ML20087G110
| ML20087G110 | |
| Person / Time | |
|---|---|
| Site: | Beaver Valley |
| Issue date: | 02/28/1995 |
| From: | DUQUESNE LIGHT CO. |
| To: | |
| Shared Package | |
| ML20087G108 | List: |
| References | |
| NSD-SGD-1081, NSD-SGD-1081-R01, NSD-SGD-1081-R1, NUDOCS 9503130149 | |
| Download: ML20087G110 (34) | |
Text
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[. ,1,5 ,e NSD SGD 1081 Revision 1 b
3 BEAVER VALLEY UNIT 1.
INTERIM PLUGGING CRITERIA RETURN TO POWER LETTER REPORT FEBRUARY 1995 3
9503130149 950302 PDR ADOCK 05000334; P PDR
5 3.- BEAVER VALLEY UNIT 1 INTERIM PLUGGING CRITERIA RETURN TO POWER LETTER REPORT FEBRUARY 1995 TABLE OF CONTENTS ' .
i 1.0 . Introduction +
2.0 Summary and Conclusions 3.0 Beaver Valley Unit-11995 Pulled Tubes 4.0 EOC-10 Inspection Results and Voltage Growth Rates 4.1 EOC-10 Inspection Results 4.2 Voltage Growth Rates 4.3 NDE Uncertainties 5.0 Data Base Applied for IPC Correlations 6.0 SLB Analysis Methods 7.0 Projected EOC Voltage Distributions 7.1 Comparison of Actur.1 and Projected EOC-10 Voltage Distributions 7.2 Projected EOC-11 Voltage Distributions 8.0 SLB Leak Rate and Burst Probability Analyses 8.1 Comparison of Projected and Actual EOC-10 Leak and Burst 8.2 Projected EOC-11 Leak Rate and Burst Probability 9.0 References 2
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BEAVER VALIRY UNIT 1 =
INTERIM PLUGGING CRITERIA D.ETURN TO POWER LETTER REPORT:
1.0 INTRODUCTION
. This report provides the Beaver Valley-1 steam generator steam line break.(SLB) leak rate and tube burst probability analysis results in support of the implementation of a 1.0 volt Interim Plugging Criteria (IPC) at end of cycle 10 (EOC 10). Information - t required by the NRC Safety Evaluation Report (SER) prior to return to power is included in this report. The analysis results are provided for SG ~A, which is the limiting SG for the actual EOC-10 and projected EOC 11 bobbin voltage distributions.
The results of the EOC-10 inspection are provided in Section 4. At EOC-10, plugs ,
were removed from previously repaired tubes, the tubes were reinspected, and tubes with indications satisfying the IPC repair limits were returned to service. The indications returned to service are included in the Cycle 11 analyses. Comparisons of the EOC-10 voltage distributions as well as leak rates and tube burst probabilities calculated for the actual distributions are compared with the projections to EOC-10 ,
. previously reported in the Beaver Valley 1 IPC technical support report, WCAP-14123, Reference 9.1. - Leak rates and burst probabilities for the projected EOC-11 voltage distributions are reported in Section 8 and compared with allowable limits.
Analysis methods are consistent with the NRC SER and WCAP-14123. The methods are described in more detail in the Westinghouse methods report, WCAP-14277, Reference 9.2.
Three tubes were pulled that provide data supporting the APC database. Eddy current data for the TSP indications on the pulled tubes are given in Section 3. ;
2.0
SUMMARY
AND CONCLUSIONS P SLB leak rate and tube burst probability analyses were performed for the actual EOC-10 and projected EOC-11 voltage distributions. SG A was found to be the limiting SG for both the EOC-10 inspection and the projected EOC-11 distributions.
For the actual EOC-10 distribution, the SLB leak rate is estimated to be 0.15 gpm and the burst probability is 3.84 x 10'8. These values are lower than projected for EOC-10 in WCAP 14123 even for an assumed probability of detection (POD) of 1.0 <
at EOC 9. The projected EOC-11 distributions with the NRC SER required POD =
0.6 result in a SLB leak rate of 0.31 gpm and a burst probability of 9.7 x 10. All results are much lower than the allowable SLB leakage limit of 6.6 gpm and the NRC .'
reporting guideline of 10 2 for the tube burst probability. f 3 ,
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s,g Comparisons of the EOC-10 projections with the actual distribution for SG A'show l that a POD = 1.0 results in an over prediction of the indications > 0.8 volt'and an !
i under prediction below 0.8 volt, while the POD = 0.6 substantially overestimates the
. actual distribution above 0.5 volt. These results show the importance of applying a voltage dependent POD and adjusting RPC NDF (no degradation found) indications left in service by the fraction ofindications that may become con 6rmed at the end of the next operating cycle. These compansons of projections with 'the actual l distribution imply a POD approaching un%y above about 1.0 volt and about 0.6 at about 0.5 volt.
A total of 1089 indications were found in the EOC 10 inspection of which 152 were RPC inspected (including all indications above 1.0 volt) and 88 were con 6rmed as Daws by the RPC inspection. The RPC confi med indications included 66 above 1.0 ;
volt. SG A had 484 bobbin indications of which 73 were above 1.0 volt and 41 of the 73 were confirmed by RPC inspection. During the inspection,242 previously plugged ,
tubes were deplugged and inspected for possible return to service based on the 1.0 volt IPC. A total of 164 tubes (80 in SG A) with 223 indications (113 in SG A) were found to satisfy the IPC repair limits and were returned to service. This resulted in a total of 1201 indications returned to service including 535 in SG A. No unexpected inspection results were found at the TSP intersections such as circumferential indications, indications extending outside the TSP or PWSCC at dented TSP t
intersections.
Thrae tubes with eight TSP intersections were pulled during the outage to provide data to support the EPRI IPC/APC correlations. Five of the eight intersections had >
Seld reported flaw indications and a sixth TSP intersection ~had a small 0.29 volt indication found by reevaluation of the field data and also found by the UT and Cecco probe inspections. The bobbin Daw voltages for these indications ranged up to 1.08 volts. One intersection had a mixed residual. signal of 1.73 volts with a flaw indication of 0.62 volt. Post pull eddy current data available at the time of this report shows modest and acceptable changes in voltage. The post pull RPC shows changes in crack indication features typical of that found previously for indications with cellular patches. The field RPC and UT data also include volumetric features typical of cellular corrosion. All indications found by the 6 eld RPC UT inspection were found by the Cecco probe and include one indication not reported by the 6 eld l bobbin and RPC inspection. The post pull bobbin data also indicate another potential small indication not identified by the pre-pull bobbin, RPC, UT or Cecco inspections, which could be indicative of a small cellular patch opened up by the tube pulling operations. The tube pulling resulted in some denting or bending at the TSP intersections but of a modest magnitude such that the indications appear to be acceptable for application to the EPRI database.
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Lab. Reevelection of Fleid Dets Fest Pelt Data T Field CaN Tube S BobMe RFC ITT Ceece BobWe RFC UT Cecco BobMe ASME BobMe Depth RPC P
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- 3 Coil 0.70 1.029 0.72 37 % 03 0.9 0.55 R22C38 1 0.64 0.26 mat 5 Coil 0.52 1.029 0.54 76% 0.I4 0.7 038 2 0.44 3 1.73* 035 MAI 3 Coil 0.60 1.029 0.62 DI 0.5 2.I 0.7 ;
0.72 039 MAI 5 Coil 0.56 1.029 0.58 66 % 0.29 DI 0.5 R28C42 1 2 1.12 0.19 sal
- 7 Coil 1.05 1.029 1.08 53 % 032 I.0 0.66 NDD NDD NDD - - 0.2 0.6 Noisy m 3 NDD NDD -
- NDD NDD mal 2 Coil 0.28 1.029 0.29 24 % NDD 0.45 R10C48 i NDD NDD NDD NDD NDD - - NDD Dent NDD 2 NDD -
Notes: 1. Field data include cross calibration of ASME standard to t?e reference id::Aj seendard
- 2. ASME calibretion represents the cross calibrasion betor foi the field ASME standard to the reference laborneory seendard and is !
applied to the !d::ej reevaluation to obtain the cervected APC volts
- 3. Bobbin voltage included mixed residual to aneurs that indecatkn was RPC ic_;::^=-3 (i.e., > 1.0 volt). Flaw component of 0.6 volt from reevaluation is reccommended for ARC databene
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,. 3.0 BEAVER VALLEY UNIT 11995 PULLED TUBES Three tubes were pulled in the EOC-10 outage. The amiated eddy current data for the eight TSP intersections on these tubes are given in Table 31. NDE data were taken in the 6 eld by bobbin, RPC, Cecco and UT probes. The results for the different probes are generally consistent except for the first TSP of R10048 which was NDD for the field bobbin and RPC probes but reported by the Cecco and UT probes as axialindications.
Laboratory review of the field data and the post-pull bobbin data indicate a small 0.29 volt bobbin indication (See Figure 3-1) for R10C48, TSP 1. In addition, laboratory review of the 6 eld RPC data indicated a possible small indication at R28C42, TSP 3, which is seen in the post pull bobbin but not identified in the field calls for either of the four probes. The laboratory review and field bobbin voltages are in reasonable agreement and, except for R22C38, TSP 3, show modest differences due to analyst interpretation of the distorted bobbin responses. The field call for R22C38, TSP 3 was '
intentionally called to include the residual bobbin signal to assure that the indication was included in the RPC program for residual signale, that could mask a bobbin signal near one volt. The flaw component of this indicatica as obtained from the reevaluation of the field data is 0.62 volt (See Figure 3 2). The laboratory reevaluation of the field voltages is recommended for the ARC voltage, consistent with prior tube pull evaluations, to minimize analyst variability in the database voltages since analyst variability is a component of the NDE uncertamty used for ARC analyses.
The RPC and UT responses indicate a high likelihood of cellular corrosion patches which can lead to some distortion in the bobbin responses. Figure 3 3 shows the pre-pull RPC inspection data for R28C42, TSP 2. The broad angular involvement shown in Figure 3-3 is typical of that found for cellular patches. Similarly, the UT results indicated the presence of many small indications in addition to the larger macrocrack associated with the peak RPC response.
The Cecco field calls are consistent with the UT response in that both probes identify '
the same TSP intersections with indications. The larger number of coils with flaw signals may also be indicative of possible cellular involvement. However, there are very limited pulled tube TSP intersections, prior to this data, to support the interpretation of Cecco probe responses.
As a result of the tube pulhng operations, the post pull bobbin data show dent signals at most of the TSP intersections. These dent signals were not present in the pre pull bobbin data. As a consequence, the post pull bobbin data have somewhat reduced
- reliability for assessing the differences between pre pull and post pull bobbin voltages although the dents are less than 5 volts, and bobbin voltages can be adequately identified. The only indication with a significant difference between pre pull and post-pull bobbin data is R22C38, TSP 3 which shows an increase from 0.62 to 2.1 volts. In addition, R28C42, TSP 3 shows a post-pull bobbin indication which was not identifiable in the pre pull bobbin data although indicated by laboratory reevaluation of the Seld RPC data, l
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4.0 EOC 10 INSPECTION RESULTS AND VOLTAGE GROWTH RATES 4.1 EOC-10 INSPECTION RESULTS In accordance with the IPC guidance provided in the SER of Reference 9.3, this End Of Cycle 10 (EOC-10) inspection of the Beaver Valley Unit 1 steam generators (SG) consisted of a complete,100% Eddy Current Test (ECT) bobbin probe full length examination of all TSP intersections in the tube bundles of the three SGs. A 0.720 inch diameter probe was used for all hot leg TSP indications where APC was applied.
Subsequently, Rotating Pancake Coil (RPC) examination was performed for all bobbin indications with amplitudes > 1.0 volt. RPC confirmed indications > 1.0 bobbin volt were repaired. In addition, an augmented RPC inspection was performed consistent with the NRC SER requirements. The augmented RPC inspection included all TSP intersections (10 intersections) with dent voltages > 5.0 volts. In addition, the '
augmented RPC program included.161 INR (indication not reportable) and INF (indication not found) calls at supports which were DSI calls at EOC-9 but not confirmed by RPC inspection at EOC 9. Any suspected artifact signals at TSP intersections that could mask a greater than one volt flaw indication were either called DSI (distorted support indication) over 1.0 volt and inspected as part of the base RPC program or called INR and RPC tested as part of the augmented program. No RPC Daw indications were found in the augmented program. There was no evidence of any unexpected eddy current results at EOC 10. There were no RPC circumferential indications, no indications extending outside the TSPs, no RPC indications with potential PWSCC phase angles, no flaw indications at dented TSP intersections of any dent voltage and no flaw indications were found in the augmented RPC inspection. All RPC responses were consistent with that expected for ODSCC at TSP intersections.
A summary of ECT indication statistics for all three steam generators is shown on Tables 4-1 and 4-2. For those tubes that were in service during cycle 10, Table 4-1 )
tabulates the number of field bobbin indications, the number of these field bobbin indications that were RPC inspected, the number of RPC confirmed indications, the number of repaired / plugged indications, the number of in-service EOC-10 indications that remain active for cycle 11 (BOC-11) and the total number of BOC-11 indications )
including deplugged tubes that were returned to service.
During this outage, some tubes that had previously been plugged were deplugged, inspected, and either returned to service or replugged, depending on inspection results.
Table 4-2 provides the same statistics for the population ofindications in deplugged tubes as Table 4-1 provides for those in service during cycle 10; together th'ey comprise the population being returned to service for cycle 11. Together, these two tables show that:
Out of a total of 1443 indications identified during the inspection (1089 from in-service EOC-10 and 354 deplugged EOC-10), a total of 1201 indications (978 from in service EOC-10 and 223 deplugged EOC-10) were returned to service for cycle 11.
Of the 1443 indications, a total of 506 were RPC inspected (152 from in-service EOC-10 and 354 deplugged EOC-10).
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Of th) 506 RPC inspected, a total cf 362 wsre RPC confirmed (88 from in service EOC 10 and 274 deplugged EOC-10).
-A total of 242 indications were removed from service (111 from in service EOC-10 ~
and 131 deplugged at EOC-10). The RPC confirmed but not removed from service l indications have bobbin amplitudes of $ 1.0 volt.
J An interesting summary of the implication of these inspection and repairs is shown on Table 4 3. A total of 94 tubes (corresponding to 111 total repaired indications) that were !
in service in cycle 10, which exhibited TSP ODSCC indications, were removed from '
service. The benefit of reevaluating prior tube plugging decisions based on implementing the 1.0 volt IPC is shown by the return to service 4164 previously ,
plugged tubes (corresponding to 223 total indications), which offsets the tubes that were l- ,
repaired for TSP ODSCC indications at EOC-10. (Upon reinspection in 1995, no indication was found in two of the deplugged tubes. Consequently, these two tube TSG B R31 CIS and SG B R40 C49) are not included in the Table '4-3 compilation. l !
Additionally, seven tubes in SG A and six tubes in SG B were deplugged but not reinspected and were replugged; these 13 tubes are not included in Table 4-3.)
i Review of Tables 4-1 and 4-2 indicates that steam generator A has more in service EOC- i 10 indications than SG B or C. Steam generators A and B have essentially equivalent deplugged EOC 10 populations included in their BOC-11 distributions, while SG C has none. Accordingly, the total BOC-11 indication population of SG A exceeds the other two .
and is considered the limiting BOC-11 SG for purposes of Monte Carlo calculations for cycle 11. !
i The data shown in Tables 4-1 and 4 2 is also shown in graphical form in Figures 41 to l 4-4. Figure 4-1 shows the bobbin voltage distribution for the 'in service EOC-10 l indications which were returned to service for cycle 11; Figure 4 2 shows the bobbin
- voltage distribution for the deplugged EOC-10 indications which were returned to service for cycle 11; Figure 4 3 shows the bobbin voltage distribution for the total population ofindications which were returned to service for cycle 11. Figure 4-4 shows the repaired population distribution for those in service EOC-10 indications which were I
plugged.
l The distribution of in-service EOC-10 indications as a function of support plate elevation, shown on Figure 4-5, confirms the presence of ODSCC predominately in the first few hot leg TSPs. This distribution indicates a strong temperaturo dependence of ODSCC at Beaver Valley-1.
4.2 VOLTAGE GROWTH RATES l
The voltage growth rates were developed from the February 1995 inspection data, ;
compared to a reeva'wtion of the same indications from the previous (1993) inspection.
The cumulative probability distributions for the Cycle 10 growth rates are presented in Table 4 4. It is seen that the maxunum growth rate for Cycle 10 is only 0.6 volt.
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Average growth rates in each SG for Cycle 10 are shown in Table 4-5. The average growth rates vary between 0% and 7% between SGs. The average growth for indications
> 0.75 volt is s 2% and significantly smaller than found for indications < 0.75 volt.
Table 4 6 compares th., average growth rates for Cycles 7 to 10. The data show a progressively decrea' mg growth rate with the implication that chemistry enhancements i have been effective in reducing the growth of ODSCC indications at the TSP !
intersections. Between cycles 9 and 10, the average growth decreased from 16% to 3%.-
The guidance of the NRC draft generic letter recommends that the more conservative growth distribution from the last two' cycles be used for projecting EOC distributions.
Table 4-7 shows the cycle 9 growth distributions which indicates larger growth rates than found for cycle 10. The largest growth value for cycle 9 was 1.2 volts. For conservatism consistent with the NRC guidance, the cycle 9 growth distribution of Table 4-7 are used for the cycle 11 projections.
4.3 NDE UNCERTAINTIES The NDE uncertainties applied for the EOC-11 voltage projections in this report are those given in the Beaver Valley 1 IPC report, WCAP-14123. The probe wear uncertainty has a standard deviation of 7.0 % about a mean of zero and has a cutoff at 15 % based on implementation of the probe wear standard. The analyst variability uncertainty has a standard deviation of 10.3% about a mean of zero with no cutoff. ;
These NDE uncertainty distributions are included in the Monte Carlo analyses used to project the EOC 11 voltage distributions. i i
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- II * * *
- II ;
- on. .. ) _ . _ _ . . . .
1]
3
)I 9
o I g, .
g
'l s i o --
ei l ii n . . is lj
> rj I l! l i l l l! ll l! Ll!! 11 8
i :: ! !) :: : , : : :::: a E
vi
$a7 !a gl 5 vi ha7 !
a 3 4,
i l
\
Table 4 3 15
Y 5
t a
f
?
Table 44 _.
Beaver Womey Unit 1 IPC Semesecs 1995 Outage cuanutstive ?._'
,- Oteerthugons for Voltage Growen -Indicellons Easneining in Service Cesninned SMB B SMB C Dette SMB A CPDF No.chs CHF No.obs CPDF No.obs Voltage No. obs CPDF 43.07 145 38.25 72 35.12 400 0 252 52.07 80.44 153 74.50 91 79.51 407 0.1 163 85.74 93.94 91.75 33 95.61 147 0.2 45 95.04 89 97.89 5 0.3 16 98.35 23 97.50 4 97.58 43 17 99.45 99.79 7 99.25 3 90.02 0.4 7 99.72 99.75 99.51 3 0.5 0 99.79 2 1 100.00 3 100.00 0.6 1 100.00 1 100.00 1 400 205 1089 484
+ -
k-e
$2 ( f!2 868
, j p :: ::: ::
i l t i s ei s s i s a
i j 555 555 555
, p, 2 : s i ss i si i
~
l l i ,
, , e . .. . . .
I i .. - , ,,
1 1 Table 4-5 j
17
l Y
l
!F a
6 Tatde 4-4 _
Beaver Vaney Unit 1 Histodcal Bobtin V@c M l vemese on=mi Aversee dy I
soc Vensee Ave. Sed.Dev Ave. Std. Dev %#Cyck Nasaber ofIneNcaGens Enere Range 1000 0.08 0.31 0.02 0.12 3%
Cycle 10 __
751 0.50 0.15 0.04 0.11- 7%
1993-1995 Vooc<.78 338 1.01 0.29 -0.01 0.15 -1%
l Vecc_>.78 Endre Range 1125 0.57 0.27 0.00 0.23 16 %
- Cycle 9
- 918 0.47 0.14 0.00 0.20 19%
1991-1993 Vecc<.75 207 1.02 0.30 0.00 0.31 6%
Weoc>.78 952 0.95 0.44 0.18 0.24 19% -
Cycle 8 Endre Range 308 0.58 0.12 0.16 0.19 28%
1989-1991 Vooc<.75 586 1.18 0.41 0.19 0.26 16 %
Vooc_>.78 918 0.08 0.31 0.29 0.27 41 %
Cycle 7 Enske Range 622 0.49 0.15 0.27 0.22 55%
1987-1989 Vecc<.75 1.01 0.28 0.34 0.33 34 %
Vooc_>.78 298 l
-- %' xgu-;
5*
- ?
5 Table 7-6 Beever Vallely Unit 1 Cumulative Probability Distributions for Vollege Growth g l l 1991 to 1993 Laboratory Re-evaluation 1987-1989 1989-1991 S/G A S/G B S/G C Combined Data
, Volta 9e # obs CPDF # obs CPDF # obs CPDF # obs CPDF # obs CPDF #obs CPDF 0 86 9.37 195 20.48 150 26.27 146 50.17 108 40.30 402 35.73 0.1 119 22.33 172 38.55 129 48.86 80 70.79 ~ 69.96 78 - 267 59.47 0.2 171 40.96 184 57.88 111 68.30 48 87.29 39 84.79 198 77.07 0.3 155 57.84 154 -74.05 62 79.16 19 93.81 23 93.54 104 86.31 0.4 150 74.18 114 86.03 45 87.04 9 96.91 11 97.72 65 92.09 0.5 82 83.12 59 92.23 22 90.89 8 98.97 4 99.24 32
' 94.93 O.6 55 89.11 28 95.17 22 94.75 2 99.66 1 99.62 ' 25 97.16 0.7 32 92.59 20 97.27 7 95.97 0 99.66 100.00 8 1 97.87 0.8 20 94.77 12 98.53 9 97.55 100.00 0 100.00 1 10 98.16 0.9 21 97.06 6 99.16 8 98.95 0 100.00 0 100.00 8 99.47 1 8 97.93 3 99.47 4 99.65 0 100.00 0 100.00 4 99.82 1.1 7 98.89 1 99.58 0 99.85 0 100.00 100.00 0 0 99.82 ~
1.2 4 99.13 0 99.58 2 100.00 0 100.00 100.00 0 2 100.00 1.3 4 99.56 1 99.68 0 100.00 0 100.00 0 100.00 0 100.00 l
G 1.4 1 99.67 1 99.79 0 100.00 0 100.00 -0 100.00 0i 100.00 1.5 2 99.89 1 99.89 0 100.00 0 100.00 0 100.00 0 190.00 1.6 0 99.89 1 100.00 0 100.00 0 100.00 0 100.00 0 100.00 ,
1.7 0 99.89 100.00 0 .100.00 0 100.00 0 100.00 0 100.00 -
1.8 0 99.89 100.00 0 100.00 0 100.00 0 -100.00 0 100.00 '
1.9 0 99.89 100.00 0 100.00 0 100.00 0 -100.00 0 100.00 <
2 0 99.89 100.00 0 100.00 0 100.00 0 100.00 0 100.00 -!
