ANEXOS

A. PARAMETROS Y COEFICIENTES UTILIZADOS EN EL TRABAJODE FALNES Y YU

ρ = 1000 kg/m3, g = 9.81 m/s2

R = 0.35 m, d = 0.63 m, Profundidad = 3 m

M = 242 kg, M∞ = 83.5 kg

Bp = 0 N/(m/s).

KA = (ρ2g3R3)0.5 = 6362.1745139

KB = ρgR2 = 1201.725

KT = (g/R)0.5 = 5.29420168

afen(1) = 1701083.8271

afen(2) = 360590.7747

afen(3) = 33535.8936

afen(4) = 1793.8286

afen(5) = 52.7302

bfen(1) = 191377.3337

bfen(2) = −2714.4023

bfen(3) = 109.0274

bfen(4) = −0.5359

bfen(5) = 0.0030.

arn(1) = 2656.1615,

arn(2) = 496.1057

arn(3) = 23.3474

brn(1) = 0.8513

brn(2) = 1.7358

brn(3) = 0.0625

a = −(M + M∞) = −325.5

Chs = ρgS = 3775.40897145

B. MASA AGREGADA Y RESISTENCIA DE RADIACION OBTENIDASDEL TRABAJO DE FALNES Y YU

Tab. B.1:Masa agregada y resistencia de radiacion adimensionales del cilındro de Falnes y Yu

ω√

g/R 2πBr/ωM Ma/M ω√

g/R 2πBr/ωM Ma/M0 0,31941924 0,48646209 0,66 0,25408348 0,32220217

0,02 0,31941924 0,46916134 0,68 0,23547396 0,321299640,04 0,31941924 0,44644543 0,7 0,21785564 0,320397110,06 0,32032668 0,43275479 0,72 0,1984783 0,32031380,08 0,32123412 0,4232852 0,74 0,18034343 0,31882810,1 0,32200195 0,41606498 0,76 0,16346503 0,319494580,12 0,32487784 0,41064982 0,78 0,14773139 0,319494580,14 0,32667877 0,40523466 0,8 0,13199776 0,319494580,16 0,32881475 0,40072202 0,82 0,11717157 0,320341570,18 0,33121597 0,39620939 0,84 0,10516543 0,321299640,2 0,33484574 0,39259928 0,86 0,09230769 0,321299640,22 0,34016473 0,3898917 0,88 0,08126483 0,322202170,24 0,34304062 0,38718412 0,9 0,07168784 0,322202170,26 0,34682396 0,38267148 0,92 0,06209689 0,323104690,28 0,35060728 0,38086643 0,94 0,05349714 0,324909750,3 0,35529806 0,37815884 0,96 0,04707525 0,325812270,32 0,36116152 0,37571508 0,98 0,04056959 0,326076090,34 0,36569873 0,37274368 1 0,03448276 0,327617330,36 0,36932849 0,36913357 1,02 0,02903811 0,327617330,38 0,37153428 0,36732852 1,04 0,02450091 0,329241880,4 0,37568058 0,36281588 1,06 0,02177858 0,329422380,42 0,37749546 0,35920578 1,08 0,01814882 0,329422380,44 0,37749546 0,3558456 1,1 0,0154265 0,331227440,46 0,37568058 0,35198556 1,12 0,01227138 0,331227440,48 0,37199498 0,34837545 1,14 0,00998185 0,332129960,5 0,36639676 0,34497361 1,16 0,00907441 0,332129960,52 0,3584392 0,34115523 1,18 0,00725953 0,333921140,54 0,34794081 0,33754513 1,2 0,00635209 0,335740070,56 0,33666062 0,33393502 1,22 0,00544465 0,335129120,58 0,32131788 0,33212996 1,24 0,00453721 0,334837550,6 0,30657545 0,32942238 1,26 0,00272232 0,33664260,62 0,28947368 0,3267148 1,28 0,00272232 0,337545130,64 0,27382382 0,32528464 1,3 0,00272232 0,33754513

C. MASA AGREGADA Y RESISTENCIA DE RADIACION OBTENIDASDEL TRABAJO DE ERIKSSON

Tab. C.1:Masa agregada y resistencia de radiacion del cilindro de Eriksson

f(Hz) Br(f) Ma(f) f(Hz) Br(f) Ma(f)0,047 289,669205 9714,08694 1,222 155,70934 6951,854990,094 637,669206 9740,942 1,269 145,32872 6969,20290,141 1499,26091 9760,01452 1,316 117,65818 6987,44920,188 2799,81173 9410,89855 1,363 122,084285 7005,43480,235 3951,5709 8779,71008 1,41 124,56747 7014,49280,282 4942,67128 8096,21739 1,457 114,18685 7032,60870,329 5519,03668 7464,89862 1,504 114,18685 7032,60870,376 5623,82287 6947,13043 1,551 114,18685 7041,66670,423 5424,60617 6564,46375 1,598 114,18685 7059,78260,47 4981,20423 6341,52174 1,645 114,18685 7068,84060,517 4278,77647 6229,95653 1,692 114,18685 7068,84060,564 3563,50999 6217,3913 1,739 114,18685 7077,89860,611 2864,53705 6253,62319 1,786 114,18685 7077,89860,658 2238,74326 6326,08696 1,833 114,18685 7086,95650,705 1746,89275 6411,88406 1,88 114,18685 7086,95650,752 1323,77024 6493,08696 1,927 114,18685 7086,95650,799 992,57854 6579,71014 1,974 114,18685 7105,07250,846 775,573693 6646,43477 2,021 114,18685 7105,07250,893 589,71073 6709,52174 2,068 103,80623 7105,07250,94 445,370232 6760,86957 2,115 103,80623 7105,07250,987 363,3218 6799,46378 2,162 103,80623 7114,13041,034 290,65744 6844,43466 2,209 103,80623 7114,13041,081 230,341913 6871,28989 2,256 103,80623 7114,13041,128 200,18819 6907,20279 2,303 103,80623 7114,13041,175 180,415092 6932,97101 2,35 103,80623 7114,1304

D. MASA AGREGADA Y RESISTENCIA DE RADIACION OBTENIDASDEL TRABAJO DE HAVELOCK

Tab. D.1:Masa agregada y resistencia de radiacion adimensionales de la esfera de Havelock

