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Renewable Energy 45 (2012) 41e50

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Renewable Energy

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Modeling passive variable pitch cross flow hydrokinetic turbines to maximizeperformance and smooth operation

L. Lazauskas a, B.K. Kirke b,*

aCyberiad, 11 Harris St, Norwood, SA 5067, Australiab Sustainable Energy Centre, University of South Australia, Mawson Lakes, SA 5095, Australia

a r t i c l e i n f o

Article history:Received 26 June 2011Accepted 3 February 2012Available online 4 March 2012

Keywords:ModelingVariable pitchCross flowHydrokinetic turbine

* Corresponding author. Tel.: þ61 8 81651608; fax:E-mail addresses: b.kirke@unisa.edu.au,

(B.K. Kirke).

0960-1481/$ e see front matter � 2012 Elsevier Ltd.doi:10.1016/j.renene.2012.02.005

a b s t r a c t

Darrieus type hydrokinetic turbines (HKTs) with fixed pitch blades, like their wind turbine equivalents,exhibit poor starting torque whether blades are straight, troposkein or helical. High solidity improvesstarting torque to some extent but reduces peak efficiency. Straight blade turbines have been observed toshake violently due to cyclical hydrodynamic forces on blades. To solve this problem, the computerprogram VAWTEX-B was used to model blade forces and to predict how variable pitch might reduceshaking while maintaining strong starting torque and high peak efficiency Cpmax. A memetic algorithmwas developed to search for optimum pitching parameters to achieve these objectives. Five optimizationobjectives were initially used, to maximize starting torque and Cpmax, to minimize downstream and theside force fluctuations causing shaking, and to minimize a function combining these four variables. Asecond optimization used a different objective function incorporating Cpmax and then minimizinga shaking force parameter while allowing a small reduction in CPmax. Modeling predicted reducedshaking while maintaining starting torque and peak efficiency well above those achievable with fixedpitch, and only slightly lower than the maxima achievable with variable pitch.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Darrieus type hydrokinetic turbines (HKTs) with straight andhelical blades are shown in Fig. 1.

Whether blades are straight, helical or troposkein, DarrieusHKTs have some advantages and some drawbacks compared withaxial flow HKTs. These have been summarized in [1]. Parameters ofparticular interest to designers of Darrieus HKTs, aside from theobvious ones of efficiency and cost-effectiveness, are (i) adequatestarting torque and (ii) smooth operation. Fixed pitch DarrieusHKTs typically fail to self-start under load, exhibit low efficiencyand unless blades are helical, they shake violently due to cyclicallyvarying angle of attack a and hence fluid dynamic forces on blades,with each blade experiencing two peaks in both radial andtangential force per revolution. The upstream peaks are typicallymuch larger than the downstream peaks, especially at higher tip-speed ratios as shown in Fig. 2, so a 3 blade turbine will typicallyexperience 3 major peaks per revolution. This problem is notrelated to mass balance of the rotor. The variation in tangentialforce, commonly referred to as “torque ripple,” affects the

þ61 8 83023380.briankirke@internode.on.net

All rights reserved.

transmission and load, while the variation in radial force affects thesupport structure, and if this frequency coincides with the naturalfrequency of the support structure it can be destructive. Duringtests in Canada reported in [2], the fixed pitch straight blade Dar-rieus HKT shown in Fig. 1 was found to shake so violently due tocyclical hydrodynamic forces on blades that tests on it had to be cutshort for fear that it would shake itself or its mounting to pieces.The helical turbine operated smoothly but its self-starting abilitywas not noticeably better than the straight blade turbine and peakefficiency was about the same e possibly slightly lower. Theseshortcomings of all fixed pitch Darrieus HKTs led to the investiga-tion into variable pitch reported in the present work.

