CHAPTER
2
ECONOMIC DISPATCH
*
2.1 INTRODUCTION
The economic dispatch problem plays an important role in power system operation.
The principal objective of ED is to obtain the minimum operating cost needed
to satisfy power balance, generator and network operating limit constraints. The
optimal operating point of a power generation system is where the operating level
of each generating unit is adjusted such that the total cost of delivered power is at
a minimum. In an energy management system (EMS), Economic Dispatch (ED)
is used to determine each generating level in the system in order to minimize the
total generator fuel cost or total generator cost and emission of thermal units while
still covering load demand plus transmission losses [1].
•Generating fuel production costs
•Generating operating limits
•Network system
•System loading condition
Optimal Generation
Scheduling
•Generating fuel production costs
•Generating operating limits
•Network system
•System loading condition
Optimal Generation
Scheduling
Fig. 2.1 The process of economic dispatch.
The cost function for each generator can be approximately represented by a
quadratic function for mathematical convenience. Mathematical programming
including gradient method, linear programming or quadratic programming (QP)
can be used to determine the ED [1-2].
*This chapter has been written with assistance from Keerati Chayakulkheeree
12 Artifi cial Intelligence in Power System Optimization
2.2 GENERATOR INCREMENTAL COST CURVE
The analysis of the problems associated with the controlled operation of power
systems contains many parameters of interest. Fundamental to the economic
operating problem is the set of input-output characteristics of a thermal power
generation unit. That is, gross input to the plant represents the total generator fuel
cost whether measured in terms of dollars per hour or tons of coal per hour or
millions of cubic feet or meters of gas per hour or any other unit. The net output
of the plant is the electrical power output (P
G
) in MW supplied to the electric
utility system.
These data may be obtained from design calculations or from heat rate tests.
When heat rate test data are used, it will usually be found that the data points do
not form a smooth curve. Steam turbine generating units have several critical
operating constraints. Generally, the minimum load at which a unit can operate
is infl uenced more by the steam generator and the regenerative cycle than by the
turbine itself. The only critical parameters for the turbine are shell and rotor metal
differential temperatures, exhaust hood temperature, and rotor and shell expansion.
A limitation to the minimum load is required to maintain fuel combustion stability
and by inherent steam generator design constraints. For example, most supercritical
units cannot operate below 30% of design capacity. A minimum fl ow of 30% is
required to cool the tubes in the furnace of the steam generator adequately. Turbines
do not have any inherent overload capability so the data shown on these curves
normally do not extend much beyond 5% of the manufacturer’s stated valve-wide-
open capacity [1].
In practice, two approximation models are widely used: quadratic approximation
as shown in (2.1) and piecewise linear approximation as shown in (2.2).
2
)(
GGG
cPbPaPF
(2.1)
°
¯
°
®
dd
d
d
43
32
21
33
22
11
,
,
,
)(
PPP
PPP
PPP
Pqr
Pqr
Pqr
PF
G
G
G
G
G
G
G
(2.2)
where
F(P
G
) generating unit operating cost,
P
G
real power generation.
and a, b, c, r
i
and q
i
with i = 1, …, 3 are cost coeffi cients.
The incremental cost of a unit is the slope (the derivative) of the unit cost
curve. The data relates $/MWh to net power output of the unit in MW. This
relation is widely used in the economic dispatching of units. Figures 2.2 and 2.3
show the operating cost and incremental costs of quadratic and piecewise linear
approximations, respectively.
Economic Dispatch 13
2.3 ECONOMIC DISPATCH PROBLEM FORMULATION
WITHOUT REGARDING LOSS
Consider a lossless system consisting of NG thermal generating units connected
to a single bus-bar serving electrical load as shown in Fig. 2.4 [1] with the aim to
minimize total power generation cost:
F
T
=
¦
NG
i
GiiNGGNGGG
PFPFPFP
1
,2211
)()(...)()(
(2.3)
Subject to the simplifi ed power balance constraint,
0)(or ...
1
,21
¦
NG
i
GiLLNGGGG
PPwPPPP
G
P
D
P
G
(2.4)
where P
D
is load demand and NG is the number of generating units. Thus, the
Lagrange function can be given as
() ()
T
LF wl=+◊
GG
PP
P
G
P
G
(2.5)
Fig. 2.2 Quadratic approximation cost function.
Fig. 2.3 Piecewise linear approximation cost function.
Cost ($/hr)
Incremental
Cost ($/MWh)
MW MW
Cost ($/hr)
Incremental
Cost ($/MWh)
MW
MW
P
1
P
2
P
1
P
4
P
3
P
2
P
4
P
3
Color image of this figure appears in the color plate section at the end of the book.
Color image of this figure appears in the color plate section at the end of the book.

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