E olu iona y echniques applied o he op imal sho - e m
scheduling o he elec ical ene gy p oduc ion
Alicia T oncoso , Jose
´ C. Riquelme , Jesu
´s S. Aguila -Ruiz ,
Jesu´s M. Riquelme San os
Abs ac
This pape p esen s an e olu iona y echnique applied o he op imal sho - e m scheduling (24 h) o he elec ic ene gy
p oduc ion. The equa ions ha define he p oblem lead o a non-con ex non-linea p og amming p oblem wi h a high
numbe o con inuous and disc e e a iables. Consequen ly, he esolu ion o he p oblem based on combina o ial me h-
ods is a he ha d. The equi ed heu is ics, in oduced o assu e he easibili y o he cons ain s, a e analyzed, along wi h a
b ie desc ip ion o he p oposed gene ic algo i hm (GA). The GA is used o compu e he op imal on/off s a us o he mal
uni s and he fi ness unc ion is ob ained by sol ing a quad a ic p og amming p oblem by means o a s anda d non-linea
In e io Poin (IP) me hod. The esul s om eal-wo ld cases based on he Spanish powe sys em a e epo ed, which show
he good pe o mance o he p oposed algo i hm, aking in o accoun he complexi y and dimensionali y o he p oblem.
Finally, an IP algo i hm is adap ed o deal wi h disc e e a iables ha appea in his p oblem and he ob ained esul s a e
compa ed wi h ha o he p oposed GA.
Keywo ds: Gene ic algo i hms; Scheduling; Op imiza ion; Feasibili y; In e io poin algo i hms
1. In oduc ion
The op imal sho - e m scheduling o he elec i-
cal ene gy p oduc ion [1] aims a de e mining which
gene a ing uni s should be online and he co e-
sponding op imal gene a ion o he mal and hyd o
uni s along he scheduling pe iod, usually 24 h, in
o de o minimize he expec ed o al cos sa is ying
he o ecas ed sys em load. The scheduling ask
leads o a non-linea mixed-in ege p og amming
p oblem. Mo eo e , his p oblem is coupled in ime
by he maximum speed ha gene a ing uni s, spe-
cially he mal uni s, a e able o change he p oduced
ene gy (known as up and down amps), and also by
he opology o he hyd oelec ic powe plan s, wi h
a delay in hou s be ween he wa e o a ese oi
being used and he a ailabili y o ha wa e in he
*
Co esponding au ho .
E-mail add esses: [email p o ec ed] (A. T oncoso), iquelme@
lsi.us.es (J.C. Riquelme), [email p o ec ed] (J.S. Aguila -Ruiz),
[email p o ec ed] (J.M. Riquelme San os).
ese oi s downs eam. A eally la ge numbe o
a iables, bo h con inuous and disc e e a iables,
is needed o p ope ly model his p oblem. Many
app oaches ha e been p oposed o he esolu ion
o his op imiza ion p oblem, anging om
Dynamic P og amming o Linea Mixed-In ege
P og amming o Lag angian Relaxa ion [2], he la -
e being he mos widely used op imiza ion me hod
in comme cial p og ams. Gene ic Algo i hms (GAs)
[3,4], a gene al-pu pose s ochas ic sea ch me hod
based on he mechanics o na u al selec ion, ha e
also been success ully applied o he elec ical
ene gy scheduling p oblem since he adap a ion is
qui e s aigh o wa d due o he combina o ial na -
u e o his p oblem. In he las ew yea s, he In e-
io -Poin (IP) o Loga i hmic-Ba ie class o
me hods has become he p e e ed nume ical
app oach o sol e non-linea op imiza ion p ob-
lems, as a consequence o i s enhanced capabili y
o deal wi h inequali y cons ain s [5,6]. Since he
ea ly de elopmen s, in ended o Linea P og am-
ming p oblems, many imp o emen s ha e been p o-
posed in o de o ex end he applica ion o IP
me hods o con ex and non-con ex non-linea
p oblems. These efinemen s ha e o do wi h s ep-
size con ol [7], line-sea ch echniques o o ce con-
e gence o a local minimum om a bi a y s a ing
poin s [8,9], use o us egions [10], among o he s.