2. Modeling fixed pitch blade forces

The VAWTEX-B model has been described in [3]. It isa momentum model based on Paraschivoiu’s DMST (doublemultiple streamtube) approach [4], developed originally to modelDarrieus wind turbines, and further developed to model passivevariable pitch and incorporating dynamic stall. In the present study,dimensions were based on those of the turbines tested in Canada,reported in [2], i.e. 3 blades of NACA 0021 section, 1.25 m long,0.14 m chord, 0.5 m turbine radius, giving solidity s ¼ 0.84, andVN ¼ 2 m/s except where otherwise indicated below.

Fig. 1. Darrieus type cross flow HKTs with straight and helical blades.

Fig. 3. Measured CP and CQ for a fixed pitch HKT compared with predictions using

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e5042

2.1. Validation of VAWTEX-B model against fixed pitch turbineexperimental data

Themodel was used to predict performance of the straight bladefixed pitch turbine shown in Fig. 1, and Fig. 3 shows good agree-ment between measured and modeled torque coefficient CQ andperformance coefficient CP without allowance for dynamic stall. Asexplained in [2], it was not practicable to measure CQ and CP at lbelow the maximum CQ because the band brake used in the testsexerted a more or less constant torque at a given setting, causingthe turbine to stall at low l. Although experimental data areavailable at VN > 3 m/s, these showed evidence of loss of perfor-mance, probably due to cavitation and/or the onset of supercriticalflow and resulting surface effects, and sites with such high flowvelocities are extremely rare, so these data are not shown in Fig. 3.

VAWTEX-B without dynamic stall.

2.2. Modeling shaking forces for fixed pitch turbines

Having validated VAWTEX-B model predictions against experi-mental data for fixed pitch HKTs, themodel was then used tomodelthe variation of radial and tangential force on a single fixed pitchblade with 0.1 m chord, but otherwise identical to the turbineshown in Fig. 1, as functions of azimuth angle and tipspeed ratio.

Fig. 2. Non-dimensional tangential thrust F*T on a fixed pitch blade as a function of sand l.

Predicted fluctuations in tangential thrust F*T are shown in Fig. 2,and a similar pattern is apparent in radial force (not shown heredue to space restrictions but shown in [5]). It is clear that both forcecomponents experience large variations with azimuth angle at alltipspeed ratios. When the contributions of multiple blades aresuperimposed there is still considerable force fluctuation at l ¼ 2,as shown in Fig. 4 for 6 blades, because peaks from different bladealmost coincide. The same is true for tangential force. Although

Fig. 4. Non-dimensional radial (shaking) force F*R as a function of azimuth angle for 6blade fixed pitch HKT at l ¼ 2.

Fig. 5. Non-dimensional radial (shaking) force F*R as a function of azimuth angle for 3blade fixed pitch HKT at l ¼ 3.

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e50 43

Fig. 5 shows that this is no longer the case at l ¼ 3, high solidityturbines achievemaximum efficiency at lw 2, as shown in Fig. 3, sothe region l w 2 is of particular interest.

It is this force fluctuation that must be minimized, and variousstrategies were investigated to achieve this aim. As a first step, itmight be expected that increasing the number of blades would

Fig. 6. Cyclical pitching with blade leading edges

help, and some commercial wind and hydrokinetic turbines have 4,5 and even 6 blades, but it can be seen from Fig. 4 that the fluc-tuation is still large. Increasing the number of blades n is self-defeating because it further increases solidity s ¼ nc/r unlesschord length c is reduced or turbine radius r is increased, andreduced c in turn reduces structural strength which could lead tostructural failure, and also reduces blade chord Reynolds numberRe, which adversely affects performance by reducing stall angle,maximum lift coefficient and lift to drag ratio. Helical blades,provided they cover the complete 360�, e.g. with 3 blades eachwrapped around a 120� arc, effectively “smear” the torque fluctu-ations evenly around the circumference, but the radial forces atopposite ends are still unbalanced so helical turbines cannotcompletely overcome the shaking problem, nor can they overcomethe problem of low starting torque. Increasing l to 3 showsa marked reduction in shaking, evenwith a 3 blade turbine, as seenin Fig. 5. This is a result of both reduced blade stall and a shift in theazimuth angle at which blades stall on the upstream pass, as seen inFig. 2. However, increasing l requires lower s, i.e. more slenderblades, which again reduces structural strength and Re. But giventhat no fixed pitch turbine can produce adequate starting torque,whether it has straight or helical blades, it was decided to modelshaking of a variable pitch turbine to see if shaking could bereduced to an acceptable level while maintaining high startingtorque and efficiency.