2.1 1 100.00 100.00 0 100.00 0 100.00 0 100.00 0 100.00 918 952 571 291 283 1125 A
EXTRACTED FROM WCAP - 14123 (SG-94-07-009)
TA BLE 4-7 .
W w'".
.. _ . w - - , , w ,, - - ,w . - . -
y ,n- ,-. v. is , - - - - - n-- <, h - r , ,.. . . , , - - . + .
4
')f
- , m.
N
=
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N a W N
<m 9 e ,
g ~
2s 8
]
" ~
- s. EO O -
5 ud.~" l b .
, > f 5_a c C w 1 mm.d 2. i e
5 l V F - ;
4e "
k E memR =3
-- 3 5 " 5 e .
e g -
3 MO v , , .
> . s 4i ii .i
> - =
o m- l' i 9
' en o
g adenumum mummme mummmmpummme o
I l. o
'-- uimmmummmmmmme o N
% o -
s E R S S S R R S Firure 41 suonntpui Jo JegwnN 20
22 2
a Figure 4-2 Beaver Valley Unit 1 Deplugged BOC-11 Bobbin Voltage Distribution 20 -
3 SG-A O SG-s
_ l [ SG-C does not have depluggext tube.
s 2 -
g 10 - = --
z L
5 - - - - ---
0 . . . . . . . . :E . :E , , , , , ,
n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.2 2.4 2.6 2.8 Bobbin AmpNeude Nolts)
- < L < .
4 4
8
_ 2 6
2 4
2 2
. 2 2
8- c-9 n G o s
1 i
o S 8 t
u . 1 b
i r
.0 g 7 i
t e m 1 D 6 e
g _ 1 t
a 5 l
V o . 1 )
n . 4 e v
d . 1 t
d t
3B o
t 3
1 m
- mp 4 1 e1 r
uC gO
- a 2 m 1
a n
i FB l
a L 1 1
w e
s t
o I1 1
T 1 9 t
i1 0 dr U ,. ll 8
y 0 l
e -
l '
7 a 0 V _
r e .
6 v 0 a
e -
5 B 0 lI lI 4
- l[lII l 0
~
3 l 0 2
1 0 1
0 0
9 0
8 0
7 - , " 0 2
0 1
0 E a ?'" .j1i!
3
t h
a Figure 4-4 Beaver Valley Unit 1 in-Service EOC-10 Repaired Indication Distribution 10
, , I
- II -
s ._
7 E i o so-a _
. O sG-c 6 = =
= 1, I l l l 1 I I I I I '
3: I I I I ll ll 2 l _
l 1 11 l l ll
, II _ J I li ll 'J I II l , II._ .I .I i
ll 1 I I I Lu .I .I I. ;.1 .I I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 .0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.2 '2.4 2.6 2.8 Bobbin Amplituale (Volts) 1
- D.
a 5
7 - - - -
3 Figure 4-5 6 Beaver Valley Unit 1 In-Service EOC-10 ODSCC Axial Distribution 350 300- -
M SG-A O so-a 250 -
CE o
200 --
5 5
o t0 a- -
A 2 150 -
E 3
~ -
100 - ---
$Q -
0 1H l
2H
- ~~
I 6t 3H i
4H 5H t
6H t -
7H support Plate Numter ,
n l
C 5.0 DATA BASE APPLIED FOR IPC CORRELATIONS j The Beaver Valley 1 SER specifies the database to be used for the IPC correlations. For l the burst pressure correlation, the SER recommended data is the same as the EPRI recommended database as described in WCAP 14123. The burst pressure correlation is also given in WCAP-14123 and is applied in the analyses of this report.
For the SLB leak rate correlation, the NRC SER recommends that Model Boiler specimen 542-4 and Plant J-1 pulled tube R8C74, TSP 1 be included ir4 the database. This database is referred to as the NRC database in WCAP 14123 and is applied for the leak rate analyses of this report. The probability of leakage correlation of WCAP-14123 is also accepted by the NRC SER and applied in this report.
1 6.0 SLB ANALYSIS METHODS Monte Carlo antlyses are used to project the EOC-11 voltage distributions and to calculate the SLB leak rates and tube burst probabilities for both the actual EOC-10 voltage distribution and the projected EOC-11 voltage distribution. These methods are consistent with the requirements of the Beaver Valley-1 NRC-SER and are described in the IPC report of WCAP-14123 and the generic methods report of WCAP-14277. l Based on the NRC SER recommended leak rate database, the leak rate data do not satisfy the requirement for applying the SLB leak rate versus bobbin voltage correlation. The NRC requirement is that the p value obtained from the regression for the slope parameter be less than or equal to 5%. For the NRC recommended data, the p value is about 6.5%
and the leak rate versus voltage correlation is not applied. The SLB leak rate correlation applied is based on an average of allleak rate data independent of voltage. The analysis methods for applying this leak rate model are given in Section 4.6 of WCAP-14277. A Monte Carlo analysis is applied to account for parameter uncertainties even though the leak rate is independent of voltaga. This method ofleak rate analysis is similar to that of draft NUREG-1477 except for the uncertainty treatment. The analyses of this report found that the Monte Carlo analyses for the SLB leak rate with the leak rate independent i of voltage results in leak rates within 10% of that obtained using the draft NUREG-1477 !
methodology.
7.0 PROJECTED EOC VOLTAGE DISTRIBUTIONS 7.1 COMPARISON OF ACTUAL AND PROJECTED EOC-10 VOLTAGES Analyses previously performed to project the EOC-10 voltage distribution are documented in Reference 9.1. Comparisons of the actual EOC-10 bobbin indication l distribution for RPC confirmed plus not RPC inspected indications with the projected distributions for probability of detection (POD) of 1.0 and 0.6 are shown on Figures 71 and 7-2, respectively. The actual 451 indications include 56 RPC confirmed and 395 not RPC inspected indications. It can be expected that RPC inspection of all 25 1
1-i >
- - indications l< 1.0 volt would. have resulted in considerably < 451 RPC confirmed indications. - The projections include 366 indications for POD = 1.0 and 747 -
indications for POD = 0.6. ;
It is seen from Figures 7-1 and 7-2 that both projections exceed the actualindications for > 1.0 volt indications. Below 1.0 volt, the POD = 0.6 projection exceeds the actual above 0.5 volt and is less than the a'ctual below 0.5 volt, while the POD = 1.0 projection is lower than the actual below 0.8 volts. These results indicate a high POD above about 0.8 volt at the EOC 9 inspection while the POD below about 0.5 volts is ,
typical of a value about 0.6. A voltage dependent POD is necessary to improve the projections over the entire voltage range. Based on the projections exceeding actuals ;
above 1.0 volt even for a POD = 1.0, it would be expected, and is shown in Section 8,-
that SLB leak and burst projections would both exceed the values obtained for the actual EOC-10 distribution.
7.2 PROJECTED EOC-11 VOLTAGE DISTRIBUTIONS The IPC indication voltage distribution for BOC-11 has been developed in Table 4-1.
SG A is the limiting SG and the total of 535 indications at BOC 11 includes the 422 EOC-10 indications from prior active tubes, as well as the additional 113 indications from tubes deplugged during the outage that satisfied the repair limits and were -
returned to service.
Growth projections are based on rates determined in Reference 9.1 for SG A during ,
t cycle 9 (Table 4-7), which are more conservative than the cycle'10 growth rates, as
- previously discussed in Section 4.2. The operating periods used in the voltage projection calculations are:
Cycle 9 - 492.75 EFPD. Cycle 10 - 435.79 EFPD. Cycle 11 - 344.00 EFPD.
Cycle 11 is projected to be a shorter fuel cycle than either cycles 9 or 10. For the .
SLB analyses, the Cycle 9 growth rates were scaled by the cycle 11 to cycle 10 EFPD l ratio of 0.789 to more conservatively predict EOC-11 conditions. l For the Monte Carlo calculations, the net total number of indications returned to i service for cycle 11 (N7 ,,3) is determined from N7,,7s = N, / POD N,,,a + Na73 ,
where ;
N, = Number of bobbin indications at EOC-10 POD = Probability of Detection N,,,,,,,a = Numb oM which i are repaired (plugged) before BOC 11 l
Nars = Number of plugged tubes returned to service at BOC-11.
The IPC voltage distribution projected to EOC-11 is shown on Figure 7-3 for POD =
1.0 and on Figure 7-4 for POD = 0.6. Per the Beaver Valley 1 SER, the POD = 0.6 distribution is applied for the reference SLB leak rate and burst analyses.
l 26 l
Dr u- -r -- -i-w -, iw--- -m- F -- -- e
, .n. .
i S Figure 7-1 a
SeaverVaHoy 1 SIG A " l 2
Cei.-;arison of Actual and Predicted EOC-10 Voltage Distribution POD = 1.0 so
- so -
i O soc-1* Predction, POD = 1.0 E acc-se Actuainecconeum.d i ,
% not nec:.u;:1 l 3 g .
I w
-a S
g 40
'istalinecations at EOC-10
$ Predicted Actual E
=
! j 4s1 l
i l' _
3 - . _
EOC-16 Actualklam.
I I
0 N
o l
v.
o 9
o 9
o e N e
y e
I; q
e
- ; ;U ;"" ;" ;" ;" .
90, e
N q n
Volts 4,
n
,r q
n q
n M q n
v n
q n
q n
T N
w W,
k l . -.
'T .. .
3, m
i Figure 7-2 3, Beaver Valley 1 StG A h Comparison of Actual and Predicted EOC-10 Voltage DI:stribution POD = 0.6 100 80 -
EOC-10 Prediction, POD = 0.5 EOC-10 Actual RPC Confirmed E Plus Not RPC inspected
- 60 -- - - - -
.3 5
S to E 00 6- o .
E E 40 -
s Z '
Total Indica'iions at EOC-10-_.
Predicw4 Actual 747 451 20 - - - - - - - - -
) EOC-10 Actual Max.
i Y
O ; . . , , , . . . . . . E; . . .
I; I; ; ;U ;n ; ; ; ; ; , ;, ,
N
- 9 9 "
N v. 9 9 N N 9 9 9 " N *. *
- o o o o N n n e 9 9 N
<- e e <- N n n n w w.
Volts i
2 3
Figure 7-3 6 Beaver Valley Unit 1 - SGA Cycle 11 BOC and EOC Voltage Distribution POD = 1.0 100 1
80 .
9- .
, E soc si C
S gg ._ _ _
g _C O
E 40 - -- - - - -- - - - - - -
20 l-0 R- . . . . . . . .
- ,
- : ; ; E: ";":~;: -
0.20.30.40.50.60.70.80.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.22.32.42.52.62.72.82.9 Bobbin Ampilhede Notts)
--__ _ _ _ _ _ ~- .. - .- ._. - _ _ _ . . .- -_ __ - - - ___ _ _ - -
i 5
7 3
, Figune 7-4 i
Beaver Valley Unit 1 -SGIA Cycle 11 BOC and EOC Voltage Distribution POD = 0.8 150 -
~
125 U BOC 11 100 - - -
E EOC 11 E
~
e l @ 75 - - - -
-. ~.. _ _ _
f l 1 l
5
=
50 _ _ _ _ _
~
25 _ _ _ _ I I .
O m . . . . . .
lrfrl;rs;1.;.___;- _ ; .
0.20.30.40.50.60.70.80.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 .1.8 1.9 2 2.12.22.32.42.52.62.72.82.9 N Amplitusse(Volts) e
_ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . . - - _ _ _ _ . _ . _______.___m__.______ _ _ _ _ _____._.
8.0 SLB LEAK RATE AND BURST PROBABILITY ANALYSES 8.1 COMPARISON OF PROJECTED AND ACTUAL EOC 10 LEAK AND BURST The calculated predictions for performance of the limiting steam generator during cycle 10 operation of Beaver Valley Unit 1 is documented in Reference 9.1. A comparison of these l predictions with the corresponding actual parameters as determined from inspection data during the EOC 10 outage and from calculations based on those data is shown on Table 8-1. In all cases, the projected SLB leak rates and tube burst probabilities are sigmficantly more conservative than that obtained from the actual EOC-10 distribution. The ,
calculated SLB leak rate for the actual EOC-10 distribution is 0.15 gpr- (based on the NRC-data base) and the tube burst probability is 3.84 x 10'*; leakage rads based on the EPRI data base are an order of magnitude lower than those based on the NRC data base.
8.2 PROJECTED EOC-11 LEAK RATE AND BURST PROBABILITY Calculations have been conducted to predict the performance of the limiting stears generator in Beaver Valley Unit I at EOC-11 mnditions. The methodology used in these predictions is described in Reference 9.2 and is essentially the same as that used in Reference 9.1 for the cycle 10 predictions. Results of the EOC-11 predictions are summarized on Table 8 2. At POD = 0.6, the projected EOC-11 SLB leak rate for S/G A of 0.31 gpm (based on the NRC data base) is much lower than the allowable limit of 6.6 gpm for the affected loop; leakage rates based on the EPRI data base are an order of ,
magnitude lower than those based on the NRC data base. Similarly, the EOC-11 SLB tube i burst probability of 9.7 x 10 (at POD = 0.6) is much lower than the NRC 1.0 x 10
threshold value reqmring further assessment as given in the Beaver Vadley 1 SER (Reference 9.3). .In addition, the actual EOC-10 SLB leak rate of 0.15 gym and burst probability of 3.84 x 10 are both lower than the allowable limits. It is therefore concluded that the actual EOC-10 and projected EOC-11 SLB leak rates and tube burst probabilities show large margins against the allowable limits. ,
A comparison of the performance of the individual steam generators is shown on Table 8-3, which further confirms that the limiting stcam generator for cycle 11 of Beaver Valley Unit 1 is SG A. The burst probability of SG B is dominated by the effect of a single IPC NDF of 2.8 volts which was left in service. The difference in tube burst probabilities between SG A and SG B is considered negligibly small and SG A is more limiting in other aspects.
9.0 REFERENCES
9.1 WCAP 14123 (SG 94 07 009), " Beaver Valley Unit 1 Steam Generator Tube Plugging Criteria for Indications at Tube Support Plates July 1994".
9.2 WCAP-14277, "SLB Leak Rate and Tube Burst Probability Analysis Methods for ODSCC at TSP Intersections", Westinghouse Nuclear Services Division, January 1995.
9.3 U.S. N.R.C. Letter, "Sciety Evaluation by the Office of Nuclear Regulation Related to Amendment No.184 to Facility Operating License No. DPR-66 Duquesne Light Company Beaver Valley Power Station, Unit No.1 Docket No. 50-334".
l 31
l TABLE 81 1
BEAVER VALLEY UNIT 1 SG A l
Comparison of Cycle 10 Performance !
i Projected vs. Actual !
IPC > 1.0 volt l l
l Projected Projected Actual POD 1.0 0.6 ---
No. of Indications 366 747 484 Max. EOC Volts 4.1 4.4 2.6 SLB Leak Rate (gpm/SG)
NRC Data base 0.22 0.38 0.15 EPRI Data base 0.021 0.038 0.012 SLB NRC Burst Prob. 1.8 x 10 4 4.7 x 10 4 3.84 x 10'8 I
e
{
32
. .- , -- = - .. .. . -- .. .
.; 4 - ,
TABLE 8 2 BEAVER VALLEY UNIT 1 SG A Prediction'of EOC-11 :
Leak Rate and Burst Probability :
IPC > 1.0 POD 1.0 0.6 Number ofIndications 535 857.67 Mcx IPC Volt BOC 2.2 2.6 ,
EOC 2.6 2.9 Leak Rate (gpm/SG)
NRC Data base 0.195 0.31 !
EPRI Data base 0.015 0.031 Total Burst Probability 4.3 x 10-8 9.7 x 10 4 4
o Single Burst 4.3 x 10 9.7 x 10-8 4.0 x 10
- 4 7.3 x 10 4 I ooThree Two Tube Tube Burst Burst 4.0 x 104
- 4.0 x 104 *
- No tube burst in 10' Monte Carlo samples. Probability limited on 95% confidence. ,
i l
e e
i 33
- i TABLE 8 3
'l'
' BEAVER VALLEY UNIT 1 Comparison of Individual S/G Performance for EOC-11 POD = 1.0 IPC > 1.0 - :
I i S/G A B C Number of Indications 535 470 196 I 1
Max IPC Volt l BOC 2.2 2.8 1.6 ;
EOC 2.6 3.2 2.1
~
Leak Rate (gpm/SG)
NRC Data base 0.195 0.164 0.072 EPRI Data base 0.015 0.012 0.004 Burst Probability 4.3 x 10'8 6.4 x 10 ' 1.45 x 10 5 !
1 i
1
)
i 1
1 l
34 l
WESTINGHOUSE NON-PROPRIETARY CLASS 3 WCAP-14277 SG 95-01-007 '
4 SLB Leak Rate and Tube Burst Probability Analysis Methods for ODSCC at TSP Intersections i
January,1995 t
l l E
WESTINGHOUSE ELECTRIC CORPORATION NUCLEAR SERVICES DIVISION P. O. BOX 158 MADISON, PENNSYLVANIA 15663-0158 l
l C 1995 Westinghouse Electric Corporation All Rights Reserved '
SAAPCCENEluCCENSECOD.WP5
., o January 27.1905 D
5 }h
I
~
j SLD Leak Rate and Ttbe Burst Probability Analysis Methods fer ODSCC ct TSP Inters:ctiras TABLE OF CONTENTS SECTION PAGE 1.0 Introduction 11 2.0 Summary and Conclusions 2-1 3.0 Methods for Projection of EOC Voltage Distributions 3-1 3.1 General Description of Methods 3-1 3.2 BOC Voltage Distributions 3-1 3.3 Voltage Growth Rates 3-2 3.4 NDE Uncertainties 3-4 3.5 Monte Carlo Methods 3-4 3.6 Projected EOC Voltage Distributions 3-6 3.7 Supplementary Considerations Relative to Growth Rates 3-7 4.0 Burst Pressure and SLB Leakage Correlations- 4-1 4.1 General Correlation Considerations 4-1 4.2 Database Used for the Correlations 4-2 4.3 Material Properties Considerations 4-2 4.4 Burst Pressure versus Bobbin Voltage Correlation 4-3 4.5 Probability of leakage Correlation 4-4 4.6 SLB Leak Rate versus Voltage Correlation 4-7 4.7 Inclusion of Future Data in the Correlations 4-9 4.8 References 4-11 1
5.0 SLB Leak Rate and Tube Burst Probability Analysis Methodology 5-1 5.1 General Methods Considerations 5-1 5.2 Deterministic Methods for Sensitivity Analyses 5-1 l 5.2.1 Deterministic Estimation of the Total Leak Rate 5-1 5.2.2 Deterministic Estimation of the Probability of Burst 5-4 5.3 Deterministic Analysis Results 5-5 i 5.4 Monte Carlo Analysis Methodology 5-5 5.4.1 Probability of Leak Simulation 5-7 5.4.2 leak Rate Versus Bobbin Amplitude Simulation 5-8 5.4.3 SLB Tube Burst Pressure Simulation 5-10 5.5 SLB Leak Rate Analysis Methodology 5-11 5.6 SLB Tube Burst Probability Analysis Methodology 5-12 I
I I
I
$MPOGENrJUCCENSEC00.WP5 i January 27.1995
x SLB Leak Rate and Ttbe Bntst Probability Analysis M thods * '
c f:r ODSCC cr TSP IntersIcti:ns TABLE OF CON TENTS (Com.)
SECTION PAGE 6.0 Er mple Analysis Results- 6-1 6.1 Simulation Code Description '61 6.2 Example Correlations and Distributions 6-2 6.3 Deterministic Analysis Results 6-2 6.4 SLB Leak Rate Analysis Results 6-3 6.5 SLB Tube Burst Probability Analysis Results 6-4 i
Appendix A: Regression Analysis A.1 Introduction A.1 A.2 The Linear Regression Model A.3 A.3 Consideration of Variable Error A.5 A.4 Detection of Outliers A.7 A.5 Selection of a Regression Coordinate System A.8 A.6 Selection of a Regression Direction A.9 A.7 Significance of the Regression A.11 A.8 Analysis of Regression Residuals A.12 A.9 References A.14 '
)
i l
I i
i swecmuucasecmwes ii 3 ry n. i,,s !
l 4 -
m-_ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _- _ -
c SLB Leak Rate and Tube Burst Probability Analysis ]
Methods' for ODSCC at TSP Intersections 1 1.0 Introduction 1
The purpose of this report is to document the methodslused for the analyses ' !
supporting application of alternate plugging criteria (APC), also known as alternate repair criteria (ARC), for the disposition of outside diameter stress corrosion
- cracking (ODSCC) indications detected in steam generator (SG) tubes at locations corresponding to the elevations of the tube support plates (TSPs). Using this report {
as a reference for the analysis of the indications in a specific plant's SGs obviates !
the need to document the analysis methods in the plant specific report. These r
' methods are intended to be in accord with the Nuclear Regulatory Commission's (NRC) generic letter entitled " Voltage-Based Repair Criteria for the Repair _of !
Westinghouse Steam Generator Tubes Affected by Outside Diameter Stress ;
Corrosion Cracking."2 F
The eddy current inspection (ECT) of the SG tubes may identify a significant number of bobbin coil indications at the intersections of the tubes with the TSPs, of -
which, several may be confirmed as being axial crack-like ODSCC indications using ;
rotating pancake coil (RPC) inspection techniques. Using traditional plugging '
criteria could result in significant tube repairs that would not be required to meet -
the NRC's draft Regulatory Guide (RG) 1.121 guidelines for tube repair. Specific '
plants may request a change to their Technical Specification to implement an alternate plugging criteria (APC)8 for the disposition of those indications.' This alternate criteria consists of a bobbin amplitude, i.e., voltage, based repair limit in lieu of a depth based repair limit.
The methodology to support the implementation of APC consists of establishing !
correlations between the expected burst pressure, the probability ofleak, and the expected leak rate to the bobbin voltage of the indication. The correlations are ther used in conjunction with a measured or calculated end-ofwele (EOC) distribution i
1' The simulation methods described in this document were previously described in WCAP-14046 (Proprietary), Revision 1, "Braidwood Unit 1 Technical Support for Cycle 5 Steam Generator Interim Plugging Criteria," Westinghouse Electric Corporation, August 1994. ;
2 The text of the proposed communication was published by the United States Nuclear Regulatory Comminaion (NRC) as a notice in the Federal Register, volume 59, number 155, on pages 41520 through 41529.
