ML19320B139
| ML19320B139 | |
| Person / Time | |
|---|---|
| Site: | Calvert Cliffs  |
| Issue date: | 12/31/1979 |
| From: | ABB COMBUSTION ENGINEERING NUCLEAR FUEL (FORMERLY |
| To: | |
| Shared Package | |
| ML19289B046 | List: |
| References | |
| CEN-124(B)-NP, CEN-124(B)-NP-PT1, NUDOCS 8007090352 | |
| Download: ML19320B139 (65) | |
Text
.
CEN-124(B) NP STATISTICAL COMBINATION OF UNCERTAINTIES PART 1 DECEMBER,1979 O
e COMBUSTION ENGINEERING. INC.
~
LEGAL NOTICE T
THIS REPORT WAS PREPARED AS AN ACCOUNT OF WORK SPONSORED BY COMBUSTION ENGINEERING, INC. NEITHER COMBUSTION ENGINEERING NOR ANY PERSON ACTING ON ITS BEHALF:
A.
MAKES ANY WARRANTY OR REPRESENTATION, EXPRESS OR IMPLIED INCLUDING THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE OR MERCHANTABILITY, WITH R ESPECT TO THE ACCURACY, COMPLETENESS, OR USEFULNESS OF THE INFORMATION CONTAINED IN THIS REPORT, OR THAT THE USE OF ANY INFORMATION, APPARATUS, METHOD, OR PROCESS DISCLOSED IN THIS REPORT MAY NOT INFRINGE PRIVATELY OWNED RIGHTE:OR B. ASSUMES ANY LIABILITIES WITH RESPECT TO THE USE OF, OR FOR DAMAGES RESULTING FROM THE USE OF, ANY INFORMATION, APPARATUS, METHOD OR PROCESS DISCLOSED IN THIS REPORT.
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3 Cell-124(B)-iip STATISTICAL C0!iBl!!ATIO!! 0F UI;CERTAltlTIES 14ETHODOLOGY PART 1: C-E CALCULATED LOCAL POWER DEtiSITY AsiD TilERI4AL fiARGIfi/ LOW PRESSUhE LSSS FOR CALVERT CLIFFS VillTS I AtlD II o
O 4
h
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ABSTRACT This report describes the methods used to statistically combine uncertainties for the C-E calculated Local Power Density (LPD) LSSS and Thermal Margin / Low Pressure (TM/LP) LSSS for Calvert Cliffs Units I and II.
A detailed description of the uncertainty probability distributions and the stochastic simulation techniques used it, presented.
The total uncertainties presented in this report are expressed in percent overpower (Pfdn, Pfdl) units, assigned to the LPD LSS3 and the TM/LP LSSS at the 95/95 probability /
confidencc limit.
O e
O G
i
_z
=. _ _
TABLE OF CONTENTS Chapter M
1.0 Introduction 1.1 Purpose 1-1
1.2 Background
1-2 1.3 Report Scope 1-3 1.4 Summary of Results 1-4 1.5, References for Section 1.0 1-4 2.0 Analysis 2.1 General 2-1 2.2 Objective of Analysis 2-1 2.3 Analytical Techniques 2-1 2.3.1 General Strategy 2-1 2.3.2 TM/LP Stochastic Simulation 2-3 2.3.3 Local Power Density Stochastic Simulation 2-4 2.4 Analyses Performed 2-5 2.4.1 TM/LP LSSS Analysis 2-5 2.4.2 Local Power Density LSSS Analysis 2-11 2.5 References for Section 2.0 2-13 3.0 Results and Conclusions 3.1 Results of Analyses 3-1 3.2 Impact on Margin to SAFDL 3-3 3.3 References for Section 3.0 3-4 O
- {%-Myg 5
.m
[
TABLE OF CONTENTS (Continued)
Appendix g
A.
Basis for Uncertainties Used in Statistical A-1 Combination of Uncertainties Program Al Axial Shape Index Uncertainties A-2 A2 Measurement Uncertainties A-25 A3 Trip System Processing Uncertainties A-29 B.
Summary of Previous Methods for Combining Uncertainties B-1 W
e 4
S y
LIST OF TABLES T ai,l e Page 1-1 NSSS Parameters Affecting fuel Design Limits 1-5 3-1 Uncertainties Associated with the Local Power Density LSSS and the TM/LP LSSS 3-5 3-2 Impact of Statistical Combination of Uncertainties on Margin to SAFDL 3-6 LIST OF FIGURES Figure Page 2-1 Stochastic Simulation Methodology 2-14 2-2 Stochastic Simulation of the D!'S Limits 2-15 2-3 Stochastic Simulation of the LPD Limits 2-16 2-4 Thermal Margin Uncertainty Analysis 2-17 2-5 Linear Heat Rate Uncertainty Analysis 2-18 e
W 9
l iv L
.. ~ -
-.~.e 2
3 DEFIf11TIOfJ OF ACR0flYMS Af1D ABBREVIATI0flS ACU Axial shape index calibration uncertainty A00 Anticipated Operational Occurrence (s)
APU,TPU Processing uncertainty ARO All rods out ASI Axial shape index after application of uncertainties i %
LSSS ASI Axial shape index af ter inclusion of the DilB LSSS uncertainties DilB ASIfR Axial shape index after inclusion of LHR LSSS Uncertainties 1
B Unless specifically defined in context as representing AT Power, 8 is used interchangeably with Q, core power.
B P
after application of uncertainties Df1B fdn B
P after application of uncertainties LHR fdl
(
B LHR overpower including uncertainties LHR h B
Power limit for LHR LSSS B
Available overpower margin op opmo Reference Bgp, for calculating the constants in the TM/LP trip equation LSSS B
Power level after inclusion of DilB LSSS uncertainties and allowances.
Df18 b
I B
Power level after inclusion of linear heat rate LSSS uncertainties t
and allowances.
4 l
B kth (hth) simulated value of overpower margin.
gp,k(h) th AB k
gth) value of sampled overpower uncertainty due to axial
~
UE*k(h) shape index uncertainties i
BMU Power measurement uncertainty BMU (h)
Value of the power measurement uncertainty sampled by SIGMA k
in trial k(h).
