multiobjective thermal generation scheduling

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MULTIOBJECTIVE THERMAL SCHEDULING 1

CHAPTER 1

INTRODUCTION

Environmental pollution i on t!e in"reae #ue to in#utrial a#van"ement$ T!ou%!

te"!nolo%& !a ma#e e"onomi" #evelopment poi'le( it at t!e ame time pro#u"e enormou)uantitie o* !arm*ul pro#u"t an# +ate$ Alo( t!e e,itin% ener%& pro#u"tion pro"ee are

not e"olo%i"all& "lean$ -or intan"e( t!ermal po+er plant pollute air( oil an# +ater$ T!e

"om'ution o* *oil *uel %ive rie to parti"ulate material an# %aeou pollutant apart *rom

t!e #i"!ar%e o* !eat to +ater "oure$ T!e parti"ulate material #oe not "aue a eriou

pro'lem in air "ontamination 'ut t!e t!ree prin"iple %aeou pollutant( o,i#e o* "ar'on

.CO,/( o,i#e o* ulp!ur.SO,/( an# o,i#e o* nitro%en.NO,/( "aue #etrimental e**e"t on

!uman 'ein%$ T!e uual "ontrol pra"ti"e i to re#u"e o**enive emiion t!rou%! pot0

"om'ution "leanin% &tem u"! a ele"trotati" pre"ipitator( ta"1 %a "ru''er( or

+it"!in% permanentl& to *uel +it! lo+ emiion potential$ 2ot0"om'ution removal

&tem re)uire time *or en%ineerin% #ei%n( "ontru"tion( an# tetin% 'e*ore t!e& "an 'e

'rou%!t online$ Hen"e( t!e o'viou alternative i %o *or *uel !avin% lo+ emiion potential(

i$e$ *rom "oal to oil3 oil i !o+ever e,tremel& e,penive an# upplie are un"ertain$ T!u(

t!ere i a !eer nee# *or optimum operatin% trate%&( +!i"! "an enure minimum pollution

level at minimum operatin% "ot$

1.1 MULTI-OBJECTIVE OPTIMIZATION

In re"ent &ear t!ere !a 'een an in"reae in reear"! on multi0o'4e"tive optimi5ation

met!o#$ De"iion +it! multi0o'4e"tive are )uite prevalent in %overnment( militar&(

in#utr&( an# ot!er or%ani5ation$ Reear"!e *rom a +i#e variet& o* #i"ipline u"! a

mat!emati"( mana%ement( "ien"e( e"onomi"( en%ineerin% an# ot!er !ave "ontri'ute# to

t!e olution met!o# *or multi0o'4e"tive optimi5ation pro'lem$ T!e ituation i *ormulate#

a a multi0o'4e"tive optimi5ation pro'lem .alo "alle# multi0per*orman"e( multi0"orre"tion or ve"tor optimi5ation/ in +!i"! t!e en%ineer6 %oal i to ma,imi5e or minimi5e not a in%le

o'4e"tive *un"tion 'ut everal o'4e"tive *un"tion imultaneoul&$ T!e purpoe o* multi0

o'4e"tive pro'lem in t!e mat!emati"al pro%rammin% *rame+or1 i to optimi5e t!e #i**erent

o'4e"tive *un"tion( G in num'er( u'4e"t to et o* &tem "ontraint$ -or e,ample(

DEPT. OF ELECTRICAL & ELECTRONICS | MSEC, BANGALORE




Ma,imi5e( f .,/ 7 8 f 9.,/(:$( f i.,/(:$( f G.,/ ;T

Su'4e"t to x ϵ X

+!ere x i an n0#imenional ve"tor o* #e"iion varia'le( X i t!e #e"iion pa"e( an# f .,/ i

a ve"tor o* G real0value# *un"tion$

In or#er to ma1e t!e pro'lem non0trivial( it i aume# t!at t!e o'4e"tive are in "on*li"t an#

in"ommenura'le$ O+in% to t!e "on*li"tin% nature o* o'4e"tive( an optimal olution t!at

imultaneoul& ma,imi5e<minimi5e all t!e "riteria i uuall& not attaina'le$ T!ere are

everal olution "alle# e**i"ient olution t!at !ave t!e propert& t!at no improvement in an&

o'4e"tive i poi'le +it!out a"ri*i"in% one or more ot!er o'4e"tive$

9$9$9 De*inition

A olution x = X i ai# to 'e e**i"ient i* *or an& x = X ati*&in% f 1 . x/ > f 1 . xe/( f 4. x/ < f 4. x

e/ *or

at leat one ot!er in#e, j ≠ k $

T!e et o* all e**i"ient olution in t!e "ontinuou "ae i 1no+n a t!e e**i"ient *rontier $An

e**i"ient olution i alo 1no+n a a non?#ominate# olution( non0in*erior olution or 2areto

optimal olution$

Man& approa"!e an# met!o# !ave 'een propoe# in re"ent &ear to olve multi0o'4e"tive

optimi5ation pro'lem$ T!ee met!o# are 'roa#l& %roupe# un#er t+o ma4or title( non0

intera"tive an# intera"tive$ In t!e non0intera"tive met!o#( a %lo'al pre*eren"e *un"tion o* t!e

o'4e"tive i i#enti*ie# an# optimi5e# +it! repe"t to t!e "ontraint$ On t!e ot!er !an#( in t!e

intera"tive met!o#( t!e #e"iion ma1er i#enti*ie a lo"al pre*eren"e *un"tion$ Sin"e DM i

involve# in t!e entire olution pro"e( intera"tive met!o# !ave 'een 'etter a""epte# in

pra"ti"e$ T!e 'et "ompromie olution i an e**i"ient olution t!at ma,imi5e t!e DM6

pre*eren"e *un"tion$

It ma& o'viou t!at tra#e o** amon% t!ee o'4e"tive are #i**i"ult 'e"aue o* t!eir #i**erent

nature$ T!i implie t!at t!e o'4e"tive are @non0"ommenura'le6$ In ot!er +or#( it i





#i**i"ult to treat repe"tive o'4e"tive un#er t!e i#enti"al "riterion$ 2rin"ipall&( met!o#olo%ie

*or olvin% multi0o'4e"tive pro'lem #i**erin% t+o +a&

.9/ T!e pro"e#ure ue# to %enerate non0in*erior olution$

./ T!e +a& an# mean ue# to intera"t +it! t!e DM an# t!e t&pe o* in*ormation ma#e

availa'le to t!e DM u"! a tra#e0o**$

T!e +ei%!tin% an# "ontraint met!o# i#enti*& t!e non0in*erior et( +it!in +!i"! t!e 'et

