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where
I
is an identity matrix.
Using
M
=
LW
, the covariance matrix of
M
can be expressed as:
C
M
=
E
(
MM
T
)=
E
(
LWW
T
L
T
)=
R
(10)
Therefore, the relevance is set in the random vector of
M
.
5 Two-Point Estimate Method
By path following interior point method [11], the POPF problem can be mathe-
matically described by a set of the simplified nonlinear equations. In the 2PEM,
[2], two concentrations (
K
=2) are used for each random variable in this scheme,
the location and weights. Taking into account the mean and standard deviation
of the bus power injection, the estimate point can be computed using the concen-
trations (or weights) located at the estimate point. Suppose that the correlation
coecient among all correlated random variables is set as 0.9 and its standard
deviation is set as 5% of its expectation value. Then, calculate the wind tur-
bine mechanical power and the electrical power using (2) and (3). The following
algorithm is given as:
1) Set
K
=1,
E
(
Z
r
)=0,
E
(
Z
r
)=0.
2) Compute the standard location
ξ
i,k
and the weight
ω
i,k
of the 2m scheme.
3) Compute the estimated points
x
i,k
(
i
=1
,
2
,...,m.k
=1
,
2) located on both
sides ofthe expectation, and obtain the sample values (
μ
x
1
,
μ
x
2
,...,
x
i,k
,...,
μ
x
m
).
4) The Path Following Interior Point method [11] is used to solve the deter-
ministic optimal power flow.
5) The slips of induction wind generator, along with the nodal voltages, are
corrected at each iteration of the POPF calculation. Then, compute the
estimated value
Z
r
of nodal voltage and nodal power.
6) Calculate the obtained results at each step using (11), and set k=k+1
E
(
Z
r
)
(
k
+1)
=
E
(
Z
r
)
(
k
)
+
ω
k
×Z
r
(
k
)
E
(
Z
r
)
(
k
+1)
=
E
(
Z
r
)
(
k
)
+
ω
k
×
(11)
[
Z
r
(
k
)]
2
7) The procedure is terminated when all the random variables are ergodic.
Otherwise, repeat the point estimate iteration procedure until all the random
variables are ergodic.
8) Compute the mean and standard deviation of nodal voltage and nodal power.
6 Numerical Example
The proposed method was tested on a modified 5-bus test system [3]. For the
5-bus test system, the equivalent wind farms were connected at node 1 of 5-bus
test system by the two step-up transformer and double circuit transmission line,
so the new nodes 6, 7 were added as shown in Fig. 1. Without specification, all
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