Abstract:
Wave energy converters incorporating the components of an induction generator that are integrated with the components of a pneumatic spring are provided. The wave energy converters have a buoy having an interior guide, motion of the buoy representing a first degree of freedom and a spar mounted within the interior guide so as to be movable therein, motion of the spar representing a second degree of freedom. There is a pneumatic spring between the spar and the buoy to provide restoring forces when the spar departs from a quiescent position relative to the buoy. There is damping element in the form of an induction generator having an armature and a stator, with one of the stator and armature fixed to the spar and the other of the stator and armature fixed to the buoy to generate power when the spar moves relative to the buoy in the presence of excitation by ocean gravity waves. The mass of the wave energy converter and dimensions of the buoy are such that in use, the buoy floats in a partially submerged position.

Description:
FIELD 
       [0001]    The application relates to wave energy converters. 
       BACKGROUND 
       [0002]    Ocean waves in moderate to high sea states present high power densities and consequently represent an attractive energy source for power generation using renewable resources. Much work on wave energy converters (WEC&#39;s) has been done and many proposals for exploiting ocean-wave power have been documented. 
         [0003]    An early approach to wave energy extraction was based on a single-degree-of-freedom system, for which an oscillating body was directly coupled to a power take-off mechanism that was suspended on the bottom. Heave-excited WEC&#39;s with a single degree of freedom generate relatively low velocity motions that are not high enough to use practical direct-drive electrical generators unless mechanical transformers are engaged. Moreover, it may be difficult to provide suitable connection hardware from the buoy to the reference platform on the sea bottom due to tidal action and the very high forces involved. Systems with two degrees of freedom overcome bottom connection problems, a necessity in deep water. 
       SUMMARY 
       [0004]    According to a broad aspect, the invention provides a wave energy converter with two degrees of freedom. The wave energy converter has a buoy having an interior guide, motion of the buoy representing a first degree of freedom; a spar mounted within the interior guide so as to be movable therein, motion of the spar representing a second degree of freedom; a pneumatic spring between the spar and the buoy to provide restoring forces when the spar departs from a quiescent position relative to the buoy; and a damping element in the form of an induction generator having an armature and a stator, with one of the stator and armature fixed to the spar and the other of the stator and armature fixed to the buoy to generate power when the spar moves relative to the buoy in the presence of excitation by ocean gravity waves. The mass of the wave energy converter and dimensions of the buoy are such that in use, the buoy floats in a partially submerged position. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIG. 1  is a side cutaway view of a wave energy converter provided by an embodiment of the invention featuring an asymmetric pneumatic spring; 
           [0006]      FIG. 2A  is a cross sectional view of another wave energy converter provided by an embodiment of the invention featuring an asymmetric pneumatic spring; 
           [0007]      FIG. 2B  is a cross sectional view of a ladder construction for an armature and a distributed armature; 
           [0008]      FIG. 3  shows the transfer function of two example wave energy converters; 
           [0009]      FIG. 4  shows cross sectional view of three buoy shapes; 
           [0010]      FIG. 5  is a plot of an example of an asymmetric pneumatic spring characteristic; 
           [0011]      FIG. 6  is a cross sectional view of a wave energy converter featuring a spar with differing diameters; 
           [0012]      FIG. 7  is a cross sectional view of another wave energy converter provided by an embodiment of the invention featuring a symmetric pneumatic spring; 
           [0013]      FIG. 8  is a plot of an example of a symmetric pneumatic spring characteristic; 
           [0014]      FIG. 9  depicts an example mooring plan; 
           [0015]      FIG. 10  is a cross sectional view of another wave energy converter provided by an embodiment of the invention featuring a symmetric pneumatic spring and a two-sided armature; 
           [0016]      FIG. 11  depicts a model of a WEC; 
           [0017]      FIG. 12  depicts an equivalent circuit model of the WEC of  FIG. 11 ; 
           [0018]      FIG. 13  contains plots of power versus frequency for WEC&#39;s having three different values of a; and 
           [0019]      FIG. 14  contains plots of relative velocity versus frequency for WEC&#39;s having three different values of a. 
