POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 1 |2016 |MARCH
An E icien Cuckoo-Inspi ed Me a-Heu is ic
Algo i hm o Mul iobjec i e Sho -Te m
Hyd o he mal Scheduling
Thang TRUNG NGUYEN1,2, Dieu NGOC VO3, Anh VIET TRUONG2, Loc DAC HO4
1Facul y o Elec ical and Elec onics Enginee ing, Ton Duc Thang Uni e si y, 19 Nguyen Huu Tho S ee ,
Dis ic 7, Ho Chi Minh Ci y, Vie nam,
2Facul y o Elec ical and Elec onics Enginee ing, Ho Chi Minh Ci y Uni e si y o Technology and
Educa ion, 01 Vo Van Ngan S ee , Thu Duc Dis ic , Ho Chi Minh Ci y, Vie nam,
3Depa men o Powe Sys ems, Facul y o Elec ical and Elec onic Enginee ing, Ho Chi Minh Ci y
Uni e si y o Technology, 268 Ly Thuong Kie S ee , Dis ic 10, Ho Chi Minh Ci y, Vie nam,
4Ho Chi Minh Ci y Uni e si y o Technology (HUTECH), 475 Dien Bien Phu S ee , Binh Thanh Dis ic , Ho
Chi Minh Ci y, Vie nam
nguyen[email p o ec ed], [email p o ec ed], [email p o ec ed], hdloc@hcmhu ech.edu. n
DOI: 10.15598/aeee. 14i1.1562
Abs ac . This pape p oposes an e icien Cuckoo-
Inspi ed Me a-Heu is ic Algo i hm (CIMHA) o sol -
ing mul i-objec i e sho - e m hyd o he mal schedul-
ing (ST-HTS) p oblem. The objec i e is o simul a-
neously minimize he o al cos and emission o he -
mal uni s while all cons ain s such as powe balance,
wa e discha ge, and gene a ion limi a ions mus be
sa is ied. The p oposed CIMHA is a newly de eloped
me a-heu is ic algo i hm inspi ed by he in elligen e-
p oduc ion s a egy o he cuckoo bi d. I is e icien
o sol ing op imiza ion p oblems wi h complica ed ob-
jec i e and cons ain s because he me hod has ew con-
ol pa ame e s. The p oposed me hod has been es ed
on di e en sys ems wi h a ious numbe s o objec i e
unc ions, and he ob ained esul s ha e been compa ed
o hose om o he me hods a ailable in he li e a u e.
The esul compa isons ha e indica ed ha he p o-
posed me hod is mo e e icien han many o he me h-
ods o he es sys ems in e ms o o al cos , o al
emission, and compu a ional ime. The e o e, he p o-
posed CIMHA can be a a o able me hod o sol ing he
mul i-objec i e ST-HTS p oblems.
Keywo ds
Cuckoo-inspi ed me a-heu is ic algo i hm, eco-
nomic dispa ch, emission dispa ch, le y ligh s,
mul iobjec i e hyd o he mal scheduling.
1. In oduc ion
The main ask o he sho - e m hyd o- he mal
scheduling (ST-HTS) p oblem is o de e mine he op-
imal powe gene a ion o he a ailable he mal and
hyd o powe plan s so as he o al uel cos o he -
mal uni s o e a schedule ime is minimized sa is ying
bo h equali y and inequali y cons ain s such as he
quan i y o a ailable wa e , powe balance, and uppe
and lowe limi s on gene a ions. In addi ion, a la ge
amoun o he elec ic powe in he wo ld is mainly
gene a ed by he mal plan s using oil, coal o na u al
gasses. The e o e, se e al con aminan s such as ni o-
gen oxides (NOx), sul u dioxide (SO2), and ca bon
dioxide (CO2) ha e been eleased in o he a mosphe e
due o he p ocess o elec ici y gene a ion om he
he mal uni s [1]. In addi ion o he uel cos objec-
i e, he gaseous emission is also ano he impo an
objec i e which needs o be conside ed in he ST-HTS
p oblem. As a esul , a mul i-objec i e ST-HTS p ob-
lem is o med. The e o e, he mul i-objec i e ST-HTS
p oblem is mo e complex han he con en ional one
since i needs o ind a se o non-domina ed solu ions
o de e mining he bes comp omise solu ion, which
is conside ed as he mos easonable one o he ac-
cep able ade-o be ween uel cos and emission ob-
jec i es.
