Analysis of Discrete Pressure Level Systems for Wave Energy Converters

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1 Analysis of Discrete Pressure Level Systems for Wave Energy Converters Rico H. Hansen Department of Energy Technology Aalborg University Aalborg, Denmark Torben O. Andersen Department of Energy Technology Aalborg University Aalborg, Denmark Henrik C. Perdersen Department of Energy Technology Aalborg University Aalborg, Denmark Abstract Within the research field of harvesting the energy of ocean waves, fluid power has been identified as a crucial technology in the Power Take-Off () design, due to the high torque densities required in Wave Energy Converters (WECs). The is the technology converting the captured wave motion into electricity. However, conventional fluid power systems are characterized by offering poor efficiencies, rendering current designs inefficient. This paper investigates the feasibility of a fluid power system based on implementing the force control of hydraulic cylinders by switching between a few fixed system pressures. The proposed design is optimized at multiple levels, as evaluating the feasibility of a solution highly depends on finding the optimum trade-off between e.g. harvested wave energy and losses in the system. P hav = ext m Point absorber P out [rad/s] [m] [MNm] Typical trajectories Wave height w s Arm velocity - forces 3 Time [s] Fig.. Point absorber type WECs and typical forces and velocities. Keywords, Fluid Power, Hydraulics, Wave Energy, WPEA, Point absorber, WEC I. INTRODUCTION A large variety of Wave Energy Converters (WECs) are being developed for harvesting the energy of the ocean waves, see e.g. [] for a survey. The focus of this paper is a group of WECs harvesting energy by directly converting the waves into an oscillating mechanical motion, e.g. point absorbers and multiple absorber systems, see Fig. and Fig.. The conversion of the captured mechanical motion into electricity is performed by the Power Take-Off (). Developing efficient systems is a big challenge as waves produce slow irregular and bi-directional motion, requiring very large forces for energy extraction, see e.g. []. An example of a irregular wave and the resulting velocities of a point absorber with a m float is shown in Fig.. The torque τ is the torque applied by the in order to extract power from the absorber. To maximise the harvested power P hav =τ ω arm, τ has to be controllable as in Fig.. How to generate the optimal τ is referred to as the Wave Power Extraction Algorithm (WPEA), see []. To implement a system, several investigations, see e.g. [3], have been performed on using linear generators driven directly by the point absorber motion. However with linear velocities in the range of. m/s, conventional permanent magnet generators becomes infeasible. In [4] it is estimated that a conventional linear generator would have a power to weight ratio of only 4kW per tonne. Instead, effort is put Fig.. Point absorber type WECs and the Wave Star 6kW WEC []. into using traditional generators, and here fluid power is the only current technology suitable for converting the slow bidirectional wave motion into the required fast uni-directional rotating motion required by generators. Unfortunately, fluid power systems are characterized by poor efficiencies, especially at part load which is an inherit property of wave power. A factor of between average and peak power is common in WECs [6]. Hence, a conventional hydraulic transmission as seen in Fig. 3a rated to absorb the full range of input power would become very inefficient in low and average power range. A complete analysis and optimisation of such a system from wave-to-grid is performed in [], showing an expected overall efficiency of only 6%, quickly dropping to 4% in smaller waves. systems related to Fig. 3b is used in [8] and [9]. However, this system may only deliver a constant resistive force, due to the accumulators. Consequently, to maintain a reasonable energy extraction from waves while limited to using a constant resistive force, WPEAs known as latching or declutching are //$6. IEEE FPM

