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Integration of fixed-speed wind Chapter | 14 369

sources ðV Sah ; V Sbh ; andV Sch Þ, the hth harmonic Thevenin impedance of the
utility side, a power transformer, an unbalanced and nonlinear load group,
which consists of a single-phase linear load and a six-pulse uncontrolled recti-
fier, a group of identical FSWECSs, a delta-connected single-tuned harmonic
filter (STF) with nonidentical branches, and a delta-connected Steinmetz com-
pensator (SC). It should be mentioned that the Thevenin impedance of the util-
ity side is represented as its short-circuit impedance.
In the following section, for the considered unbalanced and nonsinusoidal
system, dynamic modeling of FSWECS is presented; further, the proposed
algorithm is to find the PPL of the FSWECSs under the same system condi-
tions as introduced and discussed.

14.2.1 Modeling of the fixed-speed wind energy conversion
systems
The dynamic modeling of the FSWECS is provided for analyzing its permis-
sible PL under unbalanced and nonsinusoidal conditions. As shown in
Fig. 14.1, it consists of blades, gearbox, and SCIG. The mechanical power
transmitted from the gearbox to the shaft of the SCIG is written in terms of
the air density (ρ), the area swept by the rotor (A), power coefficient (C p ),
and wind speed (u) [21]:
1
3
P M 5 ρAu C p λðÞ ð14:1Þ
2
The power coefficient (C p ) depends on the tip speed ratio (λ), which is
determined as follows:

ω m R
λ 5 ð14:2Þ
u
where ω m and R are the rotor angular velocity and rotor radius, respectively.
In the analysis, SCIG is modeled using the well-known d q equivalent
circuits [14,41] shown in Fig. 14.2. Here, it should be noted that in the litera-
ture, there are several studies [42,43], which show that the results of d q
model and experiments are in close agreement for the performance analysis
of the induction machine under unbalanced supply voltages. It can also be
mentioned that the same model was considered for the transient analysis of
the induction motors and doubly fed IGs under distorted supply voltages in
many studies [44 46].
For the model, Eqs. (14.3) (14.6), the instantaneous stator voltages (V qs and
V ds ) and instantaneous rotor voltages referred to the stator side (V 0 qr andV )
0
dr
0
0
can be expressed in terms of the magnetic fluxes (ϕ qs , ϕ ds , ϕ qr ,andϕ dr ), stator
0
0
currents (i qs and i ds ), rotor currents referred to the stator side (i and i ), stator
qr
dr
resistance (R S ), rotor resistance referred to the stator side (R ), the reference
0
r
frame angular velocity (ω), and the electrical angular velocity (ω r ), thus