Howe e , despi e hese imp o emen s, i s pe o -
mance on mixed-in ege p oblems is equen ly dis-
appoin ing because he solu ion is a om he
global op imum [11]. This has aised he in e es in
compu a ionally expensi e algo i hms aken om
he a ificial in elligence a ea such as e olu iona y
me hods [12], abu sea ch [13], pa icle swa m op i-
miza ion [14], simula ed annealing [15] o an col-
ony op imiza ion [16]. Al hough con e gence o
he global op imum canno be heo e ically gua an-
eed, he abili y o escape om local minima makes
hem an a ac i e choice o many applica ions
[17,18].
In his pape , a GA applied o he op imal sho -
e m (24 h) elec ical ene gy p oduc ion scheduling
is p esen ed. Some heu is ics a e included in o de
o assu e he easibili y o he cons ain s ha
appea in he p oblem. Resul s om eal-wo ld
cases based on Spanish powe sys em a e epo ed.
The elec ic ene gy p oduc ion scheduling p esen s
a la ge numbe o a iables and cons ain s, a
non-con ex non-linea objec i e unc ion and in e-
ge and con inuous a iables. Thus, he main diffi-
cul y is o find easible solu ions and he no el y
o he pape is add essed o imp o e he easibili y.
The main con ibu ions o he encoded GA can be
s a ed as: a p ocedu e o gene a e he ini ial popula-
ion aking in o accoun he amp cons ain s;
dynamic cons ain s o minimum limi s on he
hou ly ene gy p oduc ion o he he mal uni s o
conside he s a ing and s opping pe iods as an
al e na i e o he inclusion o o he bina y a iables
o model hese s a es; a c osso e ope a o adap ed
o he ea u es o his p oblem leading o an ade-
qua e pe cen age o easible indi iduals; and spa -
si y echniques and op imal o de ing used o
educe he compu a ional o e head. In spi e o he
spa si y echniques, a po en ial limi a ion o he
GA is a he ela ed o he CPU ime when a com-
plex opology o hyd oelec ic powe plan s is
analyzed.
The pape is o ganized as ollows: Sec ion 2p e-
sen s he equa ions used o model he scheduling
p oblem, leading o a non-linea mixed-in ege
p og amming p oblem wi h a la ge numbe o
bo h con inuous and disc e e a iables. In Sec ion
3a b iefly desc ip ion o he p imal-dual IP algo-
i hm is made. Sec ion 4in oduces he p oposed
GA, and se e al implemen a ion issues ha a e
c ucial o ob ain easible solu ions a e discussed.
Finally, Sec ion 5 epo s some esul s ob ained
om ealis ic cases based on he Spanish powe
sys em, and he main conclusions o he pape
a e ou lined.
2. Fo mula ion o he p oblem
The objec i e o he scheduling p oblem is o
de e mine he on/off s a e and he ene gy p oduc-
ion o he mal and hyd o uni s a each hou o
he scheduling pe iod, in o de o minimize he o al
cos o he sys em sa is ying he o ecas ed hou ly
demand and he echnical cons ain s o he mal
and hyd o powe plan s.
The s anda d no a ion used o he scheduling o
he elec ical ene gy p oduc ion p oblem is summa-
ized in Table 1. This no a ion desc ibes fixed
pa ame e s o he he mal and hyd o uni s, indexes,
numbe o elemen s and a iables.
2.1. Objec i e unc ion
The o al ene gy p oduc ion cos o he schedul-
ing pe iod is defined by
CT¼X
n
¼1X
ng
i¼1
½CiðPi; ÞþSUiUi; ð1Ui; 1Þ
þSDið1Ui; ÞUi; 1;ð1Þ
whe e n
is he numbe o hou s o he scheduling
pe iod, n
g
he numbe o he mal uni s, each ha ing
a quad a ic cos unc ion, C
i
(P
i,
), o he ene gy p o-
duc ion, P
i,
;SU
i
and SD
i
a e, espec i ely, he s a -
up and shu -down cos o he he mal gene a o i,
and U
i,
is a bina y a iable ep esen ing he on/off
s a e o he he mal gene a o ia hou .
