Optimising power take-offs for maximizing wave energy conversions

Transcription

1 Absrac for he 0h Inernaional Workshop on Waer Waves and Floaing Bodies, Brisol, UK, h Apr Opimising power ake-offs for maximizing wave energy conversions Wanan Sheng, Raymond Alcorn and Anhony Lewis Beaufor-Hydraulics and Mariime Research Cenre, Universiy College Cork, Ireland w.sheng@ucc.ie, r.alcorn@ucc.ie,.lewis@ucc.ie) 1 Inroducion The paper presens a sudy on he differen power akeoff PTO) dampers linear and nonlinear) and heir opimizaions for maximising wave energy conversions on a poin absorber wave energy converer. To simplify he problem, a boom-fixed poin absorber and he single heave moion is considered for power conversion and analysis. For such a sysem, heoreical work has been widely carried ou in opimising he damping levels in maximising wave power conversion if he power ake-off is linear and under he assumpion of he linear hydrodynamics of wave energy conversion. I has been shown ha he relevan opimised damping can be easily obained analyically in regular waves [1, 2]. However, when i comes o he nonlinear power ake-off, he problem becomes more complicaed, and much less research work has been conduced and opimised for nonlinear power ake-offs. I has been wondered, however, wheher he nonlinear power ake-offs are beer han he linear ones, because some claims have been made ha he nonlinear power ake-offs can conver more power han hose of linear power ake-offs. Though here is limied evidence for he claims, i is no eviden wheher i is coinciden or no. For insance, if hese PTOs are no opimised, hen he comparison among he differen PTO damping coefficiens may be meaningless and even unfair in some cases. In his research work, he power conversions from he linear and nonlinear PTOs will be conduced appropriaely. More imporanly, he comparisons will be made for he opimised damping coefficiens for boh linear and nonlinear PTO damping coefficiens so ha he maximum power conversions from differen PTOs are comparable. From he sudy, i is shown ha he averaged power conversion from he opimised linear damper and nonlinear dampers can be very similar. The maximum power conversion using he nonlinear PTOs may be marginally higher han ha of he opimised linear PTO, boh in regular waves and in irregular waves. Tha is, he maximised power conversion using a nonlinear PTO may exceed heoreical maximum from he linear analysis, bu i mus be noed ha he exceedance is only marginal. One difference in opimizing he linear and nonlinear PTOs is ha he opimised nonlinear damping coefficien is boh wave period and heigh dependen, whils he opimised linear damping coefficien is only wave period dependen. 2 Dynamic equaions Figure 1 shows a schemaic drawing of he wave energy converer. Under he wave exciaion, he buoy is supposed o move up and down heave moion). When a PTO is applied o connec he buoy and he fixed reference for example, he seabed), he heave moion of he buoy can drive he PTO o conver he mechanical power ino useful energy. The generic dynamic equaion can be expressed as M A x K ) x ) d Cx 0 1) F F po where M is he mass of he device; A he added mass a infinie frequency for heave moion; K he impulse funcion; C he resoring coefficien; F he exciaion; F po he power ake-off PTO) force due o he power conversion; x he heave moion; v he heave velociy v x ). All parameers in eq. 1) excep F po can be assessed using he boundary elemen mehod for poenial flow heory in his case, WAMIT), in which he hydrodynamics of he floa has been aken as a linear dynamic sysem, hus a frequency domain can be conduced, and he relevan ime-dependen parameers can be also easily obained using a Fourier ransform. Figure 1 Seabed referenced poin absorber