ω2R/g Br(ω)πρR3ω/3

Ma(ω)πρR3/6

ω2R/g Br(ω)πρR3ω/3

Ma(ω)πρR3/6

0 0,04491363 0,85163149 1,02 0,23795585 0,445249510,03 0,09760077 0,8671785 1,05 0,23147793 0,440499040,06 0,15115163 0,87408829 1,08 0,22456814 0,436180430,09 0,18958733 0,87257677 1,11 0,2193858 0,43186180,12 0,22197697 0,86458734 1,14 0,21290788 0,428406910,15 0,25091171 0,8496881 1,17 0,20729367 0,424952020,18 0,27552783 0,82960652 1,2 0,20383877 0,423224570,21 0,29841651 0,80434261 1,23 0,19757676 0,419769670,24 0,31612284 0,78080614 1,26 0,19347409 0,416314780,27 0,33102208 0,75381478 1,29 0,18829175 0,414587330,3 0,34246641 0,73027831 1,32 0,18483685 0,412859880,33 0,35045586 0,70846929 1,35 0,18138196 0,411132440,36 0,35585413 0,68493283 1,38 0,17619962 0,409404990,39 0,35930902 0,6631238 1,41 0,17274472 0,40595010,42 0,35930902 0,64347409 1,44 0,16928983 0,404222650,45 0,35930902 0,6246881 1,47 0,16583493 0,404222650,48 0,35758157 0,60806142 1,5 0,16238004 0,400767750,51 0,35304702 0,59078695 1,53 0,15892514 0,400767750,54 0,34851248 0,57480806 1,56 0,15547025 0,399040310,57 0,34225048 0,56163628 1,59 0,15201536 0,397312860,6 0,33685221 0,54673705 1,62 0,14856046 0,395585410,63 0,33145394 0,53529271 1,65 0,14510557 0,395585410,66 0,32476008 0,52341651 1,68 0,14165067 0,393857970,69 0,31785029 0,51240403 1,71 0,13992322 0,393857970,72 0,3109405 0,50268714 1,74 0,13646833 0,392130520,75 0,30295105 0,49469769 1,77 0,13474088 0,390403070,78 0,29539347 0,4884357 1,8 0,13042227 0,390403070,81 0,28848369 0,48195777 1,83 0,12783109 0,388675620,84 0,27898273 0,47504798 1,86 0,12610365 0,388675620,87 0,27099328 0,4681382 1,89 0,12200097 0,386948180,9 0,26429942 0,46295585 1,92 0,12092131 0,385220730,93 0,25738964 0,45777351 1,95 0,11811419 0,385220730,96 0,25047985 0,45259117 1,98 0,11573896 0,38522073

E. COEFICIENTES ADIMENSIONALES DE TORQUE Y PRESION ENFUNCION DEL COEFICIENTE ADIMENSIONAL DE CAUDAL DE LA

TURBINA BIDIRECCIONAL

Tab. E.1:Coeficientes adimensionales de torque y presıon en funcion del coeficiente adimensional decaudal

ϕ CA CT

-2 15,1863656 7,33902633-1,75 14,8405274 6,82870572-1,5 14,4973808 6,20515701-1,25 13,8917265 5,30925268

-1 12,7913355 4,1049718-0,75 10,9969492 2,67940007-0,5 8,34227936 1,24272978-0,25 4,69400836 0,12825981

0 -0,04821121 -0,207604340,25 3,10850206 -0,017739760,5 6,26255206 0,999435940,75 8,98440111 2,34678523

1 11,0661863 3,723743541,25 12,4281645 4,928313861,5 13,1187125 5,857066711,75 13,3143268 6,50514015

2 13,3196241 6,96623982,25 13,5673404 7,43263881

F. ARTICULO PUBLICADO: GARCIA AGUSTIN, MONTOYA DAN ELAND DE LA VILLA JA EN ANTONIO (2010, MARCH). CONTROL OF

HYDRODINAMIC PARAMETERS OF WAVE ENERGY POINTABSORBERS USING LINEAR GENERATORS AND VSC BASED POWER

CONVERTERS CONNECTED TO THE GRID.INTERNATIONALCONFERENCE OF RENEWABLE ENERGIES AND POWER QUALITY

ICREPQ’10. EUROPEAN ASSOCIATION FOR THE DEVELOPMENT OFRENEWABLE ENERGIES, ENVIRONMENT AND POWER QUALITY

(EA4EPQ).

European Association for the Development of Renewable Energies, Environment and Power Quality (EA4EPQ)

International Conference on Renewable Energies and Power Quality (ICREPQ’10)

Granada (Spain), 23rd to 25th March, 2010

Control of Hydrodynamic Parameters of Wave Energy Point Absorbers using Linear Generators and VSC-based Power Converters Connected to the Grid

Agustín García Santana1, Dan El Montoya Andrade2 and Antonio de la Villa Jaén3

1 Department of Electrical Engineering AG Ingeniería

Phone:+0034 634 821844, e-mail: agustin.garcia@agingenieria.abengoa.com

2 Department of Electrical Engineering Venezuela Central University

e-mail: danel.montoya@ucv.ve

3 Department of Electrical Engineering E.S.I., Seville University e-mail: adelavilla@us.es

Abstract. The use of linear generators in offshore wave energy plants is one of the most promising technologies for such facilities. The first power converter stage consists of rectifying the electricity generated by the linear generator. Lower cost and easy of assembly are the main diode rectifier advantages, but these devices are not able to control the power extracted from waves. The use of power converters allows control over the instantaneous power flow, and in turn enables both power flow directions. This paper focuses on the control of the first power conversion stage using Voltage Source Converters, in order to modify the characteristics of the hydrodynamic buoy-translator system and thus, optimize the extracted energy from the waves. Simulation results are presented showing the features and advantages of the use of controlled power converters. Key words Wave energy converters, linear generators, WEC control, renewable energy, modelling. 1. Introduction Among the emerging electrical power generation choices from renewable sources of energy, the energy contained in the seas and oceans is one of the most promising. Research and industry have begun considering Wave Energy Converters (WEC) as the next new alternative energy. A diversity of prototypes has been developed during the last decades. Oscillating buoy systems are one of the systems currently in testing stage [1]. These devices use the vertical motion produced by waves on a buoy. This paper focuses in oscillating point absorber. This device consists of a buoy which is directly connected to a permanent magnet linear generator (LPMG) via a rope placed on the seabed, where storms are not dangerous. To

