Who bea s he isk? Analyzing he s a egic
in e ac ion be ween egula o s and in es o s when
se ing incen i es o enewable elec ici y
Peio Alco a Iglesias
MASTER IN ECONOMICS: EMPIRICAL APPLICATIONS AND POLICIES
Uni e si y o he Basque Coun y UPV/EHU
Facul y o Economics and Business
Ad iso s: Ma ia Paz Espinosa & C is ina Piza o-I iza
July 30, 2020
Acknowledgmen s
Fi s o all, I would like o hank p o esso s Ma ia Paz Espinosa and C is ina Piza o-I iza o he
a ious oppo uni ies ha hey ha e o e ed me beyond he w i ing o his hesis.
Now, on a much deepe and pe sonal le el, I wan o hank you Od a. You suppo has been essen ial
du ing he e y complica ed and s ange mon hs in which his hesis has been ca ied ou .
You a e he eason why hese las ou mon hs ha e been bea able.
Thank you om he bo om o my hea .
Abs ac
Ene gy policies o p omo ing in es men in enewable ene gy sou ces ha e become c ucial o deploying
di e en g een ene gy echnologies. Depending on hei design, he con en ional incen i e sys ems assign
he isk o ei he he policymake o he in es o , a ec ing he s a egic in e ac ion be ween hem when
se ing a p ice o he subsidy. Mo eo e , Feed-in Ta i s, which we e he p incipal subsidy scheme used
in Spain, we e emo ed in 2013, mainly because hei design led o an unbea able de ici . Fa ell e al.
(2017), combining op ion p icing heo y and game heo y, p opose an incen i e sys em o I ish Feed-in
Ta i s in which bo h pa ies would sha e he isk. Building on his app oach, we de elop a me hodology o
e alua e di e en op imal incen i e schemes o Spain and p esen an applica ion o 2013 and 2019. We
pe o m an ex ensi e nume ical analysis o de e mine how he di e en p oposals would wo k o Spain.
Keywo ds: Renewable Ene gy; Feed-in Ta i ; E icien Policy; G een Ene gy; Ene gy Policy;
Ene gy Economics; Op ion P icing.
Con en s
1 In oduc ion 1
2 Ma hema ical Model 4
2.1 P elimina ies ........................................... 4
2.2 Elec ici y ma ke p ice model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Game heo e ic app oach o wind ene gy in es men model . . . . . . . . . . . . . . . . . 8
2.4 Expec ed p ice o di e en FiTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 NoSubsidy ........................................ 10
2.4.2 FixedTa i ........................................ 11
2.4.3 Cons an P emium.................................... 11
2.4.4 Sha edUpside ...................................... 12
2.4.5 Cap&Floo ....................................... 13
3 Nume ical Applica ion 14
3.1 Es ima ion o how he d i and ola ili y o he VWAP depend on he ins alled wind powe 15
3.2 Calib a ion ............................................ 19
3.3 Resul s............................................... 19
3.3.1 Resul s o he yea 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.2 Resul s o he yea 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Sensi i i yAnalysis........................................ 28
3.5 Discussion............................................. 29
4 Conclusions 30
Re e ences 32
Appendix A: Sol ing Equa ion 10 i
Appendix B: Cons uc ion o he da ase i
Appendix C: Minimum o Maximum? xi
1 In oduc ion
Nowadays, one o he bigges conce ns wo ldwide is o achie e a sus ainable and clean ene gy egime.
To a g ea e o lesse ex en , mos de eloped coun ies wo ldwide ha e se he goal o igh ing global pol-
lu ion and s opped elying on ossil uel ese es, inc easingly limi ed and esponsible o he emissions
o pollu ing gases in o he a mosphe e. Al hough signi ican ad ances ha e been made in his ega d,
he e is s ill a long way o each a ully g een and sus ainable ene gy scena io. Since he deploymen o
Renewable Ene gy Sou ces (RES) is s ill mo e cos ly han ha o o he con en ional sou ces, a public
subsidy is o en equi ed o c ea e a easible and appe izing in es men en i onmen .
The e exis many suppo schemes o incen i ize g een ene gy pene a ion. These schemes include, o
example, Renewable Po olio S anda ds (i.e., a mechanism ha se s an obliga ion on elec ici y suppli-
e s o p oduce a speci ied ac ion o hei elec ici y om enewable ene gy sou ces). These Renewable
Po olio S anda ds migh be accompanied by T adable G een Ce i ica es, which a e adable asse s
p o ing ha elec ici y has been gene a ed by enewable ene gy sou ces. G een ce i ica es a e issued
o RES p oduce s, who can ade hem o con en ional ene gy supplie s so hey can ul ill he quo a
es ablished in he Renewable Po olio S anda d. In exchange, enewable supplie s ecei e ex a e enue,
and he ma ke inds he mos e icien way o mee hese goals. Ano he suppo scheme can be pa ial,
o e en ull exemp ion om some axes and le ies o g een ene gy p oduce s. Howe e , Feed-in Ta i
(FiT) schemes ha e become he p e e ed enewable ene gy suppo mechanism in many ma ke s, as hey
p o ide g ea e ce ain y o emune a ion o in es o s. Ne e heless, hei main d awback is he huge
cos s hey usually in ol e o he egula o , especially i hey a e no p ope ly designed. Tha is he main
eason why hey we e abandoned in Spain in 2013 [1].
The e a e wo a ian s o Feed-in Ta i policy schemes ha ha e been widely used, pa icula ly in
Spain: a ixed FiT, and a Feed-in P emium (FiP). A ixed FiT is a mechanism allowing he RES p oduce s
o sell he elec ici y hey supply a a ixed p ice o a speci ic pe iod. In a FiP scheme, he paymen ha
RES supplie s ecei e is based on a cons an p emium o e ed abo e he ma ke -clea ing p ice. Un il 2013,
Spanish RES p oduce s had he op ion o choose be ween ixed FiT and FiP subsidy schemes. Indeed, he
ixed a i has been he mos widely used FiT design. Howe e , he FiP has been inc easingly u ilized in
Spain, mainly o onsho e wind. Since July 2013, he Spanish go e nmen supp essed bo h ixed FiT and
FiP subsidies, being enewable ene gy auc ions he cu en mechanism used o g een ene gy p omo ion
[2]. The bene i s o FiT subsidies o enewable ene gy pene a ion ha e been o signi ican impo ance,
and hey may s ill play a ole as long as hey a e ca e ully edesigned. Indeed, app oxima ely 64% o
oday’s global wind powe has been p omo ed h ough his ype o mechanism [3].
Figu es 1 and 2 show how bo h schemes wo k. The yellow line ep esen s he e olu ion o he elec ic-
i y ma ke p ice, and he g een line shows he e olu ion o he e enue pe MWh ha a enewable ene gy
in es o ecei es unde each policy. The e o e, he g een a ea ep esen s he o al cos o he subsidy o
he policymake , whe eas he o al a ea (g een+yellow), co esponds o he o al p o i s ha he in es o s
will ecei e. As we can see in Figu e 1, a ixed a i egime emo es in es o exposu e o low ma ke
p ices, being he policymake he one who bea s he isk o ma ke p ice a iabili y. On he con a y,
cons an p emium policies emo e he policymake ’s isk. As shown in Figu e 2, unde a FiP scheme,
he g een a ea is independen o he ma ke p ice [4]. Since he p emium (deno ed as Xin he igu e)
is independen o he s ochas ic ma ke p ices, he policymake has ce ain y abou he cos pe MWh
ha he public subsidy will en ail when designing he policy. Howe e , unde a FiP scheme, in es o s a e
exposed o he ull impac o ma ke p ice luc ua ions. I is a well-documen ed ac ha he e ec i eness
o he FiTs has been a ibu ed o he educed isk hey en ail o in es o s [5]; he e o e, ans e ing all
he isk o hem migh esul coun e p oduc i e, leading o he conclusion ha adequa e managemen
o hese isks is o capi al impo ance when designing public subsidy policies o p omo e RES pene a ion.
1
Figu e 1: Fixed Feed-in Ta i scheme Figu e 2: Cons an Feed-in P emium scheme
In 2011 he e we e app oxima ely 21,000 MW o wind capaci y deployed in Spain. Tha same yea ,
he Spanish go e nmen se he a ge o achie e a o al ins alled capaci y o 35,750 MW by he yea 2020
(Plan de Ene g´ıas Reno ables 2011-2020) [6]. As Figu e 3 shows, un il FiT subsidies we e elimina ed in
2013, he le el o new ins alled capaci y was on he igh ack, bu om hen on, i s opped ab up ly.
Figu e 3: E olu ion o he ins alled wind capaci y in Spain (sou ce AEE [7])
The esea ch ques ion ha we wan o answe in his hesis is whe he incen i e schemes ha sha e
ma ke p ice exposu e be ween egula o s and in es o s can imp o e upon he usual FiT and FiP. We
p o ide an analy ical speci ica ion o incen i e s uc u es ha sha e ma ke p ice exposu e be ween eg-
ula o s and in es o s and p esen an applica ion o he Spanish ma ke . Indeed, ou esul s indica e
ha he e a e isk-sha ing incen i e schemes ha domina e FiT and FiP: hey allow eaching he same
in es men le el a a lowe cos . Ou esul s ha e in e es ing policy implica ions o u u e enewable
ene gy egula ion: incen i e schemes ha e o be ca e ully designed, aking in o accoun isk sha ing.
