calculation.

3.6.2 Intearated Power Distribution The model reliability factors for calculating power distributions are based on comparisons of integrated measured and predicted TIP trace signals obtained from normal operating core conditions.

The reliability factor (RF) is defined as a single value of ARPF/RPF, such that RPF(I,J) calculated times 1 + ARPF/RPF, has a 95% probability at a 95% confidence level of being conservative with respect to the measured RPF(I,J). The subscripts c and m will be used to denote calculated and measured values. RPF(I,J) is the integrated peaking factor determined for all I,J locations in the core. This value cannot be measured directly. What is measured by the detector system is the reaction rate in the instrument thimble. This measured reaction rate is a local value.

IRR, = &Er (measured). These values are determined at each thimble by integrating the central 22 measured axial locations.

The three-dimensional model CASMO-3/ SIMULATE-3 has been used to ]

l calculate the reaction rate in the instrument thimbles. IRR, = 4Er I (calculated) .

The observed difference distribution (ODD) has then been calculated by simply taking the relative difference of these two values ODD = (IRR, - IRR,) /IRR, for all measured locations in the core.

l The observed difference distribution determined above includes the uncertainties in the calculational model, the uncertainties in the measurement instrumentation, and the uncertainties in conversion factors from nodal power to instrument value. The calculational model uncertainty includes uncertainty in the calculation of the nodal powers as well as uncertainties in the local pin powers.

Therefore the uncertainty in the local integrated pin power can be I written as follows:

RF RPF " ORPF.95 i

where aRpr.es is determined from the ODD.

1

) The distribution of observed differences between measured and l calculated integrated instrument signals for all 68 statepoints l was determined for the CASMO-3/ SIMULATE-3 model and is shown in Figure 3.6.19. The total number of integrated observations used was 1,632.

All subsequent statistical analysis has been performed using the methods described in Appendix A.2 on the entire sample.

NSPNAD4609 Rev. 2 Page 24 of 76

The cumulative distribution function and the CDF in the region of the 95th percentile are given in Figures 3.6.20 and 3.6.21 respectively. The significant parameters calculated for this distribution are as follows:

pmc = 0.001 l ome = 0.043 l On = 1.728 ann = 0.035 Ya s ne = 0.058 IRR,n = IRR, (1 + 0.079) a npp e = 0.079 where: IRR, = Integrated reaction rate measured IRR, = Integrated reaction rate calculated For conservatism the reliability factor will remain at the value-determined for CASMO-2/NDH (reference 2) as RFapp = 0. 0 9 5 > a pp,,5 n = 0.079 No dependence of the observed difference with position was found.

Therefore, n was not reduced.

3.6.3 Gamma Scan Comparisons Gamma scan measurements are not available from Mnt cycles 11 through 15. The reliability factors for the CASMO-2/NDH methods (Reference 2) were determined from TIP comparisons which bounded the gamma scan comparisons. The greater measurement uncertainties associated with neutron TIPS results in larger measured to calculated variance as compared to gamma scan. Therefore, use of neutron TIP statistics, including the measurement uncertainty, will result in a conservative estimate of the power distribution uncertainty.

Other benchmarks of the CASMO-3/ SIMULATE-3 power distribution predictions are available for gamma scan (references 4, and 35),

critical experiments (references 4, 28, 32, 33, 34, 38, and 51), ,

fine mesh PDQ (dif fusion theory) (references 5, 37, 41, 42, 44, and 46), CASMO-3 color sets (references 5, 34, 41, 46, and 48),

gamma TIPS (references 5, 6, 40, and 45), and neutron TIPS (references 5, 37, 38, 39, 41, 45, 47, 49, and 50). These comparisons include both BWR and PWR type cores and geometries.

3.6.4 Standard Power Distribution Comnarison The following is a presentation of the power distribution using the industry standard format. Published power distribution data is usually presented in tables of axial, radial and nodal comparisons and is usually compared at the la level. Note that the entire data base is used.

3.6.4.1 Axial Power Distribution Comparisons j Table 3.6.2 presents axial peak-to-average comparisons ,

for selected statepoints from cycles 11 through 15. l 1

The following results are taken from the entire data base presented in sections 3.6.1, 3.6.2. l Simulator to measured TIP traces NSPN AD-8609 Rev. 2 Page 25 of 76

Unrodded Rodded n = 912 n = 720 y = 0.009 g = 0.010 a = 0.048 a = 0.052 This data shows excellent agreement with other published data.

3.6.4.2 Radial Power Distribution Comparisons Table 3.6.3 presents radial peak-to-average comparisons from selected statepoints from cycles 11 through 15. The following results were taken from the entire data base presented in Section 3.6.1, 3.6.2 and 3.6 3.

Simulator to measured TIP traces g = 0.001 a = 0.043 This data shows excellent agreement to other published data.

3.6.4.3 Nodal Power Distributions Comparisons Table 3.6.4 presents the nodal standard deviations for the 20 axial planes from the entire data base presented in Sections 3.6.1, 3.6.2.

Simulator to measured TIP traces y = 0.002 a = 0.071 This data shows excellent agreement to other published data.

3.7 Delaved Neutron Parameters This section deals with determining reliability factors for values which can be calculated but not measured. In these cases, an argument may be made for the general magnitude of the reliability factor without making direct comparisons between measured and predicted values.

I The importance of the reliability of the calculated values-of the delayed neutron parameters is primarily associated with the core B ,.

The uncertainties in the calculation of S ,are composed of several components, the most important of which are listed below:

a) Experimental values of 8, and A by nuclider b) Cal.ulation of the spatial nuclide inventory; c) Ca' culation of core average B, as an adjoint-flux weighted a erage over the spatial nuclide inventory.

The experimental determination of the S's and A's are assumed to be S

accurate to within 1%. The most important nuclide concentrations with respect to core S are U*, U* and pu"' . References 4 and 36 indicate that the uncertainty in the calculation of these parameters is about 0.3% for CASMO-3. Therefore, components a) and b) above are combined as ,

1.3% for CASMO-3.

