Modelling and Control of the Wavestar Prototype

Transcription

1 Modelling and Control of the Wavestar Prototype Rico H. Hansen Energy Technology, Aalborg University & Wave Star A/S Pontoppidanstraede 11, 9 Aalborg, Denmark [email protected] Morten M. Kramer Wave Star A/S Park All 35A 65 Broendby, Denmark [email protected] Abstract In the field of wave energy it is well known that control of point absorbers is essential in order to increase energy capture from waves. Correspondingly, advanced control is an integrated part of the Wavestar design. This paper presents the control method, referred to as the Wave Power Extraction Algorithm WPEA), applied to the full-scale Wavestar Prototype for maximizing energy extraction. The WPEA is optimized based on simulations of the point absorbers in different sea states. Hence, a presentation of a hydrodynamic model of the Wavestar is included in the paper. A simplified Power Take-Off PTO) is also added to the model, enabling the optimization of the WPEA to take into account the PTO constraints of PTO bandwidth and force limitations. The predicted results of the optimized WPEA are compared to real measurements from the Wavestar Prototype, showing good compliance. Index Terms Wavestar Prototype, modelling, WPEA, reactive control, hydrodynamics, optimal control. natural period of the absorbers are equal. Resultantly, point absorbers are prone to operate off-resonance and thereby nonoptimal, as the wave period varies wave-to-wave, and the average or dominating wave period can shift with a factor of two within hours. Moreover, the natural frequency of the absorbers is predispositioned to be too fast compared to waves, as light absorbers are preferred in regard to structural load. I. INTRODUCTION A wide range of Wave Energy Converters WECs) is currently under development for harnessing the energy of the sea. The range comprises a great variety of technologies like over-topping devices, oscillating water columns, attenuators, etc. [1] [3]. In this paper the focus is on WEC technologies or concepts based on point absorbers. More specifically, this paper treats the Wavestar Prototype shown in Fig. 1 [4]. A point absorber extracts energy by directly converting the waves into an oscillating mechanical motion by using a float with a smaller extension than the average wave length. The system converting the captured wave motion into electricity is referred to as the Power Take-Off PTO). The Wavestar concept is a multiple absorber system, having the advantage of increased power smoothing compared to a single absorber WEC, as the span of the converter exceeds the wave length [5]. The Wavestar Prototype is located at Hanstholm, the Westcoast of Denmark, and has been in operation since 9. It is a two-float test-section of the -float Wavestar C5 6kW converter shown in Fig. [6]. The diameter of the floats is 5m. For point absorbers it is well known that appropriate control of the applied PTO force can potentially increase the amount of energy extracted from sea waves [5]. The back-ground for this improvement is, that frequency-wise, point absorbers are often characterized by being narrow-banded with a highly under-damped resonance frequency. The energy transfer from wave to absorber is optimal when the wave period and the Fig. 1. Wavestar Prototype at Hanstholm, Denmark [4]. The fact that advanced control of the force applied by the PTO may greatly increase the performance of a point absorber dates back to the late 197 s by Salter [7], and Falnes and Budal [5]. It was suggested that by using the PTO actively, the frequency response of the point absorber can be manipulated to better match the incoming waves by moving the resonance frequency and widening the absorber s narrow frequency band [5]. However this requires the PTO to transfer energy to the absorber assisting its movement) in parts of an oscillation cycle. Due to the bi-directional energy transfer, this type of Wave Power Extraction Algorithm WPEA) has been termed reactive control and it requires a PTO capable of four-quadrant behaviour and continues force control. If bi-directional power transfer is not offered by the PTO, the reactive terms of the control are removed and a linear damping control remains. Linear damping is only optimal when the wave frequency and the absorber s resonance frequency match by other means. To improve the response of point absorbers without using reactive power a WPEA known as latching has also been

