2.1. Development and Convergence Analysis
The discrete-time observations of the unbalanced three-phase power system are modeled as [5]:
(1)
v
a
[
n
]
=
V
a
cos
(
ω
n
+
ϕ
)
+
ξ
a
[
n
]
,
v
b
[
n
]
=
V
b
cos
(
ω
n
+
ϕ
-
2
π
3
)
+
ξ
b
[
n
]
,
v
c
[
n
]
=
V
c
cos
(
ω
n
+
ϕ
+
2
π
3
)
+
ξ
c
[
n
]
,
where
V
a
,
V
b
, and
V
c
are the inequivalent amplitudes of different phase components,
ω
=
Ω
/
F
s
is the discrete frequency with
Ω
and
F
s
being the voltage frequency in radian and sampling frequency in Hz, respectively, and
ϕ
is the initial phase. The nominal value of
Ω
is
Ω
*
=
100
π
(or
120
π
)
rads
-
1
. According to [10], the noise terms
ξ
a
[
n
]
,
ξ
b
[
n
]
, and
ξ
c
[
n
]
, are independent and identically distributed additive white Gaussian noise sequences with same variance
σ
2
. The task is to find the unknown parameters, namely,
ω
,
ϕ
,
V
a
,
V
b
, and
V
c
. In this study, we apply the
α
β
-transformation [7] on (1) to achieve accurate parameter estimation. The transformed signals, denoted by
v
α
[
n
]
and
v
β
[
n
]
, are computed as
(2)
[
v
α
[
n
]
v
β
[
n
]
]
=
2
3
[
1
-
1
2
-
1
2
0
3
2
-
3
2
]
[
v
a
[
n
]
v
b
[
n
]
v
c
[
n
]
]
.
Based on (1)-(2),
v
α
[
n
]
and
v
β
[
n
]
can also be expressed as
(3)
v
α
[
n
]
=
s
α
[
n
]
+
q
α
[
n
]
,
v
β
[
n
]
=
s
β
[
n
]
+
q
β
[
n
]
,
where
(4)
s
α
[
n
]
=
(
A
cos
(
ϕ
)
-
B
sin
(
ϕ
)
)
cos
(
ω
n
)
-
(
A
sin
(
ϕ
)
+
B
cos
(
ϕ
)
)
sin
(
ω
n
)
,
s
β
[
n
]
=
(
-
B
cos
(
ϕ
)
+
C
sin
(
ϕ
)
)
cos
(
ω
n
)
+
(
B
sin
(
ϕ
)
+
C
cos
(
ϕ
)
)
sin
(
ω
n
)
,
with
(5)
A
=
6
12
(
4
V
a
+
V
b
+
V
c
)
,
B
=
2
4
(
V
b
-
V
c
)
,
C
=
6
4
(
V
b
+
V
c
)
,
q
α
[
n
]
=
2
3
(
ξ
a
[
n
]
-
1
2
ξ
b
[
n
]
-
1
2
ξ
c
[
n
]
)
,
q
β
[
n
]
=
2
2
(
ξ
b
[
n
]
-
ξ
c
[
n
]
)
.
Although both
q
α
[
n
]
and
q
β
[
n
]
contain
ξ
b
[
n
]
and
ξ
c
[
n
]
, it is easy to show that the noise terms are uncorrelated; that is,
E
{
q
α
[
n
]
q
β
[
n
]
}
=
0
where
E
denotes the expectation operator, and they have identical variance
σ
2
.
Assuming that we have
N
samples for each channel, (3) can be written in matrix form as follows:
(6)
v
=
s
+
q
,
where
(7)
v
=
[
v
α
[
1
]
v
α
[
2
]
⋯
v
α
[
N
]
v
β
[
1
]
v
β
[
2
]
⋯
v
β
[
N
]
]
T
,
s
=
[
s
α
[
1
]
s
α
[
2
]
⋯
s
α
[
N
]
s
β
[
1
]
s
β
[
2
]
⋯
s
β
[
N
]
]
T
,
s
=
G
x
,
G
=
[
H
0
0
H
]
,
H
=
[
cos
(
ω
)
sin
(
ω
)
cos
(
2
ω
)
sin
(
2
ω
)
⋮
⋮
cos
(
N
ω
)
sin
(
N
ω
)
]
,
x
=
[
A
cos
(
ϕ
)
-
B
sin
(
ϕ
)
-
A
sin
(
ϕ
)
-
B
cos
(
ϕ
)
-
B
cos
(
ϕ
)
+
C
sin
(
ϕ
)
B
sin
(
ϕ
)
+
C
cos
(
ϕ
)
]
,
q
=
[
q
α
[
1
]
q
α
[
2
]
⋯
q
α
[
N
]
q
β
[
1
]
q
β
[
2
]
⋯
q
β
[
N
]
]
T
.
