Electrical Approach to Output Power Control for a

2017 Proceedings of the 12th European Wave and Tidal Energy Conference 27th Aug -1st Sept 2017, Cork, Ireland

Electrical Approach to Output Power Control for a Point Absorbing Wave Energy Converter with a Direct Drive Linear Generator Power Take Off Irina Dolguntseva∗, Sandra Eriksson†, Mats Leijon‡ ∗†‡ Department

of Engineering Sciences, Uppsala University Box 534, Uppsala 751 21, Sweden ∗ [email protected] † [email protected] ‡ [email protected]

Abstract—Control of a wave energy converter (WEC) can provide a substantially improved output power. Control strategies proposed so far are mainly focused on mechanical models for force control such as phase control and reactive power control and implemented using boundary element method (BEM) based codes such as WAMIT to find hydrodynamic forces. This approach contains inherent constrains due to linearised wavestructure interaction and small water level variations about the draft of a floater. WEC studied in the present paper is a point absorber with a linear generator (LG) power take off (PTO). A hydro-mechanical model for wave energy absorber and an electrical circuit model for the LG are combined to investigate power absorption of the WEC using resistive loading control of the electric generator. The structure motion is found using a nonlinear numerical wave tank (NWT) based on Reynolds-averaged Navier-Stokes (RANS) equations. The solution is compared with results obtained by means of MATLAB and WAMIT. A passive control, namely resistive loading, for the LG is optimized. Relations between output power, translator velocity, and damping factor are presented for different resistive loads connected to the generator. Index Terms—point absorbing wave energy converter, power output optimization, control strategies, electric damping force, linear generator power take off.

Wave

Buoy z x rb

Connection line

Casing o End-stop spring Translator

rt

Stator End-stop spring Foundation

lf

lu

ls lt lf

ll Seabed

I. I NTRODUCTION

Fig. 1. The WEC schematic

Since 2002, the Lysekil wave power project is conducted at Uppsala University, Sweden [1]. The wave power conversion concept developed within the project is a point absorber with a direct drive linear generator power take-off anchored on the sea bed with a gravitational foundation (Fig. 1). The major parts of the wave energy converter (WEC) are a buoy placed on the water surface and a linear generator (LG) enclosed in a hull protecting the device from sea water. A permanent magnet non-salient synchronous generator is used; it consists of a stator with windings and a translator, mounted with permanent magnets. The buoy is connected to the translator via a connection wire rope transmitting the buoy motion directly to the translator. The absorbed mechanical power depends on the PTO damping force of the WEC. The damping force for the considered WEC type mainly originates from electrodynamic

forces acting in the LG when electric currents are induced in the stator windings. Within the Lysekil project, the PTO damping force was studied experimentally and numerically. The passive control strategies such as resistive loading [2], rectification and filtration [3], and resonant circuit [4] were studied experimentally. Power capture and damping force was assessed numerically for three load cases including a three phase pure resistive load, passive rectification with a filter and resonant rectification [5]. However, the translator motion was prescribed and was not affected by the applied damping force. A continuous optimum control with constraints was studied numerically in [6], [7], however a mechanical damping force was considered. Latching of a WEC, first proposed by Budal and Falnes [8],

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Copyright © European Wave and Tidal Energy Conference 2017

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consists of locking the oscillating body at certain instances and then releasing it after a certain time period [9]. Latching is a type of phase control aiming to align the wave excitation force with velocity of the device. Latching depends on an accurate wave prediction. Recently, a new control strategy, declutching, has been proposed in [10]. Declutching means that the PTO force is set to zero for certain time periods. Neither latching nor declutching has been successfully implemented with electrical damping force for linear generators in real offshore conditions [11]. In [12], [13], a hydro-mechanical model was used to investigate power variations when latching and declutching were implemented. Although latching led to an improved power absorption, the peak-to-average ratio was too large. Declutching did not demonstrate advantage compared with a constant damping force control. However, the implemented model was based on a linearized WEC motion model which may not correctly describe WECs motion [14]. A numerical wave tank should be used when strongly non-linear forces act on the WEC [15]. In [16] the importance of designing WEC as an integral system was highlighted. Several studies were done to investigate different approaches to control a point absorbing WEC with a linear generator power take off. Reactive and model predictive control with and without constrains were studied in [16]. Passive control, namely resistive loading, was studied in [17], [18]. Active control strategies, such as bottom-up hierarchical control [19] and an intelligent fuzzy logic control [20], where resistive loading was used as a reference for implemented active controls. Nevertheless, all these studies are based on a linearised models for the WEC motion and may lead to erroneous power output estimation. In the present paper, the output power from a point absorber with a LG PTO is studied numerically. The model uses purely resistive damping force control, and the WEC and translator motion are determined using a non-linear numerical wave tank (NWT) based on Reynolds-averaged Navier-Stokes (RANS) equations. The results are discussed in comparison to a conventional model where WEC’s motion is determined using a linearised approach. The rest of the paper is organized as follows. In Section II governing equations for the WEC and LG are introduced and details on their modelling are presented. Wave modelling and methods to assess absorbed power and active output power absorption are given in Section III. Results from different models are discussed in Section IV, and finally, concluding remarks are given in Section V.