2 proposed [8], [9]. Latching prolongs the natural period of the absorber non-linearly by locking the absorber s movement in parts of an oscillation cycle. Besides the latching functionality, the PTO only needs to offer a constant damping force. However, performing a proper latching control in irregular waves requires knowledge of the future incoming waves, rendering the latching method a non-causal WPEA [8], [1]. Hence, prediction of the future incoming waves is required, where the prediction horizon is in the same time-range as the natural period of the absorber [5], [8]. A WPEA dual to latching has also been proposed in [11] and [1] and is referred to as unlatching or de-clutching control. In de-clutching control the natural absorber motion is manipulated by letting the absorber move freely in parts of an oscillation cycle, and then re-engaging the PTO with full damping. However, the de-clutching control is also non-causal. In [1] it was found that de-clutching control can lead to a power capture similar to linear damping. Likewise, in [9] it was found that a simplified latching control can lead to the same power capture as the linear damping method. II. M ETHODS In order to design the reactive control for the Wavestar concept a hydrodynamic model of the point absorber combined with a dynamic PTO model is required. The hydrodynamic model is based on linear wave theory, where model parameters are calculated using the numerical tool WAMIT, which is a program for computing wave loads and motions of structures in waves [13]. The presented hydrodynamic model is for a single float, i.e. radiated waves and diffraction effects from neighbouring floats are not included. The PTO system on the Wavestar Prototype is based on a hydrostatic transmission principle, which is quite analogue to a system suggested for Salter s duck in [14]. An illustration of the PTO is seen in Fig. 3. The system consists of a symmetric cylinder operated in closed-circuit with a swashplate variable-displacement axial-piston pump/motor powering a generator. The bi-directional flow is converted to a unidirectional rotation by the closed-circuit pump/motor capable of both positive and negative swash-plate angles. The applied PTO torque τpto is generated by the cylinder force Fcyl. Fig. 3. Fig.. The Wavestar C5 6 kw converter [6]. Higher power absorption than achievable with linear damping is pursued for the Wavestar concept, thus simplified latching methods as in [9] will not be satisfactory. Additionally, implementing a proper performing discrete control such as latching highly depends on wave prediction and requires a PTO with a very high holding force in order to lock the motion of the Wavestar floats. Therefore latching is not explored for the Wavestar concept. Discrete control topologies also have a tendency to increase the stress on the system compared to continuous control. As a result, Wavestar has chosen to pursue reactive control to optimize the energy extraction capabilities of the Wavestar concept. This paper will show that a causal implementation of reactive control in the Wavestar concept gives a very good improvement compared to a linear damping strategy. Furthermore, the reactive WPEA has the advantage, that if wave prediction methods become available, the reactive control can be augmented to approach the optimal performance, i.e. complex-conjugated control [5]. The PTO system on the Wavestar Prototype. The cylinder force is controlled by adjusting the differential pressure across the cylinder, where the differential pressure is controlled by continuously adjusting the displacement of the pump/motor. The displacement control is implemented according to [15]. This paper does not cover the dynamic modelling and design of the internal control loops of the PTO. Instead a second-order approximation of the closed-loop behaviour of the cylinder-force control is used, Fcyl s) ωn = Fcyl,ref s) s + ωn ζs + ωn 1) where Fcyl,ref s) is the cylinder force reference, the bandwidth is ωn = π rad/s, and the damping factor is ζ =.7. The cylinder is limited to give a force of Fcyl = ±4 kn. The PTO of the Wavestar Prototype is able to perform fourquadrant force control. During negative power transfer the asynchronous generator will operate as a motor. In [16] a thoroughly investigation from wave-to-grid is performed on how efficient an optimized version of the PTO system in Fig. 3 will be on the Wavestar C5 6W machine. After the modelling section the reactive control strategy will be presented and optimized using the derived model. The reactive control is optimized for all sea states in which the Wavestar is in production. Examples of the optimization for some specific sea states will be given. To get an overview

