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Top-level rotor optimisations based on actuator disc theory

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Ahead of the elaborate rotor optimisation modelling that would support detailed design, it is shown that significant insight and new design directions can be indicated with simple, high-level analyses based on actuator disc theory. The basic equations derived from actuator disc theory for rotor power, axial thrust and out-of-plane bending moment in any given wind condition involve essentially only the rotor radius, R, and the axial induction factor, a. Radius, bending moment or thrust may be constrained or fixed, with quite different rotor optimisations resulting in each case. The case of fixed radius or rotor diameter leads to conventional rotor design and the long-established result that power is maximised with an axial induction factor, a=1/3. When the out-of-plane bending moment is constrained to a fixed value with axial induction variable in value (but constant radially) and when rotor radius is also variable, an optimum axial induction of 1∕5 is determined. This leads to a rotor that is expanded in diameter 11.6 %, gaining 7.6 % in power and with thrust reduced by 10 %. This is the low-induction rotor which has been investigated by Chaviaropoulos and Voutsinas (2013). However, with an optimum radially varying distribution of axial induction, the same 7.6 % power gain can be obtained with only 6.7 % expansion in rotor diameter. When without constraint on bending moment, the thrust is constrained to a fixed value, and the power is maximised as a→0, which for finite power extraction would require R→∞. This result is relevant when secondary rotors are used for power extraction from a primary rotor. To avoid too much loss of the source power available from the primary rotor, the secondary rotors must operate at very low induction factors whilst avoiding too high a tip speed or an excessive rotor diameter. Some general design issues of secondary rotors are explored. It is suggested that they may have the most practical potential for large vertical axis turbines avoiding the severe penalties on drivetrain cost and weight implicit in the usual method of power extraction from a central shaft.
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Wind Energ. Sci., 5, 807–818, 2020
https://doi.org/10.5194/wes-5-807-2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.
Top-level rotor optimisations based
on actuator disc theory
Peter Jamieson
Centre for Doctoral Training in Wind and Marine Energy, University of Strathclyde, Glasgow, G1 1XW, UK
Correspondence: Peter Jamieson (peter.jamieson@strath.ac.uk)
Received: 11 September 2019 – Discussion started: 14 October 2019
Revised: 20 April 2020 – Accepted: 7 May 2020 – Published: 23 June 2020
Abstract. Ahead of the elaborate rotor optimisation modelling that would support detailed design, it is shown
that significant insight and new design directions can be indicated with simple, high-level analyses based on
actuator disc theory. The basic equations derived from actuator disc theory for rotor power, axial thrust and
out-of-plane bending moment in any given wind condition involve essentially only the rotor radius, R, and the
axial induction factor, a. Radius, bending moment or thrust may be constrained or fixed, with quite different
rotor optimisations resulting in each case. The case of fixed radius or rotor diameter leads to conventional rotor
design and the long-established result that power is maximised with an axial induction factor, a=1/3. When the
out-of-plane bending moment is constrained to a fixed value with axial induction variable in value (but constant
radially) and when rotor radius is also variable, an optimum axial induction of 1/5 is determined. This leads to
a rotor that is expanded in diameter 11.6 %, gaining 7.6 % in power and with thrust reduced by 10 %. This is
the low-induction rotor which has been investigated by Chaviaropoulos and Voutsinas (2013). However, with an
optimum radially varying distribution of axial induction, the same 7.6% power gain can be obtained with only
6.7 % expansion in rotor diameter. When without constraint on bending moment, the thrust is constrained to a
fixed value, and the power is maximised as a0, which for finite power extraction would require R→ ∞.
This result is relevant when secondary rotors are used for power extraction from a primary rotor. To avoid too
much loss of the source power available from the primary rotor, the secondary rotors must operate at very low
induction factors whilst avoiding too high a tip speed or an excessive rotor diameter. Some general design issues
of secondary rotors are explored. It is suggested that they may have the most practical potential for large vertical
axis turbines avoiding the severe penalties on drivetrain cost and weight implicit in the usual method of power
extraction from a central shaft.
1 Introduction
Two quite different innovative rotor concepts have been con-
sidered previously. These are the low-induction rotor and the
secondary rotor.
A low-induction rotor in optimal operation is designed
to operate with lower values of axial induction than 1/3,
the ideal value according to the according to basic actu-
ator disc (AD) theory to maximise power at a fixed cho-
sen diameter. The primary motivation for the low-induction
concept is to lower the cost of energy in scenarios where
sacrificing some power in reducing design induction val-
ues leads to relatively more significant load reductions that
are of overall economic benefit to the design. Discussion
of the low-induction concept appears in Johnson (2019),
where Christopher L. Kelly of Sandia National Laboratories,
in an unpublished presentation at the Wind Energy Science
Conference of 2017, had noted that the first low-induction
design with constrained blade rotor bending moment was
due to Ludwig Prandtl and is reproduced in Tollmien et
al. (1961). Snel (2003) observed that when the power coeffi-
cient, Cp, is stationary at its maximum value associated with
an axial induction of 1/3, the thrust coefficient, Ct, is still
strongly increasing. Simple actuator disc theory determines
that dCt
da=4
3when dCp
da=0. There is therefore, for a very
Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.
808 P. Jamieson: Top-level rotor optimisations based on actuator disc theory
small power penalty, a relatively large reduction in thrust,
and there are associated bending moments to be gained from
reducing induction levels a little below the theoretical opti-
mum for maximum power, and independent blade manufac-
turers have long been aware of this. The potential benefits of
yet more radical reductions in induction to around 0.2 were
highlighted by Chaviaropoulos and Voutsinas (2013). Work
on low-induction rotors continued in the Innwind.EU project
(Chaviaropoulos et al., 2013), and structural design issues
of a low-induction rotor were reviewed by Chaviaropoulos
and Sieros (2014). Development of lower lift aerofoils that
may suit a very large rotor (Chaviaropoulos et al., 2015) was
of value in addressing the problems of stability of aerofoil
characteristics at a very high Reynolds number, a topic rel-
evant for all very large rotors, but aerofoil design require-
ments for low-induction rotors can be better defined follow-
ing the optimisation of the spanwise distribution of induc-
tion. Low-induction designs with expanded rotor diameter
continued to be explored by Chaviaropoulos et al. (2013),
Bottasso et al. (2014) and Quinn et al. (2016), but again
all of this work including associated cost of energy analy-
ses was predicated on non-optimum largely constant span-
wise distributions of induction. The Technical University of
Denmark (DTU) 10 MW reference turbine was further de-
veloped as an International Energy Agency (IEA) reference
turbine by Borlotti et al. (2019). This work involved sophis-
ticated multi-variable numerical optimisations with complex
constraints, entirely appropriate for a detailed reference tur-
bine design, but with the result that the role of reduced in-
duction could not be clearly seen in isolation. However the
rotor diameter was regarded as a free variable, and this will
be shown to be of the essence of the low-induction concept.
The secondary-rotor concept involves extracting power us-
ing a rotor generator system mounted on the blades of an oth-
erwise conventional primary turbine. The secondary rotors
operate at high speed in much-elevated relative air speeds
leading to much smaller and lighter power conversion equip-
ment than with a conventional centre-shaft-based drivetrain.
This idea emerged in designs such as the space frame turbine
of Watson (1988) and the airborne system of Jack (1992),
where the driver was to have an ultra-lightweight wind tur-
bine. Thus the motivation for secondary-rotor systems has
always been to reduce drivetrain mass and potentially also
cost. The secondary-rotor concept was considered further
by St-Germain (1992) and Madsen and Rasmussen (2008).
