E alua ion o a c oss-bo de elec ici y in e connec ion: The case o
Spain-F ance
Luis Ma ía Abadie
a
, Jos
e Manuel Chamo o
b
,
*
a
Basque Cen e o Clima e Change (BC3), Sede Building 1, 1s Floo , Scien ific Campus, Uni e si y o he Basque Coun y UPV/EHU, 48940, Leioa, Spain
b
Uni e si y o he Basque Coun y: Uni e sidad Del País Vasco/ Euskal He iko Unibe si a ea, Dp . Financial Economics II and Ins i u e o Public Economics,
A . Lehendaka i Agui e 83, 48015, Bilbao, Spain
a icle in o
A icle his o y:
Recei ed 17 No embe 2020
Recei ed in e ised o m
27 Ma ch 2021
Accep ed 4 June 2021
A ailable online 16 June 2021
JEL classifica ion:
F18
G31
L94
L98
Q41
Q56
Keywo ds:
In e connec o app aisal
Elec ici y p ices
S ochas ic models
Jumps
Tobi model
abs ac
This pape ocuses on he economics o a c oss-bo de ansmission in e connec o . The domes ic spo
elec ici y p ice is modelled as a s ochas ic p ocess wi h mean e e sion and jumps; i also includes a
de e minis ic pa ha accoun s o hou ly and daily sasonali ies along wi h non-wo king days. The wo
domes ic spo p ices a e assumed o be co ela ed. As an illus a ion o he app oach, we conside he
pa icula case o he in e connec o be ween Spain (an ‘elec ic island’) and F ance. Domes ic p ices a e
fi s calib a ed and hen used o simula ing he s ochas ic beha io o he p ice gap be ween he wo
coun ies. In addi ion, he ac ual impo /expo beha io as a unc ion o he p ice gap is cap u ed by a
Tobi model fi ed om obse ed da a. This model is hen combined wi h he simula ed p ice gaps o
compu e a mul iple se ies o hou ly p ices and expo s/impo s o elec ici y h ough he in e connec o .
D awing on hese simula ions we de i e he p obabili y dis ibu ions o e enues and expenses om
expo s and impo s, and also some isk measu es. Acco ding o ou esul s, he economics o his
in e conec o depends on di e en domes ic seasonali ies (hou ly and daily), he g owing end o he
p ice gap and some s ochas ic idiosync asies. They call o an expanded link.
©2021 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND
license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
1. In oduc ion
C oss-bo de powe in e connec ions b ing abou a numbe o
benefi s o pa icipa ing coun ies and beyond [1], among hem: (i)
enhance secu i y o elec ici y supply (SoS) by p o iding suppo
unc ions be ween in e connec ed elec ical sys ems; (ii) ensu e
he s abili y and equency o he wo sys ems; (iii) exploi p ice
di e ences h ough powe impo s and expo s hus inc easing
economic e ficiency; (i ) ha ness enewable ene gy sou ces by
allowing he ansmission o excess enewable gene a ion; ( )
de elop he In e nal Ene gy Ma ke in Eu ope.
1
This pape alls wi hin he li e a u e abou powe ansmission
expansion wi h a special ocus on in e connec o economics, i.e.
i em (iii). A numbe o models ha e been p oposed o add ess po-
we ade based on p ice di e en ials.
2
Many o hem a e applied o
Eu opean coun ies (whe he looking backwa d o o wa d in
ime), be i unde gene al o pa ial equilib ium condi ions. Typi-
cally, hey a e op imiza ion models ha aim o maximize social
wel a e o minimize sys em cos s, o ins ance. They usually
conside a single yea (o ac ions o i ) wi h daily/hou ly ime
s eps. Impo an ly, hey end o be de e minis ic; he au ho s ac-
coun o isks and unce ain ies by simula ing he models unde
se e al scena ios (e.g. wi hou and wi h a pa icula expansion o
he ansmission g id). Besides, he op imiza ion p ocess esul s in
a se ies o (daily/hou ly) powe p ices, ye hei p ope ies a e no
shown. Thus, whe he hose op imiza ion-based p ices display he
usual cha ac e is ics in ac ual powe ma ke s is all bu impossible
*Co esponding au ho .
E-mail add esses: lm.abadie@bc3 esea ch.o g (L.M. Abadie), jm.chamo o@ehu.
eus (J.M. Chamo o).
1
The e can well be conflic s among hese goals; see o ins ance Re . [32].
2
Full p ice con e gence is no an objec i e as such: i would en ail o e -in es ing
in ne wo k in as uc u es [33].
Con en s lis s a ailable a ScienceDi ec
Ene gy
jou nal homepage: www.else ie .com/loca e/ene gy
h ps://doi.o g/10.1016/j.ene gy.2021.121177
0360-5442/©2021 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
).
Ene gy 233 (2021) 121177
o ou side s o ell.
Ou pape in oduces a no el app oach ha pu s elec ici y
p ices a he o e on o he analysis. I looks a ac ual wholesale
p ices in wo specific coun ies (domes ic ma ke s) wi h a common
in e connec o . The pape p oposes a s ochas ic model whe e
(co ela ed) domes ic spo p ices display seasonali y, mean e e -
sion and jumps. The aim he e is o e alua e a pa icula c oss-
bo de in e connec o om he iewpoin o e enues. Conse-
quen ly, we also conside ac ual flows o elec ici y in bo h di-
ec ions. D awing on he la es p ice and quan i y da a publicly
a ailable we simula e he e enues o he in e connec o in he
nea u u e. Hence we no only p o ide a e age es ima es bu
p obabili y dis ibu ions (i.e. isk p ofiles) as well. We u he
illus a e ou app oach by applying i o a singula case s udy; his is
he second con ibu ion o ou pape .