BOC Beginning of Cycle i
CEA Coi. trol Element Assembly CECOR Computer code used to monitor core power distributions CETOP Computer code used to determine (he overpower limits due to thermal-hydraulic conditions l
CE-1 DNBR DNB Ratio calculated by the TORC /CE-1 correlation i
i V
(
,--w.~n==__--
n:
DBE Design Basis Event (s)
Di Value of simulation point i DNB Departure from Nucleate Boiling
' DNBR Departure from Nucleate Boiling Ratio EOC End of Cycle F
. Primary coolant flow rate f
Number of degrees uf freedom DNB s
F Coolant flow used in the generation of (Pfdn' I )
rdered pairs p
of data F,
Engineering factor on local heat flux F, F" Synthesized three-dimensional core power peak q
FE Planar radial peaking factor F
Integrated radial peaking factor R
H lleight of core I
Core average axial shape index I
External shape index I
Axial shape index for the i assembly I
Peripheral axial shape index p
50 -
QUIX-calculated core average axial shape index If QUIX-calculated I p Ih(RSF) QUIX calculated value of I using the rod shadowing factor method p
jR ROCS-calculated core average axial shape index R
I ROCS calculated Ip R
I (AWF)
ROCS power distribution based values of I using the assembly p
weighting factor method I (RSF)
ROCS power distribution based values of I using the rod shadowing p
factor method C
I I calculated by CECOR p
p Ci i calculated by CECOR L
Power in lower half of core LCO Limiting Condition (s) for Operation LilS Latin liypercube Sampling LilR Linear lleat Rate LPD Local Power Density LPD LSSS Local power density LSSS also called axial flux offset LSSS vi
(
~ ~-
1
-m k
LSSS Limiting Safety System Setting (s) 110NBR llinimum DNBR MOC liiddle of Cycle lht Megawatt (s) thermal MTC floderator Temperature Coefficient il Sample size NSSS Nuclear Steam Supply System (s)
P Reactor coolant system pressure 5(J)
Average power in axial node J P
Axially integrated power of assembly i g
P Power to the fuel design limit on fuel centerline melt fdl P
from simulation h fdt fg) h DNB P
Pressure used in calculating the (Pfdn' I )
rdered pairs of data p
Overpower from CETOP for the sampled input parameters fdn in simulation k P
Variable low pressure trip limit yy P
Variable pressure to achieve DNB at the LSSS limit ar b'"
P Variable pressure to achieve DNB at the LSSS limit including uncertainties r
PDIL Power Dependent CEA Group Insertion Limit PMU Pressure Measurement Uncertainty PU Uncertainty in predicting local core power at the fuel design limit P(x)
Normalized power level at core height x Q
Core power, auctioneered higher of flux power or AT power QUIX Computer code used to solve the 1 dimensiorul neutron diffusion equation RCS Reactor Coolant System RDT Pressure equivalent of the total trip unit and processing delay
~
time for the DBE exhibiting the most rapid approach to the SAFDL on DNBR ROCS Coarse mesh code for calculating power distributions l
RPS Reactor Protection System l
RSU Peripheral shape index uncertainty vii
f R(x)
Rod sha<'owing factor at core height x i
S Sample standard deviation SAfDL Specified Acceptable Fuel Design Limit (s)
SAU Shape annealing factor uncertainty SC Approved credit in lieu of statistical combination cf uncertainties 500 Statistical Combination of Uncertainties i
SIGMA Stochastic Simulation Code SMLS Statistically combined uncertainties applicable to the Local J
Power density LSSS T
Azimuthal tilt allowance j
AZ Reactor coolant cold leg, inlet temperature T, Tin c
B I
T Inlet coolant temperature used in the calculation of (P fdn' p) ordered pairs of data Tfb Final inlet coolant temperature for LSSS calculation n
T Reactor coolant hot leg temperature h
TMLL Thermal Margin Limit Line(s) i TM/LP Thermal Margin / Low Pressure TMU Temperature measurement uncertainty TORC /CE-l Thermal hydraulic calculational model including CE-1 critical heat flux correlation TPD Allowance for Transient Power Decalibration l
TPU Trip processing uncertainty U
Power in upper half of core VilPT Variable liigh Power Trip W
Core average linear heat rate W
Peak generated linear heat rate limit corresponding to the SAFDL on fuel cim i
centerline melt l
Wi Weighting factor of assembly i x
Axial position E
Sample mean l
th Z;
i value of a normally distributed random variable with zero mean and' unit standard deviation a
Shape annealing factor l
4 viii MM"
-u~_,
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w~--
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(u),(p),
D (y)
Coefficients in the P 84" li "
yr
(
Population mean a
Population standard deviation p
Axial shape index correction term 99 (r)
[
J R
P g
3 C
U
[
.]
Hs
[
3 X
Chi-squared deviate with f degrees of freedom f
O e
e ix n,
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1.0 litiRODUCTI0rt 1.1 FURPOSE The purpose of this report is to describe a method for statistically combining the uncertainties involved in the analog protection and monitoring system setpoints.
The following uncertainties are considered:
1.
Uncertainty in predicting integrated radial pin power 2.
Uncertainty in predicting local core power density 3.
Power measurement uncertainty 4.
Shape annealing factor uncertainty 5.
Shape index separability uncertainty 6.
Axial shape index calibration uncertainty 7.
Processing uncertainty 8.
Fluw measurement uncertainty 9.
Pressure measurement uncertainty 10.
Temperature measurement uncertainty 1-1
(.
u
l.2 BACKGROUND 1.2.1 Protection and Monitoring System The analog protection and monitoring systems in operation on the Combustion Engineering Nuclear Steam Supply Systems have been designed to assure safe operation of the reactor in accordance with the criteria established in 10 CFR 50, Appendix A.
This is demon:trated in the Final Safety Analysis Report (FSAR) and subsequent reload licensing amendments.
This is achieved by specifying:
1.
Limiting Safety System Settin;- :t tSS) in terms of parameters directly monitored by the Reactor Protection System (RPS); and 2.
Limiting Conditions for Operation (LCO) for reactor system parameters.
3.
LCOs for equipment performance The LSSS, combined with the LCO, established the thresholds for protection system action to prevent exceeding acceptable limits during Design Basis Events (DBE) where changes i.n DNBR and LHR are important, The limits addressed by the RPS are:
1.
The reactor fuel shall not experience centerline melt; and 2.
The departure from nucleate boiling ratio shall have a minimum allowable limit corresponding to a 95% probability at a 95%
confidence level that DNB will not occur'.
l The RPS trips jointly provide protection for all A00s. The RPS providing primary protection from centerline melt is the Local Power Density (LPD); LSSS.
The RPS providing primary DNB protection is the Thermal Margin / Low Pressure (TM.'LD) LSSS.
l The design of the RPS requires that correlations including uncertainties be applied to express the LSSS in terms of functions of monitored parameter..
1-2
These functions are the trip limits which are then set into the RPS.
A list of parameters which affect the calculation of limits for linear heat rate and DilB protection is shown in Table 1-1.
A more detailed discussion of C-E setpoint methodology may be found in Reference 1-1.
1.2.2 Previous Uncertainty Evaluation Procedure The methods previously in use for the application of uncertainties to the subject limits are presented in Reference 1-i and summarized in Appendix B.
As noted in Reference 1-1 these methods assume that all applicable t1 certainties occur simultaneously in the most adverse direction even though not all of the uncertainties are systematic; some a random and some contain both systematic and random characteristics.
This assumption is extremely conser-vative.
As described in References 1-2, partial credit has been allowed in view of the existence of this conservatism.
This report documents the methodology used to statistically combine uncertainties explicitly in lieu of the credit p eviously used.
1.3 REPORT SCOPE The scope of this report encompasses the following objectives:
1.
To def* e the methods used to statistically combine uncertainties applictule to the Thermal Margin / Low Pressure (TM/LP) and Local Power Density (LPD) LSSS; 2.
To evaluate the aggregate uncertainties as they are applied in the determination of the TM/LP and LPD LSSS.
~
To achieve these objectives it is necessary to define the probability distributions associated with the uncertainties defined in Section 1.1.