       
    
    
     DETAILED DESCRIPTION 
       [0020]    Embodiments of the invention provide a heave-excited WEC with two degrees of freedom. In some embodiments, the WEC is designed to have a transfer function that exhibits a bandwidth that matches or is included within the bandwidth of the average annual wave height spectrum of interest as derived from long-term wave statistics. The power take-off (PTO) is provided by an induction generator. In some embodiments, the WEC is designed to have a transfer function that also provides working velocities that are sufficiently high for the satisfactory operation of a direct drive linear electrical generator. In order to achieve this, the secondary system of the WEC includes a spring in addition to the PTO resistance and mass. 
         [0021]    WEC&#39;s are provided by embodiments of the invention in which a streamlined neutrally buoyant spar is used for the secondary mass. An induction generator, for example a linear induction generator serves as the PTO. Robustness and simplicity are very important for heave-excited wave energy converters. A direct-drive generator provides these characteristics. A linear induction generator with a distributed armature is therefore a very attractive direct-drive PTO for this application. A vector controller may be provided to control force and flux density independently during fast-response bi-directional high-power operation. 
         [0022]    A wave energy converter provided by an embodiment of the invention will be described with reference to  FIG. 1 , which provides a side cutaway view. A buoy structure is generally indicated at  100  to which is fixed an armature (elements  102 , 104  described below) of a tubular induction generator. The buoy  100  in the illustrated example generally consists of a hemispherical body  122  on which a partially submerged cylinder  121  may be mounted to increase the mass of the unit for a given buoy radius albeit at a potential penalty in output power. The armature consists of a thin-walled steel cylinder  102  inside which an aluminum thin-walled cylinder  104  having a cap  118  is flush fitted. The aluminum thin-walled cylinder  104  of the armature serves as a guide for a generally cylindrical spar  114 . The generator stator (windings)  112  is mounted on the top of the spar  114  and fitted with bearing assemblies  110  equispaced around the stator circumference at the top and bottom of the stator  112 . A piston seal is mounted on the top part of the stator  112 . Also shown are roller guides  124  to assist in maintaining the position of the spar  114 . The stator bearings maintain the specified separation between the stator  112  and the armature  102 , 104  in the presence of a magnetic attraction force between the armature and the stator when departures from the perfect geometry for zero force, which is inherent in the tubular generator configuration, occur. Such departures are introduced by manufacturing tolerance issues. The space between the top of the piston assembly and the cap  118  of the aluminum thin-walled cylinder  104  of the armature functions as a chamber  115  of a pneumatic spring. A valve  106  is provided in the cap  118  of the aluminum thin-walled cylinder  104  of the armature to adjust the mass of air in the chamber. When placed in water, the wave energy converter floats, with the designed sea level indicated at  120 . 
         [0023]    In this embodiment, the spar  114  serves as the secondary mass, the piston for the pneumatic spring and the platform for the generator stator  112 . The spar is extended to the required depth in the direction of the arrows shown in  FIG. 1 . The aluminum thin-walled cylinder  104  has a smooth inner surface that also serves as the cylinder for the pneumatic spring. In the example of  FIG. 1 , the armature is a distributed armature, but alternatively a ladder construction may be employed as detailed below. Pneumatic spring operation is realized more easily using a distributed armature because the ladder construction requires tight tolerances and careful assembly to achieve a sufficiently smooth surface for the piston seal to properly function. 
         [0024]    Any of the embodiments described herein may include a vector controller. A vector controller can be used to provide independent control of the force and the flux. In the subject application, first, energy provided by the generator in normal operation is extracted via a linear damping characteristic (force proportional to velocity). The vector controller therefore requires an accurate measurement of the relative speed between the armature and the stator. Second, the vector controller can impose a field reduction at low speeds to improve efficiency. Third, higher forces may be engaged by the vector controller to control relative displacement during extreme events associated with the random excitation. 
         [0025]    Referring now to  FIG. 2A , depicted is another wave energy converter provided by an embodiment of the invention. This embodiment differs from the embodiment of  FIG. 1  in that a stator  200  is mounted at the bottom of the buoy  202 , and the spar  204  becomes the armature in this case. 