Righ now, Spain is akin o an “elec ic island”because o i s low
in e connec ion a io o 2.8% in 2019 (compu ed as he sum o he
impo capaci ies di ided by he ins alled gene a ion capaci y). This
low a e is e y a om he EU goal o 10% o 2020 and he
minimum o 15% o 2030; ENTSO-e [2]. The Spanish elec ici y
sys em is connec ed wi h F ance, Po ugal, Mo occo and, o a lesse
ex en , Ando a. Anyway, he sho AC in e connec ion wi h F ance
is e y impo an because i gi es access o he as Eu opean
elec ici y ma ke .
3
The F ench in e connec o has a comme cial
exchange capaci y o 2800 MW.
4
A e he commissioning o a new
p ojec (c ossing he Bay o Biscay) his capaci y will inc ease up o
5000 MW; i s comme cial use is planned o s a in 2024 o 2025
[3]. This way he in e connec ion a io will ise o abou 5%. Thus,
Spain will emain an “elec ic island” o decades o come. This
condi ion lea es i especially ulne able o low- equency, high-
impac e en s, whe he o na u al, acciden al o malicious o igins.
Acco ding o ou esul s, Spanish expo s would amoun o V27
million on a e age o e he h ee-yea pe iod 2020e22, while
impo s om F ance would en ail a e age expenses o V964
million. These figu es allow jus i y subs an ial new in es men
( aking [4]; and [5]; as benchma ks).
5
One o he d i e s behind his
esul is he s onge upwa d end o Spanish powe p ice. As
expec ed, when we impose he es ic ion o no g ow h in domes ic
p ices he balance o Spain imp o es.
The pape con inues as ollows. Sec ion 2 e iews he li e a u e
on he po en ial gains om enhanced c oss-bo de in e connec o s,
p e e ably ( hough no exclusi ely) wi h a ocus on he EU. Then
Sec ion 3in oduces he heo e ical amewo k, s a ing om he
s ochas ic model o domes ic powe p ice. Sec ion 4 ocuses on he
wo coun ies in ol ed in his pape ; i p o ides backg ound da a
abou domes ic spo p ices (a he hou ly and daily ime scales)
along wi h powe expo s and impo s. The p ice model is cali-
b a ed n Sec ion 5. Nex in Sec ion 6we d aw on he ea lie
pa ame e es ima es o simula e he p ice in each coun y (and he
ensuing p ice gap) o e he pe iod 2020e2022. On he o he hand,
Sec ion 7es ima es a model o powe flows be ween F ance and
Spain. C oss-bo de flows and simula ed p ices allow simula e he
ansmission income o he in e connec o in Sec ion 8. A sensi-
i i y analysis wi h espec o he g ow h a e o powe p ices is
unde aken in Sec ion 9. Sec ion 10 concludes.
2. Li e a u e su ey
Fi s , we p oceed om a ‘mac o’ o a ‘mic o’pe spec i e: EU
na ions, egions, and indus y s akeholde s (p oduce s, consume s,
and ansmission sys em ope a o s). Then we conside some ea-
sons behind ine ficien a bi age ansmission (i.e. flows in he
‘w ong’economic di ec ion) in pai s o neighbou ing ma ke s.
Ab ell and Rausch [6]find conside able scope o wo-way
c oss-bo de ade in Eu ope (e.g. be ween Spain and F ance).-
Powe p ice di e en ials a e a om unidi ec ional. Fu he , e y
equen ly he e a e sizeable p ice gaps be ween coun ies wi h a
c oss-bo de in e connec ion (e.g. F ance and Spain). The gaps can
ce ainly a ise when ansmission cons ain s a e binding. And also
when hey a e no because o : (i) ansmission losses and/o
amping es ic ions [7]; (ii) inabili y o he in e connec o 's owne
o ake simul aneous long (i.e. pu chasing) and sho (i.e. selling)
posi ions in he wo loca ions (because ma ke liquidi y in a leas
one o hem is oo hin; [8].
G€
o ansson e al. [9] analyse he Eu opean powe sys em a he
NUTS-2 le el, which esul s in 50 egions. Thei esul s o 2020
show an annual a e age ma ginal cos a ound 50 V/MWh in he
Spanish egion ES2 and close o 30 V/MWh in he F ench one FR2.
This ‘conges ion’gap implies ha he ‘ma ginal connec ion capaci y
alue’o e 8760 h amoun s o some 173 Mill Vpe yea ( ende ing
his connec ion one o he fi e AC in e connec ions wi h he
highes alues). Fu he , hese wo egions a e a case in so-called
‘all-hou conges ion’.
Spiecke e al. [10]find u iliza ion a es o he line connec ing
F ance and Spain a ound 90% bo h wi h and wi hou g id ex ension
in 2020 (abou 2/3 o ha a e co esponds o powe flowing om
F ance o Spain, and 1/3 o e e se flow). These high a es sugges
ha bo lenecks occu equen ly. Unde he expanded g id he
a e age o absolu e p ice di e ences be ween hese wo coun ies
is cu in hal .
6
The sha e o a iable wind in eed is significan ly
highe in Spain han in F ance, which leads o mo e equen e-
e sals in he flow di ec ion. O e all, F ance is one o he majo
beneficia ies o new in e connec o s; hey ha e a posi i e impac
on p oduce su plus bu a nega i e one on consume su plus and
conges ion en . Pudjian o e al. [11] conside he pe iod 2010 o
2050. They find ha ein o cing he in e connec ion allows Spanish
consume s o access compe i i e o e s om o eign p oduce s,
which leads o lowe powe p ices and p oduce su plus. Ye no all
p oduce s su e ; o example, sola PV p oduce s gain while wind
p oduce s lose.
7
Ins ead, F ench p oduce s will mee a highe de-
mand, which esul s in an inc ease o powe p ices in F ance ( o he
de imen o F ench consume s).
A his poin , i is wo h no ing ha elec ici y does no always
flow as p ice a bi age would sugges .