Jamieson (2011) highlighted it as a possible solution to the
design challenge faced by large vertical-axis wind turbines
(VAWTs), where a very low optimum speed leads to high
drivetrain torque, weight and cost if the power is extracted
in the most usual way from the central shaft. Leithead et
al. (2019) employed secondary rotors for power take-off in
an innovative X-rotor VAWT design.
This paper shows that the low-induction and secondary-
rotor concepts have a common origin in basic optimisa-
tions derived from actuator disc theory. As already discussed,
these design concepts are not themselves new, but their fun-
damental connection to elementary actuator disc theory has
not previously been highlighted. More significantly, once this
connection is made, it much facilitates high-level analyses
that can usefully guide preliminary design. A key assump-
tion in blade element momentum theory (BEM) is that the
rotor plane may be analysed as a set of annular rings that are
regarded as mutually independent. This enables AD theory
to be generalised to deal with a spanwise variation of induc-
tion. AD theory and BEM are very long established, and the
form of equations used here often follows Jamieson (2011).
The underlying actuator disc optimisations are now pre-
sented, followed by their application to more detailed anal-
yses guiding top-level design of the low-induction rotor and
secondary rotor respectively.
2 Basic optimisations from elementary actuator
disc theory
2.1 Actuator disc equations
Actuator disc equations for power, thrust and out-of-plane
bending moment as related to ambient wind speed, U0; air
density, ρ; and rotor radius, R, are presented in Table 1. The
coefficients of power and thrust, Cpand Ct, depend only on
the axial induction factor, a, and are in widespread use. A
companion out-of-plane bending moment coefficient, Cm, is
also defined as in Jamieson (2011). The standard assump-
tion of blade element momentum theories is that each annular
ring of the actuator disc can be treated as independent. Thus,
when the axial induction varies radially, rotor area-averaged
values of the coefficients may be defined as in the right col-
umn of Table 1.
Three distinct optimisations are now considered with the
objective in each case of maximising power:
a. the rotor radius is fixed and axial induction is to be de-
termined;
b. the out-of-plane bending moment is fixed but rotor ra-
dius and axial induction are variable;
c. the rotor thrust is fixed but rotor radius and axial induc-
tion are variable.
2.2 Optimisations with radially constant induction
The optimisations are first considered in the context of an
axial induction that does not vary spanwise. Case (a) is
the familiar one where, with radius Rfixed and power
Pa(1 a)2, which is consequently maximised with a=
1/3. This represents conventional design and is the basis
of a reference design used in subsequent comparisons. In
the reference design, R=R0,P=P0,T=T0and M=
M0, where the reference values, P0,T0and M0are all
based on R=R0and a=a0=1/3. In case (b), the out-
of-plane blade bending moment is fixed and M=M0=
Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020
P. Jamieson: Top-level rotor optimisations based on actuator disc theory 809
Table 1. Basic actuator disc equations for power, thrust and out-of-plane bending moment.
Variable Actuator disc Rotor power coefficient Rotor power coefficient
equation (radially constant axial (radially variable axial
induction) induction)
Power P P =0.5ρU 3
0πR2CpCp=4a(1 a)2Cp=81
R
0
a(1 a)2xdx
Thrust T T =0.5ρU2
0πR2CtCt=4a(1 a)Ct=81
R
0
a(1 a)xdx
Moment M M =0.5ρU 2
0πR3CmCm=8
3a(1 a)Cm=81
R
0
a(1 a)x2dx
0.5ρU 2
0πR3Cm=4
3ρU 2
0πR3a(1 a)=8
27 ρU 2
0πR3
0, which
on solving for Ryields
R=(3M0
4ρU 2
0πa(1 a))1/3
.(1)
Substituting for Rfrom Eq. (1), the power equation P=
0.5ρU 3
0πR2Cp=2ρU 3
0πR2a(1 a)2becomes
P=2ρU 3
0π(3M0
4ρU 2
0πa(1 a))2/3
a(1 a)2.(2)
From Eq. (2), the power, P, now varies only with aand
Pa1/3(1 a)4/3.(3)
Differentiating Pin Eq. (3) to find a maximum leads to
(1 a)(1 5a)=0, and hence Pis maximised at a=1/5.
Comparing with a standard rotor design, when a=1/5 and
Pis maximum,
R
R0=a0(1a0)
a(1 a)1/3
=1.116,(4)
P
P0=a(1 a)2
a0(1a0)2R
Rs2
=1.076,(5)
T
T0=a(1 a)
a0(1a0)R
Rs2
=0.896.(6)
As in Jamieson (2018), general trends of R,P,Mand Trela-
tive to unit values of the standard rotor are presented in Fig. 1.
The analysis indicates that a rotor designed for an axial
induction factor of 0.2 that is 11.6 % larger in diameter can
operate with 7.6 % increased power and 10 % less thrust yet
at the same level of blade rotor out-of-plane bending moment
as the baseline design. In case (c), the thrust is maintained
at a constant value, T0. Since power R2a(1 a)2and T
R2a(1a) is constant, it is evident that the power P(1a)
and is maximised as a0. However, for the power to be
finite and positive when the axial induction and hence the
power coefficient are zero requires R→ ∞.
Figure 1. Design parameters related to axial induction.
As opposed to the conventional solution of power take-off
from a central shaft, additional (secondary) rotors are set on
the blades or other support arms at a radial distance from the
central axis of the primary rotor, thereby experiencing a high
relative wind speed. The ideal optimisation at zero induction
and hence infinite radius cannot be realised, but it will be
shown that very low induction values are feasible without un-
acceptably large secondary rotors. The secondary rotor may
be therefore be considered as an ultra-low-induction rotor. In
the system of Fig. 2, the torque reaction to the primary rotor
is provided by thrust on the secondary rotors, and a specific
value of thrust on each secondary rotor is therefore required
to optimise power extraction from the primary rotor. The sec-
ondary rotors are small, high-speed rotors, and the sum of
design torques of all secondary rotors can be much less than
the design torque associated with power take-off in the con-
ventional way from a central shaft. This property can offer
a solution to a key problem of large VAWT design where an
inherently lower shaft speed than any equivalent horizontal-
axis wind turbine (HAWT) puts a large premium on drive-
train torque, mass and cost.
https://doi.org/10.5194/wes-5-807-2020 Wind Energ. Sci., 5, 807–818, 2020
810 P. Jamieson: Top-level rotor optimisations based on actuator disc theory
Figure 2. Rotor with secondary rotors.
3 Low-induction rotor design
For a radially constant axial induction distribution and fixed
out-of-plane bending moment, M=M0, it was established
in Sect. 2 that a=0.2 maximises power, giving a 7.6 %
power gain for 11.6 % radius expansion compared to con-
ventional design. The question then arises of whether an op-
timised radially varying distribution of axial induction can
realise greater power gains or, for example, the same 7.6 %
power gain at reduced rotor expansion. Related to this is
the question of what may be a suitable, efficient generalised
model of the radially variable axial induction. A representa-
tion in the form a(x)=a(1 xn)pis found to be versatile
and highly effective. With arbitrary values of only two free
variables, nand p, a wide range of distributions can be gen-
erated (Fig. 3). This even includes approximations to con-
stant values of axial induction less than 0.333, for example,
a=0.2. The curve (yellow trace) of Fig. 3 illustrates this
although a much more accurate approximation than shown
can be obtained. More general optimisation methods could
be employed to determine optimum distributions of axial in-
duction subject to varied constraints, but the simple approach
adopted here is highly effective.
