Disco e ing Pa e ns in Elec ici y P ice Using Clus e ing Techniques
F. Ma ínez Ál a ez1, A. T oncoso2, J. C. Riquelme1, J. M. Riquelme3
1 Depa amen o de Lenguajes y Sis emas In o má icos
Escuela Técnica Supe io de Ingenie ía In o má ica. Uni e sidad de Se illa
Phone: +0034 954 552775, e-mail: [email p o ec ed], [email p o ec ed]
2 Á ea de Lenguajes y Sis emas In o má icos
Escuela Poli écnica Supe io . Uni e sidad Pablo de Ola ide
Phone: +0034 954 977522, e-mail: [email p o ec ed]
3 Depa amen o de Ingenie ía Eléc ica
Escuela Supe io de Ingenie os. Uni e sidad de Se illa
Phone: +0034 954 481274 e-mail: [email p o ec ed]
Abs ac . Clus e ing is a p ocess o g ouping simila
elemen s ga he ed o occu ed closely oge he . This pape
p esen s wo clus e ing echniques, K-means and Fuzzy C-
means, o he analysis o he elec ici y p ices ime se ies. Bo h
algo i hms a e ocused on ex ac ing use ul in o ma ion om
he da a wi h he aim o model he ime se ies beha iou and
ind pa e ns o imp o e he p ice o ecas ing. The main
objec i e, hus, is o ind a ep esen a ion ha p ese es he
o iginal in o ma ion and desc ibes he shape o he ime se ies
da a as accu a ely as possible. This esea ch demons a es ha
he applica ion o clus e ing echniques is e ec i e in o de o
dis inguish se e al kinds o days. To be p ecise, wo majo
g oups can be dis inguished hanks o he clus e ing: he i s
one ha includes he wo king days and he second one ha
includes weekends and es i i ies. Equally ema kable is he
simila i y shown among days belonging o a same season.
Key wo ds
Clus e ing, p ice o ecas ing, ime se ies model.
1. In oduc ion
I is impo an o ob ain an app oach o op imize he
bidding s a egies ca ied ou by elec ici y-p oduce
companies [1]. Consequen ly, he de elopmen o
o ecas ing echniques is becoming inc easingly ele an
in he cu en hec ic Spanish elec ici y-ma ke
de egula ion.
This wo k is ocused on ex ac ing meaning ul
in o ma ion o he p ices ime se ies by using clus e ing
echniques. Clus e ing is he basis o many classi ica ion
and sys em modelling algo i hms. The main a ge o
clus e ing is o gene a e g oupings o da a om a la ge
da ase wi h he in en ion o p oducing an accu a e
ep esen a ion o he beha iou o a sys em.
Thus, he esea ch is based on he applica ion o wo
well-known clus e ing me hods, K-means and uzzy
clus e ing [2], o inding hose g oups o p ices which
show a simila beha iou unde some pa icula
condi ions such as wo king / non-wo king days o
seasons. La e , his in o ma ion can be used o
p edic ing how he p ices will p og ess h oughou he
nex day.
O he esea che s ha e de eloped echniques o o ecas
he p ices ime se ies. Recen ly, A. J. Conejo e al. [3]
p oposed a o ecas ing model using he wa ele
ans o m and ARIMA models. Equally, R. C. Ga cía e
al. [4] p esen ed a o ecas ing echnique based on a
GARCH model. In [5] a me hod combining A i icial
Neu al Ne wo ks wi h uzzy logic is p oposed. In [6] an
adap i e non-pa ame ic eg ession app oach is applied o
o ecas he hou ly On a io ene gy p ice. In [7] a simple
model based on he Weigh ed Nea es Neighbou s
me hodology is p esen ed and i s pe o mance is
compa ed wi h o he s ecen ly published echniques.
Howe e , i can be s a ed ha he o ecas ing echniques
o he nex -day elec ici y p ices published in he cu en
li e a u e do no use p e ious clus e ing echniques.
Consequen ly, i is necessa y o disco e pa e ns in he
elec ici y p ices ime se ies o imp o e he p edic ion
models.
