spatial and model-order based reactor signal analysis methodology for bwr core stability evaluation

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Annals of Nuclear Energy 33 (2006) 1329–1338

annals of

NUCLEAR ENERGY

Spatial and model-order based reactor signal analysis methodologyfor BWR core stability evaluation

A. Dokhane a,b,*,1, H. Ferroukhi a,*, M.A. Zimmermann a, C. Aguirre c

a Paul Scherrer Institute, Laboratory for Reactor Physics and Systems Behavior, CH-5232 Villigen PSI, Switzerlandb Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland

c Kernkraftwerk Leibstadt, CH-5325 Leibstadt, Switzerland

Received 10 April 2006; accepted 2 August 2006

Abstract

A new methodology for the boiling water reactor core stability evaluation from measured noise signals has been recently developedand adopted at the Paul Scherrer Institut (PSI). This methodology consists in a general reactor noise analysis where as much as possibleinformation recorded during the tests is investigated prior to determining core representative stability parameters, i.e. the decay ratio(DR) and the resonance frequency, along with an associated estimate of the uncertainty range. A central part in this approach is thatthe evaluation of the core stability parameters is performed not only for a few but for ALL recorded neutron flux signals, allowingthereby the assessment of signal-related uncertainties. In addition, for each signal, three different model-order optimization methodsare systematically employed to take into account the sensitivity upon the model-order.

The current methodology is then applied to the evaluation of the core stability measurements performed at the Leibstadt NPP, Swit-zerland, during cycles 10, 13 and 19. The results show that as the core becomes very stable, the method-related uncertainty becomes themajor contributor to the overall uncertainty range while for intermediate DR values, the signal-related uncertainty becomes dominant.However, as the core stability deteriorates, the method-related and signal-related spreads have similar contributions to the overall uncer-tainty, and both are found to be small. The PSI methodology identifies the origin of the different contributions to the uncertainty. Fur-thermore, in order to assess the results obtained with the current methodology, a comparative study is for completeness carried out withrespect to results from previously developed and applied procedures. The results show a good agreement between the current method andthe other methods.� 2006 Published by Elsevier Ltd.

1. Introduction

During stability tests of boiling wsater reactors (BWRs),neutron flux noise signals are recorded and thereafter usedto evaluate at a given operating condition the core stabilityparameters, i.e. the decay ratio (DR) and the resonance fre-quency (RF). The noise evaluation method principally con-

0306-4549/$ - see front matter � 2006 Published by Elsevier Ltd.

doi:10.1016/j.anucene.2006.08.010

* Corresponding authors. Address: Paul Scherrer Institute, Laboratoryfor Reactor Physics and Systems Behavior, CH-5234 Villigen PSI,Switzerland. Tel.: +41 56 310 4062; fax: +41 56 310 2327.

E-mail addresses: [email protected] (A. Dokhane), [email protected] (H. Ferroukhi).

1 Now at: King Saud University, College of Science, Department ofPhysics and Astronomy, P.O. Box 2455, Riyadh 11451, SA.

sists in a time series analysis of the recorded neutron fluxsignals and is usually based on so-called system identifica-tion methods (Ljung, 1999).

A large amount of effort has in the last few decades beendevoted to the development of accurate BWR stabilitymonitoring techniques. As an example, a recent study hasbeen carried out in the framework of an OECD benchmark(Verdu et al., 2001) by different research groups using dif-ferent methods, to analyse measurements from the Fors-mark 1 & 2 NPP, Sweden. The main goal was thecomparison of different time series analysis methodsapplied for the BWR stability evaluation.

There are mainly two approaches that have been used inanalysing the neutron flux signals. In the first approach, thedynamical system underlying the signal is reconstructed

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1330 A. Dokhane et al. / Annals of Nuclear Energy 33 (2006) 1329–1338

and the stability properties of its dynamics are obtainedfrom invariant quantities (Pereira et al., 1992; Verduet al., 1997; Suzudo, 1993), e.g. the Lyapunov exponent.The other approach is based on using so-called parametric-or non-parametric methods to evaluate the decay ratio,which is considered as the main stability parameter. Fornon-parametric methods, the DR is evaluated from theautocorrelation function of the signal (Behringer and Hen-nig, 2002) while for parametric methods, it is evaluatedfrom the impulse response of the system (Van der Hagenet al., 1994) or from the effective transfer function of thesystem (Kishida, 1990; Sanchis et al., 1995).