"ompromie olution lie$ In almot all #e"iion0ma1in% pro'lem( t!ere are everal "riteria

*or 4u#%in% t!e poi'le alternative$ T!e main "on"ern o* DM i to *ul*ill t!e "on*li"tin%

%oal +!ile ati*&in% t!e "ontraint o* t!e &tem$ T!ere are t+o #i**erent approa"!e to

olve u"! pro'lem$

i$ One approa"! aume t!at t!ere e,it a utilit& *un"tion *or t!e parti"ular pro'lem$

Su"! a *un"tion i ue# to o'tain t!e 'et alternative$ii$ T!e ot!er approa"! ma1e no aumption re%ar#in% t!e e,iten"e o* an& utilit&

pro'lem 'ut provi#e t!e DM +it! t!e et o* imple 'ut e**e"tive tool to o'tain t!e

'et alternative$

Here( a *e+ met!o# *or %eneratin% e**i"ient point are revie+e#$ T!ere are t!e min0ma,

optimum( t!e +ei%!tin% met!o#( t!e 0"ontraint met!o#( an# t!e +ei%!te# min0ma, met!o#$

1.2 Weightingmeth!

T!e +ei%!tin% met!o# i#enti*ie t!e non0in*erior et( in +!i"! t!e 'et "ompromie

olution lie( alo 1no+n a t!e parametri" approa"!( !a 'een t!e mot "ommon met!o#

ue# *or olvin% multi0o'4e"tive pro'lem until re"entl&$ Multi0o'4e"tive pro'lem i

"onverte# in t!i met!o# into "alar optimi5ation a %iven 'elo+

9

9( DG

i i

i

w w=

= ≥∑

+!ere9i=

9

9( DG

i i

i

w w=

= ≥∑

+!ere +i .i79(($$G/ are t!e +ei%!tin% "oe**i"ient$





T!e approa"! &iel# meanin%*ul reult to t!e #e"iion ma1er onl& +!en olve# man&

time *or #i**erent value o* +i .i79((:$(G/$ T!ou%! ver& little i uuall& 1no+n a'out

t!e value o* +ei%!tin% "oe**i"ient( t!e DM till "!ooe t!em( preuma'l& on t!e 'ai o*

!i intution$ T!e +ei%!tin% "oe**i"ient #o not re*le"t proportionall& t!e relative

importan"e o* t!e o'4e"tive 'ut are onl& *a"tor +!i"!( +!en varie#( lo"ate point in t!e

non0in*erior et$

1." #ENERATOR OPERATIN# CO$T

T!e ma4orit& o* %enerator in e,tant &tem are o* t!ree t&pe0nu"lear( !&#ro an# *oil *uel

."oal( oil or %ae/$ Nu"lear plant ten# to 'e operate# at "ontant output level an# !&#ro

plant !ave eentiall& no varia'le operatin% "ot$ T!ere*ore( t!e "omponent o* "ot t!at *all

un#er t!e "ate%or& o* #ipat"!in% pro"e#ure are t!e "ot o* t!e *uel 'urnt in t!e *oil plant$

T!e total "ot o* operation in"lu#e t!e *uel "ot( "ot o* la'or( upplie an# maintenan"e are

*i,e# per"enta%e o* in"omin% *uel "ot t!e operatin% "ot o* t!e plant !a t!e *orm !o+n in

*i%ure $$ -or #ipat"!in% purpoe( t!i "ot i uuall& appro,imate# o* '& one or more

)ua#rati" e%ment$ So( t!e *uel "ot "urve i mo#ele# a a )ua#rati" in t!e a"tive po+er

%eneration$

B. / gi i gi i gi i F P a P b P c= + + R<!

%ig&'e 1. Operatin% "ot o* *oil0*ire# %enerator





T!e

min

gi P

i t!e minimum loa#in% limit 'elo+ +!i"! it i une"onomi"al to operate t!e unit

an#

ma,

gi P

i t!e ma,imum output limit$

T!e *uel "ot "urve ma& !ave a num'er o* #i"ontinuitie$ T!e #i"ontinuitie o""ur +!en t!e

output po+er !a to 'e e,ten#e# '& uin% a##itional 'oiler( team "on#ener( or ot!er

e)uipment$ Di"ontinuitie alo appear i* t!e "ot repreent t!e operation o* an entire po+er

tation( o t!at "ot !a #i"ontinuitie on parallelin% o* %enerator$ Fit!in t!e "ontinuit&

ran%e t!e in"remental *uel "ot ma& 'e e,pree# '& a num'er o* !ort line e%ment or

pie"e0+ie lineari5ation$

1.4 MATLABMATLAB i a !i%!0level lan%ua%e an# intera"tive environment t!at let &ou *o"u on &our

"oure +or1 an# appli"ation( rat!er t!an on pro%rammin% #etail$ It ena'le &ou to olve

man& numeri"al pro'lem in a *ra"tion o* t!e time it ta1e to +rite a pro%ram in a lo+er0level

lan%ua%e u"! a Java( C( C( or -ORTRAN$ ou "an alo ue MATLAB to anal&5e an#

viuali5e #ata uin% automation "apa'ilitie( t!ere'& avoi#in% t!e manual repetition "ommon

+it! ot!er pro#u"t$

2ro%rammin% an# #evelopin% al%orit!m i *ater +it! MATLAB t!an +it! tra#itional

lan%ua%e 'e"aue MATLAB upport intera"tive #evelopment +it!out t!e nee# to per*orm

lo+0level a#minitrative ta1( u"! a #e"larin% varia'le an# allo"atin% memor&$ T!ouan#

o* en%ineerin% an# mat!emati"al *un"tion are availa'le( eliminatin% t!e nee# to "o#e an#

tet t!em &ourel*$ At t!e ame time( MATLAB provi#e all t!e *eature o* a tra#itional

pro%rammin% lan%ua%e( in"lu#in% arit!meti" operator( *lo+ "ontrol( #ata tru"ture( #ata

t&pe( o'4e"t0oriente# pro%rammin%( an# #e'u%%in% *eature$

MATLAB !elp &ou 'etter un#ertan# an# appl& "on"ept in a +i#e ran%e o* en%ineerin%(

"ien"e( an# mat!emati" appli"ation( in"lu#in% i%nal an# ima%e pro"ein%(

"ommuni"ation( "ontrol #ei%n( tet an# meaurement( *inan"ial mo#elin% an# anal&i( an#