extract energy, springs are attached between the alternator and the foundation to pull the alternator downwards in wave troughs. The direct conversion power take-off (PTO) system provides a simple and robust way of increasing the survival possibilities, reduces the maintenance costs and may reduce the cost of electricity produced from wave energy [2]. The electric energy should be properly injected in the grid. This condition implies to keep the right wave quality at the connection point. The available primary energy features are pulsing and thus, it is necessary to use WEC systems that incorporate intermediate energy storage, to reduce the oscillations in the injected power. For this purpose, several systems have been used, such as oil or air pressure circuits. One of the most interesting proposals about connecting the electricity generated by buoy oscillating systems to the power grid, is based on using linear generators and full-scale back-to-back Voltage Source Converters (VSC) jointly [3,4,5]. Linear generators do not require mechanical linkages to convert the buoy oscillating motion into rotative generator motion [6]. Besides, the power converter enables the use of DC Link capacitor as an intermediate stage of storage. It also enables to adapt electric power output for proper grid connection [3,4]. The hydrodynamic characteristics of the WEC system made up of the buoy and the generator moving parts, largely determine the amount of energy extractable from waves. For regular waves, it is possible to calculate the optimal hydrodynamic characteristics that enable to extract the maximum amount of energy, for each type of WEC and for incident wave [7]. Controlling the auxiliary mechanical systems enables to achieve these optimal hydrodynamic conditions. However, these systems entail

more complexity in the design of the WEC and slow response times. In references [3,4] control techniques are applied to a VSC back-to-back converter using the dq frame transformation of the electrical parameters. The one used in [4] acts on the generator side converter, in order to extract the maximum power from the wave within the Archimedes Wave Swing device. This paper proposes a generalized strategy that allows optimization of energy extraction from regular waves in an oscillating system using VSC-based power converters. The WEC hydrodynamic conditions are matched to the instantaneous wave conditions in regular waves, via the electromagnetic force exerted by the generator. Controlling the first power conversion stage in the converter will be the way to implement this type of control. The optimum conditions require that during certain periods of time energy is delivered to the oscillating device [8]. It implies that the generator switches to work as motor during some periods of time. Under this operating condition, it is necessary to use a powered conversion to enable both instantaneous power flux directions. Finally, the overall balance of energy extracted using different sub-optimal control strategies have been analyzed. 2. Hydrodynamic control Hydrodynamic control in case of single oscillation frequency regular wave, aims to match the movement of the floating device to the characteristics of incident waves. Hydrodynamic control optimal strategy aims to find both elements (wave-buoy) in resonance in order to extract the maximum amount of wave energy. It is considered the simple case of a body oscillating in heave. The governing equation for the body oscillations is:

sgh fffzM ++=⋅ && (1) where M is the mass of the buoy-traslator system, is the acceleration, fh is the vertical component of the force due to water pressure on the wetted surface of the body, fg is the vertical component of the force applied on the buoy by the PTO mechanism and fs is the restoring force due to the spring attached to the translator.

z&&

If the amplitudes of the waves and body motions are small, it could be introduced the usual decomposition [9]:

hsrdh ffff ++= (2) where fd is the force produced by the incident waves on the assumedly fixed body (excitation force), fr is the

hydrodynamic force due to the body oscillation in otherwise calm water (radiation force), and fhs is the hydrostatic force. The hydrostatic force may be written as:

zSgfhs ⋅⋅⋅= ρ (3) where ρ is water density, g is acceleration of gravity and S is the buoy cross-sectional area defined by the undisturbed water free-surface. The spring force that anchors the device to the seabed is defined by the following expression:

zkfs ⋅−= (4) where k is the spring constant. In case of regular waves, it is convenient to decompose the radiation force as

zBzmf addr &&& ⋅−⋅−= (5) here madd is the added mass and B is the radiation damping coefficient. These parameters depend on the frequency of the waves and define the hydrodynamic characteristics of the buoy. In case of regular waves of frequency ωw, the excitation force is a simple-harmonic function of time t

( )tidd

weFf ⋅⋅⋅= ω Re (6) here ωw is the regular wave frequency. The amplitude of this force can be obtained by [7,9]

wd ABgF ⋅⋅⋅⋅

= 3

32ωρ (7)

where Aw is the incident wave amplitude. If linear PTO system is assumed, it follows:

zkzf gg ⋅−⋅−= &γ (8)

where γ and kg are constants. The first term represents the damping effect associated with the energy extraction, while the second is a spring effect (which may exist or not). This term represents energy exchange between the PTO and the oscillating system, similar to the one done by a spring. If spring effect exists, there will be bidirectional energy exchange between the generator and the buoy. Thus, the electrical machine will operate as motor and generator alternatively. Taking account the above considerations, the system is completely lineal and equation (1) may be written as:

( ) ( )( ) ( )ti

dg

add

weFzkkSg

zBzmM⋅⋅⋅=⋅++⋅⋅+

+⋅++⋅+ωρ

γ

Re

&&& (9)

Note the similarity between equation (9) and the one regarding to RLC series circuit fed by sinusoidal voltage. The resonant frequency of the oscillating system is

add

g

mMkkSg

+

++⋅⋅=

ρω0 (10)

To extract the maximum wave energy amount, two conditions must fulfill [7]: 1. The natural oscillation frequency of the oscillating system must match the wave frequency (resonance). 2. The damping force constant from the LPMG should be controlled equaling to the radiation damping coefficient.

wωω =0 (11)

B=γ (12)

A. Optimal strategy to extract wave energy The optimum conditions may be obtained by controlling the force that the generator applies to the oscillating system. Matching the constants kg and γ in equation (8), the two conditions for optimization may be fulfilled. If the power converter allows bidirectional power flow, the resonant frequency of the oscillating device can be modified and will depend on the control strategy applied to the generator. Thus, to extract the maximum wave energy, the force exerted by the PTO must be calculated by the constants:

Bop =γ (13)

( ) kSgmMk addopg −⋅⋅−+= ρω 20_ (14)

In the system discussed in this article, the force exerted by the PTO on the oscillating system is developed by the linear generator. Its value will depend on the type of power converter and control strategy.

