AIAA-2011-638

Performance Characterization of Cyclic Blade Pitch

Variation on a Vertical Axis Wind Turbine

D. W. Erickson, J. J. Wallace and J. Peraire

Massachusetts Institute of Technology

Department of Aeronautics and Astronautics, Cambridge 02139

This study characterizes the effects of cyclic blade pitch actuation on the efficiency andoperability of a high-solidity vertical axis wind turbine (VAWT). A VAWT utilizing a camand control rod mechanism to prescribe the pitch dynamics is constructed and tested inMITs Wright Brothers Wind Tunnel over a wide range of design and operational variables.The experiment concludes that a tuned first-order sinusoidal actuation system can achievea maximum absolute efficiency of 0.436, an increase of 35% over the optimal fixed-bladebaseline configuration, with self starting capabilities and drastically improved performanceat a wide range of suboptimal operating conditions. This performance is comparable tothat which as been previously reported for more structurally demanding variable pitchVAWT designs of lower solidity and fixed pitch designs operating at higher tip speed ratios(TSRs). Additionally, a low order order computational model is found to accurately predictthe trends and maximum efficiencies of the system.

Nomenclature

Atun Wind tunnel cross-sectional area

Aturb 2Rh Rotor swept area

c Blade chord

CpPV AWTPwind

Power coefficient (VAWT efficiency)

Cp95% Power coefficient with 95% confidence intervals

F Force on load cell

f Rotor frequency

h Blade height

I Generator circuit current

L Length of moment arm used for torque measurement

n Number of blades

p1 Upstream static pressure in wind tunnel

PVAWT VAWT shaft power

Pwind12ATV

31 Wind power

R Ideal gas constant

R Rotor radius

Professor, Associate Fellow AIAA

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American Institute of Aeronautics and Astronautics

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-638

Copyright 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Ramp Cam radius amplitude

Rcam Cam radius

Rmean Cam mean radius

t Blade thickness

T1 Upstream static temperature in wind tunnel

TSR RV Tip speed ratio

V Generator voltage

V1 Upstream velocity in wind tunnel

V Freestream velocityV Local freestream velocityV Design freestream velocity (20mph)Vrel Air velocity in blade reference frame

Vrotor R Rotor tip speed

(x, y) Coordinate system centered at rotor axis. The y axis points upstream.

Pitch angle

amp Pitch amplitude

mean Mean pitch

genIV

PV AWTGenerator efficiency

Angle to blade arm position

p1RT1 Static density in wind tunnel

n cR Rotor solidity

FL Rotor shaft torque

Phase angle (wind direction)

2pif Rotor angular velocity

I. Introduction

Vertical axis wind turbines (VAWTs) possess certain benefits that attract attention for use in applicationsill-suited for horizontal axis wind turbines (HAWTs). VAWTs are structurally simpler than HAWTs andcan potentially be scaled to larger sizes. They can be made of less expensive more conventional buildingmaterials than HAWTs and their power generators may be located at ground level for easy access. Inaddition, the tendency of an unloaded VAWT to spin out of control is inherently limited by the systemaerodynamics, adding an additional safety feature desirable for large systems and urban applications. Dueto simpler construction, modular designs, and less demanding maintenance, VAWTs could exhibit reducedlifecycle costs-per-Watt over HAWTs.

Fixed blade VAWTs do have their disadvantages, however, one of which is lower efficiency than a HAWT.Betz Law quantifies a physical limitation that exists on the maximum power that a VAWT of fixed sweptfrontal area can extract relative to the available energy in the upwind streamtube. This yields a maximumattainable efficiency of 0.593, while typical commercial VAWTs today operate with half that efficiency.1 AVAWT with low efficiency must be made larger to generate the same amount of power as a turbine withhigh efficiency, increasing materials costs and thus increasing the lifecycle cost of the VAWT.2 To keep sizeand cost to a minimum, it is important to research ways to increase the efficiency of VAWTs if they are tobecome economically viable.

Variable pitch in VAWTs has been investigated since the 1970s, and it is invariably found to providegreater efficiency than fixed pitch VAWTs. Because a VAWT blade is subject to cyclic variations in wind-speed and angle of attack, as demonstrated in Figure 1, efficiency improvements can result from corre-spondingly varying the pitch of the blades to optimize the angle of attack at each point in its cycle. Inaddition to increasing efficiency, variable pitch also increases rotor startup capabilities and improves off-design performance.3 These improvements have been validated theoretically and experimentally using both

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(a) Source of relative velocity variations (b) Relative velocity variations in blade reference frame

Figure 1: VAWTs experience cyclic variations in relative velocity and angles of attack. Figures 1a and1b show top-down views of these velocity variations in the blade absolute and relative reference framesrespectively.

forced-pitch actuation (e.g. cam-pushrod, eccentric pivot-pushrod, servo-actuation, etc.) and self-pitch actu-ation (e.g. aerodynamically-stabilized, centrifugal force stabilized, etc.) by researchers such as Vandenbergheand Dick4,5 Kirke and Lazauskas6,7 Grylls et al. (1978),8 Nattuvetty et al. (1982),9 Moran (1977),10 andPawsey and Barratt (1999).11

This study experimentally characterizes the benefits of cyclic blade angle variation over a wide rangeof VAWT configurations, characterizing efficiency improvements and sensitivities to suboptimal design pa-rameters. A high-solidity, forced-pitch VAWT was constructed utilizing a reconfigurable cam and controlrod mechanism to geometrically define the blade pitch dynamics. Measurements of turbine rotor speed andtorque were recorded in MITs Wright Brothers Wind Tunnel for fixed blade and dynamic pitch configu-rations of identical rotor geometries. Power was calculated from this data and relations were determinedbetween efficiency and three periodic blade pitch angle variables: mean pitch, pitch amplitude, and winddirection or phase angle. These experimental results are directly compared against those of a low ordernumerical model.12

II. Previous Work

Substantial work in the area of pitch actuation has been completed by Vandenberghe, Dick, Kirke, andLazauskas, with notable papers also by Grylls et al.(1978), Nattuvetty et al.(1982), and Pawsey and Barratt(1999). Each of these researchers confirm the benefits of variable pitch control, taking different approacheswith variable pitch implementation, performance modeling, and experimental validation.

