unit iv-commitment and economic dispatch

8/9/2019 Unit IV-commitment and Economic Dispatch

1/47

11

Economic DispatchEconomic Dispatch

U.T.TAMIL ARASIU.T.TAMIL ARASI

SL/EEESL/EEERECREC


2/47

2

INTRODUCTIONINTRODUCTION

Refers to the most economic loain! of theRefers to the most economic loain! of the!enerators "hich are connecte to the!enerators "hich are connecte to thes#stem an r$nnin!.s#stem an r$nnin!.

%$el cost is the principal factor in fossil f$el%$el cost is the principal factor in fossil f$elplants an is !i&en as a f$nction ofplants an is !i&en as a f$nction of!eneration.!eneration.

ED pro'lem is to e(ne the pro$ction le&elED pro'lem is to e(ne the pro$ction le&elof each plant so that the total !enerationof each plant so that the total !eneration

an transmission cost is minim$m for aan transmission cost is minim$m for aprescri'e sche$le of loas.prescri'e sche$le of loas.


3/47

3

Sol$tion Techni)$esSol$tion Techni)$es

La!ran!e M$ltiplier methoLa!ran!e M$ltiplier metho

Lam'a Iteration methoLam'a Iteration metho

*raient metho*raient metho

D#namic pro!rammin!D#namic pro!rammin!

E&ol$tionar# Comp$tation Techni)$esE&ol$tionar# Comp$tation Techni)$es


4/47

4

Inp$t+O$tp$t Cost C$r&eInp$t+O$tp$t Cost C$r&e

Use to escri'e the e,cienc# of theUse to escri'e the e,cienc# of theplants.plants.

*raphical Representation of inp$t in*raphical Representation of inp$t inRs/hr &ers$s po"er o$tp$t inRs/hr &ers$s po"er o$tp$t in

me!a"atts is calle inp$t+o$tp$tme!a"atts is calle inp$t+o$tp$t

c$r&e.c$r&e.


5/47

5

Incremental Cost c$r&e forIncremental Cost c$r&e for

thermal -o"er plantsthermal -o"er plants

A plot of the inp$t in ilocalories per ho$r &ers$sA plot of the inp$t in ilocalories per ho$r &ers$spo"er o$tp$t in me!a"atts is calle inp$t+o$tp$tpo"er o$tp$t in me!a"atts is calle inp$t+o$tp$tc$r&e.c$r&e.

The orinates of the c$r&e ma# 'e con&erte toThe orinates of the c$r&e ma# 'e con&erte toRs/hr '# m$ltipl#in! the f$el in Rs/ilocalories.Rs/hr '# m$ltipl#in! the f$el in Rs/ilocalories.

The empirical e)$ation of the this c$r&e is !i&enThe empirical e)$ation of the this c$r&e is !i&en'#'#

here a0 ' an c are constants epenin! $pon ahere a0 ' an c are constants epenin! $pon apartic$lar plant.partic$lar plant.

cbPaPC iii ++= 2


6/47

6

Incremental f$el costIncremental f$el cost

The incremental cost is e)$al to the slope of theThe incremental cost is e)$al to the slope of the

cost c$r&e. A plot of incremental cost &ers$scost c$r&e. A plot of incremental cost &ers$s

po"er o$tp$t is calle the incremental cost c$r&e.po"er o$tp$t is calle the incremental cost c$r&e.

It is sho"n in %i!.1. E)$ation 234 is of the formIt is sho"n in %i!.1. E)$ation 234 is of the form

The incremental cost is e)$al to the slope of theThe incremental cost is e)$al to the slope of the

cost c$r&e.cost c$r&e.

i

i

i

dP

dCIC =)(

=

baPi +2 ;

Rs/ MWh


7/47

7

Incremental cost curve forIncremental cost curve for

Hydro Power PlantsHydro Power Plants

The inp$t+o$tp$t c$r&e for a t#pical h#ro $nit isThe inp$t+o$tp$t c$r&e for a t#pical h#ro $nit issho"n in %i!. 5. This is o'taine '# plottin! thesho"n in %i!. 5. This is o'taine '# plottin! the

"ater inp$t or ischar!e in c$'ic meters per"ater inp$t or ischar!e in c$'ic meters persecon as a f$nction of the po"er o$tp$t insecon as a f$nction of the po"er o$tp$t in

me!a"atts. The incremental cost c$r&e for h#rome!a"atts. The incremental cost c$r&e for h#ro

$nit is sho"n in %i!. 6. This is o'taine '# plottin!$nit is sho"n in %i!. 6. This is o'taine '# plottin!

the incremental cost in r$pees per M 7 seconthe incremental cost in r$pees per M 7 secon

2Rs/M + s42Rs/M + s4

as a f$nction of po"er o$tp$t in me!a"atts 2M4.as a f$nction of po"er o$tp$t in me!a"atts 2M4.


