1152 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 4, AUGUST 2006
A Tubular-Generator Drive For
Wave Energy Conversion
Vincenzo Delli Colli, Piergiacomo Cancelliere, Member, IEEE, Fabrizio Marignetti, Member, IEEE,
Roberto Di Stefano, and Maurizio Scarano, Member, IEEE
Abstract—This paper illustrates the operation of a tubular-
machine drive as a linear generator for a heave-buoy wave energy
conversion. Linear generators, which are adopted in marine power
plants, offer the advantage of generating without introducing any
conversion crank gear or hydraulic system. The use of a tubular-
machine topology allows the electromagnetic thrust density to be
improved. This paper briefly summarizes the principles of marine
wave buoy interaction and reports the design analysis and control
of a permanent-magnet (PM) synchronous tubular linear machine
based on a scaled generator prototype and on a rotating simulation
test bench.
Index Terms—Heave buoy, linear synchronous generators,
marine energy, tubular generators.
I. INTRODUCTION
THE USE of a tubular machine as a linear generator for a
heave-buoy wave energy conversion in order to improve
the force density N/m3that can be attained is proposed in
this paper.
Linear machines are increasingly used today as generators.
In fact, some energy sources and especially marine ones exhibit
an alternating motion. One of the most promising renewable
energy sources characterized by a reciprocating motion is rep-
resented by tide and wave marine energy [1]–[4].
Linear generators are suitable for direct-drive applications,
since they permit a reduction in the number of subsequent
energy transformation steps. Coupling an electrical generator
directly with the reciprocating energy source also permits the
reduction of moving parts and simplifies the system. The overall
efficiency is thereby enhanced.
Different electrical-machine topologies have been proposed
in literature to be utilized as direct-drive linear generators [5].
One encouraging way to convert an alternating mechanical
energy into an electrical energy by means of a direct drive is
to use a linear tubular synchronous permanent-magnet (PM)
generator. The force-to-weight ratio of such machines has been
proved to be higher than the flat linear topology [6].
Section II summarizes the basic principles of the heave-
buoy mechanics, whereas, Section III outlines the design of
the proposed tubular generator and presents the results of the
finite-element analysis (FEA) together with the experimental
validation. Section IV presents the optimal buoy control, which
Manuscript received March 29, 2005; revised June 28, 2005. Abstract
published on the Internet May 18, 2006.
The authors are with the Department of Automation (DAEIMI), Fac-
ulty of Engineering, University of Cassino, 03043 Cassino, Italy (email:
Digital Object Identifier 10.1109/TIE.2006.878318
Fig. 1. Semisubmerged sphere as marine wave energy converter.
allows the sea-to-buoy power transfer to be maximized, and
Section V discusses the drive control used to perform the
electromechanical and ac/dc conversion control. Finally, Sec-
tion VI presents the test facility, and Section VII gives the
results of the experiments on mechanical to dc power conver-
sion obtained by means of a tubular prototype as well as the
results of the wave conversion obtained by a rotating simulation
test bench.
II. HEAVE-BUOY MECHANICS
In this paper, a simple heave-buoy system is considered as
a wave energy converter (WEC). There are many shapes for
the buoy as a WEC: a sphere-shaped buoy is considered in
the following. Generally, buoy WECs can be displaced in any
direction when an incident wave occurs; in this formulation, the
buoy WEC is restricted to oscillate in the heave mode only, as
shown schematically in Fig. 1.
The equation of motion for a simple heaving-buoy device is
Md2
dt2z=Fe+Fr+Fb+Fv+Ff+Fu(1)
where Mis the mass of the buoy, Feis the excitation force
due to the incident wave on the buoy, Fris the radiated force,
Fbis the hydrostatic buoyancy force, Fvand Ffrepresent
unavoidable viscous and friction effects, respectively, and Fuis
an outside force that could be externally imposed. The zheave
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DELLI COLLI et al.: TUBULAR-GENERATOR DRIVE FOR WAVE ENERGY CONVERSION 1153
coordinate represents the displacement of the center of gravity
of the sphere in the frame of the calm free-water surface.
The radiated force term Frrepresents the forces acting on the
buoy, which are due to the wave that is radiated as a result of
the buoy’s oscillation. Frconsists of an additional mass of the
buoy plus a dissipative term [7].
