Linear generator systems for wave energy conversion
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ABSTRACT: In this paper a linear switched reluctance gener-ator used in a direct drive wave energy converter is considered for modeling and model-based opti-mal control. Linear switched reluctance generators (LSRGs) are highly nonlinear electromechanical converters which must be adequately controlled in order to achieve a satisfactory energy conversion efficiency. In order to pave the way for a systematic design of a control strategy that directly addresses the wave energy converter efficiency maximization and respects system constraints, the discrete-time piecewise affine (DTPWA) hybrid model form is proposed to model the considered LSRG in the configuration with a power converter. The proce-dure of obtaining a hybrid DTPWA model itself is also pursued in a systematic way, by employing clustering-based piecewise affine fitting of nonlinear characteristics of the considered LSRG, which are in turn computed from the generator geometry using finite element analysis methods. The hybrid DTPWA model is validated on the starting nonlinear LSRG model.
Linear generator systems for wave energy conversion
H. Polinder1, M.A. Mueller2, M. Scuotto1 and M. Goden de Sousa Prado1,3
1 Electrical Power Processing Group, EEMCS Faculty, Delft University of Technology
Mekelweg 4 , 2628 CD Delft, The Netherlands
E-mail: email@example.com, firstname.lastname@example.org
2 Institute for Energy Systems, The University of Edinburgh
Kings Buildings, Mayfield Road, Edinburgh, EH9 3JL, United Kingdom
3 Teamwork Technology
De Weel 20, 1736 KB Zijdewind, The Netherlands
The objective of this paper is to review linear generator
systems for wave energy conversion and the research issues
related to this. The paper starts with a short review of wave
energy conversion, indicating that the different wave
energy conversion systems that have been presented in
literature have very different generator systems. Next, a
few state-of-the-art linear generator systems are discussed,
such as the linear generator of the Archimedes Wave Swing
(AWS) and the linear generator developed in Uppsala.
Subsequently, some remaining problems and possible
solutions that need further research are listed. The paper
concludes with some sensible directions for further
research, such as investigating an increase of the speed of
the linear motion of the wave energy converter,
investigating other generator types with higher force
densities and possibly better efficiencies (for example,
transverse flux permanent magnet machines) and
investigating generator constructions that result in cheaper
Keywords: Linear generator systems, permanent magnet
generators, ocean wave energy.
The objective of this paper is to give a short review of
linear generator systems for wave energy conversion and
the research issues related to this. It starts with a review of
linear generators and wave energy conversion. Next, state-
of-the-art linear generator systems are discussed.
Subsequently, some remaining problems and possible
solutions that need further research are listed. The paper
closes with a short summary of some issues that need
© Proceedings of the 7th European Wave and Tidal
Energy Conference, Porto, Portugal, 2007
2 Linear generators in wave energy
2.1 Wave energy conversion systems
There are different ways of classifying wave energy
conversion systems. One possible way of classification is
according to the operating principle. Four important types
are the following .
Oscillating water columns such as the Osprey.
Overtopping devices such as the Wave Dragon.
Hinged contour devices such as Pelamis.
Buoyant moored devices such as the Archimedes
Wave Swing (AWS) [2-7].
These different operating principles also require
different power take off systems:
Oscillating water columns mostly have air turbines
that drive rotating generators.
Overtopping devices mostly have hydro turbines that
drive rotating generators.
Hinged contour devices often use hydraulic power
take off systems.
Buoyant moored devices often have linear generator
Linear generator systems are only useful in applications
where the motion is linear; when there is rotating motion it
does not make sense to convert this to linear motion.
In hinged contour devices such as Pelamis, there is also
a linear power take off system. However, in this case the
forces are extremely large, while the speed of the motion
remains very low (in the order of 0.1 m/s). For these low
speeds and high forces, hydraulic power take off systems
are probably more suitable than linear generator systems.