In 1991 and 1992, Kirke and Lazauskas6 determined that an inertial system was preferable and nearlyoptimal for operability over many wind speeds. Their model showed that a peak efficiency 25% higher thana corresponding fixed pitch turbine can be achieved with variable pitch. The agreement was close betweentheir predicted and measured performance curves near the optimal configuration, but the models accuracydeviated for off-design operating conditions.

Before that, in 1987, Vandenberghe and Dick5 implemented a numerical optimization scheme for control-ling straight bladed VAWTs and conducted experiments to support their claims. Using a gear mechanism togenerate second order harmonic pitch control, which they deemed optimal, they were able to attain a 19%

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greater power coefficient when compared to the fixed blade configuration. The same authors had investigatedthe benefits of second order harmonic pitch control the previous year.4

In 1977, W.A. Moran conducted a Giromill investigation to obtain data for comparison with the Larsencyclogiro vortex theory program.10 The experiment concluded that the Giromill exhibited performanceequal to or greater than what was predicted. Using this data and theory, Moran predicts a maximum powercoefficient of 0.54 for a large gyromill of solidity = 0.125. The maximum experimental efficiencies attainedin this study are listed in Table 4

III. Numerical Model

A 2D numerical model12 of a vertical axis wind turbine was used initially to size the VAWT used inthe experiments, and later on, to compare its predictions with the experimental results. In the numericalmodel each blade is represented with a single vortex placed at the quarter-chord point. The circulation ofeach vortex is determined so that flow tangency at each of the blades control point, placed at the three-quarter-chord location, is satisfied. For local high angles of attack, a stall model13 is used to allow a leakagenormal velocity at the control points. The total velocity at the control points has contributions from thewind free stream, the blade velocity, the vortex corresponding to all the blades as well as an unsteady termaccounting for each of the blades shed vorticity. The force on each blade is calculated from the circulation,which determines the lift, and a precomputed14 airfoil drag polar. For high angles of attack, wave drag isalso introduced consistently with the stall model. In addition, there is a stream-tube expansion model toaccount for the wind flow deceleration as it passes through the turbine. The expansion characteristics of thestream-tube are determined by performing a momentum balance in the wind free stream direction. Finally,the time period is discretized into a number of small intervals (usually 80-100) and the model unknownsbecome the strength of the circulation for each blade and at each discrete time interval. The time periodicityis exploited to approximate the time derivatives using spectral differencing. The coupled system of equationsis solved using Newtons method.

The model takes in as inputs the turbine geometrical parameters, airfoil geometry and drag polar at theoperation Reynolds number, wind speed, turbine rotational speed and blade pitch motion and outputs theforces on each of the blades as a function of the blade arm position angle. In addition, it computes theaveraged power coefficient, Cp. The model also allows for fixing of some input parameters to automaticallydetermine the remaining parameters that maximize the averaged power coefficient.

Figure 13 shows the theoretical contour plot of efficiency over all of the tested operating parameters,which can be directly compared to the experimental results shown in Figure 12.

IV. Experiment

Testing was conducted to measure the performance of a VAWT utilizing a cam and control rod mechanism,which defines a first-order sinusoidal blade pitch control scheme. A total of 88 tests were performed on 78different variable pitch configurations. Testing of the VAWT was performed in MITs Wright Brothers WindTunnel over the course of three days.

In the experimental descriptions and results that follow, three specific VAWT configurations are dis-cussed. The baseline configuration is defined as the optimal fixed pitch configuration, and the optimalconfiguration is defined as the optimal variable pitch configuration. A third configuration, denoted thereference configuration, is a near-optimal configuration that was believed to be optimal during the testperiod, but was found to be suboptimal during post-test analysis of the data. This reference configurationis thus most heavily tested.

A. Variables, Measurements, and Parameters

Three main variables are characterized in this study: the mean blade pitch, mean, blade pitch amplitude,amp, and wind direction or phase angle, , as defined in Figure 2. The mean pitch is defined by settingthe lengths of the adjustable control rods. The pitch amplitude is defined for a given configuration by oneof nine interchangeable cams. Phase angle is adjusted by simply rotating the cam with respect to the winddirection. These variables fully describe the pitch dynamics of the blades according to Equation 1, where and are defined geometrically in Figure 3. The wind speed, V1, was also varied for two tests to determine

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the sensitivities of the baseline and reference configurations to off-design flow conditions. Table 1 gives thetested ranges of each variable.

= mean + amp cos( ) (1)

Table 1: Variables

Variable Adjustment Tested Range

mean Control Rod Length 5o mean 15o

amp Cam Offset 0o amp 15o

Cam Phase 75o 135oV1 V1 14 mph V1 26 mph

Early in testing it was recognized that the mean pitch and pitch amplitude had a stronger influence overthe VAWT efficiency than did the phase angle. Thus, the phase angle was deemed a low priority variableduring testing, and variations of 20o from the reference phase of 95o were only investigated once an optimalmean pitch had already been determined for each pitch amplitude.

Table 2 summarizes the measurements recorded during testing and their relations to the power output.To determine the efficiency of each VAWT configuration, shaft power is found from the product of measuredshaft torque, , and angular velocity, . Shaft torque is measured by attaching a free-spinning generatordirectly in line with the VAWT rotor shaft, mounting a load cell a known distance from the center of theshaft, L, and attaching a moment arm to the generator which applied a force, F , to the load cell. Thissystem, shown in Figure 6, essentially serves as a dynomometer. The rotational frequency, f , of the VAWTis measured directly with a tachometer and used to calculate rotor speed. Electrical power is also measuredby recording generator voltage, V , and current, I, but these measurements are not useful due to poor andvariable generator efficiency. The upstream wind speed (V1), total temperature (T1), and static pressure (p1)are recorded manually from the wind tunnel computers output parameters and used to determine power ofthe available wind tunnel flow.

