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75-3110KAZEMERSKY, Philip Michael, 1947- A COMPUTER CODE FOR REFUELING AND ENERGY SCHEDULING CONTAINING AN EVALUATOR OF NUCLEAR DECISIONS FOR OPERATION.The Ohio State University, Ph.D., 1974 Engineering, nuclear

Xerox University Microfilms , Ann Arbor, Michigan 48106

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.

A COMPUTER CODE FOR REFUELING AND ENERGY

SCHEDULING CONTAINING AN EVALUATOR OF NUCLEAR

DECISIONS FOR OPERATION

DISSERTATION

P re se n te d in P a r t i a l F u lf i l lm e n t o f th e R equirem ents fo r

th e D egree D octor o f P h ilo sophy in th e G raduate

School o f th e Ohio S ta te U n iv e rs i ty

By

P h i l ip M. Kazemersky, B .S c ., M.Sc.

» * # # *

The Ohio S ta t e U n iv e rs ity

197U

Reading Committee: Approved by

D r. R. F . Redmond Dr. S . Nakamura Dr. D. D. G lover A dviser

D epartm ent o f N uclear E n g ineering

ACKNOWLEDGEMENTS

The a u th o r e x p re s se s h i s g r a t i tu d e t o th e Tennessee

V a lle y A u th o rity and members o f i t s te c h n ic a l s t a f f f o r th e

su p p o rt p ro v id ed d u rin g th e re s e a rc h and p re p a ra t io n o f t h i s

d i s s e r t a t i o n . P a r t i c u la r l y th e a s s is ta n c e o f Raymond E. Hoskins

who p ro v id ed th e o p p o r tu n ity t o do t h i s re s e a rc h i s g r a te f u l ly

acknowledged.

D r. Douglas H. W alte rs d e se rv e s a s p e c ia l than k you fo r

h i s s t im u la t in g d is c u s s io n s , id e a s and a s s is ta n c e . H is i n t e r e s t

in and th o u g h ts concern ing t h i s work were in v a lu a b le th roughou t

th e re s e a rc h .

V irg in ia H. Simpson d e se rv e s a s p e c ia l th a n k you fo r

h e r programming a s s i ta n c e . Her a id in making t h i s model a func

t i o n a l package was v e ry h e lp fu l th ro u g h t th e re s e a rc h .

D r. R. F . Redmond and D r. S . Nakamura a re due my g r a t i

tu d e fo r t h e i r encouragem ent, g u id a n ce , and a s s i ta n c e in my

ed u ca tio n and t h i s d i s s e r a t i o n . T h e ir knowledge c o n tr ib u te d

much to t h i s work.

ii

VITASeptem ber 22, 1 9 ^ 7 ................................ Born - New Haven, C o n n ec ticu t

1969 ................................................................ B .Sc. in M athem atics, W orcesterP o ly tech n ic I n s t i t u t e , W orcester, M assachusetts

1969-1972 .................................................. Atomic Energy Commission T ra in e e ,The Ohio S ta te U n iv e r s i ty , Columbus, Ohio

1970 ................................................................M .S c., N uclear E n g in ee rin g , TheOhio S ta te U n iv e r s i ty , Columbus, Ohio

1972-197U .................................................. N uclear E n g in ee r, Tennessee V a lleyA u th o r ity , C hattanooga , Tennessee

PUBLICATIONS

"PODECKA - A Pseudo O ne-D im ensional P o in t D ep le tio n A lg o rith m ," P roceed ings o f th e N uclear U t i l i t i e s P lanning Methods Symposium, Jan u a ry 1971*. (0RNL-TM-UUU3)

"S chedu ling O ptim al System O p era tio n Using Mixed In te g e r Programm ing," P roceed ings o f th e N uclear U t i l i t i e s P lann ing Methods Symposium, Jan u ary 197^* (0RNL-TM-UU1»3)

FIELDS OF STUDY

M ajor F ie ld : N uclear E n g ineering

S tu d ie s in M athem atics: P ro fe s s o r A. M. B u o n c r is t ia n i

S tu d ie s in N uclear R eac to r Theory: P ro fe s so r R obert F.Redmond and P ro fe s so r S h o ic h iro Nakamura

S tu d ie s in I n d u s t r i a l E n g in eerin g and C o n tro l Theory: P ro fe s so r A lb e r t B. B ishop, P ro fe s so r Donald W. M il le r , and D r. D ouglas H. W alte rs

iii

TABLE OF CONTENTS\ Page

ACKNOWLEDGEMENTS....................................................................... i i

V IT A ..................................................................................................................................i i i

LIST OF F IG U R E S ..................................................... v i i

LIST OF TABLES............................................................................................................. ix

C hapter

1 . INTRODUCTION .................................................................................................... 1

2. NATURE OF PROBLEM AND RELATED WORK.................................................. 1*2 .1 In tro d u c t io n2 .2 C om plexity o f System S chedu ling

2 .2 .1 The System Model2 .2 .2 The N uclear Model2 .2 .3 The F o s s i l Model

2 .3 R e la ted Work

3. METHODS OF SOLUTION.......................................................................................163 .1 In tro d u c t io n3 .2 P o s s ib le S o lu tio n Techniques

3 .2 .1 D ire c t Search3 .2 .2 Dynamic Programming3 .2 .3 N on-Linear Programming

3 .3 L in ea r Programming — M ix ed -In teg er Programming3 .3 .1 A p p l ic a b i l i ty and S u i t a b i l i t y o f L in e a r

Programming3 .3 .2 M IP-Nuclear C a lc u la t io n a l Scheme

3.U Expansion o f Approach t o a R eal System3 .5 Comparison o f MIP to O ther Models Developed

3 .5 .1 P rev io u s Models Developed 3.5*2 R e la tio n sh ip o f CRESCENDO to O ther Models

h. DEVELOPMENT OF THE OPTIMIZATION MODEL.............................................. 29

iv

V

Page1*. 1 In tro d u c t io n

h.2 System T ra je c to ry A llo c a to r and R e fu e lin g T arg e t S ch ed u le r (STARTS)U .2.1 Dynamic L e o n tie f Model It.2 .2 M ixed -In teger Programming F orm ula tion U.2 .3 O b jec tiv e F unc tio n F orm ula tion

It. 3 Code f o r N uclear C o sting o f an Energy and R efu e lin g T ra je c to ry

U.lt F o s s i l E q u iv a len t LoadIt. 5 C o n s id e ra tio n s in Development o f th e O p tim iza tio n

ModelIt. 5 .1 U niquenessIt. 5 • 2 ConvergenceIt.5• 3 I n i t i a l and End C o nd itionsIt. 5.1+ Comments

5. APPLICATION OF CRESCENDO TO A SAMPLE SYSTEM ........................ 505 .1 In tro d u c t io n5 .2 C h a r a c te r i s t ic s o f th e Sample U t i l i t y System5 .3 A n a ly s is o f th e H y p o th e tic a l U t i l i t y System —

Case One5 .3 .1 S e le c tio n o f R e fu e lin g Times and Energy

L evels5 -3 .2 S e n s i t i v i ty A n a ly s is

5 . It A n a ly s is o f th e H y p o th e tic a l U t i l i t y System —Case Two

5-5 A n a ly s is o f th e H y p o th e tic a l U t i l i t y System —Case Three

5 .6 A n a ly s is o f th e H y p o th e tic a l U t i l i t y System —Case Four

5 .7 Comments

6 . CONCLUSIONS................................................................................................ 88

7. RECOMMENDATIONS....................................................................................... 91

A ppendices

1. F u n c tio n in g o f th e STARTS M odel....................................................... 9^A l . l In tro d u c t io nA 1.2 The Dynamic L e o n tie f Model A1.3 Summary

2. C ost C o e f f ic ie n t s ........................................................................................ 102A2.1 In tro d u c t io nA2.2 C a lc u la t io n o f Cost C o e f f ic ie n ts A 2.3 Sumnary

3. Code fo r N uclear C o stin g o f an E nergy and R e fu e lin gT r a j e c t o r y ................................................................................................................112A3•1 In tro d u c t io n A3.2 CONCERT Model A3.3 Summary

k. CRESCENDO M o d e l ....................................................................................................117AU.l D e s c r ip tio n o f th e Model Ak.2 Flow C hart o f CRESCENDO

5. CRESCENDO Model L is t in g ............................................................................... 126

6 . PODECKA D o c u m e n ta t io n ...................................................................................... l 6 lA6.1 D e sc r ip tio n o f th e ModelA6.2 M athem atical Form ulation A6.3 SummaryA6.h Q n p ir ic a l I s o to p ic and K - I n f in i ty C o r re la t io n s A6.5 C o rre c tio n F a c to rsA6.6 D e sc r ip tio n o f Sample C ases and Sample In p u t D ata

7 . L inear Programming and M ix ed -In teg er Programming Summary . . 203A7.1 L in e a r ProgrammingA7 • 2 M ixed -In teg er Programming

R eferen ces 210

LIST OP FIGURES

Figure Page2-1 System Load w ith One Low Demand P e r i o d .................................................... 7

2-2 System Load w ith Two Low Demand P e r i o d s ................................................8

3-1 B lock Diagram o f CRESCENDO............................................................................. 21

3-2 ORSIM Flow C h a r t ....................................................................................................26

1+-1 Breakdown o f P lann ing H orizon in to D is c re te TimeI n t e r v a l s ................................................................................................................31

1+-2 Dynamic L e o n tie f Model f o r One N u clea r U n it ......................................... 33

1+-3 F o rm u la tion o f a System M o d e l ............................ 38

1+-1+ CONCERT Flow C h a r t ....................................................... 1+3

5-1 Load Demand f o r Each Y ear o f th e H o r iz o n .............................................. 52

5-2 U n it Energy P ro d u c tio n L evel f o r Schedule 1 ....................................56

5-3 U n it Energy P ro d u c tio n L evel f o r Schedule 2 .................................... 58

5-1+ U n it Energy P ro d u c tio n L evel f o r Schedule 1 1 .............................. 6 l

5-5 U n it Energy P ro d u c tio n L evel f o r Y early C y c l e s ..........................67

5-6 U n it Energy P ro d u c tio n f o r Schedule 9 fo r th e Second C ase. . 77

5-7 U n it Energy P ro d u c tio n f o r Schedule 3 fo r th e T h ird Case . . 8 l

5-8 U n it Energy P ro d u c tio n f o r Schedule 5 fo r th e F o u rth C ase . . 81+

A2-1 T y p ica l Cash Flow f o r a N uclear F u e l B a t c h .......................................10l+

A2-2 E f f e c t iv e Cash Flow f o r a B atch .................................. 105

vii

viii

Figure PageA2-3 Sample o f an E f fe c t iv e Cash Flow f o r an Assumed

E q u ilib riu m R efu e lin g Schedule ........................................................... 107

A6-1 K - I n f in i ty v e rsu s Burnup fo r a Sequoyah C la ss PWR . . . . 189

A6-2 1000 MWe PWR D esign — Uranium M e ta l / I n i t i a lUranium M etal (U/Uq ) fo r A ll E n r ic h m e n ts ......................................... 190

A6-3 1000 MWe PWR D esign — E n r ic h m e n t/In it ia lEnrichm ent ( e / e Q) ...................................................................................... 191

A6-U 1000 MWe PWR D esign — F i s s i l e P lu to n iu m /In i t ia lUranium M etal ( P u /U ) ......................................................................................192

A7-1 I l l u s t r a t i o n o f a Branch and Bound T r e e ...........................................2Q8

t

LIST OF TABLES

T ab le Page

5-1 System C osts f o r Case 1 , S chedules 1 and 2 .........................................57

5-2 Breakdown o f T o ta l System C o s t s ............................................................... 60

5-3 System C osts f o r Schedule 1 1 ........................................................................ 62

5-U F u e l Cycle C o s ts Data f o r Schedule 1 1 ................................................. 65

5-5 System C osts f o r Y early C y c l e s .................................................................... 68

5 -6 I l l u s t r a t i o n o f th e S e n s i t i v i t y A n a l y s i s ............................................. 73

5-7 I l l u s t r a t i o n o f F u rth e r Economic A n a l y s e s .........................................7^

5-8 System C osts f o r th e Second C a s e ............................................................... 76

5 -9 System C osts f o r th e T h ird Case . . . . . . . .................................. 80

5-10 System C osts f o r th e F o u rth C a s e ............................................................... 85

A6-1 Energy P roduced by In p u tt in g B atch S ize andE n r ic h m e n t ................................................. 197

A6-2 B atch S ize and Enrichm ent S e le c te d from an EnergyR e q u i r e m e n t .......................................... (............................................................ 198

A6-3 F u e l Cycle C o s ts fo r C ases I-IV ........................................................ 199

A6-H I s o to p ic s f o r Cases I - I V ...............................................................................200

A6-5 I l l u s t r a t i o n o f Coastdown and E a r ly R e f u e l i n g ..............................201

A6-6 F u e l Cycle C o s ts fo r Coastdown and E a r ly R e f u e l i n g ....................202

ix

CHAPTER 1

INTRODUCTION

This vo rk p r e s e n ts a te c h n iq u e which can he used t o in v e s

t i g a t e sch ed u lin g en erg y p ro d u c tio n and n u c le a r r e f u e l in g d a te s f o r

a u t i l i t y system . The te c h n iq u e dem onstra tes th e f e a s i b i l i t y and

a p p l i c a b i l i t y o f s im u lta n e o u s ly a l lo c a t in g energ y p ro d u c tio n and

n u c le a r r e fu e l in g d a te s f o r a h y p o th e t ic a l u t i l i t y system . The

te c h n iq u e i l l u s t r a t e s se ek in g a sch ed u le which y ie ld s th e minimum

t o t a l o p e ra t in g c o s t f o r th e h y p o th e t ic a l u t i l i t y system . A pply ing

t h i s te c h n iq u e to th e sample system le a d s t o th e d e te c tio n o f

sc h e d u lin g tre n d s w hich m ight a s s i s t in so lv in g th e sch ed u lin g

c o m p le x itie s o f a l a r g e u t i l i t y system . The a b i l i t y t o r e f l e c t

c o n s t r a in t s and l i m i t a t i o n s o f th e o p e ra tin g u n i t s w h ile r e l i a b l y

m eeting th e lo a d demand i s d em o n stra ted .

Scheduling fu tu r e en ergy p ro d u c tio n and n u c le a r r e f u e l in g

d a te s f o r a number o f y e a rs (p la n n in g h o rizo n o r h o r iz o n ) f o r th e

f o s s i l , n u c le a r , and hydro u n i t s o f a u t i l i t y system i s a complex

prob lem . Of th e many f e a s ib l e sch ed u le s f o r m ee tin g th e a n t i c ip a te d

lo a d demand, th e u t i l i t y g e n e ra l ly a tte m p ts t o s e l e c t th e sch ed u le

which produces th e minimum t o t a l o p e ra tin g c o s t t o r e l i a b l y s e rv ic e

i t s f o re c a s te d lo a d demand. S chedu ling energy p ro d u c tio n o f a u t i l i t y ' s

o p e ra t in g u n i t s i s co m p lica ted when th e n u c le a r r e f u e l in g d a te s a r e

n o t known. The problem becomes a tim e-d ep en d en t coupled p rob lem .

1

The in t e r s t a g e c o u p lin g a r i s e s from th e e f f e c t o f p a s t , p r e s e n t ,

and f u tu r e o p e ra t io n on n u c le a r fu e l c y c le co s ts and on lo c a t io n o f

r e f u e l in g d a te s . The tim e h o riz o n b e in g s tu d ie d i s d iv id ed in t o

sm all t im e inc rem en ts ( i n t e r v a l s ) . The d e c is io n s made in each o f

th e se i n t e r v a l s de te rm in es th e energy p ro d u c tio n and r e fu e l in g d a te s

fo r th e o p e ra t in g u n i t s . To a c q u ire th e in s ig h t needed fo r dev e lo p in g

a p ro d u c tio n to o l t o a id in so lv in g th e sch ed u lin g problem f o r a

u t i l i t y system , a te c h n iq u e i s developed allow ing th e n a tu re o f th e

problem t o be e x p lo re d .

The te c h n iq u e i s d iv id e d in to tw o s e c t io n s : th e sy stem model

and th e n u c le a r m odel. The means o f u p d a tin g th e n u c le a r c o s ts i s

in c lu d ed i n th e te c h n iq u e s in c e an i t e r a t i v e scheme i s em ployed.

V arious c o n s t r a in t s and l im i t a t i o n s on th e system and n u c le a r u n i t s

a lso m ust be in c o rp o ra te d in t o th e m athem atical fo rm u la tio n o f th e

model. Any te c h n iq u e developed fo r th e sch ed u lin g problem o f an

a c tu a l system m ust accommodate a la r g e - s c a le problem , a problem w ith

many d e c is io n v a r ia b le s .

A s o lu t io n te c h n iq u e must be s e le c te d w hich w i l l o p e ra te

e f f i c i e n t l y , b ecau se th e problem o f sch ed u lin g energy and r e f u e l in g

d a tes i s la rg e s c a l e , even f o r a sample problem o f th r e e n u c le a r u n i t s

and two n o n -n u c le a r u n i t s . Dynamic programming i s n o t a f e a s ib le

te c h n iq u e due t o d im e n s io n a lity problem s v h lch a r i s e vhen d e c is io n

v a r ia b le s in each tim e i n t e r v a l a re co u p led to d e c is io n s in o th e r

tim e I n t e r v a l s . Due to la c k o f developm ent o f c o m p u ta tio n a lly

f e a s ib le n o n - l in e a r o p tim iz a tio n te c h n iq u e s , th i s approach i s n o t

p r a c t i c a l . However, l i n e a r programming i s a te ch n iq u e vh ich can be

e f f i c i e n t l y a p p lie d t o a l a r g e - s c a le problem . The in t e r v a l s in th e

tim e h o r iz o n may be examined s im u lta n e o u s ly by fo rm u la tin g th e

sch ed u lin g problem a s a dynamic L e o n tie f form . Combining m ixed-

in te g e r programming (MIP) w ith th e dynamic L e o n tie f form p roduces th e

system m odel which sch ed u les energy p ro d u c tio n and n u c le a r r e f u e l in g

d a te s . T h e re fo re , MIP i s a t o o l which can accommodate a l a r g e - s c a le

problem , r e f l e c t th e tim e n a tu r e o f th e problem , and employ th e b e s t

a v a i la b le s t a t e - o f - t h e - a r t o p tim iz a tio n scheme to th e problem . T h is

i s th e o p tim iz a tio n method u se d in t h i s work.

T h is work i s o rg a n iz e d in th e fo llo w in g m anner. C h ap te r 2

d e s c r ib e s th e n a tu re o f th e problem and p re s e n ts a rev iew o f p re v io u s

work on t h i s s u b je c t and r e l a t e d to p ic s . C hapter 3 d is c u s s e s th e

s o lu tio n te c h n iq u e employed and in c lu d e s a com parison to o th e r

te c h n iq u e s and m odels a v a i la b l e . C hapter H p ro v id e s th e developm ent

o f an o p tim iz a tio n model ( te c h n iq u e ) w hich may be u se d to in v e s t ig a te

th e n a tu re o f a l a r g e system problem . The system and n u c le a r submodels

a re developed a lo n g w ith th e o p tim iz a tio n c r i t e r i a and p e r t in e n t

c o n s id e ra t io n s . C hap ter 5 d em o n stra te s th e a p p l i c a b i l i t y o f th e

te c h n iq u e t o a sam ple problem . The f l e x i b i l i t y and s e n s i t i v i t y

a n a ly s is th e te c h n iq u e p ro v id e s w i l l be no ted a l s o . C hap ters 6 and 7

p re s e n t t h e c o n c lu s io n s and recom m endations o f th e w ork, r e s p e c t iv e ly .

CHAPTER 2

NATURE OF PROBLEM AND RELATED WORK

2 .1 In tro d u c t io n

W ith th e a d d i t io n o f h y d ro , pum p ed -sto rag e , a n d /o r n u c le a r

u n i t s to a u t i l i t y sy stem , a method f o r p la n n in g system o p e ra tio n

m ust be d ev eloped t o accoun t f o r th e d i f f e r e n t o p e ra t in g c h a r a c t e r i s t i c s

betw een n u c le a r u n i t s and f o s s i l u n i t s . Each u n i t ' s o p e ra t io n

in f lu e n c e s and i s in f lu e n c e d by th e o th e r u n i t s . The d i f f e r e n t ty p e s

and s iz e s o f u n i t s o f th e system a f f e c t th e number and ty p e o f

c o n s id e ra t io n s t h a t must be c o n s id e re d in p la n n in g o p e ra t io n . A

u t i l i t y system (system ) m ust o p e ra te w ith in th e c o n s t r a in t s imposed

on th e system to r e l i a b l y meet system lo a d demands.

A model which a tte m p ts t o sch ed u le system o p e ra t io n must

r e f l e c t th o s e c o n s t r a in t s d e term ined most im p o rtan t to th e sch ed u lin g

problem . The c o n s t r a in t s may be c l a s s i f i e d in to th r e e c a te g o r ie s :

( l ) sy stem , (2) n u c le a r , and (3) f o s s i l . The a b i l i t y t o s e p a ra te th e

c o n s t r a in ts in to c a te g o r ie s a llo w s a model t o be d iv id e d i n t o th re e

com ponents: ( l ) th e sy stem , (2) th e n u c le a r , euid (3 ) th e f o s s i l sub

m odels. The n a tu re o f each submodel w i l l be d is c u s s e d in d iv id u a l ly

a s w e ll a s th e i n t e r f e r i n g o f th e subm odels.

2 .2 C om plexity o f System S chedu ling

2 .2 .1 The System Model

U

5

The system model sh o u ld in c o rp o ra te th e m ajor c o n s t r a in ts

imposed on th e system . C o n s id e ra tio n sh o u ld be g iv en to th e lo c a t io n

o f a re a demand c e n te r s and g e n e ra t in g u n i t s , r e l i a b i l i t y o f th e u n i t s ,

and t e r r i t o r i a l i n t e g r i t y c o n s t r a in t s t o m eet r e g io n a l demands. In

a d d i t io n to th e c o n s id e ra tio n s ab o v e , th e r e a re e f f e c t s on th e system

due t o tra n s m is s io n l i n e lo a d in g , tra n s m is s io n l o s s e s , and s ta r tu p

c o s ts . The r e s e rv e m argin re q u ir e d t o r e l i a b l y meet system lo a d

demands sh o u ld be c o n s id e re d . A ll o f th e above c o n s id e ra t io n s sh o u ld

be in c o rp o ra te d in to a p ro d u c tio n system sc h e d u lin g m odel. For t h i s

work th e above c o n s id e ra t io n s a r e n e g le c te d . These c o n s id e ra tio n s

do n o t a f f e c t th e b a s ic fo rm u la tio n b e in g u se d . They would add

c o n s tr a in ts t o th e system m odel, b u t would n o t in f lu e n c e th e b a s ic

fu n c tio n . The purpose o f th e sy stem model i s t o seek th e " b e s t”

o p e ra t in g sch ed u le f o r th e system .

The m in im iza tio n o f t o t a l system p re s e n t-v a lu e d o p e ra t in g

c o s t i s th e c r i t e r i o n u sed to sch ed u le system o p e ra t io n . A ll d e c is io n s

a re made on th e b a s is o f c o s t , th e prim e component o f which i s f u e l

c o s t (1 , 2 ) . N uclear f u e l i s a h ig h ly p ro c e s se d , m anufactured p ro d u c t

which r e q u ir e s a s u b s t a n t i a l le a d tim e and f i n a n c ia l in v e s tm e n t. F uel

no rm ally rem ains in c o re from 2 t o U y e a r s , and when removed re q u ir e s

a d d i t io n a l tim e and f in a n c ia l in v estm en t t o p ro c e s s and re c o v e r u s e fu l

p ro d u c ts . The method o f o p e ra t io n a f f e c t s th e amount and enrichm ent

o f f u e l in s e r t e d a t any r e f u e l in g . These p a ra m e te rs , th e e f f e c t s o f

p a s t , p r e s e n t , and f u tu r e o p e ra t io n on th e c o s t o f f u e l b a tc h , and th e

l i f e t im e o f a b a tc h im ply a p la n n in g h o riz o n o f 5 t o 7 y e a r s . Thus,

n u c le a r and f o s s i l f u e l c o s ts a r e th e m ajor components o f th e system

o p e ra t in g c o s t . In a d d i t ip n to th e m in im iza tio n o f c o s t , th e

dependence o f p r e s e n t o p e ra t io n on p a s t and f u tu r e o p e ra t io n le ad s

t o a co u p lin g o f c o s ts t o p r e s e n t , p a s t , and f u tu r e d e c is io n s .

The p a s t o p e ra tio n o f th e n u c le a r u n i t in f lu e n c e s th e

p re s e n t energy c a p a b i l i ty o f th e u n i t s in c e o n ly a p o r t io n (h a tc h )

o f th e f u e l in a r e a c to r i s re p la c e d a t each r e f u e l in g . S im u ltan eo u sly

d e te rm in in g th e r e f u e l in g d a te s and a l lo c a t in g energ y p ro d u c tio n to

a system makes t h e sch ed u lin g problem tim e d ep en d en t. C onsequen tly ,

th e economic t r a d e - o f f s betw een energy p ro d u c tio n and r e f u e l in g d a te s

m ust be examined s im u lta n e o u s ly . T h is exam ination may be accom plished

by d iv id in g th e p la n n in g h o riz o n in to a d i s c r e t e number o f tim e

in t e r v a l s which may be sy n th e s iz e d in to c y c le s once th e system model

h as made th e n e c e ssa ry r e f u e l in g d e c is io n s . T hus, th e sch ed u lin g

problem i s a m u l t is ta g e tim e-d ep en d en t problem . A lthough sch ed u lin g

system o p e ra t io n i s a complex p rob lem , th e system model must meet th e

fundam ental p u rp o se o f s e rv in g th e lo a d .

The p r a c t i c e i s t o lo a d th e n u c le a r u n i t s t o r a te d c a p a c ity

and r e f u e l them d u rin g a p e r io d o f low demand ( i . e . , a c y c le le n g th o f

1 t o 1 .5 y e a r s ) . T h is p r a c t ic e h as been p o s s ib le due to th e low

p e rc e n ta g e o f n u c le a r c a p a c ity on any system , low f u e l c o s t o f n u c le a r

u n i t s , and low er req u irem en t f o r rep lacem en t pow er. F ig u re s 2-1 and

2 -2 i l l u s t r a t e a u t i l i t y system w ith one and two low-demand p e r io d s ,

r e s p e c t iv e ly . As n u c le a r c a p a c i ty in c r e a s e s , sch ed u lin g n u c le a r u n i t s

in to low-demand p e r io d s becomes d i f f i c u l t due to th e number o f u n i t s

t o be re fu e le d (3 ,). The system model shou ld s e l e c t p e r io d s fo r

r e f u e l in g which m inim ize t o t a l p re s e n t-v a lu e d o p e ra t in g c o s ts w h ile

_____IJa n

i________I_______ l_______ l_______ I------------1------------1------------1----------- 1----------- 1---------t'eb Mar Apr May Ju n J u l Aug Sep Oct Nov Dec

F ig u re 2 -1 : System Load w ith One Low Demand P e rio d

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Ja ni________i________ i________ i________l________ l________ i________ I________ i------------1-------------1-----------

Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec

F ig u re 2 -2 : System Load w ith Two Low Demand P e rio d s

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m a in ta in in g a d e s ir e d r e s e rv e m arg in . N orm ally th e se w i l l be low -

demand p e r io d s . In o rd e r f o r th e u n i t s to be o p e ra b le , m ain tenance

m ust be perfo rm ed . The assum ptions made a re t h a t n u c le a r m a in ten

ance o ccu rs d u rin g r e f u e l in g , f o s s i l m ain tenance occu rs on th e c a p a c ity

which i s n o t p roducing energy and ex cess c a p a c ity i s adequa te to

accom plish t h i s , and m ain tenance manpower i s a v a i la b le t o com plete

a l l n e c e ssa ry n u c le a r and f o s s i l m a in tenance. In a d d i t io n , b o th

n u c le a r and f o s s i l u n i t s a r e assumed t o have no fo rced o u tag es in

th e system m odel. The u n i t s in th e system model o p e ra te o r n o t based

on th e c a lc u la t io n s o f th e m odel. The fo rced ou tage r a t e s o f th e u n i t s

would be h and led by a p r o b a b i l i s t i c s im u la tio n su b ro u tin e a s p a r t o f

th e n u c l e a r - f o s s i l m odel. T hus, th e system sch ed u lin g problem can be

in v e s t ig a te d w ith o u t th e in c lu s io n o f fo rced o u tag es which do n o t

a l t e r th e b a s ic p rob lem , o n ly th e c a l c u la t io n a l com plex ity o f th e

n u c l e a r - f o s s i l m odel. F u rth erm o re , th e f o s s i l u n i t in th e s tu d y i s

a s s ig n e d a c o n s ta n t c o s t , w hich means th e fo rc e d ou tage r a t e o f th e

u n i t s would n o t change th e c o s t o f energy from th e f o s s i l u n i t .

Economic o p tim iz a tio n o f system o p e ra tio n i s a complex

problem bounded by u n i t and system c o n s t r a i n t s . The system model

shou ld e v a lu a te th e co m p e titio n betw een u n i t s f o r th e lo a d and

r e f u e l in g d a te s based on economic c o n s id e ra t io n s . I t a l s o shou ld be

a d a p ta b le t o d e te rm in in g a new o p tim a l sch ed u le a f t e r a m ajor system

p e r tu r b a t io n .

S im p lify in g assum ptions have been made abou t th e u t i l i t y

system t o f a c i l i t a t e th e developm ent o f th e te c h n iq u e . The f e a s i b i l i t y

10

and a p p l i c a b i l i t y o f s im u lta n e o u sly a l lo c a t in g energy and r e f u e l in g

d a te s can be d em onstra ted on a s im p l i f ie d h y p o th e t ic a l system w ith o u t

lo s s o f g e n e r a l i t y . To dem o n stra te th e a p p l i c a b i l i t y o f th e system

sch ed u lin g m odel, a n u c le a r model must be in c lu d e d in th e o v e r a l l

sch ed u lin g m odel.

2 .2 .2 N uclear Model

The p rim ary fu n c tio n o f th e n u c le a r model i s t o s im u la te

th e o p e ra t io n o f a n u c le a r u n i t w hich p ro v id e s th e b a tc h c o s ts when

g iven th e o p e ra t in g s t r a t e g y . The n u c le a r model m ust u se energy

p ro d u c tio n and r e f u e l in g d a te s sy n th e s iz e d f o r each n u c le a r u n i t from

th e o u tp u t o f th e system m odel. The b a tc h s iz e and enrichm ent needed

in each c y c le t o meet th e energy req u ire m en ts must be s e le c te d by th e

m odel. The burnup and d e p le tio n o f each b a tc h must be c a lc u la te d fo r

th e o p e ra t in g s t r a t e g y . These c a lc u la t io n s sure u sed to develop th e

cash flow f o r th e u n i t d u rin g th e p la n n in g h o riz o n .

The ca sh flow r e p re s e n ts a l l p r e - i r r a d i a t i o n and p o s t

i r r a d i a t i o n ch a rg e s f o r a b a tc h in s e r t e d o r d isc h a rg e d d u rin g th e

h o riz o n . A c a sh flow must be developed f o r each n u c le a r u n i t . From

th e s e cash f lo w s , each u n i t ' s c o s t c o e f f i c ie n t s f o r each in t e r v a l

sure c a lc u la te d and t r a n s f e r r e d to th e system model f o r u p d a tin g th e

c o s t fu n c tio n . These nuclesur and c o s t c o e f f i c ie n t csuLculations sure

done a t p auses in th e system m o d e l's c a l c u la t io n s . An i t e r a t i v e

p ro c e ss i s u sed betw een th e s e models t o ach iev e an economic o p tim iz a

t i o n o f th e system . An i t e r a t i v e scheme betw een th e nuclesur model

and system model r e q u ire s a r a p id ly e x e c u tin g n u c le a r model s in c e th e

11system model uses s ig n i f i c a n t amounts o f com putation tim e . T h u s, th e

n u c le a r model fu n c tio n s as a c o s t g e n e ra to r fo r th e system model and

th e m ethods used by th e n u c le a r model to de term ine c o s ts a re n o t

im p o rtan t to th e system m odel. A nother prim e component o f th e

r e a c t o r 's o p e ra tin g c o s t i s f o s s i l f u e l c o s t s .

2 .2 .3 F o s s i l Model

The prim ary fu n c tio n o f t h i s model would be to produce th e

c o s ts o f o p e ra tin g th e f o s s i l p o r t io n o f th e system and to r e p re s e n t

th e b a la n c e o f th e system as one f o s s i l e q u iv a le n t u n i t . I t w ould

be in t h i s model t h a t energy p ro d u c tio n would be a l lo c a te d to each

p la n t t o s a t i s f y th e in t e r v a l lo a d d u ra tio n cu rv e . Forced o u ta g e s

fo r a l l u n i t s would be hem d ie d in t h i s model a ls o .

To do t h i s th e f o s s i l model shou ld employ a te ch n iq u e such

as p r o b a b i l i s t i c s im u la tio n (j^, £ , 6 ) . T h is te c h n iq u e i s a compromise

between a s t r i c t l y d e te rm in is t ic te ch n iq u e and a Monte C arlo s im u la

t io n te c h n iq u e . The p r o b a b i l i s t i c s im u la tio n methods combine b o th

accu racy and c a l c u la t io n a l sp eed . Speed i s im p o rtan t s in ce th e f o s s i l

model w ould be acc essed each tim e c o n tro l p a sse s t o th e n u c le a r model.

When a l lo c a t in g energy to th e f o s s i l u n i t s , th e f o s s i l model

shou ld c o n s id e r th e energy p roduced by th e n u c le a r u n i t s which a re

reg a rd ed as e n e rg y - l im ite d re s o u rc e s t o e n su re th e f o s s i l model uses

a l l th e n u c le a r energy a ss ig n e d by th e system m odel. The f o s s i l model

a s s ig n s energy p ro d u c tio n to th e f o s s i l u n i t s on a l e a s t - c o s t b a s i s

which can be based on in c re m en ta l c o s t . From th e energy assignm ent

th e model forms one com posite c o s t c o e f f i c ie n t from th e c o s ts o f th e

12

in d iv id u a l u n i t s f o r each t i n e i n t e r v a l . These c o s t c o e f f i c ie n t s

would h e p a sse d to th e system model f o r u p d a tin g th e c o s t fu n c tio n .

An e a sy e x te n s io n w ould be t o have a number o f c o s t c o e f f i c i e n t s f o r

a number o f f o s s i l u n i t s p assed t o th e system model f o r u p d a tin g .

2 .3 R e la te d Work

The im petus f o r so lv in g th e sch ed u lin g problem i s b e in g

a b le t o ad e q u a te ly h an d le th e o u t-o f -c o re management o f n u c le a r f u e l .

The o u t-o f - c o re management i s d e f in e d t o encompass th e procurem ent

o f n u c le a r f u e l w hich i s read y f o r i n s e r t io n in to t h e r e a c to r and

th e d is p o s a l and re c o v e ry o f th e sp en t f u e l a f t e r d isc h a rg e from th e

r e a c to r .

Embodied in t h i s d e f in i t io n i s th e developm ent o f th e cash

flow f o r th e f u e l n e c e ss a ry t o meet th e chosen o p e ra t in g s t r a t e g y .

Seeking an economic minimum f o r t h i s sch ed u lin g p roblem co m p lica tes

th e developm ent o f a s o lu t io n te c h n iq u e .

C o n sid e rab le e f f o r t h as been expended in dev e lo p in g schemes

fo r o p tim iz in g th e in c o re management o f th e fu e l (l_, 2 ) • A lthough

n u c le a r co re d es ig n i s s t i l l ev o lv in g and in c o re management i s s t i l l an

im p o rtan t p a r t o f o p tim iz in g f u e l c y c le c o s ts ( l ) f o r a n u c le a r u n i t ,

in c o re management i s n o t th e concern o f th e system p la n n e r . However,

th e r e i s a need (Xt 8 ) f o r t h i s in c o re work in th e d a y -to -d a y and

n e a r - te rm o p e ra t io n o f a u n i t . The in c o re work u s in g o p tim iz a tio n

te c h n iq u e s such a s l i n e a r programming ( £ , 10,, 1 1 , 1 2 , l g ) , dynamic

programming (lU , 1 £ ) , e x h a u s tiv e se a rc h ( l 6 , IX , 1 8 ) , and p a t te r n

s e a rc h (19 ) has dem onstra ted t h a t th e a p p l ic a t io n o f o p e ra t io n s

re s e a rc h te c h n iq u e s t o n u c le a r f u e l management i s f e a s ib l e .

13

O u t-o f-c o re management w hich i s im p o rtan t in c o o rd in a tin g

o p e ra t io n o f u n i t s on a system -w ide b a s i s has r e c e iv e d l e s s a t t e n t i o n .

Only w ith in th e p a s t s e v e r a l y e a rs h as th e need f o r t h i s work become

e v id e n t (2 0 , 21 , 2 2 , 2 3 ) . The developm ental e f f o r t s t o produce an

a lg o rith m which c o u ld fu n c tio n as a n u c le a r model a s w e l l as a t o o l

f o r sco p in g s tu d ie s have r e s u l t e d i n s e v e ra l n u c le a r m odels. Most o f

th e s e models employ some o p tim iz a tio n te c h n iq u e fo r s e le c t io n o f b a tc h

s iz e s and en richm en ts to m eet c y c le e n e rg ie s and le n g th s .

Henderson (3) employs a d i r e c t se a rc h te c h n iq u e based on a

f ix e d number o f p o s s ib le b a tc h s i z e s . The d i r e c t s e a rc h t r a c e s a l l

d e c is io n p a th s th ro u g h s e v e ra l c y c le s o f o p e ra t io n to de term ine one

which y ie ld s th e minimum c o s t . S peaker and F orkner (2H, 2 5 , 26 ) u se

a p a t t e r n se a rc h method to de term ine th e b a tc h s iz e s and en richm en ts

f o r o p e ra t in g s t r a t e g y . F u rth erm o re , Specker and F o rk n er employ

l i n e a r programming t o do f u e l s h u f f l in g . Both H en d erso n 's and Specker

and F o rk n e r 's works a re 1 -d im e n s io n a l. Kearney (2 7 , 2 8 ) employs

dynamic programming fo r s e le c t io n o f b a tc h s i z e s and en ric h m en ts . In

h is 2 -d lm en sio n a l code Kearney l i m i t s h i s b a tc h f r a c t io n s to f iv e

a l t e r n a t iv e s . The m in im iza tio n o f c o s t i s th e o p tm iz a tio n c r i t e r i o n .

Omberg (29) has developed a 2 -d im en sio n a l a lg o rith m f o r c a lc u la t in g

th e in c re m en ta l c o s t o f any o p e ra t in g s t r a t e g y . The a lg o ri th m , u s in g

a 1 -1 /2 group f lu x m odel, d e te rm in es a f e a s ib le o p e ra t in g scheme to meet

th e r e q u ire d c y c le energy and le n g th . To f in d an optimum scheme, i t

i s n e c e ssa ry t o do p e r tu rb a t io n s tu d ie s on th e d e te rm in e d enrichm ents

and b a tc h s i z e s . Ontko (30) has developed a 2 -d im en s io n a l (X-Y) nodal

ih

code to c a lc u la te r e lo a d p a ram e te rs f o r PWR. The s e le c t io n c r i t e r i o n

i s based on o b ta in in g an o p tim a l t a r g e t burnup.

H enderson and Bauhs (31) d e s c r ib e an a lg o rith m f o r do ing

scop ing ty p e c a lc u la t io n s u s in g c o re average c y c le burnup as th e

t a r g e t f o r s e le c t in g p a ra m e te rs . T h is te c h n iq u e a llo w s th e developm ent

o f e i t h e r a b a tc h r e lo a d scheme o r an enrichm ent r e lo a d scheme.

Kazemersky (3 2 , 33) has developed a p seudo -one-d im ensional a lg o rith m .

The code s e le c t s b a tc h s iz e s and en richm en ts acco rd in g to a s p e c if ie d

s e t o f r u l e s .

F a r l e s s work o f a s u b s ta n t iv e n a tu re h as been done in th e

system in t e g r a t io n o r system sch ed u lin g a r e a . The sch ed u lin g problem

was reco g n ized by G i l l e la n d , e t a l (2 0 , 2 1 ) , and th e co n cep tu a l system

model was o u t l in e d by Rees and G oodrich (2 3 ) . The sch ed u lin g model

i s decomposed in to seven m odules. These would be in te r f a c e d t o form

th e com plete sch ed u lin g a lg o rith m . A lso , t h i s decom position would

a llo w each module t o be s e p a ra te ly c o n s tru c te d . H o sk in s, K oncel, and

S trau ch (22) a l s o s e t o u t a system in te g r a t io n model which i s c lo s e ly

r e l a t e d t o th e R ees’ work.

Salmon (3*0 , B en n e tt ( 3 5 ) , and Turnage (36) have c o n s tru c te d

a system in t e g r a t io n (sy stem sc h e d u le r) m odel. T h is model w il l

a l l o c a t e energy to th e u n i t s on a u t i l i t y system . The method i s based

on a s s ig n in g a lo a d in g o rd e r t o th e u n i t s based on in c re m en ta l c o s t .

I f th e r e f u e l in g sch ed u le s f o r th e n u c le a r u n i t s a r e known, th e code

w i l l i t e r a t e t o f in d th e minimum c o s t f o r th e system based on a

p a r t i c u l a r r e f u e l in g s t r a t e g y .

15

D eaton (3 7 . 3 8 . 3 9 ) has developed a system in te g r a t io n

model f o r th e a l lo c a t io n o f energy . The a l lo c a t io n i s done hy netw ork

programming based on in c re m en ta l c o s t s . Knowing a r e f u e l in g s t r a t e g y ,

D ea to n 's model i t e r a t e s t o f in d th e l e a s t - c o s t scheme o f a l lo c a t in g

energy t o th e n u c le a r and f o s s i l u n i t s . I t i s assumed th e f o s s i l

u n i t s ' c o s ts a r e alw ays g r e a te r th a n th e n u c le a r u n i t s ' c o s t s .

Wicks (UO) has o u t l in e d a scheme fo r develop ing an o p tim a l

s t r a te g y and th e te c h n iq u e s re q u ire d . The a u th o r d e s c r ib e s v a r io u s

methods t h a t m ight be f e a s ib l e fo r a system m odel. However, th e

n u c le a r p o r t io n i s r e s t r i c t e d to sm all p e r tu rb a t io n s around a

r e fe re n c e s t r a t e g y .

Rhodes ( h i %i h as i l l u s t r a t e d th e b e n e f i ts in be ing a b le

t o sch ed u le th e e n t i r e system r a th e r th a n u se nom inal r e f u e l in g

sc h e d u le s . He u se s a sam ple u t i l i t y system and a known r e f u e l in g

sc h e d u le . A f te r d e te rm in in g th e o p tim al energy a l lo c a t io n u s in g

l i n e a r program m ing, th e r e f u e l in g sch ed u le i s m anually changed t o more

f u l l y u t i l i z e th e n u c le a r u n i t s w ith th e lo w est fu e l c o s t . The n u c le a r

c o s ts were de term ined from en erg y -b u rn u p -co st c o r r e la t io n s drawn from

o f f l i n e s tu d ie s . Rhodes i l l u s t r a t e d th e r e i s a s ig n i f i c a n t m onetary

advan tage in a tte m p tin g to o p tim a lly schedu le energy on a system -w ide

b a s i s .

Kazemersky (^3) has o u tl in e d a system in te g r a t io n code which

i s under developm ent. M ix ed -in teg e r programming i s used t o seek an

o p tim al energy and r e f u e l in g schedu le ( t r a j e c t o r y ) on a system -w ide

b a s i s . The model In c o rp o ra te s a n u c le a r model f o r i t e r a t i v e l y u p d a tin g

th e n u c le a r c o s ts .

CHAPTER 3

METHODS OF SOLUTION

3.1 In tro d u c t io n

T here a re s e v e ra l o p tim iz a tio n te c h n iq u e s t h a t can be

a p p lie d t o t h i s problem . A v a ila b le o p tim iz a tio n te c h n iq u e s in c lu d e

l i n e a r , i n t e g e r , dynam ic, and n o n - l in e a r program m ing; d i r e c t s e a rc h ;

g ra d ie n t te c h n iq u e s ; and c l a s s i c a l te c h n iq u e s . However, many o f

th e s e te c h n iq u e s a re n o t a p p l ic a b le t o la rg e system s w ith many

d e c is io n v a r ia b le s because co m p u ta tio n a l c a p a b i l i t i e s and a lg o rith m

developm ent a r e n o t s u f f i c i e n t ly advanced t o a llo w th e h an d lin g o f

la r g e - s c a le p rob lem s. The te c h n iq u e a c tu a l ly employed must h an d le

la r g e and complex p rob lem s. The fo rm u la tio n o f th e problem a s a

m ath em atica l model i s dependent on th e system b e in g m odeled.

Of th e s e te c h n iq u e s , g r a d ie n t and c l a s s i c a l a r e o f l im i te d

use due to th e unknown n a tu re o f th e c o s t fu n c tio n a s a fu n c tio n o f

energy and c y c le le n g th . However, dynam ic, n o n - l in e a r , l i n e a r , and

in te g e r programming a long w ith d i r e c t s e a rc h a r e a l t e r n a t iv e s t h a t

b ea r ex am in a tio n .

3 .2 P o s s ib le S o lu tio n Techniques

3 .2 .1 D ire c t S earch

D ire c t o r e x h a u s tiv e se a rc h i s an o p tim iz a tio n method which

enum erates a l l p o s s ib le com binations o f v a r ia b le s . A f te r com pletion

16

17

o f a l l en u m era tio n s, th e s e le c t io n o f th e o p tim al com bination o f

d e c is io n v a r ia b le s i s p o s s ib le . A m ajo r d isad v an tag e i s th e number

o f enum era tions t h a t must be done. C o n sid erin g o n ly two d e c is io n

v a r i a b le s , energy p ro d u c tio n and th e r e f u e l in g in t e r v a l in d ic a te d by

in te g e r v a r ia b le s ^ , an e s tim a te i s made o f th e enum erations t h a t

m ust be done by a d i r e c t s e a rc h . The sam ple system d e sc r ib e d in

C hap ter 5 h as 126 in te g e r v a r ia b le s . T here a re 2 com binations

36o f th e s e v a r ia b le s o r app rox im ate ly 8x10 co m b in a tio n s . C o n sid e rin g

t h e energy p ro d u c tio n v a r ia b le v a r ie s co n tin u o u s ly betw een e s ta b l is h e d

l i m i t s , t h e r e i s an i n f i n i t e number o f s o lu t io n s . T h e re fo re , th e u se

o f d i r e c t s e a rc h i s in f e a s i b le fo r t h i s sch ed u lin g prob lem .

3 .2 .2 Dynamic Programming

Dynamic programming i s an o p tim iz a tio n te c h n iq u e w hich can

m arkedly d e c re a se th e com p u ta tio n a l re q u ire m en ts o f a la rg e system

o p tim iz a t io n . The re d u c t io n in com putation i s ach iev ed by t r a n s

form ing a s e q u e n tia l d e c is io n p ro cess w ith i n t e r r e l a t e d v a r ia b le s

in t o a s e r i e s o f s in g le - s t a g e d e c is io n p ro c e s se s in v o lv in g o n ly a

few v a r ia b le s . T h is tra n s fo rm a tio n i s based on th e p r in c ip le o f

o p t im a l i ty ( ;

"An o p tim a l p o l ic y has th e p ro p e r ty t h a t w hatever th e i n i t i a l s t a t e and i n i t i a l d e c is io n a r e , th e rem ain ing d e c is io n s must c o n s t i t u t e an o p tim a l p o lic y w ith re g a rd t o th e s t a t e r e s u l t in g from th e f i r s t d e c i s io n ."

T h is means t h e s ta g e s must be decoupled from each o th e r and any p a s t

d e c is io n s do n o t a f f e c t f u tu r e d e c is io n s and v ic e v e r s a .

^The v a lu e o f th e In te g e r v a r ia b le s a r e r e s t r i c t e d t o ze ro (0 ) o r one ( l ) t o in d ic a te e i t h e r o p e ra t io n o r r e f u e l in g , r e s p e c t iv e ly .

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However, f u tu r e d e c is io n s can a f f e c t th e o p t im a l i ty o f

p a s t d e c is io n s b ecau se o f th e c o u p lin g o f s ta g e s . The c o s t fu n c tio n

o f a n u c le a r u n i t i s dependent n o t o n ly on f u tu r e o p e ra t io n , b u t

a ls o on p re s e n t and p a s t o p e ra t io n . T hus, i t i s n o t p o s s ib le t o

decouple th e s ta g e s w ith o u t making some assum ptions conce rn ing th e

c o s t fu n c tio n . I f th e s ta g e s a re decoup led b y s e p a ra t in g th e r e f u e l in g

s e le c t io n from th e energy a l lo c a t io n p o r t io n o f th e c o s t f u n c t io n ,

th e dynamic program must t r a c e p a th s th ro u g h th e o p tim iz a tio n

p ro c e d u re . These p a th s le a d to d im e n s io n a lity problem s w ith in th e

dynamic programming a lg o rith m . However, th e m ajo r h in d ran ce t o u s in g

dynamic programming i s th e la c k o f developm ent o f s u i ta b le com puter

a lg o rith m s to so lv e l a r g e - s c a l e , n o n - l in e a r p rob lem s.

3 .2 .3 N on-L inear Programming

N on-linear, programming can h and le v a r io u s ty p e s o f o b je c t iv e

fu n c tio n s . Some developm ent o f a lg o rith m s d e a l in g w ith q u a d ra t ic

o b je c t iv e fu n c tio n s has been done, b u t th e developm ent o f n o n - l in e a r

a lg o rith m s has been l im i te d t o a few s p e c ia l a p p l ic a t io n s . T h is

te c h n iq u e i s o f v e ry l im ite d v a lu e a s a s o lu t io n te c h n iq u e f o r th e

system problem due t o la c k o f a g e n e ra l iz e d a lg o rith m and an e x p l i c i t

form o f th e c o s t fu n c t io n . .

3 .3 L in e a r Programming - M ix ed -In te g e r Programming

3 .3 .1 A p p l ic a b i l i ty and S u i t a b i l i t y o f L in e a r Programming

The c o n s t r a in ts o f th e system sch ed u lin g problem can b e

fo rm u la ted as a s e t o f l i n e a r e q u a l i t i e s and in e q u a l i t i e s . The

o b je c t iv e fu n c tio n fo rm u la tio n depends on th e problem . The m athe

m a tic a l form o f th e o b je c t iv e fu n c tio n i s n o t known a s a fu n c tio n o f

energy and r e f u e l in g d a te s . However, th e o b je c t iv e fu n c tio n can be

and has been fo rm u la ted (3 6 , 38 , U2) a s a l i n e a r fu n c tio n o f energy

i f th e r e f u e l in g d a te s a r e assumed known. T h is fo rm u la tio n has been

adequa te f o r p a s t in v e s t ig a t io n s . For t h i s s tu d y , i t i s assumed

th a t a l i n e a r o b je c t iv e fu n c t io n , w hich i s dependent on energy and

r e f u e l in g d a te s , i s a p ie cew ise l i n e a r approx im ation to th e a c tu a l

system o b je c t iv e fu n c tio n . T h is approx im ation may converge to

a lo c a l o r g lobal, optimum. The ty p e o f optimum may n o t be d i s

t in g u is h a b le . S in ce th e o b je c t iv e fu n c tio n s and c o n s t r a in ts a re

l i n e a r , l i n e a r programming i s used as th e o p tim iz a tio n te c h n iq u e .

L in e a r programming (U5_, U6) i s a w id e ly used o p tim iz a tio n

te c h n iq u e which can be ex tended to in c lu d e in te g e r d e c is io n v a r ia b le s .

The fo rm u la tio n o f th e c o n s t r a in ts r e q u ire s in te g e r v a r ia b le s to

in d ic a te th e s t a tu s o f n u c le a r u n i t s d u rin g each i n t e r v a l . The

e x te n s io n o f l i n e a r programming to d e a l w ith in te g e r v a r ia b le s i s

m ix e d -in te g e r programming which can be co n s id e re d one o f th e more

advanced methods a v a i la b le f o r g e n e ra liz e d a p p l ic a t io n . A m ixed-

in te g e r fo rm u la tio n on a s im i la r problem has been su g g es ted by .

Anderson e t a l ( U 7). D utton e t a l (U8) s e le c te d MIP as th e o p tim iz a

t i o n te c h n iq u e f o r a la rg e m ix e d -in te g e r problem . In view o f th e

above, MIP ap p ea rs t o be th e most p rom ising o p tim iz a tio n te ch n iq u e

and th e method a p p lie d t o th e sch ed u lin g system o p e ra tio n problem .

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3 .3 .2 MIP - N uclear C a lc u la t io n s ! Scheme

2CRESCENDO i s an in te g r a te d com puter code package designed

to se a rc h fo r an o p tim a l energy a l lo c a t io n and r e f u e l in g d a te sch ed u le .

I t i s composed o f two subm odels: STARTS and CONCERT. The STARTS

model c o n s is ts o f a system o f p ro ced u res t o d i r e c t th e flow o f a

m ix e d -in te g e r program . I t i s n e c e ssa ry t o s p e c ify o n ly th e c o r re c t

o rd e r o f c o n t ro l p ro ced u res s in c e th e IBM M athem atical Programming

System -Extended (MPSX) i s b e in g u sed . MPSX has an o p tim a l m ixed-

in te g e r package (MPSX-MIP) w hich can be r e a d i ly a c c e sse d . T ra n s fe r

o f d a ta betw een MPSX and a F o r tra n program i s accom plished by a sub

r o u t in e , READCOMM. The F o r tr a n program i s th e CONCERT model which

s im u la te s each n u c le a r u n i t ' s o p e ra t io n . The d e p le tio n o f th e u n i t ' s

co re fo r a c y c le , th p s e le c t io n o f th e b a tc h s iz e and en rich m en t, f u e l

s h u f f l in g , s e t t i n g th e p re v io u s o p e ra t in g h i s to r y o f th e c o r e , s e t t in g

th e c h a r a c t e r i s t i c s o f th e u n i t under c o n s id e ra t io n , and a n a ly s is o f

th e c o r e 's o p e ra t io n enuring th e p la n n in g h o riz o n a re fu n c tio n s o f

CONCERT.

