ABSTRACTThe paper presents three modifications for an improved performance in terms of increased power output of astraight-bladed VAWT by varying its pitch. Modification I examines the performance of a VAWT when thelocal angle of attack is kept just below the stall value throughout its rotation cycle. Although this modificationresults in a very significant increase in the power output for higher wind speeds, it requires abrupt changesin the local angle of attack making it physically and mechanically impossible to realize. Modification IIimproves upon the first by replacing the local angle of attack by the blade static-stall angle only when theformer exceeds the latter. This step eliminates the two jumps in the local effective angle of attack curve butat the cost of a slight decrease in the power output. Moreover, it requires a discontinuous angle of attackcorrection function which may still be practically difficult to implement and also result in an early fatigue.Modification III overcomes the limitation of the second by ensuring a continuous variation in the local angleof attack correction during the rotation cycle through the use of a sinusoidal function. Although the poweroutput obtained by using this modification is less than the two preceding ones, it has the inherent advantageof being practically feasible.
R = rotor radius at the equator, mr = local rotor radius, mS = rotor swept area, m2
u = interference factorV = local induced velocity, m/sV∞ = free stream wind velocity, m/sW = local relative velocity, m/sX = local tip-speed ratio, rω/VXEQ = tip-speed ratio at the equator, Rω/V∞xROyR = system of rotational cartesian coordinatesz = local turbine height, mα = local angle of attack, degαe = local effective angle of attack, α + ∆αe degαstall = local static-stall angle, degαw = atmospheric wind shear component∆αe = local effective angle of attack
Superscripts:(–) = average value(′) = downstream valueWithout (′) = upwind value
Subscripts:EQ = equatorial valuei = value at a height equal to the center of
slice i∞ = free-stream value
INTRODUCTION
The continuing oil crisis, environmental pollutionand global warming has been a matter of great con-cern for the industrialized nations and, therefore,caused a renewed interest in the field of renewableenergy resources. Amongst these resources, windenergy has been in the forefront of progress anddevelopment specifically because of its proven ef-ficiency, a growth rate of 40% and very positive en-vironmental impact [1]. Many countries, such asDenmark [2], are investing a lot of money in re-search in the field of Wind Energy that is now notonly evolving into a significant proportion of theirannual power production but is also a source of en-couragement for the wind energy community. Thisencouragement is vital for a continued research anddevelopment. The wind turbine, whether it may bea Horizontal Axis Wind Turbine (HAWT) or a Ver-tical Axis Wind Turbine (VAWT), offers a practicalway to convert wind energy into any other usefulforms of energy. The major research effort at EcolePolytechnique de Montreal, Canada, is devoted to-wards the development and improvement of the per-formance prediction of VAWTs [3–15].
In this regards, a study was carried to determineif the performance of a VAWT, in terms of the poweroutput, could be improved by operating the bladejust below stall, whether static or dynamic. Thebasic idea was to actively lower the effective an-gle of attack of a VAWT blade as it approachesstall. As such the blade never experiences stallthroughout its cycle of rotation. Moreover, it wasdesired to determine whether or not symmetry inthe tangential force component could be achievedthrough the variation of the geometric angle of at-tack of the VAWT blade during its cycle of rota-tion analogous to the concept of symmetry of liftin helicopter blades in forward flight. In order tostudy the above ideas, three modifications to theexisting CARDAAV code [11] were studied. A 2-blade straight-bladed VAWT model was used for
this study. The results were examined to determinethe potential of such modifications and their impacton the performance of a VAWT. In the sections thatfollow, some brief details of the CARDAAV codealong with a discussion on the three modificationsare presented. The paper ends with some brief con-clusions.
THE CARDAAV CODE
The first approach using a momentum model toanalyze the flow field around a VAWT was devel-oped by Templin [17] who considered the rotor asan actuator disk enclosed in a simple streamtubewhere the induced velocity through the swept vol-ume of the turbine is assumed to be constant. Anextension of this method to the multiple-streamtubemodel was developed by Strickland [18] who consid-ered the swept volume of the turbine as a seriesof adjacent streamtubes. Paraschivoiu [7, 8, 9, 10]developed an analytical model, the double-multiplestreamtube model (DMS model), that considers amultiple-streamtube system divided into two partswhere the upwind and downwind components of theinduced velocities at each level of the rotor are cal-culated by using the principle of two actuator disksin tandem. The DMS model has been implementedin CARDAAV computer code [11] in which VAWTscan be designed to meet the desired specifications.
