omplex dynamial systems applied to

the primary iruit of a nulear

power plant

Ph.D. THESIS

Csaba Fazekas

Supervisors: Prof. Katalin Hangos

Prof. György Kozmann

University of Pannonia

Faulty of Information Tehnology

Information Siene Ph.D. Shool

2008

dynamial systems applied to the primary

iruit of a nulear power plant

Értekezés doktori (PhD) fokozat elnyerése érdekében

Írta:

FazekasCsaba

Készült aPannonEgyetem InformatikaiTudományok DoktoriIskolája keretében

Témavezet®: Dr. Hangos Katalin

Elfogadásra javaslom(igen / nem)

(aláírás)

Dr. KozmannGyörgy

Elfogadásra javaslom(igen / nem)

(aláírás)

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a BírálóBizottság elnöke

A doktori (PhD)oklevélmin®sítése ...

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Az EDT elnöke

Komplex dinamikus rendszerek modellezése és modell-

kalibráiója atomer®m¶ primerkörére való alkalmazás-

sal

Adisszertáióban aszerz®egy komplex dinamikairendszernek, aPaksiAtomer®m¶

primerkörének matematikai modellezésével és a modell kalibráiójával foglalkozik.

Akifejlesztett modellminimálisegyáltalánosítottmérnöki értelemben. Amegalko-

tott modellavalósrendszerekr®lalkotott zikai tudásalapjánkészült, azaza valós

rendszerekben lejátszodó folyamatokat leíró egyenletek alapján írta fel a modell-

egyenleteket. A modellalkotás során a szerz® az ún. hétlépéses modellezési eljárást

követte.

A minimális primerköri modell a szükséges minimális elemekb®l lett felépítve.

Ez a módszer a valós rendszer el®zetes analízisét kívánja meg, mely során a szük-

séges m¶ködési egységek és azok f®bb dinamikái azonosításra kerülnek. Ezen f®bb

dinamikák modellezésével és integráiójával kapható meg végül az ered® modell. A

végeredményül kapott modellanemlineáris modellek saládjábatartozik.

Egyteljesmodellmegalkotásáhozismernikellamodellbenszerepl® paraméterek

értékeit is. Ha ezek az értékek el®re nem ismertek, akkor a valós rendszeren való

mérésekb®l modellkalibráióseljárás során besülhet®ek meg. A modellkalibráiós

eljárás egyszer¶sítéseérdekében a szerz®amodellt dekomponálta,ígyegy mérésb®l

kevesebb paramétert kellmeghatározni. A szerz® érzékenység vizsgálat alapján vá-

lasztotta ki a besülend® paramétereket és a kalibráióhoz szükséges méréseket. A

mért adatok a paraméterbeslés el®tt el®feldolgozásra kerültek, mely megnövelte a

kalibráiójóságát. Mivel a modellnemlineáris, így azoptimizáión alapuló Nelder-

Mead szimplexalgoritmuskerült felhasználásra,mintparaméterbeslésialgoritmus.

Akalibráiójóságaabesültértékekkondeniaintervallumainakvizsgálatábóllett

megállapítva.

Az el®z®leg megalkotott modellt módosította éshibaeseményekkel kib®vítette a

szerz®,hogyegyadotthiba(PRISE)érzékeléséreajánlottbiztonságieljárás formáli-

sanverikálhatólegyen. Ahibákdiszkréteseményekkéntlettekmodellezve,amelyek

indikátorváltozókéntjelennekmeg azegyenletekben. Ígyakiterjeszetett modellegy

hibrid típusúmodelllett.

Modeling and model alibration of omplex dynami-

al systems applied to the primary iruit of a nulear

power plant

The primarygoal of this dissertation isto onstrut andalibrate the onentrated

parameter,dynamiandminimalmodelsofaomplexdynamisystem: theprimary

iruit of a nulear power plant. The model building is based on rst engineering

priniples. Duringmodelonstrutiontheseven-stepmodelingproedureisfollowed.

The minimal (in a generalized sense) primary iruit model is omposed from

the minimal elements. It requires the analysis of the real system to identify its

operatingunits andtheir main dynamis. Thenthe model an be onstrutedfrom

theintegrationofthemodelsofthemaindynamis. Theresultedmodelisnonlinear.

Sine a omplete model is required for any appliation the model alibration,

using measured data from the real system, has to be performed on the developed

models. Themodelis deomposedtomakethe modelalibration easier. Sensitivity

analysiswithrespettomodelparametersisperformedheretohoosetheestimated

parameters and to determine the required measurements. The measurement data

are preproessed to assure the quality of the estimates. Sine the model is nonlin-

ear an optimization-basedparameter estimation method,the Nelder-Mead simplex

method is applied. The quality of the estimates is also investigated determining

their ondene intervals.

Thedynamimodelisextended withfaulteventsforformalveriationpurpose,

where a given safety proedure had to be veried. The faults are modeled by dis-

reteeventsandrepresentedbyindiatorvariables. Therefore theextended primary

iruitmodel isa hybrid one.

Modelado y alibraión modela de sistemas omple-

jos dinámios apliados al iruito primario de una

entral nulear

El objetivo de esta disertaión es de onstruir y alibrar los modelos. Estos son

los modelos parámetro onentrado, dinámios y mínimos de un sistema omplejo

dinámio: el iruito primario de una entral nulear. El ediio de modelo está

basado en primeros prinipios de la ingeniería. Durante la onstruión modela el

proedimientode modelado de siete-pasos es seguido.

El modelo mínimode iruito primarioesompuesto de los elementos mínimos.

Esto requiere el análisis del verdadero sistema para identiar sus unidades de

operaiones ysudinámiaprinipal. Entonesel modelopuedeser onstruido de la

integraión de los modelosde ladinámiaprinipal. El modelo pasadoesno lineal.

Yaquerequierenun modelo ompletoparaualquier uso,laalibraión modela,

usando datos de medida del verdadero sistema, tiene que ser realizado sobre los

modelosdesarrollados. Elmodeloesdesompuesto parahaerlaalibraiónmodela

más fáil. El análisis de sensibilidad en lo que onierne a parámetros modelos es

realizado a esogió los parámetros estimados y determinar las medidas requeridas.

Los datos de medida son proesados para asegurar la alidad de las estimaiones.

Yaqueelmodeloesnolineal,un métodoabasede optimizaión,elmétodosimplex

de Nelder-Mead es apliado. La alidad de estimaiones también es investigada

determinando sus intervalosde onanza.

Elmodelo dinámioesampliadoon aonteimientosde defetopara elobjetivo

de veriaión formal. Un proedimiento de seguridad de PRISE tiene que ser

veriado. Losdefetossonmodeladosporaonteimientosdisretosyrepresentados

porvariablesde indiador. Elmodelode iruitoprimarioon defetoeselhíbrido.

This thesis summarizes the ontributions of my researh work for obtaining Ph.D.

degree in Information Siene at the Department of Information Systems at the

University of Pannonia. The sienti part of the studies was mostly undertaken

at the Department of Information Systems at the University of Pannonia and the

Systems and ControlLaboratory, Computer and AutomationResearh Institute of

the HungarianAademy of Sienes.

This workwould never have been written withoutthe help, ontinuous support

andenouragementofseveral people. Firstofall,Iwanttoexpressmy sineregrat-

itudetomy supervisors,Professor KatalinHangosand ProfessorGyörgyKozmann,

for their exellent and patient guidane throughout my studies.

I would also like to thank Professor József Bokor, the head of Systems and

Control Laboratory for assuring the researh possibility. I would like to express

my sinere gratitude toDr. GáborSzederkényi for his exellent guidane and joint

work. Iamalsogratefultomyolleagues,Dr.TamásBartha,DávidCsersik,Attila

Magyar, Dr. Erzsébet Németh, Tamás Péni, Barna Pongráz, Gábor Rödönyi, Dr.

István Varga, Zsuzsanna Weinhandl for the joint work. Furthermore, I thank all

the people at the Systems and Control Laboratory for the helpful and supporting

environment.