2 Symmetric cylinder Flushing valve (a) Fig. 3. G (b) Hydraulic transmissions for systems. required. Latching optimises energy extraction by locking the motion of an absorber for brief periods during a wave period. However, for e.g. the Wavestar WEC, the locking forces required become very large, making latching undesirable. The focus of this paper is to investigate a system for single or multiple point absorber WECs, where the force control of a hydraulic cylinder is based on switching between fixed system pressures, see Fig. 4. The resulting force F is generated as, F = A p A + A p A + + A n p An [ N ] () where A i is the i th cylinder area with sign according to force direction and p Ai is the pressure of the i th cylinder chamber, which is connected to one of the system pressures p Ai {p,, }. Digital control of a hydraulic cylinder have also been investigated in [], showing promising energy saving compared to e.g. load sensing systems. The focus of this paper is to show its potential efficiency in wave energy. c = - F = F c c F F F A=A A A 4 A 3 A A F c F c Fig. 4. p 3 p p p F F Digital force control of cylinder. G p A p A p A p A p A3 p A4 The concept of the system is to allow variable cylinder force with a constant system pressures without using throttling based control, see Fig.. The decoupling of system pressure and force control allows the use of gas accumulators for energy smoothing on the system pressure. Consequently, the system is decoupled into two systems, the primary stage extracting and converting wave power to high pressure hydraulics, and the secondary stage, converting the fluid power to electricity. Due to the smoothing effect of the accumulators, the secondary stage processes power at a stable rate, allowing a second stage consisting of e.g. high efficient fixed displacement bent-axis motors combined with a generator, connected to inverters to allow variable speed control. This secondary stage will then have an approximately efficiency of: η sec = η motor η generator η conv = = 86% () Thus, to have a total efficiency between 7% and 8%, the efficiency of the primary stage has to be between 8% and 93%. A system utilizing this type of control with two asymmetric cylinders have previously been discussed in [] for the Pelamis wave energy converter, which is an attenuator type WEC. The efficiency of the primary stage was found to be between 88% and 94%, excluding the efficiency of the cylinder. In this paper the efficiency of the primary stage is investigated and optimized for point absorbers. Two systems are investigated, according to Fig. 4: A symmetric cylinder with multiple system pressures (na =and np 3). A system with two asymmetric cylinders working together, i.e. working as one cylinder with four active cylinder areas (na =4and np ). The investigation is based on a single float and arm for the Wavestar 6kW WEC seen in Fig. c, consisting of hemisphere shaped floats, each m in diameter. The advantage of a multiple point absorber system as the Wavestar WEC is the increased power smoothing, [7]. Switching circuit Primary stage Fig.. p G grid Secondary stage [m] [Nm] [Pa],ref concept using digital hydraulics. II. METHODS w p Time To give a proper evaluation of the primary stage of the proposed system, the system configuration and associated control algorithms should be optimised to yield the highest energy output. As optimising the energy output requires simulation of the WEC including for different parameters and system parameters, fast models are required, and the optimisation is performed in three levels based on the model in Fig. 6. The model consists of: An irregular wave model (including how to calculate the exiting wave torque). A dynamic model of the float and arm. A model of the for calculating losses and the applied torque. The first level of optimisation, is the WPEA algorithm. The WPEA algorithm calculates the time-varying torque τ ref to maximize the expected expected energy output of the WEC E out, taking into account the efficiency η. Hence, an initial guess of η is required. 3

3 Based on the continuous torque reference, the second level of optimization is the Force Switching Algorithm (FSA), which chooses the force step to utilize to approximate the continuous τ ref. However, each force shift is associated with power losses. Thus, based on a set of system pressures and cylinder areas, the FSA compromises between shifting losses E loss and tracking of τ ref to optimises the energy output: E out = E har E loss [ J ] (3) From (3) the efficiency of the primary stage can be identified, and a new WPEA can be found, and the FSA optimized again. This loop is then executed to the solution has converged. Being able to find a WPEA and FSA for a given system setup, a final optimisation loop can be implemented to identify the optimal system configuration e.g. pressure levels and cylinder areas, which optimises E out. This final loop is not treated in this paper. Wave ext Fig. 6. model Ploss( t) u c = [ ] u A u An Force Shifting Algorithm ext + - Float/arm ref WPEA Phar( t) Model used for analysis of primary stage of the system. III. MODELLING A. Wave Model Ocean waves are irregular waves, i.e. waves with varying frequency and amplitude. Irregular waves are described by a variance spectrum, see Fig. 7. A sea state is usually represented by two quantities, the