Numerical Modeling of the Electric Linear Generators based on the Sea Waves Energy

Constantin GHIŢĂ1, Aurel–Ionuţ CHIRILĂ1, Ioan–Dragoş DEACONU1,

Valentin NĂVRĂPESCU1, Daniel Ion ILINA1

Abstract--The paper presents the design, the calculus and a numerical simulation of the magnetic field, the electromotive force and the electromagnetic forces for a direct driven permanent magnet linear generator buoy, used to generate electric energy based on the sea wave’s energy. The magnetic field is computed using finite element method applied on a 3D model of the generator. Index Terms--Electromagnetic force, electromotive force, linear generator, permanent magnet generators, sea wave energy.

I. INTRODUCTION1 A brief but detailed presentation regarding the principles

and real devices used to obtain electric energy from sea water or oceans energy can be found in [1]. Permanent magnet linear generators can be used as a convenient method for producing electric energy based on clean resources, such as sea waves energy. These generators have special features in comparison with other types of electric generators. They take the advantage of renewable energy resources, do not pollute the environment, have to be water-proofed, and can be built in a range of various powers without having geometrical dimensions limitations. The manufacturing technologies are not complex and the implied materials are common.

The present paper deals with a generator driven by a marine buoy principle. Based on this principle there are studies and numerical simulations for the tubular form of the generator [2]-[9]. Other generators used the tidal current power and not the waves energy [10]. Studies on principles like point absorber can be found in [5], [11], oscillating water column in [5] or Archimede Wave Swing [12] just to name a few. Other studies refer to power plants of such generators [13]. A research over the environment impact is presented in [14].

The tubular generators are used more and more [15]-[17]. There are farms with interconnected generators that produce energy based on clean renewable resources [6].

A disadvantage of such generators is the low frequency of the back electromotive force. This can be partially remedied by increasing the number of magnetic poles.

The active materials used to manufacture these generators are not fully used due to the low operating frequencies.

The predicted electricity generating costs from sea wave energy converters have shown a significant improvement in the last twenty years, which has reached an average price less

1 Politehnica University of Bucharest, Electrical Engineering Faculty, Sp. Indendenţei, 313, Bucharest, ROMANIA, email: ghitac@iem.pub.ro; aurel.chirila@gmail.com; dragos.deaconu@gmail.com; valentin.navrapescu@gmail.com; ilinadaniel@yahoo.com.

than 10 c€/kWh. Compared to the average electricity price in the European Union, which is approximatively 4 c€/kWh, the electricity price produced from wave energy is still high, but it is forecasted to decrease further with the development of the technologies. [1]

II. MODEL DESCRIPTION The operating principle of the designed generator is very

simple. There is a central armature with permanent magnets on it. This armature is anchored to the sea floor, and the floater moves permanent magnets relative to the fixed armature coils to induce voltages. In Fig.1 it is present a sketch of the entire system.

. Fig. 1. Generator assembly sketch:

1 – translating armature; 2 - fixed armature with coils; 3 – shaft; 4 – fixing pillars; 5 – buoy/floater; 6 – rope; 7 – spring.

The generator is positioned above the sea level, but it is

water-proofed. Fig. 2 shows a detailed view of the generator. The translating armature of the generator is a hexagonal prism. Each face of the prism is a ferromagnetic structure serving as fixing support for the permanent magnets. The magnets are alternatively placed, so that the magnetizing directions of two contiguous magnets are antagonistic. The fixed armature has a ferromagnetic structure with slots where three-phase induced coils are placed. Its shape is a right prism with 6-side regular polygon cross-section too. The height AB of the translating armature is about three times greater than the CD height of the fixed armature. Thus, AB = 350 mm, CD = 120 mm, the length of the side of the fixed armature is 500mm and the length of the translating armature side is 450 mm.

640978-1-4244-1633-2/08/.00 ©2008 IEEE

Fig. 2. Detailed view of the hexagonal generator (items 1 and 2 from Fig. 1): (a) – one face with permanent magnets and coils with fixed ferromagnetic armature not visible; (b) – one face with permanent magnets and coils with

fixed ferromagnetic armature visible.

The translating armature forms a stiff structure from mechanical point of view.

On each face of the translating armature are placed 29 permanent magnets. The dimensions for one magnet are 322 x 10 x 5 mm (length x width x height). The distance between two adjacent magnets is 2 mm. The fixed subassembly armatures, located towards the translating armature’s exterior, forms also a stiff mechanical structure.

The modeling refers to only one phase of the generator. By one phase it is understood an electric circuit formed by 6 coils found on the each face of the fixed armatures. All these coils have the same relative position on each face. The slots of the fixed armatures have the width of 8 mm and the depth of 20 mm. The side length of one coil placed in the slot has 446 mm. The air gap between face to face armatures is denoted with δ.