CARDAAV has the capability to analyze sev-eral pre-defined or user-defined rotor shapes withstraight or curved blades (parabola, catenary, idealand modified troposkien, and Sandia shape). CAR-DAAV is also able to account for the so-called “sec-ondary effects,” such as those of rotating tower,strut, and spoiler. Dynamic stall can also be in-cluded in the analysis. Significant changes arethen observed in the aerodynamic loads and perfor-mance in the range of low tip-speed ratios. Severaldynamic-stall models are available in CARDAAV.The important ones include (a) the incidence-delaymethod originating from Boeing-Vertol which isbased on numerical correlations of the dynamic-stalldelay with pitch-rate parameter (also known as Gor-mont’s model [19]), (b) a modified Gormont’s modelbased on the turbulence level effects between theleft and right sides of the rotor, and (c) a modi-fied Gormont’s model [20] which introduced somedamping effects at very large angle of attack. Dy-namics stall results in increased peak aerodynamictorque and affects the structural fatigue of the Dar-rieus turbine. This effect significantly impacts thedrive-train generator sizing and system reliability.
CARDAAV has made it possible to design, ana-lyze and build more efficient and low cost wind en-ergy systems such as the Darrieus-rotor VAWT. TheCARDAAV code is used to determine aerodynamicforces and power output of VAWTs of any geometryat the specified operational conditions. Wind speedcan vary with height according to a power law. Theprogram output consists of the local induced veloci-ties, the local Reynolds numbers and angle of attack,the blade loads, and the azimuthal torque and powercoefficient data. Each of these is calculated sepa-rately for the upwind and downwind halves of therotor. The numerical models used by the programhave been validated through comparison with exper-imental data, obtained from laboratory tests madein wind or water tunnels, for different Darrieus-typeVAWTs, thus, making CARDAAV a very attractiveand efficient design and analysis tool.
Figures 1 and 2 show a comparison between thethe normal force coefficients and the power out-put in comparison with experiments and other nu-merical codes. In Fig. 1, the CARDAAS-1D andCARDAAS-3D are different variants of the CAR-DAAV code that take into account atmosphericturbulence through use of different scholastic windmodels. In Fig. 2, AM is an empirical constant usedto correct CL and CD for dynamic stall effects [8].
CARDAAVCARDAAS-1D turb.CARDAAS-3D turb.Experimental data
No
rma
lfo
rce
coef
fici
ent,
C
1.2
0.8
0.4
-0.4
-0.8
-1.6
2.0
-2.0
0.0
-1.2
1.6
N
Figure 1: Normal-force coefficient versus az-imuthal angle at the equator [8].
The Aerodynamic Model
Since the focus of the present study is on astraight-bladed VAWT (see Fig. 3), a simplifiedmodel for the double-multiple streamtube theory [3,
Figure 2: Comparison of the power outputpredicted by CARDAAV with experimentaldata [8].
V z( )R
H
z
zEQ
�
H
O
zR
8
Figure 3: A Sketch of the straight-bladedVAWT model used in this study.
7] applicable to straight-bladed rotor is briefly out-lined below.
For a straight-bladed VAWT rotor, the resultantvelocity vector W during the upwind half of the ro-tation cycle (−π/2 ≤ θ ≤ π/2) can be found fromthe velocity diagram of Fig. 4:
W 2 = V 2[(X − sin θ)2 + cos2 θ] (1)
The expression for the local angle of attack α can
Figure 4: The various angles, forces and ve-locity vectors in the equatorial plane.
