Fuzzy identification of transients in nuclear power plants

Nuclear Engineering and Design 225 (2003) 285–294

Fuzzy identification of transients in nuclear power plants

Marzio Marseguerraa,∗, Enrico Zioa, Andrea Oldrinia, Enrico Bregaba Department of Nuclear Engineering, Polytechnic of Milan Via Ponzio 34/3, 20133 Milan, Italy

b CESI S.p.A., Via Rubattino 54, 20132 Milan, Italy

Received 11 December 2002; received in revised form 30 April 2003; accepted 4 June 2003

Abstract

The identification of transients is of fundamental importance for the timely monitoring of nuclear plants operation. The maintarget is detecting the occurrence of the onset of a transient, in its early stages, and identifying its kind so as to be able to readilyact to fix its causes. Given the safety and economical importance of the problem, various approaches have been investigated andapplied for transient identification, and many efforts are still devoted to the improvement of the results so far obtained.

In this paper, a fuzzy-logic based method for the identification of transients is proposed. The method is ‘model-free’ in thatthe if-then rules, which constitute the heart of the approach, are inferred only from the available input-output signal data. Themethod is tested on an example of identification of reactor transients generated by four forcing functions of different nature. Thenecessary data for the identification have been simulated by the QUAndry-based Reactor Kinetics code (QUARK, distributedfreely by NEA) configured so as to model the operations of the Westinghouse Advanced Pressurized water reactor, AP600.© 2003 Elsevier B.V. All rights reserved.

1. Introduction

The early identification of unexpected departuresfrom steady state behavior is an essential step for theoperation, control and accident management in nu-clear power plants. The basis for the identificationof a change in the system is that different systemfaults and anomalies lead to different patterns of evo-lution of the involved process variables. Given the

Abbreviations: AP600, Advanced Pressurized water reactor;Bi, Big i (i = 1,2, . . . , N); CE, CEnter; COA, Center Of Area;FRB, Fuzzy Rule Base; FS, Fuzzy Set; KB, Knowledge Base;LOCA, Loss Of Coolant Accident; LWR, Light Water Reactor; MF,Membership Function; NEA, Nuclear European Agency; QUARK,QUAndry-based Reactor Kinetics;Si, Small i (i = 1,2, . . . , N);UOD, Universe Of Discourse

∗ Corresponding author. Tel.:+39-02-23996355;fax: +39-02-23996309.

E-mail addresses:[email protected](M. Marseguerra), [email protected] (E. Brega).

safety and economical importance of the problem, sev-eral approaches for transient identification have beeninvestigated and many efforts are continuously de-voted to the improvement of the results thus far ob-tained. In this work, we investigate the possibility ofusing a fuzzy logic modelling tool for efficient tran-sient identification. Fuzzy logic is a modern technique,nowadays widely applied in many different scientificand engineering fields(Zadeh, 1965; Fuzzy Sets andSystems, 1995). The approach here employed is de-rived from Wang et al. (1992)and allows generatingfuzzy rules from numerical input/output data pairs.

Given the methodological nature of the paper, noparticular class of accidents or transients (e.g. as tothe IAEA classification) is considered in the applica-tion of the method. Rather, as we shall see, transientdeviations in the system behavior are generated bysimply simulating changes in some appropriately se-lected forcing functions of the system, with no consid-erations given to concurrent accident conditions such

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286 M. Marseguerra et al. / Nuclear Engineering and Design 225 (2003) 285–294

as scram delay, timing of steam isolation valves clo-sure, etc., which may be relevant in practice, partic-ulary in the first few seconds of transient evolutions.More precisely, the method is tested with respect tothe early identification of signal trends in the opera-tion of the Advanced Pressurized water reactor AP600designed by Westinghouse(ENEL, 1992, 1993). Thisparticular reactor type was chosen as reference, al-though currently not in operation nor under construc-tion, because of convenience due to the availabilityof the computer code QUAndry-based Reactor Kinet-ics (QUARK) configured to model this type of reac-tor (Salina et al., 1994). After having introduced fewbasic concepts underpinning fuzzy logic modelling,we illustrate how the fuzzy rules are generated fromthe plant measured data and how the developed fuzzymodel can be used to provide the classification of thecause of transient associated to a given input signalvector. Then, we present the application to the abovementioned AP600 case study.