pivoting upstream to avert stall (after [13]).

Fig. 7. Torque coefficient CQ and performance coefficient CP vs. l for various pitch anglelimits.

Table 1Principal parameters (held fixed) during the optimization.

Parameter Value

Environmental Fluid density r (kg m�3) 1000Kinematic viscosity n (10�6 m2 s�1) 1.14Ambient velocity Vn (m s�1) 2

Turbine Number of blades n 3Radius r (m) 0.5Height (m) 1.2

Blades Profile NACA 0021Chord length c (m) 0.10

Model Dynamic stall YesParasitic drag No

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e5044

3. Modeling the performance of a turbine with basic passivepitch control

Pitch control mechanisms can be either active or passive.Various researchers have modeled the performance of variablepitch Darrieus wind turbines (also known as giromills or cyclo-turbines) with active pitch control, i.e. pitch is driven by a mecha-nism according to some pitching regime assumed to be optimum[6e10], and one active variable pitch HKT has achieved Cp closeto 0.5 [11]. But as far as the authors are aware, only two attempts[3,12] have been made to model turbines with passive (self-acting)variable pitch in which pitch is determined by a combination offluid dynamic and inertial forces. This is a more challenging task,and to date we are not aware of any attempts to reduce shakingeffects by modifying pitch control regimes. Given that there is nogeneral agreement as to what constitutes an “optimum” angle ofattack or pitch angle, and given the mechanical complexities ofactive mechanisms, it is our view that it is more productive to focuson simple passive pitch control systemswhich aim not for optimumperformance, but for performance much improved over fixed pitchsystems without great complexity.

Two basic systems were modeled. The first, referred to below as“stabilized free pitch,” aims to maintain zero pitch when the angleof attack a is below a certain value well below stall, say 10�, butthen to allow blades to pitch in such a way as to maintain a at orbelow value. Due to lack of space and because it is not clear howthis might be achieved simply in practice, only limited results ofmodeling of this system are shown here, but more complete resultscan be seen at [5].

The second system, referred to below as “passive eccentricpitch,” also pivots blades forward of their unstalled centre ofpressure so fluid dynamic forces tend to pitch the blade towardszero angle of attack, but uses an eccentric linked to the trailingedge of each blade to limit pitching to a sinusoidal regime asshown in Fig. 6. with a preset amplitude and offset. Although themechanism used to achieve this is similar to a simple activesystem as described in [6], it is in fact passive in that fluid dynamicforces drive the mechanism, whereas an active mechanism bydefinition drives blade pitch.

3.1. Modeling “stabilized free pitch”

“Stabilized free pitch” was modeled assuming that as q anda increase from 0, a stabilizing mechanism holds blade pitch g at0 until a0, the zero pitch angle of attack, reaches a predeterminedcut-in value ap below the stall angle in either direction (althoughthe sign of a and g changes as a blade goes through q ¼ 0 and 180�

and signs must be taken into account in the mathematics, they areignored for simplicity in the discussion below). As a0 passes ap, theblade starts to pitch so that a ¼ ap until g ¼ gmax, the limit of pitchangle permitted by the turbine geometry. If a0 continues to increasebeyond ap þ gmax the blade may stall, but if gmax is about 20 or 30�,blade stall will occur only at very low l, and it is found that thisgives ample starting torque and has no effect on peak efficiency. Asq approaches 180� and a0 decreases, g remains at gmax until a0drops below ap þ gmax and then decreases to zero when a0 ¼ ap.This sequence repeats in reverse, with opposite signs, as q goesfrom 180� to 360�. Because of the wide variation of a0 with q at lowl, gmax would have to exceed 90� to limit a to 10� at l ¼ 0.5, butneed only be as high as 30� to limit a to 10� at l � 1.5 and 18� tolimit a to 10� at l � 2. For practical reasons it is convenient to limitgmax to about 20�.

