Go inge , Hans W.
A icle
Dynamic Economic S a egies o B eede Reac o
Commi men s
Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de
Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik
P o ided in Coope a ion wi h:
Duncke & Humblo , Be lin
Sugges ed Ci a ion: Go inge , Hans W. (1985) : Dynamic Economic S a egies o B eede Reac o
Commi men s, Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de
Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik, ISSN 0342-1783,
Duncke & Humblo , Be lin, Vol. 105, Iss. 1, pp. 1-15,
h ps://doi.o g/10.3790/schm.105.1.1
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Dynamic Economic S a egies
o B eede Reac o Commi men s
By Hans W. Go inge
This app oach allows he use o analyze he e ec o policy decisions
upon he cos o elec ici y. The o wa d dynamic p og amming app oach is
adap ed o an economic assessmen o a nuclea eac o mix o allow many
e minal condi ions o be analyzed.
In pa icula , when he commi men his o y can be aced o an op imal
pa h, he li e o each uni may be compu ed and a e i emen cos may be
a ibu ed o each e minal s a e o e lec he loss incu ed by e mina ion
o a Ligh Wa e Reac o due o una ailabili y o uel be o e he planned
li e ime o he plan has been ul illed.
1. In oduc ion
In a p e ious essay, Go inge (1982), we ha e p o ided a cos -bene i
analysis o he Ligh Wa e B eede Reac o (LWBR) Sys em based on
a s a ic assessmen o uel cycle cos s. F om he poin o iew o gi en
nuclea ene gy choices, in his essay we look a dynamic aspec s o
phasing in and phasing ou nuclea echnologies in a way ha op imize
decisions o commi men s in hese echnologies. Such commi men
s a egies could cons i u e a a ional app oach o in e empo al choice
o echnologies in he case o unce ain y, Hammond (1976).
A comp ehensi e dynamic analysis o b eede eac o commi men s
in he case o he Liquid Me al Fas B eede Reac o (LMFBR) has p e-
iously been gi en by A. S. Manne (1974). In con as o he LMFBR we
mus conside he LWBR as a 'concep ual eac o '. In iew o he hypo-
he ical esou ce use si ua ion desc ibed by Go inge (1982), he ques-
ion is whe he an ad anced con e e on a Tho ium-232/U-233 uel
cycle (e.g. he p eb eede ) p oduces U-233 a a g ea enough a e o
make he LWBR sys em a mo e a o able echnology han he Ligh
Wa e Reac o (LWR). Since Tho ium-232 is se e al imes mo e abun-
dan han U anium pu suing he Th-U-233 uel cycle has a mul iplying
e ec on a ailable esou ces.
The uel cycle cos assumed o he b eede should be speci ied wi h
some ca e. While he b eede has a lowe uel cycle cos han ei he he
1 Zei sch i i Wi scha s- und Sozialwissenscha en 1985/1
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2 Hans W. Go inge
LWR o he p eb eede , he cos o ep ocessing is a signi ican con-
ibu o o he b eede uel cycle cos . Unde wha scena ios hen will
he LWER possess a o able economic p ope ies? I would appea ha
he main dis inc ion be ween he LWBR and he LWR is he li e ime
cons ain which applies o he LWR. In an en i onmen o se e ely
limi ed u anium a ailabili y, he penal y associa ed wi h he ea ly
e mina ion o LWR's as gene a ing uni s may be su icien o jus i y
he added on -end expense o addi ional uel cycle cos o es ablish
a non- e minable nuclea elec ic economy.
2. A Dynamic Cos Model
Conside a LWR ene gy sys em composed o h ee eac o ypes.
De ine he pe inen pa ame e s as ollows:
Nj ( ) is he numbe o eac o s o ype j,
j = 1,2,3 in place a ime i,
Vj
is he cos coe icien pe eac o o ype
Cj (i) is he cos a ime due o eac o s o ype j,
ej is he en ichmen equi emen o a gene a ing uni o ype
E is he o al capaci y o en iching plan s.
Cons ain : Assume ha en ichmen capaci y is sa u a ed. Tha is,
ha su icien eac o s a e buil so ha no u he en iching capaci y is
a ailable.
(1) UWje^E
3=1
Type 3 is assigned o he ligh wa e b eede ; hus es = 0 and
(2) S ^ le^ 2
ZV;.