Following he app oach p oposed by Fa ell e al. [8], we design subsidy schemes ha sha e ou
he isks associa ed wi h he s ochas ic na u e o elec ici y p ice luc ua ions among bo h in es o s and
policymake s. In his con ex , we analyze schemes based on FiT bu inco po a ing adjus able deg ees o
ma ke p ice exposu e o in es o s and egula o s. Ou analysis will depa om he assump ion ha
he s ochas ic e olu ion o he annual elec ici y p ice ollows a andom walk ha can be sa is ac o ily de-
sc ibed as a Geome ic B ownian Mo ion (GBM) p ocess. Consequen ly, he e olu ion o he payo ha
a enewable ene gy in es o ecei es will ollow he Black-Scholes equa ion bu wi h a di e en e minal
condi ion, which will depend on he pa icula design o he FiT in place. E en hough FiTs a e no
inancial asse s, no can hey be bough o sold, he Black-Scholes equa ion’s appea ance is due o he ac
ha he payo s o he co esponding subsidy will depend on he s ochas ic elec ici y ma ke p ice. This
ac allows us o ob ain analy ical solu ions o he model, which p o ides a pa h o e icien combina ions
2
o he FiT pa ame e s. By choosing di e en e icien combina ions o hese pa ame e s, we can manipu-
la e bo h policymake ’s and in es o ’s isk exposu e le els. Accu a ely quan i ying and dis ibu ing hese
isks can be an essen ial s imulus o incen i ize g een ene gy pene a ion. This is he main con ibu-
ion o Fa ell e al. (2017), who p oposed hei me hodology o a easible I ish RES deploymen scena io.
In a nu shell, ou me hodology can be summa ized as ollows:
•Fi s , o e e y FiT design unde conside a ion, we sol e he s ochas ic model o ind analy ical
solu ions o he e olu ion o each policy scheme’s expec ed payo s and cos s.
•Then, ollowing Fa el e al. (2017), we model he isk-sha ing FiT design p oblem as a s a egic
leade game, whe e he policymake (leade ) akes in o accoun he s a egic esponse o he in-
es o s ( ollowe s). Conside ing he expec ed e olu ion o payo s and cos s p e iously calcula ed,
he egula o chooses he op imal isk-sha ing FiT pa ame e s ha incen i ize he desi ed quan i y
o RES deploymen .
•Nex , we pe o m a nume ical analysis o de e mine how hese p oposals would wo k in Spain o
wind ene gy deploymen . We calib a e he model o he Spanish elec ici y ma ke in 2013 ( he
las yea wi h FiTs), and 2019 ( he las yea wi h a ailable da a).
•Since some o he needed pa ame e s o he Spanish ma ke do no appea in he li e a u e, we
ha e o es ima e hem on ou own. In pa icula , he pe cen age d i and ola ili y o he annual
olume-weigh ed a e age p ice (VWAP) o wind elec ici y, which depends on he amoun o wind
capaci y ins alled. Fo ha pu pose, we build a da ase wi h he hou ly ene gy p oduc ion by ech-
nology and he hou ly ma ching p ice o elec ici y, o e e y hou om 01/01/2014 o 31/12/2019.
•Wi h he es ima ed pa ame e s, we simula e po en ial elec ici y p ices e olu ion acco ding o ou
model, ob aining how he di e en p oposals would wo k o Spain.
•Finally, we pe o m a sensi i i y analysis, changing se e al pa ame e s and measu ing he impac
o hose a ia ions on he ele an p edic ions.
3
2 Ma hema ical Model
In his sec ion, we p esen a model o designing op imal p icing ules o he di e en subsidy schemes
we s udied. We d aw oge he he sepa a e ields o FiT policy design, game heo y, and s ochas ic
inancial calculus.
2.1 P elimina ies
One o he co ne s ones in he s udy o inancial ma ke s is he so-called E icien Ma ke Hypo hesis.
Al hough he e exis se e al di e en o mula ions o his hypo hesis, all o hem sha e he ollowing basic
ea u es [9]:
1) Ma ke s espond immedia ely o any new in o ma ion abou an asse p ice. Co ec ions o p ices
a e ins an aneous, lea ing he pa icipan s no oom o a bi age.
2) Asse p ices e lec all a ailable in o ma ion. Consequen ly, i is impossible o “bea he ma ke ”
consis en ly on a isk-adjus ed basis, since ma ke p ices should only eac o new in o ma ion.
One o he ini ial pu poses o his hypo hesis was o p o ide a gumen s in a o o he easibili y o
he Random Walk Conjec u e, which s a es ha ma ke p ices e ol e acco ding o a andom walk (p ice
changes canno be p edic ed). I is easy o see ha i he Random Walk Conjec u e holds, hen, as a
di ec consequence, he E icien Ma ke Hypo hesis mus hold.
Suppose ha a ime , he asse p ice is deno ed by S . A e an in ini esimal ime in e al d , he
asse p ice will change by an amoun dS . The mos common way o cha ac e ize he co esponding
change on he asse p ice is o decompose i in o wo di e en con ibu ions:1
dS =a(S , )d +b(S , )dW (1)
On he one hand, he e is a de e minis ic con ibu ion gi en by a unc ion a(S , ). On he o he hand,
we ha e a andom con ibu ion o he asse p ice change in esponse o unexpec ed ex e nal e ec s, gi en
by a de e minis ic unc ion b(S , ), and he di e en ial dW . The e m dW con ains he andomness,
and i is modeled as a Wiene p ocess. A andom p ocess {W }T
=0 ={W0, W1..., WT}is a Wiene p ocess
when he ollowing p ope ies a e me [10]:
•i) W ∼N(0, )−→ W0= 0
•ii) {W }T
=0 has independen inc emen s: P(W −Ws|Ws) = P(W −Ws)
•iii) Fo 0 ≤s< : (W −Ws)∼N(0, −s)
whe e N(x, y) ep esen s he no mal dis ibu ion wi h mean xand a iance y. F om hese p ope ies,
i ollows ha a Wiene p ocess is also a Ma ko p ocess.2Since we a e usually mo e in e es ed in he
ela i e change o an asse p ice han in i s absolu e change, i may be some imes mo e con enien o
exp ess Eq.(1) as
1In p ac ice, i is e iden ha he ime in e als we handle when analyzing a s ochas ic p ocess, a e always disc e e.
Howe e , he con inuous o mula ion is usually made o analy ical con enience. Once we ha e sol ed he model o
con inuous ime, he disc e e app oxima ion can always be made by aking app op ia e ime in e als.
2A sequence o andom a iables {X } o ms a Ma ko chain (o p ocess) i :
P(X +1 =x|X0, ..., X ) = P(X +1 =x|X ); ha is, i gi en he p esen , he u u e and he pas o he sequence a e
independen . In o he wo ds, i all he in o ma ion o he pas his o y o ha a iable is cap u ed in he p esen s a e [11].
4
dS
S
=µ(S , )d +σ(S , )dW (2)
I he unc ions µ(S , ) = µ∈R, and σ(S , ) = σ > 0 a e cons an , we ob ain a Geome ic B ownian
Mo ion p ocess (GBM) [12]:
dS
S
=µd +σdW (3)
I is easy o see ha he expec ed alue and he a iance 3o he LHS in Eq.(3) a e gi en by:
E[dS /S ] = µd , and V a [dS /S ] = σ2V a [dW ] = σ2d . The coe icien s µand σ, a e usually called
he d i and he ola ili y o he p ocess, espec i ely. Once we ha e cha ac e ized he GBM, i is ime o
s a e bo h a undamen al heo em and one o i s co olla ies, which will allow us o handle he andomness
in he model.
Theo em. (Feynman-Kac): Conside he s ochas ic di e en ial equa ion (1):
dS 0=a(S 0, 0)d 0+b(S 0, 0)dW 0(4)
Le h(·)be a unc ion o he s ochas ic a iable S 0. Le 0∈[0, ]be gi en, and he inal ime > 0
ixed. De ine he expec ed alue o h(S )o e he pe iod [0, ]as
ρ(S, 0) = E[h(S )|S 0=S] (5)
Then, ρ(S, 0)sa is ies he ollowing pa ial di e en ial equa ion:
∂ρ
∂ 0+a(S, 0)∂ρ
∂S +b(S, 0)2
2
∂2ρ
∂S2= 0 (6)
oge he wi h he inal condi ion
ρ(S, ) = h(S),∀S(7)
P oo : See Sh e e (2004) [13].
Equa ion (6) is known as he Kolmogo o Backwa d Equa ion.4When we know o su e ha he sys-
em will be in a ce ain s a e h(S ) a some ce ain ime in he u u e ( 0= ), he Kolmogo o backwa d
equa ion desc ibes he p obabili y o being in a s a e Sa an ea lie ime ( 0< ) [14].
Co olla y. Conside he s ochas ic di e en ial equa ion (3) desc ibing he Geome ic B ownian Mo ion:
dS 0=µS 0d 0+σS 0dW 0(8)
Le h(·)be a unc ion o he s ochas ic a iable S 0, and le o be a gi en cons an . Le 0∈[0, ]be
gi en, and he inal ime > 0 ixed. De ine he expec ed discoun ed alue o h(S )o e he pe iod [0, ]
as
(S, 0) = E[e− ( − 0)h(S )|S 0=S] (9)
Then, (S, 0)sa is ies he ollowing pa ial di e en ial equa ion:
3F om he hi d p ope y o he Wiene p ocess, i ollows ha : dW = lim
∆ ↓0{W +∆ −W }=W +d −W ∼N(0, d )
4The Kolmogo o Backwa d Equa ion is a di usion ype pa ial di e en ial equa ion ha a ises in he heo y o
con inuous- ime Ma ko p ocesses. E en hough we a i ed a his equa ion om he Feynman-Kac heo em, i was i s
s udied by he g ea Russian ma hema ician And ey Kolmogo o [15], way be o e he Feynman-Kac o mula was in oduced
by he ma hema ician Ma k Kac, and he g ea heo e ical physicis Richa d Feynman while he was s udying pa h in eg als
in quan um mechanics [16].
5
FB, (S ) = X(31)
We ind ha he expec ed discoun ed p o i s and cos s a ime a e gi en by
e− E[PB, ] = e− (X+S0eµ ) (32)
e− E[FB, ] = Xe− (33)
Following he same p ocedu e, we ind he op imal solu ion o he cons an p emium X:
X(op )=
C−
TF
X
=1 S0e(µ− ) ∂µ
∂QG +∂G
∂Q Q=QI
T1
X
=1
e− ∂G
∂Q Q=QI
(34)
2.4.4 Sha ed Upside
The payo s ha an in es o ecei es a ( ≤T1) a e desc ibed by
PC, (S ) = max{KC, ω(S −KC) + KC}=(KCS < KC
ω(S −KC) + KCKC≤S
(35)
whe e ω∈[0,1] ep esen s he sha e o he ma ke upside ecei ed by he in es o , and KC ep esen s
he p ice loo . Unde his scheme, he policymake ’s cos s will be
FC, (S ) = max{KC−S ,0}+ (ω−1) max{S −KC,0}=(KC−S S < KC
(ω−1)(S −KC)KC≤S
(36)
I ωhappened o be equal o one, hen we would ha e a FiT unde which he in es o cha ges he
ma ke p ice, bu ha ing a gua an eed minimum p ice, in case he ma ke p ice is lowe han his loo .
