Environmental Engineering Reference
In-Depth Information
15.1.2 Equivalent Structure in the Natural (abc) Frame
PI controllers are normally associated with the
dq
frame but it is possible to convert the
controller in the
dq
frame to the
abc
frame. According to (Zmood
et al
. 1999, 2001), the
PI controller implemented in the
dq
frame is equivalent to the proportional-resonant (PR)
controller implemented in the
frame at the fundamental frequency. Note that this is dif-
ferent from the harmonic compensator discussed in Chapter 16, which is tuned at harmonic
frequencies. Using the transformation techniques presented in (Zmood
et al
. 1999, 2001), the
equivalent PI controller in the
abc
frame is derived in (Twining and Holmes 2003) as
αβ
⎡
⎣
+
√
3
K
i
ω
2(
s
2
−
√
3
K
i
ω
2(
s
2
⎤
⎦
s
K
p
2
−
K
i
s
K
p
2
−
K
i
s
K
p
+
K
i
−
−
s
2
+
ω
2
+
ω
2
)
+
ω
2
)
√
3
K
i
ω
√
3
K
i
ω
K
p
2
−
K
i
s
−
s
K
p
2
−
K
i
s
+
=
.
C
PI
abc
(
s
)
−
K
p
+
K
i
−
2(
s
2
+
ω
2
)
s
2
+
ω
2
2(
s
2
+
ω
2
)
√
3
K
i
ω
√
3
K
i
ω
+
−
K
p
2
−
K
i
s
K
p
2
−
K
i
s
s
−
−
K
p
+
K
i
2(
s
2
+
ω
2
)
2(
s
2
+
ω
2
)
s
2
+
ω
2
(15.2)
It consists of cross-coupling terms, in addition to the PR controllers on the diagonal. The
equivalent control structure in the
abc
frame is shown in Figure 15.2. The cross-coupling
terms indicate that the three-phases are not independent. This could be a problem when the
system is not balanced.
The phase-lead low-pass filter (3.1) in Chapter 3 is again adopted to feed-forward the grid
voltage, which is added to the output of the current controller. This forces the output voltage
of the inverter to be the same as the grid voltage when the current references are set to be zero,
which facilitates the grid connection of the inverter. It also improves the dynamic performance
of the inverter when there are fluctuations in the grid voltage.
Figure 15.2
Equivalent control strategy in the natural frame (
abc
) for the one shown in Figure 15.1
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