Depending on the type of method that is selected, e.g.parametric or non-parametric, slightly different results interms of the stability parameters can be expected despiteusing the same recorded noise signal. Moreover, if a para-metric method is for instance selected, an additional vari-ation in the results could arise because of the possibility toemploy different types of parametric models among whichthe auto-regressive moving average (ARMA), the auto-regressive (AR) or the moving-average (MA) models arethe most common ones. Now, once a model-type isselected (e.g. ARMA), the determination of the optimalmodel order becomes the central task and usually requiresthe application of a well-defined optimisation procedure.In this context, several optimisation techniques have beendeveloped and an additional uncertainty arises thereforewith regards to the choice of the analyst upon the proce-dure to determine the optimal model order. A furtherlayer of complexity is added when it is considered thatthe evaluation of the DR/RF parameters can be per-formed using one/a few or all local power range monitor(LPRM) signals and/or using the average power rangemonitor (APRM) signals. Because among others, differentnoise levels in the recorded LPRM signals and moreoversince APRM signals consist in fact in the averaging of agiven set of LPRM signals (that additionally can belocated at different core elevations), the result of the timeseries analysis will tend to vary depending on which signalis evaluated.

Now, when considering that the results from stabilitytests, i.e. the ‘‘measured’’ DRs and RFs, are usuallyemployed to validate large coupled neutronics/thermal-hydraulic system codes applied in stability analyses, it isclear that the above mentioned difficulties induce uncer-tainties that should be taken into account in the selectedapproach to evaluate the core stability performance frommeasurements.

In the context of the above, a continuous developmentand enhancement of stability evaluation methods has dur-ing the years been developed at PSI (Hennig, 1999), princi-pally for application to Leibstadt NPP (KKL),Switzerland. Within the framework of this effort, the HPT-SAC code (Askari and Hennig, 2001) was developed tointegrate a wide range of identification methods and opti-misation procedures to be used for stability time seriesanalyses.

Based on these previous efforts and with the objective ofusing at any operating condition and during any givencycle, an identical procedure for the evaluation of theBWR core stability parameters from measurements, a sys-tematic and consistent PSI methodology was recentlydeveloped and adopted for the analysis of the Swiss BWRs.This methodology principally consists in a general reactornoise analysis where as much as possible informationrecorded during stability tests is used to determine ‘‘Best-Estimate’’ values of the stability parameters (DR, RF),referred to as the ‘‘Core Representative’’ stability parame-ters, along with an associated uncertainty range. In partic-ular, a time series analysis of all measured neutron fluxsignals, rather than only one signal, is performed and foreach signal, three different model-order optimization meth-ods are systematically employed to take into account thesensitivity upon the model-order.

The objectives of this paper are threefold. First, the PSImethodology to determine in a systematic manner core rep-resentative stability parameters along with an associatedvariation range that takes into account model-order-related uncertainties as well as signal-related uncertaintiesresulting from the impact of spatial core heterogeneitieson the local stability response is presented. Secondly, thisapproach is applied to the evaluation of the core stabilitymeasurements performed at KKL, during cycles 10, 13and 19. Finally, in order to assess the results obtained withthe current methodology, a comparative study is carriedout with respect to results from previously developed andapplied procedures.

2. Methodology

The developed PSI methodology consists principally ofthree parts: a ‘‘screening phase’’, a ‘‘neutron flux time seriesanalysis’’ and a ‘‘core stability evaluation’’.

2.1. Screening phase

This first phase is aimed at ‘‘gathering’’ as much infor-mation as possible regarding the test conditions prior toperforming the core stability evaluation from the recordedneutron fluxes. This principally consists in investigatingall recorded signals (i.e. not only neutron flux signalsbut also all other main process signals) with regard to sig-nal variations, signal-to-noise ratios and spectral/cross-spectral densities. The objective is to detect possibletrends and/or assess the eventual impact of external dis-turbances (e.g. process variations induced by control sys-tems or non-stationary test conditions) on the corebehaviour. This can indeed be useful in order to under-stand the impact of the test conditions on the spread inresults when performing the neutron flux signal analysis(Section 2.2). Moreover, during this screening step,cross-spectral and coherence analysis between all LPRMsignals is performed in order to detect out-of-phaseoscillations.

A. Dokhane et al. / Annals of Nuclear Energy 33 (2006) 1329–1338 1331

2.2. Neutron flux time series analysis

In the second phase, which represents the central part ofthe methodology, a time series analysis (TSA), using anARMA-based parametric system identification approach,is performed for ALL measured neutron flux signal, i.e.all LPRMs and all APRM signals, to evaluate the signal-specific stability parameters (DR, RF). The time seriesanalysis is performed in a systematic manner for any mea-sured neutron flux signal and is described below.

