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Energy 66 (2014) 689e698

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Energy

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Proposal for an innovative chord distribution in the Troposkienvertical axis wind turbine concept

Gabriele Bedon, Marco Raciti Castelli, Ernesto Benini*

Department of Industrial Engineering of the University of Padua, Via Venezia 1, 35131 Padua, Italy

a r t i c l e i n f o

Article history:Received 25 June 2013Received in revised form24 December 2013Accepted 2 January 2014Available online 28 January 2014

Keywords:Darrieus VAWT (vertical axis wind turbine)WOMBAT (Weatherly Optimization Methodfor Blades of Air Turbines) algorithmVariable chord distributionAerodynamic optimization

* Corresponding author.E-mail addresses: gabriele.bedon@dii.unipd.it (G.

unipd.it (M. Raciti Castelli), ernesto.benini@unipd.it (

0360-5442/$ e see front matter � 2014 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2014.01.004

a b s t r a c t

An innovative design for the Troposkien concept is introduced by means of an advanced chord distri-bution, computed using the WOMBAT (Weatherly Optimization Method for Blades of Air Turbines) al-gorithm for the performance optimization of vertical axis wind turbines. Five rotor blade architectures,characterized by a constant value of the thickness-to-chord ratio and a varying chord length along theblade span, are evaluated. The optimization process is conducted with respect to the power coefficientfor a target wind speed of 9 m/s, obtaining a consistent improvement of rotor performance with respectto the baseline blade configuration.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In the near future, VAWTs (vertical axis wind turbines) are goingto play an important role in the energy production planning. In fact,their typical characteristics of small size and lightweight allow theirinstallation also in the urban context, on the building roofs andyards. In this scenario, the Darrieus VAWT represents one the mostpromising architecture [1], thanks to its high aerodynamicperformance.

The traditional design of the Darrieus rotor involves a constantdistribution of thickness and chord along the whole blade span. Inthe present analysis, the optimal chord distribution along the bladelength is searched for increasing rotor aerodynamic performance.

The Troposkien design procedure usually involves two steps:

� the first step is conducted by considering a BE-M (BladeElement-Momentum) or Vortex based algorithm, thanks to theirrapidity to provide a reliable estimation of the rotor perfor-mance for a given parameter set;

� the second step involves the turbine simulation bymeans of CFD(Computational Fluid Dynamics) codes, which provide a very

Bedon), marco.raciticastelli@E. Benini).

All rights reserved.

deep knowledge of the rotor aerodynamics, allowing a moreadvanced power characterization.

Simulation tools based on the BE-M Theory are widely adoptedand have been proposed and improved by many authors [2e6]. Themain issue regarding these algorithms is their necessity to useairfoil aerodynamic coefficients for angles of attack over 360� [7e9]and low airfoil Reynolds numbers, consistently smaller than thosegenerally considered for aeronautical use [10].

Some aerodynamic databases for a limited number of profilesare available in literature: NACA 0012, SG6043, SD7062 and DU06-W-200 polars, characterized by an angle of attack ranging up to360� and a Reynolds number ranging from 65,000 to 150,000, wereprovided byWorasinchai et al. [11]. Sheldahl and Klimas [12] testedNACA 0009, NACA 0012 and NACA 0015 profiles, considering anglesof attacks ranging from 0� to 180� and a limited range of Reynoldsnumbers (from 350,000 to 700,000), successively extending theresults to a wider range of Reynolds numbers and also to differentairfoil geometries by means of numerical methods.

In the present work, the WOMBAT (Weatherly OptimizationMethod for Blades of Air Turbines) design algorithm developed byBedon et al. [13] is adopted. This algorithm includes a rotor aero-dynamic performance prediction tool based on the BE-M codedeveloped by Raciti Castelli et al. [14]. The simulation is based onthe aerodynamic databases provided by Sheldahl and Klimas [12]and allows also the estimation of the aerodynamic coefficients of

Fig. 1. Sandia 2 m Darrieus wind turbine installation (from [20]).

G. Bedon et al. / Energy 66 (2014) 689e698690

different NACA airfoils by means of the interpolation of such polars.Nevertheless, for the present calculations, since the profile thick-ness/chord ratio is maintained constant, this functionality is notexploited.