( ) e, = E
j=i j=i
Objec i e Func ion: The discoun ed o al cos C a ai, is assumed o be
minimized, w i en
(3) Minimize Q 0 al = /
e
- 2 ^ ( ) d
o j = l
whe e
oc
is he discoun a e.
This is in he adi ion o o mula ing he in e empo al alloca ion
p oblem.1
Conside now he b eede -p eb eede ela ionship on he basis o he
Tho ium-232/U-233 uel cycle. Assume no e i emen o p eb eede s.
i Ha ele (1975).
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Dynamic Economic S a egies o B eede Reac o Commi men s
The numbe o b eede s which may be suppo ed a he ac i a ion poin
is some unc ion o he numbe o p eb eede -yea s which ha e accu-
mula ed. Assuming a cons an y kg U-233 pe p eb eede uni , he num-
be o b eede s which may be suppo ed is
(4)
N3 (í)
=
~~~~ y
1N2( )d
a 0
whe e R is he U-233 equi emen , in kg, o one b eede uni . The
uppe limi is
—
1, assuming one yea o p ocessing and ab ica ion
o U-233 uel.
Now (i) may be assumed o ha e he o m ( ) = / [1 — e~a ],
which is a con enien bu also ealis ic assump ion on he sa u a ion
p ocess up o he en ichmen capaci y. This unc ion is easily in eg a ed
o yield
(5) / 1N2( )d = /
*
/[1 - e-«] d =//
*
d - V^d
0
/ 1N2( )d = /(i-l)-/
0 — Q - ax
a
- 1 + e-a( -1)
a
Now om he cons ain equa ion
^ i -D+^le-a«-U-l]
0 ß
(6)
Subs i u ing hese ela ionships in o he objec i e unc ion,
(7) C oUl = / e-« U- [E -
(1
-
e° )
e2] +
(1
- e-" ) 2
o I
(8)
C oui
= /- "»! ( )e~* dí + /
0 el 1
+
(1
- e-a ) 2
(í)
+
y
s
{ )
(í - 1 + ± e-a (í-l) - -ij
e-*
d
Now i is seen ha i is cons an , he objec i e unc ion may be w i -
en
(9) C 0M = ^- 1{ )e-* d + T e-*
o o
-- - 1 ( )(i-e-«9
+ (1 _
e-a )
2
(Í)
+ s
(Í)
( - 1 + — e-a (i-1) - —
R Q>
&
(10) C^-IjiTJ+ZI^T)
d
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4 Hans
W.
Go inge
When ega ded as a unc ion o he objec i e unc ion is minimized
by choosing / = 1 i h (T) is nega i e, o by selec ing / = 0 i h (T) is
posi i e.
Now in gene al he commi men pa e n o p eb eede s is no selec -
ed in ad ance. The e o e, i is desi ed ha he ime-dependen mix o
eac o s [Ni (£),
N®
( ),
IV3
( )] be choosen o minimize he objec i e unc-
ion independen ly o any pa icula algo i hm. The ela ionship be-
ween he numbe o b eede s and he p eb eede his o y is gi en, om
equa ion (4), by
(11) N3( )=-^N2( -l)
i
whe e he do deno es di e en ia ion wi h espec o ime.
F om he cons ain equa ion,
(12) N1(i)e1 + -y-JV3(i + l)e2 = E
(13) 1^(0=-^-E-yNs( + l)e2
Subs i u ing his in he cos unc ional
(14) P o oi-/«--E
( )
- -y-
W3
(i +
1) e2) Uj
(i) + -y Nz ( +
1)
2 ( )
+
N3 ( )
u3 (i) d
R
W i ing y ( ) = — N3 ( ) and expanding y ( + 1) in a Taylo se ies
abou y (i), equa ion (14) becomes, a e unca ing,
(15) C o al = / e- l±-
[E
( ) - y ( ) - y
(*)]
( )
0
I
ei
+
i e
(0 l
(*)
+ l l
*»)}
^
(Fo he sake o ealism may e e o any sui able ime uni ma ch-
ing he ac ual ' uel cycle u no e ime'.)
Now he cos s i ( ), (i) and s
( )
a e dependen upon he com-
mi men his o y, including capi al in es men , since UsOs-p ice may be
modelled as a unc ion o cumula i e consump ion.
An e ec i e me hod o he a ack o p oblems whe e cons ain s
limi he ange o admissible unc ions is ha o dynamic p og amming.