Fr=−mr(ω)d2
dt2(z−x)−Rr(ω)d
dt(z−x).(2)
The coordinate xis the displacement of the free-water sur-
face when an incident wave occurs in the frame of the calm
free-water surface.
The hydrostatic buoyancy force Fbtakes into account the
equilibrium between the weight force and the Archimedes force
and is generally indicated as follows [7]:
Fb=−Sb(z−x)(3)
where Sbis the buoyancy stiffness, in which considering small
excursions |z−x|a, is equal to [7]
Sb=πρga2(4)
for a semisubmerged sphere of radius aand mass M=
(2/3)πρa3.In(4),ρis the density of the sea water, and gis
the acceleration of gravity. The last added term in (1) takes into
account the viscous and friction effects and can be written as
follows:
Fv+Ff=−(Rv+Rf)d
dt(z−x)=−Rl
d
dt(z−x).(5)
III. TUBULAR GENERATOR DESIGN AND VALIDATION
A. Design Basics
Tubular linear machines (TLM) comprise a stator and a trans-
lator, the former being made out of a series of cylindrical metal
discs machines in order to accommodate the coils and furnish
with the necessary hole for the sliding of the translator. The
magnets and the steel back iron are canned inside a stainless-
steel sleeve, as shown in Fig. 2.
Generally, the stator is not laminated: A transverse lamina-
tion would reduce the eddy-current losses, but it would increase
the effective air gap and the cogging force of the machine if this
is a PM one.
Analytical models of the tubular machines were provided
by Marignetti and Scarano in [8], Bianchi et al. in [9], and
Zhu et al. in [10]. All the models show that, in terms of elec-
tromagnetic thrust density, buried PM machines perform better
than their surface-mounted counterparts. A general formula for
achieving the thrust per pole Tiin a tubular buried-magnet
machine has been is given in [8]
Ti=2πRBrncin
p
k
2δ
τp+k
2
σ0
σmK
(6)
where Ris the bore internal radius, Bris the remanence of the
magnets, ncis the number of turns per slot, npis the number of
slots per pole energized at the same time, kis the ratio between
Fig. 2. Stainless-steel tube with the magnets and the iron pieces.
Fig. 3. Force-per-slot plot as a function of the air gap and the magnets’ length
knormalized in relation to the pole pitch.
the length of the magnets and the pole pitch, δis the minimum
air gap, σ0is the air-gap surface, σmis the cross-sectional area
of the magnets, and Kis a coefficient derived from the finite-
element method (FEM) analysis to take the leakage fluxes into
account.
Supposing Rto be unity, for a three-phase machine with a
trapezoidal electromotive force (EMF) and one slot per pole
and per phase, (6) can be used in order to yield the force-per-
slot plot as a function of the air gap and the magnet length
normalized in relation to the pole pitch. Fig. 3 refers to a slot
current of 1-A turn and three slots per pole. A design can be
considered as being optimal when the length of the magnets is
such that the thrust per slot yields a maximum. This happens
in general when the length of the magnets is one third of the
pole-pitch length for a one-slot-per-phase machine.
B. Actual Construction
The windings were wound on a plastic spool utilizing H
thermal class enameled copper wire with a cross section of
1mm
2. The number of turns per coil is 126. The main electrical
and geometrical data of the machine are listed in Table I.
The application of (6) to the machine above yields a total
thrust of 208 N at a rated current, which was experimentally
verified. The design of the tubular linear machines was verified
by means of the FEM simulations, which were carried out in an
axial symmetry mode, since the structure presents a rotational
symmetry around the central axis.
C. FEA
In order to compute the direct- and quadrature-axis induc-
tance of the tubular motor, several FEM simulations were
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1154 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 4, AUGUST 2006
TAB LE I
TUBULAR LINEAR GENERATOR PARAMETERS
Fig. 4. Magnetic flux density along a double pole pitch with the translator
aligned on the stator q-axis.
performed, considering different values of a current density
and in different positions of the translator. More simulations
were carried out with active magnets and currents in order
to visualize the magnetic field in the air gap when the flux
sustained by the magnet is aligned with both the q-axis and the
d-axis. Fig. 4 shows the magnetic field in the air gap with the
translator in position d. Fig. 5 shows the magnetic field with
the translator in position q.
Using an integration tool [11], first, the flux was calculated,
then the inductances on the d- and q-axes were computed. The
results are listed in Table II.
