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uncertainty (Azar
2012
). Such ideas are readily applicable to the unit commitment
problem. The application of fuzzy logic allows a qualitative description of the
behavior of a certain system, the characteristics of the system, and the response of
that system without the need for exact mathematical formulation (Azar
2010a
,
2012
)
To establish our strategy, we have considered the partial derivatives of the
Lagrange function (Eq.
9
) with respect to each of the controllable variables equal to
zero.
o
P
ih
¼
o
/
i
ð
L
½
P
ih
Þ
o
P
Lh
o
P
ih
g
P
ih
U
ih
¼
0
ð
10
Þ
o
o
X
N
g
L
o
g
¼
o
P
dh
P
Lh
P
i
U
ih
¼
0
ð
11
Þ
i
¼
1
Equations (
10
) and (
11
) represent the optimality conditions necessary to solve
equation systems Eqs. (
1
) and (
4
) without using inequality constraints (Eqs.
5
and
6
).
Equation (
10
) can be written as follows:
o
/
i
ð
½
P
ih
Þ
o
P
ih
g
¼
i
¼
1
N
G
;
h
¼
1
H
ð
12
Þ
U
ih
;
; ...;
; ...;
o
P
Lh
o
P
ih
The term
o
/
i
ð
P
ih
Þ
½
o
P
ih
represents the incremental cost (IC) of each unit i and
o
P
Lh
o
P
ih
represents the incremental losses (IL). These terms occur as fuzzy variables asso-
ciated to our strategy in order to solve the Unit Commitment problem. It should be
noted that the strategy is based on the integration of a fuzzy controller to optimize
the cost of the production unit while ensuring proper planning of the production
units. In the current formulation, the fuzzy input variables associated to the Unit
Commitment problem are the load capacity of the generator (LCG), the incremental
cost (IC) and the incremental losses (IL). The output variable is the cost of pro-
duction (C
P
). The fuzzy sets related to these variables are selected and normalized
between 0 and 1. This normalized value can be multiplied by a scaling factor
chosen to accommodate any desired variable.
In the following, a brief description and explanation of the main choice of the
mentioned fuzzy variables:
Load capacity of generator LCG is considered to be fuzzy, as it is based upon
the load to be served.
Incremental losses IL is taken to be fuzzy, because the losses can lead to
changes in the total production cost and because losses varies over the holy
network architecture.
Incremental cost IC is taken to be fuzzy, because the cost of fuel may change
over the period of time, and because the cost of fuel for each unit may be
different.
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