Environmental Engineering Reference
In-Depth Information
The admittance operators are expressed using reactances and time constants. The
conventional form which contains a number of approximations is used here:
τ
qo
1 +
p
τ
do
1 +
p
τ
do
1 +
p
1 +
p
1
X
d
(
p
)
=
1
X
d
·
1 +
p
1 +
p
1
X
q
(
p
)
=
1
X
q
·
τ
d
·
;
(6.47)
τ
d
τ
q
ωτ
md
1 +
p
1 +
p
τ
dD
σ
1 +
p
G
f
(
p
) =
τ
do
·
do
τ
The coefficients in terms of the parameters of the equivalent circuit model,
Fig. 6.6b, are the transient and subtransient reactances
X
mq
X
Q
X
md
X
f
X
md
(
X
D
+
X
f
−
2
X
md
)
X
d
=
X
d
−
X
q
=
X
d
−
X
d
=
X
d
−
;
and
X
md
X
D
X
f
−
and the time constants
X
md
/
X
f
ω
X
f
ω
τ
do
=
X
D
−
X
Q
ω
τ
do
=
τ
qo
=
;
;
R
f
R
D
R
Q
τ
q
=
X
q
τ
d
=
X
d
τ
d
=
X
d
X
d
τ
do
X
d
τ
do
X
q
τ
qo
;
;
X
md
ω
X
σ
D
ω
τ
md
=
;
τ
dD
σ
=
R
f
R
D
Additionally, the armature short-circuit time constant is defined:
1
2
1
/
X
d
+ 1
/
X
q
τ
a
=
ω
R
s
The reactance operators are used for investigations in the Laplace domain or in
the frequency domain, e.g. for problems of forced oscillations,
p
= j
ν
, or asyn-
chronous operation,
p
= j
s
ω
.
6.2.4 Converter Modeling
Most of the converters used in wind energy systems are voltage source ac-ac in-
verters (VSI) with intermediate dc circuit, see 4.3.4. While two-level inverters are
standard in low-voltage systems, large WES are increasingly designed for medium
voltage, e.g. 3kV, where three-level inverters are used in order to limit currents and
reduce losses; their modelling is considered in [Ale06]. Besides, dc-dc converters
are used, either as step-up or step-down converters, see 4.3.5.
Inverter performance as described in literature is mostly either averaging input-
output behaviour or modelling the switching functions in the time domaine. Steady
state of an ac.-dc current-source inverter (CSI) is conventionally modelled assuming
sinusoidal ac-voltage and constant dc-side current, taking account of harmonics of
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