         [0026]    The armature for the embodiment of  FIG. 2A  may be a distributed armature or have a ladder construction. In this case the piston seal operates against the steel cylinder and the required smoothness of the armature is determined by the bearing assemblies on the stator instead of the properties of the seal. A comparison between an example of a distributed armature  260  and an example of a ladder armature  250  is shown in  FIG. 2B . The ladder armature  250  provides higher generator efficiency for the same stator mass. For the ladder armature  250 , shown is a stator section  258 , and alternately placed steel and aluminum rings  254 , 256  concentric with a steel yoke  252 . For the distributed armature  260 , shown is a stator section  268 , and a steel cylinder  262  inside an aluminum cylinder  264 . 
         [0027]    The embodiments of the invention shown in  FIGS. 1 and 2A  present a seawater environment to the seal and generator components. The embodiment in  FIG. 2A  offers an improved situation in this regard because the wall of the pneumatic cylinder is steel, which being less active than the aluminum in the armature will be somewhat protected. The embodiment in  FIG. 1  can expect pitting and corrosion of the aluminum thin-walled cylinder of the armature (which serves as the pneumatic cylinder) even if zinc anodes are used. Placement opportunities for the anodes are limited given that the aluminum is a working surface for the spring. 
         [0028]    For any of the embodiments described herein, the shape of the buoy, its size and the selection of the secondary components will affect the power and velocity for the generator. The equations of motion for the second order WEC describe a coupled mechanical system that has a frequency response exhibiting two peaks if the coupling impedance between the primary and secondary masses is not too high. It is desirable and possible to have the same response at the peaks and to have a relatively flat response between them. It is well known that the two-degree-of-freedom system is used for damped vibration absorber applications to reduce the vibration of machines subject to speed variations. The purpose of this device is to minimize the motion of the machine that is subject to an applied force. The absorber has a bandwidth that includes the frequency of this excitation (the speed range of the machine). The reduction of the motion of the main mass is accompanied by relatively large motions of the secondary components, which implies high losses in the damper assembly. This is the feature that is exploited in a tuned WEC. An example of the tuning procedure for the vibration absorber that can be used for the WEC design is described later. 
         [0029]    It can be shown that the velocity effective at the PTO increases with decreasing size of the secondary components but at the expense of bandwidth and therefore output power. For the purpose of this example, it is assumed that the bandwidth of the ocean wave height spectrum of interest extends from about 0.075 Hz to 0.125 Hz. Referring to  FIG. 3 , shown is a transfer function  300  for a high power machine referred to hereinafter as WEC-I that exploits this entire bandwidth by using relatively large secondary components and offering relatively low secondary velocities.  FIG. 3  depicts the power response of the WEC (kW) as a function of frequency (Hz) to excitation by gravity waves having a mean square height of 1 m 2 . 
         [0030]    Also shown is a transfer function  302  of a smaller machine referred to hereinafter as WEC-II that provides a more suitable velocity regime for the induction generator. Simulations indicate that WEC-II provides about 50% of the power of WEC-I at representative sites off the East Coast of Canada. This is not surprising considering the difference in bandwidths. 
         [0031]    In some embodiments, the viscous losses for the buoy are considered, and factored into the choice of the design of the buoy. In the illustrated example of  FIG. 1 , the buoy  100  consists of a hemispherical end-piece  122  fixed to a cylinder  121 . The buoy shape was chosen to take advantage of its streamlined projected area compared to that of a right circular cylinder with vertical axis and the same buoy displacement. This choice reduces viscous friction losses without significantly affecting the coupling to gravity waves. The drag coefficient can be reduced further using a prolate spheroid with the same buoy displacement. Three example buoy shapes are shown in  FIG. 4 . Buoy  400  has a cylindrical shape and has the highest drag coefficient of the three examples; buoy  402  has a hemispherical end-piece, while buoy  404  has a prolate spheroid end-piece that has the lowest drag coefficient. Viscous losses decrease with decreasing drag coefficient. For the specific examples of  FIG. 4 , viscous losses for buoy  400  are the largest, and are the smallest for buoy  404 , with the losses for buoy  402  falling between. 
         [0032]    In some embodiments, the viscous losses for the spar are also considered, and factored into the choice of the design of the spar. The viscous losses of the spar can be reduced by using a constant diameter spar with a streamlined tip, for example a tip having a prolate spheroid shape. The spar may benefit from additional streamlining compared to the buoy because its velocities are significantly higher than those of the buoy. 