The uncertainty in the calculation of a core average B depends on the I NSPN AD-8609 Rev. 2 Page 26 of 76 i

e relative adjoint-flux weighting of the individual assemblies in the l .

core. For demonstration purposes, consider a four region core, each with a different average burnup and average S. This is typical of advanced BWR cycles in that about a fourth of the core has seen three previous cycles, a fourth two previous cycles, a fourth one previous cycle and a fourth is the feed fuel. Typical regional 8's are given below:

1 Region 1 (fourth cycle fuel) S = 0.00543 Region 2 (third cycle fuel) S = 0.00581 Region 3 (second cycle fuei) S = 0.00633 ;

Region 4 (feed fuel) S = 0.00745 l The effect of errors in the calculated flux distribution can be !

evaluated in terms of the effect on the core average B ,. As a base case, weighting factors are all ' set to 1.0. In this case, the core 1 average 8 ,= 0.00626. Using a maximum error in the regional flux weighting of 7.0%, the worst error in the calculation of the core l average 8 ,is obtained by increasing the weight of r.he Region 1 fuel and decreasing the weight of the Region 4 fuel. It should be noted that the average relative weighting factor is unity. The revised S is calculated as follows:

S(1) x 1.07 = .00581 S(2) x 1.00 = .00581 S(3) x 1.00 = .00633 S(4) x 0.93 = .00693 S= .00622, which yields a -0.6% error for component c) above.

The sum of the errors for these four factors for CASMO is as follows:

1. 3 % (a+b) + 0. 5% (c) = 1.8%

For conservatism the reliability factor for delayed neutron parameters is set. at 4%.

3.8 Effective Neutron Lifetime An argument similar to the delayed neutron parameter argument is applied to the determination of the effective neutron lifetime (sN uncertainty.

The uncertainty components which go into the calculation of ALare as follows:

a) Experimental values of microscopic cross sections; b) Calculation of the spatial nuclide inventory; and c) Calculation of the core average effective neutron life-time as an adjoint-flux weighted average over the spatial nuclide inventory. l Uncertainties for components a) and b) are assumed to be the same as described for the calculation of Se, that is, 1% uncertainty in the experimental determination of nuclear cross section and .3% uncertainty in the determination of the spatial nuclide inventory for CASMO. The core average neutron lifetime depends on adjoint flux weighting of local absorption lifetimes. If a conservative estimate of the error in regional power sharing (7%) is used in determining the impact on the core average lifetime, the error in lifetime is on the order of 1.0%.

Combining all of these uncertainties linearly results in a total uncertainty of 1.8% for CASMO-3. Therefore, a 4% reliability factor will be applied to the neutron lifetime calculation when applied to safety related calculations.

NSPN AD-8609 Rev. 2 Page 27 of 76

__m .

o i

Table 3.6.1 Full Power Statepoints cycle cycle Power (%) Rod Density k,,

Exposure (%)

(GWD/MTU) 11 .388 99.9 11.71 0.9989

.819 100.0 8.26 0.9981 1.331 99.9 8.75 0.9980 1.759 100.0 7.85 0.9981 2.251 100.0 7.71 0.9981 2.659 100.0 7.61 0.9983 3.223 100.0 7.37 0.9985 3.716 100.0 7.30 0.9980 4.128 100.1 7.23 0.9987 4.631 100.0 6.71 0.9990 5.301 99.9 4.58 1.0003 5.624 100.0 4.17 1.0009 6.352 98.6 0.03 1.0017 6.756 92.1 0.03 1.0016 7.256 83.5 0.03 1.0015 7.764 74.0 0.03 1.0014 8.159 66.4 0.03 1.0016 12 0.535 100.0 2.89 1.0006 1.063 100.0 3.17 0.9994 1.736 99.9 5.23 0.9988 1.945 99.9 5.23 0.9993 2.478 99.9 7.82 0.9992 2.858 100.0 7.30 0.9998 4.497 99.9 7.02 1.0000 4.880 99.9 5.54 0.9998 5.478 99.8 3.03 1.0002 6.830 95.7 0.96 1.0004 7.148 91.1 0.00 0.9999 13 0.783 100.0 7.58 0.9963 1.408 100.0 8.26 0.9956 2.149 100.0 9.09 0.9952 2.672 100.0 10.47 0.9961 3.351 100.0 10.74 0.9959 3.967 99.9 10.47 0.9965 4.956 99.9 8.26 0.9966 6.165 100.0 6.06 0.9973 6.707 100.0 3.58 0.9976 7.373 99.9 0.00 0.9975 8.229 86.8 0.00 0.9970 l 8.724 77.8 0.00 0.9968 l

9.103 70.6 0.00 0.9969 l 9.729 58.9 0.00 0.9969 10.165 50.9 0.00 0.9968 14 0.751 100.0 6.20 0.9970 i

1.296 100.1 6.37 0.9962 1 1.977 100.0 7.02 0.9955 i 2.648 100.1 8.26 0.9958 I 3.325 100.1 8.68 0.9956 4.170 99.9 8.68 0.9962 5.284 100.1 8.75 0.9970 5.904 100.0 6.61 0.9969 6.530 100.0 4.58 0.9983 7.454 100.0 1.89 0.9992 8.237 92.7 0.00 0.9990 8.882 82.5 0.00 0.9988 NSPNAD-8609 Rev. 2 Page 28 of 76 l

l

.-~. . . _ .

l Table 3.6.1 Full Power Statepoints (Continued)

Cycle Cycle Power (%)* Rod Density k,n Exposure (%)

(GWD/MTU) 15 0.187 99.8 7.51 0.9979 0.954 99.9 6.23 0.9963 1.790 99.8 7.92 0.9958 2.735 99.8 8.68 0.9947 3.591 99.7 8.95 0. 9 9e.3 4.518 99.7 9.54 0.9944 5.409 99.7 9.33 0.9959 6.802 99.8 8.13 0.9971 7.636 99.8 6.30 0.9971 8.349 99.5 5.51 0.9977 9.332 99.7 2.86 0.9982 10.301 91.1 1.10 0.9972 11.197 73.4 0.00 0.9960 t

b i

I l

1 NSPNAD-8609 Rev. 2 Page 29 Sf 76

Table 3.6.2 Axial Power Distribution Comparison Peak to Peak to Rod Average Average % Difference Cycle Location TIP Calculated 11 20-29 Out 1.185 1.143 3.5 11 28-13 Out 1.272 1.297 -2.0 12 28-29 Out 1.241 1.255 -1.1 12 20-21 Out 1.182 1.192 -0.8 12 12-21 Out 1.183 1.194 -1.0 13 44 29 Out 1.405 1.406 -0.1 13 12-29 Out 1.438 1.393 3.1 13 28-21 Out 1.155 1.163 -0.6 14 20-29 Out 1.318 1.302 1.2 14 44-29 Out 1.256 1.231 1.9 14 12-37 Out 1.289 1.275 1.1 15 36-29 Out 1.161 1.193 -2.8 15 36-37 Out 1.415 1.326 6.3 15 12-37 Out 1.361 1.349 0.9 11 20-37 In 1.611 1.598 0.8 12 20-21 In 1.225 1.259 -2.7 13 28-45 In 1.173 1.115 5.0 14 20-37 In 1.500 1.434 4.4 14 12-29 In 1.324 1.333 -0.6 15 28-29 In 1.195 1.181 1.1 NSPNAD-8609 Rev. 2 Page 30 of 76

_ . _ . . _. _ _ = _ _ ._ . _ . _ . - ..

