This document discusses economic load dispatch (ELD) of power systems. ELD aims to minimize the total operating costs of generation units while meeting demand. It describes: - Traditional classifications of generation units as baseload, midload, and peaking units. - Key considerations for different generation types like thermal, hydro, wind and solar. - The mathematical formulation of ELD as a constrained optimization problem to minimize total generation costs subject to meeting demand. - Methods to solve the ELD problem like using Lagrange multipliers to equalize marginal costs of generators or iterative methods like lambda iteration when cost functions are nonlinear.

1. Economic Operation
• Demands are variable.
• By different mixes of generations (by resource
and/or by units) there may be many ways to
meet the demand of a particular instant
• Possible only with the installed generation
capacity much greater than the current load
(Flexibility)
• The idea is to consider how generally to
minimize the overall variable operating costs
1
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2. Resource Prospective
 Traditionally utilities have had three broad groups of
generators:
– “Baseload” units: large coal/nuclear; almost always on at max.
– “Midload,” ‘intermediate,” or “cycling” units: smaller coal or
gas that cycle on/off daily or weekly.
– “Peaker” units: combustion turbines used only for several
hours. during periods of high demand
2
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3. Resource Prospective with
Generation Mix
• The main types of generating units are thermal and
hydro, with wind and solar rapidly growing.
• For hydro the fuel (water) is free but there may be
many constraints on operation:
• fixed amounts of water available,
• reservoir levels must be managed and coordinated,
• downstream flow rates for fish and navigation.
• Hydro optimization (typically longer term many
months or years) also depends on PROR, Storage
• Wind and solar can be incorporated by subtracting
from load usually Non-dispatchable
• Dispatchable thermal units, looking at short-term
optimization (i.e. Units) 3
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4. Mathematical formulation for ELD
• Deals with the share of generators to meet
the demand such that the total cost of
generation would be minimum.
Assumptions:
– The generators which are ON (committed
generators) are known before the solution of
ELD.
– Consider only the real power generations.
• Mathematically ELD can be written as
Minimize total generation cost; F= Σ fi
– Subject to a number of constraints:
– will be added one by one
The first constraint could be: ΣPi = D
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5. Mathematical formulation for ELD contd.
• There could be different techniques to attempt this
problem
• The standard form of this problem is;
Maximize/Minimize y=f(x) s. t. Φ(x) = γ(x)
• Can be solved defining Lagrangian function as;
Maximize/Minimize L= f(x) ± λ{Φ(x) - γ(x)}
λ unknown penalty factor
• Also f(x) will be Maximum or minimum when
∂L/ ∂x = 0 & ∂L/ ∂ λ = 0
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6. • Considering the ELD problem in Lagrangian function;
L = Σfi ± λ(D-ΣPi )
• Applying the condition as
∂L/ ∂pi = 0 & ∂L/ ∂ λ = 0
We get
It means
– for the minimum generation cost, the incremental cost (Also
called marginal cost) of individual generators should be equal

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or
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and
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l
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1
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(
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cos
/
0
)
1
0
(
/
/

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Initially ignore
generator limits and
the losses
AKM/Pwr oprtn Contrl/lect2 6

7. Incremental Cost Example
7
2
1 1 1 1
2
2 2 2 2
1 1
1 1 1
1
2 2
2 2 2
2
For a two generator system assume
( ) 1000 20 0.01 $/hr
( ) 400 15 0.03 $/hr
Then
( )
( ) 20 0.02 $/MWh
( )
( ) 15 0.06 $/MWh
G G G
G G G
G
G G
G
G
G G
G
C P P P
C P P P
dC P
IC P P
dP
dC P
IC P P
dP
  
  
  
  
AKM/Pwr oprtn Contrl/lect2

8. Incremental Cost Example, cont'd
8
1 2
2
1
2
2
1
2
If 250 MW and 150 MW Then
(250) 1000 20 250 0.01 250 $ 6625/hr
(150) 400 15 150 0.03 150 $6025/hr
Then
(250) 20 0.02 250 $ 25/MWh
(150) 15 0.06 150 $ 24/MWh
G G
P P
C
C
IC
IC
 