I can be obse ed ha he o al p oduc ion cos
is a sum o quad a ic unc ions o he ene gy o each
he mal gene a o i he s a e o each gene a o was
p e iously s a ed by he GA. This is he case o he
p oposed echnique because he on/off s a es a e
managed by he GA. No ice ha he p oduc ion
cos is only due o he p oduc ion o he mal gene -
a o s P
i,
, i.e., gene a o s ha p oduce ene gy by
bu ning a uel o by a omic means. Hyd o uni s
p o ide ee-o -cha ge ene gy PH
h,
ha is only sub-
jec o he a ailabili y o wa e in he co esponding
ese oi s.
2.2. Cons ain s
The minimiza ion o he objec i e unc ion is
subjec o echnical cons ain s, wa e balance in
hyd oelec ic powe plan s and he associa ed ese -
oi s, and o he sys em ene gy demand and ese e
balances:
•Maximum and minimum limi s on he hou ly
ene gy p oduc ion o he he mal and hyd o
gene a o s,
Pm
i6Pi; 6PM
i;i¼1;...;ng; ¼1;...;n ;ð2Þ
PHm
h6PHh; 6PHM
h;h¼1;...;nh; ¼1;...;n ;
ð3Þ
Table 1
Defini ion o he da a and a iables o he p oblem
Da a o fixed pa ame e s
SU
i
The s a -up cos o he he mal uni i(€)
SD
i
The shu -down cos o he he mal uni i(€)
C
i
(Æ) Quad a ic cos unc ion o he he mal uni i(€)
Pm
iLowe bound o he hou ly ene gy p oduc ion o he he mal uni i(MWh)
PM
iUppe bound o he hou ly ene gy p oduc ion o he he mal uni i(MWh)
PHm
hLowe bound o he hou ly ene gy p oduc ion o he hyd o plan h(MWh)
PHM
hUppe bound o he hou ly ene gy p oduc ion o he hyd o plan h(MWh)
VHm
hLowe bound o he wa e le el o he ese oi hin e ms o ene gy (MWh)
VHM
hUppe bound o he wa e le el o he ese oi hin e ms o ene gy (MWh)
UR
i
Uppe bound o he up a e o he he mal uni i(MWh/h)
DR
i
Lowe bound o he down a e o he he mal uni i(MWh/h)
W
h
Inflow o he ese oi hin e ms o ene gy (MWh)
D
Ene gy demand a hou (MWh)
R
Gene a ing capaci y in ese e a hou (MWh)
DT
i
Numbe o hou s ha he uni imus be shu -down a e s opping
UT
i
Numbe o hou s ha he uni imus be unc ioning a e s a ing
d(k) Wa e delay ime be ween ese oi kand he nex ese oi downs eam (in h)
n(k) Nex ese oi downs eam ega ding he ese oi k
Indexes and numbe o elemen s
iThe mal uni index
hHyd o plan index
Hou index
n
g
Numbe o he mal uni s
n
h
Numbe o hyd o plan s
n
Numbe o hou s o he scheduling pe iod
Va iables
P
i,
Ene gy p oduc ion o he he mal uni ia hou (MWh)
U
i,
On/off s a e o he he mal gene a o ia hou
PH
h,
Ene gy p oduc ion o he hyd o plan ha hou (MWh)
VH
h,
S o ed ene gy o he ese oi ha hou (MWh)
whe e n
h
is he numbe o hyd o plan s, PH
h,
he
ene gy p oduc ion o hyd o plan ha hou , and
Pm
i,PM
i,PHm
hand PHM
ha e he limi s on he
hou ly ene gy p oduc ion o he he mal uni i
and hyd o plan h, espec i ely.
Eq. (2) canno be ulfilled when he mal gene a-
o s a e ei he s a ing o s opping, as s a ing
and s opping pe iods begin, espec i ely, when
he co esponding s a e changes o ON o OFF.