In buoyant moored devices with a linear motion and
speeds in the order of 1 m/s, there are different possible
power-take-off systems, such as the following
gearboxes that convert the low-speed linear motion
into rotating motion of a higher speed;
In this case linear generators are often preferred because
they are expected to be more efficient and more robust than
2.2 Linear generators
Linear generators are rarely used. When converting a
form of mechanical energy into electrical energy, mostly
rotating motion is used. Generators in conventional power
stations (coal, gas oil, nuclear), in hydro power stations, in
wind turbines, in vehicles all use rotating generators.
Linear motors or actuators are used in for example
transportation systems (including maglev trains), robotic
systems, positioning stages, and so on. Mostly, these
systems have a low power level. However, there are a few
applications with power levels comparable to the power
levels of wave energy converters, such as:
aircraft launching systems for future aircraft carriers
roller coasters driven by a linear machine [10,11].
When these linear systems have to break, the machine is
also operated in generator mode. However, in this case the
objective is not to convert energy from a mechanical from
to an electrical form in an efficient way, the objective is
just to slow down the motion or to position the moving
Linear generators in wave energy converters are
characterized by a high force (depending on the size of the
wave energy converter) and a low speed. The main other
application of generators with a high force and a low speed
is in direct-drive wind turbines. There are many
correlations between the problems in direct-drive wave
energy conversion and
conversion. However, the irregular motion in wave energy
conversion makes direct-drive wave energy conversion
more difficult than direct-drive wind energy conversion.
2.3 Requirements and characteristics
The requirements for linear generators applied in wave
energy conversion systems are
high peak force,
irregular motion, and
Other characteristics of linear generator systems in wave
energy conversion systems are the following.
There is a high attractive force between translator and
stator. This again complicates the mechanical design
and the bearing design.
The air gap between stator and rotor is mostly
relatively large. It is complicated to build a
mechanical construction for a generator with a small
air gap because of manufacturing tolerances, the
limited stiffness of the complete construction, the
large attractive forces between stator and translator,
thermal expansion, and so on.
Because of the irregular motion of continuously
varying speed, the grid connection of the wave energy
converter always has to be done using a power
electronic converter that connects the voltage of the
wave energy converter with a varying frequency and
amplitude to the grid with a fixed frequency and
direct-drive wind energy
3 State-of-the-art linear generators
There are different conventional generator types that
could be used in wave energy conversion systems, such as
linear induction machines
linear synchronous machines with electrical excitation
linear switched reluctance machines
linear permanent-magnet synchronous machines.
In literature, these generator systems have been
compared, mainly for direct-drive wind turbines, but also
for wave energy conversion . The conclusion is that
permanent-magnet synchronous machines are the most
suitable generator type for wave energy conversion.
The generator system for the AWS could be seen as a
state-of-the-art generator for a wave energy converter. It is
a permanent magnet generator with surface mounted
magnets [2-7]. It has a three-phase full pitch winding with
one slot per pole per phase. It is double-sided to balance the
attractive forces and balance the bearing loads. It is
illustrated in figures 1 and 2.
Figure 1: Sketch of a cross-section of the linear
permanent-magnet generator of the AWS.
Figure 2: Photograph of the linear permanent-magnet
generator in the AWS.
The linear permanent magnet generator developed in
Uppsala is of a comparable type [12-16].
Some very rough numbers that could be used for first
approximations of size, weight, cost and losses of
generators of this type are given in table 1. It has to be
stressed that these numbers are very rough, that these
numbers only consider magnetically active material
(copper, iron laminations, magnets and back-iron) and that
the indication of the cost is not valid for prototypes, but
could be a rough indication for series-production.
The maximum allowable force density or shear stress
depends on the cooling of the machine. At an RMS value of
the force density or shear stress of 40 kN/m2 the losses are
in the order of 6 kN/m2. It is very hard to dissipate this with
natural air cooling. With a water cooling system, higher
force densities are possible.