8/47

8


9/47

9

Economic Dispatch -ro'lemEconomic Dispatch -ro'lem

It consists of N !eneratin! $nitsIt consists of N !eneratin! $nitsconnecte to a sin!le '$s 'ar ser&in!connecte to a sin!le '$s 'ar ser&in!

a recei&in! electrical loaa recei&in! electrical loa


10/47

10

itho$t Lossitho$t Loss

letlet

NT FFFFF ++++= ...321

( )=

=N

i

ii PF1

=

==

N

i

iload PP1

0

subject to (1) energy balance equation and (11)

inequality constraints


11/47

11

La!ran!e %$nctionLa!ran!e %$nction

This is a constraine optimi8ation pro'lem thatThis is a constraine optimi8ation pro'lem thatma# 'e attace formall# $sin! a&ancema# 'e attace formall# $sin! a&ance

calc$l$s methos that in&ol&e the La!ran!ecalc$l$s methos that in&ol&e the La!ran!e

f$nction.f$nction.

In orer to esta'lish the necessar# conitionsIn orer to esta'lish the necessar# conitionsfor an e9treme &al$e of the o':ecti&e f$nction0for an e9treme &al$e of the o':ecti&e f$nction0

a the constraint f$nction to the o':ecti&ea the constraint f$nction to the o':ecti&e

f$nction after the constraint f$nction has 'eenf$nction after the constraint f$nction has 'een

m$ltiplie '# an $netermine m$ltiplier. Thism$ltiplie '# an $netermine m$ltiplier. This

is no"n as theis no"n as theLagrange function.Lagrange function.


12/47

12

La!ran!e f$nction "ith respect to theLa!ran!e f$nction "ith respect to thepo"er o$tp$t &al$es one at a time !i&epo"er o$tp$t &al$es one at a time !i&e

the set of e)$ations sho"nthe set of e)$ations sho"n

herehere is the La!ran!ian m$ltiplier.is the La!ran!ian m$ltiplier.

The necessar# conition for this optimi8ationThe necessar# conition for this optimi8ationpro'lem is tain! the (rst eri&ati&e of thepro'lem is tain! the (rst eri&ati&e of the

La!ran!e $nction "ith respect to the each ofLa!ran!e $nction "ith respect to the each of

the inepenent &aria'les an set thethe inepenent &aria'les an set the

eri&ati&es e)$al to 8ero i.eeri&ati&es e)$al to 8ero i.e

L !" #

( )0=

=

i

ii

i P

PF

P

L


13/47

13

=

i

i

P

F0

"his is called as coordination equation. i.e. the

necessary condition $or the e%istence o$ a &ini&u&

o'erating cost is that the incre&ental cost rates o$ all

the units be equal to the so&e in deter&ined alue .

=

i

i

P

F

0


14/47

14

"his is the necessary condition $or the e%istence o$ a"his is the necessary condition $or the e%istence o$ a&ini&u& cost o'erating condition $or the ther&al 'o*er&ini&u& cost o'erating condition $or the ther&al 'o*ersyste& is that the incre&ental cost rates o$ all the unitssyste& is that the incre&ental cost rates o$ all the unitsbe equal to so&e undeter&ined alue+be equal to so&e undeter&ined alue+ . ,$ course+ to. ,$ course+ tothis necessary condition *e &ust add the constraintthis necessary condition *e &ust add the constraintequation that the su& o$ the 'o*er out'uts &ust be equalequation that the su& o$ the 'o*er out'uts &ust be equal

to the 'o*er de&anded by the load. -n addition+ there areto the 'o*er de&anded by the load. -n addition+ there aret*o inequalities that &ust be satis$ied $or each o$ thet*o inequalities that &ust be satis$ied $or each o$ theunits. "hat is+ the 'o*er out'ut o$ each unit &ust beunits. "hat is+ the 'o*er out'ut o$ each unit &ust begreater than or equal to the &ini&u& 'o*er 'er&ittedgreater than or equal to the &ini&u& 'o*er 'er&ittedand &ust also be less than or equal to the &a%i&u&and &ust also be less than or equal to the &a%i&u&

'o*er 'er&itted on that 'articular unit.'o*er 'er&itted on that 'articular unit.