2. Fuzzy reasoning

A brief digression is in order to describe how fuzzyreasoning proceeds with reference to a fuzzy modelthat maps inputm-dimensional vectorsx∗ receivedfrom a plant into outputk-dimensional vectorsy∗ to beused for various purposes, e.g. process identification(Fig. 1).

The model consists of the following four main com-ponents(Fuzzy Sets and Systems, 1995).

fuzzificationmodule

fuzzy inferenceengine

defuzzificationmodule

fuzzy rulebase

A C D

B

x* x y*yPLANT

Fuzzy Inference Model

Fig. 1. General scheme of the fuzzy inference model.

A. The fuzzification module: It converts the crisp vari-ables coming from the plant into fuzzy sets, as de-fined by their membership functions MF, in orderto take into account the inherent uncertainties andimprecisions in the input data. In the present casethe input crisp values will be fuzzified as single-tons: in correspondence of the crisp valuex∗ theFSX′

p will then be defined by the MF

µX′p(xp) = δxp,x∗ (1)

δ being the usualδ-Kronecker.B. The fuzzy rule base: A fuzzy inference model is

founded on a set of if-then rules, the so-calledFuzzy Rule Base (FRB) or Knowledge Base (KB).The genericj-th fuzzy rule is made up of a num-ber of antecedent and consequent linguistic state-ments, suitably related by fuzzy connections:

if (x1 isX1j)and(. . . )and(xm isXmj)

then(y1 isY1j)and(. . . )and(yk isYkj) (2)

The linguistic variablesxp, p = 1,2, . . . , m, arethe antecedents, represented in terms of the fuzzysetsXpj of the universe of discourseXp. The lin-guistic variablesyq, q = 1,2, . . . , k are the con-sequents, represented by the fuzzy setsYqj of theuniverse of discourseYq. Obviously,Xpj is one ofthe FSsX1

p, X2p,. . . in whichXp has been divided

and analogously forYqj. The connective operator‘and’ links two fuzzy concepts and it is generallyimplemented by means of a t-norm, here chosenas the algebraic product. The rules of the FRB arejoined by the connective “else” and are generally

M. Marseguerra et al. / Nuclear Engineering and Design 225 (2003) 285–294 287

implemented by means of an s-norm, typically themaximum operator∨.

C. The fuzzy inference engine: At each (discrete) timethe fuzzy inference engine receives the (linguistic)variables, sent by the fuzzification module, whichconstitute the Fact, viz.

Fact : x1 isX′1 and . . . andxm isX′

m (3)

whereX′p ⊆ Xp, p = 1,2, . . . , m. The fuzzy

engine compares these data with those in the an-tecedents of the FRB and arrives at the Conclu-sion, viz.,

Conclusion :y1 isY ′1 and. . . andyk isY ′

k (4)

whereY ′q ⊆ Yq, q = 1,2, . . . , k.

D. Defuzzification module: Assuming, for simplicity,that there is only one consequenty, that is k =1, the output of the fuzzy inference engine mod-ule consists of a fuzzy setY ′ ⊆ Y with compactsupport(η1, η2), whose membership function isµY ′(y). However, eventually we are interested infinding a crisp numbery∗ that represents the in-formation encoded in the output fuzzy set. Thisconversion, called defuzzification, may be done inseveral ways, one commonly used being the Cen-ter of Area (COA) method:

y∗ = yCOA =∫ η2η1

yµY ′(y)dy∫ η2η1

µY ′(y)dy(5)

The crisp numbery∗ thereby obtained can be takenas the output resulting from the given inputx∗and it may used for various purposes, e.g. processidentification and control.

3. Building the fuzzy model

3.1. Training phase

Given a set ofL numerical input/output pairs(x)l, (y)l, l = 1,2, . . . , L, we wish to generate a setof fuzzy rules representative of the mapping fromthe system input space into the output one. We shalloften refer to this phase of rule construction, duringwhich both the system input and output are known,with the term ‘training’, in analogy to the procedurefor building a neural network model.

For simplicity, let us consider the case of a bidimen-sional input and a monodimensional output spaces.The universe of discourse of each of the three vari-ables, assumed to be known, is divided inton = 2N+1half overlapped fuzzy subsets, denoted asSN (smallN), . . . , S1 (small 1), CE (center),B1 (big 1),. . . , BN

(big N). To each of these, we associate a triangularnormalized membership function (MF),µ(.), havingthe peak (µ = 1) at the midpoint of the subset and thetwo vertexes of the base (µ = 0) in correspondence ofthe midpoints of the two adjacent neighboring fuzzysets (Fig. 2).