significant wave height H s and the peak wave period T p. The significant wave height is the average of the wave heights of the one-third highest waves, and T p is the period where most energy is concentrated. From a spectrum, an irregular wave can be constructed as illustrated in Fig. 7. [m s] [m] 4 Spectral density S A(f) T p H m=m T P=6s.. Wave frequency [Hz].3 Wave height h Time [s] Fig. 7. S (f) [m s] A State 3: (Large) State : (Medium) State : (Small) Spectral density H m [m]..7. T P[s] Wave Frequency [Hz] Wave spectra for sea states and an example of a corresponding wave. To evaluate the performance of the, the three sea states shown in Fig. 7 are used, which represents the range of waves in which the WEC should be able to produce power. The Pierson-Moskowitz spectrum is utilized. A wave time series has been generated for each sea state for evaluation of performance. The following section describes how the wave interacts with the float. B. Wave and Float Interaction The equation of motion for a float is given as, J mech θarm (t) =τ wave (t) τ G (t) τ (t) (4) where J mech is the inertia of float and arm, τ wave is the torque due to wave-float interaction, τ G is the torque due to gravity and τ is the torque applied by the system to the float arm. To describe the interaction between wave and float τ wave (t), linear wave theory is often applied, as it gives an adequate description in the conditions in which a WEC is producing energy, []. In linear wave theory simplified fluid dynamics is assumed in order to apply linear potential theory. Resultantly, the wave-float interaction can be described by superimposing three torques, τ wave (t) =τ rad (t)+τ Arch (t)+τ ext (t) () where τ ext (t) is the excitation torque an incoming regular wave applies to a float held fixed, τ rad (t) is the radiation toque experienced from oscillating the float in otherwise still water, and τ Arch is the torque due to the Archimedes force, i.e. buoyancy. The torque due to the radiated wave is described as, τ rad (t) = J ω arm (t) h rad (t) ω arm (t) (6) where h rad is the impulse response function from float velocity to torque, describing the hydrodynamic damping. The impulse response h rad can be viewed as a high order damping term. The inertia term J represents the added mass, which represents the effect, that when oscillating a float, it will appear to have a greater mass due to the water being displaced along with the float. The coefficients of (6) are identified by applying the numerical tool WAMIT to the float. WAMIT is a computer program for computing wave loads and motions of structures in waves [3]. WAMIT also outputs a force filter, which can be applied to η w (t) to find τ ext (t). Inserting (6) into (4) gives the equation of motion for the float, ω arm = k resθ arm h rad ω arm + τ ext τ (7) J mech + J where the sum of gravity and Archimedes term has been linearised around the draft of the float, τ res (t) =τ Arch (t) + τ G (t) θ arm (t)k res. Thus the input to float-arm subsystem are the torques τ ext and τ, and the output is the angular position and velocity of the arm. To avoid the convolution term h rad (t) ω arm (t), the impulse response has been fitted with a fifth-order system using Prony s methods [4]: τ rad (s) = a s + + a s + a b s + + b s + b ω arm (s) (8) 4

4 P [kw] out,avg P [kw] out,avg C. Wave Power Extraction Algorithm The power extracted or harvested from a wave P har is the product of the torque τ and arm velocity ω arm. Hence, τ should be controlled such that harvested energy E har is optimised: E har = τ (t)ω arm (t)dt (9) In general, to maximise (9) the natural frequency of the float in (7) should match the incoming wave frequency. However, as the dominating wave frequency varies from sea state to sea state, the natural frequency will not match. Consequently, the system is used to move the natural frequency to increase power capture. This is achieved by adding the following feedback law to (7): τ = k θ arm + b ω arm () This control law belongs to the group of WPEAs termed reactive control, because when applying () power is periodically transferred from the to the float, i.e. P har (t) is both positive and negative. Hence, four-quadrant operation must be supported by the. The optimal coefficients of () depends on the efficiency η, as the loss associated with the reactive power reduces the benefit of the increase in harvested power. To find the optimal parameters b and k as a function of efficiency η, time-series simulation has been executed for different efficiencies and sea states, to optimize the average power output, Tsim (b,k )=arg max P out (t)dt () (b,k ) T sim ( s.t. θarm (t) ω arm (t) ) T = f(θarm (t),ω arm (t),τ ext (t)) τ max τ (t) τ max where { η