8
Clemen s e al. [12]find
ins ances o elec ici y flowing om Queensland o neighbou ing
New Sou h Wales despi e he o me ha ing a highe p ice. They
show ha hese ins ances a e due o nodal ansmission cons ain s
in Queensland only (no o cons ain s ac oss egional bounda ies).
On he o he hand, Bunn and Zachmann [13] show analy ically ha
a dominan gene a o in one loca ion, unde special ci cums ances,
may choose o expo powe ( o a mo e compe i i e neighbou ing
ma ke ) agains he di ec ion o e ficien a bi age. Fu he , as hose
3
The in e connec o p ojec was fi s p oposed in 1980 ( ollowed by a second
p oposal in 2003); i s a ed ope a ion 35 yea s la e [34].
4
The ne ans e capaci y (NTC) ypically se s he comme cial ( a he han he
physical) capaci y be ween wo coun ies.
5
The po en ial alue o an in e connec o is much highe . In addi ion o hese
e enues om day-ahead coupling i comp ises he benefi s om in aday
coupling, sha ed balancing esou ces, a oided undesi able unscheduled flows, and
educed cu ailmen . Acco ding o Re . [5]; e enues om day-ahead a bi age
make up a ound 25% o o al alue.
6
The a e age o hose di e ences be ween wo egions o e a yea indica es he
wel a e e ec o a ma ginal line in es men .
7
This can be ela ed o he di e en le elized cos o elec ici y (LCOE) o hese
echnologies; Abadie and [35].
8
Unde some ci cums ances, a flow in he ‘w ong economic di ec ion’may be
socially beneficial i i s wel a e economic cos is smalle han he wel a e economic
benefi o he conges ion elie ed by such a flow; [33].
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
2
special ci cums ances do apply in he case o he Anglo-F ench
In e connec o , hey p o ide e idence ha such flow e e sions
do occu in eali y.
The abo e pape s ocus on pai s o neighbou ing coun ies o
he mos pa , and we ollow sui . None heless, his is a pa ial
pe spec i e. In an AC ne wo k, physical elec ici y flows a e ha d o
con ol and canno be di ec ed. The e o e, in highly meshed
ansmission ne wo ks (e.g. con inen al Eu ope) he di ec ly-
connec ed coun ies a e no he only playe s in de e mining
c oss-bo de flows. Changes in spa ial gene a ion/load pa e ns in
non-neighbou ing coun ies e e be a e beyond na ional bo de s
and impac o he egions and/o c oss-bo de in e connec ions in
he ne wo k. We do no conside hese e ec s beyond immedia e
neighbou s
9
; igno ing gene al equilib ium/ne wo k e ec s is no so
much o a p oblem when add essing links o isola ed sys ems like
Spain [14]. Fu he , his is no only a echnical issue. As Kunz [15]
poin s ou , he iden ifica ion o flow pa e ns has impo an e ec s
on he a ailable c oss-bo de capaci y and hence on elec ici y spo
ma ke s. We lea e his issue aside.
3. Theo e ical amewo k
Ou ul ima e goal is o simula e he ansmission income o he
in e connec o in he nea u u e ( he h ee-yea pe iod
2020e2022). We fi s in oduce a s ochas ic model o powe p ices.
This model is hen o be es ima ed wi h publicly a ailable da a.
Pa ame e es ima es allow simula e powe p ices in he wo
coun ies. Nex , i is necessa y o es ima e a model o powe
ansmission along he in e connec o . Howe e , c oss-bo de
flows a e subjec o some cons ain s; his leads o ‘censo ing’
se e al obse a ions, which in u n calls o abandoning he linea
eg ession model and eplacing i wi h a so-called Tobi model.
Upon i s es ima ion (wi h STATA), i is finally possible o simula e
powe p ices along wi h flows and de i e simula ed e enues (wi h
MATLAB).
3.1. A s ochas ic model o elec ici y p ices
As We on [16] poin s ou , he Eu opean con en ion is o e e o
he day-ahead elec ici y p ice as he ‘spo p ice’. We use spo
p ices because o hei g ea e in o ma ional con en and liquidi y.
Besides, hey eflec ma ke undamen als (as opposed o expec a-
ions abou u u e ma ke undamen als, which a e eflec ed in he
p ices o u u es and o wa d con ac s on elec ici y); Hi h [17].