       Asymmetric Spring 
       [0033]    Several configurations of the gas spring can be incorporated into the generator design with varying performance features. The pneumatic spring shown in  FIG. 1  is asymmetric in the sense that motion of the piston is hard limited in the upward direction, but unlimited in the downward direction. Such a spring can be configured to provide a specified spring constant about a quiescent operating point. 
         [0034]    More specifically, the top of the piston is limited in its upward motion by the cap  118  providing a hard spring characteristics that effectively becomes a hard upper limit on travel. Ideally, for such a spring, the spring design provides a specified spring constant at the quiescent position and also provides quasi-linear operation up to some specified displacement after which the hard spring characteristic engages over a sufficiently large distance. A displacement limit is not required in the opposite direction provided the design incorporates an armature length that is sufficient to cope with peak displacement events. 
         [0035]    In some embodiments attention is directed at reducing the Coulomb friction loss caused by the travel of the seal along the armature. Lower losses can be achieved through the proper choice of seal material and a smooth pneumatic cylinder. 
         [0036]    The valve  106  shown in  FIG. 1  is included to automatically adjust the pressure in the cavity to cope with leakage and temperature changes. The design of the spring might, for example, follow these steps:
       a) A design displacement is set to define the range of quasi-linear operation at the specified spring constant.   b) The quiescent force is chosen to set the physical length of the spring relative to the quiescent location. In a specific example, this can be set to be approximately twice the design displacement. This multiple is chosen to provide quasi-linear operation up to the design displacement and to allow an adequate distance for the hard spring characteristic to take effect without excessive shock during extreme events.   c) The area of the spar is chosen and the quiescent pressure is calculated.
 
The spring characteristics for the previously discussed WEC-II are shown in  FIG. 5  at  500 . The linear spring characteristics are shown in dotted line  502  which when compared to that of the pneumatic spring reveals a relatively weak non-linearity out to about ±5 m for this particular example.
       
 
       Atmospheric Pressure as Quiescent Condition 
       [0040]    In some embodiments, atmospheric pressure is specified for the quiescent condition. This choice avoids the need for a separate gas supply and its accompanying controls. In some embodiments in which atmospheric pressure is specified for the quiescent condition, a control system is provided that uses a pressure sensor and measurements of the position of the secondary mass to adjust the cavity pressure. Adjustments may be made to compensate for changes in cavity pressure due to temperature changes of the gas and leakage around the piston. The output of the pressure sensor when the spar is at the quiescent location is used to adjust the quiescent pressure. When the pressure is low, a specified mass of air is added to the cavity by opening the valve at the upper end of the cavity when the spar is below the quiescent location. Conversely, the mass of air in the cavity is decreased by opening the valve when the spar is above the quiescent location. 
         [0000]    Higher than Atmospheric Pressure for Quiescent Condition 
         [0041]    In some embodiments, a higher quiescent pressure is employed. A higher quiescent pressure allows for a reduced diameter for the spar. As such, it may be advantageous to use a higher quiescent pressure if the resulting diameter of the spar required for a quiescent pressure at atmospheric pressure is too large. A smaller diameter for the spar may allow for the realization of a less expensive generator and may reduce material costs. However, the advantages of a smaller diameter spar at the water line may require a spar of impractical length in order to provide sufficient secondary mass. Note that the secondary mass is mainly determined by the mass of the water displaced by the spar. In some embodiments, the diameter of the spar at the waterline is reduced compared to the diameter of the spar at depth. An example of this is depicted in  FIG. 6 , where the spar has a first diameter D1 over a portion of its length, and this changes in a streamlined manner to a second larger diameter D2 over a portion of its length. 