2.2.1. ARMA modelling and (DR, RF) stability parameters

evaluation

The underlying principle of the methodology is based ondetermining stability parameters using the parametricARMA model approach and is described here.

A given neutron flux signal is assumed to be representedby an ARMA model that relates the output (measured neu-tron flux) signal y(k) to an unknown noise signal e(k) by

yðkÞ ¼~hT~uðkÞ þ eðkÞ ð1Þ

with

~uðkÞ ¼ ½�yðk � 1Þ � yðk � 2Þ � � � � yðk � naÞ� eðk � 1Þ � � � � eðk � ncÞ� ð2Þ

being defined as the data vector and

~h ¼ ½a1a2 � � � anac1c2 � � � cnc� ð3Þbeing the unknown coefficients vector and the pair (na, nc)is defined as the ARMA model order.

Since the noise terms in Eq. (2) are not known, theymust be estimated and this is done by assuming that ifthe noise term at step k � 1 can be estimated, the predic-tion error at time step k becomes

eðkÞ ¼ yðkÞ �~hT~uðk;~hÞ ð4Þ

so that the data vector can be replaced by

~uðkÞ ¼ ½�yðk � 1Þ � yðk � 2Þ � � � � yðk � naÞ� eðk � 1Þ � � � � eðk � ncÞ� ð5Þ

For a given model order, the ‘‘best-estimate’’ values ofthe ARMA coefficients become those that minimize theprediction error equation (4). Hence, the solution isobtained by the minimisation of the expected predictionerror (EPE), called also the loss function, and defined asfollow:

V ð~hÞ ¼ 1

N

XN

k¼1

ðyðkÞ �~hT~uðk;~hÞÞ2 ð6Þ

Hence, the best estimate of~h, referred to as hN , is obtainedas

hN ¼ arg min~h

1

N

XN

k¼1

ðyðkÞ �~hT~uðk;~hÞÞ2( )

ð7Þ

The minimisation of the loss function of Eq. (6) is solvedat PSI using the HPTSAC code (Askari and Hennig, 2001),which employs first, the Box Jenkins recursive technique toestimate the noise terms, and thereafter, solves the non-lin-ear least square minimisation problem by an iterativeGauss–Newton algorithm (Krauss et al., 1994).

Once the best-estimate coefficient vector has beenobtained, the system transfer function is known and thevalues of the DR and the RF can thereafter be estimatedfrom the dominating pair of poles, i.e. closest to the unitcircle, of this transfer function since these poles will repre-sent the least damped component of the system impulseresponse. Hence, writing the dominating pair of poles as

z�ðhN Þ ¼ rðhN Þe�iuðhN Þ ð8Þthe DR and RF of the analysed neutron flux signal areevaluated as

dDR � DRðhNÞ ¼ rðhN Þ2p=u

dRF � RFðhNÞ ¼ up

F s

2

(ð9Þ

where F s ¼ 1T s

is the sampling frequency.

2.2.2. Model-order optimizationAs described in the previous section, the evaluated (DR,

RF) values will depend on the selected ARMA model order(na, nc). Therefore, a crucial step is to determine an ‘‘opti-mal’’ model order that would yield the best fit to the mea-sured time series, i.e. by producing small prediction errors,and that at the same time would contain the smallest num-ber of free parameters sufficient to adequately represent thetrue system (parsimony principle, (Soderstrom and Stoica,1989)). This latter principle implies that, out of two ormore competing models with different orders and fulfillingthe minimisation of the loss function, the model with thelowest order should be chosen.

In the PSI methodology, three optimization methods aresystematically employed for each measured neutron fluxsignal in order to take into account the sensitivity uponthe model order. The three methods, all integrated in theHPTSAC code, are the Plateau method and the AkaikeInformation Criterion (AIC) as well as the Rissanen mini-mum description length (MDL) methods (Ljung, 1999;Soderstrom and Stoica, 1989) both based on the minimiza-tion of the expected prediction error variance (EPEV).These three methods are described below:

2.2.2.1. Plateau method. The Plateau method is based onthe observations made in the framework of BWR stabilityanalyses and that indicated that beyond a certain modelorder (na,s, nc,s), the estimated decay ratio would reach asaturation value, a so-called ‘‘Plateau’’, and would remain‘‘saturated’’ even when increasing further the model order(Soderstrom and Stoica, 1989). In other words, consideringa sequence of real numbers ({DRi}, i 2 N, DRi 2 R), thissequence is characterized by a limit, so-called DR*, if andonly if the following condition is fulfilled:


limi!1

DRi ¼ DR� ð10Þ

In a mathematical sense, it can be stated that this limitexists if for each e (a positive real number), there exits aninteger number i* such that for any i greater than i*, the dif-ference between DRi and DR* is less than e , i.e.