The design procedure is conducted optimizing a given baselinegeometry. Nowadays a great number of optimization softwarebased on genetic algorithms is available on commercial platforms,allowing the solution of different problems: applications such asfluid dynamics, structural and modal analyses, as well as financialevaluations represent some examples. The best aerodynamicconfiguration for the rotor blade can therefore be found using aproper evolutive algorithm, by previously setting the chosenobjective functions. An optimum aerodynamic design model for awind turbine blade was presented by Bourguet et al. [15], whoadopted a multi-criteria optimization algorithm coupled to a CFDsimulation code, in order to gain an increase of the nominal powerproduction and the reduction of the blade weight, suggesting theadoption of the NACA 0025 profile as the best option. Carrigan et al.[16] considered a baseline rotor architecture based on a NACA 0015airfoil and increased the turbine efficiency by 6% by means of adifferential evolutionary algorithm based on a CFD simulation code.The WOMBAT algorithm adopts the multi-objective genetic algo-rithm based on work of Deb [17] and implemented in the com-mercial software Matlab [18]. The reliability of such algorithm wasproven by Bedon et al. [19] by optimizing the 2-m Sandia turbinetested by Sheldahl [20], imposing a constant evolution of bothchord and a thickness along the rotor blade span [13], achieving anincrease in performance up to 6%.

2. The case study

The blades of a Darrieus VAWT are traditionally manufacturedextruding a single profile over the whole blade length, eventuallybent to form a SCS (straight-circular-straight) shape, a practicalapproximation for the Troposkien architecture [21].

The 2-m Sandia turbine tested by Sheldahl [20] is here consid-ered as a case study. The rotor is 2 m high, the nominal radius is0.98 m and the blade shape is SCS. The main geometrical specifi-cations of the turbine are reported in Table 1. A picture of the windturbine installation in the Sandia site test is reported in Fig. 1.

This type of turbine is widely analyzed in literature, both fromthe numerical and from the experimental point of view. TheWOMBAT algorithm has been preliminary tested using such a rotorarchitecture [19]: the validation process showed a quite highsensitivity of the numerical results to the selected dynamic stallmodel, especially for lower wind speeds. As a result, the use of theGormonteStrickland model [22] determined a better predictionwith respect to the experimental data from Sheldahl [20], as can beseen in Fig. 2. For further information about the WOMBAT opti-mization algorithm and the validation of its performance predic-tion code, see Refs. [13,19].

Due to the inherent characteristics of the BEM approach (bi-dimensional simulation and streamtube independency), the pre-diction reliability is assumed not to be sensibly compromised by a

Table 1Main geometrical features of the Troposkien rotor tested by Sheldahl [20],here assumed as baseline configuration for the optimization process.

H 2 mR 0.98 mN 3Blade profile NACA 0012Blade shape Straight-circular-straight (SCS)c 58.77 mm

chord variation along the blade span. In order to confirm the con-sistency of this assumption, an additional validation is shown inFig. 3, where WOMBAT predictions are compared to experimentaldata from the Sandia 34-m wind turbine [23], characterized by avariable chord. The aerodynamic coefficients for the SAND 0018/50profile are retrieved from Ref. [24].

3. Optimization methodology

In the first optimization campaign, the thickness/chord ratio,which defines the shape of the profile, is considered uniform andequal to the value of the Sandia turbine (0.12), considered as abaseline configuration. In order to have a fair comparison, the sameair density (1 kg/m3 [25]) registered at the Sandia test site is hereconsidered.

Due to its numerical nature, the algorithm can be adapted toaccept a non uniform chord distribution. Anyway, experimentaldata for small turbines with variable chord are not available inliterature: the presented numerical results should therefore beconfirmed by further experimental activity.

As described in Ref. [13], theWOMBATalgorithm is structured asa loop which involves a genetic algorithm, a BE-M simulation code

Fig. 2. Comparison between experimental data and BE-M predictions (based on theaerodynamic database provided by Ref. [12]) without and with different dynamic stallmodels: Gormont-Strickland (GS) and Gormont-Berg (GB). Rotor angular speed is460 rpm (from [19]).

Fig. 3. Comparison between experimental data and BE-M predictions (based on theaerodynamic database provided by Refs. [12,24]) at different angular speeds for theSandia 34 m wind turbine [23].

Fig. 4. Functional scheme of the WOMBAT algorithm for wind turbine design (from[13]).

G. Bedon et al. / Energy 66 (2014) 689e698 691

and an aerodynamic coefficient predictor. This last feature is notused in the present analysis, since only the chord distribution isconsidered. The algorithm scheme is reported in Fig. 4.