"omputational 'iolo%&$ A##0on tool'o,e( +!i"! are "olle"tion o* ta10 an# appli"ation0





pe"i*i" MATLAB *un"tion( a## to t!e MATLAB environment to olve parti"ular "lae o*

pro'lem in t!ee appli"ation area$

Fit! over one million uer( MATLAB i re"o%ni5e# a a tan#ar# tool *or in"reain% t!e

pro#u"tivit& o* en%ineer an# "ientit$ Emplo&er +orl#+i#e "onitentl& report t!e

a#vanta%e o* 'ein% MATLAB pro*i"ient$

1.( %UZZ) $ET THEOR) IN POWER $)TEM

A#van"ement in "omputer te"!nolo%& ao"iate# +it! intelle"tual a"tivitie reulte# in ne+

*iel# o* in)uir& u"! a &tem "ien"e( #e"iion anal&i$ A mat!emati"al *ormulation

+it!in +!i"! t!ee variou t&pe o* un"ertaintie "an 'e properl& "!ara"teri5e# an#

inveti%ate# i no+ availa'le in term o* t!e t!eor& o* *u55& an# *u55& meaure$ In *a"t( one

o* Lo*ti a#e!6 "ontri'ution to &tem mo#elin% i t!e repreentation o* va%ue( in"omplete

1no+le#%e t!at #oe not !ave a ran#om nature an# t!ere*ore "annot 'e repreente# '& a

pro'a'iliti" approa"!$ -u55& et t!eor& provi#e# a 'ai *or t!e interpretation o* mem'er

*un"tion a poi'ilit& #itri'ution( +!i"! i a ver& ue*ul "on"ept in man& pra"ti"al

appli"ation$

Appli"ation o* *u55& et t!eor& +it!in t!e *iel# o* #e"iion ma1in% !ave( *or t!e mot part("onite# o* e,tenion or *u55i*i"ation o* t!e "lai"al t!eorie o* #e"iion ma1in%$

De"iion ma1in% un#er "on#ition o* ri1 an# un"ertaint& !a 'een mo#ele# '& pro'a'iliti"

#e"iion t!eorie$ -u55& #e"iion t!eorie attempt to #eal +it! *u55ine in!erent in

impre"ie #etermination o* pre*eren"e( "ontraint( an# %oal$ T!e intera"tion +it! t!e

#e"iion ma1er( t!e *u55& %oal o* t!e #e"iion ma1er( +a )uanti*ie# '& eli"itin% t!e

"orrepon#in% mem'er!ip *un"tion( in"lu#in% nonlinear *un"tion$

1.(.1 BA$IC$ O% %UZZ) $ET THEOR)

-u55& et are et +it! 'oun#arie t!at are not pre"ie$ T!e mem'er!ip in a *u55& et i not

a matter o* a**irmation( 'ut rat!er a matter o* a #e%ree$ F!en a i *u55& et an# , i a relevant

o'4e"t( t!e propoition K x i a num'er o* AK i not ne"earil& eit!er true or *ale( 'ut it ma&

'e true onl& to ome #e%ree( t!e #e%ree to +!i"! x i a"tuall& a num'er A$





T!ere are t!ree 'ai" met!o# '& +!i"! et "an 'e #e*ine# +it!in a %iven univeral et

9$ A et i #e*ine# '& namin% all it mem'er$ T!i met!o# "an 'e ue# *or *inite et$

Set A ( +!oe mem'er are a9( a(:::(an( i +ritten a

A7 a9( a(:::(an

$ A et i #e*ine '& a propert& ati*ie# '& it mem'er( i$e$

A7 x 2. x/F!ere t!e &m'ol i #enote# t!e p!rae u"! t!at( 2. x/ #ei%nate a propoition o*

t!e *rom , has the property 2$

$ A et i #e*ine# a *un"tion( "alle# "haracteristic function( t!at #e"lare +!i"!

element o* i mem'er o* et an# +!i"! are not$ Set A i #e*ine# '& it "!ara"teriti"

*un"tion or #i"rimination *un"tion

. / A x µ

( a *ollo+

9. /

D A

forx A x

forx A µ

∈= ∉

T!e "!ara"teriti" *un"tion map element o* to element o* t!e et ( 9( +!i"! i

e,pree# '&

A MD(9N A X µ →

T!e "!ara"teriti" *un"tion ai%n a value eit!er 9 or to ea"! in#ivi#ual in t!e

univeral et( t!ere'& #i"riminatin% 'et+een mem'er an# non0mem'er o* t!e "rip

et un#er "oni#eration$ Lar%er value repreent !i%!er #e%ree o* et mem'er!ip$

Su"! a *un"tion i "alle# a membership function( an# t!e et #e*ine# '& it i a *u55&

et$

Let 'e t!e univere o* o'4e"t +it! element x( +!ere A i "alle# a *u55& et o* $

Mem'er!ip o* x in "lai"al et @A6 "an 'e vie+e# a a "!ara"teriti" *un"tion

. / A x µ

*rom to .( 9/ u"! t!at9. /

D A

forx A x

forx A µ

∈= ∉

-or a *u55& et A o* univere ( t!ere %ra#e o* mem'er!ip o* x in A i #e*ine# a

. / A x µ

7 89( ;





F!ere

. / A x µ

i "alle# t!e membership function.

T!e value o* A µ

"an 'e an&+!ere *rom to 9( an# t!i ran%e i +!at ma1e it

#i**erent *rom "rip et$ T!e "loer t!e value o*

. / A x µ

i 9$( t!e more x 'elon% to 9$

T!u a *u55& et !a no !arp 'oun#ar&$-u55& et element are or#ere# pair in#i"atin% t!e value o* a et element an# t!e

%ra#e o* mem'er!ip( i$e$ A7 . x(

. / A x µ

/ x X ∈




MULTIOBJECTIVE GENERATION SCHEDULING

9

CHA2TER0

LITERATURE $URVE)

T!ere are everal literature revie+ in t!e area o* minimi5in% pollution level( 'ut till mu"!

pro%re i poi'le in t!e area o* e"onomi"0emiion loa# #ipat"!$ Gent an# Lamont !ave pu'li!e# a paper on minimum0emiion #ipat"! +!erein a "omputer pro%ram !a 'een

#evelope# *or online team unit #ipat"! reultin% in minimi5ation o* NO x emiion

emplo&in% t!e Ne+ton0Rap!on "onver%en"e *or "urve *ittin%$ -inni%an an# -oua# 89PQ;

!ave implemente# t+o nonlinear pro%rammin% olution pro"e#ure *or e"onomi" #ipat"!

treatin% t!e &tem pollutant a an a##itional "ontraint to t!e pro'lem$

Ca#o%an an# Eien'er% 89PQQ; treate# t!e e"onomi" environmental #ipat"! pro'lem a a

multi0trate%& pro'lem an# u%%ete# a nonlinear pro%rammin% te"!ni)ue( *or t!eimplementation o* t!e #ipat"! trate%ie$ ot!ari e,!i'ite# a "omputer oriente# te"!ni)ue