On he con a y, i ωhappened o be ze o, we would ob ain a ixed FiT wi h a ixed p ice KC. As be o e,
we ind he expec ed discoun ed p o i s and cos s a each pe iod.
e− E[PC, ] = e− KC(1 −ωΦ(d2)) + ωS0eµ Φ(d1)(37)
e− E[FC, ] = e− KC−S0eµ +ω(S0eµ Φ(d1)−KCΦ(d2))(38)
whe e Φ is he cumula i e dis ibu ion unc ion o he s anda d no mal dis ibu ion, and d1and d2
a e de ined as ollows:
d1(K, ) =
log S0
K+µ+σ2
2
σ√ d2(K, ) =
log S0
K+µ−σ2
2
σ√ (39)
12
we ind he condi ion o a easible op imal combina ion o KCand ω:
K(op )
C(ω) =
C−
T1
X
=1 ωΦ(d1(K(op )
C, ))S0e(µ− ) ∂µ
∂QG +∂G
∂Q Q=QI−
TF
X
=T1+1 S0e(µ− ) ∂µ
∂QG +∂G
∂Q Q=QI
T1
X
=1 1−ωΦ(d2(K(op )
C, ))e− ∂G
∂Q Q=QI
(40)
As i can be seen in equa ion (40), since bo h d1and d2depend on KC, we canno sol e he equa ion
o KCexplici ly. Gi en a ce ain alue o ω, he p e ious implici equa ion may be sol ed by nume ical
me hods. By pe o ming an i e a i e p ocedu e, he solu ion inally con e ges o he desi ed op imal
alue K(op )
C. Con e sely, gi en a alue o KC, we could edo he p ocedu e o ob aining an op imal
alue o ω(op ). I could be p o ed, and we will see i la e in ou p ac ical s udy, ha equa ion (40)
desc ibes a unique locus o e icien pai s o KCand ω, wi h a single e icien ω o each alue o KC(and
he o he way a ound). Mo eo e , i could be p o ed ha he e is an in e se ela ionship be ween hese
op imal alues o ωand KC.
2.4.5 Cap & Floo
The payo ha an in es o ecei es a ( ≤T1) a e desc ibed by
PD, (S ) = max{KD,min{S ,C}} =
KDS < KD
S KD≤S < C
C C ≤S
(41)
whe e KDand C ep esen he p ice loo and cap, espec i ely. Unde his scheme, he policymake ’s
cos s will be
FD, (S ) = max{K−S ,0}+ max{C−S ,0}=
KD−S S < KD
0KD≤S < C
C−S C≤S
(42)
As always, we ind he expec ed discoun ed p o i s and cos s a each pe iod:9
e− E[PD, ] = e− KD(1 −Φ(d2)) + S0eµ (Φ(d1)−Φ(d3)) + CΦ(d4)(43)
e− E[FD, ] = e− KD(1 −Φ(d2)) −S0eµ (1 −Φ(d1)) −S0eµ Φ(d3) + CΦ(d4)(44)
whe e d3and d4a e de ined as ollows:
d3(C, ) =
log S0
C+µ+σ2
2
σ√ d4(C, ) =
log S0
C+µ−σ2
2
σ√ (45)
P oceeding as in he p e ious case, he condi ion o a easible op imal combina ion o KDand C:
9Fo simplici y, when compu ing all he de i a i es o bo h Sha ed Upside and Cap & Floo schemes, we neglec he
a ia ion o Φ (di(µ(Q))) wi h espec o Q, whe e i∈ {1,2,3,4}.
13
K(op )
D(C) =
C−
T1
X
=1 S0e(µ− ) Φ(d1(K(op )
D, )) −Φ(d3(C, )) ∂µ
∂QG +∂G
∂Q Q=QI
T1
X
=1 1−Φ(d2(K(op )
D, ))e− ∂G
∂Q Q=QI
+
+
−
T1
X
=1 Ce− Φ(d4(C, ))∂G
∂Q Q=QI−
TF
X
=T1+1 S0e(µ− ) ∂µ
∂QG +∂G
∂Q Q=QI
T1
X
=1 1−Φ(d2(K(op )
D, ))e− ∂G
∂Q Q=QI
(46)
Again, i is no possible o explici ly sol e he p e ious equa ion o KD. Ins ead, o a gi en alue o
he cap C, we can sol e he implici equa ion (46) i e a i ely un il i con e ges o he op imal solu ion
K(op )
D. Con e sely, gi en a alue o KD, we could sol e i o ob ain an op imal alue o C(op ). In any
case, equa ion (46) desc ibes he locus o e icien combina ions o loo and cap. As in he sha ed upside
egime, he e is an in e se ela ionship be ween hese wo policy pa ame e s. A lowe loo implies a
highe e icien cap, and he o he way a ound. As a consequence, he e will be an in e se ela ionship
in in es o ’s and policymake ’s exposu e o ma ke p ice a iabili y.
3 Nume ical Applica ion
We apply he ma hema ical model de eloped in he p e ious sec ion o analyze how each o he a i s
unde conside a ion would wo k in Spain in he yea s 2013 and 2019. We ha e chosen hese wo yea s
because, on he one hand, 2013 was he las yea whe e he go e nmen suppo ed RES deploymen wi h
FiTs, and on he o he hand, 2019 is he las yea wi h a ailable da a. Fi s o all, in o de o analyze
each o he wo di e en scena ios, we ha e o calib a e he se e al pa ame e s o he model. In some
cases, we ind he co esponding alues in he li e a u e, bu in many o he cases, we ha e o es ima e
hem ou sel es. The p ice we mus pay o ob aining analy ical solu ions is ha we mus igno e some
echnical issues. Fo ins ance, as we men ioned in sec ion 2.2, elec ici y p ices ha e o be lowe han
e180.3/MWh by law. Howe e , his equi emen canno be included in he analy ical model, and because
o ha , and due o he high alue o he d i µwe will use, he e is a good chance ha in he las pe iods,
his equi emen does no hold. Thus, we mus cla i y ha gi en some o he assump ions and echnical
limi a ions p esen in he me hodology, he objec i e o he ollowing nume ical analysis is no o ob ain
exac igu es ega ding he bene i s and cos s ha each a i would en ail; his exe cise would equi e
a much mo e ho ough and me iculous analysis, and abo e all, wi h much mo e a ailable da a. Ra he ,
his exe cise’s p ima y pu pose is o show how he in e media e schemes beha e compa ed o he usual
FiT in he scena io ha we y, wi hin he limi a ions, o be simila o he Spanish ma ke .
In he case o µ, we a e no only in e es ed in a pa icula alue o he annual pe cen age d i o
he VWAP, bu a he on i s unc ional o m depending on he o al amoun o capaci y deployed. As
we men ioned ea lie , his is a key ea u e on ou model, since µand hus, also i s de i a i e ∂µ
∂Q
a e endogenous o he in es o ’s op imiza ion p oblem. In hei s udy, Fa el e al. es ima ed he
co esponding unc ional o m in he case o I eland using he s udy p e iously done by Dohe y and
O’Malley (2011) [22], in which hey es ima ed he p ojec ed u u e wind weigh ed ma ke p ices in I eland
as a unc ion o he deployed wind capaci y. Since he e is no such s udy done o Spain, we de ise a
di e en me hodology in o de o es ima e µ(Q). I is wo h men ioning ha Fa el e al. ea ed σas
a cons an ; howe e , in o de o be mo e igo ous, we also es ima e how he ola ili y depends on wind
ene gy pene a ion. As we will see, he es ima ed unc ions µ(Q) and σ(Q) a e almos , bu no pe ec ly,
linea , a leas in he ange o wind deploymen we a e in e es ed in. Fo ha eason, we es ima e bo h
unc ions assuming a second-o de polynomial unc ional o m in Q=Q0+QI.
14
3.1 Es ima ion o how he d i and ola ili y o he VWAP depend on he
ins alled wind powe
Fo he es ima ion o hese wo unc ions, as well as he es ima ion o he VWAP, we use a me hodology
de ised by ou sel es collec ing as quan i ies o da a and pe o ming some eg essions in R. We include
he R code used o cons uc he da ase in Appendix B. We use da a om Red El´ec ica de Espa˜na
(REE) and OMIE. The da a om REE includes he amoun o ene gy gene a ed by each echnology in
MWh: wind, sola , hyd aulic, nuclea , coal, combined cycle, es o special egime (biomass, enewable
he mal, e c.), uel/gas, in e na ional exchanges and Balea ic bond, o e e y single hou anging om 01-
01-2014 o 12-31-2019 (one ile pe day) [23]. Ou da ase also includes he ma ching p ice o elec ici y a
e e y single hou (p icei), which is aken om OMIE [24] (one ile pe day), and he nominal wind powe
ins alled up o a gi en yea (powe i), which is aken om AEE [7]. The aim is o ge an es ima ion o he
annual d i (µ) and ola ili y (σ) as a unc ion o he ins alled wind powe . Le us deno e o each yea
( ) he p edic ed annual VWAP by p . We a e conside ing disc e e annual imes eps, hus: d ≈∆ = 1.
