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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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Input - output curves Incremental cost curve
Types of Curves
Pi (MW)
/h/MWh
Pi (MW)
2. Piecewise Linear 2. Stepped
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Input - output curves Incremental cost curve
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
Unit 3: P3,max = 200 MW, P3,min = 50 MW
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
Unit 1: P1,max = 600 MW, P1,min = 150 MW
F1 = 459 + 6.48Pgen,1 + 0.00128(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
Unit 3: P3,max = 200 MW, P3,min = 50 MW
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
Unit 1: P1,max = 600 MW, P1,min = 150 MW
F1 = 459 + 6.48Pgen,1 + 0.00128(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
Unit 3: P3,max = 200 MW, P3,min = 50 MW
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.
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