Here,
T
denotes the transpose operator and 0 is the
N
×
2
zero matrix. We see that
x
corresponds to the linear unknowns, while
ω
is the nonlinear unknown in (6). Employing NLS [11, 12], the estimates of
ω
and
x
, denoted by
ω
^
and
x
^
, are
(8)
{
ω
^
,
x
^
}
=
arg
min
ω
,
x
J
(
ω
,
x
)
,
J
(
ω
,
x
)
=
(
v
-
G
x
)
T
(
v
-
G
x
)
.
Based on the Newton-Raphson procedure, the updating rule for
ω
^
is
(9)
ω
^
(
k
+
1
)
=
ω
^
(
k
)
-
∇
ω
J
(
ω
^
(
k
)
,
x
^
(
k
)
)
∇
ω
2
J
(
ω
^
(
k
)
,
x
^
(
k
)
)
,
where
(10)
∇
ω
J
(
ω
,
x
)
=
-
(
G
1
x
)
T
(
v
-
G
x
)
,
∇
ω
2
J
(
ω
,
x
)
=
x
T
(
G
1
T
G
1
)
x
+
(
G
2
x
)
T
(
v
-
G
x
)
,
with
(11)
G
1
=
[
H
1
0
0
H
1
]
,
H
1
=
[
-
sin
(
ω
)
cos
(
ω
)
-
2
sin
(
2
ω
)
2
cos
(
2
ω
)
⋮
⋮
-
N
sin
(
N
ω
)
N
cos
(
N
ω
)
]
,
G
2
=
[
H
2
0
0
H
2
]
,
H
2
=
[
cos
(
ω
)
sin
(
ω
)
4
cos
(
2
ω
)
4
sin
(
2
ω
)
⋮
⋮
N
2
cos
(
N
ω
)
N
2
sin
(
N
ω
)
]
.
Here,
ω
^
(
k
)
and
x
^
(
k
)
are the estimates of
ω
and
x
at the
k
th iteration. Once we have
ω
^
(
k
)
,
x
^
(
k
)
is easily obtained from (8) as
(12)
x
^
(
k
)
=
(
G
T
G
)
-
1
G
T
v
|
ω
=
ω
^
(
k
)
,
where −1 denotes the matrix inverse. To start the algorithm of (9) and (12), we need
ω
^
(
0
)
. Noting that
ω
should be around its nominal value
ω
*
=
Ω
*
/
F
s
, that is,
ω
∈
[
ω
*
-
τ
,
ω
*
+
τ
]
where
τ
is the maximum deviation from
ω
*
,
ω
^
(
0
)
is computed using grid search as follows. We assign
K
uniformly-spaced grid points in the range
[
ω
*
-
τ
,
ω
*
+
τ
]
where one of them is
ω
^
(
0
)
. For each possible candidate
ω
^
(
0
)
, we determine
x
^
(
0
)
according to (12). The pair
{
ω
^
(
0
)
,
x
^
(
0
)
}
which gives the minimum value of
J
(
ω
,
x
)
will be chosen as the initial guess for (9). In our study, the iterative algorithm is terminated when
|
(
ω
^
(
k
+
1
)
-
ω
^
(
k
)
)
/
(
ω
^
(
k
+
1
)
)
|
<
ϵ
, where
ϵ
>
0
is a small tolerance constant, is reached. After obtaining
ω
^
, the NLS estimates of
ϕ
,
V
a
,
V
b
, and
V
c
are straightforwardly computed from
x
^
as
(13)
ϕ
^
=
tan
-
1
(
x
^
3
-
x
^
2
x
^
1
+
x
^
4
)
,
V
^
a
=
(
6
2
x
^
1
-
6
6
x
^
3
)
cos
(
ϕ
^
)
-
(
6
2
x
^
2
+
6
6
x
^
3
)
sin
(
ϕ
^
)
,
V
^
b
=
(
x
^
4
-
2
x
^
2
)
cos
(
ϕ
^
)
+
(
6
3
x
^
3
-
2
x
^
1
)
sin
(
ϕ
^
)
,
V
^
c
=
(
x
^
4
+
2
x
^
2
)
cos
(
ϕ
^
)
+
(
6
3
x
^
3
+
2
x
^
1
)
sin
(
ϕ
^
)
,
where
x
^
m
(
m
=
1,2
,
3,4
) denotes the
m
th element of
x
^
.