STARTS and CONCERT do th e c a lc u la t io n s w h ile th e main

fu n c tio n s o f CRESCENDO a re t o c o n tro l th e flow o f d a ta and in te r f a c e

th e two subm odels. F ig u re 3-1 i l l u s t r a t e s th e c a lc u la t io n a l scheme

o f CRESCENDO. A f te r each in te g e r s o lu t io n c a lc u la te d by th e

in te g e r p o r t io n o f STARTS o r when o p t im a l i ty i s a ch ie v e d , th e

c u r re n t m ix e d -in te g e r s o lu t io n i s p assed t o CONCERT th ro u g h th e main

2CRESCENDO i s an acronym f o r : A Code f o r R e fu e lin g and EnergyS chedu ling C on ta in in g an E v a lu a to r o f N u clea r D ec is io n s f o r O p era tio n .

I n i t i a l energ y & r e f u e l in g sch ed u le in p u t te d (E xecuted o n ly a t s t a r t o f a c a se )

CONCERT ________ djgSCENDO^MAIN PR0(£R4£l)___ ___ _________

C a lc u la te n u c le a r req u ire m en ts to m eet s p e c if ie d sch ed u le

C a lc u la te h a s ic n u c le a r c o s ts a s s o c ia te d w ith sch ed u le

E d it c u r r e n t sch ed u le ( in te g e r s o lu t io n )S e t up r e f u e l in g and energy sch ed u le f o r CONCERT

1 . C a lc u la te c o s t c o e f f i c i e n t s and s e t up d a ta fo r r e v is in g c o s t f u n c tio n in STARTS

2 . E v a lu a te system c o s t fu n c tio n f o r th e c u r r e n t sch ed u le s3 . Check c u r r e n t system c o s t fu n c tio n a g a in s t h e s t p re v io u s system

c o s t fu n c tio nU. S e t NUM=1 i f c u r r e n t c o s t fu n c tio n i s > b e s t p re v io u s s o lu t io n

S e t NUM=2 i f c u r r e n t c o s t fu n c t io n i s < b e s t p re v io u s s o lu t io n5. Save c o s t fu n c tio n v a lu e

R eport G e n e ra to r |

STARTS

)

I f NUM=2 r e v is e c o s t f u n c t io n , c a l c u la te co n tin u o u s o p tim a l s o lu t io n , and save s o lu t io n . E n te r m ix e d -in te g e r program

I f NUM*1 resum e m ix e d - in te g e r c a l c u la t io n s from th e p re v io u s in te g e r s o lu t io n

E x it a t each m ix e d - in te g e r s o lu t io n

F ig u re 3 -1 : B lock Diagram o f CRESCENDO

program . CONCERT an a ly z e s th e o p e ra t io n a l scheme and p roduces th e

d a ta fo r u p d a tin g o f th e c o s t fu n c tio n . The c o s t fo r th e c u r re n t

s o lu t io n i s c a lc u la te d and checked a g a in s t th e b e s t c o s t fu n c tio n

found th u s f a r . An in d ex i s s e t to in d ic a te t o STARTS w hether to

r e v is e th e o b je c t iv e fu n c tio n o r resume c a lc u la t io n s from th e l a s t

in t e g e r s o lu t io n . The STARTS program i s e n te re d and a new mixed-

in t e g e r s o lu t io n i s so u g h t. T h is flow i s re p e a te d u n t i l o p t im a l i ty

o r a s u i ta b le convergence req u irem en t i s m et. The CRESCENDO code

f o r t h i s i t e r a t i v e scheme and sample problem r e q u ir e s 330k o f co re

and fo u r f i l e s in a d d i t io n to th e r e q u ire d f i l e s . Two f i l e s a re

needed fo r t r a n s f e r r i n g d a ta between STARTS and CONCERT. A d d itio n a l

f i l e s can s to r e th e c u r r e n t problem b e in g so lv ed and a backup f i l e .

3.U Expansion o f Approach to a R eal System

The approach o f CRESCENDO can be expanded to a reed u t i l i t y3

system . For a r e a l sy stem , severed a d d i t io n a l c o n s t r a in t s would

have to be in c lu d e d in th e fo rm u la tio n o f th e STARTS m odel. In

a d d i t io n , r e s u l t s o f f u tu r e s tu d ie s w ith th e CRESCENDO model could

p roduce in s ig h t s which le a d to in te g e r s e le c t io n r u le s . However,

d i r e c t expansion o f th e model does p r e s e n t s e v e ra l co n ce rn s .

These concerns c e n te r on th e in c re a se in d e c is io n v a r ia b le s

and co m p u ta tio n a l tim e req u irem en ts f o r each n u c le a r u n i t added.

The fo rm u la tio n needs fo u r d e c is io n v a r ia b le s —th r e e con tin u o u s and

one in te g e r — f o r each tim e in t e r v a l . The a d d i t io n o f con tin u o u s

T hese c o n s t r a in ts would in c lu d e th o se assumed n e g l ig ib le f o r t h i s s tu d y ( i . e . , fo rc e d o u ta g e , t e r r i t o r i a l , t r a n s m is s io n , e t c . ) .

d e c is io n v a r ia b le s does n o t add g r e a t ly to th e com putation tim e .

P r e s e n t ly , th e sam ple system v i t h 672 rows and 588 co n tinuous

v a r ia b le s r e q u ir e s o n e -h a lf m inu te o f com putation tim e in th e c e n t r a l

p ro c e s so r o f an IBM 370/165 f o r an o p tim a l co n tinuous s o lu t io n . Once

an o p tim a l co n tin u o u s s o lu t io n has been o b ta in e d , th e m ix ed -in teg e r

program i s a c c e s se d . In c re a s in g th e in te g e r v a r ia b le s s ig n i f i c a n t l y

in c re a s e s th e tim e re q u ire d f o r an o p tim a l s o lu t io n . A d d itio n a l

in te g e r s make th e p o s s ib le number o f com binations t h a t m ight be

s o lu t io n s 2 , w here n = in t e g e r v a r ia b le s added, and N * o r ig in a l

number o f in te g e r v a r ia b le s . As v i t h th e in t e g e r s , th e a d d i t io n o f

rows a d v e rse ly a f f e c t s th e fo rm u la tio n .

The number o f rows added t o th e fo rm u la tio n i s e q u a l to

f iv e tim es th e number o f tim e in t e r v a l s in th e h o riz o n . The in c re a s e d

number o f rows needed t o r e p re s e n t a r e a l system would in c re a s e th e

com putation tim e a l s o . S in ce CRESCENDO i s a lre a d y a la rg e problem

(from a MIP s ta n d p o in t ) , in c re a s in g th e in te g e r v a r ia b le s and ro v

c o n s t r a in ts co u ld make th e fo rm u la tio n c o m p u ta tio n a lly u n a t t r a c t iv e .

3 .5 Comparison o f MIP to O ther Models Developed

3 .5 .1 P rev io u s Models Developed

There a r e th r e e m odels o f v a r io u s d eg rees o f s o p h is t ic a t io n

which have been developed to d e a l v i t h th e system problem . These a re

th e m odels o f Rhodes (b 2 ) ; o f Salmon, B e n n e tt, T uraage , and P rin c e

(3U, 3£ , 3 6 ); and o f D eaton (3 8 ) .

2k

H is to r i c a l l y , R hodes' model v a s th e f i r s t t o i l l u s t r a t e th e

f e a s i b i l i t y o f a l lo c a t in g energy t o u n i t s on a system -w ide b a s i s .

The main f e a tu r e o f h i s RUSOP code was th e u se o f a l i n e a r program

to a l lo c a t e energ y t o each n u c le a r u n i t and a f o s s i l u n i t e q u iv a le n t .

The a l lo c a t io n o f energy i s based on th r e e c o r r e l a t i o n s : b a tc h c o s t -

b u rn u p , b u rn u p -en erg y , and c r i t i c a l i t y - e n e r g y fu n c t io n s . Each o f

th e s e c o r r e l a t io n s i s a l i n e a r fu n c tio n . The c o s t c o e f f i c i e n t s o f th e

c o r r e la t io n s w ere e s ta b l is h e d o f f - l i n e f o r th e p a r t i c u l a r n u c le a r

u n i t s b e in g exam ined.

NUSOP re q u ir e s t h a t c e r t a in d a ta be known a p r i o r i . The

m ost im p o rtan t o f t h i s d a ta i s th e r e f u e l in g sc h ed u le s fo r each u n i t .

NUSOP i s ex ecu ted fo r a known r e f u e l in g sch ed u le . A u s e r 's exam ina

t i o n o f th e r e s u l t s i s done and a re v is e d r e f u e l in g schedu le must

be d e v ise d m anually . NUSOP i s th e n re ru n to g e t th e new r e s u l t s . The

r e s u l t s o f t h i s model d em onstra ted th e economic advan tages o f

sc h e d u lin g system -w ide o p e ra t io n .

A second model (ORSIM) developed a t Oak R idge N a tio n a l

L ab o ra to ry in c o n ju n c tio n v i t h TVA and Commonwealth Edison i s a

sem iautom ated a lg o rith m t o sch ed u le system -w ide o p e ra t io n . ORSIM

em ploys an i t e r a t i v e p ro ced u re on energy a l lo c a t io n . For a g iv en

sy stem r e f u e l in g sc h e d u le , ORSIM a l lo c a t e s energy t o a l l u n i t s based

on in c re m e n ta l c o s t s .

The b a s ic code u sed i s SIMUL (£) which u s e s p r o b a b i l i s t i c

s im u la tio n t o lo a d th e system . To f in d th e p ro p er lo a d in g o rd e r fo r

th e sy stem , an in c re m e n ta l c o s t f o r each n u c le a r and f o s s i l u n i t i s

p ro v id e d and a lo a d d u ra tio n cu rve (U_, £ ) i s e s ta b l is h e d f o r each

25

p e r io d . The c a lc u la t io n s ! f lo v lo a d s a l l th e u n i t s , r e c a lc u la te s

th e in c re m en ta l c o s ts fo r th e n u c le a r u n i t s , and re lo a d s th e system

u s in g th e lo a d d u ra tio n cu rve £ ) . The p r o b a b i l i s t i c s im u la tio n

accoun ts f o r th e fo rc e d o u tag e r a t e o f each u n i t . F ig u re 3-2 i s

a f lo v diagram o f ORSIM.

For an e s ta b l is h e d r e f u e l in g sc h e d u le , ORSIM sch ed u le s

m aintenance f o r th e n o n -n u c le a r u n i t s . The system i s lo ad ed in c lu d in g

any hydro u n i t s v h ic h a re co n s id e re d energy l im i te d . T re a tin g th e

n u c le a r u n i t s a s energy l im i t e d , th e f o s s i l u n i t s a re o f f - lo a d e d and

a new lo a d in g o rd e r i s e s ta b l i s h e d . P r o b a b i l i s t i c s im u la tio n i s used

to d e term ine th e expec ted en ergy p ro d u c tio n f o r each u n i t . The t o t a l

system d isc o u n te d c o s t i s c a lc u la te d . The system i s re lo a d e d and

p r o b a b i l i s t i c s im u la tio n i s a g a in employed t o de term ine exp ec ted energy

p ro d u c tio n . The th e n u c le a r c o s ts a re c a lc u la te d by a co re s im u la to r

and th e t o t a l system d isc o u n te d c o s t i s c a lc u la te d . A nev lo a d in g

o rd e r i s e s ta b l is h e d and checked a g a in s t th e c u r r e n t lo a d in g o rd e r .

I f th e re has been no change o r th e t o t a l c o s ts have converged , ORSIM

s to p s . O th erw ise , th e scheme i s re p e a te d u n t i l convergence. P re s e n tly

ORSIM la c k s an e f f e c t iv e c o re s im u la to r . However ORSIM i s th e most

com plete system model developed .

The system model by Deaton i s s t r u c t u r a l l y s im i la r to

ORSIM. The model g e n e ra te s a s e t o f r e f u e l in g sch ed u le s (U9) fo r th e

n u c le a r u n i t s under c o n s id e ra tio n . For each r e f u e l in g sch ed u le a

n o n -n u c le a r m ain tenance sch ed u le i s d ev e loped . F or each in t e r v a l in

th e h o riz o n a p r o b a b i l i s t i c s im u la to r i s u sed to a l l o c a t e energy to th e

26

Schedule M aintenance

A llo c a te Hydro & System Energy ( P r o b a b i l i s t i c S im u la tio n )

/ K=0 ^ & Eng.Fxd v U n its

Yes A llo c a te N uclear Energy ( P r o b a b i l i s t i c S im u la tio n )

No

N uclear Core S im u la to r

T o ta l D iscoun ted C o sts

New Loading O rder

/ C h n g .^ in Loading v Order?

No E d it

Yes

/ C h n g /^ v in D iscou tdV C O StS<€ .

NoYesK=0

F ig u re 3 -2 : ORSIM Flow C hart

27

f o s s i l u n i t s and a b lo c k o f n u c le a r energy . The o p tim iz a tio n model

d e te rm in es a lo a d in g o rd e r f o r th e n u c le a r u n i t s based on th e c u r re n t

assumed r e f u e l in g sc h e d u le . The c o re model v i l l e v a lu a te th e c o s ts

and e s ta b l i s h in c re m e n ta l c o s t s . The i t e r a t i o n betw een th e o p tim iz a

t i o n and c o re m odels c o n tin u e s u n t i l th e o p tim a l a l lo c a t io n o f energy

t o th e n u c le a r u n i t s f o r th e g iv e n r e fu e l in g sch ed u le i s ach iev ed .

The same c a l c u la t io n a l scheme i s u sed fo r each f e a s ib l e r e f u e l in g

sc h e d u le . A f te r a l l r e f u e l in g sc h ed u le s have been in te r r o g a te d , th e

b e s t sch ed u le i s s e le c te d .

The system model assum es: th e in c re m e n ta l c o s ts o f n u c le a r

u n i t s v i l l alw ays be ch eaper th a n f o s s i l in c re m en ta l c o s t s , n u c le a r

energy i s n o t a s c a rc e re s o u rc e , and n u c le a r u n i t lo a d in g i s im m ate ria l

w ith re s p e c t t o th e f o s s i l u n i t s . The o p tim iz a tio n model i s a netw ork

program w hich u se s in c re m e n ta l c o s ts f o r d e c is io n m aking.

Both ORSIM and D e a to n 's model a re advanced models in

com parison w ith R hodes' m odel. However, a l l th r e e m odels m a in ta in

th e same Im p o rtan t assum ption : r e f u e l in g d a te s a re known a p r i o r i .

3 .5 .2 R e la t io n s h ip o f CRESCENDO t o O ther Models

T here a r e m ajor d i f f e r e n c e s between CRESCENDO and th e

p re v io u s th r e e m odels. CRESCENDO does n o t need to know a p r i o r i

th e r e f u e l in g sc h ed u le s f o r th e n u c le a r u n i t s . CRESCENDO s im u lta n e o u sly

a l lo c a t e s a r e f u e l in g and energy sch ed u le f o r each u n i t . P roceed ing

from a f e a s ib l e s o lu t i o n , CRESCENDO seeks th e o p tim a l s o lu t io n in an

autom ated p ro c e d u re . F u rth e rm o re , in c re m e n ta l c o s ts a r e n o t u sed as

th e b a s is o f a l lo c a t in g energy p ro d u c tio n to t h e u n i t s . The c o s t

28b

c o e f f i c i e n t s a re s t r u c tu r e d to av o id th e u se o f in c re m e n ta l c o s ts .

T here a re a ls o s i m i l a r i t i e s in th e m odels.

A ll th r e e models and CRESCENDO employ a c o n c e p tu a lly

s im i la r c a l c u la t io n a l f lo v . In th e o p tim iz a tio n ro u t in e th e s t r a t e g y

i s to a l lo c a t e th e energy a n d /o r r e f u e l in g sc h e d u le . Through th e u se

o f c o r r e la t io n s o r a co re m odel, r e v is e d ,c o s ts a re c a lc u la te d acc o rd in g

t o th e newly developed sc h e d u le . The i t e r a t i o n i s re p e a te d u n t i l

convergence ( e i t h e r m anually o r a u to m a tic a l ly ) t o cm o p tim a l o r n e a r

o p tim a l s o lu t io n .

bThe c o s t c o e f f i c ie n t s o f th e o b je c t iv e fu n c tio n a re dependent o n ly on th e d e c is io n s made a t th e tim e th e c o a ts a r e in c u r r e d . The d eco u p lin g o f th e c o a t c o e f f i c ie n t s i s d e a lr a b le i f l i n e a r program ming (netw ork programming may be fo rm u la ted a s a l i n e a r program and v ic e v e r s a ) i s u sed in th e o p tim iz a tio n m odel.

CHAPTER k

DEVELOPMENT OF THE OPTIMIZATION MODEL

U .l I n tro d u c t io n

The p e rce n tag e o f n u c le a r c a p a c ity on a u t i l i t y ' s power

system a f f e c t s th e m ethods used t o sch ed u le th e o p e ra t io n o f u n i t s .

When n u c le a r u n i t s produce a s ig n i f i c a n t p o r t io n o f th e u t i l i t y ' s

e l e c t r i c i t y , th e y canqot b e o p e ra te d in d iv id u a l ly w ith o u t re g a rd

fo r t h e i r in t e r a c t io n w ith th e b a la n c e o f th e system . The economic

o p tim iz a tio n o f a u n i t can be exam ined o n ly by a n a ly z in g th e economic

t r a d e - o f f 8 o f th e e n t i r e system . When shou ld a n u c le a r u n i t be

re fu e le d ? How much energ y should be produced and over w hat c y c le

le n g th ? These cure some o f th e q u e s tio n s t h a t m ust be e x p lo re d in th e

c o n te x t o f th e sy s te m 's econom ics. The c o m p le x itie s o f th e n u c le a r

f u e l c y c le and n u c le a r u n i t o p e ra t io n n e c e s s i t a te s th e developm ent

o f a new te c h n iq u e f o r p la n n in g system o p e ra t io n . T h is te c h n iq u e

seek s th e s t r a t e g y which m inim izes th e c o s t o f m eeting th e lo a d

demand.

A c lo se d -lo o p te c h n iq u e composed o f two s u b d iv is io n s i s

developed t o in v e s t ig a te th e problem . One su b d iv is io n i s a model

(STARTS) developed to a l l o c a t e en erg y and r e f u e l in g d a t e s . The o th e r

su b d iv is io n i s a model (CONCERT) developed to c a lc u la te n u c le a r f u e l

c o s ts b a sed on th e t r a j e c t o r y o f th e system determ ined b y STARTS.

29

30

The b a s ic s t r u c tu r e o f each submodel i s d e f in e d .

k .2 System T ra je c to ry A llo c a to r and R e fu e lin g T arg e t S ch ed u le r (STARTS)

^ .2 .1 Dynamic L e o n tie f Model

The c u r r e n t p ro d u c tio n o f energy from a n u c le a r u n i t d u rin g

any tim e i n t e r v a l i s dependent on th e p a s t and fu tu re o p e ra t io n . I f

th e r e f u e l in g d a te s fo r n u c le a r u n i t s a re n o t known, th e problem o f

de te rm in in g en erg y p ro d u c tio n l e v e l s fo r th e u n i t s i s com plica ted

because th e problem i s tim e d ependen t. The h o rizo n i s r e p re s e n te d

as a s e r i e s o f d i s c r e te tim e in t e r v a l s (F ig u re U -l) e n a b lin g th e

tim e dependency t o be in c o rp o ra te d in to th e system m odel.

The b a s ic tim e in t e r v a l d u rin g which d e c is io n s a re made

must b e sm a lle r th a n any a n t ic ip a te d cy c le le n g th . T h is p e rm its

s y n th e s iz in g th e tim e I n te r v a l s i n t o c y c le s . The eq u a l tim e in t e r v a l s

a r e l a r g e enough t o r e f l e c t th e le n g th o f a r e f u e l in g o u tag e w hich i s

assumed t o ta k e an e n t i r e i n t e r v a l '’. Excess r e a c t i v i t y in d ic a te s

when th e n u c le a r u n i t i s r e f u e le d . I f th e c h a r a c t e r i s t i c s o f a n u c le a r

u n i t a r e known, r e a c t i v i t y may be r e l a t e d t o en e rg y , and as a

consequence th e p rim ary v a r ia b le o f th e d ec ision -m ak ing p ro cess

i s en erg y .

As i s no ted in C hapter 3 , a l i n e a r o b je c t iv e fu n c tio n i s th e

o p tim iz a tio n c r i t e r i o n . L in e a r programming g e n e ra lly can examine th e

a c t i v i t i e s a t o n ly one p o in t o f t im e . I f i t i s employed to examine a

5The le n g th o f th e r e f u e l in g i n t e r v a l i s an expec ted v a lu e o f h i s t o r i c a l r e f u e l in g o u tag e d a ta . Some o u ta g es a re a s s h o r t a s th r e e weeks and some a r e as lo n g as te n m onths. The c u r re n t average i s 2 .2 m onths.

Pre-Planning Horizon Planning Horizon Post-Planning Horizon

T T " 51 f w» • • • ! • • • • • ,» • • 1 r . V i » •

L t ! s ! f* 1 1

rCycle

R e fu e lin g

C ycle Cycle C ycle C ycle

TV » • « • •

Time I n te r v a li i i .1

. w • • •

F ig u re U -l: Breakdown o f P lan n in g H orizon in to D is c re te Time I n te r v a l s

32

p la n n in g h o riz o n o f s e v e ra l y e a r s , th e system model must be fo rm u la ted

t o p ro v id e a dynamic view o f th e v h o le h o r iz o n . The fo rm u la tio n must

a ls o coup le th e p a s t , p r e s e n t , and f u tu r e o p e ra t io n o f each n u c le a r

u n i t .

The dynamic L e o n tie f model (5 0 , 51) as shown in Appendix 1

p e rm its a dynamic problem t o be fo rm u la ted as a l i n e a r program . The

p h ilo so p h y o f th e dynamic L e o n tie f model i s t o conserve m a te r ia l and

c a r r y in v e n to ry . For th e system m doel, an energy c o n s e rv a tio n b a lan ce

i s w r i t t e n f o r each tim e i n t e r v a l and i s g iv en by e q u a tio n (U - l) :

- 0 ( l , - 1 )

w here = energy a v a i la b le a t th e s t a r t o f in t e r v a l j (energyin v e n to ry ) f o r u n i t k ;

e£ , * energy produced d u rin g in t e r v a l J (energy produced) f o r u n i t k ;

* energy r e fu e le d d u rin g in t e r v a l J (energy r e fu e le d )** f o r u n i t k ;

^ * in v e n to ry energy n o t p roduced d u rin g in t e r v a l J and a v a i la b le a t th e s t a r t o f i n t e r v a l J + l f o r u n i t k .

F ig u re U-2 i l l u s t r a t e s a sm a ll p o r t io n o f th e o p tim iz a tio n model

i l l u s t r a t i n g th e co u p lin g o f a n u c le a r u n i t ' s p a s t , p r e s e n t , and fu tu r e

o p e ra t io n th ro u g h e q u a tio n ( U - l) . The dynamic L e o n tie f model (e q u a tio n

b - l ) forms one p a r t o f th e mechanism f o r s e le c t in g r e f u e l in g d a te s .

The v a r ia b le s o f th e L e o n tie f s t r u c tu r e a re c o n s tra in e d f o r each

n u c le a r u n i t . These c o n s t r a in ts c o n t ro l th e energy p ro d u c tio n l im i t s

o f n u c le a r u n i t s and p r o h ib i t th e system model from r e f u e l in g energy

and p roducing energy d u rin g th e same i n t e r v a l . These c o n s t r a in t s

c o n ta in in t e g e r v a r ia b le s r e q u ir in g m ix e d - in te g e r programming fo r

s o lu t io n .

1M 2M

+1 -1 +1 -1

+1 -1 +1 -1

+1 -1 +1

+1 -1 +1

-1

+1 -1 +1

M inim ize C = Z Z C ., E . .o j ± i j i j

S u b je c t to

Where

E — E + E — E = 0^ l j *2J 3J ^ l . J + la l l E . , > 0

i J -

E i s th e energy v a r ia b le i f o r tim e i n t e r v a l J and th e r e a re M tim e in t e r v a l s

F ig u re U-2: Dynamic L e o n tie f Model f o r One N uclear U n it

3U

U .2 .2 M ix ed -In teg er Programming F o rm ula tion

The in te g e r c o n s t r a in ts a r e th e key t o th e system t r a j e c t o r y

model b e in g cap ab le o f se a rc h in g f o r an o p tim a l energy and r e f u e l in g

t r a j e c t o r y f o r th e system . The c o n s t r a in ts l im i t in g th e energy

p ro d u c tio n in any i n t e r v a l a r e e x p re sse d as fo llo w s :

(U-2)

where

4 ) * $ 2 ? }

= maximum energy t h a t u n i t k can p roduce in i n t e r v a l J , megawatt days e l e c t r i c a l (MWDe);

= minimum energy t h a t u n i t k can p roduce in i n t e r v a l J ,^ megawatt days e l e c t r i c a l (MWDe); and

= th e in te g e r v a r ia b le (0 ,1 ) a s s o c ia te d w ith u n i t k and J i n t e r v a l J , a z e ro (0 ) in d ic a t in g u n i t k i s o p e ra t in g

and a one (1 ) in d ic a t in g u n i t k i s b e in g r e f u e le d .

In a d d i t io n , in te g e r c o n s t r a in ts a r e c o n s tru c te d t o l i m i t th e amount

o f energy r e fu e le d . These c o n s t r a in t s aare g iv e n b y e q u a tio n (U -3):

- . + ft* > 0” ” ” (U-3)

* 0u k a y i A —V

R, = maximum energy t h a t can be r e f u e le d in to u n i t k d u rin g J i n t e r v a l J ;v

R, = minimum energy t h a t can b e r e f u e le d in to u n i t k d u rin g * i n t e r v a l J .

An a d d i t io n a l c o n s t r a in t co n n ec ts th e en ergy p ro d u c tio n o f th e u n i t s in to

a sy stem . The c o n s t r a in t p ro v id e s th e c o m p e tit io n betw een u n i t s f o r

35

energy p ro d u c tio n to meet th e system lo a d demand and th e in f lu e n c e

o f a u n it on th e b a la n c e o f th e system . The c o n s t r a in t i s :

+ eJ + eJ * Dj f o r each J (M » )

where E ^ . = energy produced (d e c is io n v a r ia b le ) by u n i t k d u rin g J i n t e r v a l J (MWDe):

E . = energy produced (d e c is io n v a r ia b le ) by th e f o s s i l e q u iv a le n t u n i t d u rin g i n t e r v a l j (MWDe) ;

E* = energy produced (d e c is io n v a r ia b le ) by in te rc h a n g e u n i t d u rin g in t e r v a l J (MWDe); and

D. = energy lo a d demand (MWDe) t o b e met by th e system d u rin g i n t e r v a l J .

These c o n s t r a in t s in e q u a tio n s ( U - l) , (U -2 ), (U—3 ) , and (U-U) do

more th an s e t r e f u e l in g d a te s and impose l i m i t s on th e c a p a b i l i t i e s

o f th e n u c le a r u n i t s (s e e Appendix l ) , th e y a l s o p ro v id e m echan ical

and n u c le a r c o n s t r a in t s on th e in d iv id u a l u n i t s .

S p e c ify in g th e maximum en erg y t h a t can be r e f u e le d , th e

maximum k - i n f i n i t y o f th e n u c le a r u n i t ' s co re i s l im i te d . T h is

maximum k - i n f i n i t y i s n o t exceeded by in c lu d in g e q u a tio n (U-5) in

th e fo rm u la tio n :

J* > C -S )

where = maximum energy in v e n to ry a llow ed f o r u n i t k d u rin gJ in t e r v a l J .

S p ecify ing t h e maximum energy t h a t can b e r e f u e le d , th e maximum burnup

on any f u e l node o f th e n u c le a r model i s l im i te d . Knowing th e maximum

energy in v e n to ry in any p e r io d and th e maximum and minimum p ro d u c tio n

r a t e s o f e n e rg y , th e minimum and maximum c y c le le n g th s a r e im p l ic i t l y

in c o rp o ra te d in th e system m odel. Thu3, th e system t r a j e c t o r y

p ro v id e s secondary c o n s t r a in ts which can be o m itte d from th e n u c lea r

model.

The system model does more th a n sim ply a l lo c a te energy

and r e fu e l in g d a te s to u n i t s . The c o n s t r a in ts e x e rc is e im p l ic i t

c o n t ro l over s e v e r a l o th e r v a r ia b le s o f th e system and n u c le a r m odels.

The in c o rp o ra tio n o f an o b je c t iv e fu n c tio n com pletes th e system model.

4 .2 .3 O b jec tiv e Function F orm ula tion

The r e l a t i v e v a lu e o f o b je c t iv e fu n c tio n ^ i s th e measure o f

e f f e c t iv e n e s s o f th e o p tim iz a tio n m odel. I t m easures th e v a lu e o f

th e r e tu r n a s s o c ia te d w ith a s e t o f d e c is io n s . S p e c i f i c a l ly , i t i s

th e v a lu e o f th e c o s t c o e f f i c ie n t s i n th e o b je c t iv e fu n c tio n which

determ ines th e a c t i v i t y l e v e l s o f th e d e c is io n v a r ia b le s .

The c o s t c o e f f i c ie n t s o f th e o b je c tiv e fu n c tio n embody th e

economics o f th e sch ed u lin g problem , and th e o p tim iz a tio n c r i t e r io n

i s th e m in im iza tio n o f th e c o s t f u n c t io n . The c o s t fu n c tio n i s th e

t o t a l p re s e n t-v a lu e d system o p e ra tin g c o s t fo r th e p lan n in g h o rizo n ,

and i s re p re s e n te d as e q u a tio n (4 -6 ) :

m inim ize [C , = J J ^ * j c j Ejf ♦ j c j eJ ♦ | J b j u j ] <1,-6)

where C = t o t a l system p re s e n t-v a lu e d o p e ra t in g c o s t ; oIf

C. . = p re s e n t-v a lu e d co s t c o e f f i c i e n t f o r energy v a r ia b le s i , fo r tim e in te r v a l J , fo r n u c le a r u n i t k ;

The o b je c tiv e fu n c tio n a l s o may be r e f e r r e d to as th e c o s t fu n c tio n in t h i s work.

37

E^. = energy d e c is io n v a r ia b le i , f o r i n t e r v a l J , and n u c le a r u n i t k ;

c f * p re s e n t-v a lu e d c o s t c o e f f i c i e n t f o r i n t e r v a l J and J th e f o s s i l e q u iv a le n t u n i t f ;f

E. » energy d e c is io n v a r ia b le f o r f o s s i l e q u iv a le n t u n i t f J and in t e r v a l J ;

C* *= p re s e n t-v a lu e d c o s t c o e f f i c i e n t f o r t h e ln te rh c a n g e u n i t d u rin g in t e r v a l J ;

E* = energy d e c is io n v a r ia b le f o r in te rc h a n g e u n i t d u rin g J i n t e r v a l J ;

b = p re s e n t-v a lu e d c o s t c o e f f i c i e n t f o r t h e in te g e r v a r ia b le f o r i n t e r v a l j and n u c le a r u n i t k ;

* in te g e r d e c is io n v a r ia b le (a llo w ed v a lu e s 0 and l ) f o r i n t e r v a l J and n u c le a r u n i t k .

The dynamic L e o n tie f s t r u c tu r e and c o n s t r a in t s n e c e s s i t a te s fo u r

I n te r v a l d e c is io n v a r ia b le s f o r each n u c le a r u n i t . E q u a tio n s ( U - l ) ,

(U -2 ), (U -3 ), (U-U), (U -5 ), and (U-6) com prise a fo rm u la tio n which

a l lo c a t e s energy and t a r g e t r e f u e l in g d a te s fo r each u n i t on th e

system . F ig u re U-3 i l l u s t r a t e s th e fo rm u la tio n . Each d e c is io n v a r ia b le

r e q u ir e s a c o s t c o e f f i c i e n t .

The c o e f f i c i e n t s r e f l e c t th e a c t i v i t i e s (d e c is io n s ) o c c u rr in g

o n ly d u rin g th e p a r t i c u l a r i n t e r v a l f o r w hich th e c o s t c o e f f i c ie n t s

a re a s s o c ia te d . The num era to rs o f th e c o e f f i c i e n t s a r e p r e s e n t

v a lu ed s in c e STARTS in v e s t ig a te s th e t r a d e - o f f s betw een a c t i v i t i e s

o c c u rr in g o v e r a number o f y e a r s . I f p r e s e n t-v a lu in g w ere o m itte d ,

th e STARTS model would b e making d e c is io n s b ased on d i f f e r e n t d o l l a r s

r a th e r th a n on a cannon d o l l a r .

Nuc

lear

Pl

ant

3 N

ucle

ar

Plan

t 2

Nuc

lear

Pl

ant

1 ’ inventoryCARRYIRQMAX 4 MIR

EREROY PRODUCTION

MAX A MIRREFUELERHtOY

*IHVEHTQRY CARRYING

MAX A MIR EHERGY

PRODUCTIOR

MAX A MIRREFUELEREROY

INVENTORYCARRYING

MAX A MIRENERGY

PRODUCTION

MAX A MIRREFUEL

. EHERGYSYSTEM

CONSTRAINT

F o s s il P la n t In ta g e r V ariab les (0-1) i wl m A. , , ,— ftwi wr n*nt ?— ,,— ftslnr PXaat 3— '

r*

®2 ®1 ®2‘ +1

♦1+1+1

+1

+1♦1

+1

♦1

♦1 ♦1

♦1

♦1♦1

-1

-1♦1 ♦1♦1 ♦1 +1

♦1♦1 ♦1 ♦1

OBJECTIVE FUNCTION: Minimise C ■ I I E < £ ,e J , ♦ Z C*E? + Z * Z Z b W0 w u ^ ^ • J J . J J v , J J

■ 0 ■ 0i t ! A s!

A 0 A 0 A 0 A 0 • 0 - 0 i E iA Eii Ei A Si A 0 A 0 A 0 A 0 - 0 - 0 s Ei a s ii E |a siA 0 A 0 A 0 A 0- Ei- E l

F igu re b-3: Form ulation o f a System Model fo r Determ ining Optimum O perating S tra te g y

39

The in t e r v a l c o s t c o e f f i c ie n t s f o r each n u c le a r u n i t a re

developed in Appendix 2 . The c o e f f i c i e n t a s s o c ia te d v i t h th e energy

in v e n to ry v a lu e i s d e f in e d a s th e p re s e n t-v a lu e o f th e i n d i r e c t

ch arg es on th e f u e l v h i l e i t r e s id e s in c o re d iv id e d by th e energy

in in v e n to ry . The t o t a l i n d i r e c t c o s t i s determ ined by e v a lu a tin g

th e w orth o f th e uranium in c o re a t th e s t a r t o f any i n t e r v a l and

m u ltip ly in g by an e f f e c t iv e i n t e r e s t r a t e . The i n t e r e s t c o s t i s th e n

d iv id e d by th e megawatt days o f e l e c t r i c i t y in in v e n to ry d u rin g

t h a t i n t e r v a l . Thus,

<aj * pj ‘ *"T>V

where C . . * p re s e n t-v a lu e d c o s t c o e f f i c i e n t s .o f th e energy in v e n to ry ^ v a r ia b le f o r u n i t k i n t e r v a l j ;

k *(C_. ) = t o t a l I n d i r e c t c o s t f o r th e f u e l in c o re f o r i n t e r v a l J , ■LJ u n i t k ;

“ energy in v e n to ry f o r u n i t k a t s t a r t o f i n t e r v a l J ; and

P j * p re s e n t v a lu e f a c to r t o r e fe re n c e d a te .

The c o s t c o e f f i c i e n t a s s o c ia te d v i t h th e energy p ro d u c tio n i s

d e f in e d as th e o p e ra t in g and m ain tenance (O&M) c o s ts f o r u n i t k d u rin g

in t e r v a l j . The O&M c o s t c o n ta in s b o th a f ix e d and v a r ia b le component

w hich in c lu d e s th e c o s t o f r e f u e l in g b u t does n o t in c lu d e th e f u e l c o s t .

The v a r ia b le component i s keyed to th e in t e r v a l c a p a c ity f a c t o r s . The

in t e r v a l O&M c o s t i s d iv id e d by th e energy produced and p re s e n t-v a lu e d

t o th e re fe re n c e d a te , w hich g iv e s :

. [ak + b V [ ] PCJj " 3 3 (fc-8)

4

uowhere * p re s e n t-v a lu e d O&M c o s t f o r u n i t k and i n t e r v a l J ;

a = f ix e d component o f O&M c o s t f o r u n i t k ;

Vb = v a r ia b le component o f O&M c o s t f o r u n i t k ;

= c a p a c ity f a c to r f o r u n i t k and in t e r v a l j ; and

E^j*» energy produced d u rin g in t e r v a l J by u n i t k .

The c o s t c o e f f i c ie n t f o r th e energy r e fu e le d i s d e f in e d as

th e sum o f th e cash flow s f o r any n u c le a r f u e l d u rin g an i n t e r v a l .

When th e u n i t i s r e f u e le d , a c o s t i s in c u rre d f o r procurem ent o f a new

f u e l b a tc h and a c r e d i t i s re c e iv e d f o r th e f u e l d isc h a rg e d . These

v a lu e s a re summed, d iv id e d by th e energy r e f u e le d , th e n p re s e n t-v a lu e d

to th e r e fe re n c e d a te . Thus,

c5j - [(s!f ♦ pj (1- 9>where C_. = p re s e n t-v a lu e d c o s t c o e f f i c ie n t f o r r e lo a d f u e l f o r

u n i t k d u rin g i n t e r v a l J ;

S , ■ p re s e n t-v a lu e d c o s t o f p ro c u rin g a r e lo a d f u e l b a tc h " which i s in s e r t e d d u rin g in te W a l J in to u n i t k

k(S j i s p re s e n t-v a lu e d to i n t e r v a l j ) ;

R* = p re s e n t-v a lu e d c r e d i t f o r d isc h a rg e f u e l f o r u n i t kkand i n t e r v a l J (R. i s p re s e n t-v a lu e d to in t e r v a l J ) ;

andv

■ energy r e fu e le d to u n i t k d u rin g in t e r v a l J w hich i s J a v a i la b le f o r p roducing in i n t e r v a l J + l .

S in ce a n u c le a r u n i t i s p roducing energy d u rin g most o f th e i n t e r v a l s

in th e p la n n in g h o r iz o n , a s s ig n in g t h i s c o s t c o e f f i c i e n t to a l l in t e r v a l s

betw een th e r e f u e l in g in t e r v a l and th e i n t e r v a l b e fo re th e n e x t

r e f u e l in g i n t e r v a l does n o t a f f e c t th e v a lu e o f o b je c t iv e fu n c tio n .

T h is i s th e p ro ced u re fo llow ed in STARTS.

k l

The c o s t c o e f f i c ie n t s o f th e in t e g e r v a r ia b le s do n o t

r e p re s e n t any c o s ts in c u rre d from system o p e ra t io n , a lth o u g h th e

c o e f f i c ie n t s a r e needed to fo rm u la te th e problem . However, th e

in te g e r s a re d e c is io n v a r ia b le s and t h e i r c o s t c o e f f i c i e n t s m ust be

in c lu d e d in th e c o s t o b je c t iv e fu n c tio n . A ssign ing a zero v a lu e t o

th e c o s t c o e f f i c i e n t s o f th e in te g e r s p re v e n ts t h e i r a c t i v i t y l e v e l

from a f f e c t in g th e m in im iza tio n o f th e c o s t fu n c tio n . T h e re fo re ,

b* = 0 (U-10)J

where b , «* c o s t c o e f f i c i e n t f o r th e in te g e r v a r ia b le o f u n i t k J in i n t e r v a l J .

E q u a tio n s (U -l) th ro u g h (U-6) r e p re s e n t a fo rm u la tio n o f th e

problem o f s im u lta n e o u s ly a l lo c a t in g r e f u e l in g d a te s and energy on

a system -w ide b a s i s . M ix ed -in teg e r programming se a rc h e s f o r th e

o p tim a l energy and r e f u e l in g t r a j e c t o r y (p la n ) f o r th e system . The7

r e s u l t s o f STARTS c a lc u la t io n s a r e p a ssed to a n u c le a r c o s t code

w hich does th e c o s t e v a lu a tio n o f th e sch ed u le developed .

k . 3 Code f o r H uclear C o sting o f am Energy and R e fu e lin g T ra je c to ry

CONCERT p ro v id e s a r a p id economic and i s o to p ic a n a ly s is o f

th e n u c le a r u n i t s . T h is model w i l l be ex ecu ted once f o r each n u c le a r

u n i t p roducing th e c o s ts which a r e used to u pdate th e c o s t fu n c t io n .

CONCERT (se e Appendix 3) i s b a s i c a l ly a m od ified v e rs io n o f th e

PODECKA (3 2 , 33) w hich i s d e sc r ib e d in Appendix 6 .

CONCERT i s a pseudo-one d im en sio n a l p o in t d e p le tio n a lg o rith m .

The p su e d o -d im e n s io n a lity a r i s e s from a d iv i s io n o f th e co re in to a

7 ____The n u c le a r code i s r e f e r r e d t o a s CONCERT v h ich i s an acronym f o r COde f o r N uclear C o stin g o f an Energy and R e fu e lin g T ra je c to ry .

U2number o f equivolum e nodes which a re a ss ig n e d a nodal power f a c to r

d e r iv e d from an in p u t te d power sh ap e . A nodal p o in t d e p le tio n scheme

i s employed to c a l c u la te th e nodal burnup added to each node fo r each

i n t e r v a l . This p su e d o -d im e n s io n a lity p ro v id es a r e a l i s t i c co re bum up

p a t te r n and a bookkeeping scheme f o r f u e l s h u f f l in g .

Because STARTS makes d e c is io n s f o r a p la n n in g h o rizo n

encom passing s e v e r a l y e a rs and s e v e ra l r e f u e l in g s , CONCERT must be

a m u lt ic y c le a lg o rith m which has lo g ic to s e l e c t th e f u e l b a tch

req u ire m en ts a t eac h r e f u e l in g . The a lg o rith m m ust be a b le to

d e te rm in e th e b a tc h req u irem en ts f o r each c y c le based on th e s t a t e

o f th e f u e l in c o re and upcoming c y c le req u irem en ts (energy and c y c le

l e n g th ) , s in c e STARTS p ro v id es CONCERT w ith v a r ie d com binations o f

c y c le le n g th s and energy re q u ire m e n ts . F ix in g b a tc h s iz e and e n r ic h

ment i s n o t s u i t a b le fo r t h i s a p p l ic a t io n and th e u se o f an o p tim iz a

t i o n scheme to s e l e c t b a tc h re q u ire m en ts r e p re s e n ts a s o p h is t ic a t io n

and co m p u ta tio n a l tim e requ irem en t n o t J u s t i f i e d c o n s id e r in g th e

p u rposes o f CRESCENDO. The code s e l e c t s b a tc h req u ire m en ts acco rd in g

to a s e t o f s im p l i f ie d r u le s which make use o f th e s t a t e o f th e f u e l

a t th e end o f each c y c le .

CONCERT u se s e m p ir ic a lly -d e te rm in e d c o r r e l a t i o n s to g e n e ra te

th e is o to p ic and k - i n f i n i t y d a ta f o r th e f u e l . From t h i s d a ta and

each c y c l e 's b a tc h req u ire m en ts th e cash flow i s developed fo r each

u n i t f o r th e h o r iz o n . The c o s t c o e f f i c ie n t s a r e c a lc u la te d by

CRESCENDO from th e cash flo w s. F ig u re U—h i l l u s t r a t e s a b lo c k

c a l c u la t io n s ! flow c h a r t o f CONCERT.

P ass d a ta from CRESCENDO

- M

1*3

Set n u c le a r c h a r a c t e r i s t i c s f o r each u n i t_______

S e t s ta r tu p tim e s and c y c le le n g th s

4 . _D ep le te nodes f o r a cy c le

and c a l c u la te uranium in v e n to ry fo r each in te r v a l w ith in c y c le

D eterm ine b a tc h s iz e and enrichm ent

S h u f f le fu e l s e p a ra t in g d isc h a rg e d fu e l i n t o b a tch es

_______I n s e r t f r e s h f u e l______

A ll c y c le s

com pleted ?

C a lc u la te u n i t ' s cash flow fo r r e f u e l in g and energy

__________ schedule__________

A ll u n i t s

com pleted

Convergence o f system to an p p tim a l s t r a t e g y

?

R etu rn to>..no „ CRESCENDO fo r

r e v is io n o fc o s t c o e f f i c i e n ts

R ep o rt G en era to r o f CRESCENDO

IEND

JF ig u re k-k : CONCERT Flow C hart

kk

U.l* F o s s i l E q u iv a le n t U nit Load

The CRESCENDO model in c lu d e s a re p re s e n ta t io n o f a f o s s i l

e q u iv a le n t u n i t lo a d which r e p re s e n ts th e n o n -n u c le a r b a la n c e o f th e

u t i l i t y system . The c a p a c ity o f th e n o n -n u c le a r b a lan ce o f th e

system i s a s s ig n e d to t h i s u n i t and th e c o s t c o e f f i c ie n t s a re a

w eig h ted c o m p ila tio n o f a l l th e n o n -n u c le a r u n i t s . The c o s t

c o e f f i c i e n t s f o r th e f o s s i l e q u iv a le n t u n i t a re assumed c o n s ta n t

f o r t h i s a n a ly s is . C hapter 7 c o n ta in s some comments co n cern ing th e

req u irem en ts o f f o s s i l e q u iv a le n t u n i t .

The f o s s i l e q u iv a le n t u n i t lo a d i s assumed e q u a l in

c a p a c i ty t o th e l a r g e s t n u c le a r u n i t on th e system . The c o s t

c o e f f i c i e n t s w i l l be a d ju s te d to r e p re s e n t s e v e ra l s i t u a t io n s . S in ce

m ain tenance fo r th e n u c le a r u n i t s i s assumed to occu r d u rin g r e f u e l in g

o u ta g e s , on ly th e b a lan ce o f th e system must be sch ed u led . This non

n u c le a r m ain tenance must be sch ed u led w ith th e knowledge o f th e

r e f u e l in g sch ed u le s f o r th e n u c le a r u n i t s . C hap ter 7 c o n ta in s f u r th e r

comments on sc h e d u lin g o f n o n -n u c le a r m a in tenance.

U.5 C o n s id e ra tio n s in Development o f th e O p tim iza tio n Model

I*.5 .1 U niqueness

The STARTS model produces an energy and r e f u e l in g schedu le

fo r each n u c le a r u n i t . The CONCERT model e v a lu a te s t h i s schedu le t o

p ro v id e th e c o s t c o e f f i c i e n t s . The mapping from th e energy and

r e f u e l in g tim e sp ace (E,T) o f STARTS t o th e b a tc h s iz e and en richm ent

space (B ,e) o f CONCERT must be u n iq u e ; o r e l s e th e r e i s no a ssu ran ce

t h a t th e most econom ical com bination o f b a tc h s iz e and en richm ent i s

found fo r each p o in t o f th e (E ,T) s c h e d u le . I f th e mapping from th e

(E ,T) space to th e (B ,e ) space i s n o t u n iq u e , th e r e may e x i s t s e v e ra l

p o in ts in (B ,e ) which w i l l co rrespond to a p o in t i n (E ,T) g iv en th e

s t a t e o f th e n u c le a r u n i t ' s f u e l . T h is c o n d itio n co u ld make th e

o p tim iz a tio n problem u n so lv a b le . I f th e mapping to th e (B ,e ) space

i s n o t u n iq u e , th e r e s u l t s o b ta in e d f o r any p a th in (E ,T) may n o t be

re p ro d u c ib le and th e o p tim a l p a th in (E ,T ) would n o t always be found.

I f a p o in t in (E ,T ) has a m u l t i p l i c i t y o f p o in ts in (B ,e) and th e

mapping does n o t s e le c t th e p o in t w hich l i e s on th e co rresp o n d in g

o p tim a l p a th in ( B ,e ) , th e system model w i l l keep s e a rc h in g f o r an

o p tim a l p a th when th e o p tim a l (E ,T) p a th a lre a d y h as been byp assed .

The mapping may be made un ique by e i t h e r o f two means: an

o p tim a l i ty se a rc h o r d e te rm in is t ic s e le c t io n r u l e s . The o p t im a l i ty

se a rc h would examine each p o in t in (B ,e ) co rresp o n d in g to a p o in t in

(E ,T ). These p o in ts in (B ,e) would se rv e as b ran ch es which must be

ex p lo re d when th e mapping f o r th e n ex t p o in t in (E ,T ) i s c o n s id e re d

r e s u l t in g in a d e c is io n t r e e . D ire c t s e a r c h , l i n e a r program m ing, and

dynamic programming a re examples o f s e a rc h e s t h a t co u ld be u sed . The

u se o f an o p t im a l i ty se a rc h i s o f l im i t e d va lu e due to th e computa

t i o n a l c o m p le x itie s o f a m u ltic y c le o p tim iz a tio n a lg o rith m and

co m p u ta tio n a l tim e re q u ire m e n ts .

The o th e r a l t e r n a t iv e i s to u se d e te r m in is t ic mapping from

(E ,T ) to (B ,e ) . A mapping i s c o n s tru c te d by e s ta b l i s h in g a s e t o f

r u le s w hich a re fo llow ed every tim e th e mapping i s a c c e s se d . T h is

U6

deterministic procedure results in a path being generated in (B,e)

which is feasible but not necessarily the optimal path for the (E,T)

path. However, the procedure ensures reproducible results from the

mapping. The mapping takes a minimal computational time and allows

the feasibility and applicability of CRESCENDO to be demonstrated.

Chapter 7 contains further comments on the mapping.

U .5.2 Convergence

The system problem will have a solution. The solution may

be the result of the utility's dispatcher or unit operator taking a

nuclear unit off line for refueling because the unit can no longer

maintain criticality. Whether the system problem formulated in

CRESCENDO converges to an optimal solution (local or global) depends

on the convexity of the cost function of the system. The mixed-

integer programming used in the CRESCENDO formulation determines the

activity levels of the decision variables based on a linear extrapola

tion of the cost coefficients.

Little investigation of the cost function's nature has been

done, especially when simultaneous variation of energy and refueling

schedules is considered. If the function is convex, the MIP package

will give a global optimal solution (energy and refueling schedule).

If the cost function is not convex, the linear formulation may not be

adequate to search for a global optimal solution and the technique may

give a local optimum. It may find the global optimum but this cannot

be distinguished from a local optimum. If the cost function is not

convex, a higher order cost function formulation may be needed to

achieve global optimality, however, this would present some calcula-

tional limitations. The linear formulation used in STARTS is assumed

adequate to achieve at least a local optimum. The convergence to

an optimum is demonstrated numerically by CRESCENDO.

U.5.3 Initial and End Conditions

The nuclear model needs pre- and post-planning horizon

operating conditions so the cost coefficients in equation (b-6)

can be calculated. These conditions are necessary since the nuclear

units are batch loaded and the system should be left in an acceptable

state. If the planning horizon starts during a non-refueling interval

for any nuclear unit, CONCERT must know when the last refueling

occurred, the state of the core at that refueling, the energy refueled,

and the characteristics of the batch inserted. This information

permits the establishment of cost coefficients for those intervals

in the horizon before the first refueling is encountered. The initial

conditions for the planning horizon, energy expended, and time

elapsed in the cycle at the start of the horizon specify the state

of the nuclear units and the system's current state. From this con

figuration, seeking the optimal operational trajectory for the

system may be initiated. For the purpose of planning, the condition of

the system is projected or hypothesized.

Similarly CONCERT must be given end connditions to allow

calculation of the cost coefficients for the last refueling interval

and the succeeding intervals to the end of the planning horizon. The

U8end effects of the horizon will affect the cost coefficients which

directly influence decisions made in STARTS. Thus, it is assumed

that each nuclear unit returns to its "equilibrium" cycle length

and energy and continues on that operational scheme. The units will

return to these conditions at each unit's last refueling in the

horizon. The equilibrium cycle values for energy refueled and cycle

length are used to calculate cost coefficients. This method of

handling the end of the horizon is used in an attempt to mitigate

end effects of a finite horizon.

1+.5.U Comments

The CRESCENDO model described illustrates the feasibility

of scheduling a system as an integrated network. However, several

assumptions embodied in the model for use on the hypothetical system

in this work are: any transmission of generation losses, territorial

constraints, forced outages, and refueling constraints (manpower,

etc.) can be considered negligible. These considerations could be

incorporated into CRESCENDO at some time in the future. Further,|it

is assumed that early refueling and coastdown for the nuclear unitsj

are valuable operational tools for the dispatcher or operator in the

short term and daily operation and are not considered in the CRESCENDOj

model.Although CRESCENDO is designed to investigate the system

scheduling problem in the mid-term (i.e., from 5 to 10 years into

the future) it can be employed for short-term investigations. There

is enough flexibility in CRESCENDO to examine the optimal path of

operation from the present to several years in the future . After

the first execution, CRESCENDO has a system trajectory and a set of cost coefficients. If the state of the system at the perturbation

is set up, the current cost coefficients may be used as the starting

point for optimization of the system following the perturbation.

8This flexibility can be used to optimize the system if a major perturbation causes deviation from the originally devised system trajectory (schedule).

CHAPTER 5APPLICATION OF CRESCENDO TO A SAMPLE SYSTEM

5.1 IntroductionThe feasibility and applicability of CRESCENDO can be

demonstrated using a sample utility system. The sample system

includes the units necessary to demonstrate competition among nuclear

units, the fossil equivalent unit and an interchange unit. The

analysis of this system is given in this chapter showing how CRESCENDO

generates feasible solutions and seeks an optimal solution. Several

additional cases are illustrated and a sensitivity analysis is

included.

5.2 Characteristics of the Sample Utility System

The generating system consists of five units, four rated

at 1000 MWe and one rated at 200 Me. Three of the 1000 MWe units

are nuclear units and the fourth 1000 MWe unit represents the non

nuclear balance of the system. The 200 MWe unit represents an inter

change unit. This unit accounts for intertie arrangements with other

utilities. The three nuclear units, which ere typical of a 1000 MWe

Sequoyah class nuclear unit, provide competition for a load and

refueling times. In order to differentiate the three nuclear units

50

w hile u s in g th e c h a r a c t e r i s t i c s o f o n ly one u n i t , th re e d i f f e r e n t

e f f i c i e n c i e s a re in c lu d e d . The th e rm al to e l e c t r i c a l co n v ers io n

e f f i c i e n c i e s g ive d i f f e r e n t c o s t c o e f f i c ie n t s which make th e

n u c le a r u n i t s appear d i f f e r e n t to th e STARTS m odel. The e f f i c i e n c i e s

a re 35, 3^ , and 33 p e rc e n t f o r n u c le a r u n i t s 1 , 2 , and 3 , r e s p e c t iv e ly .