3 of the sea states, in which the Wavestar Prototype is currently tested, please refer to the annual distribution of wave climate at Hanstholm given in Tab.I. The Wavestar Prototype is designed to operate at significant wave-heights up to 3m before entering storm protection mode. Operation will also be stopped when the significant wave height goes below approximately.75 m due to low energy level. TABLE I ANNUAL DISTRIBUTION OF THE WAVE CLIMATE AT HANSTHOLM [6]. H m, Mean wave period T z [s] T p 1.17 T z) Sum [m] [%] Sum[%] A. Wave Model III. MODELLING Ocean waves are irregular waves, i.e. waves with varying frequency and amplitude. As a result, irregular waves are described by a wave amplitude spectrum, see Fig. 4. A sea state or spectrum is usually represented by two quantities, the significant wave height H m, and the peak wave period T p. The significant wave height H m, is the average wave height of the one-third highest waves and T p is the wave period where most energy is concentrated. The Pierson-Moskowitz PM) spectrum is utilized for generating waves [17]. One way of generating a time-series of waves from the spectrum is to extract the individual wave components as, η w,i t) = S A f i ) f sinπf i t) ) and then construct an irregular wave by superimposing the wave components, n η w t) = SA f i ) f sinπf i t + φ rand,i ) 3) i=1 where φ rand,i is a random phase for each component. However, depending on the number of frequency components, the random phase wave repeats itself eventually. A more random wave, representing sea waves better, is obtained by filtering white noise using proper digital filters designed according to the spectra [17]. This method is utilized instead to generate time-series of waves. The spectrum of the generated waves with this method shows fluctuation around the target Pierson- Moskowitz-spectrum. This is more consistent with real sea waves, also showing a non-smooth spectrum. The sea states presented in Fig. 4 will be used as examples when optimizing the PTO control. Being able to generate a wave η w t), the following section describes how the waves interact with the float. [m s] [m] 4 T p Spectral density S Af) H m= m T P= 6 s.1. Wave frequency [Hz].3 Wave height h Time [s] S A f) [m s] 1 5 State 3: Large) State : Medium) State 1: Small) Spectral density H m [m] T P[s] Wave Frequency [Hz] Fig. 4. Wave spectra for different sea states and an example of a corresponding wave. B. Wave and Float Interaction The equation of motion for a float is given as, J arm θarm t) = τ wave t) τ g t) τ PTO t) 4) where J arm is the inertia of float and arm, τ wave is the torque due to the wave-float interaction, τ g is the torque due to gravity and τ PTO is the torque applied by the PTO system to the float arm. An illustration is given in Fig.5a). θ arm is the angle of the float arm, where zero angle corresponds to the float position at rest. Fig. 5. Definition of float and arm quantities and wave direction. To describe the interaction between wave and float τ wave t) linear wave theory is often applied as it gives an adequate description under the conditions in which a WEC produces energy [18]. Linear wave theory is based on assuming simplified fluid dynamics. As a result, the forces in the fluid are reduced to be a conservative field and linear potential theory can be applied, which is solved for a number of boundary conditions, e.g. the particles cannot cross the seabed, submerged bodies, and the water-surface. In order to solve these equations, the equations are linearised using the assumption of low-amplitude waves relative to the height of the absorber. Numerical tools as WAMIT can then be applied to solve the fluid equation for a submerged body. The equations are solved for each of the following effects: The force exerted by an incoming wave on a float held fixed is denoted the exciting wave torque τ ext t). The effect of the distorted wave due to the body is also included. The effect of moving the float in water, i.e. the torque τ rad t) is the torque the radiated wave applies to the float. The effect of the Archimedes force τ Arch t), i.e. buoyancy.