3 This is also known as an interim plugging criteria (IPC) when submitted for implementation for a limited time period, e.g., one fuel cycle.
s arcscurmcicuszcot.wPs 1-1 n==ya.las l
- cfindic ti:n2 to estimtte tha likslihood of a tube burst and the primary-to- "
sec:ndary tottl 1:ak r:te for thm SG during a postultt:d steam lina brsak (SLB) cvant. If the probability of burst is sufficiently small, and if the total estimated l i
leak rate, at a specified confidence level, is less than acceptable limits the voltage criterion may be implemented. If either of the requirements is not met, additional tubes would be repaired until both of the requirements would be projected to be ;
met at the EOC. '
The data used in the correlations are to be based on the latest available Electric
-Power Research Institute (EPRI) database (including data obtained within approximately six months prior to the inspection outage at which the APC will be .
applied) as evaluated against those EPRI data exclusion criteria approved by the NRC. The actual database used should be referenced or documented in the plant specific APC report. Any departures from the EPRI database should be documented. An example of a modification to the database is direction from the NRC to include data which might otherwise be excluded, or newly developed data which might not be formally incorporated into the EPRI database.
The evaluations supporting the application of plant specific APC are based upon ,
the bobbin coil voltage amplitude, which is correlated with tube burst capability and leakage potential. For SLB leakage and burst analyses, the tube support plate crevices are assumed to be free span or open crevices. This assumption leads to more conservative leak rates and burst probabilities when compared to rates and probabilities associated with expected packed crevices under normal and accident conditions, If APC based on limited TSP displacement are implemented, future -
revisions of this report will include delineation of the analyses methods used to support those criteria.
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2.0. Summary and Conclusions r
This report presents methods used for the evaluation of data gathered to support the application of APC to the tubes in the SGs at nuclear power plants. The !
i methods documented for the performance of correlation analyses are based on j standard methods described in references on statistics and regression analysis. . The correlations used to support APC are based on estimating the parameters'of a .:
correlating equation based on the principles of maximum likelihood.- i
. Methods presented herein for the performance of Monte Carlo simulations reflect .
the conclusions reached from discussion with the NRC and its advisors on the appropriate techniques to be employed to properly account for the variances and - ,
covariances of the parameters of the correlations used. The simulations thus performed are expected to lead to conservative and reliable estimates of the total - '
EOC SG leak rate and of the probability of burst of EOC indications during a postulated SLB.
Per the NRC generic letter, the estimation of voltage growth rates for the next cycle of operation is to be based on the voltage growth rates for previous cycles of operation. If only one cycle of previous operating data is available, it may be used to estimate the growth during the next cycle of operation. If two cycles of previous operating data are available, the data for the cycle with the higher growth rates should be used, however, there may be technical justification for using the most '
recent growth rate ifit can be shown to be lower for cause. Growth rates may be ,
SG specific. Growth rates observed at prior cycles are used to create a cumulative i percentage distribution of growth rates. The distribution is linearly scaled to the length of the cycle to be projected. !
i EOC voltage distributions are obtained from the BOC distributions by Monte Carlo simulation of the NDE uncertainties and the voltage growth rates. The simulation of many thousands of distributions for a single SG are combined to provide a single predicted distribution to be expected at the EOC. Comparisons of predicted distributions with actual distributions after a cycle of operation has shown the ability of the simulation technique to result in conservative estimates of the EOC indications.
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3.0 Methods for Projection of EOC Voltaga Distributions EOC voltage distributions are projected from an estimated BOC voltage distribu-tion using a voltage growth model developed from tracking the growth ofindica-tions during previous cycles. This section is to describes the methods used to estimate the BOC distribution, the techniques for specifying the voltage growth '
distribution, the estimation of NDE uncertainties relative to the BOC distribution ofindications, the methods employed to simulate growth, and the development of a -
final EOC distribution from the simulation results.
3.1 General Description of Methods The progression of ODSCC indications at the TSPs is determined by reevaluation of prior inspection ECT records at the locations identified with indications in the prior ,
inspection. In most cases, some element of a precursor is identified as correspond-ing to the flaw signal reported in the prior inspection. However, it should be noted ;
that rather conservative analysis criteria are invoked to accomplish this task.' In this process, analysts are required to forego the behavior criteria they may have '
employed to screen out low signal-to-noisc indications, and to report possible flaw-lila hehavior in the TSP mix residual regardless of clarity. Review of the growth data identifies any anomalous growth data, and these are subjected to further scrutiny to eliminate spurious data. .
3.2 BOC Valtage Distributians The bobbin voltage distribution for the beginning of the next cycle (BOC) following the current outage is developed by applying a probability of detection (POD) to all indications found at the end of the previous cycle (EOC), relative to the current outage. This method of accounting for probability of detection is per the direction l
of the NRC draft NUREG-1477, the draft generic letter, and the January 18,1995, NRC/ industry meeting on the resolution of comments to the draft generic letter. l This methodology divides the EOC voltage distribution by.the POD, and then subtracts the repaired indications to define the BOC distribution. The number of i indications that are to be considered as being returned to service, N, is, . !
~
d N=Nd + Na s -Nr = Nd + Na - N, - - N' , (3.1)
POD $
POD where, l Ng = N,, + N,,i + F,,; N,,aa , (3 2 l S MPC\ GENERIC \GENSEc03.WP5 3-1 Janary n m
[and,' JN,1 = num'ber cf r:pairsd ' indications, '
. Na = number ofindicctiins n:t dstected by the bobbin inspastion, l
- POD .: = probibility of detection, specified as 0.6,
- l lNy =
number of detected bobbin indications that were confirmed by RPC, ,
m N , = number of detected bobbin' indications that were'not inspected by !
RPC, '
Fg
~
=
fraction of RPC NDD indications called NDD at one inspection and !
found to be RPC con 6rmed indications at the subsequent'inspec-tion, N
ma = number of detected bobbin indications not confirmed as flaw indica - ;
tions by RPC inspection.
The above adjustments for POD are incorporated in the BOC and EOC voltage !
distributions so that no further adjustments are required for the leakage calcula .
tion. .
t
)
The value used for the POD in equation (3.1) is 0.6 unless an alternate value or a voltage dependent POD is approved by the NRC for APC applications. This is consistent with the requirements given in the NRC generic letter. It is noted that .
when voltage indications above a few volta are found from the inspection, for whicht the POD would be expected to be significantly greater than 0.6, this methodology becomes very conservative for determining the BOC distribution. This is because it leaves 0.7 of an indication to be simulated as in service for each indication'found ;
from the inspection, independent of the voltage level of the indication. :
The value for Fm should be based on plant specific inspection results, except as noted below, when data to evaluate Fg are available for at least two cycles of operation with inspections performed using ECT analysis guidelines consistent with those used for APC inspections. The value used for Fg on a plant speci6c basis must be obtained with reasonable conseroatism. An example of reasonable conser-l vatism is to apply the largest value obtained for any SG over the last two operating cycles. A population of 250 RPC NDD hidientions is judged to be necessary to reasonably estimate Fg . If adequate plant'spectile data,are not available, the value used for Fg should be selected to bound reaults obtained from assessments of APC/IPC inspections at other domestic plants with the same tube size. Based on ,
four APC/IPC assessments conducted through 1994, a minimum, plant non-specific- '
value for Fg would be 0.25. Plant specific reports should clearly identify the value and basis for the value ofFg used in the analyses.
3.3 Voltage Growth Rates L i i
The plant specific report should identify the operational periods for which growth
. values were determined. The distribution of the growth rate data, expressed as i volts difference in the amplitude readings for two inspections, are usually tabulated in 0.1 volt bins. The width of a bin may be increased to 0.2 volts at some voltage j swesormnictorNsecos wn 3-2 m wrm.tus
1sv21,'and mzy b2 furthcr incr:ased to 0.5 volts for voltag:s above a czrtain-thruhold, up to the maxunum observ:d chengs. For each bin the number of indic:tions is entered along with ths corrcspanding cumulative probability of !
occurrence value. The bin identi6 cation value represents the highest voltage level-in the bin. For example, an amplitude of1.35 volts would be included in the next highest 0.1 volt bin, i.e.,1.4 volts. The cumulative probability is calculated as the :
sum ofindications up to and including the bin value divided by the total number of l indications. While the raw data used in developing the voltage growth distribution l may contain indications for which negative growth would be calculated, the ,
developed growth distribution is not permitted to exhibit any negative growth characteristics. This is achieved by considering all indications with calculated negative growth as zero growth. For use in the Monte Carlo analyses, the voltage growth distributions may be normalized to a growth per EFPY basis by assuming .
growth is linear with time. In general, the cycle time to be projected is similar to '
that for which the growth rates were calculated and normalization is not considered significant.
The voltage growth histograms for each of the operational periods evaluated should be presented in the plant specific APC report. These may appear as composite,'i.e.,
all SGs, for the prior cycles, but should be presented for each SG for the most recent cycle. For indications with appreciable BOC amplitude readings, i.e., greater than or equal to 0.75 volt, average growth rates may be calculated and compared to the overall growth rates. In addition, information may be presented relative to ^
growth as a function of position in the SG, e.g., the dominance of the incidence of ODSSC indications at the elevations of the lower TSPs may also be reflected in the growth rates. ;
In developing the voltage growth distribution, only NDE indications with flaw l indication characteristics in both cycles are to be included in the analysis. A mini-
[
mum of 200 indications are required to define a voltage distribution. SG specific i distributions also require a mmimum of 200 indications for application to projecting -
EOC voltages. For projections of a specific SG, the more conservative (relative to '
projecting leakage and burst) growth between the distributions for the specific SG, 4 and for all SGs collectively, should be used. If two cycles of growth distributions obtained with APC/IPC ECT guidelines are available, the larger distribution of the '
last two cycles should be used for the projections. If two cycles of data are not available, the prior cycle growth distribution may be used. If the last cycle of ;
growth is significantly lower than the prior cycle, and can be attributed to enhance-ments in secondary chemistry affecting only the last cycle, the last cycle growth !
rates may be used for the APC/IPC analyses. The plant specific APC report should >
include a discussion of the specificjustification the basis for the lower growth rates.
When an APC is being applied without 200 indications to define the growth distributions, a bounding growth distribution from other domestic plants may be applied.
SMPC\ GENERIC \GENSEc03 WP5 3-3 .wwy n, ms I
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For A' PC applic:tiens, NDE une:rtaintits must be cecountsd for in projecting the
~ ~
Ldistribution of the BOC indications to the EOC. This is accounted for by using .
[ Monte Carlo techniques. The database supporting NDE uncertainties is described
~
1 i
, in Reference 5-1, and NDE uncertainties for APC applications are given.in the
~
' EPRI repair criteria report, Reference 5-2. From Reference 5-2, the NDE uncer-tainties are comprishd of uncertainties 1 1)" due to the data acquisition technique, which is based on use of the probe ;
wear standard, and
- 2) due to analyst interpretation, which is sometimes called the analyst variability uncertainty.
- i. .
If a transfer standard is not employed, manufacturing tolerances in the probe i
calibration standards would be expected to constitute an additional source of i i.
uncertainty in the NDE results, and should be accounted for in the plant specific !'
l computations.
i i
The data acquisition, or probe wear, uncertainty has a standard deviation of 7.0%
' about a mean of zero. Variation due to probe wear is restricted to $15% of the '
I bobbin amplitude, contingent on the implementation of the probe wear standard requiring probe replacement at 15% differences between new and worn probes.
4
' ASME standards cross-calibrated against the reference laboratory standard and the i probe wear standard should be implemented during the inspection to avoid the additional consideration of manufacturing tolerances.
t The analyst interpretation (analyst variability) uncertainty has_ a~ standard devia-tion of 10.3% about a mean of zero. Typically, this uncertainty would have a computational cutoff at 20% based on requiring resolution of analyst voltage calls' differing by more than 20%, however, the NRC has not accepted the 20% limitation on the analyst interpretation uncertainty. Pending a further resolution of this ,
issue with the NRC, the analyst interpretation uncertainty is applied without a cutoff. Thus, for EOC voltage projections, separate distributions are applied for probe wear with a cutoff at 15%, and for analyst interpretation with no cutoff.
3.5 Monte Cado Methods ,
The Monte Carlo simulations to estimate the EOC voltage distribution begin with the BOC distribution as described previously. The EOC distribution ofindications ,
is calculated several thousand times, secounting for the uncertainties in the NDE '
and in the voltage growth. The cumulative distribution ofindications from all of the simulations is calculated, then , adjusted to reflect the total number ofindica- ;
tions in the BOC distribution. The methods used to account for the uncertainties i i
SAAPC\ GENERIC \oENSEC03wPS 3-4 % g ,,
- ara discussed in tha fallowing subsections. Tba calculs. tion of a single rspr:senta-tiva EOC distributi:n is discussed in ths fallowing section of this r: port.
~
3.5.1 NDE Uncertainties The method of accounting for NDE uncertainties in the Monte Carlo analyses is to adjust the field measured voltage by multiplying the standard deviation of the uncertainty under consideration by a standardized normal distribution deviate.
The uncertainty associated with probe wear is assumed to be characterized by a normal distribution with a mean of zero and standard deviation expressed as a percentage of the true, but unknown, voltage of the indication. The uncertainty of the analyst is assumed to be characterized by a normal distribution with a mean of zero and a standard deviation expressed as a percentage of the true probe voltage of the indication.
Assuming no analyst variability, the distribution of voltages read by a probe, V,, !
corresponding to a true indication voltage of V would be simulated as, 4 V, = V(1 + Z,4,) , (3.3) where V4, is the standard deviation of the probe error about the true voltage of the indication and 2,is the distribution of standard normal deviates. The measured voltages, V, , accounting for analyst variability, would then be distributed as a function of the probe voltage as, V = V,(1 + Z,(), (3.4) where V,( is the standard deviation of the analyst error about the probe voltage of the indication, and 2, is the distribution of standard normal deviates. Combin-ing equations (3.3) and (3.4), the distribution of true voltage of an indication about the measured voltage of the indication is then, V= V" .
(3.5)
(1 + Z,4,)(1 + Z,()
Thus, the simulation of the true voltage of an indication at the BOC is based on two independent draws from a standard normal distribution. To account for the L limit on probe wear, the values of Z, are limited to an abs'o lute value no larger than I 15n, i.e., 2.143.
s urcscentnicscensrcosms 3-5 .%=ry a. ms
' Altsrnetivzly, if the rnnlyst variability was assumsd to b3 distributed about the'
- j true voltags of the indieztion, equ tien (3.5) would be replaczd by, V= V* .
(3.6) !
(1 + Z,4, + 24 ) !
i The difference between the two expressions is the addition of a term,2 7 4 2,(, in !
the denominator of equation (3.5) relative to equation (3.6). Since the pr,odl with 4, is on the order of 0.007, this would be expected to be a second order effect.
1Hence, equation (3.6)is used in the Monte Carlo simulations. I 3.5.2 Voltage Growth I To account for voltage growth during an operating cycle, the cumulative distribu- ;
tion of voltage growth is entered with a random number, U,, drawn from a uniform distribution, i.e., O < Uj s 1. The growth is then obtained as a linear interpolation between the discrete values used for the cumulative growth distribution. For ,
example, if growth values were specified only at cumulative probability values of l
0.0,0.5, and 1.0, the growth corresponding to a value of U, of 0.75 would be midway '
between the growth values corresponding to 0.5 and 1.0 respectively. :
3.6 Padected EOC Voltage Distributions i Monte Carlo simulations are then performed to develop the EOC voltage distribu-
[
tions fmm the BOC distributions. The BOC voltages are increased by allowances i for NDE uncertainties and voltage growth to obtain the EOC values. In the Monte Carlo analyses, each voltage bin of the BOC distributions is increased by a random '
sample of the NDE uncertainty and growth distributions to obtain an EOC voltage ;
sample. Each sample is weighted by the number ofindications in the voltage bin. ,
The sampling process is repeated at least 100,000 times for each BOC voltage bin 3
' and then repeated for each voltage bin of the voltage distribution. Since the Monte I
i Carlo analyses yield a cumulative probability distribution of EOC voltages, a method must be defined to obtain a discrete maximum EOC voltage.value. The method adopted in this report is to integrate the _ tail of the Monte Carlo distribu- .
tion over the largest 1/3 of an indication to define a discrete value with an occur-l rence of 0.33 indication. For N indications in the distribution, this is equivalent to evaluating the cumulative probability of voltages at a probability of(N-0.33)/N.
)
The largest voltages for all distributions developed by Monte Carlo in this report have been obtained with this definition for the maximum EOC discrete voltage. i The next largest discrete EOC voltage indication is obtained by integrating the tail l of the Monte Carlo distribution to one indication and assigning the occurrence of l
0.67 indication. This process for developing the largest EOC voltage indications provides appropriate emphasis to the high voltage tail of the distribution and permits discrete EOC voltages for deterministic tube integrity analyses.
l SAAPC\GENEIUC\GENSEC03.WP5 3-6 .w-r u. ms l
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' 3.7 Cupplementary Ce==idarations Relative to Growth Rates j' It is r: cognized thtt sp:cific tetions may be takIn s.t an operating plant aimed at
?
slowing the progression of Alloy 600 ODSCC, i.e., there is a relationship between operating chemistry and ODSCC growth. For example, a plant could initiate molar ratio chemistry control and boric acid addition. Operating with elevated sodium to
.j chloride molar ratios enhances the possibility of developing caustic crevice condi-l tions conducive to initiation and propagation of Alloy 600 alkaline stress corrosion cracking. Hideout return chemistry data can be used to measure the success of the molar ratio control program in modifying the steam generator environment.
Laboratory and operating PWR plant data indicate the usefulness of boric acid as a contributor to the overall corrosion control progra2n.
The success of such efforts has not been quantified to the extent that adjustment of prior cycle growth data for chemistry enhancements is justi5ed for projecting l
[ growth rates for the next cycle. Should such data become available, their use would have to be documented and justified in a plant speci5c report, or in a future revision to this report. Current analyses do not include consideration of retarding future ODSCC growth via chemistry control.
l S.\APC\GENERJc\GENSEC03 WP5 3-7 Januur u. ms
1 i
L
.'4.0 Burst Pressure and SLB Leak Rate. Correlations The purpose of this section is to provide information and justification for all of the l
correlations developed in support of the application of alternrate plugging criteria . !
(APC) for the disposition of ODSCC indications in the SG tubes at the elevations of ' ,
.the TSPs at nuclear power plant. '
4.1 General Correlation Considerations To support the implementation of APC at nuclear power plants, correlations have been developed for tubes containing ODSCC indications at TSP locations between
-i the bobbin amplitude, expressed in volts, of those indications and the~ free-span burst pressure, the probability ofleak, and the free-span leak rate for indications that leak, References 4.1 and 4.2. In 1993, the NRC issued draft NUREG-1477, Reference 4.3, for public comment. The draft NUREG delineated a set of guidelines for criteria to be met for the application ofInterim Plugging Criteria (IPC) for
~
ODSCC indications. The criteria guidelines permitted the use of, with adequate justification, a burst pressure to bobbin amplitude correlation and a probability of leak to bobbin amplitude correlation. The criteria guidelines did not permit the use '
of a leak rate to bobbin amplitude correlation for the estimation of end of cycle (EOC) total leak rates. In essence, References 4.1 and 4.2 provided comments on the Reference 4.3 guidelines. Reference 4.4 provided an NRC response and position '
relative to resolving the differences between References 4.1 and 4.2, and Reference 4.3, along with responses to other public comments. Of significance to this report, is that Reference 4.4 indicated that a correlation between leak rate and bobbin amplitude could be employed if the correlation could be statistically justified at a 95% confidence level, and provided direction for the development of guidelines, e.g.,
Reference 4.5, that could then be employed for the identification and exclusion of i outlying experimental data. Further delineation of the NRC's position was pub-lished as a draft generic letter, Reference 4.6. NRC resolution of public comments on the draft generic letter are given in Reference 4.7. The methods of this report are intended to be consistent with the methodology of References 4.1 to 4.7 with '
emphasis on the generic letter of References 4.6 and 4.7, and with prior use of these methods as described in Reference 4.10.
Discussions with NRC personnel reviewed potential issues associated with the manner in which the leak rate to bobbin amplitude correlation was being used, and ;
questioned the ability of a deterministic model of the total leak rate to accurately account for the variability due to the uncertainties in the parameters of the '
correlation model. Thus, the potential leak rate during a postulated steam line break (SLB) is estimated by utilizing both deterministic and Monte Carlo methods.
The deterministic method is used tp screen potential leak rates, and the Monte Carlo method is used for the final determination of potential EOC leak rates. l surescrxrnicsozxssc00rP5 41 Juury 27,1995
, _ _ ~ .
Info'rmation is prusnted in the following sections on thn dr.tabue for the correla - ..*
tiens, materi:1 properties as relnted to burst pres 2ure esnsid:rz.tions, the corrsla- .{
I tion of burst pressure to bobbin amplitude, i.e., indication voltage, the correlation -
between the probability ofleak of an indication and the indication voltage, and lastly a discussion of the correlation ofleak rate to volts. The use of each of the correlations is also discussed.- A general discussion of the linear regression tech- ,
, niques employed is contained in Appendix A to this report.
All of the techniques' described for the support of APC implementation are in accord with the requirements of the NRC draft generic letter.
4.2 Da*mh== Used for the Camlatians The baseline database used for the development of the correlations should be presented or properly referenced in the plant specific report (s) supporting the ,
implementation of APC. One such database is presented and discussed in Refer-ence 4.2. Future development of APC criteria for speciSc plants may include -
additional data as it becomes available, hence, no referenceable database is includ-ed in this report. It is noted that not all additional data may be included in future correlations.
The priorities for identifying the appropriate database are a plant speci6c NRC
' Safety Evaluation Report (SER)if used for the cycle ofimplementation of an APC/
IPC, or the guidance of the NRC generic letter if applicable to the plant specific APC/IPC. When new data, such as testing results from recently pulled tube sections, are to be added to the database, the data shall be evaluated against the EPRI data exclusion criteria, Reference 4.5, as approved by the NRC at the time of I
' the APC/IPC assessment. Reference 4.7, includes the status of NRC concurrence with Reference 4.5 at the time of this report.
Any other special circumstances related to data used for the correlations should be discussed or referenced in the plant specific APC report.
CAUTION: The database used in the regression analyses performed for this '
report was randomly generated and analyzed specifically for this report.
Sample correlation results presented herein are for illustration purposes i
only and are not intended for plant specific APC evaluations.