Fig. 7 shows how the torque coefficient CQ and performancecoefficient CP vary with l for various pitch angle limits with ap ¼ 8�

and 2 m/s current speed, compared with an otherwise identical

fixed pitch turbine. It will be apparent that an even a 10� pitch anglelimit produces at big improvement in both starting torque and peakefficiency compared to a fixed pitch turbine of high solidity(s ¼ 0.6), and that there is a significant further improvement instarting torque if gmax is increased to 20�, but diminishing returnsincreasing gmax further to 30�, and no increase in peak efficiency.Increasing ap to 10 or 12� makes little difference, and plots showingthis are omitted from this report to save space. Efficiency issignificantly lower in 1 m/s current due to low blade chordReynolds numbers, but does not improve at current speeds above2 m/s. Plots showing these effects are again omitted from thisreport to save space.

Table 2Constraints on pitching parameters.

Minimum Maximum

gmin (deg) �30 0gmax (deg) 0 30a�p (deg) �16 0aþp (deg) 0 16

Table 3Pitching parameters found using the algorithm.

Objective gmin (deg) gmax (deg) a�p (deg) aþp (deg)

Maximize CQ0 �30.0 30.0 �13.4 2.4Maximize CP �16.0 13.0 �7.0 5.0Minimize JD �18.0 30.0 �3.0 2.0Minimize JS �26.9 17.7 �0.9 0.3Minimize fitness �6.5 28.2 �1.0 5.2

Table 5Pitching parameters found using the algorithm.

Objective gmin (deg) gmax (deg) a�p (deg) aþp (deg)

Maximize CP �18.4 10.9 �7.6 5.1Minimize JD �18.0 30.0 �3.0 2.0Minimize JS �26.9 17.7 �0.9 0.3

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e50 45

4. The memetic algorithm to minimize shaking

It is apparent from the foregoing discussion that a passive pitchcontrol mechanism that limits the angle of attack as describedabove can greatly improve performance over an equivalent fixedpitch turbine, assuming that the physical mechanism can achievethe pitch regime assumed in the model, but the issue of shakingremains. To assess this factor, downstream and side forces werecalculated for all q and a range of l, and hence turbine downstreamand side force coefficients were calculated. Two “shaking” param-eters were defined respectively as

JD ¼ variation of downstream force coefficient around themeanJS ¼ variation of side force coefficient around the mean

A memetic algorithm was developed to search for optimumpitching parameters to reduce shaking while maintaining strongstarting torque and high peak efficiency. Five optimization objec-tives were used within the program. The first objective was tomaximize the starting torque CQ0. The second objective was tomaximize the peak performance coefficient CPmax. The third andfourth objectives were to minimize, respectively, the downstream“shaking” parameter JD and the side force parameter JS. As anattempt at defining some combined figure of merit, a fifth objectivewas to minimize a function f combining these variables, defined as“best fitness”

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJ2

D þJ2S Þ=Cp max

q:

Many engineering design tasks can be cast in the form of opti-mization problems. One formulation, of sufficient generality formany applications, is tominimize a real-valued function f(x1; x2;.;xn), with each real parameter xi subject to (domain) constraintsai � xi � bi for some real constants ai and bi. In the context of thepresent study, f is either the performance coefficient CP or someother quantity such as the shaking parameters JD and JS.Constraints can be imposed on, among other quantities, the pitch

Table 4Values of the objective function found using the algorithm within the range0 < l < 3.0.