The parameters of this experiment are defined as constant for all tests, with the exception of wind speedfor specific tests. Table 3 shows a complete list of experimental parameters and their values.

B. Experimental Setup and Procedure

Figure 2: Graphical definitions of input variables. Figure 3: Geometric definitions of output variables

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Table 2: Measurements

Measurements Calculations

Primary Power

F = FL

f = 2pif

PVAWT =

Secondary Power

VPVAWT =

V IgenI

Flow Conditions

T1 = p1RT1p1

V1 Pwind =12AturbV

31

Table 3: Parameters

Parameter Value

Rotor Geometry

0.92

n 3

R 0.5 m

h 1 m

Aturb 1 m2

Blade Geometry

c 15.3 cm

Airfoil Profile NACA

Camber Symmetric

t/c 0.107

Pitch Actuation

Rmean 1.875 in

Measurements

L 10.70 in

Flow Conditions

V1 20 mph

Testing of the VAWT was conducted in MITs Wright Brothers WindTunnel. Figure 4 shows the setup of the VAWT with supportinginstruments, controls, and interfaces.

For each test, the resistance of the generator circuit is manuallyadjusted to maximize the trade-off between torque and rotor speed(e.g. lowering resistance increases torque and decreases rotor speed),providing control of the VAWTs shaft power. Shaft power is opti-mized for each test while observing realtime output on a LabVIEWdisplay.

The wind tunnel model is designed to be reconfigurable to anyconfiguration with regard to , mean, and amp. Setup for thewind tunnel test involves selecting the correct cam based on desiredamp. The phase angle is then set by orienting the cam with thewind direction according to marks on the cam mount. Finally, thelengths of the control rods are set to obtain a desired mean.

After setting the VAWT pitch parameters, the wind tunnel isstarted and the wind speed set by the tunnel controls. If the VAWTdoes not self start, as is true with all baseline configurations, the ro-tor is spun up by running the motor (generator) to a self-sustainablerotor speed, at which point the motor is switched back to gener-ator mode to control torque to the rotor. A resistor bank is ad-justed to manipulate the trade-off between torque and rotor speedin order to achieve the configurations maximum power coefficient,which is optimized for each test while observing realtime output ona LabVIEW display. Once the VAWT is operating at its power-maximizing steady state conditions, torque and rotor speed dataare recorded with LabView at :1 Hz for :60 s. Wind speed, tem-perature, pressure, generator voltage, and generator current wererecorded by hand in the LabView terminal at the start of data col-lection and updated if small changes occurred.

C. Description of VAWT

The VAWT consists of a 3-bladed rotor attached to a supportingbase via two roller bearings and a thrust bearing. Most of the basesupport structure, the turbine shaft, and the supporting arms aremade of aluminum (shown in Figures 5, 6, and 7). The base housesthe generator, load cell, and tachometer. A generator with the ap-propriate power curve for direct drive was not available, so an indus-trial DC motor (Imperial Electric P56-MD-008 high torque motor)was used as a generator to regulate the torque and speed of theVAWT. The motor could also be powered to start those VAWT con-figurations that were not self-starting. The motor is mounted on aturntable which allowed it to spin freely. A moment arm mounted tothe motor case impacts a load cell mounted to the base structure asshown in Figure 6, constraining the motors rotation while providingtorque data directly from the shaft. The turntable and supporting legs were attached to the floor of thewind tunnel so as not to shift during testing.

The design and sizing of the VAWT rotor primarily focuses on structural considerations for centrifugalloads. The VAWT is designed for operation at 500 rpm with a safety factor of 1.5. The blades are designedfor stiffness, constructed from Dow Blue foam core cut to fit a woven carbon tube spar and sandwichedin a fiberglass skin via wet epoxy layup. A blade cross-section is illustrated in Figure 8. Aerodynamicconsiderations additionally played a significant role in the sizing of the rotor components in order to minimizelosses at high speed. The thickness of the blades and blade arms are minimized while still maintainingstructural integrity. The blade arms are streamlined with symmetric NACA-shaped leading and trailingedges made from foam.

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Figure 4: Test apparatus. Figure 5: VAWT in wind tunnel.

Figure 6: VAWT base showing generator with moment arm and load cell.

Figure 7 shows the overall setup of the cam and control rod mechanism. The cams have circular outersurfaces secured to the base with set screws, and phase angle is adjusted by rotating the cams. The shape ofeach cam, defined by Equation 2 and Figure 9, is cut from the inside of an aluminum plate with a constantmean radius, Rmean, and unique amplitude, Ramp, corresponding to a value of amp. Cam followers roll alongthe inside of the cam, mounted to low-friction slider assemblies at the blade arm roots. The lightweight carbon

Figure 7: Cam and control rod mechanism.

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Figure 8: Turbine blade (end-view).

Figure 9: Cam profile.

fiber control rods are clamped to the sliders with shaft collars utilizing set screws for length adjustment.

Rcam = Rmean +Ramp sin() (2)

The positions of the blade pivot and control rod attachment points, shown in Figure 8 were selectedwith considerations for centrifugal loads, blade inertial moments, and blade aerodynamic moments. As seenin Figure 8, the blades center of mass is positioned between the blade pivot and control rod attachmentlocations, providing centrifugal loads to the control rods to keep them in tension. Springs are used tomaintain contact between the cam and cam follower at low rotor speeds.