Se e al app oaches ha e been de eloped o analyzing and
p edic ing elec ici y p ices; see We on [16]. So-called educed-
o m (quan i a i e, s ochas ic) models cha ac e ize he s a is ical
p ope ies o powe p ices o e ime. We e e in pa icula o
Esc ibano e al. [18]; Lucía and Schwa z [19]; Sei e and Uh ig-
Hombu g [20]; and Villaplana [21]. We use a modified e sion o
he s ochas ic model in Ma hWo ks [22] o accoun o he e ec s o
non-wo king days. Specifically, we desc ibe he (na u al loga i hm
o ) daily spo p ice p
in a gi en coun y i¼{S (Spain), F (F ance)},
unde he s a is ical measu e, as he sum o wo componen s:
lnpi
¼ ið ÞþXi
(1)
The fi s pa ,
i
ð Þ, is de e minis ic. I includes annual and semi-
annual seasonali ies ( h ough sine and cosine unc ions), a end
( ), and a dummy a iable (D
i
) o weekends and public holidays
(we conside only o ficial na ional holidays, no egional ones):
D
i
¼1 on weekends and non-wo king days, D
i
¼0 o he wise. I
also includes a cons an (
b
i
7
) along wi h 24 pa ame e s (
b
i
j
) ha
co espond o he hou ly seasonali y (H
j7;
;j¼8;…31) in each
coun y:
ið Þ¼
b
i
1sinð2
p
Þþ
b
i
2cosð2
p
Þþ
b
i
3sinð4
p
Þþ
b
i
4cosð4
p
Þ
þ
b
i
5 þ
b
i
6Di
þ
b
i
7þX
31
j¼8
b
i
jHj7;
(2)
The second pa , X
i
, is modelled as a s ochas ic equa ion
10
dXS
¼
a
S
k
SXS
d þ
s
SdWS
þJS
m
S
j;
s
S
jdqS
j(3)
dXF
¼
a
F
k
FXF
d þ
s
FdWF
þJF
m
F
j;
s
F
jdqF
j(4)
EdWS
dWF
¼
d (5)
Specifically, Equa ions (3) and (4) a e O ns ein-Uhlenbeck (OU)
mean- e e ing p ocesses wi h jumps. They include h ee e ms on
he igh hand; he fi s one is a unc ion o X
i
, while he o he wo
a e s ochas ic. Lea ing he la e aside o a momen , he equa ion
can be ew i en as dX
i
¼ð
a
i
k
i
X
i
Þd ¼
k
i
a
i
k
i
X
i
!d . Thus, he
(log) s ochas ic pa o he elec ici y p ice in coun y i ends o-
wa d
a
i
=
k
i
in he long e m, wi h a e e sion speed
k
i
.I X
i
alls
below i s long- un quilib ium alue he pa en hesis will be posi i e,
which induces an inc ease in i s alue (dX
i
>0); and con e sely: i
X
i
ises abo e
a
i
=
k
i
he pa en hesis will be nega i e, pushing X
i
downwa ds (dX
i
<0). In sum, when X
i
depa s om i s long- e m
equilib ium (due o he impac o s ochas ic shocks, namely OU
and jumps), he fi s e m ends o es o e he equilib ium (always
subjec o shocks). Besides, he highe he speed o e e sion
k
i
, he
soone X
i
app oaches i s equlib ium alue. Now, he second e m
gene a es a andom beha iou wi hou jumps. The ola ili y o he
mean- e e ing p ocess is
s
i
;dW
i
is he inc emen o a s anda d
Wiene p ocess. The hi d e m is a Poisson p ocess wi h in ensi y
l
i
( he mean a e o e en occu ence); i ime is measu ed in yea s
hen
l
i
jumps a e expec ed pe yea . The jump size is no mally
dis ibu ed wi h mean
m
i
j
and ola ili y
s
i
j
. He e dq
i
j
is a Poisson
p ocess such ha dq
i
j
¼1 wi h p obabili y
l
i
d , and dq
i
j
¼0 wi h
p obabili y 1
l
i
d . We assume ha dW
i
and dq
i
j
a e independen .
No e ha Equa ions (3) and (4) allow nega i e alues ( he loga-
i hm o some low elec ici y p ices can be nega i e).
On he o he hand, some imes bo h F ench and Spanish p ices
can mo e s ochas ically o common easons. Equa ion (5) shows
ha hese p ocesses a e co ela ed as measu ed by
. In his ega d,
he highe he p ice co ela ion, he lowe he abili y o benefi
om he p ice gap be ween coun ies and hence om he
in e connec o .
3.2. Calib a ion o he p ice model
Calib a ing he abo e jump-di usion model is ela ed o he
9
[36] adop his b oade iew bu aim a a di e en goal.
10
This second pa can be in e p e ed as a special case o he gene al s ochas ic
di e en ial equa ion o he inc emen o he (deseasonalized and de ended) spo
elec ici y p ice in Re . [16].
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
3
mo e gene al p oblem o es ima ing he pa ame e s o con inuous-
ime jump p ocesses om disc e ely sampled da a; Con and
Tanko [23] o e an excellen e iew. Es ima ion p ocedu es ha
in ol e he cha ac e is ic unc ion, such as maximum likelihood
(ML) es ima ion, a e o pa icula in e es om he iewpoin o
s a is ical soundness. Below we will p oceed in wo s eps. Fi s we
add ess he de e minis ic pa o he p ice p ocesses, and hen hei
s ochas ic pa . We s ick wi h daily powe p ices in bo h coun ies.
3.3. Mon e Ca lo simula ion o powe p ices
We simula e he s ochas ic pa o he log p ices by means o an
Eule disc e iza ion o Equa ions (3) and (4):
Xi
þ1¼Xi
þ
a
i
k
iXi
D
þ
s
iffiffiffiffiffiffi
D
p iþ
D
qi
j
m
i
jþ
s
i
jxi
j;(6)
whe e
D
q
i
j
¼1 wi h p obabili y
l
i
D
, and
D
q
i
j
¼0 wi h p obabili y
1
l
i
D
(he e
D
¼1=365). The Poisson beha iou is simula ed
wi h andom numbe s sampled om a bina y dis ibu ion wi h
jump p obabili y
l
i
D
. When he e is a jump i s size is
m
i
j
þ
s
i
j
x
i
j
,
which is simula ed wi h andom numbe s x
i
j
om independen
N(0,1) samples. This amoun s o ex ac ing he jump size om a
no mal dis ibu ion Nð
m
i
j
;
s
i
j
Þ.
Rega ding he OU componen , we gene a e andom samples o
co ela ed daily log p ices o F ance acco ding o his scheme:
S¼xS; F¼
xSþxFffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
q(7)
He e
S
deno es samples o he hi d e m in Equa ion (6) o
Spain, while
F
does so o F ance. Ins ead, x
S
and x
F
a e wo in-
dependen N(0; 1) samples.
is he co ela ion coe ficien be ween
he wo s ochas ic pa s, X
S
and X
F
.