B. Optimal strategy setting null the PTO stiffness force

In this case, the value of the constant kg is set null. There will be no energy transfer from the generator to the oscillating buoy system. Therefore, it would not be possible to change the resonant frequency of the oscillating device by acting on the generator control. Systems where the value of the constant γ may be controlled, the maximum wave energy extraction from equation (9) is achieved for:

22 )( ⎟

⎠⎞

⎜⎝⎛ +⋅⋅

−+⋅+=ω

ρωγ kSgmMB addso (15)

Systems where unidirectional power converters are implemented, work under these conditions. For instance, the ones based on uncontrolled diode rectifiers. The power extracted using this strategy is an upper limit for the power that can be obtained from systems that use non-controlled rectifiers. 3. Linear generator An important aspect of survivability is the complexity and longevity of the technology. The more complex a system is and the more moving parts it is composed of, the more likely it is that some part will fail. Furthermore, to improve the prospects of becoming commercially interesting, the need for maintenance should be kept at a minimum. To meet these requirements, the wave energy converter may use a directly driven longitudinal flux, three-phase, synchronous permanent magnet linear generator. A directly driven generator also circumvents the need of gearboxes, a component in need of regular maintenance and with a relatively high risk of failure. The generator consists of insulated cables, Nd-Fe-B permanent magnets, electroplate, structural steel, and springs. The springs are fastened underneath the translator and serve as a retracting force in wave troughs after the buoy and translator have been lifted by wave crests. Furthermore, the generator is fitted with upper and lower end stops consisting of powerful springs, whose purpose is to limit the mechanical impact on the generator in extreme sea states [2]. The voltage output varies in both amplitude and frequency parameters. Thus, conversion is necessary through a power electronic converter prior to deliver energy to grid. The dq components regarding to the linear generator model used in this paper can be expressed as follows [10]:

sqsmsd

ssdssd iLdt

diLiRv ⋅⋅−+⋅= ω (16)

fdmsdsmsq

ssqssq iLdt

diLiRv φωω ⋅+⋅⋅++⋅= (17)

λπω v

m⋅⋅

=2

(18)

where λ is the pole width of the LPMG, Rs is the stator resistance, Ls is the stator inductance and v is the linear speed of the buoy. Further Φfd is the flux linkage of the stator d-winding due to the flux produced by the permanent magnets. vsd, vsq, isq and isd are the voltage induced and the stator current, in the dq reference frame.

Fig. 1: Single phase output voltage from linear generator.

Figure 1 shows the three-phase linear permanent magnet generator voltages output, which vary in both amplitude and frequency parameters. The active power at the generator output [3] may be written as:

sqfdms ip ⋅⋅⋅= φω23 (19)

The force applied on the buoy by the generator, may be calculated by the following expression [4]:

λφπ sqfds

g

ivpf

⋅⋅⋅==

3 (20)

4. Power conversion electronics Between the different options considered in this paper, the electronics power selected includes two power converters linked by a DC link [3,4,5]. The configuration considered in this paper includes two PWM controlled VSC. The generator side converter controller match the force exerted by the generator and minimize the power losses in the generator, using the dq frame transformation. The generator force is controlled via the quadrature component of the current isq according to equation (20). Thus, the control reference of the q axis current is:

( )⋅

⋅⋅⋅−⋅−⋅

=fd

grefsq

zkzi

φπγλ3

11_

& (21)

The power losses in the generator can be reduced, equaling to zero the direct component of the current isd produced by the generator. Thus, the control reference of the d axis current is set null (isd_ref=0). The grid side converter controllers are the output active power to the power grid and the power factor. The first is controlled via the direct component of the current sent to grid iDg, and its reference is taken as the average value of the power delivered by the generator [3,4]. The second is controlled via the quadrature component of the current

sent to grid iQg, taking account a unity power factor as reference. 5. Results

A. Case study A wave power device has been developed at the Swedish Centre for Renewable Electric Energy Conversion in Uppsala. The 10 kW power point absorber combined with linear generator has been deployed in Lysekil [1]. Data regarding to oscillant system and generator concerning to this real facility, have been used in the simulations. The hydrodynamic parameters have been taken from [11]. The most important of them are shown in table I. Table II includes the main linear generator features [12].

Table I. - Main features of the point absorber

Buoy shape cylindrical Buoy radius 1.5 m Buoy height 0.8 m Buoy mass 850 kg

Spring constant 7064 N/m

Table II. - Main generator features

Nominal output power 10 kW Nominal speed 0.7 m/s

Phase-to-phase voltage r.m.s. 200 V Pole width 50 mm

Synchronous reactance 7.8 mH Stator winding resistance 0.45 ohms

The DC link nominal voltage is 1.100 V and its capacitance is 1 F. Energy is fed into the 20 kV distribution system and the connection to grid consists of 0.69/20 kV transformer and a line whose equivalent total impedance is 0.01+0.04j Ω. To perform the simulations, several typical wave states, according with the localization, have been considered and are shown in table III [13]. The chosen scenarios are characterized by an energy period Te and a significant height Hs. For each state, an equivalent regular wave state has been chosen. This equivalent scenario has the same wave power level, and its parameters are the period Te and the wave height H, which is 2 times smaller than Hs. Table III. - Parameters for the sea states used to foce the model

Scenario Te Hs H J (kW/m)

A 3.89 0.51 0.36 484 B 5.07 0.93 0.66 2099 C 7.34 2.36 1.67 19567

B. Power transfer between oscillant and PTO systems

This section discusses the characteristics of the power extracted from waves, when different references are applied to the force exerted by the electric machine according to the expression (8). The values of the constants for each strategy are defined in Table IV.

Table IV. – Control strategies

Strategy Description kg γ 1 optimum -60665 2800 2 pmax=20 kW -46032 13332 3 suboptimum 0 49031

The wave state considered in this section corresponds to scenario B. The average power extracted and the maximum instantaneous power values are listed in Table IV.

Table V. –Average and maximum power (kW) for different

control strategies in scenario B

Strategy Average power Max. power 1 13.5 250 2 5.1 20 3 1.5 2.9

In the first strategy, the optimal conditions were adopted. They were obtained from expressions (13) and (14). The oscillating system came into resonance through these conditions. The average power transfered to the PTO system (13.5 kW) is the maximum that can be extracted in this wave state. However, the operating conditions obtained are not feasible because the maximum oscillations (2.5 m/s) and the translator speed (3.1 m/s) are too large; and so does the instantaneous peak power exchanged with the PTO (250 kW). A large amount of power exchanged produces a significant increase in the electrical machine losses, and may lead to negative balances regarding to the power delivered to the grid by the system. In the second strategy, a suboptimal method is applied: the maximum instantaneous power does not have to exceed a certain value (20 kW). Figure 2 shows the instantaneous power exchanged with the PTO. It can be seen periods where the machine is operating as motor. Under these conditions, the average power extracted from waves is 5.1 kW, oscillations are around 0.7 m. and the maximum speed is 0.87 m/s, which can be considered acceptable values.