Hence, ecalling he p ope ies ollowed om equa ion 3 o he expec ed alue and a iance, we ge :
E[dp /p ] = µ, and V a [dp /p ] = σ2, espec i ely. Thus, he GBM pa ame e es ima ion is made as
ollows [25]:
µ(Q) = 1
5
2019
X
=2015
p (Q)−p −1(Q)
p −1(Q)(47)
σ(Q) =
u
u
1
4
2019
X
=2015 p (Q)−p −1(Q)
p −1(Q)−µ(Q)2
(48)
Fi s , we need o es ima e he VWAP (p ) as a unc ion o he o al wind powe (Q). As a i s s ep,
we pe o m a simple linea eg ession model o he hou ly elec ici y p ice using he 52,584 obse a ions
in ou da ase (one o each hou ).
p icei=β0+β1windi+β2sola i+β3hyd aulici+β4nuclea i+β5coali
+β6combinedi+β7speciali+β8 uelgasi+β9exchangesi+β10balea i
+
24
X
j=2
αjhou (j)
i+
7
X
m=2
χmday(m)
i+
12
X
l=2
ζlmon h(l)
i+
2019
X
k=2015
ξkyea (k)
i+εi
(49)
whe e {hou (j)}24
j=1 is a se o dummy a iables indica ing he hou o he day co esponding o he
obse a ion: o example, hou (2)
i akes he alue one i he obse a ion ico esponds o he hou 2:00,
and ze o o he wise. In he same way, {yea (k)}2019
k=2014 indica es he co esponding yea : o example,
yea (2019)
i akes he alue one i he obse a ion ibelongs o he yea 2019 and ze o o he wise, and so
on. Simila ly, {mon h(l)}12
l=1 indica es he mon h, and {day(m)}7
m=1 indica es he day o he week (1=
Monday, ... , 7=Sunday) co esponding o each obse a ion. The ob ained esul s a e shown in Table 2.
As we can see, almos e e y a iable is s a is ically signi ican , and in mos cases, wi h p- alues lowe
han he wo king p ecision o he machine. Ano he hing o no e is ha ou OLS model explains 77.3%
o he a ia ion o ou dependen a iable (R2= 0.773). As we can ell om he esul s, he e is an
in e se ela ionship be ween he hou ly p ice and he wind ene gy gene a ed. Tha is no su p ising since
we al eady discussed he me i -o de e ec [26]. The same applies o sola p oduc ion. On he o he
hand, a peak hou s, p oduc ion om con en ional sou ces is highe , and so a e elec ici y p ices. Wi h
he es ima ed coe icien s, we can ge a p edic ion o he hou ly p ices (
p icei).
15
Now, le us suppose ha o a gi en obse a ion, ins ead o ha ing he co esponding wind powe
ins alled a ha momen (powe i), we ha e a di e en gi en powe (Q1), while e e y hing else e-
mains unchanged. Then, he only di e ence would be in he wind ene gy p oduc ion a ha hou
(windi). Fo he sake o simplici y, and because we a e no ying huge wind powe di e ences o he
new capaci y, assuming a linea ela ionship wi h he co esponding new gene a ion migh be jus i ied:
windi(Q1)≈Q1
powe iwindi(powe i). Tha is, i seems easonable o expec ha i he wind capaci y Q1
had been, o example, a 10% highe han he ac ual capaci y powe i, hen he wind p oduc ion would
also ha e been a 10% highe . Using he pa ame e s we es ima ed wi h he eg ession model in Eq.(49),
we p edic o an a bi a y ins alled wind capaci y Q1, an hou ly p ice o e e y obse a ion in ou da ase :
p icei(Q1) = b
β0+b
β1
Q1
powe i
windi+b
β2sola i+b
β3hyd aulici+b
β4nuclea i+b
β5coali
+b
β6combinedi+b
β7speciali+b
β8 uelgasi+b
β9exchangesi+b
β10balea i
+
24
X
j=2 bαjhou (j)
i+
7
X
m=2 bχmday(m)
i+
12
X
l=2 b
ζlmon h(l)
i+
2019
X
k=2015 b
ξkyea (k)
i
(50)
Once we ob ain he p edic ed hou ly p ices
p icei(Q1) o a gi en capaci y o ou choice, he nex
s ep is o ob ain he co esponding annual olume-weigh ed a e age p ice p (Q1):
p (Q1) = X
yea ( )
i=1
p icei(Q1)·windi
X
yea ( )
i=1
windi
(51)
Q µ(Q)σ(Q)
10000 0.07128004 0.2756571
12500 0.07185035 0.2770892
15000 0.07242749 0.2785360
17500 0.07301158 0.2799975
20000 0.07360274 0.2814742
22500 0.07420110 0.2829661
25000 0.07480679 0.2844736
27500 0.07541993 0.2859969
30000 0.07604066 0.2875363
32500 0.07666911 0.2890919
35000 0.07730542 0.2906641
37500 0.07794975 0.2922531
40000 0.07860222 0.2938592
Table 1
Then, we ob ain µ(Q1) and σ(Q1) applying equa ions (47)
and (48), espec i ely. I we epea he p ocedu e ying di -
e en alues o he new wind capaci y, we can ob ain lis s
o alues {µ(Qλ)}λ, and {σ(Qλ)}λ om which we can make
a nonlinea i o es ima e i s unc ional o m, a leas , in he
ange o Qin which we a e in e es ed. Fo example, he in-
e al Q∈[10000,40000] migh be easonable as an in e al o
in e es . We y di e en alues belonging o his in e al. The
alues Q ha we used and he esul s o µ(Q) and σ(Q) a e
shown in Table 1. Since he alues we ob ain in bo h cases sug-
ges unc ional o ms app oxima ely linea , ollowing Taylo ’s
heo em, including e ms up o second o de may be enough o
success ully app oxima e bo h unc ions. F om hese esul s,
we es ima e µ(Q) and σ(Q) as second o de polynomials in
Q=Q0+QI:
µ(Q) = 0.069125 + 2.09513 ·10−7Q+ 6.83117 ·10−13Q2(52)
σ(Q)=0.27021 + 5.31872 ·10−7Q+ 1.47851 ·10−12Q2(53)
Plo ing he ob ained alues shown in Table 1, oge he wi h he unc ional o ms ha we es ima ed
in Eq.(52) and Eq.(53), we see ha we go a e y nice i in bo h cases.
16
Figu e 6: Adjus men o µ(Q)
Figu e 7: Adjus men o σ(Q)
As we can obse e in Figu e 6, al hough he hou ly p ices, and hence he annual VWAP dec ease as
he ins alled wind capaci y inc eases (being consis en wi h he me i -o de e ec ), he annual d i o
he VWAP is an inc easing unc ion o he deployed wind capaci y. The same hing applies o he annual
ola ili y, as shown in Figu e 7.
•No e: I is impo an o cla i y ha he de elopmen o he me hodology we ha e jus discussed has
se e al signi ican limi a ions. The mos no able one migh be he small numbe o yea s used o de e mine
he pa ame e s. This is due o he ac ha he a ailable da a in which he gene a ion by echnology
is comple ely b oken down begins in 2014. On he o he hand, ou ini ial pu pose o de e mining how
he amoun o ins alled wind ene gy a ec s wholesale elec ici y p ices was o simula e he daily ma ke
ma ching. As we ha e desc ibed in sec ion 2.2, he ma ching o he ma ke p ice ollows a de e minis ic
me hod, knowing all he daily sale and pu chase o e s, and iden i ying which o hose o e s come om
wind uni s; we would ha e a me hod ha would allow us o eliably de e mine he p ice o he daily
ma ke depending on he amoun o he ins alled wind capaci y. Un o una ely, he codes ha iden i y
he wind uni s in OMIE and REE do no coincide a all. The e o e, his al e na i e me hodology canno
be ca ied ou . I his changes in he nea u u e, he pa ame e de e mina ion p ocess would imp o e
subs an ially.
17
Table 2: Resul s o he linea eg ession model
Dependen a iable:
p ice
wind −0.000233∗∗∗
(0.000019)
sola −0.000563∗∗∗
(0.000046)
hyd aulic 0.000920∗∗∗
(0.000023)
nuclea 0.002260∗∗∗
(0.000054)
coal 0.004138∗∗∗
(0.000024)
combined 0.001001∗∗∗
(0.000024)
special 0.001450∗∗∗
(0.000040)
uelgas 0.013194∗∗∗
(0.000535)
exchanges 0.001011∗∗∗
(0.000029)
balea 0.002703∗∗∗
(0.000858)
yea 2015 4.399542∗∗∗
(0.140081)
yea 2016 0.602026∗∗∗
(0.132652)
yea 2017 9.986042∗∗∗
(0.138793)
yea 2018 17.950020∗∗∗
(0.130002)
yea 2019 18.105060∗∗∗
(0.163003)
hou 2 −1.955861∗∗∗
(0.219181)
hou 3 −3.481786∗∗∗
(0.222104)
hou 4 −4.102226∗∗∗
(0.223917)
hou 5 −4.699547∗∗∗
(0.224572)
hou 6 −4.336743∗∗∗
(0.223236)
hou 7 −2.927000∗∗∗
(0.218837)
hou 8 −1.196431∗∗∗
(0.218881)
hou 9 −0.788441∗∗∗
(0.224914)
hou 10 0.290636
(0.238500)
hou 11 0.813615∗∗∗
(0.257489)
hou 12 0.787607∗∗∗
(0.271186)
hou 13 0.892097∗∗∗
(0.279696)
hou 14 0.860672∗∗∗
(0.281255)
Obse a ions 52,584
R20.773808
Adjus ed R20.773572
Residual S d. E o 7.193956 (d = 52528)
F S a is ic 3,267.271000∗∗∗ (d = 55; 52528)
No e: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01
Dependen a iable:
p ice
hou 15 0.300588
(0.275495)
hou 16 −0.664138∗∗
(0.267760)
hou 17 −1.039772∗∗∗
(0.258639)
hou 18 −0.270835
(0.248545)
hou 19 0.871813∗∗∗
(0.244286)
hou 20 1.696003∗∗∗
(0.244304)
hou 21 1.849687∗∗∗
(0.246245)
hou 22 2.021346∗∗∗
(0.243832)
hou 23 1.257100∗∗∗
(0.230010)
hou 24 −0.197932
(0.220333)
day2 −1.125314∗∗∗
(0.118240)
day3 −1.367910∗∗∗
(0.118420)
day4 −1.163725∗∗∗
(0.118053)
day5 −0.814275∗∗∗
(0.117670)
day6 0.324673∗∗∗
(0.123201)
day7 −0.290607∗∗
(0.134090)
mon h2 −5.677778∗∗∗
(0.160314)
mon h3 −2.855130∗∗∗
(0.169959)
mon h4 −1.656027∗∗∗
(0.177717)
mon h5 2.699987∗∗∗
(0.190722)
mon h6 1.451458∗∗∗
(0.182082)
mon h7 −1.613603∗∗∗
(0.182219)
mon h8 −0.050674
(0.182902)
mon h9 −0.143878
(0.169652)
mon h10 4.706360∗∗∗
(0.169246)
mon h11 3.971814∗∗∗
(0.176132)
mon h12 2.020612∗∗∗
(0.158199)
Cons an −1.410498∗∗
(0.597689)
Obse a ions 52,584
R20.773808
Adjus ed R20.773572
Residual S d. E o 7.193956 (d = 52528)
F S a is ic 3,267.271000∗∗∗ (d = 55; 52528)
No e: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01
18
3.2 Calib a ion
The ins alla ion a ge (QI) is aken as he di e ence be ween he objec i e se by he Spanish go -
e nmen and he o al powe ins alled up o ha yea (Q0): in 2011, hey es ablished he objec i e o
35.75 GW o 2020 in Plan de Ene gias Reno ables (PER) [6], while he objec i e o 50.33 GW o 2030
is de ined in Plan Nacional In eg ado de Ene g´ıa y Clima (PNIEC) [27]. On a e age, wind u bines
a e ope a ional o 20 yea s [28]. The du a ion o he ixed-FiT and cons an -FiP gi en by he Spanish
go e nmen was 20 yea s as well [29]. We ob ained he VWAP o he ini ial yea (S0) wi h he da ase
we used o he me hodology we ha e jus explained in 3.1, bu expanding i , in o de o include 2013
oo (indeed, we included e e y hou om 2007 o 2019). Fo a gi en yea , we ook he amoun o wind
ene gy gene a ed a e e y single hou o ha yea , and he ma ke ma ching p ice o ha hou . Then,
each hou ’s p ice is weigh ed acco dingly o he amoun o wind elec ici y gene a ed a ha same hou .
