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3 Probabilistic Optimal Power Flow Model for
Grid-Connected Induction Wind Power System
The objective function of the POPF including wind farms is formulated as the
minimization of the total fuel cost for conventional generation
min
i∈S
G
(
a
2
i
P
Gi
+
a
1
i
P
Gi
+
a
0
i
)
(5)
where
S
G
is the set of power generation,
P
Gi
is the output of conventional
generator's real power in
i
th
generator,
a
2
i
,
a
1
i
and
a
0
i
are the generation cost
coecients, respectively. If wind farm is connected at
i
th
(
i
N
w
) bus, the
corresponding power flow equations for grid-connected induction wind power
system are given by
⎧
⎨
∈
P
ei
(
e
i
,f
i
,s
i
)
−P
Di
−P
i
(
e,f,t
)=0
Q
ei
(
e
i
,f
i
,s
i
)
−
Q
Di
−
Q
i
(
e,f,t
)=0
,i∈ N
w
(6)
⎩
P
mi
−
P
ei
(
e
i
,f
i
,s
i
)=0
where
P
mi
and
P
ei
are the mechanical power of wind turbine and the electrical
power of wind generator at the
i
th
bus,
N
w
is the set of the buses connected
with wind farm,
P
Di
and
Q
Di
are real and reactive load, node powers
P
i
and
Q
i
are the function of the real part
e
, imaginary part
f
of node voltage and ratio
t
, respectively. Real and reactive power generations, ratios, voltage amplitudes
and line currents are limited due to equipment and system constraints
⎧
⎨
P
min
Gi
≤
P
Gi
≤
P
max
Gi
i
∈
S
G
Q
min
Q
max
Gi
≤
Q
Gi
≤
Gi
i
∈
S
G
t
min
t
max
ij
ij
≤
t
ij
≤
(
i,j
)
∈
S
T
(7)
⎩
(
V
i
)
min
(
e
i
+
f
i
)
(
V
i
)
max
i
≤
≤
∈
S
N
I
ij
≤
(
I
ij
)
max
(
i,j
)
∈
S
L
where
S
T
,
S
N
and
S
L
are the set of the transformers, the system nodes and the
restricted line, respectively.
4 Cholesky Factorization Method
For
N
dimension random vector
M
in the power system, including generator
outputs, and load real and reactive powers, its covariance matrix
C
M
is given
by:
C
M
=
E
(
MM
T
)=
R
(8)
where
R
is a symmetric matrix. The matrix
R
can be obtained by the Cholesky
Factorization method [10], i.e.,
R
=
LL
T
, where the Cholesky Factor
L
is a
lower triangular matrix. Suppose that
W
is an
N
dimension random vector with
variance equal to one and independent mutually, i.e.,
C
W
=
E
(
WW
T
)=
I
(9)
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