arXiv:2110.10312v2 [physics.flu-dyn] 3 Dec 2021
Phase-Averaged Dynamics of a Periodically Surging Wind Tur
bine
Phase-Averaged Dynamics of a Periodically Surging Wind Tur
bine
Nathaniel J. Wei
1
and John O. Dabiri
1,2
1)
Graduate Aerospace Laboratories, California Institute of
Technology
2)
Department of Mechanical and Civil Engineering, Californi
a Institute of Technology
(*Electronic mail:
jodabiri@caltech.edu
)
(Dated: 6 December 2021)
The unsteady power generation of a wind turbine translating
in the streamwise direction is relevant to floating offshore
wind turbines, kite-mounted airborne wind turbines, and ot
her non-traditional wind-energy systems. To study this
problem experimentally, measurements of torque, rotor spe
ed, and power were acquired for a horizontal-axis wind
turbine actuated in periodic surge motions in a fan-array wi
nd tunnel at the Caltech Center for Autonomous Systems
and Technologies (CAST). Experiments were conducted at a di
ameter-based Reynolds number of
'4
=
6
.
1
×
10
5
and at tip-speed ratios between 5.2 and 8.8. Sinusoidal and t
rapezoidal surge-velocity waveforms with maximum surge
velocities up to 23% of the free-stream velocity were tested
. A model in the form of a linear ordinary differential
equation (first-order in time) was derived to capture the tim
e-resolved dynamics of the surging turbine. Its coefficients
were obtained using torque measurements from a stationary t
urbine, without the need for unsteady calibrations. Its
predictions compared favorably with the measured amplitud
e- and phase-response data. Furthermore, increases in the
period-averaged power of up to 6.4% above the steady referen
ce case were observed in the experiments at high tip-speed
ratios and surge velocities, potentially due to unsteady or
nonlinear aerodynamic effects. Conversely, decreases in me
an
power with increased surge velocity at low tip-speed ratios
were likely a result of the onset of stall on the turbine blade
s.
These results inform the development of strategies to optim
ize and control the unsteady power generation of periodical
ly
surging wind turbines, and motivate further investigation
s into the unsteady aerodynamics of wind-energy systems.
I. INTRODUCTION
New innovations in wind energy technology motivate the
study of wind-turbine performance in previously unexplore
d
operational regimes. In particular,while traditional lan
d-based
wind turbines are fixed in place, wind-energy systems such as
floating offshore wind turbines (FOWTs) and airborne wind
turbines undergo streamwise oscillations that may potenti
ally
complicate the aerodynamics of these systems. The periodic
motions of the turbine rotor in these situations introduce a
d-
ditional dynamics that can affect the power generation of the
wind turbines and the fatigue loading on their blades, there
by
impacting their contributionto global energy demands. The
re-
fore, this study investigates the dynamics ofa periodicall
y surg-
ing wind turbine through analytical modeling and laborator
y-
scale experiments.
A. Current Progress in Surging-Turbine Aerodynamics
Previous studies have generally considered surge motions
typical of wave-driven FOWTs,
1
which are of increasing rel-
evance as the emerging offshore-wind sector continues to ex-
pand. The majority of attention regarding turbine aerodyna
m-
ics has been focused on time-averaged quantities. Using a
model FOWT in a wind tunnel and wave tank,
Farrugia
et al.
2
found that the time-averaged coefficient of power,
?
, in-
creased above the steady case by 1% when oscillations in
the turbine were present. A similar increase in
?
by 1% was
observed in wind-tunnel experiments by
Khosravi
et al.
3
and
free-vortex wake simulations by
Shen
et al.
4
.
Farrugia
et al.
5
showed using free-vortex wake simulations that
?
increased
with surge frequency at tip-speed ratios above the rated val
ue
by up to 13.7%, but decreased with surge frequency at tip-
speed ratios below the rated value. Independent simulation
s
by
Wen
et al.
6
yielded similar results.
Johlas
et al.
