economic dispatch(1)

8/10/2019 Economic Dispatch(1)

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Economic Dispatch


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Economic Dispatch

Objective function

Total cost of supplying load from Nunits

Subject to load balance constraint

Subject to generator limitations

Subject to reserve requirements


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Generator Characteristics

Maximum power

Minimum power

Cannot operate regions

Auxiliaries (2 10%)


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Fuel Costs

H= heat input per hour into the unit (GJ/h)

(measured in Joules - GJ/h)

FC= Fuel cost (/GJ)

C= Per hour cost of fuel input (/h)= FCH

P = Electrical power output (MW)

E= Electrical energy output (MWh)


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Input-Output Model

Input:

heat energy cost,

Output net electrical output

Obtained from design calculations heat rate tests

Minimum load steam generator

turbine

combustion stability

design constraints


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Incremental Cost

= Incremental cost of next MWh (/MWh)F

P

Slope of input-outputcurve (derivative)

Cost of next MWh


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Input - output curves Incremental cost curve

Types of Curves

Pi (MW)

/h

1. Linear

/MWh

Costi = ai + bi pi

Pi (MW)

bi

1. Constant


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Types of Curves

Pi (MW)

/h/MWh

Pi (MW)

2. Piecewise Linear 2. Stepped


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Types of Curves

Pi (MW)

/h/MWh

Pi (MW)

3. Quadratic 3. Linear


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Economic Dispatch

Aim: Dispatch the generation to meet

demand in a least cost fashion

Recall

Input = fuel (/h) = F Output = electrical power (MW) = P

Minimise Fwhile satisfying

Pgeneration = Pdemand


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Economic Dispatch

Objective function:

Total cost of supplying the load, FT Sum of the fuel costs (Fi(Pgen,i)) for allN

machines being dispatched

,

1

F F ( )N

T i gen i

i

P=

=

1 ,1 2 ,2 ,F F ( ) F ( ) ..... F ( )T gen gen N gen N P P P= + +


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Economic Dispatch

Main constraint: load balance :

Total power generated by theNunits mustequal the load demand, Pload

,

10

N

load gen i

iP P

== =

,1 ,2 ,( ......... ) 0load gen gen gen N P P P P + + =

Constraint:


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Economic Dispatch

Minimise FT subject to load balance constraint

Neglect transmission losses

reserve constraint

operating limits, and

transmission constraints

Lagrangian functionL

Condition for extreme value of objective function results

when taking first derivative of Lagrangian wrt all

independent variables and setting them equal to zero


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Economic Dispatch

Lagrangian functionL

Add an (unknown) weighted constraint functionto the objective function

L = objective function + (lambda constraint)

To solve a Lagrangian:

DifferentiateL with respect to independent variables

Set the derivatives equal to zero, and

Solve the resulting equations

L FT = +

, ,

1 1

L F ( ) ( )N N

i gen i load gen i

i i

P P P= =

= +


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Solving the Lagrangian

Differentiate with respect toNpower outputs

(Pgen,i) and equate to 0

,

, ,

F ( )L0, 1,2,....

i gen i

gen i gen i

d Pi N

P dP

= = =

, ,

1 1

L F ( ) ( )N N

i gen i load gen i

i i

P P P= =

= +

Differentiate with respect to Lagrange multiplier and equate to zero

,

1

L0

N

load gen i

i

P P =

= = =


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Economic Dispatch Examples

Quadratic cost curves

Demand is fixed

Single instant in time

We do not consider:

Network constraints (common bus)

Reserve constraint


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ExampleDetermine the minimum operating cost and unit

schedule to meet a system demand of 850 MW. Unit 1: P1,max = 600 MW, P1,min = 150 MW

F1 = 561 + 7.92Pgen,1 + 0.001562(Pgen,1)2 /h

Unit 2: P2,max = 400 MW, P2,min = 100 MW

F2 = 310 + 7.85Pgen,2 + 0.00194(Pgen,2 )2 /h


F3= 78 + 7.97P

gen,3+ 0.00482(P

gen,3)2 /h


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Previously, just one constraint

including maximum and minimum constraints

Inequalities use Kuhn Tucker Conditions

Economic Dispatchwith Maximum & Minimum Constraints

min, , max,i gen i iP P P

,

1

0N

load gen i

i

P P=

=


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Generic Kuhn Tucker Conditions

Minimise: f (P)

Subject to: wk (P) = 0 k= 1, 2, 3 Nwgj (P) 0 j = 1, 2, 3 Ng

P = vector of real numbers, dimension =N

The Lagrange function is

1 1

L( ) f( ) w ( ) g ( )gw

NN

k k j j

k j

= =

= + + P,, P P P

Objective fn Equality constraints Inequality Constraints


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Optimal Conditions(Kuhn Tucker Conditions)

j

1. L( , , ) 0, 1,2, ....

2. w ( ) 0, 1, 2,.....

3. g ( ) 0, 1, 2,.....

4. g ( ) 0, 1,2,.....

0

P

P

P

P

o o o

i

o

k w

o

j g

o o

j j g

o

i NP

k N

j N

j N

= =

= =

=

= =

( , , ) optimal solutionP o o o

Complementaryslackness condition


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Kuhn Tucker Conditions

The Kuhn Tucker conditions

Give necessary conditions for a minimum, butnot a precise procedure to find this minimum

Experiment!

by trial and error

The minimum solution will satisfy the KT

conditions


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Example


F1 = 459 + 6.48Pgen,1 + 0.00128(Pgen,1)2 /h


F2 = 310 + 7.85Pgen,2 + 0.00194(Pgen,2 )2 /h


F3= 78 + 7.97Pgen,3 + 0.00482(Pgen,3)2 /h

Determine the minimum operating cost and unitschedule to meet a system demand of 850 MW.


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Example


F1 = 459 + 6.48Pgen,1 + 0.00128(Pgen,1)2 /h


F2 = 310 + 7.85Pgen,2 + 0.00194(Pgen,2 )2 /h


F3= 78 + 8.5Pgen,3 + 0.00482(Pgen,3)2 /h

Determine the minimum operating cost and unit

schedule to meet a system demand of 850 MW.

economic dispatch(1)

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