Environmental Engineering Reference
In-Depth Information
1
e
E
×
μ
×
1
s
−
sin
cos
v
˙
θ
n
1
˙
1
θ
θ
×
μ
2
s
s
μ
3
Figure 22.6
Sinusoidal tracking algorithm (STA) or the enhanced PLL (EPLL)
−
∂
[
J
(
ψ
(
t
)
,
t
)]
∂ψ
where
(
t
)
. The
resulting set of differential equations are then (Ziarani and Konrad 2004; Karimi-Ghartemani
and Ziarani 2003)
μ
is a diagonal matrix chosen to minimise
J
along the direction of
⎧
⎨
d
E
(
t
)
d
t
=
2
μ
1
d
sin
θ,
d
ω
(
t
)
d
t
=
2
μ
2
Ed
cos
θ,
(22.7)
⎩
d
θ
(
t
)
d
t
=
ω
+
μ
3
d
d
t
.
This is the basis for implementing the STA as shown in Figure 22.6. Since the variation of
E
is relatively small compared to the variation of
d
, the major dynamics of
ω
is from
d
and the
effect of
E
can then be combined with the proper selection of
μ
2
. In order to speed up the
response, the nominal frequency
˙
θ
n
is added to the estimated frequency
˙
θ
.
The behaviour of the frequency loop is determined by the two gains
μ
2
and
μ
3
, which forms
1
a PI controller
μ
2
directly affects the bandwidth of the loop. The larger the
μ
2
, the higher the bandwidth and therefore the faster the response. However, if a large phase
jump occurs, a large gain
μ
2
(
μ
3
+
s
), and
μ
2
can cause oscillations and it takes long time for the loop to reach
the steady state. Too large a
μ
2
could even cause instability if the phase jump is too large.
22.7 Simulation Results with SOGI-PLL and STA
22.7.1 With a Noisy Distorted Signal having a Variable Frequency
The input signal
20
√
2 and contains a noise-added harmonic
v
has the amplitude
V
m
=
component
2
√
2 sin(3
2
√
2 sin(5
v
h
(
t
)
=
ω
g
t
+
1
.
5)
+
ω
g
t
+
2
.
5)
+
n
(
t
)
,
(22.8)
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