8e > 0; e 2 R9i� 2 N j8i > i� ) jDRi �DR�j < e ð11Þ

The error criterion implemented in HPTSAC for whichthe plateau is reached is:

errorðnaÞ ¼ jDRs �DRðna; ncsÞjDRs

< e ð12Þ

for all na 2 Na and where

� DRs = DR(nas,ncs) is calculated at the beginning of theplateau,� (nas,ncs) and (nae,nce) are the model orders at the begin-

ning and the end of the plateau,� LPLATEAU = nae � nas is defined as the plateau

length,� Na = {nas+1, . . . ,nae} specifies the incremental values of

the na order within the plateau length,� DR(Na,ncs) = {DR(nas+1,ncs), . . .,DR(nae,ncs)} is the

value(s) of the DR within the plateau.

The error(na) calculates hence the normalized differencebetween the decay ratio at the beginning of the plateauand those corresponding to higher model orders withinthe plateau length.

The search procedure in HPTSAC code is performedsuch that for a given (start) value of nc, the na parameteris incremented. If a plateau has not been reached oncena = namax, the nc value is incremented by one unit andthe na-search procedure re-started. This is repeated untilthe above defined plateau is reached.

2.2.2.2. Methods for expected prediction error variance

(EPEV) minimization. The two other optimisation meth-ods, i.e. in addition to the Plateau method, systematicallyused in the PSI methodology are based on the minimiza-tion of the EPEV, W N ðhN Þ, which, for an asymptoticallyGaussian distributed hN , has the following form:

W N ¼ V NðhN Þ 1þ 2pN

� �ð13Þ

where p = dim(h) and N is the number of data points.

2.2.2.3. Akaike information criterion optimization (AIC).

The first of the two EPEV methods employed in the PSImethodology is referred to as the ‘‘Akaike Criterion’’(AIC) and is based on a method developed by Akaike onthe basis of information theory development. The AIC cri-terion is an alternative form of the EPEV minimization inthe sense that it minimises log(VN) (in the sense of the logof likehood function, (Soderstrom and Stoica, 1989)),

rather than WN of Eq. (13). Hence, the Akaike criterionto be minimized is defined as

AIC ¼ N log ðV N ðhNÞÞ þ 2p ð14ÞNoting that both approaches, i.e. Eqs. (13) and (14), are

however asymptotically equivalent.

2.2.2.4. Rissanen minimum description length (MDL). Thesecond EPEV optimisation procedure applied in the cur-rent methodology is based on the Rissanen MDL criterion,which stems from computer science theory. The aim is toselect the model that provides the shortest description ofthe data in bits, i.e. the more the model is able to compressthe data by extracting the regularities or patterns in it, thebetter is the model.

The MDL optimization criterion to be minimized isgiven by

MDL ¼ V NðhN Þ þpN

log N ð15Þ

2.2.3. Optimization procedure

The application of the three optimization procedures ismade through the HPTSAC code in the following way.First, the Plateau method is applied and once a plateauhas been reached, the maximum model order, (nae, nce),is determined. Thereafter, an assumption is made that theoptimal model order (nao,nco) for the AIC and MDL meth-ods is within the range defined by the maximum modelorder obtained with the Plateau method:

½ðnamin 6 nao 6 naeÞ; ðncmin 6 nco 6 nceÞ� ð16Þ

It should be emphasized that the AIC and MDL optimi-sation methods are thereafter applied to determine optimalmodel orders based on the model-order space defined byEq. (16).

2.3. Sensitivity upon signal and optimization method

As an outcome of the second step described previously,three pairs of stability parameters are evaluated for eachrecorded neutron flux signal. Based on this, for each signal,an average DR (and RF) value is estimated as function ofoptimization method along with an associated standarddeviation referred to as the ‘‘method-averaged’’ spread.Similarly, for each optimization method, an average DR(and RF) value is estimated for all signals along with anassociated standard deviation referred to as the ‘‘signal-averaged’’ spread. This is distinctly done for the two typesof signal categories, i.e. LPRMs and APRMs therebyallowing to assess the spread in results with regards to sig-nal type (APRM vs. LPRM), signal position (i.e. axial/radial location of LPRM signals) and optimization meth-ods (i.e. model order).