3.4. Es ima ion o powe flows along he in e connec o
D awing on his o ical hou ly da a (2016e2019, Table 1) we as-
sume a maximum ansmission capaci y o 3500 MWh. The e o e,
in ou compu a ions below, expo s and impo s a e le -censo ed
(i.e. censo ed om below) a a alue o ze o, and igh -censo ed
(i.e. censo ed om abo e) a 3500 MWh. Censo ing’means ha
we obse e he independen a iables o all cases, bu he
dependen a iable is obse ed only o e a es ic ed ange o
alues (no i s en i e ange). Censo ing does no change he sample,
bu in ol es loss o in o ma ion in a sys ema ic way. In ou case,
le -censo ed da a a e agg ega ed and included as 0s, and igh -
censo ed ones as 3,500s. Consequen ly, he s anda d Linea
Reg ession Model p o ides inconsis en es ima es o he pa ame-
e s. Ins ead, he Tobi model p o ides consis en es ima es
(assuming, as usual, ha he e o s a e no mal and homoscedas ic);
i uses all o he in o ma ion, including in o ma ion abou he
censo ing. Thus, o dina y leas squa es (OLS) mus be eplaced by
ML es ima ion; see Long [24].
4. Da a
Ou da a se includes daily and hou ly in o ma ion on domes ic
elec ici y spo p ices (V/MWh) along wi h impo s/expo s (MWh)
be ween Spain and F ance; i can be downloaded om he e-sios
da abase (h ps://www.esios. ee.es/). The sample pe iod is
2016e2019, i.e. ou yea s. In pa icula , we ha e 1461 daily p ices
and 35,064 hou ly p ices. Du ing his pe iod he comme cial
in e connec ion capaci y emained cons an a 2800 MW.
The uppe block in Table 1 shows desc ip i e s a is ics o hou ly
powe p ices. Spanish p ices a e 6.38 V/MWh highe han he
F ench ones on a e age (¼49.21e42.84)
11
; he la e a e mo e
ola ile han he o me (20.32 V/MWh s 14.35). Besides, he p ice
gap be ween Spain and F ance shows nega i e skewness (14.83),
i.e. he le ail o he dis ibu ion is longe / a e han he igh one
(in o he wo ds, he p obabili y mass is concen a ed on he igh o
he dis ibu ion). I also displays posi i e excess ku osis (741.26),
ha is, ex eme alues a e, well, mo e ex eme han in a No mal
dis ibu ion (whose ku osis is 3); his is confi med by he
maximum (68.50), minimum (810.96), and he 5% and 95% pe -
cen iles (11.07 and 25.45, espec i ely).
12
Sizeable posi i e o
nega i e p ice gaps con ibu e posi i ely o he economic alue o
he in e connec ion wi h F ance.
The lowe block p o ides in o ma ion abou ac ual powe flows.
Fo ins ance, maximum expo s om Spain o F ance each
3632.08 MWh, and 3755.34 he o he way ound. Thus, he
maximum capaci y o he in e connec o is somewha highe han
3500 MW, abo e i s comme cial capaci y (2800 MW as al eady
s a ed), because o an addi ional capaci y de o ed o SoS. Anyway,
comme cial capaci y is no exac ly cons an (see Figu e A4); i is
pe iodically ese by Red El
ec ica de Espa~
na (REE, he Spanish
ansmission sys em ope a o ). Du ing he sample pe iod he ne
balance shows elec ici y flowing om F ance o Spain (a a a e o
1597.84e399.74 ¼1198.10 MWh on a e age).
Table 2 shows some hou ly p ice and quan i y co ela ions. The
Table 1
Hou ly p ices and powe flows: Desc ip i e s a is ics (2016e2019).
Mean Minimum Maximum S anda d De ia ion Skewness Excess Ku osis Pe cen ile 5% Pe cen ile 95%
Elec ici y P ice Spain (V/MWh) 49.21 0.03 101.99 14.35 0.45 0.77 23.04 70.67
Elec ici y P ice F ance (V/MWh) 42.84 31.82 874.01 20.32 6.64 222.74 17.27 74.74
P ice gap Spain-F ance (V/MWh) 6.38 810.96 68.50 14.75 14.83 741.26 11.07 25.45
Expo s (Spain- >F ance) (MWh) 399.74 0.00 3632.08 774.97 1.91 2.25 0.10 2286.46
Impo s (F ance- >Spain) (MWh) 1597.84 0.00 3755.34 1053.14 0.26 1.21 0.00 3091.84
F ance ne impo s-expo s (MWh) 1198.10 3632.08 3636.90 1692.51 0.97 0.27 2250.88 3057.39
F ance o al impo s þexpo s (MWh) 1997.58 127.75 4291.60 744.84 0.18 0.64 698.41 3153.28
11
ACER (2020, Table 5) shows he a e age gap ac oss he Py enees in 2016 (2.9
V/MWh), 2017 (7.3), 2018 (7.1), and 2019 (8.2). This p ice di e en ial is no he
same as he ‘ma ginal alue o ansmission capaci y’in Spiecke e al. (2017),
which co esponds o he sum (o a e age) o absolu e p ice di e ences be ween
wo egions o e a yea . In ou sample pe iod, he a e age absolu e gap on his
in e connec o has been 9.78 V/MWh. As a e e ence, i was 11 V/MWh ac oss he
England-F ance in e connec o o 2011e12; [14]. In he case o Spain-F ance, ACER
(2020) p o ides yea ly es ima es in 2016 (8 V/MWh), 2017 (10.2), 2018 (10.8), and
2019 (10.1).
12
Jus o pu hese figu es in o con ex [8], analyse fi e pai s o Eu opean
neighbou ing coun ies. Absolu e a e age hou ly sp eads ange be ween 0.27
V/MWh and 15.56 V/MWh, wi h s anda d de ia ions om 17.76 o 40.75. The
maximum sp ead is 915 V/MWh (be ween The Ne he lands and UK), and he
minimum sp ead is 901 V/MWh (be ween Ge many and he Ne he lands), bo h
du ing peak hou s.
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
4
co ela ion be ween Spanish and F ench p ices is 0.6878.
13
As ex-
pec ed, Spanish impo s om F ance a e posi i ely co ela ed wi h
he p ice gap be ween hese coun ies (0.4639). A he same ime,
Spanish expo s o F ance a e nega i ely co ela ed wi h he p ice
gap (0.5261).