The development of these distributions is discussed in Appendix A.
1-3
... m
The methods presented in this report are applicable to the fo-llowing C-E reactors:
Calvert Cliffs Units I and II (Baltimore Gas & Electric Company) 1.4
SUMMARY
OF RESULTS The analytical methods presented in Section 2.0 are used to show that a stochastic simulation of uncertainties associated with the LPD LSSS and TM/LP LSSS results in aggregate uncertainties of [~
], respectively, at a 95/95 probability / confidence limit.
The total uncertainties previously applied to the LPD LSSS and the TM/LP LSSS are approximately [
'], respectively.
Therefore the use of the statistical combination of uncertainties provides a reduction in conservatism in the margin to SAFDL of approximately [
],
respectively.
1.5 REFERENCES
1-1 CENPD-199-P, "C-E Setpoint Methodology," April, 1976.
1-2 Docket No. 50-317, " Safety Evaluation by the Office of Nuclear Reactor Regulation," Calvert Cliffs Unit 1 Cycle 3, June 30, 1978.
S 1-4 p
~
-- --m
TABLE 1-1 NSSS PARA!1ETERS AFFECTIllG FUEL DESIGti LIMITS DNBR 1.
CORE POWER 2.
AXIAL POWER DISTRIBUTION 3.
RADIAL POWER DISTRIBUTION 4.
AZIMUTHAL TILT MAGillTUDE 5.
CORE C00LAt1T Il4LET TEMPERATURE 6.
PRIMARY COOLANT PRESSURE 7.
PRIMARY COOLANT MASS FLOW LINEAR HEAT RATE 1.
CpRE POWER 2.
AXIAL POWER DISTRIBUTI0tl 3.
RADIAL POWER DISTRIBUTI0t1 4.
AZIMUTHAL TILT MAGNITUDE e
1-5 m
s
.n_ _
2.0 ANALYSIS 2.1 GENERAL The following sections provide a description of the analyses performed to statistically combine uncertainties associated with the DNB LSSS and the LPD LSSS.
The technique involves use of the computer code SIGMA (Reference 2-1) to select data for the stochastic simulation of the TM/LP and LPD calculations.
The bases for the individual uncertainties are presented in Appendix A.
The stochastic simulation techniques are described below.
2.2 OBJECTIVES OF ANALY, SIS The objectives of the analyses presented in this section are:
1.
To document the stochastic simulation techniques for combining the uncertainties associated with the TM/LP LSSS and the LPD LSSS, 2.
To determine the 95/95 probability / confidence limit uncertainty factor to be applied in calculating the TM/LP LSSS and LPD LSSS.
2.3 ANALYTICAL TECHNIQUES l
l i
2.3.1 General Strategy The stochastic simulation code used for the statistical combination of I
uncertainties associated with the TM/LP LSSS and the LPD LSSS is the l
comp, uter code SIGMA.
2-1
_,x
SIGMA produces the dependent variable probability histogram for a number of Each of the independent variables has a specified independent variables.
This is illustrated in figure 2-1.
probability distribution associated with it.
The theoretical bases upon which this code depends are those involving the Monte-Carlo and Stratified Sampling Techniques.
The functional relationship between the dependent variable and the independent variables depends on the safety system under consideration.
For each independent variable a set of data points is generated corresponding to the probability distribution associated with that independent variable.
The resulting data set associated with each independent variable is then randomized.
Finally the first data point in each data set is selected and all are combined according to the appropriate functional relationship.
Combining these randomized independent variables in accordance with the appropriate functional relationship results in a calculated value of a dependent variable.
This process is continued until all data in each data set have been used and the resultant The ratio of dependant variable probability histogram has been generated.
the mean value of the dependent variable to the lower 95/95 probability /
confidence limit value is the quantity of interest for a lower limit.
The analyses considered in excess of two thousand (2000) power distributions approximately equally distributed at three times in life (BOC, MOC, EOC) for a typic'al reload cycle depletion.
These power distributions were used in the determination of the 95/95 probability / confidence limit uncertainty Power distributions were generated using xenon distributions and factors.
CEA configurations that could occur during steady state operation, load maneuvers and uncontrolled axial xenon oscillations in a manner similar to that used for determination of trip setpoints.
2-2
W' 2.3.2 TM/LP Stochastic Simulation for the Tit /LP LSSS, DNB overpower (Pfdn) is the dependent variable of in-terest. The core coolant inlet temperature, reactor coolant system pressure, RCS coolant flow rate, peripheral axial shape index and integrated radial peaking factor are the independent variable of interest.
CETOP (Reference 2-7),
which is basca on TORC /CE-1 (References 2-2, 2-3), is the model used to deter-mine the fur.ctional relationship between the dependent variable and the inde-pendent variables. The probability distributions of uncertainties associated with the independent variable are discussed in Appendix A.
Figure 2-2 is a flow chart representing the stochastic simulation of the DNB limits.
The independent variables and their uncertainties are input to SIGMA.
Each data set generated by SIGMA is evaluated with CETOP until a Pfdn prob-a bility distribution is generated. The ratio of the mean value of Pfdn to the lower 95/95 value of P is the quantity of interest for evaluating a fdn lower limit.
The core coolant inlet temperature range of interest for the DNB LSSS stochastic simulation is bounded by the loci of the core power and core coolant inlet temp-eratures corresponding to:
1.
the temperature at which the secondary safety valves open; and 2.
the temperature at which the low secondary pressure trip occurs.
The reactor coolant system pressure range of interest for the DNB LSSS stochastic simulation is bounded by 1.
the value of the high pressurizer pressure trip setpoint; and 2.
the lower pressure limit of the thermal margin / low pressure trip.
2-3
~~
v
-The details of the specific TM/LP stochastic simulations performed are presented in Section 2.4.
2.3.3 Local Power Density Stochastic Simulation For the LPD LSSS, the power to fuel design limit on linear heat rate (Pfdl) is the dependent variable of interest.
The peripheral axial shape index and 3-D peak are the independent variables of interest.
The functional relationship between the dependent variable and the independent variables is (Reference 2-4):
_ (Wclm) (100) p (p_j) fdl (Fq) (Wavg)
-where:
Wcim peak generated linear heat rate limit representing centerline fuel melt Wavg -
core average generated linear heat rate at rated power Fq synthesized core power peak.
The probability distributions of each of the uncertainties associated wi.th the independent variables are discussed in Appendix A.
Figure 2-3 is a flow chart representing the stochastic simulation of the LPD LSSS.
The independent variables and their uncertainties are input to SIGMA.
Each data set generated by SIGMA is input to the functional relation-ship defined above until a Pfdl probability distribution is generated.
The ratio'of the mean value of P to the lower 95/95 value of P is the fdl fdl quantity of interest.
The details of the specific LPD LSSS stochastic simulation performed are presented in section 2.4.
2-4 Wa
2.4 ANALYSES PERFORMED
~
2.4.1 Thermal Margin / Low Pressure LSSS Uncertainty Analysis In order to combine the uncertainties as shown in Figure 2-2 the stochastic simulation sequence shown in Figure 2-4 was used.
Distributions of the following parameter uncertainties are input to the SIGMA sampling module:
At each selected value of peripheral axial shape index (I ) the representative p
axial power distribution is read from the data file.