Prior to determining an estimation of the core represen-tative stability parameters for a given measurement, thespread in the DR and RF results as a consequence of

DR=0.57DR=0.69

DR=0.44DR=0.55

DR=0.50DR=0.55

DR=0.49DR=0.56

DR=0.53DR=0.58

DR=0.50DR=0.52

DR=0.62DR=0.67

DR=0.64DR=0.64

Loop1 Loop2 (4% to 8 % lower recirculation flow, 0.5 to 1oC lower feedwater temp.)

Fig. 1. Evaluation of bottom-level LPRM signals at two differentoperating conditions.


evaluating several signals and applying three different opti-mization methods is investigated. In this context, first, foreach signal, a method-averaged DR (and RF) is estimatedalong with a bias. This estimation allows comparing theperformance of the three different TSA methods and gives,for each LPRM/APRM, an overview of how well a methodis performing compared to the two other methods. Thisalso allows to assess the sensitivity of the (DR, RF) estima-tion upon the TSA method for each recorded neutron fluxsignal from a given stability test.

Secondly, for each method, a signal-averaged DR (andRF) is estimated by separately averaging over two catego-ries of neutron flux signals, i.e. LPRMs and APRMs,respectively. The signal-averaged estimation allows assess-ing the spread in DRs and RFs obtained from differentsignals when applying a given TSA method. In addition,since this is performed for two categories of signals, theperformance of the TSA method between global signals(APRMs) vs. local signals (LPRMs) can be compared.Finally, for each category of signal (i.e. global APRMand local LPRMs), the overall signal and method aver-aged DR (and RF) is estimated by averaging over bothmethods and signals. This allows assessing how ‘‘close’’a core stability evaluation made from LPRM signals iscompared to an evaluation made solely from APRM sig-nals. A close agreement indicates that for the given mea-surement, the uncertainty from using a specific TSAmethod or a selected signal to evaluate the core stabilityis rather limited.

2.4. Final evaluation of the core representative stability

parameter

Once the steps 1–3 (Sections 2.1–2.3) have been carriedout, the core stability evaluation consists in determining‘‘core representative’’ stability parameters along with anassociated estimate of the uncertainty range. At this stage,the proposed approach is limited to global-mode stabilityevaluation and, in this context, is only considered as a firststep towards establishing a ‘‘best-estimate’’ methodology.The proposed method, which is identical for the determina-tion of both the DR and RF values and therefore onlyillustrated below for the DR, is that for a given test, thecore representative DR is evaluated from all the APRMresults (i.e. over all signals and methods) as

DR ¼ 1

Nm � N l

XNm

m¼1

XNl

l¼1

DRðhm;lÞjAPRM ð17Þ

where m = 1, . . .,Nm is the index over methods andl = 1, . . .,Nl is the index over APRM signals.

Thereafter, the LPRM-averaged and the APRM-aver-aged deviations from the core representative DR obtainedwith Eq. (17) are computed, and the maximum value ofthese two quantities is selected as the uncertainty rangeassociated with the core representative results (noting that,in most cases, the LPRM spread is found to be larger) i.e.

UDR ¼ maxðUDR;LPRM;U DR;APRMÞ ð18Þwhere

UDR;X ¼1

N m;X � N l;X

XNm;X

m¼1

XNl;X

l¼1

ðDRðhm;lÞjX �DRÞ

¼ 1

N m;X � N l;X

XNm;X

m¼1

XNl;X

l¼1

(DRðhm;lÞjX

� 1

Nm � Nl

XNm

m¼1

XNl

l¼1

DRðhm;lÞjAPRM

)ð19Þ

and X = LPRM/APRM.

3. Trends and observations

To illustrate the motivation for selecting the approachdescribed in Section 2, some of the trends and observationsmade when analyzing the KKL stability tests are presentedhere.

As an example, when applying the TSA (Section 2.2.1)of the LPRMs, it was observed that for some tests, bot-tom-level LPRMs located in one (half) core region tendto yield larger decay ratios (by up to 0.1) compared tothe LPRMs located in the other core region. When analyz-ing feedwater, jet pump and recirculation coolant flows andtemperature signals, it was found that a slight asymmetrybetween both loops was systematically seen in all tests.These results are illustrated in Fig. 1. Hence, the observedtrend i.e. the spatial variation of the LPRM DR resultsmay be rather related to an asymmetry in the core inletsubcooling induced by process asymmetries rather thanto the core dynamical properties. From that point of view,the analysis of ALL LPRM signals as is done in the PSImethodology appears justified in order to take into account


radial dependencies (e.g. due to process variations) whenevaluating the core stability.