4.3 Mataial Pumperties Ca-n=3tions The variation of material properties between tubes is a significant factor in determining the burst pressure. The rupture of tubes fabricated from Alloy 600 material is an elastic-plastic fracture process. A rigorous analysis of the process L would require knowledge of the strain hardening characteristics of the material. '
i~
However, reasonably accurate predictions may be obtained by empirically correlat-swesaturarctornseco4 wrs 4-2 >=-r n ms
4 ing the burst pr:ssur:s obtain d from tubes with differ:nt material prop:rties by
- n
- rmalizing tha ruults to the fisw strength, Sf, of the metcriel. The concept of a
- . flow strzs allows the material to be approximated by elastic perfectly plastic . ;
. behavior, i.e., at some critical pressure the flanks of the crack deform without . ;
bound and the tube ruptures. For a material with no strain hardening capability ,
the flow stress would correspond to the yield strength, S y , of the material. In-
~
practice the flow strength is taken as some value between the yield stress and the ultimate tensile strength, Sv. For Alloy.600 SG tube material, a flow stress of one-' '
half of the sum of the yield and ultimate strengths has been widely used, thus,.
Sf - f(Sy + SU)- (41) i Alloy 600 material typically' exhibits a flow stress on the order of 75 kai at ambient conditions, hence, test results are usually adjusted to this value for the presenta-tion of the data and the development of the regression equation. Once the correla-tion has been obtained, it can be scaled by the flow stress to estimate the burst-pressure at other temperatures, e.g.,
Sf l P3 l =P3 600'F l .
(4.2) 600*F 70*F S f-70l*F Tube material properties for APC applications are summarized in Table 4-1. While the values presented are not from on a randomized database, they are represen-tative of Westinghouse mill annealed tubing only. .
4.4 Bumt Pmesure Venus hhhin Voltage Canelathm The bobbin coil voltage amplitude and burst pressure data presented in'the EPRI ,
. database, Reference 4.2, have been used to estimate the degree of correlation between the burst pressure and bobbin voltage amplitude. The details of perform-ing the regression analysis to determine the degree of correlation and to estimate the parameters of a log-linear relationship between the burst pressure and the L
' bobbin amplitude, are provided in the EPRI database report. General techniques for the performance of regression analysis are described in Appendix A. to this' '
report. The evaluations examined the scale factors for the coordinate system to be :
employed, e.g., linear versus logarithmic, the detection and treatment of outliers, the order of the regression equation, the potential influence of measurement errors in the variables, and the evaluation of the residuals following the development of a l relation by least squares regression analysis. The results of the analyses indicated L
that an optimum linear, first order relation could be obtained from the regression of l the burst pressure on the common logarithm (base 10) of the bobbin voltage i amplitude.
SMPC\GENEMC\GENSECRWP5 4-3 a ry n. im i
1
7 ,
- The cquztion form rol-ting the bur::t pr:szura, Pa , ofindication i to the logarithm '
of th2 bobbin amplituda, V,, is given by, P3, = a1 + a2l og(Vi ), (4.3) ~
~
where ai and a2 are least squares estimates of coefficients ta and as that would be '
obtamed if the entue population of tubes with indications were tested. Here, the -
burst pressure is usually. measured in kai and the bobbin amplitude is in volts. .
A typical value for the index ofdetermination of the regression of the burst pressure on the bobbin amplitude is 80%. The corresponding correlation coefficient 1 would be 0.90, which is significant at a >99.999% level. This means that the-typicalp value for the slope of the line is < 0.001%. Hence, equation (4.3) provides an excellent functional form for the prediction of the burst pressure from the bobbin 4
- amplitude.
The estimated standard deviation of the residuals, i.e., the error of the estimate, s of the burst pressure is typically on the order of ~0.95 kai. Examination of the residuals from the regression analyses for 3/4" and 7/8" diameter tubes indicated 1 that they are normally distributed, thus verifying the assumption of normality inherent in the use ofleast squares regression.
A typical format for reporting the results of the regression analysis is illustrated in
- Table 4-2. The database used for the analysis and the regression results are shown on Figure 4-1.
Using the regression relationship, a lower 95% prediction bound for the burst pressure as a function of bobbin amplitude is then developed. These values are -
further reduced to account for the lower 95%/95% tolerance bound for the Westing-house database of tubing material properties at 650*F. Using the reduced lower prediction bound curve, the bobbin amplitude corresponding to a free span burst ,
- pressure of 3657 pai is found.1 This is the structurallimit as reported in plant ,
specific APC supporting analyses. A typical value for 3/4" diameter tubes is on the order of 5V. An additional limit corresponding to the actual SLB differential pressure is also calculated and reported.
4.5 Penhahmty ofImakage N Historically, the probability ofleakage has been evaluated by segregating the model ,
boiler and field data into two categories, i.e., specimens that would not leak during i a SLB and those that would leak during a SLB. These data were analyzed to fit a i 1
The value of 3657 psi results from considering a SLB differential pressure of i 2560 psi divided by 0.7 in accord with the guidelines of RG 1.121, Reference 4.8.
saecmmmicmassco4m 44 January 27,1995
, 1
- sigmoid typs cquetien to establish an elgsbraic r lationship between the bobbin:
I
- amplitudo and the probtbility ofIrak. Tha sp::ciSc algsbraic form used to date hr.s
~
been the logistic function with tlie common logarithm of the bobbin amplitude employed ~ as the regrassor variable, i.e., letting' P be the probability ofleak, and considering a logarithmic scale for volts, V, the logistic expression is:
1' P(leak lV) = ,
(4,4) 1 , - 10 + Eslor(V)]
. This is then rearranged as:
In =
1+ 2l og(V), (4.5)
,1-P,
_' to permit an iterative, linear, least squares regression to be performed to find the .
maximum likelihood estimators, by and bs, of the coefficients, p; and 3 The use of the logistic function for the analysis of dichotomous data is standard in many fields. The differential form assumes that the rate of change of the probabili-ty ofleak is proportional to the product of the probability ofleak and the probabili-ty of no leak. As noted, the function is sigmoidal in shape, and is similar to the '
cumulative normal function, and likewise similar to using a probit model (which is .
a normal function with the deviate axis shifted to avoid dealing with negative values). In principle, any distribution function that has a cumulative area of unity could be fit as the distribution function, a limitless number of possibilities. Trying -
to identify a latent, or physically based, distribution for the probability ofleak would be considered to be unrealistic and unnecessary. For most purposes the logistic and normal functions will agree closely over the mid-range of the data being fitted. The tails of the distributions do not agree as well, with the normal function !
approaching the limiting probabilities of 0 and-1 more rapidly than the logistic function. Thus, relative to the use of the normal distribution, the use of the logistic function is conservative. -
In addition, consideration was given as to whether the bobbin amplitude or the logarithm of the bobbin amplitude should be used. Since the logistic, normal and Cauchy distribution functions are unbounded, the use of volts would result in a finite probability ofleak from non-degraded tubes, and would be zero only for V=- .
By contrast, the use of the logarithm of the voltage results in a probability ofleak-
, for non-degraded tubes of zero. Clearly, the second situation is more realistic than the first, especially in light of the fact that a voltage threshold is a likely possibil-ity.
The log-logistic function falls into a category of models referred to as Generalized Linear Models (GLMs). This simply means that the model can be transformed into l
SMPC\GEMRIC\GENSEcRWP5 4-5 Jan=rr n, ms
_ ~ _ _ ___ ,
a linsar farm, e.g., squztion (4.5). The 1sft sida of equation (4.5) is rsferred to as '
tha link function for the mod::1. The p:ramst:rs of the cquition are estimcted by fitting the dzta using an iterative least squares technique, resulting in the maxi- -
mum likelihood estimate of the parameters.
The results of a typical regression analysis are summarized in Table 4-3. The coefficients of the equation are provided along.with the elements of the variance-covariance matrix for the coefficients. In addition, the deviance for the solution is also given. One accepted measure of the goodness of the solution or fit for GLMs is the deviance, given by, D=2{ g.i P, in - P-L. + (1 - Pg)ln P(V,)
1 - P'-
'. (4.6) '
, 1 - P(V,) ,
where Pg is the probability associated with data pair i and P(V) is the calculated probability from V.g The deviance is used similar to the residual sum of squares in linear regression analysis and is equal to the error, or residual, sum of squares (SSE) for linear regression. For the probability ofleak evaluation Pg is either zero or one, so Equation (4.6) may be written A
D = -241{ P,In[P(vg)) + (1 - P,)ln[1 - P(u,)]} . (4.7)
Since the deviance is similar to the SSE, lower values indicate a better fit, i.e., for a lower the residual sum of squares, more variation of the data is considered to be explained by the regression equation. Prior to the preparation of this report, analyses were performed to investigate the NUREG-1477 recommended forms for the POL function. The differences in the deviances from the analyses performed were judged to be not numerically significant relative to selecting the best form of a fitting function.
A significant outcome of performing the additional regression analyses was the finding that, taken in cordunction with the leak rate versus voltage correlation, the choice of a probability ofleak function is relatively unimportant. For typical APC/
IPC voltage distributions, the final total leak rate values obtained using all of the functions tend to differ by only a few percent across the spectrum of POL functions.
An example of the format of reporting the results from the POL regression analysis is provided as Table 4 3. The Pearson standard deviation, om , in the table is discussed in Section 5.0 of this report. A plot of the POL database used for the example regression is shown on Figure 4-2, along with the regression curve obtained from the GLM analysis of the data.
4 S.MPC\ GENERIC \GENSECMWP6 4-6 hwy 27,1995 4
4.6 SLB Imak Rate Versus Voltage Correlation i
- The bobbin coil and leakage data previously reported were used to determine a correlation function between the SLB leak rate and the bobbin amplitude voltage. ,
- Since the bobbin amplitude and the leak rate would be expected to be functions of -
the crack morphology, it is to be expected that a correlation between these vari- {
ables would exist. Previous plots of the data on linear and logarithmic scales )
indicated that a linear relationship between the logarithm of the leak rate and the logarithm of the bobbin amplitude would be an appropriate choice for establishing a correlating function via least squares regression analysis. Thus, the functional form of the correlation is log (Q) = b3 + b4 log (V), (4.8) where Q is the leak rate, V is the bobbin voltage, and b3 and b, are estimates obtained from the data of some coefEcients, E3 and p4 The Snal selection of the
~
form of the variable scales, i.e., log-log, was based on performing least squares regression analysis on each possible combination and examining the square of the correlation coefEcient for each case. The results of the analyses, using the EPRI database, indicated the appropriate choice of scales to be log-log.
A format for reporting the results from the regression analysis is provided as Table 4-4. The data used for the analysis and regression curve obtained from the analysis are illustrated on Figure 4-3. The example value of r2 of 59.4% is signifi-cant at a level of >99.99% based on an F distribution test of the ratio of the mean square of the regression to the mean square'of the error. This can also be inter-preted as the probability that the log of the leak rate is correlated to the log of the bobbin amplitude. The p value for the illustrated slope parameter is 1.510 4 . The conclusion to be drawn from these results is that it is very likely that the variables -
are correlated. Per the draft generic letter, the validity of the regression is judged ;
by the p value associated with the slope. Since this is significantly less than the ~
0.05 value stipulated in the draft generic letter, the regression would be concluded to be valid, and the use of the linear regression results would be acceptable.
The expected, or arithmetic average (AA), leak rate, Q, corresponding to a voltage ,
level, V, was also determined from the above expressions. Since the regression was performed as log (Q) on log (V) the regression line represents the mean oflog(Q) as a function of bobbin amplitude. This is not the mean of Q as a function of V. The residuals oflog(Q) are expected to be normally di9tributed about the regression line. Thus, the median and mode of the log (Q) rMduals are also estimated by the regression line. However, Q is then expected to be distributed about the regression line as a log-normal distribution. The regression line still estimates the median of Q, but the mode and mean are displaced. The corresponding adjustment to the normal distribution to obtain the AA of Q for a log-normal distribution is suresomsnicsaussenwn 4-7 amor n. im i
m '---*-*Y- - '
- pt- wm e- em i t- __ e- --- * -.________ ___ m--__-.+-__v---__--r -- --.___ _ _--__.__ - -
A ,
d
^ -
6a o 6.logt vu WWe2
-(4,g) j Q - ElQ lV) = 10 2 i,'
~
2 for a given V, where o is the estimated variance oflog(Q) about the regression -
line. The variance of the expected leak rate about the regression mean is then l obtained from ;
~
~
s Var (Q) = Q 2 yowmo: -1. (4.10) ,
To complete the analysis for the leak rate, the expected leak rate as a function of l'
log (V) was determined by multiplying the AA leak rate by the probability ofleak as a function oflog(V). The results of this calculation for an example database are -
also depicted on Figure 4-3 for a steam line break differential pressure of 2560 psi.
~
{
Leak Rate Model when the p-Value is Greater than 5% i 1
- The NRC generic letter requires that the dependence of the leak rate to the bobbin "
amplitude be demonstrated by a rigorous statistical analysis. If the p value ob-tained from the regression for the slope parameter is less than or equal to 5%, the dependence of the leak rate on the bobbin amplitude is considered to have been demonstrated. There is the potential for the 7/8" tube data to exhibit a p value in excess of 5% depending on the level of application of the data exclusion criteria -
permitted for a plant speciSc APC implementation. Thep value can be thought of as the probability that the true slope of the correlation is zero even though the i
value obtained from the regression analysis is other than zero. If the p value is !
greater than 5%, it must be assumed that a correlation does not exist between the leak rate and the bobbin' amplitude. 4 If the leak rate is considered to be independent of the bobbin amplitude, the leak rate model is,
~
Qg = 3+e, (4.11) !
j- i where Qg again represents the common logarithm of the leak rate. The mean of the l .
data in the database, b3 , is used to estimate 3, and e is the estimated error about
' the mean of the logarithm of the leak rates in the database. Again, e is assumed to i be distributed such that it has a mean of zero. The standard deviation'of the err
' is estimated by the standard deviation of the data. For both databases, statistical-i analyses have been performed that demonstrate, at a greater than 95% confidence ;
level, that the leak rate data are log-normally distributed, independent of correla-l tion considerations. Hence, the erro:s about the mean of the log leak rate are i
assumed to be log-normally distributed. i, SAAPCiGENEEC\GENSEC04.WP5 4-8 January 27,19D5 i
Leak Rate versus Bobbin Amplitude when the p-Value is greater than 5%
' If the p value from the regression analysis is greider than 5%, it is still possible to rigorously demonstrate a dependence of the leak rate on the bobbin amplitude. In this case, it is assumed that the model is either invalid or may not apply over the entire range of the data. Analyses have been performed by segregating the data at the median value of the voltage range. Using the model of equation (4.11), it has been shown at a greater than 95% confidence level that the leak rates for the lower half of the voltages are not from the same population as the leak rates for the upper half of the voltages. This has been demonstrated for the 3/4" and 7/8" tube data. Thus, it would be possible to select an upper bound voltage to be expected at the EOC, and to use an equation (4.11) model based only on the leak rate data from indications with bobbin amplitudes less than the upper bound. While not currently approved for use if the p value is greater than 5%, this approach may be pursued in the future, in which case it should be documented as a revision to this report.
4.7 Indusian of Future Data in the Cormlations The initial analyses performed for the implementation of APC verified the validity of the regressions for the burst pressure, the POL, and the leak rate. For each of the correlations, additional veri 6 cation of the appropriateness of the regression was obtained by analyzing the regressioh residuals, i.e., the actual variable value minus the predicted variable value from the regression equation. Plots of the residuals as a function of the predicted values was found to be nondescript, indicating no appar-ent correlation between the residuals and the predicted values. Cumulative probability plots of the residuals on normal probability paper approximated a straight line, thus verifying the assumption inherent in the regression analysis that the residuals are normally distributed. Based on the results of the residuals scatter plots and the normal probability plots, it was concluded that the regression curves and statistics could be used for the prediction of the burst pressure, the POL, and leak rate as functions of the bobbin amplitude of the indications, and for the establishment of statistical inference bounds.
As additional data become available they may be incorporated into the reference database utilizing the approved outlier criteria. Verification of continued use of the speci5ed equations may be based on a visual inspection of the data relative to the l database and the correlation equations. Analyses of additional data since the l original determination of the regression equations has revealed no circumstances where the form of the equations should be questioned. In these cases, the analyses of the residuals does not have to be repeated tojustify the use of the equation forms specified in this report. As new data is added to the database and new correlation parameters calculated for the implementation of APC at specific plants, the appropriate tests on the data relative to incorporation should be performed and documented in the report for that plant.
swcscar.asesomsecoms 4-9 Anuary 27,199$
.w- . -. _
f $..' Sample nsw burst dnta and tha regrcerion curve obtain d by including the data in ~ ~
the dnt b:ca ars shown en Figura 4-1. Ths r; ults'of ths regression c.nalysis performed with the new data included in the database are summarized in Table 4-5, and the results of the regression analysis of the new data only are' summarized in Table 4-6. The new data may bejudged by inspection to fit with- '
the reference database. It is also noted that the slope of the reference database regression curve is only about one standard deviation different from the slope of the line using the new data only. Hence, the new data could be statistically demon-strated at a high level of confidence to be from the same population as the data in the reference database.
Sample new POL data and the regression curve obtained by including the data in - -
the database are shown on Figure 4-2. The results of the regression analysis performed with the new data included in the database are summarized in Table.4-7. By inspection, the. data are similar to that in the reference database.
The regression curve obtained by including the new data in the analysis is not signi5cantly different from that obtained using the reference data only.
Sample new leak rate data and the regression curve obtained by including the data in the database are shown on Figure 4-3. The resulte of the regression analysis performed with the new data included in the database are summarized in Table 4-8. The new data would be judged by inspection to belong to the same population as the data from the reference database. The inclusion of the data in the regression analysis has an insignificant effect on the regression line. !
It should be noted that for the eramples analyzed herein, demonstrating that the new data should be included in the reference database was straightforward. This may not always be the case with real data. Thus, additional analyses may have to be performed for plant specific reports if the reference database is to be expanded. -
F s.urcsormuucsornsrcums 4 - 10 m a ry n m s
y . - - - _. .~ _ _. . . . _ . . _. ..
~ 7, o
- 4.10 : Refer-naa
. a c
The refarene:s us:d in tbo ,areparaticn of this report section were:
..t 4.1 ' TR-100407, Revision'l-(draft), "PWR Steam Generator Tube Repair Limits - - :
Technical Support Document for Outside Diameter Stress Corrosion Crack at -
Tube Support Plates," Electric Power Research Institute, August'1993. :
4.2 . ' NP-7480-L, Volume 2, " Steam Generator. Tubing Outside' Diameter Stress Corrosion Cracking at Tube Support Plates '- Database for Alternate Repair / :
. Limits, Volume 2: 3/4 Inch Diameter Tubing," Electric Power Research Insti -
3 tute, October,1993.
4.3 ' NUREG-1477'(draft), " Voltage-Based Interim P'ugging Criteria for Steami s Generator Tubes - Task Group Report," United States Nuclear Regulatory l Commission (NRC), June 1,1993. :
a 4.4 [ United States Nuclear Regulatory Commission] Meeting with EPRI, i
NUMARC, " Resolution of Public Comnients on Draft NUREG-1477," United
' States Nuclear Regulatory Commission, February 8,1994. i 4.5 EPRI Letter, " Exclusion of Data from Alternate Repair Criteria (ARC) Data . '
bases Associated with 7/8 inch Tubing Exhibiting ODSCC," D. A. Steininger (EPRI) to J. Strosnider (USNRC), April 22,1994 [to become Appendix E of
'l ;
Reference 4.2].
~
l 4.6 Draft Generic Letter 94-XX, " Voltage-Based Repair Criteria' for the Repair of ' >I Westinghouse Steam Generator Tubes Affected by Outside Diameter Stress Corrosion Cracking," United States Nuclear Regulatory Commission, Federal Register, Vol. 59, No.'155, August 12,1994, pp. 41520-41529. .i 4.7 USNRC/ Industry Meeting, " Resolution of Public Comments, NRC Draft
- Generic Letter 94-XX,"' January 18,1995. '
4.8 Regulatory Guide 1.121 (draft), " Bases for Plugging Degraded PWR Steam -
Generator Tubes," United States Nuclear Regulatory Comminaion, issued for comment in August,1976.
4.9 Docket STM-50-456, " Safety Evaluation by the OfEce of Nuclear Reactor Regulation Related to Amendment No. to Facility Operating License No.
J NPF-72 Commonwealth Edison Company Braidwood Station, Unit No.1,"-
United States Nuclear Regulatory Comminaion, May,1994. -
4.10 WCAP-14046 (Proprietary), Revision 1, "Braidwood Unit 1 Technical Support for Cycle 5 Steam Generator Interim Plugging Criteria," Westinghouse l Electric Corporation, August 1994.
STAPC\ GENERIC \GENSEC04 WP5 4 11 Juury M. M5 i
t
b 4
- m. .,
5 i
Table 4-1: Tube Materal Properties for APC Applications (Westinghouse)
"Wy Value at RT Value at 650*F Alloy 600 Mill An== led 3/4" x 0.043" SG Tubes Sample Size 635 627
- Yield Strength Mean 53.05 45.78 -
Yield Strength St. Dev. 4.8602 3.9081.
Tensile Strength Mean 101.29 97.35-Tensile Strength St. Dev. 4.2173 3.9676 =
Flow Stress Mean 77.17 71.57 ;
Flow Stress St. Dev. .4.1422 ~ 3.5668 -
95%/95% LTL Flow 69.925 65.325-Alloy 800 Mill An===1=d 7/8" x 0.050" SG Tubes >
Sample Size 361 360 Yield Strength Mean ' 50.98 41.89 I Yield Strength St. Dev. 4.2068 3.5856 Tensile Strength Mean 99.96 95.67 Tensile Strength St. Dev. 3.6123 3.4196 Flow Stress Mean 75.47- 68.78 Flow Stress St. Dev. 3.5002 ~ 3.1725 95%/95% LTL Flow 69.225 63.115 I
- S \APC\ GENERIC \GENSEC04.WP5 4 - 12 3.=uar727. toes i i
..e ;
t l
e
}
Table 4-2: Regression Analysis Results -
Burst Pressure vs. log (Bobbin Amplitude) '
Alloy 600 MA SG Tubes (Reference or = 75 kai) '
t CAUTION: Fannh- dm*mhame used, for illustration only. t Parameter Value Value Parameter r by -3.126 7.832 bo SE by 0.168- 0.129 SE bo .
8 r 81.2% 0.946 SE P3 F 346.5 80 DoF SS, 310.04 71.58 SS, P Pr(F) 8.3E-31 31.72 SSgg :
p3-value 8.3E-31 8.5E-69 po-value 1
)
l l
S.\APC\ GENERIC \GENSEC04.W6 4 13 a ,, n, an 1
l
- l 1
0 1
i Table 4-3: Results of POL Regression Analysis
@ 620'F and AP = 2560 psi CAUTION: Randam datahame, fbr illustration only.