4 These effects are then superimposed to yield the total wavefloat interaction. The three effects are illustrated in Fig. 6. Note that for clarity the illustrations are shown in heave-mode, whereas the Wavestar system is calculated as the rotational system around the arm-pivot, e.g. an exciting wave torque is identified instead of an exciting wave force. Gain [db] Magnitude plot of force filter τ ext jω) / η w jω) θ w = o θ = 45 o w θ = 45 o w Frequency [Hz] Fig. 7. Magnitude plot of the exciting wave torque filter. where V disp is the volume of the submerged part of the float, ρ water is the density of sea water, d f,arm is the floats moment arm, and g is the gravitational acceleration. If the torque due to the gravitational force on the float and arm is denoted τ g, then the hydrostatic restoring torque τ res is the combination of τ Arch and τ g, Fig. 6. Illustration of the three effects which are superimposed to yield the total wave-to-float interaction. 1) Exciting Wave Torque: The torque an incoming wave applies to a float held fixed is referred to as the exciting wave torque τ ext and is illustrated in Fig. 6a). Using WAMIT the fluid equations are solved for a regular wave passing a fixed submerged float, yielding the pressures along all submerged surfaces as a function of time. These pressures are then integrated into resulting forces and torques on the absorber. By solving the equations for each wave frequency, a filter for converting a wave into the exciting wave torque τ ext is established. Furthermore, this is performed for each incoming wave angle θ w. The angle θ w is defined in Fig. 5b). A magnitude plot of the resulting filter τexts) η ws) is shown in Fig.7 for three different wave directions. The filter s magnitude response resembles a 1st order filter with a cut-off frequency at approximately.18 Hz. As seen, a positive incoming wave angle gives a slight increase in gain from wave to exciting wave torque. This occurs because, that at positive wave angles, the waves approach the float arm from behind, causing the horizontal force exertion on the float to generate useful torque τ ext. However, at negative wave angles, the horizontal force components counteract the desired movement of the float. ) Hydrostatic Restoring Torque: The Archimedes force illustrated in Fig. 6b) is equal to the weight of the displaced water, τ Arch = V disp θ arm )ρ water gd f,arm θ arm ) 5) τ res t) = τ Arch t) τ g t) 6) where τ res is linearised around the draft of the float defined as the position where τ Arch =τ g. Hence, the torque can be described as: τ res t) τ res = k res θ arm t) 7) θarm= θ arm The spring constant or hydrostatic restoring coefficient k res is given as: k res = ρ water ga w d f,arm, where A w is the crosssectional area of the float at the draft line. 3) Radiated Wave Torque: When moving the float in water it will radiate waves as illustrated in Fig. 6c). If a massless float is forced to oscillate with a frequency ω w in otherwise calm water, the torque applied to the float by the radiated waves τ rad can be described as: τ rad t) = J add ω w ) ω arm t) b hyd ω w )ω arm t) 8) The parameter J add ω w ) is referred to as the added mass or inertia and b hyd ω w ) is the hydrodynamic damping coefficient. The damping term is due to the power being dissipated by the float to radiate waves. The added mass inertia) term is due to the effect that when oscillating a float, it will appear to have a greater mass due to the nearby water being displaced along with it. Using WAMIT these parameters have been identified for the Wavestar float and are shown in Fig. 8. The two parameters are frequency dependent, hence for an irregular wave, Eq. 8) it is not suitable for time simulations.