V. Data Reduction

Three sources of error are identified in the VAWT experimentation: bearing degradation, wind tunnelblockage, and random error. The bearing degradation problem was discovered and repaired prior to the lastof three days of testing, resulting in a 3% to 4% improvement in efficiency over the previous days for the sameconfigurations. Thus, a correction is performed to the torque and rotor speed data to allow for comparisonsof efficiencies between different test days and correspondingly different degrees of bearing torque. Bearingtorque correction factors, assumed to be constant with respect to rotor speed, are used in conjunction withthe measured torque and rotor speed to generate linear approximations to corrected power curves, whichare optimized to determine the true maximum power capability of the VAWT. Day 3 data is used as theuncorrected standard against which efficiencies on other days are compared, and constant bearing torquecorrection factors are chosen for days 1 and 2 such that differences in corrected efficiencies between tests ofthe same configuration are minimized across all days. The bearing torque correction is described in detailin Appendix A.

The effect of the second source of systematic error, wind tunnel blockage, is also present in the data.Because the tunnel restricts the flow to a constant cross-sectional area 5.1 times larger than the sweptfrontal area of the VAWT, the air downstream of the VAWT does not return to upstream pressure. This net

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pressure difference imparts extra energy to the VAWT, resulting in higher efficiencies than those that wouldresult from unrestricted outdoor testing. Conversely, the blockage decreases the static pressure measuredby the transducers located along the tunnel wall 5 ft upstream of the VAWTs axis, increasing the apparentflow velocity and wind power and thus decreasing the calculated efficiency. The static pressure of the flowdownstream of the VAWT was not measured in this study. Appendix B presents a modified Betz criterionthat accounts for tunnel blockage. It is apparent from this analysis that for the levels of blockage in ourexperiment, the optimal efficiency obtained in the wind tunnel can be quite different from that measured inopen air. Although the effect of blockage is estimated here for awareness, all results are reported withoutblockage corrections.

Random error is considered in two categories: random error occurring between tests resulting from con-figuration setup discrepancies, and measurement error varying within each test. The ability to observe errorin these two independent categories allows for estimation of the variance in efficiency without concern for thespecific source of the error. For those configurations that were only tested once, the error occurring betweentests was not measured and thus it is approximated from the configurations that were tested multiple times.Estimates of the variance are used to calculate 95% confidence intervals for the true mean efficiency of eachconfiguration. A detailed description of the error analysis and confidence interval calculations may be foundin Appendix C. We point out that the number of tests carried out is too small to perform a comprehensivestatistical analysis of the results. Our analysis is aimed at providing an idea of the repeatability of the dataand of the level of uncertainty in the reported results.

VI. Results

In this section, conclusions regarding the VAWTs optimal and sub-optimal performance are discussed.The results of two experiments in which the power curve and windspeed effects are measured is also used tocompare the reference and baseline configurations. Third, comparisons are drawn between the experimentalresults and the results predicted using the numerical model. Finally, the results of this study are comparedwith those of Kirke and Lazauskas, Vandenberghe and Dick, and Moran.

A. Characterization of Pitch Variable Effects

Over the course of three days, a total of 88 tests were performed to characterize the effects of mean pitch,pitch amplitude, and phase angle on the efficiency of a VAWT. Of these 88 tests, 52 characterized the effectsof mean pitch and pitch amplitude at a phase angle of 95o. The experimental results are plotted in Figure12 for a phase angle of 95o and can be directly compared with the theoretical results shown in Figure 13.Small blue circles on the contour represent configurations that were tested or modeled, while the efficienciesof all other configurations are interpolated from the data using a cubic surface approximation. The optimalconfigurations are indicated on each plot with black stars. These experimental results are shown with 95%confidence intervals in Figure 10.

1. Optimal Performance and Improvement

The optimal VAWT configuration and corresponding performance is an interesting topic of discussion. Asignificant improvement in efficiency over the baseline is, in fact, observed in the data reflected in Figures 12,13, and 11. A comparison between the two configurations demonstrates an increase in efficiency from 0.323(54.5% of the Betz limit) to 0.436 (73.5% of the Betz limit), a total improvement of 35.0%. Even in a worstcase scenario in which the true mean efficiency of the baseline is at the upper bound of its 95% confidenceinterval and the true mean efficiency of the optimal configuration is at the lower bound of its 95% confidenceinterval, the dynamic pitch scheme still performs with 32.9% higher efficiency than the baseline (see Figure11).

For a phase angle of 95o, the numerical model accurately predicts a similar efficiency improvement overthe baseline with less than 2% efficiency overestimations of the baseline and optimal VAWT configurations.The model also accurately predicts the optimal combination of mean pitch and pitch amplitude for a phaseof 95o, with only a 2o discrepancy in estimating the mean pitch and a 1o discrepancy in estimating the pitchamplitude. However, the region within a few degrees of the optimal configuration is insensitive to the pitch

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variables such that the measured efficiency of the predicted optimal configuration is only 1% lower than thatof the experimental optimal configuration. Each of these errors are well within the confidence intervals andmodel accuracy.

2. Sensitivities and Trends

Figure 14 shows sensitivities of the VAWTs performance to perturbations in each variable about the referenceconfiguration (note that since the reference configuration is suboptimal, small improvements in efficiency areachieved by these perturbations). As expected, a relative insensitivity in efficiency to the phase angle isobserved compared with the mean pitch and pitch amplitude. Looking at the experimental results (Figure12), it is also apparent that the VAWTs overall performance is slightly more sensitive to changes in meanpitch than to changes in pitch amplitude, although there are local regions where the opposite is true. Thenumerical model also predicts these trends.

Figure 10: Experimental VAWT efficiencies with 95% confidence intervals, = 95o.

Predicted Experimental

Reference* Optimal

Baseline Configuration

Cp95% 0.329 0.323 0.002mean [deg] 12 12

Variable Pitch Configuration

Cp95% 0.441 0.436 0.004mean [deg] 11 9

amp [deg] 8 9

[deg] 95 95

*Predicted reference data is not predicted optimal: phase was fixed at 95o

Figure 11: Predicted and experimental efficiency improvements.