4 a

Table 3.6.3 Radial Power Distribution Comparisons Cycle Location Exposure TIP Calculated % Difference 11 20-37 0.388 1.118 1.103 1.4 11 36-13 1.331 1.151 1.185 -3.0 11 28-37 2.251 1.179 1.157 1.9 11 20-29 3.223 1.141 1.122 1.7 11 12-21 4.631 1.230 1.242 -1.0 11 12-29 7.764 1.228 1.176 4.2 12 36-29 1.063 1.159 1.139 1.7 12 36-29 2.478 1.187 1.150 3.1 12 12-21 4.880 1.155 1.131 2.1 12 28-21 6.830 1.131 1.199 -6.1 12 28-29 6.830 1.190 1.208 -1.6 12 28-37 7.148 1.159 1.208 -4.2 13 20-21 2.150 1.217 1.278 -5.0 13 28-37 2.672 1.136 1.149 -1.2 13 36-29 6.165 1.117 1.150 -3.0 l

13 36-29 6.707 1.125 1.147 -1.9 l l

13 28-21 7.374 1.155 1.163 -0.7 l 13 28-37 10.165 1.120 1.127 -0.6 14 12-29 1.297 1.129 1.144 -1.3 14 36-29 1.977 1.117 1.168 -4.5 14 20-37 3.325 1.153 1.121 2.7 14 28-21 5.904 1.121 1.130 -0.8 14 28-13 7.454 1.110 1.130 -1.8 14 20-37 8.882 1.181 1.193 -1.0 15 12-29 0.954 1.111 1.168 -5.2 15 36-13 2.735 1.181 1.157 2.0 15 28-21 5.409 1.148 1.118 2.7 j 15 20-37 8.349 1.168 1.151 1.5 l i

15 36-21 10.301 1.140 1.199 -5.2 4 1

15 20-37 11.197 1.233 1.201 2.6 NSPNAD-8609 Rev. 2 Page 31 of 76

4 Table 3.6.4 Power Distribution Standard Deviations in 20 Axial Planes anar Standard Planes Deviation 3 0.062 1

4 0.061 5 0.056 6 0.054 J 7 0.055 8 0.054 1

9 0.056 l 10 0.060 11 0.058 12 0.060 13 0.062 14 0.064 15 0.062 16 0.060 17 0.066 18 0.061 19 0.060 20 0.062 21 0.063 22 0.074 NSPN.G 8609 Rev. 2 Page 32 of 76 i

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Comparison !

8000 , , , , , , i l t

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i Error-(in standard deviations frorn the meon) l NSPN AD-8609 Rev. 2 Page 48 of 76

a Figure 3.6.17 Cumulative Distribution Function (CDF)

Comparison 1.0 , !

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+

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NSPNe') 3:: 4 Pw. 2 Page 49 of 76

- _ _ - - _ _ _ - - _ _ _ _ - - _ _ _ _ _ _ _ _ _ _ - _ _ - - _ _ _ _ _ _ _ _ - - _ _ _ _ _ _ _ - _ - _ _ - _ _ - - - _ _ _ _ _ - _ - _ - - _ - _ = _ _

i Figure 3.6.18 :

CDF in the Region of -the 95th Percentile i Model Comparison :

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Error (in standard deviations from the meon) !

NSPNAD-8609 Rev. 2 Page 50 of 76

k Figure 3.6.19 Observed Differences Density Function Integrated Reaction Rates Comparison l

350 , , , , ,

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NSPNAD-8609 Rev. 2 Page 51 of 76

=

Figure 3.6.20 Cumulative -Distribution Function (CDF)

Integrated Reaction Rates Comparison 1.0 ,

f

~

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NSPN AD-8609 Rev. 2 iage 52 of 76

Figure 3.6.21 CDF in the Region of the 95th Percentile For Integrated Reaction Rates !

0.956 , i i i i i i l

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NSPNAD-8609 Rev. 2 Page 53 of 76

_ - - - _ _ - - - - - _ - - - - - - - - - - _ _ -i

Figure 3.6.22 Standard Deviation '

vs Measured Instrument Response Absolute Differences (Meas-Calc) 0.10 i i , , , i i 3 0.09 -

0.08 -

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/!

0.05

%R a 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Measured Response Relative Differences (Meas-Calc)/ Meas 0.10 i i i , , , , i 0.09 - -

0.08 -

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O.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 18 2.0 Mecsured Response N$PNAD-8609 Rev. 2 Page 54 of 76

4 4.0 MODEL APPLICATIONS TO REACTOR OPERATIONS This section describes the methods used in applying the reliability factors and biases to reactor operations. It is not the intent of this section to define the procedures used. However, some aspects of these procedures are presented in order to clarify the approach taken in applying the model reliability factors and biases.

The model will be applied to reactor operations in two primary modes, predictive and monitoring. Cold critical comparisons, including few rod and in-sequence criticals; and hot criticals at power are given below to verify this mode of application.

In the monitoring mode, process computer support and isotopic inventory calculations must be considered.

4.1 Predictive Acolications 4.1.1 Cold Criticals NSP has predicted few rod cold criticals around the high worth rod for each cycle of operation in order to verify the predicted model. The resultant cold critical kg for all few rod criticals calculated for cycles 11 through 15 is:

km = 0.9929 .0023 NSP has predicted in-sequence withdrawals to cold critical for each cycle of operation to verify the rod withdrawal pattern and to prevent the withdrawal of a high notch worth rod that could scram the reactor.

The resultant cold critical k ,for all in-sequence criticals calculated for cycles 11 through 15 is:

kg = 0.9922 - 0028 The combined statistics of few rod and in-sequence criticals calculated for cycles 11 through 15 is:

km = 0.9923 i .0027 Table 4.1.1 gives the detailed information for each critical.

Figure 4.1.1 gives the graphical representation of the criticals for each cycle.

4.1.2 Hot Full Power Criticals NSP has predicted the hot at-power critical condi? ions throughout each cycle.

The resultant hot critical kg for all criticals calculated for cycles 11 through 15 is:

kg = 0.9979 i .0019 Table 3.6.1 gives the detailed information for each critical.

Figure 4.1.2 gives the graphical representation of the criticals for each cycle. Circled points indicate coastdowns.