In o de o a oid his p oblem, his equa ion is
modified o he mal uni s ha a e ei he being
s a ed-up o shu -down,
06Pi; 6PM
i;i¼1;...;ng; ¼1;...;n :ð4Þ
Mo eo e , he ene gy p oduced by he mal uni s
du ing pe iods o shu ing-down (U
i,
= 0) is ou
o he op imal scheduling. Consequen ly, penal y
e ms p opo ional o his ene gy a e added o
he objec i e unc ion as ollows:
C0
T¼CTþX
n
¼1X
ng
i¼1
CpPi; ð1Ui; Þ:ð5Þ
•Maximum up and down amps o he mal uni s.
The he mal uni s can no inc ease o dec ease
he p oduc ion o ene gy a consecu i e hou s
by mo e han a gi en maximum a e,
DRi6Pi; Pi; 16URi;i¼1;...;ng;
¼1;...;n ;ð6Þ
whe e UR
i
yDR
i
a e, espec i ely, he maximum
up and down a es o he he mal gene a o i,
usually known as amp limi s.
•Limi s on he a ailable wa e . The hyd o uni s
use wa e o gene a e elec ical ene gy and wa e
is a limi ed esou ce. Thus, he ene gy p oduced
by a hyd o uni is limi ed by he olume o a ail-
able wa e in he associa ed ese oi . In conse-
quence, ese oi le els a e subjec o capaci y
limi s,
VHm
h6VHh; 6VHM
h;h¼1;...;nh;
¼1;...;n ;ð7Þ
whe e VH
h,
is he s o ed ene gy o ese oi ha
hou , co esponding o he hyd o uni h;VHm
h
and VHM
ha e espec i ely he minimum and max-
imum limi s on he s o ed ene gy imposed by he
maximum and minimum possible wa e le el o
ese oi h.
•Hyd aulic coupling be ween ese oi s. Time cou-
pling exi s due o cascaded ese oi s, since he
wa e used o p oduce ene gy in a hyd o uni will
be a ailable la e o he nex hyd aulic uni down-
s eam wi h a ce ain delay, ob iously when he
wa e has a i ed o he co esponding ese oi .
VHh; ¼VHh; 1PHh; þX
nðkÞ¼h
PHk; dðkÞþWh;
ð8Þ
whe e d(k) is he wa e delay ime in hou s be-
ween ese oi kand he nex ese oi down-
s eam, n(k), ha is supposed o be ese oi h,
and W
h
is he na u al inflow o ese oi h.
•The o al hou ly ene gy p oduc ion mus be
equal he o al ene gy demand a ha hou , D
,
which has been p e iously o ecas ed.
X
ng
i¼1
Pi; Ui; þX
nh
h¼1
PHh; ¼D ; ¼1;...;n :
ð9Þ
•The o al ene gy ha can be p oduced a each
hou mus exceed he o ecas ed demand by a
specified amoun , R
, i.e., he gene a ing capaci y
in ese e o be used i an unexpec ed e en such
as he ailu e o a plan o a la ge e o on he
o ecas ed demand happens.
X
ng
i¼1
PM
iUi; þX
nh
h¼1
PHM
hPD þR ;
¼1;...;n :ð10Þ
•Minimum up and down imes o he mal uni s.
The minimum up ime, UT
i
, is he minimum
numbe o hou s ha he uni imus be unc ion-
ing a e s a ing. Besides, he minimum down
ime, DT
i
, is he minimum numbe o hou s ha
he uni imus be shu -down a e s opping.
X
DT i1
k¼0
ð1Ui; þkÞPDT i
i uni iis shu -down a hou ð11Þ
X
UT i1
k¼0
Ui; þkPUT i
i uni iis s a ed a hou :ð12Þ
S a -up and shu -down cos s o ealis ic cases end
o educe he numbe o shu -downs and s a -ups
o a minimum, making he minimum- ime con-
s ain s useless in mos cases. Mo eo e , he inclu-
sion o hyd aulic gene a ion acili a es he
ulfillmen o he he mal uni cons ain s because
he hyd o uni s a e as e in esponse and p oduce
ene gy a no cos , i.e., he hyd aulic ene gy will be
s a egically dis ibu ed among he hou s o he
scheduling ho izon in o de o a oid he s a ing
o mo e he mal uni s han he s ic ly equi ed.