For a generic triplet of numerical input/outputdatax1(i), x2(i), y(i) we wish to determine the cor-responding ‘if . . . then’ fuzzy rule which describestheir relation. We first determine, for each of thethree values separately, the ‘degree of membership’,µ(.), to the corresponding existing fuzzy sets. Be-tween the two fuzzy sets fired byx1(i), i.e. havinga non-zero membership value, in the rule we asso-ciate tox1(i) the linguistic term bearing the largestmembership value. Analogously, we proceed forx2(i) and y. Fig. 3 (left) illustrates an example inwhich x1(i) = −1.35, x2(i) = 2.1 andy(i) = 1.15.The first inputx1(i) fires the linguistic termsS3 andS2, with S2 having the largest membership valueµS2(x1(i)) = 0.65; the second inputx2(i) fires B1and B2, with µB1(x2(i)) = 0.9 > µB2(x2(i)) =0.1; finally, the outputy(i) fires the fuzzy sets CEand B1, with CE having the largest membershipµCE(y(i)) = 0.85. Then the triplet of data activatesthe rule:

If x1 isS2 andx2 isB1 theny is CE (6)

During the process of rule generation it may occurthat triplets of input/output data give rise to contra-dictory rules, i.e. rules sharing the same antecedents(same ‘if’ part) but differing in the consequents (dif-ferent ‘then’ part). To resolve the contradiction, eachtime a new rule is created we assign it a ‘degree ofbelief’, D(rule), and among those in conflict we keepin the FRB the one with largest value ofD. Moreprecisely, the degree of belief of a rule is defined asthe product of the membership functions of the lin-guistic variables characterizing both its antecedentsand consequents, evaluated in correspondence of thecrisp values which have generated the rule.Fig. 3shows an example of two conflicting rules,i (left)


-3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

S3

S2

S1

CE B1

B2

B3

Fig. 2. Partitioning a variable’s domain in fuzzy sets.

and j (right), with D(rulei) = 0.65 × 0.9 × 0.85 >

D(rulej) = 0.6 × 0.9 × 0.6. In this case, betweenthe two conflicting rulesi andj we keep in the FRBrule i.

3.2. Using the fuzzy model: test phase

At the end of the training phase, once the fuzzymodel constituted byR rules is established, we feedit with the FACTS, each consisting of an input vectorx∗ = (x∗

1, x∗2, . . . , x

∗m).

Each of them crisp values of the Fact, after fuzzi-fication as a singleton, belongs to two fuzzy sets. Dif-ferently from the training phase, now both fuzzy setsare retained so that each fact can fire at most 2m rules.To determine the corresponding output vectory∗ =(y∗

1, y∗2, . . . , y

∗k ), we compute the ‘strength’ of each

of the R rules in the FRB as the product of its an-tecedent membership values in correspondence of thecrisp inputs. For example, for the genericl-th rule, thestrengthsl(x∗) is given by:

sl(x∗) =

m∏p=1

µXpl(x∗p) (7)

whereXpl is the name of the fuzzy set in thep-thantecedent of thel-th rule. Obviously, thel-th ruleis activated, that is its strength is not zero, only ifx∗p ∈ Xpl,∀p. If the number of fuzzy rules fired is

Rf > 0, we proceed as follows. We denote byylqthe midpoint of the fuzzy setYql pertaining to theq-th component of the output, which appears as con-sequent in thel-th activated rule. The crisp value ofthe q-th output is finally determined as a weighedsum of theq-th outputsylq of theRf active rules, theweights being the strengths of the rules, normalized tounity:

y∗q =

∑Rf

l=1 sl(x∗)ylq∑Rf

l=1 sl(x∗)

(8)

On the contrary, it may happen that no rules are firedby the incoming Fact. In other words, a newcominginput may be unable to fire any of the rules constructedduring training, and thus the fuzzy model would beunable to map it to any output value. To overcome thisproblem, we introduce a concept of ‘distance’ (not tobe interpreted in a strict mathematical sense) betweena given input vectorx∗ = (x∗