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Figure 12: Experimental VAWT efficiencies, = 95o.

Figure 13: Theoretical VAWT efficiencies, = 95o.

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B. Characterization of Torque and Rotor Speed Trade-offs

To characterize tradeoffs between the VAWTs torque and rotor speed, two configurations were selectedthebaseline and reference configurationsand the generators electrical load was adjusted incrementally. Testswere performed at a wind speed of 20 mph and data was recorded in rotor speed increments of :30 rpmfrom the minimum sustainable speeds to the maximum aerodynamically limited speeds. Results are shownin Figure 15 for both configurations.

The clearest result of this experiment is that the reference configuration is capable of operation througha much larger range of torques and rotor speeds than the baseline. The few TSRs that are plotted for thebaseline are the only TSRs that could be sustained (the baseline was not self-starting), and increasing thetorque slightly from the minimum sustainable rotor speed is enough to stall the blades, quickly stoppingrotation. The reference configuration on the other hand has a startup torque of 1.62 ft/lbf and operates evenat low rotor speeds. The aerodynamically limited upper TSR bound is approximately the same for both thebaseline and reference configurations at 3.1 to 3.2.

C. Characterization of Windspeed Effects

For this experiment, the baseline and reference configurations were tested over a range of wind speeds. Eachconfiguration was first brought to the test speed of 20 mph, and the electrical load was adjusted to optimizethe power at this wind speed. The performance of the VAWT was then recorded at a range of wind speedswhile holding the electrical load constant to simulate off-design operating conditions. The efficiency is plottedagainst wind speed in Figure 16. The highest tested wind speed was limited to prevent rotor speeds greaterthan 550 rpm due to structural considerations.

Again the most noticeable aspect of these results is that the reference configuration is capable of efficientoperation over a much larger range of wind speeds, specifically at wind speeds lower than the configurationsdesign speed. As windspeed is decreased, rotor speed drops more rapidly. This lowers TSR and increases theincidence angle on the blades until they stall. For a given wind speed, the optimal fixed-pitch configurationsits on the verge of stall and operation is not sustainable below this wind speed. However, for the same windspeed the optimal variable pitch configuration has stall margin, and even after stalling the VAWT maintainsenough torque to turn the generator. For the reference configuration, the efficiency remains within a modest:4% range for wind speeds between 0.83 and 1.31 times the design speed of 20 mph, and there is a suddendrop-off in efficiency between 0.75 and 0.83 times the nominal wind speed after the blade has stalled. For highwindspeeds, the baseline demonstrates the same insensitivity to the wind speed as the reference configuration.

D. Comparison of Experimental Results to Historical Results

The experimental VAWT used in this study performed very well on an absolute performance scale, and itstands competitively near previous experimental data. The direct comparisons of model parameters and

Figure 14: Performance sensitivity to variable perturbations about reference configuration.

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Figure 15: Non-dimensional torque vs. TSR, electrical load varied.

Figure 16: Efficiency vs. wind speed, electrical load fixed.

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results between this study and those by Vandenberghe & Dick, Kirke & Lazauskas, and Moran are shown inTable 4, with primary items for comparison being solidity, maximum efficiency, and tip speed ratio. Althoughhigh solidity VAWT designs are traditionally thought to be less efficient than designs with low solidity, theperformance of the VAWT in this study compares well with the others, reaching a maximum efficiency of0.436 at a low TSR of 2.35. It should be noted that obtaining high efficiency with high solidity and low TSRrepresents a significant structural advantage since not only the load requirements are reduced but lighter,stiffer structural solutions can be devised.

It should be pointed out that two of the experiments shown here were tested in open wind tunnels,so tunnel blockage effects were not present. For the other studies, results are presented as measured andcorrections are not made for blockage effects.

Table 4: Experiment Comparisons

Erickson, Wallace, Kirke, Vandenberghe,

Parameter Peraire Lazauskas Dick Moran

h [m] 1 1 0.6 1.5

R [m] 0.5 1 0.5 1.05

c [m] 0.153 0.1 0.1 0.21

n 3 3 3 3

Blade Profile NACA0011 NACA0018 NACA0012 NR

V [m/s] 8.9 10 8 NRTunnel Dimensions 3.05 m x 2.13 m ellipse 2 m x 3 m rectangle Open tunnel 4.6 m x 6.1 m open jet(

AtunAturb

)5.10 3 n/a n/a

0.92 0.3 0.6 0.6

Cpmax 0.436 0.22 0.45 0.39

Optimal TSR 2.35 2.6 2.7 2.2

*NR: not reported

VII. Conclusion

In this study, a vertical axis wind turbine of high-solidity design utilizing a forced sinusoidal blade pitchactuation scheme was successfully tested over a wide range of design and operational variables. The primarylimitations of this study stem from the three identified error sources. First, random variations in efficiencyoccurring between tests cannot be fully characterized because few configurations were tested more thanonce. Another limitation arises from the bearing degradation problem, which requires that a theoreticalcorrection be made to the measured results for comparison across different test periods. Finally, the windtunnel blockage effect is thought to significantly boost the reported results, although the exact effect is notknown with accuracy because downstream flow conditions were not measured during the test.

Despite these limitations, the results presented in this paper detail a number of successes. First, theperformance of a wide range of VAWT configurations utilizing cyclic blade pitch actuation are quantified andcharacterized. Second, it is shown that variable pitch VAWTs have a number of advantages over fixed pitchdesigns. A 35% efficiency improvement over a fixed blade VAWT is achieved, and variable pitch VAWTsare shown to have self-starting capabilities and improved performance at off-design operating conditions.Additionally, variable pitch VAWTs operate optimally at lower TSRs than fixed-pitch designs, decreasingblade centrifugal loading and allowing for more cost-effective designs. Third, it is shown that a basic 2Dtheoretical VAWT model can accurately predict and optimize the performance of a VAWT utilizing cyclicblade pitch variations to within a few percent error. Future VAWTs may be reliably developed using sucha model, and sensitivities could be considered to increase robustness of a design. Finally, the high-solidityVAWT tested in this experiment is recognized to perform with comparable efficiencies to those reportedof lower solidity VAWTs in other studies. Because high solidity VAWTs may be more easily designed and

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constructed with tolerance for large centrifugal forces, such designs may offer further opportunity to lowerthe life-cycle cost of a VAWT.