As an example, Table 2 shows he numbe o
cons ain s, bina y and con inuous a iables o he
abo e p oblem o a es sys em comp ising 49 he -
mal uni s, wo hyd o uni s and he scheduling ho i-
zon emb acing 24 h.
3. P imal-dual IP algo i hm
Among he dis inc i e ea u es o he abo e op i-
miza ion p oblem, he mos impo an a e: la ge
numbe o a iables and cons ain s in p ac ical
cases; non-con exi y o he objec i e unc ion; and
p esence o in ege and con inuous a iables. The
IP me hods only can be applied when all he a i-
ables o he p oblem a e con inuous. An al e na i e
equen ly used in p ac ice consis s in elaxing he
disc e e na u e o U
i,
and imposing ins ead he nex
cons ain :
06Ui; 61:ð13Þ
This leads o a simplified model wi hou disc e e
a iables which equi es ha a heu is ic p ocedu e
be applied du ing o a he end o he i e a i e p o-
cess in o de o de e mine he bes in ege alue o
e e y U
i,
. Thus a mixed-in ege p og amming p ob-
lem is conside ed om a con inuous pe spec i e. In
addi ion, e e y disc e e a iable can be handled as a
con inuous a iable p o ided ha he ollowing
quad a ic cons ain is added:
Ui; ð1Ui; Þ¼0:ð14Þ
Howe e , his p ocedu e inc eases he non-linea i y
and non-con exi y o he ini ial p oblem.
Based on he abo e commen s and p ac ical
expe ience, he ollowing p ocedu e has been chosen
o sol e he op imal sho - e m scheduling o he
elec ical ene gy p oduc ion.
(1) The p oblem is sol ed adding he cons ain
(Eq. (13)).
(2) Using he ob ained solu ion in he be o e s ep
o ini ialize he IP algo i hm, sol e he p ob-
lem including he cons ain (Eq. (14)).
A b ie desc ip ion o a classical IP algo i hm is
p o ided nex . This me hod in oduces auxilia y
posi i e slack a iables in o de o u n inequali y
es ic ions in o equali y cons ain s:
xj6xj,xjþsj¼xjsjP0;ð15Þ
whe e x
j
ep esen s any a iable subjec o a limi ,
and s
j
is he co esponding posi i e slack a iable.
In o de o gua an ee he posi i eness o he slack
a iables, loga i hmic penal y e ms a e included in
he objec i e unc ion by means o a penal y ac o
l ha is p og essi ely educed h oughou he i e -
a i e p ocess [5].
0ðxj;sj;lÞ¼ ðxjÞlX
j
ln sj:ð16Þ
The main s eps o he IP algo i hm a e he
ollowing:
(1) Ini ialize he a iables so ha he slack a i-
ables a e posi i e.
(2) Ini ialize he penal y ac o lso as o make he
loga i hmic e ms domina e o e he o iginal
objec i e unc ion.
(3) The minimiza ion o he co esponding
Lag angian unc ion is pe o med by sol ing
he non-linea op imali y equa ions using an
one-s ep New on’s algo i hm, and he op imal
inc emen o p imal and dual a iables is
compu ed.
(4) The s ep-leng h ais educed, i necessa y, so
ha he slack a iables emain posi i e.
Lag ange mul iplie s associa ed o equali y
cons ain s a ising om he in oduc ion o
auxilia y slack a iables mus also emain
posi i e because op imali y condi ions lead
o equa ions o he o m:
sjzj¼l;ð17Þ
whe e z
j
is he Lag ange mul iplie associa ed
wi h he slack a iable s
j
.
(5) Upda e p imal and dual a iables aking in o
accoun he necessa y s ep-leng h limi a ion.