14
Thus, he p ice gap is a majo d i e o powe flow
along he in e connec ion wi h F ance.
Now, Table 3 shows desc ip i e s a is ics o daily elec ici y
p ices. The a e age p ice gap wi h F ance emains simila as wi h
hou ly p ices (6.47 V/MWh). No su p isingly, p ice gap ola ili y
(10.79 V/MWh) is lowe han wi h hou ly ones (14.75). Daily p ices
also show posi i e excess ku osis. The a e age daily ne impo
om F ance is 28,754.37 MWh.
As seen in Table 4, he co ela ion be ween Spanish and F ench
daily p ices (0.7549) is a bi highe han wi h hou ly p ices; his
lowe hou ly co ela ion can be explained by di e en seasonali y
in hese coun ies (e.g. di e en hou ly habi s in consume
beha iou ). The co ela ion be ween he p ice gap and powe flows
is 0.6709 o Spanish impo s and 0.7500 o expo s o F ance,
bo h s onge han wi h hou ly p ices.
Fig. 1 displays daily p ices in bo h coun ies. Mos o he ime
F ench p ices a e cheape han Spanish ones. Besides, in bo h
coun ies p ice ola ili y is high. In Spain, he minimum p ice is
lowe han he a e age less h ee imes he s anda d de ia ion
(1.94 <49.21e312.86 ¼10.63), while he maximum p ice in
F ance is highe han he a e age plus fi e imes he ola ili y
(125.67 >42.83 þ516.31 ¼124.38). Usually, whene e he e is
an abno mal peak (o he opposi e) he s a ing p ice is mo e o less
no mal and hen e u ns owa d a no mal le el in he ollowing
day.
Fig. 2 shows he daily p ice gap be ween hese coun ies. The e
is a seasonal pa e n, wi h wide gaps in he summe and na owe
ones in win e . Fu he in o ma ion ex ac ed om ou sample da a
is a ailable in Appendix A.
5. Es ima ion o he p ice model
Rega ding he fi s se en pa ame e s o he de e minis ic pa ,
om Equa ion (2) and applying OLS we de i e he es ima es in
Table 5. Some es ima es a e ele an o he alue o he de e -
minis ic componen .
b
i
1
and
b
i
2
in pa icula e eal a g ea e impac
o annual seasonali y o F ance. O he s, such as
b
S
3
, ha e li le
influence.The es ima es o he end coe ficien s,
b
S
5
¼0.0728 and
b
F
5
¼0:0454, sugges ha he p ice gap has been widening o e
ime. This in u n ansla es in o an inc ease in he economic alue
o he in e connec ion.
We ha e also de i ed nume ical es ima es o he pa ame e s
in ol ed in hou ly seasonali y:
b
i
8
,…,
b
i
32
, wi h i¼S,F. Fo his
pu pose we calcula e he di e ence be ween he (log) p ice in each
hou o a day and he (log) p ice in ha day. Thus, since he sample
comp ises 1461 days, we ha e 1461 di e ences o each o he 24 h.
The a e age o hose 1461 di e ences o , say, he fi s hou o he
day, is he seasonali y o ha hou . The p ocess is epea ed o each
o he emaining hou s and sepa a ely o F ance and Spain. Thus,
he numbe s in Table 6 a e o be in e p e ed wi h espec o he
daily p ice (in a gi en day): a posi i e ( espec i ely, nega i e) figu e
means an hou ly p ice abo e ( esp. below) he o e all daily p ice
(no e ha we use log p ices).
As can be seen in Fig. 3, hou ly seasonali y shows wide a ia-
ion in F ance, wi h peaks and oughs u he away om each
o he han in Spain. Maximum hou ly p ices end o happen a ound
20:00 in F ance and 22:00 in Spain; he minimum p ices a e usually
eached abou 5:00 in bo h coun ies. In F ance below-a e age
p ices a e ound om 1:00 ill 7:00; in Spain hey un un il 8:00.
These di e en hou ly pa e ns can impac bo h expo and impo
powe flows be ween he wo coun ies.
Upon es ima ion o
i
ð Þwe can b eak he p ice p ocess in o i s
wo componen s: de e minis ic and s ochas ic. The uppe panel in
Fig. 4 shows he (na u al loga i hm o ) powe p ice in Spain, lnðp
S
Þ,
alongside i s de e minis ic pa ,
S
ð Þ. The lowe panel, ins ead,
displays lnðp
S
Þwi h
S
ð Þ emo ed, i.e. he s ochas ic pa , X
S
.
Simila ly o F ance, he uppe panel in Fig. 5 shows lnðp
F
Þ,
alongside
F
ð Þ. Ins ead, he lowe panel displays he s ochas ic
componen , X
F
.
Conce ning he s ochas ic pa o he (na u al loga i hm o )
powe p ices, X
i
, we ollow maximum likelihood es ima ion (see
Appendix B) and ob ain he pa ame e es ima es in Table 7. On he
o he hand, he co ela ion coe ficien be ween X
S
and X
F
is
¼
0:6570. This is somewha di e en om he one ob ained wi h
daily p ices (0.7549), because i e e s only o he s ochas ic pa s o
he (log) p ices.
In Spain an a e age o 32.15 jumps a e expec ed pe yea and
37.88 in F ance; hus, he daily (
D
¼1=365) jump p obabili ies a e
l
i
d ¼0.0881 and 0.1038, espec i ely. Jumps in Spain ollow a
no mal dis ibu ion Nð
m
S
j
;
s
S
j
Þ¼Nð-0.1347; 0.5462). In F ance hey
beha e acco ding o Nð-0.0973, 0.4431); his sugges s nega i e, less
p onounced, and less ola ile jumps. Howe e , in he absence o
jumps, he log p ice in F ance is mo e ola ile (2.4374) and ends o
e u n as e (67.4638) o i s long- e m equilib ium alue.