Parameter Values '
b; -4.947 b, 8.337 V;; 1.369-V;3 -1.932 l V22 3.106 Deviance 40.37 Pearson o_, 0.77 l
s:urcscarmcsozuszcos.wes 4 - 14 aowy st. tees i
\
i Table 4-4: Regression Analysis Results:
log (Leak Rate) vs log (Volts) for Alloy 600 SG Tubes -
' @ 620*F and .AP = 2560 psi CAUTION: Random datahame, for illustration only.
Parameter Value Value Parameter b, 3.259 -2.000 b3 SE b, 0.421 0.410 SE b3 2
r 59.4% 0.707 SE log (Q)
F 60.1 41 DoF SS, 30.01 20.48 SS,,,
Pr(F) 1.5E-09 2.825 SSun pi-value 1.5E-09 1.67E-05 po-value i
S.\APC\ GENERIC \GENSEC04.Wn 4 - 15 Jano ryav,isss l
t
i t
Table 4-5: Regression Analysis Results -
Burst Pressure vs. log (Bobbin Amplitude)
NEW DATA ADDED Alloy 600 MA SG Tubes (Reference or = 75 kai) '
CAUTION: Bandr== databsoe used, fbr illustration only.
Parameter Value Value Parameter "
bf -3.145 7.832 bo SE b, 0.153 0.116 SE bo .
r8 83.0 % 0.915 SE P3 F 423.3 87- DoF SS. 354.51 72.86 SS, '
Pr(F) 3.5E-35 35.83- SS un pi-value 3.5E-35 6.2E-77 po-value SMPC\ GENERIC \GENSEC04.WP6 4 - 16 January 27,1996 a
~
h i
Table 4-6: Regression Analysis Results -
Burst Pressure vs. log (Bobbin Amplitude)
-NEW DATA ONLY -
Alloy 600 MA SG Tubes '
(Reference or = 75 kai)
CAUTION: Ba=lr= datahama used, ibr illustration only.
Parameter Value Value Parameter b, -3.420 7.749 bo SE by 0.246 0.166 SE bo r# 97.5 % 0.437. SE Pg i F 194.03 5 DoF t SS, 36.98 0.953 SS, '
Pr(F) 3.4E-07 3.16 SSwg i p1-value 3.4E-07 8.4E-08 po-value i
STDC\ GENERIC \GENSEC04.WP5 4 - 17 Juury 27,1995
{;
m ..
t )
4 Table 4-7: Resulta' of POL Regression Analysis with New Data Added
@ 620*F and AP = 2560 psi -
i CAUTION: R==dr= d=*mhama, for
- l. Illustation only. ,
e
[ Parameter Values bi -5.091 i 63 8.582 V;1 1.399 V,3 -1.987 V23 3.197 Deviance . 41.14 Pearson o_ 0.80 1
6 1
SAAPC\ GENERIC \GENSEC04.WP5 4 . }g January 27,1995
_ _ _ _ . _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ - - - - - - - - - - - ~~
t l':4 .
f; i
L ,. , , -
l
-r I
i i
Table 4-8: Regression Analysis Results: ;
log (Leak Rate) vs log (Volts). ,
with New Data Added i sr Alloy 600 SG Tubes. !
.@ 90*F and AP = 2560 psi l
CAUTION: Randam d=*mha=, i
. for illustruthm only.
Parameter Value Value Parameter !
b, 3.244 -1.981 b3 SE b, 0.408 0.397 SE b 3 !
3 r 59.6% 0.691 SE log (Q)
F 63.2 43 DoF SS, 30.19 20.53 SS, t
Pr(F) 5.5E-10 2.868 SS m I pi-value 5.5E-10 .1.03E-06 po-value ;
~
4 ,
t b
t f
SAAPC\GMF.RIC\GESECH.WP5 4 - 19 a .r, n, ms
Figure 4-1: Burst Pressure vs Bobbin Amplitude 3/4" x 0.043" Alloy 600 MA SG Tubes (Random Database @ 650 F) 12.0 l 4 6 i ii a Data adjusted to 650*F LS Regression 10'0 ~~~
aN "
El
~-
' u ------95% Prediction & LTL s
\ %
D
's s -0 -- -
SLB RG I.121 Limit 8.0 -., '
N '
3 ---- SLB Pressure u
' " "c n .
~., s A New Data j . c a . ._
v .,' ., N, '
u D 13 + LS w/New Data -
, t E-n ,
3 U
A '
! ---1 ___ f --J -
gC h
m 6.0
- s.
s
- s. oB I 3
\ %
0 o\
o Reference o, = 75 ksi 5 ., % a g
a s .
" a 'a\ w U U n g
's, o "gh pp -- - -
4.0 3.657 ksi _. -
o goo. a+ n
_._._.__ _ . . _ . . . . _ . __ __ __ . s
_._. . _ _._._. _ _ . . . _. 3, s, ci 2.560 ksi g
s x% D N -
.I_
._.. _ . ._..... ... . _ . . _ . . _ .._......_ ... . .... "tc T,
~N 2.0 '
- '.'. ,s '+
I g
.s-.
CAUTION: Random database, used for . i_.
illustration only.
._ _ _ _. .1
_ _ _U J N. s_- _
I 4.5 0.0 ' ' ' ' .
V . l11.4 V O.I 1.0 10.0 100.0 Bobbin Amplitude (Volls)
[34PV_ GEN.XLSj 34PV. Gen
" ' " * * ' ' ' ' ~ ' ~
Figure 4-2: Probability of Leak for 3/4" SG Tubes @ 650 F, AP = 2560 psi Comparison of New Data with Random Database 1.0
, , , ,,;,,; = = :r - .............. -c o Random Database 0.9 J
/
4 New Data -
Log Logistic 0.8 - - Log Logistic w/New Data J v3 0.7 E
'g 0.6 Q
I j% 0.5 CAUTION: Random database, used ;
y% o for lilustration only.
I 0.4 I
E g 0.3 7
0.2 0.1
(
0.0 3-o-o-- . - - . . - - - - . . - - - - -
m o- ar -c 0.01 0.8 1.0 10.0 100.0 Bobbin Amplitude (Volts)
[34PL_ GEN XLS] Pol. Gen WF 1 r*8 '"^ "-' ' "
_ _ - s
Figure 4-3: SLB Leak Rate vs. Bobbin Amplitude, Random Database 3/4" x 0.043" Alloy 600 SG Tubes @ 650 F, AP = 2560 psi 10000
-Z ~ ___ _ _
__7b;_4/l ,
o Random Database o ' ' -
7 ' ~
4.[-
- Regression Fit -- - ~ '
1000. --- Arithmetic Average (AA) '
-7
. . . . . 95% Confidence on AA
~-
/ /
f ~7 7 _ - . _ _ _ .
AA Leak Rate
- Pr(Ixak)
~ ~ ~ ~ ~ .
. "- f-~ -- ~ ~ ~ ~ ~ ~ ~1 ~ ~~ ~
~
a New Data
^
+ LS w/New Data
.~/V
^
$. 100-
",do u
y u -
== :--_ :- ... ~
=
'5.
g m
4 [ C-
_ b T --
3
&,a1L. !
a n +n
//
s/
@ g n' '
A
- o o O s
, 4 _" .e __ z ZZ Er T ~
E -
-D
- i
,'/g. __.a-g y ,..-
, D .D z-- :
j ,. ,' -__ - -
.' /
g . _ _ _ _ _
$ 1.
e
/ . _ _ _ . . _ . . . _ .
m _
,- , ,c a .
2
._ _;.3 =.
5= =-. ; 2 p' -
. _ . 2.z= : -n- -
f f . - , .- ._ -
-/ ,'
0.1 CAUTION: Random database used,
' ~
- ,- ,/ ___ _ _ _
for illustration only.- - "
/ .' -
-- n -
O.01 < / '
i 10 l(0 Bobbin Amplitude (Volts)
[340V GEN.XLSIOvsV. Gen ~
firK 1r/fm. ? '4 MA
5.0 SLB' Leak Rate & Tube Burst Probability Analysis Methodology 5.1 General Methods Considerations The purpose of this section is to provide information on the use of the correlations described in Section 4.0 of this report in support of the application of APC to specific indications in tubes in SGs at nuclear power plants. .Information is presented on the use of deterministic models used for sensitivity analyses, and on the use of statistical simulation methods, i.e., Monte Carlo analyses, to estimate the total leak rate of all indications in the SG and the probability of burst of one or more of the indications in the SG.
The NRC generic letter requires calculations of SLB SG total leak rates and tube burst probabilities from both the actual, i.e., as measured, EOC voltage distribution and for the projected next EOC voltage distribution. The methods of this section may be applied to either voltage distribution.
5.2 Deter =inistic Methods for Sensitivity Analyses The leak rate versus voltage correlation can be simulated in conjunction with the EOC voltage distributions obtained by Monte Carlo methods, or by applying the POL and leak rate correlations to the EOC voltage distribution obtained by Monte Carlo methods as applied for the draft NUREG methodology. This second approach is a hybrid that joins Monte Carlo and deterministic calculations. Parallel analyses verified that the full Monte Carlo leak rates and the direct application of the correlations to the EOC voltage distribution yield essentially the same results.
Thus, it is adequate to apply the correlations to the EOC voltage distributions.
5.2.1 Deterministic Estimation of the Total Leak Rate The determination of the end of cycle leak rate estimate proceeds as follows. The beginning of cycle voltages are estimated using the methodology described in Section 3.0 of this report. The distribution ofindications is binned in 0.1V incre-ments. The number ofindications in each bin is divided by the POD. The result-ing number ofindications in each bin is reduced by the number ofindications plugged in each bin. The final result is the beginning of cycle distribution used for the Monte Carlo simulations. The NDE uncertainty and growth rate distributions are then independently sampled to estimate an end of cycle distribution, also l reported in bins of 0.1V increment. Given the EOC voltage distribution the !
calculational steps to obtain an estimate of the total leak rate are as follows- l l
l (1) For each voltage bin, the leak rate versus bobbin amplitude correlation.is
- l used to estimate an expected, or average, leak rate for indications in that 1 bin.
l- S.\ APC\ GENERIC \GENSEC05 WP5 5-1 .w.n 2s, ms l
l l
7 f(2); Th2 pro $ ability cf13akaga c:rr:lcticn is th:n used to cetimsts the mes.n^
~
?
- L prob:bility cfIf:ak for thz indic ti:ns in c:ch bin.-
( '
.i (3) !The relationships derived in Appendix C of draft NUREG 1477 for the variance of the' product of the probability ofleak with the leak rat'e and ]
for the total leak rate are then used to estimate the expected total leak-age and variance for the' sum of the indications in each bin as a function
~ '
o
.-probability f the correlation of leak. means~and estimated ~ variances for the leak' rate
. Recall from Section 4.0 that the expression used for the probability ofleak of an indication,'i, with a bobbin amplitude V,is, P(leak lV ) =
g
'l
.g'n.
1+e d
- 1 * * * '** ( H.
v and the expression for the expected, or AA,-leak rate, Q4, from indication i as a function of volts, Vg , is given by, l t
63+ 6 log (V,)+ In(10)
Q, = E(Q4 lVg) = 10 2 -
{
with a variance of the expected leak rate about the regression mean of, ;
2 I Var (Q4) = Qg 10 "IIO)U- 1, . (5.3)
To account for the variances of the coefficients of the regression equation for the leak rate, the o used in equations (5.2) and (5.3) is that from the predictive distri-g bution for the logarithm of the leak rate as a functior: of bobbin amplitude, i.e., for each voltage, V , an effective standard deviation of the regression error, o , is 4
calculated as g ;
1 5
g og = o, 1+i+ log (Vi )-log (V)
N -' 2
, (5.4) ,
E j
log (Vj )-log (V) '
where N is the number of data pairs in the regression analysis, and o, is an unbiased estimate of a for the population. '
i s upciornzmicsczxsecos wPs 5-2 m.,y a. ms '
t
' j '
Ths cxpected total Isak rate from cll of the indications in'all of the bins is - <;
7
, N,.
63+ b, logW, ) + Inf 10) ,'2 - l 9
a..
' T =fi 1 , , - 16 + .6n 1or 2 W.H 10 .2
.(5.5) where N3is the number ,sf bins, and n, is the number ofindications in the bin, which is not necessarily an integer number, with bobbin amplitude V,. Thus, the. -
expected total leakage for the entire. distribution is obtained as th'e sum' of the '
expected leak rates for each bin.
i In order to es.timate an upper confidence bound for the total leak rate an expression -
is needed for the variance of the total leak rate. There are two sources of variance to be considered, the variance about the predicted expected value and the variance of the predicted expected value; the estimated total variance about the predicted expected value being the sum of the two. Moreover, the variance of the probability - '
ofleak must be considered in the variance about the predicted value. The variance '
of the total leak rate about the predicted expected value, including consideration of ,
the variance of the probability ofleak, is Na '
V(T) = { ng P, Q,2 , Wlo>o.'- 1, 2
+ q p (1_ p )f , (5.6) 41 I
where Pg is the probability ofleak from equation (5.1). Equation (5.6)is based on an application of the standard expression for the biased estimate of the variance of ,
a product. As noted, an additional variance term is added in order to estimate the ,
contribution to the variance from the correlation between the individual leak rates, i.e., from the covariance, which arises as a consequence of using the regression ,
equations. Thus, the second term accounts for the variances of the positions of the regression equations. A linearized approximation (via Taylor's Theod of the variance of the mean of the regression prediction, T,, is given by
<r '
[ Coo ( i, 2)]
y 0 0 dT -
0 dT V(Tp ) = E n,< 0 -
(5.7)
[ Coo (ps, ,))
i-1 ' dQj d p>. .
0 0 V(o,2) where the derivative of the total leak rate vector contains five elements forj=1,...,5, and the Covariance Matrix is a square 5x5 matrix consisting of the estimated :
variances and covariances of the estimated individual regression coefficients and '
oj . Note that here [ Coo ( 1,02 )] and (Cov( 3, p4 )] are each 2x2 matrices, where surcscrutarcsorwsrcos wps 5-3 .ro.,y x ms
tha p's arb estim ted by y b through b,, and rzc:ll thtt o, is an estimete of 3 The -
. varianca cf tha varitnea is e:timet:d as 20,1 V(o,2) , (5 8) n-2, where n is the number of data pairs used in the leak rate regression analysis. The standard deviation of the total leak rate is then taken as the square root of the variance of the total leak rate. The upper bound 95% confidence limit on the total leak rate is then obtained as the expected total leak rate plus 1.645 times the standard deviation of the total leak rate. The results obtained with this approach have been compared to results obtained from the Monte Carlo simulations without significant differences being observed for total leak rates at a 95% confidence level when leak rates in excess of 1 GPM are predicted. At higher confidence levels, e.g.,
99%, the differences could be significant. Because of this uncertainty the determin-istic analyses are limited to sensitivity studies.
5.2.2 Deterministic Estimation of the Probability of Burst as a Function of Volts Using BOC or EOC distributions and the regression results for the correlation of burst pressure to bobbin amplitude, an estimate of the probability burst of one or more tubes in the SG can be estimated. The regression curve was given in Sec-tion 4.0 as, Ps, = a1 + a2l og(Vi ) (5.9) l where the burst pressure is measured in kai and the bobbin amplitude is in volts. 1 Here, a; and af are estimates of unknown parameters at and a2 of the relation.
The value obtained from equation (5.9) applies to tubes with a flow stress equal to the reference flow stress, Syf, used in estic: . ting the equation parameters. A normalized value of the burst pressure cc 2en be found as, Pa' .
(5.10)
Py, = S,7 The burst pressure for any single indication is then given by, P4 - Py, Sf. (5.11) i surcsotxtarcsotustcos wrs 5-4 a.-ry 28.1995
~ , .. . -
~
- and the varienca of ths bur t przszura accounting for the variance of the rssiduals.
ebout ths regres2 ion curva and ths varir. tion in S7can b3 calculated as
'1 ,
7 2
V(P3_ )= P3, y ( g f) y ,2 V(P6er) - VCP6ar)V(St) ,
(5.12)
, ms 1
where V is used to represent an unbiased estimate 'of the variance of the respective variable ~ in parentheses. The standard deviation of the burst pressure is then taken -
i as the square root of the variance of the burst pressure. For any voltage level, the number of standard deviations difference between the predicted burst pressure and .
the SLB differential pressure can be calculated. The probability of burst, Pr,, is l then obtained from a Student's t distribution. For n indications in a voltage bin, the probability that none of the indications in the bin burst is then, Pra = (1 - Prg )" . (5.13)
For all of the indications in all of the bins, the probability of burst of one or more .
indications is then, N
Pr = 1 - [ (1 - Pr a)N , G.W k =1 where n3 is the number ofindications in bin k, not necessarily an integer number, and Pr4 is the probability of burst of a single indication in bin k.
5.3 Deterministic Analysis Results An example of the results from a deterministic analysis of a sample data set is presented in Section 6.0 of this report. .
5.4 Monte Carlo Analysis Ma*hadningy The estimated, total end of cycle leak rate can also be calculated using Monte Carlo techniques, e.g., the method documented in the EPRI ODSCC report (TR-10047, Rev.1). In the Monte Carlo analysis, the variation in the parameters, i.e., coeffi-cients, and the variation of the dependent variable about the regression line are simulated. A 95% con 6dence bound on the total leak rate from the SG is calculated using a Monte Carlo simulation. The results from the deterministic analyses are used as an order of magnitude verification of the Monte Carlo results. The ap-proach used for the simulation is different from that discussed in the EPRI ODSCC report (see the Section 4.0 reference list). While both methods simulate the varia-swesornraictozusscos wPs 5-5 J==rra ms
'i
$ tiin of ecch paramster cf the c:rrelction equstions, the msthod 'discu:std hsrein- _
~
$ ' cl:o^simulct:d tha effect cf thm covarianca cf ths individuti indication laak rates.
- In ordsr to simplify ths di:cuesien of ths Menta Carlo techniques, different nomen- ,
clature is used from that of the previous section, i.e., Q, is used to represent the common logarithm of the leak rate, and V, is used to represent the common logarithm of the bobbin amplitude. Thus, the following model is used to describe a working relationship between the logarithm of the leak rate and the logarithm of the bobbin amplitude, Q4 = ba + bi V; + e , (5.15) where e is the estimated error of the residuals, assumed to be from a population that has a zero mean, and a variance that is not dependent on the magnitude of V.g The coefficients, b3 and b, are the estimates from the regression analysis of some true coefficients, 3 and 4, representing the intercept and slope of the equation, -
respectively.
The method used by Westinghouse for simulating the total leak rate is the outcome of a series of technical discussions held with the NRC. The method differs from that reported in some prior WCAP reports, wherein the predictive distribution was simulated and covariance terms were ignored. It is noted that, although both methods yielded similar results (within ~3%) for one domestic plant analyzed, the .
method described herein is more statistically accurate. This small difference in the total leak rate results is because the contribution of the covariance terms relative to the variance terms is relatively small for the correlations used herein. In summa-ry, random versions of the POL and leak rate correlations are generated and used to calculate the sum of the leak rates for all of the indications in a SG to obtain a single simulated value of the total leak rate. This process is repeated to obtain a distributiori of the total leak rate from at least 10,000 simulations of the correlation equations. A non-parametric 95% confidence bound on the total leak rate is then estimated from the distribution of total leak rates.
At the start of each SG simulation, i.e., the calculation of a single total leak rate, a random value for the standard deviation of the errors for the population is calculat-2 ed from the x distribution, the degrees of freedom from the data, and the standard deviation of the regression errors. This is used to calculate random values for the parameters of the regression equation, which remain constant for the entire SG simulation. The variation of the regression predictions are accounted for by randomly estimating the POL from a uniform distribution, and by adding the product of a random normal deviate and the standard deviation of the errors for the population to the predicted logarithm of the leak rate, for each individual indication in the SG distribution. The total leak rate for the SG simulation is calculated as S \APC\ GENERIC \GENSEC06 WP5 5-6 a = =ry a, m s
'l N
C gM " ; s.. '
t .. .
tha sum ~of thfliak~rst:s from all of the indications in the SG. The expressica for d tha total Isak rats is-a . i N
1T = E ,Rj(pi, 2)Qi(@s, @4, Ps), (5.16) -
e t.1
- where N = the total number of whole or partial in'dications in the SG at-EOC,.
. q, = the proportion of the indication, e.g.,1 for a whole indication, y
( Rj (pi, p2) = ' O or 1 is the POL for a single indication,- i, in'a tube, ,
Q,(p3,p4, 3) =: is the conditional leak rate ofindication i, i.e.,'the leak rate if '
.the indication is leaking, 1, p2 = ,the coemeients of the POL equation, .
p3, 4 = : the coemeients of the leak rate versus bobbin amplitude equa-tion, and 3= the standard error of the log of the leak rate about the correla-
' tion line, also referred to herein as o.
To simulate the total leak rate from all of the indications in the generator, random coemeients for the probability ofleak, POL, and leak rate correlation equations are generated, and then those coemeients are used to simulate the POL and leak rate -
for each indication. . The POL, Rj , for each indication, i, is simulated as, -
R (p) = 1 if U < l Ki t(@i + @2l og(Vi ))
4 i
(5.17) !