The f o s s i l u n i t re p re s e n ts th e b a lan ce o f th e n o n -n u c le a r system and

th e sp in n in g r e s e rv e , and fo r th e se a n a ly se s th e c o s t c o e f f i c ie n t s

a re assumed c o n s ta n t f o r th e h o riz o n . F u rtherm ore , th e f o s s i l

c o e f f i c ie n t s sure assumed a r b i t r a r i l y la rg e ( s e v e ra l o rd e rs o f m agnitude)

when compared to th o s e o f th e n u c le a r u n i t s . T h is p e rm its th e

maximum n u c le a r u n i t co m p e titio n fo r th e lo a d w ith o u t c o m p e titio n

from th e f o s s i l e q u iv a le n t u n i t . A lthough t h i s s i t u a t io n i s

a r t i f i c i a l , i t p e rm its dem o n stra tio n o f th e f e a s i b i l i t y o f th e method

developed in t h i s work.

The in te rc h a n g e u n i t r e p re s e n ts energy a v a i la b le from

a d jo in in g system s. I t i s assumed t h a t i t s c o s t c o e f f i c i e n t s a re

la rg e (same o rd e r o f m agnitude as th e f o s s i l c o e f f i c i e n t s ) and c o n s ta n t

b u t s l i g h t l y le s s th a n f o r th e f o s s i l u n i t . The in te rc h a n g e u n it i s

assumed ch eap er th a n th e f o s s i l s in c e energy i s swapped r a th e r than

purchased as rep lacem ent pow er. The in te rc h a n g e u n i t i s assumed

a v a i la b le fo r th e e n t i r e h o rizo n a lth o u g h CRESCENDO has a p ro v is io n to

r e s t r i c t o p e ra t io n o f th e u n i t as f o r l im i te d a v a i l a b i l i t y d u ring a

summer o r w in te r demand peak .

The lo a d demand f b r one y e a r o f th e 7 -y e a r h o riz o n i s g iven

in F ig u re 5-1 • T h is lo a d p a t te r n i s assumed c o n s ta n t and r e p e t i t i v e

fo r each y e a r o f th e h o riz o n . The tim e in t e r v a l s a re e q u a l and two

Load

(xlO

MWDe

)

25

20

15

10

5

Jan-Feb Mar-Apr May-Jun Jul-Aug Sep-Oct Nov-Dee

Figure 5-1: Load Demand for Each Year of the Horizonvnro

months in le n g th . The f i r s t f iv e y e a rs o f th e h o rizo n can be used

fo r p la n n in g o f sch ed u le s and th e rem ain ing tv o y e a rs a s a b u f f e r

b e fo re th e end o f th e h o rizo n i s en co u n te red . The lo a d p a t te r n has

s e v e ra l low-demand in t e r v a l s advantageous to r e f u e l in g . Any lo a d

p a t te r n can be accommodated ( t h a t i s , lo ad grow th) as lo n g as th e r e

i s enough c a p a c ity t o meet th e h ig h e s t demand. T here must a ls o be

in t e r v a l s to a llo w r e f u e l in g th e n u c le a r u n i t s . I f th e s e c o n s tr a in ts

a r e n o t m e t, th e model s t a t e s t h a t no f e a s ib le s o lu t io n was found.

The en ergy i s a l lo c a te d to each u n i t as an energy b lo ck which must be

p roduced . Any lo a d v a r ia t io n s ( th a t i s , d a l l y , w eekly , m onthly lo ad

demand c u rv e s ) o f a s h o r te r d u ra tio n th a n th e tim e in t e r v a l a re

su p p ressed and must be d e a l t w ith s e p a ra te ly . The a d d i t io n o r

r e t i r e m e n t o f c a p a c ity i s n o t co n s id e re d in th e sample problem .

However, this condition can be Incorporated in CRESCENDO.A ll th e u n i t s a re assumed t o have a fo rced o u tage r a t e o f

z e ro . T hus, th e n u c le a r u n i t s must e i th e r be o p e ra tin g o r r e f u e l in g .

The f o s s i l and in te rc h a n g e u n i t s a r e 100 p e rc e n t a v a i la b le , how ever,

r e q u ire d m ain tenance fo r th e f o s s i l e q u iv a le n t u n i t o ccu rs c o n tin u o u s ly .

The c a p a c i ty which i s n o t in s e rv ic e d u rin g an I n te r v a l i s on

m a in tenance . The in te rc h a n g e u n i t i s assumed n o t to need m ain tenance.

M aintenance f o r th e n u c le a r u n i t s i s assumed t o occu r d u rin g r e f u e l in g

and to ta k e a f u l l I n t e r v a l .

A wide ran g e o f i n i t i a l s t a t e s f o r th e system msy be used

w ith CONCERT c o n ta in in g th e n u c le a r d a ta and STARTS th e system

d a ta . T h is f l e x i b i l i t y a llo w s th e fo rm u la tio n t o seek a new optimum

i f a m ajor p e r tu r b a t io n occu rs on th e system . T h is s i tu a t io n

w i l l be i l l u s t r a t e d l a t e r in t h i s c h a p te r .

5 .3 A n a ly s is o f th e Sample U t i l i t y System

CRESCENDO d e te rm in es th e energy a l lo c a t io n and r e f u e l in g

sch ed u le f o r a sample system . The i n i t i a l s t a t e o f th e system

c o n s is ts o f th e e q u ilib r iu m c y c le f o r each n u c le a r u n i t . The

e q u ilib r iu m c y c le p a ram ete rs a r e 1000 MWe produced p e r c y c le , one

y e a r cy c le le n g th and an 85 p e rc e n t c a p a c ity f a c t o r . The c o s t

c o e f f i c i e n t s were developed by in p u t t in g e q u ilib r iu m c y c le s in to

CONCERT. These c o e f f i c i e n t s a re t r a n s f e r r e d t o STARTS.

F or t h i s a n a l y s i s , th e i n i t i a l s t a t e s o f th e n u c le a r u n i t s

a re assumed t o b e : ( l ) u n i t 1 h as expended t h r e e - f i f t h s o f i t s

a v a i la b le en ergy f o r th e c y c le and t h r e e - f i f t h s o f i t s y e a r ly c y c le ,

(2 ) u n i t 2 h as expended f o u r - f i f t h s o f i t s a v a i la b le energy and fo u r -

f i f t h s o f i t s y e a r ly c y c le , and ( 3 ) u n i t 3 h as expended o n e - f i f t h o f

i t s a v a i la b le energy and o n e - f i f t h o f i t s y e a r ly c y c le le n g th .

The end c o n d itio n s f o r th e n u c le a r u n i t s a re assumed t o be th e

e q u ilib r iu m c y c le . At th e l a s t r e f u e l in g w ith in th e h o r iz o n , th e

r e s p e c t iv e n u c le a r u n i t i s p la c e d on th e e q u ilib r iu m c y c le . The r e tu rn

to e q u i lib r iu m c o n d itio n s le a v e s th e system in an a c c e p ta b le s t a t e .

The c o n s t r a in t s in c o rp o ra te d in to CRESCENDO l im i t each u n i t 's

p ro d u c tio n . The maximum a n u c le a r u n i t o r f o s s i l u n i t may produce i s

1000 MWe. The minimum power produced by a n u c le a r u n i t i s 500 MWe.

The maximum th e in te rc h a n g e u n i t may produce i s 200 MWe. The f o s s i l

55and interchange units have a lower limit of zero. The nuclear units

may reload energy for 18 months of full power operation; the minimum

energy refueled is equivalent to six months of full power operation.

The maximum energy potential in core cannot exceed an 18-month full

power cycle.

5.3.1 Selection of Refueling Times and Energy Levels -

Case One

The first analysis demonstrates the feasibility of simultaneous

selection of refueling dates and energy schedules for the hypothetical

utility system. The first feasible schedule determined by CRESCENDO9is shown in Figure 5-2 . The three nuclear units' costs for the 7-year

planning horizon and the horizon's total system cost (in $1000) for

all units are listed in Table 5-1* The first system schedule

determined is generally not near the optimal cost schedule. Using the

calculational scheme indicated in Chapter 3, CRESCENDO generates more

system schedules which are progressively less expensive. The second

schedule produced is shown in Figure 5-3 and the costs in Table 5-1.O

The system cost of this schedule is $7,088 x 10 million less than the

initial schedule. Examining Figures 5-2 and 5-3, it is apparent

that nuclear unit 2 absorbs some of the load swings of nuclear unit 3.

Nuclear unit 1 operates at essentially full load. The second schedule

has reduced the amount of fossil and interchange energy as illustrated

9The asterisks in the figures indicate the unit's energy production level (MWDe) for each interval of the planning horizon. Asterisks on the 0.0 line indicate refueling intervals. The ordinate is linear from 3.0U to 6.08 MWDe(xl0,000) while the ordinate from 0.0 to 3.0U is compressed.

«UDE NUCLEAR PLANT 1 NUCLEAR PLANT 2IX I n . 0 0 1__________________________________________________________ CX_1000p_»__________________________________

6 .0 6 * * * * * * * * * * * * * » » « » * * * » * * • • « » * • * • • • * 6 .0 6 ** * * * * * * * * * * * * * * * * * * * * * * * * * » * • * * . **»1 1

•11

11

5 .3 2 ♦ 5 .3 2 ♦

1 1

6 .5 6 ♦

1 |

• » 9 ♦ —

i 1 i

1 1

1 1

3 .8 0 ♦ 1

3 .8 0 •

1 •

13 .0 6 ♦

13 .0 6 ♦ •

b 1C l b 2C 25 30 35 60 5 10 15 2 0 25 30 35 60

I N T E R V A L 1 N T E R V A L

MMOE NUCLEAR PLANT 3IX 100001 _ .............................................................. ........... ._

____ 6 .C 8 * ** * * * * * * * * * * * •1I • • •I1 .. ..................................................................... • ............................... . . .

5 .3 2 ♦I ........ ... . . . . . . . . ........................ .........1 • •1 *

6 .5 61 • ♦

. . . . ---------------------------------- --------- ---------------------- ----------------------

11 _ ____________ . ...................................... ...................... . . . . . . .

11

3 . BO ♦1 . . . ...................................... _ . . . .

_______________________ __ ________ „ _

| • * • •1. . . . ........................... . . ____________ - ________ .1* •

. . .

3 .0 6

. . . . . . _ _ -------------------- . . . ------------- ____0 .05 10 15 20 25 30 3 5 60

. .

FIGUREI N T E R V A L

5 - 2 : UNIT ENERGY PRODUCTION LEVEL FOR SCHEDULE 1

Table 5-1

Schedule

1

2

System Costs for Case 1, Schedules 1 and 2 (Thousand $)

Nuclear Fossil Interchange Total System Costs

$301,383 $110,1+09,600 $53,768,1+16 $l61+, 1+79,399

290,563 105,772,672 51,327,328 157,390,563

vn-a

HUbE IX 1 0 .0 01

NUC MriOEj x _ioogo_t.

NUCLEAR PLANT 2

$.32

3 .6 0

3.06 ♦JU<r. LO !•

I N f

20 2$

E~R V A I

6 .C 8 ♦ * • • • « * * • * •I

. . . I . *I

L____________5 .3 2 ♦. . I ___________

I *I . ...........................I

_ 4 .5 6 _ * _______________I

. I _______ .II

3 .8 0 ♦ I ____________

III

. 3 . 0 4 ♦

0 .0 ♦ - » ► ----------

•• •

30 35 40 10 15 2 0 25

I N T E R V A L

- * ♦ -

30"' 35 40

HUOE IX 100001

NUC

II

5 .3 2 «

I.I4 .5 6 *

I I

I I

3 .8 0 * II I

I_ 3 .0 4 _ «

o.o — ♦ » — * 10 IS 20 25

1 N T E R V A L

30 35— ♦ -

40

FIGURE 5 - 3 i UNIT ENERGY PRODUCTION LEVEL FOR SCHEDULE 2

ui03

59by the reduction in their cost components while the total nuclear

costs have decreased from the first schedule.

Continued improvement of the cost function is achieved by

providing sufficient computation time. Starting with schedule 2,

subsequent schedules sure found by CRESCENDO. Table 5-2 summarizes

the costs for each schedule. As these schedules indicate, the

minimum total system operating cost is decreasing by substantial,

dollar amounts. This is produced by the artifically large cost

coefficients assigned to the fossil and interchange units which cause

a substantial decrease in the cost function for a small change in

energy produced by these units.

The best schedule (schedule 11) generated by CRESCENDO has

a system operating cost of 13.6 x 1010 dollars. This schedule is

shown in Figure 5- . The cost components of this schedule are listed

in Table 5-3. Nuclear unit 3 is base loaded at 1,000 MW electrical

for cycle lengths of 12 months10, lU months, 18 months, 12 months,

16 months, and 1*» months, respectively. The levelized fuel cycle cost

for this unit is 2U.3**6 cents/MBtu. Table 5- lists the schedule's

fuel cycle cost for all three nuclear unit3. Nuclear unit 2 is

essentially base loaded at 1,000 MW electrical for the planning horizon. The cycle lengths for this unit are 12 months, ll» months, l*t months,

lU months, 18 months, 12 months, and 10 months. The levelized fuel

cycle cost is 23.98U cents/MBtu. The fuel cycle cost data is listed

in Table 5-b* The schedule shows that nuclear unit 1 absorbs most of

10The first cycle length for each nuclear unit is the result of energy produced and time elapsed prior to the start of the planning horizon and thq extension of the cycle into the planning horizon.

Table 5-2Breakdown of Total System Costs (Thousand $)

hedule Nuclear Fossil Interchange Total System Costs

3 $298,236 $100,957,552 $1+9,01+9,520 $150,305,3081+ 301,718 95,809,661+ 53,768,1+16 11+9,879,7985 287,588 105,201,728 1+3,213,120 11+8,702,1+366 305,292 103,629,280 1+3,11+0,81+8 11+7,075,1*207 292,270 98,789,^0 1+5,858,976 11+1+, 9U0,6868 302,1+89 96,397,968 1+1+ ,591 • 581+ 11+1,292,01+19 308,1*95 86,983,90U 53,1+1+8,576 1U0,71+0,97510 306,793 89,176,816 1+7,978,880 137,1+62,1+89

OSo

NUCLEAR PLANT 1IX 100001 _____ _ _ _____ ____________

6.06 ♦»* *♦•**• ••• • * *•* ••••••** *»*••

HWOE IX 100001

NUCLEAR PLANT 2

5 .3 2

4 .5 6

3 .0 0

3 .0 4 •

0 .0 ♦ -

6 . 0 0

5 .3 2

4 .5 6

3 .8 0

3 .0 4 ♦

5 10 15 20 25 30 35 " 40

I N T E R V A L

. . NUCLEAR PLANT 3IX 100001

0 .0 ♦»»"' • >■ ■■■ —»—♦5 10 15 20 25 30 35 40

I N T E R V A L

5 .3 2

4 .5 6

3;so

3 . 0 4 * ____________ ____ ______

0 . 0 4 - " ♦ ------- 4 » ~ —— . ------------------. --------- I —

5 10 15 20 25 30 35 40

I N T E R V A L

FIGURE S - 4 : UNIT ENERGY PRODUCTION LEVEL FOR SCHEDULE 11

TABLE 5-3:

INT date energy prod ' ENERGY REFLO INV ENERGYKMO X 10000 MWO X 10000 MWO X 1Q0C

1 7 8 . 3 3 3 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 72 7 8 . 5 0 0 6 . 0 8 3 3 0 . 0 6 .0 8 3 33 7 8 . 6 6 7 0 . 0 3 6 .5 0 0 0 0 . 04 7 8 . 8 3 4 6 . 0 8 3 3 0 . 0 3 6 . 5 0 0 05 7 9 . 0 0 1 6 . 0 8 3 3 0 . 0 3 0 .4 1 6 66 7 9 . 1 6 8 6 . 0 8 3 3 0 . 0 2 4 . 3 3 3 37 7 9 . 3 3 5 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 08 7 9 . 5 0 2 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 79 7 9 . 6 6 9 6 . 0 8 3 3 0 . 0 6 .0 8 3 3

10 7 9 . 8 3 6 0 . 0 4 8 . 6 6 6 6 0 . 011 8 0 . 0 0 3 4 . 5 2 1 1 0 . 0 4 8 . 6 6 6 612 8 0 . 1 7 0 6 . 0 8 3 3 0 . 0 4 4 .1 4 5 513 8 0 . 3 3 7 6 . 0 8 3 3 0 . 0 3 8 . 0 6 2 214 8 0 . 5 0 4 6 . 0 8 3 3 0 . 0 3 1 .9 7 8 815 8 0 . 6 7 1 4 . 7 0 7 3 0 . 0 2 5 .8 9 5 516 8 0 . 8 3 8 6 .0 8 3 3 0 . 0 2 1 .1 8 8 217 8 1 . 0 0 5 3 . 4 0 6 7 0 . 0 1 5 .1 0 4 918 8 1 . 1 7 2 5 .6 1 4 9 0 . 0 1 1 .6 9 8 219 8 1 . 3 3 9 6 . 0 8 3 3 0 . 0 6 . 0 8 3 320 8 1 . 5 0 6 0 . 0 1 8 .2 5 0 0 0 . 021 8 1 . 6 7 3 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 022 8 1 . 8 4 0 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 723 8 2 . 0 0 7 6 . 0 8 3 3 0 . 0 6 .0 8 3 324 8 2 . 174 0 . 0 4 8 . 6 6 6 6 0 . 025 8 2 . 3 4 1 6 . 0 8 3 3 0 . 0 •*8.666626 8 2 . 5 0 8 6 . 0 8 3 3 0 . 0 4 2 . 5 8 3 327 8 2 . 6 7 5 6 . 0 8 3 3 0 . 0 3 6 .5 0 0 028 8 2 . 8 4 2 6 . 0 8 3 3 0 . 0 3 0 . 4 1 6 629 " 8 3 . 0 0 9 6 . 0 8 3 3 0 . 0 2 4 . 3 3 3 330 8 3 . 1 7 6 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 0

SYSTEM COSTS FOR SCHEDULE UNUCLEAR PLANT 1

COST ENERGY COST ENERGY COST INV TOTAL COST CUM COSTPROO S/NWO REF S/KWD 8/MWO THOUSAND S THOUSAND $

8 . 8 1 1 0 3 4 . 5 5 1 4 3 . 9 0 2 5 1 0 1 0 .8 0 0 1 0 1 0 .8 0 08 . 6 9 6 7 3 4 .1 0 3 3 7 . 6 8 1 6 9 9 6 .3 4 5 2 0 0 7 .1 4 50 . 0 3 7 .2 7 5 9 0 . 0 1 3 6 0 5 .6 6 4 1 5 6 1 2 .8 2 8

1 3 .7 8 3 9 3 6 .7 9 2 5 2 . 2 8 0 1 1 6 7 0 .7 4 0 1 7 2 8 3 .5 6 68 . 3 6 2 8 3 6 .3 1 5 5 1 .3 4 6 0 9 1 8 . 1 4 1 1 8 2 0 1 . 7C78 . 2 5 4 3 3 5 . 8 4 4 6 1 . 6 5 5 5 9 0 4 . 9 7 9 1 9 1 0 6 .6 8 48 . 1 4 7 3 3 5 .3 7 9 8 2 . 1 7 2 1 8 9 2 . 0 4 0 1 9 9 9 8 .7 2 38 .C 4 1 7 3 4 .9 2 1 0 3 . 2 0 6 5 8 7 9 . 3 2 0 2 0 8 7 8 .0 3 97 . 9 3 7 4 3 4 .4 6 8 2 6 . 3 1 1 6 8 6 6 .8 1 5 2 1 7 4 4 .8 5 20 . 0 3 9 .2 6 2 2 0 . 0 1 9 1 0 7 .5 7 0 4 0 6 5 2 .4 2 2

1 5 .9 3 0 3 3 8 .7 5 3 1 1 . 5 9 7 0 1 4 9 7 .4 3 9 4 2 3 4 9 .6 5 97 . 6 3 2 6 3 8 . 2 5 0 6 0 . 8 6 6 7 8 4 6 .9 1 9 4 3 1 9 6 .7 7 77 . 5 3 3 7 3 7 . 7 5 4 6 0 . 9 8 8 9 8 3 4 .6 8 5 4 4 0 3 1 .4 6 17 . 4 3 6 0 3 7 .2 6 5 0 1 . 1 5 8 0 8 2 2 .6 6 1 4 4 8 5 4 .1 2 18 . 6 8 4 5 3 6 .7 8 1 8 1 . 4 0 7 0 7 7 3 .1 5 6 4 5 6 2 7 .2 7 37 . 2 4 4 4 3 6 .3 0 4 9 1 .6 9 3 2 7 9 9 .4 6 8 4 6 4 2 6 .7 3 8

1 0 .6 7 2 3 3 5 .8 3 4 2 2 . 3 3 7 3 7 1 6 .6 2 1 “ 4 7 1 4 3 .3 5 57 . 4 2 6 8 3 5 .3 6 9 5 2 . 9 7 3 9 7 6 4 . 9 0 7 4 7 9 0 8 .2 6 26 . 9 6 6 2 3 4 . 9 1 0 9 5 . 6 2 9 7 7 6 6 . 2 5 0 4 8 6 7 4 .5 0 60 . 0 2 6 .6 6 5 2 0 . 0 4 8 6 6 .3 9 1 5 3 5 4 0 .8 9 8

1 1 .0 4 1 1 2 6 .3 1 9 4 3 . 4 5 8 8 1 3 0 2 .8 9 5 5 4 8 4 3 .7 9 36 . 6 9 8 7 2 5 .9 7 8 2 2 . 5 5 2 8 7 1 8 .1 0 0 5 5 5 6 1 .6 9 16 . 6 1 1 9 2 5 .6 4 1 3 5 .0 2 4 9 7 0 7 . 9 0 1 ' “ 5 6 2 6 9 .7 6 90 . 0 3 2 .7 3 6 6 0 . 0 1 5 9 3 1 .7 9 7 7 2 2 0 1 .5 6 3

1 0 .4 7 9 5 3 2 .3 1 2 1 1 .2 6 1 2 1 2 5 1 .2 6 6 7 3 4 5 2 .8 1 36 . 3 5 8 0 3 1 .8 9 3 1 0 . 7 0 9 0 6 8 8 . 6 7 6 7 4 1 4 1 .4 3 66 . 2 7 5 5 3 1 . 4 7 9 6 0 . 8 1 3 8 6 7 8 .7 8 2 7 4 8 2 0 .1 6 86 . 1 9 4 2 3 1 . 0 7 1 4 0 . 9 6 0 8 6 6 9 .0 5 3 7 5 4 8 9 .1 6 86 . 1 1 3 9 3 0 .6 6 8 5 1 . 1 8 1 8 6 5 9 . 4 8 8 7 6 1 4 6 .6 2 56 . 0 3 4 6 3 0 .2 7 0 8 1 . 5 5 0 6 6 5 0 .0 8 3 7 6 7 9 8 .6 8 8

ONfU

TABLE 5-3 (CONTINUED)NUCLEAR PLANT 2

I NT' DATE ENERGY PROO ENERGY REFLD INV ENERGY COST ENERGY COST ENERGY COST INV ' TOTAL COST CUM COSTMWD X 10000 MWO X 10000 MHO X 10000 PROO S/MWO REE S/MWD S/MWD THOUSAND S THOUSAND S

1 7 8 . 1 6 6 6 . 0 8 3 3 0 . 0 6 .0 8 3 3 8 . 8 1 1 0 3 5 .9 7 1 9 7 . 8 7 2 5 1 0 1 4 .9 1 1 1 0 1 4 .9 1 12 7 8 . 3 3 3 0 . 0 3 6 .5 0 0 0 0 . 0 0 . 0 3 8 .5 8 0 8 0 . 0 1 4 0 8 1 .9 8 4 1 5 0 9 6 .8 9 53 7 8 . 5 0 0 6 . 0 8 3 3 0 . 0 3 6 . 5 0 0 0 1 3 .9 6 4 9 3 8 .0 8 0 6 2 . 3 6 0 6 1 7 1 1 .1 6 6 1 6 8 0 6 .0 5 9A 7 8 . 6 6 7 6 . 0 8 3 3 0 . 0 3 0 . 4 1 6 6 8 . 4 7 2 6 3 7 .5 8 6 8 1 . 3 9 3 4 9 3 9 . 2 4 5 1 7 7 4 7 .3 0 15 7 8 . 8 3 4 6 . 0 8 3 3 0 . 0 2 4 .3 3 3 3 8 . 3 6 2 8 3 7 .0 9 9 4 1 .7 1 3 6 9 2 5 .7 2 0 1 6 6 7 3 .0 2 06 7 9 . 0 0 1 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 0 8 . 2 5 4 3 3 6 .6 1 8 3 2 . 2 4 8 2 '9 1 2 .4 2 9 1 9 5 b 5 .4 4 57 7 9 . 1 6 8 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 7 8 . 1 4 7 3 3 6 .1 4 3 5 3 . 3 1 8 4 8 9 9 .3 6 6 2 0 4 8 4 .8 0 98 7 9 . 3 3 5 6 . 0 8 3 3 0 . 0 6 .0 8 3 3 8 .0 4 1 7 3 5 .6 7 4 9 6 . 5 3 1 4 8 6 6 .5 2 7 2 1 3 7 1 .3 3 29 7 9 . 502 0 . 0 3 6 .5 0 0 0 0 . 0 0 . 0 3 6 .3 2 7 5 0 . 0 1 3 2 5 9 .5 3 1 3 4 6 3 0 .8 6 3

10 7 9 . 6 6 9 6 . 0 8 3 3 0 . 0 3 6 .5 0 0 0 1 2 .7 4 5 6 3 5 .8 5 6 5 2 .0 9 6 8 1 5 4 0 .6 6 3 3 6 1 7 1 .5 4 311 7 9 . 8 3 6 ' 6 .0 8 3 3 0 . 0 3 0 .4 1 6 6 7 . 7 3 2 9 3 5 .3 9 1 5 1 .2 3 7 6 8 4 6 .6 6 6 3 7 0 1 8 .4 0 612 8 0 . 003 6 . 0 8 3 3 0 . 0 2 4 . 3 3 3 3 7 . 6 3 2 6 3 4 .9 3 2 6 1 . 5 2 2 1 8 3 4 .6 8 6 3 7 6 5 3 .0 9 013 8 0 . 1 7 0 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 0 7 . 5 3 3 6 3 4 .4 7 9 7 1 .9 9 6 8 8 2 2 .7 1 5 3 6 6 7 5 . 6C1I* 8 0 . 3 3 7 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 7 7 . 4 3 60 3 4 .0 3 2 6 2 . 9 4 7 4 6 1 0 .9 4 8 3 9 4 6 6 .7 4 615 8 0 . 5 0 4 6 . 0 8 3 3 0 . 0 6 .0 8 3 3 7 . 3 3 9 5 3 3 .5 9 1 3 5 . 8 0 1 0 7 9 9 . 3 6 2 4 0 2 8 6 .1 2 516 8 0 . 6 7 1 0 . 0 3 6 .5 0 0 0 0 . 0 0 . 0 3 2 .6 7 6 9 0 . 0 1 1 9 2 7 .0 5 5 5 2 2 1 3 .1 8 017 8 0 . 8 3 8 6 . 0 8 3 3 0 . 0 3 6 .5 0 0 0 1 1 .6 3 2 8 3 2 .2 5 3 2 1 .9 2 9 7 1 4 1 2 .0 0 5 5 3 6 2 5 .1 8 418 8 1 . 0 0 5 6 . 0 8 3 3 0 . 0 3 0 . 4 1 6 6 7 . 0 5 7 7 3 1 .8 3 5 0 1 .1 3 9 1 7 7 5 .8 1 2 5 4 4 0 0 .9 9 219 8 1 . 1 7 2 6 .0 8 3 3 0 . 0 2 4 .3 3 3 3 6 .9 6 6 2 3 1 .4 2 2 2 1 .4 0 0 9 7 6 4 . 6 6 2 5 5 1 6 5 .6 5 220 8 1 . 3 3 9 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 0 6 . 8 7 5 9 3 1 . 0 1 4 7 1 .8 3 7 9 7 5 3 . 7 0 2 5 5 9 1 9 .3 5 221 8 1 . 5 0 6 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 7 6 . 7 8 6 7 3 0 .6 1 2 6 2 . 7 1 2 9 7 4 2 . 9 3 1 5 6 6 6 2 .2 8 122 8 1 . 6 7 3 6 . 0 8 3 3 0 . 0 6 .0 8 3 3 6 . 6 9 6 7 3 0 .2 1 5 6 5 .3 3 9 8 7 3 2 .3 4 2 5 7 3 9 4 .6 2 123 8 1 . 8 4 0 0 . 0 4 7 . 1 0 4 4 0 . 0 0 . 0 3 3 .6 5 6 6 0 . 0 1 5 8 5 4 .6 7 2 * 7 3 2 4 9 .2 5 024 8 2 . 0 0 7 6 . 0 8 3 3 0 . 0 4 7 . 1 0 4 4 1 0 .6 1 7 1 , 3 3 .2 2 2 1 1 . 3 8 7 6 1 2 9 9 .4 9 5 7 4 5 4 8 .6 8 825 8 2 . 1 7 4 6 . 0 8 3 3 0 . 0 4 1 .0 2 1 1 6 .4 4 1 5 3 2 .7 9 1 3 0 . 7 8 3 7 7 1 3 . 3 4 0 7 5 2 6 2 .0 0 026 8 2 . 3 4 1 6 . 0 8 3 3 0 . 0 3 4 .9 3 7 8 6 . 3 5 8 0 3 2 .3 6 6 1 0 . 9 0 5 2 7 0 3 .0 4 3 7 5 9 6 5 .0 0 027 8 2 . 5 0 8 6 . 0 8 3 3 0 . 0 2 8 . 8 5 4 4 6 . 2 7 5 5 3 1 .9 4 6 5 1 . 0 7 8 4 6 9 2 . 9 2 2 7 6 6 5 7 .8 7 528 8 2 . 6 7 5 6 .0 8 3 3 0 . 0 2 2 .7 7 1 1 6 . 1 9 4 2 3 1 .5 3 2 2 1 .3 4 4 5 6 8 2 . 9 7 4 7 7 3 4 0 .8 1 329 8 2 . 8 4 2 4 .5 2 1 1 0 . 0 1 6 .6 8 7 8 7 . 4 3 8 1 3 1 . 1 2 3 4 1 . 8 0 5 3 6 3 7 . 5 5 7 7 7 9 7 8 .3 1 33 0 8 3 . 0 0 9 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 7 6 . 0 3 4 6 3 0 .7 1 9 8 2 . 4 3 8 7 6 6 3 . 8 0 6 7 8 6 4 2 .0 6 3

ONU>

TABLE 5-3 (CONTINUED)NUCLEAR PLANT 3

INT OATE ENERGY PROO ENERGY REFLO INV ENERGY COST ENERGY COST ENERGY COST INV TOTAL COST CUM COSTMHO X 10000 MWO X 10000 MWO X 1COOO PROO S/MWD REF S/MWO S/MWO THOUSAND S THOUSAND S ,

1 7 8 . 6 6 6 6 . 0 8 3 3 0 . 0 2 4 .3 3 3 3 8 . 8 1 1 0 3 7 .4 0 7 8 2 .0 C 9 4 1 0 2 4 .9 4 9 1 0 2 4 .9 4 92 7 8 . 8 3 3 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 0 8 . 6 9 6 7 3 6 . 9 2 2 7 2 . 6 3 5 9 1 0 1 0 .1 0 8 2 0 3 5 .0 5 73 7 9 . 0 0 0 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 7 . 8 . 5 8 3 9 3 6 . 4 4 3 9 3 . 8 9 0 5 9 9 5 . 5 3 0 3 0 3 0 .5 8 66 7 9 . 1 6 7 6 . 0 8 3 3 0 . 0 6 . 0 8 3 3 8 . 4 7 2 6 3 5 . 9 7 1 4 7 . 6 5 6 8 9 8 1 . 2 0 9 4 0 1 1 .7 9 55 7 9 . 3 3 6 0 . 0 3 6 . 5 0 0 0 0 . 0 0 . 0 3 8 .1 5 6 3 0 . 0 1 3 9 2 7 .0 4 3 1 7 9 3 8 .8 3 66 7 9 . 5 0 1 6 . 0 8 3 3 0 . 0 3 6 .5 0 0 0 1 3 .4 2 8 7 3 7 .6 6 1 5 2 . 3 2 5 7 1 6 6 5 .8 0 8 1 9 6 0 4 .8 4 17 7 9 . 6 6 8 6 . 0 8 3 3 0 . 0 3 0 . 4 1 6 6 8 . 1 4 7 3 3 7 . 1 7 3 2 1 .3 7 2 7 9 1 3 . 1 5 6 2 0 5 1 7 .7 9 38 7 9 . 8 3 5 . 6 . 0 8 3 3 0 . 0 2 4 .3 3 3 3 8 . 0 4 1 7 3 6 .6 9 1 2 1 .6 B 8 1 3 9 9 .9 6 0 2 1 4 1 7 .7 5 09 8 0 . 0 0 2 6 . 0 3 3 3 0 . 0 1 8 .2 5 0 0 7 . 9 3 7 4 3 6 . 2 1 5 4 2 . 2 1 4 5 8 8 6 . 9 9 5 2 2 3 0 4 .7 4 2

10 8 0 . 1 6 9 6 . 0 8 3 3 0 . 0 1 2 .1 6 6 7 7 . 8 3 4 5 3 5 .7 4 5 8 3 . 2 6 8 4 6 7 4 .2 5 5 2 3 1 7 8 .9 9 611 " 8 0 . 3 3 6 6 . 0 8 3 3 0 . 0 6 . 0 8 3 3 7 . 7 3 2 9 3 5 .2 8 2 3 6 . 4 3 2 7 8 6 1 .7 3 6 2 4 0 4 0 .7 3 012 8 0 . 5 0 3 0 . 0 4 8 . 6 6 6 6 o . o 0 . 0 4 1 . 1 0 5 0 0 . 0 2 0 0 0 4 .4 3 8 4 4 0 4 5 .1 6 813 8 0 . 6 7 0 6 . 0 8 3 3 0 . 0 48 «6t>66 1 2 .2 5 6 2 4 0 .5 7 2 0 1 . 5 5 2 9 1 5 0 1 .3 1 3 4 5 5 4 6 .4 8 016 8 0 . 8 3 7 6 . 0 8 3 3 0 . 0 4 2 . 5 8 3 3 7 . 4 3 6 0 4 0 .0 4 6 0 0 . 8 7 2 8 8 2 4 . 0 1 4 4 6 3 7 0 .4 9 215 8 1 . 0 0 6 6 .0 8 3 3 0 . 0 3 6 . 5 0 0 0 7 . 3 3 9 5 3 9 .5 2 6 7 1 . 0 0 1 6 8 1 2 .0 7 8 4 7 1 8 2 .5 7 016 8 1 . 1 7 1 6 .0 8 3 3 0 . 0 3 0 .4 1 6 6 7 . 2 4 4 4 3 9 .0 1 4 2 1 . 1 8 2 4 8 0 0 .3 4 8 4 7 9 8 2 .9 1 817 8 1 . 3 3 8 6 . 0 6 3 3 0 . 0 2 4 . 3 3 3 3 7 . 1 5 0 4 3 8 .5 0 8 3 1 .4 5 4 1 7 8 8 .8 2 1 4 6 7 7 1 .7 3 618 8 1 . 5 0 5 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 0 7 . 0 5 7 7 3 8 . 0 0 9 0 1.9C 77 7 7 7 . 4 9 1 4 9 5 4 9 .2 2 719 8 1 . 6 7 2 6 . 0 8 3 3 0 . 0 1 2 . 1 6 6 7 6 . 9 6 6 2 3 7 .5 1 6 1 2 . 8 1 5 7 7 6 6 .3 5 6 5 0 3 1 5 .5 8 220 8 1 . 8 3 9 6 . 0 8 3 3 0 . 0 6 .0 8 3 3 6 . 8 7 5 9 3 7 .0 2 9 7 5 . 5 4 1 9 7 5 5 . 4 1 2 5 1 0 7 0 .9 9 221 8 2 . 0 0 6 0 . 0 3 0 .4 1 6 6 0 . 0 0 . 0 3 1 .3 8 8 6 0 . 0 9 5 4 7 .3 7 1 6 0 6 1 3 .3 6 322 8 2 . 1 7 3 6 . 0 8 3 3 0 . 0 3 0 . 4 1 6 6 1 0 .8 9 7 9 3 0 .9 8 1 6 2 . 3 7 4 4 1 3 8 5 .1 6 3 6 2 0 0 3 .5 2 3

"23 " 8 2 .3 6 0 " 6 . 0 8 3 3 0 . 0 2 4 .3 3 3 3 6 . 6 1 1 9 3 0 .5 7 9 9 1 .4 5 9 9 '* 7 5 7 .4 6 2 ' 6 2 7 6 0 .9 8 426 8 2 . 5 0 7 6 . 0 8 3 3 0 . 0 1 8 .2 5 0 0 6 . 5 2 6 1 3 0 . 1 8 3 4 1 .9 1 5 1 7 4 6 .5 1 1 6 3 5 0 7 .4 9 225 8 2 . 6 7 6 6 .0 8 3 3 0 . 0 1 2 .1 6 6 7 6 . 4 4 1 5 2 9 .7 9 2 0 2 . 8 2 6 5 7 3 5 . 7 5 3 6 4 2 4 3 .2 4 226 8 2 . 8 6 1 6 . 0 8 3 3 0 . 0 6 . 0 8 3 3 6 . 3 5 8 0 2 9 .4 0 5 7 5 .5 6 2 9 7 2 5 .1 8 5 6 4 9 6 8 .4 2 627 8 3 . 0 0 8 0 . 0 4 2 . 5 8 3 3 0 . 0 0 . 0 3 0 .5 4 2 6 0 . 0 1 3 0 0 6 .0 6 6 7 7 9 7 4 .4 3 828 8 3 . 1 7 5 6 . 0 8 3 3 0 . 0 4 2 . 5 8 3 3 1 0 .0 7 7 1 3 0 .1 4 6 6 1 . 4 1 5 0 12 1 5 .5 6 3 7 9 1 9 0 .0 0 029 8 3 . 3 6 2 6 . 0 8 3 3 ' 0 . 0 ' 3 6 . 5 0 0 0 6 . 1 1 3 9 2 9 .7 5 5 7 0 . 8 1 1 9 6 6 8 . 2 5 7 7 9 8 5 8 .2 5 030 8 3 . 5 0 9 6 . 0 8 3 3 0 . 0 3 0 . 4 1 6 6 6 . 0 3 4 6 2 9 .3 6 9 9 0 . 9 5 8 4 6 5 8 . 6 0 2 8 0 5 1 6 .8 1 3

CUMULATIVE TOTAL COST FOR THIS SOLUTION 2 3 5 9 5 7 .5 6 3

TOTAL SYSTEM COST FOR THE PLANNING HORIZON OF 62 INTERVALS (THOUSAND S)NUCLEAR COSTS « 2 9 3 7 8 8 .6 3 * - -FOSSIL COSTS * 9 2 0 2 8 1 6 0 . COINTERCHANGE COSTS » 6 3 8 5 1 5 0 6 .0 0

TOTAL COST * 1 3 6 1 7 3 6 6 0 .0 0

Table 5-1+F u e l C ycle C ost D ata fo r Schedule 11 (Cents/MBTU)

Jycle Nuclear Plant 1 Nuclear Plant 2 Nuclear Plant 3No Fuel Levelized Fuel Levelized Fuel Levelized

Cost Fuel Cost Cost Fuel Cost Cost Fuel Cost

2 k . 921 21+.921 21+.072 21+.072 21+.539 21+.5395 26.21+9 25.618 23.765 23.911 27.076 25.8696 25.^56 25.555 2U.179 23.998 21+.197 25.2237 25.9^3 25.601 2 6.1U0 21+.U86 23.756 2U.9608 22.870 21+.995 23.601+ 21+.301 23.050 21+.6039 22.1+02 24.707 22.953 21+.151+ 22.219 2U.3I+610 22.295 23.981+

O NVI

66

the load swings which are due to varying load demands. Nuclear unit

l's level of production varies to absorb reduced demand requirements.

The unit operates at various levels with cycle lengths of 12 months,

lU months, 20 months, 8 months, 18 months, and 20 months. The

levelized fuel cycle cost is 2U.707 cents/MBtu. The fuel cycle costs

are presented in Table 5-*+.

From the levelized fuel cycle costs, it can be seen that

unit 2 is the cheapest unit to operate, unit 3 follows next, and

unit 1 is the most expensive. However, unit 1 has the highest

efficiency, unit 2 the second best, and unit 3 the lowest. Thus the

operating schedule of a unit is as important in determining the fuel

cycle costs as the unit's efficiency and a more efficient plant may

not be the least expensive to operate. This is further demonstrated

by Figure 5-5 and Table 5-5 which is an equilibrium cycle schedule.

Each nuclear unit is operated at the equilibrium energy

(1000 MWe) and cycle length of 12 months. The levelized fuel cycle

costs are 22.517, 22.370, and 22.217 cents/MBtu for unit 1,2, and

3. It is apparent that the levelized fuel cycle cost for each unit is

less than the same unit in schedule 11. However, the total system

operating cost for the equilibrium cycle case is 19.6 x 10 million

dollars more expensive than for the schedule 11. This difference

results from schedule 11 reducing the fossil and interchange cost

components while the nuclear costs increased. Unit 3 has the lowest

fuel cycle cost since the reactor operating data corresponds to a

unit with a 33 percent efficiency and 1000 MWe rating.

MWOE (X IOOOOI

NUCLEAR PLANT 1 MWUb IX lOOOO)

NUCLEAR PLANT 2

6*08 ♦ ** » •* « * • • * • * • • * * * * • * * * ***

5 .3 2 5 .3 2

* .5 6 * .5 6

3 .8 0 “3 .6 0

3 .0 * ♦ 3 .0 * ♦

0.0 ♦ - < . *—♦ ——* . — . ♦. ♦— * ♦ . — ♦--i i o 1? 20 25 36 SIT *5~

" T T T 't T R v A l

5 ro 15— TO-----25----- 36----- 35-----55 “

— : 1 n t e i r r * l ---------------------------

KWDE IX LOOOO)

6 .0 8

NUC

5 .3 2

* .5 6

^.80

3 .0 * *

0*0 ♦-♦ — *4-' 5 10 15 20 25 30 35

— ♦ * -

*0

FIGURE 5 - 5r N T E R V A I '

UNIT.ENERGY PRODUCTION LEVEL FOR YEARLY CYCLES

C\-q

68

Table 5-5

System Costs for Yearly Cycles

Nuclear Fossil Interchange Total System Costs(Thousand $) (Thousand $) (Thousand $) (Thousand $)

$267,508 $106,9.93,168 $1*8,1*90,128 $155,750,8oU

Levelized Fuel Cost

Nuclear Plant 1 Nuclear Plant 2 Nuclear Plant 3 ( /MBtu) (tf/MBtu) (<£/MBtu)

22.517 22.370 22.209

The comparison of the equilibrium cycle case and schedule 11

shows the importance of scheduling nuclear unit operation on a system-

wide basis. Furthermore, although an operating schedule developed for

an individual unit may be the cheapest possible, that schedule may not

be the cheapest for system operation. This is illustrated by Table 5-2.

The nuclear component of the system cost is seen to vary considerably

from schedule to schedule and the least expensive schedule does not

result in the most economical system cost. Thus optimizing nuclear unit

operation does not guarantee optimizing system operation. Likewise

minimizing fossil and interchange costs do not result in the most

economical operating schedule. The minimization of system cost requires

an examination of various alternatives and a balancing of these

alternatives.

A schedule which improved upon schedule 11 was not found

after 3 hours of additional computation time (CPU). Thus, Table 5-2,

and Table 5-3, and calculations between; revisions of the cost function

imply the calculational scheme is converging. The expenditure of

3 hours of CPU without finding an improved schedule indicates that

schedule 11 is approaching an optimum schedule. From an analysis of

the refueling intervals selected, more computation time is needed to

obtain an improved schedule^ or to show a better optimum cannot be

reached. The mixed-integer program is sensitive to the values of the

cost coefficients. This sensitivity affects the calculational time

required for an analysis and the results obtained. The values of the

^This case used 7 hours of computation time.

cost coefficients describe a point on the system energy and time cost

surface. Depending on the starting point and the shape of the system

cost function it may not be possible to reach a global or a local

optimum which is more economical than the one previously found from

the particular starting point. The cost surface appears irregular

(not convex). Thus, the linear function could step through valleys

where local optima are located resulting an a schedule which is

higher than the previous one, especially after a revision. Thus, two

objective functions are necessary: one is used in the mixed-integer

program and one in the FORTRAN program. The FORTRAN program retains

the best system cost to date and allows the cost coefficients in STARTS

to be updated only when a better FORTRAN system cost function is found.

This technique tends to force the STARTS program to continuously search for a global optimum.

The nuclear units supplied almost 85 percent of the energy

throughout the planning horizon at a cost of $293.8 million. The

total nuclear cost is minimal compared to the system operating cost.

The nuclear units 1, 2, and 3 were utilized to provide 32.5, 33.2, and

3**.3 percent of the nuclear energy and 27.6, 28.2, and 2 9 .1 percent oft

the system energy throughout the planning horizon. The fossil unit

supplied almost 10 percent of the system energy. This corresponds to

an average loading of 300 MW electrical per interval. The interchange

unit produced approximately 5 percent of the system energy. The average

loading of this unit per interval is 156 MW electrical. Comparing

this to schedule 10, the fossil unit produced 2.U x 10** MWD electrical

71less energy and the cost was 2.85 x 10^ dollars less in schedule 10.

1*Similarly, the interchange unit produced 3.76 x 10 MW1) electricali 9more energy and the cost was '♦.l x 10 dollars more in schedule 10.

The average interval production level for the fossil unit is 291 ?IW

electrical and for the interchange unit 171 MW electrical. The large

changes in the cost function from schedule to schedule are the result of small changes in unit loading, especially the fossil and inter

change. With more realistic cost coefficients for the non-nuclear

units, the cost function would change less with changes in the

non-nuclear energy produced implying the schedules' costs are converging.

5.3.2 Sensitivity Analysis

A sensitivity analysis can be obtained through any of six

post-optimal procedures of the MP5X package. Fixing the values of the

integers for the particular schedules, an analysis of the effects of

cost changes on the optimum activity levels is available. The levels

to which an activity can increase or decrease, its unit cost effect

on the objective function, the cost level at which another activity will

enter the solution, and what the level of the new activity will be are

outputted. This information indicates when a valuable cost savings

is available, how to reduce costs, and when a new system operating

schedule should be determined because of cost changes.

A sensitivity analysis of the first case provides insights

into how to reduce system costs. An obvious way to reduce the cost

function is to lower the load demand. The decrease in the cost

function results from less utilization of fossil and interchange units

with their large energy costs. Table 5-6 is an illustration of the

output from the sensitivity analysis. Examining number 658 row

R050l*27» decreasing the activity (interval demand) to the lower

activity of 13*38331 leads to a unit cost decrease of 119*99992 and activity C05270U would leave the solution. Activity C05270U corresponds

to the fossil unit in interval 27. Similarly increasing the interval

demand to the upper activity of 19.^6661 would increase the cost

function because the fossil unit produces more energy during interval

27. Number 659» row R050U28 is similar to number 658 except the

interchange unit is affected as the fossil unit was in number 658.

Number 660 row R050li29 indicates decreasing the demand to a lower

activity of 15.20831 decreases the cost function because less energy

would be produced in interval 29 by nuclear unit 2, R021229.

Similarly, increasing to the upper activity of I8.2U998 increases the

cost function since more energy would have to be reloaded for unit

2 interval 23. This analysis can be carried further by tracing the

affected activity to find how it interacts with the other activities.

Thus, the interaction of the activities can be traced.

Utilizing the maximum amount of the most economical energy

reduces the cost function by displacing more expensive energy.

Table 5-7 is a portion of the sensitivity analysis indicating maximum

utilization. Number 755 column C012802 shows increasing the upper

cost to 109.95893 does not affect the qnergy produced from nuclear

unit 1 during interval 28. Since the nuclear unit is needed to meet

Table 5-6

I l l u s t r a t i o n o f th e S e n s i t iv i ty A nalysis

HUMBER ...ROW.. AT . . .ACTIVITY.. . SLACK ACTIVITY . .LOWER LIMIT. LOWER ACTIVITY . . .UNIT COST.. . .UPPER COST.. LIMITING AT..UPPER LIMIT. UPPER ACTIVITY . . .UNIT COST.. . .LOWER COST.. PROCESS. AT

658 R050U27 EQ 16.87393 • 16.87393 13.38331 119.99992- C052701* LL16.87393 19.1*6661 119.99992- C052701* UL

659 R0501*28 EQ 18.93253 • 18.93253 1 8 . 2L998 110 . 0000U- c 062805 LL18.93253 18.21*998 110.00001* C062805 UL

660 R050it29 EQ 16.68778 e 16.68778 15.20831 .01*81*0- R021229 LL16.68778 18.2U998 .01*81*0 R020323 UL

—3U)

Table 5-7

I l lu s t r a t io n o f Further Economic A nalyses

HUMBER .COLUMH. AT . . .ACTIVITY.. . . .INPUT COST.. ..LOWER LIMIT. ..UPPER LIMIT.

LOWER ACTIVITY UPPER ACTIVITY

...UNIT COST..

...UNIT COST....UPPER COST.. ..LOWER COST..

LIMITINGPROCESS.

AT# n A -

755 C012801 BS 30.1*1665 .00096M6.66663

28.851*1*530.1*1665

.001*62

.00031*.00558.00062

R01032U C 0121*01

LLT.T.T.

753 C012702 BS 6.08333 .00628HONE

6.083336.08333

119.95357INFINITY

119.95985INFINITY-

R010227NONE

UL

1099 c 060605 BS .71*825 109.999991.21666

5.33501*-3.78993

10.00000109.91*51*6

119.99999.051*53

C050601*R020206

LLUL

lUMBER . . .ROW.. AT . . .ACTIVITY.. . SLACK ACTIVITY ..LOWER LIMIT. ..UPPER LIMIT.

LOWER ACTIVITY UPPER ACTIVITY

. . .UNIT COST..

...UNIT COST....UPPER COST.. . .LOWER COST..

LIMITINGPROCESS.

ATAT

526 R031221 BS 3.01*165 • 3.01*165NONE

3.01*1653.0U165

119.96857INFINITY

R030221NONE

UL

75demand and it produces cheaper energy than the fossil or interchange

units, a decrease in the cost to minus infinity does not affect the

activity level of the nuclear unit. Number 753 column C012702,

number 1099 column C060605, and number 526 row R031220 support

maximum utilization of the most economical energy.This analysis can be extended to investigate the load

demands, the cost function, and columns and rows. The conclusions to be drawn from this brief analysis are to increase nuclear energy

and decrease non-nuclear energy. In addition, loosening of the

energy production and energy refueling limits tends to decrease the

cost function.

5.U Analysis of the Hypothetical Utility System - Case Two

The second configuration of the hypothetical utility system

analyzed is similar to case one, except the fossil cost coefficients

were revised to be lower than the interchange unit's coefficients, but

not quite competitive with the nuclear cost coefficients. This system

configuration converged more rapidly and smoothly than case one.

Table 5-8 gives the schedules' cost generated by CRESCENDO. The total

co3t is decreasing by smaller amounts due to the magnitude of the fossil

cost coefficients. These schedules were generated in two hours of

CPU time compared to 7 hours of CPU time in case one.The best schedule found is shown by Figure 5-6 and has a

total system cost of 1.06l x 10^ dollars. The nuclear units account

for $313.M million or approximately 30 percent of the total, the

Table 5-8System Costs for the Second Case (Thousand $)

;dule Nuclear Fossil Interchange Total System Costs

1 $2#+, 700 $1,559,803 $1,91+6,021+ $3,770,5272 278,7U9 1,1+29,150 1,832,319 3,51+0,2183 301,191 813,871 198 1,115,2601+ 308,071+ 771,1+12 165 1,079,6515 307,771 771,1+12 165 1,079,31+86 305,712 761,501+ 165 1,067,3817 316,975 71+6,086 198 1,063,2598 316,975 71+6,086 198 1,063,2599 313,1+1+7 71+7,1+27 198 1,061,072

—io\

NWOE iv ifinrtAi

NUCLEAR PLANT 1 HWOE NUCLEAR PLANT 2iv in n n n i

i i

i i

5 .3 2 ♦ 5 .3 2 ♦

1A

1

1

1 1

1 1

3 .8 0 ♦ |

3 .8 0 ♦ |

1 1

13 .0 4 ♦ ----- ----------------------------------- 3 C '

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

I N T E R V A L•w o e

- -------------------- IX 100001 -

11

I N T E R V A LNUCLEAR PLANT 3 *

----------------- ------- - -

--------------------- - --------------------1

5 .3 2 ♦ |

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1I1

A- CA -1

- - II

1-3 .8 0 ♦

1_

— ----------------------------------------------------------3 -0 4 —* .

. . . 0.0 -♦ — ____________

5 10 15 20 25 30 3 5 40

I N T E R V A LFIGURE 5 - 6 : UNIT ENERGY PRODUCTION FOR SCHEDULE 9 FOR THE SECOND CASE

f o s s i l u n i t ' s co s t i s $Y**7.**?7 m i l l io n o r ap p ro x im ate ly 70 p e rc e n t

o f th e t o t a l , and th e in te rch an g e u n i t ' s c o s t i s $.3-90 m ill io n fo r

th e p lan n in g h o rizo n . T h is s i t u a t io n i l l u s t r a t e s th e r e v e r s a l o f

th e f o s s i l and in te rc h a n g e u n i t s ' p ro d u c tio n from ca se one. F o r t h i s

c a s e , th e f o s s i l u n it always p roduces b e fo re th e in te rc h a n g e u n i t .