7
suggested
that the increases in average power with surge velocity can
be described by a simple quasi-steady model, where the term
“quasi-steady” refers to effects for which successive insta
nces
in time can be considered as being in independent states of
local equilibrium. Since the model is derived from the cubic
dependence of power on the incident inflow velocity at the ro-
tor, the relative power gains over the steady value of
?
from
surge motions in the upwind direction outweigh the relative
losses from surge motions in the downwind direction. Their
model agrees well with the time-averaged power results from
their numerical simulations of a surging turbine, but the va
lid-
ity of the model has not yet been evaluated over a wide range
of surge velocities and operating conditions. A fully chara
c-
terized, quantitatively accurate explanation for the obse
rved
increases in
?
thus remains elusive.
The time-resolved dynamics of a turbine in surge have been
explored as well. These unsteady dynamics determine the un-
steady loads on the turbine and its supportstructures,and t
here-
fore inform the design of FOWT control systems. The presence
of fluctuations in turbine thrust, torque, and power at the sa
me
frequency as the imposed surge motion is well-documented
in the literature.
2
,
4
–
8
These fluctuations increase in amplitude
as the surge frequency is increased.
5
Mancini
et al.
9
, how-
ever, showed in wind-tunnel experiments with a surging tur-
bine that the relationship between torque amplitude and sur
ge
frequency increases above the prediction of their linear qu
asi-
steady model at high frequencies. They attributed this to
mechanical resonance, and not to a breakdown of their lin-
earization of power as a function of the surge velocity or the
influence of unsteady aerodynamics. By contrast, torque am-
plitudes measured in wind-tunnel experiments by
Sant
et al.
10
Phase-Averaged Dynamics of a Periodically Surging Wind Tur
bine
2
were much lower than those computed by quasi-steady and
dynamic-inflow codes.
The phase response of the turbine similarly lacks a single
consistent characterization in the literature. The model d
e-
veloped by
Johlas
et al.
7
predicts that the instantaneous power
from the turbine will be in phase with the surge velocity. The
linear quasi-steady model of
Mancini
et al.
9
supports the same
prediction. Some computational
5
,
6
,
8
,
11
and experimental
9
re-
sults, however, have shown phase differences in excess of 90
◦
,
while others displayed close to zero phase offset.
4
,
12
,
13
The dis-
crepancies in the literature regarding the amplitude and ph
ase
of the torque and power output of surging turbines motivates
the current study.
The lack of consensus with respect to mean quantities and
their amplitude and phase stems in large part from unan-
swered questions regarding the relative importance of quas
i-
steady and unsteady effects. The models of
Johlas
et al.
7
and
Mancini
et al.
9
can be classified as purely quasi-steady models
that neglect unsteady effects. Other models have incorporat
ed
unsteady effects directly. For example,
de Vaal
et al.
14
com-
pared the results of different dynamic inflow models, which
include corrections for time-varying inflow velocity and ac
-
celeration, that were paired with blade-element momentum
(BEM) simulations of a surging turbine. They concluded that
these engineering models were capable of capturing global
forces on FOWTs in typical offshore conditions. In a differ-
ent approach,
Fontanella
et al.
15
derived a state-space model
that maps linearized turbine aerodynamics and wave dynam-
ics to the time derivatives of the kinematic parameters of th
e
turbine. The model was shown to perform well both in sim-
ulations and as the basis for control systems.
16
In addition
to these models, others have suggested various unsteady flow
phenomena that could influence the turbine dynamics. For
instance, several of the aforementioned studies have consi
d-
ered the effects of airfoil stall, particularly at the blade r
oot,
on time-averaged and fluctuating quantities.
5
,
6
,
8
,
11
In addition
to blade stall,
Sebastian and Lackner
17
postulated the forma-
tion of unsteady recirculation regions in or downstream of
the rotor plane during turbine surge, as a result of slip-str
eam
violations. Furthermore,
Wen
et al.
11
attributed the phase dif-
ferences observed in their simulations to added-mass effect
s,
blade-wake interactions, and unsteady aerodynamics. Thes
e
unsteady flow phenomena may affect the structure, dynamics,
and recovery of the wake of a surging turbine.
8
,
18
–
22
However,
it still remains to be seen which (if any) of these unsteady ef
-
fects must be accountedfor in a modelto capture the torquean
d
power production of real surging turbines, or whether exist
ing
quasi-steady models are sufficient for this purpose.