Objective CQ0 CPmax JD JS f

Maximize CQ0 0.157 0.417 0.140 0.316 0.829Maximize CP 0.092 0.470 0.141 0.276 0.659Minimize JD 0.131 0.312 0.072 0.245 0.818Minimize JS 0.121 0.166 0.156 0.135 1.246Minimize fitness 0.103 0.426 0.112 0.217 0.574

Optimum values shown in bold typeface.

angle limits, the length of radial arms, position of the blade pivot,the blade polar moment of inertia, etc.

Many techniques exist for solving optimization problems suchas the one described above, but they vary greatly in efficiency andin the quality of the final solution for a given number of functionevaluations. No single technique is best for all design problems.Gradient-based methods work well with smooth, unimodal func-tions, but may yield local optima for multimodal functions.Heuristic algorithms can increase search efficiency, but at theexpense of guaranteed optimality e they do not always find theglobal optimum.

Genetic Algorithms (GA) are adaptive search methods that useheuristics inspired by natural population dynamics and theevolution of life. According to [14], they differ from other searchand optimization schemes in four main respects:

- Search proceeds from a population of points, not from a singlepoint

- They use a coding of the parameters, not the parametersthemselves

- Objective function values guide the search process e they donot use gradients or other problem-specific information

- State transition rules are probabilistic, not deterministic.

While they are quite effective techniques for global optimiza-tion, GA can sometimes take an intolerably long time to converge.In short, both GA and local search techniques can spend dispro-portionate amounts of effort examining infeasible regions of the(often enormous) search space. GA sometimes search too broadly,local search methods too narrowly.

Many different methods have been devised combining GA withother search techniques in an attempt to improve their overallperformance. The evolutionary biologist Richard Dawkins coinedthe word “meme” as a term for non-material entities that arecapable of transmission or imitation, such as ideas, tunes, andcatch-phrases [15]. memetic algorithms are GA that incorporatelocal search techniques.

4.1. Computer techniques

“VAWTEX”, Author 1’s computer program, implements a non-traditional GA similar to Eshelman’s [16] “CHC” algorithm. Thesystem has been enhanced with additional heuristics such ascataclysmic restarts, and incest prevention, described in Eshelmanand Schaffer [17]. VAWTEX’s general operation can be describedquite succinctly: create and evaluate new (candidate) designs untilsome termination criterion is met. In the present study, a run isterminated when a certain fixed number of designs have beenevaluated. There are no theoretical results that can be used to

Table 6Values of the objective function found using the algorithm within the range2.0 < l < 3.0.

Objective CPmax JD JS

Maximize CP 0.471 0.161 0.262Minimize JD 0.312 0.072 0.245Minimize JS 0.165 0.155 0.081

Optimum values shown in bold typeface.

Table 7Pitching parameters found using the algorithm to minimize JR subject toconstraints on CPmax.

Objective gmin

(deg)gmax

(deg)a�p(deg)

aþp(deg)

a�p0(deg)

aþp0(deg)

Maximize CP �18.4 10.9 �7.6 5.1 0.0 0.0Minimize

JR: 0.95�10.5 14.4 �9.8 8.3 �2.6 3.5

Minimize JR: 0.90 �11.8 14.2 �9.4 9.3 �3.9 4.5

Table 8Values of the objective function found using the algorithm tominimizeJR subject toconstraints on CPmax.

Objective CPmax JD JS JR

Maximize CP 0.471 0.187 0.262 0.322Minimize JR: 0.95 0.452 0.124 0.147 0.192Minimize JR: 0.90 0.429 0.115 0.123 0.168

Optimum values shown in bold typeface.