τ (t)ω arm (t) ; τ (t)ω arm (t) > P out (t) = () η τ (t)ω arm (t) ; τ (t)ω arm (t) The problem in () is solved using a simplex-based optimisation algorithm, simulating a sequence T sim of T P for each iteration. Note that a saturation limit has been added to the torque of τ max =MNm in (), as it has been found to be a reasonable limit in regard to harvested power versus requirements of the. The limit have been found by multiple simulations, where the limit have gradually decreased. The results for the optimal values of k and b for sea state and sea state is seen in Fig. 8 as a function of efficiency. The expected power outputs are also shown. D. Hydraulics Model The purpose of the hydraulic model is to describe the losses associated with force control of a hydraulic cylinder by switching chamber pressure. As the optimisation rely on long time simulations, it is undesirable to have a hydraulic model included with e.g. flow-continuity equations, as each switch yields very fast transients, slowing a simulation. Instead, -k [N] b [kgm ] e6 b Sea state e6 Sea state k P out,avg b k efficiency [%] efficiency [%] P out,avg Fig. 8. Solution of () as a function of efficiency η. the idea is to calculate the losses of shifting and flow losses without simulating the hydraulic equations. Hence the loss at each discrete shift instant is calculated and then summed. Instant force change as the pressure build is fast compared to the float dynamics. To avoid cavitation, the lowest system pressure is always set to bar. At bar the effective bulk modulus is already above 9bar if % free air is assumed. As the valve for shifting is assumed fitted directly to the cylinder, the bulk modulus is not limited by e.g. hose flexibility. As a result, a fixed value of = bar is used. In the following the shifting losses and flow losses for a single cylinder chamber is analysed. Losses for a primary stage with multiple chamber is then afterwards described by summing the losses of the individual chambers. When changing a pressure in a chamber, the shifting loss due to the compressibility of the fluid can be calculated as: Theorem : Given a volume of fixed size V with a constant effective bulk modulus and an initial pressure of p (see Fig. 9a), the power loss from connecting the volume to a fixed pressure supply is: E -loss = ( p ) V 3 (3) Proof: First the required volume of flow V required for compressing the pressure in the volume p V (t) from p to is found using the flow-continuity equation: ṗ V (t) = V Q(t) p V(t) = V As lim t p V (t) =, the volume V is given as: = V Q(t)dt+p V = Q(t)dt + p (4) Q(t)dt =( p ) V () If the energy out of the pressure supply is denoted E p, and the energy supplied and saved as potential pressure energy in V is denoted E V, the energy loss is given as: E -loss = E p E V = Q(t)dt p V (t)q(t)dt (6) Denote V Q (t) = Q(t)dt, then p V (t)= V V Q(t)+p. Note that

5 dv Q dt = Q(t). Inserting into (6): ( ) E -loss = Q(t)dt V V Q(t)+p Q(t)dt = V p Q(t)dt V Q (t)q(t)dt V = V p V V Q (t) dv Q(t) dt (7) V dt =( p )V [ ] V V Q(t) =( p )V V V = ( p ) V Note that in the proof no assumption was made on how the flow Q(t) is transferred to the volume. Hence, the energy loss due to compression is independent of the opening area and opening time of the valve, i.e. (3) represents an inevitable loss when connecting a volume to a fixed pressure, when the pipe inductance can be neglected. p Q(t) Q(t) V p V(t) p p V(t) V() t A A na p A V() t p A Manifold Manifold t t t 3 t 4 (a) (b) (c) Fig. 9. Force control by switching pressure in chambers. Hence the shifting/compression losses can be calculated without simulating the hydraulic equations. To calculate the compression losses for a series of the shifts, the following is used: Theorem : When connecting a fixed pressure supply to a volume of fixed size V with a constant effective bulk modulus and an initial pressure of p, the energy E p transferred to the fixed pressure supply is: E p = (p ) V Proof: According to (6) and (7): p A p A (8) E p = Q(t)dt = V = (p ) V Hence given a single cylinder chamber, which is shifted between different fixed pressures at the time instances t shift = {t,,t ns }, the total loss due to compression can be calculated as: ns E tot,-loss = p new,i (p V (t i ) p new,i ) V (t i) (9) i= where V (t i ) and p V (t i ) is the size and pressure of the chamber at t i and p new,i is the pressure shifted to at a time t i, i.e. p new ={p new,,,p new,nt } Note that (9) includes the energy recovered to a pressure line when decompressing. As a cylinder chamber is always connected to one pressure line through a valve/control edge, and if all valves used for that given chamber have an opening area A o, the throttling losses P v (t) can be calculated