A series of simulation trials (500-1000) is run at this I.
Each simulation trial uses one sampled p
value from each parameter distribution.
2.4.1.1 Sampling Module SIGMA The values of input parameterg selected for simulation trials are represen-tative of the actual distribution of parameter values.
The SIGMA sampling module performs this data selection using Latin Hypercube Sampling (LHS).
(Refere ce 2-5)
LHS is a stratified sampling scheme that covers the range of the independent variables with a minimum of simulation data points.
Distributional charac-teristics are input to SIGMA [
].
In LHS the range f parameter variation is divided into equal probability intervals.
In each interval a point is selected at random from the distribution.
i l'
I i
2-5 l
L'
~
~,
The specific sampling procedure used in this analysis is discussed.
e s
9 b
2-6
The specific sampling procedure used in this analysis is discussed.
The sampled values for each interval are stored in an array.
To generate sets of input values, SIGMA selects intervals at random from each variable using each interval only once in a simulation.
2-7
- 4,,,_
2.4.1.2 Axial Shape Index Calculation The axial shape seen by the excore detectors is related to the core average axial shape provided by QUIX (Reference 2-6) by several factors.
These factors are obtained by calculation or measurement and are subject to some uncertainty.
A 20-node core average axial shape is selected from [
.].
The core average axial shape index, i, is calculated from this shape.
20
- L-U I
1 L+U U=
P(J)
(2-9)
J=ll (2-10) 10 j_)P(J)
(2-11)
L=
To relate this to the peripheral shape index inferred by the excores, the following relation is used:
(2-12) 9 9
= = =
l I
2-8 1
~ - _ - _
Y
~~~~7
~
g cer____
['
] have uncertainties associated with them.
These uncertainties were used in SIGMA to generate representative values of [
J.
Using these values, corresponding values of Ip are computed to obtain a distribution of Ip.
Uncertainties in Ip affect the margin calculation by affecting the trip point selected by the on-line calculators.
To account for this, the standard deviation of the distribution of Ip is converted to overpower units using a conservative value of the sensitivity of overpower to Ip.
~
Thus the. standard deviation in overpower, o(Bopm) is (2-13)
This uncertainty in overpower due to shape index uncertainties is combined with other factors as detail,ed under Combination of Uncertainties (2.4.1.5).
2.4.1.3 Processing Uncertainties The Thermal Margin / Low Pressure (TM/LP) trip calculator receives inputs of hot and cold leg temperatures and Ip.
It uses these values and the precal-culated setpoint relation to produce a low pressure trip point.
[
] methodology is used to estimate the uncertainty due to electronic processing in this result.
This estimated standard deviation in the low pressure trip point is calculated for mean values of hot and cold leg temperatures and Ip.
To produce the pressure equivalent of the processing uncertainty, pressure values are sampled from [
.] the processing uncertainty for the low pressure trip.
2.4.1.4 Overpower Calculation with Respect to DNBR
~
Overpower limits due to reactor thermal-hydraulic conditions are determined by the code CETOP (Reference 2-7), which uses the TORC /CE-1 correlation.
2-9 b
CETOP accepts values of pressure, inlet temperature, axial shape, core coolant flow, and radial peaking factor, and returns an overpower limit.
In the simulation sequence, the input array produced by SIGMA containing values of CETOP input parameters is modified by adding an adjustment to the pressure value.
[
].
The modified pressure value, along with the other parameter values, are input to CETOP, and the resultant overpower value is available for combination with other overpower modifiers.
2.4.1.5 Combination of Uncertainties During each simulation trial k, the value of DNB overpower produced by CETOP is modified by additional uncertainty values to produce a final overpower value.
The final value is given by After all simulation trials are run a distribution in overpower is produced for each specific axial power distribution under study, incorporatinr1 all uncertainties under consideration.
2-10
2.4.2 Local Power Density LSSS Uncertainty Analysis The stochastic simulation procedure shown in Figure 2.5 was used to implement the calculational sequence outlined in Figure 2.3.
The following distributions of parameter uncertainties are input to SIGMA:
The SIGMA sampling module is described in Section 2.4.1.1.
2.4.2.1 Overpower Calculation with Respect to Linear Heat Rate For this calculation, ordered pairs of P and i values are input to the fdl code.
These are obtained from the lower bound of all the " flyspeck" points of the QUlX calculation.
[
simulation run, Pfdl, is h
(2-15)
Ite value of [
] is obtained from SIGMA for each simulation trial.
- 2. 4. 2. 2 ASI Calculational and Processing Uncertainties Tr e I used in the linear heat rate simulation is converted to a peripheral sNpe index Ip as outlined in Section 2.4.1.
If this Ip were generated fro 7 the excore detector signals, it would be subject to clactronic processing uncertainties.
The uncertainty in the simulated value of Ip is 2-11 v.
~,, _
w----
,__n__
,,~
e evaluated by a [
] methodology to estimate the uncertainty due to processing.
Values of Ip and mean hot and cold leg temperatures are evaluated to produce a one standard deviation value in Ip due to processing uncertainties.
This calculation from i to Atl is performed once for each simulation opm trial.
2.4.2.3 Combination of Uncertainties For each simulation trial, [
P
] the modified overpower value fdl.
Thus, the LilR overpower h
including uncertainties, BLilR, is h
(2-16)
Over many simulation trials, the required distribution on overpower is built up for each value of ASI incorporating the uncertainties under consideration.
2-12
~ ~ ~ - -
.. _... ~.
2.5 REFERENCES
2-1 F. J. Berte, "The Application of Monte Carlo and Bayesian Probability Techniques to Flow Prediction and Determination,"
TIS-5122, February, 1977.
2-2 " TORC Code:
A Computer Code for Determining the Thermal Margin of a Reactor Core", CENPD-161-P, July, 1975.
2-3 " TORC Code:
Verification and Simplified Modeling flethods",
CENPD-206-P, January, 1977.
2-4 CENPD-199-P, "C-E Setpoint Methodology," April, 1976.
2-5 McKay, M. D., et al., " Report on the Application of Statistical Techniques to the Analysis of Computer Codes," LA-NUREG-6526-MS, Los Alamos Scientific Laboratory, October, 1976.
2-6 System 80 PSAR, CESSAR, Volume 1, Appendix 4A, Amendment No.
3, June 3, 1974.
2-7 C. Chiu, J. F. Church, "Three Dimensional Lumped Subchannel Model and Prediction-Correction Numerical Method for Thermal Margin Analysis of PWR Cores," TIS-6191, June, 1979.
e 2-13
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UNCERTAINTIES ON o
INDEPENDENT VARIABLES O
FUNCTIONAL RELATIONSHIP PROBABILITY v
4 SAMPLED BETWEEN
. DISTRIBUTION M
INDEPENDENT SIGMA DEPENDENT AND OF DEPENDENT VARIABLES VALUES INDEPENDENT VARIABLES mg VARIA BLES
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UNCERTAINTIES ON i
i n0 INDEPENDENT VARIA BLES
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0 SAMPLED
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37 INDEPENDENT VALUES m
P i
H VARIABLES SIGMA CETOP
>P fdn
--+
fdn
[
8 SPANNING f
I RANGE OF INTEREST 95/95 m
Pfdn i
8 S3 J
7 E
P
'_Sd MAXIMUM.