*or t!ermal po+er %eneration "!e#ulin% +!i"! reult in NO x emiion$ E**luent #ipat"!in%

i t!e operation o* a %eneration &tem o t!at t!e po+er plant at !i%!er emiion rate

%enerate le t!an t!e& +oul# un#er leat "ot #ipat"!in%$

o1o&ama 89P; !ave e,pertie an e**i"ient al%orit!m to o'tain t!e optimal po+er *lo+ in

po+er &tem operation an# plannin% p!ae '& olvin% a multi0o'4e"tive optimi5ation

pro'lem$ Minimum inter*eren"e in t!e environment +a one o* t!e o'4e"tive in #e"i#in% t!e

optimal &tem operation$ Ho'' 89PP; !ave tate# a mo#el *or evaluatin% t!e "ot an#

emplo&ment impa"t o* e**luent #ipat"!in% an# t!e *uel +it"!in% a mean *or re#u"in%

emiion *rom po+er plant$ T!e& !ave pro#u"e# tra#e0o** "urve( +!i"! !o+ t!e "ot an#

"oal *iel# 4o' loe ao"iate# +it! re#u"tion in SO emiion$ In t!e "onte,t o* in"reain%

pu'li" a+arene o* t!e environmental ituation an# t!e plea *or "lean air( t!e US "on%re

!a pae# a"i# rain le%ilation 8Ho''(9PP;$ In an e**ort to alleviate a"i# rain( Title IV o*

t!e 9PP U$S$ "lean a"t amen#ment man#ate# a retri"tion o* over in ele"tri"al utilit&

SO emiion '& t!e &ear $ Ho'' !a *urt!er ummari5e# a met!o# *or in"lu#in%(

un#erutili5ation "ontraint in pro'a'iliti" pro#u"tion "otin% mo#el an# real0time

#ipat"!$





1

CHA2TER0

PROBLEM %ORMULATION

$9 ECONOMIC LOAD DIS2ATCH

Let u aume t!at it i 1no+n a priori( +!i"! are t!e %enerator to 'e run to meet a parti"ular

loa# #eman# on t!e tation$ Suppoe t!ere i ituation +it! NG %enerator "ommitte# an# t!e

a"tive po+er loa# #eman# 2D i %iven( t!e real po+er %eneration 2%i *or ea"! %enerator !a

to 'e allo"ate# o a to minimi5e t!e total "ot$ T!e optimi5ation pro'lem "an t!ere*ore 'e

tate# a

Minimi5e

9

. / . / NG

gi i gi

i

F P F P =

= ∑

.9/

Su'4e"t to .9/ t!e po+er 'alan"e e)uation

9

. / NG

gi

i

P P =

=∑

!$9/

an# ./ t!e ine)ualit& "ontrain

min ma,

gi gi gi P P P ≤ ≤.i79((::$NG/ .$/

+!ere

gi P

i t!e #e"iion varia'le( i$e$ real po+er %eneration

P

i t!e real po+er #eman#

NG

i t!e num'er o* %eneration plant





11

min

gi P

i t!e lo+er permii'le limit o* real po+er %eneration

ma,

gi P

i t!e upper permii'le limit o* real po+er %eneration

F i!P gi "# i t!e operatin% *uel "ot o* t!e i t! plant an# i %iven '& t!e )ua#rati"

e)uation

B. /i gi i gi i gi i F P a P b P c= + + R<!

.$/

T!e a'ove "ontraine# optimi5ation pro'lem i "onverte# into an un"ontraine# optimi5ation

pro'lem$ La%ran%e multiplier met!o# i ue# in +!i"! a *un"tion i minimi5e# +it! i#e

"on#ition in t!e *orm o* e)ualit& "ontraint$ Uin% t!i met!o#( an au%mente# *un"tion i

#e*ine# a

9

. ( / . / . / NG

gi gi gi

i

$ P F P P P λ λ =

= + − ∑./

A ne"ear& "on#ition *or a *un"tion F i!P gi "#u'4e"t to ener%& 'alan"e "ontraint to !ave a

relative minimum at point 2%i i t!at t!e partial #erivative o* t!e La%ran%e *un"tion #e*ine#

'& L7L.2%i(λ

/ +it! repe"t to ea"! o* it ar%ument mut 'e 5ero$ So( t!e ne"ear&

"on#ition *or t!e optimi5ation pro'lem are

( ). ( /D

gi gi

gi gi

F P $ P

P P

λ λ

∂∂= − =

∂ ∂.i79((::$NG/ ./

an#

( )9

(D

NG gi

gi

i

$ P P P

λ

λ =

∂= − =

∂ ∑ ./





12

-rom E)$ ./

( ) gi

gi

F P

P λ

∂=

∂ .i79((::$NG/ .W/

+!ere

( ) gi

gi

F P

P

∂

∂ i t!e in"remental *uel "ot o* t!e it! %enerator .R<MF!/

Optimal loa#in% o* %enerator "orrepon# to t!e e)ual in"remental "ot point o* all t!e

%enerator$ E)$ .W/( "alle# t!e "oor#ination e)uation num'erin% NG( are olve#

imultaneoul& +it! t!e loa# #eman# to &iel# a olution *or La%ran%e multiplierλ

an# t!e

optimal %eneration o* NG %enerator$ Coni#erin% t!e "ot *un"tion %iven '& E)$ .$/( t!e

in"remental "ot "an 'e #e*ine# a

. /B

gi

i gi i

gi

F P a P b

P

∂= +

∂

.Q/

Su'titutin% t!e in"remental "ot into E)$ ./( t!i e)uation 'e"ome

B i gi ia P b λ + = .i79((::$NG/ ./

Rearran%in% E)$ ./ to %et 2%i

B

i gi

i

b P

a

λ −=

.i79((::$NG/ .P/

Su'titutin% t!e value o* 2%i in E)$ . /( +e %et

9 B

NGi

i i

b P

a

λ

=

−=∑

.9/





13

T!u(λ

"an 'e "al"ulate# uin% E)$9 an# 2%i "an 'e "al"ulate# uin% E)$P$ No+ "oni#er

t!e e**e"t o* t!e %enerator limit %iven '& t!e ine)ualit& "ontraint o* E) .$/ i* a parti"ular

%enerator loa#in% 2%i rea"!e t!e limit

min

gi P

or

ma,

gi P

( it loa#in% i !el# *i,e# at t!i value

an# t!e 'alan"e loa# i !are# 'et+een t!e remainin% %enerator on an e)ual in"remental "ot