In he case o p eb eede commi men , such an app oach is ac able
since he abili y o he nuclea indus y o suppo he U-233-Th uel
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Dynamic Economic S a egies o B eede Reac o Commi men s 5
cycle is limi ed by lack o ep ocessing acili ies. La ge capi al equi e-
men s o cons uc ion o ep ocessing plan s would no mally be as-
sumed o ac as a limi ing in luence upon he a e o g ow h o ep o-
cessing acili ies.
As a means o allowing o conside a ion o limi a ions upon p oduc-
ion capabili y, he Dynamic P og amming o mula ion o he minimiza-
ion p oblem allows he inpu o a bi a y limi s upon he sys em mix.
These cons ain s e lec assump ions upon he capaci y o he nuclea
se ice sec o o mee needs a ising om a ious p opo ions o ligh
wa e eac o s, p eb eede s, and b eede s. Al e na i ely, hey could
e lec policies adop ed by ei he he Go e nmen o indus y g oups o
suppo ligh wa e b eede eac o de elopmen . This me hod o a ack
yields an e ec i e app oach o he minimiza ion p oblem de ined by
equa ions (1) and (3).
3. Dynamic Fo mula ion o Reac o Plan Selec ion
I is desi ed o de e mine he op imal pa h o eac o commi men
based upon ime-dependen alues o capi al cos , ope a ing cos , and
uel cycle cos . The s age a iable in his analysis is ime; he s a e is
he ec o consis ing o he eac o s o a ious ypes.
(16) x = [Xl X2,Xz]
whe e
Xn
is he commi men o eac o ype n.
The con ol
u
applied a ime
is he ec o o plan addi ions. Fo
con enience, he elemen s o he x and u ec o s shall be aken o be
GW
(e) o nuclea -elec ic addi ions. The analysis may be expanded o
include plan s o o he ypes such as coal- i ed and na u al gas acili ies.
We adop cons ain s o help educe he compu ing ime o he p ob-
lem a hand. In he i s case, we a e limi ed by u anium o e a ail-
abili y. Thus, he numbe o u anium-consuming eac o s is limi ed.
Tha is, he yea ly consump ion o u anium mus be consis en wi h he
capaci y o he indus y o p o ide i . A second cons ain conce ns
a ailabili y o en iching se ices. The sepa a i e wo k equi ed in any
yea mus be less han o equal o a ailable o p ojec ed a ailable
capaci y. Fuel ab ica ion capabili y may also be employed as a con-
s ain . Howe e , i is gene ally aken o be a simpli ying assump ion
ha ab ica ion capabili y is no a limi ing es ic ion. I is p obably
mo e p ecise o say ha en iching se ices a e a mo e es ic i e con-
s ain han is he a ailabili y o uel ab ica ion acili ies. Rep o-
cessing a ailabili y will be a cons ain o some eac o ypes bu no
o o he s.
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6 Hans W. Go inge
The s anda d Ligh Wa e Reac o s, he PWR and he BWR, a e no
en i ely dependen upon a closed loop uel cycle o hei ope a ion. On
he o he hand, he Ligh Wa e B eede Sys em is dependen upon
ep ocessing capabili y, and i should be obse ed ha a a ional deci-
sion-make would no commi his company's esou ces o a b eede
unless he we e ce ain o ep ocessing a ailabili y. The a gumen s may
be ex ended o include a ious measu es o social cos and ad e se
en i onmen al e ec s as elemen s o cos compu a ion o cons ain o -
mula ion. The only equi emen in such a case is ha one has quan i i-
able ela ionships be ween cos s and bene i s, o mo e p ecisely, be-
ween he measu able impac s and he esul ing componen o cos .
Such an unde aking is equen ly impossible when dealing wi h ma -
e s ela ing o social cos s, and e en when such a emp s a e made, he
esul s may no be accep ed by he o he schola s in he a ea. As a sim-
pli ica ion o a dynamic p og amming app oach, i is he case he e ha
he social cos s a e aken o be equal o all eac o ypes.
We wish o minimize he o al discoun ed cos o socie y due o he
ins alla ion o nuclea -elec ic powe . I we deno e by ( ) he o al
cos o powe o socie y a ime i, we may use Bellman's p inciple c
op imali y o es ablish he i e a i e equa ion o Dynamic P og amming
(17) ( ) = Min [k ( ,
)
+ a
( ) A
+ ( +
A
)]
Reducing his o he disc e e o m
(18) ( ) = Min
[7c
( ) + a( )A + exp (- <x ) ( + A )]
whe e
k
( )
is he cos o adding new uni s
a ( ) is he ou pu a e o he cu en uni s.