Lastly, since nearly all existing work on surging-turbine
aerodynamics has been conducted in the context of surge os-
cillations typical of FOWTs under normal operating condi-
tions, the dynamics of wind turbines surging through larger
amplitudesor higherfrequenciesremain relatively unexpl
ored.
Larger surge oscillations would be relevant not only to FOWT
s
in more extreme conditions, but also to airborne wind turbin
es
mounted to aircraft or crosswind kites.
23
Crosswind kites gen-
erally fly through large periodic orbits with length scales m
uch
greater than the size of the aircraft itself.
24
Turbines mounted
to these kites would therefore undergo surge motions at am-
plitudes far larger than those experienced by FOWTs. In addi
-
tion,
Dabiri
25
recently suggested that streamwise unsteadiness
could be leveraged to increase the efficiency of wind-energy
systems above the theoretical steady limit. Since increase
s in
time-averaged power have already been observed at the rel-
atively low levels of unsteady motion typical of FOWTs, an
investigation of higher surge amplitudes and frequencies c
ould
provide insights toward the practical realization of these
theo-
retical efficiency gains.
B. Research Objectives
This study aims to address several open questions regard-
ing the time-resolved dynamics of a wind turbine in surge.
The amplitude and phase of torque and power relative to the
surge motions are investigated in wind-tunnel experiments
.
Trends in the data are parameterized by a model that accounts
for quasi-steady aerodynamic torques and unsteady generat
or
torques. The model is first-order in time and linear in the tur
-
bine surge velocity and rotor speed; thus, it shall hencefor
th
be referred to as a first-order linear model. An important fea
-
ture of the model is that its coefficients can be computed from
measurements obtained under steady conditions; no data fro
m
actual surge tests are required to obtain time-resolved tor
que
and power predictions. The experiments span higher levels o
f
unsteadiness than previous studies in the literature, with
scaled
amplitudes up to
∗
=
/
=
0
.
51 and nondimensional surge
velocities up to
D
∗
=
5
/
D
1
=
0
.
23, where
is the surge
amplitude,
5
is the surge frequency in radians per second,
D
1
is the free-stream velocity, and
is the turbine diameter.
By contrast, the highest values reported in the literature a
re
∗
=
0
.
13 (
Tran and Kim
8
) and
D
∗
=
0
.
42 (
Wen
et al.
11
) in
simulations, and
∗
=
0
.
15 (
Sant
et al.
10
) and
D
∗
=
0
.
079
(
Mancini
et al.
9
) in experiments. The findings in this study
may thus be generalized to FOWTs operating under extreme
conditions, as well as novel airborne wind-energy systems a
nd
other emergent technologies. The combined analytical and e
x-
perimental results presented in this work provide a foundat
ion
upon which questions regarding the influence of unsteadines
s
and nonlinearity, including the dependence of the mean torq
ue
and power on surge kinematics, may be more comprehensively
investigated in future work.
The paper is structured as follows. In Section
II
, a first-order
linear model is derived that enables a disambiguation betwe
en
aerodynamic and generator torques. Its amplitude and phase
characteristics are also analyzed. In Section
III
, the experimen-
tal apparatus is described, and methods for computing the co
-
efficients of the analytical model from measurements in stead
y
conditions are given. Phase-averaged results from experim
ents
with sinusoidal and trapezoidal surge-velocity waveforms
are
presented in Section
IV
, and the results are compared with
model predictions. Finally, a discussion regarding model c
a-
pabilities and limitations, nonlinear and unsteady effects
, and
application to full-scale wind turbines is provided in Sect
ion
V
.
Phase-Averaged Dynamics of a Periodically Surging Wind Tur
bine
3
II. ANALYTICAL MODEL
In this section, we derive a model for the torque generated by
a surging horizontal-axis turbine from an ordinary differen
tial
equation that is first-order in time and linear in the turbine
surge velocity and rotor speed (or rotation rate). We linear
ize
the aerodynamic torque with respect to the inflow velocity an
d
rotor speed, and combine it with a model for the generator
torque to obtain a differential equation for the rotor speed o
f
the turbine. We then derive transfer functions in the freque
ncy
domain to characterize the amplitude and phase relative to t
he
surge-velocity waveform of the aerodynamic and generator
torque. A notable advantage of this model is that the model
coefficients can be extracted directly from torque and rotati
on-
rate measurements of the turbine in steady conditions (i.e.
without surge motions); these methods will be described in
Section
III D
for the turbine used in these experiments.