I also would liketo thank all the people at the Department of Information Sys-

tems at the University of Pannonia (Gábor Balázs, Tibor Dulai, Balázs Gaál, Dr.

Feren Leitold, SzilárdJaskó, András Jókuthy, Dr. Zoltán Juhász, KrisztiánKunt-

ner, Anna Medve, Emil Mógor, Dániel Muhi, Krisztián Poós, Szabols Póta, Dr.

KatalinTarnay,Shné TamásnéTóthMária,Dr.IstvánVassányi, BalázsVégs®) for

the equally helpful and supporting environment.

I alsowould like tothankallthe peopleatthe Department ofBioengineeringat

the Researh Institute for Tehnial Physis and Material Siene: Andrea Bolgár,

Kristóf Haraszti and KrisztinaSzakolzai.

Finally,I amgrateful to my familyfor supporting my studies in many ways for

suha long time.

1 Introdution 1

1.1 Motivation and goals . . . 1

1.2 The layout of this thesis . . . 2

2 Basi notions 4 2.1 The seven-step modelingproedure . . . 4

2.2 The omplexity of dynamimodels . . . 7

2.3 Identiability . . . 8

2.4 Model alibration . . . 9

2.5 Fault modelingand analysis . . . 12

2.6 The primaryiruit of a nulear power plant . . . 13

2.6.1 The struture and the operating units inthe primaryiruit . 14 2.6.2 Literature review onmodeling nulear power plants . . . 16

2.6.3 Literature review onthe ontrol of nulearpower plants. . . . 17

2.6.4 PRISEand the appliationof the oloured Petri net. . . 19

3 Model development 20 3.1 Model buildingof the primaryiruit . . . 20

3.1.1 Modeling goal . . . 20

3.1.2 Overall modeling assumptions . . . 21

3.1.3 Measured signals . . . 22

3.1.4 The simpliedoperatingunits and theirdynami models . . . 22

3.1.5 The state-spae model of the system . . . 31

3.1.6 Desriptionof the simulator . . . 33

3.2 Summary . . . 34

4 Model alibration 35 4.1 Calibration of the primary iruit model . . . 35

4.1.1 Measured variables . . . 36

4.1.2 Applieddata . . . 36

4.1.3 Model deomposition . . . 37

4.1.4 Sensitivity analysis . . . 37

4.1.5 Identiability analysis . . . 39

4.1.6 Estimationmethod . . . 39

4.1.7 Results. . . 40

4.1.8 Model integration and veriation . . . 51

5 Fault modeling in the primary iruit 55

5.1 The PRISEevent and itssafety proedure . . . 55

5.1.1 The PRISEfault event and itsproess onsequenes. . . 55

5.1.2 The PRISEsafety proedure . . . 56

5.2 Modelingfor safety proedure veriation. . . 58

5.2.1 Operatingunits and balane volumes . . . 59

5.2.2 Simplifyingassumptions . . . 60

5.2.3 Continuous time model equations . . . 61

5.2.4 Safety proedure onditions . . . 65

5.2.5 The properties of the dynami engineering model . . . 66

5.3 Summary . . . 67

6 Conlusion and future work 68 6.1 New results . . . 68

6.2 Diretions for future researh . . . 70

6.3 Publiations . . . 70

6.3.1 Publiationsdiretly related tothis thesis . . . 70

6.3.2 Otherpubliations . . . 71

A I

A.1 Parameters and variables of the primary iruit model. . . I

A.2 Identiability of the nonlinear state-spae system . . . III

A.2.1 Bakground ondierentialalgebra . . . III

A.2.2 Denitionof identiability . . . IV

A.2.3 Identiability and harateristi sets . . . V

A.2.4 Identiability with a priori informationon initialonditions . VII

A.2.5 Identiability analysis of the primary iruit model . . . .VIII

A.3 Results of parameter estimation of the primary iruit. . . .XIX

A.3.1 Reator . . . .XIX

A.3.2 Liquidin the primary iruit. . . .XXII

A.3.3 Steamgenerator. . . .XXIII

A.3.4 Model integration . . . .XXIV

A.4 PRISE parameters . . . .XXVI

Bibliography XXVI

Introdution

Amodelisanimitationoftherealityandamathematialmodelisapartiularform

of the representation. In the proess of model buildingwe translate our real world

problemintoanequivalentmathematialproblemwhihwesolveand thenattempt

to interpret. We do this togain insight into the original real world situation or to

use the model forontrol, optimization, safety et. studies.

The models we deal with inthis thesis are mathematial. Mathematialmodels

attempttoapture, in the formof equations,ertain harateristisof a system for

aspeiuse (modelinggoal) of that model.

Themodelingexeriselinkstogetherapurposewithasubjetorphysialsystem

and the system equations whih represent the model. A series of speial "pseudo"-

experiments an then be applied to the model in order to answer questions about

the system from the point of viewof the purpose.

The appropriatelevelof detailsofmodelingisdeterminedby the modelinggoal.

A frequentmistake isto assumethat a moredetailed model isneessarily superior.

Beausemodelsatasbridgesbetweenlevelsofunderstanding,theymustbedetailed

enough to make ontat with the lower level yet simple enough to provide lear

results at the higher level. It means that a good model satises the modeling goal

and it is as simple as possible to make the understanding easier. Suh a model is

alled minimal model.

Thus,appropriatemodelshelpusunderstandtherealworld. Theyanbeapplied

in dierent kinds of sienti areas thereby modeling and simulation support the

researh of other areas,suh asnuleartehnology.

1.1 Motivation and goals

The goal of this thesis is to onstrut minimal models, in a generalized sense, of

speialomplexsystemsbased onrstengineeringpriniples inordertoapply these

models for system analysis, parameter estimation, ontroller design and fault anal-

ysis. The approah of model building applied in this thesis is to use our physial

knowledge about the system to onstrut its model. This means that the dynami

equations originate from onservation balanes, Newton's law, et. omplemented

by suitable algebraiequations.

nami and minimal models are needed. A speial model onstrution proedure

(the model omposing from minimal elements) is presented on the primary iruit

of anulearpower plant.

The primary iruitof a nulear power plant transfers theheat generatedinthe

reatortotheseondary iruitandoolsthereatorontinuously. Itisonstruted

byhumans,thus, itsoperationis-inpriniple-ompletelyknown. Themoderniza-

tion of nulear power plants requires a omplete redesign of some of its parts, suh

as ontrollers of the primary iruit. To design good ontrollers we need a quite

aurate, dynami and yet simple model of the primary iruit. Therefore, the aim

here has been to develop a minimal, in a generalized sense, primary iruit model

for ontroller design purposes. Parts of the model must have physial meaning and

the modelmust represent all of the main dynamis of the real system.

A omplete model is required for any appliation. It means that not just the

model equations but the values of model parameters must beknown. Howeverit is

veryrare toknowapriorithe exatvalues ofeahoftheparameters. Therefore,the

model alibration proedure also has to be ompleted during modeling that yields

theestimatedmodelparametersusingmeasureddatafromtherealsystem. Itisnot

a trivial exerise for nonlinear models. To build omplete primary iruit model,

an important aim has been to realize the model alibration proedure for nonlinear

model.

In ase of highly safety-ritial and omplex systems, suh as a nulear power

plant,theveriationand validationofthe safety proedures isofgreatimportane

beause it an provide the orret responses of the system to the possible faults.

A nulear power plant is alled a safe system, if it an respond orretly to the

faultsin a predened time limit. Beause of the large number of variables and the

omplexity of the plant and its dynamial behavior, however, one needs to apply

formalmethodsforthistask. Therefore, theaimherehasbeentodevelopa primary

iruitmodelforfaultanalysisandveriationofthe reommendedsafetyproedure.

1.2 The layout of this thesis

Thethesisonsistsof5hapters(notontainingthisIntrodution)andanAppendix.

Eah hapter begins with a motivation and introdution part that desribes the

loal problem statement and the aim of the orresponding hapter. The hapters

are nished with a summary where the loalonlusions are drawn. The layout of

the thesis is desribed below.

Chapter 2 This hapter gives a briefsummary about the model building,model

redution,modelalibrationandfaultmodelingtehniques. Inthishapterthebird-

eye view desription of the investigated system (the primary iruit of the nulear

power plant)an befound with literature reviews about their modeling.