O VALUE OF z
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P P
fdn pfdn H
pg fdn j
95/95 E
95/92 p
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SP fdn m
gg9 Cd I
P l
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0 v
(W Pfdf pf _
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INDEPENDENT SAMPLED P
VARIABLES 4 SIGMA VALUES >
fd (F ) (Wavg) 95/95 q
P o
fdl 5
I m
P
-o MAXIMUM 5
VALUE OF s
-O Pfdf z
a pf 95/95
[
fd P
E Pfdf fdl 4
r-
- s l
2 x
95/95 i
'N P
r-fdf m
N I
P
,7 2
BALTIMORE I
'.9""
GAS & ELECTRIC CO.
THERMAL MARGIN UNCERTAINTY ANALYSIS
~
2-4 Calvert Cliffs Nuclear Power Plant
_ g
r-i l
i 1
I BALTIMORE Fi GAS & ELECTRIC CO.
LINEAR HEAT RATE UNCERTAINTY ANALYSIS 2 qure 5
Calvert Cliffs Nuclear Power Plant
-y w
m
t 3.0 RESULTS AND CONCLUSTIONS 3.1 RESULTS OF ANALYSES i
The analytical methods presented in Section 2 have been used to show that a stochastic simulation of uncertainties associated with the local Power Density LSSS and the TM/LP LSSS results in aggregate uncertainties of [
), respectively, at a 95/95 probability / confidence limit.
Table 3-1 shows the values of the individual uncertainties which were statistically combined to yield the above aggregates.
Appendix A contains a further discussion of the bases for these individual uncertainties.
l The aggregate uncertainties are in units of percent overpower (P and fdl fdn) and are applied in the generation of the LPD and TM/LP LSSS as P
discussed below.
3.1.1 Local Power Density LSSS The fuel design limit on linear heat rate corresponding to fuel centerline melting is represented by the ordered pairs (Pfdl' I ).
A lower bound is p
drawn under the " flyspeck" data such that all the core power distributions analyzed are accommodated.
This lower bound is reduced by the applicable uncertainties and allowances to generate the LSSS as follows:
l (3-1)
(3-2) where:
LSSS B
P wer limit for LHR L5SS LHR 3-1 7 --
Statistically Combined Uncertainties Applicable to the Local
- SMLS Power Density LSSS Allowance for Transient Power Decalibration TPD S
b b ASI Axial shape index associated with B 3.1.2 TM/LP LSSS The fuel design limit on DNBR for the TM/LP LSSS is represented by a combination of the ordered pairs (Pfdn' I ) and the DNB thermal margin limit p
lines. A lower bound is drawn under the " flyspeck" data such that all the core power distributions analyzed are accommodated.
This lower bound is reduced by applicable uncertainties as follows:
(3-3) i (3-4) where:
Bopm -
Available overpower margin SMDS
- Statistically Combined Uncertainties Applicable to the TM/LP LSSS Axial shape index associated with B ASI opm*
DNB i
Both components of the TM/LP LSSS can be represented by the followinq eauations:
h im (3-5) i r
(3-6) 1 (3-7) 3-2
~
-~-
- - - - =.
.-.;-~.... -, - - _. -,
(3-8) where:
a.D.Y-Coefficients B
Core power, % of rated power
~
-Variable pressure to achieve DNB at the LSSS Limit including uncertainties var RDT -
Pressure Equivalent of the Total Trip Unit Processing Delay Time for the DBE Exhibiting the Most Rapid Approach to the SAFDL on DNBR.
LSSS B
Power level after inclusion of DNB LSSS uncertainties and DNB allowances.
TPD -
Allowance for Transient Power Decalibration LSSS DNB LSSS,0NB T
-Core inlet temperature associated with P in var DNB T
~
Inlet coolant temperature used in the calculation of (Pfdn' IP) in ordered pairs of data.
3.2 IMPACT ON MARGIN TO SAFDL The motivation for using a statistical combination of uncertainties is to improve NSSS performance through a reduction in the analytical conservatism in the margin to the SAFDL.
This section contains a discussion of the margin obtainable through a reduction in this conservatism.
Table 3-2 lists the uncertainty values previously used on the plants included in this analysis.
The approximate worth of each of these uncertainties in terms of percent overpower margin (Pfdl, Pfdn) is also shown.
3-3
The total uncertainties previously applied to the Local Power Density LSSS and the TM/LP LSSS are approximately [
], respectively.
The uncertainties resulting from the application of the statistical combination of uncertainties program are approximately [
].
The use of the-statistical combination of uncertainties provides a reduction in conservatism in the margin to SAFDL of approximately [
], respectively.
Although the conservatism in t h margin to SAFDL has been reduced, a high degree of assurance remains that the SAFDL will not be violated.
3.3 REFERENCES
3-1
" TORC Code:
A Computer Code for Determining the Thermal Margin of a Reactor Core", CENPD-161-P, July, 1975.
3-2 " TORC Code:
Verification and Simplified flodeling Methods", CENPD-206-P, January, 1977.
3-3' CENPD-199-P, "C,E Setpoint Methodology," April, 1976.
G O
3-4
- - - - ~ ~
TABLE 3-1 UNCERTAINTIES ASSOCIATED WITil THE LOCAL POWER DENSITY LSSS AND THE IM/LP LSSS Uncertainty
+ 2%
+ 2%
Primary coolant mass flow (% design)
NA Primary coolant pressure (psid)
NA Core coolant inlet temperature ( F)
NA Power distribution (peaking factor) 7%
6%
1.
Separability (asiu)
See Table 1 of Appendix Al 2.
Calibration (asiu) 3.
Shape Annealing (asiu) 4.
Monitoring system processing ((asiu)
Notes:
- For complete description of these uncertainties, see Appendix A.
- [
] values e
O 3-5
TABLE 3-2 IMPACT OF STATISTICAL COMBINATION OF UNCERTAINTIES ON MARGIN TO SAFDL Approximate Values of Equivalent Operpower Margin (%)
DNB LPD Uncertainty Value LSSS LSSS Pcwer 2% of rated Core coolant Inlet Temperature 2 F React or coolant system Pressure 22 psid Axial shape index:
Separability
[
]
Shape Annealing
[
]
Calibration
[
]
[
]
Peaking factors 6% DNB, 7% LPD Equipment processing:
[
]
LPD LSSS
[
]
Total Less credit for statistics Total Uncertainty Applied Previously Total Uncertainty Statistically Combined Net Margin Gain 3-6 j
--- - ~ ~~-
. - - ~ - = - - ~
APPEilDIX A Basis for Uncertainties Used in Statistical Combination of Uncertainties 4
O A-1
Al Axial Shape Index Uncertainties 4
6 o
A-2
.=
LIST OF TABLES 1.
Uncertainty [
] components for the Evaluation of the peripheral shape index.
2.
[
]
3.
[
]
4.
Measured Values of Shape Annealing Factors.
5.
[-
] Standard Deviation of the Shape Annealing Factor for Euch Channel.