also be found from geometric considerations:
α = arcsin
[cos θ√
(X − sin θ)2 + cos2 θ
](2)
Assuming that the free-stream velocity profile isgiven by the relation:
V∞i/V∞ = (zi/zEQ)αw (3)
Then by applying the blade element theory andequating the change in momentum to the drag onthe rotor for each streamtube, one finds:
f(V/V∞)2 = πη(V/V∞) [(V∞i/V∞) − (V/V∞)] (4)
or, in terms of interference factor,
fu = πη(1 − u) (5)
where η = r/R ≡ 1 and f is the upwind function [3,7] that characterizes the upstream half-cycle of therotor on the blade element rotating in this zone.The upwind function is given by the equation:
f =Nc
8πR
∫ π/2
−π/2
CN cos θ + CT sin θ
| cos θ|(
W
V
)2
dθ (6)
where
CN = CL cos α + CD sin αCT = CL sinα − CD cos α (7)
The blade section lift and drag coefficients, CL
and CD respectively, are obtained by interpolatingknown experimental data using both the local bladesection Reynolds number Reb = Wc/ν∞ and thelocal angle of attack α. Using Eq. 1, Reb is givenas a function of the induced upwind velocity V foreach blade element in rotation. Defining the turbineReynolds number as Ret = Rωc/ν∞, one obtainsthe relationship between Reb and Ret:
Reb = (Ret/X)√
(X − sin θ)2 + cos2 θ (8)
The Computational Procedure
For a given rotor geometry and rotational speedω, a value of the local tip-speed ratio X is chosenby assuming that the interference factor u is unity.Thus, Reb and α are evaluated as first approxima-tion and the airfoil CL and CD characteristics areinterpolated from test data, (the blade section typeis usually a symmetrical NACA airfoil). Then, withEq. 7, the normal and tangential force coefficients ofthe blade section are estimated while Eq. 6 allowsthe upwind function f to be evaluated. With thefirst value of f , Eq. 5 is used to calculate anothervalue for the interference factor and the iterationscontinue until successive sets of u are reasonablyclose. Convergence is rapid with an error of lessthan 10−4. Once the final value of the upwind zoneV has been calculated, the local relative velocity Wis determined from Eq. 1 and the local angle of at-tack α from Eq. 2. A similar procedure is adoptedfor the downwind half cycle.
The geometric characteristics as well as operat-ing conditions of the straight-bladed VAWT aero-dynamic model chosen for this study are listed inTable 1 below.
Table 1: The geometric characteristics andthe operating conditions used in this study.
Figure 5: Local angle of attack as a functionof the azimuthal angle.
Azimuthal angle, θ (deg)
Tan
gent
ialf
orce
,FT
(N)
-90 0 90 180 270-200
0
200
400
600 Static stall onlyStatic and dynamic stallmod1
Figure 6: Tangential force as a function of theazimuthal angle.
MODIFICATIONS
In general, the modifications were mainly per-formed in the aerodynamics module of the CAR-DAAV program, and, in particular, in the way itcalculates the local angle of attack α. Overall, threedifferent modifications were studied that are de-scribed below along with the associated results.
Modification I
In modification I (mod1), the local blade angle ofattack α is kept just below the local stall angle atall points along the rotation cycle. This is accom-
Azimuthal angle, θ (deg)
Tor
que
coef
fici
ent,
CQ
-90 0 90 180 270-0.04
0.00
0.04
0.08
0.12
0.16
0.20
0.24 Static stall onlyStatic and dynamic stallmod1
Figure 7: Torque coefficient as a function ofthe azimuthal angle.
Equatorial wind speed, VEQ (m/s)
Pow
er,P
(kW
)
0 5 10 15 20 250
5
10
15
20
25Static stall onlyStatic and dynamic stallmod1
Figure 8: Power produced as a function of thewind speed at the equator.
plished by introducing an effective angle of attackαe(= α + ∆αe) term as an additional step in thecomputations. For mod1, ∆αe = αstall − α. Thus,essentially the local geometric blade angle of attackα is increased/decreased to the local blade static-stall angle value and as such the blade never expe-riences stall since it avoids the stall region, whetherstatic or dynamic, altogether. Hence the lift and thetangential force only increase with the wind speedand the power output is not affected by stall effects.
Figure 5 illustrates the variation of the local an-gles of attack α and αe during a complete rotation.Here the α is replaced by αe (solid line) in the
Azimuthal angle, θ (deg)
Ang
leof
atta
ck,α
(deg
)
-90 0 90 180 270-20
-10
0
10
20
30 αe = α + ∆αe ≤ αstall
α∆αe
Figure 9: Local angle of attack as a functionof the azimuthal angle.