A second observation was that performing a TSA notonly for all measured LPRMs but also for all APRM sig-nals could be advantageous in order to take into accountpossible axial-dependencies of the results. As an illustra-tion, the average DR results evaluated from the APRMsare compared to the average LPRM results at several oper-ating conditions (indicated by the power-to-flow ratio) inFig. 2. The figure clearly shows that as the core becomesincreasingly stable for a given power/flow ratio (e.g. P/F =1.5 in Fig. 2), a larger average DR is obtained whenthe evaluation is made from the APRMs compared to anLPRM-based evaluation. Similarly, for stable core condi-

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.3

0.4

0.5

0.6

0.7

0.8

Test 27Test 26

Test 21

Test 28

Test 24

Test 20

Test 22

Test 25

Test 23

Average DR over 8 LPRMsAverage DR over 8 APRMs

Dec

ay R

atio

Power/Flow

Test 19

Fig. 2. Average APRM and LPRM results as function of power/flowconditions of KKL C19 stability tests.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

DR=0.4

DR=0.43

DR=0.35

DR=0.20

DR=0.13

Test Test 20 Test 13 Test 2 Test 1

Ave

rage

Spr

ead

T

Fig. 3. Average spread in decay ratio evalu

tions (DR < 0.5), as the P/F ratio is decreased, increasinglylarger DRs are obtained from the APRM evaluation com-pared to the LPRM evaluation.

Now noting that only bottom-level LPRMs wererecorded during the tests while the recorded APRMs alsoinclude the contribution from LPRMs located higher upin the core, the results of Fig. 2 indicate a certain axial var-iation in the local stability parameters particularly at stablecore operating conditions. When applying in addition thethree optimization methods (Section 2.2.2) for the TSA,it was observed that the ‘‘signal-averaged’’ spread tendsto be larger than the ‘‘method-averaged’’ spread forLPRMs, indicating a higher sensitivity of the evaluationresults upon the signal location. This is illustrated inFig. 3 where the spread in results is illustrated both as func-tion of signal and method for both categories of signals(noting that the tests shown in the figure are ordered byincreasing DR values). Conversely, the method-relatedspread for LPRMs is larger than the signal-related spreadonly at very stable core conditions (DR < 0.2). A furtherobservation is that as the core becomes less stable, themethod-related spread decreases but the signal-relatedspread on the other hand increases up to a certain DRrange (DR > 0.55) and thereafter decreases again. Con-cerning the APRMs, the same decreasing trend of themethod-related spread is seen as the core becomes less sta-ble. For the signal-related spread however, no clear trendcan be identified. Moreover, both the signal-related andthe method-related spreads are found to be smaller forthe APRMs compared to those obtained for LPRM sig-nals. The former become very small, both with respect tothe method and the signal. The main conclusion is thatas the core becomes very stable, the method-related sensi-tivity increases and becomes the dominant factor for the

DR=0.97

DR=0.71

DR=0.64

DR=0.59DR=0.55

9

Test 10 Test 28 Test 26 Test 25 Test 2422

est

over Methods, for LPRMs

over LPRMs

over Methods, for APRMs

over APRMs

ation from LPRM and APRM signals.


overall estimation uncertainty. At intermediate DR values,the signal-related sensitivity dominates the overall estima-tion uncertainty principally due to the impact of axial/radial core heterogeneities on the local LPRM detectorresponse. As the core approaches the instability limit, theinherent core behaviour becomes so strong that neitherthe method-related nor the signal-related sensitivity playsany important role, i.e. the estimation uncertaintydecreases significantly. A further conclusion is that in allcases, the estimation derived from the APRM signalsappears to be more robust i.e. similar signal/method sensi-tivities and much smaller than those for LPRMs.

4. Application to KKL nuclear power plant

4.1. KKL stability measurements

For the KKL stability tests performed at beginning-of-cycles 10, 13, 19, a similar test procedure was employedat the plant: the stability tests were carried out after havingoperated the core at full power for several days in order to

Table 1Overview of KKL stability tests during cycles 10, 13 and 19

Cycle Date Core loading

10 September 1993 (1) 50% GE fuel bundles (including 8 · 8, 10(2) 50 % SVEA-96 fuel (10 · 10 design)

13 September 1996 100 % SVEA-96 10 · 10 Core

19 September 2002 (1) 40% SVEA-96 10 · 10(2) 60 % SVEA-96 Optima with PLR

Note: PLR, part length rod.