Acknowledgments

The authors would like to thank Mark Drela from MIT and David J. Willis from U. Massachusetts atLowell for many useful comments and suggestions regarding the work presented in this paper.

Appendices

A. Bearing Torque Correction

Bearing Torque Correction Nomenclature

C Number of configurations tested on two or more different days

Cpm Measured power coefficient

Cpcor Power coefficient corrected for bearing degradation(Cpcor

)xyz

Corrected power coefficient of configuration x, day y, and test z

Dx Number of days configuration x was tested

Pa a Power along actual power curve

P a am Actual power generated by VAWT at measured power-maximizing rotor

speed

P cor cor

cor Actual maximum power VAWT was capable of producing

Pm m Power along measured power curve

P m mm Measured power at measured power-maximizing rotor speed

Txy Number of tests performed on configuration x during day y

b a m Difference in bearing torque between measured and actual power curves Rotor angular velocity

cor Actual power-maximizing rotor angular velocitym Measured power-maximizing rotor angular velocitya Actual torque generated by VAWT

a Actual torque at measured power-maximizing rotor speedcor Torque at actual power-maximizing pointm Component of actual torque that was measured (i.e. not removed by

bearing)

m Measured torque at measured power-maximizing rotor speed

A series of tests were performed on the VAWT over the course of three days. A bearing degradationproblem was discovered and repaired prior to the final day of testing, resulting in a 3% to 4% improvement inefficiency over the previous days for the same configurations. Thus, a correction is performed to the torqueand rotor speed data to allow for comparisons of effciencies between different test days and correspondinglydifferent degrees of bearing torque. Day 3 data is used as the uncorrected standard against which efficiencieson other days are compared, and constant bearing torque correction factors are chosen for days 1 and 2such that differences in corrected efficiencies between tests of the same configuration are minimized acrossall days.

A linear approximation of actual power curve created at ( = m, a = a ) is given by Equation 3, and

a constant approximation to the bearing torque offset is assumed.

a a +dad

a=a

( m) (3)

b Constant (4)

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Figure 17: Measured and optimal power locations on corrected and uncorrected power curves.

The general relation between a, m, and b is given by Equation 5. This can be seen in Figure 17.

a m + b (5)Assuming the torque and rotor speed measurements were taken at the power maximizing location along

the measured power curve, the components of the power gradient at Pm = Pm are zero with respect to

and m. This assumption results in the relation given by Equation 6 using the gradient with respect to (note that this is the same result that may be obtained by using the gradient with respect to m).

Pm

Pm=Pm

=(m)

=m

= m + m

dmd

m=m

0

= dmd

m=m

= m

m(6)

Taking the derivative of Equation 5 at a = a and substituting in Equations 4 and 6 gives Equation 7,

which is the slope of the actual torque curve at the measured rotor speed.

dad

a=a

=dmd

m=m

+dbd

= dad

a=a

= m

m. (7)

By substituting Equations 5 and 7 into Equation 3, the linear approximation of the actual torque is givenby Equation 8 in terms of the measured variables and correction factors.

a = b + m

mm

( m)

= a = b + 2m mm

(8)

Maximizing the actual power, Pa, using this approximation yields equations for the corrected torque,rotor speed, and maximum efficiency.

Pa

Pa=Pcor

=(a)

=cor

= b + 2m 2

mm

cor 0

= cor = m(

1 +b2m

)(9)

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cor = b + 2m

mm

m

(1 +

b2m

)

= cor = m(

1 +b2m

)(10)

P cor = cor

cor = CpcorPwind P

m =

mm = C

pmPwind

= Cpcor = Cpm(

1 +b2m

)2(11)

Each day is assigned a different value of b, which is equal to the relative bearing torques between thatday and the final day of testing, for which the data was not corrected. The values of b were selected suchthat the sum of the squared residuals between VAWT efficiencies of the same configuration was minimizedacross all three days of testing. This was performed by iterating over combinations of two tests which wereperformed on the same configuration during different days of testing, as shown mathematically in Equation12. The constraint in Equation 13 is also applied to reflect the supposition that the bearing torque degradedas the week progressed (except for the final day of testing).

min

{ Cm=1

Dm1n=1

Dmp=n+1

Tmnq=1

Tmpr=1

[(Cpcor )mnq (Cpcor )mpr

]2}(12)

subject to (b

)dayn

(b)dayn1 (13)The optimization was implemented using MATLABs fmincon function, a gradient-based numerical opti-mizer that accepts constraints.

Figures 18 and 12 show the VAWTs uncorrected and corrected efficiencies respectively. The bearingtorque correction smooths the plot significantly, and it results in a slight change in the optimal configurationand its corresponding efficiency.

Figure 18: Uncorrected efficiencies, = 95o. Figure 19: Corrected efficiencies, = 95o.