Chapter 3 The minimalmodel of the primary iruitof the nulear power plant

isdevelopedhere. Theminimalmodelisreatedfromminimalelements,identifying

ing priniples. Then the obtained model equations are transformed into intensive

and state-spae form.

Chapter 4 In this hapter model alibration isompleted on the primary iruit

model. The model is nonlinear, therefore nonlinear parameter estimation methods

are applied. First, sensitivity analysis is ompleted to determine the estimated

parametersandmeasureddatathataresuitablefortheparameterestimation. Then,

themeasureddataareinvestigated,andtheirsignalproessingisdesribed. Finally,

theparameterestimationproeduresareompletedandtheirsolutionsaredesribed.

Chapter 5 In this hapter the faultmodelingof the primary iruit ispresented.

It is performed by the modiation and extension of the primary iruit model

developed in Chapter 3. The aim of the fault modeling has been to investigate

(verify and validate)a safety proedure dened by the experts in the Paks Nulear

PowerPlant (NPP).

Chapter 6 This hapterontains the nalonlusions ofthe thesis and desribes

the possible diretions of the future researh.

Appendix A This hapter ontains simulation results and data tables, suh as

variable and parameter lists, their values et.

Basi notions

Inthishapterthestepsofmodelbuilding,modelalibrationandfaultmodelingare

summarized. Thereafter, the investigated omplex system model lass, the nulear

power plantmodels are briey introdued.

2.1 The seven-step modeling proedure

Themathematialmodelingofsystemsandtheanalysisofthemodelsleadtomath-

ematial problems of various types. It is useful to formulate these mathematial

problems in a formal way speifying the inputs to the problem, the desired output

or question to be solved and indiate the proedure or method of solution. This

formaldesription of amathematial problemis alledproblem statement.

The modeling goal speies the intended use of the model. The modeling goal

hasamajorimpatonthelevelofdetailandonthemathematialformofthemodel

whih willbe built. Thus, amodel is determinedby the system itdesribesand by

itsmodeling goal.

Constrution of mathematial models of omplex systems is generally based on

rst engineeringpriniples, i.e.the dynami equationsoriginatefromthe basi laws

of physis, hemistry, biology, et. suh as onservation balanes, Newton's law,

reationkinetis.

A reommended model development proedure has been developed [22, 43, 44℄

and used here that onsists of seven steps:

1. Problemdenition. Thisdenes thesystem, themodelinggoalandthevalida-

tion riteria. The denition of system means the denition of system bound-

aries and the way of interations between the system and its environment

together with the desription of the internal struture of the system itself.

Any model is developed for a spei use or uses that is determined by the

modelinggoal. Validationriteriadetermine when the modelingdevelopment

yle should terminate.

Theproblemdenitionxes thedegreeof detailrelevanttothemodelinggoal

andspeiese.g.thepossibleinputsandoutputs,thesales,andtheneessary

rangeand aurayof the model.

*Problem * *definition*

*Evaluate the * *problem data*

*Identify * *controlling factors*

*Construct the * *model equations*

*Solve the * *models*

*Verify the * *solution*

*Calibrate and * *validate the model*

Figure1: Seven-step modeling proedure.

2. Identify ontrolling fators. Here the proesses and phenomena are olleted

taking plae in the system relevant to the modeling goal, suh as operating

units, balane volumes, physial and hemial laws, onservation laws, heat

transfers,ows et.

3. Evaluatethe problem data. The aprioriknown data andparameter valuesare

investigated here that are used in the model e.g. we determine the values of

measured variables and parameters together with their preision. This step

ontainstheinvestigationwhatdataofthesystemaremeasurableanduniquely

determinesthe model inputsand outputs based onthemodelinggoaland the

measurements.

4. Construt the model equations. The model equations are developed in this

step that an be either dierential oralgebraiones. Its sub-steps are:

4.1 Determine the partsof the system (subsystems) where the equationsare

established, suh asbalane volumes in the proess system (balane vol-

ume determinesthe regioninwhihtheonserved quantity isontained).

Dene the system and the subsystem boundaries.

uniquely desribe the behavior of the investigated subsystem from the

modeling goal point of view.

4.3 Establish the dierentialequations (e.g. balane equationsthat desribe

the onservation balanes in the balane volume) ineah subsystem ap-

plying the identied ontrolling fators of the subsystem.

4.4 Developtheonstitutive(algebrai)equationsofeahsubsystemapplying

the identied ontrollingfators of the subsystem.

4.5 Dene the onstraintsand the onnetionsbetween the subsystems.

5. Solvethemodels. Asolutionproedureisfoundandimplementedforthemodel

equations in this step. The solution method depends on the mathematial

form of the model equations. We must ensure that the model is well posed,

i.e. degrees of freedom are satised. We also try to avoid ertain numerial

problems suhas high index system.

6. Verify the solution. Here we determine whether the model behaves orretly.

Model veriation inludessyntax hekingandsemantisheking,aswellas

the well-posedness heking of the model inmathematialsense, and analysis

of omputationaland dynami properties.

It is possible or required that some exerises of the model veriation are

performed before the model solution beause their results an inuene the

solutionmethodand algorithm.

7. Calibrate and validate the model. In this nal step we estimate the unknown

parts of the model from measurements (alibration) and hek the quality of

the resultant model against independent observation or assumption (valida-

tion). The atual validationmethod strongly depends on the system, on the

modeling goal and on the possibilities of getting information for validation.

The validationan be performed by e.g. omparing the model behavior with

thesystem behavior and/oromparingthe model diretlywith the data. The

validation results indiate that the developed model is proper or how to im-

prove it.

Model developmentisiterativeinitsnature. One must returnandrepeat anearlier

step in ase of any problems or if the urrent model does not satisfy the modeling

goal.

This general model development proedure an be applied to develop mathe-

matialmodel for any kind ofphysial systems. However, this proedure should be

extended or modied a little if the required model has speial properties, suh as

inherentmulti-salenature. Wehaveusedamodiedversionoftheseven-stepmod-

elingproedure[22℄todevelopabiomehanial,multi-salearmmodelfordiagnosti

purposes [O3, O6℄.

Asit hasbeen mentionedbefore, developingamodel isaniterativeproedure. The

nal and good model must satisfy the modeling goal and should be as simple as

possible. The simplerthemodel,the easieritsappliation. Minimalrepresentations

are known to have no redundant elements. Thus an engineering model should be

minimalwith respet to itsperformane, size and other measures.

Models are alled funtional equivalent models if all of them satisfy the same

modeling goal [66℄. Other denition is that two models are equivalent if they give

rise to the same input-output behavior. This denition implies that the modeling

goal an nowbegiven by aurately presribing the desired input-outputbehavior.

If one wants to ompare funtionally equivalent models with respet to their

simpliity, a suitable quality or size norm that reets simpliity should be rst

dened. One andeneinteger-valuedindieswhihharaterizethe generalsize of

a set of funtionally equivalent proesses, i.e. the generalized size index assigns an

integertoeahmodel. Theseouldbethedimensionofthestatevariable,themodel

omplexity measure or nonlinearity measure, the relative degree of the model, the

dimension of the ontrollability subspae ordistribution, et.

Severaldierentsizeindies(e.g.

## ξ _{i}

^{,}

## i = 1, . . . , n

^{)}

^{ould}

^{be}

^{dened}

^{for}

^{the}

^{models}

thatare olleted inavetor

## ξ =

## ξ 1 , . . . , ξ n

## T

. Inthis asethe size norm isdened

asavetornormof

## ξ

^{,}

^{suh}

^{as}

## ν . (ξ) = |ξ| .

^{.}

^{Based}

^{on}

^{the}

^{notion}

^{of}

^{size}

^{norm,}

^{one}

^{an}

ompare andorder twofuntionallyequivalentproess models: one of themodels is

"smaller"or more simple than the another one if itssize norm is less than the size

norm of the another one.

