5.3 the Optimal Power Flow Problem

8/14/2019 5.3 the Optimal Power Flow Problem

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OPF IntroductionThe idea of minimizing the total generation cost under full consideration of the

power flow equations goes back to the decade between 1960 and 1970. The

mathematical formulation and first solutions of this optimization problem have been

given by Carpentier (1962) and Tinney, Dommel (1967).

5.3.15.3 The optimal power flow problem

Looking back to the Lagrangian function which has been used in economic dispatch;

N1i)PPP()P(FLi

GilossLGii

i K=++=

We can realize, that the power flow in the network has been reduced to one simple

equality constraint: =+

iGilossL 0PPP


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The idea of Carpentier was to replace the simplifying constraint

load plus losses equals generation by the power flow equations for each node in

the network. This formulation of the minimization problem including the power flow

equations is called Optimal Power Flow (OPF).

Important applications of the OPF today:

Calculation of the optimum generation pattern to achieve the minimum total cost

of generation. Calculation of the optimum generation pattern to minimize air pollution by thermal

generating units.

Using the network losses as objective, the OPF algorithm can find the optimum

reactive power injections of generators, optimum settings of transformer taps and

switched capacitors.



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PG2PL2 =100 MW

V2 = 224 kV

PG1PL1 =100 MW

V1 = 224 kV; 1=0

PG3PL3 = 600 MW

V3 = 224 kV

1 2

3

~

~ ~L12

L23L13

OPF Example:Find the operating pattern with minimum total fuel costs if the total load of 800 MW is

distributed on the nodes PL1 = 100 MW, PL2 = 100 MW, PL3 = 600 MW.

( ) [ ]( ) [ ]

( ) [ ]h$2

3G3G3G3

h$2

2G2G2G2

h

$2

1G1G1G1

P012.0P162000PF

P01.0P141500PFP008.0P121000PF

++=

++=

++=

Line impedances:( )+=== 6164841412

2 31 31 2

.j.ZZZ

Line shunt admittances:

0=+ ikik jBSGS

Network voltage:

kV224VVV 321 ===


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Solution:

2

33

2

22

2

11

01201620000101415000080121000 GGGGGG P.PP.PP.PF ++++++++=

Objective function:

Equality constraints (power flow equations, active power):

( ) ( )[ ] 32102 ,,iPPsinBcosGVVGV LiGiiiiiiiii ==+++

( ) ( ) ( ) ( ) ( ) 01001311 3311 3211 2211 21 31 2

=++++ GPsinbcosasinbcosaaa

( ) ( ) ( ) ( ) ( ) 01002322 3322 3122 1122 12 32 1


( ) ( ) ( ) ( ) ( ) 06003233 2233 2133 1133 13 23 1


MW.aaaaaa 42473 22 33 11 32 11 2

======

MW.bbbbbb 99683 22 33 11 32 11 2

======


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Variables:

Lagrange function:

( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ]600xxxsinbxxcosaxsinbxcosaaa

100xxxsinbxxcosaxsinbxcosaaa

100xxsinbxcosaxsinbxcosaaa

x012.0x162000x01.0x141500x008.0x121000L

34532453253153132313

25423542342142123212

151351341241213121

233

212

211

+++++

+++++

+++++

++++++++=

3

2

1

3

2

3

2

1

3

2

1

5

4

3

2

1

=

G

G

G

P

PP

x

x

x

xx


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Necessary conditions for extremum:

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]

( ) ( )[ ] 0453 2453 23

542 3542 342 142 12

41 241 21

=+

++++

xxcosbxxsina

xxcosbxxsinaxcosbxsina

xcosbxsina:x

L0

4

=

:x

L0

5

=

:x

L0

1

=

0016012

11

=+ x. (1)

:xL 0

2

= 002014

22

=+ x. (2)

:x

L0

3

=

0024016

33

=+ x. (2)

(5)

(4)

( ) ( )[ ]

( ) ( )[ ]

( ) ( ) ( ) ( )[ ] 0xxcosbxxsinaxcosbxsina

xxcosbxxsina

xcosbxsina

453245325315313

542354232

5135131

=++++

+


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:L

0

1

=

( ) 0

5411

=x,x,xh

(6)( ) ( ) ( ) ( ) ( ) 0100151 351 341 241 21 31 2 =++++ xxsinbxcosaxsinbxcosaaa

:L

0

2

=

(7)

( ) 05422

=x,x,xh

( ) ( ) ( ) ( ) ( ) 01002542 3542 342 142 12 32 1

=++++ xxxsinbxxcosaxsinbxcosaaa

:

L0

3

=

(8)

( )0

5433

=x,x,xh

( ) ( ) ( ) ( ) ( ) 06003453 2453 253 153 13 23 1

=++++ xxxsinbxxcosaxsinbxcosaaa


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000

X

h

X

h100

000X

h

X

h010

000X

h

X

h001

X

h

X

h

X

h

XX

L

XX

L

000

X

h

X

h

X

h

XX

L

XX

L000

10000024.000

010000020.00

0010000016.0

0G

GL

5

3

4

3

5

2

4

2

5

1

4

1

5

3

5

2

5

1

55

2

54

24

3

4

2

4

1

45

2

44

2

T

2

=


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[ ] [ ] [ ]

[ ] [ ] [ ]( )

( )

( )5433

5422

5411

321

321

33

22

11

024016

020014

016012

x,x,xh

x,x,xh

x,x,xh

x.

x.

x.

h

L

++

++

+

+

+

=

Vector of right

hand side:


10/12


63901

.x =

Result of

iterative

procedure:

62372

.x =

91953

.x =

052104

.x =

242405 .x =

250181

.=

752182 .=

701203

.=

MW.PG 63901 =

MW.PG 62372 =

MW.PG 91953 =

= 98722

.

= 889133 .

MWh$.25018

1=

MWh

$

.75218

2

=

MWh$.70120

3=

h

$

.F9617892=


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PG1=390.6 MW

V1 = 224 kV ; 1=0

PL1 =100 MW

1=18.250 $/MWh 1 2

3

~

~ ~

Ploss=24.06 MW

F = 17892.96 $/h

PG2=237.6 MWV2 = 224 kV ; 2=-2.987

PL2 =100 MW

2=18.752 $/MWh

PG3=195.9 MW

V3 = 224 kV ; 3=-13.889

PL3 =600 MW3=20.701 $/MWh

50.82 50.15

239.81

225.35

187.72

178

.79


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Locational Marginal Price (LMP)

Cost to supply the next unit of energy in the most economic way at a particular location in

the network

PG1 = 0.351; PG2 = 0.248; PG3 = 0.420PL3=1

PG1 = 0.320; PG2 = 0.414; PG3 = 0.249PL2=1

PG1 = 0.471; PG2 = 0.256; PG3 = 0.234PL1=1

LMPPGi (result of OPF calculation)PLiNode

1

1

25018 ==

MWh

$

L

.P

F

2

2

75218 ==

MWh

$

L

.P

F

3

3

70120 ==

MWh

$

L

.P

F

1

2

3

Date post:	30-May-2018
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5.3 the Optimal Power Flow Problem

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