The capi al and ope a i e cos s o onsho e and o sho e wind ene gy a e di e en . In gene al o sho e
wind- a ms a e much mo e expensi e bo h o build and o main ain han he onsho e ones. Fo ha
eason, we es ima e Aas a weigh ed a e age o he onsho e and o sho e wind capi al cos s, weigh ed
acco ding o he espec i e p opo ions o onsho e and o sho e deploymen on he ins alla ion a ge QI.
We ollow he same p ocedu e o es ima ing he ope a i e cos s. Fo he discoun a e ( ), we use he
Weigh ed A e age Cos o Capi al (WACC). Gene ally, he isk associa ed wi h ene gy p ojec s which a e
inanced by he go e nmen is conside ably lowe han hose which a e en i ely p omo ed by he p i a e
sec o . Thus, he discoun a e used in each o he wo cases is usually di e en . WACC, which is a
weigh ed a e age cos o he i m’s equi y and deb [30], does no di ec ly conside he isk in ol ed in
he in es men . This is why public subsidized RES p ojec s o en use his me hod o discoun u u e
cash lows and conside he inancial easibili y o he in es men . All he calib a ed pa ame e s and hei
espec i e sou ces a e ou lined in Table 3.
Table 3: Calib a ed pa ame e s and sou ces
Pa ame e Value 2013 Sou ces 2013 Value 2019 Sou ces 2019
TF20 IDAE (2011) [28] 20 IDAE (2011) [28]
T120 BOE [29] 20 BOE [29]
Ae1,400,000/MW Own es ima ion (da a: PER 2011-2020 [6]) e1,150,000/MW Own es ima ion (da a: PNIEC 2021-2030 [27])
O0.035AOwn es ima ion (da a: PER 2011-2020 [6]) 0.035AOwn es ima ion (da a: PER 2011-2020 [6])
Q022,960 MW AEE [7] 25,704 MW AEE [7]
QI12,790 MW PER 2011-2020 [6] 24,626 MW PNIEC 2021-2030 [27]
u· ·h2100 PER 2011-2020 [6] 2355 PNIEC 2021-2030 [27]
Qmax 332,000 MW IDAE [31] 332,000 MW IDAE [31]
γ4.05 ·10−6Own es ima ion (da a: REE [23]) 3.6·10−6Own es ima ion (da a: REE [23])
a1.0207 Own es ima ion (da a: REE [23]) 1.0185 Own es ima ion (da a: REE [23])
µ00.0691 Own es ima ion (da a: REE [23] & OMIE [24]) 0.0691 Own es ima ion (da a: REE [23] & OMIE [24])
µ12.095 ·10−7Own es ima ion (da a: REE [23] & OMIE [24]) 2.095 ·10−7Own es ima ion (da a: REE [23] & OMIE [24])
µ26.831 ·10−13 Own es ima ion (da a: REE [23] & OMIE [24]) 6.831 ·10−13 Own es ima ion (da a: REE [23] & OMIE [24])
σ00.270 Own es ima ion (da a: REE [23] & OMIE [24]) 0.270 Own es ima ion (da a: REE [23] & OMIE [24])
σ15.319 ·10−7Own es ima ion (da a: REE [23] & OMIE [24]) 5.319 ·10−7Own es ima ion (da a: REE [23] & OMIE [24])
σ21.479 ·10−12 Own es ima ion (da a: REE [23] & OMIE [24]) 1.479 ·10−12 Own es ima ion (da a: REE [23] & OMIE [24])
0.10 Noo hou e al. [32] 0.071 CNMC [33]
S0e39.80/MWh Own es ima ion (da a: REE [23] & OMIE [24]) e45.63/MWh Own es ima ion (da a: REE [23] & OMIE [24])
3.3 Resul s
3.3.1 Resul s o he yea 2013
Fi s o all, we check o each o he ou subsidy schemes, whe he he singula poin s de ined by he
equa ions (29), (34), (40) and (46), a e maximum poin s o in es o ’s p oblem in Eq.(19) o he gi en
ins alla ion a ge QI. As we men ioned in sec ion 2.4, hese equa ions cha ac e ize only hose poin s in
which i s -o de necessa y condi ions a e sa is ied. The e o e, o concluding whe he hose poin s a e
19
maximum o no , we ha e o check second-o de condi ions as well. As shown in appendix C, o he
pa ame e alues co esponding o he yea 2013, he analyzed poin s in he cases o Fixed Ta i , Sha ed
Upside, and Cap & Floo policies a e maximum poin s o he ins alla ion a ge Q=QI. On he con-
a y, in he case o Cons an P emium, he alue gi en by equa ion (34) o he cu en pa ame iza ion
co esponds o a minimum poin .
No subsidy: Wi hou any subsidy, he expec ed p o i s ha in es o s will make acco ding o equa-
ion (23) a e e-2,287.8M. Tha is, in es o s a e expec ed o lose mo e han 2 billion eu os i hey only
e ie e he ma ke p ice.
Fixed Ta i (FT): In he case o a ixed-FiT, he isk-neu al decision make will ind i op imal o
in es in he deploymen o exac ly QIuni s o wind ene gy, i , acco ding o he solu ion gi en by equa-
ion (29), he policymake se s a FT o : K(op )
A=e87.838/MWh, which leads o he ollowing expec ed
bene i s and cos s:
ΠF T =e607.52 M ΛF T =e2,895.29 M
Cons an P emium (CP): As we ad anced abo e, o he gi en alue o he pa ame e s, i does
no exis a p emium X o which in es o s will ind op imal o deploy exac ly QInew uni s in his un e-
s ic ed op imiza ion p oblem.10 In his case, equa ion (34) co esponds o a minimum poin . The e o e,
we mus se ano he c i e ion o es ablish an adequa e p emium Xbo h o in es o s and egula o s.
Since in Spain, be o e he emo al o he FiTs, he Spanish go e nmen was indi e en be ween o e ing
o a RES in es o he op ion o a FT and o a CP, by choosing an adequa e alue o he p emium (X),
bo h he expec ed in es o ’s p o i s and policymake ’s cos s a e iden ical o hose gi en by he op imal
FT we jus se abo e. The alue o ha chosen CP would be X∗=e10.746/MWh, leading o bene i s
and cos s iden ical o hose o he ixed a i (e607.52M and e2,895.29M, espec i ely). E en hough
bo h bene i s and cos s a e iden ical unde he wo p e ious policies, as we will see, he isk exposu e o
in es o and policymake will be o ally asymme ic.
Sha ed Upside (SU) and Cap & Floo (C&F): In hese wo cases, since we ha e wo deg ees o
eedom when designing each o he FiT, he op imal con igu a ion is now de ined by a locus o e icien
combina ions (KC, ω) o he SU, and (KD, C) o he C&F egimes. These locus a e ep esen ed in
Figu es 8 and 9. In bo h cases, o each possible alue o a p ice loo K, he e exis s an op imal alue o
he o he pa ame e (and he o he way a ound). Fo any poin belonging o his locus, unde he co -
esponding FiT scheme, he in es o will ind op imal o in es in he ins alla ion o exac ly QInew uni s.
As i can be seen in Figu es 8 and 9, he e is a nega i e ela ionship be ween KCand ω, as well as
be ween KDand C. This is no a su p ise since i is qui e in ui i e ha he e mus be some ade-o
om he policymake ’s pe spec i e: i he o she is willing o o e a highe gua an eed minimum p ice o
he in es o , we can expec ha he o she will demand a highe sha e o he po en ial upside (1 −ω) o
he SU egime, and a lowe cap C o he C&F, which would imply mo e equen emune a ion o him
o he (as e e y hing exceeding he cap goes o he egula o ). Fo he gi en alues o he pa ame e s,
he easible alues o he p ice loo anges om app oxima ely e47 o e87 pe MWh unde he Sha ed
Upside and om 64 o 87 o he Cap & Floo .11
10I could be possible o ind an op imal poin i we add speci ic es ic ions o he p oblem. Fo example, we could bound
policymake ’s budge .
11The easible alues a e hose ul illing some sensible equi emen s, i s o all, hey mus co espond o maximum poin s.