Therefore, basedontheorderingofthefuntionallyequivalentmodelsamodelis

alledminimal inthis abovegeneralizedsense ifitssize normistheminimalamong

the size norms of the other funtionally equivalent models [66℄.

In this thesis the simpliity of the model, i.e. its size norm is measured by the

numberofstatevariablesand bysomekindofmeasure ofnonlinearityofthemodel.

We use the word "minimal" from an engineering point of view. It means that

the model is minimal in this generalized sense, if it omputes all of the important

variableswiththerequiredpreision,fromthemodelinggoalpointofview, assimple

aspossibleand doesnot ontain any unneessary orredundant elements.

Model redution

Themost eetivewayoffousingonapartofadynamisystemrelevanttoour

purposes is to apply model redution or model simpliation tehniques. Minimal

models inthe abovegeneralized sense an beonstruted by twomethods:

## •

^{Iterative}

^{model}

^{redution:}

^{Here}

^{one}iteratively redues the number of state variables as far as the auray of the redued model satises the modeling

goal and the redued model beomes the minimal one. Suh a method was

appliedduring the development of the multi-salearm model [O6℄.

## •

Construtingompositemodelsfromminimalelements: Usingsystemanalysis the mainoperating unitsof the system are determined that inuene the sys-tem dynamis from the aspet of the modeling goal. Then the proesses and

the system behavior (from the point of view of the modeling goal). Finally,

the mathematialmodels of the determined mehanisms and proesses of the

operatingunits are onstruted.

Ifthe overall model doesnot satisfy the modelinggoal then the model an be

extended by additionalelementsthat are originallynegleted.

2.3 Identiability

Identiability onernsuniqueness of the modelparameters determinedfrominput-

output data, under ideal onditions of noise-free observations and error-free model

struture. Identiabilityisafundamentalprerequisiteformodelidentiation(model

alibration) and qualitativeexperimentaldesign. It allows usto distinguish among

those experiments that annot sueed and those that mightensure the estimation

of the unknown parameters.

In ase of linear models the identiability analysis is well understood. Some of

the most popular methods are the transfer funtion method, the transfer funtion

topologial method, the modal matrix method and the similarity transformation

method [13, 36℄. However, many hallenges still exist in the area of non-linear

identiability. Several approahes for identiability analysis have appeared in the

literature. When thereexists aninput-outputoperator,the loalstateisomorphism

theorem [110℄ allows us to test identiability. Other methodsare based on the lin-

earizationproedure[37℄orthe seondhandmemberidentiabilityapproah[30℄or

the power series expansion [97℄ of the solution, or on the similarity transformation

approah [25, 111℄. Reently dierential algebra tools have been applied to study

identiabilityofnonlinear systems (nonlinearityappears asapolynomialoraratio-

nal funtion in the right side of the state equations) [12, 70, 72, 102, 117℄. These

methodsallexploit theharateristiset ofthe dierentialidealassoiatedwiththe

dynami equationsof the system (see details insetion A.2).

However, it has been observed that even with the new algorithms, problems

an arise in testing identiability for systems started atgiven initialonditions. In

ordertoguarantee theorretnessoftheidentiabilitytestbasedonaharateristi

set, one has to hek that some strutural onditions hold, whih are related to

the spei initial ondition. A natural strutural ondition whih guarantees the

validity of the identiability test is the aessibility of the system from the given

initialondition[12,102℄. Whenthesystemisnotaessible, anewdierentialideal

desribing the solutions of the system has to be onstruted (see details in setion

A.2).

Globalidentiabilityoflinearandnonlinearmodelsisdiulttotestsine,what-

ever the method used, itrequires solving a system of nonlinear algebraiequations

whose omplexity inreases very fast with the number of unknown parameters, the

numbers of input-outputvariables, the degree ofnonlinearity and the model order.

Model alibration plays an important role in the seven-step modeling proedure

[69℄ inuening more steps (see the feedbaks in Fig. 1)and itmay generate anew

iterationduring the model building.

Inpratialases,weoftenhave aninompletemodel afterthe rstfoursteps of

the seven-step modeling proedure [43℄. This is beause we rarely have a omplete

model together with all the parameter values. We want to obtain these model

parameters using experimental data. Beause measured data ontain measurement

errorsweanonlyestimatetheunknownorpartiallyknownmodelparameters. This

sub-step is alled model alibration.

The sub-steps of model alibrationare [43℄:

1. Analysisof model speiation.

2. Resampling of measured data (and the model if it isneeded).

3. Dataanalysis and preproessing.

4. Model parameter estimation.

5. Evaluationof the quality of the estimate.

Analysis of model speiation

Analysisofmodelspeiationdetermineswhihparametersneedtobeorshould

be estimated.

Sensitivity analysis with respet to model parameters is alsoperformed here to

investigate the eet of their hanges on the model output and behavior. This

tool helps to selet the important parameters with respet to the urrent applia-

tion goal. The model parameters an be divided into three groups aording to

the knowledge and ondene of their values, as "known", "partially-known" and

"unknown". Parametri sensitivity assessment isperformedontheparameterswith

partially-known and unknown values. The goal is to identify members that have

negligible impat on the model output and to remove them from the andidate

parameters to be estimated by simply setting them to an arbitrary value in their

expeted range.

Data analysis and preproessing

The quality of the estimates depends ritially on the quality of the measure-

ment data. Therefore, before making any use of the measurements for parameter

estimation, its data quality is assessed to qualify whether it is appropriate for this

purpose and remove data of unaeptable quality. Almost every parameter estima-

tion method has assumptions with respet todata qualityand property.

Harmful deviations in the data might take the following forms: data with bias,

gross error, poor sampling,outliers, jumps, trends, measurement errors, et. Data

sreeningmethodsare appliedtohekthe measureddataquality. The mostsimple

and eetive way of data sreening is visual inspetions. This is done by plotting

the olleted set of measured data againsttime, frequeny orone another, et.

The model parameter estimation problemstatement is the following [43℄.

Given:

## •

^{A}parametrizedexpliit system model in the form

## y ^{(M} ^{)} = F (x, p ^{(M} ^{)} )

withthe model parameters

## p ^{(M)} ∈ ℜ ^{ν}

^{being}

^{unknown,}

^{the}vetor-valued inde- pendent variable

## x ∈ ℜ ^{n}

^{and}vetor-valued dependent variable

## y ^{(M)} ∈ ℜ ^{µ}

^{.}

## •

^{A}

^{set}

^{of}

^{measured}

^{data}

## D[0, k] = {(x(i), y (i)) | i = 0, . . . , k}

where

## y(i)

^{is}

^{assumed}

^{to}

^{ontain}measurement error and

## x(i)

^{may}

^{ontain}

additionalmeasurementerrors.

## •

^{A}

^{suitable}

^{signal}

^{norm}

## k.k

^{to}

^{measure}

^{the}

^{dierene}

^{between}

^{the}

^{model}

^{output}

## y ^{(M)}

^{and}

^{the}

^{measured}independent variables

## y

^{to}

^{obtain}

^{the}

^{loss}

^{funtion}

^{of}

the estimation:

## L(p) = ky − y ^{(M)} k

Compute: An estimate

## P ˆ ^{(M)}

^{of}

## p ^{(M} ^{)}

^{suh}

^{that}

## L(p)

^{is}

^{minimal,}

^{i.e.:}

## ky − y ^{(M)} k → min

One of the most popular methods for parameter estimation of linear systems

and linear-in-parameters systems is the least squares (LS) estimation method. It

reeivesits namefrom the square signal norm

## k · k 2

^{whih}

^{is}

^{applied}

^{for}

^{omputing}

the lossfuntion:

## L(p) = ky − y ^{(M} ^{)} k 2

One of the advantages of the LS method is that the estimation values an be

expressed from the measured data analytially [43℄. Therefore, these methods an

be analyzed muh easier, e.g. the ovariane matrix of the estimates an also be

omputed analytiallyand the quality of the estimates an beinvestigated easier.