Fig. 4. Power-flow map and stability measurem

achieve Xe-equilibrium conditions. A 10-min recordingwas made after the targeted operation conditions have beenreached for any given set of neutron flux and process sig-nals using the same data acquisition system. (A lower sam-pling frequency was used for the cycle 10 tests.) A total of42 signals are measured during each test, including thetotality of eight APRM signals, eight LPRM signalslocated at the bottom part of the core and a variety of pro-cess signals such as the core flow, steam flow, feedwaterflow, etc.

However, the objectives for performing the stability testswere slightly different between these three cycles. The mainobjective of the KKL cycle 10 core stability tests was toinvestigate the stability characteristics of a very heteroge-neous core (with regards to fuel bundle design and fuel ven-dor) at different conditions in the power-flow map(3138 MW at that time). The cycle 13 tests were carriedout with the primary goal to validate the exclusion regionsup to the permitted uprated power level of 112% power andto check the performance of the online stability monitoringsystem that had been installed at the plant. The cycle 19

Stability tests number Sampling frequency (Hz)

· 10 designs) 1–10 10

11–18 40

19–28 40

ents performed during cycles 10, 13 and 19.

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

Test Number (-)

DR-Cycle 10

DR-Cycle 13

DR-Cycle 19

Dec

ay R

atio

(-)

0 5 10 15 20 25 30

0.5

0.6

0.7

0.8

0.9RF-Cycle 10

RF-Cycle 13

RF-Cycle 19R

eson

ance

Fre

quen

cy (

-)

Test Number (-)

a

b

Fig. 5. PSI results for KKL stability tests in cycle 10, 13 and 19: (a) dfecayratio and (b) resonance frequency.

0.0

0.2

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Fig. 6. Comparison of PSI results with ‘‘separate approach’’ for cycle 10.


tests were performed in order to validate the maximumextended operating domain (MEOD) line in the updatedoperating map as well as the analytically defined bound-aries for stability monitoring, i.e. monitoring and excludingregions, concluding the power uprate of the plant to3600 MW. The investigated cycles are summarized in Table1 below, noting that the location of the different tests in theKKL operating domain are shown in Fig. 4.

4.2. PSI analysis results

This section presents the nominal results, i.e. the corerepresentative stability parameters, obtained when apply-ing the PSI reactor noise analysis methodology to the sta-bility tests described in Section 4.1. The objective is toprovide confidence by applying a consistent and systematicapproach in the database of stability measurements sincethese results will primarily serve as basis for the validationof coupled neutronics/thermal-hydraulic system codes usedat PSI for BWR stability analysis.

The evaluated core representative decay ratios and theassociated uncertainty ranges are shown in Fig. 5a, notingthat for each cycle, the test results are presented in ascend-ing order with regard to the decay ratio (i.e. decreasing sta-bility). It is clear from Fig. 5a that the KKL core is foundto be stable in all the investigated operating points of eachcycle, except for one operating point in cycle 10 whichyields a DR value close to 1 indicating a possible limit cycleoscillation. The results for the resonance frequency are dis-played in Fig. 5b, in which the uncertainties are found to bevery small compared to those for the DRs, except at onemeasurement at very stable core conditions where a0.05 Hz uncertainty is obtained. This can be explained bythe fact that, for a very stable core, no distinct value ofRF (spectral peak) can be obtained. This is due to theincreased difficulties during the identification processthe identification process in detecting a unique value forthe core resonance frequency even when using differentoptimization methods.

4.3. Assessment against earlier analyses

As a next step, an assessment of the PSI approach is car-ried out against results prepared for some of the KKL sta-bility tests by independent organizations using otherapproaches, referred to as ‘‘separate approaches’’. Theobjective is to provide confidence, on a qualitative andquantitative basis, in the obtained results when applyingthe PSI methodology. As an example, the PSI results forthe cycle 10 measurements are compared in Fig. 6 to thoseprovided by a fuel vendor. The latter were evaluated basedon the analysis of only one neutron flux signal using onlyone parametric ARMA model. As can be seen, the (nomi-nal) results are in good agreement. More interesting to noteis that the ‘‘separate approach’’ results are in most caseswell inside the PSI variation range. Since one objective inthe PSI approach is to determine an uncertainty range that


captures the effect of methods and signals, the resultsshown here tend to lend confidence in the adoptedapproach. From the perspective of code validation, theadvantages of determining such a variation range areclearly illustrated in Fig. 6, for instance for Rec5: the sep-arate approach indicates a DR close to 0.35, whereas thePSI approach indicates a rather similar value, but in addi-tion provides an estimate of its uncertainty, in this case sug-gesting that a DR around 0.5 is also plausible.