5 5 4 Fig. 8. hydrodynamic parameters given as added inertia and an hydrodynamic damping coefficient. Instead, the radiation torque can be expressed using a convolution integral [18], [19], τ rad t) = J ω arm t) k r t) ω arm t) t = J ω arm t) k r t τ)ω arm τ)dτ } {{ } τ rad,kr where k r is the impulse response function where added inertia at infinite frequency is omitted) from float velocity to radiated torque, and the inertia term is given as: J = 9) lim J addω f ) 1) ω f The convolution term can be viewed as a high-order hydrodynamic damping term, which will be denoted τ rad,kr. The impulse response k r t) is actually the inverse Fourier transform of, K r jω) = b hyd ω) + jω J add ω) J add ) ) 11) and can be calculated by WAMIT from the hydrodynamic damping coefficient as follows [18]: k r t) = π b hyd ω) cosωt)dω 1) The resulting impulse response is seen in Fig. 9. To avoid performing the cumbersome convolution τ rad,kr = k r t) ω arm t), which still only can be approximated when having an infinite impulse response, the convolution is instead approximated as a system of Ordinary Differential Equations ODE). This can be performed using Prony s method [19], []. Prony s method decomposes the impulse response into a response of a sum of complex exponential functions, which are the solutions of linear ODEs. The number of exponential functions is equal to the order of the corresponding linear ODE. Using the Prony s method implemented in MATLAB, which bases on [], different orders of ODEs have been tested to fit the impulse response, see Fig. 9. By comparing the responses it is concluded that the 5th order model is Impulse response k r t) [MNm] 3 1 Zoomed view Real system nd order ODE 3rd order ODE 5th order ODE Time [s] Fig. 9. The impulse response function k r t) being approximated. satisfactory, thus the convolution term can now be replaced with the following transfer function by Laplace transforming the 5th order ODE: τ rad,kr ω arm s) = K rs) = a 5s a 1 s + a b 5 s b 1 s + b 13) C. Complete Float Model The complete model is obtained by superimposing the effects of the exciting wave torque, hydrostatic restoring torque Eq. 7), and radiated wave Eq. 9) and Eq. 13) into Eq. 4): where J arm ω arm =τ ext k res θ arm J add, ω arm τ rad,kr τ PTO ω arm t)= τ extt) k res θ arm t) τ rad,kr t) τ PTO t) J arm + J add, 14) τ rad,kr s) = a 5s a 1 s + a b 5 s b 1 s + b ω arm s) 15) In open loop without PTO force feedback) the model Eq. 14) is linear and the response from exciting torque to float position is given by the transfer function: θ arm s) τ ext s) = 1 J arm + J add, )s 16) + K r s) s + k res A bode-diagram of Eq. 16) is given in Fig. 1 with the parameters in Tab. II. The bode diagram shows that the system has a resonance peak at f r =.85 Hz, corresponding to an eigen-period of 3.51 s. Hence, the Wavestar point absorber will have optimal absorption at a wave period of 3.51s, which corresponds to the fastest waves in which the WEC will be in production. In the next section the complete model including PTO is summarized.

6 Magnitude [db] f r =.85 Hz 18 Phase [deg] 9 18 Fig Frequency [Hz] Bode diagram of Eq. 16), θ arm s)/τ ext s). TABLE II PARAMETER VALUES FOR THE WAVESTAR PROTOTYPE. hydrodynamic model parameters: Inertia of arm and float w. ballast J arm [kgm ] water) Hydrostatic restoring torque coefficient k res [Nm/rad] Added-inertia J add ω) for ω J add, [kgm ] Transfer-function coefficients for K r s): a5, a 4, a 3, a, a 1, a ) =.1,.144,.64, 8.16, 13.1, 1.44 ) 1 6 b5, b 4, b 3, b, b 1, b ) =.1,.96, 1.67, 6.31, 13.3, 9.18 ) D. Complete Model for Optimization of the WPEA The PTO system applies torque using the hydraulic cylinder, which is able to produce ±4kN. The moment arm d a from cylinder force F cyl to τ PTO is found from Fig. 11 to be, where x c = c + Fig. 11. d a = a b sinθ a α ) x c + c ) 17) a b cos θ arm α ) + a + b ) 18) Moment arm of the PTO cylinder. The result is shown is Fig.11. The complete model used for optimizing the WPEA algorithm is displayed in Fig. 1 and has been implemented in SIMULINK. Fig. 1. Complete model for optimizing the WPEA. IV. OPTIMIZATION OF THE WPEA For the Wavestar concept a WPEA performing adequately