Inaseofnonlinear modelsornonlinear-in-parameters modelstheparameteres-

timationproblemis usuallysolved by asuitableoptimizationalgorithmsuhasthe

Levenberg-Marquardt method or the Nelder-Mead simplex method [43℄. It means

thatthereisnogeneralsolutionmethodforparameterestimationofnonlinearmod-

els. However, inthis asethe estimatehaslostitseasieranalyzableproperties. The

problem is muh more diult if there are onstraints for the parameter values to

be estimated.

TheNelder-Meadsimplexalgorithm [88℄isone ofthe mostpopularoptimization

methods applied for parameter estimation for nonlinear models. It is a heuristi

method to nd the extreme values of an arbitrary nonlinear funtion, i.e. it is not

guaranteed that all of the extreme values of the funtion are found and that the

global optimum is obtained. Therefore, it is ruially important that the initial

of this method needs to be veried thoroughly. The reason for its popularity is

that it does not need any analyti or numeri gradient input information of the

partiular funtion. Moreover, the solutionis obtained even insuh ases whenthe

funtion to be minimizedor maximizedis not smooth or highlydisontinuous. On

the other hand there are only few theoreti results about its onvergene even for

low dimensionalases [65℄.

Thebriefalgorithmofthe diretsearhmethod(e.g. Nelder-Meadmethod)isas

follows (see detailsin[23℄). If

## n

^{parameters}

^{of}

^{the}

^{model}

^{are}

^{going}

^{to}

^{be}

^{estimated,}

a non-degenerated simplex in the n-dimensional parameters spae is haraterized

by the

## n + 1

^{distint}

^{vetors}

^{that}

^{are}

^{verties}

^{of}

^{the}

^{simplex.}

^{It}

^{means}

^{that}

^{the}

algorithm starts from

## n + 1

^{initial}

^{values}

^{for}

^{the}parameters. The method nds a minimum of the error funtion by evaluating the error funtion at the point of

the verties of the simplex. At eah step of the searh, a new point in or near the

urrentsimplex isgenerated. The funtionvalue atthenew pointisomparedwith

the funtion values at the verties of the simplex and, usually, one of the verties

is replaed by the new point, giving a new simplex. This step is repeated until

the diameter of the simplex is less than the speied tolerane. The basi steps of

ndinga new vertex are the reetion step, the expansionstep and the ontration

step.

It is important to note that the simplex searh methodis similar to other non-

linearoptimizationmethodsinthat itdoesnot guaranteethat theobtained pointis

aglobalminimaof thefuntioninthe wholeparameterdomain. Therefore itisru-

iallyimportantthat theinitialparametervetorislose tothe globalminima. Itis

pratialtotunethe parameters manuallybefore startingtheoptimizationproess.

Wealsohavetonotethattheseoptimizationbasedmethodsantakeintoaount

the onstraints of the estimated values while LSmethodsannot.

Methods for evaluating the quality of the estimateThe ondeneinter-

valsofthe estimatedparametersgiveinformationaboutthe qualityof theestimate.

In the ase of linear-in-parameter systems, the ovariane matrix of the estimate

ontains information about the ondene intervals [69℄. The empirial ovariane

matrix an be omputed with the least squares (LS) method. In ase of linear-in-

parameter systems the model output (

## y(k|θ)

^{)}

^{an}

^{be}

^{written}

^{in}

^{the}

^{form}

^{[69℄:}

## y(k|θ) = θ ^{T} · φ(k)

^{(1)}

where

## θ

^{is}

^{the}

^{vetor}

^{of}

^{the}

^{estimates}

^{and}

## φ(k)

^{is}

^{the}

^{regression}

^{vetor}

^{in}

^{the}

## k ^{th}

time instane. The regression vetor ontains the measured signals. Based on the

LS method the empirial ovariane matrix is omputed by the following equation

[69℄:

## cov = 1/N X N

### k=1

## φ(k) · φ ^{T} (k)

## ! −1

(2)

where

## N

^{is}

^{the}

^{number}

^{of}

^{the}

^{time}

^{instanes.}

The eigenvalues and eigenvetors of the ovariane matrix ontain information

aboutthe ondeneintervalsoftheestimates. The eigenvetorsshowthediretion

orresponding to the eigenvetor shows the length of that axis. The shape of the

ontour plot of the ondene interval is lose to the irle if the orresponding

eigenvalues are lose toeah other.

However, inthease ofnonlinearparameter estimationtheestimated parameter

vetor does not have the nie statistial properties as in ase of linear parameter

estimation[43℄. Therefore wean onlyapplyanapproximativeapproahifwe want

to estimate the parameter ondene intervals for the estimated oeient. This

approahrequiresustodepittheerrorfuntion(

## e(p)

^{)}

^{as}

^{a}

^{funtion}

^{of}

^{the}

^{parameter}

## p

^{to}

^{be}

^{estimated.}

^{Let}

## e _{opt}

^{be}

^{the}

^{minimal}

^{value}

^{of}

^{the}

^{error}

^{funtion.}

^{Then}

^{the}

value ofthe errorfuntionorrespondingto the

## α

^{ondene}

^{level}

^{an}

^{be}

^{omputed}

using [43℄:

## e α = e opt

## 1 + v

## k − v F ^{−1} (v, k − v, 1 − α)

(3)

where

## k

^{is}

^{the}

^{number}

^{of}measurements (datapoints),

## v

^{is}

^{the}

^{number}

^{of}

^{estimated}

parameters and

## F ^{−1} (v, k − v, 1 − α)

^{is}

^{the}

^{inverse}

^{Fisher}distribution value for de- greeoffreedom

## (v, k − v)

^{and}

^{ondene}

^{level}

## α

^{.}

^{The}

^{ends}

^{of}

^{the}

^{ondene}

^{interval}

of the estimated parameter an be dened as the absissas of the error funtion

where the value of the error funtion is equal to

## e α

^{.}

We an obtain information from the shape of the ontour plots of the error

funtion, too. The estimation is of good quality if this funtion has got unique

minimum in the value of estimated parameters and the shape of ontour plot is

approximately irular. The more ellipti the shape of the ontour plot (i.e. there

isavalleyinthe errorfuntion)is,the more unreliablethe estimated values are. In

thisase thewidthofthe valleygivesusinformationaboutthe auray. The wider

the valley the lower the auray.

2.5 Fault modeling and analysis

Faultsgenerallyappear asdisrete events inthe mathematialmodelsbeausethey

havedisretenature,i.e.afaultoursordoesnotour. Howeverthemathematial

model of a physialsystem isontinuous in time in most of the ases. Therefore, a

mathematialmodel extended with faultsis generallya hybrid-type model.

Nulear power plants are highly safety-ritial and omplex systems where the

veriationandvalidationofthe safetyproeduresareofgreatimportane. Beause

of the large number of variables and the omplexity of the plant and its dynami-

al behavior, one needs to apply formal methods for this task. At the same time,

the dynamis of a nulear power plant is also hybrid innature aused by the state

dependent swithing modes and by the disrete ontrol and the safety proedures

themselves, therefore the methodology and tools for hybrid systems should be ap-

plied.

Theneedtoapplyformalveriationmethodsforsafetysystemsinsafety-ritial

areas have long been reognized [77℄. However, the majority of the safety-ritial

industrial studies do not onsider dynamial information for the formal analysis,

but restrit themselves to hazard and operability analysis (HAZOP) and/or fault

information,see e.g. [38, 57℄.

From the methodologialaspet, there are two entirely dierent approahes to

desribeandanalyzehybridsystemsfromformalveriationpointofview. Oneway

istoembedthe disrete valuedtime-dependentvariablesintoanexistingdynamial

model [113℄, for example into a state-spae model. The other way is to extend the

disrete event system tehniques [24℄ withthe ontinuous dynamialinformationin

the formofwaiting orexeutiontimestoget atimed automatonorPetri net inthe

simplest ase [116℄, or to dene some more or less simple dynamis assoiated to

eah state and/or state transition. Driven by the atual goal of modeling,analysis

and/or ontrol, further approximations an beor should be made totransform the

desription to a homogenous disrete event system model form [68℄. This allows

using, for example, the well-established methods for model analysis developed for

disrete event systems.