15/47

15

These conitions an ine)$alitiesThese conitions an ine)$alitiesma# 'e s$mmari8e as sho"n in thema# 'e s$mmari8e as sho"n in the

set of e)$ations N e)$ationsset of e)$ations N e)$ations

=

i

i

P

F

&a%+&in+ iii PPP

=

=N

i

Loadi PP

1


16/47

1616

Economic ispatch "ithEconomic ispatch "ith

lossloss


17/47

17

INTRODUCTIONINTRODUCTION

F1

F2

F3


18/47

18

ith Lossith Loss

O':ecti&e %nO':ecti&e %n

%%TT;%;%>=%=>>>=%NNS$':ecte toS$':ecte to

===+

N

iiLossLoad

PPP1

0


19/47

19

Usin! La!ran!ian %$nctionUsin! La!ran!ian %$nction

L ; %T=

here ?@ is the la!ran!ian m$ltiplieri.e.0 a the constraine f$nction ? ??to the o':ecti&e f$nction after the

constraine f$nction is m$ltiplie '#an $netermine m$ltiplier


20/47

20

Tain! the eri&ati&e of la!ran!ianTain! the eri&ati&e of la!ran!ianf$nction "ith respect to an# one off$nction "ith respect to an# one of

the N &al$es of -the N &al$es of - iiis sho"n asis sho"n as01 =

=

i

loss

i

i

i P

P

P

F

P

L

=

+

i

loss

i

i

P

P

P

F

0

1

=+ =

N

i

ilossload PPP


21/47

21

*eneral approaches to the*eneral approaches to the

sol$tion of the pro'lemsol$tion of the pro'lem

Step


22/47

22

Iteration methoIteration metho

One of the metho to sol&e the economicOne of the metho to sol&e the economicispatch pro'lem '# ne!lectin! the losses.ispatch pro'lem '# ne!lectin! the losses.

E!.E!. consier three $nits s#stem0 to (nconsier three $nits s#stem0 to (n

the optimal economic operatin! pointthe optimal economic operatin! point

"itho$t losses"itho$t losses

+ ass$me an incremental cost rate an+ ass$me an incremental cost rate an(n the po"er o$tp$ts of these three $nits.(n the po"er o$tp$ts of these three $nits.


23/47

23

-roce$re-roce$re

Is an iterati&e t#pe of comp$tation anIs an iterati&e t#pe of comp$tation anstoppin! r$les are $se0stoppin! r$les are $se0

2


24/47

24

ase point an -articipationase point an -articipation

factorfactor

ED- is sol&e repeatel# '# mo&in! the !enerators fromED- is sol&e repeatel# '# mo&in! the !enerators fromone economicall# optim$m sche$le to another as theone economicall# optim$m sche$le to another as theloa chan!es '# a reasona'l# small amo$nt.loa chan!es '# a reasona'l# small amo$nt.

Ne9t0 the sche$ler ass$mes a loa chan!e anNe9t0 the sche$ler ass$mes a loa chan!e anin&esti!ates ho" m$ch each !eneratin! $nit nees to 'ein&esti!ates ho" m$ch each !eneratin! $nit nees to 'emo&e ie0 participate in the loa chan!e in orer thatmo&e ie0 participate in the loa chan!e in orer thatthe ne" loa 'e ser&e at the most economic operatin!the ne" loa 'e ser&e at the most economic operatin!point.point.

Ass$me that 'oth the (rst an secon eri&ati&es in theAss$me that 'oth the (rst an secon eri&ati&es in thecost &ers$s po"er o$tp$t f$nction are a&aila'le 2ie0 'othcost &ers$s po"er o$tp$t f$nction are a&aila'le 2ie0 'othan 4. As the $nit loa is chan!e '# an amo$nt 0 thean 4. As the $nit loa is chan!e '# an amo$nt 0 thes#stem incremental cost mo&es from to . %or a smalls#stem incremental cost mo&es from to . %or a smallchan!e in po"er o$tp$t on this sin!le $nit.chan!e in po"er o$tp$t on this sin!le $nit.