Many con en ional me hods ha e been applied o
sol ing he ST-HTS p oblem such as he me hod based
on Lag ange mul iplie heo y [2], lambda-gamma i e -
a ion me hod (LGM) [3], dynamic p og amming (DP)
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F1=
M
X
m=1
N1
X
i=1
masi +bsiPsi,m +csiP2
si,m +|esi ×sin( si ×(Pmin
si −Psi,m))|.(1)
[4], Lag ange elaxa ion (LR) me hod [5], and decom-
posi ion and coo dina ion me hod [6], and a i icial in-
elligence based me hods such as pa icle swa m op-
imiza ion (PSO) [7], o p eda o -p ey op imiza ion
echnique (PPO) [8]. Gene ally, hese con en ional
me hods ha e a common cha ac e is ic ha hey can
be applicable only o op imiza ion p oblems wi h di -
e en iable objec i e and cons ain s. In ecen yea s,
se e al a i icial in elligence based me hods ha e been
implemen ed o sol ing he mul i-objec i e ST-HTS
p oblem. Simula ed annealing-based goal-a ainmen
(SA-BGA) me hod [1] has been success ully applied
o he p oblem, bu he me hod has coped wi h long
execu ion ime. In [9], gamma based me hod (γ-
PSO) ha e been demons a ed o be supe io o con-
en ional PSO bu he imp o ed e sion canno deal
wi h sys ems wi h noncon ex uel cos unc ion. Ge-
ne ic algo i hm is one o he ea lies a i icial in elli-
gence me hods bu i s applicabili y on complex sys-
ems is s ill compe i i e [10] and i is slowe han
PSO[11]. The e o e, se e al imp o ed e sions o i
ha e been in oduced such as non-domina ed so ing
gene ic algo i hm-II (NSGA-II) [12], imp o ed gene ic
algo i hm (IGA) [13], mul iplie upda ing and he -
cons ain echnique (IGA-MU) [13]. P eda o -p ey
op imiza ion and Powell sea ch (PPO-PS) me hod [14]
is complica ed o implemen o he p oblem; howe e ,
i s achie emen can sa is y esea che s since i is mo e
e icien han all imp o ed e sions o GA in [12], [13].
Augmen ed Lag age Hop ield ne wo k (ALHN) me hod
[15] is e y as o con e gence wi h high accu acy;
howe e , i s applica ion also ends a sys ems wi h non-
con ex uel cos unc ion simila o γ-PSO. In gene al,
he a i icial in elligence based me hods can ind nea
op imum solu ion o non-con ex op imiza ion p ob-
lems wi h non-di e en iable objec i e and cons ain s.
Howe e , since he a i icial in elligence based me hods
a e gene ally based on he andom sea ch o a popula-
ion in he p oblem space, hey need o be un se e al
imes o ob ain he bes solu ion.
In his pape , he CIMHA, i s de eloped by Yang
and Deb in 2009 [16], is p oposed o sol ing he mul-
iobjec i e ST-HTS p oblem conside ing powe losses
in ansmission sys ems and al e poin loading e ec s
in uel cos unc ion o he mal uni s. The p oposed
me hod has been es ed on di e en sys ems wi h di -
e en numbe s o objec i e unc ion, and he ob ained
esul s ha e been compa ed o hose om o he me h-
ods a ailable in he li e a u e.
2. P oblem Fo mula ion
Conside an elec ic powe sys em ha ing N1 he mal
plan s and N2hyd o plan s scheduled in Msubin e -
als. The goal o he mul iobjec i e ST-HTS p oblem
is o simul aneously minimize he uel cos and gaseous
pollu an emission le el o he mal plan s while sa is-
ying a ious ope a ional cons ain s o a sys em and
he mal and hyd o uni s.
2.1. Fuel Cos Objec i e
The uel cos unc ion o he mal uni s is ep esen ed
as Eq. (1) [10], whe e asi,bsi,csi,esi, si a e uel cos
coe icien s o he mal plan i;Psi,m is powe ou pu
o he mal uni ia subin e al m; mis he du a ion
o subin e al m; and N1is o al numbe o he mal
plan s.
2.2. Emission Objec i e
The emission o he mal uni s including sul u diox-
ides (SO2), ca bon dioxides (CO2), and ni ogen ox-
ides (NOx) eleased in o he ai by ossil- ueled he mal
plan s is ep esen ed as ollows [9]:
NOsi,m =α1si +β1siPsi,m +γ1siP2
si,m,(2)
SOsi,m =α2si +β2siPsi,m +γ2siP2
si,m,(3)
COsi,m =α3si +β3siPsi,m +γ3siP2
si,m,(4)
and he o al emission can be combined as ollows[9]:
F2=w1NOsi,m +w2SOsi,m +w3COsi,m,(5)
whe e w1,w2, and w3a e posi i e weigh ing ac o s o
di e en gaseous emissions con ibu ing o he emission
objec i e; α1si,β1si, and γ1si a e emission coe icien s
o NOx;α2si,β2si, and γ2si a e emission coe icien s
o SO2; and α3si,β3si, and γ3si a e emission coe i-
cien s o CO2.
In addi ion, he amoun o emissions om each he -
mal uni can also be exp essed in a o m o a quad a ic
and exponen ial unc ion as Eq. (6) [12], whe e αsi,βsi,
and γsi,ηsi, and δsi a e emission coe icien s o he mal
uni i.