NSPNAD-8609 Rev. 2 Page 55 of 76

2 a

4.2 Monitorina Aeolications 4.2.1 Process Computer The General Electric 3D-Monicore System recently installed at Monticello will be retained. NSP is currently evaluating several options for support of this system for cycles 18 and beyond. GE will supply support for cycle 17. The support options are as follows:

1. Continue to have GE supply all support. ,

2. NSP will support with system as installed.

3. NSP will support with system modified by replacing Panacea with an approved core model (i.e. SIMULATE-3 or NDH).

4.2.2 Isotocic Inventory The isotopic inventory calculation will be performed by NSP if either option 2 or 3 is decided upon in Section 4.2.1. The calculation of the isotopic inventory for Monticello is based upon a two-dimensional, CASMO-3 calculation. This is the same model as is used to calculate the TIP trace design input. Therefore, the l accuracy of the burnup distribution can be verified by the agreement of the measured and calculated reaction rates which is used to evaluate the measurement uncertainties, see Section 3.6 above. The accuracy of the isotopics versus local exposure is a described in references 4 and 36 based on measurements at Yankee l Rowe.

l l

l

)

I I

NSPNAD-8609 Rev. 2 Page 56 of 76

e TABLE 4.1.1 Few Rod and In-sequence Cold Criticals Cycle Cycle Temperature F = Few Rod k,y Exposure (*F) S = Sequence (GWD/MTU) 11 0.000 85 F 0.9921 0.000 106 F 0.9936 0.000 106 S 0.9948 0.000 113 S 0.9936 12 0.000 129 F 0.9964 0.000 128 S 0.9928 0.000 128 S 0.9938 3.256 141 S 0.9903 6.260 203 S 0.9896 13 0.000 91 F 0.9905 0.000 91 F 0.9904 0.000 91 S 0.9897 0.000 91 S 0.9908 8.853 201 S 0.9876 9.764 164 S 0.9851 14 0.000 109 F 0.9907 0.000 111 S 0.9913 J 0.000 118 S 0.9936 j 4.569 122 F 0.9924 l 4.569 123 S 0.9919 I 5.811 152 S 0.9895 6.647 209 S 0.9905 8.510 154 S 0.9923 8.510 142 S 0.9920 15 0.000 108 F 0.9933 0.000 108 F 0.9963 l 0.000 113 S 0.9939 0.000 107 S 0.9963 0.093 200 S 0.9979 0.093 147 S 0.9962

{ 1.616 182 S 0.9939 l 4.151 137 S 0.9922 6.202 129 S 0.9928 Statistics l

j Type N Mean a 3

\ 1 Few Rod 9 0.9929 0.0023 Sequence 24 0.9922 0.0028 Combined 33 0.9923 0.0027 NSPN AD-8609 Rev. 2 Page 57 of 76

__j

Figure 4.1.1 Cold Criticals vs Core Average Exposure 1.000 ! , , ! ! ! ,

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0.998 --

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t Figure 4.1.2 Hot Criticals 1.010 , , , , , , , , ,

1.009 --

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O co etee "

$ 1.003 - - - - - - -

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-- t - --- --- ---

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I :

i ' ' ' '

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- , - - . . . . ~ .

d 5.0 MODEL APPLICATIONS TO SAFETY EVALUATION CALCULATIONS This section describes the methods used in applying the reliability factors and biases to the results of safety related physica calculations. It is not the intent of this section to define the procedures to be used in performing the physics calculations. However, some aspects of these procedures are presented in order to clarify the approach taken in applying the model reliability factors and biases.

In such applications, the question is generally: Will the reload core maintain a safe margin to established safety limits (i.e., peak linear heat generation rate, minimum CPR, shutdown margin, etc.) under normal and non-normal or accident conditions? The question is usually answered by performing cycle specific safety analyses for the limiting transients and accidents.

For each parameter of interest, RFx and Biasx are given in Table 3.0.1.

The application of the RFx and Biasx for each parameter of interest is shown below.

5.1 Linear Heat Generation Rate (LHGR and APLHGR)

The Linear Heat Generation Rate (LHGR) and the Average Planar Linear Heat Generation Rate (APLHGR) are calculated directly in SIMULATE-3.

The model reliability factor and bias listed in Table 3.0.1 are then applied as follows:

LHGR = LHGR(model) (1 + Bias + RFypp)

APLHGR = APLHGR(model) (1 + Bias + RFrr) i where model signifies the best estimate value directly calculated with the 3D simulator.

5.2 Critical Power Ratio (CPR)

I

! The Critical Power Ratio is defined as the ratio of the bundle power required to produce onset of transition boiling somewhere in the bundle (critical power) to the actual bundle power, i.e.

CPR(I,J) = Pc (I, J) / P(I,J) l where:

l Pc (I, J) is the critical bundle power in assembly (I,J)

P(I,J) is the actual bundle power in assembly (I,J)

The minimum critical power ratio, MCPR, is defined as the minimum value of CPR in the core, i.e.:

MCPR = (Pc(I,J) / (P(I,J))

The model reliability and bias listed in Table 3.0.1 are then applied as follows:

CPR = [Pc(I,J) / P(I,J)) (1 + Bias + RFan))

5.3 Control Rod Worth Rod worth are calculated using the three-dimensional nodal model. Worth are determined by varying the rod position while the independent core parameters such as core power, flow, and void distribution are held constant.

l NSPNAD-8609 Rev. 2 Page 60 of 76

The model reliability factor and bias listed in Table 3.0.1 are then l applied as follows: ;

AKnoo = Mgop(MODEL) (1 + Bias) (1 RFgoo)

The reliability factor is either added or subtracted, whichever is most conservative for each particular application.

5.4 Void Reactivity The model reliability and biases listed in Table 3.0.1 are applied to Ak/ AU as Ak/dU (1 4 Bias) (1 RF )

The reliability factor is either added or subtracted, whichever is most conservative, for each application.

5.5 Fuel Temperature (Donoler) Coefficient The Doppler coefficient is a measure of the change in neutron multiplication associated with a change in fuel temperature. Reactivity is changed mainly due to Doppler broadening of the U-238 parasitic resonance absorption cross section due to increases in fuel temperature.

The model reliability factor and bias listed in Table 3.0.1 are then applied at each point as follows:

Ak / At," (1 + Bias) (1 i RFo)

Again, the reliability factor is either added or subtracted, whichever is most conservative for each particular application.

5.6 Delayed Neutrons The delayed neutron constants; Sai and A , are assumed to be constant in s

time during a transient. The use of constant delayed neutron constants corresponding to the initial conditions is justified by the results in Reference 29 which show that Em does not change significantly during a transient until the scram is over. Adjoint flux weighting is used to obtain these constants.

The reliability factor listed in 3.0.1 is applied as shown:

Bai (model) (1 + Bias) (1 RFs) 4 5.7 Prompt Neutron Lifetime The prompt neutron lifetime ALis assumed to be constant in time.