The inductance was also experimentally measured using a
methodology based on the current response to a sinusoidal
voltage on the d- and q-axes of the machine. This methodology
is described in [12] and [13]. It can be noticed that the value
Fig. 5. Magnetic flux density along a double pole pitch with the translator
aligned on the stator d-axis.
TAB LE II
DIRECT AND QUADRATURE INDUCTANCE CALCULATION
Fig. 6. Linear tubular machine prototype.
calculated by means of the FEM method and the experimentally
measured ones (see the values of Table I) are quite comparable.
Fig. 6 shows the linear tubular-machine prototype.
In all electrical machines, roughly forces grow linearly
with the volume. The weight of the translator increases at
the same rate. The machine used in this paper was designed
and built to be cooled by a natural air convection. The force
density of an air-cooled linear tubular machine is in the range
of 1.6×105N/m3(for slot-less tubular SPM machines) to
2.4×105N/m3(for slotted tubular IPM machines) [9]. The
amplitude of the force in a typical 100-kW wave buoy converter
is in the order of 100 kN [18], which results into a generator
of volume ranging from 0.4 to 0.6 m3, which means that the
generator weighs about 3 to 4 tons.
However, it must be considered that the thermal exchange in
submerged electrical devices is facilitated, hence, their power
density can be further improved.
IV. HEAVE-BUOY OPTIMAL ENERGY CONVERSION
The WEC considered is a sphere that is semisubmerged in its
equilibrium position. As soon as an incident wave occurs, the
sphere starts moving in the heave direction, and also a relative
movement between the sphere and the free-water surface of the
wave begins.
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DELLI COLLI et al.: TUBULAR-GENERATOR DRIVE FOR WAVE ENERGY CONVERSION 1155
An incident wave of the height Hs=2A and the angular
frequency ωproduces a heave displacement of the free-water
surface equal to
x(t)=Asin(ωt).(7)
Taking into account equations (2)÷(5), the dynamic (1) for a
buoy restricted to oscillating in the heave mode only becomes
(M+mr(ω)) d2
dt2z+(Rr(ω)+Rl)d
dtz+Sbz=Fe+Fu
(8)
where the excitation force includes the following terms
Fe=Sbx+(Rr(ω)+Rl)d
dtx+(M+mr(ω)) d2
dt2x. (9)
In terms of complex amplitudes, (8) may be written as
−ω2[M+mr(ω)] + iω [Rr(ω)+Rl]+Sbˆz=ˆ
Fe+ˆ
Fu
(10)
where the superscript ∧indicates the phasors of the quantities.
The introduction of the mechanical impedance [7]
˙
Zm(ω)=Rr(ω)+Rl+iω(m(ω)+M)−Sb
ω(11)
modifies the complex (9) into
˙
Zm(ω)ˆu=ˆ
Fe+ˆ
Fu(12)
where ˆuis the phasor of the speed of the oscillating buoy ( i.e.,
u(t)=d/dtz(t)).
According to the theory on the converted power energy
from a heave-buoy system [7], the maximum useful power is
obtained when the external force Fusatisfies the following
relationship:
ˆ
Fu=−˙
Z∗
m(ω)ˆu(13)
where the symbol “∗” denotes the complex-conjugate quantity.
This optimum condition is reported in the literature as a reactive
control or complex-conjugate control and was experimentally
tested for sinusoidal waves [14]. Condition (13) implies that
the optimal load impedance is equal to the complex conjugate
of the intrinsic impedance. As remarked in [7], due to the need
for involving a mechanical reactive power in order to achieve
the optimum condition, the load and control machinery, which
supplies the load force Fu, should be capable of returning some
energy to the buoy during part of the oscillation cycle.
The external force Fuwill be the electromagnetic thrust of
the linear tubular generator. The reference signal for the linear
direct drive will be represented by the optimal Fuexpressed
by (13).
V. T UBULAR GENERATOR CONTROL
A voltage-source converter (VSC) connects the TLM to a
dc source. The dynamic electrical model of the generator in
the synchronous reference frame is described by the following
equations:
vd=−R·id−dψd
dt +v.t ·π
τp
·ψq
vq=−R·iq−dψq
dt −vt·π
τp
·ψd
ψd=Ld·id−ψPM
ψq=Lq·iq
F=3
2·π
τp
·[id·iq·(Ld−Lq)−ψPM ·iq].(14)
In (14) vtrepresents the linear speed of the TLM generator,
coinciding with the speed of the buoy; τpis the pole pitch, and
ψPM is the flux due to PMs.