Fig. 2: Power (W) exchanged between the oscillating system

and generator applying the strategy 2 in scenario B

Finally, in the third strategy, the optimal conditions for zero stiffness, as shown in equation (15), are used. Figure 3 shows the instantaneous power exchanged with the PTO. This time, the power always flows from generator to the PTO, and the average power extracted is reduced to 1.5 kW.

Fig. 3: Power (W) exchanged between the oscillating system and generator applying the strategy 3 in scenario B

To implement the three strategies described above, the power converter configuration has to allow the control over the force developed by the electrical machine. The first and second strategies also require bidirectional power flow. The results in the third strategy are an upper limit for the power that can be obtained from systems that use uncontrolled rectifiers.

C. Power fed into the grid. This section presents the results of simulations carried out incorporating models PTO made up by the linear generator and the power converter, according to the configuration described above. Simulations have been developed taking account the wave scenarios shown in Table III. Table VI shows the average power delivered to the PTO and the average power fed into the grid when control strategies 2 and 3 are applied.

Table VI. – Powers (kW) fed into the grid

Strategy 2 Strategy 3 scenario Pav to PTO Pav to grid Pav to PTO Pav to grid

A 1.0 0.9 0.5 0.4 B 5.1 2.5 1.5 1.1 C 9.7 6.9 7.1 4.6

Comparing results in Table VI, it is clear the advantage of applying a suboptimal strategy which lies in incorporating the effect stiffness. Scenarios A and B show that through the proposed strategy, the average power fed into the grid could double. Thus, the energy extracted by a power converter which incorporates proposed control in the first stage is larger than systems using non-controlled rectifiers. 6. Conclusions This paper focuses on the way to improve the hydrodynamic features of the WEC by controlling rightly the power converters. It has been taken into account that the force applied to the oscillant system by the PTO depends, not only on its velocity, but also on its position. Using this control strategy, it has been shown how to reach resonance and extract the maximum power from the wave. Simulations show that applying optimal control strategy is not feasible. Suboptimal strategies are compared. Results indicate that strategies including stiffness effects controlled in the first conversion stage, increase the energy extracted from waves. Acknowledgements This work is part of the project ENE2007-63306/CON, financed by the Spanish Ministry of Science and Education. The authors are also grateful for the financial support provided by Focus-Abengoa Foundation. References [1] C. Boström, E. Lejerskog, M. Stålberg, K. Thorburn and M. Leijon, “Experimental results of rectification and filtration from an offshore wave energy system”, Renewable Energy (2009), Vol.34 Nº.5 pp.1381-1387. [2] M. Leijon, R. Waters, M. Rahm, O. Svensson, C. Boström, E. Strömstedt, J. Engström, S. Tyrberg, A. Savin, H. Gravråkmo, H. Bernhoff, J. Sundberg, J. Isberg, O. Ågren, O. Danielsson, M. Eriksson, E. Lejerskog, B. Bolund, S. Gustafsson, and K. Thorburn, “Catch the wave to energy”, (2009), IEEE power & energy magazine, Vol. 7, pp50-54. [3] F.Wu; X. P. Zhang, P. Ju, M. J. H. Sterling, “Modelling and Control of AWS-Based Wave Energy Conversion System Integrated Into Power Grid”, IEEE Trans. Power System (Aug. 2008). Vol. 23, nº 3, pp. 1196-1204. [4] F. Wu, X. Ping, P. Ju and H. Sterling, “Optimal Control for AWS-Based Wave Energy Conversion System”, IEEE Trans. Power System (Nov. 2009). Vol. 24, nº 3, pp. 1747-1755.

[5] M. Molinas, O. Skjervheim, P. Andreasen, T. Undeland, J. Hals, T. Moan, B. Sørby, “Power electronics as grid interface for actively controlled wave energy converters”, (2007), IEEE. [6] M.A. Mueller, “Electrical generators for direct drive wave energy converters,” IEE Proceedings, Generation, Transmission and Distribution (Jul. 2002), Vol. 149, nº. 4, pp. 446-456. [7] Johannes Falnes, Ocean waves and oscillating systems, Cambridge University Press, Cambridge (2002), pp.51-52. [8] J. Falnes, “Principles for capture of energy from ocean waves. Phase control and optimum oscillation”, (1997), Departament of Physics, NTNU, N-7034 Trondheim, Norway. [9] A.F. de O. Falcao, “Modelling and control of oscillating-body wave energy converters with hydraulic power take-off and gas accumulator” Elsevier Ltd (2007), Ocean Engineering vol. 34, pp. 2021–2032. [10] A. Schacher, “A Novel Control Design for a Wave Energy Converter”, Thesis for the degree of Master of Science in Electrical and Computer Engineering, Oregon State University, Oregon (2007) pp.55. [11] M. Eriksson, J. Isberg and M. Leijon, “Theory and Experiment on an Elastically Moored Cylindrical Buoy”, (2006), IEEE Journal of Oceanic Engineering; Vol. 31, nº 4, pp.959–963. [12] O. Danielsson, M. Eriksson and M. Leijon, “Study of a longitudinal flux permanent magnet linear generator for wave energy converters”, (2006), International Journal of Energy Research; Vol. 30, pp.1130–1145. [13] J. Engström, M. Eriksson, J. Isberg and M. Leijon, “Wave energy converter with enhanced amplitude response at frequencies coinciding with Swedish west coast sea states by use of a supplementary submerged body”, in journal of applied physics 106, 064512 (2009).

