5.1 Economic Dispatch Of Thermal Units

5.1 Economic dispatch of thermal units

5.1.1

System of N thermal generating units: PG1

F1 1

G PG2

F2 2

G PL PGN

FN N

G

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 busbar serving a total load PL.

5.1 Economic dispatch of thermal units

5.1.2

Typical production cost curve for a steam turbine generating unit: Fi $ h   ε

dFi = tan ε dPGi

=Incremental Cost

0 0

PGimin

PGimax

PGi [MW]

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 Economic dispatch of thermal units

Economic dispatch as a problem of constrained optimisation: Objective function: N

F = ∑ Fi (PGi ) = F1(PG1 ) + F2 (PG2 ) + L + FN (PGN ) i =1

Minimize: N

F = ∑ Fi (PGi ) i=1

Subject to the equality constraint N

∑P

Gi

= PL

i=1

and to the inequality constraints PGimin ≤ PGi ≤ PGimax

5.1.3

5.1 Economic dispatch of thermal units

5.1.4

Example with three generating units: Unit 1

Unit 2

Coal fired steam plant

Coal fired steam plant

max PG1

= 500 MW

PG2 = 500 MW

PG1 = 100 MW

PG2 = 100 MW

Input-output curve:

Input-output curve:

max min

min

2

2

H1 = 952.4 + 11.429 PG1 + 0.00762 PG1 [GJ/h] H2 = 1428.6 + 13.333 PG2 + 0.00952 PG2 [GJ/h] Fuel cost coal:

FC1 = 1.05

[$/GJ] 2

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

Fuel cost coal:

FC2 = 1.05

[$/GJ] 2

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

5.1 Economic dispatch of thermal units

5.1.5

Unit 3 Oil fired steam plant max

PG3 = 500 MW min

PG3 = 100 MW Input-output curve: 2

H3 = 2105.3 + 16.842 PG3 + 0.01263 PG3 [GJ/h] Fuel cost oil:

FC3 = 0.95

[$/GJ] 2

F3 = 2000 + 16 PG3 + 0.012 PG3

[$/h]

5.1 Economic dispatch of thermal units Unit 1

5.1.6 Unit 2

Unit 3

F1 [$/h]

F2 [$/h]

F3 [$/h]

10000

10000

10000

5000

5000

5000

0

0

0

0

100 200 300 400 500

PG1 [MW]

0

100 200 300 400 500

PG2 [MW]

0

100 200 300 400 500

PG3 [MW]

5.1 Economic dispatch of thermal units dF1 dPG1

Unit 1

dF2 dPG2

5.1.7 Unit 2

dF3 dPG3

Unit 3

 $   MWh    30

 $   MWh    30

 $   MWh    30

20

20

20

10

10

10

0

0

0

0

100 200 300 400 500

PG1[MW]

0

100 200 300 400 500

PG2[MW]

0

100 200 300 400 500

PG3[MW]

5.1 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 x1

Variables:

PG1

x 2 = PG2 x3

PG3

Objective function: F = 1000 + 12x1 + 0.008x12 + 1500 + 14x 2 + 0.01x 22 + 2000 + 16x 3 + 0.012x 32 Equality constraint: x1 + x 2 + x 3 = PL

5.1.8

5.1 Economic dispatch of thermal units

5.1.9

Lagrange function: L = 1000 + 12 x1 + 0.008x12 + 1500 + 14x 2 + 0.010x 22 + 2000 + 16 x 3 + 0.012x32

+ λ (PL − x1 − x 2 − x 3 )

Necessary conditions for an extremum are: ∂L =0 ∂x1

; 0.016 x1 + 12 − λ = 0

(1)

∂L = 0 ; 0.02 x 2 + 14 − λ = 0 ∂x 2

(2)

∂L = 0 ; 0.024 x 3 + 16 − λ = 0 ∂x 3

(3)

∂L =0 ∂λ

(4)

; PL − x1 − x 2 − x 3 = 0

5.1 Economic dispatch of thermal units

5.1.10

With PL = 800, equations (1) ... (4) can be solved directly for the unknowns x1, x2, x3, and λ . x1 = 432.4 x2 = 245.9 x3 = 121.6 = 18.919

PG1 = 432.4 MW PG2 = 245.9 MW PG3 = 121.6 MW

F = 17 354.9 $/h

= 18.919 $/MWh

Dispatch with minimal cost is achieved, when all units operate at equal incremental costs dF1 dF2 dF3 = = dPG1 dPG2 dPG3

λ=

and their individual production PGi add up to the total load PL 3

∑1 P i=

Gi

= PL

5.1 Economic dispatch of thermal units dF1 dPG1

Unit 1

dF2 dPG2

 $   MWh    30

5.1.11 Unit 2

dF3 dPG3

Unit 3

 $   MWh    30

 $   MWh    30

20

20

10

10

10

0

0

0

λ = 18.919

0

100 200 300

500 432.4

PG1[MW]

0

100 200 300 400 500 245.9

800 MW

PG2[MW]

0

200 300 400 500 121.6

PG3[MW]