The adopted optimization tool is the multi-objective geneticalgorithm developed by Mathworks, based on the work of Deb [17]and implemented as gamultiobj function in the software Matlab[18]. The algorithm includes the following functionalities:

� elitarism selection based on the genotype or phenotypedistance;

� roulette wheel parent selection;� random or uniform crossover;� mutation function based on the Gaussian.

A total of 50 individuals are evaluated and evolved for 100generations. Among the individuals of the first generation, thebaseline Sandia profile, is included.

The number of decisional parameters np provided by the geneticalgorithm is equal to half the number of vertical divisions. Twodifferent individual creation methods are tested and implementedin theWOMBATalgorithm, as shown in Fig. 5. The Troposkien shapeof rotor blade mean line is assumed to be fixed, whereas thedecisional parameters are considered as:

1. independent numerical values of the chord (np genes), definedas “Free Optimization”;

2. control point coordinates for a curve (np/2 abscissas and np/2ordinates) that describe the trend of the chord, defined as“Spline Optimization”.

The two methods would ideally provide the same optimal so-lution but, since the considered number of generations is finite, the“Free Optimization” approach does not guarantee a smooth trend,thus the possibility for a physical realization. On the contrary, the“Spline Optimization” approach always considers constructiblegeometries. The performance can however be limited because ofthe distribution shape constraint.

Traditionally, the main purpose of the optimization activity is tomaximize an efficiency parameter for the target operating condi-tions. The test report by Sheldahl [20] does not deal with the designof the turbine and, therefore, the target conditions are not clearlystated. The turbine optimization can be conducted considering anywind velocity, since the algorithm would provide the optimal

geometry for the given speed. A wind speed equal to 9 m/s isconsidered for the proposed computations. The efficiency param-eter considered in the optimization process is the power coefficientCP: it can be considered as an efficiency of the conversion system[10] and is defined as

G. Bedon et al. / Energy 66 (2014) 689e698692

CP ¼ P0:5$r$A$U3 (1)

N

being P the power production, r the air density, A the swept area ofthe rotor and UN the freestream wind speed.

Thewind turbine is assumed to be coupledwith a variable speedgenerator and would ideally operate at the rotational speed whichmaximizes the power coefficient.

The gamultiobj algorithm tends to a solution that minimizes thefitness value. The fitness of the profile is thus computed consideringthe following expression:

fitness ¼ 1CP;Individual;9

¼ 1max

�CP;Individual;9;k

� (2)

where CP,Individual,9 is the power coefficient of the consideredconfiguration for a wind speed of 9 m/s and CP,Individual,9,k is thepower coefficient of the considered configuration, computed for awind speed of 9 m/s and for the kth rotational speed.

Additional optimization campaigns are successively conduced,varying the thickness/chord ratio. In fact, the optimization resultsfrom the analysis by Bedon et al. [13] for a blade architecturecharacterized by a constant chord distribution, highlighted anoptimal thickness/chord ratio equal to 0.15, differently fromSandia baseline configuration (which is based on the NACA-0012airfoil) [20]. Different airfoil profiles are therefore additionallyconsidered, NACA-0015, -0018, -0021 and -0025, assuming auniform chord equal to the Sandia turbine case for the baselineconfigurations.

Fig. 5. Frontal view of the Troposkien rotor blade: the decision variables can be interpretedtrend over the blade length (b).

4. Optimization results

Fig. 6 shows the optimal chord distribution provided by theWOMBAT algorithm, considering a thickness-to-chord ratio equalto 0.12, 0.15, 0.18, 0.21 and 0.25.

The results show that the optimal chord distribution, consid-ering a fixed airfoil thickness/chord ratio, is not uniform. Theoptimization convergence is not however complete: the resultingarchitectures are in fact affected by discontinuities in thegeometrical parameters that will eventually need to be fixedbefore rotor blade manufacturing. The same trend is approxi-mately registered for all the configurations, presenting a chorddistribution with two (for t/c ¼ 0.15 and 0.18) or three (for t/c ¼ 0.12, 0.21 and 0.25) maximum values, at the middle height(only for t/c ¼ 0.12, 0.21 and 0.25) and at two intermediate bladesections, comprised between the equatorial plane and the bladetips (for all the considered thickness-to-chord ratios). The curvestiffness corresponding to the middle height peak can be veryhigh (especially for t/c ¼ 0.12): in the practical realization thispeak needs to be neglected due to static and fatigue concernsregarding blade geometries characterized by sudden changes inthe section area.