The inal goal is o p o ide se e al p edic ions o he
p ice e olu ion cu e o he subsequen day. This
in o ma ion would be used in op imiza ion models in
o de o help o ma ke agen s o gene a e hei op imal
bidding s a egies. Thus, he i s objec i e is o
de e mine a p e ious clus e ing o e eal p ices
popula ion cu es in o de o ob ain g oups o days
acco ding o he p ice o he elec ici y hou by hou .
F om his di ision, he connec ion be ween belonging o
a ce ain clus e and he model o p edic ion o each
clus e could be ound. The e o e, i would be chosen he
clus e o which he cu en day i belong in o de o
p edic he p ices cu e o he ollowing day. Finally, he
p edic ion would be gene a ed by means o he
co esponding model o his clus e .
The no el and main con ibu ion o he pape is o apply
clus e ing echniques o he elec ici y p ices ime se ies
o disco e simila pa e ns. The pa e ns p o ide use ul
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in o ma ion o imp o e he o ecas ing echniques. The
ime se ies is he a ia ion o he p ice o he elec ici y
h oughou he day. The mo e days conside ed in he
da ase , he mo e p ecise will be he p edic ion.
The es o he pape is o ganized as ollows. Sec ion 2
de ails he wo clus e ing echniques applied o ind
pa e ns in ime se ies. As selec ing he numbe o
clus e s esul s a key p ocess, Sec ion 3 explains he
mo i a ion o choosing he numbe o clus e s in bo h
algo i hms. Sec ion 4 p esen s all he esul s ob ained, as
well as i compa es bo h echniques. A desc ip ion o he
da ase used is also shown in his sec ion. Finally, Sec ion
5 expounds he conclusions achie ed and he u u e
wo k.
2. Me hodology
Clus e ing is a p ocess o g ouping an unlabeled se o
examples in o a numbe o clus e s such ha a simila
pa e n is associa ed o e e y clus e , ha is o say,
clus e ing ope a es on a se o examples ha mus be
pa i ioned acco ding o some no ion o simila i y.
Clus e analysis echniques ha e been classi ied in o wo
majo me hods:
1) C isp clus e ing (o ha d clus e ing) in which
he bounda y be ween clus e s is ully de ined.
2) Fuzzy clus e ing in which he bounda y be ween
clus e s can no be clea ly de ined (such is he
case o many eal cases).
Bo h app oaches p esen a la ge se o algo i hms, mos
o hem designed o speci ic p oblems. In his pape wo
di e en clus e ing-based echniques ha e been used in
o de o iden i y pa e ns o beha iou in he p ices
cu es: K-means, ep esen ing he c isp clus e ing and
he Fuzzy C-means (FCM) ep esen ing he uzzy
clus e ing.
A. k-means algo i hm
K-means is a as me hod o pe o m clus e ing. The
basic in ui ion behind K-means is he con inuous
eassignmen o objec s in o di e en clus e s so ha he
wi hin-clus e dis ance is minimized.
I uses an i e a i e algo i hm di ided in wo phases o
minimize he sum o poin - o-cen oid dis ances, o e all
k clus e s.
In he i s phase, each i e a ion consis s o eassigning
poin s o hei nea es clus e cen oid and hen i
ecalcula es he clus e cen oids.
In he second phase, poin s a e indi idually eassigned i
doing so educe he sum o dis ances; clus e cen oids
a e ecompu ed a e each eassignmen . Each i e a ion
consis s o one pass h ough all he poin s. Bo h phases
a e summa ized in Table I, which desc ibes he k-means
in e ms o i s basic s eps.
Table I. – Ou line o he k-means algo i hm
STEP DESCRIPTION
1 Decide a alue o k
2 Ini ialize he k clus e cen es
3 Assign an example o he nea es clus e cen e
4 Re-calcula e he k clus e cen es assuming
ha he membe ships ound in s ep 3 a e
co ec
5 Exi i no example changes o clus e in he
las i e a ion. O he wise go o s ep 3.
B. Fuzzy C-means algo i hm
The Fuzzy C-means clus e ing, whe e C is he numbe o
clus e s o classi y, he da a is a echnique whe ein each
da a belongs o a clus e o some deg ee speci ied by a
membe ship g ade. I p o ides a me hod ha shows how
o g oup da a poin s ha popula e some mul idimensional
space in o a speci ic numbe o di e en clus e s.