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e5046

determine the optimum number of evaluations that should beperformed. VAWTEX begins the optimization process by creatingan initial population of real-valued design vectors and calculatingCP and other quantities for each design. In the present study, initialdesigns are randomly generated. Genetic operators and otherheuristics are used to create candidate designs. Genetic operatorscreate new (offspring) vectors from two parent vectors in the

Fig. 8. CQ and CP for optimization to minimize the shaking force parameter JR subjectto constraints on CPmax.

population, using heuristics inspired by the recombination of DNA.The primary genetic operator in VAWTEX is based on an operatorgleaned from fuzzy set theory as described in Voigt [18]. VAWTEXuses “binary tournament selection” to select parent vectors fromthe population. In this method, two individuals are selectedwithout replacement from the population. The individual with thehigher CP becomes the first parent. A second binary tournamentdetermines the other parent. After evaluating the CP of theoffspring, VAWTEX replaces another individual in the populationwith the offspring if the offspring’s CP is larger. This replacementstrategy guarantees that the best individual in the population isnever replaced by an inferior individual.

Stochastic search techniques can place great demands oncomputer resources. It is important to run a sufficient number ofcases, each time varying only the initial population, in order toenable some confidence that an optimal solution has been found, orat the least that it has beenwell-approximated. Just as importantly,each run must be allowed to continue for long enough to be able tofind the optimal solution, but not for so long that resources arewasted. Unfortunately, there are no theoretical results available thatallowreasonable estimates to bemade for the requiredduration, norof the numberof runs required to guarantee that anoptimal solutionhas been found. To put the size of the present design problem intoperspective, imagine that we wanted to do an exhaustive search tofind the six angles defining the present pitching system, each towithin 0:1. This brute force method would require evaluation of

Fig. 9. Non-dimensional tangential force coefficient F*T for all 3 blades combined.

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e50 47

about 300 � 300 � 160 � 160 � 160 � 160 z 6 � 1013 individualdesigns. At 1 s perdesign evaluation on a3.2GHzPC, thatwould takeabout two million years.

4.2. Initial optimization results

The principal physical and model parameters are shown inTable 1. And the constraints on pitching parameters are shown inTable 2. The optimum pitching parameters found using the algo-rithm are shown in Table 3, and the corresponding values of theobjective function are shown in Table 4 (these values apply only to0 < l < 3.0, the range of tipspeed ratios used during the search). Itwas found that optimization for any single criterion alone (exceptCPmax) involves considerable sacrifices to CPmax, but the “bestfitness” curves show only a modest loss in performance for all l upto about 3. The “best fitness” curves show a considerable reductionin variation of CDS and CSF compared to fixed pitch, but not muchreduction compared to the “good CPmax” curve. In summary, start-ing torquewas more than adequate for all conditions, but there wasstill much room for improvement in minimization of shaking.

4.3. Optimization for l from 2 to 3

The initial optimization was then repeated for the narrowerrange of l from 2 to 3, optimizing for only three objective functions,omitting CQ0 since it was more than adequate for all conditions. The

Fig. 10. Non-dimensional radial force coefficient F*R for all 3 blades combined.

principal physical andmodel parameters are the same as in Tables 1and 2, while the values found during the search, summarized inTables 5 and 6, differ slightly. However it was found that mini-mizing JD or JS individually led to unsatisfactory CPmax. So thefocus was redirected to minimize shaking while retaining highCpmax.

4.4. Minimizing shaking while retaining high Cpmax

The aim in the next optimization study was to determinewhether shaking can be reduced significantly by allowing a slightreduction in CPmax. A different objective function was used, incor-porating CPmax at all times. Optimization was done for the range oftipspeed ratios 1.75 � l � 3, since peak efficiency occurs in thisrange. The first objective was to maximize CP. The pitchingparameters and output quantities found for this objective areidentical to those given in Tables 5 and 6. The second and thirdobjectives were to minimize the “resultant” shaking force param-eter JR subject to constraints on CPmax. The “best” variable pitchturbine found in the first optimization has CPmax ¼ 0.47, although itshould be noted that parasitic drag has not been allowed for andthis will reduce actual CP significantly. The first constraint imposeddemands that CPmax � 0.95� 0.47, i.e. wewill accept a 5% reductionin CPmax, as long as we reduce JR. The second constraint demandsthat CPmax� 0.9� 0.47, i.e. we are prepared to trade a 10% reductionin CPmax for improved shaking characteristics.