0 otherwise p
where Uj is an independent draw from a uniform distribution. The step of deter ' ,
mining an integer value for the POL accounts for the variation of the distribution 'of probabilities about the log-logistic regression line. Discussion of the generation of-1 and $2i s left until after the discussion of the coescients for the leak rate equation. '
I 5.4.1 Probability of Leak Simulation The generation of the coemeients of the POL relation to be used in the simulation of the total leak rate proceeds in the same manner as for the 'coescients of the leak j rate relation. The elements of the covariance matrix are obtained from the GLM -
regression analysis and used with the estimated coescients in equations like (5.21) and.(5.22) to obtain $1 and $2f or a random population POL equation. However, for the simulation of the POL, there is no term of the form 2, o in the simulation of the total leak rate. This exception is due to the fact that the data are binary. In effect, j this additional term is beinr simulated through the use of the random sampling to determine if R; is 0 or 1 in equation (5.17). l I
S.\APC\ GENERIC \GENSEC06.WP5 5-7 Ja=ry x. ms '
.It is ncted thzt the cirmentslef the covariance metrix obtain d from the GLM'.
regr:csien ara scaled to a mzen squara error (m:s) of1. This is because the mse .
fer the binary variabl:s is asymptotically _1. ' A check of this assumption can be. -
- made by. calculating an estimate of the square root of the mse, also referred to as- -
the Pearson~' standard deviation, from the regression results as 1
- 3. 1 - (Ji - Mi )2 (5.18) n-2{g pj(1 pg) ,
where the yf s are the observed probabilities ofleak, either zero or one, from the *
. leak and burst testing, and the pj's are the calculated probabilities ofleak from the logistic regression equation. A significant departure from 1 for this quantity.could
. be indicative of an inadequate model.
i 5.4.2 Leak Rate versus Bobbin Amplitude Simulation -
To simulate the leak rate from the regression line, random coefficients p3 4and L must be simulated. Each of these has a variance that is dependent on the variance -
of the error of the log of the leak rate about the regression line. _Thus, the first step is to simulate a random error variance by picking a random x2 deviate for n-2 degrees of freedom and.then calculating a random error variance, a2 , for the correlation equation from the regression error variance as c2, (n - 2) 62 = fy b 2>
(5.19)
% (n - 2),inadom where n is the number of data pairs used to calculate the regression coefficients, and fy is defined by equation (5.19). This is now one possible variance for the population oflog-leak rates about a correlation equation. Thus, it is appropriate to - -
use the normal distribution to obtain random . values for the parameters of the -
correlation equation. The distribution of $3 and p4 will be bivariate normal. Since they are correlated, although each is normally distributed marginally, they are not free to vary independently. If a value for the slope is determined first, then the distribution of the intercept values will be conditional on that value of the slope. i The degree of correlation is indicated by the off-diagonal entry in the parameter covariance matrix calculated from the regression analysis. The' entries of the covariance matrix of the parameters, V fy , V23
, and V,,, for the correlation equation to be used for a SG simulation are obtained from the corresponding estimated -
matrix obtained from the regression analysis as Vfj = fy 9fj , (5.20)
SMPC\GMEMC\GMSEC06.WP5 5-8 J u w y ss.1995
- whtra tha caret; "^", is used to indies.ts an estimate from the regr:s2 ion data. A bivaricto normal int:rc:pt for tha simuletion corr:lation is th:n calculatzd from the i regression e'quation intercept as Ep3 = 3b + Z ]V11 i ,. (5.21) and the bivariate normal slope is calculated from the regression slope as l
1 y12 y*2* i 4=b+Zi 4 +2-2 V22 ,
(5.22) j S Y 11
/V11 where 21 and 2 'are 2 random univariate normal deviates, i.e., from a population with a mean of zero and a variance of one. We now have a> 04, and a for use in simulating all of the leak rates from each of the indicatio is in the SG for one simulation of the total leak rate. For each simulation of an individual indication, i, the leak rate from an indication with a proportion, (,, of unity will be, Qg( ) = 10 E8
- M 8W ' E84, (5.23) with 2, representing the i* value from N independent draws from a standard normal distribution. Once the probabilities ofleak have been calculated, the total leak rate for one simulation is then calculated using equation (5.16). It is noted that each simulation of T requires the generation of one vector, N binomial variates Rj , and a marimum of N log-normal variates Q,. In practice, a value for the leak rate only needs to be generated for each indication that is leaking, i.e.,
when R, = 1.
Simulation When the p-Value is Greater than 5%
If the p value from the regression analysis is greater thafi 5%, it is assumed that the leak rate is independe.ot of the bobbin amplitude and the leak rate model, as discussed in Section 4.0, is, L
Qt=b 3 +c, (5.24) where Q, again represents the common logarithm of the leak rate, and ba is the mean of the leak rate data. The simulation proceeds similar to that when the leak rate correlation is used. For each simulation of total leak rate from a SG, a random estimate of the population standard error, p3, is obtained using equation l
l l
SMPC\ GENERIC \GENSEC05.WP5 5-9 .r.nwy u. ms
(5.19) with tha numarctor cf the frtction and ths dtgrces'of frscdom for the rendom *
' ccl:ction cf%2 being (nk1). A' rand:m valus of 3 is thsn calcult.ted from b es, 3
' Pa = ba + 2 ,
(5.25)
. y'n
.where 2 is a random normal deviate, - N(0,1).~ An individual simulation of the leak rate from indication i with a proportion of unity is then given by,.
0 Qf = 10 8
- ZE5 (5.26)
The total leak rate from all of the indications in the SG is then calculated as per equation (5.16), except that the leak rate is a function of and 3 only.
a 5.4.3 SLB Tube Burst Pressure Simulation The simulation of the burst pressure is performed in a manner similar to that for simulating the leak rate. For the burst pressure, however, an additional simulation must be made of the flow strength of the material of the tube containing the '
indication. This is because the correlation of the burst pressure to the bobbin amplitude was performed for test burst pressures which were adjusted to corre-spond to a reference flow stress.-
For each simulation of the SG, a random estimate of the standard deviation of the .
residuals of the regression errors about the regression line is generated. This is I followed by generating random estimates of the parameters of the regression equation. The burst pressure, Pyf, for each individual indication in the SG is then calculated as, Pyf = cto + ail og(V,) + Z,a3, (5.27) where a3 is the estimated standard error of the population of the residuals, a and o l
L a tbeing the bivariate normal estimates of the parameters of the correlation, and Zg being a random standardized normal variate. The result thus obtained is valid for the reference flow stress, Syf, of the adjusted data used to calculate the estimates of the parameters of the burst pressure correlation. A random estimate of the flow '
stress of the tube materialis then made as, Sg = S, + t as , (5.28) where S, and og are the mean and standard deviation rspectively of the flow stress from a database of the materials of fabrication of the tubes in the SG or from swescrmtnicscr.xsrcosms 5 - 10 a.m u. ms
y
' e c
9 # cn expanded detabeco for tubes from a population of SGs.SThe final estimate of the- - +
L burst pressura is thsn calculated as, x
Pf = P,,f SI .
(5.29)
S,,f The value of the burst pressure is 't hen compared to the SLB differential pressure to determine if the tube ~would be'likely to burst during a postulated SLB event.
5.5. SLB imak Rate Analysis Methodology Once the simulations have been performed it is necessary to estim'a te the total leak -
rate from the SG during a postulated SLB event. The ouput from the' simulations is a distribution of total leak rates that might be expected. The' current accepted -
m'ethodology is to estimate a 95% confidence bound on the total leak rate and -
compare that value to accepted limits. The total leak rate values from the' Monte -
Carlo simulation are ordered from the lowest to the highest. A one-sided distribu-tion-free 95% confidence bound for the 95* percentile of the population of total leak rates is then calculated. Thus, there is a 95% confidence that at least 95% of the potential population of total leak rates will be less than the estimated value.
A confidence interval for value,4,, bounding the 100pthpercentile of the distribu-tion from a drawn sample of size n is constructed by means of the binomial distribution. A'one-sided conservative upper 100-(1-a)% confidence bound for the 100 p'h percentile of the sampled population is obtained as 4, o= x , where u is chosen as the smallest integer such that '
u-Ir ,
" l P(xj > 4,) = { p /(1 p)a -1 21-a. (5.30) j.o r J s .
Since the binomial distribution is computationally difHeult for the extremely large number of simulations performed, an equivalent approach using the F-distribution is to find the smallest value of a such that, 1
1 P
(5.31) 1 + N- n + 1 F I- a,2(N- a + 1),2n l
where N is the total number of simulations performed. Note that for equation ;
(5.31), the index is found such that the desired confidence is maintained constant j while the actual percentile will be greater than or equal to that desired. The result l
obtained using equation (5.30) maintains the percentile constant so the actual confidence level will be greater than or equal to that desired. For example, if the swescrunnicsocusscos wPs 5 - 11 un-y x. ms
4 .e4r- -- ' _ _ .- .. - - -
number cf SG simulaticns performed was 1,000, a onosidtd upper 95% confidence
- bound en tha 95* pere:ntile cf the tots.1 leak ratsa of tha population of possible leak rat:s would be givsn by the 962"d ' ordered (from smallest to largest) total leak rate. '*
This value is a 95.7% confidence value for the 95.0 percentile by equation (5,30), or a 95% confidence value for the'95.1 percentile by equation (5.31)..
5.6 SLB Tube Burst Penhahmty Analysis Methodology During the simulation of the burst pressures, the number of tubes, base'd on the .
indication proportion, with burst pressures less than the SLB differential pressure is counted. The value for each SG simulation is retained. Thus, at the end'of the simulation the number of SGs experiencing zero bursts is known, the number of ;
SGs experiencing one burst is known, et cetera. . The method of accounting for fractional tubes is to add the fractional part to the next integer number of tubes.
Thus, if 1.3 bursts result from the simulation, it is reported as 1 occurrence of 1 burst tube, and 0.3 occurrences of two burst tubes. Using these results, confidence bounds are determined for the likelihood of one, two, or more bursts to occur, i.e., a one-sided 100-(1.a)% upper confidence bound for the Monte Carlo results is found from the following equation:
1 PrU =
N-n . (5.32)
(n + 1)F - 1a,2(a + 1),2(N- = >
where N is the total number of Monte Carlo trials, n is the number of observed occurrences of P 3 s Psta , i.e., predicted bursts, and F is from the F-distribution for the specified number of degrees of freedom for the numerator and denominator respectively. For zero occurrences in the Monte Carlo simulation, equation (5.32) can still be used to find an upper confidence bound on the probability. The value of the upper confidence bound relative to the observed fraction of occurrences is larger when fewer occurrences are predicted. If nN is significant, e.g., on the order of 10 2 ,
and n is large, the upper bound might be only a few percent higher than the mean result. However, if nN is not significant, say 104, and n is very small, the '
upper bound could be an order of magnitude greater than the mean estimate.
Since the probability, i.e., relative frequency, of multiple ruptures is expected to be very low, the upper confidence bound on the probability will be relatively higher than that for a single burst.
S3APC\ GENERIC \GENSEC05 WP5 5 - 12 an=ry u. m5
- 7 Fer inf
- rmetion, sinca it would rarely be ofinterent r:letiva to ths probebility of-
. burst, e 100-(1-a)% cno-sided low:r confidInca bound on ths Monto Carlo results
] :can:be found as 1
Prt = .
~'
1+ ' . N - n + 1 ' F 1- a,2(n - n .13,2n (5.33 ) -
For zero occurrences in the Monte Carlo simulation, the limit of equation (5.33) is '
also zero. "
i
\
S.\APC\ GENERIC \GENSEC05.WP5 5 - 13 J.no.ry 2s, m5 ,
i e
Figure 5-1: Prcb:bility ef Burst vs. B::bbin Amplitude .
3/4" OD x 0.043" Thick, Alloy 600 MA, SG Tubes @ 650 F 1.0E-02 . ..
___._._. _ _ _ . . e
_ , . ..-._,s. _ . _ _ .
x Z Random Database '
~ ~~_~1~_ T _ .
1.0E-03
= = . = = . - --
- .. = ;.:.:: = = =. . : = --- -=- :. -
.=-:
1.0E44
. . _ . ._ _ . . _. . _ .1. _: nr .r.
- n . nn% --
, . . . _ . . _ . . S
/
1.0E-05 _-
- ==
__ -__ .7 . _ - _ . - - ..
u 6
= 1.0E-06 -
m _ _ . _ . . . _ . . _ _ _ _ .
'~= ,
g - , _ . . _ .
_._ 2.c. n
.A _ _ _ . . I _I. .
= ..
_ i $
3n . -
e 1.0E-07 c.,
y -
I 1.0E-08 .
_z.7_ -_ _ ,.
- :. = =
.. 17 ^7.
~-- ..
.--*...ar -
.h,,.e-e.ee, ..g -.. .
1.0E-09 .__._._.,- . - -
_ . ~ . . . . . .
CAUTION: Random database esed 1.0E-10 . . _ . , - . . - - -
r . for lilustration only. -
. - . ~ . _ . - - . . - . . .
1.0E-11 0.1 1.0 10.0 Bobbin Amplitude (Volts)
[34PV GEN.Xt.S)PoB. Gen 5 - 14 RFx: v2s/95,2:sa pu
' 6.01 Example Analysis Results .
The purpose _of this section is to provide illustrative example results from the analyses described in the previous sections. Since the database used for the examples is not to be kept up to date by continuously revising this document, the -
. ' numerical results are not applicable to any one plant and should not be used to perform plant specific APC analyses.
i l 6.1.S' imulation Code Desmiption In order to estimate the probability of burst (PoB) and the total' leak rate during a -
postulated SLB event two Westinghouse proprietary computer codes, EOC_VQB and LEAKTOTL, were initially written. These were later combined into a single Westinghouse proprietary code named SIMCYCLE to perform the same calcula-tions. Input to the code consists of a begmning of cycle distribution ofindications, the length of the cycle to be simulated, a distribution ofindication growth rates, and the length of the cycle used to determine the growth rate distribution. The .
parameters of the burst versus volta correlations (for 3/4" and 7/8" diameter tubes),
limiting material properties, the POL versus volts correlations, and the leak rate versus volts correlations are coded into the program. The regression standard error, the values of the variance-covariance matrices, and other values pertinent to the regressions are also contained within the code. '
Several options for running the code are also provided, e.g., inputting the EOC distribution ofindications, the plugging distribution, followed by the application of !
the POD as described in section 3.0. In practice, the program may be first run to i develop a projected EOC distribution ofindications. This is achieved by simulating i the EOC ' distribution of the indications in the SG several thousand times. ~ Once the EOC distribution is obtained, it is used as a basis for simulating the total leak rate
~
and for estimating the probability of burst during a postulated SLB event. The simulation methodology is described in Section 5.0 of this report.
~
An alternative to specifically developing the EOC distribution followed by the simulation of the total leak rate and the distribution of the burst pressures is to base the entire simulation on the BOC distribution ofindications. In this case the simulation proceeds as follows. The parameters of the POL, the leak rate, and the burst presure correlations to the bobbin amplitude are randomly established using the methodology described in Section 5.0. Once the set of correlating equations is established, the entire distribution ofindications in the SG is sampled to obtain for each indication, a random BOC voltage, a random NDE uncertainty, a random growth, a random POL, a random leak rate if the POL is 1, and a random burst pressure. The total leak rate for the entire SG is calculated as the sum of the leak rates of all of the indications in the SG. Again, the leak rate from a partial indications is weighted by the proportion of an indication being simulated. The swescruzarcscruszcoswrs 6-1 m-y a. ms
t j number cfindieztions eihibiting an EOC burst pr:szure 1:ss than the SLB diffsren- '
tiel pressura is counted. The r:2ultant values from each SG simulction are r:tained.1 ,
1 6.2 - Example Com&ations and Di.Lhtions
~
Regression analyses results were given in Section 4.0 for randomized data sets.
For those examples, a reference set of data was randomly adjusted to develop new data sets. The results of the analyses were that the burst pressure can be predict-ed as a function of the bobbin amplitude by the relation,-
i P3, = 7.832 - 3.1261og(V f) . (6.1)
Likewise, the POL of an indication as a function of the bobbin amplitude was estimated as, P(leak lVg) = 1 y , , -[-4.947 + 8.3371og(V,)] , (6.2)'
and the equation for predicting the leak rate as a function of bobbin amplitude was estimated as,
-2.000 3.2591ag(V,) + l*( "I d Q,=10 2 u i where of is estimated from the predictive distribution as discussed in Section 5.0.
The other parameters associated with the estimated correlating functions were provided in tables in Section 4.0.
A typical BOC distribution ofindications is illustrated as Figure 6-1. A typical growth distribution for the indications is illustrated as Figure 6-2. Finally, a :
typical EOC distribution ofindications is illustrated as Figure 6-3.
6.3 Detenninistic Analysis Results '
As discussed in Section 5.0 of this report, deterministic analyses may be performed for screening purposes The estimation of total leak rate from the SG at a 95%
confidence level may be expected to be reasonably accurate, e.g., within about 5% of 1
The total number of simulations of the SG is a function of the particular result desired, e.g., for the total leak rate the number of simulations might be on the order of 50,000 to 100,000. The number of simulations to characterize the burst i
pressure distribution would likely be significantly larger.
L s w esczurmcsornsec"
- 6-2 January 26.1995 i
~. -~ ,- -. - . . ..
the valu:s obtained from Monto Carlo simulations, however, the estimmtes for -
+ I highsr confid:nca 1:vals m:y diverga rapidly from tho:e ~obtained by simulation.
' The deterministic values are based on only.a first order approximation of the covariance that exists between the estimated leak rates for the individual indica-tions. Releatively good agreement with the simulation results at a 95% confidence-level has been observed for all analyses performed to date. There is,' however, no rigorous theoretical justification for the efficacy of the results obtained, hence, the restriction on usage to scoping analyses only.
A BOC deterministic leak rate estimate is illustrated in Table 6-1, and an EOC estimate is illustrated in Table 6-2. For the case illustrated the estimated BOC !
leak rate during a postulated SLB is 2.6 gpm and the' estimate EOC leak rate is 1 4.6 gpm. For this level ofleak rate, the deterministic estimate at a 95% confidence ' '
level would be expected to match the Monte Carlo result within about 25 to 30%.
For some speci6c distributions the agreement has. been within 3%. If the EOC leak i rate was predicted to be on the order of 0.5 gpm, the deterministic estimate could '
be in error on the order of 50% or more. For even lower values the error can . l increase to an order of100%. However, the relative magnitude of the error j becomes less important for the lower values since they would be expected to.be.
significantly less than the allowable leak rate.
An EOC deterministic estimate of the probability of burst of one or more tubes is illustrated in Table 6-3. For the example shown the POB is greater than 110-2 , l thus, the result would require reporting to the NRC ifit was obtained from a i
Monte Carlo simulation. For low probabilities of burst, as in the example, the POB obtained from a simulation using the same data will generally result'in an estimate about two orders of magnitude less. This is because the random estimate of the i POB is the product of two assumed normal distributions. Th~e resulting distribu- ,
tion is not normal, although it is treated as such for the deterministic estimate. An ;
examination of the actual distribution as obtained from Monte Carlo simulations indicates it to be skewed right, i.e., the upper tail is longer than the lower tail. !
Hence, the probability of a burst pressure being lower than a specified value is less in the tail than would be estimated assuming the distribution to be symmetric.
6.4 SLB Leak Rate Analysis Results i
The output from the Monte Carlo simulation code consists of a check of the input l data, a deterministic estimate of the total leak rate (a check value), statistics of the - ;
simulated NDE uncertainties and voltage growth rates, statistics of the simulated leak rate distrigution and the 95% confidence estimate of the 95* percentile value-of the leak rate. An example of the deterministic check of the analysis in provided i L
l as Table 6-4. The descriptive statistics of the simulated SG total leak rate distribu- 'l tion is provided as Table 6-5. A check value of the 95% confidence value of the total leak rate during a postulated SLB event is calculated based on assuming a log-normal distribution. The example cited this value was 2.37 gpm. The mini-ST APC\ GENERIC \GENSEC06 WP5 6-3 Anwy n;ms
mum and maximum total 12:k ret:s simulat:d are rsported ca chown in Table 6-6
- clong with the cppropricto certed ind:x number and ths 95% confid:ncs total Inak rat 2.
Th:ss rc:pr:s;nt the extrema tails of the distribution and are not statistically .
i significant. For the case shown, the leak rate from the simulation,2.87 gpm, was about 30% less than the deterministic estimate. Finally, a representative listing of the sorted leak rates is printed, see Table 6-7, to gain an understanding of the magnitude of the leak rates in the upper tail of the distribution. If the values in the tail are found to vary signiScantly, it would be an indicator that the analysis should be repeated with the number of simulations increased. The example results presented do not indicate that a repeat of the analysis is necessary.