without relying on wave prediction has been the focus, however, the chosen topology should be able to utilize wave prediction when it becomes available in the future. Consequently, reactive control of the form shown in Eq. 19) has been investigated as it may be extended to complex-conjugated control. τ PTO,ref t) = k PTO θ arm t) + B PTO ω arm t) + J PTO ω arm t) 19) τ PTO,ref s) 1 = H PTO s) = k PTO ω arm s) s + B PTO + J PTO s) ) The coefficients k PTO, B PTO and J PTO are not real physical quantities but control parameters. Hence, τ PTO,ref is generated as a linear feedback from measurements of the float movement. This is a causal implementation of reactive control, as the parameters are not changed wave-to-wave, but tuned according to the current sea state. The current sea state is found by continuously measuring the wave height near the float, where the measurements are averaged over a window of approximately 1 minutes to identify the mean wave period and the significant wave height. To argue how the WPEA is implemented, the WPEA is first discussed for regular waves, and then treated for irregular waves afterwards. A. Reactive Control in Regular Waves For a regular wave with known frequency ω, a WPEA of type Eq.) is able to obtain maximum power extraction from waves [5]. When inserting the feedback law in Eq. ), the transfer function from exciting wave torque to float angular velocity becomes, ) 1 τ ext s)= J arm +J add )s+b hyd +k res ω arm +H PTO s)ω arm s) s } {{ } } {{ } τ PTOs) Z mech s) ω arm s) τ ext s) = 1 Z mech s) + H PTO s) 1)

7 where Z mech s) is referred to as the intrinsic impedance. Note that J add and b hyd are taken at the frequency of the regular wave. The average harvested power P avg for a regular wave of frequency ω can then be computed as: P avg = 1 τ ext,amp Re{H PTO jω)} Z mech jω) + H PTO jω) ) B PTO = 1 τ ext,amp ) + bhyd +B PTO ImZmech jω))+imh PTO jω)) ) The average power of Eq.) is maximized if the imaginary part reactance) ImH PTO jω)) cancels ImZ mech jω)) and B PTO =b hyd. The imaginary part of Z mech, ImZ mech ) = ωj arm + J add ) k res 3) ω can be cancelled at the given frequency both using pure position feedback Eq. 5) or pure acceleration feedback Eq. 4): ImH PTO ) = ωj PTO k res ω = ImZ mech) J PTO = J arm + J add ) + k res ω, k PTO = 4) k PTO = J arm + J add )ω k res, J PTO = 5) The natural frequency of the simplified system in Eq. 1) from τ ext to θ arm is given as: k res + k PTO ω N = 6) J arm + J add + J PTO Thus both Eq.4) and Eq.5) move the system s resonance frequency to: k res + k PTO ω N = = ω 7) J arm + J add J arm + J add ) + kres ω Hence, the control moves the resonance frequency to the wave frequency as expected. As the system s normal resonance frequency is too fast, the control will always be used to decrease the resonance frequency, implying that either k PTO < or J PTO >. However, even though the same resonance frequency with Eq. 4) and Eq. 5) is obtained, the frequency response of the float is different for the two cases. This is shown in Fig. 13, where the closed-looped behaviour of the float is seen when matching a regular wave with period 4.5s. The figure shows that the two reactive strategies manipulate the un-compensated response black) to have the same resonance peak, namely at 1/4.5=. Hz. However, the spring compensated system red) has a broader frequency band compared to the inertia compensated system green). Hence, the spring-compensated system will harvest more energy from neighbouring wave frequencies, i.e. it will perform better in irregular waves. The optimized linear damping blue) has to over-dampen the system in order to move the damped natural frequency closer to the wave period. This is at the cost of a general lower responsiveness towards the waves. The optimal solution for linear damping, i.e. when k PTO = J PTO =, is the damping coefficient given as [5]: B PTO = Z mech = b hyd + J arm + J add )ω + k ) res 8) ω As the position signal is of higher quality compared to the angular acceleration, combined with the fact that the position gives a better frequency response, the following WPEA is used for the Wavestar in irregular waves: τ PTO,ref t) = k PTO