The schedule i l l u s t r a t e s n u c le a r u n i t 1 o p e ra t in g e s s e n t i a l l y

b a se -lo a d e d a t 1000 MW e l e c t r i c a l f o r th e p la n n in g h o r iz o n . The u n it

o p e ra te s w ith cy c le le n g th s o f 12 m onths, 16 m onths, 18 m onths, 18

m onths, 18 m onths, and lU m onths, r e s p e c t iv e ly . The le v e l iz e d fu e l

c y c le c o s t f o r th e p la n n in g h o rizo n i s 25.**97 cents/M B tu. N u clea r

u n i t 2 ab so rb s se v e ra l lo a d swings due to v a ry in g lo a d demands.

O therw ise , u n i t 2 o p e ra te s a t 1000 MW e l e c t r i c a l w ith cy c le le n g th s

o f 12 m onths, 1** m onths, l6 m onths, 18 m onths, 18 m on ths, and 16 m onths.

The le v e l iz e d fu e l c y c le c o s t f o r t h i s u n i t i s 25.169 cents/M B tu.

N uclear u n i t 3 o p e ra te s s im i la r t o n u c le a r u n i t 1 w ith cy c le le n g th s o f

12 m onths, 18 m onths, 18 m onths, 18 m onths, 1** m onths, and Ik m onths.

The le v e l iz e d fu e l c y c le co s t f o r t h i s u n i t i s 25.772 cents/M B tu.

F o r t h i s s c h e d u le , th e n u c le a r u n i t s supp ly ap p rox im ate ly

8 6 .5 p e rc e n t o f th e en erg y th roughou t th e p la n n in g h o r iz o n , w h ile

acco u n tin g f o r approx im ate ly 30 p e rc e n t o f th e c o s t . N uclear u n i t s 1 ,

2 , and 3 p roduced 3 3 .5 . 33 . 0 , and 33 .5 p e rc e n t o f th e t o t a l n u c le a r

energy and 2 9 . 0 , 28 . 6 , and 29.0 p e rc e n t o f th e system lo a d demand

th roughou t th e p lan n in g h o rizo n . The f o s s i l u n it su p p lie d 13.**

p e rc e n t o f t h e re q u ire d system energ y a t an average lo a d in g o f Uo6 MW

e l e c t r i c a l p e r in t e r v a l . The in te rc h a n g e u n i t p roduced e s s e n t i a l l y

z e ro energy f o r th e p la n n in g h o r iz o n .

79

The a n a ly s is o f t h i s case p ro v id es r e s u l t s com patib le

w ith th e f i r s t case a n a ly z e d , and i l l u s t r a t e s th e same t r e n d s tow ard

lo n g c y c le s , s u b s t a n t i a l n u c le a r energy c o n tr ib u t io n s to th e system

re q u ire m e n ts , and convergence o f th e o b je c t iv e fu n c tio n . S im i la r ly ,

an exam ination o f th e r e f u e l in g in t e r v a ls shows th e m a jo r ity o f

r e f u e l in g s a re lo c a te d in th e low est demand in t e r v a l (33 p e rc e n t o f

th e r e f u e l in g s ) and th e t h i r d low est demand in t e r v a l (Uo p e rc e n t o f

th e r e f u e l in g s ) . Only 20 p e rc e n t o f th e r e f u e l in g s o ccu r d u rin g th e

second low est demand in t e r v a l . T h is s i t u a t io n im p lie s a more

econom ical sch ed u le m ight be r e a l iz e d i f a d d i t io n a l CPU tim e i s

devo ted to t h i s c a se .

5 .5 A n a ly s is o f th e H y p o th e tic a l U t i l i t y System - Case Three

For th e t h i r d a n a ly s i s , th e f o s s i l u n i t ' s c o s t c o e f f i c ie n t s

were s e t a t t h e i r v a lu e in case o n e , and th e in te rc h a n g e u n i t ' s c o s t

c o e f f i c i e n t s were r e v is e d to make th e in te rc h a n g e u n i t c o m p e titiv e

w ith th e n u c le a r u n i t s ; o th e rw is e , th e system c o n f ig u ra tio n i s th e

same as in case one. The a n a ly s is o f t h i s ca se r e s u l te d in schedu le

3 o f T ab le 5-9* T his schedu le was g e n e ra te d r a p id ly and th e

rem ain ing com putation tim e was sp en t t r y in g to improve t h i s sch ed u le .

The c o s ts p re s e n te d in T ab le 5-9 w ere g e n e ra te d from 3 ho u rs o f

com putation .

The most econom ical sch ed u le i s g iv en by F ig u re 5-7 and

has a t o t a l system c o s t o f 96 .82 x 10^ d o l l a r s . The la rg e m agnitude

can be a t t r i b u t e d to th e h igh c o s t o f f o s s i l en e rg y . The n u c le a r

u n i t s ' c o s t i s $29**.6U2 m i l l io n and i s l e s s th a n 1 p e rc e n t o f th e

Table 5-9System C osts f o r th e T h ird Case (Thousand $)

Schedule K uclear F o s s i l In te rc h a n g e T o ta l SystemC o sts

1 $290,^11 $11*2,371,856 $76,61*9 $11*2,738,9162 307,932 97 ,371 ,272 6k ,261 97,780,1*653 296,915 96,1*50,576 72 ,519 96 ,820 ,010

C DO

NWDE IX 100001

N U C L E A R P L A N T 1 MHOE IX 1 0000)

N U C L E A R P L A N T 2

*• ••••••

5 .3 2

4 .5 6

3 .8 0

• • •

3 .0 * ♦ •

0 .0 * — * »— ♦—5 10 15 20 25 30 35 * 0

6 .0 8

5 .3 2

* .5 6

3 .8 0

3 .0 * *

0 , 0 _ ♦ - » — ♦ - - ■ -5 10 15 20 25 3 0 35 * 0

I N T E__R_V A L HWOE

IX 10000)

I N T E R V A LNUC ANT 3

6 .0 8 « • • * • • • • • • • • • • • • • • « *

* _ _ *

I5 .3 2 ♦

* .5 6

3 .8 0

3 .0 * *

0.05 10 15 20 25 3 0 35 *0

I N T E R V A LFIGURE 5 - 7 : UNIT ENERGY PRODUCTION FOR SCHEDULE 3 FOR THE THIRD CASE

00

82

system c o s t . T h is case does i l l u s t r a t e th e c o m p e titio n between th e

n u c le a r u n i t s and a n o n -n u c le a r u n i t f o r th e lo a d on th e b a s is o f c o s t .

N uclear u n i t 1 o p e ra te s w ith c y c le le n g th s o f 12 m onths,

16 m onths, 18 m onths, 20 m onths, 6 m onths, and 20 m onths. The

le v e l iz e d f u e l c y c le c o s t f o r th e p la n n in g h o riz o n i s 2 5 . 1+95 c e n ts /

MBtu. F ig u re 5 -7 shows n u c le a r u n i t 1 a t v a r io u s energy p ro d u c tio n

le v e ls w hich r e s u l t s from co m p e titio n f o r th e lo a d . N u c lea r u n i t 2 i s

b ase lo a d e d a t 1000 MW e l e c t r i c a l th ro u g h o u t th e p la n n in g h o riz o n .

The u n i t o p e ra te s w ith c y c le le n g th s o f 12 m onths, 16 m onths, 8 m onths,

lU m onths, 16 m onths, 12 m onths, and 16 m onths. The le v e l iz e d fu e l

c y c le c o s t f o r u n i t 2 i s 23 .531 cen ts/M B tu . N u clea r u n i t 3 o p e ra te s

a t v a r io u s energy p ro d u c tio n l e v e l s th ro u g h o u t th e h o r iz o n s im ila r

to u n i t 1 . The c y c le le n g th s f o r t h i s u n i t a re 12 m onths, 20 m onths,

16 m onths, 18 m onths, and 22 m onths. The le v e l iz e d f u e l c y c le c o s t

i s 25 .606 cents/M B tu.

The n u c le a r u n i t s produced 2 7 .6 , 28.1+, and 2 7 .3 p e rc e n t o f

th e t o t a l h o r iz o n 's energy req u irem en t f o r a t o t a l o f 8 3 .3 p e rce n t

o f th e system en erg y . The n u c le a r u n i t s 1 , 2 , and 3 p roduced 3 3 .1 ,

3U .0, and 32.9 p e rc e n t o f th e t o t a l n u c le a r en e rg y . The in te rch an g e

u n i t c o n tr ib u te d 6 .3 p e rc e n t o f th e t o t a l system en e rg y . This

p e rc e n ta g e co rresp o n d s t o an average lo a d in g o f 189 .2 MW e l e c t r i c a l

p e r i n t e r v a l . T h is i s c lo s e t o th e 200 MW e l e c t r i c a l maximum imposed

on th e in te rc h a n g e u n i t . F u rth erm o re , F ig u re 5 -7 shows n u c le a r u n i t s

1 and 3 b e in g o f f - lo a d e d s in c e th e in te rc h a n g e u n i t p ro v id e s le s s

expen siv e en erg y .

S3

This analysis shows a mixture of long and short cycle lengths

which is different from the two previous cases. The low demand

intervals are utilized for refueling better than in the previous cases

with 33 percent of the refuelings during the lowest demand interval,

26 .7 percent of the refuelings during the second lowest demand interval,

and 26 .7 percent of the refuelings during the third lowest demand interval.

5.6 Analysis of the Hypothetical Utility System - Case Four

This analysis illustrates the development of a new schedule

if the system is subjected to a major forced outage. The interchange

unit's cost coefficients are made competitive with the nuclear units'

and the fossil unit is more expensive than any other unit. The

capacity rating on the fossil unit is increased to 2000 MW electrical

for intervals 2, 3, and U, otherwise the capacity rating is 1000 MW

electrical. To have a feasible schedule the system must have enough

capacity to meet the load; this is the reason the fossil unit's

capacity is increased. Nuclear unit 1 is out of service for intervals

2, 3, and U. Nuclear unit 2 is scheduled to be refueled during

interval 2. Hence, this case is similar to case three analyzed in

section 5. .The most economical schedule generated by CRESCENDO is

r. 9illustrated in Figure 5-8 and has a total system cost of 111 x 10

dollars which corresponds to schedule 5 of Table 5-10. The major cost

component of schedule 5 is the fossil cost.. The nuclear units and

interchange units have minimal effect on the results. Nuclear unit 1,

MUOE IX lOOOOI.

NUCLEAR PLANT I MUOEIX 1 3 0 0 0 1

NUCLEAR PLANT 2

9 .3 2

4.56

3 .00

3 .0 4 ♦

5 .3 2

4 .5 6

3 . 8 0

3 .C 4 ♦

0.0 . ♦— - — ♦ -♦ —. ♦5 10 IS 20 25 30 35 40

c .c — — ♦* — * • ---------------- * . -5 1 0 15 2 0 2 5 3 0 3 5 4 0

l i l t X V 4 L MUOE IX 100001

I N T E R V A LNUCLEAR PLANT 3

6 . CO

5 .3 2

4 .5 6

3 . 0 0

3 .0 4 ♦

0.0 - * --------------♦ ------- * ♦ -

5 10 15 20 2 5 30— * ♦ * -

3 5 4 0

FIGURE 5 - 0I N T E R V A L

UNIT ENERGY PRODUCTION FOR SCHEDULE 5 FOR THE FOURTH CASE

OO- t"

Table 5-10

System Costs for the Fourth Case (Thousand $)

dule Nuclear Fossil Interchange Total System Costs

1 $276,035 $156,971,81*0 $76,61*9 $157,32U,52l*2 290,238 11*2,371,888 76,61*9 11*2,738,7753 295,238 127,685,696 62,915 128,01*3,9991* 297,537 121,621,776 70,691* 121,830,0075 29 ,61+2 111,221*, 208 58,181 111,577,031

CoV I

which no rm ally would be re fu e le d d u rin g in t e r v a l 3 , o p e ra te s a t

f u l l power fo r in t e r v a l s 1 and 5 . The u n i t i s r e fu e le d b o th d u rin g

th e o u tage (energy eq u a l to a lU month f u l l power c y c le ) and d u rin g

in te r v a l 6 . The u n i t o p e ra te s a t f u l l power fo r in t e r v a l 5 , so th e

u n it has enough energy f o r an 18 month f u l l power c y c le . A f te r th e

fo rced o u ta g e , u n i t 1 o p e ra te s w ith c y c le le n g th s o f 12 m onths, 18

m onths, 16 m onths, 18 m onths, U m onths, and 18 m onths. U nit 1 i s

e s s e n t i a l l y b ase lo ad ed a t 1000 MW e l e c t r i c a l and has a le v e l iz e d

fu e l c y c le c o s t o f 25.222 cents/M B tu f o r th e p la n n in g h o riz o n .

N uclear u n i t 2 i s b ase lo ad ed a t 1000 MW e l e c t r i c a l th ro u g h o u t th e

p lan n in g h o riz o n . The u n i t o p e ra te s w ith c y c le le n g th s o f 12 m onths,

18 m onths, 12 m onths, 18 m onths, 8 m onths, 18 m onths, and 12 m onths.

The le v e l iz e d f u e l c y c le c o s t i s 2U.775 cents/M B tu. N uclear u n i t 3

i s base loaded a t 1000 MW e l e c t r i c a l f o r th e h o rizo n and o p e ra te s

w ith c y c le le n g th s o f 12 m onths, 18 m onths, lU m onths, 12 m onths,

16 m onths, 12 m onths, and 12 m onths. The le v e l iz e d f u e l c y c le c o s t

i s 2i t .066 cents/M B tu.

The n u c le a r u n i t s produced 83 p e rc e n t o f th e p lan n in g

h o r iz o n 's energy req u irem en ts and u n i t s 1 , 2 , and 3 c o n tr ib u te d 2 6 . 6 ,

2 8 .2 , and 28 .2 p e rc e n t o f th e t o t a l energ y . U n its 1 , 2 , and 3 produced

32 .0 , 3 it .0 , and 3^ .0 p e rc e n t o f th e n u c le a r en e rg y , r e s p e c t iv e ly . The

f o s s i l u n i t p ro v id ed 12 p e rc e n t o f th e system energy and th e i n t e r

change u n i t accoun ted fo r 5 p e rc e n t. A lthough th e p e rc e n t o f n u c le a r

energy produced in t h i s case and case th r e e i s about e q u a l , t h e r e a re

more r e fu e l in g s f o r t h i s s i t u a t i o n , 20 v e rsu s 15. However, th e

d i s t r i b u t io n o f r e f u e l in g s rem ained v e ry s im i la r t o case th r e e .

F u rth erm o re , th e l e v e l iz e d f u e l c y c le c o s t f o r each u n i t i s cheaper

th a n th e co rre sp o n d in g u n i t i n c a se th r e e which r e s u l t s in th e n u c le a r

u n i t s b e in g more econom ical t o o p e ra te th a n th e in te rc h a n g e .

The in te rc h a n g e u n i t h a s an av erage lo a d in g o f 152 MW

e l e c t r i c a l / i n t e r v a l , an average d ec re a se o f 38 MW e l e c t r i c a l / i n t e r v a l

from th e p re v io u s c a s e . T h is in c re a s e can be a t t r i b u t e d t o th e e x t r a

p ro d u c tio n re q u ire d t o meet system energy demands d u rin g th e fo rc e d

o u tag e . The n e t r e s u l t i s an in c re a s e in t o t a l system c o s t o f

lU .76 x 10^ d o l l a r s .

5 .7 Comments

The CRESCENDO model p ro v id e s economic a n a ly s is d a ta ,

e q u iv a le n t c o s t a l t e r n a t iv e s c h e d u le s , and f l e x i b i l i t y t o examine a

v id e range o f system c o n d itio n s in a d d i t io n to th e most econom ical

o p e ra tin g sc h e d u le . However, th e fo rm u la tio n i s s e n s i t i v e t o th e c o s t

c o e f f i c ie n t s and s t a r t i n g c o s t c o e f f i c i e n t s . T h is s e n s i t i v i t y a f f e c t s

th e com putation t im e , th e most econom ical sch ed u le found , and

p ro g re s s io n to convergence. F u rth e rm o re , th e r e s u l t s from each case

a re com patib le w ith th e rem ain in g ca se s an a ly z e d . When n o n -n u c le a r

energy i s c o m p e tit iv e w ith n u c le a r e n e rg y , th e p r e f e r r e d o p e ra t in g

sch ed u le d e v ia te s from th e n o n -c o m p e titiv e s i t u a t i o n .

CHAPTER 6CONCLUSIONS

Thi3 work has dem onstra ted th e need and economic in c e n tiv e

fo r develop ing methods to p la n fu tu re power system o p e ra t io n . A

model was developed which s im u ltan e o u sly a l lo c a te d r e fu e l in g d a te s

and energy p ro d u c tio n to th e n u c le a r u n i t s , and energy p ro d u c tio n

to th e n o n -n u c le a r u n i t s on a system -w ide h a s i s . A b a s ic component

developed and in c o rp o ra te d in t h i s model was a r e a c to r co re model

which r a p id ly s tim u la te d f u e l management d e c is io n s fo r a v a r i e ty o f

o p e ra t in g sc h e d u le s . A m u lti-d im e n s io n a l and dynamic system model

was developed to ad eq u a te ly re p re s e n t n u c le a r u n i t o p e ra tio n and

to a l lo c a t e energy and R efu e lin g tim e s . The CRESCENDO model in c lu d ed

a m ix e d -in te g e r program fo rm u la ted as a dynamic L e o n tie f model

(system m ode l), th e co re m odel, and s e v e ra l fundam ental u n i t and

system o p e ra t in g c o n s t r a in t s . The a p p l ic a t io n o f t h i s model to th e

h y p o th e tic a l power system dem onstra ted th e f e a s i b i l i t y and a p p l i c a b i l i t y

o f in te g r a te d power system sc h e d u lin g . The in v e s t ig a t io n s perform ed

on th e h y p o th e tic a l u t i l i t y system le a d to th e fo llo w in g c o n c lu s io n s .

The o p tim iz a tio n o f th e system m ust be done s im u lta n e o u sly

f o r th e e n t i r e system . The m in im iza tio n o f a n u c le a r u n i t ' s o r a

group o f n u c le a r u n i t s ' c o s t does n o t m inim ize th e t o t a l system c o s t .

88

The n u c le a r c o s t component o f th e system c o s t fu n c tio n v a r ie d

c o n s id e ra b ly from sch ed u le to sch ed u le as th e system c o s t fu n c tio n

was im proved. S im ila r ly th e f o s s i l and in te rc h a n g e u n i t s ' c o s ts

v a r ie d b u t w ere l e s s th a n th e n u c le a r c o s t v a r ia t io n s .

The n u c le a r u n i t s m ust be scheduled to o p e ra te on v a ry in g

c y c le le n g th s from 12 to 18 months and not on c o n s ta n t e q u ilib r iu m

(y e a r ly ) c y c le le n g th s . The lo n g e r c y c le s were more expensive th an

fo r a u n i t no rm alized t o a y e a r ly eq u ilib r iu m c y c le , bu t th e

d if f e re n c e was o f f s e t by few er r e fu e l in g s and reduced n o n -n u c le a r

energy req u ire m en ts .

Energy g e n e ra te d by th e n u c le a r u n i ts d is p la c e d a s much

f o s s i l g e n e ra te d energy as p o s s ib le as long as n u c le a r c o s ts were l e s s

th a n f o s s i l c o s t s . The u n i t w hich produced th e ch eap est energy was

loaded to i t s maximum p ro d u c tio n l i m i t and th e o th e r u n i t s were

loaded p ro g re s s iv e ly acco rd in g to th e c o s t o f t h e i r energy and

r e f u e l in g re q u ire m e n ts .

The system c o s t s u r fa c e appeared to be an i r r e g u l a r s u rfa c e

making a l i n e a r o b je c t iv e fu n c tio n v a l id w ith in a l im ite d a r e a . Two

o b je c t iv e fu n c tio n s w ere n e c e ssa ry to use l i n e a r programming. One

fu n c tio n was u sed in th e system model f o r th e LP-MIP to o p tim iz e .

The second one was u sed in th e n u c le a r model to d e term ine i f th e c o s t

c o e f f i c ie n t s w ere u p d a ted . These two fu n c tio n s a ssu re d convergence

to a t l e a s t a lo c a l optimum.

A d d itio n o f a tim e v a r ia b le to th e en erg y sch ed u lin g problem

g r e a t ly in c re a s e d co m p u ta tio n a l tim e re q u ire m e n ts . A llow ing th e

90movement o f r e fu e l in g d a te s meant a t l e a s t s e v e ra l hours o f

12com putation on an IBM 370/165 com puter ' .

T his model hap dem onstra ted th e f e a s i b i l i t y o f one approach

to th e system sch ed u lin g problem . T here would be la rg e co m p u ta tio n a l

tim e req u irem en ts i f t h i s model were s c a le d up t o handle a la rg e

u t i l i t y system . However, th e model cam do p a r t i a l system a n a ly s is

as w e ll as many s e n s i t i v i t y an a ly se s f o r a la rg e system , i f a

r e fu e l in g schedu le o r a p a r t i a l r e f u e l in g sch ed u le i s known.

12I f th e r e fu e l in g schedu le i s known, an o p tim al s o lu tio n can be o b ta in e d w ith in 1 m inute o f com putation tim e on an IBM 370/165 com puter.

CHAPTER 7

RECOMMENDATIONS

The CRESCENDO model developed in t h i s work i s though t

t o be a unique f i r s t approach to s im u ltan e o u sly a l lo c a t in g fu tu re

energy req u ire m en ts and r e f u e l in g d a te s on a system -w ide b a s i s . The

model has dem onstra ted f e a s i b i l i t y b u t th e re a re s e v e ra l a re a s where

e x te n s io n s can be made.

F u r th e r in v e s t ig a t io n s need to be done to examine th e

b eh av io r and t r e n d s o f th e h y p o th e t ic a l system to v a r io u s c o n d it io n s .

These co u ld in c lu d e :

131 . p e r tu rb a t io n a n a ly se s ;

2 . a d d i t io n o f new c a p a c ity ;

3 . lo a d growth d u rin g th e p la n n in g h o r iz o n ;

U. v a r io u s c o n f ig u ra tio n s o f a sy s te m 's i n i t i a l c o n d i t io n s ;

5 . v a r io u s i n i t i a l c o s t c o e f f i c i e n t s ; and

6 . v a r io u s in te g e r s e le c t io n r u le s .

The l a s t two item s cou ld m arkedly a f f e c t th e r e s u l t s and com p u ta tio n a l

tim e f o r th e system b e in g c o n s id e re d . In a d d i t io n , th e c a lc u la t io n a l

^ ^ P o ss ib le p e r tu rb a t io n s would be a n u c le a r u n i t fo rc e d o u t o f s e rv ic e fo r an ex tended p e r io d , d e la y in th e a d d i t io n o f new c a p a c i ty , fo rc e d ou tage o f a n u c le a r u n i t f o r an i n t e r v a l , and a la rg e in c re a s e in demand f o r an i n t e r v a l .

91

9?

method used f o r th e c o s t c o e f f i c i e n t s a f f e c t s th e r e s u l t s . I f a l t e r

n a te schemes co u ld be d e v is e d , perhaps a sch ed u le b e t t e r s u i te d t o a

u t i l i t y ' s need would be found .

A s tu d y o f 1^he system c o s t s u r fa c e a s a fu n c tio n o f energy

and r e f u e l in g d a te s shou ld be done. T h is would g iv e in s ig h t in to

i t s n a tu r e , i . e . , th e shape o f th e s u r fa c e formed by lo c a l and g lo b a l

o p tim a . Knowledge o f th e s u r fa c e would perm it a d e te rm in a tio n o f th e

adequacy o f a l i n e a r fu n c tio n to a c h ie v e a globed, optimum. The e x p lo ra

t i o n o f th e c o s t su r fa c e would r e q u ir e e x te n s io n s o f CRESCENDO,

a lth o u g h th e s e shou ld be un d ertak en even w ith o u t th e e x p lo ra t io n .

S ince th e CONCERT model has th e f l e x i b i l i t y t o hand le

n u c le a r f u e l p a ra m e te rs , th e m odeling o f a b o i l in g w ater r e a c to r (BWR)

shou ld be in c o rp o ra te d in t o CONCERT. C o n s id e ra tio n shou ld be g iv en to

m odify ing CRESCENDO and CONCERT to accommodate e a r ly r e f u e l in g and

coastdow n.

A f o s s i l e q u iv a le n t u n i t m odel shou ld be in c lu d ed in

CRESCENDO. I t sh o u ld have a m ain tenance sc h e d u le r which should d e v ise

a minimum c o s t m ain tenance p la n knowing th e n u c le a r u n i t s ' r e f u e l in g

s c h e d u le . A p r o b a b i l i s t i c s im u la to r shou ld be u sed to d e term ine th e

ex p ec ted energy c o n t r ib u t io n from each o p e ra t io n a l u n i t t o th e non

n u c le a r system energy re q u ire m e n t. The expec ted energy i s c a lc u la te d

hy f i t t i n g th e u n i t s t o an in t e r v a l lo a d d u ra tio n cu rv e . The n u c le a r

u n i t s would be c o n s id e re d energy l im i te d . The s im u la to r shou ld t r e a t

fo rc e d o u tage r a t e s f o r each u n i t . F in a l ly th e f o s s i l model shou ld

c a l c u la te a com posite c o s t c o e f f i c i e n t f o r r e v is io n o f th e c o s t

c o e f f i c i e n t . A tte n tio n shou ld be ad d re ssed to th e p o s s i b i l i t y o f

o s c i l l a t i o n s betw een th e n u c le a r u n i t s and f o s s i l m odel. T h is cou ld

a r i s e due t o th e p e n a l ty o f o p e ra t in g th e f o s s i l system in a

p a r t i c u l a r mode.

S e v e ra l lo n g e r ran g e e x te n s io n s shou ld in c lu d e th e i n t e r

fa c in g o f CRESCENDO w ith d a i ly and lo n g -ra n g e o p e ra t io n a l p la n n in g ,

u s in g a more s h o p h is t ic a te d n u c le a r model t o op tim ize b a tc h s i z e and

en richm en ts o f f u e l b a tc h e s , in s e r t in g c o n s t r a in ts n e g le c te d f o r

t h i s w ork, and expanding th e model t o a l a r g e r system .

APPENDIX 1

FUNCTIONING OF THE STARTS MODEL

A l . l In tro d u c t io n

Any m ath em atica l model desig n ed t o sch ed u le system

o p e ra t io n i s dependent on th e o p tim iz a tio n te c h n iq u e s e le c te d .

Because o f th e tim e dependency o f o p e ra t io n and r e f u e l in g s , th e

o p tim iz a tio n model must b e dynam ic. The p o te n t i a l f o r c u r re n t

p ro d u c tio n o f en ergy i s a l s o dependent on p a s t and fu tu r e o p e ra t io n

o f th e u n i t . T h e re fo re , i t i s im p o rtan t t h a t any model r e f l e c t

th e s e i n t e r a c t io n s .

L in e a r programming (LP) i s g e n e r a l ly .a s t a t i c o p tim iz a tio n

te c h n iq u e w hich i s d es ig n ed t o examine a problem a t one p a r t i c u l a r

in s t a n t o f tim e f in d in g th e o p tim a l com bination o f d e c is io n v a r ia b le s .

I f th e r e f u e l in g sch ed u le f o r th e n u c le a r u n i t s i s known, LP g e n e ra lly

w i l l be a p p l ic a b le to th e problem . As m entioned in C hapter 3 ,

th e s iz e and com plex ity o f th e system problem su g g e s t th e use o f LP.

A dynamic L e o n tie f fo rm u la tio n i s a p p l ic a b le to t h i s

problem . S e c tio n A1.2 i l l u s t r a t e s th e b a s ic n a tu re o f th e L e o n tie f

model and d e s c r ib e s th e e x te n s io n o f th e b a s ic L e o n tie f model t o

s u i t th e system sc h e d u lin g prob lem . S e c tio n A1.3 c o n ta in s a b r i e f

summary.

9U

95

A1.2 The Dynamic Leontief Model

The dynamic Leontief model (££, 51.) is a method of

input-output analysis which is widely used in economic planning.

The Leontief model provides the basic dynamic philosophy used in

the model formulation for STARTS. The philosophy is the dynamic

balancing of producing, carrying, and refueling energy in each time

interval. The output from each activity flows to meet system

demands of the time interval or to sustain the internal level of

other activities. Additionally, the model can add new capacity

(refueling energy) to the system. For each nuclear unit, let

Ej, = the activity level of energy inventory at the start of interval J for unit k;

E? = the activity level of energy production during interval J for unit k; and

E_, = the:activity level of energy refueled duringinterval J for unit k.

A conservation of energy equation may be rewritten for

each interval:

4 ) - ♦ 4 ) - < J+i = ° t o - 1 '

The equation is identical to ( -l). Writing an equation (Al-l) for

each interval completely couples the operation of the nuclear unit in each interval with every other interval in the horizon. Figure

k-2 illustrates this set of equations.Combining the set of equations for each nuclear unit in the

sample system provides the foundation for the STARTS model. However,

the units are not yet interfaced for the optimization model to examine

96

th e in f lu e n c e one u n i t 's o p e ra tio n e x e r ts on th e o th e r u n i t s .

F u rth erm o re , e q u a tio p (A l- l ) does n o t p re c lu d e th e p o s s i b i l i t y

o f p roducing and r e f u e l in g energy in th e same tim e i n t e r v a l . The

a d d i t io n o f c o n s t r a in t s and in te g e r v a r ia b le s c o r r e c ts th e above

s i t u a t io n s .

The in te r f a c in g o f each u n i t w ith o th e r u n i t s i s accom plished

w ith th e fo llo w in g c o n s t r a in ts :

L ♦ e5 = d j (ai-2)where E^j * energy produced by u n i t k d u rin g in t e r v a l J ;

fE, = energy produced by f o s s i l u n i t e q u iv a le n t d u rin g

J i n t e r v a l j ;

E* = energy produced by in te rc h a n g e u n i t d u r in g in t e r v a lJ J ; and

Dj = system demand d u rin g in te r v a l J .

P a r t o f th e c o n s t r a in ts p ro h ib i t in g p ro d u c tio n and r e f u e l in g

a r e :

(A l-3)

4l * 5j "3 s 4where th e v a r ia b le s have th e same d e f in i t i o n i s in e q u a tio n ( U—2 ) .

T h is s e t o f c o n s t r a in ts fu n c tio n s n o t on ly t o l i m i t energy p ro d u c tio n

l e v e l s , b u t a lso t o in d ic a te r e f u e l in g s when combined w ith eq u a tio n

(A l-2 ) .

97

I n te g e r v a r ia b le s have been in tro d u c e d in to th e form ula

t i o n . The in te g e r v a r ia b le s do n o t a f f e c t th e L e o n tie f model

fo rm u a ltio n b u t they do a f f e c t r e f u e l in g s . The in te g e r s in c lu d ed

in th e c o n s t r a in t s as d e c is io n v a r ia b le s a re in c lu d ed in th e

o b je c t iv e fu n c tio n as i l l u s t r a t e d in eq u a tio n ( h - 6 ) .

The c o n s t r a in t (A l- l) fu n c tio n s a s an energy b a la n c e f o r

each u n i t . However, th e c o n s t r a in ts (A l-3) m ust be s a t i s f i e d a t

th e same tim e . L im itin g th e in te g e r v a r ia b le t o 0 o r 1 g iv e s :

i f = 0 , th e n

(Al-U)

and i f U* = 1 , then

UJ - EJ - *24 ’ 0(A l-5)

> o

T h u s , energy p ro d u c tio n w i l l be a llo w ed d u rin g in t e r v a l s

when a 0 . There i s no p ro d u c tio n during in t e r v a l s when = 1 .

F u rth erm o re , i f th e u n i t i s to p roduce (lJ^ = 0 ) , then th e u n i t must

be capab le o f d e l iv e r in g a base amount o f e n e rg y . H ence, th e

c r i t e r i o n f o r r e f u e l in g i s : i f th e u n it can n o t produce du ring

th e i n t e r v a l , i t must be re fu e le d . The u n i t a l s o can be tak en o u t

o f s e rv ic e b e fo re th e r e fu e l in g c r i t e r i o n i s re a c h e d . T h is i s done

98

t o avo id r e f u e l in g in a d isadvan tageous in t e r v a l—i . e . , s a t i s f y in g

c o n s t r a in t (A l-2 ) in th e most econom ical way.

The r e f u e l in g o f a u n i t w h ile p roducing energy has no t y e t

been p ro h ib i te d . S im u ltan eo u sly p roducing and r e f u e l in g energy i s

p rev en ted by in tro d u c in g th e c o n s t r a in ts

(Al-f?)

where th e v a r ia b le s have th e same d e f in i t io n s as c o n s t r a in ts ( U—3).

For exam ple, i f Uj = 0 , then

^3J " - j UJ - ° ^ E3J “ 0

(A l-7 )

and i f = 1 , then J

-■4 ♦ 4 4 5 0 - *4 s 4(A l-8 )

The fu n c tio n in g o f th e s e c o n s t r a in ts i s in c o n t ra s t to

th e fu n c tio n in g o f c o n s t r a in ts (A l-3 ) . T h is i s seen by n o tin g th a t

when Uj = 0 , th en

4 S 4 * 4

from (Al—1*) w h ile

E4-°from (A l-7 ) .

99A lso , i t can b e no ted t h a t when = 1 , th e n

from (A l-5) and

4s s 4from (Al—8 ) .

T hus, th e c o n s t r a in t s o f (A l-2 ) i n t e r f a c e th e o p e ra tio n o f

each u n i t w ith a l l o th e r u n i t s . The c o n s t r a in ts ( A l - l ) , (A l-3 ) ,

and (A l-6) s e rv e to in t e g r a te th e i n t e r v a l s in to a co n tinuous

h o riz o n and t o s e le c t r e f u e l in g d a te s f o r th e n u c le a r u n i t s . The

c o n s t r a in ts ( A l- l ) and (A l-6 ) a r e n o t needed fo r th e f o s s i l e q u iv a le n t

u n i t s in c e th e f u e l c o s t f o r o p e ra tin g th e u n i t from one i n t e r v a l to

th e n ex t i s assumed t o b e uncoupled . The f u e l f o r t h i s u n i t i s

used alm ost im m ediately and does n o t have a long h i s to r y and energy

dependence a s does n u c le a r f u e l . C o n s tra in t (A l-3 ) i s imposed on

th e f o s s i l u n i t w ith o u t th e in t e g e r v a r ia b le s .

The c o n s t r a in t s i l l u s t r a t e d have a b ro a d e r im p lic a tio n th a n

t h e i r u ses shown above. The v a lu e d en o tes th e maximum th e rm al

power r a t i n g on th e c o re . I t a l s o c o n tro ls th e r a t e o f energy

d e l iv e ra b le ( r a t e o f d e p le t io n o f r e a c t i v i t y ) d u rin g an i n t e r v a l .

The v a lu e d en o tes th e p o in t where th e u t i l i t y w ould d ec id e n o t J

t o co n tin u e o p e ra t io n w ith o u t r e f u e l in g . The maximum energy r e f u e le d ,

R j, c o n tro ls th e maximum excess r e a c t i v i t y in th e u n i t ' s c o re . TheIr

Rj v a lu e r e p re s e n ts th e l i m i t below which th e u t i l i t y c o n s id e rs i t

in f e a s ib le t o r e f u e l l e s s energy . The c o n s t r a in t s (U-5) se rv e th e

100**ksame pu rp o se on th e energy In v en to ry v a r ia b le s a s th e Rj v a lu e does

d u rin g th e r e f u e l in g i n t e r v a l . The v a lu e s Rj and l i m i t th e

le n g th o f o p e ra t io n u n t i l th e n ex t r e f u e l in g . The q u a n t i ty d e r iv e d

from

*5 - (fiJ * ^ ' 4r e p re s e n ts a maximum number o f in t e r v a l s o f o p e ra t io n f o r u n i t k

b e fo re r e f u e l in g i s n e c e ssa ry i f Rj i s r e fu e le d d u rin g in t e r v a l J .

L ik ew ise , th e q u a n t i ty t h a t r e s u l t s from

T^ » (rJ + 1^) /“J T J Jr e p re s e n ts th e minimum number o f in t e r v a l s o f o p e ra t io n f o r u n i t k

b e fo re r e f u e l in g i f Rj i s r e fu e le d d u rin g in t e r v a l J .

S in ce 4 > 4 ' said Rj a r e in p u t te d , th e c o re model does

n o t r e q u ir e c o n s t r a in ts w ith re g a rd t o r a t e o f power p ro d u c tio n ,

r e a c t i v i t y d e p le t io n , maximum b u m u p , e t c . S p e c i f i c a l l y , maximum

—k kbum up i s c o n t ro l le d by R. and I . . These q u a n t i t i e s l i m i t th e bum upJ J

in any c y c le . The maximum burnup on a b a tch r e s u l t s from s e v e ra l

—k k •*k kc y c le s o f o p e ra t io n a t th e maximum o f Rj and I j . I f Rj and Tj a r e

de term ined from o f f - l i n e r e a c t i v i t y c a l c u la t io n s , th e maximum bum up

on any f u e l i s no t exceeded .

A 1.3 Summary

The dynamic L e o n tie f model and o th e r c o n s t r a in ts dem onstrate

th e f e a s i b i l i t y o f s e le c t in g r e f u e l in g in t e r v a l s a t th e same tim e

energy i s b e in g a l lo c a te d t o each u n i t under c o n s id e ra t io n . The

c o n s t r a in t s a r e l i n e a r fu n c tio n s r e a d i ly in c o rp o ra te d in to an LP model.

101

The c o n s t r a in ts im p l ic i t l y l im i t and c o n t ro l o th e r param eters which

o th e rw ise would r e q u ir e more c o n s t r a in ts in th e n u c le a r m odel.

T h e re fo re , a com pact, e f f i c i e n t system model i s fo rm u la ted f o r use

in CRESCENDO.

APPENDIX 2

COST COEFFICIENTS

A2.1 Introduction

The method of calculating the cost coefficients for the

objective function influences the solutions found by STARTS. The

simplex method optimizes the objective function by bringing into

the basis (solution) those decision variables which most improve

the objective function. The relative magnitudes of the cost

coefficients determine the variables, and the constraint values

establish the activity levels of the entering basic variables.

The cost coefficients reflect the cost of decisions

made during each interval. The traditional method of calculating

a nuclear fuel cycle cost is inadequate. This inadequacy arises

from the combination of pre-irradiation and post-irradiation charges

into a single total. This total is then divided by the energy

produced by the batch. The batch costs are combined to give a fuel

cycle cost. Linear programming cannot relate future decisions about

refueling dates and energy production to present decisions and the

cost coefficients should reflect only the costs of present decisions.

Section A2.2 presents the method used to calculate the

cost coefficients. Their relationship to the nuclear fuel cash flow

102

103

i s p re s e n te d . S e c tio n A 2.3 p re s e n ts a b r i e f summary conce rn ing th e

c o s t c o e f f i c i e n t s .

A2.2 Calculation of Cost Coefficients

Chapter U presented the calculational scheme of the

CRESCENDO model. It is important to remember the calculational

flow. The decisions made by STARTS are always based on the cost

coefficients for the best solution of the previous pass through

STARTS. The decisions minimize the present value of the total

system operating cost for the horizon.

The LP cannot calculate a present-valued objective

function unless the cost coefficients entered are present-valued.

The costs must be present-valued to ensure all decisions are based

on the same dollars, otherwise incorrect system trajectories would

result. The decisions being made are the energy activity levels.

The LP cannot present-value its decision variables. Therefore, the

energy variables are the actual activities at each time interval.The total system cost function is present-valued if the cost

coefficients contain a present-value cost and an energy variable

which is not present-valued.

The same total system cost for a given system trajectory

must result for any cost coefficient scheme employed. Making the

cost coefficient calcu3* ’ *ons match the cash flows of the trajectory

assures the proper system cost. Figure A2-1 is the typical cash

flow for a nuclear fuel batch. All pre-irradiation charges may be

lumped at the time of discharge of a batch. Figure A2-2 illustrates

the effective cash flow for a batch. For the sample system trajectory,

Mining & Milling

Enrichment Insertion

Conversion

L.I_L

FabricationPayment

Ii

Discharge Reprocessing & Recovery

i Cycle 1 i Cycle 2 i Cycle 3 i

3 months 3 months2 months

8 months6 months

Cred ts

Figure A2-1: Typical Cash Flow for a Nuclear Fuel Batch

All batch costs present- Reprocessing and creditsvalued to insertion present-valued to discharge

I II Cycle 1______[ Cycle 2_______ | Cycle 3______ j

Figure A2-2: Effective Cash Flow for a Batch

105

the effective cash flow during the horizon based on an assumed

refueling schedule is depicted by Figure A2-3.

The objective function equation ( k -6 ) requires four nuclear

cost coefficients for each interval and one fossil cost coefficient.

It is assumed that the fossil costs remain constant for this analysis

and can be modified in future work on the code. The cost coefficients

for the integer variables can be established readily. The purpose

of the integers is to indicate and locate the refueling intervals.

There are no operating costs associated with these variables. The

cost coefficients are defined as:

where b, = integer cost coefficient for unit k duringinterval J.

The remaining three cost coefficients concern nuclear energy

production. The nuclear unit's cost coefficients are inventory,

production, and refueling. The inventory and refueling coefficients

deal with the cost of the fuel. The production cost coefficient is

associated with the operating and maintenance (O&M) expenses of

operating the unit. The yearly O&M cost which includes the refueling

cost but not the fuel costs is:

C = a + b * cap

where a = yearly fixed component of O&M;

b ® yearly variable component of cost; and

cap ■ yearly capacity factor.

Planning Horizon

Unit 1

Cost input batch

I j Cycle

Cost input batch 5

+5

etc.

tX

IJ_

ICredit discharge

batch 2 ♦I

etc.Credit discharge

batch 3

Unit 2

Cost input Cost input batch U batch 5

♦ ti Cycle U I______§_ICredit disposal batch 2

iI

Credit disposal batch 3

etc.

tX ♦etc.

♦X

Unit 3 ___

Cost input batch U

4 i Cycle 4

Cost input etc.batch 5

I ♦ !________ 5_______L

IX

ICredit disposal batch 2 Credit disposal

batch 3

Figure A2-3: Sample of an Effective Cash Flow for an Assumed Equilibrium Refueling Schedule 101

108The in t e r v a l p ro d u c tio n c o s t i s e s ta b l i s h e d u s in g th e

i n t e r v a l 's c a p a c ity f a c to r and re q u ir in g th e y e a r ly c y c le c o s t to

equal th e y e a r ly sum o f th e i n t e r v a l 's c o s ts . In t h i s sample system

th e re a r e s ix in t e r v a l s p e r y e a r . I t i s assumed th e u n i t p roduces

fo r f iv e in t e r v a l s and does n o t produce fo r one in t e r v a l . The

in t e r v a l p ro d u c tio n c o s t i s found by so lv in g :

5 a ' = a

b (cap ) = 5 b '( ic a p )

5 ( i c a p )■ = cap

where a 1 = in t e r v a l f ix e d component o f O&M;

b ' = i n t e r v a l v a r ia b le component o f c o s t ;

ic a p = in t e r v a l c a p a c ity f a c to r ; and

c a p , a , b , a r e d e f in e d in (A 2-2).

The in t e r v a l j p ro d u c tio n c o s t f o r u n i t k i s g iven by:

Cj = | + | ( ic a p ) (A2-3)

The energy p ro d u c tio n c o s t c o e f f i c i e n t fo r u n i t k in t e r v a l J i s eq u a l

to th e p re s e n t v a lu e o f th e O&M c o s ts d iv id e d by th e energy produced:If

C x PV ck = i J

2J J c (A2-M

where . * e l e c t r i c a l energy produced d u rin g i n t e r v a l J byu n i t k and

PVj = p re s e n t-v a lu e f a c to r from i n t e r v a l to re fe re n c e tim e .

109The in t e r v a l in v e n to ry c o s t c o e f f i c i e n t i s th e p r e s e n t

v a lu ed c o s t o f c a r ry in g charges ( th a t i s , t a x e s , i n t e r e s t , e t c . )

d iv id e d by th e energy in v e n to ry . The c o e f f i c i e n t i s c a lc u la te d by

th e fo llo w in g method:

a . th e v a lu e o f th e c o re i s e s ta b l is h e d f o r each in t e r v a l ;

b . th e v a lu e i s m u l t ip l ie d by th e e f f e c t iv e i n t e r e s t r a t e ; and

c . th e r e s u l t in g v a lu e i s d iv id e d by th e energy in v e n to ry .

T hus, th e in v e n to ry c o s t c o e f f i c ie n t f o r u n i t k and in t e r v a l

J i s g iv en a s :

c j = CORIN(J) x BMRATE x PVj / (A2-5)

where E.. = e l e c t r i c a l energy in in v e n to ry f o r u n i t k d u rin gin t e r v a l J ;

CORIN(J) = n e t va lu e o f uranium in c o re du rin g i n t e r v a l j f o r u n i t k ; and

BMRATE = e f f e c t iv e b im on th ly i n t e r e s t r a t e .

The v a lu e o f th e co re i s c a lc u la te d as a fu n c tio n o f burnup .

Each b a tc h o f f u e l in s e r t e d in to th e co re has a n e t v a lu e . T his n e t

v a lu e i s determ ined from th e c o s t o f p ro c u rin g th e b a tc h p r e s e n t

va lued to th e tim e o f in s e r t io n and any c r e d i t s re c e iv e d from f u e l

d isc h a rg e d d u rin g t h a t i n t e r v a l . F ig u re A2-2 i l l u s t r a t e s th e cash

flow s f o r th e sample sy stem . T h is method o f d e te rm in in g n e t b a tc h

v a lu e p re se rv e s th e b a s ic cash flow o f th e t r a d i t i o n a l b a tc h c o s t .

S ince i t i s assumed an in s e r t e d b a tc h has one enrichm ent and th e n u c le a r

code u se s equivolum e n o d e s , a n e t u n i t c o s t f o r th e b a tc h may be

d e f in e d a s :

UNIN0K(JK,K) = UNICPL(NB) / Wl(NB) (A2-6)

110

where UNINOK(JK,K) = n e t u n i t c o s t f o r node K, c y c le JK,($1000/MTU);

UNICPL(NB) - n e t v a lu e o f a f u e l b a tc h , NB, ($1000); and

Wl(NB) = w eigh t o f f u e l b a tc h NB.

Each c y c le has s e v e r a l b a tc h e s in c o re . Each b a tc h h as a

c a lc u la te d n e t u n i t c o s t and com prises s e v e ra l nodes o f th e c o re .

F or each in t e r v a l th e co re in v e n to ry i s c a lc u la te d on th e f r a c t io n

o f i n i t i a l uranium rem a in in g . The in v e n to ry v a lu e may be summed

g iv in g :KMAX

CORIN(J) = 5 UNINOK(JK.K) * URANIA(J,K) (A2-7)k = l

where CORIN(J) = v a lu e o f in v e n to ry uranium f o r in t e r v a l J ;

URANIA(J,K) * uranium (MTU) a t node K d u rin g i n t e r v a l J ; and

KMAX = number o f nodes composing th e c o re .

S u b s t i tu t in g e q u a tio n (A 2-7) in to e q u a tio n (A2-5) p roduces a c a lc u la

t i o n a l scheme f o r d e te rm in in g th e in v e n to ry c o s t c o e f f i c ie n t f o r each

r e a c to r f o r each i n t e r v a l .

The rem ain ing c o s t c o e f f i c i e n t i s th e r e f u e l in g c o e f f i c i e n t .

I t u ses th e n e t b a tc h v a lu e UNICPL(NB). The n e t v a lu e o f th e b a tc h i s

d iv id e d by th e energy r e fu e le d a t t h a t r e f u e l in g i n t e r v a l . The

re f u e l in g c o s t c o e f f i c i e n t , ( £ . , f o r u n i t k , in t e r v a l J i s :i U

C^j *= UNICPL(NB) * PVj / (A2-8)

where E^, = th e en ergy r e f u e le d t o u n i t k , in t e r v a l J .

A2.3 SummaryThe cost coefficients outlined are consistent with the cash

flow of the traditional fuel cycle cost calculation. The coefficients

do include the operating and maintenance expenses which normally are

not included with fuel cycle costs.

The objective function is the total present-valued operating

cost. Coupled with the iterative scheme where the cost coefficients

a re alw ays one i t e r a t i o n beh ind th e energy a l l o c a t i o n s , th e energy

lUused in th e c a lc u la t io n o f th e c o s t c o e f f i c ie n t s i s n o t p re s e n t-v a lu e d .

The c o s t o f p ro c u rin g new f u e l , e q u a tio n (A 2-7 ), i s assumed

to be p ro ra te d o v e r th e n ex t c y c l e 's p lanned en erg y . T h is i s done

to decouple p re s e n t d e c is io n s from fu tu re d e c is io n s . The same i s t r u e

fo r c a lc u la t in g a n e t b a tc h c o s t . These two p ro ced u res make th e c o s t

c o e f f i c ie n t s f o r th e r e s p e c t iv e in t e r v a l dependent on ly on th e d e c is io n s

made in th e i n t e r v a l . T h is i s im p o rtan t s in c e l i n e a r programming does

no t have th e c a p a b i l i ty to d e a l w ith fu tu re d e c is io n s i n t e r a c t in g

w ith p re s e n t d e c is io n s .

Decisions are made to minimize total present-valued operating cost.

APPENDIX 3CODE FOR NUCLEAR COSTING OF AN ENERGY AND REFUELING TRAJECTORY

A 3.1 In tro d u c t io n

The i t e r a t i v e scheme fo r t h i s problem r e q u ir e s a m ethod o f

r e v is in g th e c o s t c o e f f i c i e n t s . The r e v is io n p e rm its th e model to

p ro ceed se e k in g th e o p tim a l energy and r e f u e l in g t r a j e c t o r y . I f th e

r e v is io n i s n o t done, th e n th e CRESCENDO model would seek th e o p tim a l

s o lu t io n b ased on th e wrong c o s t c o e f f i c i e n t s . The upd a tin g o f th e

c o s t c o e f f i c ie n t s csm be done by s e v e ra l m ethods. The m ajor c o n s id e ra

t i o n s a re accu racy and e x e c u tio n tim e .

The STARTS model re q u ire s s e v e ra l hours o f com putation

tim e . T h e re fo re , th e n u c le a r u n i t model shou ld use a minimum o f

e x e c u tio n tim e c o n s is te n t w ith m a in ta in in g a f a i r d eg ree o f accu racy .

A pseudo one-d im ensional p o in t d e p le tio n a lg o rith m (PODECKA) was

developed and m odified f o r th e purpose o f r e v is in g th e c o e f f i c i e n t s .

PODECKA was w r i t t e n as a m u ltip u rp o se co re m odel. The main

purpose o f th e model i s to fu n c tio n as p a r t o f a system sch ed u lin g

co de . PODECKA c a lc u la te s th e n u c le a r f u e l cy c le c o s ts acco rd in g to

c u r r e n t ly a c c e p te d m ethods. I t s accu racy i s w ith in U p e rc e n t o f

112

113more d e ta i le d c a lc u la t io n s o f th e n u c le a r fu e l cy c le c o s t . The

d e s c r ip t io n o f t h i s code , flow c h a r t , and t e s t runs a re in c lu d ed

in Appendix 6 .

S ince PODECKA i s a s ta n d -a lo n e m odel, i t re q u ire d some

m o d if ic a tio n s t o in t e g r a te i t in to CRESCENDO. The m o d if ic a tio n s w ere

made to accommodate d a ta t r a n s f e r betw een STARTS (MIP) and CONCERT

(PODECKA) and c a l c u la t io n a l re q u ire m e n ts . The m o d if ic a t io n s a re

p re s e n te d in s e c t io n A 3-2. S ec tio n A3-3 c o n ta in s a b r i e f summary.

A 3.2 CONCERT Model

CONCERT i s b a s i c a l l y th e PODECKA co d e . The m o d if ic a t io n s

in t e r f a c e th e code w ith STARTS. CONCERT com piles th e c o s ts o f

m ee tin g a t r a j e c t o r y p roduced by STARTS. In a d d i t io n , s e v e ra l o f th e

o p tio n s o f PODECKA a re n o t u sab le in CONCERT.

The d a ta in p u t-o u tp u t su b ro u tin e (DATIN) o f PODECKA has

been removed and re p la c e d by th e main program o f CONCERT. The purpose

o f th e main program ib t o r e ta in th e c h a r a c t e r i s t i c d a ta f o r each

n u c le a r u n it o f th e sample system . The main program s e t s each u n i t ' s

c h a r a c t e r i s t i c s which a r e co n ta in ed in DATA s ta te m e n ts . T his method

o f m a in ta in in g th e d a ta i s n ece ssa ry s in c e th e in c o re s to ra g e a re a s

o f th e MPSX package and CONCERT o v e r la p . C onsequently a l l knowledge

o f v a r ia b le v a lu e s in one code i s l o s t when th e o th e r code i s a c c e sse d .

PODECKA has been fu r th e r m o d ified t o c a lc u la te th e m e tr ic

to n s o f uranium a t each r e a c to r node f o r each tim e i n t e r v a l . The

m e tr ic to n s o f uranium a t th e b e g in n in g o f each in t e r v a l a r e :

URANIA(L,K) = MTU[A11+A12(BURNAP(K)/1000)

+A13(BURNAP(K)/10002 ] (A3-1)

Ill*

where MTU = m e tr ic to n s o f uranium m e ta l a t node K a t th es t a r t o f in t e r v a l L;

A l l , A12, A13 = e m p ir ic a lly d e r iv e d f i t t i n g c o e f f i c i e n t s ; and

BURNAP(K) = burnup o f node K a t th e s t a r t o f i n t e r v a l L.

The cum ula tive burnup o f any node f o r any i n t e r v a l i s found from :

BURNAP(K) = BURNAP(K)+EXPO(K)/MTU (A3-2)

and EXPO(K) = PAWER(K)+THEEMA(J,L) (A3-3)

where EXPO(K)/MTU = th e bum up re c e iv e d by node K d u rin g ani n t e r v a l ;

PAWER(K) = a b s o lu te power f a c to r f o r node K as determ ined in PODECKA and m a in ta in ed th ro u g h o u t th e p la n n in g h o r iz o n ; and

THERMA(J,L) a th e rm al energy (MWD) produced d u ring in t e r v a l L by u n i t J .

E quation (A3-1) p roduces th e uranium m e ta l lo a d in g s which

a re used t o c a lc u la te th e v a lu e o f th e co re in v e n to ry . S ince th e

uranium m e ta l i s d e p le te d as a fu n c tio n o f b u rn u p , th e in v e n to ry va lue

d u rin g any c y c le w i l l d e c re a se . T h is means in v e n to ry charges a re a

fu n c tio n o f energy p ro d u c tio n .

