EconomicDispatch

8/7/2019 EconomicDispatch

1/67

Economic Dispatch

Antonio J. Conejo

UNIV. CASTILLA - LA MANCHA

2002


2/67

Antonio J. Conejo 2

Economic Dispatch

Basic economic dispatch: no generating limits,

no losses

Generating limits, no losses

Losses

Losses and bus balances

Network constrained economic dispatch

Optimal power flow


3/67

Antonio J. Conejo 3

Basic Economic Dispatch

Generation Units

( )Gii PC( )Gii PC

GiP GiPminGiP

max

GiPmax

GiPmin

GiP


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Antonio J. Conejo 4


( ) ( )

=

=

==

n

1i

totalDGi

n

1i

GGii

PPtoSubject

PCPCMinimize


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Antonio J. Conejo 5


[ ]( ) ( )

=

=

=

==

n

1iGi

total

D

n

1i

GiiG

T

Gn2G1GG

PP

PCPC

P,...,P,PP


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Antonio J. Conejo 6


No Generation Limits, No Losses

( ) ( )( )

( ) ( )Gi

GiiGii

n

1i

Gitotal

D

Gii

Gi

n

1i

n

1i

totalDGiGiiG

dP

PdCPCI

0PP)(

n,...,1i;0PCI

P

)(

PPPC,P

====

===

=

=

= =

L

L

L


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Antonio J. Conejo 7



( ) ( )( )

( )total

D

G

n

1i

totalDGi

n

1i

Gi

n

1i

GiGii

n

1i

GiiG

dP

PdC

dPdPdP

dPPCI

PdCPdC

====

==

==

=

=


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Antonio J. Conejo 8



( ) 2GiiGiii0Gii Pb2

1PaCPC ++=

[ ][ ][ ]

[ ][ ][ ]TDn2D1DD

T

Gn2G1GG

T

Tn21

T

n21

T

n002010

P,...,P,PP

P,...,P,PP1,...,1,1e

(b)diagBb,...,b,bb

a,...,a,aa

C,...,C,CC


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Antonio J. Conejo 9


No Generation Limits, No LossesTotal cost( ) GTGGT0TG BPP

2

1PaCePC ++=

Power balancetotal

DGT

PPe =Optimality Conditions

totalDD

TG

TG

PPePe

eBPa == =+{Solution

eBe

aBeP

aBeBP

1T

1TtotalD

11G

+==


10/67

Antonio J. Conejo 10


No Generation Limits, No LossesAlso += totalDG PP

totalDG dPdP =

where

eBeeB 1T

1

=( )

aBeBe

aBeeB 11T

1T1

=

Note that

1eT =


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Basic Economic DispatchNo Generation Limits, No Losses

Example 1

30000.10252002

40000.05201001

b(/MW2h)a(/MWh)C0(/h)Unit )MW(PmaxG)MW(P

minG

MWh/30

650PtotalD +=

MW

3

100P

3

13

100P32

Ptotal

D

totalD

G

+=


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Basic Economic DispatchNo Generation Limits, No Losses

Example 1

increases linearly with total demand Load allocated to generators in different

proportions

The least expensive generator gets more load

Feasibility for 550P100 totalD


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2.4. Ejemplos Ejemplo 1 (1)

Datos de entrada

Solucin

Lmites de factibilidad:

Generador C0 [/h] a [/MWh] b [/MW2h]

min

GP [MW]max

GP [MW]

1 100 20 0.05 0 400

2 200 25 0.1 0 300

MW

3

100

P3

13

100P3

2

PMWh/30

650P

total

D

total

D

G

total

D

+=

+=

MW550P100 totalD


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Potencia generada


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Evolucin de


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Coste total del sistema

1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 00 .2

0 .4

0 .6

0 .8

1

1 .2

1 .4

1 .6

1 .8x 1 0

4

DEM ANDA D E PO T ENCIA (MW )

COSTETOTAL

(EUROS)


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Datos de entrada

Lmites de factibilidad:

Generador C0[/h] a[/MWh] b[/MW2h] PG

min [MW] PGmax [MW]

1 40 2 0.0350 0 800

2 50 3 0.0450 0 7253 60 4 0.0525 0 650

4 75 6 0.0625 0 575

5 100 7 0.0750 0 500

6 150 9 0.0850 0 4507 200 10 0.1000 0 350

8 275 12 0.1250 0 275

9 300 14 0.1500 0 225

10 350 15 0.2000 0 175

MW3358.6P1249.5 totalD


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Resultados

MW

129

5732444

23345743485

167

6491107

363474

87537

67259

10425655

4531452

6111

P

928

33232

1158033

464

331972

165116

11290

33406

55348

55812

165

P totalDG

+

= /MWh4640

51

43837

P33

total

D+=


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Potencia generada


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Evolucin de


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Coste total del sistema

1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 3 5 0 01

2

3

4

5

6

7x 1 0

4

D EM AN D A D E PO T EN C IA ( M W )

COSTETOTAL

(EUROS)


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Economic Dispatch

Generation Limits, No Losses

( ) ( )( )( )( )

=

=

=

= =

== ===

=

n

1i

Gi

total

D

min

i

max

iGii

Gi

n

1i

min

GiGi

min

i

n

1i

max

GiGi

max

i

n

1i

n

1i

total

DGiGiiG

0PP)(

n,...,1i;0PCI

P

)(

PP

PP

PPPC,P

L

L

L


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Economic Dispatch


min

GiGi

min

i

minGiGi

mini

maxGiGi

maxi

maxGiGi

maxi

PPif0

PPif0

PPif0

PPif0

>==


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Economic Dispatch


( )( )( ) maxGiGimaxiGii

maxGiGi

minGiGii

min

GiGi

min

iGii

PPifPCI

PPPifPCI

PPifPCI

=+== =+=

( ) validtillsdP

PdCtotal

D

G=


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Economic Dispatch


CICI2 CI1

A

BC

MW


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Economic DispatchGeneration Limits, No Losses

Example 2

19300454540200400(max)600D2

821731.6731.6731.6766.7233.3300C2

667530303050200250B2

11402225220(min)4040A2

C(/h)(/MWh)CI2(/MWh)CI1(/MWh)PG2(MW)PG1(MW)Case )MW(PtotalD


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3.4. Ejemplos - Ejemplo 1 (1)

Datos de entrada

Resolucin

Generador C0 [/h] a [/MWh] b [/MW2h]

min

GP [MW]max

GP [MW]

1 100 20 0.05 0 400

2 200 25 0.1 0 300

CasoTotal

DP

[MW]

1GP

[MW]

2GP

[MW]

1CI

[/ MWh]2CI

[/ MWh]

[/ MWh]

C

[/h]

A 40 40 0 22 25 22 1140B 250 200 50 30 30 30 6675

C 300 233.3 66.7 31.67 31.67 31.67 8217

D 600 400 200 40 45 45 19300


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Caso D: Interpretacin de 1) Resultados obtenidos

2) Clasificacin de la generacin

CasoTotal

DP

[MW]

1GP

[MW]

2GP

[MW]

1CI

[/ MWh]2CI

[/ MWh]

[ / MWh]

C

[/h]

D 600 400 200 40 45 45 19300

32

max

Gi3

libre

Gi2

min

Gi1

I1G;I2G

PI;PI;PI

Generador [i] maxi [/MWh] mini [/MWh] Ci [/h]

1 -5 0 12100

2 0 0 7200

3 4 Ej l Ej l 1 (3)


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3) Nuevo problema

4) Nuevos resultados

=

++=

:PPP

.a.s

Pb2

1Pac)P(CMinimizar

max

1G

total

D2G

2

2G22G202G

Caso

Total

DP

[MW]

2GP

[MW]

2CI

[/ MWh]

[/ MWh]

mx

2

[/ MWh]

mn

2

[/ MWh]

C2

[/h]

D 200 200 45 45 0 0 7200

3 4 Ej l Ej l 2 (1)


