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Table 3 Amount of load required
Hour
3
6
9
12
15
18
21
24
Demand (MW)
259
200
300
450
527
610
480
320
In this paper, we considered 8 successive periods in order to establish the
temporal evolution of the power demand. Each period lasts 3 h, hence, the total
period is about 24 h (Table
3
).
The IEEE 4th order state model has been adopted for all the
five machines as
written in (Eq.
19
), (Abbassi and Chebbi
2012
).
8
<
dE
0
q
dt
¼
1
E
0
q
þð
X
d
Þ
T
do
½
E
fd
X
d
I
d
dE
0
d
dt
¼
T
qo
½
1
E
0
d
þð
X
q
X
q
Þ
I
q
ð
Þ
19
:
d
dt
¼
1
M
½
P
m
P
e
D
ðx
1
Þ
d
dt
¼
x
1
where
E
0
d
;
E
0
q
d and q axis transient emf
'
X
d
, X
q
d and q axis reactance
s,
X
d
;
X
q
'
d and q axis transient reactance
s,
w Mechanical speed,
M Moment of inertia,
P
m
Maximum available power extracted by the turbine,
P
g
, Q
g
Active and reactive powers supplied by each machine,
D
Friction coef
cient,
δ
Load angle,
V
r
Voltages at bus i.
After calculating of the instantaneous (d, q) components of voltages of the
generator
s terminals and currents as detailed in (Abbassi et al.
2012
), the active
and reactive powers supplied by each generator are chosen to be:
'
8
<
E
0
q
V
r
sin
ðd
r
Þ
X
d
1
1
1
V
r
P
g
¼
þ
2
ð
X
q
X
d
Þ
sin
ð
2
d
r
Þ
ð
20
Þ
E
0
q
V
r
cos
ðd
r
Þ
X
d
:
1
1
X
d
Þ
V
r
1
1
1
X
q
Þ
V
r
1
Q
g
¼
þ
2
ð
X
q
cos
ð
2
d
r
Þ
2
ð
X
d
Figure
6
illustrates the total production cost of various optimization methods for
solving the unit commitment problem. Compared to the algorithms of Wei and Cai
(Wei and Li
1999
; Cai and Cai
1997
), we find that these optimization methods
present high performance since they improved to win in the production cost.
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