In addi ion, o he SU a i , hey mus ul ill (0 < ω < 1), whe eas o he C&F egime: (K < C).
20
Figu e 8: Op imal Locus (Sha ed Upside, 2013) Figu e 9: Op imal Locus (Cap & Floo , 2013)
I is ema kable ha in bo h cases i he p ice loo is se a a alue e87.838/MWh, unde he SU
scheme he alue o ω( he e icien sha e ha he in es o would pe cei e o he upside exceeding he
loo ) becomes ze o, and unde he C&F, he e icien cap Cbecomes e87.838/MWh as well. In bo h
cases, we would ob ain he e icien FT we discussed in he i s place, which is a esul ha suppo s he
consis ency o ou model.
We wish o ep esen he expec ed bene i s and cos s o he ou policies we discussed, as well as he
isk exposu e. In he case o SU and C&F, he expec ed bene i s and cos s depend on he pa icula poin
o hei espec i e locus ha we use o design he a i . In o he wo ds, depending on he p ice loo K
(as se ing he alue o Kuniquely de e mines he e icien pai ed pa ame e ω/C). In o de o quan i y
he di e en isk exposu es o each a i , we will de ine he Value a Risk (VaR) as a ce ain pe cen ile o
he dis ibu ion o po en ial bene i s(cos s) ha in es o s(policymake s) migh ob ain(incu ) unde he
unp edic able elec ici y p ice e olu ion. In ou case, we will se he VaR a he 10 h pe cen ile o in-
es o ’s bene i s as a measu e o exposu e o unde - emune a ion, and a he 90 h pe cen ile o egula o ’s
cos s as a measu e o exposu e o cos o e un. The e o e, in es o s will p e e subsidies o e ing highe
alues o VaR, whe eas in es o s will p e e policies o e ing lowe VaR alues. We say ha in es o s a e
exposed o isk i hei VaR is lowe han hei expec ed p o i s. In he same way, egula o s a e exposed
o isk when hei VaR is highe han he expec ed cos s.
In o de o ob ain a alue o he VaR, we i s ha e o es ima e he p obabili y dis ibu ion o each
a i o po en ial p o i s and cos s. Fo achie ing ha goal, wi h he pa ame e s gi en in Table 3, we
simula e he s ochas ic GBM desc ibed in Eq.(15) a g ea numbe o imes (in ou pa icula case, we
pe o med 500,000 ials). All he simula ions a e ca ied ou using Wol am Ma hema ica. Fo each
ial, we ob ain a pa icula s ochas ic e olu ion o he VWAP (S ), and hen, we use ha alue o
calcula e he p o i s and cos s each FiT would imply, and in he case o SU and C&F, we do i o e e y
easible p ice loo . Finally, wi h he ob ained sample o esul s, and he equency o each esul , we can
easily es ima e he VaR o in es o ’s bene i s and policymake ’s cos s. Nex , we show o each a i , he
expec ed bene i s (Figu e 10), he expec ed cos s (Figu e 11), he in es o ’s VaR (Figu e 12), and he
policymake ’s VaR (Figu e 13).
21
3.4 Sensi i i y Analysis
Gi en he alues o he pa ame e s ha we ha e been able o p edic o es ima e, he solu ions ha we
ha e ob ained o he bene i s and cos s unde each subsidy scheme a e op imal in expec a ion. Howe e ,
he possibili y ha some pa ame e s ha depend on he ma ke migh no e ol e as expec ed should be
ega ded. Knowing he sensi i i y o each policy’s expec ed bene i s and cos s wi h espec o de ia ions
in he alues o hese ma ke -depending pa ame e s can be a ac o conside o bo h in es o s and
egula o s when designing hei op imal policies.
Le us conside ha once each a i is designed, wi h he police pa ame e s (K, X, ω, C) se , i is
e ealed ha some o ou model’s pa ame e s did no e ol e as we an icipa ed. Thus, he expec ed
p o i s and cos s will be comp omised. We show below in Table 6 he elas ici ies (E) o expec ed p o i s
(Π) and cos s (Λ) wi h espec o some ma ke -depending pa ame e s. Speci ically, we analyze de ia ions
in he d i (µ) and ola ili y (σ) o he GBM, in he discoun a e ( ), in he o al in es men cos pe
ins alled MW (C), and in he elec ical gene a ion in each pe iod (G ). Whe e, as usually, elas ici ies o a
a iable αwi h espec o o he a iable β, a e de ined as: Eα,β =∂α/α
∂β/β . This means ha i he a iable
βinc eases by 1%, hen he a iable αinc eases app oxima ely by an amoun o Eα,β%. The e o e,
lowe exposu e o unexpec ed changes o a gi en pa ame e will be gi en by elas ici ies lowe in absolu e
alue. La ge elas ici ies mean ha i unexpec ed changes occu , he expec ed esul s can ei he inc ease
o dec ease d ama ically. Thus, he close he elas ici ies o ze o, he sa e he in es o o egula o will
be. Fo illus a ion, we b ie ly discuss some o he ob ained esul s.
2013 FT CP SU
(K=55)
C&F
(K=55)
SU
(K=67.5)
C&F
(K=67.5)
SU
(K=80)
C&F
(K=80)
P o i s
EΠ,µ 0 25.97 140.19 74.80 30.67 17.31 6.99 3.99
EΠ,σ 0 0 43.42 6.51 11.50 -1.36 3.00 -0.88
EΠ, -22.35 -30.40 -204.13 -120.28 -62.26 -45.19 -31.46 -27.61
EΠ,C -37.95 -37.96 -269.30 -161.61 -88.54 -66.60 -49.47 -44.56
EΠ,G 38.95 38.96 270.30 162.61 89.54 67.60 50.47 45.56
Cos s
EΛ,µ -5.45 0 -1.59 -2.10 -3.06 -3.72 -4.55 -4.89
EΛ,σ 0 0 1.57 0.38 1.18 -0.18 0.51 -0.16
EΛ, 0.95 -0.73 -0.48 -0.34 0.05 0.27 0.61 0.74
EΛ,C 0 0 0 0 0 0 0 0
EΛ,G 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2019 FT NS SU
(K=42)
C&F
(K=42)
SU
(K=55)
C&F
(K=55)
SU
(K=62)
C&F
(K=62)
P o i s
EΠ,µ 0 2.30 49.44 7.30 6.06 2.19 0.75 0.33
EΠ,σ 0 0 7.24 -7.22 1.32 -2.03 0.19 -0.30
EΠ, -9.92 -1.73 -61.43 -17.06 -16.27 -12.08 -10.71 -10.24
EΠ,C -21.90 -1.58 -116.61 -35.98 -33.51 -26.12 -23.34 -22.53
EΠ,G 22.90 2.58 117.61 36.98 34.51 27.12 24.34 23.53
Cos s
EΛ,µ 2.47 - 1.65 2.06 2.11 2.30 2.41 2.44
EΛ,σ 0 - -0.10 0.33 -0.07 0.13 -0.01 0.02
EΛ, -1.09 - -0.91 -1.03 -1.01 -1.07 -1.08 -1.09
EΛ,C 0 - 0 0 0 0 0 0
EΛ,G 1.00 - 1.00 1.00 1.00 1.00 1.00 1.00
Table 6: Elas ici ies o expec ed p o i s and cos s wi h espec o di e en pa ame e s
Rega ding unexpec ed changes in µ, in es o s unde a Cons an P emium bea he en i e deg ee o
unce ain y compa ed o hose unde a Fixed Ta i . Fo he in e media e schemes, he highe he alue
o K, he lowe he elas ici ies wi h espec o he VWAP d i pa ame e . On he con a y, policy cos s
a e insensi i e o such changes unde CP schemes, and egula o s unde FT a e he ones who bea he
isk, while o he SU and C&F schemes, he highe he alue o he p ice loo , he highe he sensi i i y
(in absolu e alue). Expec ed p o i s and policy cos s a e insensi i e o changes in ola ili y unde bo h
FT and CP egimes, as he expec ed alue o he GBM (S0eµ ) does no depend on σ. Unde a SU
28
o C&F schemes, a highe alue o K(and hus a lowe alue o ωo C) dec eases bo h in es o ’s and
egula o ’s sensi i i y o changes in he alue o σ. As expec ed, egula o s a e insensi i e o changes in
o al ins alla ion cos s pe MW (C). As can be seen, he p o i ’s elas ici ies wi h espec o Gand C
di e exac ly in one uni , bu wi h he opposi e sign. Fo bo h pa ame e s, he highe he alue o K,
he lowe (in magni ude) he in es o ’s sensibili y unde SU o C&F egimes. Fo any policy, egula o s
sha e he same exposu e o changes in he elec ici y gene a ion a each pe iod G.
3.5 Discussion
Ou me hodology elies on he assump ion ha in es o s a e isk-neu al, which migh be jus i iable.
When in es o s a e isk-neu al, i is because hey can di e si y hei in es men s in o he sec o s. In he
bes -case scena io, hey will ha e in es men s nega i ely co ela ed wi h he one we a e discussing, so
ha , i hey lose some o he in es men s in enewables, hey could s ill make money. This assump ion
allows us o ob ain analy ical solu ions o he op imal expec ed p o i s and cos s associa ed wi h each
subsidy scheme. Ne e heless, in ligh o he esul s ob ained p e iously, we can push u he ou analy-
sis. Le us suppose ha in he i s place, in es o s seek o maximize expec ed p o i s, and policymake s
seek o minimize expec ed egula o y cos s. Then, once he op imum is de e mined o each FiT scheme,
isk can be aken in o conside a ion by choosing he mos app op ia e subsidy ype, oge he wi h an
adequa e alue o he FiT pa ame e s o SU and C&F designs (i.e., by choosing he p ope p ice loo K).