Now there can never be benefit in a > 1/3 as the bend-
ing moment would be increased and power decreased. Also
as x0, the bending moment M0, and so in the limit
x0, which is approaching the shaft centre, it is logical
that a1/3 in any design that seeks to constrain only bend-
ing moment. In the following analyses, a,nand pare all
treated as free variables although, as expected, the value de-
termined for ais usually very close to 1/3. This tends to
confirm that the optimisation, although in effect having only
two free variables, nand p, is quite accurate. Polynomial
representations by comparison are far inferior. A quadratic,
for example, a2x2+a1x+a0, with a0=1/3, would have two
free variables, a2and a1, but could only represent linear or
parabolic shapes. In order to have results that are likely to be
realistic for typical rotors with small finite blade numbers, a
Figure 3. Distributions of axial induction for arbitrary choices on n
and p.
tip loss effect is introduced using the Prandtl tip loss factor,
F(x)=2
πacosne(1x)
2(1a)o. The question of an overall max-
imum in power regardless of required diameter expansion is
now addressed. Using the generalised forms of Cpand Cm
from Table 1, the power is expressed as
P(a, n, p)=
4ρπ U 3
0M
2
3
0
1
R
0
a(1xn)p1a(1xn)p2xF (x)dx
(4ρπ U 2
0
1
R
0
a(1xn)p1a(1xn)px2F(x)dx)2/3.(7)
Using a maximisation routine such as available in PTC Math-
cad 15, an overall maximum in power P(a,n,p) is obtained,
with values a=0.331, n=1.504 and p=1.125 giving an
axial induction distribution as in Fig. 4. The gain in power
(see Fig. 4, Pmax) is found to be 11.9 %, which is much
greater than the 7.6 % for a radially constant axial induc-
tion but requiring a radial expansion of 34%. This is too
large a radial expansion to be of practical benefit considering
the implications in increased tip speed or drivetrain torque.
In the next analysis the radial expansion is constrained (see
Fig. 4, Pcon) to a value such that the power gain is 7.6 % as
for optimum constant induction. The associated axial induc-
tion distribution has parameters a=0.333, n=0.417 and
p=0.136 as illustrated in Fig. 4. Note that all the distri-
butions of Fig. 4 maintain the same constant value of out-
of-plane bending moment at the shaft centre line. The strik-
ing result however is that this same power gain of 7.6 %
is realised with a radius expansion of only 6.7 % (diamond
marked point of Fig. 5) as opposed to the 11.6 % (triangular
marked point of Fig. 5) required with a constant axial induc-
tion of 0.2.
Also shown in Fig. 5 is the ratio of power gain to expan-
sion which maximises around 3 % expansion. Above this low
level, the required rotor expansion rises more rapidly than the
gain in power although the most economic benefit will prob-
Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020
P. Jamieson: Top-level rotor optimisations based on actuator disc theory 811
Figure 4. Axial induction distributions giving rise to the same out-
of-plane bending moment, M0, at rotor centre.
Figure 5. Power gain related to rotor radius expansion ratio.
ably arise with power gains and rotor expansions in a 5% to
10 % range.
Comparing (see Fig. 4) the optimum axial induction distri-
bution (for 7.6 % power gain) with the constant value of 0.2,
it is evident that more power is being obtained over most
of the span except near the blade tip. Consistent with these
higher power levels, there is only a 3.5% reduction in ax-
ial thrust for the radially variable axial distribution (Pcon in
Fig. 4) as opposed to approximately 10 % reduction for con-
stant induction at a=0.2. If cost of energy modelling sug-
gests that there is benefit say from reduced wake impacts in
a thrust reduction greater than 3.5 %, say at the same power
gain of 7.6 %, with appropriate constraints on the power max-
imisation procedure, the necessary rotor expansion can then
be related to thrust reduction as in Fig. 6.
Tip loss has no effect in comparing distributions where the
axial induction is constant radially because it cancels in the
power, moment and thrust ratios, provided the low-induction
rotor is compared with a reference rotor having the same
number of blades. It has a small effect (Fig. 7) for designs
with rotor expansions below about 15% and a more notice-
able effect at large expansion ratios which however may be
of little practical interest.
The distributions in Fig. 7 are very similar and that is what
matters most. On account of the sensitivity of the power law
relationships, the associated values of nand pwill often dif-
fer considerably. For no tip loss n=0.416 and p=0.136,
Figure 6. Rotor expansion related to thrust reduction for a fixed
power gain (7.6 %).
Figure 7. Effect of tip loss on optimum axial induction distributions
for a power gain of 7.6 %.
with tip loss n=0.295 and p=0.112. Another main issue of
practical relevance is that blades are never aerodynamically
active near the shaft centre line. They may become cylindri-
cal near the root contributing only drag and connect to a hub
having a conical cover or spinner. To approximate the loss of
aerodynamic performance in the hub area, some of the anal-
yses were repeated, with lower limits on integrals such as in
Eq. (7) changed from 0 to 0.15. As with tip loss, effects were
only very noticeable at large (impractical) expansion ratios.
Figure 8 compares the results for maximum possible power
gain with and without exclusion of the first 15 % of span.
Table 2 presents data relating to axial induction distribu-
tions of Fig. 8. Although the power gains differ only 1 %,
there is a noticeable difference in the axial induction distribu-
tions of Fig. 7 and a large difference in the rotor expansions at
34 % for the complete span being aerodynamically active and
25 % when the innermost 15 % of span is excluded. When
designs in a more realistic range of parameters are consid-
ered, for example, as in Fig. 7 with power gain restricted to
7.6 %, there is no significant difference between cases with
and without exclusion of the inner 15 % of the rotor.
https://doi.org/10.5194/wes-5-807-2020 Wind Energ. Sci., 5, 807–818, 2020
812 P. Jamieson: Top-level rotor optimisations based on actuator disc theory
Figure 8. Axial induction distributions for maximum power gain.
Table 2. Parameters of the distributions for maximum power gain.
Fraction of span inactive aerodynamically 0 0.15
a0.331 0.333
n1.504 1.130
p1.125 0.674
Radius expansion factor 1.343 1.246
Power gain 1.119 1.109
4 Secondary-rotor design
4.1 Power extraction using secondary rotors
Secondary rotors near the tip of HAWT blades will experi-
ence a higher relative flow velocity and may thus be smaller
in diameter than those of a VAWT of similar rated power.
However the tip region of a large HAWT is subject to large
deflections and a torsional stiffness that is relatively reduc-
ing with upscaling. Thus reacting the total edgewise load of
a blade near the tip may pose problems for aerodynamic sta-
bility and structural stiffness. Even more problematic may
be preserving alignment of secondary rotors on a pitching
blade. The classic issues with VAWTs which had led to
them being uncompetitive historically are (a) an intrinsically
lower optimum speed leading to factors of 2 or 3 on driv-
etrain torque, weight and cost and (b) reduced power per-
formance associated with intrinsically lower average lift-to-
drag ratios per cycle of rotation leading to maximum power
coefficients 0.4 when large HAWTs have power coeffi-
cients 0.5. Power take-off using secondary rotors may
avoid the torque penalty intrinsic in a conventional VAWT
design, providing a more effective drivetrain solution that
may breathe new life into VAWT technology.