θ arm t) + B PTO ω arm t) 9) However, to further show the advantage of position feedback compared to acceleration feedback, the difference of the inertia and spring compensated WPEAs will also be shown for irregular waves in the following section. Magnitude [db] Phase [deg] H PTO = H PTO =Z mech ω arm H PTO =b hyd ω arm +J PTO sω arm H PTO =b hyd ω arm +k PTO θ arm f r =.85 Hz Frequency [Hz] Fig. 13. Bode diagram for an un-compensated and a compensated system θ arm /τ ext. B. Reactive Control in Irregular Waves In order to optimize the WPEA in irregular waves, where PTO bandwidth and force limit are included, simulations of the system in Fig.1 on page 6 with an irregular wave input is used. The optimal coefficients k PTO and B PTO for a sea state are then the coefficients maximizing the average harvested power P avg of the model. The coefficients are optimized for all relevant significant wave heights H m, and mean wave periods T z, thereby constructing a look-up table of control parameters. The Wavestar Prototype then chooses the optimal control parameters from the look-up table based on the estimated sea state. 1) Optimization of Reactive Control in Irregular Waves: From simulations it has been found that the average harvested power settles after approximately 3 times the mean wave period. Consequently, this is used as the simulation length in

8 the optimization. On a decent laptop the model can execute approximately 1 s simulation in 1 s. The output of the simulation is the average harvested power, hence the optimization problem may be formulated as, max P avg xt), k PTO t), B PTO, η w H m,, T, )) 3) k PTO,B PTO s.t. ẋt) = fxt), τ PTO t), η w t)) where the constraint ẋt)=f ) means that the solution must comply with the dynamic system of Fig. 1. Eq. 3) is solved for each combination of significant wave height and mean period. The optimization is based on the simplex based algorithm given in [1]. However, for illustration purposes, brute-force has been applied for sea state and the results are shown in Fig. 14. Sea state was defined in Fig. 4. The results are for the following three cases: Spring-compensated WPEA without PTO force limit in model. Spring-compensated WPEA with PTO force limit in model. Inertia-compensated WPEA with PTO force limit in model. The simulations are performed for parameter settings of B PTO, k PTO and J PTO in the range: B PTO [.1; 8][Mkgm /s] k PTO [ 1.3; ][MNm] J PTO [; 1.3][kgm ] In Fig. 14 it is seen that in all cases an optimum can be found. Comparing to the non-force limited PTO version, the force limit greatly flattens the curve in a neighbourhood around the optimum. The optimal linear damping performance can be seen as the curve where k PTO and J PTO equal zero. For each case the optimum values of k PTO or J PTO are identified, and the curve going through this point as a function of damping is shown in Fig. 15. Hence, the PTO force limit greatly changes the optimal parameter values of the control, emphasizing the importance of the taking into account the PTO limits. Moreover, the inertia implemented reactive control harvest less energy compared to the spring-based reactive control as expected. Finally, the results show the benefit of using reactive control compared to linear damping. V. RESULTS With the WPEA in Eq. 9) the parameters k PTO and B PTO have been optimized for all relevant sea states, yielding the power matrix in Tab. III, which shows the expected average harvested power in [kw]. For each entry the corresponding k PTO and B PTO have been saved. For completeness and to show the effect of reactive control, the corresponding power matrix for optimal linear damping has been extruded from the results and is shown in Tab. IV. Finally, the power values for linear damping have been divided in Tab. V with the corresponding average power values for P P avg [kw] P avg [kw] avg [kw] x 1 6 Sea State Optimization with force limit B PTO x k PTO Sea State Optimization without force limit. B PTO 8 6 x k PTO x x 1 6 Sea State Optimization inertia with force limit 3 1 B PTO J PTO 15 x 1 6 Fig. 14. Three cases of average harvested power P avg as a function of WPEA parameters. A contour plot is also shown.

