Dynamic Modeling and Optimum Load Control of a PM Linear Generator for Ocean Wave Energy Harvesting Application

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1 Dynamc Modelng and Optmum Load Control o a PM Lnear Generator or Ocean Wave Energy Harvestng Applcaton Haoe Luan, Omer C. Onar, and Alreza Khalgh Energy Harvestng and enewable Energes Laboratory, Electrc Power and Power Electroncs Center (EPPEC), Electrcal and Computer Engneerng Department, Illnos Insttute o Technology, 3301 S. Dearborn St. Chcago, IL 60616; Tel: (31) ; Fax: (31) ; Emal: khalgh@ece.t.edu; UL: Abstract Ths study presents a permanent magnetc lnear generator based system or ocean wave energy harvestng applcatons. The use o lnear generator as energy converter enables a drect couplng o the moton o the buoy to the energy harvestng system. Mathematcal model governng the system s presented; requency doman analyss reveals that converson rate vares wth respect to load resstance. Detaled numercal results ncludng average power, system ecency are obtaned usng MATLAB, Smulnk, SmPower-Systems. The optmum load value s ound based on the ully physcs smulaton o the system. To ths end, current control applcaton to the boost converter s presented. Presented results show that the controller successully regulates the current. The proposed system s desgned to operate wth the optmum load value. Keywords üocean wave energy harvestng, permanent magnet lnear generator, boost converter, current control technque. I. INTODUCTION Interests n energy harvestng rom renewable sources have been ncreased due to envronmental and economcal ssues. Among renewable energy harvestng technologes whch are stll beng nvestgated through varous ndustral and academc groups, ocean energy harvestng technology has already shown to be easble, snce oceans cover almost 70% o earth s surace [1]. Varous methods have been purposed or convertng ocean wave energy nto practcally usable electrcal energy. In [], authors proposed a system consstng o a loatng buoy wth an ar chamber and an ar drven generator. The electrcal energy s extracted rom rotatng the turbne by varyng pressure n the ar chamber, when the waves ht the body. However, ths requres good mechancal nsulaton through the ar chamber and the ventlatng generator to acheve ecency, whch brngs desgn complexty and addtonal cost to the system. In [3] and [4] the authors propose a Salter Cam desgn, whch rolls around a xed nner cylnder by the actvaton o an ncomng wave. Power can be captured through the derental rotaton between the cylnder and the cam. The problem wth ths desgn s the luctuaton o the ecency rangng rom 0-75% depends on the power ratng [5]. By 008, hundreds o prototype devces had been purposed but only 0 o them stepped up the second phase and some o them are really close to the nal stage and commercal deployment [6]. In ths paper, a lnear generator based ocean wave energy harvestng system s proposed. The dstnct property o ths lnear generator s exstence o varable requency current and voltage. In order to extract the maxmum output power rom the proposed structure, an AC/DC recter ollowed by a DC/AC converter s used to provde the lexblty o actvely controllng the power. Accordng to the dynamc model o the proposed system, we could observe that the power and ecency o the generator depend on the load resstance. For urther mprovement o the ecency o the proposed system, an optmum load resstance value s numercally speced or the proposed system. Ths dea, however, s applcable to other desgns and s valdated by smulaton results. Current control technque helps to mantan ths value regardless o actual load varaton. The rest o ths paper s organzed as ollows. The system operaton prncple, modelng and ecency equatons are gven n Secton II. The optmum load s dened and conrmed through analytcal analyss and smulaton n Secton III. The power condtonng technque or keepng system operatng on the optmum load s presented n Secton IV. Fnally, some concludng remarks are gven n Secton V. II. SYSTEM DESCIPTION AND OPEATION A system level dagram s shown n Fg. 1. The system conssts o a permanent magnet lnear generator, an AC/DC converter, and a boost DC/DC converter to control the optmal power extracton o the generator. Fg.. Lnear generator /09/$ IEEE 739