'ai$

$ O2TIMAL GENERATION SCHEDULING

-rom t!e unit "ommitment ta'le o* a %iven plant( t!e *uel "ot "urve o* t!e plant "an 'e

#etermine# in t!e *orm o* a pol&nomial o* uita'le #e%ree '& t!e met!o# o* leat )uare *it$ I*

t!e tranmiion loe are ne%le"te#( t!e total &tem loa# "an 'e optimall& #ivi#e# amon%

t!e variou %eneratin% plant uin% t!e e)ual in"remental "ot "riterion o* E)$ ./$ It i

!o+ever( unrealiti" to ne%le"t tranmiion loe parti"ularl& +!en lon% #itan"e

tranmiion o* po+er i involve#$ A mo#ern ele"tri" utilit& erve over a vat area o*

relativel& lo+ loa# #enit&$ T!e tranmiion loe ma& var& *rom to 9 per"ent o* total

loa#$ T!ere*ore it i eential to a""ount *or tranmiion loe +!ile #evelopin% an

e"onomi" loa# #ipat"! poli"&$

T!e e"onomi" #ipat"! pro'lem #e*ine# a t!at +!i"! minimi5e t!e total operatin% "ot o* a po+er &tem +!ile meetin% t!e total loa# plu tranmiion loe +it!in %enerator limit$

Mat!emati"all&( t!e pro'lem "an 'e #e*ine# a

B

9

. / . / NG

gi i gi i gi i

i

F P a P b P c=

= + +∑

Minimi5e .99/

Su'4e"t to .9/ t!e po+er 'alan"e e)uation

9

NG

gi $

i

P P P =

= +∑ .9$9/





14

./ an# t!e ine)ualit& "ontraint

min ma,

gi gi gi P P P ≤ ≤

.i79(:::$$NG/ .9$/

+!ere

ia

,ib

an#ic

are t!e "ot "oe**i"ient

P

i t!e loa# #eman#

gi P

i t!e real po+er %eneration an# +ill a"t a #e"ri'e# varia'le

NG i t!e num'er o* %eneration 'ue

$ P

i t!e tranmiion po+er loe$

One o* t!e mot important( imple 'ut appro,imate met!o# o* e,prein% tranmiion lo

a a *un"tion o* %enerator po+er i t!rou%! B0"oe**i"ient$ T!i met!o# ue t!e *a"t t!at

num'er un#er normal operatin% "on#ition( t!e tranmiion loe )ua#rati" in t!e in4e"te#

'u real po+er$ T!e %eneral *orm o* t!e lo *ormula uin% B0"oe**i"ient i

9 9

NG NG

$ gi ij gj

i j

P P % P = =

= ∑∑ MF .9/

+!ere

gi P

an#

gj P

are t!e real po+er %eneration at t!e i t! an# 4t! 'ue( repe"tivel&

ij %

are t!e lo "oe**i"ient +!i"! are "ontant un#er "ertain aume# "on#ition

NG i t!e num'er o* %eneration 'ue$





15

( )9

( . / . / NG

gi gi $ gi

i

$ P F P P P P λ λ =

= + + − ∑T!e a'ove "ontraine# optimi5ation pro'lem i

"onverte# into an un"ontraine# optimi5ation pro'lem$ La%ran%e multiplier met!o# i ue# in

+!i"! t!e *un"tion i minimi5e# +it! i#e "on#ition in t!e *orm o* e)ualit& "ontraint$

Uin% La%ran%e multiplier( an au%mente# *un"tion i #e*ine# a

.9/

+!ereλ

i t!e La%ran%ian multiplier$

( ). ( /9 D

gi gi $

gi gi gi

F P $ P P

P P P

λ λ

∂ ∂ ∂= + − = ÷ ÷∂ ∂ ∂

Ne"ear& "on#ition *or t!e optimi5ation pro'lem are

.i79((::$NG/

Rearran%in% t!e a'ove e)uation(

( )9

gi $

gi gi

F P P

P P λ

∂ ∂= − ÷ ÷∂ ∂

.i79((::$NG/ .9/

+!ere

( ) gi

gi

F P

P

∂

∂ i t!e in"remental operatin% "ot o* t!e it! %enerator .R<MF!/

$

gi

P

P

∂∂

i t!e in"remental tranmiion loe$

( )

9

(D

NG gi

$ gi

i

$ P P P P

λ

λ =

∂= + − =

∂ ∑E)$ .9/ i 1no+n a t!e e,a"t "oor#ination e)uation( an#

.9W/





16

E)uation .9/ ( t!e o0"alle# "oor#ination e)uation( num'erin% NG( i olve# imultaneoul&

+it! E)$.9W/ to &iel# a olution *or La%ran%e multiplier λ

an# t!e optimal %eneration o* NG

%enerator$ B& #i**erentiatin% t!e tranmiion lo e)uation( E)$.9/ +it! repe"t to 2%i( t!e

in"remental tranmiion lo "an 'e o'taine# a

D

9

B NG

$i ij gj

j gi

P % % P

P =

∂= +

∂ ∑ .i79((::$$NG/ .9Q/

an# '& #i**erentiatin% "ot *un"tion E)$.99/ +it! repe"t to 2%i ( t!e in"remental "ot "an 'e

o'taine# a

. /B

gi

i gi i

gi

F P a P b

P

∂= +

∂ .i79((:::NG/ .9/

E)$ .9/ "an 'e re+ritten a

. /

9

gi

gi

$

gi

F P

P

P P

λ

∂∂

=

∂− ∂

or

( ) gi

i

gi

F P $

P λ

∂ ÷ = ÷∂

.i79((::$$NG/ .9P/

+!ere

9

9i

$

gi

$ P

P

=∂−∂

i "alle# t!e penalt& *a"tor o* t!e i t! plant$




( ) D

9

B 9 B NG

i ii gi i ij gj i

j j i

a % P % % P bλ λ =≠

+ = − − −

∑


17

To o'tain t!e olution( u'titute E)$ .9/ an# .9P/ into E)$ .9W/

D

9

B 9 B NG

i gi i i ij gj

j

a P b % % P λ =

+ = − − ÷

∑

.i79((:::NG/

Rearran%in% t!e a'ove e)$ to %et 2%i( +e !ave

.i79((::$NG/

T!e value o* 2%i "an 'e o'taine# a

( )