A. Aerodynamic Torque Model
A first-order linear model for the aerodynamic torque can
be derived using a local linearization with respect to the in
flow
velocity and rotor speed:
g
04A>
≈
g
0
+
mg
mD
D
=
D
1
,l
=
l
(
D
−
D
1
)+
mg
ml
D
=
D
1
,l
=
l
(
l
−
l
)
,
(1)
where
D
=
D
1
+
*
(
C
)
is the instantaneous inflow velocity rel-
ative to the turbine,
*
(
C
)
is the turbine surge velocity in a
stationary frame of reference,
l
is the rotor speed, and
g
0
is
the steady aerodynamic torque, i.e. the mean torque measure
d
on a stationary turbine at a wind speed of
D
1
. In this work,
bars denote time averages over a single streamwise-motion
oscillation period for time-dependent variables in the cas
e of
unsteady streamwise motion, while the subscript 0 denotes t
he
value of a variable in the reference case corresponding to a
steady inflow at speed
D
1
. We then define the performance
coefficients
ℓ
=
mg
mD
D
=
D
1
,l
=
l
(2)
and
3
=
−
1
'
mg
ml
D
=
D
1
,l
=
l
,
(3)
where
'
is the radius of the turbine. These coefficients qual-
itatively correspond to lift and drag terms in a blade-eleme
nt
expression for aerodynamic torque. Values for these consta
nts
can be obtained empirically from measurements of the turbin
e
torque taken with a stationary turbine over a range of wind
speeds and loading conditions (cf. Section
III D
). Simplifying
the above expression yields the following model:
g
04A>
≈
ℓ
*
−
3
'
(
l
−
l
) +
g
0
.
(4)
The accuracy of this aerodynamic model depends on
whether
g
is sufficiently linear in
D
and
l
in the neighbor-
hood of the steady operating condition (
D
=
D
1
and
l
=
l
).
Since the model is inherently quasi-steady, its accuracy wi
ll
also depend on whether any unsteady effects such as dynamic
stall on the turbine blades are present.
B. Generator Torque Model
The torque applied by the generator (
g
64=
) in opposition
to the aerodynamic torque (
g
04A>
) represents the torque that
is converted to usable power at each instant in time. It thus
also represents the mechanically measurable torque on the
turbine shaft (
g
<40B
). It is important to note that the generator
torque is not necessarily equal to the aerodynamic torque in
the case of unsteady rotation, with any difference between th
e
two inducing a change in the angular velocity of the rotor.
The equations of motion for a permanent-magnet generator
are identical in principle to those for a permanent-magnet
motor, which is frequently modeled as a first-order ordinary
differential equation in time:
26
g
64=
=
g
<40B
=
2
3l
3C
+
1
l
+
0
,
(5)
where
2
is the moment of inertia of the generator about
its rotational axis,
1
is the generator constant, and
0
is
an empirical zero-speed offset. Since the generator torque
is proportional to the current through the generator coils,
1
and
0
scale inversely with the resistive load applied to the
generator.
26
Hence, a higher resistive load applied to the gen-
erator corresponds to a lower physical load on the turbine.
This formulation assumes that the generator is directly dri
ven
by the turbine; a gear-ratio scaling could be incorporated t
o
map the rotor speed to the generator rotation rate for turbin
es
with gearboxes.
It should also be noted that the generatormodel in its curren
t
form does not include any effects of control, such as tip-spee
d
ratio control systems that are typically present in utility
-scale
wind turbines. For the purposes of this study, the use of
a direct-drive generator with fixed resistive loading and no
speed control simplify the modeling of the turbine dynamics
and subsequent model validation against experimental data
.
However, the linear form of the model means that linear or lin
-
earized tip-speed ratio controllers can readily be incorpo
rated
using classical analytical techniques.