Our hoie for the desription formalism of the safety system is the Coloured

Petri Net (CPN) [50, 51, 52℄. CPN is an extension of Petri nets: most important

dierenes are that plaes an ontain oloured tokens (i.e. multi-sets) that an

symbolize thedata ontent indata owmodels,and thatCP nets arehierarhially

strutured using substitutiontransitions and subnets.

A CPNis a unifying modeling tool that allows modeling, and simulation-based

veriation of safety-ritial proedures in nulear power plants. It is one of the

most popular tehnologies for the formalveriation of safety proedure beause:

## •

^{CPN}

^{is}

^{one}

^{of}

^{the}

^{most}

^{useful}

^{tools}

^{for}

^{modeling}

^{and}

^{simulation}

^{of}

^{disrete}

event systems [50,51, 52℄,

## •

^{CPNs}

^{have}

^{been}

^{suessively}

^{used}

^{for}reliability analysis of hybrid systems [104℄,

## •

^{a}

^{proess}

^{model}

^{in}

^{a}qualitative dierential algebrai equation (DAE) form an be represented as aCPN [35℄,

## •

^{there}

^{exists}

^{a}

^{powerful}

^{tool}

^{[1℄}

^{that}

^{supports}

^{the}

^{modeling}

^{of}

^{our}

^{plant}

^{and}

^{its}

safety proedure in the form of a joint CPN and performs the veriation by

using CPNanalysis proedures.

2.6 The primary iruit of a nulear power plant

Nulearpowerplantsaresomeofthemostsafetyritialengineeringsystems. Their

proper operation, ontrol and maintenane are very important issues. To design

good ontrollers for nulear power plants we need its quite aurate and simple

model. Sinethere are pressurizedwater reators in thePaksNulear PowerPlant,

Hungary,only this type of reatorhas been investigated.

A nulear power plant is a omplex system beause a great number of dierent

proesses take plae there suh as thermo-nulear proess, liquid and vapor ow,

heattransfer,evaporation,eletriitygeneration,ooling,et. Its omplexityisalso

inreased by the multilevelsafety mehanism.

ation of the power plant more safe and more eetive. This goal is supported by

two importantfators: rstly, the signiantdevelopment of proess modeling and

ontrol theory in the last deades (see e.g. [42, 43, 49, 89, 112℄) and seondly, the

improvingqualityof thehardware and software environment[75℄ providingthene-

essaryamountofmeasureddata. However, thediagnostiandontrolsolutionsthat

were designed and implemented atthe buildingtime ofthe powerplantusually did

not take intoonsideration the detailed nonlinear dynamis of the operating units.

This means that there isa lotof spae for the improvementof ontrolloops.

2.6.1 The struture and the operating units in the primary

iruit

One of the most importantparts of the nulear power plant isthe primary iruit.

The goal of the primary iruit is to transfer the heat generated in the reator to

the seondary iruitand to oolthe reatorontinuously.

Fig. 2 shows the main operatingunits (the reator, the steam generator(s), the

main irulating pump(s), the pressurizer) and their onnetions. The sensorsthat

provide on-line measurements are also indiated in the gure by small blak ret-

angles. The ontrollers are denoted by double retangles, their input and output

signals are shown by dashed lines.

The reatoristhe mainoperatingunitinthe primaryiruitthatats primarily

as an energy soure. The ontrolled thermo-nulear proesses take plae in the

reator [40, 54℄. The liquidin the tubes of the primary iruit inluding the liquid

in the reator, in the primary side tubes of the six steam generators and in the

pressurizer is irulated by a high speed, and it is under high pressure in order to

avoidboiling. Theenergygeneratedinthereatoristransferredbytheliquidinthe

primaryiruittotheliquidinthesteamgeneratormakingitboiling. Itmeansthat

the steam generators transferthe energy generated by the reatorto the seondary

steam ow. There are six steam generators in a primary iruit. The generated

seondary iruit vapor isthen transferred tothe turbines.

The goal of the pressurizer as an operating unit is twofold: it regulates the

pressure in the primary iruit by heating its water ontent and also serves as an

indiatorfor the primary iruitinventory ontrollerby itswater level.

The steady-state values of the system variables in the normal

## 100 %

^{power}

^{op-}

eratingpointare also indiated inFig. 2

Currently applied ontrollers in the Paks Nulear Power Plant are the lassi-

al proportional (P), proportional-integral (PI) or proportional-integral-derivative

(PID) ontrollers. Their performanes are aeptable lose to the operating point.

However, they annot over the whole operating domain of the power plant and

they are tuned independently. Sometimesthis results inundesired transients under

ertainoperating onditions.

### Pressurizer:

### Level controller Pressurizer:

### Pressure controller

### Base signal

### Correction signal

### Base signal

### m SG,out

### l SG

### m SG,in

### Steam generator

### SG

### 46 bar, 260°C 450 t/h, 0,25%

### Steam

### Valve position

### Inlet secundary water 222°C l PR

### Valve position

### 297°C Pressurizer, PR

### 123 bar 325°C

### P PR

### Heating power

### T PC,HL

### Main hidraulic

### pump 220 - 230°C

### T PC,CL

### Preheater

**Reactor,R**

### Reactor power controller

### Steam generator:

### Level controller P SG

### N

### v

2:Proessowsheetwiththeoperatingunitsofthesimpliedmodel

15

Therearegooddynamimodelsavailableforpressurizedwaterreators (PWRsand

VVERs) forequipment design, safety and riskassessment and/or operatortraining

[2,34℄. The thermal-hydraulipart aloneanbewellmodeledby usinge.g. APROS

[8℄ or [41, 40℄, but usually oupled neutron kineti/thermal odes are used for a

dynami analysis or simulation study [78, 114℄. These models and the omputer

odes behindthem are,however, toomuh detailedfor ontrolstudiesbeausethey

ontain too many state variables (they are of too high degree) and their struture

does not allow us to design model-based feedbaks diretly based on them. Of

ourse, these high-delity odes are indispensable when the designed ontrol loops

are later ne-tuned and tested.

Thereareafewpapersintheliteraturethatreportondevelopingsimpledynami

modelsforboilingwaterorpressurizedwaterreatorsforvariouspurposes. Asimple

model was developed by [53℄ for the thermal-hydraulis part of a boiling water

reator that was used for stability analysis of the reatorunder dierent operating

onditions. Another simplied reator model was developed in [107℄ to thermo-

hydraulianalysisandanotheronewasapplied[21℄toestimatethereatordynamis.

Simplied reator models were also applied for ontroller design purposes. These

models eitherontain the equationof neutronsand the equationof delayed neutron

emittingnulei[101℄orontaintheseequationsandtemperaturefeedbaks[4,9,55℄

orontainthe previouslymentionedequationsandtheequationsofxenonpoisoning

[10, 56℄.

Adynamimodelofprimaryiruitwasdevelopedin[95℄fortheanalysisofow

indued vibrations. Arelativelysimpledynamimodelused inatrainingourse for

simulationpurposes is reportedin [2℄.

Therearealsoafewsimpledynamimodelsavailablefortheindividualoperating

units of the primary iruit. Dynamial models of the steam pressurizer [2, 3, 61,

64, 87℄ and the steam-gas pressurizer [62℄ were desribed to analyze the transient

behaviorof the pressurizer and in[115℄to designpressure ontroller[108℄.

Detailedtheoretialmodelsforthe steamgenerators,basedonfundamentalon-

servation equationsand thermodynamipriniples are typiallyused prior toplant

liensing for simulating aident onditions [45℄, for operator training [118℄ and to

validate ontroller performane prior to implementation on the plant [76℄. These

models, unfortunately, are too omplex for use in ontroller design. The most used

model of steam generator for water levelontroller design wasdeveloped in[48℄. It

wasalinearparameter varyingmodeldesribing thesteamgeneratordynamisover

the entire operating power range. Its state spae form representation was given by

[59℄. Others [121℄ developed a group of linear models to desribe the water level

inthe steamgenerator in wide operating range. A steam generator model was also

developed in [98℄ for sensor fault detetion. In [40℄ and [2℄ a physial based model

of the steam generator was developed. This model aounts for a one-dimensional

dynami desription of the interation between the primary oolant system and

seondary steam system and it is generallyappliable for transient simulations.