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Datos de entrada

Resultados

Generador C0 [/h] a [/MWh] b[/MW2h]

min

GP [MW]max

GP [MW]total

DP [MW]

1 200 25 0.05 250 1500

2 200 20 0.10 150 15003 150 15 0.07 50 1000

3000

Generador

[i]

PGi

[MW]

[ / MWh]

max

i [/MWh]

min

i [/MWh]

CIi

[ / MWh]

Ci

[/h]

1 1300 0 0 90 74950

2 700 0 0 90 38700

3 1000

90

-5 0 85 50150

3 4 Ej l Ej l 2 (2)


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Interpretacin de

=

+++

++=

:PPP

.a.s

Pb2

1Pac

Pb2

1Pac)P(CMinimizar

max

3G

total

D2G

2

2G22G202

2

1G11G101G

Total

DP

[MW]

1GP

[MW]

2GP

[MW]

1CI

[ / MWh]

2CI

[/ MWh]

[/ MWh]

C

[/h]

2000 1300 700 90 90 90 113650

Economic Dispatch


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Economic DispatchGeneration Limits, No Losses

Lambda Iteration

1. Approximate by k2. Compute output power

3. If , stop, optimal solution found.

Otherwise, continue in step 4

4. Update k , , and go to step 2

( )( )

( )i

i

k

Gi

k

Gii

max

GiGi

kmax

Gii

minGiGi

kminGii

b

aPthen,PCIifelse

PPthenPCIif

PPthenPCIif

===

1kk1k

+

+=

total

D

n

1iGi PP ==

2


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Economic Dispatch

No Generation Limits, Losses

( ) ( )( )

=

=

+==

n

1i

DGlosstotalDGi

n

1i

GGii

P,PPPPtoSubject

PCPCMinimize


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Economic Dispatch


( ) ( ) ( )

( )( )

( )s

Gi

loss

Gii

n

1i

GiDGloss

total

D

s

Gi

lossGii

Gi

n

1i

n

1i

DGloss

total

DGiGiiG

P

P1

PCI

0PP,PPP)(

n,...,1i;0P

P1PCI

P

)(

P,PPPPPC,P

=

=+= ==

=

=

=

= =

L

L

L


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Economic Dispatch


( )s

Gi

loss

Gii

P

P1

PCI

=

The notation indicates that the slack bus is bus ss


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Economic Dispatch

No Generation Limits, Losses, Bus Balances

( )= PPP DGVoltage magnitude in all buses assumed to be 1

P() is the vector of power injectionsP() dimension is n dimension is n-1 ( =0 for reference bus)

Economic Dispatch


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Economic DispatchNo Generation Limits, Losses, Bus Balances

Example 3

00.1032

00.1031

00.1021

b (pn)x (pn)r (pn)to busFrom bus

( ) ( )( ) ( )( ) ( )213D3G

2122D2G

1211D1G

0sin100sin10PP

0sin10sin10PP

0sin10sin10PP

+= +=+=

E i Di t h


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Economic Dispatch


( )( )

( )( ) =

=

====

dPedP

PeP

dP

dPdPdP

PPPP

0Tloss

T

loss

0DG

DG

also

0 is a linearization point

E i Di t h


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Economic Dispatch


( )

0P is n x (n-1)

sGP is PG eliminating slack bus entry; its dimension is n-1

is PD eliminating slack bus entry; its dimension is n-1sDP

( )

s

0P ( )

0Pis eliminating the row corresponding

to the slack bus; its dimension is

(n-1) x (n-1)