25/47

2525

UnitUnit

CommitmentCommitment


26/47

26

-"R,0"-,-"R,0"-,

The $nit commitment economicall# sche$lesThe $nit commitment economicall# sche$les!eneratin! $nits o&er a short+term plannin!!eneratin! $nits o&er a short+term plannin!

hori8on s$':ect to the satisfaction of emanhori8on s$':ect to the satisfaction of eman

an other s#stem operatin! constraints.an other s#stem operatin! constraints. It in&ol&es eterminin! start+$p an sh$t+It in&ol&es eterminin! start+$p an sh$t+

o"n sche$le of $nits to 'e $se to meet theo"n sche$le of $nits to 'e $se to meet the

forecaste eman 0 o&er a f$t$re short termforecaste eman 0 o&er a f$t$re short term

perio.perio. 'elon!s to the class of comple9 com'inatorial'elon!s to the class of comple9 com'inatorial

optimi8ation pro'lems..optimi8ation pro'lems..


27/47

27

An E9ampleAn E9ample Consier 1 $nits sho"n 'elo"Consier 1 $nits sho"n 'elo" Unit


28/47

28

%easi'le Com'ination%easi'le Com'ination

Ma9 Gal$es Ma9 Gal$es 050 3 M.050 3 M.UnitUnit


29/47

29

Meetin! Garia'le emanMeetin! Garia'le eman

An E9ample of -ea+&alle# loaAn E9ample of -ea+&alle# loapatternpattern1200 MW

500 MW

Time

Total Load


30/47

30

Unit commitment sche$leUnit commitment sche$le

600 MW

Time

Total Load

1200 MW

UNIT 1

UNIT 2UNIT 2

UNIT 1 UNIT 1


31/47

31

Constraints in UnitConstraints in Unit

CommitmentCommitment Spinnin! Reser&eSpinnin! Reser&eDescri'es the total amo$nt of !enerationDescri'es the total amo$nt of !eneration

sche$le a&aila'le from all $nitssche$le a&aila'le from all $nitss#nchroni8e on the s#stem min$s presents#nchroni8e on the s#stem min$s presentloa an losses 'ein! s$pplie.loa an losses 'ein! s$pplie.

Loss of one or more $nits sho$l not ca$seLoss of one or more $nits sho$l not ca$sea too far rop in fre)$enc#.a too far rop in fre)$enc#.

Reser&e m$st 'e capa'le of main! $p theReser&e m$st 'e capa'le of main! $p theloss of most hea&il# loae $nit in a !i&enloss of most hea&il# loae $nit in a !i&enperio of time.perio of time.


32/47

32

Units

1,2 and 3Units 4

and 5550

MW

UnitUnit UnitUnitcapacitycapacity

UnitUnitoutputoutput

SpinningSpinningReserveReserve


33/47

33

Constraints 2Contin$ation4Constraints 2Contin$ation4

Thermal ConstraintsThermal Constraints Thermal $nits $ner!o onl# !ra$alThermal $nits $ner!o onl# !ra$al

temperat$re chan!es.temperat$re chan!es.

Minim$m Up timeMinim$m Up time Once the $nit is r$nnin!0 Once the $nit is r$nnin!0it sho$l not 'e t$rne oK immeiatel#it sho$l not 'e t$rne oK immeiatel# Minim$m Do"n timeMinim$m Do"n time Once the $nit is Once the $nit is

ecommite0 there is a minim$m time 'eforeecommite0 there is a minim$m time 'eforeit can 'e recommitte.it can 'e recommitte.

Cre" ConstraintsCre" Constraints No eno$!h cre" mem'ers No eno$!h cre" mem'ersto atten t"o or more $nits "hile startin! $p.to atten t"o or more $nits "hile startin! $p.


34/47

34

Start $p costStart $p cost

Ener!# e9pene to 'rin! the $nit on+line.Ener!# e9pene to 'rin! the $nit on+line.

Coolin!Coolin!

Allo"s the $nits 'oiler to cool o"n anAllo"s the $nits 'oiler to cool o"n an

then heat 'ac $p to operatin!then heat 'ac $p to operatin!temperat$re in time for a sche$le t$rn+temperat$re in time for a sche$le t$rn+

on.on.

anin!anin!

Re)$ires that s$,cient ener!# 'e inp$t toRe)$ires that s$,cient ener!# 'e inp$t tothe 'oiler to :$st maintain operatin!the 'oiler to :$st maintain operatin!

temperat$retemperat$re


35/47

35

Start 7$p cost "henStart 7$p cost "hen

coolin!coolin!

f

t

c cFc +

)1(

cc

F

fc

t

!e"e

= #old$sta"t #ost %M&t'(

= )'el #ost

=)i*ed #ost

= T!e"mal time #onstant +o" t!e Unit

= time %!( t!e 'nit as #ooled


36/47

36

Start+$p cost "hen 'anin!Start+$p cost "hen 'anin!

ft cFtc +

tc = #ost %M&t'!( o+ maintainin- 'nit

at o.e"atin- tem.e"at'"e

!e"e


37/47

37

Time epenent start+$pTime epenent start+$p

costcost

&an/in-

oolin-ta"t '.