1, x∗2, . . . , x

∗m) and the


-2 0 2 40

0.5

1

x1

-2 0 2 40

0.5

1

-2 0 2 40

0.5

1

x2

-2 0 2 40

0.5

1

-2 0 2 40

0.5

1

y

-2 0 2 40

0.5

1

(-1,35 ; 2.1 ; 1.15) ( -1,4 ; 2.1 ; 0.4)

Fig. 3. Conflicting rules: left, MF for thei-th triplet (−1.35; 2.1; 1.15); right, MF for thej-th triplet (−1.4; 2.1; 0.4).

antecedents of a given rulel:

dl(x∗) =

√√√√m∑

p=1

(xlp − x∗p)

2wp (9)

where,xlp is the midpoint of the fuzzy setXpl in thel-th rule and 0< wp ≤ 1 is the ‘importance weight’that the analyst is free to assign to give more or lessimportance to the various antecedents of the rule. No-tice that since the input variables may very well beof different nature (mm, MPa,◦C), their values mightneed to be normalized ‘a priori’.

In case of one single output variable, denoting byR

the total number of rules in the FRB, we take as crispoutput value the following expression:

y∗ =∑R

l=1 yl/dl(x

∗)∑Rl=1 1/dl(x∗)

(10)

which gives more weight to the outputs of the rulescloser to the input under consideration.

4. A case study: identifying the causes oftransients in the Westinghouse AP600 reactor

The pressurizzed water reactor AP600 belongs tothe latest generation of nuclear reactors for civil useproduced by Westinghouse, with a low power densitycore and a power rate of 1933 MWth, at full power.A more detailed description of the reactor design andfunctioning can be found in(ENEL, 1992, 1993).

In our work a simulation code was used to re-produce the evolution of the underlying nuclear andthermal–hydraulic processes under various conditionsand operations modes. Given the methodological na-ture of the work, the role of the specific code used isnot decisive for the fuzzy identification method pro-posed. Yet, in order to obtain an accurate simulation of


realistic reactor transients, including the inter-relationbetween neutronic and thermal–hydraulic effects, wechose to use the computer program QUARK, a detaileddesign code developed by ENEL/CISE and distributedfreely by Nuclear European Agency (NEA) for re-search purposes(Salina et al., 1994). The code allowsto simulate the neutronic and thermal–hydraulic tran-sient behavior of a three-dimensional Light Water Re-actor (LWR) core, starting from either a critical steadystate (eigenvalue problem) or a subcritical steady state(fixed source problem). In the code, the two-groupneutron diffusion equation is solved by the Analyt-ical Nodal Method(Smith, 1979), while the corethermal–hydraulic model is made-up of an upgradedversion of the COBRA code(Basile et al., 1987), athermal–hydraulics code that resolves the mass andenergy conservation equations of the flowing coolantfor a given reactor geometry. The code has been ex-tensively validated against other detailed design codesand experimental data. In any case, possible inaccu-racies in the process evolution prediction by the code,due for example to an inadequate or incomplete inputdeck or the nodalization and code uncertainties, areonly partially relevant to the scope of the present work,provided that the training of the fuzzy model and its

-1 0 1 2 3 4 5520

525

530

535

540

545

550

555

Fig. 4. Example of a sigmoidal change in the forcing functionF = Tin from the reference valueF0 = 548 K to the final valueFfin = 530 K.The other parameters ofEq. (11)are τ = 2.5 andm = 5/τ.

use are performed coherently, i.e. on data coming fromthe same source. Hence, for our purposes, the output ofthe code is taken as the correct (pseudo)experimentalvalues of the process variables to be used in the fuzzymodel training and identification tasks.

From the point of view of our application, we seethe reactor as a “black box” and consider the follow-ing four possible forcing functions which can origi-nate the transients: primary pressure in the corePp,primary core inlet temperatureTin, core inlet flow rateΓin and boron concentrationCB. For our purposes,the choice of the forcing functions is somewhat arbi-trary and it has no influence on the proposed method.Yet, these particular forcing functions were chosen be-cause they allow the simulation of various transients:(1) leakage of the pressurizer (Pp); (2) pressure in-crease (Pp); (3) increase of coolant inlet temperature(Tin); (4) decrease of coolant inlet temperature (Tin);(5) LOCA (Γin); (6) rise in boron concentration (CB);(7) decrease in boron concentration (CB). The genericforcing functionF(t) is assumed to vary according tothe following sigmoidal shape:

F(t) = F0 + Ffin − F0

1 + e−m(t−τ)(11)


Table 1Forcing functions: range of variability of parameterFfin

Minimum Maximum ReferenceF0 Units

Pressure 100 190 155 barTin 531 567 548 KΓin 3.02× 103 6.05× 103 6.05× 103 kg/sCB 1150 2118 1728 ppm

whereF0 is the initial value at timet = 0 s,Ffin is thefinal value,τ is a delay time andm is the characteris-tic frequency of the transient. An example of forcingfunction is given inFig. 4.

