5.1.15.1 Economic dispatch of thermal units

System of N thermal generating units:

1 G

PG1F1

2 G

PG2F2

N G

PGNFN

Input for each thermal unit is the fuel cost per hour Fi with the dimension [$/h].The electrical outputs PGi are connected to a single busbarserving a total load PL.

PL

5.1.2

Typical production cost curve for a steam turbine generating unit:

minGiP max

GiP PGi [MW]

h$

Fi

ε ε= tandPdF

Gi

i

00

5.1 Economic dispatch of thermal units

=Incremental Cost

Fi: Fuel cost per hour [$/h]; PGi: Net electrical power [MW]The Fi(PGi) characteristic shown is idealized as a smooth and convex curve.

5.1.35.1 Economic dispatch of thermal units

Economic dispatch as a problem of constrained optimisation:

Objective function:

( ) ( ) ( ) ( )GNN2G21G1

N

1iGii PFPFPFPFF +++==∑

=

L

Minimize:

( )∑=

=N

1i

GiiPFF

Subject to the equality constraint

∑=

=N

1i

LGiPP

and to the inequality constraintsmax

GiGi

min

GiPPP ≤≤

5.1.45.1 Economic dispatch of thermal units

Example with three generating units:

Unit 1

Coal fired steam plant Coal fired steam plant

Unit 2

Input-output curve:

Fuel cost coal: FC1 = 1.05 [$/GJ]

Input-output curve:

Fuel cost coal: FC2 = 1.05 [$/GJ]

F1 = 1000 + 12 PG1 + 0.008 PG1 [$/h]2

H1 = 952.4 + 11.429 PG1 + 0.00762 PG1 [GJ/h]2

H2 = 1428.6 + 13.333 PG2 + 0.00952 PG2 [GJ/h]2

F2 = 1500 + 14 PG2 + 0.01 PG2 [$/h]

PG1 = 500 MWmax

PG1 = 100 MWmin

PG2 = 500 MWmax

PG2 = 100 MWmin

2

5.1.55.1 Economic dispatch of thermal units

Oil fired steam plant

Unit 3

Input-output curve:

Fuel cost oil: FC3 = 0.95 [$/GJ]

H3 = 2105.3 + 16.842 PG3 + 0.01263 PG3 [GJ/h]2

F3 = 2000 + 16 PG3 + 0.012 PG3 [$/h]2

PG3 = 500 MWmax

PG3 = 100 MWmin

5.1.65.1 Economic dispatch of thermal units

0 100 300 400200

Unit 2

PG2 [MW]

F1

0

5000

10000

0

Unit 1

PG1 [MW]

[$/h]

100 300 400200

F2

[$/h]

500 500 0 100 300 400200

Unit 3

PG3 [MW]

[$/h]

500

F3

0

5000

10000

0

5000

10000

5.1.75.1 Economic dispatch of thermal units

0

0 100 300 400200

Unit 2

PG2[MW]

0

10

20

30

0

Unit 1

PG1[MW]100 300 400200

10

20

30

500 500

0

0 100 300 400200

Unit 3

PG3[MW]

10

20

30

500

MWh$

dPdF

G1

1

MWh$

dPdF

G2

2

MWh$

dPdF

G3

3

5.1.85.1 Economic dispatch of thermal units

• All three units are committed• Lower and upper limits of generating units 1, 2, 3 are not considered • Find the operating point with the minimal fuel cost when a total

load PL = 800 MW has to be served

Objective function: 233222211 01201620000101415000080121000 x.xx.xx.xF ++++++++=

Equality constraint:

LPxxx =++ 321

Variables: 321321 G

G

G

P

P

P

x

x

x

=

5.1.95.1 Economic dispatch of thermal units

Lagrange function:

233

222

211 0.012xx0x.xx.xL ++++++++= 1620000101415000080121000

)xxx(PL 321 −−−λ+

Necessary conditions for an extremum are:

0xL =

∂∂ 1 0120160 1 =λ−+x.; (1)

0xL =

∂∂ 2 014020 2 =λ−+x.; (2)

0xL =

∂∂ 3 0160240 3 =λ−+x.; (3)

0L =λ∂

∂ 0321 =−−− xxxPL; (4)

x1 = 432.4

x2 = 245.9

x3 = 121.6= 18.919

PG1 = 432.4 MWPG2 = 245.9 MW

PG3 = 121.6 MW

= 18.919 $/MWh

5.1.105.1 Economic dispatch of thermal units

With PL = 800, equations (1) ... (4) can be solved directly for the unknowns x1, x2, x3, and .λ

F = 17 354.9 $/h

Dispatch with minimal cost is achieved, when all units operate at equal incremental costs332211 GGG dP

dFdPdF

dPdF ===λ

and their individual production add up to the total load GiP LP

∑=

=31i

LGi PP

5.1.115.1 Economic dispatch of thermal units

0

0 100 300 400200

Unit 2

PG2[MW]

0

10

30

0

Unit 1

PG1[MW]100 300200

10

20

30

500 500

0

0 300 400200

Unit 3

PG3[MW]

10

20

30

500

MWh$

dPdF

G1

1

MWh$

dPdF

G2

2

MWh$

dPdF

G3

3

91918.=λ

432.4 245.9 121.6

800 MW