LIST OF FIGURES 1.
[
]
Millstone II, Cycle 1.
2.
[-
]
St. Lucie I Cycle 2.
3.
[
]
Calvert Cliffs 1. Cycle 3.
m G
A-3
- ~ -...
-m..-,.,_=
p..-
Appendix Al Al.1 Objectives of this Analysis The four peripheral shape index uncertainties which are incorporated into the setpoint analyses are:
- 1) the Separability Uncertainty,
- 2) the Calibra-tion Uncertainty,
- 3) the Shape Annealing Factor Uncertainty, and 4) the Processing Uncertainty (uncertainties in the electronic procest.' 1 of excore detector signals).
Prior to the development of the metho.
y to combine these uncertainties statistically, they were combined addicively to yield a not uncertainty (Reference Al-1).
The purpose of this part of the SCU program is to develop the data base necessary to support a pro-cedure for statistically combining these four components of the axial shape index uncertainty.
Table 1 shows the values of the uncertainties developed in this program.
A1.2 General Strategy Each of the components of the axial shape index uncertainty is investigated in this Appendix in order to justify their statistical combination.
The Separability Uncertainty accounts for the difference between the core average axial shape index and the peripheral axial shape index.
This uncertainty has four components:
1.
[
]
2.
[
]
3.
[
]
4.
[
]
The Calibration Uncertainty accounts for errors introducted into the protection system when the excore' detector system is periodically adjusted to match measured parameters of the core's power distributon.
l A-4
4 t
The Shape Annealing Factor Uncertainty accounts for the error in the measurement of the shape annealing factor.
i The Processing Uncertainty accounts for the uncertainty in Ip calculated by the protection system.
This uncertainty is taken into account by its explicit representation in the stochastic simulation procedure used to statistically combine al' the uncertainties.
+
A1.3-Specific Uncertainty Evaluations
]
i A1.3.1 Separability Uncertainty The Separability Uncertainty is a calculational uncertainty.
It is the uncertainty associated with inferring a peripheral shape index, Ip, from a 4
given known core average shape index i.
The one dimensional shape analysis l
used in the development of setpoints correlates the power to centerline melt (Pfdl) and the power to DNB, (Pfdn) to the core average axial shape.
Since the excore detectors respond only to the power distribution near the 4
periphery of the core, a calculated relationship is needed between i and l
Ip.
This relationship, represented in the setpoint development by incorporation of the rod shadowing factors in QUIX (Reference Al-2), is currently calculated by means of the three dimensional code ROCS (Reference Al-3).
The' uncertainty in this calculation is the Separability Uncertainty.
'I The Separability Uncertainty consists of four components:
[
.]
The J
l components of the Separability Uncertainty are discussed in detail below.
1;-
j A1.3.1.1
[
]
Definition of the first component of the separability uncertainty.
A-5 e
_ _ _ _... _ _ _, _ _ _ _. _ _.. _. ~..
sF-
.--..-,v.,....
Rod Shadowing Factor Method The peripheral axial shape index, Ip, is defined in the following manner:
D
~U L
U 1
(AI~I) p DL+DU H
where D = f dx R(x) P (x)
(Al-2)
U H/2 H/2 D = f dx R(x) P (x)
(Al-3)
L 0
where D,D are the powers at the periphery of the upper and lower U
L half of the core, respectively.
P (x) is the core average power distribution R(x) is the rod shadowing factor for the rod configuration inserted at position x.
H is the height of the core.
The rod shadowing fcctors are derived from the product of rodded and unrodded 2D power distributions and the assembly weighting factors, which account for the contribution of each assembly to the excore detector response to a given power distribution.
6 A-6 w-
=c--
Assembly keighting Factor Method The Assembly Weighting Factor (AWF) method consists of the following calculation of Ip:
lN P I i i 5
i (Al-4)
I
=
1W P P
i i i
where P; is the axially integrated power of fuel assembly i I
is the axial shape index of assembly i j
W is the weighting factor of assembly i 9
The W values are computed for those core edge assemblies which are the 5
principal source of the excore detector's response.
The result of this procedure is [
].
O A-7 I
' ~ ~ ~ - -
.~.
- - - -. - ~
m-
- n____,
Analyses have determined this uncertainly and have shown it,to be essentially
[
] This component of the separability uncertainty is as shown in Table 1 along with the other components.
A1.3.1.2
[
]
Definition of the second component of the separability uncertainty.
[
] A review of previous cycles shows that [
] Ip is dependent on rod bank insertion.
The [
] is rod bank insertion dependent.
A[
] fit of the calculated data was performed to determine the mean which is shown in Table 2.
An error analysis performed on the difference between the calculated data and the mean shows that [t,
.] (see Table 1).
Al.3.1.3
((
]
The third component in the Separability Uncertainty consists of [.
].
The AWF method is described in section A1.3.1.1.
O A-8
_w--
m___-
. ~ ~ ~..
=-
Definition of the third component of the separability uncertainty.
A1.3.1.4
[
]
The fourth component of the Separability Uncertainty consists of the [
] the uncertainty in the calculated power distribution also results in a component of the Separability Uncertainty.
Definition of the forth component of the separability uncertainty.
e e
A-9
[
.].
The result is as follows:
(Al-5)
Since the above result also [
].
A1.3.2 Uncertainty on Ip Calibration of the excere detectors relative to the axial shape index as measured by [
] The components of this measurement uncertainty consist of the uncertainty in [
] modeling the reactor power distribution.
lae calibration is performed [
] This calibration is done near an ASI of zero so that accuracy of the shape annealing factor has minimal impact on the calibration result.
A-10 c
~ _ _
The measurement uncertainty on i is analyzed herein by [
] Differences between i [
] were studied to determine uncertainties statistically.
The mean and standard deviation of the respective differences for each cycle were calculated, af ter which the data were examined to determine whether the cyle ;y cycle data could be pooled.
Description of data used.
Results of analysis.
Table 3 shows the standard deviations of the [
] comparison of 5.
The pooled cycles whi,ch formed the basis of the above uncertainty data is also indicated in Table 3.
A1.3.3 Shape Annealing Factor Uncertainty The shape annealing factor, a, is an experimentally measured value which relates the external axial shape index l to the peripheral axial shape e
index.
I = al (Al-6) p e
This factor accounts for the fact that the excore detectors respond to the power in both the upper and the lower portion of the core.
This signal mixing yields shape annealing factors which are larger for detectors which are far from the periphery than for detectors which are near the periphery.
The theoretical lower limit of a is unity.
A-11 N
J
-=_.
The shape annealing factor is measured [
] by inducing a xenon oscillation in the core and measuring the external shape index of the jth l
J excore channel (Ie ) along with the internal axial shape index } as measured by the CECOR system using incore instruments.
The [
]
slope of i versus l j is the shape annealing factur.
At the beginning of e
lifeiisassumedtobeequaltoI.[
p
.i
] as discussed above.
Measured values of the shape annealing factor are shown in Table 4 for various C-E operating reactors.
4 l
An error analysis was performed on this data to determine the deviation of each value of a from the average values for a given plant and a given l'
channel.