6. Mon e Ca lo simula ion o powe p ices
Ou nume ical applica ion o he scheme in Equa ion (7) gen-
e a es 10,000 co ela ed andom samples wi h
¼0.6574, e y
close o he es ima ed alue o 0.6570. We un 10,000 simula ions
o h ee yea s (2020, 2021 and 2022), i.e.1096 days, unde he eal-
wo ld p obabili y measu e.
Table 2
Hou ly p ices and powe flows: Co ela ion coe ficien s (2016e2019).
Elec ici y P ice Spain Elec ici y P ice F ance P ice gap Spain-F ance Expo s (Spain- >F ance) Impo s (F ance- >Spain)
Elec ici y P ice Spain 1.0000
Elec ici y P ice F ance 0.6878 1.0000
P ice gap Spain-F ance 0.0251 0.7084 1.0000
Expo s (Spain- >F ance) 0.0104 0.3747 0.5261 1.0000
Impo s (F ance- >Spain) 0.0874 0.2751 0.4639 0.7075 1.0000
13
[29] es ima e a co ela ion o 0.6524 based on 81 mon hly p ice obse a ions
be ween 2004 and 2011. Bo h figu es a e simila o he co ela ion (0.67) be ween
hou ly p ices in F ance and he UK om No embe 2001 h ough June 2009 ound
by Re . [8].
14
These signs a e consis en wi h esul s in Re . [36]. D awing on mon hly da a o
29 Eu opean coun ies hey find ha powe p ice (as an explana o y a iable o ne
expo s) is s a is ically significan in mos o hei specifica ions; i has a nega i e
impac , i.e. dec easing domes ic p ices make ne expo s mo e appealing. In e -
es ingly o ou case, hey also find ha , on a e age, neighbou ing coun ies (Spain)
o “la ge”coun ies (F ance) a e posi i e ne impo e s (admi edly, a low o de s o
magni ude).
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
5
S a ing wi h Spain, ini ially we simula e he s ochas ic daily
pa , X
S
using Equa ion (7). In a second s ep we add he de e -
minis ic daily pa ,
S
ð Þ, acco ding o Equa ion (1), wi h he annual
and semi-annual seasonali ies, end, e ec s o weekend and non-
wo king days, and a cons an . Finally, we ans o m he log p ices
in o absolu e p ices (in V/MWh). Fig. 6 shows he his o ical pa h o
he daily log p ice o e 2016e2019 along wi h a simula ed pa h o
2020e2022 and he de e minis ic pa alone.
As be o e, we add he de e minis ic pa o X
F
and finally come
up wi h simula ed pa hs o daily powe p ices o F ance. Thus, we
ha e 10,000 simula ed pa hs o u u e daily p ices in each coun y.
Hence we can compu e 10,000 daily p ice gaps be ween hese
coun ies o e e y single day o e he pe iod 2020e2022. Fig. 7
displays he a e age o 10,000 daily gaps in any day du ing his
pe iod. The gap shows a seasonal beha iou ; he same applies o
he obse ed p ice gap (see Fig. 2).
Nex , we ans o m he simula ed daily log p ice se ies, lnðp
i
Þ,
in o hou ly se ies by applying he hou ly seasonali y coe ficien s
(Table 6) o each o he o me se ies ( hus ob aining 24 log p ices
o each day); he log p ices a e u he ansla ed in o absolu e
p ices (V/MWh). Finally, we compu e he hou ly p ice gaps o e
2020e22, namely 24 (366 þ365þ365) ¼26,304 hou ly gaps o
each o ou 10,000 simula ions, i.e. 10,000 hou ly pa hs o 26,304
alues each. Fig. 8 shows he esul ing p obabili y dis ibu ion. The
10% pe cen ile is 13.63 V/MWh while he 90% pe cen ile is 39.19
V/MWh. The a e age is 13.65 and he median a bi highe , namely
13.88 V/MWh. The dis ibu ion displays nega i e skewness.
7. Powe flows along he Spain-F ance in e connec o
Based on he 35,064 hou ly obse a ions o Spain, we es ima e
a Tobi model o expo s o F ance and ano he one o impo s
Table 3
Daily p ices and powe flows: Desc ip i e s a is ics (2016e2019).
Mean Minimum Maximum S anda d
De ia ion
Skewness Excess Ku osis Pe cen ile 5% Pe cen ile 95%
Elec ici y P ice Spain (V/MWh) 49.21 1.94 91.88 12.86 0.52 0.87 25.41 68.01
Elec ici y P ice F ance (V/MWh) 42.83 2.66 125.67 16.31 0.99 1.98 20.86 70.89
P ice gap Spain-F ance (V/MWh) 6.47 68.04 50.87 10.79 0.99 5.41 9.53 21.90
Expo s (Spain- >F ance) (MWh) 9593.75 0.00 71,261.77 15,472.64 1.80 2.22 2.60 47,360.50
Impo s (F ance- >Spain) (MWh) 38,348.12 0.00 78,997.68 21,106.50 0.23 0.99 1369.82 70,036.63
F ance ne impo s-expo s (MWh) 28,754.37 70,673.92 78,994.02 34,564.26 0.95 0.08 44,301.98 69,326.31
F ance o al impo s þexpo s (MWh) 47,941.86 15,596.54 81,699.32 13,231.98 0.19 0.63 27,997.47 71,082.50
Table 4
Daily p ices and powe flows: Co ela ion coe ficien s (2016e2019).
Elec ici y P ice Spain Elec ici y P ice F ance P ice gap Spain-F ance Expo s (Spain- >F ance) Impo s (F ance- >Spain)
Elec ici y P ice Spain 1.0000
Elec ici y P ice F ance 0.7549 1.0000
P ice gap Spain-F ance 0.0465 0.6171 1.0000
Expo s (Spain- >F ance) 0.0605 0.4457 0.7500 1.0000
Impo s (F ance- >Spain) 0.1420 0.3266 0.6709 0.7805 1.0000
Fig. 1. Daily spo elec ici y p ices in F ance and Spain, 2016e2019.