The discoun ac o is he ecip ocal o (1 + i), whe e i is he in e es
a e. Fo quan ized and uni alue o A , e-« equals (1 + i)-1. This
equa ion says ha , i one knows he ideal combina ion o uni s o gen-
e a e powe om yea + 1 o T, he e minal yea in he analysis, hen
i is desi ed o de e mine he ideal combina ion o uni s which akes
he sys em om he beginning o he inal yea . Now i is o be obse ed
ha he unc ion { + A ) is a eal- alued unc ion o he ec o x, as
we a e ul ima ely a emp ing o de e mine he alues o he s a e a i-
able x and he con ol a iable k as well as ha o he minimum cos .
Cons ain s upon he p oblem may assume a numbe o o ms, and in
he simple analysis o be pe o med ini ially i shall be aken ha he
cons ain s be o mula ed in e ms o s a e- ec o quan i ies a he
han con ol ec o quan i ies. Deno ing he en ichmen cons ain by
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Dynamic Economic S a egies o B eede Reac o Commi men s 7
e ( ), a eal quan i y, and he a e age en ichmen equi emen o each
eac o by he ec o E, he i s cons ain equa ion may be w i en
E' X <e (£) whe e he p ime deno es anspose. Simila ly, i he ec-
o u deno es U3O8 p oduc ion equi ed o suppo each eac o ype, he
o mula ion
LT
X <,u ( ) may be used o speci y he u anium p oduc-
ion cons ain . The basic equa ion o he eac o deploymen model is
(19) x ( + 1) = x
( )
+ k
( )
which s a es ha he capaci y dis ibu ion a ime + 1 is ha a ime
plus any addi ions which migh occu be ween and + 1. The e is one
u he cons ain , an equali y cons ain , which s a es ha he o al
capaci y a ailable in pe iod mus be equal o he demand schedule
d ( ). Fo n plan ypes
(20) 2 Xi( ) = d( )
i = 1
Now he con ol ec o k (i) may assume an unlimi ed numbe o
ep esen a ions, since he mix o eac o ypes may be con inuously
ep esen ed.
4. Exempli ica ion o he Dynamic P og amming App oach
A s age-inc emen echnique can be de eloped o sol e he op imiza-
ion p oblem u ilizing echniques o Dynamic P og amming. The un-
damen al equa ion sugges s ha we may apply each con ol o all exis -
ing s a es in o de o de ine he s a es a he succeeding s age. Addi-
ional assump ions a e equi ed o educe he numbe o s a es kep
du ing he sea ch o an op imum in he dynamic p og amming al-
go i hm.
Two app oaches we e combined o yield a ac able solu ion. The i s
is a unnel cons ain , he o he is ela ed o he quan iza ion o he
s a es hemsel es. Whe e a calcula ed s a e is iden ical o an exis ing
s a e a some s age , he p inciple o op imali y equi es ha he s a e
ha ing minimum cos be e ained.
A s a e is assumed o be a slowly- a ying unc ion o i s pa ame e s,
i may be assumed he a nea by s a e is e lec i e o he same p ope ies
as a gi en s a e x. I mus be bo ne in mind ha he s a e x is an o de ed
n- uple which, o he case o he LWBR sys em, includes as one o i s
componen s he numbe o p eb eede -yea s. The e o e, wo s a es
which a e close o each o he in pa icula ha e he same o nea ly he
same numbe o p eb eede -yea s, and hence ha e gene a ed app oxi-
ma ely he same quan i y o U-233. The me ic u ilized o es ablish
closeness is he sum o he absolu e alues o he de ia ions. In gene al,
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8 Hans
W.
Go inge
one migh use d (x, y) = 2
I
xi
— Hi I
as he dis ance unc ion o he
i = l
e alua ion o closeness o wo s a es. Fo he sys em consis ing o he
3
LWBR and he LWR, he equa ion educes o d(x, y) = 2 xi
—
Ui -
i = i
Gi en
e
> 0 and a poin y, one may say ha x is in a neighbo hood o y
i d (x, y) <
e.