The FCM algo i hm ocuses on minimizing he alue o
an objec i e unc ion which calcula es he weigh ed
wi hin-g oup sum o squa ed e o s. To measu e he
quali y o he pa i ioning i compa es he dis ance om
an example o he cu en candida e clus e cen e wi h
he dis ance o o he candida e clus e cen es. Table II
shows he summa ized s eps ollowed in he algo i hm.
Table II. – Ou line o he Fuzzy C-means algo i hm
STEP DESCRIPTION
1 Decide a alue o C
2 Ini ialize he clus e cen e ma ix, W( =0)
3 Ini ialize he membe ship ma ix, U( =0)
4 Inc ease by one and compu e W( )
5 Compu e U( )
6 I (U( ) – U( -1)) is lowe han a gi en e o s op.
O he wise go o s ep 4.
While hese wo algo i hms a e ypically used in he
li e a u e ela i e o clus e ing app oaches in ime se ies,
hey p esen a well-known sho coming: he numbe o
clus e s mus be speci ied in ad ance. The choice o his
pa ame e will be jus i ied in he subsequen sec ion.
3. Selec ion o he numbe o clus e s
The numbe o clus e s selec ed is one o he mos c i ical
decisions in clus e ing echniques. The ac o choosing a
la ge numbe o clus e s does no necessa ily imply ha e
a be e quali y o in o ma ion. On he con a y, esul s
could be unclea and could muddle he pa e n
ecogni ion up. This limi a ion can be mi iga ed by
es ing all alues o K o C clus e s wi hin a la ge ange.
Fu he s a is ical es can, hen, be used o de e mine
which alue o K o C i s be e . Sec ions 3.A and 3.B
show a me hodical way o selec he op imal numbe o
clus e s o bo h echniques.
A. Numbe o clus e s in K-means
The silhoue e unc ion in Ma lab p o ides a measu e o
he clus e s sepa a ion. I s alue a ies be ween –1 and
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+1, whe e +1 deno es clea clus e sepa a ion and –1
ma ks poin s wi h ques ionable clus e assignmen . A
success ul clus e ing has a mean silhoue e alue highe
han 0,6 o all clus e s. Howe e , in eal ime se ies i is
almos impossible o each his alue and no ha ing
nega i e alues in he igu e is usually enough o decide
how many clus e s ha e o be chosen.
Figu es 1.a, 1.b and 1.c show he plo ed silhoue e
unc ion o 4, 5 and 6 clus e s espec i ely o he p ices
o he elec ici y o he yea 2005. The me ic used was
squa ed Euclidean dis ance since cosine me ics ga e
wo se esul s. Fo u he analysis, 4 clus e s ha e been
chosen due o ha only one clus e has nega i e alues
and i s g aphical ep esen a ion p o ides sa is ac o y
esul s.
B. Numbe o clus e s in Fuzzy C-means
The FCM clus e ing algo i hm is sensi i e o he si ua ion
o he ini ializa ion and easy o all in o a local minimum
o a saddle poin when i e a ing. To sol e his p oblem
se e al o he echniques ha e been de eloped ha a e
based on global op imiza ion me hods [8]. Howe e , in
many p ac ical applica ions he clus e ing me hod ha is
used is FCM wi h mul iple es a s o escaping om he
sensibili y o ini ial alue.
The subclus unc ion in Ma lab inds clus e cen es and
i is commonly used in o de o ob ain he op imum
numbe o clus e s in i e a i e op imiza ion-based
clus e ing me hods such as FCM.
This unc ion es ima es he clus e cen es in a se o da a
by using he sub ac i e clus e ing me hod. I assumes
ha each da a poin is a po en ial clus e cen e and
calcula es a measu e o he likelihood ha each da a poin
would de ine he clus e cen e, based on he densi y o
su ounding da a poin s. A e he execu ion o his
algo i hm, i was ound ha 6 is he op imum numbe o
clus e s.
Fig. 1.a. Silhoue e alues wi h 4 clus e s.
Fig. 1.b. Silhoue e alues wi h 5 clus e s.
Fig. 1.c. Silhoue e alues wi h 6 clus e s.