The p o s s i b i l i t y t h a t CRESCENDO may s e le c t t o r e f u e l a

r e a c to r b e fo re th e u n i t has reach ed d e p le tio n o f r e a c t i v i t y e x i s t s .

To accoun t fo r t h i s , CONCERT d e p le te s each node o f th e co re by th e

amount o f energy t h a t th e u n i t has produced d u rin g a c y c le ( th e s e

c y c le s a re sy n th e s iz e d in CRESCENDO). CONCERT s e le c t s th e b a tc h s iz e

and en richm en t f o r th e upcoming c y c le based on th e c y c le le n g th and

th e amount o f energy s e le c te d t o be r e fu e le d by STARTS. S ince th e

115

d e p le tio n i s b ased on th e p ro d u c tio n energy and r e f u e l in g based on

th e r e f u e l in g en e rg y , CONCERT accoun ts f o r th e c a r ry in g o f r e a c t i v i t y

i f f u l l d e p le t io n does n o t o c c u r . The c a r ry in g o f r e a c t i v i t y i s

r e f l e c te d by a d e c re a se in th e c o s t o f f u e l re q u ire d f o r th e upcoming

c y c le .

The su b ro u tin e NCOST (52) i s used t o c a lc u la te n u c le a r f u e l

c y c le c o s t s . A lthough th e s e numbers a re n o t u sed by CONCERT, NCOST

does c a lc u la te th e t o t a l p re s e n t-v a lu e d p r e - and p o s t - i r r a d i a t i o n

ch arg es f o r each d isc h a rg e f u e l b a tc h . As n o te d in Appendix 2 ,

th e s e numbers a re im p o rtan t in c a lc u la t in g th e c o s t c o e f f i c i e n t s .

T h e re fo re , th e s e numbers a re e x t r a c te d and p la c e d in a common d a ta

b lo c k .

CONCERT r e q u ir e s th e same d a ta a s PODECKA. However, th e

f l e x i b i l i t y o f PODECKA i s r e s t r i c t e d in CONCERT. As m entioned

p re v io u s ly , coastdow n o p e ra t io n i s n o t a v a i la b le in CONCERT. E arly

r e f u e l in g i s a v a i la b le o n ly t o th e e x te n t t h a t STARTS in d ic a te s th e

u n i t i s r e fu e le d e a r l y . CONCERT always p la n s t o d e l iv e r th e energy

s p e c if ie d by th e r e f u e l in g energy v a r ia b le s o f STARTS. T h is im p lie s

e a r ly r e f u e l in g i s n o t r e a l l y d e a l t w ith a s an o p tio n i n CONCERT and i t

i s in PODECKA.

CONCERT a s s e s se s an en ergy and r e f u e l in g t r a j e c t o r y . Thus,

th e o p tio n o f in p u t t in g a b a tc h s iz e and enrichm ent to produce a c y c le

energy i s n o t a v a i la b le . T h is o p tio n co u ld be a v a i la b le i f p ro v is io n s

a re adop ted t o a p p ro p r ia te ly r e v i s e STARTS. CONCERT a l s o excludes

in p u t t in g new w e ig h tin g f a c t o r s . S im i la r ly , th e r e l a t i v e power shape

and th e rm a l le ak ag e c o r r e c t io n f a c to r cannot be m od ified f o r an a n a ly s i s .

F u rth erm o re , d a ta must be p ro v id ed t o g ive th e s t a t e o f th e u n i t a t

th e r e fu e l in g im m ediately p re c e d in g th e s t a r t o f th e p lann ing h o r iz o n .

F ig u re 1»-U i s a b lo c k d iagram o f th e c a lc u la t io n a l flow o f CONCERT.

A d e ta i le d flow c h a r t o f PODECKA i s in c lu d ed in Appendix 6 .

A3.3 Summary

CONCERT i s an a d a p ta tio n o f PODECKA w hich was developed as

a s ta n d -a lo n e c o re m odel. PODECKA's developm ent was undertaken

p r im a r i ly f o r th e purpose t h a t CONCERT s e rv e d . The PODECKA code i s a

f a s t- e x e c u t in g a lg o rith m w ith s u i t a b le accu racy f o r i t s p rim ary

p u rp o se . CONCERT has been expanded t o r e p re s e n t th r e e n u c le a r u n i t s

in an e n t i r e l y s e l f - c o n ta in e d package. The code a t p re se n t r e p re s e n ts

p re s s u r iz e d w a te r r e a c to r s o f th e 1 ,000 MW e l e c t r i c a l c la s s . B o ll in g

w a te r r e a c to r s a re n o t p r e s e n t ly m odeled.

APPENDIX U

CRESCENDO MODEL

AU.l D e sc r ip tio n o f th e Model

CRESCENDO i s th e m a ste r FORTRAN program l in k in g STARTS and

CONCERT. I t encom passes th e MPSX s u b ro u tin e READCOMM and a s e r i e s o f

d a ta -h a n d lin g p ro c e d u re s . The s t r u c tu r e o f CRESCENDO on th e com puter

r e q u ir e s c o n c a te n a tin g CRESCENDO, CONCERT, READCOMM, and th e FORTRAN

P a r t i t io n e d D ata S e t w ith th e MPSX P a r t i t io n e d D ata S e t . The r e s u l t i s

th e in c o rp o ra tio n o f CRESCENDO and CONCERT under t h e c o n tro l o f MPSX.

H ovever, CRESCENDO s t i l l m a in ta in s i t s fu n c tio n as th e l i a i s o n between

STARTS and CONCERT.

The m ost re c e n t in te g e r s o lu t io n i s p a sse d to CRESCENDO by

u se o f th e READCOMM s u b ro u tin e . CRESCENDO s y n th e s iz e s th e i n t e r v a l

d a ta in to c y c le le n g th s and energy req u ire m en ts . The i n i t i a l s t a t e

o f th e n u c le a r p o r t io n o f th e sam ple system i s s p e c i f i e d by DATA s t a t e

m ents. T h is d a ta in c lu d e s th e energy produced to d a te in th e c u r re n t

c y c le , th e e la p se d tim e s in c e th e s t a r t o f th e c y c l e , th e s t a r t u p

tim es f o r th e p re v io u s c y c le s , and th e e q u ilib r iu m energy and cy c le

le n g th . CRESCENDO u se s t h i s d a ta t o lo c a te r e f u e l in g d a te s and energy

p ro d u c tio n . The e l e c t r i c a l en ergy produced each i n t e r v a l i s co n v erted

117

t o th e rm al en erg y by:

THERMA(K,J) * ELECT(K,J)/EFF(K) (AU-l)

w here ELECT(K,J) * e l e c t r i c a l energy produced by u n i t K d u rin gi n t e r v a l J ;

EFF(K) a e f f ic ie n c y o f co n v ers io n o f th e rm a l energy t o e l e c t r i c a l en ergy f o r u n i t K; and

THERMA(K,J) ■ th e rm a l energy p roduced by u n i t K d u r in g in t e r v a l J .

The th e rm al energy r e f u e le d f o r u n i t K f o r c y c le Lf ENRG(L), i s :

ENRC-(L) » REFUEL(K,J)/EFF(X) (kk-2)

w here REFUEL(K,J) ■ e l e c t r i c a l energy r e f u e le d d u rin g i n t e r v a lJ f o r u n i t K.'

The u se o f a v a r ia b le v l t h a s in g le s u b s c r ip t , e q u a tio n (kb -2 ) ,

i s s u i ta b le s in c e CRESCENDO t r e a t s one n u c le a r u n i t com pleting

a l l c a lc u la t io n s b e fo re doing any c a lc u la t io n s f o r any o th e r n u c le a r

u n i t . Each tim e th e r e f u e l in g en erg y v a r i a b le , REFUEL(K,J), h a s a

n o n -ze ro v a lu e , a new c y c le i s e s ta b l i s h e d f o r th e p a r t i c u l a r u n i t

u n d e r c o n s id e ra tio n .

CRESCENDO tr a n s m its t h e th e rm a l energy p roduced and th e rm a l

en e rg y r e f u e le d a r ra y s t o CONCERT from STARTS. U sing th e s e a r r a y s ,

CRESCENDO c a lc u la te s a c y c le c a p a c i ty f a c to r d e f in e d a s :

CFCYC(J) - (T JT (J)-B )/T JT (J) (AU-3)

w here CFCYC(J) ■ c a p a c ity f a c t o r o f c y c le J f o r u n i t u n d e rc o n s id e ra t io n ;

T JT (J) ■ nuriber o f days in c y c le J in c lu d in g r e f u e l in g o u ta g e ; and

B ■ 60 .833 days in a r e f u e l in g o u ta g e ;

and eui I n te r v a l c a p a c ity f a c to r d e f in e d a s :

CFINV(LL) - ELECT(K,LL)/VERTA (AU-U)

where CFINV(LL) ■ c a p a c ity f a c to r o f in t e r v a l LL f o r u n i tunder c o n s id e ra t io n ; and

VERTA * maximum e l e c t r i c a l energ y th a t can be produced du ring an i n t e r v a l ( u n i t r a t in g K d a y s ) .

Having e s ta b l is h e d c y c le s t a r t d a t e s , c a p a c ity f a c t o r s ,

energ y re q u ire m e n ts , and e q u ilib r iu m c y c le c h a r a c t e r i s t i c s , CRESCENDO

g iv e s c o n tro l to CONCERT. A f te r e v a lu a tin g th e t r a j e c t o r y f o r each

u n i t , CRESCENDO c a lc u la te s th e c o s t c o e f f i c ie n t s d is c u s s e d in

Appendix 2. The c o s t su b ro u tin e o f CRESCENDO a s s o c ia te s th e a p p ro p r ia te

ca sh flow s d e term ined in CONCERT f o r any c y c le w ith th e co rresp o n d in g

in t e r v a l s in t h a t c y c le . The c o s t c o e f f i c i e n t s a re c a lc u la te d and

p re s e n t-v a lu e d from th e cash f lo w s . A d a ta s e t in th e a p p ro p r ia te

form i s e s ta b l is h e d f o r th e s e r e v is e d c o s t c o e f f i c i e n t s . A f te r

com ple tion o f th e t r a j e c t o r y a n a ly s i s and c o s t c a lc u la t io n f o r each

u n i t in d iv id u a l ly , CRESCENDO r e tu r n s c o n t ro l to STARTS which r e v is e s

th e c o s t fu n c tio n and b eg ins se e k in g a new o p tim a l s o lu t io n .

AU.2 Flow C hart o f CRESCENDO

120

R e tr ie v e r e s u l t s o f most r e c e n t m ixed- in te g e r s o lu t io n from MPSX o u tp u t th ro u g h READCOMM su b ro u tin e and p la c e v a lu e s in th e fo llo w in g v a r ia b le s :ENRINV(k , i ) = In v e n to ry energy fo r p la n t

k in in t e r v a l i ELECT(k,i) = Energy produced by p la n t k

in in t e r v a l i REFUEL(k,i) = E nergy r e fu e le d by p la n t k

in in t e r v a l i NOWUP(k , i ) = In te g e r v a r ia b le f o r p la n t

k in in t e r v a l i

LENGTH = Number o f in t e r v a l s in t h i s s tu d y EFF(k) = Thermal e f f ic ie n c y o f p la n t k NPLANT - Number o f n u c le a r u n i t s ________ ■r

Begin u n i t lo o p , k

ISet s t a r t u p d a te s fo r

p re -p la n n in g h o riz o n c y c le s■- tBegin in t e r v a l lo o p , i

n te rv a

S e t s ta r tu p d a te o f new c y c le a t t h i s i n t e r v a l : STRT(J) ' __________

Accum ulate energy produced in SUMCYC

No Have

C a lc u la te le n g th o f l a s t c y c le : TJT(.l)J -P la c e accum ulated e le c t r i c a l energy produced in E L E C ( J ) ____________— t rP la c e number o f i n t e r v a ls s in c e l a s t r e f u e l - in g in INTER(j)

C a ll COSTER

C a ll CONCERT

P la c e th e rm a l energy re fu e le d d u rin g c y c le J in ENRG(j)

E s ta b l i s h c a p a c ity f a c to r f o r cy c le J : CFCYC(j) = (T JT (J ) -6 0 .8 3 3 ) /T J T ( j)

C onvert energy produced by p la n t k in in t e r v a l i to th e rm a l energy : ______________ THERMA(k.i)_______________

S e t v a lu e s o f T J T ( j ) , STRT(j), ELEC(J) and INTER(J) f o r p o s tp la n n in g h o rizo n (e q u il ib r iu m ) _____________ c y c le s_____________

E s ta b l i s h c a p a c ity f a c to r fo r in t e r v a l i , p la n t k :CFINV(i) = ELECT(k,i)/maximum

p o s s ib le energy produced in an in t e r v a l ______

1 p la n t com plete

?

■ laa__________________P la c e re v is e d c o s ts fo r o b je c t iv e fu n c tio n : C O ST PV (k,i,l), C 0ST PV (k,i,2 ), C 0ST PV (k,i,3), on f i l e fo r a c c e ss by STARTS

STARTS

SUBROUTINE COSTER 122(P la n t k , l a s t cy c le in p lan n in g h o riz o n , number o f nodes in c o re a r e p a ssed from

________ __________ MAIN.)__________________

C a lc u la te UNICPL(m) f o r m = 1 th ro u g h a l l b a tc h e s f o r t h i s r e a c to r :UNICPL(m) = p re s e n t-v a lu e d procurem ent

c o s t o f b a tc h m + p re s e n t-v a lu e d d i s p o sa l v a lu e o f b a tc h e s d isch a rg e d d u rin g th e cy c le p re c e d in g th e c y c le in which m i s in s e r te d

Begip cy c le lo o p , J , f o r a l l c y c le s in p lan n in g

h o rizo n

3IZZBegin b a tc h lo o p , m

Accumulate nodes in c o re in c y c le J in NODE. S e t m in LOCATE (MUM) and in crem ent MUM.

as b a te in c o re d u r in

cy c le J ?

A ll b a tc h e co n s id e re d

?

NODE= KMAX

Loop number o f nodes in b a tc h m in co re d u rin g c y c le J .UNIN0K(J,NAP) = procurem ent cost/MTU fo r

f u e l lo c a te d a t node NAP d u rin g c y c le j - UNICPL(m)

I n i t i a l i z e CORIN(i) * co re in v e n to ry v a lu e fo r a l l in t e r v a ls i

Begin cy c le loop» j±

Begin in te r v a l lo o p fo r a l l i n t e r v a l s i i n J

±Loop f o r a l l nedes

n in co re=

CORIN(i) * CORIN(i)+UNINOK(j,n)»URANIA(i,n) where URAHIA(i,n) i s MTU a t node n a t b eg in n in g o f in t e r v a l i

M -

COBFRO(i) + o p e ra t io n and m ain tenance expense f o r i n t e r v a l i a

336 + (200 * c a p a c ity f a c t o r f o r in t e r v a l i )

A l l in te r v a l in cycle j considered

h o rizo n

D

12U

BMRATE = b im onth ly i n t e r e s t r a t e

F in d b a tc h e s , m, in s e r t e d in f i r s t cy c le o f p la n n in g ho rizo n :

TIMPA = UNICPL(m)*Wl(m) i s accum ulated f o r a l l m

TAMPA = T IM P A /e lec trica l en ergy r e fu e le d ______ i n t h i s c y c le _________________ _

B egin c y c le lo o p , J , f o r a l l c y c le s in p lan n in g

h o riz o n

IB egin in te r v a l loop fo r a l l

i n t e r val, s i in .1

s t h r e fu e l in g n te rv a l

?

C0STUM(k,i,3) * TAMPA C0STUM(k,i,2) + CORPRO(i)/

e l e c t r i c a l en ergy produced in c y c le i

COSTUM(k,i,l) - CORIN(i)* BM RA TE/electrical energy in in v e n to ry i n cy c le i

COSTUM(k,i,l) = 0 C0STUM(k,i,2) » 0 CORIN(i+ 1 ) « CORIN(i+l)

+ CORIN(i)CORPRO(i+l) =

CORPRO(i+l)+336

(Batch lo o p ,_b a tch m

STAR - 1 .0 /(1 .0 +BMRATE)

PVD(l) = 1 .0PVD(i) = P V D (i-l)*

STAR f o r a l li n t e r v a l s i

a s t h i s b a tch in s e r te d

u rin g c y c le ?

Accumulate UNICPL(m) in TEMPO

1 b a tc h e co n s id e re d

?

C0STUM(k,i,3) = TEMPO/ e l e c t r i c a l energy re fu e le d in cy c le J

TAMPA = C0STUM(k,i,3)

| Begin in t e r v a l lo o p , i ~1

COSTPV(jc»i»l) = FVD(i)*COSTUM(k,i,l) ■ c o s t o f energy in in v e n to ry fo r p la n t k in in t e r v a l i

C0STPV (k,i,2) = FVD(i)*C0STUM(k,i,2) = c o s t o f energy produced by p la n t k in i n t e r v a l i

C0STFV (k,i,3) - PVD(i)*C0STUM(k,i,3) = c o s t o f energy r e fu e le d by p la n t k in i n t e r v a l i

No

RETURN

APPENDIX 5

CRESCENDO LISTING

126

cC_________________________ DEFINITION OF VARIABLES * * * _______________________________CC MULTIPLE SINGLE _ _ j___ __ _ _C PLANT PLANTC A1AINI-A9AINI A1-A9 FITTING COEFFICIENTS FOR K-INFINITT CURVC AIOA(N) - AlO -C A19AINI___________A19____________FITTING_COEFFICIENTS FOR U235 CURVE_______C A21AIN) - A21 - FITTING COEFFICIENTS FOR FISSILEC A29AIN) ___A 2 9 __________ PLUTONIUM CURVEC ~ " AVGBURN AVERAGE BURNUP OF DISCHARGED BATCHC _ AVNRCH AVERAGE ENRICHMENT OF DISCHARGED BATCHC BASE!NI ELECTRICAL ENERGY EXPENDED PRIOR TOC __________________________________PLANNING HORIZON _ _________________ _c b e g i n i n Yj j "date of boc j for plant n for PRE-C_ __ _ _ PLANNING HORIZON CYCLESC " BENRCH ......... ENRICHMENT OF A BATCH BEING REMOVEDC FROM COREC BMRATE " BIMONTHLY INTEREST RATEC___________________ BURN ___ FULL POWER DAYS_______________ _ _ ______________C BURNINIKI BURNUP ON FUEL AT NODE K AT BOCC BURNUP!Kl BURNUP ON FUEL AT NODE K AT EtiCC C1AINI-C9AIN) C1-C9 ENRICHMENT FITTING COEFFICIENTSC ____ _ CFCYCIJI _ CAPACITY FACTOR FOR CYCLE J FOR PLANNINGC " HORIZONC___________________ CFINVIU_________ CAPACITY^ FACTOR FOR INTERVAL L _____________C COMPAR DESIRED TOTAL EOC K-INFINITYC COR I NIL I _ INVENTORY VALUE OF FUEL INCORE INC INTERVAL LC CORPROILI OPERATION AND MAINTENANCE EXPENSE FORC ~ INTERVAL LC COSTPVIN>L» 1 1_____________________ COST OF ENERGY IN INVENTORY FOR PLANT NC IN iNTERVAL LC COSTPVI NfL't 2 1 _ _ COST OF ENERGY PRODUCED BY PLANT K INC INTERVAL LC.COSTFVIN,L,3l COST OF ENERGY REFUELED BY PLANT K INC - INTERVAL LC_____________________CTP_____________ CORE THERMAL POWER__________________ ;__________C DUKI 'ENERGY PROOUCED BY NODE KC OISPAINI _ TIME EXPENDED IN CURRENT CYCLE UP TOC " START OF'PLANNING HORIZONC ECFAIN.NNI ECFINNI SW WORK MULTIPLE ENRICHMENT CORRECTION FC FOR PLANT NNC EFFINI _____ _______ THERMAL EFFICIENCY OF PLANT N__ ___________C EIGHT(NtLI WEIGHtfLI ' WEIGHTING FACTOR FOR NODE LC EKCYCLINI CYCLE LENGTH FOR PLANT N IN EQUILIBRIUMC CYCLEC EKEFFAIN«JI EKEFF IJI DESIRED EOC K-EFFECTIVE FOR CYCLE JC EKENRGINI ~ ENERGY PROOUCED BY PLANT N IN EQUILI-C _ ______ __________________ BRIUM CYCLE _ _ _ __C " ELEClJI ELECTRICAL ENERGY PRbDUCED IN CYCLE "J "C ELECTINfL)_________________ ______ ELECTRICAL ENERGY PROOUCED BY PLANT NC IN INTERVAL LC ENAINtLI ENILl ENRICHMENT AT NODE L AT BOCC ENAAIN*Jl ENOAIJI ENRICHMENT OF NEW FUEL BATCH IN CYCLE JC ___________ ENRGIJl__________ENERGY REFUELED IN CYCLE J _____________

128

MAINC ENRINVIN.L) ELECTRICAL ENERGY IN INVENTORY AT PLANTC N IN INTERVAL L

"C EOCYCAIN) EOCYCL CYCLE LENGTH OF CHARACTERISTIC CYCLEC EOENRAI N) EQENRG ENERGY PRODUCTION FOR CHARACTERISTICC CYCLEC EQNRCAIN) EQNRCH ENRICHMENT OF CHARACTERISTIC CYCLEC ERGAIN.J.L) ERGIJ.L) ENERGY PRODUCED IN CYCLE J BY FUEL ATC NODE LC EXPO(K) ENERGY REFUELED IN CYCLE JC FACTOR(J) CAPACITY FACTOR FOR CYCLE J FOR ALLC CYCLESC FC1CJ) LEVELIZED FUEL COST FOR CYCLE JC FDEBTAIN) FDEBT TOTAL CAPITAL PROVIDED BY DEBTC FDNRCH FEED ENRICHMENTC FLFAIN.NN) FLFINN) FABRICATION LOSSES FOR PLANT NNC FR FEED TO PROOUCT RATIOc ................................ GAM IK) REACTOR GEOMETRIC CONSTANT FOR NODE KC HRLEKAIN) THRLEK THERMAL LEAKAGE CORRECTION FACTORC IMAXAIN) IMAX "NUMBER OF SETS OF UNIT COST DATAC INTER!J) NUMBER OF INTERVALS IN CYCLE JC IYEARAIN) I YEAR YEAR WHICH CORRESPONDS TO FIRST SET OFC UNIT COSTSC J l l A I N ) J l l FIRST CYCLE OF PLANNING HORIZONC J22AIN) J2.2 LAST CYCLE OF PLANNING HORIZONC KCTIJK) nod£ of fuel with re l at i ve order ofC REACTIVITY JKC KMAXAIN) — KMAX NUMBER OF NODES IN COREC LATEAIN.J) LATE(J> SIGNALS OPTION TO HAVE DELAYED STARTUPC OR COASTDOWNC LENGTH NUMBER OF INTERVALS IN PLANNING HORIZONC MTU METRIC TONS URANIUM IN EACH FUEL REGIONC NBAT NUMBER UF BATCHESC NINPAIN) NINP DEBUGGING OUTPUT OPTIONC NJ1IM) CYCLE NUMBER BATCH M WAS INSERTEDC NJ2IMI CYCLE NUMBER BATCH M WAS DISCHARGEDC NKEAPIN.J) NKEEP(J) NUMBER QF NODES TO BE KEPT IN CORE AFTERC SHUFFLING OF FUEL DURING CYCLE JC NNOBAT NUMBER OF NODES IN A BATCHC NODESIM) NUMBER OF NODES IN BATCH MC NOWUPIN.L) REFUELING FLAG FOR PLANT N, INTERVAL LC NREMOV NUMBER QF NODES TO BE DISCHARGEDC NRMAXAtN1 NRMAX NUMBER OF REACTORS FOR WHICH ECONOMICC DATA IS INPUTTEDC NRMV!J) BATCH SIZE INUMRER OF NODES) FOR CYCLE JC OTMTUAIN) TOTMTU METRIC TONS URANIUM INCOREC PAWERAIN,L1 PAMERILI RELATIVE POWER FACTOR FOR NODE Lc PHIO(K) INITIAL FLUX GUESS FOR FUEL AT NODE KC POWER(K) POWER LEVEL OF NODE KC PVA(N»1 • MM) PV(NN,I,MM) FEED MATERIAL EFFECTIVE TIME PERIODC PVAIN,2,MM) PVINN» 2 *MM) SW EFFECTIVE TIME PERIODC PVAIN,3,MM) PVINN,3 , MM) FABRICATION EFFFCTIVE LEAD TIME PERIODC PVAIN.A.MM) PVINN,A,MM) REPROCESSING EFFECTIVE TIME PERIODC PVAIN,5,MM) PVINN,5 ,MM) PLUTONIUM EFFECTIVE TIME PERIODC PVA(N,6,HM) PVINN,6 , MM) DISCHARGED URANIUM EFFECTIVE TIME PERIODc PVC1IM) PROCUREMENT COST OF FUEL FOR BATCH Mc PVC2IM) SALVAGE VALUE OF FUEL FOR BATCH M

129

MAINC RATE FIXED CHARGE RATE FOR FUEL CYCLE WORKINGC ____ ____________ ____________ CAPITAL ____________ __________________________t RATEDAIN) RATED INTEREST RATE ON DEBT CAPltALC RATEEAIN) RATEE RATE OF RETURN ON ECUITY CAPITAL ______C RATE I AIN) RATE) PROPERTY INSURANCE RATEC RATEPA(N) RATEP PROPERTY TAX RATEC KEFUELIN,L) ELECTRICAL ENERGY REFUELED BY PLANT NC ____________________________________ IN INTERVAL_L ________________________C R LFAIN«NN) RLFINN) REPROCESSING LOSSES FOR PLANT NNC _ _ RQKIN TOTAL K-INFINITY FROM CLD FUEL INCOREC - STRT(J) ' DATE OF REFUELING IN CYCLE JC _ SUNKIKI TOTAL K-INFINITY FOR A NEW FUEL BATCH ATC NODE KC TAXIFA(N) TAXIF FEDERAL INCOME TAX RATE______________________C TAXI SAI N) TAXIS STATE INCOME tAX RATEC TAXSAIN) TAXS GROSS REVENUE TAX ON POWER SALESC THERMAtN.L) THERMAL ENERGY PRODUCEO BY PLANT N INC INTERVAL LC TIINA(N.L) TIINIL) LOADING TIME OF FUEL AT NODE LC TINAIN,I,MM) _ TUIIMM, l._MM)_ FEED MATERIAL LEAD TIME BEFORE INSERTIONC~ ' " ' “ FOR PLANi" NNC TINAIN,2 , MM) TININN,2 * MM) SW LEAD TIME BEFORE INSERTIONC TINAIN,3 , MM) TININN,3 , MM) FABRICATION LEAO TIME BEFORE INSERTIONC TINAIN,A,MM) TININN,A,MM) REPROCESSING LAG TIME AFTER DISCHARGEC TINAIN,5 , MM) TININN,5 , MM) PLUTONIUM LAG TIME AFTER DISCHARGEC TINAIN,6 , MM)__TININN,6 , MM)____ DISCHARGED URANIUM LAG AFTER DISCHARGEC •' ........... T I P " THERMAL ENERGY EXPENDEO PRIOR TO PLANNINC HORIZONC TJIJ) CYCLE LENGTH OF CYCLE JC TJ1AIN) TJ1 FIRST YEAR OF DEPLETION OF INITIAL FUELC LOADINGC UCAlN,1 , 1 ) UC I1 , I ) ________ COST OF NATURALURANIUM FEED MATERIALC" ~ ' ’ ~ FOR YEAR T ‘C UCAIN,2 , i ) UC I2 . I ) COST OF SW FOR YEAR IC UCAlN,3 • I ) UCI3 , I ) COST OF GENERAL FABRICATION FOR YEAR IC UCAlN,A,I) UCIA,I) COST OF RECOVERY FOR YEAR IC UCAlN,5 , I ) UCI5 , I) VALUE OF RECOVERED PLUTONIUM FOR YEAR IC ____________ _____ UEC Jl J)__ FUEL COST_FOR CYCLE J IN CENTS PER M8TU_C~ UECNIM) “ FUEL COST PER BATCH IN CENTS PER MBTUC UFAIN,J• I ) U F I J . I ) COST OF FABRICATION FOR JTH REACTOR FORC YEAR IC UNICPLIM) PRESENT-VALUED PROCUREMENT COST OFC BATCH M ♦ PRESENT-VALUED DISPOSAL VALUEC OF BATCHES DISCHARGED DURING THE CYCLEC “ ' ' PRECEDING THECYCLE IN WHICH M IS INSERTEC UNINOKIJ,K) PROCUREMENT COST OF FUEL AT NODE K INC .............................. CYCLE JC URANIAIL,K> MTU AT NODE L AT BEGINNING OF INTERVAL KC URNUPAIN,L) BURNUPIL) BURNUP AT NOOE L DURING AND EOCC ______________ VERT _ _______ MAXIMUM THERMAL^ENERGY_PROOUCED_ IN_AN__C INTERVALC VERTA MAXIMUM ELECTRICAL ENERGY PRODUCEO IN ANC " INTERVALC WIIM) MTU LOADED IN BATCH M _ _ _ _ _ _C W2IM) MTU DISCHARGED IN BATCH MC_____________ W3IM)_________ KG FIS SILE PLUTONIUM DISCHARGED IN____________

130

M A I N

C ~ BATCH MC ___._______HR ____________SHU PER KG ENRICHED U R A N I U M __________C XllHI U235 ASSAY LOADED IN BATCH MC X2(Ml U235 DISCHARGE ASSAY IN BATCH MC XF FEED ENRICHMENTC XTAIN) XT U235 TAILS ASSAYC YET(N) CAPACITY FACTOR FOR POST-PLANNINGC _ HORIZON CYCLE K_ _ _ REAL *8 NAME,COLUMN!600).VALUES!6001,Y(6001,Z1600),FUN,XNA,WORD

DIMENSION KINO(600),Nl14) __ .DATA FUN/'FUNCTION•/DATA LETTER/'C'/COMMON/ERNIE/THERMA!3,42)»NOWUP!3*42>/CAPFAC/CFCYCI151/STARTS/

1STRTI15)/PRflD/ENRG(20»/CF1N/CFINV!42)/COSTS/COSTPV!3,42,6)/QR10N/2INTERI151/CONVGR/EFF(3>..... .COMMON/ZEE/TCP _ ______________CUMMON/ENINV/ENRINV!3,42)COMMON/PLAN/ENRGA!20) _ _ _ _ _ __COMMON /GRAF/ELECT!3,42)DIMENSION R£FUEL!3,42),EKCVCLI3),EKENRG!3),01SPA!3I,______________

' 1BASEI3I,ELEC!20),TJT!20)DIMENSION BEGIN!3*4)DATA BASE/192520.,243360.,60840./.DISPA/.666,.833,.333/DATA EKENRC/3*304200./,EKCYCL/3*1.167/DATA BEGIN/78.333,78.166,78.666,79.5*79.333,79.833,BO.667,60.5,

181.0,81.833,81.666,82.166/_________;________________________________IFILE*1

C**** THIS SECTION RETRIEVES THE RESULTS OF THE MOST RECENT C**** MIXED-INTEGER SOLUTION FROM MPSX THROUGH THE FOLLOWING C**«* READCOMM SUBROUTINES

F0S=0.0 CHANGE = * 0 . 0 __________________________________________________

CALL GETARGINUM) CALL "POSI TNI IF ILE, INDIC J _ ____________ ______________

CALL ARRAY!IFILE,IND,NAME ICALL COLNAM! IFILE,KIND,COLUMN,NUMB! _______ __ _ __ _CALL VECTOR(IFILE,INDI,VALUESIDO 16 N=1,NUMBIFICOLUMNlNI.EQ.FUNI GO TO 15

16 CONTINUE 15 CALL POSITNIIFILE,INDIC,3)

CALL ARRAY1IFILE,IND,NAME)1 - 0 ______

2 4 l » l * lCALL VECTOR!IFILE,IND,VALUES) IF!INO.LE . 1) GO TO 21XNA-VALUESIl) ___CALL BCDBINIXNA.NI)Y( I>*VALUES(3)ZII)-VALUES!4IIF!Nil4 ).E0.1IENRINV(N1!2),N1I3I)-Y!I>*10000.IF I N I ! 4 ) . E 0 . 2 ) E L E C T ( N II 2) ,N 1I 3) ) S Y I I > * 1 0 0 0 0 . " IF ( NlI 4 ) .EQ. 3 ) REFUEL!Nil2 > , N 1 I 3 ) ) » Y I I 1*1000 0 . IFIN1I4) .EQ.4>FOS*FOS+Y!I> * 1 0 0 0 0 . * 1 2 0 .IFINl14).EQ.51 CHANGE«CHANGE»Y11>*10000.*110.

131

M A I N

IF m m .E0.6IN0WUPIN112) .NIC 31 0X0+0010C0NTI N O E ________ ____________ _____________________________

GO TO 24"C _ _ _

21 1*1-1C________________________________________________C

_ WRITE! 6. 1000) !(ELECT(Il,JJI,lI=l,3),.JJ=lt4 2 l __________" 1000‘FORMAT! 6D20.71

WRITE(6,10001 (I REFUEL!II,JJI,11 = 1.31,JJ = l,421 WRITE(6,1001) ({NOWUP(IL,JJ)tII=l,3)»JJ=l,42)

1001 FORMAT! 611$ I _ ______Cj l .________________________________________________;_______

LENGTH=42EFF (11 = • 35 _ _ __ _______________ __________EFF( 2) = •34EFF (31 *. 33_______________________________________ ______NPLANT=3

C***» THE 1 LOOP DOES ALL OF THE NUCLEAR CALCULATIONS FOR EACH C**** PLANT INDEPENDENTLY

00 1 MOP*I,NPLANT K0LD=1

C**** THE 17 LOOP SETS THE STARTUP OATES FOR THE PRE-PLANNING C**** HORIZON CYCLES.

_00 17 J = l,4 _ - ________________________17 STRT!j| =OEGIN!NOP,J)

J*4 _____STRTIJ*l)=STRT(J)*DI SPA (MOP I TIP*BASEIMOP)/EFF(MDP|

C**»* THE 2 LOOP CALCULATES ENERGY PRODUCFD!ELEC). CYCLEC**** LENGTH! TJTI. INTERVAL OF REFUELING!INTER I AND__________C***'* STARTUP DATE! STRT) FOR EACH CYCLE INPLANNINGC***» HORIZON 142 TWO-KONTH INTERVALS! ___

SUMCYC = BASE1 HOP)00 2 K=l,LENGTH _ ___IF!NOWUP(MOP.KI.EO.OI GO TO 3

IF(K.EO.l) GO TO 1 8 ___________________________________IF(NOWUP1MOP.K-1I.EQ•1) GO TO 19

18 ELEC(J)=SUMCYCTJT!J1=60.833*IK-KOLDIINTER!Jl=K-KOLO _ _______ _______

’ KOLD=K______ SUMCYC*0.0_________________________________________________

j*j*iSTRT!J*1I*.167*STRTIJI__________ __ __ __ ____GO TO 2

19 STRT!J*l)=STRT!J*iI*.167 _ ____GO TO 2

3_ SUMCYC=SUMCYC*ELECT 1MOP.KI________________________________STRT!Jt1)=STRT(J*1)*.167

2 CONTINUE _C**** VALUES ARE SET FOR ELEC, TJT, INTER'AND STRT FOR C**»* POST-PLANNING HORIZON CYCLES — J22 IS LAST CYCLE C**** IN PLANNING HORIZON______ IF1N0WUPIM0P.LENGTH!.E0.1IG0 TO 4 _________________________

132

M A I N

ELLCIJ)*EKENRG(MOP) STRTIJ*1 ) = STRT(J J+EKCYCLI M O P I __________ __________________________

TJT C J) = EKCVCL<MOP I*365.1NTER(J)=A2-K0LD+1

A J22=J1FINOWUPIHOP,LENGTH).EQ.l)CLEC(J)»EKENRG(MOP> VERT=60.8333*1000./EFF(MOP)

_______ VERTA=60.8333*1000. ____ ________________________________________DO 9 LL=1,LENGTHTHERMAIMOP,LL)=ELECT(MOP,LL)/EFFIMOPI____________

9 CFINV(LL)=ELECT(MOP»LL)/VERTA DO 11 J-4,J22ENRGA(J)=ELEC(JI/EFF(MOP)

11_ CFCYC(JI=CLEC(J)/(TJT(J)*1000.) _ ______________________ ____CF CYC < A )=ELEC(A ) /((TJT(A )+DISPAIMOP I*365.)* iOOO.)J=5DO 12 LL=1.LENGTHIF(NOWUP(MOP,LL).EQ.O) GO TO 12IF(NOWUP(MOP»LL-1).EO.1)G0 TO 13

ENRG( J)=REFUEL(MOP,LL_)/EFF(MOP)____________________________________J=J*i GO TO 12

13 ENRG(J-l)*ENRG(J-l)*REFUEL(MOP,LL)/EFF(MOP)12 CONTINUE

ENRG(A)=ENRGA(A) ENRG(J22)=EKENRG(MOP)/EFF(MOP) _________ ____

CFCTC(J22)=EKENRG(MOP)/(EKCYCL(MOP)*365.*1000.)9999 CONTINUE _

CALL COCERT(MOP,J22,MARK)CALL COSTER I MOP ,J22,MARK)

1 CONTINUE WRITE (9,52)_______________ ____________ _______________________

52 FORMAT(•NAME•.TlS.'TEST^'/* COLUMNS*/• MODIFY*)C**** THE COST COEFFICIENTS RETURNED BY CONCERT AND COSTER C***» ARE PASSED BACK TO MPSX

00 50 11=1,3 __DO 50 I2=1»h 2

______DO 50 13=1,3 ______________________________________________CALL REVERS ~(WORD,LETTER , 11,12,13)WRITE (8,51) W0RD,C0STPV(Ii,I2,I3) _ _ _ _ _____

51 FORMAT)T5,AB,T15,*0BJl*,F18.6)50 CONTINUE _ _ _______ _

WRITE (8,53) 53 FORMAT('ENOATA')_____________________________________________________

~ENDFILE iff .REWIND 9

£**** THE FOLLOWING SECTION PRINTS OUT THE INTERVAL AND SYSTEM .......C***» COSTS FOR THE CURRENT SOLUTION ______ _

SYSCST*0.0 . F T C D S T * 0 . 0 _____________________________________________________

DO 101 1 = 1,3CTC0ST=0.0 _ _______ __ ____WRITE(6,94)I

94 FORMAT)1H1.59X,'NUCLEAR PLANT*,I2///3X,*INTRVL OATE ENERGY PRO ID ENERGY KEFLD INV ENERGY COST ENERGY COST ENERGY COST IN

______2V TOTAL COST CUM COST'/19X.«MWD X 10000 MWD X 10000 MWD

133

MAIN3 X 10000 PROD CV/MWD) REF C i / H W 0 r ' (i /MHD) * I THOUSAND $) CTHOUA SAND I ) • 1 _____________________________________________________________________

DO 102 J= l« 30DATE-BEGINI I , 1 l + J* . 1 6 7 - . 167 _ _ _________________ ______ ________FICOST-COSTPVCI,J.1 1 * 1 00 0 .COSTP-COSTPV( I , J * 2 1 * 10 0 0 . __ ___ _______COSTR-COSTPVCI,J, 3 1 * 1 0 0 0 .

TCOST- I ENRINV ( I , J 1 *C OSTPV ( l_, J , 1 II ♦lELECTC I .J 1 *COSIPVI I , J . 21 )_♦_______ICREFUELCI,J1*C0STPV<I,J,3)1

EL ECT(I»JI=E LECT(I .J ) /10000 . _______REFUEL( 1 ,J1-REFUEL( I • J I / I 0 0 0 0 .ENRINVII,Jl -ENRINVII• J l / 1 0 0 0 0 . __________ ______CTCOST=CTCnST*TCOST

MRITEC6,9 5 ) J . DAT E,ELECT!I .J l , REFUEL I I , J 1 , ENRINV1 1. J 1,COSTP,COSTR,__i f iccisT, t c o s t , ctcost ............

95 FORMAT!5X,I 2 . 3 X , F 6 . 3 , 3 X . F 9 . A , 5 X . F 9 . A . 6 X . F 9 . A . 5 X . F 8 . A , 5 X , F 8 . A , A X , 1 F 8 . A « 3 X , F 1 0 . 3 , 3 X , F 1 2 . 3 )

102 CONTINUEFTCOST-FTCOST+CTCOSTDO 110 J - 3 l , A 2 __ _

110 SYSCST-SYSCST* CENRINVII, J)*COSTPVII , J , 11) + CELECT( I , Jl*COSTPV( I . J . 2 l ) l+C REFUELC t ,J )*C0STPVC I,J ,3 ) l _

101 CONTINUEMRITEI6 . 5 0 2 1FTC0ST

502 FORMAT!1H0.67X,'CUMULATIVE TOTAL COST FOR THIS SOLUTION*. F 1 5 . 31_ SYSCST-SYSCST+FTCO_ST_____________________________ ________________ ___________

CSTNUC-SYSCSTSYSCST-SY SCST + FOS+CHANGC______________________MRITEI6 . 5 0 3 1 C STNUC «FOS.CHANGE• SYSCST

5 03 F0RMATC1H0,IX,'TOTAL SYSTEM COST FOR THE PLANNING HORIZON OF *,1*42 INTERVALS */5X,'NUCLEAR COSTS - ' , A X , F 2 0 . 2 / 5 X , • FOSSIL COSTS • •2 , 5 X , F 2 0 . 2 / 5 X , J INTERCHANGE COSTS - * , F 2 0 . 2 / / 1 X , ' TOTAL COST « ' , 1 1 X , __3 F 2 0 .2 I ............... ..

CALL'PCTUREIF (NUM.Ell.01 GO TO 77REAOI2 , 7 8 ) ZTCOST _ __REMIND 2

_______ IF IZTCOST.LE.SYSCST) GO TO 79________________________________________________MRI TEC 2 . 781 SYSCST”REMIND 2 _ _ _ ____NUM-2CALL PUTARGCNUMI _______ ____ _____MRITE( 6 , 7 6 )

76 FORMATIIH1,IX,'OBJECTIVE FUNCTION REVISEO1 )_______________________________GO TO 31

77 MRITEI2.78I SYSCST78 FORMAT( F 2 0 . 31 "

79 NUM-1 _ ________CALL PUTARGCNUMI '

31 RETURN______________________ _______________________________________________________END

13U

C ** ** ** COCERT ******C**«* REACTOR PARAMETERS ARE INPUTTED PLUS CONVERSIONS TO _____ __________C**** PLACE VARIABLES SUBSCRIPTED BY PLANT INTO VARIABLES C**** NOT SUBSCRIPTED BY PLANT FOR USE IN THE NUCLEAR C**** CALCULATIONS OF NUCREX. THESE VARIABLES CORRESPOND TO C***« THOSE OF PODECKA

DOUBLE PRECISION D.DVAL, ORATE.DHRATE_ REAL KINF.MXPPFC.MTU.NEWKNF.NDLKNF ______________________________________

INTEGER AOF.EIGHTA COMMON DVAL.DRATE.DMRATECOMMON A l , A 2 , A 3 , A9, A5, A6 ,A7 , A8,A9, A l l , A 1 2 . A 1 3 , A 1 9 , A 1 5 , A 1 6 , A 1 7 , A 1 8 ,

I A 19 .A 20 .A 21 , AZ2, A23, A29 , A25, A2h,A27 , A 2 8 ,A 2 9 , C 1 , C 2 ,C 3 , C 9 , C 5 , C 6 , C 7 , 2C8,C9,CTP»T0TMTU,MTU»TJl«FDNRCH« EQNRCH,EQENRG,EQCYCL

COMMON KMAX,NR,Nl.Jl , J11 , IMAX,NJM_AX, NREMOV, IYEAR.JB.JE. ___1NMAX,J2.NIHP.NCLST, IBUG.KEIGOT,IFLAGCommon nrmv<20i , m i u i o ) , mi r i i o i , kcti30i , aofi30ICOMMON PlIRN INI 3 0 ) , BUR NUPI 30) . PH 101 3 0 1, ENI 30) , EI 6 0 0 1, K INF I 30 > ,

IT I INI 3 0 1,ERG( 1 5 , 3 0 ) . W 2 I 3 0 I , ECF11 0 ) , FLF( 1 0 ) ,RLFC1C I , T I N ! 3 , 6 , 2 1 ,2M3(3 0 ) , PVI3 , 6 , 2 > , UC( 6 , 3 0 > ,UF( 1 0 , 3 0 ) , POWER! 3 0 1 , PAWERI3 0 ) , WEIGHTI 30 3> ,E KE FF(20) ,ENBU RN (30) ,FFCl(30) f X l ( 3 0 ) , X 2 ( 3 0 ) , 0 1 ( 3 0 ) ,GAM(30)

COMMON TAX,TAXIF,TAX IS,TAXI,RATE.FDEBT.RATEE,RATED,RATEV,RATE I , 1RATEVM,TAX2,TAXS«RATEP,THRLEK,XT,LATE(30)

COMMON NKEEPAI20).EN0AI20)COMMON /ERNIE/THERMA(3,92)»NOWUPI 3 , 4 2 1/BUYS/PVC 1 13 0 ) , PVC213 0 ) /

lDEPLET/URANIAI9 2 , 3 0 ) /CAPFAC/CFCYCI1 5 ) /TIME/TJ( 3 0 ) /CYCLES/NJ1( 3 0 ) ♦_ 3NJ2(30)/URAN/W1(30J/CONS/D ,NBAT____________ ..

COMMON/STARTS/StR T( 1 5 ) / PROD/ENRG 12 0 )~DIMENSION NtNPA(3) ,NBLSTA(3) ,IBUGAI3) ,KMAXA(3), EIGHTAI3 ) • OTMTUA

1 ( 3 ) , HRLEKAI3 ) , A 1 A ( 3 ) , A 2 A ( 3 ) , A3A( 3 ) , AAA( 3 ) , A5A< 3 1, A6A(3 ) , A7A( 3 ) , 2 A B A ( 3 ) , A 9 A ( 3 ) , A l l A t 3 ) , A 1 2 A ( 3 ) , A l 3 A 1 3 ) , A 1 9 A ( 3 ) , A 1 5 A ( 3 ) , A 1 6 A ( 3 > , A 1 7 A 3 ( 3 ) , A 1 0 A < 3 ) , A 1 9 A ( 3 ) , A 2 1 A ( 3 ) , A 2 2 A ( 3 ) . A 2 3 A ( 3 ) ,A2AA( 3 ) , A25A( 3 ) , A26A 9 ( 3 > , A 2 7 A ( 3 ) , A 2 B A ( 3 ) , A29A( 3 1 ,C 1A(3 1 ,C2A(3) ,C3A( 3 ) ,C9A(3 ) ,C 5A (3 ) ,C6A 5 1 3 ) , C 7 A ( 3 ) , C B A ( 3 ) ,C9A( 3 ) , LATEA( 3 , 2 0 ) , FACTOA( 3 , 2 0 ) , EKEFFA{3 , 2 0 ) ,6 EIGHT ( 3 , 3 0 ) , URNUPA( 3 , 3 0 ) • ENA(3 , 3 0 ) . T11NA( 3 • 2 0 I . ERGA(3,3 ,30)

DIMENSION IMAXAI3 ) .NRMAXA( 3 1 , 1YfcARA( 3 ) , RATEDA( 3 ) , RATEEA(3 ) , FOEBTA 1 ( 3 ) , TAX If A'(3),TAXISA(3) . RATE I A( 3) , RATEPAI 3) , T AX SA( 3) ,XT A ( 3 >, TJ1A 2 ( 3 ) , U C A ( 3 , 6 , 3 0 ) , U F A ( 3 , 1 0 , 3 C ) , M I I A ( 3 , 1 0 ) , M 1 R A ( 3 . 1 0 ) , E C F A ( 3 , 1 0 ) , F L F A3 ( 3 , 1 0 1, R L F A ( 3 , 1 0 ) . T I N A ( 3 ,6 , 2 ) , P V A ( 3 ,6 , 2 ) , E O E N R A I 3 ) , EJ0CYCAI3), _______9 E 0 N R C AI 3) , J1A (3> .J1 1A I3I , J 2 2 A I 3 ) , PAWERAI3 , 3 0 )

DIMENSION EFF(3),EKENRG(3I»EKCYCLI3)DIMENSION NKEAP(3,20)»ENAA(3,20)»YET(3)»FACTOR(15)DATA NKEAP/60*6/ ,ENAA/60*3 .2 0 / , Y ET /3 * .6 9 /DATA J l A / 3 * l / » J l l A / 3 * 9 / , J 2 2 A / 3 * 1 0 /DATA E F F / . 3 5 , . 3 2 5 , . 3 0 / , E K E N R G / 3 * 1 0 0 0 . 0 / , E K C Y C L / 3 * 1 . 1 6 7 / _____DATA NINPA/3*6/«NBLSTA/3*1 / , IBUGA/3*0/ ,KMAXA/3*18/ , EIGHTA/3*1/ ,

1 0TMTUA/3*B6.2/ , HRLEKA/3*.9 5 / . A 1 A / 3 * . 8 2 3 3 5 / , A 2 A / 3 * . 2 2 5 3 3 7 / , A3A/3* 2 - 0 . 0 2 5 1 8 9 / , AAA/3 * - . 3 9 9 3 7 / , A 5 A / 3 * - . 0 3 9 1 5 7 / , A 6A /3 * . 0 0 7 8 9 8 / , A7A/3*3 . 0 1 5 8 8 3 / , AHA/3 * - . 0 0 1 1 2 2 / , A 9 A / 3 * - . 0 0 0 2 1 / . A l l A / 3 * l . / , A 1 2 A / 3 * - . 0 1 7 7 0 2 9 / , A 1 3 A / 3 * . 0 0 1 0 6 5 / . A1AA/3 * 0 . 0 / . A 1 5 A / 3 * 0 . 0 / , A16A/3 * 0 . 0 / , A 1 7 A / 3 * 0 . 0 / ,

5 A 1 8 A / 3 * O . 3 / . A 1 9 A / 3 * 0 . 0 / , A 2 1 A / 3 * 2 . 1 3 9 5 / , A 2 2 A / 3 * - i . 1 9 1 3 3 9 / , A23A/3*6 . 1 3 5 0 9 3 / , A2 9 A / 3 * - l . 0 9 9 3 5 9 / , A25A/3 * - . 5 0 1 9 6 8 / , A26A/3 * . 1 3 5 8 9 l / , A 2 7 A / 3

_ 7 * 5 . 2 5 3 9 6 / , A 2 8 A / 3 * l . 5 9 7 0 5 9 / , A 2 9 A / 3 * - . 2 9 5 8 9 2 /' DATA C 1 A / 3 * . 9 1 / , C 2 A / 3 * . 0 5 3 1 1 5 / , C 3 A / 3 * - . 0 0 8 9 7 / , C 9 A / 3 * - . 8 7 1 8 / , C 5 A / 3 *1 . 2 6 8 2 9 6 / , C 6 A / 3 * - . 0 2 9 6 8 / , C 7 A / 3 * . 1 7 5 6 6 / , C 8 A / 3 * - . 0 7 8 1 1 7 / , C 9 A / 3 * . 0 1 0 1 9 2 9 /

DATA LATEA/1 5 * 0 , 9 5 * 0 / , FACT0A/60 * 1 .0 / ,E K EF F A /6 0 * 1 . 0 / , EIGHT / 9 0 * 1 . 0 /

135

COCERTI t URNUPA/3*3551B.2 , 3 * 3 5 0 7 7 . , 3 * 3 3 8 6 6 . 5 , 3 * 3 2 6 5 2 . , 3 * 3 1 7 7 7 . 6 , 3 * 2 9 8 9 3 .