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F2=
M
X
m=1
N1
X
i=1
mαsi +βsiPsi,m +γsiP2
si,m +ηsiexp(δsiPsi,m).(6)
2.3. Sys em and Uni Cons ain s
1) Load Demand Cons ain
The o al powe gene a ion om he mal and hyd o
uni s mus sa is y he load demand and powe losses
in ansmission lines:
PN1
i=1 Psi,m +PN2
j=1 Phj,m −PL,m −PD,m = 0,
m= 1, . . . , M,
(7)
whe e he powe losses in ansmission lines a e calcu-
la ed using K on’s o mula as ollows:
PL,m =PN1+N2
i=1 PN1+N2
j=1 Pi,mBij Pj,m+
+PN1+N2
i=1 B0,iPi,mB0,0,
(8)
whe e N2is o al numbe o hyd o plan s; Phj,m is
powe ou pu o hyd o uni j a subin e al m;PD,m
and PL,m a e o al sys em load demand and o al
ansmission loss a subin e al m, espec i ely; Pi,m
is powe ou pu o hyd o o he mal uni i; and Bij,
B0i,B00 a e ma ix coe icien s o ansmission powe
losses.
2) Wa e A ailabili y Cons ain s
The o al wa e discha ge o each hyd o uni du ing
he scheduled pe iod is limi ed by an a ailable amoun
o wa e o ha uni :
M
X
m=1
mqj,m =Wj, j = 1, . . . , N2,(9)
whe e he wa e discha ge qj,m o hyd o uni j a
subin e al mis de e mined by:
qj,m =ahj +bhjPhj,m +cjP2
hj,m,(10)
whe e ahj,bhj ,chj a e wa e discha ge coe icien s o
hyd o uni j; and Wjis he olume o wa e a ailable
o gene a ion by hyd o plan jdu ing he scheduled
pe iod.
3) Gene a o Ope a ing Limi s
The powe ou o he mal and hyd o uni s is limi ed
be ween hei uppe and lowe limi s:
Psi,min ≤Psi,m ≤Psi,max,
i= 1,2, . . . , N1, m = 1,2, . . . , M,
(11)
Phj,min ≤Phj,m ≤Phj,max,
j= 1,2, . . . , N2, m = 1,2, . . . , M,
(12)
whe e Psi,max,Psi,min a e maximum and minimum
powe ou pu o a he mal uni i, espec i ely; and
Phj,max,Phj,min a e maximum and minimum powe
ou pu o hyd o plan j, espec i ely.
3. Cuckoo-Inspi ed Me a -
Heu is ic Algo i hm o
Mul iobjec i e ST-HTS
P oblem
3.1. Cuckoo-Inspi ed Me a-Heu is ic
Algo i hm
The o e all CIMHA me hod is summa ized in he h ee
main p incipal ules [16] including 1) a cuckoo bi d pu
i s egg in o he bi d’s nes ; 2) he Cuckoo egg is ha ched
and con inues o lay hei egg, 3) The Cuckoo egg is
disco e ed by hos bi d and i is abandoned.
Among he ules, he i s one is applied o build an
ini ial popula ion o nes s whe eas he second ule and
he hi d ule enable he CIMHA o p oduce new solu-
ions, which is ega ded as a special poin and ad an-
age o he CIMHA compa ed o o he me a-heu is ic
algo i hms.
3.2. Calcula ion o Powe Ou pu o
Hyd o Uni s and Slack The mal
Uni
In he conside ed hyd o he mal scheduling in he pa-
pe , he e a e wo se s o equali y cons ain s consis ing
o powe balance cons ain Eq. (7) and wa e a ailabil-
i y cons ain Eq. (9). In o de o sa is y he equali y
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cons ain s, wo se s o co esponding slack a iables
will be used including he i s he mal uni gene a ion
in each o Msubin e als Ps1,m (m= 1,2, . . . , M)and
wa e discha ge o each hyd o uni a M h subin e -
al qj,M (j= 1,2, . . . , N2). Consequen ly, he powe
ou pu o each hyd o uni is i s calcula ed om i s
co esponding wa e discha ge in each subin e al and
he slack he mal uni is hen ob ained by using equa-
ion Eq. (7). The de ailed calcula ion o slack a iables
can be ound in [17].
3.3. Implemen a ion o
Cuckoo-Inspi ed Me a-Heu is ic
Algo i hm
The s eps o implemen a ion o he CIMHA me hod
o sol ing he mul iobjec i e ST-HTS p oblem a e de-
sc ibed as ollows:
1) Ini ializa ion
In he CIMHA, each egg ep esen s a solu ion
which is andomly gene a ed in he ini ializa-
ion. A popula ion o Nphos nes s is ep e-
sen ed by X= [X1, X2, . . . , XNp]T, in which each
Xd(d= 1, ..., Np) ep esen s a solu ion ec o o
a iables gi en by Xd= [Psi,m,d, qj,m,d], whe e
Psi,m,d is he powe ou o he mal uni i a subin-
e al mco esponding o nes dand qj,m,d is
he wa e discha ge o hyd o uni ja subin e -
al mco esponding o nes d. The e o e, ec o
Xdo nes dis ep esen ed in de ail by Xd=
[Ps2,m,d, Ps3,m,d, ..., PsN1,m,d, q1,m,d, q2,m,d, ..., qN2,m,d],
which includes he he mal uni s om 2 o N1 o
Msubin e als and wa e discha ges o hyd o uni s
om 1 o N2 o he i s (M−1) subin e als.
