Economic Dispatch: Formulation
The goal of economic dispatch is to determine the generation dispatch that minimizes the instantaneous operating cost, subject to the constraint that total generation = total load + losses
Initially we'll ignore generatorlimits and thelosses
2
Unconstrained MinimizationThis is a minimization problem with a single
equality constraint
For an unconstrained minimization a necessary (but not sufficient) condition for a minimum is the gradient of the function must be zero,
The gradient generalizes the first derivative for multi-variable problems:
( ) f x 0
3
Minimization with Equality ConstraintWhen the minimization is constrained with an
equality constraint we can solve the problem using the method of Lagrange Multipliers
Key idea is to represent a constrained minimization problem as an unconstrained problem.
That is, for the general problem
minimize ( ) s.t. ( )
We define the Lagrangian L( , ) ( ) ( )
Then a necessary condition for a minimum is the
L ( , ) 0 and L ( , ) 0
T
x λ
f x g x 0
x λ f x λ g x
x λ x λ 4
Economic Dispatch Lagrangian
G1 1
G
For the economic dispatch we have a minimization
constrained with a single equality constraint
L( , ) ( ) ( ) (no losses)
The necessary conditions for a minimum are
L( , )
m m
i Gi D Gii i
Gi
C P P P
dC
P
P
P
1
( )0 (for 1 to )
0
i Gi
Gi
m
D Gii
Pi m
dP
P P
5
Economic Dispatch Example
1 2
21 1 1 1
22 2 2 2
1 1
1
What is economic dispatch for a two generator
system 500 MW and
( ) 1000 20 0.01 $/h
( ) 400 15 0.03 $/h
Using the Lagrange multiplier method we know:
( )20 0.0
D G G
G G G
G G G
G
G
P P P
C P P P
C P P P
dC P
dP
1
2 22
2
1 2
2 0
( )15 0.06 0
500 0
G
GG
G
G G
P
dC PP
dP
P P
6
Economic Dispatch Example, cont’d
1
2
1 2
1
2
1
2
We therefore need to solve three linear equations
20 0.02 0
15 0.06 0
500 0
0.02 0 1 20
0 0.06 1 15
1 1 0 500
312.5 MW
187.5 MW
26.2 $/MW
G
G
G G
G
G
G
G
P
P
P P
P
P
P
P
h
7
Lambda-Iteration Solution Method
The direct solution using Lagrange multipliers only works if no generators are at their limits.
Another method is known as lambda-iteration
– the method requires that there to be a unique mapping from a value of lambda (marginal cost) to each generator’s MW output:
– for any choice of lambda (marginal cost), the generators collectively produce a total MW output
– the method then starts with values of lambda below and above the optimal value (corresponding to too little and too much total output), and then iteratively brackets the optimal value.
( ).Gi
P
8
Lambda-Iteration Algorithm
L H
1 1
H L
M H L
H M
1
L M
Pick and such that
( ) 0 ( ) 0
While Do
( ) / 2
If ( ) 0 Then
Else
End While
m mL H
Gi D Gi Di i
mM
Gi Di
P P P P
P P
9
Lambda-Iteration: Graphical ViewIn the graph shown below for each value of lambda there is a unique PGi for each generator. This
relationship is the PGi() function.
10
Lambda-Iteration Example
1 1 1
2 2 2
3 3 3
1 2 3
Consider a three generator system with
( ) 15 0.02 $/MWh
( ) 20 0.01 $/MWh
( ) 18 0.025 $/MWh
and with constraint 1000MW
Rewriting generation as a function of , (
G G
G G
G G
G G G
Gi
IC P P
IC P P
IC P P
P P P
P
G1 G2
G3
),
we have
15 20P ( ) P ( )
0.02 0.01
18P ( )
0.025
11
Lambda-Iteration Example, cont’dm
Gi
i=1
m
Gii=1
1
H
1
Pick so P ( ) 1000 0 and
P ( ) 1000 0
Try 20 then (20) 1000
15 20 181000 670 MW
0.02 0.01 0.025
Try 30 then (30) 1000 1230 MW
L L
H
mL
Gii
m
Gii
P
P
12
Lambda-Iteration Example, cont’d
1
1
Pick convergence tolerance 0.05 $/MWh
Then iterate since 0.05
( ) / 2 25
Then since (25) 1000 280 we set 25
Since 25 20 0.05
(25 20) / 2 22.5
(22.5) 1000 195 we set 2
H L
M H L
mH
Gii
M
mL
Gii
P
P
2.5
13
Lambda-Iteration Example, cont’d
H
*
*
1
2
3
Continue iterating until 0.05
The solution value of , , is 23.53 $/MWh
Once is known we can calculate the
23.53 15(23.5) 426 MW
0.02
23.53 20(23.5) 353 MW
0.01
23.53 18(23.5)
0.025
L
Gi
G
G
G
P
P
P
P
221 MW
14
Generator MW Limits
Generators have limits on the minimum and maximum amount of power they can produce
Typically the minimum limit is not zero.
Because of varying system economics usually many generators in a system are operated at their maximum MW limits:
Baseload generators are at their maximum limits except during the off-peak.
16
Lambda-Iteration with Gen Limits
,max
,max
In the lambda-iteration method the limits are taken
into account when calculating ( ) :
if calculated production for
then set ( )
if calculated production for
Gi
Gi Gi
Gi Gi
P
P P
P P
,min
,min
then set ( )
Gi Gi
Gi Gi
P P
P P
17
Lambda-Iteration Gen Limit Example
G1 G2
G3
1 2 31
In the previous three generator example assume
the same cost characteristics but also with limits
0 P 300 MW 100 P 500 MW
200 P 600 MW
With limits we get:
(20) 1000 (20) (20) (20) 10m
Gi G G Gi
P P P P
1
00
250 100 200 1000
450 MW (compared to 670MW)
(30) 1000 300 500 480 1000 280 MWm
Gii
P
18
Lambda-Iteration Limit Example,cont’dAgain we continue iterating until the convergence
condition is satisfied.
With limits the final solution of , is 24.43 $/MWh
(compared to 23.53 $/MWh without limits).
Maximum limits will always caus
1
2
3
e to either increase
or remain the same.
Final solution is:
(24.43) 300 MW (at maximum limit)
(24.43) 443 MW
(24.43) 257 MW
G
G
G
P
P
P
19
Inclusion of Transmission Losses
The losses on the transmission system are a function of the generation dispatch.
In general, using generators closer to the load results in lower losses
This impact on losses should be included when doing the economic dispatch
Losses can be included by slightly rewriting the Lagrangian:
G1 1
L( , ) ( ) ( ) m m
i Gi D L G Gii i
C P P P P P
P
20
Impact of Transmission Losses
G1 1
G
The inclusion of losses then impacts the necessary
conditions for an optimal economic dispatch:
L( , ) ( ) ( ) .
The necessary conditions for a minimum are now:
L( , )
m m
i Gi D L G Gii i
C P P P P P
P
P
1
( ) ( )1 0
( ) 0
i Gi L G
Gi Gi Gi
m
D L G Gii
dC P P P
P dP P
P P P P
21
Impact of Transmission Losses
th
( ) ( )Solving for , we get: 1 0
( )1
( )1
Define the penalty factor for the generator
(don't confuse with Lagrangian L!!!)
1
( )1
i Gi L G
Gi Gi
i Gi
GiL G
Gi
i
iL G
Gi
dC P P P
dP P
dC P
dPP P
P
L i
LP P
P
The penalty factorat the slack bus isalways unity!
22