Fo he dynamic p og amming p oblem, all quan i ies
a e in ege s and he e o e he neighbo hood should be speci ied in e ms
o an in ege . The subs i u ion c i e ion he e o e becomes d (x, y)<K
whe e K is an in ege . Fo ini ial dynamic p og amming s udies, K has
been aken o be equal o 2. This app oxima ion may esul in p op-
aga ed e o , since he algo i hm is ha i d (xj, xn) < K, j = 1,..., n
—
1
hen he s a e wi h minimum cos is e ained and he o he one is dis-
ca ded. This is a sequence dependen p ocedu e. Obse e, o example,
ha he sequence o iples (12,10,11), (12,10,12), (12, 10,13), (12,10,14)
will esul in he s o ing o he single s a e (12, 10, 14), while he same
ou s a es in he o de (12,10,11), (12,10,14), (12,10,13), (12,10,12) will
esul in he e en ion o s a es (12,10,14) and (12,10,12).
E en wi h he imposi ion o cons ain s and simpli ying app oxima-
ions, he p ocedu e o inding an op imal ajec o y o e a sui able
planning pe iod, say wen y yea s, is a o midable ask. The p oblem
may hen be scaled o a manageable size in o de o yield, in a eason-
able amoun o ime, a solu ion which will bea some esemblance o he
op imal pa h o he mo e elabo a e o m. Fo example, a s age may be
aken o be wo yea s ins ead o one, and powe addi ions may be
assumed o be in inc emen s o wo GW (e) a he han one. I would be
desi able o in es iga e he e ec o policy cons ain s upon he cos
o powe , o mo e app op ia ely, he cos o nuclea -elec ic powe . I
is no necessa y o u ilize absolu e cos alues; ela i e cos s may be
used o in es iga e he economic p ope ies o he eac o sys em.
5. Compu a ional Me hod
I is necessa y o de e mine whe he he e will be a ne bene i o
socie y de i ing om he exis ence o he ligh wa e b eede eac o
sys em. I has p e iously been no ed ha he esea ch and de elopmen
cos o he LWBR sys em a e an icipa ed o be small ela i e o hose
o a eac o ype no in p oduc ion. Thus, i is qui e alid o compa e
he ligh wa e b eede sys em wi h he ligh wa e eac o sys em,
since i may be assumed ha p oduc ion capaci y and ope a ional
cha ac e is ics a e simila in he wo cases. I is desi ed ha quan i ica-
ion be made o he discoun ed cos di e en ial be ween ligh wa e
eac o s and he ligh wa e b eede sys em. To de e mine he e ec
o his cos di e en ial, a Dynamic P og amming app oach is employed.
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Dynamic Economic S a egies o B eede Reac o Commi men s
ene gy-gene a ing cos calcula ion as a a ia ional p oblem. The implemen a-
ion o hese concep s in compu a ional o m is done ia dynamic p o-
g amming. An algo i hm is de eloped in which he numbe o he eac o
ypes and powe equi emen s a e assumed and he op imal plan com-
mi men schedules a e gene a ed o any se o hypo hesized economic
condi ions. The applica ion o his algo i hm o he sys em con aining Ligh
Wa e Reac o , P eb eede s and B eede s is made and cos s a e gene a ed.
Zusammen assung
In E wei e ung eine bishe du chge üh en s a ischen Kos en-Nu zen-
Analyse au den Fall eine dynamischen Ressou cenalloka ion ü ein
nuklea es Ene giesys em wi d in diesem Bei ag ein Ansa z o geschlagen,
de die Be echnung eine op imalen Reak o kombina ion als die Lösung
eines Va ia ionsp oblems mi Hil e de dynamischen Op imie ung e möglich .
Re e ences
Conaes (Commi ee on Nuclea and Al e na i e Ene gy Sys ems) (1980),
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cisco,
Go inge , H. W. (1982), The Economics o B eede Reac o s wi hin a Nuclea
Ene gy Regime. Angewand e Sys emanalyse 3 (4), 167 - 175.
Hä ele, W. (1975), Objec i e Func ions. II AS A Wo king Pape . Schloß Laxen-
bu g, Aus ia, 75 - 125.
Hammond, P. J. (1976), Changing Tas es and Cohe en Dynamic Choice.
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Manne, A. S. (1974), Wai ing o he B eede , Symposium on Exhaus ible
Resou ces. Re iew o Economic S udies 41, 47 - 65.
Na ional U anium Resou ce E alua ion (1976), P elimina y Repo , Repo
GJo-III (76). U.S. Ene gy Resea ch and De elopmen Adm. (ERDA).
G and Junc ion, Colo.
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DOI h ps://doi.o g/10.3790/schm.105.1.1 | Gene a ed on 2023-04-04 12:07:10