For these reasons the focus in the following analyses is
on secondary rotors for a primary rotor of VAWT design al-
though much of the analyses are directly relevant or easily
adapted to HAWT design. The secondary rotors are always
assumed to be HAWTs. In the following analyses upper case
symbols refer to a primary rotor and lower case to a sec-
ondary rotor. Where there are multiple secondary rotors, the
parameters of 1 of nrotors will have the subscript n. The
aerodynamic torque of the primary rotor is reacted by the to-
tal thrust of the secondary rotors acting (under the present
simplified assumptions) with a moment arm at the maximum
radius R0of the primary rotor. The relative wind speed inci-
dent on the secondary rotors is equal to the tip speed of the
VAWT 40 m s1, and as a further simplification, the ambi-
ent wind speed which is small in comparison is ignored. The
power generated by the primary rotor is then
P=ntnR0.(8)
The total power, p, extracted by the secondary rotors is then
p=ntnR0(1 a).(9)
Now with the usual assumption that each annular ring of
the actuator disc can be analysed independently, then Eq. (9)
applies to the elemental power and thrust contributions of
each annulus, and a radially varying axial induction, a(x),
will have exactly the same performance as a constant induc-
tion of a, the area-averaged value of a(x). For this reason
only radially constant values of axial induction are consid-
ered although, in a detailed design embracing all aspects of
structure and loads, there may be some benefits from radi-
ally varying axial induction. This result is of course quite
different from the case of the low-induction rotor where the
bending moment is constrained and radial variation of axial
induction is very significant. As an example, to focus dis-
cussion of secondary-rotor design issues, parameters as in
Table 3 are selected for a VAWT rated at 5MW.
4.2 Sizing of secondary rotors
The power produced by the primary (VAWT) rotor is P=
0.5ρU 3
0(2R0L)CP, and the total power extracted by nsec-
ondary rotors is p=npn=P(1a). For each secondary ro-
tor, pn=0.5ρ(R0)3πr 2
n4a(1a). Hence the ratio of radius
of one of nsecondary rotors to that of the primary rotor can
be expressed as
rn
R0=LCP
2λ3a(1 a)R00.5
.(10)
The ratio of secondary- to primary-rotor radius defined by
Eq. (10) is shown in Fig. 9 as based on the data of Table 3.
The curve is symmetrical about a=0.5 although this is not
obvious as a logarithmic scale is employed in order to show
more clearly the variation of rn/R0at very low axial in-
duction values. The vertical line of Fig. 9 marks a=0.333.
There is no interest in greater values of a, and the optimum
design value for an effective system will certainly be much
less than 0.333 as this would imply a sacrifice of 1/3 of
primary-rotor power. In the data of Table 3 the number of
secondary rotors is chosen as six, which may be two on each
of three blades or three on each of two. A value of aof 0.05
is chosen for further illustration of secondary-rotor design is-
sues. This implies a sacrifice of 5 % of primary-rotor power,
Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020
P. Jamieson: Top-level rotor optimisations based on actuator disc theory 813
Table 3. Parameters of a primary Htype VAWT rotor and of secondary HAWT rotors.
Primary Secondary Unit
Ambient wind speed U0m s1
Rated power P5/{η(1 a)}pn5/n MW
Design tip speed Vt40 vt160 m s1
Rotor power coefficient CP0.4 Cp4a(1 a)2
Rotor thrust coefficient CTCt4a(1 a)
Rotor radius R065 rnm
Rotor angular speed  ωnrad s1
Design tip speed ratio 34λ ωrn/ R0=4
Blade length L100 m
Drivetrain efficiency η
Number of rotors 1 n6
Rotor thrust T tnN
Rotor torque Q nR0tnqnpnnN m
Blade chord cnm
Figure 9. Ratio of secondary-rotor radius to that of primary rotor.
and the associated radius fraction is 0.084. Each secondary
rotor then has a radius (5 m) that is 8.4 % of the primary-
rotor radius (60 m).
4.3 Torque benefit of secondary rotors
A major issue with large VAWTs especially is a very high
level of drivetrain torque. In a conventional drivetrain solu-
tion with power take-off from a central shaft, the torque, Q,
of the primary rotor would drive mass and cost of the drive-
train. To assess the benefit in secondary-rotor power take-off
the ratio of the sum of secondary-rotor torques to Qis now
compared.
nqn
Q=pn
ωnQ=(1 a)
n=(1 a)rn
λR0
(11)
For a design with a=0.05 and parameters otherwise as
in Table 3, the torque ratio nqn
Q=(1a)rn
λR0has a value
0.95×0.084
4=0.02 showing that the sum of secondary-rotor
torques is 1/50th of primary-rotor torque. As a power
take-off system, each secondary-rotor system comprises both
bearings and generator but also an aerodynamic rotor sys-
tem. The estimates of secondary-rotor diameter and torque
reduction factor are realistic, provided it is accepted that at
a=0.05 the fraction of available primary-rotor power ex-
tracted will be less than 95 % to an extent, depending on the
effect of parasitic drag losses. For conventional large HAWTs
and possibly more so for VAWTs, rotor cost is generally less
than the drivetrain cost, but even at ratios 2/50th, 3/50th or
much more, there are potentially very large savings in cost
and weight of power conversion with secondary rotors. The
further benefits of multiple rotors are in rnreducing as 1/n,
with the torque ratio of Eq. (11) similarly reducing.
4.4 Design characteristics of secondary rotors
Does the design of the secondary rotor differ much from con-
ventional HAWT designs considering the unusually high rel-
ative wind speed and unusually low design levels of axial
induction? This is initially assessed by deriving an equation
for rotor solidity. From Jamieson (2011) a non-dimensional
lift distribution, with Cldas design lift coefficient (lift value
at maximum lift-to-drag ratio), is determined as
cnCld
rn=8πa(1 a)F(x)
(1 +á)p(1 a)2+λ2x2(1 +á)2.(12)
In Eq. (12) the tangential induction factor, á, is determined
as
áá(x)=4a4a2+λ2x20.5λx
2λx .(13)
Considering an annular ring of the rotor swept area of span-
wise width, dr, the local solidity is the sum of planform el-
emental areas of Bblades within the ring as a ratio of the
complete swept area of the ring. Thus the local solidity at
radius ris given as
https://doi.org/10.5194/wes-5-807-2020 Wind Energ. Sci., 5, 807–818, 2020
814 P. Jamieson: Top-level rotor optimisations based on actuator disc theory
σn(r)=Bcndr
2πrdr=Bcn
2πr ,(14)
and the solidity of the whole rotor is then
σn=2
πr2
n
rn
Z
0
πn(r)dr=2
1
Z
0
n(x)dx
=2
1
Z
0
4a(1 a)F(x)
Cld(1 +á)p(1 a)2+λ2x2(1 +á)2dx . (15)
The right-hand side of Eq. (15) is obtained using Eq. (12) to
substitute for cnin Eq. (14). A tip loss factor, F(x), appropri-
ate to a three-bladed rotor is applied; the inner rotor region
where the solidity would become infinite is omitted; and a
typical aerofoil design lift coefficient, Cld, of 0.8 is assumed.
An estimate of secondary-rotor solidity with a=0.05 and
otherwise consistent with the values of Table 3 is determined
as
σn=
1
Z
0.15
8a(1 a)F(x)
Cld(1 +á)p(1 a)2+λ2x2(1 +á)2dx=0.072.(16)
The dependence of rotor solidity on aerofoil design lift coef-
ficient is illustrated in Fig. 10. An aerofoil such as NACA 63-
418 has been used on wind turbines and (with some variation
according to data sources) may provide a lift-to-drag ratio
of 125 at Cld1. According to Fig. 10 this may yield a
solidity 6 % at a design axial induction 0.05, which is
only a little higher than values of 4 %–5 % most common
in large HAWT designs. Thus the secondary rotor need not
differ much from conventional designs of large HAWTs in
respect of solidity. Light loading from a very low design ax-
ial induction value and very high relative flow velocities have
mutually compensating impacts on rotor solidity, whereas a
secondary-rotor design for the usual design values of axial
induction, a1/3, would have solidity 30 %.