D99 B

B

NG

i ij gj i j j i

gi

i ii

% % P b

P a %

λ

λ

=≠

÷

− − − ÷ ÷ =

+∑

.i79((::$$NG/ ./

I* t!e intial value o* 2%i .i79(:(NG/ an#λ

are 1no+n( t!e a'ove e)uation "an 'e olve#

iterativel& until E)$.9W/ i ati*ie# '& mo#i*&in%λ

$ T!i te"!ni)ue i 1no+n a u""eive

appro,imation$ T!e tep +ie pro"e#ure i e,plaine# 'elo+$ -or impli"it& it i "oni#ere#t!at t!e olution remain +it!in limit$

"." M&*ti-+,etie P'+*em %'m&*/tin

In t!e multi0o'4e"tive pro'lem *ormulation( *our important non "ommenura'le o'4e"tive in

an ele"tri"al t!ermal &tem !ave 'een "oni#ere#$ T!ere are e"onom& an# environmental

impa"t 'e"aue o* NO, (SO ( an# CO %aeou pollution$

".".1 Enm0 +,etie

T!e *uel "ot o* a t!ermal unit i re%ar#e# a an eential "riterion *or e"onomi" *eai'ilit&$

T!e *uel "ot "urve i aume# to 'e appro,imate# '& a )ua#rati" *un"tion o* a %enerator

po+er output 2%i a





18

( )B

9

9

NG

i gi i gi i

i

F a P b P c=

= + +∑ R<! .9/

F!ere ai ( 'i (an# "i are "ot "oe**i"ient an# NG i t!e num'er o* %enerator$

".".2 Eni'nment/* +,etie

T!e emiion "an 'e #ire"tl& relate# to t!e "ot "urve t!rou%! t!e emiion rate( +!i"! i a

"ontant *a"tor *or a %iven t&pe o* *uel$ T!ere*ore( t!e amount o* NO, emiion i %iven a a

)ua#rati" *un"tion o* %enerator output 2%i ( i$e$

( )B

B 9 9 9

9

NG

i gi i gi i

i

F & P e P f =

= + +∑ 1%<!

./

+!ere #9i (e9i (an# * 9i are NO, emiion "oe**i"ient$

Similarl&( t!e amount o* SO emiion i %iven a a )ua#rati" *un"tion o* %enerator output

2%i (i$e$

( )B

E B B B9

NG

i gi i gi ii F & P e P f == + +∑

1%<! ./

+!ere #i (ei (an# * i are SO, emiion "oe**i"ient$

T!e amount o* CO emiion i alo repreente# a )ua#rati" *un"tion o* %enerator output

2%i (i$e$

( )B

R E E E

9

NG

i gi i gi i

i

F & P e P f =

= + +∑ 1%<! ./

+!ere #i (ei (an# * i are CO, emiion "oe**i"ient$

Onl& NO, emiion o'4e"tive i "oni#ere# in t!i tu#&$





19

"."." Cnt'/int

9$ To enure a real po+er 'alan"e( an e)ualit& "ontraint i impoe#( i$e$

( )9

D NG

gi $

i

P P P =

− + =∑

./

+!ere

2D i t!e po+er #eman#

2L i t!e tranmiion loe( +!i"! are appro,imate# in term o* B0"oe**i"ient a

9 9

NG NG

$ gi ij gj

i j

P P % P = =

= ∑∑

MF .W/

$ T!e ine)ualit& "ontraint impoe# on %enerator output are

min ma,

gi gi gi P P P ≤ ≤ .i79((:(NG/ .Q/

F!ere

min

gi P

t!e lo+er i limit( an#

ma,

gi P

i t!e upper limit o* %enerator output$

T!e o'4e"tive *un"tion i #e*ine# a minimi5e

B B9 B

9 9

. / . / NG NG

i gi i gi i i gi gi i

i i

w a P b P c w & P eiP f = =

+ + + + +∑ ∑

./

T!i "an 'e olve# '& met!o# o* La%ran%e Multiplier$ T!e La%ran%ian i #e*ine# a

9 9 B B

9

. / NG

$ gi

i

$ w F w F P P P λ =

= + + + − ∑

.P/ T!e optimalit& "on#ition "an 'e o'taine# '& partiall& #i**erentiatin% t!e

La%ran%ian +it! repe"t to t!e #e"iion varia'le an# e)uatin% t!e reult to 5ero$

9 B< .B / .B / . < 9/ D gi i gi i i gi i $ gi $ P w a P b w & P e P P λ =∂ ∂ + + + + ∂ ∂ − =

./





2

9

< D NG

$ gi

i

$ P P P λ =

∂ ∂ = + − =∑

.9/

E)$ ./ "an 'e impli*ie# to

9 B

9

. / . / . B 9/ D NG

i gi i i gi i ij gj

j

w a P b w & P e % P λ =

+ + + + − =∑

./

E)$ ./ "an 'e rearran%e# to o'tain e)uation *or po+er allo"ation

9 B 9 B

9

8 . B /; < .B B B / NG

ij gj i i ii

j j i

Pgi w bi w ei % P w a w & %λ λ λ =≠

= − + + + +

∑ i 79((:$NG

./

".". Initi/* /*&e

Initial value o* X an# po+er allo"ation "an 'e *oun# E)$ ./ '& aumin% loe e)ual to

5ero$

Initial value o* X i

9 B B

9 9

. . < // < .9< / NG NG

i i

P t t t λ = =

= + ∑ ∑

./

An# t!e initial po+er allo"ation are

9 B 9 B. / < .B B / gi i i i i P w b w e w a w & λ = − + +

./





21

". %&330 Deiin m/4ing

Coni#erin% t!e impre"ie nature o* #e"iion ma1er6 4u#%ment( it i natural to aume t!at

t!e #e"iion ma1er ma& !ave *u55& or impre"ie %oal *or ea"! o'4e"tive *un"tion$ T!e

*u55& et are #e*ine# '& e)uation "alle# mem'er!ip *un"tion$ T!ee *un"tion repreent

t!e #e%ree o* mem'er!ip in ome *u55& et uin% value *rom to 9$ T!e mem'er!ip

value ( in#i"ate in"ompati'ilit& +it! t!e et( +!ile 9 mean *ull "ompati'ilit& '& ma1in%

a""ount o* minimum an# ma,imum value o* ea"! o'4e"tive$

-un"tion to%et!er +it! t!e rate o* in"reae o* mem'er!ip ati*a"tion( t!e #e"iion ma1er

mut #ete"t mem'er!ip *un"tion

. /i F µ

in a u'4e"tive manner$ Here it i aume# t!at

. /i

F µ

i tri"tl& a monotoni" #e"reain% an# "ontinuou *un"tion #e*ine# a

min

ma,min ma,

ma, min

ma,

93

. / 3

D3

i i

ii i i i

i i

i i

F F

F F F F F F

F F

F F

µ

≤

−= < < − ≥

.i79((:$(M/ .W/

T!e value o* mem'er!ip *un"tion u%%et !o+ *ar .in t!e "ale *rom to 9/ a non0in*erior