The reliability factor listed in Table 3.0.1 is applied as follows:

.Almodel) (1 + Bias) (1 RF3 )

NSPN AD-8609 Rev. 2 Page 61 of 76

a of

6.0 REFERENCES

1. NSP Topical " Qualification of Reactor Physics Methods for Application to PI Units", NSPNAD-8101P, Rev. 1, December, 1982

2. NSP Topical " Qualification of Reactor Physics Methods for Application to Monticello", NSPNAD-8609, Rev. 1, April, 1992.

3. NSP Topical "Monticello Nuclear Generating Plant Safety Evaluation Methods," NSPNAD-8608, Rev. 1, August, 1988.

4. A. S. DiGiovine, K. B. Spinney, D. G. Napolitano, J. Pappas, "CASMO-3G Validation and Verification", YAEC-1653-A, Yankee Atomic Electric Company, 1990.

5. A. S. DiGiovine, J. P. Gorski, M. A. Tremblay, " SIMULATE-3 Validation and Verification", YAEC-1659-A, Yankee Atomic Electric Company, 1990. ,

i

6. B. Y. Hubbard, D. J. Morin, J. Pappas, R. C. Potter, "MICBURN-3/CASMO-3/ TABLES-3/ SIMULATE-3 Benchmarking of Cycles 9 Through 13," YAEC-1683-A, Yankee Atomic Electric Company, 1990.

7. M. Edenius, and B. H. Forssen, "CASMO-3 A Fuel Assembly Burnup '

Program User's Manual", Studsvik AB, NFA-89/3 Rev.2, March, 1992.

8. M. Edenius, H. Haggblom, and B. H. Forssen, "CASMO-3 A Fuel Assembly Burnup Program Methodology", Studsvik AB, NFA-89/2 Rev.1, January, 1991.

9. M. Edenius, A. Ahlin, and H. Haggblom, "CASMO-2 User's Manual, Studsvik AB, NR-81/3, 1981.

10. M. Edenius, and C. Gragg, "MICBURN-3 Microscopic Burnup in Burnable Absorber Rods User's Manual", Studsvik AB, NFA-89/12, November, 1989.

11. J. A. Umbarger, A. S. DiGiovine, K. S. Smith, and J. T. Cronin,

" SIMULATE-3 Advanced Three-Dimensional Two-Group Reactor Analysis Code Users Manual", Studsvik of America, SOA-92/01 Rev.0, April, 1992.

12. K. S. Smith, J. T. Cronin, and J. A. Umbarger, " SIMULATE-3 Advanced Three-Dimensional Two-Group Reactor Analysis Code Methodology", Studsvik of America, SOA-92/02 Rev.0, April, 1992.

13. J. A. Umbarger, and K. S. Smith, " TABLES-3 Library Preparation Code for SIMULATE-3", Studsvik of America, SOA-92/03 Rev.0, April, 1992.

14. "S3 POST SIMULATE-3 Summary File Postprocessor", Studsvik of America, SOA-91/04, 1991 H. R. Freeburn, et al, "ESCORE-The.

=

15. I. B. Fiero, M. A. Krammen, EPRI Steady-State Core Reload Evaluator Code: General Description", Electric Power Research Institute, EPRI NP-5100-L-A, April, 1991,

16. M. A. Krammen, H. R. Freeburn, et al, "ESCORE-The EPRI Steady-State Core Reload Evaluator Code Volume 1: Theory Manual",

Electric Power Research Institute, EPRI NP-4492-CCMP Volume 1, August, 1986.

17. M. A. Krammen, R. B. Fancher, N. T. Yackle, et al, "ESCORE-The EPRI Steady-State Core Reload Evaluator Code Volume 2: User's Manual", Electric Power Research Institute, EPRI NP-4492-CCMP NSPNAD4609 Rev. 2 Page 62 of 76

. ~ - _ . .

9 Volume 2, August, 1986.

18. M. A. Krammen, R. B. Fancher, M. W. Kennard, et al, "ESCORE-The EPRI Steady-State Core Reload Evaluator Code Volume 3:

Programmer's Manual", Electric Power Research Institute, EPRI NP-4492-CCMP Volume 3, August, 1986.

19. D. B. Jones, "ARMP-02 Documentation Part II, Chapter 7-MICBURN-E Computer Code Manual Volume 1: Theory and Numerics Manual",

Electric Power Research Institute, EPRI NP-4574-CCM, Part II, Ch.

7 Volume 1, December, 1986.

20. D. B. Jones, "ARMP-02 Documentation Part II, Chapter 7-MICBURN-E Computer Code Manual Volume 2: User's Manual", Electric Power Research Institute, EPRI NP-4574-CCM, Part II, Ch. 7 Volume 2, December, 1986.

21. D. B. Jones, "ARMP-02 Documentation Part II, Chapter 7-MICBURN-E Computer Code Manual Volume 3: Programmers Manual", Electric Power Research Institute, EPRI NP-4574-CCM, Part II, Ch. 7 Volume 3, December, 1986,

22. M. Edenius, and H. Haggblom, " Benchmarking of CASMO Resonance Integrals for U-238 Against Hellstrand's Measurements", Studsvik of America, SOA-91/05, December, 1991.

23. M. A. Edenius, " Benchmarking of CASMO Resonance Integrals for U-238 Against Hellstrand's Measurements. Comparison between CASMO-3 Versions 4.4 and 4.7", Studsvik of America, SOA-93/04, March, 1993.

24. M. A. Edenius, "CASMO Doppler Coefficients versus MCNP-3A Monte Carlo Calculations", Studsvik of America, SOA-93/06, October, 1993.

25. M. Edenius, " Studies of the Reactivity Temperature Coefficient in Light Water Reactors," AE-RF-76-3160, AB Stomenergi, 1976.

26. M. Edenius, " Seminar on U-238 Resonance Capture," S. Pearlstein, Editor, page 87, BNL-NCS-50451, 1975.

27. M. Edenius, " Temperature Effects in Thermal Reactor Analysis,"

Internal Report presented to Oskarshamnuerkets Kraf tgrupp AB(OKG) ,

Stockholm, Sweden, employed by AB Stomenergi Studsvik, Sweden.

28. M. Edenius, and A. Ahlin, "CASMO-3: New Features, Benchmarking, and Advanced Applications," Nuclear Science and Engineering, 100, No. 3, p. 342, November, 1988.

29. J. M. Holzer, et.al. "A Code System to Produce Point Kinetics Parameters for LWR Calculations," ANS Trans, 19, 946-7, 1981.

30. M.G. Kendall, A. Stuart, "The Advanced Theory of Statistics," Vol.

1, 5th ed., Hafner Publishing Co. N.Y., 1987.

31. D.B. Owen, " Factors for One-Sided Tolerance Limits and for Variables Sampling Plans" Sandia Corporation, March 1963.