The next consideration for secondary-rotor design is the
range of Reynolds number, Re. For a solidity 0.07 as in
Eq. (16), the chord at around 80 % span will be
cnσnrn
0.8B=0.072 ×5
0.8×3=0.15,(17)
and the associated Reynolds number is
Re =0.8ρvtcn
µ=0.8×1.225 ×160 ×0.15
1.8×105=1306667.(18)
Considering the high tip speed of the secondary rotor, us-
ing vtas the resultant velocity in the estimate of Eq. (18), and
by implication neglecting the ambient wind speed, will give
a good approximation. Equation (18) shows that Re values
of the secondary rotor will be in a normal range for medium
to large HAWTs although the rotor diameter is small 10 m.
Figure 10. Rotor solidity related to design axial induction and de-
sign lift coefficient.
Another important design consideration is the level of op-
erational loads on the secondary rotor. Assuming a rated
wind speed of Ur=11 m s1, and a relative wind speed for
the secondary rotors of 160 m s1, then, compared to a con-
ventional rotor of similar diameter, rotor thrusts and out-of-
plane bending moments are both in the same ratio:
tn
T0=mn
M0=v2
ta(1 a)
U2
ra0(1a0)=1602×0.05 ×0.95
112×0.333 ×0.667 =45.2.(19)
This is a huge increase in steady operational loading com-
pared to conventional design. Also the steady and turbulent
components of the ambient wind speed will introduce cyclic
and random disturbances to secondary-rotor inflow, which
may increase available power (Leithead et al., 2019) but will
inevitably introduce fatigue loading. Now it is vital for the
secondary rotors to minimise parasitic drag in the hub region
as torque from this will absorb power from the primary rotor
that cannot be recovered. It is of no benefit to have a spin-
ner that may deflect the central flow outwards, augmenting
flow over the inboard blade sections, and, equally, it is of no
benefit to have ducted secondary rotors that produce any flow
augmentation. This is because any augmentation contributes
to added thrust (drag) on the spinner or the duct that will con-
sume irrecoverable primary-rotor power. This suggests that
the secondary-rotor system may benefit from having blades
of more ideal profile than is usual near the hub centre line not
because any very significant gain in secondary-rotor power
can be obtained but in order to minimise drag in that area. In
this scenario the blades would twist to near 90out of plane,
bringing the blade roots very close each other and to the axis
of rotation. The large chord widths nearly parallel to the axis
would be exploited for structural strength of the whole rotor,
which would most probably use a lot of carbon in its con-
struction and have titanium leading-edge erosion protection.
Another idea aiming to reduce parasitic drag, perhaps too far-
fetched, would be to engineer a rotor generator system with
a hollow centre although there would still be issues of drag
on the internal surfaces.
Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020
P. Jamieson: Top-level rotor optimisations based on actuator disc theory 815
Figure 11. Twin rotor secondary-rotor system.
4.5 Secondary rotors on a common axis
Returning to actuator disc theory, the idea of twin rotors
counterrotating on a common axis enabling a doubling of
relative velocity at the generator air gap has been considered
(Shen, 2017, and Rosenberg et al., 2014). According to sim-
ple actuator disc theory, the ideal maximum Cpwith the twin
rotors in series, assuming they are sufficiently apart for com-
plete pressure recovery near the downstream rotor, increases
from the Betz limit of 0.593 only to 0.64 (see Newman, 1986)
or decreases to 0.32 if the swept area of both rotors is ac-
counted for. The situation is very different for very lightly
loaded secondary rotors (Fig. 11) where the downstream ro-
tor may operate almost as efficiently as the upstream. The
potential benefit of a secondary-rotor pair in a series arrange-
ment is not only that the design torque and weight of the
power train may be reduced compared to a single equiva-
lent rotor but perhaps that a slimmer generator and hence a
slimmer centre body with less parasitic drag may be realised.
Any kind of multi-rotor, secondary-rotor system has obvi-
ous advantages in torque and weight reduction, but having a
physical arrangement of support structure and connection to
the primary rotor that minimises parasitic drag will be very
important.
Based on wind tunnel tests on actuator discs represented
as porous screens, Newman (1986) concluded that his theory
for multiple actuator discs in series, in the particular case of
two actuator discs, became inaccurate only at spacings closer
than a disc radius. This gives confidence that at the very low
disc loadings applicable to a pair of secondary rotors in se-
ries, spaced about a diameter apart, there should be complete
pressure recovery between the rotors. A single rotor of radius
5 m could be replaced by two rotors side by side as in a multi-
rotor arrangement of radius 5/2=3.536 m. When the ro-
tors are twins in series on the same axis, the radius required
to have the same total thrust at an equivalent axial induction
of 0.05, thereby extracting 95 % of primary-rotor power, is
related to the velocity recovery approaching the downstream
twin. In the analysis following, pressure recovery is assumed
and the velocity approaching the downstream turbine is taken
as the far wake velocity of the upstream turbine, R0{1δ},
where the velocity deficit ratio is δand would be 2a=0.1
for a single ideal actuator disc in inviscid flow. The axial in-
duction factors are selected in an optimisation constrained so
that the twin rotors provide the specific total thrust required
for primary-rotor power extraction and also extract the same
total power as a single secondary rotor with the design axial
induction value, a=0.05. This is accomplished as follows.
The thrust, t1, on a single rotor that would be replaced by the
twin system is proportional to the square of the radius, r1; the
square of the relative velocity, Vt=R0; and a thrust coef-
ficient based on the axial induction, a=ae=0.05. Consider
now the equivalent twin rotor system, with axial induction au
on the upstream turbine, adon the downstream turbine, rela-
tive velocity Vton the upstream turbine and Vt{12(1z)au}
on the downstream turbine. The wake velocity deficit ratio is
δ=2(1 z)au, where zis a factor measuring the extent of
velocity recovery being 0 when, as for a single actuator disc
far wake, the deficit is 2aand 1 if there is complete velocity
recovery. For the twin to produce the same total thrust as the
single rotor with thrust, t1, requires
r2
1ae(1ae)=r2
uau(1au)+r2
dad(1ad)
{12(1 z)au}2.(20)
In addition, if the same total power is required, then, with
power being proportional to the square of the radius, to the
power coefficient and to the cube of the relative velocity,
r2
1ae(1ae)2=r2
uau(1au)2+r2
dad(1ad)2
{12(1 z)au}3.(21)
For given values of z, Eqs. (20) and (21) are solved with the
additional assumption that the upstream and downstream ro-
tors have the same radius, ru=rd, that is to be minimised.
The results in Fig. 12 show the variation of secondary-
rotor radius; upstream rotor induction factor, au; and down-
stream rotor induction factor, ad, with velocity recovery fac-
tor, z. Conventional wake models, such as assessed in a com-
parative study of velocity deficit by Luong et al. (2017), sug-
gest little velocity recovery will take place between rotors 2
to 3 radii apart. However such models may be conservative
and it is also difficult to gauge their applicability. The very
high relative wind speed would imply a very low turbulence
intensity, which would not assist velocity recovery. However,
the loading on the secondary rotors is necessarily very light
to avoid too much loss of primary-rotor power, and the weak
wake may be skewed by centrifugal force. Quite close spac-
ings 1 radius may be beneficial because of the interaction
of the rotating wake which is not accounted for in any sim-
ple actuator disc modelling. A considerable amount of re-
search into various counterrotating rotor systems has taken
place since Newman (1986). Tests on a small 6 kW contra-
rotating rotor discussed in Shen et al. (2017) indicated that,
at the relative high loadings of conventional turbines, 30 %
more power (as opposed to 8 % on the basis of an ideal Cp
of 0.593 rising to 0.64) can be obtained. Numerical mod-
elling (also Shen et al., 2017) of a counterrotating pair of
https://doi.org/10.5194/wes-5-807-2020 Wind Energ. Sci., 5, 807–818, 2020
816 P. Jamieson: Top-level rotor optimisations based on actuator disc theory
Figure 12. Secondary-rotor radius for no power loss related to ve-
locity recovery factor.