       The Symmetric Spring 
       [0042]    The embodiments of  FIG. 1  and  FIG. 2  feature an asymmetric spring that permit the salt-water environment to impact the stator and the piston materials. Specifically, the spar  114  interacts directly with the water. A second configuration of the gas spring that takes the piston (and the generator) out of the seawater will now be described with reference to  FIG. 7  which provides a side cutaway view. This configuration also provides improved accessibility to the secondary components. A buoy structure is generally indicated at  700 . A generally cylindrical guide  732  having a top  734  is mounted within the buoy structure for guiding a spar  714 . The spar is supported by bearings  710  and a roller guide  715  and is extended to the required depth in the direction of the arrows shown in  FIG. 7 . A pneumatic cylinder  702  ( 702 A,  702 B) having top  703  and bottom  733  is connected to the top  734  of the guide  732 ; the pneumatic cylinder also functions as an armature, and includes a steel outer cylinder  702 A inside which an aluminum cylinder  702 B is flush mounted. Inside the aluminum cylinder of the pneumatic cylinder  702 , there is a piston  716  connected to the spar  714  by a piston rod  730  that passes through a seal in an opening in the bottom of the pneumatic cylinder  702 . The piston rod  730  and the companion seal/opening in the top of the guide may have a cross section that locks the rotation of the spar to that of the buoy, for example elliptical. A stator  712  is formed on the piston  716 . A first valve  707  is provided in the top  703  of the pneumatic cylinder  702 , and a second valve  706  is provided at the bottom of the pneumatic cylinder  702 . The space  740  between the top of the piston  716  and the top  703  of the pneumatic cylinder functions as a first chamber of a symmetric pneumatic spring. The space  742  between the bottom of the piston  716  and the bottom  733  of the pneumatic cylinder  702  functions as a second chamber of the symmetric pneumatic spring. When placed in water, the wave energy converter floats, with the designed sea level indicated at  720 . While the top  734  of the guide  732  and the bottom  733  of the pneumatic cylinder  702  are described as separate components, they could alternatively be formed of a common component. 
         [0043]    In this embodiment, the spar  714  serves as the secondary mass but does not function as the piston for the pneumatic spring; instead a separate piston  716  is provided for the pneumatic spring; the separate piston  716  provides the platform for the generator stator  712 . The armature consists of a steel thin-walled cylinder  702 A inside which the aluminum thin-walled cylinder  702 B is flush-fitted. The aluminum cylinder has a smooth inner surface that serves as the cylinder for the symmetric pneumatic spring. The previously discussed ladder construction could be used to achieve greater generator efficiency if tolerances are set to achieve the required smoothness. 
         [0044]    The resulting symmetric pneumatic spring also includes functional efficiency. The stator  712  is included in the piston  716 , the pneumatic cylinder  702  serves as the armature and a displacement limiting function is built in. Specifically, the top  703  of the pneumatic cylinder limits the upward mobility of the piston, and the bottom  733  of the pneumatic cylinder limits the downward mobility of the piston. Atmospheric or a higher pressure (depending on the diameter that is chosen for the spring) determines the quiescent operating point. The automatic tuning of each chamber  740 , 742  via valves  707 ,  706  may follow the procedure described for the asymmetric spring. In some embodiments, pressure is supplied via a tank instead of from the atmosphere. 
         [0045]    The guide  732  for the spar  714  serves to guide the spar  714 . The roller guide  715  placed a suitable distance below the buoy  700  lines up the spar to properly engage the bearings  710 . Seals and the attention to tolerances for fitting the spar  714  in the buoy  700  to satisfy the generator requirements can be eased. The main action takes place above the spar and its guide. The motion of the spar  714  is coupled to the piston  716  in the pneumatic spring through a seal contained in the bottom  733  of the pneumatic cylinder. The lengths of chambers  740 , 742  may, for example, be equal to the length calculated for the asymmetric spring but the diameter can be reduced (assuming the same quiescent pressure) because the springs in the upper and lower chambers are acting in parallel. 
         [0046]    Piston seals are located above and below the piston  716  to achieve symmetry for both chambers  740 , 742 . The lower chamber  742  may be slightly longer than the upper chamber  740  to compensate for the volume of the piston rod  730 . Preferably, the piston rod  730  is designed to have a specified column strength and low thermal resistance. It is the main conductor of heat out of the stator and into the heat sink (the spar). 
         [0047]    The spring characteristics for a simulated example implementation of the embodiment of  FIG. 7  are depicted in  FIG. 8  at  800  along with those for the linear spring at  804  and the asymmetric spring  802 . 
         [0048]    An example of a mooring plan for a line of WEC&#39;s with stators mounted in the spars is shown in  FIG. 9 . A series of WECs  900 , 902 , 904  are depicted. Three are shown for the sake of example only. The WECs  900 , 902 , 904  are interconnected through tether cables  901 , 903 . Also shown are tether cables  905 , 907 , either to other WECs, not shown, or to a moored end buoy that may or may not be submerged. The power is taken out from the bottom of the spar of each WEC  900 , 902 , 904  and connected via a respective slack wire tether  910 , 912 , 914  which may include placement of small floats at intervals along negatively buoyant electrical cable to a respective junction box  930 , 932 , 934  on a bottom-mounted transmission cable  936 . The view of each WEC in  FIG. 9  shows that the spar is acting like a sea anchor and together with proper distribution of the ballast in the buoy should considerably reduce the roll of the buoy. In some embodiments, the WEC designs in  FIGS. 1 and 7  employ this approach. 