Finally, we examine the convergence of (9). According to [13], global convergence with quadratic rate is guaranteed when
M
|
ω
-
ω
^
(
0
)
|
<
1
is satisfied, where
M
is as follows:
(14)
M
=
sup
ω
∈
{
-
π
,
π
}
|
∇
ω
3
J
(
ω
,
x
)
∇
ω
2
J
(
ω
,
x
)
|
.
To determine the value of
M
, we first relate
G
,
G
1
, and
G
2
as follows:
(15)
G
1
T
G
1
+
G
2
T
G
=
c
E
,
G
1
=
N
1
G
F
,
G
2
=
N
2
G
,
where
(16)
E
=
[
1
0
0
0
1
0
0
2
0
]
,
F
=
[
0
-
1
0
0
1
0
0
0
0
0
0
-
1
0
0
1
0
]
,
c
=
N
(
N
+
1
)
(
2
N
+
1
)
6
,
N
1
=
diag
(
[
1
2
⋯
N
1
2
⋯
N
]
)
,
N
2
=
diag
(
[
1
2
2
⋯
N
2
1
2
2
⋯
N
2
]
)
.
Based on (10)-(11),
∇
ω
2
J
(
ω
,
x
)
can be expressed as
(17)
∇
ω
2
J
(
ω
,
x
)
=
x
T
(
c
E
-
G
2
T
G
)
x
+
x
T
G
2
T
(
v
-
G
x
)
≥
x
T
G
T
N
2
v
-
2
x
T
G
T
N
2
G
x
≥
x
T
G
T
N
2
v
-
2
N
2
x
T
(
G
T
G
)
x
.
Equality holds if and only if
x
1
2
+
x
2
2
+
2
x
2
x
3
=
0
,
ω
=
m
π
with
m
=
-
1
,
-
1
/
2
,
0
,
1
/
2
,
1
and
(18)
∇
ω
3
J
(
ω
,
x
)
=
x
T
G
1
T
N
2
v
=
x
T
F
T
G
T
N
3
v
,
where
N
3
=
diag
(
[
1
2
3
⋯
N
3
1
2
3
⋯
N
3
]
)
. Substituting (12) into (17)-(18) yields
(19)
∇
ω
2
J
(
ω
,
x
)
≥
v
T
(
G
(
G
T
G
)
-
1
G
T
)
N
2
v
-
2
N
2
v
T
(
G
(
G
T
G
)
-
1
G
T
)
v
≥
v
T
W
1
v
,
∇
ω
3
J
(
ω
,
x
)
=
v
T
W
2
v
,
where
(20)
W
1
=
G
(
G
T
G
)
-
1
G
T
(
N
2
-
2
N
2
)
,
W
2
=
G
(
G
T
G
)
-
1
F
T
G
T
N
3
.
We can then write
M
as
(21)
M
=
sup
ω
∈
{
-
π
,
π
}
{
|
∇
ω
3
J
(
ω
,
x
)
v
T
v
|
|
1
(
∇
ω
2
J
(
ω
,
x
)
)
/
(
v
T
v
)
|
}
=
|
λ
2
λ
1
|
,
where
λ
1
and
λ
2
are the minimum eigenvalue of
W
1
and the maximum eigenvalue of
W
2
, respectively. It is easily shown that
λ
1
=
N
3
and
λ
2
=
N
2
. Hence
M
=
N
. As a result, if the initial estimate is chosen such that
N
|
ω
-
ω
^
(
0
)
|
<
1
is satisfied, global solution will be obtained.