The range of variations of the forcing functions pa-rameter (Table 1) have been set on physical grounds.

A generic transient is originated by the variation ofa single forcing function from its reference valueF0at timet = 0 s, to the final valueFfin sampled at ran-dom in the corresponding range ofTable 1. The pos-sible forcing functions were coded as integer numbers1, 2, 3, 4, in the order as they appear inTable 1. The(pseudo)measured signals (computed by the QUARKcode) upon which the transient classification is to bebased are: the pressure drop in the core,PDrop, the out-put coolant temperatureTOut, and the coolant heatingHCool (power transmitted to coolant, relative to nom-inal fission power).

In the training phase, the FRB was constructed bysampling, as above explained, a large number of forc-ing functions and by computing the corresponding pat-terns for the fuzzy classifier.Fig. 5 reports the threemeasured signals for 1000 transients (250 for eachforcing function), lasting 20 s each. We note that inmost cases the relevant part of the variation of themeasured variables occurs in the first 10 s. In order totest the potential for early diagnostics, the data col-lected after 7 s were not used (this is indeed a “veryearly” diagnostics if compared to the current practiceof 1800 s with no operator action needed: although notdone here, the signals recorded at the successive timesmay be used to confirm the initial diagnosis).

The fuzzy classifier receives in input the values at-tained by the three relevant variables at the followinginstants, found by trial and error, after the start of theforcing function:PDrop: 1, 3, 6 s;TOut: 2, 4, 6 s;HCool:2, 3, 5, 7 s (Fig. 6), and provides as output the class(1–4) denoting the forcing function. Thus the rules arecomposed by 10 antecedents and one consequent. An

0 2 4 6 8 10 12 14 16 18 2032

34

36

38

40

42

44

46

48

t

KP

a

Pressure Drop in the core

0 2 4 6 8 10 12 14 16 18 20550

555

560

565

570

575

580

585

t

K

Coolant Exit Temperaure

0 2 4 6 8 10 12 14 16 18 200.2

0.4

0.6

0.8

1

1.2

1.4

t

Mw

/Mw

NO

M

Total Coolant Heating

Fig. 5. Trend of the input signals during the first 20 s of 1000random transients.


Fig. 6. The fuzzy classifier.

example of a rule is the following:

if [PDrop(1) is B1] and [PDrop(3) is B3] and[PDrop(6) is S3] and [TOut(2) is S1] and [TOut(4)is S3] and [TOut(6) is B1] and [HCool(2) is S2]and [HCool(3) is S3] and [HCool(5) is S1] and[HCool(7) is B1]

thenForcing Function is 2

Concerning the input, we partitioned the range (Uni-verse of Discourse, UOD, in fuzzy logic terminology)of each input variable in 51 half-overlapped FSs, each

0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 7. Distribution of the defuzzified numbers around their integer.

covering 1/25 of the UOD (the reason behind this par-titioning will become clear shortly). Concerning theoutput, the defuzzification procedure results in a realnumber which must be suitably truncated to obtain theinteger number which, according to the fuzzy clas-sifier, codes the forcing function responsible of theconsidered transient. To perform the truncation, we in-troduced an ambiguity band between each pair of adja-cent integers. All the defuzzified values falling outsidethe ambiguity band are assigned to the nearest integer,whereas those falling within the ambiguity band areclassified as “don’t know”, or ambiguous. Of course,misclassification, that is wrong identifications of tran-sients, may occur among the non-ambiguous values.It turned out that a compromise for a reliable classi-fication might be a value of 0.2 for the width of theambiguity band. In this case all the transients givingrise to defuzzified output values falling, for example,in the interval (1.6, 2.4) are classified as due to theforcing function number 2.