Consequen ly, nes d only con ains he mal uni s
om 2 o N1a subin e al M. Ce ainly, he
uppe and lowe limi s o each nes a e espec i ely
Xdmin = [Psimin, qjmin]and Xdmax = [Psimax, qjmax].
The powe ou pu o he he mal uni s and wa e
discha ges in he Npnes s a e andomly ini ial-
ized sa is ying Psi,min ≤Psi,m,d ≤Psi,max and
qj,min ≤qj,m,d ≤qj, max. Each elemen in nes do
he popula ion is andomly ini ialized as ollows:
Psi,m,d =Psi,min + and1·(Psi,max −Psi,min),
i= 2, ..., N1, m = 1, ..., M,
(13)
qj,m,d =qj,min + and2·(qj,max −qj,min),
j= 2, ..., N2, m = 1, ..., M −1,
(14)
whe e and1and and2a e uni o mly dis ibu ed an-
dom numbe s in [0,1].
Based on he ini ial alue o nes s, he i ness
unc ion including objec i es unc ions oge he wi h
penal y e ms o he slack he mal uni o all M
subin e als and slack wa e discha ge o all hyd o
uni s a subin e al Mco esponding o each nes o
he p oblem is calcula ed by:
FTd=PM
m=1 PN1
m=1(w·F1(Psi,m,d)+
+(w−1) ·F2(Psi,m,d))+
+KsPM
m=1(Ps1,m,d −Plim
s1)2+
+KqPN2
j=1(qj,m,d −qlim
j)2,
(15)
whe e 0≤w≤1is weigh ing ac o o a combina-
ion o objec i es [18]; Ksand Kqa e penal y ac o s
o he slack he mal uni and a ailable wa e , espec-
i ely; Ps1,m,d is powe ou pu o he slack he mal uni
1 a subin e al m co esponding o nes din he popu-
la ion; qj,M,d is he wa e discha ge o all hyd o plan s
a subin e al Mco esponding o he nes din he
popula ion.
The limi s o he slack he mal uni and wa e dis-
cha ge a subin e al Min Eq. (15) a e de e mined as
ollows:
Plim
s1=








Psi,max i Ps1,m,d > Ps1,max
Psi,min i Ps1,m,d < Ps1,min,
(m= 1, ...M)
Psi,m,d o he wise
(16)
qlim
j=








qj,max i qj,m,d > qj,max
qj,min i qj,m,d < qj,min,
(j= 1, ...N2)
qj,m,d o he wise
(17)
whe e Ps1,max and Ps1,min a e he maximum and mini-
mum powe ou pu s o he slack he mal uni 1, espec-
i ely; qj,max and qj,min a e he maximum and mini-
mum wa e discha ge o he hyd o plan j.
The ini ial alue o nes s in he popula ion is se o
he bes alue o each nes Xbes d(d= 1, . . . , Nd)and
he nes co esponding o he bes i ness unc ion in
Eq. (15) is se o he bes nes Gbes among all nes s
in he popula ion.
2) Gene a ion o New Solu ion ia Le y
Fligh s
The new solu ion is calcula ed ia Le y ligh s based
on exchanging in o ma ion be ween he p e ious bes
nes s and each p e ious nes . In he p oposed me hod,
he op imal pa h o he Le y ligh s is calcula ed by
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Man egna’s algo i hm [19]. The new solu ion by each
nes is ob ained by:
Xnew
d=Xbes d+α× and3×∆Xnew
d,(18)
whe e α > 0is he upda ed s ep size; and3is a no -
mally dis ibu ed s ochas ic numbe ; and ∆Xnew
dis an
inc eased alue [9].
In case he new ob ain solu ion iola e he limi s,
hey will be ede ined as below:
Xnew
d=(Xd,max i Xnew
d> Xd,max
Xd,min i Xnew
d< Xd,min.(19)
Using Sec ion 3.2. , he powe ou pu o N2hyd o
uni s and he slack he mal uni a e ob ained. The
i ness alue is hen calcula ed using Eq. (15) and each
nes is se o Xbes . The nes wi h he bes i ness
unc ion Gbes is no equi ed o de e mine because i s
in o ma ion is no used o ob ain he new solu ion in
he nex sec ion.
3) Alien Egg Disco e y and Randomiza ion
The second phase o new solu ion gene a ion in his
sec ion is o imp o e he quali y o he p e iously ob-
ained solu ion. Like he Le y ligh s, he ac ion o
alien eggs disco e y in he nes s wi h a p obabili y o
pa can also gene a e a new solu ion o an op imiza ion
p oblem. The new solu ion is c ea ed by:
Xdis
d=(Xd+ and(X 1−X 2) i and < pa
Xdo he wise.
(20)
The newly ob ained solu ions also need o be ede-
ined using Eq. (20) in case hey iola e uppe and
lowe limi s. The i ness alue is calcula ed using equa-
ion Eq. (15) and he nes co esponding o he bes
i ness unc ion is se o he bes nes Gbes .