A further comparison has been performed in the frame-work of the cycle 19 measurements evaluation. For thiscycle, the PSI results are compared in Fig. 7 to those calcu-lated by the KKL online stability monitoring system,which employs a recursive identification method to esti-mate the DR and RF values. It is seen that, a good agree-ment is obtained since all the KKL results, except for Rec4,are well inside the PSI variation range.

Finally, with regards to assessing the uncertainty rangeobtained with the adopted method, the PSI range obtainedfor the three cycles is shown as function of the DR in Fig. 8.

0.2

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Fig. 7. Stability-evaluation results comparison between PSI methodologyand on-line system for cycle 19.

0

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Test 23

Fig. 8. PSI variation range vs. NEA uncertainty range.

Here, it is compared to the uncertainty that would beobtained for a given DR when applying the noise analysisuncertainty range specified in an earlier NEA benchmark(Johansson et al., 1994; Lefvert, 1996) in which onlymodel-order uncertainty effects were taken into account.The figure shows that, qualitatively, the PSI approach pro-vides results, which are consistent with findings reported in(Lefvert, 1996), e.g. the uncertainty range decreases as func-tion of the DR. This is because, as seen in Section 2, the PSImethodology also shows a larger sensitivity to the modelorder (i.e. method optimization) as the core becomes morestable. However, quantitatively, the results of Fig. 8 indicatethat at very stable core conditions, the PSI uncertainty rangeis much narrower which is probably because the NEA rangeis based on a maximum, rather than a standard deviation.While for intermediate DR value cases, it is clearly seen thatfor some tests, the uncertainty obtained with the PSIapproach (which is based on both method-order and spatialuncertainty effects) is significantly larger than that obtainedwhen applying the NEA range (which only takes intoaccount method-order-related uncertainties). In fact, thisdiscrepancy is not surprising since at such DR ranges, itwas found that the uncertainties due to spatial effects playa dominant role in the overall uncertainty (see Section 3).As an example, for tests 19 and 23 shown in Fig. 8, theDR value averaged over the APRMs is found to be signifi-cantly larger than the one averaged over the LPRMs (seeFig. 2). And since the DR uncertainty range is in the PSIapproach based on the average deviation between theLPRM and the APRMs results, a larger uncertainty canbe expected for these tests.

5. Summary and conclusions

A systematic and consistent methodology for the evalu-ation of BWR core stability properties, i.e. DR and RF,from reactor noise signals measured during stability testshas been developed. This methodology principally consistsin a general reactor noise analysis where as much informa-tion as possible recorded during the tests is investigatedprior to estimate core representative stability parametersalong with an associated uncertainty range.

The core representative stability parameter and theuncertainty range are estimated after having applied a sys-tematic time series analysis that principally consists in eval-uating, using a parametric identification method, allmeasured neutron flux signals and applying for each signal,three different model-order optimization methods. Whenusing this approach, it is found that as the core becomevery stable, the method-related spread (due to the choiceof the optimization method) becomes the major contribu-tor to the overall uncertainty range while for intermediateDR values, the uncertainty due to the choice of the signalbecomes dominant. However, as the core stability deterio-rates, the method-related and signal-related spreads havesimilar contributions to the overall uncertainty, and areboth found to become quite small. Furthermore, it is found


that the APRM evaluation appears more robust (i.e. smal-ler method- and signal-related spreads compared to theLPRMs). Hence, although it is generally acknowledgedthat the evaluation uncertainty increases as the corebecome more stable, the observations made with the PSImethodology are helpful in order to increase the knowledgeconcerning the different contributions to this uncertainty.

The proposed methodology is thereafter applied for aconsistent and systematic evaluation of the core stabilitymeasurements performed at KKL during cycles 10, 13and 19. The results show that using such a consistentapproach is indeed valuable in the perspective of validatingcoupled neutronics/thermal-hydraulic BWR stabilitycodes.

Finally, a comparative study is carried out between thePSI results for the KKL stability tests with those obtainedby other organizations using different procedures. Thiscomparison shows a good agreement, providing therebyconfidence in the adopted approach, and moreover bring-ing out the advantage of using this method to take intoaccount not only method-related but also signal-relateduncertainties.

Acknowledgement

This research has partially been supported by Kernk-raftwerk Leibstadt AG, CH-5325 Leibstadt, Switzerland.

References

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spatial and model-order based reactor signal analysis methodology for bwr core stability evaluation

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