C. Governing Equation and its Transfer Functions
The dynamics of a turbine under the influence of competing
aerodynamic and generator torques are given by the swing
equation,
27
3l
3C
=
g
04A>
−
g
64=
,
(6)
Phase-Averaged Dynamics of a Periodically Surging Wind Tur
bine
4
where
is the moment of inertia of the turbine, its shaft assem-
bly, and the generator about the axis of rotation, and is thus
in
practice much larger than
2
. Deviations of the instantaneous
aerodynamic torque away from equilibrium, if not immedi-
ately matched by the generator torque, will lead to a change
in the rotor speed until the generator torque overcomes iner
tia
and restores the torque balance. This implies that the torqu
e
measured by a torque transducer, and consequently the power
measured from the generator, will lag behind the aerodynami
c
torque.
Substituting Equations
4
and
5
into the above relation, we
arrive at the equation of motion
3l
3C
=
(
ℓ
*
−
3
'
(
l
−
l
) +
g
0
) −
(
2
3l
3C
+
1
l
+
0
)
.
(7)
In the limit of equilibrium, in which all time-derivatives a
re
zero, the steady aerodynamic torque
g
0
must equal the gener-
ator torque, i.e.
g
0
=
1
l
+
0
. We can therefore simplify the
model into a more informative form:
3l
3C
=
ℓ
*
−
3
'
(
l
−
l
) −
2
3l
3C
−
1
(
l
−
l
)
.
(8)
The resulting model is conceptually similar to that of
Fontanella
et al.
15
, though in this work the linearization for
the aerodynamic torque is obtained differently and the gener
-
ator torque is an output, rather than an input, to the system.
An additional benefit of the model in Equation
8
is that it
requires no data from unsteady surge experiments to make
time-resolved predictions, since all of its coefficients can
be
computed either from measurements in steady flow or from
geometric properties of the turbine and generator. The mode
l
has the form of a linear time-invariant (LTI) system, which
allows transfer functions of the aerodynamic and measured
torques to be computed in order to quantify the phase and am-
plitude behavior of the system. Taking the Laplace transfor
m
of Equation
8
with respect to an arbitrary surge velocity
*
(in-
put) and the resulting rotor speed
l
(output) yields the transfer
function
l
(
B
)
*
(
B
)
=
ℓ
(
+
2
)
B
+
1
+
3
'
.
(9)
This transfer function has the form of a first-order low-
pass filter with critical frequency
5
2
=
1
+
푑
'
+
2
. Using this
transfer function, we can also derive transfer functions fo
r the
aerodynamic and generator torques:
g
04A>
(
B
)
*
(
B
)
=
ℓ
−
3
'
l
(
B
)
*
(
B
)
=
ℓ
(
+
2
)
B
+
1
(
+
2
)
B
+
1
+
3
'
(10)
and
g
64=
(
B
)
*
(
B
)
=
(
2
B
+
1
)
l
(
B
)
*
(
B
)
=
ℓ
2
B
+
1
(
+
2
)
B
+
1
+
3
'
,
(11)
which share the same critical frequency
5
2
. The frequency
response can be computed from these transfer functions usin
g
the imaginary part of the Laplace variable
B
, i.e. Im
(
B
)
=
5
.
Phase and amplitude predictions from the model can thus be
obtained analytically, and the mean torque is given by the
steady-state value
g
0
. Power can then be computed as
P
=
gl
. The linear form of the model dictates that, for periodic
surge motions with zero net displacement, the period-avera
ged
mean torque and power are not functions of the surge motions.
According to this model, then, unsteady surge motions will
not affect the period-averaged power generation of the turbi
ne.
The model thus forgoes the ability to predict time-averaged
quantities in favor of an analytical formulation of the time
-
resolved turbine dynamics. The consequences of this tradeo
ff
will be discussed in Section
V B
.
The transfer functions suggest that the relevant nondimen-
sional parameters for the surge dynamics are the nondimen-
sional surge velocity,
D
∗
=
5
/
D
1
, and the normalized surge
frequency,
5
∗
=
5
/
5
2
. The analysis suggests that the ampli-
tude of the unsteady torque oscillations scales directly wi
th
D
∗
, with a frequency dependence dictated by
5
∗
. Either the
reduced frequency
:
=
5
/
D
1
or the nondimensional surge
amplitude
∗
=
/
would complete the nondimensional pa-
rameterization by including the length scale of the turbine
, but
in contrast to suggestions in the literature,
6
,
19
these parame-
ters do not appear to follow directly from the transfer-func
tion
formulation of the model.