Fig. 11. Sinusoidal pitch: CQ for maximum CPmax (top) and for minimum JR with 5%loss of CPmax (bottom).

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e5048

The values of the pitching parameters found using the algorithmtominimizeJR subject to constraints on CPmax are shown in Table 7.For this optimization an “extended” pitch control model was intro-ducewith two furtherpitchingparameters:a�p0 andaþp0. Itwas foundthat shaking could be reduced by pitching the blade to maintaina¼ 0 until a0 reaches a small angle designated a�p0, a

þp0. The values of

the three objective functions found during the search are summa-rized in Table 8. The underlined values correspond with the partic-ular objective function. Thus, for example, the underlined value inthe first row is the value of CPmax found when the objective was tomaximize CP, etc. The other two objectiveswere tominimizeJR. Theresults show that they are each better than the JR achieved whenusing the CPmax objective, but by definition they have a worse CPmax(by about 5% and 10%, respectively), as shown in Fig. 8, in whichcurves are labeled according to their respective objective functions.For example, the turbine labeled “JR: 0.95” was optimized forminimumJR subject to the constraint CPmax � 0.95 � 0.471.

Figs. 9 and 10 show, respectively, the (non-dimensional)tangential force and the (non-dimensional) radial force for all threeblades combined. The variable pitch systems do an excellent job atsmoothing out the oscillations apparent in the curves of F*T and F*Rfor the fixed pitch variant.

4.5. Modeling “passive eccentric” pitch

The foregoing indicates that a turbine with “stabilized freepitch” can in principle achieve much better starting torque,

Fig. 12. Sinusoidal pitch: CP for maximum CPmax (top) and for minimum JR with 5%loss of CPmax (bottom).

efficiency and smoothness of operation than an equivalent fixedpitch turbine. However at the time of writing this has not yet beenachieved in practice, and it was decided to model a turbine with“sinusoidal” pitch control, as it appeared that this would be easierto build, and since the modeling was done, such a turbine has beenbuilt and tested, and shaking has been observed to be acceptablymild when in variable pitch mode, and violent when pitch is fixed.

Since a cam can in principle produce any pitch regime, anoptimization was performed for pitching including up to 5harmonics rather than limiting the study to a simple sinusoidalpitch variation. Optimizationwas done for VN¼ 2.0m/s and l¼ 2.5.The pitch system is defined by

gðqÞ ¼ c1 þXM

j¼2

cj sin½ðj� 1Þq�

where the ci are constants to be found by the optimization processand results are computed for values of M from 2 to 6. Thus, forM ¼ 2 we have

gðqÞ ¼ c1 þ c2 sin q

which is identical to the system examined in [6]. Results are shownfor pitching systems optimized for maximum CPmax at tipspeedratio l ¼ 2.5. These are labeled S2, S3, . S6 in the graph legends.The last digit is the value of M, i.e. the number of free parameters.Results labeled P2, P3,.P6 are optimized for minimumJR with 5%loss of CPmax. These can again be considered as compromise

Fig. 13. Sinusoidal pitch: downstream shaking parameterJD for maximum CPmax (top)and for minimum JR with 5% loss of CPmax (bottom).

Fig. 14. Sinusoidal pitch: side force shaking parameter JS for maximum CPmax (top)and for minimum JR with 5% loss of CPmax (bottom).

