Environmental Engineering Reference
In-Depth Information
V
d
V
q
=
0. Moreover, the estimated amplitude
E
of the input
v
is realised as
E
=
+
V
q
.
As a result, the frequency, the phase and the amplitude of the signal
v
are all available.
22.6 Sinusoidal Tracking Algorithm (STA)
This method was introduced with several different names, e.g. the sinusoidal tracking algorithm
(STA) (Ziarani and Konrad 2004), the amplitude phase model (APM) and amplitude phase
frequency model (APFM) (Karimi-Ghartemani and Ziarani 2003), the enhanced PLL (EPLL)
(Karimi-Ghartemani and Iravani 2001, 2002). In this topic, it is referred to as the STA in order
to avoid confusion. The STA is able to extract the fundamental component of a sinusoidal
signal and, at the same time, to estimate its amplitude, phase and frequency.
A typical voltage
v
(
t
) has the general form of
∞
v
(
t
)
=
θ
gi
+
n
(
t
)
V
mi
sin
i
=
0
where
V
mi
and
are the amplitude and phase of the
i
-th harmonic component of
the voltage, and
n
(
t
) represents the noise on the signal. The objective of a PLL can be regarded
as extracting the component
e
(
t
) of interest, which is usually the fundamental component,
from the input signal
θ
gi
=
ω
gi
t
+
δ
v
(
t
). Assume that the estimated signal is
E
(
t
) sin(
t
e
(
t
)
=
0
ω
(
τ
) d
τ
+
δ
(
t
))
,
˙
where
E
(
t
) is the estimated amplitude,
ω
(
t
)
=
θ
(
t
) is the estimated frequency and
θ
(
t
)
=
t
0
ω
(
τ
) d
τ
+
δ
(
t
) is the estimated phase of
e
(
t
). Define the state vector of the PLL to be
=
E
(
t
)
(
t
)
T
. Then the problem of designing a PLL can be formulated as finding
ψ
(
t
)
ω
(
t
)
δ
the optimal vector
ψ
(
t
) that minimises the cost function
d
2
(
t
)
e
(
t
)]
2
J
(
ψ
(
t
)
,
t
)
=
=
[
v
(
t
)
−
,
where
d
(
t
)
=
v
−
e
(
t
) is the tracking error. There are several methods to solve this problem.
The method applied in (Karimi-Ghartemani and Ziarani 2003; Ziarani and Konrad 2004) is
the gradient descent method (Giordano and Hsu 1985), which is outlined below. In order to
solve the optimisation problem, formulate
(
t
)
d
ψ
(
t
)
d
t
=−
μ
∂
[
J
(
ψ
(
t
)
,
t
)]
∂ψ
(
t
)
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