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t
=[
t
of f
i
,
begin
,
t
of f
u
i
= 0
for
i
,
end
]
(7)
where
t
of f
i
,
begin
and
t
of f
i
,
end
are beginning and ending time of
i
th online must-off unit.
Ramp up/down rates for online units:
P
i
−
P
t
−
1
i
≤
UR
i
(8)
P
t
−
1
i
P
i
≤
−
DR
i
(9)
where
DR
i
and
UR
i
are down and up ramp rate limits of
i
th unit.
System power balance:
N
unit
i
=1
P
i
u
i
=
P
t
D
,
t
= 1
,
2
,···,
T
(10)
where
P
t
D
is the system load demand at time
t
.
The spinning reserve of the power system:
N
unit
i
=1
P
i
max
≥
P
t
D
+
P
R
(11)
where
P
i
max
is obtained by
P
i
max
= min
P
i
max
,
P
t
−
1
= 10
min
,which
is 10 minutes maximum active power of the
i
th unit with up ramp rate limit. And
P
R
is
the system reserve at time
t
.
Minimum up/down time:
{
+
τ
UR
i
}
with
τ
i
T
i
≥
T
i
≥
MU T
i
if
0
(12)
T
i
≤−
T
i
<
0
MDT
i
if
(13)
where
T
i
represents the continuous ON/OFF status duration of the
c
th cycle of the
i
th
unit.
MU T
i
and
MDT
i
are minimal up time and minimal down time respectively.
3
Two-Layer Hierarchical Market Competition Algorithm
(THMCA)
The two-layer hierarchical market competition algorithm (THMCA) is inspired by com-
petitions among enterprises in economic activities. The THMCA begins with an initial
population called perfect competition companies. Some of the best companies that have
the best objective fitness function values are selected to become the holding companies.
The rest become the subsidiary companies which are then divided among the holding
companies based on their power. The power of a holding company is positively propor-
tional to its fitness value. The holding companies and their subsidiary companies form
different conglomerates. Then subsidiary companies move toward their relevant hold-
ing companies and the position of the holding companies will be updated if necessary.
In the next step, the market competition among the conglomerates begins, and the weak
conglomerates are eliminated. This is called monopolistic competition procedure. Then
the oligopoly procedure begins. The market competition will gradually lead to an in-
crease in the power of strong conglomerates (oligopolies) and a decrease in the power
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