In Fig. 3 it is presented a cross-section of a plane that cuts two face to face armatures. The double-headed arrow line shows the oscillating direction of the translating armatures under the sea wave’s oscillations.

Fig. 3. Cross-section through two face-to-face armatures: 1 – translating ferromagnetic armature; 2 – permanent magnet; 3 – fixed

ferromagnetic armature; 4 – coils.

The air gap δ is maintained constant with the help of a system made of 12 guiding straight metal bars (not present in Fig. 1). On the sides of each face of the translating armature are fixed two bars with bearings. One of these bars is presented in Fig. 4. Each bar slides on a sliding path, placed on the fixed assembly.

Fig. 4. Detail on the bar with bearings. Fig. 5 shows a view in perspective of the fixed

ferromagnetic armature, with 6 slots.

Fig. 5. Fixed ferromagnetic armature.

In Fig. 6 it is depicted a perspective of the armature three-phase winding, made of 3 identical coils, being placed in the 6 slots of the fixed ferromagnetic armature.

Fig. 6. The three-phase winding coils shape, placed within the slots.

Because the height of the translating armature is greater than the height of the fixed armature, the electromotive force of the coils is not diminished when the translating armature is found at the end of the lift. In other words, the end-effects of the generator are neglected. The shift of the translating armature is alternative with low frequency equal to the sea wave frequency. However, the obtained voltage has a greater frequency due to the increased number of magnetic poles.

III. THE STUDY OF MAGNETIC FIELD

The study refers to numerical modeling of the generator’s excitation magnetic field repartition produced by the permanent magnets. The computations are performed taking into account only two face to face armatures of the generator. The cross-section of these face to face armatures is depicted in Fig. 3. The excitation magnetic field is computed using 3D finite element method, with the help of specific software [18].

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A. Hypotheses

The computation has been performed supposing the following simplifying hypotheses:

- the magnetizing curve of the fixed and translating ferromagnetic armatures is non-linear;

- the magnetic hysteresis of the ferromagnetic armatures is neglected;

- the reaction magnetic field that would be created by the electric currents traveling through three-phase coils placed on the fixed armature is neglected.

The air gap δ has been modified from 1mm to 4mm. The magnetizing characteristic of the ferromagnetic materials used for the armatures is presented in Fig. 7.

The permanent magnets are NdFeB type and have the following parameters: coercivity Hc = -827600 A/m; relative magnetic permeability µr = 1.1; resistivity ρ =1.5·10-6 Ωm.

0

0.5

1

1.5

2

2.5

0 5000 10000 15000

Magnetic field H (A/m)

Flux

den

sity

B (T

)

Fig. 7. The magnetizing characteristic of the ferromagnetic materials.

After solving the numerical problem, all components Bx,

By, Bz of the flux density are obtained. The magnitude of the resultant flux density B is obtained with the following relation:

222zyx BBBB ++= (1)

B. The influence of the air gap on the inductor magnetic field The study is performed taking a mean position of the fixed

armature with respect to the translating one. The air gap has been set to 1mm, 1.5mm, 2mm, 2.5mm,

3mm and 4mm. For each case the inductor magnetic field has been determined using (1).

In Fig. 8, 9 and 10 are presented the variations of the air gap flux density along the length AB, for the following air gaps: 1mm, 2mm and 4mm respectively. The represented flux densities refer to the points of the middle air gap.

As a result, the air gap field is influenced by the air gap value. If the air gap is greater than 2 mm then the flux density, the electromotive force and the generated electric power diminish. The maximum air gap flux density values vary from around 0.9 T for 1 mm air gap, to 0.55 T for a 4 mm air gap.

Table I shows the variation of the maximum flux density for the fixed armature teeth with the air gap value. Thus, when the air gap increases 4 times the maximum flux density value decreases only 1.67 times.

-1-0.8-0.6-0.4-0.2

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1

0 50 100 150 200 250 300 350

Height AB (mm)

Flux

den

sity

B (T

)

Fig. 8. The air gap flux density variation with the height for δ = 1 mm.

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

0 50 100 150 200 250 300 350

Height AB (mm)

Flux

den

sity

B (T

)

Fig. 9. The air gap flux density variation with the height for δ = 2 mm.

-1-0.8-0.6-0.4-0.2

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1

0 50 100 150 200 250 300 350

Height AB (mm)

Flux

den

sity

B (T

)

Fig. 10. The air gap flux density variation with the height for δ = 4 mm

TABLE I

THE VARIATION OF THE MAXIMUM FLUX DENSITY WITHIN THE FIXED ARMATURE TEETH WITH THE AIR GAP

IV. THE ELECTROMOTIVE FORCE The electromotive force induced in the three-phase coils,

placed in the slots of the fixed armature depends mainly on the following parameters: number of turns, the physical geometry of the coils, the induction flux density and the speed of the translating armature with respect to the fixed one. All these

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parameters must be known in order to compute the electromotive force of the coils.