(6) Reduce he penal y ac o l. The p opo ional
ela ionship be ween he penal y ac o and
he duali y gap (dugap) defined by Eq. (17)
Table 2
Dimension o he p oblem o a es sys em
Numbe o cons ain s Numbe o a iables
Bina y Con inuous
(2 Æn
g
+3Æn
h
+2)Æn
+2Æn
g
n
g
Æn
(n
g
+2Æn
h
)Æn
2642 1176 1272

p o ides he mos common app oach o
educe his penal y ac o :
l¼cPnl
j¼1sjzj
nl
;ð18Þ
whe e c61andn
l
is he numbe o inequali y con-
s ain s o he o iginal p oblem. S eps 3–6 a e i e a-
i ely epea ed un il op imali y condi ions a e
sa isfied and he penal y coefficien l, and conse-
quen ly he a e age duali y gap, is small enough.
Mo e sophis ica ed e sions o s eps 3 and 4, includ-
ing line sea ches, modified Hessians, e c. [8–10]
could be needed in he non-con ex case, in o de
o a oid di e gence o con e gence o unaccep able
poin s. Howe e , such efinemen s ha e no been
ac ually implemen ed because he beha iou o he
IP me hod has p o en good enough o he applica-
ion es ed.
The applica ion o New on’s me hod o sol e he
non-linea op imali y equa ions yields a e y la ge,
spa se linea sys em, specially when amp and
hyd aulic couplings a e conside ed. Consequen ly,
spa si y echniques and op imal o de ing [19] mus
be used o educe he compu a ional o e head.
Fig. 1 shows he fill-ins gene a ed when sol ing a
small example (fi e hyd o plan s, fi e he mal plan s
and a 5-hou scheduling ho izon), wi h and wi hou
op imal o de ing. I can be no ed ha he fill-ins a e
educed a 50% app oxima ely when an op imal
o de ing is made. Table 3 shows ela i e execu ion
imes o he IP algo i hm o wo ealis ic p oblems
(73 he mal plan s, 24 h, 8 and 30 ese oi s, espec-
i ely). As can be no iced, execu ion imes g ow
conside ably wi h he numbe o a iables, specially
when s anda d o de ing is pe o med. Finally,
Fig. 2 shows he ela i e o e head o he diffe en
p ocesses comp ising he IP algo i hm.
4. The p oposed gene ic algo i hm
As p esen ed in he p e ious sec ion, he op imal
scheduling o he elec ic ene gy p oduc ion is a
non-linea , non-con ex, combina o ial, mixed-in e-
ge and e y la ge p oblem. Hence, he e is no ech-
nique ha would always lead o he op imal
solu ion o he p oblem o ealis ic cases. In he las
yea s, echniques based on heu is ics, dynamic p o-
g amming, linea mixed-in ege p og amming and
lag angian elaxa ion ha e been applied o his pa -
icula p oblem. Techniques based on heu is ics ely
on simple ules ha depends on he knowledge o
powe plan ope a o s. Cons ain s o ealis ic p ob-
lems a e no p ope ly modelled by dynamic p o-
g amming app oaches, and he numbe o
equi ed s a es inc eases exponen ially, hus leading
o excessi e compu a ion imes. Linea p og am-
ming app oaches canno p ope ly model nei he
he non-linea objec i e unc ion no he non-linea
cons ain s, and c ude app oxima ions a e equi ed.
Finally, he use o heu is ic echniques is equi ed
by lag angian elaxa ion app oaches o calcula e
easible solu ions, de e io a ing he quali y o he
ob ained solu ions.
Consequen ly, new me hods a e s ill needed o
ob ain mo e op imal solu ions o ealis ic p oblems.
In his pape , a GA [20,21] has been used o sol e
he scheduling p oblem due o i s abili y o deal wi h
non-linea unc ions and in ege a iables.
The p oposed GA algo i hm is used o compu e
he op imal on/off s a es o he mal uni s, i.e., he
bina y a iables, while he op imal con inuous a i-
ables, i.e., he hou ly ene gy p oduc ion o hyd o
and commi ed he mal uni s, a e calcula ed sol ing
a ypical quad a ic p og amming p oblem by a clas-
sical IP op imiza ion algo i hm in which he on/off
s a es o he mal uni s a e known.
Con e gence cha ac e is ics o GA depend on
se e al key implemen a ion issues ha a e discussed
in he es o his sec ion.
4.1. Codifica ion o he indi iduals
Each indi idual is ep esen ed by he on/off s a es
o he mal gene a o s du ing he scheduling pe iod.