based on the orifice equation: P v (t) = Δp v (t)q v (t) = Q v(t) ρ A ocd Q v(t) = v3 c A 3 c ρ A ocd () where A c is the cylinder area, v c is the cylinder velocity and C d is the discharge coefficient. Given a system as illustrated Fig. 9b with system pressure {p, } and active cylinder areas {A, A na }, the energy output of the primary stage becomes, E out = E har E tot,-loss E tot,v = Tsim τ (t)ω arm (t)dt j= E tot,-loss,j Tsim j= P v,j (t)dt () where t shift = {t,,t ns } is the time instances of force shift as illustrated in Fig. 9c, and τ (t) = k= p V,k (t)a k d arm (t) () where d arm (t) is the cylinders moment arm. The efficiency of the primary stage is then given as η prime = Eout E har. IV. FORCE CONTROL ALGORITHM The purpose of the FSA is to choose the appropriate force levels to approximate F ref, where F ref is cylinder force yielding τ ref. Given a vector of system pressures p=[ ] and cylinder areas A=[A A na ], the number of force steps nf is, nf =(np) na (3) and the vector F steps containing the nf forces can be found as A F steps = M p A T p A =.. (4). A na where M p is a matrix of all possible pressure combination for the na areas. Let F denote a vector, which is F steps sorted from smallest to largest force. For describing the pressure combination of each force, a control vector u i is defined as the vector of pressure indexes for the i th force. A. FSA- The simplest force control algorithm is to always choose the force closest to the continues force reference, e.g. the force k minimizing the tracking error, { } F (t) =F [k] k = arg min F ref (t) F [k] () where the vector-notation v[k] denotes the k the element of v. To avoid possible jittering when F ref is equally close to two forces, a hysteresis effect can for example be added, or a limit on how frequently switching is allowed. The latter is 6

6 chosen as this parameter also may be tuned to prevent doing an unnecessary amount of shifts. Hence, when a shift has been made, () is disabled for a period of T min. The algorithm is illustrated in Fig.. B. FSA- The algorithm FSA- does not decide based on the cost of different force shifts. To avoid doing very expensive force changes, this proposed FSA- strategy calculates the energy expense of possible force-shifts, and makes a compromise between tracking and energy-cost. For each force i of F, a control vector u i is defined as the vector of pressure indexes for the i th force. E.g. if F = [ p ][A A A 3 ] T then u =[]. The loss from switching from force x to y is then given by summing the losses (3) for pressure changes in the individual volumes: E sh (x, y) = j= ( p [ u y [j] ] p [ u x [j] ]) V j (t) (6) Note E sh (x, y) =E sh (y, x). The idea now is to define a maximum allowed tracking error F b, meaning that F must stay within a band of ±F b about F ref. However, within the band, the FSA may choose the force steps with the lowest shift cost. A fixed time limit T min on how frequently switching is allowed is also added. If the current force is denoted k, the control law can be stated as { F (t) =F [k] k = arg min F ref(t) F [k] } (7) k {k,k,k + } where k and k + are the two cheapest forces to shift to within the band ±F b : k + =arg min k S + E sh (k,k), S + = { k F ref (t)<f [k]<f ref (t)+f b } k =arg min k S E sh (k,k), S = { k F ref (t) F b <F [k]<f ref (t) } (8) (9) Hence, the tracking band is always defined around the value of F ref. The algorithm is illustrated in Fig.. V. RESULTS/DISCUSSION The two systems illustrated in Fig. 4 are selected for analysis. The system parameters in Tab. I are used for an initial evaluation of the losses. The areas are selected such that the configuration can deliver ±4 kn (corresponding to the earlier mentioned MN). The force distribution is seen in Fig.. The valve opening areas for each chamber are chosen such that the pressure across them are 3bar at nominal flow. Nominal flow is in this context defined as the flow from a cylinder area travelling at. m/s. TABLE I INITIAL SYSTEM PARAMETERS FOR SYSTEM AND. p [bar] A [m ] System : [ ] [ ] System : [ 4] [ ] The two systems are tested with both the FSA- and FSA- algorithm for different settings of maximum tracking-error F b and minimum time T min between shifting. The results are shown in Tab. II and Tab. III for sea state. From the results it is evident, that the FSA- achieves a better average power output compared to FSA-. Taking the optimum of the results (indicated with bold red), the average harvested power, shifting loss and valve losses are shown in Tab. IV. It is seen that the shifting losses dominates the losses. From the table it can also be concluded that the FSA- algorithm reduces shift-losses, consequently only this control is used in the evaluation of the