A. Computation algorithm for the electromotive force The electromotive force varies with time on a quite

complicated law. However, the waveform of the electromotive force can be numerically determined starting from the flux density values.

The general expression of the electromotive force of one coil is given by:

])([')(2])([)(2)( ' txBlltxBltvwte ssss ⋅−+⋅⋅⋅= (2)

where ws is the number of turns; 'sl is the length of one side of

one coil (placed in the slot), ls is the length of one permanent magnet; v(t) is the instantaneous speed of the translating armature (imposed by the sea wave); ])([ txB is the flux density along the length ls, and x is the shift defined in Fig. 2;

])([' txB is the flux density over the 'ss ll − zone, and t is the

time moment when the electromotive force is computed. In the simulation, the wave speed is considered ideal, that is

a sinusoidal variation given by: )π2sin()ωsin( tfVt Vv(t) vmvm == (3)

where Vm is the amplitude of the instantaneous speed, ωv is the pulsation of the sea wave, and fv is the sea wave frequency. The shift of the translating armature can also be computed by integrating the speed:

ovv

mvm xtf

fVdttfV x(t) +−=⋅= ∫ )π2cos(π2

)π2sin( (4)

where xo is the initial shift of the translating armature with respect to the fixed one. The chosen value for xo is zero.

Taking into account (3) and (4) the final expression for the electromotive force for one coil is:

+−⋅−+

+

+−⋅⋅⋅=

ovv

mss

ovv

mss

xtff

VBll

xtff

VBltvwte

)π2cos(π2

')(2

)π2cos(π2

)(2)(

' (5)

The numerical algorithm used to compute the electromotive force is the following. The initial time is t = 0. It is chosen a fraction of time variation ∆t. For example, for one spatial period of the flux density, the time step could be taken at least 40 equal time intervals. In this way, the flux densities ])([ txB and ])([' txB become from functions of time and space to simple functions depending only on time. It is set-up a table with discrete values for the magnetic flux density so that the variation can be known at every time-step, given by:

ttt kk ∆+= −1 (6) In the end, the discrete variation of the electromotive force is computed with (5).

B. Study case

For the worked example, the following parameters were considered:

- air gap δ = 2 mm; - sea wave frequency fv = 0,9 Hz ; - maximum speed of the sea wave crest Vm = 0,9 m/s ;

- ws = 100 turns; - the length of the coil side placed in the slot 'sl = 0,446 m; - the permanent magnet length ls = 0,322 m; - the discrete time-step ∆t = 0,001 s; Fig. 11 and 12 present the sea wave speed and the shift x of

the translating armature with time, respectively.

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1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Time t (s)

Spee

d v(

m/s

)

Fig. 11. The sea wave speed.

050

100150200250300350

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Time t (s)

Shift

x(m

m)

Fig. 12. The shift of the translating armature.

Using the algorithm, the instantaneous flux density value is

computed on the symmetry axis of the air gap. The time variation of the flux density is presented in Fig. 13.

In Fig. 14 is presented the time variation of electromotive force for one coil of the fixed armature.

The coils of one phase can be connected in series or in parallel. In the first case the voltage would be 6 times greater than the voltage obtained from a single coil situated on a single face.

If the 6 coils corresponding to the 6 faces of the fixed armature are connected in parallel the electromotive force is the same but the electric current is 6 times greater. Anyway, the 6 coils on the 6 faces are not identical because the electric and magnetic fields are different due to technological reasons. Thus, circulation currents can exist and so the preferred connection is in series.

Analyzing the waveform presented in Fig. 14, the generated voltage has changing amplitude. The envelope frequency is equal to the sea wave frequency. The envelope period is 1.11 seconds, while the voltage period is 26 times lower, the average value being 1.11/26 = 0,0427s. In other words, the average frequency of the voltage is f = 1/0.0427 ≈ 23.4Hz.

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-0.8-0.6-0.4-0.2

00.20.40.60.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Time t (s)

Flux

den

sity

B(T

)

Fig. 13. The magnetic flux density variation in the air gap.

-50-40-30-20-10

01020304050

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Time t (s)

Coi

l vol

tage

e(V

)

Fig. 14. The waveform of the electromotive force e of one coil.

The electromotive force )(te given by (5) contains two

terms. The first term gives the voltage generated on the length sl of the coil and the second term gives the electromotive

force obtained along the length ss ll −' . As a result of the simulation, the numeric value of the second term is insignificant (below 1% with respect to the value of the first term).