4. Resul s
A. Da ase desc ip ion
The da a sou ce is he p ices o he elec ici y ma ke o
mainland Spain o he yea 2005 (OMEL) [9].
Be o e ope a ing wi h he elec ici y p ices, da a
no maliza ion was ca ied ou wi h he aim o a oiding
he e ec s o he g ow h o he in a-annual p ices. The
no maliza ion was pe o med by di iding he hou ly
p ices by he a e age p ice o he whole day.
B. K-means
Figu e 2 shows he yea 2005 classi ied in o 4 clus e s, as
jus i ied in sec ion 3.B, ia he k-means algo i hm. Wi h
jus a quick look, i can be clea ly di e en ia ed wo
kinds o clus e s: clus e s 1 and 2 g oup all he wo king
days and clus e s 3 and 4 he weekends. Ne e heless,
he e a e some days ha ha e an appa en ly disco dan
beha iou . Table IV shows he pe cen age o days
classi ied in o he 4 clus e s.
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Table IV. – G ade o membe ship o days o clus e s
Clus e 1 Clus e 2 Clus e 3 Clus e 4
Monday 36,54% 51,92% 3,85% 7,69%
Tuesday 31,48% 57,41% 3,70% 7,41%
Wednesday 30,77% 63,46% 3,85% 1,92%
Thu sday 32,69% 59,62% 5,77% 1,92%
F iday 28,85% 59,62% 3,85% 7,69%
Sa u day 11,32% 0,00% 39,62% 49,06%
Sunday 0,00% 0,00% 44,23% 55,77%
The e a e 22 wo king days ha ha e been g ouped in
clus e s 3 o 4. A me iculous analysis e eals ha mos o
hese days we e holiday. A de ailed lis o his ac is
summa ised in Table V.
Table V. – W ong classi ica ion o wo king days
Nº OF DAY DATE FESTIVITY
6 06-01 Epiphany
70 11-03 None
75 16-03 None
77 18-03 F iday p e-Eas e
82 23-03
83 24-03
84 25-03
Eas e
87 28-03 Monday pos -Eas e
98 08-04 None
122 02-05 Wo king es i i y
123 03-05 Mad id es i i y
125 05-05 Long weekend 01-05
126 06-05 Long weekend 01-05
227 15-08 Assump ion o Ma y
231 19-08 None
235 23-08 None
285 12-10 Columbus Day
304 31-10 Long weekend 01-11
305 01-11 All Sain s
340 06-12 Spanish Cons i u ion Day
342 08-12 Immacula e Concep ion
360 26-12 Monday a e Ch is mas
One commen has o be done abou he i s week o
May. The eal holiday o he Wo king Day is he 1s May
and o he Mad id Fes i i y he 2nd May. Howe e , 1s
May 2005 was Sunday and bo h es i i ies we e
pos poned one day.
Wi h e e ence o weekends, he e a e six Sa u days ha
ha e been g ouped ha ha e been g ouped as i hey
we e wo king days, conc e ely, in clus e 1. A de ailed
lis o hese Sa u days is shown in Table VI.
Table VI. – Sa u days classi ied in w ong clus e s
NUMBER OF DAY DATE
169 18 h June
176 25 h June
183 2nd July
197 16 h July
204 23 d July
211 30 h July
Fig. 2. Days classi ied in o 4 clus e s ia K-means.
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Fig. 3. Cha ac e is ic cu es o clus e s ob ained by K-means algo i hm in yea 2005.
No e ha almos all six Sa u days a e consecu i e and
belong o summe , excep o he 9 h July ha has been
classi ied in o clus e 4.
The whole yea is di ided in o 261 wo king days and 104
weekends o es i i ies. In Table V, i e days we e
imp ope ly classi ied (11 h Ma ch, 16 h Ma ch, 8 h Ap il,
19 h Augus and 23 d Augus ). Hence, he a e age e o in
wo king days is 1,92% (5 days ou o 261).
Wi h ega d o weekends and es i i ies, he e a e 6
Sa u days which ha e been imp ope ly g ouped (18 h
June, 25 h June, 2nd July, 16 h July, 23 d July and 30 h
July). Gi en ha , he a e age e o o weekends and
es i i ies is 5,77% (6 days o ou 104), he o al e o is
3,01% (11 days ou o 365).