2 If nonlinear effecs are considered, for insance, a nonlinear power ake-off, hey are only exernal forces, raher han he hydrodynamic forces. When we consider he wave energy conversion, he wave heighs may be medium, hence he nonlinear hydrodynamic effecs may no be eviden. Hence in his research, linear hydrodynamics is assumed. Under he assumpion of he linear hydrodynamics, he dynamic equaion 1) is correc whils he PTO force can be considered o be nonlinear or even piecewise ype, like in laching conrol see Sheng e al. []). Overall, his convenion will be applied hroughou his research. For a linear PTO, a pure damper PTO can be simply expressed as a linear relaion beween he PTO force and he moion velociy as F po b v ) 2) 0 where b 0 is he consan damping coefficien of he PTO, and v he velociy of he device in heave i.e., v x ). For he nonlinear PTOs, we will examine differen ypes of PTO. The firs ype is inspired by he nonlinear air urbine, for example, he impulse urbine see Falcao e al.[4]), in which he PTO force can be expressed as a nonlinear funcion of he velociy as, 2 F po b v sign v ) ) 1 where b 1 is he nonlinear damping coefficien, and he PTO force is proporional o he velociy squared, * means an absolue value. The second ype of nonlinear PTO is inspired by he relaion of he newly invened bi-radical urbine see Falcao e al. [5]), in which he PTO force can be expressed as F po b v sign v ) 4) 2 where b 2 is he nonlinear damping coefficien, and he PTO force is proporional o he velociy square roo. Once he dynamic equaion 1) is solved, he power conversion is simply calculaed as P ) F )* v ) 5) po he corresponding average power is given by T P 1 P d 6) T 0 where T is he ime inerval for calculaing he average power. Resuls and analysis.1 Power conversion in regular waves Figure 2 shows he averaged power conversions using linear and nonlinear PTOs in he regular waves of a heigh H=2m and a period T w=8s. In he calculaions, ime-domain simulaions and averaged power conversion have been conduced using he procedure shown in he previous secion. I can be seen ha he linear PTO has an averaged power conversion close o never larger han) he heoreical maximum in he frequency domain analysis, i.e., kw. Using he opimized damping coefficien, he linear PTO could exrac he maximal power close o he heoreical maximum. I can be seen ha away from he opimised damping coefficien, he capured power is decreased when he damping coefficien is eiher increased or decreased solid line in Figure 2). When he nonlinear PTOs are considered in he forms of Eqs. ) and 4), he maximised power conversions can be slighly larger han ha of he linear PTO. Figure 2 Damping level for regular waves H=2m and Tw=8s) The opimised damping coefficiens are b 0= kn*s/m, b 1=596.5 kn*s 2 /m 2 and b 2=189.8 kn*s 1/2 /m 1/2 for he respecive linear and nonlinear PTOs. I mus be noed ha he opimised nonlinear PTO coefficiens given above are based on boh he specific wave heigh H=2m and period, T w=8s, whils for he linear PTO, he opimised damping coefficien is only decided by he wave period. Figure o Figure 6 show he ime series of he simulaions in he specific regular wave. I can be seen ha he moions for differen opimised PTOs are very similar, only small differences can be discerned in he peaks and roughs Figure ). Relaively, he velociies of he heave moions for differen PTOs are quie differen in ampliude Figure 4). The PTO forces are very close again in he magniudes, and no large difference can be seen Figure 5), whils as a combinaion of he PTO force and he velociy, he power conversions are quie differen in peaks. I mus be noiced ha hough he difference in peaks in he power conversion, heir average power conversions are very similar, 40.77kW, 41.92kW and 42.01kW for b 0, b 1 and b 2 respecively. The nonlinear PTO could exceed he maximal power conversion given by he linear PTO by 2.82% and.04% respecively. For a reference, he raio of he maximal power over he average power is for he nonlinear PTO b 2), for he nonlinear PTO b 1), compared o he case wih a linear PTO, which is a consan of 2 Figure 6).

3 Figure Moion H=2m &Tw=8s) Figure 4 Velociy H=2m & Tw=8s) Figure 5 PTO force H=2m & Tw=8s) Figure 6 Power conversion H=2m & Tw=8s).2 Power conversion in irregular waves Figure 7 shows he averaged power conversions using linear and nonlinear PTOs in he irregular wave of a significan heigh H s=2m and a peak period T p=8s for a Breschneider specrum). From he calculaions, i can be seen ha he linear PTO has a maximal averaged power conversion for he opimised damping coefficien based on he wave energy period, T e=6.86s, ha is, b 0= kn*s/m in his case. The corresponding maximal power conversion for he linear PTO is kw. Away from he opimised damping coefficien, he capured power decreases whenever he damping coefficien is eiher increased or decreased solid line in Figure 7). When he nonlinear PTOs are used, he maximised power conversions can be slighly larger han ha of he linear PTO. Based on he simulaions, he opimised damping coefficiens for he irregular waves are b 0= kn*s/m, b 1=505.5 kn*s 2 /m 2 and b 2=12.79 kn*s 1/2 /m 1/2 for he respecive linear and nonlinear PTOs. And all opimised linear and nonlinear PTO coefficiens given above are based on he wave condiion of a significan heigh H s=2m and a peak period, T p=8s. Figure 7 Damping level for irregular waves Hs=2m & Tp=8s) Figure 8 o Figure 11 show he ime series of he simulaions for he specific irregular waves. I can be seen ha he moions in differen opimised PTOs are very similar, hough some differences can be seen in he peaks and roughs Figure 8). Similarly, he velociies of he heave moions for differen PTOs are differen, again in peaks and roughs Figure 9). Figure 10 shows he differences of he PTO forces in he magniudes Figure 10). Though he power conversions in ime series are quie differen in peaks, bu he averaged power conversion are very similar, kw, 18.05kW and kw respecively. The nonlinear PTOs may increase power oupu by 1.6 % and 1.24% for b 1 and b 2 respecively. Again, a very small increase of he power conversion can be only possible using he opimized nonlinear PTOs. For his paricular case, he raio of he maximal power over he average power is wih he nonlinear PTO b 2), 9.12 wih he nonlinear PTO b 1), compared o he linear PTO, which is Figure 11). These saisic values are based on he simulaions for abou 150 wave cycles. Figure 8 Moion Hs=2m & Tp=8s) Figure 9 Velociy Hs=2m & Tp=8s) Figure 10 PTO force Hs=2m & Tp=8s)