         [0049]    In some embodiments, the WEC design in  FIG. 2  with stator mounted on the buoy incorporates electrical conductors in an armoured tether cable, thereby dispensing with the need for the bottom transmission line and separate electrical cables such as cables  910 , 912 , 914  depicted in  FIG. 9 . 
       The Double-Sided Induction Generator 
       [0050]    In some embodiments, a double-sided tubular construction for the induction generator is employed. This can have the effect of reducing the mass of the generator and obtaining improved performance. For example, in some implementations leakage inductance almost vanishes and magnetic attraction forces between the iron-free armature and the stators vanish. Bearing stresses are not eliminated because departures from the perfect geometry in the tubular generator configuration are present due to manufacturing tolerances. This saving in mass arises because the armature uses only an aluminum thin-walled cylinder instead of the concentric aluminum/steel cylinders that are used by the single-sided configurations shown in  FIGS. 1 ,  2 , and  7 . An example of a WEC featuring this type of construction will now be described with reference to  FIG. 10 . 
         [0051]    A buoy structure is generally indicated at  1000 . A generally cylindrical guide  1032  is mounted within the buoy structure for guiding a spar  1014 . The spar is supported by bearings  1010  and a roller guide  1015  and is extended to the required depth in the direction of the arrows shown in  FIG. 10 . A pneumatic cylinder  1002  having top  1003  and bottom  1033  is connected to the top of the guide  1032 ; the pneumatic cylinder  1002  includes an aluminum cylinder. Inside the aluminum cylinder of the pneumatic cylinder  1002 , there is a piston  1016  connected to the spar  1014  by connecting rod  1030  that passes through an opening in the bottom of the pneumatic cylinder and/or the top of the guide  1032 . An inner stator  1013  is formed on the piston  1016 . An outer stator  1012  is held by connecting rods  1054  that are connected to the spar  1014  that pass through openings in the top of the guide  1032 . A first valve  1007  is provided in the top  1003  of the pneumatic cylinder  1002 , and a second valve  1006  is provided at the bottom of the pneumatic cylinder  1002 . The space between the top of the piston  1016  and the top  1003  of the pneumatic cylinder functions as one chamber  1040  of a symmetric pneumatic spring. The space  1042  between the bottom of the piston  1016  and the bottom of the pneumatic cylinder functions as the other chamber of a symmetric pneumatic spring. When placed in water, the wave energy converter floats, with the designed sea level indicated at  1020 . 
         [0052]    In this embodiment, the spar  1014  serves as the secondary mass but does not function as the piston for the pneumatic spring; instead a separate piston  1016  is provided for the pneumatic spring; the separate piston  1016  provides the platform for the inner stator  1013 . 
         [0053]    The inner stator  1013  is connected to the spar  1014  by connecting rod  1030  and the outer stator  1012  is connected to the spar using, for example, three or four rods  1054  between the outer stator and the top of the spar  1014 . The rods  1054  may be equispaced in angular space to avoid troublesome moments. The connecting rods  1054  from the spar  1014  to the outer stator  1012  may pass through bearings/guides  1052  in the top of the guide  1032  that guide the motion of the spar  1014  inside the buoy  1010 . These bearings lock the angular position of the spar to that of the buoy and this eliminates the requirement for high power slip rings for the electrical tether and reduces wear on the seals of the pneumatic spring. With a single stator such as described in previous embodiments, the stator has a yoke of steel that turns the flux from one pole and returns it to the other (of opposite polarity along a pole length of the yoke). The steel in the armature sandwich is the yoke. In the double-sided construction the flux passes straight through the aluminum conductor from one stator to its mirror image pole on the other stator that has opposite polarity. The yoke is not needed. 