III. EXPERIMENTAL METHODS
In this section, the experimental apparatus used to study
the torque and power production of a wind turbine in periodic
surge motions is described. First, the wind tunnel and turbi
ne
apparatus are described. Then, the parameter space explore
d
in these experiments is presented, and the experimental pro
ce-
dure is outlined. Finally, methods for empirically determi
ning
values for the scaling coefficients of the analytical model de
-
rived in the previous section and an overview of sources of
uncertainty are provided.
A. Experimental Apparatus
Experiments were conducted in a large open-circuit fan-
array wind tunnel at the Caltech Center for Autonomous Sys-
tems and Technologies (CAST). The fan array was composed
of 2,592 computer fans arranged in two counter-rotating lay
-
ers within a 2
.
88
×
2
.
88 m frame, mounted 0.61 m above
the floor of the facility (cf. Figure
1
). The open-air test sec-
tion downstream of the fan array vented directly to the atmo-
sphere, while the other three sides and ceiling of the arena
were enclosed with walls or awnings to mitigate the effects
of atmospheric disturbances. The experiments in this study
were carried out at a free-stream velocity of
D
1
=
8
.
06
±
0
.
16
ms
−
1
, corresponding to a diameter-based Reynolds number
of approximately
'4
=
6
.
13
×
10
5
. The relevance of this
study to Reynolds numbers typical of utility-scale wind tur
-
Phase-Averaged Dynamics of a Periodically Surging Wind Tur
bine
5
bines is discussed in Section
V C
. The turbulence intensity in
the tunnel, represented by the standard deviation of the vel
oc-
ity fluctuations normalized by the average streamwise veloc
ity,
was measured to be 2
.
20
±
0
.
17%. These measurements were
obtained with an ultrasonic anemometer (Campbell Scientifi
c
CSAT3B) placed at the hub height and streamwise zero posi-
tion of the wind turbine. Because the facility was exposed to
the atmosphere and experiments were conducted over a range
of atmospheric conditions and times of day, temperature and
relative-humidity readings were recorded with measuremen
t
precisions of
±
1
◦
C and
±
5% from a portable weather station
(Taylor Precision Products model 1731) so that the air densi
ty
could be calculated accordingly.
The turbine apparatus was constructed on an aluminum
frame (80/20 1515 T-slotted profile) that was bolted to the
floor and secured with sandbags. The frame was 2.00 m long,
0.69 m wide, and 0.87 m tall. Two 2-m long rails with two
ball-bearing carriages each (NSK NH252000AN2PCZ) were
mounted on top of the frame, parallel to the streamwise di-
rection and spaced 0.65 m apart in the cross-stream directio
n.
A traverse was mounted on the ball-bearing carriages, which
supported a 0.99-m tall, 0.038-m wide turbine tower. The
wind-turbine shaft assembly was placed on top of this tower a
t
a hub height of 1.97 m above the floor. The origin of the surge
motions of the turbine was located 3.09 m downstream of the
fan array, and the rails afforded a maximum surge stroke of
1.52 m upstream of the origin. A schematic of the apparatus
and its position relative to the wind tunnel is given in Figur
e
1
.
A three-bladed horizontal-axis wind turbine (Primus Wind
Power AIR Silent X) with a rotor diameter of
=
1
.
17 m and
hub diameter of 0.127 m was attached to a 25.4-mm diameter
steel shaft supported by two axially mounted shaft bearings
(Sealmaster NP-16T). The blade chord ranged from 100 mm at
the root to 32 mm at the tip. The blades were constructed from
a laminated carbon-fiber composite. A rotary encoder (US
Digital EM2-2-4096-I) with a resolution of 4096 counts per
revolution was attached between the shaft bearings. A rotar
y
torque transducer (FUTEK TRS300, 20 Nm torque rating) was
connected to the turbine shaft on one end and to the turbine
generator (Primus Wind Power AIR 30, 48V) by means of a
floating 19.0-mm diameter steel shaft on the other. The shaft
assembly was surrounded by a 4.76-mm thick acrylic nacelle
(0
.