The error analysis was performed on [
] The data is presented in Table 5 for all plants except for BG&E Unit 2.
For BG&E Unit 2 only one test has been performed and therefore a specific deviation from an average cannot be defined.
l j
This data was analyzed for pooling csing the Bartlett test, and for nor-mality using the W test.
It was found that the pooled standard deviation
[
] and that the corresponding Bartlett statistic [
] This is to be compared with a theoretical Bartlett statistic'at the upper 5% significance' level equal to[
].
This means that the above data i
is consistent with tiie assumption that all are samples from the same parent population.
[
4
]
[
3 l
A-12 i
-,,.. =, -. - -
.-l L
~
J X :.
Sin'e the assumption of pooling has been shown to be warranted,'[
c
]
tolerance limit can be evaluated.
Results show that [
] This K factor times the above standard deviation yields a 95/95 tolerance limit
[
]
A1.3.4 Processing Uncertainty The Processing Uncertainty is discussed in Appendix A3.
Al.4
[
.] of the Peripheral Shape Index Uncertainties The following [
] have been identified in the development of peripheral shape index uncertainties.
DiscJssion of the components of the peripheral shape index uncertainties.
l A-13 7
.:-.-,...,..---.-~
Discussion of the components of the peripheral shape index uncertainties.
Equation Al-10 is an identity.
Equation Al-ll follows from the assumption that [
.].
Equation Al-12 and the results summarized in Table 1 are used in the stochastic simulator described in Section 2.4 of this report.
A-14 m
A1.5 References Al-1 "C-E Setpoint Itethodology," CEf1PD-199-P, April,1976.
Al-2 System 80 PSAR, CESSAR, Volume 1, Appendix 4A, Amendment tio. 3, June 3, 1974.
Al-3 BG&E Application for Cyria 4 Reload, AE Lundvall (BG&E) to R. W. Reid (f1RC), Februc, 23, 1979.
Al-4 " INCA, Method of Analyzing In-Core Detector Data in Power Reactors," CEllPD-145-P, April, 1975.
Al-5 " Evaluation of Uncertainty in the riuclear Form Factor Measured by Self-Powered Fixed In-Core Detector Systems,"
CEllPD-153, August, 1974.
9 O
e A-15
l Tabic 1 Uncertainty [
] Components for the Evaluation of the Peripheral Shape Index(I)
Ko 95/95 (asiu)
K(f)(2)
[
_]
I.
Separability Uncertainty II.
Calibration Uncertainty (")
III. Shape Annealing Uncertainty (")
IV.
Processing Uncertainty (")
Notes On Table 1 (1) All components of the peripheral shape index have been tested for normality, [
]
(2) f = degrees of freedom.
(3)
[
]
(4) This Ko95/95 is for consistent sets of input data used by the uncertainty processors.
A-16
Table 2 Rod Bank Insertion
[ Generic QUIX Bias, asiu]
All Rods Out (AR0)
Reg Bank 1 (20%)
Reg Bank 1 (40%)
Reg Bank 1 (60%)
Reg Bank 1 (80%), Reg Bank 2 (20%)
Reg Bank 1 (100%), Reg Bank 2 (40%)
Reg Bank 1 (100%), Reg Bank 2 (60%)
Reg Bank 1 (100%), Reg Bank 2 (80%), Reg Bank 3 (20%)
Reg Bank 1 (100%), Reg Bank 2 (100%), Reg Bank 3 (40%)
Reg Bank 1 '(100%), Reg Bank 2 (100%), Reg Bank 3 (60%)
O e
A-17
~
Table 3
~
Mean Standard Number of
- Value, Deviation, Reactor Data Points asiu asiu 1.
St. Lucie I Cycle 1 2.
St. Lucie I Cycle 2 3.
Calvert Cliffs I Cycle 1 4.
Calvert Cliffs I Cycle 2
- 5. ' Calvert Cliffs I Cycle 3 6.
Calvert Cliffs II Cycle 1 7.
Calvert Cliffs II Cycle 2 8.
Millstone II Cycle 1 9.
Millstone II Cycle 2 O
]
A-18
s_
. ~,......
Table 4 Measured Values of Shape Annealing Factors St. Lucie 1 Cycle 1 Cycle 1A Cycle 2 Cycle 3*
June 1976 Jan 1977 June 1978 June 20, 1979 C3innel 50%
Power 50% Power 80% Power 80%
Power
- Note that a new streaming shield was placed in St. Lucie I at E0C2.
This new streaming shield changed the shape annealing factors.
Calvert Cliffs Unit 1 Cycle 1 Cycle 2 Feb 1975 April 7, 1977 Channel 80% Power 50% Power k
S A-19
a:
Table 4 (Continued)
Calvert Cliffs Unit 2 Cycle 1 Dec 27, 1976 C ha n.ne_l_
50% Power Millstone Point 2 Cycle 1 Cycle 1 Feb 6-9, 1976 March 11, 1976 Channel 50% Power 80% Power lameuum M
A-20
~
Table 5 L
3 Standard Deviation of the Shape Annealing factor for Each Channel
[
]
Plant &
Number of Standard Deviation per Channel Channel Degrees of Freedom f
St. Lucie 1 Calvert Cliffs 1 Millstone Point 2 L_
t 1
l A-21
-.. - - Q
o z
c t
- e. n
? ?- D' O.03 i
i i
i i
i
,e l:
F: ;:;5 s n-3 i
2=50 v.
,,a n g
/
n o
i a
P 0.02 i
c I
I 1
e f
0.01 t
c.70.0 i
I~
i o
l
-0.01
-0.02 l!
l i
1:
-0*03 O
2000 4000 6000 8000 10000 12000 14000 16000
>a 7i BURNUP, MWDIT
~ n
phares e
e e
Z E
c
- 2. n
- o1 5
,' s P,, r-O p
Eny Q=EO 3,= n g i
o n
E O
I I
e 0.01 i
i i
i i
i i
- 0. 0 e.
H l~
0.01
-0.02 I
-0.03 i
i i
i i
i i
i i
I O
1 2
~
3 4
5 6
7 8
9 BURNUP, GWDIT
> m i
~o i
d C
t-O Z
c
- e. n *
'l 0O pa cp>
.a :r r~
a r-r gagg
<n-a=ao s =. n =
m n
a.
o 1
i 1
I f
0.02 I
I p
U.
0.01 1.
I ;
O. 0 a
I I
1-I' i
-0.01 I
r
-0.02 I
i i
i i
i i
-0.03 0
1 2
3 4
5 6
7 8
9
>m 7i BURNUP, GWDIT wa v
/.
A2 Measurement Uncertainties e
l l
l A-25 l
l
_______..w w w,_-
Appendix A2 A2.1 Basis for Flow Uncertainty The flow rate was determined by an evaluation,of calorimetric data taken from the Calvert Cliffs Nuclear Power Plant at approximately 100% reactor power.
Uncertainty in that flow rate was evaluated by examining the uncertainties in each input parameter used in the flow determination.
The inputs include hot and cold leg RTD temperatures, system pressure, and core
~
thermal power.
The core thermal power is based on a secondary side calorimetric measurement.
Each component uncertainty was first evaluated and then the net effect of all instrumentation inaccuracies on calculated flow rate was determined [
-].