Fh,z Fcen Fcor

Fd

Fw,t

Fe

mtg

Fh,x Fw,b m bg Buoy

Translator

Fig. 2. Forces acting on the WEC

A. Governing Equations for Structure Motion The motion of the buoy and translator is given by equations [15] mb ¨rb = mb g + Fh + Fw,b + Fcen + Fcor mt ¨rt = mt g + Fw,t + Fd + Fe Ib θ˙b = Nh

(1) (2) (3)

where mb and mt are the mass of the buoy and translator, respectively; rb and rt are the buoy and translator displacements, respectively; g is the gravitational acceleration; Fh is the net hydrodynamic force acting on the buoy; Fw,b is the wire force from translator acting on the buoy and Fw,t is the wire force from the buoy acting on the translator, |Fw,b | = |Fw,t |; Fcen and Fcor are the centrifugal force and Coriolis force, respectively; Fd is the total damping force; Fe is the upper/lower end-stop spring force; Ib is the moment of inertia of the buoy; θb is the buoy pitch angle; Nh is the moment of the hydrodynamic force acting on the buoy with respect to the buoy center. All forces are illustrated in Fig. 2. Although the motion of an incompressible fluid is described by the conservation of mass and momentum equations, the fluid motion and the fluid-structure interaction can be modeled differently. In the present study, we utilize two methods: BEM and NWT. Definitions of all forces depend on the fluidstructure interaction modeling method and can be found in [15] for the NWT-based method and in [12] for the BEM-based method. It is worth noting that the NWT-based simulations are performed in three degrees of freedom (DOF): surge, heave and pitch, and the BEM-based modelling is implemented in two DOF: surge and heave. The generator damping force is implemented using a linear generator (LG) electric model as it is described below. We also shortly recall special aspects of the hydrodynamic force modelling. B. Linear Generator Modelling

II. WAVE E NERGY C ONVERTER M ODELING In this chapter, the WEC model is presented. Only essential details of the model are presented in the section. For more details, the reader is referred to [15] for the NWT-based modeling and to [12] for the BEM-based modeling.

The linear generator consists of a four-sided translator, moving linearly up and down, mounted between a four-sided cablewound stator. The translator poles consist of surface-mounted NdFeB magnets, mounted between aluminium wedges. The stator consists of laminated stator steel with slots with PVC insulated copper cables. The most important generator parameters can be found in Table I [21].

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TABLE I WEC S IMULATION PARAMETERS

Parameter

Value

Buoy type Buoy radius r Buoy height h Buoy draft d Mass of buoy mb Buoy moment of inertia Ib Water depth D Mass of translator mt Length of translator lt Length of stator ls End stop spring coefficients ks Wire rope spring coefficient kw Free stroke length lf Maximum stroke length upwards lu Maximum stroke length downwards ll Distance between the caser and buoy’s CoG L0 Rated translator speed vn Pole pair width τp Induced voltage (emf, l-l, rms) El−l Generator resistance Rg Generator inductance Lg Generator losses damping force Floss

cylinder 1.947 m 1.05 m 0.64 m 3,000 kg 6,000 kg · m2 25 m 5,000 kg 2.132 m 1.199 m 270 kN/m 833 kN/m 0.622 m 1.612 m 1.106 m 14.6 m 0.7 m/s 79.5 mm 257 V 0.64 Ω 20 mH 1 kN