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e50 49

solutions where we are trading maximum performance for lowershaking. Figs. 11 and 12 show CQ and CP for maximum CPmax and forminimum JR with 5% loss of CPmax. Although CQ0 is less for thisform of pitch control than for the “stabilized free pitch” systemshown in Fig. 8 (about 0.05 compared with 0.08), it is still muchbetter than the fixed pitch turbine, and CPmax is about the same asfor the “stabilized free pitch” system, about 50% higher than forfixed pitch. Additional harmonics appear to offer little if anyadvantage over a simple first order sinusoidal pitch regime shownin bright red, so the higher orders will not be discussed further.Figs. 13 and 14 show downstream and side force shaking parame-ters JD and JS for maximum CPmax and for minimum JR with 5%loss of CPmax. Comparing the “5% loss of CPmax” curves for the 2 pitchcontrol systems shown in Fig. 9 with Figs. 13 and 14, JD is signifi-cantly higher (i.e. worse) in the sinusoidal system, but still muchbetter than for fixed pitch (about half), andJS in the range 2< l< 3is marginally better for the sinusoidal system.

5. Conclusions

Modeling using VAWTEX-B and a memetic algorithm to opti-mize pitch parameters for two basically different passive pitchcontrol systems predicts that performance can be greatly increasedover that of an equivalent fixed pitch turbine. Peak efficiency ispredicted to be about 50% higher, starting torque several timeshigher and shaking can be reduced by factors of 2e3. It remains tobe seen if these improvements can be achieved in practice.

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Notation

Ab: Blade planform areac: Blade chord lengthCDS: Turbine downstream force coefficientCL: Blade lift coefficientCP: Turbine performance coefficient ¼ P=ð0:5rAtV3

NÞCPmax: Best value of CPCQ: Turbine torque coefficient ¼ Cp/lCQ0: Turbine starting torque coefficient, i.e. CQ at l ¼ 0CSF: Turbine side force coefficientCT: Coefficient of tangential thrust ¼ CL sin a � CD cos a

f: “best fitness” parameter ðto be minimizedÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJ2

D þJ2S Þ=Cp max

q:

FD: Drag force ¼ CDqrelAbFL: Lift force ¼ CLqrelAbFN: Centripetal hydrodynamic force ¼ CNqrelAbF*R: Non-dimensional blade radial force ¼ FN/qNAbFSF: Blade side forceFT: Tangential thrust ¼ CTqrelAbF*T : Non-dimensional blade tangential thrust ¼ FT/qNAbGA: Genetic algorithmHb: Blade heightHKT: Hydrokinetic turbineM2, ..M6: pitch amplitude harmonicsn: Number of bladesP: Instantaneous power delivered by turbineP2, ..P6: pitch amplitude harmonics for minimum JR allowing 5% reduction in CPmaxqrel: Dynamic pressure based on blade relative velocity ¼ 0:5rV2

relqN: Dynamic pressure based on ambient fluid velocity ¼ 0:5rV2

N

r: Turbine radiusRe: Reynolds numberS2, ..S6: pitch amplitude harmonics for best CPmaxVb: Blade tangential velocity ¼ urVrel: Relative velocity between blade and surrounding fluid

L. Lazauskas, B.K. Kirke / Renewable Energy 45 (2012) 41e5050

VN: Ambient fluid velocitya: Blade angle of attack ¼ a0 þ ga0: Blade angle of attack for zero pitcha�p , aþp : Pitch angle parametersa�p0 , a

þp0: Pitch angle cut-in parameters

g: Blade pitch anglegmax: Maximum allowable blade pitch anglegmin: Minimum allowable blade pitch angleq: Blade azimuth angle

l: Blade tipspeed ratio ¼ Vb/VN

n: Fluid molecular kinematic viscosityr: Fluid densitys: Turbine solidity ¼ nc/rJD: Variation of downstream force coefficient around the meanJS: Variation of side force coefficient around the mean

JR: “Resultant” variation of force coefficients ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJ2

D þJ2S Þ

q

u: Angular velocity