CHEN
et al.
: POWER SYSTEM CAPACITY EXPANSION UNDER HIGHER PENETRATION OF RENEWABLES CONSIDERING FLEXIBILITY
6245
where
s
i
t,k
and
u
i
t,k
must be no larger than the total aggregated
nameplate capacities for unit group k:
0
≤
s
i
t,k
, u
i
t,k
≤
¯
I
i
k
(28)
After introducing the above continuous variables, flexibility
constraints for the unit group could be formulated equivalently
as follows.
The aggregated power output
p
i
t,k
for the
i
th
category of
generation units at
k
th
region satisfies:
μ
−
i
k
·
¯
p
i
t,k
≤
p
i
t,k
≤
¯
μ
i
k
·
¯
p
i
t.k
(29)
where
μ
−
i
k
and
¯
μ
i
k
are the minimum and maximum output ratios
for the
i
th
category of thermal units in region
k
at hour
t
, respec-
tively. For instance, the values of
μ
−
i
k
and
¯
μ
i
k
for the coal fired
units in China are 0.5 and 1 respectively. The power output for
p
i
t,k
will range between 50% and 100% of the online capacity
for
¯
p
i
t,k
at time
t
.
Incorporating the continuous variables
p
i
t,k
,
s
i
t,k
and
u
i
t,k
, the
ramping constraints for the unit group are formulated as in (30)
and (31):
p
i
t,k
≤
¯
μ
i
k
·
(¯
p
i
t,k
−
s
i
t,k
−
u
i
t
+1
,k
) +
V S
i
k
·
s
i
t,k
+
V D
i
k
·
u
i
t
+1
,k
(30)
⎧
⎨
⎩
p
i
t,k
−
p
i
t
−
1
,k
≤
RU
i
k
·
(¯
p
i
t,k
−
s
i
t,k
) +
V S
i
k
·
s
i
t,k
−
μ
−
i
k
·
u
i
t,k
p
i
t
−
1
,k
−
p
i
t,k
≤
RD
i
k
·
(¯
p
i
t,k
−
s
i
t,k
)
−
μ
−
i
k
·
s
i
t,k
+
V D
i
k
·
u
i
t,k
(31)
where
RU
i
k
and
RD
i
k
denote the ratios of upward/downward
ramping for the
i
th
category of thermal units in region
k
, re-
spectively.
V S
i
k
and
V D
i
k
are the start-up and shut-down ramp
limits for the
i
th
category of thermal units in region
k
. For sim-
plicity of the analysis, we assume hereafter they both are equal
to the minimum power output level
μ
−
i
k
. In this case, the power
output for the second time period after startup will be equal to
the minimum output level.
Employing the above continuous variables, minimum on/off
time constraints are formulated as follows:
0
≤
u
i
t
+1
,k
≤
¯
p
i
t,k
−
∑
t
−
1
τ
=0
s
i
t
−
τ,k
t
= 1
· · ·
UT
i
k
−
1
0
≤
u
i
t
+1
,k
≤
¯
p
i
t,k
−
∑
U T
i
k
−
2
τ
=0
s
i
t
−
τ,k
t
=
UT
i
k
· · ·
T
−
1
(32)
0
≤
s
i
t
+1
,k
≤
¯
I
i
k
−
¯
p
i
t,k
−
∑
t
−
1
τ
=0
u
i
t
−
τ,k
t
=1
· · ·
DT
i
k
−
1
0
≤
s
i
t
+1
,k
≤
¯
I
i
k
−
¯
p
i
t,k
−
∑
DT
i
k