The three-phase generator is modelled using the one-phase equivalent circuit for a non-salient generator and the generator parameters from Table I. Assuming a cosinusoidal magnetic flux 2π Φ = Φ0 cos( zt + ψ) (4) τp where Φ0 is the amplitude of the magnetic flux, τp is the pole pair width, zt is the translator displacement, and ψ is the phase shift, ψ = 0, ±2π/3, the induced emf e in the generator is calculated for each time-step according to e = −N

dΦ 2π = Ep sin( zt + φ) dt τp

(5)

where Ep = 2πN Φ0 z˙t /τp is the no-load voltage amplitude, and N is the total number of turns of the stator windings. The translator is longer than the stator, but it can still partly leave the stator at large waves, affecting the active area of the generator. This is accounted for by multiplying the induced voltage with the active area factor Aa , calculated from  1, if |zt | 6 21 (lt − ls )  1 0, Aa = (6)  if |zt | > 2 (lt + ls )  1 1 (l + l ) − |z | , otherwise t s t ls 2

where lt is the translator length, and ls is the stator length. The damping force Fd is calculated from the total active generator power divided by the generator speed. The total active generator power consists of the active output power from the generator to the resistive load and the generator losses. The generator losses can be divided into electromagnetic losses and friction losses. The electromagnetic losses are

resistive losses inside the generator windings and iron losses such as hysteresis losses and eddy current losses. The iron losses are frequency dependent and for the low frequencies considered here, hysteresis will dominate the iron losses [22]. For simplicity, the iron losses are therefore assumed to vary linearly with the speed. The frictional losses are also speed dependent and here are assumed to vary linearly with the speed. Therefore, the total damping force is found by Fd =

2 3Erms R 1 · + Floss R2 + X 2 z˙t

(7)

√ where Erms is the rms no-load voltage, Erms = Ep / 2, R and X are the total resistance and reactance per phase, R = Rg + Rl , X = 2πLg z˙t /τp , and Floss is the constant damping force due to the iron and friction losses. C. Linear Potential Wave Theory Within the linear potential wave theory the fluid is assumed incompressible, inviscid and irrotational. Then the total hydrodynamic force Fh can be decomposed into the excitation force due to the incident and diffracted waves Fi , radiation damping force Fr , and the hydrostatic stiffness force Fhs . In the time domain, these forces are obtained as follows. Fi = fi (t) ∗ η(t)

(8)

where fi (t) is the impulse response function of the excitation force, and η(t) is the fluid surface elevation. Fr = A(∞)¨ zb +

Z

t 0

K(t − τ )z˙b (τ )dτ

(9)

where A(∞) is the added mass at the infinite frequency, K(t) is the impulse response function of the radiation damping force in heave. According to [23] and [24], K(t) can be assessed using the frequency-domain added mass and radiation damping coefficients via the Fourier transform as Z 2 ∞ B(ω) cos ωtdt (10) K(t) = π 0 Z ∞ 2 = ω[A(ω) − A(∞)] sin ωtdt (11) π 0 where A and B are the frequency dependent added mass and radiation damping coefficients, and ω is the angular frequency. Fhs = Czb (t)

(12)

where C is the hydrostatic stiffness coefficient. Frequency dependent hydrodynamic coefficients A(ω), A(∞), B(ω), C are often found using a boundary element method (BEM) based software such as WAMIT. Applying a state-space representation of the convolution term in (9), the structure dynamics is implemented using an ode solver, e.g. ode45 in MATLAB.

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D. Reynold-Averaged Navier-Stokes Equations In general, the total hydrodynamic force Fh is given by Z Fh = (T − pI) · ns ds (13) S

where S is the wetted buoy surface; T is the viscous shear stress tensor; p is the pressure; I is the unit tensor; ns is the outward unit normal vector to the surface element ds. The moment Nh is obtained by Z Nh = (rs − rb ) × (T − pI) · ns ds (14) S

where rs is the position vector of the point on the buoy surface; rb is the position vector of the buoy center. The motion of an unsteady incompressible flow field is described by the mass and momentum conservation equations ∇ · v = 0, (15) 1 1 dv = − ∇p + ∇ · T + S (16) dt ρ ρ where v is the fluid velocity field, ρ is the fluid density, S is the body force per unit mass (e.g. gravity). The fluid and the buoy motions are fully coupled via forces and velocities at the fluid-structure interface S. The boundary conditions on S are given by vs = v on S,