The modeling and identiation of a drum boiler in a boiling water reator is

reported by [11℄.

The appropriate ontrol of the parts of a nulear power plant is of great pratial

importaneand is ahallenging researharea atthe same time.

The simpliedmodel of a pressurized water reator (PWR) shows that the be-

havior of this system is nonlinear and its parameters vary with power level, burn

up, and othertime varying fators [29℄. "Therefore, onventional and onstanton-

trollersdo not demonstratea good performane under alloperatingonditions and

duringthe fuelyle" [10℄.

Thepowerontrolofreatorisabasitaskinanulearpowerplant. Amultivari-

able ontrol algorithm was omposed [119℄ of non-linear time-varying feedforward

and feedbak ontrol signals, a referene model of the nulear reator and a dy-

namiobserver. Thenon-linearproportional-integral (PI)feedbakontrollerfores

the nulear reator to follow the response of the referene model. A state feedbak

assistedlassialontrolforPWRpowerontrolwasdeveloped in[9℄whilearobust

ontroller was designed using

## H ∞

optimization in [14℄. The main disadvantage of the lassialproportional(P) and PI ontrollersis that they onlywork properlyinaneighborhoodoftheseletedoperatingpoint. Itmeansthatseveral ontrollersare

requiredfor thesame purposetoontrolthe physialsystem onitswholeoperating

regime.

Nowadays, soft omputingtehnologiesare appliedtodesignreator poweron-

trollers. Fuzzy logi ontrollers were developed by several researhers. Fuzzy on-

trollers were only used for generating adaptively the gain of a PI ontroller [56℄ or

the fuzzy ontroller an be the reator power ontroller itself. Fixfuzzy rule based

approah wasapplied for power ontroller in [4, 101℄,however in[4℄ separate fuzzy

ontrollers were used in ase of steady state and in ase of transients. Adaptive

fuzzy rules were applied in [73℄. A model-based fuzzy ontrol system ombining

withgenetialgorithmwasdeveloped [74℄todriveaplanttoadesiredreferenetra-

jetorywhilea fuzzymodel preditive powerontrollerwasdesigned in[83℄. In[46℄

the feedwater ontrolsystemof thereatorwasrealizedwithfuzzylogi. The fuzzy

ontrollers an be robust, i.e. they are often not very sensitive to the disturbanes

and measurement errors, but the global stability of the losed loop system annot

be analyzedor proved easily.

Neural network reator power ontrollers were also developed. An inverse on-

troller was applied in [10℄. This inverse ontroller was adapted to the hanging

plantirumstanes byaneuralnetwork. In[55℄severalmulti-layerpereptronneu-

ral networks were applied to realize the ontroller. In [19℄ a neural network and a

fuzzy system was ombined with a heuristi ontrol algorithm to form an on-line

intelligent ore ontroller for load following operations. This ontroller used the

good approximation ability of reurrent neuralnetworks (RNN) in identiation of

nonlinearomplexdynamimultiple-inputandmultiple-output(MIMO)plants,i.e.

theRNN hasbeentrainedwithanauratethree-dimensionalreatorore dynami

odetopredittheoredynamibehaviors. Thefuzzylogionsiders allofthe pos-

sible ontrolrod groupsmanoeuvres,and proposesthe optimum rodmanoeuvre. A

similaronstrutionwasappliedin[18℄exeptforthatneuralnetworkstruturewas

modied. However, purely neural network based models do not reet the under-

qualityforthem ifthe disturbanesandinputslargely dierfromthe trainingdata.

Therefore, it an be problemati to apply them in pratie in suh safety ritial

systems asnulearpower plants.

Some optimalontrolmethods for the reatorpowerwere alsodesigned in [103,

120℄.

It was determined that about

## 25%

^{of}

^{all}

^{reator}

^{trips}

^{(when}

^{the}

^{nulear}

^{power}

plantisunavailable)were initiatedbyproblems relatedtofeed-watersystems ofthe

steam generator [63, 93℄. More speially, improper water level ontrol of steam

generators at start-up and low-power operation was ited as a major operational

problem. Therefore, one of the widest investigated ontrol problems of a nulear

power plantis the water level ontrolof the steam generator.

Some researhers have attempted to resolve the low-power water-level ontrol

problem by digital redesign of PI ontrollers and by supplying additional sensor

information [94, 105℄. Nevertheless, the stability robustness problems inherent in

onstant-gain PI ontrollers are still present, resulting in poor overall system per-

formane.

Many of the modernontrolmethodshavebeen appliedto resolve this problem

withsomesuess. In[47℄amodelrefereneadaptiveproportional-integral-derivative

(PID) water-level ontroller, based ona linear parameter-varying model desribing

the steam generator dynamis was proposed. Design of suboptimal ontroller us-

ing linear output feedbak ontrol was reported in [33℄. A PI-type ontroller that

uses a model-based observer to estimate the steam generator water inventory [28℄

and an adaptive observer-based ontroller [84, 85℄ were reported. In [58℄ a model-

based PI ontroller was proposed that o-sets the inverse steam generator water-

level response. A gain-sheduled linear quadrati Gaussian/loop transfer reovery

(LQG/LTR)ontrollerusing alinearized versionof anonlinear validatedmodelwas

also presented [76℄. In [81℄ a steam generator water-level ontroller based on the

estimation of the owerrors was developed.

A gain-sheduled ontroller with guaranteed

## L 2

^{performane}

^{[59℄}

^{and}

^{a}

^{gain-}

sheduled

## H 1

^{ontroller}

^{[93℄}

^{were}

^{also}

^{designed.}

^{In}

^{[7℄}

^{an}

## H 2

^{-optimal}

^{ontroller}

^{was}

proposed, whereas in [17℄ the

## H ∞

^{method}

^{for}

^{a}

^{robust}

^{steam}

^{generator}water-level ontrollerwas used.

Some researhers applied fuzzy logi to ontrol the water level of the steam

generators. The parameters of PI ontroller were adaptively modied by fuzzy

logi in [71, 99℄. In [80℄ a stable ontrol strategy of the water level during low

power operationand transientstates wasstudied. A pratialself-tuningalgorithm

based on the ontrol performane was also suggested and applied to tuning the

membership funtion sale of the ow error. In [90℄ a fuzzy ontrolalgorithmwith

learningfuntion was investigated toredue the tuningworks of fuzzy ontroller of

the steam generator. This learning algorithmould modify the rule base and tune

the membership funtionsautomatially based onthe measured data.

There were several experiments to ombine neural networks and fuzzy logi in

theontrolofwaterlevelofthesteamgenerator. In[26℄aneurofuzzylogiontroller

was developed whih was implemented by using a multilayer neural network with

speialtypesoffuzzysystem. ThestabilityofthisontrollerwasprovedbyLjapunov

the water level ontrol. A fuzzy system was used as gain-sheduler to hoose the

best linearization and ontroller. In [79℄ an adaptive neurofuzzy ontroller of the

water levelwas developed.

There were experiments to ombine of the modern ontrol tehniques with soft

omputing methods. In [27℄ a robust fuzzy gain-sheduler designed based on the

synthesis of fuzzy inferene and

## H ∞

^{tehnique}

^{was}

^{desribed.}

^{A}

^{novel}

^{arhiteture}

for integrating neural networks with industrial ontrollers was proposed in [92℄. In

[32℄ an adaptive fuzzy model based preditive ontrol of nulear steam generators

was developed.

Someresearhersdevelopedontrollersusingmodelpreditiveontroltehnology.

Forexample,asteamgeneratorwater-levelontrollerbasedonanextensionofmodel

preditiveontrolpriniples[63℄wasproposed. In[82℄anauto-tunedPID ontroller

using a model preditive ontrol method was presented while in [86℄ an adaptive

preditive ontroller was designed.

The physial models help us understand the underlying proesses and design

suh advaned ontrollers that an handle state or input onstraints and model

unertainties.