     
     
   
   
Therefore not optimal
AKM/Pwr oprtn Contrl/lect2

9. Intuition behind the solution
• At the solution, both generators have the same
marginal (or incremental) cost, and this common
marginal cost is equal to λ.
• Intuition behind solution:
– If marginal costs of generators were different, then by
decreasing production at higher marginal cost generator,
and increasing production at lower marginal cost
generator we could lower overall costs.
– Generalizes to any number of generators.
• If demand changes, then change in total costs can be
estimated from λ
• fi must be available in terms of Pi
9
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10. Systematic approach to achieve equal
marginal cost (THERMAL)
• To optimize generation costs we need to develop cost
relationships between net power out and operating
costs.
• Between 2-10% of power is used within the generating
plant; this is known as the auxiliary power.
Auxiliary power.
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11. Operating Characteristics of thermal and hydro
generators
 Relatively more input for large output (Non-linear
I/P~O/P characteristics). Why so?
 Any curve fitting technique could be applied to get the
mathematical representation of this curve
Fuel
Input/Cost
Pg
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12. • The characteristic could be represented either
polynomial, exponential or combination of these
• The simplest (mathematically) can be a second
order polynomial:
fi = ½ ai P i
2 + bi Pi +ci
• Applying ELD conditions as
∂fi/ ∂Pi = ai Pi + bi = λ (Ng No. of eqn first condition of ELD)
Σ Pgi – D = 0 (1 eqn second condition)
• NG+1 linear equations and same number of
unknowns can be solved
Solution of ELD problem
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13. Solution of ELD problem contd.
For 2nd order the incremental change in fuel is
given by;
• Example
2
2
2
2
2
1
1
1
03
.
0
15
400
01
.
0
20
1000
g
g
g
P
Pg
C
P
P
C






Incremental
fuel cost
Pg
AKM/Pwr oprtn Contrl/lect2 13

14. Example solution
14
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

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 
AKM/Pwr oprtn Contrl/lect2

15. • General expression for generator power output
could be derived
– if all the cost characteristics are represented by
quadratic equations
– and there are no other constraints.
• The procedures are as follows;
• Let us assume the cost function of ith generators
are given as;
• differentiating above equation with respect to Pi
i
i
i
i c
Pi
b
P
a
fi 

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2
5
.
0
i
i
i
i
i b
P
a
P
f 


 .
/
AKM/Pwr oprtn Contrl/lect2 15

16. 
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Just change the D get new λ and P
AKM/Pwr oprtn Contrl/lect2 16

17. Considering generator maximum and
minimum power limit
This expression can be modified by taking
generator max & min power limit as
Set Pi = Plimit and PD’ = PD- Σ P limit
J
( for those generators which crosses their max and
min limit)

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load
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es
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dF
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, _
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....
..........
_
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/ 
int
1
.
..........
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max
max
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constra
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P
forP
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P
Pi
forP
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AKM/Pwr oprtn Contrl/lect2 17

18. Lambda iteration method
• Suppose the cost characteristics is expressed
other than quadratic , for example
– a cubic or higher order polynomial
– exponential function
– combination of exponential & polynomial
• In this case some iterative method is used
• One such method is lambda iteration method
• The idea behind the method is;
– If generation is greater than demand , λ is to be
reduced.
– If generation is smaller than demand , λ is to be
increased AKM/Pwr oprtn Contrl/lect2 18

19. Steps to be followed for λ iteration
method
1. Assume a suitable value of λ
2. Calculate P gi
3. Calculate ε = Σ P gi – PD
4. If ε is within the tolerance, stop
5. If ε > 0 (Σ P gi > PD) , decrease λ & go to step 2
6. If ε < 0 (Σ P gi < PD) , increase λ & go to step 2
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20. Assignment
For the two generators
Solve the ELD using lamda iteration method if
the demand(s)
i. 500 MW
ii. 520 MW
Iii. 700 MW
2
2
2
2
2
1
1
1
03
.
0
15
400
01
.
0
20
1000
g
g
g
P
Pg
C
P
P
C






AKM/Pwr oprtn Contrl/lect2 20