The previous section shows how the desired force depends
on the speed and other wave parameters. Here, it can be treated
as an exogenous reference.
Considering the speed and the PM flux to be known, the
model is a system of five linear differential equations with the
following unknowns Id,Iq,Ψd,Ψq,vd, and vq.
In order to make the system solvable, one needs to add an
additional equation, namely, the generator control algorithm. A
possible choice for this equation is based on the control of the
phase currents tracking the phase angles of the generated EMF
profile [15] and is also termed field-oriented control (FOC). In
a constant flux operation, the FOC implies that the direct axis
current be zero
iref
d=0.(15)
The thrust output by the optimal complex-conjugate control
of the wave buoy (10) represents the LTM reference force,
hence, the quadrature current becomes
iref
q=−Fu
3
2·π
τpψPM
.(16)
The adopted strategy commonly uses two proportional plus
integral (PI) regulators in the synchronous reference frame to
control the currents.
The control scheme requires the EMF as a function of the
position of the translator to be known, in this case, only the
angle of the magnet flux vector in a fixed frame; a linear encoder
was therefore used.
There are many ways to implement current regulators, and
the easiest of these is to use a hysteresis band controller.
These however generate an inconstant-switching frequency
that causes a nonoptimal thermal exploitation of the power-
switching devices. This algorithm works very well with low
EMFs, but with higher EMFs, the current controller will satu-
rate in part of the cycle, and fundamental frequency related har-
monics will appear. In this condition, the fundamental current
will be less, and its phase will lag in relation to the command
current [16].
Brooking et al. [17] used an approach based on the propor-
tional regulators in a fixed frame. PI controllers can be used
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1156 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 4, AUGUST 2006
Fig. 7. Control algorithm block scheme.
Fig. 8. Test bench A: tubular generator connected to an external PMSM drive.
to track a current command, and three similar controllers can
be implemented, one for each phase. This is a simple control
method, but there are a number of problems: Due to the limited
bandwidth of the control system, the actual current will have
a phase lag and magnitude error, which will increase with
frequency. The ripple affecting the generated voltage command
may cause undue zero crossings in the modulation [16]. Instead,
two PI regulators are used in the synchronous reference frame
to control the currents.
The control block scheme using the EMF phase reconstruc-
tion is reported in Fig. 7.
VI. TEST FACILITY
A test facility comprising two separate test benches was
set up in the “Giovanni D’Angelo” Laboratory of Industrial
Electronics at Cassino University:
1) test bench A based on the tubular-generator prototype
(Subsection A);
2) test bench B based on a rotating heave-buoy simulator
(Subsection B).
A. Test Bench A Based on the Tubular-Generator Prototype
Test bench A is designed to test the generating capabilities of
the tubular generator of Section III with the FOC of Section V
regardless of the buoy mechanics. Test bench A is illustrated
in Fig. 8.
B. Test Bench B Based on a Rotating Heave-Buoy Simulator
Test bench B intends to test the control laws of Sections IV
and V, taking into account the buoy mechanics of Section II
and some of the actual characteristics of the drive. This bench
is based on a rotating unit composed of a PM synchronous
generator replacing the tubular generator, and a PM synchro-
nous motor (PMSM) drive controlled in order to track in real
Fig. 9. Test bench B: Block scheme.
time the dynamics of a WEC buoy. The unit operates with a
tangential speed 2.6 times the linear speed of such a WEC
and with a torque 1/20 000 times the force of the WEC. The
test facility combines the real-time simulation, hardware-in-
the-loop simulation, and actual motor drive to achieve the
computation and actuation of the wheel dynamic in real time.
Test bench B consists of the following items as shown also in
the block scheme of Fig. 9.
Items under test (on the left of Fig. 9):
1) control hardware;
2) dc/ac power converter;
3) PM generator.
Buoy real-time simulator (on the right of Fig. 9):
1) control hardware for the real-time computation of the
buoy dynamic;
2) power inverter;
3) PMSM.
This facility can impose, in reduced force scale and slightly
augmented speed scale, arbitrary dynamic buoy parameters and
can modify the “sea conditions” when running. The simulator
imposes on the PMSM the speed foreseen by the buoy equa-
tions. Because of its torque and bandwidth performances, the
simulator is capable of imposing the axle speed in any type of
operating conditions considered.
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