The resulting overall [
] uncertainty was found to be
[,
) of the flow rate.
A2.2 Monitored Thermal-Hydraulic Parameter Uncertainty Distributions The uncertainty distributions previously used to characterize the inputs to the safety analyses and setpoint thermal-hydraulics modules were based on highly conservative assumptions.
Table 1 outlines these distributions.
It is now possible to refine these distributions using more detatled system analysis and observed plant data.
Updated distributions representing more detailed system analysis and measured data from the Calvert Cliffs Nuclear Power Plant have been examined to define specific contributors to the total uncertainty and dependencies between parameters.
The uncertainty distributions shown in Table 2 represent the results of this detailed systems analysis.
^
l l
l l
l A-26
[~
-. _ _ =
Measurement of these parameters' uncertainties show both random and I
nonrandom component' which are so small that their most adverse cont. rib-utions are fully covered by the uncertainties of Table 2.
The degree of i
I dependency found is so small that, in conjunction with the size of the evaluated uncertainties, the assumption of independence amoung the
}
parameters of Table 2 is justified.
Therefore, for the purposes of the,
1
- statistical contribution of uncertainties evaluation reported herein, the uncertainties of Table 2 can be used in the stochastic simulation model.
4 A.2.3 Power Peaking Factor Uncertainties l
The 3D Power Peaking Factor Uncertainty (F ) and the Integrated Radial Power q
Peaking Factor Uncertainty (F ) are currently being re evaluated in response R
to NRC questions regarding C-E's uncertainty topical report (Reference A2-1).
Pending resolution of these questions and approval of the topical report, C-E will continue to use the values listed in Table 3.
These values are used in the stochastic simulator described in this report.
References A2-1 " Evaluation of Uncertainty in the Nuclear Form Factor Measured by Self-Powered Fixed In-Core Detector Systems" CENPD-153, August 1974.
6 A-27 y_
7
TABLE 1 Previously Assumed Uncertainty Distributions on Monitored Thermal / Hydraulic Parameters Parameter Distribution O
=
l em Note:
[
]
- This value includes measurement and processing system uncertainties.
TABLE 2 Results of Detailed Syst(ms Analysis of Monitored Thermal /Hydraelic Parameters Parameter Distribution Note:
[
]
- This value is a measurement uncertainty only.
Processing system uncertainties are handled separately.
TABLE 3 Peaking Factor' Uncertainties Peaking Factor Uncertainty (% of Power)
F R 6.0 Fq 7.0 A-28
A3 Trip System Processing Uncertainties 91 e
A-29
,,_ _.-,_ _ _ _m,.-,-
Appendix A3 A3 Trip System Processing Uncertainties Two types of instrument errors are considered in this analysis.
First are those errors that are random in nature.
The basic accuracy of an instrument or component falls into this category as it is dependent upon such fact' ors as manufacturing tolerances, etc.
Second are those errors that are deter-ministic and present in approximately the same degree in any equipment built to.a given design.
Examples of this type of error are changes due to temperature, changes under force loads etc.
The reason for considering two types of errors is that the mathematical techniques for combining errors from several sources differs for each type of error.
The deterministic errors are combined using the governing equations and the techniques of ordinary algebra, while the random errors are best combined usirg probabilistic methods.
The method of determining the random error of an instrumentation loop is based upon two approximations.
The first approximation is that the errors of the v.'.rious pieces of equipment are independent.
The second approx-imation that is used in the analysis is that the equations which define the relationships between the variables in the instrumentation loop can be approximated by the linear terms of a Taylor series expansion.
This is a good approximation because the errors are very small in relation to the overall range of the quantities in question and cause only small perturba-tions about the nominal value.
^
The procedure followed in calculating the variance consists of obtaining the partial derivatives of the system or instrument equation with respect to each of the variables and evaluating them at the nominal values.
These partial derivatives are then used to calculate the variance.
A-30
This method of determining the variance of a function of several variables was arrived at without placing any restrictions on the probability distri-butions of the variables involved, hence the method is generally applicable.
Having obtained the variance, its significance can only be interpreted in terms of the distribution to which it applies.
The probability distribution of a function that is dependent upon several variables is dependent upon the distribution of those variables.
However as the number of variables increases (such as that obtained by using the previously described method),
the resulting distribution tends to a normal curve (this is the Central Limit Theorem).
If the probability densities of the variables are reasonably concentrated near the nominal values [
w*
The instrument errors are calculated in the stochastic simulation procedure.
In this computerized error analysis, a subprogram is used for each type of' module -(i.e., power supply, mul tiplier/ divider, adder /subtracter, etc. )
Each subprogram accepts the input voltages and errors (in volts) for its module and determines the outputs of the module and their associated errors.
The simulation then goes through the calculator, module by module.
As each module is reached, the appropriate subprogram is called.
The module inputs are obtained from the outputs of the modules which feed it.
e A-31
=
APPEt1 DIX B Summary of Previous flethods for Combining Uncertainties 5
h e
B-1
Appendix B The methods previously used for the application of uncertainties to the LSSS are presented in Reference B-1 and are summarized in this Appendix.
B.1 Limiting Safety System Setting on Linear Heat Rate (LPD LSSS)
The fuel design limit on linear heat rate at fuel centerline melt is represented by the ordered pairs (Pfd1' I ).
A 1 wer b und is drawn under p
this " flyspeck" data such that all the core power distributions analyzed are accommodated.
Using the previous methodology this lower bound was reduced by the applicable uncertainties and allesances to generate the Local Power Density LSSS as follows:
(B-1)
(B-2) where:
T
- Azimuthal Tilt Allowance AZ PU - Uncertainty in predicting local core power at the fuel design linit BMU - Power measurement uncertainty SAU - Shape annealing factor uncertainty RSU - Shape index separability uncertainty ACU - Axial shape index calibration uncertainty APU - Processing uncertainty B.2 Limiting Safety System Setting on DNBR (TM/LP LSSS)
The fuel design limit for the TM/LP trip on DNBR is represented by a the DNB TMLL.
A lower combination of the ordered pairs (P p) fdn' bound is drawn under the " flyspeck" data such that all the core power B-2 I
'I distributions analyzed are accommodated.
Using the previous methodology this lower bound was reduced by applicable uncertainties and allowances as follows:
[
(B-3) i 9 (B-4) here:
w SC - approved partial credit for conservatism in uncertainty application.
Both components e( the DNB LSSS were then represented by the following equations:
[
(B-5)-
(B-6)
I i
I (B-7) i (B-8)
I where:
RDT Pressure equivalent of the total trip unit and processing delay time for the 087.xhibiting the most rapid approach to the SAFDL on DNBR i*
i PMU -
Pressure measurement uncertainty i
TPU Processing uncertainty l
B-3 i
L_
.__.m.m_~,.,,-----
w__m-_.
=, -.
. w --
=m
y Power r.ieasurement uncertainty BMU TMU -
Temperature measurement uncertainty REFEREf4CE B-1 CEf4PD-199-P, "C-E Setpoint Methodology," April, 1976 6
4 S
b B-4
_ _ _.. _. _ _ _.. - - _.