4) S opping C i e ia
In his esea ch, he p oposed algo i hm is s opped
when he maximum numbe o i e a ions is eached.
3.4. Bes Comp omise Solu ion by
Fuzzy-Based Mechanism
In a mul iobjec i e p oblem, he e o en exis s a con-
lic among he objec i es. To deal wi h his issue, a
se o op imal non-domina ed solu ions is ound ins ead
o only one op imal solu ion. In his pape , he bes
comp omise solu ion om he se o non-domina ed so-
lu ions is ound using he uzzy sa is ying me hod [18].
4. Nume ical Resul s
The p oposed CIMHA has been es ed on h ee sys ems
including wo sys ems wi h quad a ic uel cos unc ion
and one sys em wi h noncon ex uel cos unc ion o
he mal uni s. The p oposed CIMHA is coded in Ma -
lab pla o m and un on a 1.8 GHz PC wi h 4 GB o
RAM.
4.1. Selec ion o Pa ame e s
By expe imen s, he numbe o nes s in his pape is
se om 20 o 50 depending on he sys em size and he
maximum numbe o i e a ions Nmax is chosen om
300 o small sys ems o 2 500 o la ge-scale sys ems.
Unlike Npand Nmax, he alue o he p obabili y pa
has no in luence on execu ion ime bu he inal op imal
solu ion. Di e en op imal solu ions can be ob ained
co esponding o di e en p ede e mined alues o pa.
The e o e, he alue o pahas o be selec ed in u n
in he ange om 0.1 o 0.9 wi h a s ep o 0.1 in his
pape .
4.2. Sys ems wi h Quad a ic Fuel
Cos Func ion o The mal Uni s
1) The Fi s Sys em wi h Two Objec i e
Func ions
The es sys em wi h wo hyd o and wo he mal uni s,
in his case, includes a o al cos unc ion and one
emission unc ion [12]. The sys em is scheduled in a
24 hou pe iod in h ee subin e als wi h eigh hou s
o each. The p oposed CIMHA is applied o ob ain-
ing he op imal solu ions o he economic, emission
and economic-emission dispa ches.
The numbe o nes s and he maximum numbe o
i e a ions o his sys em a e espec i ely se o 20 and
300 in ad ance o each alue o w. The alue o p oba-
bili y pa is chosen as ollows. Fo he case o economic
dispa ch (w= 1) and emission dispa ch (w= 0), he
alue o he p obabili y pachanges in he ange om
0.1 o 0.9 wi h he s ep o 0.1. As a esul , he bes so-
lu ion o economic dispa ch and emission dispa ch can
be ob ained a he same alue o pa= 0.9. The alue
o pa= 0.9is hen used again o pe o m he p oposed
CIMHA me hod 20 independen uns o es o alues
o wwhich is di e en om 1 and 0 co esponding o
economic dispa ch and emission dispa ch. The e ha e
been 20 non-domina ed solu ions ob ained. By using
he uzzy mechanism o de e mina ion o he bes com-
p omise o his case, he weigh ac o is de e mined
a w= 0.07.
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Tab. 1: Resul compa ison o he i s sys em wi h quad a ic uel cos unc ion o he mal uni s.
Me hod Economic dispa ch Emission dispa ch Comp omise dispa ch
Cos ($) CPU (s) Emission (lb) CPU (s) Cos ($) Emission (lb) CPU (s)
RCGA [12] 66031 21.63 586.14 20.27 - - -
NSGA-II [12] - - - - 66331 618.08 27.85
MODE [12] - - - - 66354 619.42 30.71
SPEA-2 [12] - - - - 66332 618.45 34.87
PSO-PM [14] 65741 18.25 585.67 18.00 65,821 620.78 18.98
PSO [14] 65241 18.32 579.56 18.31 65731 618.78 19.31
PPO-PM [14] 64873 16.14 572.71 15.93 65426 612.34 16.53
PPO [14] 64718 15.99 569.73 15.18 65104 601.16 16.34
PPO-PS-PM [14] 64689 15.98 568.78 15.92 65089 600.24 16.15
PPO-PS [14] 64614 15.89 564.92 15.45 65058 594.18 16.74
CIMHA 64606 0.7 564.81 0.65 65,055 593.97 0.76
Tab. 2: Resul compa isons o he second sys em wi h quad a ic uel cos unc ion o he mal uni s.
Me hod LGM [9] EPSO [9] γ-PSO [9] CIMHA
Economic dispa ch Fuel cos ($) 53053.791 53053.793 53053.790 53051.476
CPU (s) - - - 50.1
Emission dispa ch
NOx 21739.271 21739.270 21739.185 21370.479
Emission (kg) SO2 74131.817 74131.817 74131.681 73924.733
CO2 373122.569 373122.568 373121.273 368209.983
CPU (s) - - - 50.5
Combined economic and
emission dispa ch
Fuel cos ($) 54337.014 54337.027 54336.888 54333.564
Emission (kg)
NOx 21745.127 21745.138 21745.021 21540.195
SO2 74114.989 74115.007 74114.821 73868.9859
CO2 373165.020 373165.186 373163.420 370203.756
CPU (s) - - - 49.9
To al CPU ime o h ee dispa ch cases 12.26 100.65 49.01 150.5
Tab. 3: Resul compa ison o he sys em wi h al e poin loading e ec s o he mal uni s.