G. ARTICULO PUBLICADO: PEREIRAS B., MONTOYA D., DE LAVILLA JA EN A.,EL MARJANI A. AND RODRIGUEZ M. (2010,

OCTOBER). CONCEPTION OF A RADIAL IMPULSE TURBINE FOR ANOSCILLATING. THIRD INTERNATIONAL CONFERENCE ON OCEAN

ENERGY ICOE 2010

3rd International Conference on Ocean Energy, 6 October, Bilbao

Conception of a Radial Impulse Turbine for an Oscillating Water Column (OWC)

B. Pereiras1; D. Montoya2; F. Castro1; A. de la Villa3; A. el Marjani4; M. A. Rodríguez1

1 Energy and Fluid Mechanics Engineering Department. University of Valladolid. Paseo del Cauce 59, 47011, Valladolid, Spain. E-mail: castro@eis.uva.es

2 School of Electrical Engineering. Venezuela Central University. , Caracas, Venezuela. E-mail: danel.montoya@ucv.ve

3 Department of Electrical Engineering. E.S.I., Seville University, Seville, Spain. E-mail: adelavilla@us.es

4 Labo. de Turbomachines, Ecole Mohammadia d’Ingénieurs (EMI). University of Mohammed V Agdal. Av Ibn Sina, B.P. 765

Agdal Rabat, Morocco. E-mail: marjani@emi.ac.ma

Abstract

This work deals with the conception, from a mechanical and electrical point of view, of an impulse axial turbine for an oscillating water column (OWC) plant. Air turbines for wave energy conversion have a special feature to be taken into account: these turbines are self-rectifying and work with a cyclically bidirectional air flow alternatively as an inflow/outflow turbine.

The turbine geometry proposed in the bibliography has been used. A previously developed numerical CFD (Computer Fluids Dynamics) model has been used to improve aerodynamically the internal geometry of the turbine. With a better efficiency, the impulse turbine is now attractive for the industry. In order to start the industrial design, some specifications expected for the turbines have been underlined. The nominal power will be 20kW.

To carry out a global mechanical design of the turbine, the first step was principally to propose a feasible structure, easy to assemble and guarantee easy maintenance. The interaction between the turbine and the generator is an important factor, which has a direct influence on the global efficiency of the OWC plant. Therefore, it has been necessary to study both the dynamic and electrical behaviour in order to control the turbine operations. Specifically, the evolution of the power provided by the swell over the time is taken into account. Keywords: Wave Energy, OWC, Turbine, Mechanical Conception.

1. Current status of OWC and air turbines

OWC is one of the more interesting wave energy conversion systems. It is capable of transforming wave energy into pneumatic energy and this into electrical

energy. The OWC plant consists of a chamber where the water free surface goes up and down as a function of the waves which arrive at the chamber. When a wave reaches the chamber, the oscillating water free surface rises and the air inside the chamber is forced to go out of the chamber through the turbine. At the moment in which the wave moves away from the chamber, the air pressure inside it decreases and the air comes into the chamber through the turbine. To take advantage of this bi-directional flow, self-rectifying turbines are used. The main characteristic of these turbines is that they have to rotate in the same direction independently of the flow direction.

Until now, the most common turbine used in OWC was the Wells turbine [1]. But Setoguchi et al. [2] made a study of this turbine and found that the Wells turbine has many drawbacks, such as low efficiency far from the best efficiency point, stall when the flow rate is too high, a very high rotation velocity and the long time in reaching operational velocity. In order to avoid these problems, many modifications have been proposed, including guide vanes [3], variable-pitch angle blades [4], several rotors [5], contra-rotating rotors, using different chord blades and geometry ratios [6]. One of the alternatives to the Wells turbine is the impulse turbine: axial [7] or radial [8].

Reports which compare the Wells and axial impulse turbines with fixed guide vanes [2] show that impulse turbines have a better behaviour than the Wells one in running and starting characteristics under irregular flow conditions.

The radial impulse turbine, proposed by McCormick, has lower maximum efficiency but is able to maintain a high efficiency in a wide range of operational conditions. An experimental work which studied the behaviour of the radial impulse turbine [9] shows that this sort of turbine has a relatively high mean efficiency and there is no oscillating axial thrust. Other important benefits of this kind of turbine are its low cost and high torque. On the other hand, this turbine causes a great damping on the OWC system.

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3rd International Conference on Ocean Energy, 6 October, Bilbao

Over the last five years, our group has been investigating the hydrodynamic behaviour of the radial impulse turbine. This knowledge has enabled us to create a new geometric design [10] from an initial design taken from the bibliography [9]. This new design considerably improves the efficiency of the previous geometries [9], [11]. In [10] the optimization of the turbine design was carried out by means of a numerical model. This numerical model, previously validated [12], allowed us to study in depth the flow through the machine and its sources of energy loss [13; 14]. This new geometry has meant an advance in turbine performance.

The objective of this paper is to propose a semi-industrial prototype by taking into account mechanical and electrical aspects. Therefore, this paper shows the first steps about how a previous configuration, numerically developed [10], is turning into reality. This work can be divided into two parts: mechanical and electrical design. Nevertheless, firstly it is necessary to determine the size of the turbine. The mechanical conception about the manufacturing process is shown. The second part analyzes the performance of the radial impulse turbine in an electrical system in a stand-alone mode. This structure, isolated from the grid, allows us to study the coupling between the turbine and the generator with more flexibility in the frequency and the magnitude of the voltage produced. Analyzing the performance of the turbine-generator system under non-stationary flow conditions is the aim of the electrical study

2. Mechanical conception This turbine has been conceived on a large scale,

with low-cost manufacturing which is easy to use and maintain and with a long life. As this turbine will be placed on the coast, it should also be able to resist harsh environmental conditions such as corrosion, humidity, etc.

The turbine operating conditions were previously imposed: the nominal power is 20kW. To guarantee that the turbine works as long as possible in high efficiencies, this nominal power should be reached at a higher flow rate than the optimal one. Therefore, the nominal power will be obtained at a flow rate coefficient (φ) of 1.2.

From a dimensionless analysis, the following can be obtained:

( )1243 +⋅⋅⋅⋅= ϕΩρ RTturbine rhCW

where W is the power extracted by the turbine, CT is the torque dimensionless coefficient, ρ the density, h is the blade height, Ω the rotation speed and rR is the mean radii of the turbine. From this expression we can extract a relation between the size and the rotation speed (for a given nominal power and flow rate coefficient), as seen in Figure 1. A rotation speed of around 50 rd/s and a rR of 0.48 m is a balanced situation. Greater sizes will be a problem from the point of view of installation (great weight), manufacturing

cost and the connection to the generator (due to the low speed), although it could have the advantage of reducing the rotation speed and, therefore, vibrations and noise will be reduced. On the other hand, smaller sizes involve higher rotation speed, which is better for the generator and the installation. But this is not advisable from the point of view of vibrations and noise.