610
×
0
.
152
×
0
.
152 m) with a slanted rear section intended
to reduce bluff-body separation effects in the turbine wake.
The estimated blockage of the swept area of the turbine and al
l
support structures, relative to the fan-array surface area
, was
14%.
The load on the turbine was controlled electrically with re-
sistors. The three-phase alternating current produced by t
he
generator was converted to DC by a bridge rectifier (Comchip
Technology SC50VB80-G) and passed to a bank of resistors.
Different combinations of fixed 10-Ohm resistors (TE Con-
nectivity TE1000B10RJ) in series or parallel and an 8-Ohm
rheostat (Ohmite RRS8R0E) were used to achieve a range of
loading conditions that spanned the operational envelope o
f
the turbine. An emergency short-circuit switch built from a
solid-state relay (Crydom D1D12) and a 12-A fuse (Schurter
4435.0368) were included in the circuit for safety purposes
.
The turbine traverse was driven in the streamwise direc-
tion by a piston-type magnetic linear actuator (LinMot PS10
-
70x320U-BL-QJ). Its sliding cylinder was attached to the tr
a-
verse at rail height and its stator was mounted to the down-
stream end of the frame. The motions of the traverse were con-
trolled by a servo driver (LinMot E1450-LU-QN-0S) with an
external position sensor (LinMot MS01-1/D) mounted along
one of the two rails for increased repeatability. Power for t
he
system was provided by a step-up transformer (Maddox MIT-
DRY-188). Motion profiles were loaded onto the driver, and a
TTL pulse was used to start each successive motion period. In
these experiments, the maximum surge velocity was
*
=
2
.
40
ms
−
1
and the maximum surge acceleration was
3*
3C
=
23
.
7
ms
−
2
, while the average absolute position error was 0.68 mm
and the average absolute velocity error was 8
.
33
×
10
−
3
ms
−
1
.
Data were collected from the rotary encoder and torque
transducer using a data acquisition card (National Instrum
ents
USB-6221), as well as an amplifier (FUTEK IAA100) for
the raw torque voltage signals. Data collection occurred at
a
sampling rate of 1 kHz. A LabVIEW control program coor-
dinated data collection and triggering for the linear actua
tor.
It was also used to adjust the resistance of the rheostat re-
motely via a stepper motor (Sparkfun ROB-13656). After
each experiment, the program converted the raw voltage sig-
nals from the amplifier to dimensional values by interpolati
ng
between calibrated points, and it performed a fourth-order
central-difference scheme on the angular measurements from
the encoder to obtain a rotation-rate signal. Furthermore,
the
measured torque signals were filtered using a sixth-order Bu
t-
terworth filter with a cutoff frequency of 100 Hz to reduce the
influence of electrical noise.
B. Experimental Parameters
The apparatus described in the previous section was used
to investigate the unsteady torque and power production of
a wind turbine actuated in surge motions. Surge amplitudes
between
=
0
.
150 and 0.600 m (
∗
=
/
=
0
.
128 and
0.514) were tested in combination with motion periods be-
tween
)
=
0
.
5 and 12 s. These combinations resulted in
reduced frequencies
:
=
5
/
D
1
between 0.079 and 1.821 and
nondimensional surge velocities
D
∗
=
5
/
D
1
between 0.039
and 0.234. The specific combinations of
and
)
explored in
this study, and their respective values of
D
∗
, are given in Table
I
. Sinusoidal and trapezoidal surge-velocity waveforms ser
ved
as motion profiles for these experiments. The trapezoidal
waveforms consisted of alternating segments of constant ac
-
celeration and constant velocity. The relative duration of
the
constant-acceleration phases was parameterized by
b
, defined
as the total time of nonzero acceleration in a single cycle di
-
vided by the cycle period. Hence,
b
=
0 corresponded to a
square wave, while
b
=
1 corresponded to a triangle wave.
The types of waveforms used in these experiments are shown
in Figure
2
.
The wind turbine was tested with resistive loads of 7.39,
7.48, 9.80, 10, 20, and 40 Ohms. The first three loads were