An interpretation for the chord trend is presented as follows:the minimum value of the chord at the rotor top and bottom leadsto a lower aerodynamic resistance of the blade sections where thepower production is very low; the chord length towards the centeris higher in order to increase the aerodynamic load where theradius is increasing, thus enhancing the power production; thesuccessive chord reduction represents a compromise between theincrease of the aerodynamic load and the decrease of the

as independent chord values (a) or coordinates for a spline curve describing the chord

Fig. 6. Baseline (solid line), free optimized (dash-dot line) and spline optimized (light dashed line) chord distributions with respect to total blade height.

G. Bedon et al. / Energy 66 (2014) 689e698 693

Fig. 7. Evolution of the power coefficient as a function of the tip speed ratio for the baseline rotor configuration (solid line), the free optimized blade architecture (dash-dot line) andthe spline optimized one (light dashed line).

G. Bedon et al. / Energy 66 (2014) 689e698694

G. Bedon et al. / Energy 66 (2014) 689e698 695

interference factor, that would determine an excessive slowdownof the current if the chord kept on increasing.

As can be clearly seen, slightly different configurations areregistered for the “Free Optimization” and the “Spline Optimiza-tion”, due to the constraint imposed in the second method. Theconstraint to create a spline curve, imposed on the decisional pa-rameters, provides in fact a smoother distribution, thus simplifyingrotor blade construction.

The main performance parameter, considered in order to eval-uate the rotor blade performance, is the power coefficient. This isanalyzed with respect to the tip speed ratio, defined as:

l ¼ uRUN

(3)

being R the maximum rotor radius.The comparison between the optimal solutions and the baseline

configuration is shown in Fig. 7.The power coefficient is increased in all the optimized archi-

tectures with respect to the baseline configuration. The “FreeOptimization” reaches higher peak values than the “Spline Opti-mization”, because the chord distribution is not limited by anyshape constraint. However, as explained before, this could preventthe physical realization of the rotor. Moreover, it can be observedthat the peak is moved towards lower values of the tip speed ratio,determining a reduction of the optimal rotor angular velocity and,thus, a decrease of the inertial loads on rotor blades.

Table 2 summarizes the computed values of the peak powercoefficient for the considered blade configurations, as well as thecorresponding angular velocities.

5. Discussion of the numerical results

The performance increase obtained by all the optimized rotorblade configurations needs to be clarified by additional analyses.For this purpose, the local power coefficient along the blade spanCP,y is computed for every vertical blade subdivision. Local values ofthe power coefficient with respect to the blade height are reportedin Fig. 8.

It can be observed that the local power coefficient of the opti-mized rotors is higher than the baseline configuration for almostevery vertical blade subdivision. In particular, the higher increase inperformance is registered at the center of the blade, where theradius is maximum. This subsequently leads to a consistent in-crease in the overall power production in the central portion of therotor blade, affecting the overall efficiency of the rotor, due to the

Table 2Peak values of the power coefficient and corresponding rotational speeds for thebaseline rotor configuration and the two optimized blade architectures.

Airfoil profile Configuration CP [e] n [rpm]

NACA 0012 Baseline 0.3129 450Free optimization 0.3484 400Spline optimization 0.3269 350

NACA 0015 Baseline 0.3265 400Free optimization 0.3701 350Spline optimization 0.3556 300

NACA 0018 Baseline 0.3108 400Free optimization 0.3481 300Spline optimization 0.3414 300

NACA 0021 Baseline 0.2912 400Free optimization 0.3445 300Spline optimization 0.3319 300

NACA 0025 Baseline 0.2619 400Free optimization 0.3727 300Spline optimization 0.3623 300

high lever-arm of such blade stations with respect to the turbineshaft.

Fig. 9 shows the evolution of instantaneous torque coefficientwith respect to a single rotor blade, defined as:

Ctb ¼ Tb0:5rU2

NAR(4)

where Tb is the torque generated by a single rotor blade, as afunction of its azimuthal position.

It can be observed that the blade torque is sensibly increased,especially during the upwind portion of the revolution. A peak inthe torque coefficient is registered in the very same angular posi-tion where it is generated by the baseline configuration, but thevalue is considerably increased (up to three times) for the opti-mized blade architectures. On the contrary, in the downwind sec-tion, the computed torque is almost the same as in the baselineconfiguration, leading to similar aerodynamic performance.

The upwind increase in rotor efficiency for the optimized bladearchitectures can be further highlighted considering the variationin the interference factor a, defined as:

a ¼ 1� Vblade;i

UN(5)

where Vblade,i is the flow velocity at blade section (i can be up ordown).