E i Di t h


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Economic Dispatch

No Generating Limits, Losses, Bus Balances( )( ) =

==

dP

edP

dP

dPdPdP

0Tloss

s0

sDsGs

Combining the above equations

( ) ( ) ( )( ) ( )

s

1

s00T

loss

sDsG

1

s00T

loss

dPPP

edP

dPdPPP

edP

=

=

E i Di t h


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Economic Dispatch

No Generating Limits, Losses, Bus Balances

( ) ( )( ) ( )

ePP

P

dPITL

PPeP

dP

T0

1T

s0

sloss

1

s00Tsloss

==

=

Economic Dispatch


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Economic Dispatch


The total loss differential can be computed as

( )DiGisn1i i

lossis

n

1i i

lossloss dPdP

P

PdP

P

PdP

=

=

==

Economic Dispatch


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Economic DispatchNo Generating Limits, Losses, Bus Balances

Total Cost Differential

Gi

n

1i

s

Gi

lossGi

n

1i

i dPP

P1dPCIdC ==

==

For fixed loads

s

i

losss

Gi

loss

P

P

P

P =Then

= == Gin1i silossn1i Gi dPPPdPdC

Economic Dispatch


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pNo Generating Limits, Losses, Bus Balances

Total Cost DifferentialTaking into account that

loss

n

1iDi

n

1iGi dPdPdP += ==

The cost differential becomes

+= == Gin1i silosslossn1i Di dPPPdPdPdCUsing the expresion for the total loss diferential

( )

+= == = Gisn1i ilossDiGisn1i n1i ilossDi dPPPdPdPPPdPdC

Economic Dispatch


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Total Cost Differential

Di

n

1i

s

i

loss

dPP

P1dC = =

The marginal cost at every bus becomes

== si

loss

Di

i

P

P1

dP

dC

Economic Dispatch


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Slack Bus Indifference

loss

n

1i

i dPdP ==i

n

1i

s

i

lossloss dP

P

PdP = =

Therefore

0dPP

P1 i

n

1i

s

i

loss =

= For slack s

0dPP

P1 i

n

1i

r

i

loss = = For slack r

Economic Dispatch


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s (slack)

pNo Generation Limits, Losses, Bus Balances

0dPP

P1...1...dP

P

P1...dP

P

P1 ns

n

lossrs

r

loss1s

1

loss =++++++Dividing by

r(New slack)

s

r

loss

P

P1

0

P

P1

P

P

1...

P

P1

1...1...dP

P

P1

P

P

1

s

r

loss

sn

loss

s

r

loss1

s

r

loss

s1

loss

=

++

++++

r

i

loss

ri

loss

s

i

loss

P

P1

P

P

1

P

P1

=

Therefore

Economic Dispatch


49/67


Example 4~

1 2

~ G1 G2

3

D

00.10.0232

00.10.0231

00.10.0221

b (pn)x (pn)r (pn)To busFrom bus

Economic Dispatch


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Example 4

838332.9232.0831.7270.8234.4300C4

678630.9830.0230.0053.3200.4250B4

C(/h)3(/MWh)2(/MWh)1(/MWh)PG2(MW)PG1(MW)PD(MW)CasoLosses

821731.6731.6731.6766.7233.3300C2

667530.0030.0030.0050.0200.0250B2

C(/h)3(/MWh)2(/MWh)1(/MWh)PG2(MW)PG1(MW)PD(MW)CasoNo losses

Network Constrained Economic Dispatch


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Network-Constrained Economic Dispatch

No Generating Limits, No Losses

( ) ( )( )

( ) maxFDGTmaxFn

1i

DiGi

n

1iGGii

PPPP

0PPtoSubject

PCPCMinimize

=

==

=



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No Generating Limits, No LossesOnly one line with limited capacity!

( ) iGii PCI +=( )

( )( ) =

=

=

=

:0PP

:PPPP

PPP

n

1i

DiGi

max

FDG

Tmax

F

n

1i

DiGiiF



53/67



Example 5

918160.0743.0126.00180.06119.94300C5

674237.4932.9928.5079.94170.06250B5

C(/h)3(/MWh)2(/MWh)1(/MWh)PG2(MW)PG1(MW)PD(MW)CasoNo losses, Transmission limit in line 1-3: 140 MW

821731.6731.6731.6766.7233.3300C2

667530.0030.003050200250B2

C(/h)3(/MWh)2(/MWh)1(/MWh)PG2(MW)PG1(MW)PD(MW)CasoNo losses, No transmission limits