6.5 SLB Tube Burst Probability Analysis Results The output from the simulation of the burst pressures of the tubes for the example case is illustrated in Table 6-8. The results are based on estimating the probabili-ties of one, two, three, etc., bursts in a single SG as the fraction of occurrences divided by the number of simulations. N95% upper confidence bound on the respective probabilities of burst is also calculated and reported. Finally, the statistics of the simulated burst pressure distribution are calculated and reported as shown in Table 6-9. For the example distribution, the 95% upper confidence bound probability of a single tube burst was found to be 2.910 2 The correspond-2 ing estimate from the deterministic analysis was 3.610 . The probability of two bursts in the same SG during a SLB was estimated to be 7.310 4 . Finally the probability of three bursts was 7.8104 at an upper bound 95% confidence. This was however for 0.3 tubes, hence the probability would be expected to be about 1/3 of this value for a whole indication. '
i 1
swescruzarcscruszcos wPs 6-4 J.=ry 2s.1995 l
m e
. 1 Table 6-1: Example of SG BOC Volts & Deterministic Leak Rate REGRESSION Leak Rate Calculation BOC Number of Number POD No. POL Expected Variance of Vamnce of Volts Indications Plugged N, P, Leak Rate Leak Rate ' O' N, V(P, QJ P, Q, 0.2 3 1 4.0 7.38E-06 3.45E-04 2.21E-05 1.02E-08 1.64E-10 6.57E-10 0.3 31 0 51.7 4.69E-05 1.29E-03 1.85E-04 3.13E-06 8.78E-09 4.54E-07 0.4 83 1 1373 1.59E-G4 3.16E-03 7.97E-GS 6.89E-05 1.28E-07 1.76E-05 0.5 112 6 180.7 3.94E-04 6.26E-03 2.46E-03 4.45E-04 9.84E-07 1.78E-G4 t 0.6 107 6 1723 8.14E-04 1.09E-02 6.19E-03 1.52E-03 5.14E 06 8.85E-04 0.7 93 12 143.0 1.49E-03 1.73E-02 136E-02 3.69E-03 2.07E 05 2.96E-03 0.8 69 1 114.0 2.50E-03 2.59E-02 2.70E-02 737E-03 6.91E 05 7.87E 03 0.9 38 1 623 3.93E-03 3.69E-02 4.97E-02 9.03E-03 2.00E-04 1.2SE-02 1.0 34 1 55.7 5.86E-03 5.07E-02 8.61E-02 1.66E-02 5.20E-04 2.90E-02 1.1 41 35 33 3 8.40E-03 6.76E-02 1.42E-01 1.89E-02 1.23E-03 4.1IE-02 1.2 18 17 13.0 1.16E-02 8.80E-02 2.26E-01 133E-02 2.72E-03 3.53E-02 1.3 14 13 103 1.57E-02 1.12E-01 3.46E-01 1.81E-02 5.63E-03 5.79E-02 1.4 9 9 6.0 2.06E-02 1.40E-01 5.16E-01 1.74E-02 1.10E-02 6.62E-02 1.5 6 5 5.0 2.65E-02 1.73E-01 7.51E-01 230E-02 2.07E 02 1.03E-01 1.6 4 3 3.7 335E-02 2.llE-01 1.07E+00 2.62E-02 3.72E 02 138E-01 1.7 6 6 4.0 4.17E-02 2.54E-01 1.49E+00 4.23E-02 6.47E-02 2.59E-01 1.8 3 2 3.0 5.llE-02 3.02E-01 2.04E+00 4.63E-02 1.09E-01 3.26E-01 1.9 4 4 2.7 6.18E-02 3.57E-01 2.76E+00 5.95E-02 1.78E-01 4.80E-01 2.0 3 3 2.0 738E-02 4.17E-01 3.67E+00 6.16E-02 2.83E-01 5.66E-01 2.2 4 4 2.7 1.02E-01 5.60E-01 6.27E+00 1.54E-01 6.68E-01 1.80E+00 23 1 1 0.7 1.18E-01 6.42E-01 8.07E+00 530E-02 9.95E-01 6.96E-01 2.5 1 1 0.7 1.54E-01 8.31E-01 130E+0! 8.96E 02 2.09E+00 1.46E+00 2.6 1 1 0.7 1.74E-01 9.38E-01 1.63E+01 1.14E-01 2.95E+00 2.07E+00 2.8 1 1 0.7 2.17E-01 1.18E+00 2.49E+01 1.79E-01 5.64E+00 3.95E+00 2.9 1 1 0.7 2.39E-01 132E+00 3.06E+01 2.21E-01 7.64E+00 535E+00 3.2 1 1 0.7 3.11E-01 1.79E+00 5.44E+01 3.90E-01 1.76E+01 1.23E+01 33 1 1 0.7 336E-01 1.97E+00 6.52E+01 4.64E-01 2.28E+01 1.60E+01 3.7 1 1 0.7 435E-01 2.82E400 1.29E+02 8.61E-01 5.79E+01 4.06E+01 3.9 1 1 0.7 4.83E-01 333E+00 1.76E+02 1.13E+00 8.80E+0! 6.16E+01 4.0 1 1 0.7 5.07E-01 3.61E+00 2.05E+02 1.28E+00 1.07E+02 7.51E+01 43 1 1 0.7 5.72E-01 4.54E+00 3.I8E+02 1.82E+00 1.87E+02 131602 5.I 1 1 0.7 7. I4E-01 7.81E+00 9.04E+02 3.91E+00 6.58E+02 4.61E+02 8.9 1 1 0.7 9.50E-01 4.74E+01 3.15E+04 3.15E+01 3.00E+04 2.10E+04 10.5 1 1 0.7 9.72E-01 8.16E+01 9.41E+04 5.56E+01 9.17E+04 6.42E+G4 Regression Equations Analysis CAUTION: Equations used for this Sutn[ N, E(Q,) P, ) = 98.081 example were based on a randomized Surn[ Var + Cov ] = 8.88E+04 data M Effective Standard Deviation = 2.93E+02 Confidence =
t 95.0 %
Z - Deviate = 1.645 Q total (LPH) = 580.42 Q total (GPM) = 26 tsceoceoc.nst so soc 6-5 = mis, w m l
l 1
'8 Tcble 6-2: Example e f SG EOC Volts & Det:rministic Leak Rate L EOC Cumulauve Tubes with Expected Q, Vanance of
~
Volts Probabihry N' P' 3p Vanance of
>= Volts (#2560 pu '
Q. P, Q, 0.2 0.000984 1016.00 1.0 7.38E-06 3.45E 04 3.30E-05 2.55E-09 2.45E- 10 0.3 0.010827 101533 10.0 4 69E-05 1.29E 03 232E-04 6.06E-07 1.10E-08 04 0.049213 1005.09 39 0 1.59E44 316E 03 9 27E-04 196E 05 l .49E-07 0.5 0.123031- 966.28 75.0 3.94E-04 6.26E 03 i
2.74E-03 1.85E 04 1.10E-06 '
06 0.222441 891.35 101.0 8.14E-04 1.09E-02 6.74E-03 8.94E-04 5.58E-06 0.7 0334646 790.25 114.0 1.49E-03 1.73E-02 1.45E-02 2.94E 03 2.2tE 05 0.8 0 447835 675.79 115.0 2.50E-03 2.59E-02 2.85E-02 7.44E-03 730E-05 0.9 0.550197 561.22 104.0 3.93E 03 3.69E-02 5.21 E-02 1.51 E-02 2.10E-04 1.0 0 639764 456.55 91.0 5.86E-03 5.07E-02 8.97E-02 2.71E-02 5.41E-04 1.1 0.712598 366.30 74.0 8 40E-03 6.76E-02 1.47E-01 4.21 E-02 1.286 03 1.2 0.771654 291.82 60.0 1.16E-02 8.80E-02 233E-01 6.15E-02 2.80E-03 1.3 0.817913 231.92 47.0 1.57E-02 1.12E-01 3.56E-01 8.27E-02 - 5.78E-03 1.4 0.853346 184.% 36.0 2.06E-02 1.40E-01 5.29E-01 1.04E-O L 1.13E-02 1.5 0.880906 148.63 28.0 2.65E-02 1.73E 01 7.67E-01 1.29E-01 2 llE-02 1.6 0.902559 120.69 22.0 335E-02 2.llE-01 1.09E+00 1.56E-01 3.80E-02 I .7 0.919291 99.09 17.0 4.17E-02 2.54E 01 1.52E+00 1.80E41 6.58E-02 1.8 0.932087 82.01 13.0 5.IIE-02 3.02E-01 2.08E+00 2.01E-01 1.IOE 01 1.9 0.942913 69.04 11.0 6.18E-02 3.57E 01 2.80E40 2.42E-01 1.80E-01 !
2.0 0.950787 58.50 8.0 73bE-02 4.17E 01 3.72E+00 2.46E 01 2.87E-01 2.1 0.956693 50.34 6.0 8.72E-02 4.85E-01 4.89E+00 2.54E-0! 4.45E-Ol 2.2 0 961614 43.86 5.0 1.02E-01 5.60E-01 635E+00 2.85E 01 6.76E-01 ~
23 0.965551 38.83 4.0 1.18E-01 6.42E-01 8.16E+00 3.03E 01 1.01E+00 2.4 0 968504 34.97 3.0 135E-01 732E-01 1.04E+01 2.97E-01 1.47E+00 2.5 0.971457 31.96 3.0 1.54E-01 831E-01 1.31E+01 3.84E-01 2.l lE+00 2.6 0.973425 2939 2.0 1.74E-01 938E-01 1.6tE+0! 3.26E-01 2.98E+00 2.7 0.975394 27.19 2.0 1.95E-01 1.05E+00 2.04E+01 4.llE-01 4.14E+00 2.8 0.976378 25.24 1.0 2.17E-01 1.18E+00 2 51E+01 2.56E-01 5.68E+00 2.9 0.978346 23.55 2.0 2.39E-01 132E+00 3.08E+01 6.31E-01 7.69E+00 3.0 0.979331 21.99 1.0 2.638-01 1.46E+00 3.75E+01 3.85E-01 1.03E+01 3.1 0.981299 20.58 2.0 2.87E-01 1.62E+00 4.54E+01 9.31E 01 1.36E+01 3.2 0.982283 19.34 1.0 3.llE-01 1.79E+00 5.47E+01 5.58E-01 1.77E+01 3.3 0.983268 18.25 1.0 336E 01 I.97E+00 6.56E+0! 6.63E-01 2.29E+01 3.4 0.984252 17.35 1.0 3.61E-01 2.17E+00 7.82E+01 7.82E-01 2.93E+01 3.6 0.985236 15.74 1.0 4.llE-01 2.59E+00 1.10E+02 1.06E+00 4.67E+01 3.7 0.986220 15.01 I.0 435E-01 2.82E+00 1.29E+02 1.23E+00 5.82E+01 3.9 0.987205 13.56 1.0 4.83E-01 333E+00 1.77E+02 1.61E+00 8.83E+01 4.0 0.988189 12.83 1.0 5.07E-01 3.61E+00 2.06E+02 1.83E+00 1.08E+02 4.1 0.989173 12.08 1.0 5.29E-01 3.90E+00 - 239E+02 2.07E+00 130E+02 4.3 0.990157 10.55 1.0 5.72E-01 434E+00 3.19E+02 2.60E+00 1.88E+02 4.4 0.991142 9.71 1.0 5.93E-01 4.88E+00 3.67E+02 2.89E+00 2.23E+02 4.5 0.992126 8 94 1.0 6.12E-01 5.24E+00 4.205*02 3.21Edo 2.64E+02 4.7 0.993110 7.51 1.0 6 50E-01 6.02E+00 5.4SE+02 3.91E+00 3.64E+02 4.8 0.994094 6.90 1.0 6.67E-01 6.44E+00 6.24E+02 429E+00 4.25E+02 5.1 0.995079 5.62 1.0 7.14E-01 7.81E+00 9.06E+02 5.58Edo 6 60E+02 5.7 0.996063 4.60 1.0 7.90E 01 -1.12E+01 1.81E+03 8.81E+00 1.45E+03 7.4 0 997047 3.52 1.0 9.07E-01 2.59E401 9.48E+03 235E+01 8.65E+03 8.9 0.998031 2.52 1.0 9.50E-01 4.74E+01 3.15E+04 4.50E4Cl 3.00E+04 9.7 0.999016 1.58 1.0 9.63E-01 6.29E441 5.55E+04 6.05E+01 536E+04 10.5 0.999705 0.65 0.7 9.72E-01 8.16E+01 9.41E+04 5.56E+01 9.16E+04 11.2 1.000000 033 03 9.7EE-O f I.0lE+02 1.45E+05 2.97E401 1.42E+05 Regression Enanons Analysis Sum [ N, E(QJ P. ] = 261314 MM: 4 sMW Sum [ Var + Cov ] =
Effecove Standartl Deviation =
2 23E+05 this example were bened on a 4.72E+02 Confidence = 95 0 %
Z Deviare = 1.645 Q total (LPif) = 1038.2 Q total (GPM) = 4.6 Isceocrocasisasoc 6-6 m- mw a is m
Tcble 6-3: Example EOC Deterministic Pr:b:bility cf Burst Estimate 0.750" x 0.043" Alloy 600 MA SG Tubes Predicted Burst Adjusted Deviates Volts . . Burst Pressure Burst Probability of Indications Above SLB Pressure Vanance Variance Pressure 0.20 1 10.018 0.943 1.041 7.166 1.70E 10 0.30 10 9.467 0.932 1.024 6.743 1.10E-08 0.40 39 9.076 0.926 1.013 6.430 1.68E-07 0.50 75 8.773 0.921 1.005 6.I80 9.52E-07 0.60 101 8.526 0.918 0.999 5.971 3.12E-06 _
0.70 114 8.317 0.916 0.994 5.791 7.53 E-06 0.80 115 8.135 0.914 0.990 5.632 1.47E-05 0.90 IN 7.975 0.912 0.986 5.491 2.38E-05 1.00 91 7.832 0.911 0.983 5.363 3.52E-05 1.10 74 7.703 0.910 0.980 5.246 4.60E-05 1.20 60 7.585 0.909 0.978 5.139 5.74E-05 1.30 47 7.476 0.909 0.976 5.039 6.69E-05 1.40 36 7.375 0.908 0.974 4.946 7.39E-05 1.50 28 7.282 0.908 0.972 4.859 8.09E-05 1.60 22 7.194 0.907 0.970 4.777 8.73E-05 1.70 17 7.112 0.907 0.%9 4.699 9.09E-05 1.80 13 7.034 0.907 0.967 4.626 9.20E-05 1.90 11 6.%I 0.906 0.966 4.556 1.01E-N 2.00 8 6.891 0.906 0.%5 4.490 9.47E 05 2.10 6 6.825 0.906 0.%3 4.426 9.00E-05 2.20 5 6.762 0.906 0.962 4.366 9.39E-05 2.30 4 6.701 0.906 0.%I 4.308 9.30E-05 2.40 3 6.644 0.906 0.960 4.252 8.56E-05 2.50 3 6.588 0.906 0.960 4.198 1.04E-04 2.60 2 6.535 0.906 0.959 4.140 8.36E-05 2.70 2 6.484 0.906 0.958 4.096 1.00E-04 2.80 1 6.434 0.906 0.957 4.048 5.94E-05 2.90 2 6.387 0.906 0.956 4.001 1.40E-N 3.00 1 6.341 0.906 0.956 3.955 8.21E-05 3.10 2 6.2 % 0.906 0.955 3.912 1.91E-04 3.20 1 6.253 0.906 0.955 3.869 1.1IE-04 3.30 1 6.211 0.906 0.954 3.828 1.28E-04 3.40 1 6.171 0.906 0.953 3.787 1.47E-04 3.60 1 6.093 0.906 0.952 3.710 1.91E-04 3.70 1 6.056 0.906 0.952 3.673 2.16E-OS 3.90 1 5.984 0.906 0.951 3.601 2.74E-04 4.00 1 5.950 0.906 0.950 3.567 3.07E-N 4.10 1 5.916 0.906 0.950 3.533 3.43E-04 4.30 1 5.852 0.907 0.949 3.468 4.23E-04 4.40 1 5.820 0.907 0.949 3.437 4.69E-04 4.50 1 5.790 0.907 0.948 3.406 5.17E-04 4.70 1 5.731 0.907 0.948 3.346 6.26E-04 4.80 1 5.702 0.907 0.947 3.317 6.85E-04 5.10 1 5.620 0.908 0.946 3.233 8.90E-04 5.70 1 5.469 0.908 0.945 3.079 1.42E-03 7.40 1 5.115 0.911 0.941 2.713 4.08E-03 8.90 1 (864 0.913 0.939 2.452 8.18E-03 9.70 1 4.747 0.914 0.939 2.330 1.12E-02 10.50 0.7 4.639 0.915 0.938 2.217 1.03E-02 11.20 0.3 4.552 0.916 0.937 2.125 5.54E.03 Total: 4.71 E-02 rsosoceoc.nsi sa nos 6-7 am imas. a 22 pu
. , _ ~ _ _ . . . _ - . ~ . __ ._ _- .
J .,
ep,d' s .
Table 6-4: Deterministic Estimate of the Total Leak Rate I Parameter Value '
E N: E(QJ P, 1.4851021ph 2 VI N i V( P, Qi ) ] 2.0021051ph 2 Cov{ Ni V( Pi Qi ) ] 1.935104 1ph2:
V + Cov . 2.1951051ph 2 Standard Deviation ' 4.686102 1ph Z - Deviate 1.645 Qw 919.31ph (4.05 gpm) .
Covariance Contribution 3.8 %
Table 6-5: Descriptive Statistics of the Monte Carlo EOC Leak Rate Distribution Parameter Value '
Maximum Qsum 91.62D gpm !
Minimum Qsum 6.475 104 gpm Sum of Qsum 4.112104 gpm l
P
! Average Qsum ~ 0.8223 gpm '
Std Dev Qsum 2.716 gpm i
Avg log (Qsum) -0.4297 StD log (Qsum) -0.4885 Approx. Bound 2.366 gpm i
f surctourmescuszcos.wPs 6-8 Jawyn.ms i.
y.;
~
l Table 6 6i Monte Carlo Total Leak Rate 50,000 Simulations of the SG -
~ 95% Confidence.
f Minimum Total Leak Rats .6.50 103 epm
. Maximum Total-Leak Rate 91.4 gpm - ' !
P
~
Confidence Index Number 47580 Bounding Total Leak Rate 2.87 gpm Table 6-7: Monte Carlo Estimates of the '
Total Leak Rate Bin Sorted Number Leak Rate Index Index. in Bin (GPM) .
649 47566 48 -2.871 650' 47614 25 2.900 -
651 47639 49 2.929- !
652 47688 25. 2.959 653 47713- 73 2.988 654- 47786 24 3.019 ;
655 47810 24 3.049 656 47834 74 3.079 657 47908 24 3.110 l 659 47932 74 3.173 660 48006 24 3.205 !
, 661 48030 25 3.237 663 -48055 1 3.303 665 48056 72 3.370 '
666. 48128 24 3.403 I 669 48152 48 3.507 l
surctornrarcscruszcos.wPs 6-9 J ourse. m i
l
, .e- <
^
I Table 6-8: Monte Carlo Results for the Simulation of Tube Burst
_ (Upper bound at 95% confidence.)" !
Ntunber of Frequency of Probability of Upper Bound Upper Bound !
Bursts Occurrence Burst ~ Pr(Burst) . Cumulative - ,
- Pr(Burst)
- 1.- 1397.8 2.796 10 2 2.920 10 2 2.920 10 2, 2 26.6 5.320 10d
'7.356 104 2.974 10 2-3 0.3 6.000 104 -7.851'104 2.975 10 2 !
Table 6-9i Statistics of Simulated Burst -
Pressures-Parameter . Value (ksi)
Minimum - .0.582- '
Maximum -14.854 Mean 7.677 '
Mode 7.650 Median - 7.750 Standard Deviation 1.187 Mode to SLB Margin 5.090- '
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. Appendix A: Regression Analysis A.1 Intmiudion i The analysis of the relationship between two variables is generally termed either regression analysis or correlation analysis; In addition, one.may also find the term i
conBuence analysis in the literature n.2.ans.n. For each, the objective is to establish a mathematical model describing a predictive relationship between the variables. The use of the term regression is frequently interpreted to imply that "
some sort of causal relationship exists while correlation has been reserved for non-causal relationships. Other differentiations between the two terms involve the '
nature of the variables, i.e., whether or not one or both is stochastic. In addition, r
the term regression is also frequently used to mean the process by which the l parameters of a relationship are determined.
For the purposes of the evaluations reported herein the name regression analysis is used in the broad sense of covering the aspects of the fitting of a curve, i.e, equa-tion, referred to as the regression curve or line, to observed data points, where concern is with the slope and position of the curve that best fits the data, and to the analysis of how well the data points can be represented by the curve, i.e., the correlation analysis. The' correlation analysis has two aspects, one is a measure of the degree of covariability between two variables, and the second is as a measure of- .
the closeness of fit of a regression line to the distribution of the observations. The L statistical analysis is performed for the purpose of establishing a stochastic depen-dence, and does not, nor does it have to, demonstrate the existence of a causal dependence.
For the analyses dealing with the APC it is desired that models be developed
' relating the burst strength and leak rate of degraded tubes to the morphology of the degradation. Unfortunately, the degradation morphology is only known exactly for tubes which have been destructively examined. However, a third variable, based or; the non-destructive ernmination of the tubes, is available which is also directly related to the morphology of the degradation. Each degradation state is .
taken to correspond to a set of quantifiable characteristics or variables, such as the burst strength (measured by a burst pressure test), the leak rate (measured as a function of differential pressure), and a non-destructive ernmination (NDE) response, e.g., eddy current bobbin coil signal amplitude in either an absolute or differential mode. Since the field eremination of the tubes is based on the NDE response it is appropriate to avamine the relationships between the first two variables and the third.
i GENSECOA.WP5 A.1 ov2wes l
I
q
-. u The exp:rimsntal:and 5sid dzta for outside diamster atro corrosion cracking-
" 1(ODSCC) st tubs support pletzs (TSPs) consists cf bobbin coil voltages and. mea- .)l
" c sured tube burst pressure, and leak rates at. differential pressures corresponding to
" ' normal operating conditions and steam line break (SLB) conditions? As noted,. i
'these data are correlated, but not causally related. For example, high burst !
pressures correlate with low voltages but high burst pressure does not cause low Lvoltage.1Similarly, low leak rates are correlated with low voltage, but low leak
, . rates do' not cause low voltage ~.
3 The degradation process determines the magnitude of the evolution of e'ach vari-q able, however, the degradation process is complex and the morphology and time history will vary even under conditions which would normally be termed identical.
- Thus, it is expected that the correlation between any pair of the^three variables may have signiScant scatter. This is expected even if each 6f the variables is.
measured with perfect accuracy and contains no measurement error. ,
J t
. In order to predict burst pressures'and leak rates under postulated conditions for !
' degraded tubing, confirmed by field inspection by eddy current test, it is necessary i to develop regression lines which relate average burst pressure to measured voltage =
and average leak rate to measured voltage. The " conventional" regression lines are usually determined by considering the variable which is to be predicted in the future, e.g., burst pressure, as the regressed variable, and the variable which will-j
- be measured in the future, i.e., voltage, as the regressor variable. : While regression lines can also be established to predict voltages from measured burst pressures or leak rates, there is no particular reason to do so as these " inverse" correlations do i not usually provide useful information beyond that which is obtained by the conventional regression lines.
It is to be noted that the causative factor relative to the magnitude of each variable-is the crack morphology, and that none of the three characteristic variables can be considered to be the cause of the other. This means that for.any. pair', either may !
be treated as the predictor and the remaining variable treated as the response. l Once 'a correlating relationship has been established, eitber variable may be used to predict an expected value for the other. For example, a correlating relationship i
may be mathematically determined using burst pressure as the response and l
bobbin amplitude as the predictor. Once the relationship is known,' a mean bobbm -
amplitude associated with a given burst pressure 'or leak rate can be' calculated.
1 I
Confidence limits for predicted burst pressure or for predicted leak rate can then be 1
established about the regression line using standard statistical methods. The confi-
'i r
dence limits which are determined directly from the regressions of burst pressure or leak rate on voltage will be narrower, for a fixed probability level, than the corresponding limits which could be deduced from the inverse regression lines.
these correlations can then be used to determine high confidence values for the '
GENSEC0A WP5 A.2 ot/27ss '
I
. . _._ __-____________--________-_______a
. . _. ._ . . ..m _ .
.structurd limit or leak ras, corresponding to the postulated SLB diffsrential pres-sur3.
s A.2 The Linear Regression Model The general, linear (meaning linear in the coefficients), first order regression analysis model relating two variables is given by y, . = ao + atxg + c , (A.1) >
i where yi s taken here as the response or predicted variable, and xi as the predictor, ;
or regressor. The e, or error, term accounts for deviations from a perfect predie-tion. In order to establish confidence and prediction limits on yg, the error is assumed to be normally distributed with a mean value of zero and a variance that '
is uniform over the range ofinterest. An analysis is then performed to determine -
the best values of ao and a1 to use in equation (A.1). Three methods are commonly used for the analysis, maximum likelihood estimation, least squares (LS), and weighted least squares (WLS). For maximum likelihood analysis the values of a o
and a1are found that maximize the probability of obtaining the observed responses.
The use of marimum likelihood analysis is formally correct, however, if the errors -
are normally distributed, the maximum likelihood estimators (MLE) will be identical the estimators obtained using least squares. If both variables are stochas-tic and the errors are normally distributed, the application ofleast squares still leads to the marimum likelihood estimatorso of a and at.