B. Tunnel Blockage Estimation

The Wright Brothers Wind Tunnel at the Massachusetts Institute of Technology was used to collect thedata presented in this paper. The wind tunnel has an elliptical cross section with major and minor axis

17 of 23


Tunnel Blockage Estimation Nomenclature

A Cross-sectional area of channel or wind tunnel (see Figure 20)

Aw Cross-sectional area of wake (see Figure 20)

F Aerodynamic force acting on VAWT rotor

p1 Upstream static pressure (see Figure 20)

p2 Downstream static pressure (see Figure 20)

S Swept frontal area of turbine (see Figure 20)

v Velocity through turbine (see Figure 20)

v1 Upstream velocity (see Figure 20)

v2 Downstream velocity outside wake (see Figure 20)

vw Downstream velocity inside wake (see Figure 20)

V2v2v1

Ratio of downstream and upstream velocities outside wake

Vwvwv1

Ratio of downstream and upstream velocities inside wake

SA Geometric ratio of turbine and channel areas

AwS Ratio of wake area and turbine area

Air density in wind tunnel

Figure 20: Wind tunnel flow and station designations.

dimensions of 10 ft and 7 ft respectively. This yields a total cross sectional area of :5.1 m2, 5.1 times theswept frontal area of the VAWT.

Unlike a wind turbine operating outdoors, the blockage effect inside a wind tunnel prevents downstreamair from returning to its original upstream static pressure. This net pressure difference imparts extra energyto the VAWT. Counteracting this effect is the tendency of the mass flow through the turbine to decreasein the wind tunnel as the downstream blockage forces more air around the turbine. The combined effect ofthese two forces is to increase the maximum amount of extractable power in the flow over that which maybe extracted during outdoor operation. Ultimately, it is shown that an ideal VAWT operating in a windtunnel with area ratio = 0.20 is limited by a power coefficient of 0.916, as opposed to the traditional BetzLimit of 0.593.

Equations 14 through 19 are derived using 1D mass, momentum, and energy conservation principlesthrough two streamtubes shown in Figure 20. All calculations are performed assuming incompressible flow,a uniform upstream pressure and velocity distribution, and a uniform downstream pressure distribution.

From continuity:

(AAw)v2 +Awvw = Av1 (14)Sv = Awvw (15)

From momentum conservation, two relations are found to describe the force on the turbine:

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F = Spt = S

(1

2(v21 v2w) + (p1 p2)

)(16)

F = A(p1 p2) + Av21 (AAw)v22 Awv2w (17)Using Bernoulli, the downstream and upstream pressures may be related through the outside streamtube:

p1 +1

2v21 = p2 +

1

2v22 (18)

Finally, the power is found fromP = Fv (19)

Proceeding to solve these equations, the input blockage parameter is first defined

=S

A

and for convenience we also define

=AwS, V2 =

v2v1, Vw =

vwv1

Combining Equations 15, 16, 18, and 19 we can write

P = Fv = FAwSvw =

1

2Aw(v

22 v2w)vw (20)

and

CP =P

12Sv

31

= (V 22 Vw V 3w) (21)

Equation 14 can be written as,(1 )V2 + Vw = 1 (22)

Finally, Equation 17 can be combined with 16 and 18 to give,

(1 + 2)V 22 + (2 )V 2w = 1 (23)Now, given and Vw, Equations 22 and 23 can be numerically solved for and V2, which in turn can be

used to determine CP . In other words, Equations 21-23 define implicitly the function CP (, Vw). The resultsare shown in Figure 21 for a range of area ratios and wake velocities, and the maximum power coefficienttaken at Vw =

13 is plotted in Figure 22. For the reported VAWT test geometry, = 0.20, the limiting power

coefficient is 0.916.

Figure 21: Effect of blockage on efficiency. Figure 22: Effect of blockage on maximum efficiency.

19 of 23


C. Random Error Quantification

Random Error Quantification Nomenclature

Cpi Power coefficient of an individual measurement, i

Cpj Power coefficient of an individual test, j

Cpk Power coefficient of configuration, k

dk Power coefficient confidence interval dimension for configuration k

Ni Total number of power coefficient measurements for a given configuration

Nj Total number of power coefficient measurements within test j

Nk Total number of tests performed on configuration k

Cpi Cpi j Difference between power coefficient of measurement i and average powercoefficient of test j

j j k Difference between average power coefficient of test j and average powercoefficient of configuration k

j Mean power coefficient of test j

j Estimated mean power coefficient of test j

k Mean power coefficient of configuration k

k Estimated mean power coefficient of configuration j

1k E[Var(Cpi)j ]Quantity that characterizes power coefficient variations within a test

1k Estimate of quantity that characterizes power coefficient variations within atest

2k Var[E(Cpi)j ]Quantity that characterizes power coefficient variations between tests

2k Estimate of quantity that characterizes power coefficient variations betweentests

j Standard deviation of power coefficient for test j

j Estimated standard deviation of power coefficient for test j

k Standard deviation of power coefficient for configuration k

k Estimated standard deviation of power coefficient for configuration k

Cumulative distribution function (CDF) for a Gaussian distribution

For a given configuration, the error in the estimated average power coefficient is influenced by twocategories of random error. The first category, random error occurring within a single test, is present for alltested configurations. This category includes instrument noise, fluctuations in flow conditions and measuredquantities, and human error in estimating and recording the flow conditions as they fluctuated. For a giventest, data was recorded at a frequency of 1 Hz for a duration of approximately 1 min. The second sourceof random error, configuration setup error occurring between tests, is only present for those configurationsthat were tested more than once. This category of error includes inaccuracies in manually adjusting thecam phase angle and control rod lengths, setting the tunnel freestream velocity to the desired 20 mph, andadjusting the generators electrical load for the power-maximizing combination of torque and rotor speed.Only a handful of configurations were tested more than once, and most of these were only tested two times.

The mean efficiency measured during a single test, j, is calculated according to Equation 24. The meanefficiency of a specific configuration, k, is estimated as the average of the mean efficiencies calculated fromeach test, with equal weight being given to each test according to Equation 25.

E(Cpj ) = j j =1

Nj

Nji=1

Cpi (24)

E(Cpk) = k k =1

Nk

Nkj=1

j (25)

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Using the law of total probability, given in general form by Equation 26, the variance of the efficiency ofconfiguration k may be calculated as the sum of two terms: the expected variance in efficiency within tests,which quantifies the first category of random error, and the variance of the average efficiencies calculatedbetween tests, which quantifies the second category of random error. In this way, error resulting from eachcategory of random error may be calculated separately.

