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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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