Thus, indi iduals a e ep esen ed by 0/1 ma ices,
wi h columns co esponding o ime scheduling
in e als and ows associa ed wi h he mal uni s.
I he elemen (i,j) is equal o one, he s a e o he -
mal uni idu ing ime in e al jis on. Simila ly, i
he elemen (i,j) is equal o ze o, he s a e o he mal
uni idu ing ime in e al jis off.
Fig. 3 shows he ep esen a ion o a ce ain indi-
idual o he popula ion. I can be obse ed ha he
he mal uni 1 is on om 1am o 4am and he es o
hou s is off; he he mal uni 2 is off om 1am o
6am and he es o hou s is on; he he mal uni 3
is on du ing all scheduling ho izon; he he mal uni
4 is only on om 11am o 4pm, e c.
4.2. Ini ial popula ion
Up and down amp cons ain s o he mal uni s
(Eq. (6)) a e a key ac o in he con e gence o he
GA: i he ini ial popula ion is s ic ly andomly
selec ed, amp cons ain s lead o many in easible
indi iduals in he ini ial gene a ion, which makes
successi e gene a ions suffe om poo di e si y,
and he GA may con e ge p ema u ely. To assu e
ha he ini ial popula ion con ains an adequa e
0 20 40 60 80 100
0
20
40
60
80
100
Columns
Rows
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
Columns
Rows
Fig. 1. S uc u e o he linea sys em including fill-ins.
pe cen age o easible indi iduals, ini ial on/off
schedulings a e andomly selec ed bu modified o
accoun o he minimum s a -up and shu -down
imes imposed by amp cons ain s. Fo example,
i gene a o g, wi h a maximum down amp equal
o 100 MWh, is on a hou 3 p oducing an ene gy
o 400 MWh, his gene a o would equi e 4 hou s
o shu -down and, consequen ly, he gene a o a
hou s 4, 5 and 6 should be on. The s a e U
g,3
is
s ic ly andomly gene a ed bu he s a es o he
ollowing hou s, U
g,4
,U
g,5
and U
g,6
, a e gi en by
Ug;3¼1)Ug;4¼Ug;5¼Ug;6¼1:ð19Þ
4.3. Fi ness unc ion
The fi ness unc ion e alua es he quali y o an
indi idual o he popula ion. In his case, he unc-
ion is he in e se o he o al p oduc ion cos o he
indi idual. The o al p oduc ion cos is ob ained
sol ing a quad a ic p og amming p oblem by using
a non-linea In e io Poin me hod [22,23].An
ex a-high-cos fic i ious gene a o is included o
sa is y he sys em demand (Eq. (9)). This fic i ious
gene a o gene a es he necessa y ene gy ha he
es o gene a o s canno p oduce o sa is y he
demand o he cus ome s. A penal y e m p opo -
ional o he defici in ese e equi emen s is added
in he cos unc ion aiming a sa is ying he ese e
cons ain . Penal y e ms only apply o in easible
indi iduals, which a e consequen ly elimina ed
h oughou he e olu iona y p ocess.
4.4. Selec ion ope a o
To p oduce a new gene a ion, pa en s a e an-
domly selec ed using a oule e wheel selec ion ech-
nique ha selec s he bes indi iduals o
ep oduc ion. The p obabili y o a pa icula indi-
idual being selec ed is in p opo ion o i s fi ness
unc ion, aking in o accoun ha he o al gene a-
ion cos , including possible penaliza ions, is being
minimized. The indi iduals chosen o be pa en s
a e included in he ollowing gene a ion.
4.5. C osso e ope a o
Offsp ing is ob ained by adding he bina y s ings
ha esul s om andom pa i ions o each ow, as
shown in Fig. 4a. A column-pa i ioning p ocedu e
may also be applied (Fig. 4b). This c osso e ope -
a o is a pa icula case o he mul i-poin c osso e
ope a o whe e he numbe o poin s is equal o he
numbe o ows o columns, espec i ely.