Me hod Economic dispa ch Emission dispa ch Economic emission dispa ch
Cos ($) CPU (s) Emission (lb) CPU (s) Cos $) Emission (lb) CPU (s)
SA-BGA [1] 70718 - 23200 - 73612 26080 1492
RCGA [12] 66516 40.36 23222 41.98 - - -
NSGA-II [12] - - - - 68333 25278 45.42
MODE [12] - - - - 68388 25792 46.76
SPEA-2 [12] - - - 68392 26005 57.02
GA-MU [13] 67751 90.15 23223 78.27 68521 26080 96.10
IGA-MU [13] 66539 51.63 23223 42.87 68492 26080 53.54
PSO-PM [14] 66349 33.14 23167 33.63 67994 25902 34.11
PSO [14] 66223 32.15 23112 32.34 67892 25773 34.52
PPO-PM [14] 65912 21.03 23078 21.18 67211 25606 22.04
PPO [14] 65885 21.45 22966 21.56 67170 25601 22.11
PPO-PS-PM [14] 65723 21.12 22912 24.74 67092 25600 24.90
PPO-PS [14] 65567 22.00 22828 21.98 66951 25596 22.76
CIMHA 64989 16.4 22817 16.8 66530 25247 16.30
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Fig. 1: Fi ness unc ion con e gence cha ac e is ic o economic
dispa ch o sys em 1.
Fig. 2: Fi ness unc ion con e gence cha ac e is ic o emission
dispa ch o sys em 1.
The cos o economic dispa ch, he emission o
emission dispa ch, and he cos and emission o eco-
nomic emission dispa ch om he p oposed CIMHA
me hod ha e been compa ed o hose om o he me h-
ods as in Tab. 1. Ob iously, he p oposed me hod can
ob ain be e solu ions han all compa ed me hods o
he h ee dispa ch cases. Mo eo e , he compu a ional
ime o pe o ming he CIMHA me hod is signi ican ly
sho e han ha om o he me hods. No e he me h-
ods in [12] ha e been implemen ed on a Pen ium-IV
3.0 GHz PC. The e is no compu e epo ed o he
me hods in [14].
Figu es 1, Fig. 2, and Fig. 3 espec i ely show
he i ness unc ion con e gence cha ac e is ic o eco-
nomic dispa ch, emission dispa ch, and combined eco-
nomic and emission dispa ch in addi ion o he Pa e o-
op imal on depic ed in Fig. 4.
Fig. 3: Fi ness unc ion con e gence cha ac e is ic o comp o-
mise dispa ch o sys em 1.
Fig. 4: Pa e o-op imal on o uel cos and emission o sys-
em 1.
2) The Second Sys em wi h Fou Objec i e
Func ions
The combined objec i e o his sys em wi h wo hy-
d o and wo he mal uni s includes a o al cos unc-
ion and h ee emission unc ions o NOx, CO2and
SO2. The sys em is scheduled in a 24 hou pe iod wi h
one hou o each subin e al. The emission da a o
he sys em is om [20] and he es o da a is om
[2]. The p oposed CIMHA is implemen ed o ob ain
he op imal solu ion o he cases o economic dis-
pa ch (w= 1, w1=w2=w3= 0), emission dispa ch
(w= 0, w1=w2=w3= 1/3), and he comp omise
case (w= 0.5, w1=w2=w3= 0.5/3). The numbe
o nes s, maximum numbe o i e a ion, and alue o
he p obabili y pa a e espec i ely se o 40, 1800 and
0.9 o he h ee cases. Fo each case o dispa ch, he
CIMHA me hod is un 20 independen ials. The e-
sul compa ison o he h ee cases om he p oposed
CIMHA wi h o he me hods including LGM, EPSO,
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and γ-PSO in [9] a e gi en in Tab. 2. As obse ed,
he p oposed CIMHA me hod can ob ain be e so-
lu ion han LGM, EPSO, and γ-PSO in [9] o he
h ee dispa ch cases. The compa ison o o al compu-
a ional ime has indica ed ha he p oposed CIMHA
me hod is slowe han all me hods in [9]. Howe e , he
h ee me hods canno deal wi h p oblems wi h noncon-
ex uel cos o he mal uni s, leading o di icul y o
dealing wi h complex sys ems. This ad an age o he
CIMHA me hod o e he h ee me hods will be demon-
s a ed in Sec ion 4.3. The e is no compu e epo ed
o he me hods in [9].