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1

0 50 100 150 200 250 300 350

Height AB (mm)

Flux

den

sity

B (T

)

(1 )(2 )(3 )

Fig. 15. The flux density variation at different distances from the ends of the

permanent magnets, supposing δ = 2 mm. As a consequence, the second term can be neglected. The explanation is that the flux density changes rapidly over the coil zones situated in the slot, but outside the zone of the permanent magnets. This phenomenon is depicted in Fig. 15 where are presented the waveforms of the air gap flux density at three different distanced from the edge of the permanent

magnets. Curve 1 is at 0 mm, curve 2 at 5 mm and curve 3 at 10mm, respectively from the ends of the permanent magnets. The curve 3, placed at 10 mm from the permanent magnets edges shows that the flux density is almost zero.

V. THE FORCES ACTING OVER THE TRANSLATING ARMATURE

The computation of the electromagnetic forces is performed using the generalized force method for magnetic fields, with the following general definition:

=

∂

∂= =

zyx

WF constim

k

,,ξξ .ξ (7)

where: ξ is the generalized coordinate, that can be any of the three rectangular directions x, y or z, shown in Fig. 16; Fξ is the resultant force that acts in the direction ξ; Wm is the magnetic energy enclosed in the volume between the armatures, where the force is wanted to be computed; ik are the electric currents traveling through the fixed coils and that are supposed constant when the magnetic energy derivative is computed.

Fig. 16. The definition of rectangular directions x, y, z. The expression of the magnetic energy is written as

follows:

dxdydzBBB

dxdydzzyxBdvBHdvwW

Vzyx

VV Vmm

⋅++⋅=

=⋅=⋅=⋅=

∫

∫∫ ∫

)(µ21

µ2),,(

2

222

2

where µ is the magnetic permeability of the environment; Bx, By, Bz, are the components of the flux density over the three rectangular directions shown in Fig. 16.

Using (7), the forces over the three directions can be computed:

∂∂= = .consti

mx kx

WF ,

∂

∂= = .constim

y kyWF ,

∂∂= = .consti

mz kz

WF (9)

Using (9) and the Infolytica’s MagNet software [18], forces

zyx FFF ,, are computed. Table II presents the values of these forces.

(8)

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TABLE II

NUMERICAL VALUES OF THE FORCES EXERTED OVER THE TRANSLATING ARMATURE

The most important force component is the one over the y direction that is perpendicular on the faces. In Fig. 17 it is present the variation of Fy with the air gap.

The Table II and Fig. 17 show that Fy is the greatest and is of attraction type; the forces along the x and z directions can be neglected. The forces Fy generates a powerful mechanical stress if the fixed and the moving armatures would be singulars. The Fy forces of the 6 faces are canceling each other due to the hexagonal shape of the two armatures that are symmetrical. These forces must be taken into account when the mechanical structure of the entire generator is designed.

01000200030004000500060007000

0 1 2 3 4 5

Air Gap δ (mm)

Forc

e F y

(N)

Fig. 17. The Fy force variation with the air gap.

VI. CONCLUSIONS In the paper are studied the inductor magnetic field, the

electromotive forces and the electromagnetic forces that exert on the armatures of an electric linear generator, operated by the sea waves energy. By choosing different number of the permanent magnets of the translating armature, it is possible to increase the frequency of the generated voltage.

The electromotive force amplitude is not constant. The time variation of the waveform shows that the envelope has a low frequency equal to the sea wave frequency.

The electromagnetic force Fy acting over the translating armature is the greatest. The values of these forces are useful for the mechanical design of the generator.

The numerical study performed within the paper helps the manufacturers and the users of permanent magnet linear generators with information regarding the most stressed parts magnetically and mechanically. Before the manufacturing of the linear generator it is recommended to perform numerical simulations in order to improve the prototype.

VII. REFERENCES [1] European Commission, Centre for Renewable Energy Sources, Ocean

Energy Conversion in Europe - Recent advancements and prospects, Published in the framework of the Co-ordinated Action on Ocean Energy, EU project under FP6 Priority: 6.1.3.2.3, 2006.

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[3] A.M. Eid, K. Y. Suh; K. J. Choi; H. D. Han; H. W. Lee; M. Nakaoka, “A Unique Starting Scheme of Linear-Engine Tubular PM Linear Generator System Using Position Feedback Controlled PWM Inverter”, 37th IEEE Power Electronics Specialists Conference, PESC 2006, 18-22 June 2006, pp. 1 – 5.

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[13] S. Gustafsson, O. Svensson, J. Sundberg, H. Bernhoff, M. Leijon, O. Danielsson, M. Eriksson, K. Thorburn, K. Strand, U. Henfridsson, E. Ericsson, K. Bergman, "Experiments at Islandsberg on the West coast of Sweden in preparation of the construction of a pilot wave power plant", 6th EWTEC conference in Glasgow , 28th of August to 3rd of September 2005.

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