Azimuthal angle, θ (deg)
Tan
gent
ialf
orce
,FT
(N)
-90 0 90 180 270-200
0
200
400
600 Static stall onlystatic and dynamic stallmod1mod2
Figure 10: Tangential force as a function ofthe azimuthal angle.
computations. Figures 6 and 7 not only indicatean overall increase in the area under the tangentialforce and torque coefficient curves for mod1 but alsoa smoother variation along the rotation cycle. Asa result of this modification, the power output in-creases linearly with wind speed (Fig. 8) . Moreover,the increase in power output at high wind speeds issignificantly higher than the case where static ordynamic stall or both are encountered by the blade.However, a major drawback of this modification arethe two jumps in the local effective angle of attackat θ = −π/2 and π/2 that may be physically andmechanically difficult to implement as well as resultin an early fatigue. In order to overcome this diffi-
Azimuthal angle, θ (deg)
Tor
que
coef
fici
ent,
CQ
-90 0 90 180 270-0.04
0.00
0.04
0.08
0.12
0.16
0.20
0.24 Static stall onlyStatic and dynamic stallmod1mod2
Figure 11: Torque coefficient as a function ofthe azimuthal angle.
Equatorial wind speed, VEQ (m/s)
Pow
er,P
(kW
)
0 5 10 15 20 250
5
10
15
20
25Static stall onlyStatic and dynamic stallmod1mod2
Figure 12: Power produced as a function ofthe wind speed at the equator.
culty, a slightly different strategy is applied and isdiscussed next.
Modification II
In modification II (mod2), the local effectiveblade angle of attack αe is kept at the blade stallangle during the rotation cycle only when the geo-metric angle of attack exceeds the static-stall anglevalue αstall. This change eliminates the two jumpsin the local effective angle of attack at θ = −π/2and π/2 as shown in Fig. 9. As a result of thismodification, the tangential force and torque coeffi-cient curves also show a more gradual change or a
Azimuthal angle, θ (deg)
Ang
leof
atta
ck,α
(deg
)
-90 0 90 180 270-20
-10
0
10
20
30αe = α + ∆αe
α∆αe
αstall
Max. amplitude
Stall regions
Figure 13: Local angle of attack as a functionof the azimuthal angle.
Azimuthal angle, θ (deg)
Tan
gent
ialf
orce
,FT
(N)
-90 0 90 180 270-200
0
200
400
600 Static stall onlyStatic and dynamic stallmod2mod3 - max. amplitude
Figure 14: Tangential force as a function ofthe azimuthal angle.
smoother transition at θ = −π/2 and π/2 as evidentfrom Figs. 10 and 11, respectively. The decrease inthe area under the tangential force and torque co-efficient curves as a result of this modification isreflected in a slightly less power output than mod1as seen in Fig. 12. In this case as well, the poweroutput increases linearly with wind speed since thestall regions are avoided during the rotation cycle.
From Fig. 9 we can see that mod2 does not com-pletely eliminate the drawback of mod1 since theevolution of the correction function for the local an-gle of attack curve (∆αe in Fig. 9) is still discontin-uous at some points which may not be practical to
Azimuthal angle, θ (deg)
Tor
que
coef
fici
ent,
CQ
-90 0 90 180 270-0.04
0.00
0.04
0.08
0.12
0.16
0.20
0.24Static stall onlyStatic and dynamic stallmod2mod3 - max. amplitude
Figure 15: Torque coefficient as a function ofthe azimuthal angle.
Equatorial wind speed, VEQ (m/s)
Pow
er,P
(kW
)
0 5 10 15 20 250
5
10
15
20
25Static stall onlystatic and dynamic stallmod1mod2mod3 - max. amplitudemod3 - 50% of max.
amplitude
Figure 16: Power produced as a function ofthe wind speed at the equator.
implement and also cause early fatigue due to theabrupt dynamical loads. A continuous correctionfunction, such as a sinusoidal function, will over-come this limitation. This idea is explored next.