2 , 3 * 2 2 8 3 2 . 7 , 3 * 2 2 3 9 1 . 5 , 3 * 2 1 1 7 9 . , 3 * 1 9 9 6 6 . 5 , 3 * 1 9 0 9 1 . 5 , 3 * 1 7 2 0 7 . 6 , 3 * 1 0 1 *3 7 . 2 , 3 * 9 7 0 6 . , 3 * 9 1 5 5 . * , 3 * 8 3 8 6 . 1 , 3 * 7 9 5 0 . 8 , 3 * 6 6 1 9 . 1 , 3 6 * 0 . 0 / , ENA/56*3 .2 * , 3 6 * 0 . 0 / , T I I N A / 7 8 . 3 3 3 , 7 8 . 1 6 6 , 7 8 . 6 6 6 , 7 8 . 3 3 3 , 7 8 . 1 6 6 , 7 8 . 6 6 6 , 7 8 . 3 3 3 , 5 7 8 . 1 6 6 , 7 8 . 6 6 6 , 7 8 . 3 3 3 , 7 8 . 1 6 6 , 7 8 . 6 6 6 , 7 8 . 3 3 3 , 7 8 . 1 6 6 , 7 8 . 6 6 6 , 7 8 . 3 3 3 ,6 7 8 . 1 6 6 . 7 8 . 6 6 6 . 7 9 . 5 0 0 . 7 9 . 3 3 3 . 7 9 . 8 3 3 . 7 9 . 5 0 0 . 7 9 . 3 3 3 . 7 9 . 8 3 3 . 7 9 . 5 0 0 ,7 7 9 . 3 3 3 . 7 9 . 8 3 3 . 7 9 . 5 0 0 . 7 9 . 3 3 3 . 7 9 . 8 3 3 . 7 9 . 5 0 0 . 7 9 . 3 3 3 . 7 9 . 8 3 3 . 7 9 . 5 0 0 ,

8 7 9 . 3 3 3 , 7 9 . 8 3 3 , 8 0 . 6 6 7 , 8 0 . 5 0 0 , 8 1 . 0 0 0 , 8 0 . 6 6 7 , 8 0 . 5 0 0 , 8 1 . 0 0 0 , 8 0 . 6 6 7 ,9 8 0 . 5 0 0 , ' 8 1 . 0 0 0 , 8 0 . 6 6 7 , 8 0 . 5 0 0 , 8 1 . 0 0 0 , 8 0 . 6 6 7 , 8 0 . 5 0 0 , 8 1 . 0 0 0 , 8 0 . 6 6 7 , 1 8 0 . 5 0 0 , 8 1 . 0 0 0 /

DATA ERGA/3*4B763 . ,3 * 6 0 9 6 1 . , 3 * 6 0 9 6 1 . , 3 * 4 6 6 4 3 . , 3 * 6 0 9 6 1 . , 3 * 6 0 9 6 1 . . 1 3 * * 3 9 9 7 . , 3 * 5 7 7 8 0 . , 3 * 6 0 9 6 1 . , 3 * * 0 2 9 0 . , 3 * 5 * 7 7 8 . , 3 * 6 0 9 6 1 . , 3 * 3 8 2 0 8 . , 3 *2 5 3 5 3 9 . . 3 * 6 0 9 6 1 . , 3 * 3 1 8 0 8 . , 3 * 5 0 8 8 * . , 3 * 6 0 9 6 1 . , 3 * 0 . 0 , 3 * * 8 7 6 3 . , 3 * 6 0 9 6 1 . 3 , 3 * 0 . 0 , 3 * * 6 6 * 3 . . 3 * 6 0 9 6 1 . , 3 * 0 . 0 , 3 * * 3 9 9 7 . , 3 * 5 7 7 8 0 . , 3 * 0 . 0 , 3 * * 0 2 9 0 . , 3 * * 5 * 7 7 8 . , 3 * 0 . 0 , 3*382087; 3 * 5 3 5 3 9 . , 3 * 0 . 0 , '3* 31 808 . • 3 * 5 0 8 8 * . , 3 * 0 . 0 , 3 * 0 . 0 5 , 3 * * 8 7 6 3 . , 3 * 0 . 0 , 3 * 0 . 0 , 3 * * 6 6 * 3 . , 3 * 0 . 0 , 3 * 0 . 0 , 3 * * 3 9 9 7 . , 3 * 0 . 0 , 3 * 0 . 0 , 6 3 * * 0 2 9 0 . , 3 * 0 . 0 , 3 * 0 . 0 , 3 * 3 6 2 0 8 . , 3 * 0 . 0 , 3 * 0 . 0 , 3 * 3 1 8 0 8 . , 1 0 8 * 0 . 0 /

DATA lMAXA/3*22/»NRMAXA/3*l/ , IYEARA/3 * 7 5 / , RATEDA/3 * 8 . 5 / .RATEEA/3 l * 1 4 . / , F D F R T A / 3 * . 6 / , T A X I F A / 3 * 4 8 . / » T A X I S A / 3 * 4 . / , R A T E I A / 3 * . 5 / » RATEPA/

2 3 * 0 . 0 / , T A X S A / 3 * 0 . 0 / , X T A / 3 * . 0 0 2 / , T J I A / 7 8 . 3 _ 3 3 , J 8 . 1 6 6 , 7 8 . 6 6 6 / _________DATA E0ENRA/3*961*10. / ,

3E0CYCA/3*1.1 6 7 / , EQNRCA/3*3.2 / . PAKERA/24*1.1 5 , 3 * 1 . 0 9 , 3 * 1 . 0 5 , 3 * 1 . 0 1 , * 3 * . 9 6 , 3 * . 9 2 , 3 * . 8 8 , 3 * . 8 3 , 3 * . 7 6 , 3 * . 7 2 , 3 * . 6 , 3 6 * 0 . 0 /

DATA Ml I A / 3 * 1 , 2 7 * 0 / , MIRA/3 * 3 0 , 2 7 * 0 / , ECFA/3 * 1 . 0 , 2 7 * 0 . 0 / , F L F A / 3 * . 0 1 , 1 2 7 * 0 . 0 / , R L F A / 3 * . 0 2 , 2 7 * 0 . 0 /

DATA TINA/3*13 . 8 , 3 * 1 2 . 0 , 3 * 9 . 0 , 3 * 7 . 0 , 3 * 9 . 0 , 3 * 1 1 . 0 , 3 * 1 3 . 8 , 3 * 1 2 . 0 , 3 * 9 * 1 . 0 , 3* 7 . 0 , 3 * 9 . 0 , 3*11. 0 / »~PVA/3* 1 3 . 8 , 3 * 1 2 . 0 , 3 * 9 . 0 « 3 * “ 7 . 0 » 3 * - 9 . 0 , 3 * “ i i 2 . 0 /

DATA U C A/ 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . . 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 1 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 02 . 0 . 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3

3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * B . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3A l 7 , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 05 . 0 . 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 *6 8 . . 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 7 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 t 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 08 . 0 . 3 * 3 5 . 5 , 3 * 6 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3

_ 9 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 31 1 . . 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 . 3 * 3 5 . 5 , 3 * 8 . , 3 * 02 . 0 . 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 3 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 *3 8 . . 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 * 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 05 . 0 . 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 36 2 . . 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 3 1 . , 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 , 3 * 37 1 . . 3 * 3 2 . , 3 * 0 . 0 , 3 * 3 5 . 5 , 3 * 8 . , 3 * 0 . 0 /

OATA U F A / 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . 1 , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 3 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 6 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 *5 0 . 0 . 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 . 0 , 3 * 9 0 . , 2 7 * 0 6 . 0 /

EQUIVALENCE ( CFCYC( 1 ) , FACTOR(1 1)60 FORMAT!•1, , 5 X , • KMAXA( MOP> = • , 12 , 5X, • KEIGHTA(MOP» * • , 12 , 5 X , • TOTMTUA

l ( M 0 P ) = ' , F 8 . 6 , 5 X , , THRLEKA(M0P|*«,F8.5»62 FORMAT!»0»,«A1A(MOP)THRUA9A(H0P1. Al l A ! MOP!. A12A1M0P1»/ 1 1 F 1 2 . 6 ) ___

136

C O C E R T

43 FORMAT( • 0 * « • A13A!M0P) THRUA19AI MOP), A21A!MOP)THRUA24A|MOP I * / 1 U F 1 2 . 6 )

44 FORMAT!• 0 • , • A25A(MOP) TNRUA29A( MOP),C1A( MOP)THRUC6ACMOP) • 7 11F1 2 . 6 ) 90 FORMAT!'O'.*C7A(M0P).CHA(MOP),C9A|M0P)*/3Fl2.6)61 FORMAT!*0ERG*/*p*,9F12.4 )47 FORMAT!• 0*» 3X,'BURNUP*,9X, • ENRICH* , 5X« ' TIME IN*)46 FORMAT!• • , F1 2 . 1 , 2F 12 . 4 )

48 FORMAT!*0 (MAX NRMAX I YEAR* ,3X,•RATED*, 5X» *RATEE* »5 X , * FDEBT*1 5X ,*T AXI F' , ,5X ,*T AXIS* .5X, • RATE I • , 5 X , • RATEP*,6X, • TAXS* . 8X ,* X T ' ,7 X , 2 * T J 1 * / 3 I 6 , 1 0 F 1 0 , 4 )

52 FORMAT! *OUC S • , 4F10 . I / ' C'Uf S* , 5F1 0 . 11 53 FORMAT!*0*»2X»*NN*, 3X, • Ml I A( MOP.NNI• , 3 X , • MIRA!MOP.NN) • , 3X,

1*ECFA!M0P,NN)• , 3 X , • FIFA!MOP,NNI • , 3 X , *RLFA(MOP,NN) ' / ' 0 * , 1 4 , 5 F 1 0 . 2 )54 FORMAT!*0TIN S * , 6F1 0 . 2 / ' OPV S * , 6 F 1 » . 2 ) ______ _______55 FORMAT ( • 0* , 2X, • NR * »3X, ' N l ' * 3X* ' J l * *2X« *'Jll • , 2X» * J22* / * 0* •

1 1 4 , 4 1 5 / / ! "JENRG S * , 7 F 1 0 . 1 ) )56 FORMAT! *0 tQENRGA(MOP)• , 4 X , • EQCYCLACMOP) • » 4 X , * EQNRCHA!MOP) */ ' O ' ,

1 F 9 . 0 . 2 F 1 0 . 4 )00 5 KKK=1,20NKEEPA!KKK)*NKEAP(MOP,KKK)

' 5 ENOA( KKK) =CNAA( MOP, KKK)NlNP*NINPA(HOP)NBLST=N8LST A( MOP)IBUG=IBUGA!MOP)_____________________ ____KMAX*KMAXA!MOP)

MARKsKMAX ;______________________________________KEIGHT = E IGHT AI MOP)T01HTl)= OTMTUA ( MOP) _____TMRLEK= HRLEKAI MOP).................................................. ..........................................................IF1N1NP.E0.1) V«RIT£(6,40) KMAX«KEIGHT, TOTMTU,THRLEK Al=AlA!MOP)

A2=A2A!MO®1 _ _______________________________________________________________A3=A3A(MOP)~A4=A4A{ MOP) ______ ____________ _________________ ____________A5=A5A!MOP)A6-A6A! MOP) __ _ _A7=A7A1 MOP) "" ' ....... ....... .

A8=A8A!MOP) ___________________________________________________________________A9=A9A'|M0P|“Al lsAl lA(MOP) _______ __________________ _______ __________A12*A12A!MOP)A13*A13A ( MOP) _A14»A14A!M0P)

A15*A15A(M0PI __________________________________________________________________A16°A16A(MOP)A17«=A17AI MOP)A18*A18A(MOP) ~ ~ ' "...................A19-A19AIM0P)A21*A21A!MOP)A22 = A22A(MOP I A23«A23A!M0P)A24=A24A1M0P)A25-A25AIMOP)A26-A26AIM0P)A27-A27AIM0P)

A28-A28AIM0P)_____________________________________________________________ •

137

COCERT

A29=A29A( MtlPI............................................................... .....................................................________ CI=C1A( MOP)________ ______________________________ _____________________________

C2=C2ACMOPIC3=C3A(MOP) _______ ____CA*=C4AC MOP)CS=C5A( MOP) _ _ ___C6=C6A(MOP)

________ C7*C7A( MOP I______________ ;_______________________________________C8=tPA(HOP)C9=C9A(M0P|IF€NINP.EQ.1 I KRITEC6.A2)A1, A2 . A 3 , A* .A 5 , A6 ,A7 , A8 , A9,A11 , A12 IF( NI NP. EO. l ) W RI TE <6 ,A3 IA 13 , A1 ' . ,A15 .A16 ,A17 ,A18 .A19 ,A21 ,A22 ,A23 ,

1 A2A______ J F ( N I N P .E O . l ) WRITEI6.AAIA25 .A26,A27 ,A2B,A29.C1,C2 .C3 ,CA.C5.C6

IFTn INP.EO.11 WRITEI 6*901C7*C8,C9NR*1 ! _ _ __________ _______N l=iJ1=JIAC MOPI _ __________J 11 = J 11 A( MOP IDO 80 J=J 11 • J 2 2 __________ _____________________________________________LATE!JI=LATEA(HOP»J)FACTOR!J)=CFCYC(J)

80 EKEFF!J)=EKEFFAIMOP*J)NY=J22+1 _ _ _________________MASS=J22+3

DO 81 PAINE=NY,MASS _ ____________________________________________________EKEFF( MAINEV= EKEFFA( MOP.MMNE )FACT0RIMAINEI=YET1M0P> _ _ _____

81 CONTINUESUMP0W=0.0 ___DO 82 L=1.KMAX

__ BURNUPC L »= URNUPA! MOP, L) _ _________________________________________ :_________ENILI=ENA CMOP» LI TIINILI=TIINA!VOP,l l WEIGHT!LI=EIGHT! MOP* LI PAWERILl=PAwERAtMOP,LI

222 SUMPOW=SUMPOW*PAWER!LI ....................82 CONTINUE ___ _____________________________________________________________

C**«* NORMALIZE POWfR CURVEDO 973 I = l,KMAX_______________________ _____ __

9 7 3 PAHERCIl=PAWER(Il/SUMPOWJ J1=JIIAIM0P)-1 ___DO 83 J = J 1 . J J 1

_ DO 83 KK=l,KMAX ;______________8 3 ERG! J » KKI=ERGAI MOP » j , KKI

IF( NI NP. EQ. l )1WR1TEI6 , 6 1 1 ! IERGIJ.KK), KK=1, KMAXI, J * J l , J J 1 J

IMAX=IMAXA(MOP)NRMAX=NRMAXAIMOP I ........................................................ ........................

_______IYEAR=IYFARAIMOP)_______ ________ ___________________________________________RATED=RATGf)A! MOP I RATEE=RATEEA(MOP|FDEBT=FDE«TA(MOP)TAXIF*T AX IFA(MOP I _TAX IS“ TAX ISA!NOP I RATEI=RATEIA(MOP)

138

C OC E RT

RATEP*RATFPAI MOP)_ TAXS=TAXSAI MOP) ________________________________________ ________

XT=XTA(MOP)TJ1=T JlAIMOP)

C**»* COMPUTE TAX RATESTAX = TAX I F / 1 0 0 . 0 + ( X.O-TAXIF/100 .01*1 TAX I S / 1 0 0 . 01 TAX 1 = 1 . 0 - TAXRATE=t1.0-FOEBT)»(RATEE/100.0)»TAX1*FDEBT»(RATED/100.0>

' RATEV=IRATEH-RATEP)/100.0RATEVM = 0.083333*RATEV _ ____TAX2=TAX1*(1.0-TAXS/10 0 . 0 ) ' ................. .

50 WRITE(6,47)______________________________________ ____00 84 K*1*XMAX

WRITE16 , 4 6 ) BURNUP(X),ENIK),TIINIK) _________________________84 CONTINUE

IF( NI NP. EQ. l ) WRITE(6,48) 1 MAX,NRMAX, I YEAR.RATED,RATEE, 1FDEBT,TAXIF,TAXIS,RATEI,RATEP.TAXS,XT,TJ1

00 85 I - I t I MAX ______________________ _______DO 88 M*l»6

_ 8 8 UCIM.I^UCAIMOP.M, I) ______________________________________NN=1UFINN»I)=UF A(MOP.NN, 11IF ( N I N P .E O . l ) W R ITE I6 .52) (UCIM , l l ,M= l ,2> , (UC IM, 11 ,

1M=4,5 ) » (UFINN,1 ) ,NN*l,NRMAX)________________ ______85 CONTINUE

00 86 NN=l,NRMAX _ _______________________________________ M i l l NNI*M11AI MOP, NN)~~

M1RINN)=M1RAIMOP*NN)ECFINN)*ECFA(MOP»NN)FLF(NN)*FLFA(MOP,NN) _RLF(NN)»RIFA(MUP,NN)

IFI NINP.EO. 1) WR1TEI6,53)NN,H1I( N N ) , HIR(NN),ECF(N N ) . _1FLF(NN), RLF(NN)

MM=MII(NN) ____ ______RLF(NN)*1.0-RLF|NN)FLF(NN)*1.0*FLF(NN) _ ________ ________DO 8b N1=1»MM

_______ 00 87 X * 1 , 6 _______________________________________________TIN!NN,K,Ml) = T INAI MOP,K,Ml)

87 PVINNfKfMl) = PV A(MOP* K,M1)IF ININP.EO.1 ) WRITE( 6 , 5 4 ) ( TININN,K,MlI ,K=1 , 6 ) ,

IIPVINN»K»M1),K*1»6) _ _________ ___C***« COMPUTE PRESENT-VALUE FACTORS _______O R A T E * R A T E _________________________________________________

D=DLOG(1,0+ORATE)OMRATE = DEXP10 / 1 2 . 0 ) - 1 . 0 _____________________________ _____

..... DO 86 J = l , 6_ TI N IN N ,J ,M I )= T I N ( N N ,J ,M 1) /1 2 . 0 _____________________________

DVAL=PV(NN,J,Ml)_______ DVAL*ID*DVAI. ) / 1 2 . 0 ________________________________________________

DVAL*DEXP(DVAL) ~ PVINN,J »M11=DVAL ______ _____________________________

86 CONTINUEIFI NINP.EO. 1 )WRITE(6,55)NR,N1, J l j i J l l , J 2 2 _________J J2 «J 22 *1

________EQCYCL=EQCYCA( MOP)____________________ ___________________________

139

COCERT

EONRCH=EQNRCA( MOP > • .................................... .......EQENRG^EQENRAiKOP) __ ____ __________I H N I N P . F G . U t»R I TE ( 61 56) EQE NRG tEOCYCL • EONRCHCALL NUCREKCMOP,J22) __RETURNEND _ _ _ ________ ___________

lUO

C ****** NUCREX ******C***« NUCREX USES THE DATA PASSED TO IT BY STARTS AND COCERT ____________'C**** TO DO THE NUCLEAR CALCULATIONS NECESSARY TO REVISEC*«** THE COST COEFFICIENTS IN STARTS ___

DOUBLE PRECISION D,DVAL,ORATE,l)MRATEREAL KINF tMXPPFC.MTUtNEWKNF *NDLKNF _INTEGER AOFCOMMON DVAL,D«ATE,DMRATF _COMMON A1, A 2«A 3,A4,A 5 « A6,A 7«A 8 < A9,A1ltA12•A13 « A14,A i 5,AI6,A 17» A16 , 1A19,A20,A21,A22,A23,A24.A25,A26,A27,A28,A29,CI,C2,C3,C4,C5,C6,C7, 2C8.C9*CTP,T0TMTU»MTU ,TJi,FDNRCH, EQNRCH,EQENRG,EQCYCL COMMON KMAX.NR.N1.J1.J11, IMAX.NJMAX, NREMOV,IYEAR,JB,JE, -INMAX*J2«NINP,NttLST , I BUG.KE If.HT ,IFLAG COMMON NRMV(20),M1II10),M1R(10)»XCT130)»AOFI 30)

' COMMON BURN INI 30), BURNUP ( 30 ) , PHIOI 30) ,FNI 301 , El 6_001 ,X INF I 30 f,IT I INI 30),ERGI15,30),W2130),ECFI10I,FLFI10),RLFI10),TINI 3,6,2),2U3I30),PVI 3,6,2),UC(6,30),UF(10*33)>POWER!30I.PAWERI30),WEIGHTI 30 3I.EXEFF120),EN0URN130),FFC1130),XI130),X2130 I,D1130),GAM(30)COMMON TAX,TAXIF,TAXIS,TAXI,RATE,FOEBT.RATEE,RATED,RATEV.RATEI,

_ 1RATEVM,TAX2,T AXS .RATEP,THRLEX,XTjLATEI 3 0 ) ___________ ______________COMMON NKEEPAI20),EN0A(20)COMMON /ERNIE/THERMA13,42).N0HUPI3,42>/BUYS/PVC1130),PVC21 30>/ IDEPLET/URANIAI42,30)/CAPFAC/CFCYCI1511 TIME/TJI 301/CYCLES/NJ11 301 , 3NJ2130 I/URAN/WII30)/C0NS/0 ,NBATCOMMON /STARTS/STP. TI20) /PROD/E NRG 120)/FUEL/ZTU

COMMON/PLAN/ENRGA120) _ _ _ _____ _DI MENSI ON TRGI 15,30>,E TI 30), TTINI 30),TERG 115,30),TBRNUPI301,ITTEN130),TTI INI 30)__________________DIMENSION FACTOR 115)EQUIVALENCE ICFCYCU ) .FACTORIDI _NALT-1

J J 1 = J U - 1 _ _____ _______________________ ____________ ____________DO 30 k‘k=l,KMAx'

5 TBRNUPIXX) =6URNUPIKX) _ _ _ ____________TTENIXK)=ENIXK) " ' ~ '........."TTIIN(KK)*TIINIKK) _00 30 J=Jl,JJl

30 TEKSIJ,KK)=EKGIJ,KK)_______________________________J112=JU+1J113=J11*2 ______ __________________JJ1=J11-1DO 31 XX»1,XMAX _ ____BURNUPIXX)=TBRNUPIXX)

ENIXX)=TTENIXK) ___________________________________________________flINIXX)=TTlINIXX)DO 31 J-J1,JJ1 _ _ • ________

31 ERGIJ»KK)=rERG(J,KK)C**** COMPLETE DATA FOR EQUILIBRIUM CYCLES _ _

NBAT-0 jb= t ________________________ ;_______________ ____________________

NJMAX=0STRTIJ22*2)=STRTIJ22+l)*1.0 STRT!J22+3)=STRT!J22*l»*2.000 11 NX* 1,600 . ____________ _________

11 EI NX I »0,0 ... NRMVI J22*l)xXMAX/3_________________________ _________________________

lUl

NUCREXNRMV!J2242)=NRMV!J22+lI ..

NRMV! J22+3l-NRfW_(J22 + l) ___________________________________J2=J22+3ENRG!J22*l»=EQENRG ;ENRG!J22*2I=G0ENRG ~ENRG1J22+3)=EOENRG _ ______ENRGAIJ22+1l=El)ENRG

______ ENRGAI J 22+2)SECENRG________________________________ _________ENRGAIJ22+3MEOENRGTJIJ22*1)-EOCYCL _______ ______ ________TJIJ22+2»=EOCYCLTJIJ22+3)aEQCYCL __________ ___ __DO 50 JK*J1,J22

SO TJ( JKJrSTRU JK+ 1)-STRT ( JK> _______________________________>JEN0=J22 + 3

_MTU=TOTMTU/KMAX ____ _ _ZTU*MTUENRGAI3I-I.G _ _ _ _ENRGI3)“1.0

C*»»« THE 1000 LOOP DOES ,CYCLE_ CALjCULAJKINS___________________ _00 1000 J = J11« JENDIF ILATEIJI.E0.il CALL PARMXIJI ______

C**♦* COMPUTE CORE THERMAL POWERIFIENRGIJI.GE.ENRGa ! Jl) GO TO 60CTP=1ENRGIJ-1)-ENRGA!J-1)+ENKGIJI I/(TJ(J)*365.*FACTOR!JI)

_______GO TO 61___________ ____ ____________________________________60 CTP = ENRG( jV/ITJI J)'*365.*FACT0R« J)l61 SUMCTP=0.0 ___

00 883 K-1,KMAXPOWERIK)*PAWCRIK)*CTP _ ___

883 SUMCTP»SUMCTP+POWERIKJ .........AO IF ILATEIJ)«EQ.11 GO TO 1A9________________________________

CALL GENATR(J.J22)CTP-ENRGAIJI/!TJIJ)*365.*FACT0R!JJI SUMCfP*0.0DO 88A K»l,KMAX _ _ _ __________POWER!X)=PAWER(KI*CTP ....... ... .

88A SUMCTP*SUMCTP+POWERIK>______________________________________1A9 NREMbv=NRMV|J)

IF (J.E0.J11I CALL SORT __ ___CALL OATGENIJ,J22)

51 NROLD*NREMOV_________________ __ _______ ____________CALL SORT...........

__150 CALL SHUFL ___________________________________________KGND=KMAX-9REM0V

C**«* 200 LOOP PLACES PARAMETERS OF FUEL REMAINING INCORE C***P IN TEMPORARY STORAGE AFTER SHUFFLING

DO 200 JLvl.KEND KLOC-KCTC A(1F(JL) )BURNIN!JL)*»URNUP(KLOCI ETljL)*ENIKLOC>TTIN(JL) = niN(KLOC)

JFIN*J-l00 200 JCaJltJFIN TRG!JC,JL)=ERGIJC.KLOC)

200 CONTINUE ’

N U C R E X

C***« 76 LOOP MOVES FUEL FROM TEMPORARY STORAGE INTOCj**** CENTRAL REGION OF CORE _______________1

DO 76 JKL=l,KENO ENT JKLI=EIIJKL )TIINIJKL)=TTIN(JKL)00 76 JCT=J1»JFINEKGIJCT *JKL)=TRGIJCT, JKL)

76 CONTINUE _ _________________________‘ KLOO»KMAX-MREMUV*i

C**** INITIALIZE PARAMETERS OF FUEL IN CORE PERIPHERY DO 300 KL=KLOD,KMAX BUKNINIKLI=0f0 TIIN(KL)*STRTIJ)

ENI KL) =FDNRCH ___ ____________________________' 'DO 300 JC=JI, JFTnERGIJC,KL1=C.0 _ _____

300 CONTINUECALL 0EPCALIJ.J22.NUD) _

C**»* COMPUTE ENERGY DELIVERED DURING CYCLE JDO 1000 K=1,KMAX _ _________ERGIJ.K) = (IIURNUP(K)-BURNIN(K))*MTU

1000 CONTINUEC

J = JEND 1 NRHVIJ) = KMAX

CALL DATGENtJ.J22)__ ___________________________NMAXxNBATCALL NCOSTIJ22 I _ _RETURNEND

11*3

C ****** SORT ******C****_SORT SORTS THE FUEL IN ORDER ACCORDING TO ITS REACTIVITY___________C***'* AS'PRESENTLY CODED, SORT IS A DUMMY SUBROUTINE

DOUBLE PRECISION D.DVAL,DRATE,DMRATE __ _REAL KINF,MXPPFC,MTU,NEWKNF,NOLKNFINTEGER AOF _ _COMMON DVAL»QRATf, DMRATE

COMMON A1,A2,A3,A4,A5,A6,A7,A8,A9,A11,A12*A13,A14,A15,A16,A17,A18_,1A19,A20,A21,A22,A2 3,A24.A25.A26,A27,A28,A29,Cl,C2.C3,CAtC5,C6,C7, 2C8,C9,CTP,rOTMTU,MTU,TJl,FDNRCH,FQNRCH,EQENRG,EQCYCL _COMMON KMAX.NR.NI,Jl,Jll, IMAX.NJMAX, NREMOV.IYEAR.JB,JE, .

INMAX, J2 ,Nt Nl'tiNBLST .1 BUG.KE IGHT . I FLAG COMMON NRMVI20) .Ml I(10)»MIR(10),KCT(30I, AOF(30)

COMMON BURNIN(30),BURNUPI30)»PHIO(301.ENI30)t E (6001»KINJFI3 0 ) . _____ifnN(3bl,ERG(15.30),W2(30),ECF(10),FLF110).RLFI10I.TINC 3,6,21,2W3I301,PVI 3,6,21,UC(6,30),UF(10,301,POWER!30».PAWERI301,HEIGHTI 30 3 1 ,EKEFF120J,ENBURNI 30),FFC11 301«XII 30),X2130),01(301,GAM I 301 COMMON TAX,TAX IF.TAXIS,TAXI,RATE,FDEBT.RATEE,RATED,RATEV.RATEI,

IRATEVM, TAX2,TAXS,RATFP,THRLEK,XT COMMON /FRNIE/THERMA(3,42),NOWUP(3,421/BUYS/PVC1(301,PVC2(30)/

- lDEPLEf/URAMI A (42,30)/CAPFAC/CFCYC(I 5)/TIME/TJ(30J/CYCLES*NJI(30) , 3NJ2 (301/URAN/W1(30 I/CONS/O ,NBATCOMMON /STARTS/STRT(20)/PROD/ENRG(20)DO 83 KA2»I,KMAX

83 KCT(KAZ)=KAZ RETURN____________________________________________________________________

END

jM

c ****** SHUFL ******C**** SHUFL SIMULATES THE INCORE FUEL MANAGEMENT DECISIONS ___________C**** ASSOCIATED WlTH THE OUT- INSCATTER REFUELING PHILOSOPHY

DOUBLE PRECISIQN D.DVAL,DRATE.DMRATE REAL KINF,MXPPFC,MTU,NEWKNF,NDLKNF INTEGER AOFCOMMON OVAL.DRATE.OMRATECOMMON AltA2,A3,AA,A5,A6,A7,A8,)V9.All,Ai2.Al3.AlA,A15,A16,A17,A18,

'1A19,A20,A21.A22.A23,A2A,A25,A26,A2T,A2b,A29,Cl,C2,C3.CA.C5,C6,C7, 2C8, C9.CTP,TOTMTU.MTU.TJI.FDNRCH,EQNRCH.EQENRG,EOCYCL COMMON KMAX.NR.Nl.Ji•Jll, IMAX.NJMAX, NREMOV.IYEAR.JB,JE, .1NMAX.J2.NINP.NBLST.1BUG,KEIGHT,IFLAG COMMON NRMV(2Q),M1I(IQ),M1R|10)«KCTI30)«AOFI 301

COMMON BURN INI30)»BURNUP(301•PHI0< 301,EN(301,E1600),KINF(39I,l'THNI3ni,FRG(15,30J,W2I 30 J , ECF110) ,FLF 110I.RLFI 101, TIN! 3,6,2),2W3I30)» PVI 3,6,2),UC16,30),UF110,30),PCWERl30),PAWERI30).WEIGHT! 30 3),EXEFF(2C>.ENBURN130),FFC11301,XII 301,X2130),D1130),GAM!30)COMMON TAX,TAXIF,TAX IS,TAXI.RATE,FDEBT,RATEE,RATEO.RATEV,RATEI,

1RATEVM,TAX2,TAXS»RATEP.THRLEK,XT COMMON /ERNIE/THERMA(3,42),NOWUPI3,42)/BUYS/PVC 1130),PVC21 30)/

" 1OEPLET/URAMI A!42 »30)/CAPF AC/CFCYC!15)/TIME/TJ(30)/CYCLES/NJ11 30), 3NJ21 30)/URAN/W11 30)/CONS/D .NBATCOMMON /STARTS/STRTI20)/PROD/ENRG1201

3 NMRMV-KMAX-NREMOV _ _ __K*NREMOV* I - ■ • -......— - . DO 31 JK= 1,NMRMV, 1__________________ ____________________________ ___

AOF!JK)*K 31 K=K + 1

41 RETURNEND _

1U5

C ****** DATGEN ******’c**«* DATGEN EVALUATES THE FUEL BEING REMflVEO AT THE END OF A____________C**** CYCLC," SORTS IT INTO APPROPRIATE BATCHES" IN ACCORDANCE C***« WITH ITS HISTORY. AND GENERATES THE ENERGY AND ISOTOPIC C***« HISTORY ESSENTIAL FOR THE ENERGY COST CALCULATION

DOUBLE PRECISION D.DVAL,ORATE,OMRATE REAL KINF,MXPPFC ,MTU,NEWKNF«NDLKNF

INTEGER AOF______ _______________ _____________________________________COMMON OVAL,DRAfE,DMRATECOMMON A1,A2, A3,AA,A5,A6,A7,A8,A9,All,A12,A13,A14,A15,A16.A17,A18, 1A19,A20,A2I,A22,A23,A24.A25,A26,A27,A2e,A29.C1,C2.C3*CA,C5,C6.C7, 2CB,C9,CTP,TOTMTU,MTU,TJl,FONRCH,EONRCH,EQENRG,EQCYCL COMMON KMAX,NR,N1.J1,Jll, IMAX.NJMAX, NREMOV,IYEARtJB,JE,INMAX»J2»NI'IP,NBLST , I RUG, KE IGHT, I FLAG __ _____________________

"COMMON NRMVI20),M1II10),M1R110),KCTI 30),AOFI 30>COMMON BURN IN(30),BURNUP(30),PHIO(30),EN(30),E(600),KINF(30),

1T1IN(3DJ,ERG(15,30),W2(30),ECF(10) ,FLF(101,RLF(10),TINI 3,6,2), 2W3I30),PVI 3,6,2),UC(6,30),UF110,301,POWERl30),PAWERI 301.WEIGHT!30 3),EKEFF(20),ENBURN(30),FFCI(30),XII 30),X2(30),D1(30),GAM(30)

_____ COMMON TAX, TAXIF,TAX IS,TAXI, RATE ,FDEBT , RATE E, RAT ED*RATEV*RATEI,____YRATEVMiTAX2.TAXS,RATEP,THRLEKVXT COMMON /ERNIE/THERMAI 3,A2),NOWUPI3- A21/BUYS/PVC1(30),PVC2(30)/ 1DCPLET/URANIA|A2,30)/CAPFAC/CFCYC115)/TIME/TJ130)/CYCLES/NJ11 30), 3NJ2130)/URAN/W1I30)/CONS/D , NOATCOMMON /STARTS/STRTI20)/PROD/ENRGl20)

COMMON /POINTS/NODES(30)_____________________________________________DIMENSION X130)IFSQ=999 _ ■ _ _JJ*J-i ' ' ........... "JST=J1NODEZ=NRMV( J )

BENRCH=EN(KCT(_l) ) _ ___________________ ______ _____________________"29 "IGLOO*i 22 NNDHAT=0

SMBRN=0.0 SMNRCH=0.0 JULIP*0

£***« SORT_NODES INTO_ DISCHAKGE BATCHES ACCORDING TO ________________C**** TIME-IN AND INITIAL ENRICHMENT " ’

DO 25 JK=15L00»N0DEZIFIABSITIINIKCTIJK))-STRT(JST)l.GT,0.005) GO TO 25 IFSOsJKIFIENIKCTlJK)).NE.BENRCH) GO TO 38"

NNDBAT=NNDRAT + 1 ____________________________________________IFINNOBAT.EQ.il' NBAT*NBA T♦1IF(NBAT.EO.l) JE*J-JST _•____BENRCH=ENIKC TIJKI)IFINBAT.NE.lJ GO TO 10 JT=JST

t * * * * CALCULATE FUEL ENERGY HISTORY BY BATCH AND CYCLE __________________00 12 JN*J3.JEElJN)*E(JN)*ERGIJT.KCTlJK))*8.1912E-5 ______ ____ ______JT=JT*1 .

12 CONTINUE_____________________________________CO TO 30

10 IF INNDBAT.NE.1) GO TO 13_____________________________________________

1U6

DATGENJH»JE+1J E = J B * J J - J S T ____ ________ _________

13 JT*JSTDO 14 JN=JB,JEE CJN)*E C JN) +ERGC JT »KCT(JK))*B. 1912E-5 JT=JT+l

14 CONTINUE30 JN1=JE _ _ _

C**«* CALCULATE SUM OF BURNUP AND ENRICHMENT ON EACH NODE SMbRN=SM8RN+0URNUPCKCTCJK)I IF(J.NE.Jli) CO TO 24 XIKCTCJKI)=BURNUPCKCT(JK)I/IOOCO.ENBURNCKCTC JK) 1 *CC1+C2*ENCKCTC JK M *C3* (ENIKCTI JK I)> 4*2+1 C4*C5*EN IK

_ 1CTIJK)) +C6* CENC KCT(JKI))»*2I»X(KCTIJKI!♦<C7+C8*ENIKCTIJK))*C9*CENI2KCTCJK)))**2l*CXCKCT(JK)))**2l*CNCKCTfJK>)

24 J>KNRCH*SKNRCM+ENBURN(KCT(JKII _____ __ _ __ _25 CONTINUE " ....

GO TO 39 _ ___________ _____38 JULIP-139 IFI NNDBAT• EQ.O l_GO_ T0_ 28______ ______________________________________

NJ1CN8AT)=JSTNJ2INBATI-J-1_______________________________NODESINBAT)=NNDBAT............ ................. .........

C*«"** CALCULATE MTU LOADED _MilNBATI*NNDBAT*MTU

C«"*** SET U-235 ASSAY LOADED XIINBAT)=BEMRCH/100•0AVGBRN=SMBRN/NNDBAT _ ____AVNRCH* SMNRCH/NNDBAT - -BSC*BENRCH**2 AVGFR«AVGBRN*.IE-3U25D*Wl(N«AT)*IAJl + A12*AVGFR+A13*AVGFR**y *AVNRCH'‘iPi______________

C**'** CALCULATE KG FISSILE PU DISCHARGEDPLUTOD=HlCNDAT)*C (A21 + A22*IIENRCH'*-A2.3*BS0)♦( A24+A25______ _______

1*BENRCH+A26*BSC)*AVGFR+CA27«-A28*BENRCH+A29*DSQI2*AL0GCAVGFR*1.0) I _J.____________ ____W3CNBAT)=PLUTOD

C**»* CALCULATE METRIC TONS U DISCHARGED________ _________________________W2CNBATI=W11NBATI-111.17*PLUT001/LOOO.C)-C C Wi <NBAT)

1*BENRCH*10.0-U25D)/1000.01__________ ______________________ ____C*«** CALCULATE U-235 DISCHARGE ASSAY

X2CNBAT)=U25D/(W2CNBAT)*1000.0IIF I irS0.LT.Nd0E2.AND.IFS0.NE.999) BENRCH*EN(KCTC1FS0+11 I.........

______ IFCJULIP.NE.l) GO TO 27______________________________________________IGLOO* IF St)BENRCH*EN(KCT(IF SO 1)IFSO-OGO TO 22 _ _____ _________ _____

27 NJMAX*NJMAX»J-JST ... . 28 _ IF CJST .EO. J22) RETURN____________________________________________

IFCJST.CO.J-lI RETURN JST-JSTUGO TO 29 ” “ 'END

ll*7

c ****** Ncnst ******c***« MCDST GENERATES ENERGY COST BY CYCLE _AND A CUMULATIVE_______________"C****"LEVELIZEO COST FOR EACH CYCLE OF THE PLANNING PERIOD.

DOUBLE PRECISION 0,DVAL,DRATE,PMRATE __ ___REAL KINF,MXPPFC,MTU,NEWKNF,NDLKNFINTEGER AOF _ _ _ ___COMMON DVAL,ORATE,DMRATE

_ COMMON A1,A2,A3,A4,45, A6, A7, AH. A9, All, A12 ,Al 3,A14,A15, A16, A17, A18jr__lVl9,A20,A21,A22,A23,A24,A25,A2G,A27,A2e,A29,Cl,C2,C3,C4,C5,C6,C7,2C6,C9,CTP.T0TMTU,MTU,TJ1»FDNRCH,EONRCH,EOENRG,EQCYCL___________ ____COMMON KMAX.NR,N1, Jl, Jl 1, IMAX.NJMAX, NREMOV. IYEAR, JBtiJE,1NHAX,J2,NINP,NBLST,IBUG,KEIGHT,IFLAG _COMMON NRMV(20),MiI(10)*M1R(101,KCT(30),AOF(30)COMMON BURNIN(30).DURNUPI30)tPIIIOI30).EN(30),E(6001,KINF<30), l"T.IINC30I,FRGH5,3O).H2(3C),ECF(IOI,FLFClO'),RI.F( 10),T I N T 3,6,2),2W3I30),PVI 3,6,2),UC(6,30),UF( 10,301,POWER!30),PAWERI30),HEIGHT(30 3),EKEFF(20),ENBURN(30),FFC1(30),XI(30),X2(301,01(30),GAM(30)COMMON TAX,TAXIF,TAX IS,TAXI,RATE,FDEBT,RATEE,RATED,RATEV,RATEI,

IRATEVM,TAX2,TAXS«RATEP»THRLEK,XT COMMON /ERNIE/THENMA(3,42).NOVJUPI 3,42)/BUYS/PVC1(30),PVC2(30)/ ____

iOEPLET/URAMIA(42,30)/CAPFAC/CFCYCII5 f/TIME/TJ(30J/CYCLES/NJ1(301, 3NJ?(30)/URAN/Wl(30)/CONS/D .NBA,COMMON /STARTS/STRT(2G)/PR00/EN«G(20)DIMENSION PV1J(30),PV2J(30),U(6),UECN(30),

1FC(30),UECJ(30),FC1( 30)DIMENSION U0(6),DC1(30),0C2(30).DUECNI30),CE(31»_____________________JJ=J2-J1+2 ...........JJ1=JJ-1 _ _DO 22 1 = 1,1 MAX

22 UC(3,I)=UF(NR,I)_________________________XF=.00711

VF = 0.9B578*DL0G(139.647D0)______________________________ __________miBUG.NE.l) GO TO 301 WRITE(6,71)

71 FORMAT!•ORATCH COST DATA•/'O'.T55,•UNIT COST*)C***« ENRICHMENT PARAMETERS________________ _____301 VT=(1.0-2.0*XT)*ALOG(l.O/XT-l.O)

XFT=XF-XT______________________________________________________________VTF=VT-VFPV1J(1)=1.0 _ _ ; _ _DVAL=TJ(1)DVAL=-D*DVAL ____DVAL=DEXP(DVAL)PVV=DVAL

~ P V 2 J <H = < <I.0+OHRATE )**(12.0*TJ(1))-i.C )/(12.0*TJ(I)*DMRATE*(T.0T10MRATE)**( 12.0*TJ( 1) ) I _ _____ _ _J=*2

412 CONTINUE 106 PV1J(J)=PV1J(J-1»*PVV

IF(J-JJ)105,21*105___________________________________________________105 DVAL-TJ(J)

DVAL=-D*DVAldval«dexpcdval) '.~ ........................'PVV-DVAL

C**** CALCULATE PRESENT WORTH OF ONE UNIFORMLY DISTRIBUTEDC****0VERCYCL E ___________________ J____________________________ ___________

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NCOST400 FORMAT!' U< MI • . 15,F14.4I

M = ;___________________________________________IF!M-61 52,55,52

55 TP=TD+TIN1NR,M,MII _ _ . _ . ____CTP+l-IYEARVP*!1.0-2.9*X2!N))*ALOG(l.O/X21NI-l.O) __ __ _________FR=(X2I.NI-Xn/XFT

_____ WR^FR*VTF-VT *-VP_____________________________________________________TP1=TP- lYEAR+i-IUO!6)=I I UC( l«l)+IUC!l,l+l l-UC (1*1) I*TPll*FR_ ___ _________1*-!UCI2, I >MUC(2,H-1»-UC(2,I I)*TP1)*WRIU(6 1=U0!6I*PV!NR,M,M1J __ __________IF(IBUG.EO.1) WRITE (6,401) U(6)

401 FORMAT!' U(6)*,F14.4)______________ _________________________________DC2(N l = W21NI * 1UD! 4 I -UDI6 l'*RLF (NR) »~W3« Nl*UO (5i *RLF (NR I

C*«** CALCULATE VALUE OF DISCHARGE FUEL RELATIVE _ _______ _ _C****. TO DISCHARGE DATE

PVC2(N)=W2(NI*(UI4|-U16)*RLF!NKII-W3!NI*U(5)*RLF1NR)NNN=>NJ1(N)

NN=NJ2INC1 ____ ;____________PVC=PVC UN) *P V1 JINJl (NIC PVC2 (Nl ♦PV t j (NJ2 (NI ♦ 11SUM1=0.0 _ ____ ___ ___SUM2*0.0NNl=NJ2(N) _DO 360 L=NMN,NN1

_STMP*i2.C*TJ!L)*PV2JJLJ____________ .________________________________SUK1=SUM1*STMP

360 SUM2=SUM2+STMP*CE(LISl=2.0*(1.0*DRATE)**(-TL+rjl+0.0833lS2=12.D*TIN(NR,4,M1I*(1.0+DRATE|4*!-T0*TJ1+0.5*TIN(NR,4,M1I)SS2=S2*EEN

______ PVB=DClIN»*(SUMl+Sl*S2)-(DCHiij*nC2INi>/EEN*(SUM2tSS2»_____________PVTAXS=RATEVM*PVB

C**** CALCULATE UNIT COSTS _DUECN!N) = !(DC1 INCDC2(Nn*PVEN|/EENUECN!N)=(PVC*-TAXl*PVTAXS-TAX*DUECN(N)I/(TAX2*PVEN*10.0)IF!IBUG.EO.1IWRITE(6,234)SUM2.SUMI,SI*S2

234 FORMAT! •0SUM2,SUMl,S1,S2•/4F14.51 _ _ ___IF!IBUG.EO.1) WRITE(6,402 IN»EEN,DClIN)*DC2!NI,PVC,PVfAXS,DUECN(N)

402 FORMATC N,EEN,DCl,DC2,PVC,PVTAXS,DUECN',15,6F12.31 IF(IBUG.NE.l) GO TO 100 WRITEI6,72)

72 FORMAT!•OBA TCH•,T12*•FEED•,T23.•ENRICHMENT*,T37,•FABRICATION*,T53, 1'RECOVERY*,T66.1 PLUTONIUM',T81* * URANIUM*,T92,'PROCUREMENT',T109,__

2’ SALVAGE* * T124*'FUEL *1 “ - -WRITE!6,731

' 7 3 FORMAT!* *,* NO*,T12,•i/KG•,T26,*t/KG*,T40,*S/KG*,T54,•A/KG*,T67,• 1 S/GRAM*,T81,*S/KG *,T95,•* 1000*»Tl10,*S1000*,Tl2i,'C6NTS/M6TU*IWRITE(6,T4)N,U(1),U(2),U(3),U(4),U!5),U(6),PVC1IN),PVC2IN),UECN(N)

100 CONTINUE ________________________________________________ _______996 CONTINUE

C**»* CALCULATE COST BY CYCLES FCX * 0.0FEJ - 0.0 _ _

74 FORMAT!* •,13,1X.9F14.5I.... 00 500 J _= JU,J22__________________________________ _________________

150

NCOSTrj * o.o

_ UECJ(J) » 0.0 __ _____________________________ ______________DO 420 N=1,NMAXIF!NJliN).GT.J .OR. NJ2IN).LT•J) GO TO 420K = 0DO 440 L»i«N

440 K=K * NJ21L ) -NJ11L ) + 1_____ K=K - INJ2IN)-J) ____ ________________________________________________

UECJIJ )=UECJIJ)+UECN!N)*Elk)EJ * EJ + E! KI ___ __________ _________

' 420 CONTINUEUEC JIJI * UECJIJ l/EJ _ _ __ __________

C^*** CALCULATE LEVELIZED COST PVEJ = E J*PV2 J (J) ________________________ _________________________

FCIJ)=FCX ♦ PVEJ*UEC JU)FEJ = FEJ «• PVEJ __________________________ _______FC1IJ)*FCIJ)/FLJIFIJ.E0.J22I GO TO 70 _ __FCX = FCIJ)

500 CONTINUE _■______. ____________________________________________70 IF!IBUG.NE.l) GO TO 110

WRtTEtb,75>75 FORMAT!'OBATCII COST DATA'»

HR I TE (6i 76176 FORMAT!• OBATCH•,TIO,'CYCLE'.T20,'CYCLE'.T29,'U LOAOEO',T44,•IN IT A

_ _1SSAY',T56,'U DISCHRSD'.T70,'FINAL ASSAY',T85.'PLUTQNIUM')___ _____HR ITE16* 771

77 FORMAT!' ',' NO*,T9,•LOADED*,T19,•DISCHRG'J32,•KTU*,T59,'MTU',TB8 1,'KG'IDO 110 N=1,NMAX _X1INI«X1!N|*100.0

_____ X2( N)=X2!NI*1000.0__________ _________________________ ________________WRITE!6,78!N.NJ11N1.NJ2INI, Wl IN I * XIIN) , W21 N> ,X~2 IN) , W3I N>

78_ FORMA.T! ' •, I 3, 2X ,2 (I 7, 3X) ,F 12 . 3,2X,F 11 .4,3X .F12 .3 , 2X, F10.4, 4X, F11. 13)

j XII N)«X1INI/100.0 _X21N)=X2!N)/1000.0

110_ CONTINUE. ____________________________________________________ ___WRITEI6,82~)

82 FORMAT I'OFUEL CYCLE COST DATA*»HKITE(6,P 3).............................. .

83 FORMAT!'OCYCLE•,T14,•FUEL',T24,•LEVELIZED•)WKITE i 6,84)

84 FORMAT!'O',• NO*,T11,1CENTS/MBTU'.T24,'CENTS/MBTU')_________________DO lil j-JliVj22WRITEI6,81)J,UECJIJ),FC1(J)

'81 FORMAT!• •,I 3,4X.2F11.3)111 CONTINUE999 CONTINUE ....... "

RETURN_____ ______________________________________________________END

151

****** PARMX ♦****•'ARMX READS IN BATCH _SJZE AND ENRICHMENT FOR_______________________

C***» CYCLE J AND CALCULATES ENERGY TO~B'c PRODUCED_DOUBLE PRECISION 0,OVAL,DRATE.DMRATE_________ __ ___ ___REAL KINF,MXPPFC,MTU,NEWKNF,NULKNFINTEGER AOF ' _ _____COMMON OVAL,ORATE,DMRATE

_______ ^COMMON A I, A2.A3, A4,A_5,A6, A7, A8, A9, All,A12 ,A13,A14,A15,A16,ALT,A18,1A19,A20,A2i, A22.A23,A2R.A25,A26,A2T,A2H,A29,Cl,C2.C3,CR,C5,C6,C7, 2C8,C9»CTP »TOTMTU»MTU,T Jl.FDNRCH,EQNRCH,EQENRG,EOCYCL COMMON KMAX,NR,N1,J1,J11, IMAX.NJMAX, NREKCV,IYEAR,JB,JE,

INMAX,J2,NlNP,NHLST,IBUG,KEIGHr,IFLAG _ _ • COMMON NRMVI20),M11(1Q)«M1R(10)«KCT(30),AOF(30)

COMMON BURN IN(30),BURNUP(30),PH 10130»,EN C 30),E(6001,KINF(30),IT.11N( 331 ,ERG (15, 30) , W2(30) ,ECF110),FLF<10). RLFI IO) , TINI 3,6,2), 2W3I39I,PV( 3,6,2),UC(6,30),UF(10,30),POWER!30),PAWER(30),WEIGHTI 30 3),EKEFF(2G),ENtiORN(30)«FFCl(3G)*Xl(30),X2(30),01(30)«GAM(30)COMMON TAX,TAX IF,TAX IS,TAXI,RATE,FDEBT.RATEE,RATED,RATEV,RATE I,

1RATEVM,TAX2,TAXS,RATEP,THRLEK,XT,LATE(30)_______COMMON NKEFPAI20),ENOA(20) _ ___ ________

COMMON '/ERNIE/THERMAI 3,42),NOWUPI 3.42)/BUYS/PVC1!30 V,PVC2(3017 10EPLET/URANIAI42,30) /CAPFAC/CFCYCU!,)/TIME/TJ(30)/CYCLES/NJ1(30) , 3NJ2(30)/URAN/H11 30)/CONS/D ,NBATCOMMON /STARTS/STRTI20)/PROO/ENRGI20)NKEEP=NKEEPAC J )

________ ENO = ENOA(J»______________________________________________________________IFLAG=I

. C**** ESTIMATE OF ENERGY SET AT EQUILIBRIUM ENERGY_____EST=EQENRGNUP=KMAX-N<EEP _N0P=KMAX-NKEEP+1 ................. ....... . ..........

________ RMTUrMTU ;___________WANT=EKFFFIJ)*KMAX/THRLEK"

14 QKIN-0.0_________________________________________ _____C**** LOOPS.1 AND 2 CALCULATE K-INFINITY (QKIN)

00 2 NOX*l,NUP____________________________ __ ___EBURN= EST*PAWCR(NOX)/RMTU ~...........BURP= I BURNUP INOX+NKEEP)+EBURN) *1.0E-4CUP=A1 + A2+EN(N0X4NKEEP)+A3+E N(NOX+NKEEP)**2+1 A4+AS*ENINOX+NKEEP )

1+AI*ENINOX+NKEEP)**2)*BURP+(A7+A8*ENINOX+NKEEP)+AS*EN(NOX+NKEEPI 2**2)*BURP**2

2 OKIN=QKIN+CUP*WE IGHT INOX) __ _DO 1 LOX«=NOP,KMAX

_______ EBURN=(EST*PAWER(LOX)j/RMTU _ _ J _______________________________________BURP=EBURN*l.0E-4

_ CUP»Al+A2*ENO+A3*ENO**2+(A4+A5*EN0+A6*EN0**2)*BURP _1+IA7+AB*EN0+A9*EN0**2)+BURP**2

C**** COMPUTED K-INFINITY COMPARED TO DESIRED K-INFINITY.C*+** IF NOT WITHIN .0 0 0 1 ESTIMATE OF ENERGY INCREMENTEDC**** BY LINEAR EXTRAPOLATION ____________________________ ______________

1 OKIN=QKIN+CUP*WEIGHT(LOX)IF CARSIOKIN-WANT).LE.0.001) GO TO 10 IF (IFLAG.EQ.l) GO TO 11ESTA=IIWANT-SAVEQ)/IQKIN-SAVEU))*IEST-SAVEE>+SAVEE SAVEQsQKIN

C**** STORE ENERGY AND BATCH SIZE

152

PARMXSAVEE=EST ~ ..............

E $1=6ST4 ___________________________________________________________GO TO 14

11 SAVEQ=QKIN SAVEE=ESTIF (OK IN-WANT) 12il0,13

12 EST=EST-0.1*EST IFLAG* IFLA3*1______________________________ _________________________

GO TO 1413 EST=EST*0.1*EST _ _ _

IFLAG=IFLAS*1GO TO 14

10 ENRGIJ)*ESTFDNRCH=ENO________ _________________________________________ ________NRMV(J)=NKEEPWRITE(6,101JJ,ENRGIJ»,TJ(Jl *NRMVCJI» FDNRCH

101 FORMAT!‘O',4X,'CYCLE NUMBER = ', 13/10X,•ENERGY SCHEDULED =',F14.3/ UOX.'CYCLF LENGTH »• .F12.3/10X,'BATCH SIZE =•.I 3/10X.•FEED ENRICHM 2ENT *',FB.4I

_ _ RETURN __ ______________________________________________________END

153

C ****** SENATR ******..............._C**** Gf.NATR_MAKES THE EXTERNAL REFUELING DECISIONS CONCERNING___________C****'BATCH SIZE AND ENRICHMENT ......