Nordtank 500 kW wind turbines using the EllipSys3D code
developed at the Technical University of Denmark (DTU)
with reference to a particular site predicted 43.5 % more en-
ergy than for a single turbine. None of the existing literature
considers the very light loadings appropriate to a pair of sec-
ondary rotors, but experiments and computational fluid dy-
namics (CFD; Koehuan et al., 2017) analyses generally pro-
vide encouragement that performance in real flow will ex-
ceed, sometimes greatly exceed, the performance predicted
by simple actuator disc inviscid flow models. Even with little
velocity recovery where the required diameter of the twin ro-
tors approaches that of a single equivalent rotor, there may be
still be net advantage from lighter blade loading, lower gen-
erator torque and reduced generator diameter with associated
reduced centre body drag. The velocity recovery that may oc-
cur is evidently speculative and may only be better assessed
by CFD modelling of a specific design arrangement.
5 Concluding remarks
Three quite distinct design directions have emerged from op-
timisations relating to basic loads predicted by actuator disc
theory. These are (a) conventional design with rotor radius
predetermined, which has been used as a reference; (b) the
low-induction rotor arising from constraint on out-of-plane
bending moment; and (c) the secondary-rotor concept aris-
ing from constraint on rotor thrust loading.
In comparison to conventional design, the design chal-
lenges in realising a low-induction rotor are not radically
new. The present work highlights that the power gain in rela-
tion to required rotor expansion (a cost) and thrust reduction
(a benefit for turbine loads and wind farm wake impacts) is
sensitive to the radial distribution of axial induction and dis-
cusses optimisation around these factors. In particular it is
shown that the same power gain of 7.6% with an optimum
radially constant axial induction of 0.2 that required a rotor
expansion of 11.6 % can be achieved with an expansion of
only 6.7 % when the axial induction varies radially and is
optimised. The modelling developed here enables the defini-
tion of a space of all self-consistent combinations of power
gain, rotor expansion and thrust reduction with each associ-
ated axial induction distribution. This could enable a prelim-
inary determination of an overall optimum axial induction
distribution using a combined wind turbine and wind farm
cost of energy model.
An expanded rotor of standard design could be operated
at low induction using pitch control, thereby restricting the
steady-state blade root bending moment, but this would not
be satisfactory. It is vital to contain all loads of the expanded
rotor, steady state, dynamic and loads when idling in extreme
wind conditions by limiting the lift and drag of the rotor to
the levels of the non-expanded reference rotor. This calls for
lower lift aerofoils or reduced solidity or both. There is much
less of a design challenge in the low-induction rotor with a ra-
dially varying optimised axial induction distribution (Fig. 4)
as compared to the constant induction of 0.2. The required ro-
tor expansion is much less, and the progressive reduction of
axial induction towards the blade tip is sympathetic to blade
structural design with a natural taper in strength and solidity
from rotor to tip. The graded reduction in spanwise axial in-
duction is also much more favourable than a global reduction
to 0.2 for limiting tip deflection to maintain acceptable tower
clearance without having undue added cost in stiffening the
blade.
Overall the results suggest there may be great value in
treating the axial induction distribution and rotor diameter as
free variables in a basic system optimisation for the lowest
cost of energy where direct power gains, rotor loading and
reduced wake effects from thrust reduction can all be traded
in the design optimisation.
Secondary rotors have not been used on an operational
wind turbine although a design is now being developed (Leit-
head et al., 2019). The main aim in using secondary rotors is
to have a drivetrain with much reduced design torque com-
pared to the usual transmission system based on power take-
off from a central shaft. That can certainly be achieved, with
torque reduction of 1 to 2 orders of magnitude being pos-
sible depending on specific design choices. Although the
design of secondary rotors is much more demanding than
that of conventional rotors of the same diameter, the design
torque reduction is so great that it seems certain that sub-
stantial savings in drivetrain cost can be realised. The fo-
cus of the secondary-rotor design exploration is on VAWTs
as the primary rotor rather than HAWTs because it solves
a key problem with VAWTs of relatively low shaft speed
leading to high drivetrain torque and expensive drivetrains,
whereas, as applied to HAWT design, it could introduce ma-
jor problems for primary-rotor blade design. It emerges that
the radial distribution of axial induction is not critically im-
portant for secondary-rotor design as all distributions with
the same area-averaged axial induction will lead to the same
size of secondary rotor. The high relative wind speed com-
pensates for relatively small rotor diameter and very low de-
Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020
P. Jamieson: Top-level rotor optimisations based on actuator disc theory 817
sign axial induction in a way that for primary rotors in the
multi-megawatt range maintains a Reynolds number 106
and suggests a solidity only a little higher than that typi-
cal of large HAWTs is required. However with secondary
rotors, very high tip speeds are desirable to limit drivetrain
torque and to limit the overall scale of the rotor and generator
system. Also steady-state operational loads are exceptionally
high in relation to rotor diameter. Having multiple secondary
rotors (more than one per blade) has the usual benefits of
multi-rotors (Jamieson, 2011) in reducing net torque, weight
and cost of secondary-rotor systems, but, as was mentioned,
it is particularly important with secondary rotors to minimise
losses from parasitic drag or degradation of primary blade
performance depending on their physical mounting arrange-
ment. The idea of realising multi-rotors as a twin set in series
on a common axis looks promising considering the very low
axial induction levels required of secondary rotors to avoid
wasting primary-rotor power. Whether this is a particularly
good idea cannot be resolved without evaluating specific de-
sign arrangements and developing a greater understanding of
the flow field around the twin secondary-rotor system.
The preliminary evaluation of the X-rotor VAWT design
(Leithead et al., 2019) suggests that use of secondary rotors
will lead to more competitive VAWT designs. Another in-
novative VAWT design, the DeepWind VAWT of Paulsen et
al. (2015), has major savings through integration of the ro-
tor blade shaft and support structure into a single element.
On the other hand, substantial challenges remain for the de-
sign and maintenance of the underwater electrical generating
system. Could an adapted variation of this design with modu-
lar secondary rotors that can form a more economical power
train to be accessed and maintained above sea level be ad-
vantageous?
In summary, three quite different rotor optimisations are
shown to arise naturally from long-established actuator disc
equations and can usefully guide high-level design of the in-
novative rotor systems described as the low-induction rotor
and the secondary rotor.
Data availability. No data sets were used in this article.
Supplement. The supplement related to this article is available
online at: https://doi.org/10.5194/wes-5-807-2020-supplement.
Competing interests. The author declares that he has no conflict
of interest.
Special issue statement. This article is part of the special issue
“Wind Energy Science Conference 2019”. It is a result of the Wind
Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.
Financial support. The author acknowledges the support of the
Centre for Doctoral Training in Wind and Marine Energy of the
University of Strathclyde, Glasgow.
Review statement. This paper was edited by Katherine Dykes
and reviewed by two anonymous referees.