Var(X) = E[Var(X|Y )] + Var[E(X|Y )] (26)

Var(Cpk) = E[Var(Cpk |Setup = setupj)] + Var[E(Cpk |Setup = setupj)] (27)= Var(Cpk) = E[Var(Cpi)j ] + Var[E(Cpi)j ] (28)

The quantity E[Var(Cpi)j ] is estimated for a configuration, k, by averaging the variances of efficiencywithin each test, j, of that configuration according to Equation 29 and 30. The quantity Var[E(Cpi)j ] isestimated for configurations that were tested more than once configuration by taking the variance of theaverage efficiencies calculated from each test, as shown in Equations 31 and 32. All tests are weighted equally.Adding these quantities gives the estimate for the configurations total variance, Equation 33.

Var(Cpi)j = 2j 2j =

1

Nj 1Nji=1

(Cpi j)2 (29)

= E[Var(Cpi)j ] = 21k 21k =1

Nk

Nkj=1

2j (30)

E(Cpi)j = j j (31)

= Var[E(Cpi)j ] = 22k 22k =1

Nk 1Nkj=1

(j k)2 (32)

= Var(Cpk) = 2k 2k = 21k + 22k (33)

For configurations that were only tested once, the second term in the variance expression, 22k, is notknown. It is therefore estimated from the configurations tested more than once. Figure 23 shows a histogramof the natural log of 22k for all configurations tested multiple times. Although there is not much data todraw from, it is assumed that this quantity is normally distributed. Thus, for those configurations onlytested once, 22k is modeled as a log-normal distribution with probability density function f(

22k).

Figure shows a distribution of Cpi = Cpi j for all configurations, which reflects differences in efficien-cies measured within tests. Similarly, Figure shows a distribution of j = j k for all configurations,which reflects differences in efficiencies measured between tests. From this, it can be seen that both categories

of random error are normally distributed and thus k is normally distributed along N(k,2kNi

). Ninety-five

percent confidence intervals for the location of the true mean value of Cp are thus calculated as follows.Assuming a value of 22k,

0.95 = P (|k k| dk) = 1 2(k dk

2kNi

). (34)

For configurations tested more than once where 22k is known, Equation 34 is directly solved for theconfidence interval size.

dk =1.96kNi

(35)

For configurations tested only once where 22k is estimated from a distribution, Equation 34 is integratedover all possible values of 22k.

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Figure 23: Log variation in 22k between configurations.

Figure 24: Variation in efficiency within tests. Figure 25: Variation in efficiency between tests.

0.95 =

[1 2

(k dk

2kNi

)]f(22k)d2k (36)

Equation 36 is numerically integrated using Simpsons Rule, and the value of dk is iteratively solved forusing MATLABs fsolve.

References

1Rogers, A., Wind Power: Are Vertical Axis Turbines Better? February/March 2008, pp. 2, URLhttp://www.motherearthnews.com/Renewable-Energy/2008-02-01/Wind-Power-Horizontal-and-Vertical-Axis-Wind-Turbines.aspx, (visited November 6 2009).

2Malcolm, D. J., Market, Cost, and Technical Analysis of Vertical and Horizontal Axis Wind Turbines Task 2: VAWTvs. HAWT Technology, Tech. rep., Global Energy Concepts, LLC, Kirkland, WA, May 2003.

3Kirke, B., Evaluation of Self-starting Vertical Axis Wind Turbines for Stand-alone Applications, Ph.D. thesis, GriffithUniversity, Queensland, Australia, April 1998.

4Vandenberghe, D. and Dick, E., A Theoretical and Experimental Investigation into the Straight Bladed Vertical AxisWind Turbine with Second Order Harmonic Pitch Control, Wind Engineering, Vol. 10, No. 3, 1986, pp. 122138.

5Vandenberghe, D. and Dick, E., Optimum Pitch Control for Vertical Axis Wind Turbines, Wind Engineering, Vol. 11,No. 5, 1987, pp. 237247.

6Kirke, B. and Lazauskas, L., Experimental Verification of a Mathematical Model for Predicting the Performance of aSelf-Acting Vertical Axis Wind Turbine, Wind Engineering, Vol. 17, No. 2, 1993, pp. 5866.

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7Kirke, B. and Lazauskas, L., Enhancing the Performance of Vertical Axis Wind Turbine Using a Simple Variable PitchSystem. Wind Engineering, Vol. 15, No. 4, 1991, pp. 187195.

8Grylllls, W., Dale, B., and Sarre, P.-E., A Theoretical and Experimental Investigation into the Variable Pitch VerticalAxis Wind Turbine, Papers presented at the second International Symposium on Wind Energy Systems, held in Amsterdam,Netherlands, October 3rd-6th, 1987., 1987, pp. E9101 E9118.

9Nattuvetty, V. and Gunkel, W., Theoretical Performance of a Straight-Bladed Cycloturbine Under Four DifferentOperating Conditions, Wind Engineering, Vol. 6, No. 3, 1982, pp. 110130.

10Moran, W., Giromill Wind Tunnel Test and Analysis, October 1977.11Pawsey, N. and Barratt, A., Evaluation of a variable-pitch vertical axis wind turbine, Wind Engineering, Vol. 23, No. 1,

1999, pp. 2330.12Asher, I., Drela, M., and Peraire, J., A Low Order Model Model for Vertical Axis Wind Turbines, 28th AIAA Applied

Aerodynamics Conference, 2010.13Drela, M., Integrated Simulation Model for Preliminary Aerodynamic, Structural, and Control-Law Design of Aircraft,

Paper AIAA 99-1394.14Drela, M., Subsonic Airfoil Development System, Paper AIAA 99-1394, 1999.

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