2.6.4 PRISE and the appliation of the oloured Petri net

ThePRImary-to-SEondaryleaking (PRISE)safetyproedure,speiedbythePaks

NulearPowerPlantpersonnel, ontrols thedrainingoftheontaminatedwater ina

faultysteamgeneratorwhenaprimarytoseondaryiruitnon-ompensableleaking

ours. A simple low dimensional nonlinear dynami model of the primary iruit

inthe Paks Nulear PowerPlant being a VVER-type nulear power planttogether

with its relevant safety proedures desribing all of the major leaking type faults

is developed. This model is applied to ahieve the model based veriation of a

PRISE safety proedure that is reommended by the experts of the nulear power

plant. The engineering model belongs to a onentrated parameter hybrid model

lass sine the faults are presented as disrete events.

CPNs have suessfully been applied in the area of reliability analysis, as well

as for modeling and veriation of safety-ritial software and ontrol omponents

innulearpowerplants,too. A CPN-based integrated knowledgebasedevelopment

tool for the veriation of the dynami alarm system has been reported in [91℄. A

safety-ritialsoftwarerequirementsveriationusingombinedCPNandPrototype

Veriation System methods is desribed in [106℄. Fuzzy CPNs have also been

appliedinanautomatedoperatingproeduresystemin[67℄. Eventhehumanfator,

i.e.thepropertiesanddynamisofoperatorpereptionandationsanbedesribed

using CPNs [60℄.

Model development

Inthishaptertheminimalprimaryiruitmodelhas beendeveloped fromminimal

elements,identifyingthemainpartsoftherealsystem,theirharateristidynamis

and their onnetions. The model building isbased onthe neutronux, energy and

massonservationpriniples. Toget theformofthesystem ofequationssuitablefor

modelalibrationandontrollerdesign,themodelequationshavebeentransformed

intostate-spae formontainingintensive variables. This work ispublished in[P1℄.

3.1 Model building of the primary iruit

The modeling goal is toonstrut a dynami minimalmodel of the primary iruit

of the Paks Nulear Power Plant for ontrollerdesign purpose.

The model will be onstruted from minimal elements based on rst engineer-

ing priniples. The basis of this approah is to onstrut the model based on the

important onservation balanes for onserved extensive quantities suh as overall

mass, internal energy, omponent masses, number of neutrons with given energy

level et. supplemented with algebrai onstitutive equations. The proedure in-

ludes the expliit speiation of the modeling goal, the identiation of balane

volumes, the speiation of modeling assumptions and the evaluation of available

data before onstruting the model equations. To get a suitable model form for

ontroller design, the system of equations are transformed into state-spae form

applyingintensivevariables.

3.1.1 Modeling goal

The modeling goal is to onstrut a simplest, i.e. a minimal in generalized sense

dynamimodel ofthe primaryiruitof anulearpowerplantforontrollerdesign.

It means that alow order lumped (onentrated parameter) model is requiredthat

aptures the dynami input-output behavior of the system. At the same time it

is advantageous if the variables and parameters of this model have lear physial

meaningbeause suh a model is more transparent to the operatingpersonnel and

itis more easy touse engineering judgement duringits veriation.

The model minimalityis driven by its appliation, the ontroller design. There

are several ontroller design methods but the majority of them require a simple

generally, toodetailedforontrolstudiesbeausethey ontain toomanystate vari-

ables(theyareoftoohighdegree)and/ortheirstruturedoesnotallowustodesign

model-based feedbaks diretly based onthem. Of ourse, these high-delity odes

are indispensablewhen the designed ontrolloops are later ne-tuned and tested.

At present, in the Paks Nulear Power Plant eah operating unit is ontrolled

separatelyandtheirontrollersworkseparately. Therefore,ifwewouldliketosimply

redesign their ontrollers we do not have tomodel the whole primary iruit, only

the modeling of eah operating units would be neessary. However, the eieny

of the power plant an be inreased if their ontrollers work together. It requires

a design of a so-alled "supervisor" ontroller. The design of supervisor ontroller

makes the modeling of the whole primary iruit neessary beause the supervisor

ontrollerhas toknow how the operatingunits interat with eah other.

Thedomainofthissimpliedmodelinludesthedynamibehaviorinthenormal

operatingmode together with the load hanges between the day and nightperiods.

These riteria (minimal model, the modeling of the whole primary iruit and

the domain) make the purposed model unique. In the literature (see setion 2.6.2

and2.6.3)onlyanoperatingunitorapartoftheprimaryiruithavebeenmodeled

with widerdomain,and only the ontrollerof anoperatingunit has been designed.

3.1.2 Overall modeling assumptions

In order to apply the priniple "the minimal model omposed from minimal ele-

ments" and to obtain a low dimensional dynami model, the simplest possible set

of operating units is onsidered in their simplest funtional form. Part of the pri-

mary iruit with lear funtionality is onsidered as an operating unit (like the

pressurizer). An operating unit may ontain more than one physial units (pipes,

ontainers, valves, et.) but it is then regarded as a primary balane volume over

whih onservationbalanes an be onstruted. The overallmodelingassumptions

speify the onsidered operating units and their generalproperties.

G1 The set of operating units onsidered in the minimaldynami model inludes

the reator, the liquid in the primary iruit, the pressurizer and the steam

generator (they are the ontrolling fators of the real plant).

G2 The dynami model of the operating units is derived from simplied mass,

energy and neutron balanes onstruted for a single balane volume that

orresponds tothe individualunit.

G3 The ontrollers are not modeled expliitly, however, their eets are taken

into aount applying the suitable measured signals (inputs and outputs of

ontrollers). The designed ontrollers are the pressure ontroller, the level

ontroller of the pressurizer, the power ontroller of the reatorand the level

ontrollerin the steam generator.

Thisassumptionmeansthatthedevelopedmodelhastoomputethemeasured

inputsand outputs of ontrollers.

Identier Variable Type:(state, input,

output,disturbane)

## N

^{R}

^{neutron}

^{ux}

^{s}

## v

^{R}

^{ontrol}

^{rod}

^{position}

^{i}

## W R

^{R}

^{reator}

^{power}

^{o}

## m in

^{PC}

^{inlet}

^{mass}

^{ow}

^{rate}

^{i}

## m out

^{PC}

^{purge}

^{mass}

^{ow}

^{rate}

^{d}

## T P C,I

^{PC}

^{inlet}temperature d

## T P C,CL

^{PC}

^{old}

^{leg}temperature (s)

## T P C,HL

^{PC}

^{hot}

^{leg}temperature (s)

## p P R

^{PR}

^{pressure}

^{o,(s)}

## T P R

^{PR}temperature s

## ℓ P R

^{PR}

^{water}

^{level}

^{o,(s)}

## W _{heat,P R}

^{PR}

^{heating}

^{power}

^{i}

## m SG,in

^{SG}

^{water}

^{mass}

^{ow}

^{rate}

^{i}

## m SG,out

^{SG}

^{steam}

^{mass}

^{ow}

^{rate}

^{d}

## T _{SG,SW}

^{SG}

^{inlet}

^{water}temperature d

## p SG

^{SG}

^{steam}

^{pressure}

^{o}

G4 The domain of the model inludes the dynami behavior innormal operating

mode together with the load hanges between the day and the night periods.

In other words, failures and faulty mode transitions annot be desribed by

this simpliedmodel.

3.1.3 Measured signals

Values of extensive variables depend on the extent of the system, suh as energy,

mass,volume,et. Intensivevariables showintensivepropertiesthatdonot depend

on the extent of the system. Intensive variables are the temperature, pressure,

density et.

Sine nulear power plants are safety ritial systems, a large number of mea-

surements isavailable. Most of the intensive variablesin the system are measured.

Themeasured valuesare highlyreliable,however, theyontain measurementerrors.

The measured variables of the primary iruit relevant for our model an be found

inTable 1

3.1.4 The simplied operating units and their dynami mod-

els

Fromthe pointof viewof theirdynamisand the typeof theirdependene onother

operating units, the units of the minimal dynami model are lassied into three

groups (see Fig.3):

## •

^{The}

^{reator}

^{whih}

^{has}

^{a}

^{fast}

^{dynamis}

^{ompared}

^{to}

^{the}

^{other}

^{operating}

^{units}