(17)

Ts + Tf = 0 on S

(18)

where vs is the velocity on the wetted surface of the buoy; Ts is the vector of traction acting on the buoy on the wetted surface of the buoy; Tf is the vector of stresses exerted on fluid at the fluid-structure interface. Velocity vector vs is determined as follows: vs = vb + θb × (rs − rb )

(19)

where vb is the velocity of the buoy center. The finite volume method for incompressible viscous flows is used here to discretize the governing equations of the fluid motion, resulting in the RANS equations written in their integral form (see, e.g. [25]). In the RANS equations, the shear stress tensor for incompressible Newtonian fluids is defined as T′ = (µ + µt )(∇v + (∇v)T )

(20)

where µ is the dynamic fluid viscosity; µt is the turbulent viscosity computed by a turbulence model of eddy-viscosity type (SST K-ω) as a function of the turbulence kinetic energy K and the specific rate of dissipation ω [26]. The pressure and velocity fields are coupled by means of PISO method [27]. The scalar quantities K and ω satisfy the conservation equation for scalar fields [25]. The liquid-gas interface is tracked by the volume of fluid (VOF) model [28], which assumes that the two phases present in a control volume share velocity and pressure. The fluid motion is simulated using the commercial finite volume method package ANSYS FLUENT. The structure motion is defined by a User Defined Function (UDF) in ANSYS FLUENT as explained in [15].

Fig. 3. The damping factor as a function of the translator velocity for different resistive loads (the effective damping factor is plotted with solid curves, the total damping factor is plotted with dashed curves)

III. M ETHOD In both cases, the BEM-based model and the NWT-based modelling, monochromatic waves are used. In the BEM-based model, the incident wave is given by η(t) = A0 sin(ωt)

(21)

where A0 is the wave amplitude, and ω0 is the wave angular frequency, ω0 = 2π/T0 , T0 is the wave period. In the NWTbased modelling, the fifth-order Stokes waves are used of the given amplitude A0 and wave period T0 . The total absorbed mechanical power is then given by p(t) = Fd z˙t .

(22)

The average absorbed power Pavg is investigated for different loads and different sea states. Z 1 T p(t)dt (23) Pavg = T 0 IV. R ESULTS

AND

D ISCUSSION

A. Steady State Modelling When the WEC is connected to a resistive load Rl , the effective electric power p˜ delivered to the load is determined by the effective damping force 2 ˜ d = 3Erms Rl · 1 . F 2 R + X 2 z˙t

(24)

The damping factor is the coefficient proportionality between ˜ d and the translator velocity z˙t . Due to the damping force F the phase reactance X, the damping factor is not constant for different values of the translator velocity, see Fig. 3. The effective damping force varies with the translator velocity reaching its maximum at z˙t = Rl τp /(2πLg ) (Fig. 4) 2 ˜ max = 1.5π(N Φ0 ) · Rl . F d τp Lg R

(25)

With an increasing translator velocity the effective

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Fig. 4. The damping force as a function of the translator velocity for different resistive loads (the effective damping force is plotted with solid curves, the total damping force is plotted with dashed curves)

Fig. 6. The power as a function of the load resistance for different sea states (the effective power is plotted with solid curves, the total absorbed power is plotted with dashed curves)

and NWT-based methods. B. BEM-based modelling

Fig. 5. The power as a function of the translator velocity for different resistive loads (the effective power is plotted with solid curves, the total absorbed power is plotted with dashed curves)

output power increases reaching its limiting value of 1.5(N Φ0 )2 Rl /L2g (Fig. 5). The damping factor, damping force and absorbed power are plotted in Figs. 3–5 to illustrate relations described above for load resistances of 1, 2 and 3 Ω to preserve clarity. The dashed lines are used to demonstrate the effect of resistive, iron and friction losses of the generator for different resistances. The translator velocity depends on the buoy velocity that is a consequence of the flow-induced forces acting on the WEC. At the same time, the buoy motion affects the fluid flow and its own motion. Moreover, the translator motion is reciprocal bounded by the upper and lower end-stops, which results in a periodic translator velocity changing its sign twice per wave period. The translator speed shall not exceed 2 m/s due to the structural strength of the WEC [13]. Thus, the power absorption is limited and significantly reduced when a passive pure resistive load control is applied. Nevertheless, an optimum resistive load can be determined for a particular sea state. Below, we discuss how different resistive loads influence the WEC motion in different sea states using both BEM-based