Optimal Power Flow


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Optimal Power Flow

( )( )

( ) maxFFmaxGG

minG

DG

n

1i

Gii

PP

PPP

PPPtoSubject

PCMinimize

==

Optimal Power FlowExample 6


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Generation limits, Losses, Transmission constraints

C(/h)3(/MWh)2(/MWh)1(/MWh)PG2(MW)PG1(MW)PD(MW)Caso114222.1322.0722.010(min)40.1140A6

687439.7833.7028.3287.0166.4250B6

946964.1743.9325.78189.31115.55300C6Caso infactible600D6

821731.6731.6731.6766.7233.3300C2

667530303050200250B2

11402222220(min)4040A2

19300454545200400(max)600D2

918160.0743.0126.00180.06119.94300C5

674237.4932.9928.5079.94170.06250B5

C(/h)3(/MWh)2(/MWh)1(/MWh)PG2(MW)PG1(MW)PD(MW)CasoGeneration limits, No losses, Transmission constraints

C(/h)3(/MWh)2(/MWh)1(/MWh)PG2(MW)PG1(MW)PD(MW)CasoGeneration limits, No losses, No transmission constraints

Optimal Power Flow


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pPiecewise linear cost

Ci (PGi) (/h)

PGi (MW)min

GiPmax

GiP

s1

s2

s3

max1

GiPmax2

GiPmax3

GiP

1

GiP2

GiP3

GiP

Optimal Power Flow


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3Gi3

2Gi2

1Gi1i

max3Gi

3Gi

max2

Gi

2

Gi

max1Gi

1Gi

3Gi

2Gi

1GiGi

PsPsPsC

PP0

PP0

PP0

PPPP

++=

++=

Optimal Power Flow


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iPsC

i,bPP0

iPP

b

b

n

1b

b

Gibi

maxb

Gi

b

Gi

n

1b

b

GiGi

=

=

=

=

Optimal Power Flow


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pNo losses, Linear cost

( )( )

iPPP

k,iPBP

i:PPBtoSubject

CMinimize

maxGiGi

minGi

imaxFikkiik

maxFik

k

iDiGiikik

n

1i

i

i

=+

=


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Optimal Power FlowExample (2/3)


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Example (2/3)

( )( )( )

( ) ( )( ) ( )( ) ( ) 85.000.305.3

0P5.200.30P5.205.3

5.005.35.04.000.34.0

3.05.23.0

4.0P10.0

6.0P15.0toSubject

P76PMinimize

21

G2212

G1121

1

2

21

G2

G1

G2G1

=+=++ =++

+

Optimal Power FlowExample (3/3)


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Example (3/3)

833333.7

0.7

0.6

116667.0

142857.0

284524.0P

565476.0P

3845240.5Cost

3

2

1

2

1

2G

1G

===

====

=

Optimal Power Flow


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Locational marginal prices

Di

iP

cost=Production cost increment as a result of an

increment in the demand of bus i

Optimal Power Flow


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Losses, Linear cost

( ) ( )[ ]( )

iPPP

k,iPBP

i:cos1GPPBtoSubject

CMinimize

maxGiGi

minGi

imaxFikkiik

maxFik

k

i

k

ikikDiGiikik

n

1i

i

i i

+=+

=

Optimal Power FlowL Li t


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Losses, Linear cost

( ) ( )( )

iPPP

k,iPBP

i:2

GPPBtoSubject

CMinimize

max

GiGi

min

Gi

i

max

Fikkiik

max

Fik

k

i

k

2

ikik

DiGiikik

n

1i

i

i i

+=+

=

Optimal Power FlowLi i ti f l


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Linearization of losses

=

=

==

=

b

b

n

1b

b

ikik

kiik

n

1b

b

ik

b

ikikLik GP

Optimal Power FlowLi i ti f l


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Linearization of losses

0

0

ik

ik

ikik

ikik

ki

ikki

=+==

+

+

+

EconomicDispatch

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