Figure 1. The relationship between the size of the turbine

rotor and rotational velocity (W=20 kW, φ=1.2)

3. Mechanical design One problem referred to in many papers is the

oscillatory bi−directional axial thrust which appears on the bearings in axial turbines but which does not appear in radial turbines. Nevertheless, to obtain a balanced stress-distribution on the structure (mainly on the bearings), a vertical configuration has been adopted. The global configuration is shown in Figure 2. With this configuration, all the weight is transmitted vertically to the floor through the peripheral elements.

The turbine has a rotor and a stator, which is composed of two rows of guide vanes: Inner Guide Vanes and Outer Guide Vanes. The guide vanes are joined to a disk by pivots which are fixed from the opposite side to the disk. The rotor has a row of fixed blades which are screwed to the hub. Both vanes and blades are fixed by two pivots to prevent any possible misalignment during performance. The geometry profiles of both vanes and blades are shown in [10]. The rotor is fixed to the shaft by means of a cone.

With this first design, the weight of the turbine was estimated approximately. Indeed, because of environmental requirements, the structure has to resist corrosion, wind, vibrations, overload, etc. As a consequence, metals like stainless steel or aluminium have been chosen for the construction. The estimated weight of the rotor and the stator manufactured in aluminium or stainless steel is approximately 500 kg and 900 kg, respectively. It is a considerable weight to be supported by the structure.

One of the main problems is the transmission of the weight of the moving parts: the rotor, gearbox and generator. These parts are placed in the upper part and are covered by an external case. This outer case supports the generator weight and isolates the moving parts from the environment.

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3rd International Conference on Ocean Energy, 6 October, Bilbao

a)

b)

c)

Figure 2. General view of the turbine: a) cross section, b) stator, c) side view of rotor

The rotor is composed of a disk where the blades are mounted and a cone which is used to guide the flow. The rotor is fixed to the shaft through the cone, which is connected with the generator. All the weight of the rotor rests on a bearing which is connected to the external upper case by four ribs. The second bearing is used to ensure the alignment of the rotor and the stator.

The weight which is held by the upper external case is supported by the outer guide vanes which are fixed to the upper case by various pivots, Figure 2b. All the weight is transmitted to the floor by the lower case.

4. Dynamic conception The main problems of turbines for OWC devices are

their operating conditions, as mentioned before: the bi-directional and variable flow rate.

Self-rectifying turbines is a possible solution for bidirectional flows. Once the first problem is solved, the problem of the flow rate variability is faced. A variable flow rate means that the torque is also variable. However, the rotation speed of the turbine should be as constant as possible to guarantee the correct performance of the generator. An inertia wheel, which allows us to maintain the rotation speed in a desired range when the flow rate is close to zero, can be used to solve the variability of the flow rate. Nevertheless, it should be carefully designed because an inertia wheel can harm the self−starting process of the turbine. Should it be necessary, it would be possible to use the generator as a motor during the start-up.

Another important point is the variability of the sea conditions. To guarantee the high efficiency of the OWC plant, it is necessary to adapt the damping made by the turbine. This can be achieved by modifying the rotation speed of the ensemble.

Determining the electrical performance of the turbine-generator ensemble is the first point to be carried out. This is done by stationary simulations at a constant flow rate. Next, the coupling turbine-generator is analyzed under a non−stationary flow rate. In this work, the turbine is connected to an isolated electrical system, without grid connection. This specification allows us to analyze the performance with more flexibility in the frequency and the voltage magnitude of the electricity produced.

4.1. Electrical System

An electrical system converts the mechanical energy in the shaft into electrical energy. In this application, electrical energy is dissipated on a load. In consequence, the electrical system is isolated and the frequency and voltage magnitude can be variable.

These specifications imply greater degrees of freedom in the turbine-generator control system, allowing a wide range of working speeds. In this way, the rigidity that grid connection involves (for instance, strict frequency and voltage connection) is avoided.

In this paper, the electrical machine used is a synchronous generator of 20 kVA rated power. Generator rated power agrees with the power extracted from the turbine at the flow coefficient, corresponding to the maximum efficiency. On the other hand, this kind of electrical machine allows more control over the power take-off of the electrical system in an isolated grid. The main generator specifications are shown in Table 1.

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3rd International Conference on Ocean Energy, 6 October, Bilbao

4

In order to adapt the turbine rotation speed to the generator a gearbox is used.

Control over the generator is made through the electrical load and the field excitation. Figure 3 shows the torque-speed features of the synchronous generator at different electrical load connected and a field current of 25 A. This field current corresponds to nominal power and nominal speed.

Feature Value

Rated Power 20 kVA Voltage 415 V Frequency 60 Hz Speed 1800 rpm Stator resistance 0.577 Ω Xd 19.23 Ω Xq 9.95 Ω Xl 1.29 Ω Coeff. of inertia 1.39 kg.m2 Pole pairs 2

4.2. System model

Figure 4 shows the connection scheme used in the simulations. The model has been developed in Matlab-Simulink software.

A first module allows us set up the flow across the turbine. The second one is the turbine module that determines its torque (Tt) through the speed and the air flow rate. This module solves the dynamical equation of the turbine-generator ensemble:

Table 1. Generator characteristics.

dtdΩJTT gt ⋅=−

where Tt y Tg are torques developed by the turbine and electrical system, respectively, J is the inertia of the rotor, and Ω is the shaft speed. The last module has the electrical system model, which calculates the generator torque in accordance with the rotation speed and the electrical load connected.

A specific air flow in the turbine needs values of electrical load and synchronous generator excitation to work in a stable manner. If the electrical system is not adjusted, the mechanical energy from the flow is not extracted, but it is converted into kinetic energy. It leads to an unstable performance characterized by a rising speed. Figure 3. Torque-speed at different load values.

Figure 4. Simulink model of the system.

3rd International Conference on Ocean Energy, 6 October, Bilbao

4.3. Constant air flow tests

These tests allow us to evaluate the performance of the turbine while air flow is constant. The aim is to obtain, for a specific flow rate, different operation points of the turbine-generator system. These operation points allow us to determine the turbine torque-speed curve. The final equilibrium stage of the turbine-generator system is defined by the equilibrium between the torque and the joint speed.