The BEM-based model is used to find relation between different resistances Rl and the average absorbed power and effective electric power delivered to the load. The considered sea states are (A0 , T0 ) = (0.5m, 5s), (A0 , T0 ) = (1m, 6s), and (A0 , T0 ) = (1.5m, 7s). A range of resistive loads from 1 Ω through 10 Ω have been tested to investigate how the power absorption and WEC’s behavior are affected by different resistive loads in different sea states. The results are plotted in Fig. 6. The total absorbed power for different sea states is plotted with dashed curves, the effective electric power — with solid curves. For smaller resistances of the load, the difference between the total and effective electric power is larger than for greater load resistances due to dominating generator losses. With an increasing resistance of the load, the currents in the stator windings decreases resulting in a larger portion of power delivered to the load. For each sea state, there exists an optimum resistive load, when the average effective power is maximal. An increasing amplitude of the incident wave leads to an increasing translator velocity for all resistances while the wave amplitude is not greater than the generator maximum stroke length. Increase of the wave amplitude beyond the generator stroke length does not lead to considerable increase of the rms no-load voltage Erms , but the generator impedance X becomes larger due to larger translator velocity variations. The effective damping force and power as functions of the translator velocity are plotted in Figs. 7–8. The steady state and the BEM-based models agree well when the translator moves upwards and is above the middle of the generator, otherwise the damping force lags the translator velocity. Therefore, the damping coefficient depends on both the translator position and velocity. As a result, less power is absorbed when the damping force lags the translator velocity (Fig. 8). If a passive damping force control is used, then alignment of the damping

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Fig. 7. The effective damping force as a function of translator velocity for A0 = 1.5m, T = 7s and Rl = 2Ω

Fig. 8. The WEC performance under the sea state (A0 , T ) = (0.5m, 5s) while connected to a resistive load of 2.3Ω

force with the translator velocity can help to improve the power absorption of the WEC. C. NWT-based modelling The generator model has been implemented in the NWT and has been tested for a wave with an amplitude of 0.5 m and a period of 5 s when a resistive load of 2.3Ω is applied. The translator velocity, total power and the damping force are plotted with respect to time in Fig. 9. More results on the NWT modelling with different loads and sea states will be presented at the conference. V. C ONCLUSION A complete model ’wave-to-wire’ of a point absorbing wave energy converter with a direct drive LG has been implemented using both the BEM-based and the NWT-based methods. Relations between the power, damping force, damping factor and the translator velocity have been studied numerically and analytically using a steady state model assuming the WEC to be connected to a pure resistive load. It has been concluded that the damping force has upper bound with respect to translator velocity and connected resistive load, which particularly means that the damping factor and absorbed power are also limited in this case. Unlike a mechanical approach to the damping force representation, the damping force may lag the translator velocity, perhaps, due to the presence of

Fig. 9. The WEC performance under the sea state (A0 , T ) = (0.5m, 5s) while connected to a resistive load of 2.3Ω

reactive components in the PTO. Possible solution to it can be a reactive power control which can align the damping force with the translator velocity and improve the power absorption capacity of the WEC. Resistive loads can be used when the generator is not gridconnected, see e.g. [3], [29]. In reality, very few loads can be considered as pure resistive. Power absorption can further be improved if the WEC is actively controlled. Moreover, power conversion is required since WEC’s PTO is a direct drive producing output power of varying amplitude and frequency. Therefore, power converters and transformers are needed and their presence will further reduce output power. Another limitation of the model is that the resistive load was assumed to be connected right to the generator, i.e. power transmission cable effects are not considered here. ACKNOWLEDGMENT This work was conducted within the STandUP for Energy strategic research framework and financially supported by the Swedish Energy Agency P42243-1, Uppsala University, ˚ Milj¨ofonden and AForsk 17-550. R EFERENCES [1] R. Waters, M. St˚alberg, O. Danielsson, O. Svensson, S. Gustafsson, E. Str¨omstedt, M. Eriksson, J. Sundberg, and M. Leijon, “Experimental results from sea trials of an offshore wave energy system,” Applied Physics Letters, vol. 90, no. 3, pp. 1–4, 2007.

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