.non0#ominate#/ olution !a ati*ie# t!ei F

o'4e"tive$ T!e a%%re%ate or avera%e o*

mem'er!ip *un"tion value

. /i

F µ

.i79((:(M/ *or all t!e o'4e"tive "an 'e "ompute# in

or#er to meaure t!e accomp'ishment o* ea"! olution in ati*&in% t!e o'4e"tive$

".( Ite'/tie /*g'ithm

ALGORITHM -OR MULTI0OBJECTIVE GENERATION SCHEDULING

9$ Rea# #ata( namel& "ot "oe**i"ient .ai('i("i/( NO, emiion "oe**i"ient .#i(ei(* i/(

B0"oe**i"ient .i79((:(NG ( 479((:$(NG/( "onver%en"e toleran"e( tep i5e(

ma,imum iteration ( +ei%!t "om'ination et"(





22

$ Coni#er a +ei%!t ve"tor

$ Compute initial value o* X uin% E)$ ./ an# initial value o* po+er uin% E)$ ./

$ Set iteration "ount IT79

$ Compute 2%i uin% E)$ ./

W$ Compute 2L7 2t B 2

Q$ Compute Y27 2D2L0 Z 2i

$ X 7 X mult[ Y2

P$ IT7IT9

9$ I* Y2 > t!e %o to tep ( ele ne,t tep

99$ C!e"1 %eneration limit$ I* t!ere are no violation t!en ne,t tep ele ma1e t!e

mo#i*i"ation

I* 2%i \ 2%i min t!en 2%i 7 2%i

min

I* 2%i > 2%i ma, t!en 2%i 7 2%i

ma, an# %o to tep

9$ Compute total "ot 7 Z .ai 2%i 'i 2%i "i/

9$ Compute total emiion 7 Z .#i 2%i ei 2%i * i/

9$ I* no more +ei%!t ve"tor t!en ne,t tep ele %o to tep

9$ Lo"ate ma,imum "ot an# ma,imum emiion

9W$ Compute *u55& mem'er!ip *un"tion o* total "ot an# total emiion uin% E)$

9Q$ Lo"ate t!e 'et "ompromie olution

9$ 2rint reult

9P$ Stop

CHA2TER

TE$T $)$TEM AND $IMULATION$

.1 Tet $0tem

A i,0%enerator &tem i "oni#ere#$ T!e *uel "ot an# NO, emiion are %iven in Ta'le





23

$9 an# Ta'le $ repe"tivel&$ Tranmiion lo "oe**i"ient are %iven in Ta'le $$T!e

po+er #eman# i 9 MF

TABLE .1 %&e* t 5R6h7 e8&/tin

B

99 9 9B

9B B B

B

9E E E

B

9R R R

B

9U U U

B

9W W W

D$DDBDEU T$REBDU TU$WERT

D$DDETWW W$R9DE9 EDE$QQTD

D$DDB9TB Q$RBTPD TRQ$9RTR

D$DD9ERU T$ED9UR BQR$BBR9

D$DDB9TB Q$RBTPD TRQ$9RTR

D$DDUPWE W$P9UUP BDB$D

F P P

F P P

F P P

F P P

F P P

F P P

= + +

= + +

= + +

= + +

= + +

= + + BUT

TABLE .2 NO9 emiin54g6h7 e8&/tin

B

B9 9 9

B

BB B B

B

BE E E

B

BR R R

B

BU U U

B

BW W W

D$DDWEBE D$ET9BT TD$PD9P

D$DDWRTE D$QPDBQ BT$TBRP

D$DDE9QR 9$EWDW9 EBR$9QQU

D$DDWQEB B$EPPBT W9D$BUEU

D$DDE9QR 9$EWDW9 EBR$9QQU

D$DDW9T9 D$EPDQQ UD$ETD

F P P

F P P

F P P

F P P

F P P

F P P

= − +

= − +

= − +

= − +

= − +

= − + T





24

TABLE ." LO$$ COE%%ICENT$

$ $9 $9 $ $ 0$$9 $ 0$ $9 $9 $9

$9 0$ $9 0$9 $9 $

$ $9 0$9 $9 $W $$ $9 $9 $W $ $

0$ $9 $ $ $ $9





25

.2 MATLAB PRO#RAM

"l"3

"lear all3

*p7*open.]#ipat"!$out](]+]/3

Data

mult7$9W3

reult75ero.99(9/3

Cot "oe**i"ient

a978$ $WW $9 $9 $9 $PW;3

'978$ W$99 Q$P $9 Q$P W$P9P;3

"978$W $QQ Q$9 Q$9 Q$9 $;3

NO, emiion "oe*i"ient

a78$W $W $9Q $WQ $9Q $W99;3

'780 $9 0$QPQ 09$WW9 0$PP 09$WW9 0$PQQ;3

"78$P9P $P $9QQ W9$ $9QQ $;3

Lo "oe**i"ient

B78$ $9 $9 $ $ 0$3

$9 $ 0$ $9 $9 $93

$9 0$ $9 0$9 $9 $3

$ $9 0$9 $9 $W $3

$ $9 $9 $W $ $3

0$ $9 $ $ $ $9;3

Ma,imum<minimum limit o* %enerator





26

2ma,789 9 9 9 9 9;3

2min78 ;3

Num'er o* %enerator

n7W3

a*a"tor79$W3

epilon7$93

p#793

p#97p#3

itma,7

#elp75ero.9(9/3

#793

+ta'le7 8 9$ $3

$P $93

$ $3

$Q $3

$W $3

$ $3

$ $W3

$ $Q3

$ $3 $9 $P3

$ 9$;3

57i5e.+ta'le/3

5975.9(9/3

*or "ount7959 t!e loop i repeate# *or ea"! +ei%!t ve"tor[[[[[[[[[[[[[9

+97+ta'le."ount(9/

+7+ta'le."ount(/

Computin% initial value o* lam#a

um973





27

*or i79n

t97+9['9.i/+['.i/3

t7$[+9[a9.i/$[+[a.i/3

um97um9.t9<t/3

en#

um73

*or i79n

t7$[+9[a9.i/$[+[a.i/3

um7um.9$<t/3

en#

lam#a7.p#um9/<um3

Computin% initial value o* lam#a en#

Computin% initial value o* po+er

*or i79n

2.i/7 .lam#a0+9['9.i/0+['.i//<.$[+9[a9.i/$[+[a.i//3

en#

Computin% initial value o* po+er en#%tatu78 ;3

*la%73

it73

+!ile *la%77 ^^ it \ itma, Iteration tart !ere[[[[[[[[[[[[[[[[[[[[[[[[

Computin% po+er

*or i79n

i* %tatu.i/77

um73

*or 479n





28

i* 4_7i

um7um$[B.i(4/[2.4/3

en#

en#

nr7lam#a[.9$0um/0+9['9.i/0+['.i/3

#r7[+9[a9.i/[+[a.i/[lam#a[B.i(i/3

2.i/7nr<#r3

en#

en#

Computin% po+er en#

Computin% loe

2L73

*or i79n

*or 479n

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29

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31

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32

CHA2TER0

RE$ULT$ AND DI$CU$$ION$

(.1 RE$ULT$

Ta'le $9 Fei%!t "om'ination an# "orrepon#in% po+er allo"ation

:1 :2 P1 P2 P" P P( P;

