economic dispatch control

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Economic Dispatch without loss

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INTRODUCTION

• Refers to the most economic loading of thegenerators which are connected to the systemand running.

• Fuel cost is the principal factor in fossil fuelplants and is given as a function of generation.

• ED problem is to define the production levelof each plant so that the total generation andtransmission cost is minimum for a prescribedschedule of loads.

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Solution Techniques

• Lagrange Multiplier method

• Lambda Iteration method

•

Gradient method• Dynamic programming

• Evolutionary Computation Techniques

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Incremental Cost curve for thermal Power

plants

• A plot of the input in kilocalories per hour versuspower output in megawatts is called input-outputcurve.

•

The ordinates of the curve may be converted toRs/hr by multiplying the fuel in Rs/kilocalories.

• The empirical equation of the this curve is given by

• Where a, b and c are constants depending upon aparticular plant.

cbP aP C  iii  

2

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•  Incremental fuel cost

The incremental cost is equal to the slope of the cost

curve. A plot of incremental cost versus power

output is called the incremental cost curve. It is

shown in Fig.3. Equation (2) is of the form

=

The incremental cost is equal to the slope of the cost

curve.

i

i

i

dP 

dC  IC    )(

=

baP i  2

  ;

Rs/ MWh

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Input-Output Cost Curve

• Used to describe the efficiency of the plants.

• Graphical Representation of input in Rs/hr

versus power output in megawatts is called

input-output curve.

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Incremental cost curve for Hydro Power

Plants

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Lagrange Function

• This is a constrained optimization problem that may

be attacked formally using advanced calculus

methods that involve the Lagrange function.

• In order to establish the necessary conditions for anextreme value of the objective function, add the

constraint function to the objective function after

the constraint function has been multiplied by an

undetermined multiplier. This is known as the 

Lagrange function. 

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• Lagrange function with respect to the poweroutput values one at a time give the set ofequations shown

Where   is the Lagrangian multiplier.

• The necessary condition for this optimizationproblem is taking the first derivative of the Lagrangeunction with respect to the each of the independentvariables and set the derivatives equal to zero i.e

L = FT +   

0

i

ii

i  P 

 P  F 

 P 

 L

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i

i

 P 

 F 0

i

i

 P 

 F 

0

This is called as coordination equation. i.e. the necessary

condition for the existence of a minimum operating cost is that the

incremental cost rates of all the units be equal to the some in

determined value .

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• This is the necessary condition for the existence of a

minimum cost operating condition for the thermal power system is that the incremental cost rates of allthe units be equal to some undetermined value, .

• Of course, to this necessary condition we must add

the constraint equation that the sum of the poweroutputs must be equal to the power demanded by theload. In addition, there are two inequalities that must

 be satisfied for each of the units.

• That is, the power output of each unit must be greaterthan or equal to the minimum power permitted andmust also be less than or equal to the maximum

 power permitted on that particular unit.

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• These conditions and inequalities may besummarized as shown in the set of equations

N equations

i

i

 P 

 F 

max,min,   iii  P  P  P   

 N 

i

 Load i  P  P 

1

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Economic dispatch with loss

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