DOUBLE PRECISION D.DVAL,ORATE,DMRATF _ _____ __^ REAL KINF,MXPPFC,MTU,NEWKNF,NDLKNF

INTEGER AOF __COMMON DVAL,ORATE,DMRATECOMMON A1,A2,43,A4.A5,A6,A7,A8,A9,4ll,A12,A13,A14,A15,A16.A17, A18,_

" 1A19, A20, A21 , A2.2 »A23,A24,A25,A26,A27,A2 8» A29,Cl,C2«C3« C4,C5,C6,C7, 2CB.C9,CTP,T0TMTU,MTU,TJ1,FDNRCH,EUNRCH,EQENRG,EOCYCL

COMMON KMAX,NR,N1«J1,J11< I MAX,NJMAX, NREMOV,I YEAR,JB,JE,1NM.AX, J2,NINP,NRLST,I BUG,KEIGHT,IFLAG COMMON NRMV12P),Ml II101,MIR(101,KCT130),AOF(301 COMMON BURNINI 30),BURNUPI 30),PH10130),ENI30),E1600),KINF130),1T1 INI 3C’i, ERG I IB, 30), W2130) ,ECF110),FLF110),RLFI1G) ,TINl 3,6,2),2W3I301,PVI 3,6,2),UC16,30),UF110,30),POWERI30),PAWERI30),WEIGHT!30 3),EKEFFI20),ENI)URNI30),rFClI 30),XlI 30»,X2130),Dl<30),GAMI30)COMMON TAX, TAX IF,TAX IS,TAX 1,RATE.FDEBT,RATEE,RATED,RATEV,RATE I,

1RATEVM,TAX2,TAXS,RATEP,THRLEK,XT COMMON /ERNIE/THERMAI3.42),N0t,UPI3.42)/BUYS/PVC1130),PVC2130)/ _lbEPLET/URANI Al 4 2»30) /CAPF AC/CFCYC I.lr» )/T IME/TJI 30)/CYCLES/NJ11 30) , 3NJ2I30)/URAN/W1I30)/C0NS/D .NBATCOMMON /STARTS/STRTI2C')/PR0D/ENRGI20)DIMENSION XI30I .XXI30) .SUNKI30)DIMENSION FACTOR!15)

FOUIVAL ENCE ICFCVC 11).FAC TOR 11 H ______________________________________CALL SORT"RMTU=MTU ______________ ____XKMAX=KMAXCOMPAR={EKEFFIJ)*XKMAX)/THRLEK BURN=TJIJ)*365.*FACT0RIJ)

______EZERO*EQNRCH ___________ ____________________________________________LASTBA=KMAX

C**** COMPUTE K-INFINITY FOR EACH NODE OF NEW FUEL 00 1 KL = 1,L ASTHA/EXBO=I POWER(KL)*BURN)/|RMTU*10000.I

1 PHIOIKL )= ' 'A Al*A2»EZER0+A3»EZER0**2»U4+A5*EZER0+A6»t.ZER0*»2)»EXB0+l__1A 7 ♦ A 8 * E Z E RO ♦ A9 * E Z E RQ * * 21 * E X B 0 ** 2 .... .SUNKI1)=PHI0IKMAX)*WEIGHTIKMAX) _

C**** FIND CUMULATIVE K-INFINITY OF NEW FUEL" ............DO 2 KL=2,LASTBA

2 SUNK I KL ) =SUNKIKL-1I + PHIOIKMAX-KL+1)*WEIGHTIKMAX-KL+1) I Y=0 _ _________________________________________________C***» ESTIMATE FIRST BATCH SIZE

NHOLD= IENRGIJ)/EOENRG)*NRMVIJ22*3) _ __ ___ ____A NNN=KMAX-NHOLO

ROKIN=O.C _411 CONTINUE

C***» SHUFFLE AND OEPLETE FUEL THROUGH PRESENT CYCLE _________________” 00 45 K*1,NNNEX6=|P0WER|K)*B0RN)/RMTU _ _ _ ___ _ •DELTA*IBURNUP!K+NHOLD)+EXB)/10000.0ANKINF*Al*A2*EN(K+NHOLD)♦A3*ENIK*NH0L0)**2*IA4* A5*ENIK+NHOLD)*A6*I

1ENIK+NHOLD)l**2)*DELTA*IA7*A8*EN(K*NH0L0)*A9*IENIK*NHOLD))**2)*0EL 2 T A * * 2 ___ ___ _________________________________________________________

GENATR45 ROK IN=RQK IN>ANKI NF*WE IGHT (KI

410 CONTINUE _______ ______________ ____________________________IF IJ.GT.J22) GO TO 6

C**** COMPARE EOC SUM OF K-INFINITIES OF NEW AND OLD __C***« rUfcL TO EOC K-EFFECTIVG FOR PRESENT CYCLE. INCREASEC«*** OR DECREASE BATCH SIZE DEPENDING ON RESULTS OF TFST

GAMINHDLD)=RQKIN*SUNKINHOLO) IFIIV.EO.lt GO TO 7__________________________________ _______________

IFCIY.EQ.2I GO TO 5IF( IY.EO* 31 GO TO 6____________ _______________________________I FIGAM(NHOLDI-COMPAR) 7,6,5

7 i y=i : _ ___IF I(GAM(NHOLD)-COMPARt.GE.O.OI GO TO 6NH0LD=NH0LD+1GO TO 4

5 I Y=2 _____________________IFIIGAMINHOLDI-COMPARI.LE.O.O) GO TO 8NHOLD=NHOLO-1 _ _ _ ______I FINHOLD.EO•2) IYa3

_ GO TO 4___________ __________________________________________________8 ~ NI10LD=NH0LD* I

I Y= 3 _____GO TO 4

6 NRMVIJI*NHOLD _LLL=KMAX-NRMVIJl+l "" "

E Z E R 0=E ONR CH.______________________________________________________]__JEM-0

46 GUESS=0.0C**4* CALCULATE EXPOSURE AND EOC K-INFINITY FOR EACH C***4 NEW FUEL NODE

DO 47 LAL=LLL« KMAXEXBO=(POWERILAL)*BURNI/IRMTU*10000.)ANK IN=Al«-A2*EZeRO+A3*EZERO**2VlA4+A5*E ZERO+A6*EZERO**2Y*EX'BO+U7«-A

_ 18*E ZER0+A9*EZER0**21*EXB0**2 _ _ _ __ ____________47 GUESS=GUESS*ANKIN*WEIGHT(LALI

C**P* COMPUTE EOC K-FFFECTIVE AND COMPARE TO DESIRED EOC _ ___________C*4>** K-EFFECTIVE

AVKEF = IIR0KIN»GUESS)/FL0ATIKMAX))*THRLEK__________________________998 CONTINUE

IF IABSlEKEFFIJI-AVKEF t.LE.0.001) GO TO 4 8 ______ _________EZERO=EZERO«-EZERO*l EKEFFI J t-AVKEF 1*4.

C*«** JEM IS COUNTER TO STOP ITERATIONS ON ENRICHMENT CONVERGENCE.C**4* IF ENRICHMENT VALUE IS NOT FOUND IN 20 PASSES, THIS LOOP ISC*»** EXITED. _____ ____________________________________________________

I F IEZcRO.LF .2.2) GO TO 79IF1EZER0.GF.4.3I GO TO 89ir-IJEM.EQ.20) GO TO 48

997 CONTINUE _JcM=JEM+1 ' ..........GO TO 46

‘C**** IF CALCULATED AND DESIRED K-EfTECTIVE ARE WITHIN ~C**«* TOLERANCE RANGE, SET ENRICHMENT 79 EZERO«2.2 '

JEM*>79 GO TO 48 ‘ ” — -

89 EZER0a4.3_____________________________________________________________

155

GENATRJEM«89

A8. _FDNRCH*EZERO ___________ ;______________________________________WRITE(6,5531 JEM

555 FORMAT(2Xt•JEM EOUALS PASSES TO CONVERGE JEM*'»I3)WRITE(6,10JIJiENRG(J) .TJ(J) ,NRMVIJI,FONRCH

100 FORMATPO'.AX,'CYCLE NUMKER =',I3/10X,•ENERGY SCHEOULED ^'.FIA.SZ UOX,'CYCLE LENGTH *•tF12.3/10X,•BATCH SIZE *•,I 3/JOX,•FEED ENRICHM

2ENT =' , F 8 . M ___________________________________________________________________________________RETURN

_ END __ __________ __________ ______________ ______________

156

c ****** dep ca l ******c***« OEPCAL depletes the core in a point-wise_manner and __________c**** CALCULATES CORE URANIUM 1NVENTORY AT THE START OF C**** EACH TIME INTERVAL _

DOUBLE PRECISION D.DVAL,DRATE.DMRATE REAL KINF,MXPPFC,MTU,NEWKNF,NDLKNF INTEGER AOFCOMMON DVAL,DRATE,DMRATECOMMON A1,A2,A3,AA,A5,A6, A7, AO, A9, A U , A12 , A13, A1A, A15', A16, A17, A18, 1A19,A20,A21,A22,A2 3,A2A.A25,A26.A2 7,A28,A29,Cl,C2,C3,CA,C5,C6,C7, 2C8»C9,CTP,T0TMTU»MTU»TJ1, FONRCH,EUNRCH,EQENRG,EQCYCL COMMON KMAX,NR,Nl,Jl,Jil, IMAX.NJMAX, NREMOV,IYEAR,JB,JE,INMAX,J2,NINP,NBLST,I BUG,KEIGHT.IFLAGCOMMON NRMV|23),M1I(10)«M1R(13)«KCT(30),AOF(30) __ ___COMMON BURN[NI30I,BURNUP!301,PH10(301,EN!30 ),6(6001,KINF!30),

ltlIN!3C),ERG115,30),W2!30).ECF|10),FLFI10),RLF<10),TIN! 3,6,21, 2W3I301,PVI 3,6,2),UC(6,30I,UF|10,301,POWER!30),PAWERI3G),WEIGHTI 30 3I,EKEFF(20), ENBURN 130) ,FFCl(3'j),Xl!30) ,X2(30) ,01 ( 301 ,GAM( 30 I COMMON TAX,TAXIF.TAXIS,TAXI,RATE,FOEBT,RATEE,RATED,RATEV,RATE I,

1RATEVM,TAX2.TAXS.RATEP,THRLEK,XT _ _ ___' COMMON /ERNIE/THERMA13.A2), NOWUPI 3, 42) /RUYS/PVC1 I 30)”»PVC2 ( 30) / lDcPLET/URANIA(A2,30)/CAPFAC/CFCYC115)/TIME/TJI 301/CYCLES/NJ11 30), 3NJ2I30I/URAM/W11 30)/CONS/D .NBATCOMMON /STARTS/STRTI 20)/PROD/cNRGI20)/ORION/INTER!15)COMMON/ZEE/TIP

COMMON/PLAN/ENRGAI23) __________________________________..... DIMENSION EMS()l30),Fldl30),S0RSI301,

10ET AI 30),CAPGAMI30),OMEGI 30 I,EXPOI 30),2U25I30),X( 30),PLUTO!30),EI6I30),HINF(30),SQEMI 30),FLUCI 30) DIMENSION RljRNAPt 301 DATA LENGTH/A2/.BURNAP/30*0.0/DIMENSION FACrORIlS)________________ ________ ___________ ____EQUIVALENCE ICFCYC11),FAC TOR I 1)1

500 FORMAT!• •,AOX,I 2,6X,F7.1,5X,F5.3.5X,F7.A,7X,F6.A)550 FORMAT!*0',39X,'NODE*,faX,'BURNUP*,6X,'KINF',7X,'U235',6X,

1'PLUTONIUM*)860 FORMAT!' •,38X,12,11X,F7.A,12X.F8.2,I3X,F5.3)

8 DO 9 K= 1, K M A X ________________________________________________9 BURNUP!K)=3URNIN|K)

RMTU=MTU. _ __ _DAYP = DAYS ................ ................

C**** CALCULATE 30C STATE OF FUEL INCORE DO 1A K=1,KMAXXIK)=BURNU'»IKI/10000.0 ___ ____________ ______ ___ __

' KINFIK) = A1 ♦ A2*EN IK ) VA3*EN Ik ) **2*TaA+A5*EN i K ) +A6*ENTk ) **2) *X"l K ) + 11AT*A8*EN!K)*A9*ENIK)**2)*X!K)**2 _U25(K) = l AlUAl2*XlK)+A13*XIK)**2l*ENiK)

1A CONTINUE__________________________ ____ __C**** DEPLETION CALCULATION CYCLE*TJIJ1*365.Q*FACTORIJ)_______________________ ___________________

DU 113 KYM.KMAX BURNAP!KY)=BURNUPIKY)EXPOlKY)=POWER!KY)*CYCLE/MTU BURNUPIKY)=BURNUP(KY)+EXPOIKY)D11KY)=P0WER!KY)*CYCLE

110 CONTINUE _________________________________________________________

157

DEPCALIF IJ .EQ.JU) MIP=I ~NAP=NIP+INTERIJI-l ______________________________________________IF Ij.NE.Jll I GO TO 200 3 JIB = 1<UMAX _ ______EXPDlJIB)=TIP*PAWER(JIDI/MTU

3 BURNAPI JIB)=nURNAP(JIBHEXPOI JIB) _ _____ ________2 IF IJ.GT.J22) GO TO 113

_00 1 1 7 _LL=NIP,NAP _________ _________________________________________6o’ 114 K = l, KMAXURANIA(LL,X)=(A11+A12*BURNAP(K»/10000.+A13*(BURNAP«K1/10000.>**2)

' 1*MTUEXPOIK) = PAWERIK WH E R M A I N U B . L L ) _

114 8URNAP!K)=HURNAPIK)«-EXI*0!KI/MTU 117 C O N T I N U E ______;_______________________________________ ____________________

113 NIP=NAP*1DAYS = DAY? ___ ____DELTE=ABS(CTP*CYCLE-ENRGAIJI)/ENRGAIJIENRGY = CTP*CYCLE _ _ ________________________ _____FLL*0.0

______ FLX=0.0_______________________________________________________________________FXIN=b.O'

2C0 DO 210 K=1,KMAXXIK)=BURNUPIKJ/10000.KlNFIK)=AUA2*EN(K)fA3*ENIK)**2+IA4«-A5*ENIK)+A6«'EN(KI**2)*XIK) +

II A 7*A8*ENU )+A9*ENtK)**2>*XtK)**2PLUTOIX')=4?L*A22*EN(K>*A2 3*ENIK)**2+<A24+A25*DNIK) + A26*ENIX)**2)__

1*X(X»♦(A27»A2n*ENIK>+A29*CNIK>**2)*ALOGlXIK)-H.O)ENBURNIKI = IC1<-C2*6N(K)*C3*IEN(KJ )**2* I C4+C5*ENI K )*C6*I ENI K > ) **2 > *X

'l(K)fr(C7+C8*EN(K>«C9*(EN(K>>**2)*(X(K>)**2)*ENtK) U25!K)=(A1H-A12*XIX)*A13*XIK)**2)*ENBURNIK)FKIN=FXIN+XINFIK)

210 C O N T I N U E ________________________________________ ____________________________" Y19=KMAX '

FXIN*FXIN/Y19 C**** OUTPUT EOC STATE OF FUEL

WkITE16,482 > FKIN 482 FORMAT!2X, • AVERAGE XINF-NO WEIGHT=*,F 6 .4)

ENRGD=CTP*CYCLE _________________________________________________________bAYS=0AY>WRITE I 6,3361 ENRGD

336 FORMAT!10X,'ENERGY DELIVERED =',F10.1)700 FORMATI'O',14X, 'KEFF = •,F 6.4,10X,•CYCLE LENGTH = *,F5.l)650 FORMATI• •,II X, 12,9X,F7.4,HX,F8.2,9X,F5.3 ,6X,F8.2,6X,F7.3,19X,F 7 .3

II900 FORMAT!'O',10X, 'NODE• ,«X,• ENRICHMENT', 8X. • BURNUP'. (IX,'KINF'/bX,

l'ENERGY',8X,'P0WER',BX,'FINAL ENRICHMENT') _311 RETURN

END

158

C ****** COSTER ******C**** COSTER EOCTS THE COSTS CALCULATED BY NUCREX AND ITS ___________C**** COST COEFTICI ENTS TO STARTS

DOUBLE PRECISION DCOMMON/CRNIF/THERMAI3»42 ItNOWUPIIt42 I/BUYS/PVC1 I 30>tPVC21 30>/

1DEPLET/URANI A(42,3D)/CAPFAC/CICYC 115 I/CFIN/CFINV142»/COSTS/COSTPV 213t 42t6»/MME/TJC 301/C YCLE S/N JI ( 301 ,NJ2I 30)/URAN/W11 30)/CONS/30 tNBAT/ORION/INTFRI15)/PRUU/ENRGI 20)/C0NV6R/EFFI 31/FUEL/ZTU_COMMON /POI NTS/NODES(301 COMMON/EMINV/ENRINV13t42)DIMENSION UNICPL I 30),CORIN142 J.CORPROI42).UNINOK120t30»,

ICOSTUMI3.A2.6I, LOCATE(20>•MAC(201tPVD(42ILENGTH=42

C**** MAXP,AT = NUMnER OF BATCHES LOAOEO IN STUDY PERIOD______________________C*«** NJl(N»=CYCLE IN WHICH BATCH N IS LOADEDC**** NJ2IN)=CYCLE IN WHICH BATCH N IS DISCHARGED _

MAXBAT=NBAT IRIS=C

C**** UNICPLINBATCHI= PRESENT-VALUED PROCUREMENT COST OFC**** BATCH NBATCH * PRESENT-VALUED DISPOSAL VALUE OF __________________C**** BATCHES DISCHARGED DURING THE CYCLE PRECEDING THE'C**** CYCLE IN WHICH NBATCH IS INSERTED

DO 20 NBATCH3 l.MAXBAT UNICPLt Ne.UCH!=PVCII NBATCHI 1RMA=NJ11 ND A TCH) — 1IF I 1RMA.NF .IRIS) GD TO 21_______ _____________ _________________________UNICPLINBATCH)-PVC1INBATCHI GO TO 2 2

21 DO 23 NCO-1,NBATCHIF CNJ2INCDI.NE.IRMAI GO TO 23UNI CPL I NB ATCH )= UN I CPI I NBATCH I PVC2 (N C O )

23 CONTINUE _ _ ___ __________ ___ ____________ ___________ 22 IRIS3 IRMAC**** W1=MTU LOADED IN BATCH

20 UNICPLI NRATCH)=UNICPLINBATCHI/Kl (NBATCH) '■C***» THE 3 LOOP CALCULATES UNINOKIJt MAPI“ PROCUREMENT C**** COST/MTU FOR FUEL LOCATED AT NUDE NAP DURING CYCLE J

DO 3 JCYCLE=4,J22___________ ___________ __________________________....... N0dE=0

MUM=1 _DO 2 NUBA=l,MAXBAT ' ........IF (NJ1(NUBA).L E .JCYCLE.AND.JCYCLE.L E .NJ2(NUBA)) GO TO A GO TO 2

4__N00E=N0DE*NUDE SINURA)_____________________ _______________________________LOC ATE I MUMI=NUHA

C**** LMAX=NUMBER OF NODES IN CORE ___ ____IF (NODE.6 0 -LMAX) GO TO 5MUM=MUM*1 1 _ _ __ __

2 CONTINUE_ 5_N I GH=0 _____ ___________________________________________ ______________

DO 7 LM=1,MUM LAP=NODESI LOC ATE (LM)I

DO 8 NAB=1»LAP " .......NAP*NAB+NIGH

8 UNINOMJCYCLE*NAP)=UNICPL(LOCATEILH)I 7 NIGH3N1GH*N0DES(L0CATEILM) >___ ____ _____________________________________

159

COSTER3 CONTINUE

995 CONTINUE _ _ _ ________ ________COR IN(MOO) = CORE INVENTORY VALUE FOR INfERVAL MUO DO 16 MUD=1,LENGTH

16 COR INIMUO)=0•0 MAU = 0MOD = 1

_00 13 JIG=4,J22___________________________ __________________MID =IN TER IJIGI ~MAO=MAD+MID _ ___MAC IJIG)=MID

C***« URANIA!MUO,NASI = MTU AT NODE MUD AT BEGINNING OF C**** INTERVAL MUOC**»* CORPROIMUD) = OPERATION AND MAINTENANCE EXPENSE FOR

INTERVAL MUDDO 15 wUD=MOD,MAD __DO 1A NAG-1,LMAX

14 COR INI MUD)=CORINIMUO)♦UNINOKIJIG,NAG)*URANIAIMUD.NAG)15 CORPROIMUD)=336.♦200»*CFINVIMUD I13 MOD = MAD+1 ___________________ _________

C**** BMRATE =" 81-MONTHLY INTEREST" RAT F. ~BMRATE=0FXPlD/6.0)-l.0

C**»* THE REMAINDFR OF THIS ROUTINE CALCULATES THE PRESENT-VALUED C**** COSTS FOR EACH INTERVAL. COSTUM=CUSTS BEFORE PRESENT-VALUING.

MUD= I_ 45 UMPA = 0.0 ____ _______________ j________________________________

...... DO 50 MIX=1,MAXBATIF (NJ1 (MIX) .El).A) TIMPA=.TIMPA»UNICPL(MX)*W1(MIX)

50 CONTINUETAMPA=TIMPA/(ENRGI A)*EFF(NUB)I DO 19 J IN=4*J22MOD=MUD+INTCRIJIN)-! ___ ____ _ __

C****~~IF THIS IS A REFUELING INfERVAL,SET INVENTORY AND"ENERGY C*«** PRODUCTION COSTS TO 0.

DO 17 1NDY=MUD,MODIF I NOWUPI NUB, INDY). EO. 0) GO TO 18 _ _ __C0STUMINU3. INDY, 1)=*9.0

______ COSTUMINu3»INDY»2)=0.0 _____ _________________________COR I N( I NO Y^ I) =C0R I NI IND Y* l") +C OR I N11 NDY)CORPROI INDY^1»=C0RPR0( INDY+U + 316. _ ____ _______TEMPO*0.0DO 24 IRA=1.MAXBAT _ _IFINJ1IIRAI.NE.JIN) GO TO 24 TEMPO=TEMPO^UNICPL(IRA)*W11 IRA)

24' CONTINUE ............ .COSTUMI NUB, INDY,3)=TEMPO/1ENRGIJIN)*EFF(NUB))TAMPA=COSTUM(NUB »INDY * 3)GO TO 17

18 COSTUMI NUB,INDY,3)=T AMPACOSTUMINUP,1NDY,2)=:0RPR0(INDY)/1THERMAINUB,INOY ) *EFFI NUB)) COSTUMI NUB,INDY,I)=CORIN(INDY)*BMRATE7ENRINV(NUB,INOY)

17 CONTINUE19 MUD=M00+1

STAk*1.0/ll.0*HMRATE>PORT-1.0PVDI1)=PORT

COSTER

00 30 L=2,42 30 PVDIL)=PVD(L-i ) * s m

£***♦ COSTPVINUB.MIT»1) = PRESENT-VALUED COST OF INVENTORY ENERGY FOR C***♦ UNIT NUB IN INTERVAL MITC**»* C0STPVINUB.MIT.2)=PRESGNT-VALUE0 COST OF ENERGY PRODUCTION C***» COSTPV(NUB.MIT.3)=PRESENT-VALUG0 COST OF ENERGY REFUELEO

DO 31 MIT=I.42COSTPVINUB.MlT»1) = PVO(MITI*C0STUMI NUB.MIT»1)__________________

..... COSTPVI NUB.MIT»2)=PVO(MIT)*COSTUMlNUB»MlT«2)31 COSTPVINUB.MIr,3)»PVDlMIT)*C0SrUMlNUB,MIT,3>

RETURNEND

APPENDIX 6

PODECKA DOCUMENTATION

A6.1 D e s c r ip tio n o f th e Model

The co re p h y s ic s cure com prised o f a p o in t d e p le t io n scheme

w ith p se u d o -d im e n s io n a lity added by d iv id in g th e co re i n t o a number

o f equivolum e n o d e s , each node be in g d e p le te d in one s te p fo r each

c y c le . A t th e end o f a c y c le d e p le t io n , a l l c a lc u la t io n s fo r

de te rm in in g b a tc h s i a e and en richm en t o r energy p ro d u c tio n f o r th e

n e x t c y c le a r e e x e c u te d . F o r bookkeeping p u rp o se s , th e f u e l i s

th e n s h u f f le d under an o u t - in management scheme. T his does n o t

n e c e s s a r i ly re p re s e n t th e t r u e geom etry , b u t a flow o f f u e l on a

sy s te m a tic p a t te r n th ro u g h th e c o re . The flow p e rm its r e p r e s e n ta t io n

o f any ty p e fu e l management scheme by in p u t t in g th e d e s i r e d power

shape w hich i s used ip th e d e p le t io n .

Each node o f th e c o re i s a s s ig n e d a r e l a t i v e power f a c t o r ,

which i s in p u t te d . The s e t o f f a c to r s rem ains unchanged f o r each

com plete c a se end a r e com piled to o b ta in an a b so lu te power f a c t o r .

From t h i s f a c t o r , power i s a l lo c a te d t o each node and th e burnup i s

a s c r ib e d f o r a p a r t i c u l a r c y c le .

The code m a in ta in s an acc o u n tin g o f th e cu m ula tive burnup

f o r each node. The in c re m e n ta l burnup re c e iv e d by a node f o r a

161

162

p a r t i c u l a r c y c le i s added to th e p rev io u s c y c l e 's cum ula tive burnup

o f t h a t node. C um ulative nodal burnup i s one o f th e key p a ram ete rs

in d e te rm in in g th e b a tc h s iz e and enrichm ent o f th e new f u e l b a tc h

which i s in s e r t e d f o r th e upcoming c y c le .

The p re s e n t c o re burnup and c o s tin g model need th e is o to p ic

d e n s i t i e s , c u r re n t en ric h m en t, and k - i n f i n i t y o f th e f u e l . PODECKA

u ses e m p ir ic a lly d e r iv e d cu rv es f o r t h i s p u rp o se . The c o e f f i c i e n t s

f o r th e s e cu rves a re d e r iv e d from le a s t - s q u a r e s app rox im ations to

d a ta g e n e ra te d from more d e ta i le d codes. These c o e f f i c i e n t s a re

in p u t te d t o PODECKA so th e co rresp o n d in g s p e c i f ic ty p e o f r e a c to r may

be re p re s e n te d . Each cu rve i s a fu n c tio n o f burnup and i n i t i a l

en rich m en t. The cu rv es f o r k - i n f i n i t y , c u r re n t en ric h m en t, and

uranium m e ta l d e p le tio n a re ex p ressed as q u a d ra tic fu n c tio n s o f b o th

i n i t i a l en richm en t and burnup. P lutonium p ro d u c tio n i s re p re s e n te d as

a q u a d ra tic fu n c tio n o f en richm ent and a l i n e a r lo g a r i th m ic fu n c tio n o f

burnup.

The c o s tin g su b ro u tin e i s c a l le d NCOST ( 52) . T h is package

i s a s im p l i f ie d v e rs io n o f th e n u c le a r econom ics code NUCBID (£ 3 ) .

NCOST employs b a tc h d a ta g e n e ra te d from a n o th e r PODECKA s u b ro u t in e ,

BATDAT, which u ses th e e m p ir ic a l c o r r e la t io n s to g e n e ra te th e b a tc h

is o to p ic d e n s i t i e s .

A m ajor o p tio n o f PODECKA fu rn is h e s th e u s e r w ith th e

a b i l i t y t o e i t h e r s p e c ify th e d e s ir e d c y c le energy and have th e program

s e l e c t th e b a tc h s iz e and en ric h m en t, o r p p e c ify th e b a tc h s iz e and

en richm ent and l e t th e energy l e v e l be s e le c te d .

163The s e le c t io n o f b a tc h s iz e and en richm ent to meet a g iven

energy demand i s accom plished under a s e t o f r i g i d r u le s :

1 . S e t i n i t i a l enrichm ent to " tq u il ib r iu m " v a lu e .

2 . F ind b a tc h s iz e so th e k - e f f e c t iv e o f th e new co re i s g r e a te r

th a n o r eq u a l t o d e s ire d e n d -o f-c y c le k - e f f e c t iv e .

3. S et b a tc h s iz e t o th a t found in s te p 2 .

k. I t e r a t e on enrichm ent u n t i l k - e f f e c t iv e o f th e new co re eq u a ls

th e d e s ir e d e n d -o f-c y c le k - e f f e c t iv e .

F or any c y c le , th e a lg o rith m s e t s th e enrichm ent eq u a l to

th e " e q u ilib r iu m " c y c le v a lu e as a f i r s t g u e s s , and th e b a tc h s iz e

i s de term ined . T h is i s done by making an i n i t i a l guess o f th e b a tc h

s iz e . The new fu e l lo a d in g i s tak en th rough a s im u la te d c y c le

d e p le tio n . The e n d -o f-c y c le (EOC) k - e f f e c t iv e i s compared to th e

d e s ire d in p u t te d e n d -o f-c y c le k - e f f e c t iv e . I f th e s e a re n o t e q u a l ,

a new e s t im a te o f th e b a tc h s iz e i s made b ased on th e r e la t io n s h ip o f

th e c a lc u la te d k - e f f e c t iv e to th e d e s ir e d k - e f f e c t iv e . T h is c a lc u la -

t i o n a l sequence o f inc rem en ting th e b a tc h s i z e i s re p e a te d u n t i l a

b a tc h s iz e i s reach ed to make th e e s tim a te d e n d -o f -c y c le k - e f f e c t iv e

g r e a te r th a n o r eq u a l to th e d e s ire d e n d -o f-c y c le k - e f f e c t iv e . At th is -

p o in t th e b a tc h s iz e i s f ix e d to th e v a lu e j u s t found. The enrichm ent

now i s a llow ed to change u n t i l th e c a lc u la te d EOC k - e f f e c t iv e eq u a ls

th e d e s ire d EOC k - e f f e c t iv e . The en richm ent i s f ix e d when k - e f f e c t iv e

a t e n d -o f -c y c le e q u a ls d e s ire d EOC k - e f f e c t iv e . T hus, th e b a tc h s iz e

and en richm ent a re s e t fo r th e c y c le . The c o re i s s t i l l in th e

c o n f ig u ra tio n o f th e p re v io u s c y c le ( i . e . , f u e l has n o t been s h u f f le d ,

16U

only th e new fu e l b a tc h s iz e and enrichm ent have been d e te rm in e d ).

The f u e l i s s h u f f le d and re p la c e d l a t e r in a n o th e r s e c tio n o f th e

program .

I f th e energy p ro d u c tio n fo r th e n e x t c y c le i s t o be

c a lc u la te d , th e c a lc u la t io n p roceeds along much th e same l i n e s as

o u t l in e d above. The co re undergoes a s im u la te d s h u f f l in g and in s e r t io n

o f a new fu e l b a tc h acco rd ing to th e in p u t b a tc h s iz e and en rich m en t.

An i n i t i a l guess i s made fo r th e energy to be produced d u rin g th e n ex t

c y c le . The energy p ro d u c tio n i s increm ented u n t i l convergence

betw een th e d e s ire d EOC k - e f f e c t iv e and th e c a lc u la te d EOC k - e f f e c t iv e

i s reac h ed . The a c tu a l s h u f f l in g and d e p le tio n o f th e new f u e l lo a d in g

i s done l a t e r in th e program once th e energy has been s e t . T h is

p rocedu re i s cap ab le o f h an d lin g two b a tc h s iz e s and two en richm en ts

as i n p u t .

In d e te rm in in g th e b a tc h s iz e and en richm ent o r energy

p ro d u c tio n le v e l f o r a c y c le , PODECKA com pares th e c a lc u la te d EOC

k - e f f e c t iv e t o th e in p u t te d k - e f f e c t iv e . K -e f fe c t iv e i s c a lc u la te d by

summing th e w eigh ted k - i n f i n i t y a t each node and d iv id in g by th e

number o f nodes to o b ta in a co re average k - i n f i n i t y . T his r e s u l t i s

m u l t ip l ie d by a lump leakage t e r m ^ to o b ta in k - e f f e c t iv e . In c a lc u la

t io n o f an average co re k - i n f i n i t y each node has a w eig h tin g f a c t o r .

This f a c to r may be used to w eigh t th e k - i n f i n i t y p r o f i l e as d e s ir e d .

The leakage term accoun ts f o r a l l leakage e f f e c t s : f a s t , th e rm a l ,r a d i a l . The term i s d e riv e d by a d ju s t in g th e le ak ag e v a lu e u n t i l PODECKA r e s u l t s a re no rm alized to a more d e ta i l e d n u c le a r code.

165

The laser a lso h a s th e o p tio n o f changing th e w e ig h tin g f a c to r s fo r

each c y c le .

At p r e s e n t th e re a r e no m echanical l im i t a t i o n s im posed on

th e c o re . T here i s a range imposed on th e en richm en t v a lu e . One

h a tc h s iz e and enrichm en t w i l l be s e le c te d f o r a g iven en ergy le v e l .

The b a tc h s iz e i s n o t l i m i t e d , th e e n t i r e co re may be r e p la c e d . A

power peak ing c o n s t r a in t w ould be m eaning less s in c e a r e l a t i v e power

3hape i s r e q u ire d a s in p u t .

PODECKA i s segm ented in to s e v e ra l s u b ro u tin e s w hich makes

changing a p a r t i c u l a r p o r t io n r e l a t i v e l y e a sy . In a d d i t io n , th e

u se r needs t o be f a m i l ia r w ith on ly one p a r t i c u l a r su b ro u tin e to

do m o d if ic a tio n s from PODECKA's in te n d e d a p p l ic a t io n .

In o rd e r t o show th e m athem atica l fo rm u la tio n o f PODECKA,

i t s flow i s t r a c e d . This w i l l show n o t on ly th e g e n e ra l g roup ing o f

th e b lo c k s o f c a l c u la t io n s , b u t a ls o th e flow o f th e code.

A6.2 M athem atical F orm ula tion

PODECKA needs a p a s t h i s to r y o f th e c o re to s t a r t th e

e x e c u tio n o f any a n a ly s is w hich i s p ro v id ed by th e u s e r . The p a s t

h i s to r y may c o n s i s t o f a number o f p a s t o p e ra t in g cy c les o r J u s t one

p a s t c y c le . I f t h e u s e r w ish es to s t a r t from a c le a n c o re , he must

in p u t a dummy c y c le b e fo re s t a r t i n g th e p la n n in g p e r io d . The h i s to r y

th e u s e r s u p p lie s must be in p u t te d w ith th e r e a l i z a t i o n t h a t b e fo re

th e f i r s t c y c le i s e x e c u te d , some o ld f u e l w i l l b e d isc h a rg e d , th e

rem ain ing f u e l moved in w ard , and new f u e l in s e r t e d in th e o u te r nodes.

C onsequen tly , t h e u s e r sh o u ld in p u t h i s f i r s t c y c le b a tch s i z e and

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enrichm ent t o g iv e th e p ro p e r c o n f ig u ra tio n f o r a c le a n c o re . T h is

w i l l g iv e a f i r s t - c y c l e energy s t a r t i n g from a c le a n c o re . I f th e

u s e r w ishes* th e code may he used t o s e l e c t th e f i r s t c y c le h a tc h

s iz e and en richm en t f o r a c le a n co re hy in p u t t in g th e ahove dummy

c o n f ig u ra tio n and d e s ir e d f i r s t - c y c l e en e rg y .

A f te r th e n e c e s sa ry d a ta h as heen in p u t te d th ro u g h th e

DATIN s u b ro u t in e , th e main program e s ta b l i s h e s th e v a r ia b le s i t needs

t o s t a r t th e c y c le c a lc u la t io n s .

To c a lc u la te f u e l c y c le c o s ts o v e r th e number o f c y c le s

(p lan n in g p e r io d ) d e s i r e d , knowledge o f p r e - and p o s t-p la n n in g p e r io d

o p e ra tio n m ust be a v a i la b le . The p o s t-p la n n in g p e r io d o p e ra t io n i s

assumed. "E q u ilib riu m " c y c le le n g th and energy a re assumed. E q u il

ib rium b a tc h s iz e i s u se d . Enrichm ent i s de term ined by PODECKA to

s a t i s f y th e d e s ir e d EOC k - e f f e c t iv e . The p o s t-p la n n in g p e r io d c y c le s

a re m u l t ip le s o f th e "e q u ilib r iu m " c y c le .

T hese v a r ia b le s may be r e p re s e n te d as

NRMV(J) - KMAX/3

ENRG(J) = EQENRG f o r J = J22+1 th ro u g hJ22+3

T J (J ) ■ EQCYCL

STRT(J) = STRT(J - l )*EQCYCL f o r J = J22+2 and J22+3

where J22 = l a s t c y c le o f p la n n in g p e r io d ;

NRMV(J) = b a tc h s iz e ( in nodes) f o r c y c le J ;

ENRG(J) = en ergy f o r c y c le J ;

T J (J ) = c y c le le n g th o f c y c le J ;

STRT(J) = decim al y e a r i n w hich c y c le J s t a r t s ;

KMAX = number o f nodes in th e r e a c to r c o re ;

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EQENRG and EQCYCL = " e q u il ib r iu m " cy c le e n e rg y and le n g thr e s p e c t iv e ly .

The c y c le le n g th s f o r th e p la n n in g p e r io d a r e de te rm in ed from th e

in p u t te d s t a r t t im e s :

TJ(K) = STRT(K+1) - STRT(K) f o r K » J l , . . J22

where J l = i n i t i a l r e a c to r c y c le number.

I f th e u s e r d e s i r e s , PODECKA can r e f l e c t e a r l y r e f u e l in g

a n d /o r o p e ra t io n beyond r e a c t i v i t y l i m i t a t i o n by in p u t t in g a s e t o f

v a lu es f o r STRTN(J). These v a lu e s a re d i f f e r e n t frcm th e v a lu es

in p u t te d f o r STRT(J). From th e v a lu e s o f STRTN(J) a. s e t o f c y c le

le n g th s , T JN (J) , w i l l b e c a l c u la te d :

T Jlf(J) = STRTN(j-fl) - STRTN(J) f o r J = J l , . . . , J22 .

The a lg o ri th m alw ays u se s T J(J ) when s e le c t in g th e en e rg y fo r th e

in p u t te d b a tc h c h a r a c t e r i s t i c s o r s e le c t in g b a tc h c h a r a c t e r i s t i c s fo r

in p u t te d en e rg y . The f u e l w i l l b e d e p le te d acco rd in g t o th e TJM (J);

c y c le le n g th and c o s ts w i l l be de term ined from STRTK(J) and T JN (J).

Thus, by making TJN (J) lo n g e r o r s h o r te r them T j ( j ) ( th ro u g h STRTN(J)),

e a r ly shutdown o r coastdow n may b e em ployed. A tte n tio n must b e p a id

to th e f a c t t h a t th e code always u se s th e c y c le le n g th T J (J ) f o r

p la n n in g p u rp o se s . As an i l l u s t r a t i o n , i f coastdown i s m odeled,

th e n STRTH(J+l) STR T(J+ l), th u s TJN (J) T J ( j ) . More energy w i l l be

p roduced , and th e c o re d e p le te d beyond r e a c t i v i t y s in c e T J (J ) was

used in p la n n in g c y c le J . For p la n n in g th e n e x t c y c le , J + l , th e

a lg o rith m u ses TJ(J-KL) a s th e c y c le le n g th . Thus, a f t e r th e abnorm al

c y c le th e p la n n in g goes back to t h e o r ig i n a l cy c le schem e. I f , a f t e r

an abnorm al c y c le i s s im u la te d , a d i f f e r e n t c y c le scheme i s d e s i r e d ,

168

t h i s sh o u ld "be In p u tte d i n STRT(J+2), e t c . , t o a llo w th e code t o p la n

fo r a new scheme a f t e r an abnorm al o c c u rre n c e .

When th e s e v a r ia b le s a re c a lc u la te d , th e main program moves

in to th e c y c le c a lc u la t io n s . At th e end o f each c y c le , th e new and

r e ta in e d in c o re f u e l i s s h u f f le d and th e d isc h a rg e d f u e l i s s e p a ra te d

in to b a tc h e s . The c o s t c a lc u la t io n i s done in to t o o n ly a f t e r

com pletion o f th e p la n n in g p e r io d and p o s t-p la n n in g c y c le s . At t h a t

t im e , c o s t f o r a l l th e b a tc h e s and a l l c y c le c o s ts a re c a lc u la te d .

S u b ro u tin e SETPOW i s u sed o n ly once t o d e te rm in e th e

a b so lu te power f a c to r s which a r e r e ta in e d f o r u se in a l l c y c le s . SETPOW

re a d s th e r e l a t i v e power f a c t o r PAWER(l) f o r each node. The a b s o lu te

f a c to r i s g iven by

pA w n ,(i) * S J - f* 6- 1 '

KMAXwhere SUMPOW = 2 PAWER(I) i s c a lc u la te d p r i o r t o e q u a tio n

1=1 (A 6-1).

Having e s ta b l i s h e d a n o rm alized power sh a p e , th e c y c le

c a lc u la t io n lo o p i s e n te re d . T here a r e fo u r v a r ia b le s t o s p e c ify th e

s t a t e o f th e f u e l a t each node: ( l ) k - i n f i n i t y , (2 ) r a t i o o f c u r re n t

en richm ent t o i n i t i a l en ric h m en t, (3 ) r a t i o ^ 3 5 m e ta l t o i n i t i a l

m e ta l lo a d e d , and (U) p lu to n iu m p ro d u c tio n .

These v a r ia b le s a r e e x p re sse d by th e fo llo w in g cu rv es f i t t e d

by a l e a s t sq u a re s ap p ro x im atio n :

koo - V a2eo+a3eo+(V a5eo+a6eo )b+(a7+a8eo+a9eo )b2(A6-2)

where k M = k - i n f i n i t y ;

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e ~ i n i t i a l en rich m en t; ob = bum up (MWD/MTU) s c a le d down b y 10 ;

a ^ , . . . , a^ = in p u t te d c o n s ta n ts .

e / e o = c 1+c2eo+c3e 2+(c u+cueo+c6e 2 )b+ (c7+c 8eo+c9e 2 )b2

a ^ , a 1 2 , a13 = in p u t te d c o n s ta n ts ;

U = i n i t i a l uranium m e ta l; o

b as in e q u a tio n (A 6-2 ).

(A6-3)

where e = c u r r e n t en rich m en t;

e and b a s i n e q u a tio n (A 6 -2 ); o

c ^ , , . • , Cj = in p u t te d c o n s ta n ts .

U/Uq = a l l +a12b+a13b2 (A6-U)

where U = c u r r e n t uranium m e ta l;

Pu/Uo * a 21+a22eo+a23e o+^a21t+a25e o+a2ueo ) >+

(a27+a28e o+a29eo )1 °6 (l + b ) (A6-5)

where Pu = p lu ton ium p roduced ;

U = i n i t i a l uranium m e ta l; o

e and b a s in e q u a tio n (A 6 -2 ); o

a21*‘ * ** a 29 * in Pufcted c o n s ta n ts .

F or each p ass th ro u g h th e p la n n in g p e r io d ( i . e . , c y c le s

J l l t o J22 ) th e a lg o rith m d e c id e s w hether t o c a lc u la te th e d e s ir e d

energy o r b a tc h s iz e and en richm en t f o r each c y c le . I f i t i s t o c a l

c u la te th e en e rg y , i t c a l l s PARMX; o th e rw is e , i t p a sse s t o th e n e x t

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s ta te m e n t. PARMX reads in th e b a tc h s iz e and enrichm ent f o r t h a t

c y c le . Two b a tc h s iz e s and two en richm en ts can be in p u tte d a t each

r e f u e l in g . An e s tim a te o f th e energy i s s e t a t th e e q u ilib r iu m cy c le

v a lu e . Knowing t h i s , th e bum up f o r each node i s found from:

EBURN = EST*PAWER(K)/RMTU

where EBURN = nodal burnup (MWD/MTU);

EST = e s tim a te d energy ;

PAWER(K) = a b so lu te power f a c to r o f node K;

RMTU = nodal m e tric to n n ag e .

In PARMX a s im u la te d s h u f f l in g and in s e r t io n o f new fu e l

i s done ( th e a c tu a l s h u f f l in g i s done l a t e r ) . The i s c a lc u la te d

f o r each node and th en summed. Thus,

b1 * [b(K+n)+EBURN]*10-lt

k ^ (K) = f f b 1 , EN(K+n))P

Q = 2 koo (K)K=1

1 - hwhere b * burnup*10 th a t f u e l a t node K+n w i l l have a f t e rs h u f f l in g and d e p le t io n f o r c y c le o f i n t e r e s t ;

n = number o f nodes b e in g re p la c e d ;

b(K) = burnup o f node K up to th e s t a r t o f th e p re s e n tc y c le ;

EBURN = burnup o f node K fo r th e p re s e n t c y c le (see p re v io u s d e f in i t i o n ) ;

^oo ^ = k - i n f i n i t y o f node K (c a lc u la te d fo r EOC)c a lc u la te d from c o r r e la t io n (A 6-2);

EN(K) = i n i t i a l enrichm ent o f f u e l lo c a te d a t node K;

p * KMAX-n = number o f nodes o f f u e l t h a t w i l l rem ain in c o re th ro u g h th e p re s e n t c y c le ;

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and f o r th e new f u e l

kgj, (K) « f(BBURH.ENO)

KMAXQ = Q + 2 koo (K)

K*p+1

where EBURN i s a s above;

ENO = enrichm en t o f new f u e l ;

Q = t o t a l k - i n f i n i t y f o r a l l f u e l in th e r e a c to r a t th e end o f th e p re s e n t c y c le .

At t h i s p o in t Q i s checked a g a in s t th e d e s ir e d EOC k -

e f f e c t iv e :

- EKEFF( J )*KMAX WJU"1 THRLEK

where EKEPP(J) = d e s i r e d k - e f f e c t iv e f o r c y c le J ;

THRLEK = le ak ag e c o r r e c t io n ( in p u t te d ) .

I f th e c a lc u la te d EOC k - e f f e c t iv e i s w ith in a to le r a n c e o f 0.0001

w ith th e d e s i r e d EOC k - e f f e c t i v e , PARMX e x i t s t o th e main program .

I f th e to le r a n c e i s n o t m e t, a new energy i s e s tim a te d by l i n e a r

e x t r a p o la t io n , and PARMX i s ex ec u ted a g a in . T h is c a l c u la t io n a l scheme

i s re p e a te d u n t i l th e to le r a n c e i s m et.

W hether PARMX i s ex ec u ted o r n o t , th e n e x t s te p i s t o f in d

th e co re power l e v e l . Knowing th e energy ( e i t h e r in p u t te d o r c a l

c u la te d i n PARMX) t o b e p roduced d u rin g th e p re s e n t c y c le , th e power

i s g iven by

CTP « ENRG(J)/[TJ(J)*365*FACTOR(J)]

where ENRG(J) * th e rm a l energy o f c y c le J ;

FACTOR ( J ) ■ c a p a c i ty f a c t o r o f c y c le J ;i

CTP * c o re th e rm a l pow er.

172Once CTP i s fo u n d , th e power can be d i s t r i b u t e d t o each node by

POWER(K) = CTP»PAWER(K) f o r K = 1 , KMAX.

The a lg o rith m now d e c id e s w hether s u b ro u tin e ALTGEN sh o u ld be

e x ec u ted . I f PARMX was e x e c u te d , ALTGEN i s b y passed and v ic e v e r s a .

ALTGEN employs much th e same te c h n iq u e u sed i n PARMX, b u t

ALTGEN s e le c t s b a tc h s i z e and en richm en t knowing c y c le energ y .

ALTGEN's f i r s t a c t io n s a r e to e s t a b l i s h s e v e r a l o f th e c o n s ta n ts

u sed r e p e a te d ly i n t h i s s u b ro u tin e .

COMPAR = (EKEFF(J)#KMAX)/THRLEK = d e s i r e d t o t a l EOCi n f i n i t y

BURN * T J(J)* 3 6 5 #FACT0R(J) = f u l l power d ay s .

The en richm ent i s s e t a t th e e q u i l ib r iu m c y c le v a lu e ;

EZERO = EQNRCH.

The burnup and k - i n f i n i t y th e new f u e l h as a t t a in e d a t

th e end o f a c y c le i s de te rm in ed by

EXBO * (POWER(KL)»BURN)/(RMTU*101*)

PHIO(KL) = f(EZERO,EXBO)

w here KL = 1 , . . LAST;

LAST = KMAX-U

EXBO = burnup*10 ;

PHIO * k - i n f i n i t y c o r r e l a t i o n (A 6-2).

T hus, th e EOC k - i n f i n i t y f o r each node o f new f u e l f o r th e

c o re i s found . The t o t a l k - i n f i n i t y o f th e new f u e l i s found by

SUNK(l) - PHIO (1)»WEIGHT (KMAX)

SUNK(KL) ■ SUNK (KL-1)+PHIO(KMAX-KL+1) • WEIGHT(KMAX-KL+1)

w here SUNK(KL) * t o t a l k - i n f i n i t y f o r a new f u e l b a tc hconsisting of KL nodes;

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WEIGHT(K) = k - i n f i n i t y w e ig h tin g f a c t o r f o r node K ( in p u t te d ) .

Now th e f i r s t r e lo a d b a tc h s iz e i s e s t im a te d by

NHOLD = » NRMV(J22+3).

As in PARMX, th e f u e l t o rem ain in c o re (NNN=KMAX-NHOLD) i s concep

t u a l l y s h u f f le d and d e p le te d th ro u g h th e p re s e n t c y c le . T hus,

EXB = POWER (K)*BURN/RMTU

DELTA = [BURfflJP(K+NH0LD)+EXB]»10“ l*f o r K * 1 , . . . , NNN

ANKINF = f (EN(K+NHOLD) .DELTA)(A6-6)

RQKIN = RQKIN+ANKINF*WEIGHT(K)

where EXB = exposure a t node K;

DELTA - t o t a l burnup a t th e end o f th e p re s e n t c y c le on f u e l now a t node K+NHOLD b u t t o b e lo c a te d a t node K f o r p re s e n t c y c le ;

ANKINF = k - i n f i n i t y c o r r e la t io n (A 6-2 );

RQKIN = t o t a l k - i n f i n i t y from th e o ld f u e l in c o re (NNN n o d e s ) .

RQKIN i s combined w ith th e f i r s t c a l c u la t io n t o g iv e th e t o t a l k -

i n f i n i t y o f th e co re a t th e end o f th e p re s e n t c y c le i f NHOLD nodes

a r e r e p la c e d .

GAM(NHOLD) = RQKEN+SUNK (NHOLD) . (A6-7)

I f GAM(NHOLD) i s l e s s th a n , eq u a l t o , o r g r e a te r th a n

COMPAR, th e flow b ran ch es t o th e a p p ro p r ia te s e c t io n o f th e s u b ro u tin e .

I f GAM (NHOLD) = COMPAR, th e n ALTGEN's flow p a s s e s t o an enrichm en t

s e le c t io n s e c t io n . I f GAM(NHOLD) i s l e s s th a n COMPAR, th e n NHOLD i s

inc rem en ted by p lu s one node (NHOLD = NHOLD+l) and e q u a tio n s (A6-6)

and (A6-7) a r e re p e a te d . GAM (NHOLD) i s ag a in com pared to COMPAR and

17*»

i f GAM(NHOLD) i s s t i l l le s s th a n COMPAR, NHOLD i s increm en ted and

(A 6-6) and (A6-7) re p e a te d . T h is scheme i s i t e r a t e d u n t i l th e f i r s t

o ccu rren ce o f GAM(NHOLD) be ing g r e a t e r th a n COMPAR. At t h i s p o in t

NRMV(j) i s s e t :

NRMV(J) = NHOLD.

C o n tro l i s t r a n s f e r r e d to an enrichm ent s e le c t io n s e c t io n .

I f th e i n t i t a l NHOLD g iv e s GAM(NHOLD) g r e a t e r th a n COMPAR,

NHOLD i s increm ented by minus one node (NHOLD = NHOLD-l) and

e q u a tio n s (A6-6) and (A6-7) i t e r a t e d , in c rem en tin g NHOLD each tim e ,

u n t i l th e f i r s t o ccu rren ce o f GAM(HHOLD) l e s s than COMPAR. At t h i s

p o i n t ,

NRMV(J) = NHOLD = NIIOLD+1

and a f in a l p ass i s made th rough (A 6-6). C o n tro l i s th e n p a sse d to

th e enrichm ent s e le c t io n s e c t io n .

N o tic e th e p h ilo so p h y in v o lv ed in th e s e le c t io n o f b a tc h

s i z e i s no t one o f o p tim iz a tio n . Two assum ptions a r e made: ( l )

th e enrichm ent equals th e e q u ilib r iu m v a lu e and (2 ) th e b a tc h s iz e

•which g ives an e q u a l i ty o r s l i g h t l y la r g e r EOC c o n d itio n i s chosen .

The second assum ption i s made on th e b a s is th a t i t i s cheaper t o

buy and f a b r ic a te an a d d i t io n a l node r a th e r th an buy more s e p a ra t iv e

work and fe e d u n its t o in c re a s e en richm en t i f a s m a lle r b a tc h s iz e

i s s e le c te d . I t i s th e s e t t in g o f th e en richm ent t h a t fo rc e s a

d e te rm in a tio n o f th e b a tc h s iz e in s e r t e d i n t o th e r e a c to r .

The b a tch s i z e as de term ined above i s p a s se d to th e e n r ic h

m ent s e c t io n . H ere, b a tc h s iz e i s f ix e d and en richm ent i s a llow ed

to f l o a t t o b r in g

GAM(NHOLD) » COMPAR

w ith in a 0 .0001 to le r a n c e . The en richm en t i s s e t a t th e e q u i lib r iu m

v a lu e . The exp o su re i s found f o r each new f u e l n o d e ,

EXBO » (POWER (LAL )*BURN/RMTU*10**) (A6-8)

and k - i n f i n i t y a t EOC f o r each new node i s found from c o r r e l a t i o n

(A 6 -2 ),

ANKIN = f(EZERO,EXBO). (A6-9)

A lso ,

GUESS » GUESS+ANKIN*WEIGHT(LAL) (A6-10)

(A 6 -8 ), (A 6-9 ), (A6-10) a re done f o r LAL = KMAX-NRMV(J)+1 t o KMAX.

Then k - e f f e c t iv e i s d e te rm in ed :

AVKEF » * THRLEK. (A6-11)KMAa

AVKEF i s checked a g a in s t EKEFF(J). I f AVKEF i s v i t h i n 0 .0001 o f

EKEFF(J), th e en rich m en t i s s e t . I f t h e a b s o lu te v a lu e o f AVKEF-

EKEFF(J) i s g r e a t e r th a n 0 .0001

EZERO » EZERO+EZERO* ( EKEFF (J ) -AVKEF ) *1+,

th e n (A 6-8), (a6-9 )» and (A6-10) a re ex ecu ted a g a in . T h is I t e r a t i o n

i s fo llo w ed u n t i l convergence i s re a c h e d . The en richm en t i s s e t ,

FDNRCH * EZERO, and c o n t ro l i s r e tu rn e d t o MAIN.

T here p re l i m i t s im posed on w hat th e en richm en t may b e .

The o n ly r e s t r i c t i o n on b a tc h s i z e i s t h a t i t can n o t b e g r e a te r th a n

th e number o f nodes o f th e c o re .

A f te r c o n t r o l i s r e tu rn e d t o MAIN, i f J e q u a ls J l l , th e

BATDAT s u b ro u tin e i s sk ip p e d . T h is i s due t o t h e f a c t t h a t th e f i r s t

176

s e t o f nodes does n o t e n te r in t o c o s t c a lc u la t io n s in th e p la n n in g

p e r io d . BATDAT i s d is c u s s e d h e re s in c e i t i s ex ecu ted in t h i s lo c a

t i o n f o r a l l c y c le s ex cep t J l l .