9$ $ 9$WP $QQQ $9 Q$W 9$WPP 9PQ$9

$P $9 $P P$P $WW W$W 9Q$WQ 9PQ$P99

$ $ $WQ Q$9 9$W P$ $9P 9P$9

$Q $ $ QP$WP 9Q$W $Q9 P$QQ 9P$9Q

$W $ W$WQ Q$ 9$W9Q $9W WW$P 9PQ$P9<.(< <.(< 22<.=><> 2;2.<=?> (2.=">? "(.=>1> "?".?2?= 1=>.("<2

$ $W 9$9 $9 Q$WP $9 9$WP 9PW$P9PW

$ $Q 9$WW $PWP $ $P $9 9PW$99Q

$ $ $999 $WQQ $WQ $W9Q P$9Q 9P$99Q

$9 $P $ $9 $ $99Q $P 9P$P9P

$ 9$ 9P$P 9$ W$9PPP P$9WQ QP$Q9 9P$9

Ta'le $9 %ive t!e po+er allo"ation *or +ei%!t "om'ination var&in% *rom 89 ; to 8 9;(

+!ere +9 repreent t!e +ei%!t atta"!e# to "ot *un"tion an# + i t!e +ei%!t atta"!e# to

emiion *un"tion o* NO,$

Ta'le $ lit t!e o'4e"tive *un"tion +!i"! are "ot an# emiion o* NO,$ T!e ta'le

alo lit tranmiion loe( value o* lam'#a( mem'er!ip *un"tion o* o'4e"tive *un"tion

value an# *itne value *or ea"! +ei%!t "om'ination$ Ta'le $ lit t!e overall

ati*a"tor& olution$





33

Ta'le $ Fei%!t ve"tor an# "orrepon#in% *itne value

:1 :2 %1 5R.6h7 %2

54g6h7

PL

MW

L/m+!/

R.6MWh

@5%17 @5%27 %itne

9$ $ 9Q9$P $PW 9$9Q 9$WQ 9$ $ $

$P $9 9Q$ $Q 9$WQQ P$ $PP9 $9 $QW

$ $ 9QP$W $Q 99$9 P$ $PW9 $Q $WW

$Q $ 9QP$PP 9QQ$WQ 9$QQ $ $P9P $PQ $QW

$W $ 9Q$WP 9$P 9$QWQQ Q$Q $P $WP $QP

<.(< <.(< 1?>>".= 212;.?? 1"(.""?1 ;.(" <.>>1< <.>""" <.>(22

$ $W 9QP$ 9Q$9P 9Q$QP $WP $WWW9 $ $QW

$ $Q 9Q$ P9$ 9P$W $ $ $PP $Q9P

$ $ 9W$9P QP$ 99$P $ $W $P $WQ

$9 $P 9P$9 Q$W 9$W $9 $Q $P $P9

$ 9$ 9P$Q Q$9 9$9 $ $ 9$ $

Ta'le $ Overall ati*a"tor& olution

:1 :2 P1 P2 P" P P( P;

$ $ $PQQ W$PQ $PQ $PQ9Q $P 9PQ$

:1 :2 %1

5R.6h7

%2

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PL

MW

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@5%17 @5%27 %itne

$ $ 9QQ$P 9W$ 9$9 W$ $QQ9 $Q $QA

(.2 DI$CU$$ION

-itne value i "ompute# *or ea"! +ei%!t "om'ination$ T!e *itne value i t!e ma,imum

*or t!e +ei%!t ve"tor 8$ $;$ T!i !o+ t!at t!e +ei%!t "om'ination 8$ $; reult

in t!e overall ati*a"tor& olution$





34

CHA2TER W

CONCLU$ION

In t!i +or1( -ormulation o* olution met!o# to o'tain t!e optimum olution o* multi0

o'4e"tive optimal po+er #ipat"! .MO2D/ pro'lem !a 'een implemente# u""e*ull&

uin% +ei%!tin% met!o#$

T!e *o"u o* t!i +or1 i on minimi5ation o* t+o o'4e"tive "ot o* %eneration an#





35

emiion o* NO,$ Multi0o'4e"tive optimal po+er #ipat"! .MO2D/ pro'lem !a 'een

*ormulate# '& uin% +ei%!tin% met!o#$ Non0in*erior olution are %enerate# '&

parametri"all& var&in% +ei%!t atta"!e# to t!e o'4e"tive$ -rom t!e et o* non0in*erior

olution o'taine#( t!e 'et "ompromie olution i i#enti*ie# uin% -u55& Car#inal

2riorit& Ran1in%$

RE%ERENCE$

89;$ OTHARI D$2 an# Na%rat! I$J( b ()*+N P),*+ --/*( ANA$-0- K( r#

e#ition MC GRAF0Hill( Ne+ or1 $

8;$ UMA RAO ( b1)(P2/*+ /*13N042*- AN ()*$- 0N P),*+

--/*(5(9t e#ition( I$ International 2u'li!in% Houe$ 2VT$ LTD( Ne+ Del!i Q$





36

8;$ OTHARI D$2$ an# DHILLON J$S$( b P),*+ --/*( )P/0(06A/0)N K( n#

e#ition 2HI Learnin% 2rivate Limite#( Ne+ Del!i 99$

8;$ J$G$VLACHOGIANNIS( b F266 $)G01 APP$01A/0)N 0N $)A F$),

-/20*- K$IEE 2ro"0Generation(Tranmiion(Ditri'ution(Vol$9(No 9(Januar& 9$

8;$ JOHN EN( REA LANGARI( A te,t 'oo1 o* b F266 $)G01 K Intelli%en"e(

Control( an# In*ormation$

multiobjective thermal generation scheduling

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