F om he esul s ob ained in sec ion 3.3.1, we can conclude ha i is unlikely ha in es o s in 2013
decided o in es in he ins alla ion o he wind capaci y equi ed o mee he go e nmen ’s objec i e
se in 2011 o he yea 2020, wi hou any public subsidy suppo . Fu he mo e, he ac ha wind
ene gy in es men oze a e subsidies we e elimina ed in 2013 s ongly suppo s ou indings. Ou e-
sul s sugges ha om in es o s’ pe spec i e, he bes way o c ea ing a easible appe izing in es men
en i onmen in 2013 was o o e hem a Fixed Ta i suppo scheme, o ins ead, in o de o allow some
isk-bea ing om in es o ’s pa and elie ing some isk exposu e om he egula o , o e ing a Sha ed
Upside o Cap & Floo schemes as close as necessa y o he Fixed Ta i (i.e., wi h he loo Kclose o
he op imal FiT p ice e87.838/MWh). As we ha e seen in Figu es 10 and 11, he highe he alue o
he p ice loo , he highe he expec ed bene i s and he lowe he isk exposu e o unde emune a ion
he in es o has o bea . I in es o s ha e o choose be ween he wo in e media e schemes, hey will
p e e he Cap & Floo subsidy egime wi h he highes possible p ice loo . F om he policymake ’s
pe spec i e, he policy con igu a ion ha leads o he lowes expec ed cos s and lowe exposu e o cos
o e un would be he Sha ed Upside scheme wi h a p ice loo as low as possible. Indeed, no only he
Sha ed Upside, bu he Cap & Floo scheme also p o ides be e expec ed esul s o he policymake
han he adi ional subsidies. We canno s ess his inding enough: i is possible o design in e media e
schemes ha sha e he isk exposu e be ween egula o s and in es o s, and mo e impo an ly, hey do i
mo e e icien ly, leading o he same le el o in es men bu in ol ing lowe expec ed go e nmen cos s.
On he con a y, ega ding he esul s ob ained in sec ion 3.3.2, we can conclude ha an in es o in
2019 would decide o in es in he ins alla ion o he wind capaci y equi ed o mee he go e nmen ’s
objec i e in 2030 wi hou he need o any public subsidy. Fo ins ance, ou analysis sugges s ha a isk-
neu al in es o seeking o maximize expec ed p o i s, will p e e o ecei e p ecisely he ma ke p ice
a he han any suppo ing policy we ha e conside ed, con a y o wha happened in 2013. This inding is
consis en wi h he ac ha in all he ene gy auc ions held in he las couple o yea s, he discoun a e o
which he auc ion winne s bid was ze o. Tha is, hey we e all willing o in es in he ins alla ion o wind
ene gy, ecei ing only he ma ke p ice as paymen . Th ee main easons can explain his phenomenon as
a as we a e conce ned. Fi s ly, he cos s associa ed wi h he ins alla ion o a wind a m ha e plumme ed
compa ed o 2013. I is inc easingly cheape o in es in wind ene gy deploymen . Second, since he an-
nual d i o he ma ke p ice is endogenous o he ins alled capaci y a ge and an inc easing unc ion o
i , as his ins alla ion a ge is so la ge o he nex decade, his pa ame e inc eases conside ably, and
he e o e, he ma ke p ice ends o be highe as well. Mo eo e , he p ice o he ini ial pe iod (S0) is
29
conside ably highe in 2019 han in 2013. Finally, in ecen yea s since he end o he inancial c isis,
he WACC has been dec easing, and he e o e, u u e paymen s o in es o s a e discoun ed a a lowe a e.
As we ha e seen, any o he h ee schemes analyzed in ol e nega i e expec ed cos s (i.e., ea nings
o he egula o ) due o he high ma ke p ices. Al hough his esul may seem un ealis ic, we mus
emembe ha his hesis’s objec i e is no o ob ain comple ely eliable and accu a e esul s o he
u u e Spanish ene gy ma ke ; a he , ou objec i e is o compa e how in e media e schemes beha e wi h
espec o he usual FiT. The e o e, i is c ucial o poin ou ha , as happened in 2013, he policymake ’s
expec ed cos s o he in e media e schemes a e again lowe han hose co esponding o he Fixed-Ta i
o o he absence o any subsidy. The same le el o in es men (QInew uni s o wind powe ) is encou -
aged mo e e icien ly, e idence o an indispu able gain compa ed o he usual FiT schemes.
We ha e seen how he in e es s o he policymake and hose o he in es o end o be opposed.
Bo h pa ies no only ha e o conside p o ec ing hemsel es agains he isk o low elec ici y ma ke
p ices, bu hey mus also conside he isk posed by he possibili y ha he ma ke -depending pa am-
e e s o which public subsidies a e designed, do no e ol e as expec ed. Thus, he exis ing ade-o
be ween highe p o ec ion agains low ma ke p ices and highe (lowe ) expec ed bene i s(cos s) ha in-
es o s(go e nmen s) ha e o ace, equi es an exhaus i e and in-dep h analysis which should be handled
ca e ully, and will e en ually depend on he pa icula in e es , budge , and he isk ha each o he wo
pa ies is willing o assume.
4 Conclusions
Un il hey we e elimina ed in 2013 mainly due o he high cos s ha hey implied o he go e nmen ,
commonly used Feed-in Ta i s ha e been he main ins umen o suppo in es men in enewable ene gy
in Spain. Since hen, wind ene gy in es men has been ozen o se e al yea s, making i impossible o
achie e he ins alla ion a ge se o ha egula o y pe iod ending in 2020. Ano he d awback o he
ypical FiT schemes is ha only one o he pa ies mus assume all he isk. Inspi ed by he inno a ion
in oduced by Fa el e al., by combining sequen ial game heo y, FiT design, and s ochas ic calculus, we
ha e been able o design and o es in e media e policies ha allow isk-sha ing be ween in es o s and
go e nmen s, and mo e impo an ly, doing i mo e e icien ly.
In his hesis, we ha e had o de ise ou own me hodology o p edic he d i and ola ili y o he
olume-weigh ed a e age p ice o elec ici y as a unc ion o he amoun o ins alled wind capaci y. E en
hough in ou pa icula case he esul can be comp omised wi h a conside able e o , due o he ac
ha in Spain he e is no enough openly a ailable da a, we belie e ha his me hod could be use ul in
o he s udies beyond he pa icula use ha we ha e gi en o i in his p ojec . Mo eo e , i could be
used o di e en enewable echnologies. Nex , we ha e un s ochas ic simula ions o de e mine how
hese schemes would beha e in a scena io ha ies o app oxima e he Spanish en i onmen o he
yea s 2013 and 2019. We ully cha ac e ized he dis ibu ion o po en ial bene i s and cos s unde each
scheme. Finally, we ca ied ou a sensi i i y analysis o analyze he e ec s on he expec ed bene i s and
cos s unde each policy i di e en ma ke -depending pa ame e s do no e ol e as expec ed.
Acco ding o he esul s we ob ained, i is possible o design Sha ed Upside and Cap & Floo schemes
ha allow achie ing he same le el o in es men in enewables a a lowe cos o he go e nmen . Then,
in es o s and egula o s can adjus hese mo e e icien schemes o modi y he alue o he expec ed cos s
and bene i s, as well as he isk each o he pa ies has o assume. Besides, ou esul s e lec ha he e
is an una oidable con lic be ween he in e es s o in es o s and go e nmen s, which end o be opposed.
Fu he mo e, each o he pa ies aces hei own ade-o : in he case o in es o s, hey migh ha e o
choose be ween highe expec ed p o i s and lowe isk o unde emune a ion. In u n, egula o s migh
be o ced o choose be ween lowe expec ed cos s and lowe isk o public cos o e uns. Fu he mo e, we
ha e shown how he sensi i i y analysis can o e some in e es ing insigh o bo h in es o s and egula o s
when designing each policy.
30
Ou esul s ha e in e es ing policy implica ions o u u e enewable ene gy egula ion. Al hough he
me hodology pe o med may seem complex o manage a i s glance, he i up ion o echnologies such
as a i icial in elligence, machine lea ning, and big da a, allow his ype o analysis o be au oma ed and
ex ended. I would be easible ha such ools ha could be p og ammed nowadays wi hou oo many
di icul ies and ou lay, ca y ou hose simula ions o all he possible alues o he e icien pa ame e s
o he FiT, and ha gi en he p e e ences o he in es o s, and some echnical es ic ions, such as, he
budge o he go e nmen o enewable ene gy suppo , would e u n o he policymake he op imal
a i and pa ame e choice ha acili a es he desi ed in es men . Also, unlike in Spain, whe e he
pa ame e s o he s anda d FiT and FiP we e e iewed e e y se e al yea s, his au oma able p ocedu e
would allow upda es o he pa ame e s o he FiTs in place much mo e equen ly, which would be an
undoub ed imp o emen in e iciency o bo h pa ies.
As we ha e poin ed ou on se e al occasions, he pu pose o his hesis is no o cha ac e ize o p edic
wi h accu acy he inancial esul s ha a pa icula policy would ha e, bu a he , o es how each o he
analyzed policies beha es compa ed o he es . This is because he me hodology ha we ha e de eloped
has se e al limi a ions in o de o conside he ob ained igu es as eliable. Fi s ly, he limi a ions men-
ioned abo e in es ima ing he pa ame e s µand σ. Secondly, he ob ainmen o analy ical solu ions o
e icien con igu a ions o each a i does no come wi hou a cos , he high alue ob ained o he d i o
he GBM has he consequence ha when we ca y ou he simula ions a la ge numbe o imes, in some
o hose ials he legal maximum e/180.3 MWh is exceeded. Finally, he a ious app oxima ions made
o ob ain closed- o m solu ions, such as aking he iden ical cos s pe MW (C) o he en i e p ojec , o
assuming cons an wind gene a ion (G ) o e ime, will comp omise he esul s.