The distribution of the interference factors along the swept areaof the rotor is represented in Fig. 10 for the t/c ¼ 0.18 baseline andoptimized cases.

The distribution of the interference factor on the swept area ofthe rotor is very similar for all the three examined configurations,but its magnitude is sensibly higher for the optimized architectures,due to the increased power production: this means that a greateramount of energy is extracted from the air, decreasing its velocity.As previously highlighted, the increase in power coefficient isexperienced especially at the equatorial plane of the rotor, wherethe radius is maximum. As a matter of fact, a dramatic increase ofthe interference factor is registered at the equatorial plane for thetwo optimized blade architectures.

As observed in the previous section, being not limited by anyshape constraint, the free optimized blade geometry registershigher values of the interference factor with respect to the splineone, due to its higher efficiency.

6. Conclusion and future work

A variable chord distribution for the blade of a Troposkien ver-tical axis wind turbine is proposed, leading to increased rotorperformance with respect to the baseline configuration, charac-terized by a uniform chord distribution along the whole span.

Five optimization campaigns are conducted, consideringdifferent thickness-to-chord ratios for the blade section, deter-mining a similar evolution of the chord length along the blade span.The registered increment in rotor performance is appreciable andcan reach up to 6% with respect to the considered baselineconfiguration. The highest increases are achieved with both NACA0015 and NACA 0025 profiles. Moreover, the performance increaseoccurs with a lower optimal angular velocity than the one regis-tered on the original configuration, leading to a reduced bladeloading due to inertial forces. The registered enhancement in rotorperformance is mainly located in the upwind zone, where theaerodynamic torque is increased up to three times with respect tothe baseline solution. Finally, the performance is strongly increased

Fig. 8. Local values of the power coefficient with respect to the blade height for the baseline rotor configuration (solid line), the free optimized blade architecture (dash-dot line)and the spline optimized one (light dashed line).

G. Bedon et al. / Energy 66 (2014) 689e698696

Fig. 9. Evolution of the torque coefficient relative to a single rotor blade with respectto its azimuthal position (wind is coming from the upper side of the figure) for thebaseline rotor configuration (solid line), the free optimized blade architecture (dash-dot line) and the spline optimized one (light dashed line).

Fig. 10. Distribution of the interference factor on the swept area for the baseline, freeoptimization and spline optimization configurations, t/c ¼ 0.18.

G. Bedon et al. / Energy 66 (2014) 689e698 697

G. Bedon et al. / Energy 66 (2014) 689e698698

close to the equatorial plane of the rotor, where the higher lever-arm allows to maximize the energy extraction.

A reduction in the chord length at the blade tips is registered forthe optimized geometries, determining a decrease in the aero-dynamic drag in those blade stations which generate a reducedcontribution to the overall rotor torque, being placed close to therotational axis. Approaching to the equatorial plane of the rotor, anincrease in the chord length allows enhancing the torque contri-bution of the blade sections characterized by higher radius, whilethe registered chord reduction at the equatorial plane should befurther investigated, in order to fully understand its contribution tothe energy extraction. It could be assumed that a too high chordvalue at the middle height of the rotor determines a relevantslowing of the air stream, thus dramatically reducing the flux ofkinetic energy through the central sections of the swept area, beingproportional to the cube of the wind velocity. A full CFD campaignof analysis could enlighten this assumption through the compari-son between a rotor blade characterized by a constantly increasingchord towards the middle of the rotor and a reduced chordconfiguration, as proposed in the present work.

Nomenclaturea [�] interference factorA [m2] rotor swept areac [m] airfoil chordCD [�] airfoil drag coefficientCL [�] airfoil lift coefficientCP [�] rotor power coefficientCP,y [�] local power coefficient relative to a vertical blade

subelementCtb [�] torque coefficient relative to a single rotor bladeCP,Individual,9 [�] power coefficient of the considered configuration

for a wind speed of 9 m/sCP,Individual,9,k [�] power coefficient of the considered configuration

computed for a wind speed of 9 m/s and for kthrotational speed.

h [m] height of a blade elementH [m] rotor total heightn [rpm] rotor angular speednp [�] number of genesN [�] number of bladesP [W] power produced by the turbiner [m] radius of a blade elementR [m] wind turbine maximum radiust [m] airfoil thicknessTb [Nm] torque generated by a single rotor bladeUN [m/s] wind speedVblade,i [m/s] flow velocity at blade section (i can be up or down)l [�] Tip speed ratior [kg/m3] air density (assumed 1 [kg/m3])

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