The ollowing ask consis s in explaining when a wo king
day belongs o clus e 1 o o clus e 2 as well as when
es i i ies belong o clus e 3 o o clus e 4: he e a e
h ee zones clea ly di e en ia ed in Figu e 2 o bo h
wo king days and es i i ies. F om he 1s Janua y un il
he 18 h May (day numbe 144), mos o he wo king days
belong o clus e 2. F om his day un il he 20 h
Sep embe (day numbe 263) hey belong o clus e 1.
Finally, om he 21s Sep embe (day numbe 264) un il
he yea ends he wo king days belong again o clus e 2.
In es i i ies he e is a simila si ua ion. F om he 1s
Janua y un il he 27 h Ma ch (day numbe 86) mos o he
es i i ies and weekends belong o clus e 3. F om his
weekend un il 30 h Oc obe (day numbe 303) hey
belong o clus e 4. Finally, om his weekend un il he
yea ends he es i i ies and weekend belong o clus e 3.
Consequen ly, a seasonal beha iou can be obse ed in
he ene gy p ices ime se ies.
The cha ac e is ic cu es o each clus e a e depic ed by
Figu e 3. Especially ema kable is ha cu es associa ed
o clus e s 3 and 4 (weekends and es i i ies) ha e
s a ing and ending p ices highe han he ones associa ed
o he wo king days (clus e s 1 and 2). The i s ones
show hei highe alues in he la e a e noon. I is due o
people consuming mo e elec ici y all he nigh long
du ing weekends. On he o he hand, he second ones
ha e hei peak p ices a midday when indus ies,
comme ce and en e p ises a e ully unc ioning.
C. FCM.
Figu e 4 p esen s he six pa e ns ound by he FCM
algo i hm o he ene gy p ices o he yea 2005. I can be
no ed ha hese pa e ns a e no e y di e en o he
pa e ns ob ained by using he K-means app oach. Fo he
ep esen a ion o hese cu es he ollowing me hodology
has been used. Fi s , he clus e wi h he maximum g ade
o membe ship was assigned o e e y day. Then, he
ep esen a ion was pe o med like in K-means algo i hm
as i has depic ed in Figu e 4.
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Focusing on Figu e 4, i can be clea ly di e en ia ed wo
kinds o clus e s: clus e s 2, 3, 4 and 5 g oup all he
wo king days while clus e s 1 and 6 he weekends.
Ne e heless, he e a e some days ha ha e an appa en ly
disco dan beha iou . Table VII shows he pe cen age o
days classi ied in o he 6 clus e s.
Table VII. – G ade o membe ship o days o clus e s
CLUSTER 1 CLUSTER 2 CLUSTER 3
Monday 7,69% 15,38% 32,69%
Tuesday 0,00% 23,08% 28,85%
Wednesday 0,00% 28,85% 26,92%
Thu sday 3,85% 25,00% 26,92%
F iday 5,77% 25,00% 26,92%
Sa u day 66,04% 3,77% 5,66%
Sunday 53,85% 0,00% 0,00%
CLUSTER 4 CLUSTER 5 CLUSTER 6
Monday 38,46% 1,92% 3,85%
Tuesday 44,23% 0,00% 3,85%
Wednesday 40,38% 0,00% 3,85%
Thu sday 40,38% 0,00% 3,85%
F iday 36,54% 1,92% 3,85%
Sa u day 3,77% 0,00% 20,75%
Sunday 0,00% 0,00% 46,15%
The e a e 19 wo king days ha ha e been g ouped in
clus e s 1 o 6. Table VIII summa izes he es i i ies
ound in hese days.
