Environmental Engineering Reference
In-Depth Information
dC
dP
λ
P
Am
P
A
P
AM
P
Bm
P
BM
P
Cm
P
CM
P
Dm
P
DM
P
B
Figure 7.7
Economic distribution among four generators. (Based on fi gure in Neuenswander, J.R.,
Modern Power Systems
, Intertext Books, 1973)
mathematical optimization technique of Lagrange multipliers. Note that another term for
incremental cost is
marginal cost
.
7.3.4 OED with Several Units and Generation Limits
Steam driven generators have not only an upper limit of loading that corresponds to their
nameplate rating but also a lower limit that is imposed by cavitation, which occurs in the
turbine if the steam throughput falls below a certain critical level. This lower limit may be
anything between 30 and 50% of the generator rating.
Differentiation of quadratic cost curves gives a linear function. Figure 7.7 shows
incremental cost curves for a four-generator power system (with upper and lower limits
for each generator) supplying a demand
P
d
=
P
A
+
P
B
+
P
C
+
P
D
. It can be seen that the
whole range of incremental costs of generator
C
are below that of the other units; hence
C
should be loaded up to its full capacity
P
CM
. The new demand
d a CM
has to be
distributed among the other three generators according to the equal incremental cost
criterion.
Note that unit D, with a higher range of incremental costs, might be shut down (if units A
and B are capable of providing the required reserve when partly loaded) or might also be
required to provide reserve, in which case it will be loaded at its lower limit
P
Dm
. Assume
that enough reserve can only be provided if unit D is loaded at its lower limit. The new
demand then is
PPP
′ =−
PPP
d d Dm
which has to be optimally shared between units A and B.
Now choose a horizontal line (shown as
′′ =
′ −
in the fi gure). Suppose that the height of that
line is such that it intercepts the incremental cost curves so that
λ
+=′′
This is the graphical solution to the optimal economic dispatch problem in which the incre-
mental costs of the generators operating between their limits are the same and equal to
PPP
ABd
.
The practical inadequacy of the merit order scheduling should now be obvious. The difference
λ