To develop simulations supposing a constant flow rate, each work point is obtained by modifying the electrical load and the field excitation on the synchronous generator.

Figure 5 and Figure 6 show the turbine torque-speed and shaft power-speed curves obtained at constant flow rates of 2 m3/s, 5 m3/s and 7.4 m3/s.

Figure 5. Turbine torque-speed curve.

Figure 6. Turbine shaft power-speed curve

4.4. Variable air flow tests

In this section, simulations obtained while the flow rate varies sinusoidally are shown:

) (ωQQ(t) max tsin=

where Qmax is the amplitude of the flow rate. During simulations, the electrical system is kept constant with an excitation field of 25 A and a load resistance of 95 Ω. Figure 7, Figure 8 and Figure 9 show the

simulation results obtained while turbine flow rate has a maximum value of Qmax = 7.4 m3/s with a period of 5 s.

In Figure 8, it can be noted that the turbine power sometimes takes instantaneous values of 20 kW and at other times the turbine power is zero. However, in Figure 9 it can be observed that the generator supplies a non zero oscillating power due to the system inertia.

Figure 7. Turbine speed obtained at Qmax = 7.4 m3/s

Figure 8. Turbine power obtained at Qmax = 7.4 m3/s

Figure 9. Electric power obtained at Qmax = 7.4 m3/s.

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3rd International Conference on Ocean Energy, 6 October, Bilbao

6

5. Conclusions This work shows the first steps towards developing a

turbine prototype for an OWC plant. The work is focused on the mechanical conception and the electrical performance of the system.

Obtaining a resistant and compact design has been the target of the mechanical study. A vertical configuration has been adopted to obtain a balanced distribution of the mechanical loads. It reduces the mechanical requirements of the bearings, which could be a problem from the maintenance point of view. It has been necessary to design a system to guarantee the alignment of the rotor and the stator. A solution for the rotor support is shown where the outer guide vanes have been used to bear the rotor and generator weight.

Numerical simulations in Simulink have been carried out to analyze the electrical performance of the turbine-generator ensemble. The ensemble has been analyzed under sinusoidal flow conditions with a stable model developed in steady conditions.

The results show that the variability of the rotation speed and the power extracted are less than 10%. This is caused by the inertia of the system which compensates for the lack of energy on the blades when there is no air flow through the turbine.

6. Acknowledgements This research has been carried out by two research

teams. The team from University of Seville is supported by project ENE2007-63306/CON, financed by the Spanish Ministry of Science and Education. The second group is composed of Fluid Mechanics and Turbomachinery research teams, respectively, of the University of Valladolid (Spain) and the University of Mohammed V-Agdal (Morocco). This group is supported by an AI project of the Agencia Española de Cooperación Internacional.

7. References [1] Raghunathan S., Tan C. P. and Ombaka 0.0 (December

1985). Performance of the Wells self-rectifying air turbine. Aeronautical Journal. pp. 369-378.

[2] Setoguchi T. and Takao M. (2006). Current Status of Self Rectifying Air Turbines for Wave Energy Conversion. Energy Conversion and Management, 46, pp. 2382-2396.

[3] Setoguchi, T., Santhakumar, S., Takao, M., and Kim, T.. (2001). Effect of guide vane shape on the performance of a Wells turbine. Renewable Energy, Vol. 23, pp: 1-15.

[4] Setoguchi, T.; Santhakumar, S.; Takao, M.; Kim, T.H.; Kaneko, K. (2003). A Modified Wells Turbine for Wave Energy Conversion. Renewable Energy, Vol.28, pp. 79−91.

[5] Gato L.M.C. and Curran R. (1996). Performance of the Biplane Wells Turbine. Journal of Offshore Mechanics and Arctic Engineering (OMAE), Vol. 118. pp. 210-215.

[6] Govardhan M. and Dhanasekaran T.S. (2002). Effect of guide vanes on the performance of a self-rectifying air

turbine with constant and variable chord rotors. Renewable Energy. 29, pp. 201-219.

[7] Maeda, H., Santhakumar, S., Setoguchi, T., Takao, M., Kinoue, Y., and Kaneko, K. (1999). Performance of an Impulse Turbine with Fixed Guide Vanes for Wave Energy Conversion. Renewable Energy, 17, pp. 533-547.

[8] McCormick M.E. and Cochran B. (1993). A performance study of a bi-directional radial turbine. Proceedings of the European Wave Energy Symposium. Edimburgh, pp. 443-448.

[9] Setoguchi, T.; Santhakumar, S.; Takao, M.; Kaneko, K. (2002) A performance study of a radial impulse turbine for wave energy conversion. Journal of Power and Energy, Vol. 216 (A1), pp. 15-22.

[10] Pereiras B., Castro F. and Rodríguez M. A. (2009). Development of a New Radial Impulse Turbine Design for OWC Proceedings of the 19th International Offshore and Polar Engineering Conference, Vol. 1. - ISSN 1098-6189.

[11] Pereiras, B., Castro, C., Marjani, A., and Rodríguez, M.A. (2008). Radial impulse turbine for wave energy conversion. A new geometry. Proceedings of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering (OMAE). - Estoril, June 15-20,.

[12] Marjani, A. e., Castro, F., Rodríguez, M., and Parra, M. (2008). Numerical Modelling in Wave Energy Conversion Systems. Energy, Vol. 33, pp. 1246-1253.

[13] Pereiras B. (2008). Estudio de una turbina de impulso radial para el aprovechamiento de la energía del oleaje. Doctoral thesis. University of Valladolid.

[14] Pereiras B., Castro F. and Rodríguez M. A. (2009). Tip Clearance Effect on the Flow Pattern of a Radial Impulse Turbine for Wave Energy Conversion. Proceedings of the 19th International Offshore and Polar Engineering Conference. Vol. 1. ISSN 1098-6189.

8. Nomenclature AR Characteristic area at rR

( ) RR2R

2R

T

rAuvρ21

TC

+=

Torque coefficient

h Blade height Q Flow rate rR Mean radius Tt Output mechanical torque Tg UR=ω rR Circumferential velocity at rR vR=Q/AR Mean radial velocity ρ Air density φ= vR/uR Flow coefficient Qmax Flow coefficient amplitude; Q= Qmax

sin (2πt/T) Ω Rotational speed

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