2 The dynamc model o the wave actvated lnear generator has been derved usng the orce equatons o the lnear generator n [7]. These orce equatons are moded and ther physcal meanngs are expressed here. As n Fg., ths two degree o reedom mechancal system wll generate power by electro-magnetc nducton. The generator s enclosed n a buoy and xed to the buoy base through a sprng. The generator has a rectangular wre loop o N turns, drectly supported by the buoy rame and a system wth two permanent magnets that are connected by helcal sprngs to the buoy. The wre loop moves between the magnets and electro-magnetcally nduces power. By targetng at both the buoy and the wre loop, we could get a par o second order equatons descrbng the moton o the system:..... BNL ( M+ Ma) Z ρgkpaacos( ωt) ( Z X) K( Z X) bz ρgaz (1).... BNL Mm X ( X Z) K( X Z) () where M s the mass o the buoy (mass o the sprng supported magnetc system s not ncluded), M a s the added mass, M m s the mass o the wre loop, b s the lud vscosty at temperature 0 C, ρ s the water densty, K p s the pressure response actor, a a s the wave ampltude, K s the total sprng constant, ω s the wave requency, Z s the buoy dsplacement, X s the wre loop dsplacement, L s the wre length, and N s the number o wre loops. The system wth permanent magnets s excted by the orce that s transmtted through two helcal sprngs. The electrc power generaton can be expressed as, N B L.. P ( Z X ) (3) The average power, P over one wave perod T can be obtaned by ntegratng the power equaton gven n (4). 1 T P P d t T (4) 0 The power whch s carred by an ocean wave can be expressed as: P ρ g a T D / 8 π (5) W Fg.1. System level conguraton o the lnear generator and power converters. The converson rate o the system can be expressed as the rato o the electrc power generaton over the total wave power whch s also the system ecency. η π ρ P / Pwave 8 N B L ( Z X ) dt / g a T D & & (6) III. OPTIMUM LOAD ANALYSIS From the ecency equaton, t can be observed that the system ecency s a uncton o both relatve speed o buoy and wred loop and the load resstance. In practce, both lud vscosty and sprng constant are small. They exert dumpng orce way less than that o the nducton to the heavy buoy. However, or a lghter wred loop, these eects are almost not neglgble. Takng these acts nto account, moton equatons can be smpled to the ollowng state space descrpton: x& 1 ρga ( BNL/ ) BNL/ x1 0 0 x& M+ Ma M+ Ma M+ M a x 1 ρgka p a + x& x 3 0M+ Ma x4 K BN L / K BNL / x4 0 Mm Mm Mm Mm cos( ωt ) (7) & where the ollowng substtuton apples: x1 z, x z&, x3 x, x4 x& ; The general solutons to these lnear equatons n requency doman are gven by: 1 1 x( ω) ( ω A) x(0) + ( ω A) BU( ω) (8) Where A s the state matrx, B s the nput and x (0) represents the ntal condtons [8]. The rst term descrbes the transent response o the system and t s not eectve n the steady-state response. The second term descrbes system behavor under partcular perturbaton. In ths case, t takes the orm o snusodal wave as the assumpton that ocean wave moves n ths pattern. Consequently, the soluton would consst o a pure snusodal wave at the same requency as the nput sne wave. The ampltude and phase determned by the system s requency response at that requency s gven by: Gt () G( ω)cos( ωt+ ϕ ) (9) s s /09/$ IEEE 740

3 One reasonable desgn o ths PM lnear generator s gven n [6], the assocated physcal parameters are: M 1344Kg, M 119.5Kg, L.64m, K 1, B 1.4T a, A 4.86m, L.64m, N 30. Furthermore, we assume, D.5m, K p 0.9, aa 1.5m and ω π( rad / s). Wth mplementaton o these parameters, the state-transton matrx can be wrtten as: A Accordng to the above analyss, the roots o the moton equatons are calculated as ollow: x G ( π) 9cos( πt+ G ( π)) (10) G s s ( π ) 4 3 s s s s ss ( s ) ( π ) 4 3 s π G s s s s s π 61.47s G3 ( π ) 4 3 s s s s s π s(61.47s ) G4 ( π ) 4 3 s s s s s π Fg. 4. Average power and ecency varaton versus load resstance. Accordng to the results, the maxmum power and maxmum ecency values can be acheved or a partcular value o the load resstance. Ideally, the system should operate at ths pont. In order to acheve ths goal, a current regulaton s provded usng a current controlled DC/DC boost converter. IV. POWE CONDITIONING The generator power and generator voltage are both n snusodal orm as shown n Fg. 5 and Fg. 6, respectvely. The generator power has a based snusodal waveorm. These solutons are substtuted wthn the system ecency equaton (3); then t s observed that system ecency s a complex uncton o resstance. It s also notced that by pckng a partcular, the system ecency may be maxmzed. Snce the analytcal soluton o system ecency results n very complcated expressons, the system s dynamc model s mplemented and the solutons o ts moton are analyzed numercally usng MATLAB and Smulnk. The numercal solutons are n good accordance wth our analytcal solutons. Once these dsplacements are known, they are used to observe other system dynamcs and calculate other varatons. Here, the ecency (Energy converson rate) s used as our perormance crtera. In the smulaton envronment, a varable resstance s used to present the eect o the load resstance on extracted power and ecency. A snusodal source s used to represent the ocean waveorm. The resstance-average power and resstance-ecency curves are shown n Fg. 4, whch share the same x axs. Fg.5. Generator power. Fg.6. Generator voltage. In order to regulate the generator s output power and voltage, the output o the lnear generator termnals s rst converted /09/$ IEEE 741