4 4 Conclusions Figure 11 Power conversion Hs=2m & Tp=8s). Maximised power conversion in irregular waves Figure 12 shows he maximised power conversions for differen significan wave heighs in irregular waves peak period T p=8s). I can be seen ha if he damping coefficiens are opimised, he linear and nonlinear PTOs can exrac very similar maximised powers from waves. The maximised power conversions are generally proporional o he wave heigh squared wih slighly differen coefficiens for each PTO. From Figure 1, i is ineresing o noe ha for he linear PTO, for he specific wave period, T p=8s, he opimised damping coefficien is a consan, regardless of he wave heighs. Bu for he nonlinear PTOs, he opimised damping coefficiens are boh wave period and heigh dependen. To reach opimised power conversions for differen wave heighs, he opimised damping coefficien b 1 decreases wih he increase of he wave heigh, whils he opimised damping coefficien b 2 increases wih he increase of he wave heigh. Figure 12 Opimised damping levels wih he wave heigh in irregular waves Tp=8s) Figure 1 Power conversion wih he opimised damping levels for irregular waves Tp=8s) In his invesigaion, some comparisons have been made for he linear and nonlinear PTOs in convering wave power ino useful energy in which he nonlinear power ake-offs are inspired by he pracical PTOs. From he invesigaion, he following conclusions can be drawn: - For maximising power conversion for he linear and nonlinear PTOs, he damping coefficiens mus be opimised. Under he opimised damping coefficiens b 0, b 1 and b 2), he averaged power conversions are very similar for he linear and nonlinear PTO dampers. The nonlinear PTOs may exrac he maximised power more han ha of he linear PTO, by 2-4% in regular waves, and 1-2% in irregular waves, respecively. - For linear PTOs, he opimisaion of he damping coefficien is only based on he wave period in regular waves and in irregular waves, regardless of he wave heigh. For he nonlinear PTOs, he opimised damping coefficiens are based boh on he wave period and wave heigh. For a specific wave period, he opimised damping coefficien decreases wih he increase of he wave heigh for he nonlinear PTO b 1), and he nonlinear PTO b 2) has an opposie rend wih regard o he nonlinear PTO b 1). Acknowledgemens This maerial is based upon works suppored by he Science Foundaion Ireland SFI) under he Charles Parsons Award a Beaufor -Hydraulics and Mariime Research Cenre HMRC). Saisics and daa were correc a he ime of wriing he aricle; however he auhors wish o disclaim any responsibiliy for any inaccuracies ha may arise. References 1. Falnes, J., Ocean Waves and Oscillaing Sysems: Linear Ineracion Including Wave-Energy Exracion. 2002: Cambridge Universiy Press. 2. Sheng, W. and Lewis, A., Assessmen of wave energy exracion from seas: numerical validaion. Journal of Energy Resources Technology, Dec. 2012).. Sheng, W., Alcorn, R., and Lewis, A., On improving wave energy conversion, par II: developmen of laching conrol echnologies. Renewable Energy, hp://dx.doi.org/ /j.renene Falcao, A. and Gao, L.M.C., eds. Air Turbines. Comprehensive Renewable Energy, ed. A. Sayigh. Vol , Elsevier: Oxford Falcao, A., Gao, L.M.C., and Nunes, E.P.A.S., A novel radial self-reecifying air urbine for use in wave energy converers. Renewable Energy, : p Sheng, W., Alcorn, R., and Lewis, A., 201, Laching conrol for improving wave energy conversion, submied o Renewable Energy for publicaion.

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