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
6
om F ance. Tables 8 and 9 show he esul s om STATA,
espec i ely.
Rega ding Spanish expo s, he likelihood a io (LR) chi-squa e
wi h p- alue ¼0.0000 in o ms us ha he Tobi model is signifi-
can ly be e han an emp y model. The coe ficien s a e s a is ically
significan . The/sigma s a is ic (671.7724) is he es ima ed s anda d
e o o he eg ession. Wi h he nume ical es ima e o he p ice
gap coe ficien (28.06604) we simula e powe expo s o F ance
unde unce ain y: Expo s (S/F) ¼564.46e28.06 simula ed
gap þN(0; 671.77).
As o Spanish impo s, again he LR ells us ha he Tobi model
is significan ly be e han an emp y model. The coe ficien s a e
s a is ically di e en om ze o. The es ima ed s anda d e o o he
eg ession in his case is 958.8273. We use he gap coe ficien
(53.39729) o simula e powe impo s om F ance unde
unce ain y.
8. Simula ed ansmission income o he in e connec o
Powe flows om one coun y o he o he gi e ise o a mon-
e a y income (conges ion en ) he size o which depends on he
p ice gap be ween hem. A he same ime, we assume ans-
mission cos s o 5 V/MWh (as in Re . [25]; o [8]. Thus, some imes
he ne income can be nega i e because o his ansmission cos .
None heless, i can also be nega i e because he e can be expo s
when he gap p ice is posi i e (i.e. powe flows om Spain o F ance
despi e i s highe p ice in Spain), he same way ha he e can be
impo s when he gap is nega i e ( ha is, Spain pu chases powe
Fig. 2. Daily powe p ice gap be ween Spain and F ance, 2016e2019.
Table 5
Pa ame e es ima es o p ice p ocesses (2016e2019, daily da a): de e minis ic pa
i
ð Þ, as shown in Eq. (2).
Pa ame e Spain (i¼S) F ance (i¼F)
b
i
1
0.1288 0.1597
b
i
2
0.0350 0.2081
b
i
3
0.0007 0.0329
b
i
4
0.0662 0.0206
b
i
5
0.0728 0.0454
b
i
6
0.1770 0.3233
b
i
7
3.7569 3.6939
Table 6
Pa ame e es ima es o p ice p ocesses (2016e2019, daily da a): de e minis ic pa
i
ð Þ, hou ly seasonali y in Eq. (2).
Spain F ance
Hou
b
S
h
Hou
b
S
h
Hou
b
F
h
Hou
b
F
h
1:00 0.0237 13:00 0.0471 1:00 0.0906 13:00 0.0551
2:00 0.1154 14:00 0.0401 2:00 0.1990 14:00 0.0056
3:00 0.1838 15:00 0.0084 3:00 0.2747 15:00 0.0675
4:00 0.2274 16:00 0.0324 4:00 0.3868 16:00 0.0941
5:00 0.2463 17:00 0.0438 5:00 0.4277 17:00 0.0735
6:00 0.2075 18:00 0.0090 6:00 0.3143 18:00 0.0455
7:00 0.1163 19:00 0.0385 7:00 0.1271 19:00 0.1802
8:00 0.0177 20:00 0.0887 8:00 0.0432 20:00 0.2281
9:00 0.0334 21:00 0.1206 9:00 0.1187 21:00 0.1640
10:00 0.0693 22:00 0.1324 10:00 0.1315 22:00 0.0804
11:00 0.0713 23:00 0.0872 11:00 0.1014 23:00 0.0884
12:00 0.0575 24:00 0.0153 12:00 0.0861 24:00 0.0132
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
7
om F ance in spi e o highe p ices he e). This beha iou ema-
na es om he ac ual beha iou eflec ed in he Tobi model
(excluded 1073 þ3433 nega i e obse a ions; he 96 obse a ions
abo e 3500 a e censo ed). Fig. 9 displays he p obabili y dis ibu-
ion o cumula i e e enues and expenses ( om he iewpoin o
Spain) o e he simula ion ho izon.
Table 10 shows a ew basic desc ip i e s a is ics o he 3-yea
ansmission income. Powe expo s gene a e an a e age o V27
million o e 2020-22 while impo s en ail expenses o V964
million on a e age du ing he same pe iod. The alue o bila e al
Fig. 3. Log p ice o elec ici y in F ance and Spain (2016e2019, hou ly da a): hou ly seasonali y ( he a e age o he di e ence be ween he log p ice in each hou o a day and he log
p ice in ha day).
Fig. 4. Log elec ici y p ice in Spain (2016e2019, daily da a): de e minis ic and s ochas ic pa s.
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
8
Fig. 5. Log elec ici y p ice in F ance (2016e2019, daily da a): de e minis ic and s ochas ic pa s.
Table 7
Pa ame e es ima es o p ice p ocesses (2016e2019, daily da a): s ochas ic pa X
i
, Eqs. (3) and (4).
Pa ame e Spain F ance
Value 95% confidence in e al Value 95% confidence in e al
a
4.3161 2.2018e6.4304 3.7019 0.8469e6.5568
k
56.3486 46.2128e66.4844 67.4638 56.9641e77.9635
m
j
0.1347 0.2378e0.0316 0.0973 0.1842e0.0105
s
1.8261 1.7266e1.9205 2.4374 2.2788e2.5862
s
j
0.5462 0.4547e0.6244 0.4431 0.357e0.515
l
32.1534 23.7531e40.5537 37.8801 24.1242e51.636
Fig. 6. Ac ual daily log p ice o elec ici y in Spain (2016e2019) and a simula ed s ochas ic pa h (2020e2022).
L.M. Abadie and J.M. Chamo o Ene gy 233 (2021) 121177
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