concepts from here on. Comparing system and they harvest approximately the same amount of power, however the shifting losses for system are three times as big as for system. Even though system has larger volumes, the smaller steps in pressure reduces the losses. This is shown in the following. If the required force for a cylinder chamber A c is F req, the same force is achieved by scaling pressure and area as, F req =(αa c ) Δp (3) α where α is a scaling parameter. The cylinder volume is given by V c = A c αx stroke. Inserting this into (3) yields: E -loss = ( ) Δp A c αx stroke = Δp A c x stroke (3) α α Hence, increasing α to increase cylinder area and decrease the pressure reduces the shifting losses. However, the flow F T min T min F k + F nf F nf F nf- F nf- k + k - F nf-4 F nf-4 F nf-6 F nf-6 k - F b FSA- Time FSA- Time Fig.. Illustration of the FSA- and FSA- algorithm. Fig.. Distribution of forces for systems in Tab. I. 7

7 is increased, which gives larger -components and flow losses, which has to be taken into consideration. Including an extra pressure line in system will thus decrease shifting losses. TABLE II AVERAGE OUTPUT P OUT IN [KW] OF SYSTEM AND USING DIFFERENT PARAMETERS FOR THE FSA- ALGORITHM. System : (P out [kw]) System : (P out [kw]) F b [kn] T min [ms] TABLE III AVERAGE OUTPUT P OUT IN [KW] FOR SYSTEM AND USING DIFFERENT T MIN FOR THE FSA- ALGORITHM. T min [ms] Sys. : Sys. : TABLE IV AVERAGE HARVESTED POWER, SHIFTING LOSSES, VALVE LOSSES AND EFFICIENCY OF PRIMARY STAGE. P har [kw] P shift [kw] P v [kw] η[%] Sys. (FSA-) Sys. (FSA-) Sys. (FSA-) Sys. (FSA-) In Tab.V the results of system and for the three different sea states are shown. The control is optimised for each sea state. System maintains a good efficiency at all sea states, which is a key property for the concept, as conventional hydraulic transmissions are unable to maintain efficiency for all load cases, see []. System is less convincing at maintaining the efficiency in small waves. However, if the system pressure can be reduced when operating in small waves, the efficiency may be improved as the largest forces are not required in small waves. In Tab. VI system has been modified by adding an extra third pressure line of 3bar. As a result, the modified system achieves a performance similar to system. TABLE V SYSTEM AND EVALUATED FOR THREE SEA STATES. System : System : Sea P har P shift P v η P har P shift P v η state [kw] [kw] [kw] [%] [kw] [kw] [kw] [%] SS SS SS TABLE VI SYSTEM MODIFIED WITH AN EXTRA PRESSURE LINE EVALUATED FOR THREE SEA STATES. Sea P har P shift P v η state [kw] [kw] [kw] [%] SS SS SS VI. CONCLUSION The efficiency of different configurations of a hydraulic system for wave energy has been analysed, where the is based on performing force control of cylinders by switching between fixed system pressures. From the analyse it is shown that the dominating loss in such a system is due to the compressibility of the fluid, e.g. the energy loss associated with compression when switching pressures. This loss is shown to be independent of valve opening characteristics. Hence, the main subject of this feasible study has been to show, if this loss can be kept at an acceptable low level. The systems with more than two system pressures perform best in the analysis, however, the remaining system will become more complicated and expensive, also, as energy has to be extracted from more pressure lines. Suggestion for doing this is a part of the future work that will be performed. To improve switching efficiency it is also desirable to have larger cylinders and lower pressures, however this will most likely have a negative impact on the reaming system, e.g. larger components. By proposing algorithms to avoid doing energy-expensive force shifts, it is concluded that efficiency of this switching stage can be kept between 88%-94%. It is discussed that to have an overall efficiency of 7%-8%, the switching stage should have an efficiency between 8% and 93%. Hence, based on this analysis the concept is concluded to be feasible. REFERENCES [] A. Muetze and J.G. Vining. Ocean wave energy conversion - a survey. In Industry Applications Conference, 6. [] J Cruz. Ocean Wave Energy: Current Status and Future Perspectives, 8, Green Energy and Technology Series, ISBN: [3] M.A. Mueller, H. Polinder and N. Baker. Current and novel electrical generator technology for wave energy converters. IEEE International Electric Machines & Drives Conference, 7. [4] E. Spooner and J. Grimwade. SnapperTM : an efficient and compact direct electric power take-off device for wave energy converters. World Maritime Technology Conference IMarEST 4th MAREC Conf., 6. [] R. Hansen, T.O. 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