32, K. S. Smith, " SIMULATE-3 Pin Power Reconstruction: Benchmarking Against B&W Critical Experiments, " Trans. Am. Nuc. Soc. , 56, p.

531, San Diego, CA, June, 1988.

33. M. Edenius, "CASMO-3 Benchmarking," Trans. Am. Nuc. Soc., 16, p.

536, San Diego, CA, June, 1988.

34. K. R. Rempe, K. S. Smith, and A. F. Henry, " SIMULATE-3 Pin Power NSPNAD-8609 Rev. 2 Page 63 of 76 ,

I l

Reconstruction: Methodology and Benchmarking," Nuclear Science and Engineering, 103, No. 4, p. 334, December, 1989, i 1

35. T. Uegata, E. Saji, and H. Tanaka, " Verification of the CASMO- i 3/ SIMULATE-3 Pin Power Accuracy by Comparison with Operating Boiling Water Reactor Measurements," Nuclear Science and Engineering, 114, No. 1, p. 81, May, 1993. j

36. P. J. Rashid, "CASMO-3 Benchmark Against Yankee Rowe Isotopics",

Studsvik of America, SOA-86/05, September, 1986,

37. " Nuclear Design Methodology Using CASMO-3/ SIMULATE-3P," Duke Power Company DPC-NE-1004A, November, 1992.

38. D. J. Edwards, L. E. Kostynak, F. A. Monger, R. M. Rubin, and C.

E. Willingham, " Steady State Reactor Physics Methodology," Texas Utilities Electric Company, RXE-89-003-NP, July, 1989.

39. R. Y. Chang, C. W. Gabel, "PWR Reactor Physics Methodology Using CASMO-3/ SIMULATE-3," Southern California Edison Company, SCE-9001-A, September, 1992.

40. M. Edenius and P. J. Rashid, " Benchmarking of the Gamma-TIP Calculation in CASMO Against the Hatch BWR," Trans. Am. Nuc. Soc.,

12, p . 4 31, Boston, MA, June, 1985.

41. K. S. Smith and K. R. Rempe, " Testing and Applications of the QPANDA Nodal Mode 1,

Nuclear Science and Engineering, 100, No. 3,

p. 324, November, 1988.

42. A. S. DiGiovine and D. G. Napolitano, " SIMULATE-3 Pin Power Reconstruction and Comparison to Fine-Mesh PDQ," Trans. Am. Nuc.

Soc. , 11, p. 3 61, Dallas, TX, June, 1987.

43. K. R. Rempe and K. S. Smith, " SIMULATE-3: Power Distributions and Detector Response Modeling, " Trans. Am. Nuc. Soc., 11, p. 355, Dallas, TX, June, 1987.

44. D. G. Napolitano, A. S. DiGiovine, K. R. Rempe, and K. S. Smith,

" SIMULATE-3: Pin Power Reconstruction Applied to Seabrook Station, " Trans. Am. Nuc. Soc., 11, p. 590, Los Angeles, CA,

~

November, 1987.

45. R. Hakanson and E. Kurcyusz-Ohlofsson, "Forsmark 1 Core Analysis with the Studsvik Code Package," Proceedings of the 1968 International Reactor Physics Conference, Jackson Hole, NY, September, 1988.

46. A. S. DiGiovine, J. P. Gorski, and M. A. Tremblay, "Verific-tion of the SIMULATE-3 Pin Power Distribution Calculation.," Nuclear Science and Engineering, _1_0.3, 3 , No. 4, p. 324, December, 1989.

47. E. Kurcyusz-Ohlofsson, " Analysis of Advanced PWR Corec '.+ith CASMO-3/ SIMULATE-3," PHYSOR 90, Vol.2, p. XIV-1, Marseille, Fra nce ,

April, 1990.

48. K. S. Smith and K. R. Rempe, " Mixed-Oxide and BWR Pin Power Reconstruction," PHYSOR 90, Vol.2, p. VII-11, Marseille, France, April, 1990.

49. H. Grubel, R. Rippler, G. Skoff, B. Wikes, and G. Wupperfeld, "Umstellung der nuklearen Kernausleng fnr das KKW Mnlheim-K&rlich (KMK) auf das Studsvik-Programmsystem CASMO/ SIMULATE,"

Tagungsbericht Proceedings, Jahrestagung Kerntechnik '91, p. 3, Bonn, Germany, May, 1991. ,

NSPNAD-8609 Rev. 2 Page 64 of 76 .

v 6 .

50. Y. Wang, J. Yang, Y. Yeh, and S. Yaur, "Neutronic Model Verification for Maanshan Power Plant with Advanced In-core Fuel i Management Package," Proceedings of the 1992 Topical Meeting on l Advances in Reactor Physics, p.1-13, Charleston, SC, March, 1992.

51. A. Jonsson, D. R. Harris, R. Y. Chang, O. J. Thomsen, " Analysis of Critical Experiments with Erbia-Urania Fuel," Trans. Am. Nuc.

Soc. , fji, p. 415, Boston, MA, June, 1992.

1

)

NSPNAD-8609 Rev. 2 Page 65 of 76 -

~a ..

w APPENDIX A Statistical Methods for the Determination and Aeolication of Uncertainties The purpose of using statistical methods is to determine the value Xc (calculated) such that there is a 95% probability at the 95% confidence level that Xe will be conservative with respect to Xr (true value) when applying the calculational methods to safety related reactor analyses.

The first step is to determine whether or not a distribution is normal.

If it is, the methods described in Section A.1 are used If the distribution cannot be treated as normal, but the distributions are known, then the methods described in Section A.2 are used.

If neither of the above methods apply, then the parameter in question is conservatively bounded.

NSPNAD-8609 Rev. 2 Page 66 of 76

A.1 Application of Normal _D_istribution Statistics Separation of Measurement and Calculational Uncertainties Comparison of measured and calculated reactor parameters includes the i effects of both the measurement and calculational uncertainties.

Methods used in this report to isolate the calculational uncertainties are described below in terms of the following definitions:

X, = true reactor parameter X, - measured reactor parameter X, = calculated reactor parameter e ,,, = (X, - X,) / X, = measurement error e, = (X, - X,) / X, = calculation error e, = (X, - X,) / X, = observed differences D

[ ej p . 21 n

o=( (ej -pj)') / (N- 1) ) * = standard devia tion If e, and e, are independent, then the follcwing relationships exist.

(Note that these relationships apply for non-normal distributions as well).

o*c " o*,- o*

Pc

Mc- P ,c once the o, and p, have been calculated from historical data, they could be used to apply conservatism to future calculations of reactor parameters, X,, as follows:

X, - X, (1 + pe) (i t K,0,)

The factor rs is defined as described in Table A.1.14 to provide a 95%

probability at the 95% confidence level that X, is conservative with ;

respect to the true value, X,. !