Table VIII. – W ong classi ica ion o wo king days
Nº OF DAY DATE FESTIVITY
6 06-01 Epiphany
70 11-03 None
75 16-03 None
77 18-03 F iday p e-Eas e
82 23-03 Eas e
83 24-03 Eas e
84 25-03 Eas e
87 28-03 Monday pos -Eas e
98 08-04 None
122 02-05 Wo king es i i y
125 05-05 Long weekend 01-05
126 06-05 Long weekend 01-05
136 16-05 None
227 15-08 Assump ion o Ma y
304 31-10 Long weekend 01-11
305 01-11 All Sain s
340 06-12 Spanish Cons i u ion Day
342 08-12 Immacula e Concep ion
360 26-12 Monday a e Ch is mas
Wi h e e ence o weekends, he e a e six Sa u days ha
ha e been g ouped as i hey we e wo king days,
conc e ely, in clus e 3. I is shown in Table IX.
Table IX. – Sa u days classi ied in w ong clus e s
NUMBER OF DAY DATE
176 25 h June
183 2nd July
190 9 h July
204 23 d July
211 30 h July
330 26 h No embe
Fig. 4. Days classi ied in o 6 clus e s ia FCM.
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Fig. 5. Cha ac e is ic cu es o clus e s ob ained by K-means algo i hm in yea 2005.
No e ha almos all six Sa u days a e consecu i e and
belong o summe , excep o he 16 h July ha has been
classi ied in o clus e 1. The dis ance om 16 h July o
clus e 1 is 0,7910 ( he clus e o which i belongs) while
o clus e 2 is 0,7970 ( he clus e o which i should
belong, assuming ha all he Sa u days in summe
beha e as i hey we e a wo king day).
The whole yea is di ided in o 261 wo king days and 104
weekends o es i i ies. In able VIII, ou days we e
imp ope ly classi ied (11 h Ma ch, 16 h Ma ch, 8 h Ap il,
16 h May). Hence, he a e age e o in wo king days is
1,53% (4 days ou o 261).
Wi h ega d o weekends and es i i ies, he e a e six
Sa u days which ha e been imp ope ly g ouped (25 h
June, 2nd July, 9 h July, 23 d July, 30 h July and 26 h
No embe ). The e is also a es i i y, Columbus Day,
which has been g ouped in clus e 2. Gi en ha , he
a e age e o o weekends and es i i ies is 6,73% (7
days o ou 104), he o al e o is 3,01% (11 days ou o
365).
In con as o wha i happened wi h K-means clus e ing,
i is no ob ious o de e mine clea pe iods o he yea o
days o belong o a speci ic clus e .
The cha ac e is ic cu es o each clus e a e depic ed by
Figu e 5. Especially ema kable is ha cu es associa ed
o clus e s 1 and 6 (weekends and es i i ies) ha e
s a ing and ending p ices highe han he ones associa ed
o he wo king days (clus e s 2, 3, 4 and 5). The i s ones
show hei highe alues in he la e a e noon. I is due o
people consuming mo e elec ici y all he nigh long
du ing weekends. On he o he hand, he second ones
ha e hei peak p ices a midday when indus ies,
comme ce and en e p ises a e ully unc ioning.
5. Conclusions
I has been p o en ha u ilising clus e ing echniques in
he p ices ime se ies is as powe ul as use ul. Two
algo i hms ha e been used in o de o classi y he
elec ici y p ice cu es o he Spanish Ma ke : K-means
and Fuzzy C-means. The clus e analysis ca ied ou ia
bo h K-means and Fuzzy C-means algo i hms yielded
e y ele an in o ma ion: wo king days ha e beha iou
diame ically opposi e o weekend and es i i ies. The
a e age e o commi ed in hei classi ica ion was
3,01%. Only 11 days o he yea 2005 we e imp ope ly
g ouped which means a g ea deg ee o accu acy o bo h
echniques wi h ei he 4 (K-means) o 6 (Fuzzy C-means)
clus e s.
The esul s ob ained may be ex emely p o i able. Fu u e
wo ks will be di ec ed in he p edic ion o day-ahead
p ices once known he p e ious clus e ing. In sho , N
clus e s will be ob ained and di e en models will be
applied o e e y clus e o imp o e he quali y o he
ma ke p ice o ecas ing.
Acknowledgemen s
The au ho s would like o acknowledge he inancial
suppo om he Spanish Minis y o Science and
Technology, p ojec s TIN2004-00159 and ENE-2004-
03342/CON, and om he Jun a de Andalucía, p ojec
P05-TIC-00531.
h ps://doi.o g/10.24084/ epqj05.245
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