4.3. Sys em wi h Noncon ex Fuel
Cos Func ion o The mal Uni s
and wo Objec i e Func ions
The sys em consis s o wo hyd o plan s and ou he -
mal plan s wi h noncon ex uel cos and emission unc-
ions om [13] scheduled in ou subin e als wi h 12
hou s o each. The numbe o nes s and he maxi-
mum numbe o i e a ion a e se o 50 and 2500 and
meanwhile Pais in ange om 0.1 o 0.9 o all eco-
nomic, emission and economic-emission dispa ches. As
a esul , he bes alue o Pa o he h ee dispa ch
cases is ob ained a 0.9. The esul compa ison wi h
o he me hods o he economic, emission, and comp o-
mise dispa ches is gi en in Tab. 3. As obse ed om
he able, he CIMHA can ob ain be e o al cos and
emission han o he me hods o he case o economic
dispa ch, emission dispa ch and economic emission dis-
pa ch. Mo eo e , he compu a ional ime om he p o-
posed me hod is also as e han ha om he o he
me hods. The o al compu a ional ime o inding so-
lu ions o economic dispa ch, emission dispa ch and
economic emission dispa ch by SA-BGA [1] is 24 min-
u es and 52 seconds based on he Ma lab 6.0 pla o m
and a Pen ium 3 PC. The compu a ional imes o GA-
MU and IGA-MU in [13] we e om a PIII PC. The e
is no compu e epo ed o he me hods in [14].
5. Discussion
5.1. S opping C i e ia
Gene ally, he s opping c i e ia o me hods sol ing
op imiza ion p oblems a e usually based on he i e a-
i e e o o wo consecu i e i e a ions, cons ain mis-
ma ch, and maximum numbe o i e a ions. In ac ,
depending on he applied solu ion me hods, he s op-
ping c i e ia may be used in di e en ways as long as
he inal solu ion is a easible one. In his pape , he
s opping c i e ia o he p oposed CIMHA me hod a e
only based on he maximum numbe o i e a ions like
o he me a-heu is ic sea ch me hods since he equali y
cons ain s o he p oblem a e always sa is ied by he
slack a iables. Mo eo e , he i e a i e e o o wo
consecu i e i e a ions is also no conside ed since he
p oposed me hod is a popula ion-based me hod using
andom sea ch and i may happen ha he ob ained
solu ion a e se e al i e a ions is no imp o ed. Tha
means, he solu ion ob ained a e se e al i e a ions is
s ill he same and i canno be used as s opping c i e ia
since i may lead o he e mina ion o he algo i hm
wi h non-op imal solu ion. In ac , he s opping c i e-
ia o popula ion based me hods a e always based on
he maximum numbe o i e a ions. Howe e , a small
numbe o i e a ions may lead o a non-op imal solu-
ion. On he con a y, a la ge numbe o i e a ions
will lead o he excessi e ime consump ion. In addi-
ion, he alue o he maximum i e a ion is dependen
on he scale and he complexi y o conside ed sys ems.
The e o e, a p ope selec ion o maximum numbe o
i e a ions o each sys em is based on expe imen s.
5.2. Con e gence Analysis
Al hough he e is no ma hema ical ela ionship be-
ween CIMHA me hod and he mul i-objec i e ST-
HTS p oblem, he CIMHA can p ope ly deal he mul i-
objec i e ST-HTS p oblem based on he p oblem o -
mula ion as an op imiza ion p oblem. Fo implemen a-
ion o CIMHA me hod o he mul i-objec i e ST-HTS
p oblems, each nes in he popula ion ep esen s he
powe ou pu s o he mal uni s and he wa e discha ge
o hyd o uni s. Du ing he sea ch p ocess, he quali y
o each nes will be e alua ed ia a i ness unc ion
which is de ined as a combina ion o objec i e unc ion
and penal ies o iola ed cons ain s. The nes wi h
lowe i ness unc ion alue has be e quali y han ha
wi h highe i ness unc ion alue. The inal solu ion
will be he nes wi h he lowes i ness unc ion alue
a he end o he i e a i e p ocess. Fo each sys em,
he p oposed me hod is un wen y independen ials
and he a e o success is 100 %. The con e gence cha -
ac e is ics o es sys em 1 ha e been gi en in o m o
he nume ical esul s in he pape . The ob ained e-
sul s ha e shown he app op ia eness and e ec i eness
o he CIMHA me hod o he mul i-objec i e ST-HTS
p oblem.
5.3. Di e si y o he Sea ch Space
Be o e ob aining he op imal solu ion, he CIMHA
me hod pe o ms an i e a i e sea ch p ocess whe e wo
imes new solu ions a e gene a ed a each i e a ion con-
sis ing o he i s new solu ion gene a ion ia Le y
ligh s as in Sec ion 3.3.2) and he second gene a ion
ia he ac ion o alien eggs o be abandoned as in Sec-
ion 3.3.3). In each s ep, i he new ob ained solu ion
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is ou o he limi s o a iables, he solu ion will be
ixed in he limi s. In ac , he second gene a ion e-
ines he esul ob ained om he i s gene a ion ia
Le y ligh s. Fo each es sys em in each case, he ob-
ained esul s including maximum cos , minimum cos ,
a e age cos and s anda d de ia ion will e eal he so-
lu ion quali y o he p oposed me hod o he p oblem.