         [0054]    Referring now to  FIG. 11 , a model of a WEC is depicted. Shown is a buoy having mass m 1  that is equal to the sum of the buoy displacement and the hydrodynamic mass. Displacement of the buoy is represented as y 1 , and velocity is v 1  (not shown). A secondary mass m 2  is connected to the primary mass through a spring having spring constant k 2 . Motion of the secondary mass is represented as y 2 , and velocity is v 2  (not shown). Damper d represents damping in the secondary system; this is damping caused by the generator. For this example, the buoy has a cylindrical top portion the submerged portion of which has a height, L, with a hemispherical bottom portion having radius a. 
         [0055]    Other parameters that do not directly relate to the physical hardware are present. A gravity wave of amplitude, A, produces forces that move the buoy in heave. The motion of the buoy yields additional mass for the primary system (hydrodynamic mass) and it introduces a damping coefficient, b (not shown), that corresponds to the wave making energy dissipated by the oscillating buoy. Buoyancy forces acting on the buoy yield an effective spring constant, k 1  (not shown), in the primary system. Non-linear viscous damping also acts on the heaving buoy. 
         [0056]    An equivalent circuit model is depicted in  FIG. 12 . The velocities v 1  and v 2  of the masses m 1  and m 2  equate to currents in the equivalent circuit model. The secondary circuit (the second mesh in the circuit diagram) can be interpreted as an impedance that varies with frequency and is at its highest in the vicinity of parallel resonance. This condition reduces the primary velocity (as required by the vibration absorber) and generates high velocities through both reactive components of the secondary system. The reactive components of these velocities are nearly in anti-phase and for this reason are sometimes referred to as circulating currents in the electrical analog. The power dissipated in the resistor, d, at these high velocities corresponds to the output of the WEC. The design procedure for the damped vibration absorber is directly applicable to the determination of the parameters of the secondary system. Small adjustments to this procedure can be made to deal with the frequency dependence of several parameters and to provide for a relatively flat frequency response in the pass band. The latter adjustment follows from the fact that the flat bandwidth for the vibration absorber applies to the displacement of the primary mass and not to the WEC variable of interest, which is the relative velocity between the armature and the stator (v 1 −v 2 ). The inclusion of viscous damping can be modeled through simulation techniques. 
         [0057]    A parameter α=ratio of secondary mass to primary mass=m 2 /m 1  has a significant effect upon the transfer function of the WEC. The transfer function of a small WEC with bandwidth higher than desired for useful energy extraction was modeled to show the dependence of the transfer function and velocity response on α. Referring to  FIG. 13 , shown is a transfer function plotted for three values of α (the ratio of secondary mass to primary mass), namely α=0.05, α=0.1, α=0.2. In  FIG. 13 , the vertical axis is power absorbed by the WEC as a function of frequency given excitation by gravity waves having a mean square height of 1 m 2 . It can be seen that a WEC with a smaller α has a higher response, but over a narrower bandwidth compared to a WEC with a larger α. In other words, the bandwidth increases as the mass of the secondary is increased, but as the secondary mass decreases, a higher response is achieved over a smaller bandwidth. 
         [0058]    Referring now to  FIG. 14 , shown are plots of relative velocity as a function of frequency for the same three WEC&#39;s modeled and discussed with reference to  FIG. 13 . It can be seen that the velocity is higher for smaller alpha, and lower for higher alpha. A higher velocity is better for the generator. Recall, a smaller alpha can be realized by having a smaller secondary mass relative to the primary mass. However, from  FIG. 13 , it can be seen that a smaller alpha results in a transfer function with less bandwidth. As such, it can be seen that there is a tradeoff between velocity and bandwidth. 
         [0059]    With appropriate selection of the parameters in the model, the bandwidth can be matched to a spectrum of interest. Ideally, the resulting bandwidth would offer a high enough velocity regime to use a direct drive generator. If not, the velocity level can be increased at the expense of bandwidth. The reduced bandwidth can be placed in the upper part of the spectrum to obtain a smaller overall unit. This is the scenario that is described by  FIG. 3 . A high power output that corresponds to the response curve  300  is obtained at the expense of low relative velocities between the spar and the buoy. The response curve of a smaller unit and smaller α  302  settles for lower power to obtain velocities high enough for a practical induction generator. 
         [0060]    Numerous modifications and variations of the present disclosure are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the disclosure may be practiced otherwise than as specifically described herein.