This hesis demons a es how he FiT schemes used in Spain un il 2013 a e imp o able in nume ous
aspec s. Howe e , he design o he op imal suppo scheme o a speci ic yea has no been ca ied ou ,
gi en ha such a ask equi es a much longe ime and ex ension han wha is unde s ood o comp ise
his mas e ’s hesis, and especially, a e y accu a e calib a ion. Ne e heless, we belie e ha his exe cise
can be ca ied ou in he o eseeable u u e. We p opose ha he nex s ep in he di ec ion o his esea ch
is o gi e mo e weigh o nume ical me hods and o play down he impo ance o ob aining closed- o m
solu ions. By doing so, we belie e ha some o he men ioned limi a ions could be o e come, such as he
equi emen o bounded ma ke p ices, and allowing he possibili y o elaxing some o he assump ions
in which we ha e based ou wo k. The mos ema kable imp o emen could be o cha ac e ize in es o s’
p e e ences by a cons an ela i e isk a e sion (CRRA) u ili y unc ion, which would make he ob ained
esul s much mo e gene al. Ano he ad an age o a ully compu a ional app oach could be o include
non-cons an gene a ion o e ime (G ) and a iable in e sion cos s (C). Fu he mo e, we belie e ha a
way o o e come he limi a ions imposed by he ob ained alues o µand σcould be ins ead o conside ing
he VWAP as he ma ke p ice, o use he wholesale p ice. Since he VWAP is usually cheape han
he pool p ice bu ollowing simila ends, he wholesale p ice could be used as an app oxima ion when
s udying he p ices ecei ed by in es o s in enewables. A simila app oach is ollowed by Blazquez e
al. (2018) [19]. Finally, he nume ically ound op imal scheme could be compa ed o some o he o he
public suppo sys ems o enewables ha a e o ha e been commonly used, such as T adable G een
Ce i ica es and Ene gy Auc ions [35]. These asks a e le as open p oblems o u u e esea ch.
31
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33
Appendix A: Sol ing Equa ion 10
∂
∂ 0+µS ∂
∂S +σ2S2
2
∂2
∂S2− = 0 (S, 0) = h(S) (54)
Fi s , we de ine he ime o ma u i y τ= − 0. Then we ha e: ∂
∂τ =−∂
∂ 0. Subs i u ing in Eq.54
and mul iplying he whole equa ion by 2
σ2:
−2
σ2
∂
∂τ +2µS
σ2
∂
∂S +S2∂2
∂S2−2
σ2 = 0 ∂2
∂S2− = 0 (S, τ = ) = h(S) (55)
In o de o eplace a iables and µ, we de ine a=2
σ2, and b=2µ
σ2, which yields o
−2
σ2
∂
∂τ +bS ∂
∂S +S2∂2
∂S2−a = 0 (S, τ = ) = h(S) (56)
We now eplace Sand τin he abo e equa ion by de ining he a iables uand :
=σ2(b−1)2τ
2u= (b−1) log S
K+ (57)
Acco ding o he chain ule o u(S, τ) and (S, τ):
∂
∂S =∂
∂S ∂
∂ +∂u
∂S ∂
∂u (58)
∂
∂τ =∂
∂τ ∂
∂ +∂u
∂τ ∂
∂u (59)
F om equa ion 57, sol ing o Sand τ, i is s aigh o wa d o ob ain:
S(u, ) = Ke(u−
b−1)(60)
τ(u, ) = 2
σ2(b−1)2(61)
We ind he Jacobi ma ix (J) o he coo dina e ans o ma ion (S(u, ), τ(u, )):
J=
∂S
∂u
∂S
∂
∂τ
∂u
∂τ
∂
=
S
(b−1) −S
(b−1)
02
σ2(b−1)2
(62)
F om he gene al p ope ies o he Jacobi ma ix, i is easy o p o e ha he pa ial de i a i es o
he in e se ans o ma ion (u(S, τ), (S, τ)) a e gi en by J−1. Tha is:
i
∂u
∂S
∂u
∂τ
∂
∂S
∂
∂τ
=J−1=
b−1
S
σ2(b−1)2
2
0σ2(b−1)2
2
(63)
which yields o
∂
∂S =b−1
S∂
∂u (64)
∂
∂τ =σ2(b−1)2
2∂
∂u +∂
∂ (65)
In addi ion, om Eq.64, we ob ain:
∂2
∂S2=∂
∂S b−1
S∂
∂u=−(b−1)
S2
∂
∂u +(b−1)2
S2
∂2
∂u2(66)
Subs i u ing hese h ee las exp essions in equa ion 56, we ob ain:
−∂
∂ +∂2
∂u2−a
(b−1)2= 0 (67)
Wi h he addi ional change o unc ion (S, 0) = (S, −τ)−→ g(u, ), de ined by:
(S, −τ) = e− τ g(u, ) = e−a
(b−1)2g(u, ) (68)
which yields o:
∂
∂u =e−a
(b−1)2∂g
∂u
∂
∂ =e−a
(b−1)2∂g(u, )
∂ −ag(u, )
(b−1)2(69)
Subs i u ing ,∂
∂u and ∂
∂ in 67, inally yields o a e y well known PDE: he hea equa ion.
∂2g(u, )
∂u2=∂g(u, )
∂ g(u, 0) = g0(u) (70)
In o de o sol e his equa ion, we use he Fou ie T ans o m me hod o sol e PDEs. The Fou ie
ans o m (F) o he unc ion g(u) o −∞ < u < ∞is gi en by
F[g(u)] = F(z) = 1
√2πZ∞
−∞
g(u)e−izudu (71)
Thus, i is easy o e i y ha he ans o ma ion o he pa ial de i a i es ∂2g(u, )
∂u2, and ∂g(u, )
∂ a e
gi en by he ollowing o mulas
F∂2g
∂u2=−z2F[g] (72)
F∂g
∂ =∂
∂ F[g] (73)
ii
Hence, deno ing F[g] = G( ), and F[g0] = G0(Z), equa ion 70 becomes
dG( )
d =−z2G( )G(0) = G0(z),(74)
Thus, he solu ion o his p oblem is ob iously gi en by:
G( ) = G0(z)e−z2 (75)
Now, inding he in e se ans o m (F−1), i can be e i ied ha he solu ion o equa ion 70 is de ined
by he ollowing exp ession:
g(u, ) = F−1[G0(z)e−z2 ] = 1
2√π Z∞
−∞
g0(z)e−(u−z)2
4 dz (76)
Finally, eplacing u(S, τ), (S, τ), g0and subs i u ing z= (b−1) log x
K, we ob ain he o iginal
e minal condi ion h(x):
(S0, ) = e− τ
σ√2πτ Z∞
0
h(x)
xexp −(log x
S0−(µ−σ2
2)τ
2σ2τ!dx (77)
•All he me hods applied du ing hese calcula ions can be checked ou in:
Riley, K. F., Hobson, M. P., & Bence, S. J. (2006). Ma hema ical me hods o physics and enginee ing:
a comp ehensi e guide. Camb idge uni e si y p ess.
iii
Appendix B
Cons uc ion o he da ase
Peio Alco a Iglesias
2020-07-23
m(lis =ls())
yea s=2014:2019
o (a in 1:leng h(yea s)){
yea =yea s[a]
se wd("/Use s/peio/desk op/TFM_o denado/R")
n=365
i (yea ==2008|yea ==2012|yea ==2016|yea ==2020){n=366}
h=n*24
o iginda e=pas e(as.cha ac e (yea ),"-01-01",sep="")
ile.1=pas e("/Use s/peio/desk op/DATOS/REE/",as.cha ac e (yea ),"/Cus om-Repo -",
as.Da e(0,o igin = o iginda e),"-Es uc u a de gene ación (MW).cs ",sep="")
da a ile.1 = ead.cs ( ile.1,heade =F, sep=",")
da a ile.1=da a ile.1[-c(1:21),]
da a ile.1=da a ile.1[-c(146:163),]
m1=leng h(da a ile.1[,2])
m2=leng h(da a ile.1[10,])
o (j in 2:m2) {
da a ile.1[,j]=as.nume ic(pas e(da a ile.1[,j]))
}
i (yea >2015){
aux8 o9=da a ile.1[,8]
aux9 o10=da a ile.1[,9]
da a ile.1[,8]=da a ile.1[,12]+da a ile.1[,13]
da a ile.1[,11]=da a ile.1[,10]+da a ile.1[,11]
da a ile.1[,9]=aux8 o9
da a ile.1[,10]=aux9 o10
}
colnames(da a ile.1)=c("Ho a","Eolica","Nuclea ","Fuel/Gas","Ca bón","Ciclo Combinado",
"Hid aulica","Res o.Reg.Especial","In e cambios","Enlace Balea ","Sola ")
da a ile.1=da a ile.1[,c(1:11)]
da a_ho a io.1=da a. ame("Ho a" = c(1:24),
"Eolica"=c(1:24),
"Nuclea "=c(1:24),
"Fuel/Gas"=c(1:24),
"Ca bón"=c(1:24),
"Ciclo Combinado"=c(1:24),
"Hid aulica"=c(1:24),
"Res o.Reg.Especial"=c(1:24),
"In e cambios"=c(1:24),
1
Appendix B: Cons uc ion o he da ase (R code)
i
Appendix C: Minimum o Maximum?
We check o each o he ou subsidy schemes, whe he he singula poin s de ined by he equa ions
(29), (34), (40) and (46), a e maximum poin s o (19) o he ins alla ion a ge QI. Fo bo h 2013 and
2019, he analyzed poin s in Fixed Ta i , Sha ed-Upside, and Cap & Floo policies a e maximum poin s
o he ins alla ion a ge Q=QI. On he con a y, in he case o Cons an P emium, he alues gi en
by equa ion (34) o he pa ame iza ion o bo h yea s, co espond o minimum poin s. Fu he mo e, we
see how o SU and C&F schemes, he singula poin s can ei he be maximum o minimum depending
on he pa icula alue o he e icien p ice loo (K) chosen.
2013
Fixed Ta i
Solu ion o Eq.(29): KA= 87.838
Figu e A1: FT 2013
Cons an P emium
Solu ion o Eq.(34): X= 8.1963
xi
Figu e A2: CP 2013
Sha ed Upside
KC anging om 55 o 85
Figu e A3: SU 2013
Cap & Floo
KD anging om 55 o 85
xii
Figu e A4: C&F 2013
2019
Fixed Ta i
Solu ion o Eq.(29): KA= 63.4575
Figu e A5: FT 2019
xiii
Cons an P emium
Solu ion o Eq.(34): X=−43.2159
Figu e A6: CP 2019
Sha ed Upside
KC anging om 25 o 65
Figu e A7: SU 2019
xi
Cap & Floo
KD anging om 5 o 65
Figu e A8: C&F 2019
x