Reliability Factors It is the objective to define reliability factors which are to be used to increase / decrease calculated results to the point where there is a 95% probability at the 95% confidence level that they are conservative with respect te actual reactor parameters.

For any given application, there is cencern only with one side of the component; that is, if the calculated value is too large or too small.

NSPN AD-8609 Rev. 2 Page 67 of 76 j

i 4

Therefore, one-sided tolerance limits based on normal distributions may be used to find a K, which will give a 95% probability at the 95%

confidence level to the reliability factor defined by:

RF = K c0c Numerical values of N for various sample sizes used to calculate a, are provided on Table A.I.

I l !

J-a l

l

[

i Page 68 of 76 .

NSPNAD-8609 Rev. 2

l TABLE A.1 Single-Sided Tolerance Factors (Reference 31) n V, 2 26.26 3 7.66 4 5.15 )

5 4.20 l 6 3.71 7 3.40 6 8 3.19 9 3.03 10 2.91 11 2.82 l 1

12 2.74 15 2.57 20 2.40 25 2.29 30 2.22 40 2.13 60 2.02 100 1.93 200 1.84 500 1.76 co 1.645 {

f n = Number of data points used for a l

l l

1 l

1 i

NSPNAD-8609 Rev. 2 Page 69 of 76

i A.2 ADDlication of Non-Normal Distribution Statistics If a distribution is determined to be other than normal, the requirement is that there is a 95% confidence level that X, will be conservative with respect to the true value X . (In the following, the notation used is consistent with that defined in Section A.1). It is thus required that a 95% upper confidence limit be determined for the 95th percentile j of the distribution of errors. ,

l In the calculation, a set of error observations (e,) are determined.

The mean ( g.) and the standard deviation (o.) are calculated using the following forTnulation:

n

[ ei p,c = '"

n o,c = ( (f (e - sp,c) 2) j (n _1) ) se 1*1 Note that the e, above are determined from the following:

e, = e,c = (x,-x c) /x, = observed diiferences Generally, the e, are taken f rom several cycles of operation; thus, they represent the true distribution, The ei are then transformed to standard measure by the following formula:

Zi= o ,c and the resulting variates (Z) are sorted into ascending order and the kth (such that k a.95n) variate is chosen as an estimate of the 95th percentile of the distribution (See reference 30, page 50-51). This gives a 95th percentile of Z to be Qe. This implies that 95% of the errors are likely to be less than oe.

It remains to calculate a 95% confidence interval on Oe. (The formula for this calculation is taken from reference 13 page 236-243 (See references section 6.0).

Var Qn = GS1~GI nf 1 where: q = the quantile (.95) n = number of independent observations in the sample f = ordinate of the density function of the distribution of observed differences at abscissa q It is necessary to determine if the observations are independent. If they are not independent, it is necessary to reduce the sample size to account for the dependence in the determination of the 95% confidence level.

NSPN AD-8609 Rev. 2 Page 70 of 76

+

Di D2 D3 l

D4 Ds l Ds D,

i Ds )

l Figure A.2.1. Differences for Nearby Positions j To set notation, let 6,5 be the population 95th percentile.for the observed differences, that is P [D, s 6 ,,5) = .95. We wish to determine a 95% upper confidence limit for 6.,5 when some of the differences are dependent. For differences observed at adjacent positions, the appropriate measure of association for our analysis can be shown to be C(1) = P[D3 s 6,93 and D a s 5,,3] - (.95)2 We also consider the association of differences observed at locations two apart C(2) = P[D1 s 8,,3 and D 3 s 0,,3] - (.95)2 and, more generally,

! C(k) = P[D1 s 6.ss and D2 .x s 0.,3] - (.95)2 l

for k = 1,2,3,4,5,6,7 locations apart. In this example, there are 8 differences D,, 7 adjacent pairs ( D, , D,. i ) , 6 pairs with indices two apart (D,, D,+2) , 1 pair D Ds. i Let d,3 be the sample 95th percentile with s selected to be the smallest i

l integer not less than .95n. The large sample distribution of d,3 g l depends on that of i

T(x) number of differences, D,, that are less than or equal x.

~

=

t Even with dependence among the D,,

1

- -- ( T(x) -nF(x) )

l T(x) -nF(x) , {E

s. d. [T(x) } 1

--s. d. [ T(x) ]

[E will be approximately standard normal. Here F(x) = P [D, s x] and f (x) is the probability density function for the observed differences.

NSPNAD-8609 Rev. 2 Page 71 of 76 l

l It follows that l P[8(d ,3 t -6,,5) s z] = 1 -P[T(6,,3 + n '*z) s s-1]

,y_, -f(6.,5) z l 1 s.d [T(6,,5)]

l where i

1 7 6 2 ggg7)) j nC(2) + ...

[1 s.d. [ T(6 ,5)] ] 2 n [n(.95) ( 05) + 2 8 nc(1) + 2 8 8 g

1

= ( .95) ( . 05) + C(1) + C(2) + C(3) + ... C(7 )

Under independence 0 - C(1) = C(2) = . . = C(7) and this expression reduces to its customary value (.95) ( .05) . If the differences are dependent, the variance of d,a is

( .95) ( .05) y. 2 (8-k) C(k) nf2(6,,5) m 8 ( .95) ( .05)

In order to apply this result, we estimate C(1) by

, number of adjacent pairs (D ,D i i.1) where both S d<,> _ ( 95y 2 Total number of adjacent pairs The estimate of C(2) is number of pairs (D ,D i i.2) Where both S d<,, _ ( 95y 2 Total number of pairs (D ,D i i.2) and

, number of pairs (D ,D i i .y) where both S d;,, _ ( 95y 2 Total number of paits (D ,D .y) i i for k = 3,4,5,6,7. The value of 1 2(6.,5) can be estimated as previously suggested. Then, the large sample upper 95% confidence limit for 6.,3, adjusted for depende,ri.ee cong differences by location is given by 3 "4

+ 1.645 ( .95) ( .05) y 2 (8-k) C(k) d(*)

@ f 2(6,,3) ( { 8 (.95) (.05) m One interpretation of this confidence limit, or the variance expression, is that the total sample size n is effectively reduced by the dependence. We estimatu the effective sample size to be NSPN AD-8609 Rev. 2 Page 72 of 76

W n

1 2 (8-k) C(k)

~

1 8 ( .95) ( . 05)

If only two terms are used, the effective sample size is estimated to be n ( .95) ( . 05) l

( .95) ( .05) + C(1) + C(2)

It is necessary to obtain an estimate of fi (.95) on a short interval of the cumulative distribution function of Z in the region of the 95th percentile. The slope of the cumulative distribution function is an estimate of the ordinate of the density function since the density function is simply the derivative of the cumulative distribution function. Thus ok, = Var 0,5 This value then allows an estimate of the 95% confidence limit on On.