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Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020
... Figure 4 shows the axial induction for various operational points as a result from BEM simulations. Jamieson [17] took the idea of the low induction rotor [18] one step further and derived an optimal axial induction distribution over blade span which allows for greater power gain with modest increase in rotor diameter, compared to a constant low induction factor of 1/5. In fact, the decreasing axial induction towards blade tip in the strong wind mode (see Figure 4, yellow line) shows a similar trend as derived by Jamieson which is beneficial for rated load reduction. ...
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A novel wind turbine rotor optimization methodology is presented. Using an assumption of radial independence it is possible to obtain an optimal relationship between the global power- (CP) and load-coefficient (CT, CFM) through the use of KKT-multipliers, leaving an optimization problem that can be solved at each radial station independently. It allows to solve load constraint power and Annual-Energy-Production (AEP) optimization problems where the optimization variables are only the KKT-multipliers (scalars), one for each of the constraint. For the paper two constraints, namely the thrust and blade-root-flap-moment is used, leading to two optimization variables. Applying the optimization methodology to maximize power (P) or Annual-Energy-Production (AEP) for a given thrust and blade-root-flap-moment, but without a cost-function, leads to the same overall result with the global optimum being unbounded in terms of rotor radius (R~) with a global optimum being at R~ → ∞. The increase in power and AEP is in this case ΔP = 50 % and ΔAEP = 70 %, with a baseline being the Betz-optimum rotor. With a simple cost function and with the same setup of the problem a Power-per-Cost (PpC) optimization resulted in a Power-per-Cost increase of ΔPpC = 4.2 % with a radius increase of ΔR = 7.9 % as well as a power increase of ΔP = 9.1 %. This was obtained while keeping the same flap-moment and reaching a lower thrust of ΔT = −3.8 %. The equivalent for AEP-per-Cost (AEPpC) optimization leads to an increased cost efficiency of ΔAEPpC = 2.9 % with a radius increase of a ΔR = 17 % and an AEP increase of ΔAEP = 13 %, again with the same, maximum flap-moment, while the maximum thrust is −9.0 % lower than the baseline.
Chapter
Die wesentliche Komponente einer Windenergieanlage (WEA), die die Energie des Windes in eine mechanisch nutzbare Energieform umwandelt, ist das Rotorblatt. Es stellt im Gesamtkontext der Anlage einen bedeutenden Kostenfaktor dar und hat gleichzeitig heraus­ ragenden Einfluss auf den Ertrag der Anlage. Beim Entwurf eines Rotorblattes handelt es sich um einen komplexen, iterativen Prozess, der aufgrund des strukturellen Gesamtkonzepts des Rotorblattes einige Herausforderungen für den Entwicklungsingenieur mit sich bringt. Wesentlich für einen erfolgreichen Entwicklungsprozess ist ein hinreichendes Systemverständnis, weshalb die Einbettung des Rotorblattkapitels in dieses Buch außerordentlich hilfreich ist. Ausgehend von dem betrachteten System lassen sich Vereinfachungen einführen, die eine mathematisch-physikalische Erfassung und Beschreibung des Problems ermöglichen. Dadurch wird das Entwurfsproblem einer Lösung zugänglich gemacht. Kern des Kapitels ist die Betrachtung der Rotorblattstruktur für WEA mit horizontaler Achse. Daraus ergibt sich, dass sich das Rotorblatt als Kragbalken betrachten lässt, der an der Rotornabe über das Blattlager, auch Pitchlager (engl. pitch bearing) genannt, fixiert ist. Dort ist er einerseits aerodynamischen und andererseits eigenmasseinduzierten Belastungen, kurz Lasten genannt, ausgesetzt. Insbesondere der zweite Lastanteil führt dazu, dass eine Verstärkung der Struktur in aller Regel mit einer Erhöhung der Masse und damit der Belastung einhergeht. Um diesem sogenannten Schneeballeffekt entgegenzuwirken, ergibt sich die Notwendigkeit für den Leichtbau. Um sich der Problematik in geeigneter Weise nähern zu können, werden in diesem Kapitel zunächst die normativen Anforderungen für die Entwicklung und den Nachweis der Gebrauchstauglichkeit und Betriebssicherheit von Rotorblattstrukturen dargestellt. Daraufhin wird auf die auf das Rotorblatt einwirkenden Belastungen sowie auf die Eigenschaften verwendeter Materialien eingegangen. Weiterhin werden Strukturmodelle zur Nachweisführung von Rotorblättern beschrieben. Ferner wird auf Auslegungskriterien für die Topologie und auf gängige Fertigungsverfahren sowie auftretende Abweichungen eingegangen. Nicht zuletzt findet eine Betrachtung der Kosten im Rahmen des Auslegungsprozesses und deren Einfluss auf die Gestalt der Rotorblattstruktur statt, da die Minimierung der bezogenen Stromgestehungskosten (engl. Levelized Cost of Energy (LCoE)) das Hauptoptimierungsziel bei der Auslegung von WEA ist. Das Kapitel soll einen fundierten Überblick über relevante Themenbereiche für den sich mit dem Rotorblatt auseinandersetzenden Ingenieur geben. Für den Einstieg in Detail­themen sei auf die Literaturverweise hingewiesen.
Article
Full-text available
A novel wind turbine rotor optimization methodology is presented. Using an assumption of radial independence it is possible to obtain an optimal relationship between the global power (CP) and load coefficient (CT, CFM) through the use of Karush–Kuhn–Tucker (KKT) multipliers, leaving an optimization problem that can be solved at each radial station independently. It allows solving load constraint power and annual energy production (AEP) optimization problems where the optimization variables are only the KKT multipliers (scalars), one for each of the constraints. For the paper, two constraints, namely the thrust and blade root flap moment, are used, leading to two optimization variables. Applying the optimization methodology to maximize power (P) or annual energy production (AEP) for a given thrust and blade root flap moment, but without a cost function, leads to the same overall result with the global optimum being unbounded in terms of rotor radius (R̃) with a global optimum being at R̃→∞. The increase in power and AEP is in this case ΔP=50 % and ΔAEP=70 %, with a baseline being the Betz optimum rotor. With a simple cost function and with the same setup of the problem, a power-per-cost (PpC) optimization resulted in a power-per-cost increase of ΔPpC=4.2 % with a radius increase of ΔR=7.9 % as well as a power increase of ΔP=9.1 %. This was obtained while keeping the same flap moment and reaching a lower thrust of ΔT=-3.8 %. The equivalent for AEP-per-cost (AEPpC) optimization leads to increased cost efficiency of ΔAEPpC=2.9 % with a radius increase of ΔR=17 % and an AEP increase of ΔAEP=13 %, again with the same, maximum flap moment, while the maximum thrust is −9.0 % lower than the baseline.
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The following paper provides an overview of a novel wind turbine concept known as the X-Rotor Offshore Wind Turbine. The X-Rotor is a new wind turbine concept that aims to reduce the cost of energy from offshore wind. Cost reductions are achieved through reduced capital costs and reduced maintenance costs. The following paper includes results from an early feasibility study completed on the concept. In the feasibility study exemplary designs were created and structural analyses were carried out. Turbine capital costs and maintenance cost of the X-Rotor concept were then roughly estimated. X-Rotor turbine costs and O&M costs were compared to four existing wind turbine types to investigate potential cost savings from the X-Rotor concept. Results show that the X-Rotor has potential to reduce O&M costs by up to 55% and capital costs by up to 32%. The combination of the capital cost and O&M cost savings show potential to reduce the CoE by up to 26%.