As ows a e associa ed wi h he he mal uni s, he
fi s app oach yields mainly he in easibili y o new
indi iduals in e ms o minimum up and down imes
(Eqs. (11) and (12)), while he second app oach has
an effec bigge on he cons ain o he demand
(Eq. (9)) and he ese e (Eq. (10)). This abo e s a e-
men is shown in he nex example. Le be a es
Table 3
Rela i e execu ion ime and i e a ion numbe .
Numbe o a iables Na u al o de ing Op imal o de ing
I e a ions Time I e a ions Time
2376 33 4.00 34 1.00
3960 27 140.98 34 2.45
O he s
23%
Op imal
o de ing
6%
Linea
sys em
building
10%
Linea
sys em
solu ion
61%
Fig. 2. Rela i e o e head o he diffe en p ocesses.
Hou s o he Schedulin
g
Ho izon
The mal Uni s
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0
.....
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
Fig. 3. Rep esen a ion o an indi idual o he popula ion.
sys em comp ising ou he mal uni s and a 4-hou
scheduling ho izon. The minimum up and down
ime a e 2 hou s o all he mal uni s. Two indi id-
uals selec ed o be pa en s a e ep esen ed in Fig. 5.
Fig. 6 shows he child en ob ained using a c osso e
ope a o by ows om he andom pa i ion
(1,2,1,2). I can be obse ed ha he child en do
no sa is y he Eqs. (11) and (12) (gene a o h ee)
while he Eqs. (9) and (10) can be easible. Fig. 7
shows he child en ob ained using a c osso e ope -
a o by columns om he same pa i ion. I can be
obse ed ha he indi idual on he igh does no
sa is y he Eqs. (9) and (10) since all gene a o s
a e off a hou s 3 and 4. Howe e , in his case he
Eqs. (11) and (12) a e ulfilled.
The c osso e p obabili y has been se o one, i.e.,
wo indi iduals ha ha e been selec ed o be pa en s
a e always combined o ob ain a new indi idual.
In he final e sion o he GA, he c osso e by
ows has been chosen because s a -up and shu -
down cos s o ealis ic cases, along wi h he inclu-
sion o hyd aulic gene a ion, end o educe he
numbe o shu -downs and s a -ups o a minimum,
making he minimum- ime cons ain s useless in
mos cases. All he ows a e always combined o
ob ain a new indi idual, hough p obabili ies migh
ha e been used o de e mine which ows should be
combined.
4.6. Mu a ion ope a o
A e he c osso e p ocess, he indi iduals o he
popula ion a e mu a ed o in oduce some new
gene ic ma e ial acco ding o a p e-defined mu a-
ion p obabili y p. Consequen ly, he pe cen age o
mu a ed indi iduals o a gene a ion is equal o
100p%. The mu a ion o an indi idual means he
mu a ion o an only gene. The gene o be mu a ed
1 2 3 1 2 3
11
22
33
..
..
..
gg
1 2 3 1 2 3
11
22
33
..
..
..
gg
PARENTS
CHILDREN
123 123
11
22
... ...
gg
123 123
11
22
... ...
gg
CHILDREN
PARENTS
ab
Fig. 4. C osso e Ope a o : (a) andom pa i ions o ows and (b) andom pa i ions o columns.
h1 h2 h3 h4 h1 h2 h3 h4
g1 1 1 1 1 g1 1 1 0 0
g2 1 1 0 0 g2 1 1 0 0
g3 0 0 0 0 g3 1 1 1 1
g4 1 1 0 0 g4 0 0 1 1
Fig. 5. Pa en s.
h1 h2 h3 h4 h1 h2 h3 h4
g1 1 1 0 0 g1 1 1 1 1
g2 1 1 0 0 g2 1 1 0 0
g3 0 1 1 1 g3 1 0 0 0
g4 1 1 1 1 g4 0 0 0 0
Fig. 6. Child en by pa i ions o ows.
h1 h2 h3 h4 h1 h2 h3 h4
g1 1 1 1 1 g1 1 1 0 0
g2 1 1 0 0 g2 1 1 0 0
g3 1 1 1 1 g3 0 0 0 0
g4 0 0 1 1 g4 1 1 0 0
Fig. 7. Child en by pa i ions o columns.