Concerning the misclassifications, most of them oc-cur due to mild forcing functions which generate sig-nals characterized by small variations with respect totheir initial values: in this case the generating forcingfunction remains undistinguishable. Then, we decidedto exclude from the analysis the patterns characterized


0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

N.Mfs

% E

rror

Fig. 8. Classification error vs. number of MFs with a thresholdS∗th = 0.005.

by almost constant input signals. To do this, we char-acterized each signal by a classificability thresholdSth:

Sth = smax − smin

s̄(12)

wheresmax, smin and s̄ are the maximum, minimumand average value of the signal in the transient, re-spectively. Transients such that all the signals haveSth indexes less or equal than a pre-defined thresholdS∗

th are considered unclassifiable in the sense that ei-ther they are too small for affecting the measured sig-nals or because the measured signals are insensitive tothem.

A set of 1000 new transients, different from thoseused in the training phase although still generated bysigmoidal forcing function of the kind ofEq. (11)sampled at random from the ranges ofTable 1, wasused to test the classification capabilities of the de-vised fuzzy classifier. In case of a thresholdS∗

th =0.005, we obtained 0.58% misclassifications (0.45%errors and 0.13% ambiguities), while forS∗

th = 0.009the misclassifications increased to 2.24%, but all ofthe “don’t know” kind (no classification errors wereobserved). In the latter case, the distribution of the de-fuzzified outputs around their corresponding true inte-

gers is reported inFig. 7. Notice that these thresholdvalues might well be within the statistical fluctuationsof the signals, here not considered.

An important parameter for the performance ofthe classifier is the number of membership functionsused in the partitioning of the inputs’ universes ofdiscourse.Fig. 8 refers to the caseS∗

th = 0.005 andreports the fraction of classification errors made bythe fuzzy engine as a function of the number of mem-bership functions used to partition the inputs. Wecan see that the error attains its minimum around 40.To avoid error fluctuations, present in the range of20–40, we used for our classifier, 51 MFs as abovesaid.

5. Conclusions

The early identification of the transients occurringin a nuclear power plant is a difficult task because ofthe high complexity and non-linearity of the systembehavior. In this paper, we have presented a methodbased on a fuzzy logic modelling tool. The proposedapproach stems on a method for automatically generat-ing the FRB starting from plant input/output data pairs.During operation, the identified set of rules provides


the mapping from measured data to system behaviorrequired for the transient identification. The procedureallows taking into account also the possibility of someambiguities in the classification.

An example of application of the method has beenprovided with respect to the task of identifying whichone of four chosen forcing functions is responsible forthe onset of given transients generated by simulationof the operation of an AP600 reactor: based on thesignals measured in the first 7 s after the beginning ofthe transient, the responsible forcing function is read-ily classified with a small percentage of errors. Indeed,filtering out those signal patterns characterized by nor-malized variations below 0.005, approximately 99.4%of transients are correctly classified, with most of theremaining transients being assigned to the wrong forc-ing function; lowering the signal variation thresholdto 0.009 leads to close to 98% of the transients beingcorrectly classified, the remaining 2% of the patternsbeing classified in the ‘don’t know’ category.

References

Basile, D., Beghi, M., Chierici, R., Salina, E., 1987. COBRA-EN,an upgraded version of the COBRA-3C/MIT Code forthermal–hydraulic analysis of LWR fuel assemblies and cores,Synthesis Srl. Report 1010/1, Milan.

Fuzzy Sets and Systems, 1995. Special Issue on Fuzzy LogicApplications in Nuclear Science and Technology. vol. 74.

ENEL/DSR/CRTN Report, 1992. Westinghouse AP600 Data forReactivity Accident analysis.

ENEL/DSR/CRTN, 1993. Generation of AP600 equilibrium corenuclear data library. Report. vol. 1.

Salina, E., Alloggio, G., Brega, E., 1994. QUARK: a computercode for the neutronic and thermal–hydraulic space- andtime-dependent analysis of light water reactor cores byadvanced nodal technique. ENEL/DSR/VDN Report.

Smith, K.S., 1979. An Analytical Nodal Method for Solving theTwo-Group, Multidimensional, Static and Transient NeutronDiffusion Equations, PhD Thesis, Department of NuclearEngineering, Massachusetts Institute of Technology.

Wang, L.-X., Mendel, M., 1992. Generating fuzzy rules by learningfrom examples. IEEE Trans. Sys., Man Cyber. 22 (6), 1414–1427.

Zadeh, L., 1965. Fuzzy Sets, Information and Control 8, 338–353.

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Fuzzy identification of transients in nuclear power plants