In ac , he di e ence be ween he maximum cos and
minimum cos o economic dispa ch is e y low and
he s anda d de ia ion is close o ze o o se e al cases
o he sys ems. Consequen ly, i can be s a ed ha he
quali y o solu ion is e y high.
5.4. E iciency
In his pape , he pe o mance o he CIMHA is ali-
da ed by es ing on h ee es sys ems whe e he chal-
lenges a e no only he la ge-scale o sys em bu also
he complexi y o he objec i e unc ions including
noncon ex uel cos objec i e unc ion and exponen-
ial emission objec i e unc ion oge he wi h many
objec i e unc ions including h ee emission objec i e
unc ions and one uel cos objec i e unc ion o es
sys em 2. The compa ison o uel cos , emission and
execu ion ime be ween he p oposed CIMHA me hod
and o he me hods in he li e a u e ha e indica ed ha
he CIMHA me hod can ob ain be e solu ion qual-
i y han he o he me hods wi h as e compu a ional
ime, especially o sys ems wi h complica ed objec-
i e unc ions. The esul compa ison as epo ed in
he nume ical esul s sec ion has shown he e iciency
o he p oposed CIMHA me hod o each es case.
Consequen ly, he CIMHA me hod is e y e icien o
sol ing he mul i-objec i e ST-HTS p oblem.
5.5. Measu emen o Robus ness
The op imal solu ion by he p oposed CIMHA me hod
depends on many pa ame e s, such as numbe o nes s,
maximum numbe o i e a ions, and p obabili y o alien
egg disco e y. The quali y o he ob ained solu ions by
he p oposed CIMHA is e alua ed ia he s anda d de-
ia ion whe e he smalle s anda d de ia ion o he ob-
ained esul s e lec s he be e solu ion quali y o he
solu ion me hod. By expe imen s, hese pa ame e s
ha e been selec ed o each es sys em. Fo ob ain-
ing op imal solu ion o he es sys ems, he p oposed
CIMHA me hod is a subjec o wen y independen i-
als. The s anda d de ia ion o he sys em in each case
is e y low (close o ze o o jus sligh ly highe han
ze o) and he solu ion quali y is, he e o e, conside ed
e y high.
6. Conclusion
In his pape , he p oposed CIMHA me hod has been
success ully applied o sol ing he mul iobjec i e ST-
HTS p oblem. The e ec i eness o he CIMHA me hod
is based on wo main ea u es including he Lé y ligh s
and p obabili y o disco e y o a s ange egg in a hos
bi d’s nes . The ad an age o he CIMHA me hod
is ha i is e ec i e o inding he op imal solu ion
wi h ew con ol pa ame e s. The p oposed me hod
has been es ed on h ee hyd o he mal sys ems wi h
di e en numbe s o objec i e unc ions. The esul
compa ison has indica ed ha he p oposed me hod
can ob ain be e solu ion quali y wi h sho e com-
pu a ional ime han many o he me hods o he es
sys ems. The e o e, he p oposed CIMHA can be an al-
e na i e me hod o dealing wi h mul iobjec i e sho -
e m hyd o he mal scheduling p oblems. In he u u e
esea ch, he CIMHA will be implemen ed o sol ing
he mul i-objec i e a iable-wa e head sho - e m hy-
d o he mal scheduling p oblem whe e he hyd o gene -
a ion is a unc ion o wa e discha ge and ese oi ol-
ume because he wa e head is no a cons an . Mo e-
o e , a mo e complex hyd o he mal scheduling p ob-
lem wi h a se o cascaded ese oi s o hyd opowe
plan s can also be conside ed o e i y he e iciency o
CIMHA me hod o di e en hyd o he mal scheduling
p oblems.
Re e ences
[1] BASU, M. A simula ed annealing-based goal-
a ainmen me hod o economic emission load
dispa ch o ixed head hyd o he mal powe sys-
ems. In e na ional Jou nal o Elec ical Powe .
2005, ol. 27, iss. 2, pp. 147–153. ISSN 0142-0615.
DOI: 10.1016/j.ijepes.2004.09.004.
[2] RASHID, A. H. A. and K. M. NOR. An E icien
Me hod o Op imal Scheduling o Fixed Head Hy-
d o and The mal Plan s. IEEE T ansac ions on
Powe Sys ems. 1991, ol. 6, no. 2, pp. 632–636.
ISSN 0885-8950. DOI: 10.1109/59.76706.
[3] WOOD, A. J. and B. F. WOLLENBERG. Powe
Gene a ion, Ope a ion and Con ol. New Yo k:
John Wiley & Sons, 1996. ISBN 0-471-58699-4.
[4] YANG, J. and CHEN, N. Sho Te m Hyd o he -
mal Coo dina ion Using Mul i-Pass Dynamic P o-
g amming. IEEE T ansac ions on Powe Sys ems.
1989, ol. 4, no. 3, pp. 1050–1056. ISSN-0885-
8950. DOI: 10.1109/59.32598.
[5] SALAM, M. S., K. M. NOR, and A. R. HAMDAN.
Hyd o he mal Scheduling Based Lag angian Re-
laxa ion App oach o Hyd o he mal Coo dina-
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