The LS method is based on minimizing the sum of the squares of the errors, also -
referred to as residuals, between the observed and predicted. values, thus, the best values of ao and at are those that make +
n
{ (Yg - ?g )2 (A.2) i1 :
a minimum, where the caret indicates the predicted value, fg = a o+ a1x, , (A.8)
Expression (A.2)is differentiated with respect to ao and at and the resulting expressions set equal to zero and solved for the coefficients. For WLS, the same expression for the errors is established by considering the error term, c, to be '
weighted non-uniformly, i.e., the error distribution is e, - N(0,I f c 2) (A.4) ;
c uszen wes A.3 omms
and the expression to be minimized becomes "
h wj(Y, - Pg )2, (A.5) l i-1 where the I iand hence the wj are known. In situations where the variance of the response is not uniform it is possible to find appropriate weights such that the resulting estimators are MLEs. Such a case is the dependence of the probbility of leak on the bobbin voltage. In this case the response is either 0 for no leak, or 1 for leak. A predictive model based on the logistic function can be fitted by transform-ing the variables and iteratively solving the resulting weight least squares problem.
For the unweighted LS analysis the slope of the regression or correlation line is found to be _
xg - 2 )(yf - )
a1 = { ([ (xj -1)2 , (A.6) where the summation limits are understood. The intercept is then found as a0 "I ~ Ul i (A.7) ify has been regressed on x. If x is regressed on y the slope will be
[ (xg -1)(yg - )
(A.8)
[ (y, - S )2 relative to the ordinate, or y, axis. If this is reckoned to the x axis, i.e., the abscissa of the original coordinates, the slope is
- ' ~
(A.9) a1 = { (x, - 2 )(yg - f )
If the data used for the analysis contains significant scatter the values found by (A.6) and (A.9) can be quite different. A rough visualization of this can be obtained by picturing the smallest ellipse that can be drawn that envelopes all of the data points. A line connecting the largest and smallest abscissa values of the ellipse will approximate the regression of the y variable on the x variable, while the line connecting the marimum and minimum ordinate values will approximate the regression of the x variable on the y variable.
GENSECOA.WP5 A.4 omms
_ _ -. w. s= , em - s m o a m ,a t su a m m swamu mwrwwerMTM'EWWhNM
A, Fcr ths APC analysts, thnlobjtetive is'to rsis.ta burst przszura and leak rate to 1 bobbin voltsgs. This msans that bobbin amplituds is 'dspicted cs the cbscissa x : variable while burst pressure and leak rate are depicted as ordinate variables-respectively. For the conventional regression analysis, these are the corresponding choices for the regressor and regressed . variables. However, if conditions dictate, an inverse regression may be performed, thus the depiction does not necessarily imply the direction of the regression analysis performed. The considerations. discussed in - i
- the introduction indicate that the inverse regression is only worthwhile if addition-al useful information can be gained from such an analysis.
- i The expansion of the model to include more terms, e.g., considering burst pressure - i to be related to the logarithm of the bobbin amplitude by a second order polynomial
is still linear regression analysis. If the assumption of constant variance of the residuals is verified, the application ofleast squares still results in the maximum '
likelihood estimators of the coefficients of the equation. 'If the prediction equation is non-linear in the coefficients, e.g., exponential, a transformation may be made to result in a linear equation, or non-linear regrersion techniques may be necessary.
The use of a logarithmic transformation is common, and may result in a stabiliza- -l tion of the variance, i.e., a non-uniform variance before the transformation may.
become uniform as a result of the transformation. Consideration of a non-linear regression model, e.g., logistic regression 113,14,151, which can be transformed into a linear model, is contained in the body of this report relative to determining the . !
probability ofleak as a function of bobbin amplitude.
A.3 C=miamstian of Variable Emr !
If the values of the regressed variable, sayy, are subject to error, but the regressor, x, is free from error, no bias will be introduced into the mean, i.e., regression pre- .
' dicted mean value ofy for a given x, although the variance will be greater due to a
the errors in the measurement ofy. The calculated values ofy are then unbiased !
estimates of the true values ofy,' assuming the error to also be normally.distribut-ed. The only effect of the errors in the measurement ofy'is to increase the vari-ance of the residuals and render the estimate ofy less reliable,i.e., the estimate
~
1 will have larger inference bounds. If now x is also subject to measurement error -l the regression will be of observed values on observed values instead of true values I
on true values. If there is measurement error present in the predictor variable, the j slope obtained from the regression analysis will be biased [2,3,s,7,8,9,10,111, but the regression line will still pass through the centroid of the data. The standard regression analysis assumes that the regressor variable is known without error and that the regressed variable is a measured value subject to uncertainty. '.Thus, for example, the regression of burst strength, P on the logarithm of the bobbin ampli-tude, log (V), estimates the mean value P3 for which the observed value of bobbin amplitude is log (V3 ). If the bobbin examination and evaluation technique were to cruszcaws A.5 omms
, , ~ , . .. .. , _. . - ..-.- , -.
. be changed in;ths future to reduca thm miasuremsnt errors ths cerralations based on curr nt technology might have to be reparted. ..
1
~ If there is significant error present in the measurement of the variables, the'-
regression analysis may be performed using what is termed as the error in vari-ables form model. In this case' it'is assumed that the data measurements are of the t
X.= x + q and' Y_ = y + S , - (A.10) -
where X and Y are the measurements corresponding to the true values ofihe -
variables x and y, and y and 6 are their corresponding errors of measurement. For -
the predictor, say X, the total variance will be
2 ag , g2 ,2,, (A.111 It can be shown that'when the measurement error is independent of the true value, the expected value of the calculated slope, ai, will be U1 ai =
c"2 , . (A.12) 1+
2 Oz where a i, is the true value of the slope, or the vidue that would result if no mea-surement error was present, and
- 2 E (*i - * )* ,
.(A.13) n-1 It is noted that a1 would be found from equation (A.7) as before. A key point to note is that the calculated slope under predicts the true slope (without measure- '
ment error). If the measurement error is known, and is uniform, its effect on the analysis slope can be calculated directly and the appropriate slope to be used for prediction would be .
1
' 2 a3=a3 1+ . (A.14)
, 0, x '
When the error variance is known and can be expressed as a fraction of the variable variance the slope will be affected by a like amount.
causzcawes A.6 ouzws
Wh:n ths error varianes is n:t known, which is usually the esse, an estimcte of the i trua slope can b3 mrds using ths partitioning t:chniqua d velop:d by Wald (101 and subsequently improved upon by Bartlett IU3. The technique consists of partitioning the data into three groups based upon the ordered regressor variable. The line joining the centroids of the upper and lower groups is an unbiased and consistent estimator of the true slope. If the slope thus found is close to the slope determined without considering measurement errors then the measurement errors are consid-cred to be not significant. The application of this technique must be done with caution since the order of the true values of the regressor variable (s)is not known, only the order of the measured variables. For the APC analyses the application of the Wald-Bartlett technique is restricted to estimating whether or not significant measurement error is present. For the correlations examined through the writing of this report,it has been concluded that the measurement errors are not signifi-cant. Moreover, it may ' assumed that the measurement errors are not signifi-cant, and a standard regression analysis performed. If the residuals are normally distributed about the regression line, inference bounds may be determined using the standard inference methods.
It is noted that if the magnitudes of the measurement errors associated with each of the variables, or their ratio, is not known, an " errors-in-variables" analysis does not lead to a criterion for the selection of the best regression direction. In general, the need for performing an " inverse" regression can be based on the determination of whether or not useful information beyond the conventional regression analysis will result.
A.4 Detection of Outliers If the errors are normally distributed the application of LS to determine the coefH-cients of the regression equation minimizes the variance of these estimators. The coefficients are also the MLEs. A drawback of the LS technique is that it is not very robust. This means that the fitted line may not be the best estimator of the correct relationship because it can be significantly influenced by potentially outlying data. In addition, the resulting fit may be such'that potential outliers become hidden if examined after the analysis is performed. There are established methods for identifying influential data that may result in a distortion of the regression line. Such methods fall into the categories of regression diagnostics and robust regression. Robust regression methods are designed to be insensitive to potential outliers, and can be used to identify outliers based on the residual errors from the robust regression line. A rather simple example ofimproving the robust-ness of the fit would be simply minimize the sum of the absolute values of the l residuals instead of the sum of the squares. This provides significant improvement if the outlier is in the y-direction for a y on x regression, but is not resistant to outliers in the x-direction.
l l
l GENSEC0A.WP5 A.7 ovm3
Ons very robust techniqua is tsrmed the "Isamt medien of squares," or LMSl *
. regrsasion. Thn best r:gr:srion lino'(or polynomiel) is tha ona for which the median of the squared residuals is a minimum. The drawbacks to this techn. - '
. are that there is no closed form solution and techniques for the determination of inference regions would be difficult to apply. However, the determination of a reasonable solution is quite' easy using a computer. The algorithm proceeds by drawing sub-samples of a given size from the data set. For each sub-sample, regression line coefficients and the median of the squared residuals are calculated.
The coefficients of the minimum median solution are ' designated as the LMS solution. A median based scale estimate (analogous to the standard deviation) is determined for the identification 'of outliers at'a two-sided 98% confidence level, or a one-sided 99% confidence level.
The data for the APC were examined using the LMS robust regression program PROGRESS by Rousseeuw and Leroy. It is noted that the application of robust regression is not intended to be used for the justification of the deletion ofimproba-ble data points, only for the identification of potential outliers. The rejection of any data must be based on an evaluation of the circumstances surrounding the data collection to search for possible sources of error.
A.5 Seledica of a Regression Coonlinate System For the analysis of continuous variable data four, alternatives were examined for each correlation. These choices are listed in Table A.I. For each case, the correla-Table A.1: Fitting Options Considered for LS Regression Abscissa Ordinate Relation Linear Linear y, = ao + a, x,
, Logarithmic Linear y, = ao + a, log (x,)
Linear Logarithmic log (y, ) = ao + a, x, Logarithmic Logarithmic log (y, ) = ao + a, log (x,)
' tion coefficient, r, measuring the " goodness-of-fit" of the regression line was calculated. The correlation coefficient is a measure of the variation of the data explained by the regression line, thus the largest value is indicative of the best fit.
cruszcawn A.8 omms
- . -- . _ - -= ._ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _-_-
, - . .-. - -- . .. -- - -.. .. ~
The exprs:sion for the square of the corralation coefficient,.known as the index of; determination,'is .
.y r
2,[SI .
(A.15)
{(yt- )2
. The index of determination is the proportion of the total variation about the mean of the predicted variable that is explained by the regression line. The scale combi- s nation yielding ths largest index of determination, and, hence, correlation coeffi-cient, was selected for the analysis. In the event that the predicted variable for the '
regression is the logarithmic transformation of a physical variable, the above calculation is performed on the untransformed variable. It is readily apparent, however, that for data with a range of several orders of magnitude, e.g., bobbin amplitudes ranging from O(0.1 'volt) to O(100 volts), the use of a logarithmic scale is appropriate. It is .also to be expected.that the variation of observed voltages :
would be normally distributed about the log of the voltage. The same is true for the leak rate which ranged from O(0.021/hr) to O(5001/hr) for specimens for which leaking was observed. It is to be noted that the use of a logarithmic transformation is commonly used for data with a large range as a variance stabilizing technique.
If the dependent variable has be logarithmically transformed, regression line predictions will be of the expected median of future values, not the expected mean of future values.
t A.6 Seledian of a Regmeston Dimetion As noted in the introduction to this appendix, the bobbin amplitude does not cause the observed burst pressure and vice versa. The same is true for the relation between bobbin voltage and leak rate. Thus, the regression direction is not
, specified by the choice of variates.
The objective of performing the regression analysis is prediction. For all practical '
purposes the bobbin voltage will be used as a predictor of burst pressure and leak ,
rate. However, the intended use does not automatically dictate the designation of the predictor and response variable roles for the regression analysis. The LS fit simply finds the line such that the variance of the responses is minimized relative to the regression line. As previously noted, once the LS fit has been performed ,
either variable can be predicted from the other. In addition, inference regions or bands established for prediction in one direction may be similarly use in the reverse direction (although the terminology is changed to discrimination).
1 I
For a regrecsion ofy on x, the mean of future values ofyo for a given xois bounded (confidence) with a level of confidence of 2(1-a) 100% by l
cruszcoA.wn A.9 omms
o , -,
2 2 -
[Jo-a o - atxo f 2 t ;a/2,n- 2 3 8 +
(A.16L
.n . { (x, _3 )2 2
Where s is the" standard error of regression," i.e.,
T
,2, (Ji-8i) -(A.17) n-2 and t 1-a/2, n.2 i s found from the Student's t-distribution. ' Similarly, an individual '
future value ofyo for a given x ois bounded (prediction) with a level of confidence of 2(1-a)100% by 2 -2
[yo - ao - a1xo]2 3 f 1- a/2,n - 2 8 1 + -. + , (A.18) n { (x, _3 )2
' However, for a given yo, the bounds on xo , referred to as discrimination bounds, are found by solving equation (A.18) for the values of ox that satisfy the equality, although care must be taken relative to the solution, since real roots of equation (A.18) may not exist depending on the results of the data analysis, If the scatter of the data is small, as for the burst pressure to bobbin amplitude correlation, the regressions of x on y and y on x will yield slopes that are similar.
However, for APC analyses the data exhibit signi6 cant scatter for the leak rate to bobbin amplitude correlation and the two regression lines have significantly different slopes. In this case it is appropriate to select the regression line based on non-statistical considerations. Such considerations may be known end points of the regression line, e.g., burst pressure for non-degraded tubes, or comparison of the slope with theory based results. For either regression direction, inference regions can be determined.
As noted, equation (A.18) can be used to determine inferbnce bounds regardless of the direction of the regression. In general the magnitudes of the inference bounds will not be identical. However, the confidence level statements are true for both bounds, i.e., one bound is not invalidated by the other. Thus, if a 2(1-a) 100% lower bound on the burst pressure from the regression of the burst pressure on the logarithm of the bobbin amplitude is higher than the corresponding lower bound from the inverse regression, it simply means that the confidence level of the inverse -
regression is >(1-a)100%. Thus,if the residuals are verified to be normally distributed, the lower prediction bound may be taken as the higher of the predic-tion bounds established by performing the regression analysis in each directions.
cENSEC0AWP5 A.10 owns
___._____________________________.__---w..-r - - - - - -
,. . - - - - . - - - - . ~ . . - - . -.
~n ~ . , ,
s,
. A.7- Signifimace of the Ragressian The significance of the regression is evaluated by calculating the improvement in -
the estimate of the predicted variable based on knowledge of the regressor variable.
t c . For the APC analyses this is the same as determining whether or not the estimate-of the burst pressure or leak rate for a tube is improved by knowing the bobbin coil voltage amplitude. For a linear,1" order regression this is the same as testing to -
determine if a zero slope is probable. If the confidence interval for the slope -
includes zero then the relationship between the predicted variable ^and the regres-
-sor could be accidental, i.e., due to random error. The actual determination may be .,
made by calculating the confidence interval on the slope, at , to see ifit includes zero, or by testing the null hypothesis that the true slope is zero. In practice this is stated as 1
Ho : a1 = 0 ' '
(A.19)
R 1: a1 x 0.
If the null hypothesis, H o, is true, then ratio of the mean square due to regression (SSR) to the.mean square due to error (SSE), i.e., the mean square of the residuals, follows an "F" distribution with the regression degrees of freedom (DOF) in ths numerator and the residuals degrees of freedom in the denominator. For a linear analysis with k regressor variables and n data points, then . i MS Regression
-F I -"'*'"~*~l' (A.20)
MS Error i
where 100-(1-a)% is the associated confidence level. (Note that a is the area in the tail of the distribution.) If the true value of the slope is zero then both mean-square (MS) Regression and MS Error are independent estimators of the true value of the error variance. Since they are both estimates they 'would not be expected to be exactly equal, however, it would be expected that they would be nearly equal so that their ratio would not be too far from unity. It is noted that the F ratio and the 2
Index of Determination, r , are both calculated from the sums of squares of the variables, so SSR/k , r2 n-k-1 L yI ~ "' A'" ~ * ~ I , SSE/(n - k - 1) 1-r 2 ' (A.21) l-k 2
and a critical value of r for a selected critical a can be found as kF1 _,,3,,_3_1 -
i- 7"2 22)
" , (n - k - 1) + k F 1_ ,,3,, . 3 1 l
L
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7 q~ ' LIf th's valus of r f$und free the regre:sica is 'gratter than ths critical value from . t .
L(A.22) tha null hypoth::is; !!n, i.e, thtt the slopa is z:ro, would bn rsj:ctec!. and the 3)
' ' alternate hypothesis, H 1, that the true slope is not zero would be accepted.lFor.
example, consider 2 k=1,' n=15, and a=0.01. Then,'from equation (A.22) we. find a
' : critical value 2
of r of 0.411' and a critical value of r of 20.641. If the regression -
valus of r exceeds'0.411 the regression is significant at ai level greater than 99%.-
For a-1" order regression, equation (A.21) can be rearranged as .
I I- a/2, n - 2 ' *
- _7 2 (A.2m s n-2
. i.e., a t distribution with n-2 DOF's. ' Given a value of r from the regression . .
. analysis, a value of t can be calculated and a significance level determined. For the
[..
2 same example as above, we consider r =0.411 and n=15. From equation (A.23)'we T
find t=3.013, and a' significance level of100.(1-a)=99.1%, which agrees with the~
above determination. It is to be noted that for a small number of regressor variables and a large number of data points, the square of the correlation coefE-cient does not have to be very close to one to reject the null hypothesis and accept . i the alternate hypothesis that the slope is not equal to zero, thus implying that a '
correlation does exist. This test is identical to testing the hypothesis of equation (A.19) by calculating t as 63 f l- a/2, n - 2 == ,
-(A.24) ;
P where the denominator is the estimated standard deviation of the slope parameter, and then determining the probability of obtaining t at random. The probability thus obtained, i.e., the probability that the slope of the equation for the entire :
population is zero, is referred to as the p value for the coefficient. If the p value is less than a selected criterion value, e.g.,5%, use of the regreaion equation is considered justified. ,
A.8 Analysis of Regression Ba=inhi=1= 1 To use the results of the least squares analysis, it is assumed that" 1.
the expectation function is correct, that the response is given by the expectation function plus a disturbance, 2.
that the disturbance is independent of the response function, crwsscuws A.12 omas
y m < .
}
1 i.- thzt c:ch di:turbance hea a normal distribution about the response func- l tion vdua, that erch disturbince has z:ro mean,'
- 4. that the disturbances (or. weighted disturbances) have equal variances, and l p i L 5. that the disturbances are independently distributed.
The purpose of analyzing the residuals, i.e., the differences between the actual variable value and the predicted variable value,is to verify each of the assumptions inherent in performing the least squares analysis. There are a variety of plots that L can be used for the analysis of the residuals, although not all may be judged necessary for each analysis. A plot of the residual values against the predicted values should be nondescript since the residuals should not be correlated with the predicted values. Such results indicate that the variance is approximately constant (as assumed), that there is no systematic departure from the regression curve, and that the number of terms in the regression equation is adequate. A frequency plot (histogram) of the residual values should appear to be similar to a normal distribu-tion. A plot of the ordered residuals on normal probability paper should approxi-mate a straight line. Any of these plots may be used to verify that the regression residuals are normally distributed, although the results are not obvious from the scatter plot. The normal probability plot offers the advantages that it can easily be used to determine if the mean is approximately zero, and a reasonable estimate of the standard deviation of the residuals may be read directly.
To prepare the cumulative normal probability plot, the residuals are sorted in ascending order and then plotted against an ordinate cumulative percent probabili-
- ty value given by
. 100 ,
(A.25) n where n is the number of data points used in the regression and i is an index ranging from 1 to n. If a small number of outliers have been omitted from the i regression analysis, but the depiction of their residuals is desired, n may be taken !
as the total number of data points and the residuals of the outliers included I accordingly. This has the effect of compressing the spread of the outliers along the probability axis, but generally will not affect the conclusions relative to the linear-ity of the plot. The rationale for the cumulative probability values used is if the .
unit area under the normal curve is divided into n equal segments, it can be expect-ed, if the distribution is normal, that one observation (residual) lies in each section.
Thus, the i
- observation in order is plotted against the cumulative area to the 1 i
I middle of the i* section. The factor of 100 is used to convert the scale to percent '
probabilities.
caszcawes A.13 ov ws l .
- If tha plotted r:siduals approximate a visuallE fitted straight line, it'may be
, concluded thzt they are ncrmally distributed about the regression curve. The .
residual value where the line crosses the 50% probability value is an estimation of the mean of the residuals, and can be used to verify that the mean is approximately zero. The residual distance from the 50% point to the 84% point is an approxi-i mation for the standard deviation of the residuals. If the residuals for the outliers L have been included in the plot they will distort the results obtained for the mean and standard deviation, with the mean value being less affected. For this type of plot, the outliers in the data, if any, will tend to appear on the far left in the lower half of the residual normal plot and on the far right in the uppar half, i.e., large negative and positive residual values. The results from the normal probability plot' may be used to determine the need for preparing any of the other plots,~i.e., it may f be apparent that no additional information would be available from a scatter plot.
L A.9 References
- 1. Yamane, T., Statistics An Introductory Analysis,2nd Edition, Harper & Row, New York (1967).
- 2. Draper, N.R., and Smith, H., Anplied Rm- ion Analysis. Second Edition.
Second Edition. John Wiley & Sons, New York (1981).
- 3. Hald, A., Statistical Theory with Enrinaarina Annlications John Wiley & Sons, New York (1952).
- 4. Weisberg, S., Annlied Linear Raara== ion, John Wiley & Sons, New York (1985).
- 5. Lipson, C., and Sheth, N. J., Statistieml Dani-n and Analysis of Enainee-ina Erneriments, McGraw-Hill, New York (1973).
- 6. Deming, W. E., Statistical Adiustment of Data. Dover Publications, New York (1964). ~
- 7. Mandel, J. The Statistical Annivais of Ernerimental Data, Dover Publications, New York (1984).
- 8. Davies, O. L. (Editor), Statistical Methods in Research and Production, Oliver and Boyd, London, England (1957).
- 9. Fuller, W.A., Measurament Error Models, John Wiley & Sons, New York (1987).
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- 10. Wald, A. Fitting a Straight Line When Both Variables are Subject to Error, Ann Is cf Meth::mntical Statistics, Vcl.11, pp. 284-300 (1940). )
- 11. Bartlett, M.S., Fitting a Straight Line When Both Variables are Subject to i Error, Annals of Mathematical Statistics, Vol. 5, pp. 207-212 (1949).
- 12. Rousseeuw, P.J., and Leroy, A.M., Robust Reeression and Outlier Detection, John Wiley and Sons, New York (1987). '
-q 13.
Hosmer, D.W.~ ,
and Lemeshow, S., Anplied inaistic Regression, John Wiley & i Sons, New York (1989).
j
- 14. CSS Statistica Users Manual, Statsoft (1991).
- 15. SAS/ STAT User's Guide, Revision 6, Fourth Edition, Volume 2, Chapter 27, 1 The Logistic Procedure, SAS Institute, Inc., Cary, North Carolina (-1990).
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