Even though nothing is known about the distribution of 0% , the distribution can be shown to be normal using the following derivation.

P[D1 s 6,,5 and D, s 6,,5) where 6e is the 95th percentile of the distribution of differences. If the differences D and D; are independent 3

P[D1 s 6,,3 and D2 5 0.ssl " PID 5 6,,5) P[D2 5 0.ssl 1

= ( .95) ( .95) = ( .95)2 The difference ,

P[D1 s 6,,5 and D, s 6,,5) - (,95) 2 is a measure of association (dependence) from position to adjacent position. Note that if

' 1 if D1 s 6 ,,

I(D1 s 6,,5) = b0 if 4 > 6'm 1 if D 2 s6 I(D2 5 0.ss) " 0 if D > 6, 2 then the covariance is ,

C(1) = Cov(I(D s 6,,5) ,I(D 25 0.9s)) = P[D2 s 6,,3 and D2 s 6,,3] - (.95)2 3

We assume the same covariance for I(D 5 0.n ) and I(D3 s 6,,) .

2 I (D, s 6.e) and I (D, s 6.e) .

There are about n 7/8 such pairs among the whole set of n observed l

NSPNAD-8609 Rev. 2 Page 73 of 76

?

9 8e differences.

Let d,be the sample 95th percentile where s is the smallest integer not less than n(.95). When n is large n'imber of pairs (D ,D i i.1) where both s d<,, _(9Sy, Total number of pairs (D ,D 3 .1) i is a good estimate of C(1) . Similarly, for the approximately 6n/8 pairs

( D, , , D. + 2 )

C(2) = cov[I(D s 6 ,3) , I(D 3s 5.,5) ]

1 is estimated by number of pairs (Dj,D i.2) vhere both s d<,; _ ( 95)2 Total number of pairs (Ds,D .2) i and number of pairs (D i,Di .x) where both s d<,, _ ( 95y, Total number of pairs (D ,D i .g) i Let us now see how to modify the proof that d) o is asymptotically normal in order to account for the dependence among adjacent differences. It is still true that (8) ##

~

( 8) # * ~ #~ #' '# #'#

(A1) = 1-P[T(x) < s}

where T(4 = J(Dj s x) = # differences Di s x. Moreover, T(x) - nF(x) has mean 0 and, for large samples, is approximately normal under a wide range of dependence structures. Consequently, the sums b J(D s x) are independent of one another and each has the same 3

fel distribution. Since T(x) is just the sum of these group sums, the central limit theorem gives

" is appr ximately standard normal.

s {T(

Consequently, from (A1) and the normal approximation P[fri(d<,3 - 6,,5) s 2] = P[d ,3 t s 5,,3 + n ~*zl

= 1-P[ T(6,,3 + n ~42) < s]

(A2) s-nF(b.,3 + n-"z)

s. d. [ T(5,,3 + n -*z) ]

Now, note that NSPN AD-8609 Rev, 2 Page 74 of 76

1 (s-nF(6,,3 + n ~*z) ) = 1 (s-nF(6,,5) - nf(6,,5) n'* z+0 (1) )

= 1 (s-n( .95) - n*rf(6 ,5)) + 0 (1)

= -zf(6,,5) + 0 (1)

Furthermore, I

Var [ T(6,,3 + n~* z)) = Var [I(D 2 s 6,,3 + 2n *))

~

2 (8 2) ,I(D3 .3 s 6,,3n *z)]

5 0.95 +n~

} -

+

8 COVIII 2 which converges to 2 ( 8 -k) (P[D s 6,,3, D ., s 6 ,5) - ( . 9 5 ) 2}

F(6,,5) -F2 (6,,3) + 2 1

= ( .95) ( 05) + 2_18g

-k) C(k) = limf Var [T(6 ,5))

Therefore, by (A2),

P[@ ( d ,, - 6,,3) $2)-1-@

g ,"

- s. d. [ T(6 ,5) )

or v'n (d (s) - 6.,5) is approximately normal with mean 0 and variance

( .95) ( .05) + 2 (8-k) C(k) /8 1_ x1 n f 2 (6,,5)

As was indicated above, the C(k) may be estimated by C(k) and the large sample normality will still hold. Therefore using Table A.1 to obtain F:

w Ag, = K, ( Var 0,5)

Thus it is 95% certain that 095 lies in the interval 0,3 s 0,3 + A00,3 therefore it is safe to say that we are 95% confident that the 95th percentile of the differences is:

p ,+oy,,3 s p, + ( 0,3 + A o p,3 ) o,e NSPN AD-8609 Rev. 2 Page 75 of 76

4 APPENDIX B Computer Code Summary Description COMPUTER CODE DESCRIPTION CASMO-3 CASMO-3 is a multigroup two-dimensional transport theory code for depletion and branch calculations for a single assembly.

It calculates the cross sections, nuclide concentrations, pin power distributions, and other nuclear data used to calculate input to the SIMULATE-3 program. Some of the characteristics of CASMO-3 are:

1. 40 energy group cross section library.

2. 7 energy groups are used during the two-dimensional transport calculations. 1

3. Gadolinium effective cross sections are generated by the MICBURN-3 program.

4. The predictor-corrector approach is used for depletion.

5. Effective resonance cross sections are calculated individually for each pin.

ESCORE ESCORE is a steady-state fuel performance code capable of modeling the thermal and mechanical response of LWR fuel and used to provide fuel temperature inputs.

MICBURN-3 MICBURN-3 calculates the burnup of a fuel pin containing gadolinium and generates 40 group effective cross sections as a function of number density for gadolinium to be input to CASMO-3.

SIMULATE-3 A two-group 3D nodal program based on the QPANDA neutronics model. Some of the features of SIMULATE-3 are:

1. Explicit reflector cross model.

2. Pin power reconstruction.

3. Fourth order expansion of intranodal flux distribution.

4. No input normalization from higher order calculations or benchmark results. j

! SPM Roccives input from S3 POST of the predicted, measured, and difference of the TIP reaction rates and calculates the biases and reliability factors.

S3 POST Reads output SIMULATE-3 and generates summaries and comparieons to measured incere TIP response. Modified by NSP to generate a file containing measured and predicted incore TIP response for input to the SPM program.

TABLES-3 Reads CASMO-3 output files and generates the input tables and curve fits for each fuel type for the SIMULATE-3 computer :

i program.

NSPNAD-8609 Rev. 2 Page 76 of 76

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