9 TABLE V POWER HARVESTED WITH LINEAR DAMPING INTAB. IV RELATIVE TO REACTIVE CONTROL RESULTS IN TAB. III. THE NUMBERS ARE IN [%]. H m, Mean wave period T z [s] T p 1.17 T z ) Fig dimensional curves from Fig. 14 of P avg B PTO ) for k PTO or k JTO fixed at the optimum values. harvested power using reactive control. This shows how the linear damping compares to the reactive control percent-wise. The table shows that reactive control especially improves performance at slower waves as expected. It is also seen, that the difference is less at larger waves, which is due to the force limit of the PTO, i.e. at large waves the PTO becomes more frequently saturated and does not benefit from reactive control. TABLE III HARVESTED POWER MATRIX FOR THE WAVESTAR PROTOTYPE IN [KW] USING REACTIVE CONTROL. period distribution in Tab. I. Also,the measured data in the Wavestar Prototype is the hydraulic power coming out of the cylinder. Thus the power matrix has been scaled with a cylinder efficiency of 95% to give the blue curve in Fig. 16, which overall shows a good agreement with the measurements. However, the waves at Hanstholm are more steep due to the near coast waters, giving a slightly lower performance of the float, especially at waves above 1.75m. At this point the waves begin to break, increasing slamming effects instead of energy production. The measured and calculated performance curves thus begin to divert at this point. Some of the diversion is also due to an increased violation of the linear wave theory at larger waves, i.e. the original assumption of small amplitude waves compared to float height. H m, Mean wave period T z [s] T p 1.17 T z) TABLE IV HARVESTED POWER MATRIX FOR THE WAVESTAR PROTOTYPE IN [KW] USING LINEAR DAMPING. H m, Mean wave period T z [s] T p 1.17 T z ) Fig. 16. Measured results from the Wavestar Prototype. [] The optimized control parameters have been applied to the Wavestar Prototype for the period from September 1 to February 11. In this period the average harvested power for 1 minutes windows have been calculated and logged together with the sea state, see [] for more info. The measured results are seen in Fig. 16 as a function of significant wave height. Consequently, some of the scatter is due to the different wave periods. The expected power curve is constructed from the power matrix in Tab. IV. The vales have been averaged to a function of wave height by weighting according to the wave

10 VI. CONCLUSION The paper has shown how the Wave Power Extraction Algorithm WPEA) for the Wavestar Prototype has been optimized in irregular waves based on a rigorous model of its hydrodynamics. The Prototype consists of two absorbers of 5m in diameter, however in this paper the absorbers were each treated as a single absorber system. Essential PTO features as bandwidth and force limitation of the PTO were also added to the model to enable optimization with respect to these constraints. The model showed the essential characteristics of the Wavestar absorbers, e.g. an eigen-period resonance frequency) of 3.51s. The model was implemented and shown to be fast executing, enabling using long time simulation for optimizing the WPEA parameters. The WPEA algorithm utilized is reactive controlled, and by inspecting the closed-loop behaviour of different implementations of reactive control, it was concluded that a position feedback gives a better absorption compared to an acceleration feedback. I.e. it is better to use a negative spring-term in the PTO control compared to adding extra inertia to the system. This difference was also shown in simulations with irregular waves. A causal implementation of reactive control was chosen, i.e. the control parameters are not changed wave-to-wave, but tuned according to the current sea state. The current sea state was found by continuously measuring the wave height upfront of the float, where the measurements are averaged over a window of approximately 1 minutes to find the mean wave period and significant wave height. Should wave prediction become available, the reactive control can then be extended to the non-causal complex-conjugated control. The influence of the PTO force limits on the optimal setting of the WPEA parameters was also investigated by performing optimization on a system with and without the force limit. This showed a very large difference in the found optimum values for the PTO damping coefficient. Hence, it is important to include these PTO limits in the optimization. Finally, a power matrix has been derived for the system and compared to the real measurements from the Wavestar Prototype, showing good agreement. Comparing the power matrices for reactive control and linear damping also clearly demonstrates the increased power production using reactive control, especially at medium sized and slow waves. At larger waves, the force limit reduces the benefits of reactive control. REFERENCES [1] A. Muetze and J. Vining, Ocean wave energy conversion - a survey, in Industry Applications Conference, 6. 41st IAS Annual Meeting. Conference Record of the 6 IEEE, vol. 3, 6. [] B. Drew, A. Plummer, and M. 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