BATDAT s e p a ra te s th e d isch arg e d no d es, as de term ined hy

ALTGEN o r PARMX, in to b a tc h e s . The f u e l b a tc h th a t comes o u t a t a

p o in t in tim e does n o t n e c e s s a r i ly co rrespond t o a f u e l b a tc h I n s e r te d

s e v e ra l c y c le s p re v io u s ly . The in s e r t io n b a tc h i s th e q u a n t i ty and

en richm en t o f f u e l charged in t o th e co re a t th e s t a r t o f a g iven

c y c le . The f u e l t h a t i s d isch a rg e d i s composed o f nodes t h a t m ight

have been in s e r t e d d u rin g s e v e ra l d i f f e r e n t c y c le s . BATDAT s e p a ra te s

th e d isc h a rg e d fu e l in to h a tc h e s acco rd in g to i n i t i a l enrichm en t and

tim e o f in s e r t io n . T hat i s , a l l nodes in a d isch a rg e q u a n t i ty o f

f u e l w ith th e same tim e o f in s e r t io n and i n i t i a l en richm ent w i l l

form one b a tc h . T h is s o r t in g i s done f o r each c y c le in th e p la n n in g

h o r iz o n .

BATDAT dete rm in es th e average burnup and av erag e d isc h a rg e

a ssay f o r each b a tc h . T h is i s done by

SMBRN * SMBRN + BURNUP(KCT( JK ))

SMNRCH ■ SMNRCH + ENBURN(KCT(JK))

where BURNUP(KCT(JK)) * burnup o f d isch a rg e d node KCT(JK);

ENBURN(KCT)JK)) ■ U235 a ssa y o f d isch a rg e d node KCT(JK).

Both o f th e above a re summed o v er th e number o f nodes in each b a tc h

as s e t in BATDAT. The average d isch a rg e c h a r a c t e r i s t i c s a r e determ ined

by th e number o f nodes in a b a tc h . T hus,

AYGBRN * SMHRN/NNDBAT

AVNRCH » SMNRCH/NNDBAT

where NNDBAT ■ number o f nodes i n a b a tc h ;

AVGBRN - av erage burnup o f a b a tc h ;

AVNRCH » average en ricnm ent a s s a y .

The is o to p e s f o r a b a tc h a r e c a lc u la te d from

U25D 3 W1(NBAT)*(U235/U)»AVNRCH

PLUTOD a Wl(NBAT)*f(HENRCH,AVOFR)

w here U25D a d isc h a rg e d U235 in b a tc h NBAT;

Wl(NBAT) a m e tr ic w eigh t o f b a tc h NBAT;

U235^J * c o r r e l t t t io n (A6-U);

PLUTOD a m e tr ic w eigh t o f p lu ton ium c r e a te d ;

f (BENRCH,AVGFR) * p lu ton ium c o r r e l a t i o n (A 6-5 ); and

AVGPR a AV0BRN*10“1*.

The d isc h a rg e a s sa y o f i s d e term ined f o r b a tc h NBAT by

X2(NBAT) a U25D/(W2(NBAT)*1000)

where W2(NBAT) a W l(NBAT)-[(l.l7»PLUT0D/1000] -[Wl(NBAT)(BENRCH*10-U25D) /1 0 0 0 ]

These c a lc u la t io n s a r e re p e a te d u n t i l a l l d isc h a rg e nodes have been

s e p a ra te d in t o b a tc h e s and t h e i r b a tc h i s o to p ic s d e te rm in e d . C on tro l

i s r e tu rn e d t o MAIN.

MAIN n e x t c a l l s SORT. SORT was o r ig i n a l l y d e s ig n e d t o s o r t

th e nodes in t o an o rd e r o f in c re a s in g r e a c t i v i t y . In p r e s e n t usage

SORT i s a duany s u b ro u tin e i n t h a t th e f u e l i s n o t r e a l l y s to r e d .

The f u e l i s j u s t p la c e d in th e same r e a c t i v i t y o rd e r th e nodes a re

i n : t h a t i s ,

KCT(KAZ) - KAZ f o r KAZ - 1 .................KMAX

178

where KCT(K) = f u e l re g io n o f f u e l v i t h r e l a t i v e o rd e r o fr e a c t i v i t y K.

MAIN c a llB SHUFL. SHUFL was in te n d e d t o move th e f u e l no t b e in g

d isc h a rg e d acco rd in g t o a f ix e d p a t t e r n . Thus,

AOF(L) ■ K f o r L - 1 , . . (KMAX-NRMV(J))

where AOF(L) * r e l a t i v e r e a c t i v i t y o rd e r number o f f u e l t o bep la c e d in node L;

K = NRMV(j)+l;

K = K+L.

C o n tro l 1b r e tu rn e d t o MAIN.

Both SORT and SHUFL cure r e ta in e d . A lthough a sim ple sh u f

f l i n g scheme i s used and th e s e two s u b ro u tin e s a re n o t r e a l ly needed,

th e y a re in c lu d e d so th e u s e r may m odify th e fu e l management i f he

d e s i r e s .

The n ex t b lo ck o f s ta te m e n ts ( in c lu d in g su b ro u tin e s SORT

and SHUFL) in MAIN s h u f f le th e f u e l . T h is s e c t i o n 's methods a re

r e ta in e d I n t a c t from NUCSIM. T his e x p la in s p a r t i a l l y why SORT and

SHUFL were r e ta in e d : t h i s s e c tio n d id n o t need r e w r i t in g . T h is

r e te n t io n a ls o a llow s th e u s e r to i n s e r t any s o r t in g a n d /o r s h u f f l in g

schemes d e s ir e d . As m entioned e a r l i e r , PODECKA's f l e x i b i l i t y f o r

m o d if ic a t io n i s w idened by keep ing SORT and SHUFL a c t iv e . The

s ta te m e n ts fo llo w in g th e s h u f f l in g b lo c k i n i t i a l i z e th e param eters o f

th e new f u e l . T h is com pletes th e p r e p a ra t io n o f th e co re so MAIN may

p ass c o n t ro l t o TRASIM, th e c y c le d e p le t io n s u b ro u tin e .

TRASIM d e p le te s th e co re f o r th e p re s e n t c y c le by

burnin(k ) - Burnup(k )

CYCLE - T J(J)*365#FACT0R(J)

EXPO(K) « POWER(K)*CYCLE/MTU f o r K * 1 , KMAX

BURNUP(K) = BURNUP(K)+EXPO(K)

D1(K) = POWER(K)*CYCLE

where BURNIN(K) = burnup on f u e l a t node K a t s t a r t o f p re s e n tc y c le ;

BURNUP(K) = burnup on f u e l a t node K a t EOC (new f u e lp lu s f u e l rem ain ing in c o re a f t e r d isc h a rg e a t end o f l a s t c y c le ) ;

EXPO(K) = c y c le exposure a t node K;

BURNUP(K) = EOC burnup a t node K;

Dl(K) = energy produced b y node K.

The EOC k - i n f i n i t y i s c a lc u la te d u s in g c o r r e la t io n (A 6-1).

From t h i s a c a lc u la te d k - e f f e c t iv e i s found and o u tp u t te d a s i s th e

d e s ire d k - e f f e c t iv e .

The s e t o f com putations s t a r t i n g v i t h th e p la n n in g h o riz o n

DO loop and going th ro u g h TRASIM i s re p e a te d f o r each c y c le i n th e

p la n n in g h o riz o n p lu s th r e e p o s t-p la n n in g p e r io d c y c le s p ro d u c in g th e

e q u ilib r iu m en e rg y , u s in g e q u il ib r iu m c y c le le n g th and b a tc h s iz e

o f o n e - th i r d o f th e c o r e , and s e le c t in g th e p ro p e r en rich m en t. T his

com pletes th e n e u tro n ic s and MAIN e n te r s th e c o s t in g s u b ro u tin e .

NCOST i s a s im p l i f ie d , e f f i c i e n t n u c le a r f u e l c o s t sub

ro u t in e u t i l i z i n g m ethods w hich approx im ate th e d e ta i l e d methods

employed i n th e com prehensive n u c le a r econom ics code NUCBID.

The s u b ro u tin e c a lc u la te s procurem ent and d is p o s a l c o s t s ,

in c lu d in g c a r ry in g c h a rg e s , f o r each f u e l b a tc h r e l a t i v e t o th e fu e l

lo a d in g and d isc h a rg e d a te . From th e above c o s t c a l c u la t io n s , an

e f f e c t iv e u n d isco u n ted cash flow i s developed . A n u c le a r f u e l u n i t

180c o s t i n c e n ts p e r m i l l io n B tu in c lu d in g in d i r e c t ch arg es i s c a l

c u la te d f o r each o p e ra t in g c y c le . The su b ro u tin e a l s o c a lc u la te s

a le v e l iz e d f u e l c o s t over a number o f cy c le s o f t h e p la n n in g p e r io d .

The n u c le a r f u e l u n i t c o s t i n c e n ts p e r m il l io n B tu i s

c a lc u la te d as a fu n c tio n o f f iv e n u c le a r fu e l component u n i t c o s ts

a s fo llo w s :

U C (1 ,I) a n a tu r a l uranium d if fu s io n p la n t feed m a te r ia l c o s t i n $ /kg o f c o n ta in e d uranium as UFg a t y e a r I .

U C (2 ,l) B en richm en t s e rv ic e s c o s t i n $ /kg u n i t o f s e p a ra t iv e work a t y e a r I .

U C (3 ,I) a f a b r ic a t io n c o s t i n $ /kg o f uranium lo a d e d in to th e r e a c to r a t y e a r I . (T h is in c lu d e s a l l c o s ts f o r w ithd raw al o f e n r ic h e d uranium t o d e l iv e r y o f f u e l t o th e r e a c to r s i t e . )

U C (b ,l) a re c o v e ry c o s t i n $ /kg o f uranium c o n ta in e d in th e d isc h a rg e d b a tc h a t y e a r 1 . (T h is in c lu d e sa l l c o s ts in c u r re d from t h e tim e th e sp e n t f u e li s removed from th e co re u n t i l v a lu e i s r e c e iv e d from th e re c o v e re d uranium and p lu to n iu m .)

U C (5 ,I) a f i s s i l e p lu ton ium v a lu e i n $/gm o f f i s s i l e p lu to n iu m a t y e a r I .

The v a lu e s o f t h e above u n i t c o s ts a r e in p u t a s a fu n c tio n o f I ,

where I r e f e r s t o th e i n t e g e r y e a r . P q r exam ple, 1970 would b e I a

70. T hus, th e u n i t c o s ts in c lu d e th e e f f e c t s o f p r ic e ad ju stm en t

f o r fo re c a s te d e s c a la t io n . The in p u t v a lu e s a r e th e e f f e c t iv e u n i t

c o s ts a t th e b e g in n in g o f each y e a r . As many v a lu e s as r e q u ir e d a re

in p u t t o in c lu d e th e tim e span b e in g s tu d ie d . L in e a r in t e r p o la t io n

betw een y e a rs i s tp e d . T h is p e rm its f l e x i b i l i t y i n s p e c ify in g u n i t

c o s t and th e u se o f d i r e c t acc e ss th ro u g h in d e x in g r a th e r th a n u s in g

181t a b le lo o k -u p p ro c e d u re s . F a b r ic a t io n u n i t c o s ts a r e in p u t s e p a r a te ly

f o r each r e a c to r a s U F(H R ,l), where HR i s th e p a r t i c u l a r n u c le a r u n i t

i d e n t i f i c a t i o n num ber. When th e NCOST s u b ro u tin e i s c a l l e d , th e v a lu e

o f HR i s s p e c if ie d and th e UF(NR,I) a r r a y i s t r a n s f e r r e d t o th e

U C (3 ,l) a r r a y . The p ro ced u re i s in d ic a te d a s : U C (3 ,l) * U F(H R ,I).

The f ix e d change r a t e on f u e l v o rk ln g c a p i t a l , RATE, i s

used t o c a lc u la te a l l p re s e n t w orth v a lu e s u s in g th e fo llo w in g

r e la t io n s h ip :

p re se n t w o rth f a c t o r = E X P (-D *(t-tQ))

where D = Ln(l.O+RATE);

t = tim e a t w hich payment i a made on v a lu e r e c e iv e d ;

t “ tim e t o w hich th e amount i s t o b e d isc o u n te d , o

S ince th e le a d o r la g tim e u sed f o r u n i t p r ic e s e le c t io n

may n o t r e p re s e n t t h e a c tu a l cash flow tim e s due t o th e p a r t i c u l a r

te rm s o f payment in v o lv e d , an e f f e c t iv e d is c o u n tin g p e r io d ,

PV(HR,M,Ml), i s in p u t f o r each f u e l c y c le ite m f o r each r e a c to r . These

v a lu es a r e in p u t i n months b u t a re im m ediate ly c o n v e rte d t o p r e s e n t -

v a lu e f a c to r s and a r e s to r e d in th e same a r ra y p o s i t io n as th e o r ig i n a l

in p u t a s fo llo w s: PV(HR,M,Ml) = EXP(-D*PV(HR,M ,M l)/l2.0).

Each r e a c to r has s e p a ra te l e a d tim e p a ra m e te rs , TIH(NR,M,Ml)

f o r each o f th e procurem ent and d is p o s a l s te p s c o n s is te n t w ith th e

c o n t r a c tu a l arrangem ents f o r t h a t p a r t i c u l a r r e a c to r . From th e c y c le

le n g th s and s t a r t i n g t im e , T J1 , f o r th e i n i t i a l c y c le , th e tim e s in

decim al y e a rs a t w hich th e b a tc h e s a r e i n s e r t e d , TL, and rem oved, TD,

a re d e te rm in ed . The tim e a t w hich th e u n i t p r i c e f o r f u e l c y c le

item M b a tc h H i s t o be s e le c te d i s g iv e n by TP ■ TL ■ TIH(HR,M,Ml)

162f o r f u e l procurem ent Item s o r TP » TD+TIN(1IR,M,M1) f o r f u e l d is p o s a l

i te m s . The y e a r in d ex f o r u n i t c o s t s e le c t io n i s d e te rm in ed by

t r u n c a t in g TP, a f lo a t in g p o in t number* a c ro s s th e e q u a l s ig n t o an

in te g e r a s fo llo w s : I * (TP-IYEAR+l). The p re s e n t w orth u n i t c o s t

f o r item M a t tim e TP i s th e n g iv e n by U(M) ■ [UC(M,l) + (UC(M,I+1) -

UC(M ,l)) • (TP-(lY EA R +I-l))] FV(NR,M,M1) w here th e q u a n t i ty

( IYEAR+I-1) i s th e whole number y e a r j u s t p re c e e d in g TP.

I d e a l cascade th e o ry i s u sed t o compute th e v a lu e o f

uranium a s a fu n c tio n o f i t s en richm en t u s in g a f ix e d t a i l s a s s a y .

In th e p a s t* t h i s method h a s ag re e d v i t h t h e p u b lish e d AEC's S tan d a rd

T ab les o f E n ric h in g S e rv ic e s and i s f a s t e r th a n a t a b l e lo o k -u p

p ro ced u re and r e q u ir e s l e s s c o re s to r a g e . The p re s e n t e f f e c t iv e

u n i t c o s t o f th e I n i t i a l uranium r e l a t i v e t o th e f u e l lo a d in g d a te

f o r b a tc h U i s g iv en by

PEVC « U(1 )#FR+U(2 )*WR

where FR ■ fe e d t o p ro d u c t r a t i o ■ (X1(1*)-XT)/(XP-XT);

Xl(N) * av erag e e n ric h m en t;

XT * t a i l s e n ric h m en t;

XP * fe e d en ric h m en t;

WR = s e p a ra t iv e v o rk u n i t s p e r kg o f e n r ic h e d uranium ;

WB » [FR*(VT-VF)+(VP-VT)]»ECR(NR);

where VT* VF, and VP a re th e s e p a ra t io n p o te n t ia l* fe e d stream s* and

p ro d u c t s tream s o f an .enrichm ent p la n t w hich a re g iv en a s a fu n c tio n

o f th e r e s p e c t iv e s tream enrichm ent* X:

V(X) - ( l .0 -2 .0 * X ) L n ( l .0 /X -1 .0 )

183

ECR(NR) ■ s e p a ra t iv e work c o r r e c t io n f a c to r f o r r e a c to r HR. T h is f a c t o r accoun ts f o r th e u se o f m u l t ip le en richm en ts when a p p l ic a b le .

The p re s e n t v a lu e o f th e d isch a rg ed uranium f lu o r id e r e l a t i v e

t o th e d isc h a rg e d a te i s c a lc u la te d i n a s im i la r fa s h io n .

The v a lu e o f th e f u e l in s e r t e d in to t h e r e a c to r in c lu d in g

p re -g e n e ra tio n c a r ry in g ch a rg es up t o th e tim e o f in s e r t io n i s g iv e n

by

FVCl(N) = W1(N)*[U(3)+(U(1)*FR+U(2)*(WR))»PLP(HR]

v h e re Wl(N) = m e tr ic to n s o f uranium lo a d e d ;

FLF(NR) = f a b r ic a t io n lo s s f a c to r « 1 .0 /( l .0 -F L F (H R )) .

The v a lu e o f th e d isc h a rg e d f u e l r e l a t i v e t o th e d is c h a rg e d a te i s

g iven by

PVC2(N) - W2(N)*(U(lf)-U(6)*RLP(HR))-W3(N)*U(5)*RLF(HR)

v h e re W2(N) ■ m e tr ic to n s o f uranium d isc h a rg e d ;

W3(N) * kg o f f i s s i l e p lu ton ium d is c h a rg e d ;

RLF(NR) * re p ro c e s s in g lo s s f a c to r “ l.O-RLF(NR).

The u n i t energy c o s t i n c e n ts p e r m i l l io n B tu f o r a f u e l

b a tc h in c lu d in g c a r ry in g ch a rg e s i s based on th e fo llo w in g assu m p tio n s :

1 . Each B tu h as th e same va lu e r e g a r d le s s o f when i t i s d e l iv e re d .

2 . The p re s e n t v a lu e o f a l l c o s t m ust eq u a l th e p re s e n t v a lu e o f a l l c r e d i t s and b e n e f i t s r e c e iv e d .

3 . The b e n e f i t r e c e iv e d i s th e en erg y d e l iv e re d tim es th e u n i t energy c o s t .

iT hus, th e c o s t b a la n c e e q u a tio n i s a s fo llo w s : p re s e n t-v a lu e c o s t *

p re s e n t v a lu e o f c r e d i t s + p re s e n t v a lu e o f th e en ergy tim e s th e

en ergy v a lu e . T h e re fo re , t h e u n i t c o s t i s g iv e n by

UECN(N) * (PVC1(NW1J(HNN)4PVC2(N)*FV1J(NN))/pven*i o . o )

v h e re NNN =* N Jl(N ) *= th e b e g in n in g o f th e c y c le a t v h ic h b a tc hN i s in s e r t e d ;

NN = NJ2(N)+1 = th e b e g in n in g o f th e cy c le a t v h ichb a tc h N i s d is c h a rg e d , v h e re NJ2(N) i s th e l a s t c y c le i n v h ic h b a tc h N i s u sed .

PVEN i s th e p re s e n t v a lu e o f th e en ergy d e l iv e re d by b a tc h N r e l a t i v e

t o th e s t a r tu p ( i . e . , s t a r tu p o f c y c le J l ) . The energy p roduced by

a b a tc h f o r each c y c le i s assumed t o b e u n ifo rm ly d i s t r i b u t e d over

each c y c le f o r a l l c y c le s th e b a tc h i s p ro d u c in g en erg y .

NJ2(H)PVEN = (E (N ,J)# P V 2 J(J))

J«N Jl(N )

v h e re J * c y c le number;

E (N ,J) ■ th e en erg y in 1012 B tu 's d e l iv e r e d by b a tc h N d u rin g c y c le J .

PVU(NNN) i s th e p re s e n t w orth o f 1 from th e b e g in n in g o f c y c le NNN

t o th e b eg in n in g o f c y c le J l g iven by :

P V U (J) - P V U (J-1)*FW

PW = EX P(-D *TJ(J)) .

PV2J(NN) i s th e p re s e n t w orth o f 1 u n ifo rm ly d i s t r i b u t e d o v e r cy c le

NN t o th e b e g in n in g o f c y c le 1 and i s g iv en by :

FV 2J(J) = P V U (J)* ( ( (l.0+D M RATE)**(l2.0*TJ(Jl) ) - 1 .0 ) /(12.0*TJ(J)*DMRATE*(1.0+DMRATE)«*(12.0«TJ(J)) ) )

v h e re DMRATE * pEXP (D /12 .0 ) - 1 .0 .

The u n i t q o s t o f energy d e l iv e r e d d u r in g a f u e l c y c le i s

th e w eig h ted av erag e c o s t o f th e b a tc h e s t h a t c o n tr ib u te d t h e energy

d u rin g th e c y c le :

185NJ2(J)

2 UECH(N)«E(N,J)

5 E (N ,J)N -N JI(J)

vh ere N J1(J) th ro u g h N J2(J) a r e th e b a tc h numbers o f th e f u e l b a tc h e s i n th e r e a c to r d u rin g c y c le J .

L e v e l l e d f u e l c y c le c o s t from c y c le J l l t o c y c le J22 I s

g iven b y : J22

2 UECJ(j)*PVEJ(J)J=J11 ______PC1(J) J22

2 FVEJ(J) J - J l l

vh ere FVEJ(J) i s th e p re s e n t v a lu e o f t h e energy d e l iv e re d d u rin g

cy c le J r e l a t i v e t o th e b eg in n in g o f c y c le J l l ; t h a t i s ,

N J2(J)PVEJ(J) * PV 2J(J) * 2 E (N ,J)

N*NJ1(J)

The c o s t c a lc u la t io n acco u n ts f o r incom e, s a l e s , and

p ro p e r ty ta x e s and in su ra n c e paym ents th ro u g h th e use o f an e f f e c t iv e

d isc o u n t r a t e :

i » (1-b ) i^ b i^ -k b i^

v h ere b ■ bond f r a c t i o n ;

i . ■ I n t e r e s t r a t e p e r d is c o u n tin g p e r io d on bo rro v ed mpney;

i ■ minimum r a t e o f r e tu r n on e q u i ty funds p e r d is c o u n tin gp e r io d ;

k * k .+ ( l - k . )k = canibined f e d e r a l and s t a t e income t a x8 r a t e v h e re s t a t e ta x e s a r e assumed

d e d u c t ib le b e fo re f e d e r a l ta x e s ;

186

kj. a f e d e r a l income ta x r a t e ;

k ■ s t a t e income ta x r a t e , s

A6.3 Summary

PODECKA i s a com puter code d es ig n ed to r e p re s e n t ( in a

s im p l i f ie d sen se ) th e n e u tro n ic s and co n seq u en tly th e econom ics o f a

n u c le a r power r e a c to r . In f a c t , th e m o tiv a tio n f o r developm ent was

to have an a lg o rith m which would p ro v id e a means o f d e te rm in in g w ith

re a so n a b le accu racy th e c o s ts o f o p e ra tin g a n u c le a r u n i t under

v a rio u s o p e ra t in g schemes. The c o s ts g e n e ra te d by PODECKA would be

employed in th e o b je c t iv e fu n c tio n o f a system in t e g r a t i o n m odel.

The code fu n c tio n s p r im a r i ly a s a f a s t , s im p le and e a s i l y

a p p lie d a lg o rith m g e n e ra tin g th e n e c e ssa ry is o to p ic d a ta f o r c a lc u la

t io n o f a n u c le a r f u e l c y c le c o s t . PODECKA makes u se o f t h i s d a ta to

determ ine n u c le a r f u e l c y c le u n i t c o s ts in cen ts p e r m i l l io n B tu

in c lu d in g in d i r e c t ch a rg e s . PODECKA w i l l a lso c a l c u la te a le v e l iz e d

fu e l c y c le c o s t .

The following assumptions govern the calculations of PODECKA.

1 . A ll n eu tro n le ak ag e and a b s o rp tio n may be re p re s e n te d in a s in g le c o r r e c t io n te rm a p p lie d t o k - i n f i n i t y .

2 . The power d i s t r i b u t io n rem ains c o n s ta n t th ro u g h a c y c le .

3. A psuedo one-d im ension ad e q u a te ly r e p r e s e n ts a n u c le a r r e a c to r c o re .

h. No maximum power p eak in g , b u rnup , o r ex cess k - i n f i n i t y c o n s t r a in ts a r e re q u ire d .

5. The co re may be d e p le te d th rough a c y c le in one c a lc u la t i o n w ithou t c o n s id e r in g th e power shape changes due to i s o to p ic d e p le tio n ( i . e . , assum ption l ) .

6 . The d e s ire d EOC k - e f f e c t iv e i s known.

T. The e q u ilib r iu m energy and c y c le le n g th a r e known.

8 . The n u c le a r r e a c to r r e tu r n s t o t h r e e c y c le s o fe q u ilib r iu m en erg y p ro d u c tio n and c y c le le n g th a f t e r th e p la n n in g p e r io d .

These assum ptions r e q u ir e a c e r t a in amount o f in p u t . There

a re some b a s ic v a r ia b le s w hich m ust be in p u t te d . These a r e :

1 . The EOC k - e f f e c t iv e t h a t th e c o re model u ses in making d e c is io n s a s t o en erg y p ro d u c tio n l e v e l o r b a tc h s iz e and e n ric h m en ts .

2 . The d e s ir e d en ergy t o be p roduced in each c y c le —t h i s may be a dummy v a r ia b le depending upon o p tio n s u se d .

3. The c y c le le n g th f o r each c y c le o f th e p la n n in g p e r io d .

U. The c o r r e c t io n f a c t o r t o acco u n t f o r n e u tro n le ak ag et o t r a n s l a t e k - i n f i n i t y t o k - e f f e c t iv e .

5. The e q u ilib r iu m en ergy p ro d u c tio n and c y c le le n g th .

6 . A p a s t o p e ra t in g h i s to r y o f th e r e a c to r .

7 . Economic d a ta f o r c o s t in g .

8 . C o e f f ic ie n ts f o r l e a s t sq u a re s app rox im ation t o d e term ine i s o to p ic d e n s i t i e s .

The m ain o u tp u ts o f PODECKA a re th e f u e l c y c le c o s t f o r

each c y c le i n th e p la n n in g p e r io d and a cu m u la tiv e le v e l iz e d f u e l c y c le

c o s t . A sunmary o f th e f u e l f o r eaph c y c le i n th e p la n n in g p e r io d i s

g iv e n . T h is summary in c lu d e s th e e n ric h m en t, cum ula tive burnup and

k - i n f i n i t y f o r e&cb node f o r th e b e g in n in g o f c y c le , and th e e n r ic h

m en t, cum u la tiv e b u rn u p , k - i n f i n i t y , energy p ro d u ced , power l e v e l and

f i n a l en richm en t f o r each node a t th e end o f c y c le . In a d d i t io n , a

summary o f th e b a tc h d a ta (uranium lo a d e d , i n i t i a l a s s a y , uranium

d is c h a rg e d , f i n a l a s sa y and p lu to n iu m c o n te n t) may b e o b ta in e d f o r

each b a tc h . A u n i t c o s t summary f o r each b a tc h f o r each s te p i n th e

f u e l c y c le o f a b a tc h i s g iv e n .

188

A6 .U E m p iric a l I s o to n ic and K - I n f in i ty C o r re la t io n s

The c o e f f i c i e n t s u sed in th e n e u tro n ic s o f PODECKA were

determ ined by a l e a s t sq u a re s f i t . The o r ig i n a l d a ta u sed t o g e n e ra te

th e cu rves came from LEOPARD c a lc u la t io n s . These c a lc u la t io n s were

done f o r a t y p i c a l 1 ,000 MW e l e c t r i c a l FWR. R e s u lts w ere th e n

p lo t t e d (se e F ig u re s A 6-1, A6-2, A 6-3, A6—U) f o r a ran g e o f i n i t i a l

en richm en t.

F o r each en ric h m en t, th e curve was f i t acc o rd in g t o

T * a 1 + a2b + a 3b2 (A6-12)

where T = v a lu e r e a d from c u rv e ;

a . , On* a . a r e c o e f f i c i e n t s de term ined from l e a s t sq u ares f i t ;

b = burnup (MWD/MTU) * 10-1\

A f te r do ing t h i s f o r each c u rv e , th e r e s u l t i n g c o e f f i c ie n t s w ere f i t

acco rd in g t o

a1 = a^ + a2e 1 + a^e2 (A6—13)

where e = i n i t i a l en ric h m en t;o

a . hew th e v a lu e s co rresp o n d in g t o each a d e term ined fo ra g iv en e from th e f i r s t f i t ; o

a ^ , a2 , a^ de te rm in ed from f i t .

The second f i t i s done f o r each c o e f f i c i e n t determ ined by th e f i r s t

f i t ( i . e . , a ^ , a 2 , a^ i n A 6-12). The r e s u l t i s

V ■ c^ + Cgb + c^b2

C1 3 S1 +S2eo

+ 2S3eo

C2 " 8U +s 5e o

+ s 6eo

C3 “ aT+

s 8eo+ V o

K-I

nfi

nit

y

l .U

1 .3

1.2

1.1

U.003 .753 .5 03 .2 5 3 .002 .7 52 .5 02 .2 5 2 .0 0

1.0

ItOOO 360002U000 280008000 160000 320002000012000Burnup (MWD/MTU)

F ig u re A 6-1: K - I n f in i ty v s . Burnup f o r a Sequoyah C la ss PWR — E q u ilib riu m XE and SM

00vo

Kg U/

Kg

U In

itia

l .

1.00

.99

.98

.97 -

.96

• 95

2U000 280008000 12000 16000 360000 20000 32000

Burnup (MWD/MTU)

F ig u re A 6-2: 1000 MWe PWR D esign Uranium M e t a l / I n i t i a l Uranium M etal (U/U ) f o r A l l E nrichm ents

190

Gram

s Fissile

Pu/K

g U

Initial 0 . 8 -

0.6 -

0.U

3 .3 0 Enr2 .8 0 Enr 2 .2 5 Enr0 . 2 -

l 6000 20000Burnup (MWD/MTO)

Uooo 2^000 28000 360008000 3200012000

F ig u re A6-3: 1000 Mtfe PUR D esignE n r ic h m e n t / I n i t ia l Enrichm ent ( e /e )

3 .3 0 Enr 2 .8 0 Enr 2 .2 5 Enr6

5

k

3

2

1

0 2U000 28000Uooo 16000 320008000 20000120000Burnup (MWD/MTU)

F ig u re A6-U: 1000 MWe FWR D esign — F i s s i l e P lu to n iu m / I n i t ia l Uranium M etal (Pu/U)

193where th e s ' s a r e determ ined in f i t t i n g (A6-13) to c o e f f i c ie n t s

o f (A 6-12).

Thus,

2 2 2 2 V = s ,+ s_ e +s_e + (s ,+ s_ e +8^ )h + (s_+s0e +sne )b 1 2 o 3 o 4 5 o 6 o 7 o o 9 o

The f i t t i n g i s done f o r th e d a ta re p re se n te d in F ig u re s A6-1, A 6-2,

and A6-3 as d e sc r ib e d above. F ig u re A6—U i s done in th e same

fa sh io n b u t th e form o f th e f i t t i n g eq u a tio n i s d i f f e r e n t . The

c o e f f i c ie n t s fo r each enrichm ent curve a re f i t t o

V = a1 + a 2b + a ^ lo g f l+ b ) .

The r e s u l t in g c o e f f i c i e n t s f o r each enrichm ent a re th e n f i t to

1 1 1 2 a = a l + a 2eo + a 3eo .

A6.5 C o rrec tio n F a c to rs

THERML(J) may be c o n s id e re d an e f f e c t iv e n e u tro n leak ag e

f a c t o r . THRLEK i s s e t eq u a l t o THERML(J) in th e a lg o rith m . T his

f a c to r i s used as a m u l t ip l ie r to co n v e rt th e k - i n f i n i t y o f th e c o re

to k - e f f e c t iv e w hich i s used as th e EOC s t a t e . T his f a c to r encom

p a s s e s a number o f e f f e c t s such as f a s t and th e rm a l le ak ag e and any

a x i a l and r a d ia l b u c k lin g e f f e c t s .

THERML(j) may be de te rm in ed in s e v e ra l ways. The f i r s t i s

to do an o f f - l i n e c a lc u la t io n u s in g sm other c o re model code. From

th e co re a v e r s e k - i n f i n i t y o f t h i s ru n and knowing th e EOC k - e f f e c t iv e ,

THERML(J) i s found by sim ple d iv i s io n . This method g iv e s a good

approx im ation to THERML(J). The a c tu a l v a lu e o f THERML(J) may be

a d ju s te d s l i g h t l y a s needed a s PODECKA i s u sed .

A second method i s t o s e t up a run o f PODECKA m odeling a

r e a c to r fo r which in fo rm a tio n i s on hand . THERML(J) i s in p u tte d a s

19U

a g u ess and a d ju s te d a s needed t o n o rm alize PODECKA t o th e d e t a i l e d

r e s u l t s on hand . A s t a r t i n g p o in t t h a t appears re a s o n a b le from work

done t o d a te i s THERML(J) ■ .9 ^5 .

Due t o th e n a tu re o f PODECKA, i t i s d i f f i c u l t f o r t h i s code

to m odel a r e a c to r s t a r t i n g up from a c le a n c o re . T h is r e s u l t s

b eca u se th e e a r ly c y c le s d i f f e r s i g n i f i c a n t l y from th e e q u ilib r iu m

c y c le s . In t r a c k in g th e e a r ly c y c le s i n t h i s c a s e , i t i s n e c e ssa ry

t o u se d i f f e r e n t v a lu e s o f THERML(J) f o r each c y c le . Once th e

e q u ilib r iu m c y c le s a r e re a c h e d , a c o n s ta n t THERML(J) w i l l s u f f i c e .

The n o d a l w e ig h tin g f a c to r s a re an im p o rta n t s e t o f v a lu e s

in PODECKA. These f a c to r s r e l a t e th e r e l a t i v e im portance o f each

n o d e 's c o n tr ib u t io n t o d e te rm in in g th e co re av e rag e k - i n f i n i t y . T hat i s

KMAX

2 k*, (K) * W(K)

K»1CAKINF = ------------------------------------

KMAX

w here k ^ (K) = n o d a l k - i n f i n i t y ;

W(K) = n o d a l w e ig h tin g f a c t o r ;

KMAX = number o f c o re n odes;

CAKINF = c o re av erage k - i n f i n i t y .

The u se o f th e s e v a lu e s a llo w s th e u s e r t o a s s ig n th e

im portance o f each node. The u s e r can e f f e c t iv e l y employ th e s e f a c to r s

t o com pensate f o r th e im portance o f nodes co rre sp o n d in g t o v a r ia t io n s

o f t h e power shape .

A r e l a t i v e power shape i s u sed f o r d e p le t io n i n PODECKA. A

v a lu e i s in p u t te d f o r each node. From th e n o d a l v a lu e s an a b s o lu te

shape i s d e te rm in ed .

195

Power (K) = Power(K)R el Power (K) “ KMAX Avg. Core Power F a c to r

2 Power(K)K«1

KMAX

where Power (K) = a c tu a l power f a c to r a t node K;

R el Power (K) * In p u tte d r e l a t i v e power f a c to r a t node K.

Thus*

R e la t iv e Power(K)

KMAX 2 Power(K)

K=1______________ = KMAXKMAX

2 Power(K)K»1

KMAX

where Power (K) ■ R el Power(K) f o r K » 1 , . . . , KMAX.KMAX

A6 .6 D e sc r ip tio n o f Sample Cases and Sample In p u t D ata

The d a ta u sed f o r th e sam ple c a se s i s t y p i c a l o f th e p re s e n t

c l a s s o f 1000 MW e l e c t r i c a l PWR's. The economic p a ra m e te rs , e m p ir ic a l

c o r r e l a t i o n c o e f f i c i e n t s , r e l a t i v e power sh a p e , and r e a c to r c o n s ta n tsI

a re k e p t th e same i n a l l th e sam ple c a s e s . These ca se s a re d esig n ed

t o i l l u s t r a t e th e u se o f v a r io u s o p tio n s o f PODECKA, and th e e f f e c t

w hich th e u se o f th e s e o p tio n s has on th e en erg y sc h e d u le , i s o to p i c s ,

and n u c le a r f u e l c y c le c o s ts p roduced by th e co de . A ll o f th e cases

a re compared t o th e r e s u l t s from th e NUCBID co d e , a com prehensivei

f u e l economic co d e , i n th e r e s p e c t iv e t a b l e s .

Case I : In t h i s c a s e , th e re q u ire d b a tc h s i s e and e n r ic h

ment a re in p u t te d . Two en richm en ts (and th e r e f o r e two b a tc h s iz e s )

a r e u se d . The en erg y in p u t te d i s ig n o re d by th e code . The en e rg y ,

196

i s o to p lc s and c o s ts r e s u l t in g from t h i s scheme a r e shown in T ab les

a 6—1 , A 6-3, and A6—U.

Case I I : The o n ly change made i n th e in p u t d a ta f o r t h i s

ca se from Case I , i s t o i n s e r t an average en richm en t (and th e r e f o r e

one h a tc h ) in p la c e o f th e two in Case I . B atch s iz e o f 9 and a

w eigh ted average o f th e two en richm en ts a r e shown. See T ab les

A6-1, A 6-3, and A6-U f o r th e r e s u l t in g e n e rg y , i s o to p ic s and c o s t s .

Case I I I : For t h i s c a s e , PODECKA u se s th e en ergy in p u t te d .

The energy in p u t te d i s t h a t p roduced from e x e c u tio n o f Case I . The

r e s u l t s a r e d e ta i le d in T ab les A 6-2, A 6-3, and Ab-U.

Case IV: T h is case i s i d e n t i c a l t o Case I I I , ex cep t th e

energy in p u t te d i s t h a t r e s u l t in g from Case I I .

Case V: The d a ta in p u t te d f o r t h i s c a se i s th e same as

th a t f o r Case IV , ex cep t th e o p tio n i s u sed t o in p u t a second s e t

o f r e f u e l in g tim e s t o i l l u s t r a t e e a r ly r e f u e l and ex ten d ed o p e ra t io n

(coastdow n). The new r e f u e l in g tim e s a re in p u t te d a s w e ll as th e

p la n n in g r e f u e l d a te s . The r e s u l t s o f t h i s run a re compared w ith

th o se o f Case IV in T ab les A6-5 and A6- 6 .

Case V I: T his ca se i s th e same a s Case I I , e x ce p t t h a t

new r e f u e l in g tim e s a re in p u t te d a s i n Case V. The r e s u l t s a re

compared w ith th o s e o f ca se I I .

197

Table A6-1Energy Produced by In p u tt in g B atch S iz e and Enrichm ent

Case I I NUCBIDMWD MWD

PODECKA C ycle #

Case I MWD

2 1,273,»*053 933,9631* 1 ,0 5 1 ,8 7 95 1,035,1*906 1 ,082 ,6237 1 , 088,0328 1 , 090,7689 1 ,088 ,817

10 1,089,1*3711 1 , 089,36112 1 ,0 89 ,31313 1 ,08 9 ,3 5 6Ik 1 ,0 89 ,33815 l , 06l , 090a16 l , 06l , 090a17 1 , 061 , 090*

1,273,1*05 1,275,61*991*0,117 9U8.U08

1 , 059,632 1 , 062,0101 , 011,818 1,010,1*751 , 069 , 81*2 1 , 061,8221,061* ,1*73 1 ,0 61 ,0901 , 071,051 1 , 061,0901,068,1*75 1 ,0 6 1 ,0 9 01 , 068,970 1 , 061,0901 , 069,012 1 , 061,0901 , 068 , 91*6 1 , 061,0901 , 068,976 1 , 061,0901 , 068,976 1 , 061,090l , 06l , 090a 1 ,0 6 1 ,0 9 0l ,0 6 l ,0 9 0 a 1 ,061 ,090l , 06l , 090a 1 ,061 ,090

a P o s t-p la n n in g h o rizo n c y c le s : EQEHRG u se d , b a tc h s iz e and e n r ic h m ent a llow ed to f l o a t .

198

Table A6-2B atch S iz e and Enrichm ent S e le c te d from an Energy Requirem ent

3DECKA Case I I I Case IV HUCBIDrc le # B atch U-235 B atch U-235 B atch U-235

S ize Assay S ize A ssay S iz e A ssay

2 6 3 .30 6 3 .3 0 6 3 .303 9 3 .17 9 3 .2 0 9 3 .22k 10 3.25 9 3 .27 9 3.275 10 3 .16 9 3 .37 9b 3.366 10 3 .26 9 3 .37 9 3 .3 67 10 3 .28 9 3 .37 9 3 .366 10 3 .21 9 3 .3 7 9 3 .369 10 3.27 9 3 .37 9 3 .36

10 10 3.23 9 3 .3 7 9 3 .3611 10 3.25 9 3 .3 7 9 3 .3612 10 3 .25 9 3 .3 7 9 3 .3613 10 3.21* 9 3 .37 9 3 .36ll* 10 3.25 9 3 .37 9 3 .3 615 8C 3.79 8C 3 .5 9 9 3 .3 616 8C 3.U2 8c 3.U9 9 3 .3617 8C 3.21 8c 3 .35 9 3 .36

A c tu a lly ( f o r e q u ilib r iu m c y c le s ) 23 a sse m b lie s o f 2 .8 e n r ic h m ent, k9 a s se m b lie s o f 3.65 en richm en t.

c B atch s iz e i s assumed to be o n e - th i rd o f co re i n th e l a s t th r e e c y c le s .

T ab le A6-3

F u e l C ycle C osts f o r Cases I-IV (Cents/M Btu)

PODECKA C vcle #

Case I Case I I Case I I I Case IV NUCBID

2 19.5 1 9 . k 1 9 .5 19 . k 23.73 18.7 18 .8 1 8 .9 1 8 .8 19 .5k 1 8 .6 18 .8 19 .2 1 8 .8 18 .55 18.7 1 8 .9 1 9 . u 1 8 .9 1 8 .06 18.8 1 9 .0 1 9 .6 1 9 .0 1 8 .U7 18 .9 1 9 .1 19 .7 1 9 .1 18 .78 1 9 .0 1 9 .2 1 9 .9 1 9 .2 1 9 .09 19 .1 19.33 2 0 .0 1 9 .3 19 .3

10 1 9 .2 19. k 20 .1 19.U 19.511 19 .3 1 9 .6 2 0 .3 1 9 .6 19 .612 19. k 19 .7 20. U 1 9 .7 19 .813 19 .5 19 .8 20 .5 1 9 .8 20.1Ik 19 .6 19 .8 2 0 .5 1 9 .8 20.5

200

T able A6-1*

D ischarge Uranium A ssay

I s o to p ic s fo r C ases I-IV

D ischarge F i s s i l eP lu tonium - J g .

o Case Case Case Case NUC- Case Case Case Case NUC-A I I I I I I IV BID I I I I I I IV BID

1 .96 .96 .9 6 • 96 1 .0 0 163.1* 163.1* 163.1* 163.1* 16U2 .79 .78 .79 .78 .88 196 .6 196.7 196.6 196 .7 1993 .81 .80 .81 .81 .91 ll*2 .9 ll*3.1 ll*3.0 ll*3 .1 ll*31* • 95 1 .23 1 .2k 1 .2 3 .98 6 2 .9 61*.1 81*.7 61*.1 61*5 .92 .80 .8 0 .80 .95 ll*3.2 11*0.5 116.6 11*0.5 ll*l*6 .90 1.21* 1 .2 6 1.21* • 96 6 3 .8 61*. 9 107.1* 6U. 9 61*7 .93 .80 .81 .80 .96 11*5.8 11*2.2 9^.3 ll*2 .2 11*68 .88 1 .30 1 .1 5 1 .3 0 .98 61*.2 65.5 129.6 6 5 .6 659 1 .02 .85 .71* .85 .93 11*8.7 1*3.8 93.9 ll*3 .9 ll+8

10 .85 1 .27 1 .1 9 1 .27 1 .02 6U. 6 66 .0 131.3 6 6 .0 6511 1 .0 0 .83 .7 9 .83 .93 1U9 .2 ll*l*.2 95 .0 ll*l*.2 ll*812 .81* 1 .27 1 .2 0 1 .27 1 .0 2 61*.7 66 .0 131.6 6 6 .0 6513 1 .0 0 .83 .8 0 .8U .93 ll*9 .2 ll*l*.2 95 .1 ll*l*.2 ll*8Ik .81* 1.27 1 .1 5 1 .27 1 .02 61*.7 66.1 131.1 6 6 .1 6515 1 .00 .83 .75 .83 .93 11*9.3 ll*l*. 2 9^ .5 ll*l*.3 11*816 .8U 1.27 1 2 .0 1 .27 1 .0 2 61*.7 66 .0 131.5 6 6 .1 6517 1 .00 .83 .79 .83 .93 11*9.2 ll*l*.2 95.1 ll*l*.2 11*818 .81* 1 .27 1 .1 7 1.27 1 .02 61*.7 66.1 131.2 6 6 .1 6519 1 .0 0 .83 .77 .83 .93 11*9.2 1UU.2 91*.7 ll*l*.3 ll*820 .81* 1 .27 1 .1 8 1.27 1 .02 61*.7 66.1 131.1* 66 .1 6521 1 .0 0 .83 .78 .83 .93 11*9.2 l k k . 2 9 k . 9 ll*l*.3 11*822 .81* 1 .27 1 .1 8 1.27 1 .0 2 61*.7 66.1 131 .3 66 .1 6523 1 .0 0 .83 .78 .83 .93 ll*9.2 ll*l*.2 9k.9 ll*l*. 3 ll*8

201

T able A6-5

I l l u s t r a t i o n o f Coastdovn and E a r ly R e fu e lin g

Case V Case IV Case VI Case I IPODECKA B atch U-235 B atch U-235 MWD MWDCycle # S iz e A ssay S ize Assay

2 6 3 .30 6 3 .3 0 1,273,1*01* 1,273,1*053 9 3 .20 9 3 .20 91*0,117 9^0,117hi 9 3 .27 9 3.27 971,95** 1 ,0 5 9 ,6 3 2

9 3 .19 9 3 .37 1.05U.995 1 , 011,8186<* 10 3 .2 0 9 3.37 1.11*6,655 1.069,81*27 10 3.37 9 3.37 1 ,0 22 ,163 1,061* ,1*738 9 3 .37 9 3.37 1 ,0 82 ,776 1 ,071 ,0519 9 3 .37 9 3.37 1 ,0 67 ,683 1,068,1*75

10 9 3 .37 9 3.37 1 ,067 ,993 1 ,0 6 8 ,9 7 011 9 3 .37 9 3.37 1 ,069 ,585 1 ,069 ,01212 9 3 .37 9 3 .37 1 ,068 ,768 1 , 068 , 91*613 9 3 .37 9 3.37 1 ,068 ,982 1 ,0 68 ,976ll* 9 3 .37 9 3 .37 1 ,0 6 8 ,9 8 9 1 ,0 6 8 ,9 7 6

d P o s t-p la n n in g h o riz o n c y c le s : EQEHRG u s e d , b a tc h s iz e and enrichm ent allow ed t o f l o a t .

202

Table A6-6

F u e l Cycle C osts f o r Coastdovn and E a rly R e fu e lin g(Cents/M Btu)

PODECKA Cycle #

Case V Case VI

2 19 .5 19 .53 19 .0 19 .0l»e 19.2 19 .05 1 9 .3 19 .06e 19 .2 18 .87 19.1 1 8 .98 19 .2 19 .29 19.1+ 19 .3

10 19.5 19-511 19-7 19 .612 19 .8 19 .713 19 .9 1 9 .9ll* 20 .0 1 9 .9

e P o s t-p la n n in g h o riz o n c y c le s : EQEHRG u se d , b a tc h s iz e and en richm en t a llow ed t o f l o a t .

APPENDIX 7LINEAR PROGRAMMING AND MIXED-INTEGER

PROGRAMMING SUMMARY

A7.1 Linear Programming

Linear programming is an optimization technique for a linear

function of decision variables which are constrained by a set of

linear equations and linear inequalities. There is a powerful solution

technique which can be applied to linear programs. This technique,

developed by George Dantzig and called the simplex procedure, explores

the extreme points of the convex set defined by the linear constraints.

The optimal solution which is guaranteed to be found with a finite

number of evaluations is reached when movement from an extreme point

to an adjacent extreme point does not result in an improvement to the

objective function (linear function of decision variables).

The simplex procedure starts with a basic feasible solution

using two conditions, optimality and feasibility. The optimality

condition is used to select the non-basic variable which enters the

basis (solution set). The non-basic variable is selected from the

objective function variables such that the greatest improvement will

be realized in the value of the objective function. The feasibility

condition selects the variable that leaves the basis such that the

solution remains basic feasible. The leaving variable corresponds to

203

20U

the smallest ratio of the values of the current solution (excluding

the objective function) divided by the positive constraint coeffi

cients of the entering variable.

The computation steps of the simplex procedure can be defined

as follows:

1. Express the problem in the simplex tableau.

2. Select a basic feasible solution to start.

3. Determine a basic feasible solution based on the optimality

and feasibility conditions.

An example illustrates the simplex procedure:

minimize x = -3x. - 2x„ - 5x„ o x d isuch that

x1 + 2x 2 + x^ < 1*30

3x^ + 2x^ < U6o

x1 + Hx2 < 1*20

Xj > 0 , x2 > 0 , x^ > 0 .

Starting tableau using slack variables to make inequalities equalities:

Obj. Value Decision Variables Slack Variables RIISCons

1 3 2 5 0 0 0 0

S1 0 1 2 1 © 0 0 1*30

S2 0 3 0 2 0 © 0 1*60

S3 0 1 It 0 0 0 © 1*20

where x + 3x. + 2x_ + 5x0 + OS.. + 0S„ + 0S_ = 0o 1 2 3 J- r. 1

^1* ^2* ^3 ^orm tasic feasible solution with an objectivevalue, x , of zero, o ’

Choose x^ as the entering variable as it will cause the

greatest improvement of the objective value (optimality condition):

Current basic solution Ratio of coefficientsof x^

S1 = V30 1+30/1 = 1+30

Sg = 1+60 1+60 /2 = 230

S3 = 1+20 1+20/0 = -----

S2 is leaving variable (feasibility condition).

The new tableau:

x x, x_ x_ S. S S_ SolutionBasis ----- 2------ -1----- §----- 3------- 1-----2 ----- 3-------------

1 - 9 /2 2 0 0 - 5/2 0 -1150II . ■ 1 I 1 I 1 1 I, . » ■ ■ 1 I

S1 0 -1/2 2 0 1 -1/2 0 200

X3 0 3/2 0 1 0 1/2 0 230

S3 0 1 1+ 0 0 0 1 1+20

Choose Xg as the entering variable and

Current basic solution Ratio of coefficientsof x2

200/2 = 100

230/0 = ---

1+20/1+ = 105

is the leaving variable.

The new tableau becomes:

S1 « 200

X3 = 230

S = 1+20

206

x xn x_ x_ S, S_ S_ S o lu tio nB asis 2 1 2 1 2 1 _ --------------

_______________1______ -It 0 0______ -1 -2 __ 0 -1350

X2 0 -iA 1 0 1/2 -l/H 0 100

X3 0 3/2 0 1 0 1/2 0 230

S3 0 2 0 0 2 1 1 20

Exam ining th e ta b le a u shows th a t no o th e r v a r ia b le w i l l improve

th e o b je c t iv e v a lu e s in c e a l l th e c o e f f i c ie n t s f o r s e le c t io n o f a

new e n te r in g v a r ia b le axe n e g a tiv e ( th e m in im iza tio n case and v ic e

v e rs a fo r m a x im iza tio n ). T h e re fo re , th e o p tim a l s o lu t io n i s :

XoS -1350

X1= 0

X2 = 100

X3 = 230

S1= o, s2

A7.2 M ix ed -In teg er Programming

M ix e d -in te g e r programming i s u sed to o b ta in th e o p tim a l

s o lu t io n t o a l i n e a r program w hich c o n ta in s in te g e r v a r ia b le s . There

a r e s e v e ra l methods f o r m ix e d -in te g e r programming such as Gomory's

c u t t in g p la n e , b ranch -and -bound , and B a la s 1 ze ro -o n e a lg o rith m . The

method d e s c r ib e d and u sed in t h i s work i s th e b ranch-and-bound method.

The problem i s f i r s t so lv ed as a l i n e a r program a llo w in g th e

in te g e r v a r ia b le s t o be con tin u o u s between zero and one ( f o r t h i s

w ork). I f t h e in te g e r v a r ia b le s a r e a l l i n t e g e r s , th e o p tim a l s o lu

t i o n i s fo u n d . I f a p a r t i c u l a r in te g e r component i s no t z e ro o r one,

207

two linear programs are solved. Figure A7-1 is used to illustrate

the method and is referred to in this discussion. After obtaining

the continuous optimal solution Zq=H, the first non-integer is selected,

1" 1 SSt ^ rst zero the linear program solved with thisconstraint. This results in an objective function value Z^=10.5.

X^ is then set to one and Z^=8.5. Tlie method selects the minimum of

the two Z^'s and branches from this node setting another integer, X ,

to zero and then one solving two more linear programs. In this illus

tration, X,>=0 is chosen and Z^=9> Branching off this node gives

Z^=19 and Zy=l8. Since the method has already found a node better

than these values, the method retraces its path bade to 3 =8.5.

Selecting X^=l branch from this node to Z =10, the branches XÔ

and X^=l are evaluated giving infeasible solutions. The method

retraces its path to find the previous best solution, the path X =0,

ZÎO.5, recalling that this branch was initally abandoned.

Pursuing this branch, the tree that is generated is XÔ,

3 =11; X^=l, Z =12; X =l, Z^=12 is abandoned and X^=0, Z^=13 selected;

X^=0, Z^=lU; and Xgr=0, ?-g=17. This tree results in a feasible mixed-

integer solution of 17. This value and tree is stored and the branch-

and-bound algorithm searches for the best node in the tree which has

not been explored. In this illustration the branch from Z^ll where X,,=0 is selected and pursued. Each branch is pursued, i.e., Z^

branch, until an infeasible solution is reached, the value of the

objective function exceeds the best integer solution found, or all

branches of the tree have been exhausted and optimality of a mixed-

208Cont, Opt

Z =10.5

Z.=10 Z =9

Z =20I n f I n f

=12

Z. =12

Z7=l6I n f .

<Z_«1U Z =22 I n f

FeasS o l.

I n fI n f

In f

F ig u re A7-1: I l l u s t r a t i o n o f a Branch and Bound Tree

integer solution proved.

'Phis method is computationally time consuming because it

can be called an implicit direct search. The major consumption of

time lies in the proof of optimality of the mixed-integer solution.

However, this is one of the best general purpose, mixed-integer

algorithms available for use.

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