Modification III
In modification III (mod3), the evolution of thelocal angle of attack correction ∆αe for the localangle of attack curve is replaced by a smooth con-tinuous function such as a sinusoidal function rep-resented by a dotted line in Fig. 13. The maximumamplitude of the sinusoidal correction function is setequal to the maximum difference between the local
geometric angle of attack α, and the blade static-stall angle αstall. As a consequence, the local effec-tive angle of attack αe curve also follows a sinusoidalchange. For this case, the blade does experience dy-namic stall at a few locations during the rotationcycle as evident from Fig. 13. As a consequence,abrupt changes in the tangential force and torquecoefficient curves are predicted by the program onlywhen the blade experiences dynamic stall as seen inFigs. 14 and 15, respectively. This can, however, beeliminated by decreasing the maximum amplitude ofthe correction function till the dynamic stall regionsare avoided all-together yielding a smooth continu-ous variation in the tangential and normal forcesand the torque. Moreover, a smooth and contin-uous correction function for ∆αe is more practicalsince it is physically and mechanically feasible toimplement.
A drawback of mod3 is that the power output, al-though significantly higher than the one without anymodification, is much less than in the case of mod1and mod2 as shown in Fig. 16. Another interestingpoint to note in Figs. 16 and 17 is that by usinghalf the magnitude of the maximum amplitude, thepower output is seen to increase. This increase isdue to the fact that the local effective angle of at-tack αe value is higher at each azimuthal locationfor the case when 50% of the maximum amplitudeis used. Thus, it suggests that an optimum value forthe maximum magnitude must be determined thatyields the maximum power output. Figure 17 showsthe variation of the normal force during the rotationcycle after applying modifications II and III.
Azimuthal angle, θ (deg)
Nor
mal
forc
e,F
N(N
)
-90 0 90 180 270-2000
-1000
0
1000
2000 mod2mod3 - max. amplitudemod3 - 50% of max. amplitude
Figure 17: Normal force as a function of theazimuthal angle.
CONCLUSIONS
In this study, three modifications for an improvedperformance in terms of increased power output ofa straight-bladed VAWT by introducing a variationin the blade pitch have been presented along withresults. These modifications include:
(a) Modification I examines the performance of aVAWT when the local angle of attack is kept just be-low the stall value throughout its cycle of rotation.This is accomplished by adjusting the local geomet-ric angle of attack of the blade. This modificationresults in a very significant increase in the poweroutput for wind speeds above 10 m/s. However,this modification requires sharp changes (jumps) inthe local angle of attack making it physically andmechanically impossible to realize.
(b) Modification II improves upon the first by re-placing the local geometric angle of attack by the lo-cal profile stall angle, obtained in the same manneras for modification I, but only when the former ex-ceeds the latter. As a consequence, this modificationeliminates the two jumps in the local effective angleof attack curve but at the cost of a slight decrease inpower output. Moreover, it also renders the angleof attack correction function as discontinuous whichmay be practically difficult to implement and resultin an early fatigue. A remedy for this limitation isto introduce a smooth and continuous variation inthe local angle of attack correction.
(c) Modification III overcomes the limitation ofthe second by ensuring a continuous variation in thelocal angle of attack correction during the rotationcycle through the use of a sinusoidal function. Theamplitude of the sinusoidal function is set equal tothe maximum difference between the local geomet-ric angle of attack and the blade static-stall angle.The difference is only calculated when the formerexceeds the latter. Although the power output ob-tained by using this modification is less than thetwo preceding modifications, it has the inherent ad-vantage of being practically feasible.
Although the above modifications suggest an in-crease in the power output of a straight-bladedVAWT, the scheduling of blade angle of attacknegates some nice features of VAWTs, i. e. theirinsensitivity to wind direction as well as the naturalfeathering effect. It would, therefore, require spe-cial wind-direction sensors, blade pitch mechanismand power generation and transmission systems de-signed for heavier loads to achieve the desired per-formance.
Acknowledgements
The authors would like to acknowledge the sup-port of Natural Sciences and Engineering ResearchCouncil (NSERC), Canada, through their GrantNo. 1442.
References
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[14] Brochier, G., Fraunie, P., Beguier, C., andParaschivoiu, I., “Experimental Study of DynamicStall on Darrieus Wind Turbine Blades,” Proceed-ings of the 20th Intersociety Energy Conversion En-gineering Conference, Miami, Florida, USA, Au-gust 18–23, 1985.
[15] Desy, P., and Paraschivoiu, I., “Aerodynamics ofSmall-Scale Vertical Axis Wind Turbines,” Pro-ceedings of the 20th Intersociety Energy Conver-sion Engineering Conference, Miami, Florida, Au-gust 18–23, 1985.
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