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Investigation of the dual rotor counter-rotating wind turbine (CRWT) performance using non-dimensional parameters of the rotor diameter ratio and the rotor axial distance ratio against the characteristics of power coefficient with tip speed ratio (TSR) as input parameters have been successfully carried through CFD simulation. CFD simulation used k-e turbulence realizable with hexahedral meshing to predict the CRWT performance to the rotor diameter ratio of D1/D2< 1, D1/D2 = 1 and D1/D2> 1 and rotor axial distance ratio with the s826 airfoil that has been applied to the single rotor wind turbine. The best CRWT performance obtained on the rotor diameter ratio of D1/D2 = 1.0 with the peak power coefficient of 0.5219 or increased to ∆Cp, max = 16.49% from the single rotor. CRWT performance through the addition of rotor axial distance ratio showed the power coefficient of the front rotor continued to rise closely to the single rotor performance while the rear rotor will continue to decline. However, the overall CRWT performance were relatively stable after the ratio of the distance Z/D1 = 0.5 with the peak power coefficient of 0.5348 or increased to ∆Cp, max = 19.37%.
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In this report, the details of an investigation into the effect of low induction rotor (LIR) wind turbines on the Levelised Cost of Electricity (LCoE) in a 1GW offshore wind farm is outlined. The 10 MW INNWIND.EU conventional wind turbine and its low induction variant, the 10 MW AVATAR wind turbine, are considered in a variety of 10x10 layout configurations. The Annual Energy Production (AEP) and cost of electrical infrastructure were determined using two in-house ECN software tools, namely FarmFlow and EEFarm II. Combining this information with a generalised cost model, the LCoE from these layouts were determined. The optimum LCoE for the AVATAR wind farm was determined to be 92.15 e/MWh while for the INNWIND.EU wind farm it was 93.85 e/MWh. Although the low induction wind farm offered a marginally lower LCoE, it should not be considered as definitive due to simple nature of the cost model used. The results do indicate that the AVATAR wind farms require less space to achieve this similar cost performace, with a higher optimal wind farm power density (WFPD) of 3.7 MW/km2 compared to 3 MW/km2 for the INNWIND.EU based wind farm.
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DeepWind has been presented as a novel floating offshore wind turbine concept with cost reduction potentials. Twelve international partners developed a Darrieus type floating turbine with new materials and technologies for deep-sea offshore environment. This paper summarizes results of the 5 MW DeepWind conceptual design. The concept was evaluated at the Hywind test site, described on its few components, in particular on the modified Troposkien blade shape and airfoil design. The feasibility of upscaling from 5 MW to 20 MW is discussed, taking into account the results from testing the Deepwind floating 1kW demonstrator. The 5 MW simulation results, loading and performance are compared to the OC3-NREL 5 MW wind turbine. Finally the paper elaborates the conceptual design on cost modelling.
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The design methodology and performance verification of a family of low lift airfoils is presented in this paper. High performance low lift profiles are well suited to low power density rotor designs. Such designs are suitable for increasing the energy yield of mufti-MW offshore wind turbines using larger diameter rotors under moderate loading conditions. For the design of the profiles numerical optimization techniques are used for maximizing a suitable performance–based cost function which is evaluated using XFOIL. The resulting shapes are assessed by means of high fidelity CFD tools. The main uncertainty characterizing the performance of such profiles, at the relative low flow angles where the maximum performance occurs, comes from transition modelling. This uncertainty is even higher at the very large Reynolds numbers encountered in the 10+ MW rotors of our interest. To handle the uncertainty our design options are selected on the conservative side.
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A new generation of multi-MW offshore rotors that are under development deviate from the established design trends, displaying high tip-speed, low solidity and larger than expected rotor diameters. A study was undertaken to quantify the relative merits of this approach, associated to low induction – high swept area rotors. We are using BEM analysis of such solutions and comparing the results to those of a reference wind turbine at the 10MW scale designed for the InnWind.EU project, in order to identify the possibilities for a significant reduction of the cost of energy, especially in large offshore wind farms.
Book
An updated and expanded new edition of this comprehensive guide to innovation in wind turbine design. Innovation in Wind Turbine Design, Second Edition comprehensively covers the fundamentals of design, explains the reasons behind design choices, and describes the methodology for evaluating innovative systems and components. This second edition has been substantially expanded and generally updated. New content includes elementary actuator disc theory of the low induction rotor concept, much expanded discussion of offshore issues and of airborne wind energy systems, updated drive train information with basic theory of the epicyclic gears and differential drives, a clarified presentation of the basic theory of energy in the wind and fallacies about ducted rotor design related to theory, lab testing and field testing of the Katru and Wind Lens ducted rotor systems, a short review of LiDAR, latest developments of the multi-rotor concept including the Vestas 4 rotor system and a new chapter on the innovative DeepWind VAWT. The bookis divided into four main sections covering design background, technology evaluation, design themes and innovative technology examples. Key features: Expanded substantially with new content. Comprehensively covers the fundamentals of design, explains the reasons behind design choices, and describes the methodology for evaluating innovative systems and components. Includes innovative examples from working experiences for commercial clients. Updated to cover recent developments in the field. The book is a must-have reference for professional wind engineers, power engineers and turbine designers, as well as consultants, researchers and graduate students.
Conference Paper
This work investigates an integrated free-form approach for the design of wind turbine blades, and its application to low induction rotors. In the free-form methodology, the airfoil shapes are treated as unknowns and optimized together with the other blade design parameters including chord, twist and the thickness of the structural elements. As the design of the airfoils automatically adapts to the evolution of the blade, it is possible to better explore the solution space and to obtain improved solutions that find better compromises between the aerodynamic and structural points of view. This process is here employed to evaluate low induction rotors as a possible way of designing very large wind turbines, and demonstrated through case studies concerning wind turbines up to 10 MW. © 2015, American Institute of Aeronautics and Astronautics Inc. All rights reserved.
Chapter
Für einen Tragflügel gegebenen Auftriebes erhält man bei vorgeschriebener Spannweite den geringsten induzierten Widerstand, wenn man den Auftrieb nach einer Halbellipse verteilt. Die Nebenbedingung, daß die Spannweite vorgeschrieben ist, ist dabei aber durchaus wesentlich, und es ist also die Behauptung durchaus unzulässig, daß die elliptische Auftriebsverteilung die beste schlechthin sei. Der induzierte Widerstand ist um so kleiner, je größer die Spannweite gemacht wird. Wenn in einem Sonderfall die Spannweite des Flugzeuges durch die Forderung begrenzt wird, daß das Flugzeug durch ein bestimmtes vorgegebenes Hallentor geschoben werden kann, so ist es am Platz, innerhalb der so vorgeschriebenen Spannweite den Auftrieb elliptisch zu verteilen. Wenn aber eine derartige Begrenzung nicht vorliegt, dann wird man sich nach anderen Gesichtspunkten richten müssen. Eine beliebige Vergrößerung der Spannweite verbietet sich durch das in diesem Fall allzu stark anwachsende Holmgewicht. Eine den flugtechnischen Belangen gerecht werdende Formulierung der Aufgabe wäre wohl die, daß nicht das dem Auftrieb gleiche Gesamtgewicht, sondern das Gewicht der nichttragenden Teile als vorgegeben anzusehen ist, und daß nun diejenige Gestaltung des Flügels gesucht wird, durch die der gesamte Flügelwiderstand (induzierter plus Profilwiderstand), in dem sich das Holmgewicht mit auswirkt, ein Minimum wird. Es wäre sehr schwierig, diese Aufgabe etwa als Variationsproblem zu formulieren.