4 to the DC voltage and then ltered. Consderng the large oscllatons n the output, a large capactor s requred to acheve satsactory voltage regulaton perormance. A boost DC-DC converter operatng n the current control mode s employed, snce the nput voltage to boost converter s the output voltage o the recter and that s xed. Current control mode [9] helps optmum loadng o the lnear generator snce the equvalent Thevenn s mpedance s determned by the converter nput voltage and current. The advantages o current control ncludes: (1) Falures due to the excessve swtch current can smply be prevented by montorng the data obtaned va measurng the nductor current and () Smplcty o desgn eedback control because o the removal o one pole rom the characterstc equaton o the system. The duty rato o the PWM generator s controlled by a PI controller as: D K pε + K ε dt (11) where ε(t) s the error that s the derence between the reerence and measured current and calculated as ε( t) e (1) Assumng a lossless converter, the power at the nput and output o the converter are equal. Hence, the reerence current s calculated as the rato o DC voltage over optmum resstance value: vt () e () t (13) opt The two gans parameters are obtaned based on the averagng model o the boost converter [10]. In our smulated topology, the output voltage o the AC-DC recter s xed at 70V. In order to get the maxmum power at ths condton, the optmal resstance value equals to 17Ω whch yelds 4A reerence current approxmately. It s worth mentonng that ths computaton only provdes a reerence to ths desgn. In actual smulaton, the nstantaneous resstance value s derved rom the rato o DC voltage over the controlled current assocated wth the maxmum power Fgs.7 and 8 show the reerence current trackng and power perormance o the system wth derent actually load. From these gures, we could observe that the two systems have almost dentcal responses. Under ether condton, the systems are capable o outputtng 1,000 W o power and track the desred reerence current wthn 3s. Ths ndcates that the proposed system s robust to load change and s able to drectly support actual load. Thereore, regardless what s connected at the output, by controllng the nput current o the boost converter under xed nput voltage, you can control the equvalent resstance. The xed nput voltage or the converter s provded by the recter and capactor connected at the generator termnals. e Fg.7. eerence current trackng and power output or a 000Ω load. e Fg.8. eerence current trackng and power output or 100Ω load. The optmum load value(equvalent mpedance) s shown n Fg.9, where t s seen that the system ndeed operates on the desred pont. Fg.9. Optmum Load Value. V. CONCLUSION In ths study, dynamc model and control o permanent magnetc generator based ocean wave energy harvestng system has been presented. In order to extract maxmum energy rom the ocean wave and mprove energy converson rate, the optmum load value s ound. System dynamc, requency doman analyss and smulaton result conrms that operaton o system n ths pont leads to the maxmum /09/$ IEEE 74

5 converson ecency. An AC/DC recter ollowed by a boost converter s used to condton power as well as track the optmum load control value. EFEENCES [1] Ocean energy, eport o the US Department o Interor, Mnerals Management Servce, 007. [] S. H. Salter, Wave power, Nature, vol. 49, June 1974, pp [3] S.H. Salter, D.C. Jerey, and J..M. Taylor, The archtecture o noddng duck wave power generators, The Naval Archtect, London, Jan. 1976, pp [4].E. Dngwell, Predctons o power producton by a cam type wave energy converter or varous locatons, Master s thess, Massachusetts Insttute o Technology, [5] A.T. Jones, A. Westwood, Economc orecast or renewable energy technology development, IEEE power engneerng socety 005 meetng panel sesson, June 005 [6] P.Mesen, T,Hammons, Harnessng the untapped energy potental o the oceans: tdal, wave, currents and otec, IEEE Power engneerng socty energy development and power generaton commttee, September 15,005 [7] K, Thorburn, M, Leon, Farm sze comparson wth analytcal model o lnear generator wave energy converters, Techncal note Ocean engneerng 34 (007) [8] S. Goyal,. Kumar, and. A. Gupta, Smulaton and analyss o current controlled PFC converter-nverter ed SM drve, IEEE Internatonal Conerence on Industral Technology, Dec. 005, pp [9] J.Gary, Lnear system undamentals: contnuous and dscrete, class and modern, New york: McGraw-hll,(c)1983 [10].Pndado, A modelng and closed-loop control method or DC-DC PWM converters by local average technques Proceedngs o the IEEE Internatonal Symposum on ndustral Electroncs, /09/$ IEEE 743