#ost

Time


38/47

38

Other ConstraintsOther Constraints

M$st r$n constraint Some $nits areM$st r$n constraint Some $nits are!i&en a m$st r$n stat$s $rin! all the!i&en a m$st r$n stat$s $rin! all the

times.times. %$el Constraints Some $nits ha&e%$el Constraints Some $nits ha&e

limite f$el or ha&e constraints thatlimite f$el or ha&e constraints that

re)$ire them to '$rn a speci(ere)$ire them to '$rn a speci(eamo$nt of f$el.amo$nt of f$el.


39/47

39

Di,c$lties in Sol&in! UnitDi,c$lties in Sol&in! Unit

Commitment -ro'lemsCommitment -ro'lems

Let there 'e M perios an N $nits.Let there 'e M perios an N $nits.

The total n$m'er of com'inations at each ho$rThe total n$m'er of com'inations at each ho$risis

The ma9im$m n$m'er of possi'leThe ma9im$m n$m'er of possi'lecom'inations is 23com'inations is 23NN+


40/47

40

Sol$tion MethosSol$tion Methos

-riorit# list schemes-riorit# list schemes

D#namic -ro!rammin!D#namic -ro!rammin!

La!ran!e Rela9ationLa!ran!e Rela9ation


41/47

41

-riorit# List Metho-riorit# List Metho

It consists of priorit# list of $nits toIt consists of priorit# list of $nits to'e committe.'e committe.

%$ll loa a&era!e pro$ction cost of%$ll loa a&era!e pro$ction cost ofeach $nit is calc$late.each $nit is calc$late.

Then in the orer of ascenin! costs0Then in the orer of ascenin! costs0

the $nits are arran!e forthe $nits are arran!e forcommitment.commitment.


42/47

42

An E9ampleAn E9ample

Consier 1 $nits sho"n 'elo"Consier 1 $nits sho"n 'elo" Unit


43/47

43

Calc$lation of %LA-CCalc$lation of %LA-C

UnitUnit %$ll loa%$ll loaa&era!ea&era!e

pro$ctionpro$ction

cost2R/Mhcost2R/Mh

44

Min MMin M Ma9 MMa9 M


44/47

44

Optimal Com'inationsOptimal Com'inations

Com'inatioCom'inationn

Min MMin M Ma9 MMa9 M

3=


45/47

45

Limitations of -riorit# ListLimitations of -riorit# List

SchemeScheme

No loa costs are 8ero.No loa costs are 8ero.

Unit inp$t+o$tp$t characteristics areUnit inp$t+o$tp$t characteristics are

linear 'et"een 8ero o$tp$t an f$lllinear 'et"een 8ero o$tp$t an f$llloa.loa.

There are no other restrictions.There are no other restrictions.

Start $p costs are a (9e amo$nt.Start $p costs are a (9e amo$nt.


46/47

46

D#namic -ro!rammin! forD#namic -ro!rammin! for

UCUC

Ass$mptionsAss$mptions A state consists of an arra# of $nits "ithA state consists of an arra# of $nits "ith

speci(e $nits operatin! an the restspeci(e $nits operatin! an the restoK+line.oK+line.

The start $p cost of a $nit isThe start $p cost of a $nit isinepenent of the time it has 'een oK+inepenent of the time it has 'een oK+line 2i.e.0 it is a (9e amo$nt4.line 2i.e.0 it is a (9e amo$nt4.

There are no costs for sh$ttin! o"n aThere are no costs for sh$ttin! o"n a$nit.$nit.

There is a strict priorit# orer0 an inThere is a strict priorit# orer0 an ineach inter&al a speci(e minim$meach inter&al a speci(e minim$mamo$nt of capacit# m$st 'e operatin!.amo$nt of capacit# m$st 'e operatin!.


47/47

Rec$rsi&e Al!orithm toRec$rsi&e Al!orithm to

comp$te the minim$m costcomp$te the minim$m cost

)+()+1(cos)+3+1(cos

)+(cos)+(cos

)+(cos)+(cos

)4+1(cos)+3+1(cos)+(cos5&in)+(cos

IKtoLKstatefromttransitionIKLKtS

IKstatefortproductionIKtP

IKstateatarrivetottotalleastIKtFwhere

LKtFIKLKtSIKtPL

IKtF

=

=

=

++=

unit iv-commitment and economic dispatch

Documents