In order to overcome the high attractive force produced
by Maxwell stress, Mueller and Baker [17-21] have
investigated air-cored machines, in which, the coils are
suspended in a non-magnetic material. The electromagnetic
performance is not as good as a conventional iron-cored
machine similar to those used in the AWS or by Uppsala,
but the mechanical and structural design is simpler.
However, this generator type has not yet been applied in
wave energy converters.
4 Problems and further research
conversion have some disadvantages:
their efficiency is physically limited
they are huge and expensive
the bearing load are large and the bearings are not
More research is necessary to solve these problems.
The next sections will discuss these problems and
discuss further research to solve these problems.
4.1 Limited efficiency
For machines as depicted in figure 1, the voltage
induced per unit of length of a conductor in a slot of a
permanent magnet generator can be calculated as
B is the air gap flux density and
v is the relative speed between stator and translator.
It is important to realize that this expression is not
always valid, and can not be used in for example transverse
flux machines. However, in most conventional machines
(among which the permanent magnet machine of figure 1)
it can be used. For air gap flux densities with an amplitude
of 1 T and for a speed of 1 m/s, this results in an induced
voltage in the conductor of 1 V/m. At the same time, there
is also a resistive voltage drop in the conductor if the
generator is loaded. This resistive voltage drop per unit of
length of the conductor can be calculated as
The state-of-the-art linear generators for wave energy
ρCu is the resistivity of the conductor material (mostly
J is the current density in the conductor.
For values of the current density in the order of 5
A/mm2, this results in a resistive voltage drop in the order
of 0.1 V/m. This resistive voltage drop is not only present
in the slots, but also in the end windings and cable
Maximum RMS value of the force density or
shear stress (kN/m2)
Loss density at an RMS value of the shear stress
of 40 kN/m2 (kW/m2)
Weight of active material (kg/m2)
Cost of active material in series production
Table 1: Rough numbers characterising a linear
generator for a wave energy converter
Figure 3: Average generator efficiency as a function
of wave height and wave period.
Figure 4: Average generator system efficiency
(including cable and power electronic converter) as a
function of wave height and wave period.
From these equations, a few conclusions can be
For low speeds, the efficiency is physically limited.
For speeds in the order of 0.1 m/s, the use of this type
of linear generators is questionable.
By increasing the speed of the motion, the efficiency
also increases. This is true as long as the speed is so
low that iron losses are negligible. At high speeds iron
losses may become dominant.
By decreasing the current density, the efficiency can
be increased. However, decreasing the current density
implies that the generator has to become larger and
more expensive for the same force.
These two equations do not tell the complete story,
mainly because iron losses are neglected. However, they
give an important trend.
Sensible directions for further research are the
It should be investigated if the speed of the linear
motion of the wave energy converter can be increased.
It could be investigated if there are materials with a
lower resistivity than copper. However, copper already
is a very good material compared to others. Only
superconducting materials can do much better and it
could be investigated if superconducting materials are
a realistic option for wave energy conversion.
It makes sense to investigate other generator types
where this direct relation between speed and
efficiency is eliminated, such as transverse flux
permanent magnet machines [3,17,19,20].
4.2 Electromagnetic Forces
There are two main electromagnetic forces in an
electrical machine as shown in Figure 4:
The torque or thrust producing force, FS, acting
tangential to the rotor surface.
The normal force, FM, attracting the two iron surfaces.
These forces are given by equations
B (T) represents the air gap flux density and
K (A/m) is the electric loading.
In linear machines as the machine of figure 1, the shear
force density (3) is limited. The amplitude of the air gap
flux density is limited to around 1 T because of saturation.
The amplitude of the current loading is limited because
current loading produces heat, and the heat dissipation is
limited. For machines that produce a constant force and
have air cooling, the resulting force density is limited to
about 25-50 kN/m2. With a good water cooling system, the
force density can be increased further. The force of a linear
generator in a wave energy converter is continuously
varying. Therefore, higher peak force densities may be
flux density, for a typical electric loading of 50kA/m.
The fact that the force density is limited to a certain
value implies that the active surface area of the generator is
proportional to the force, which has serious implications in
terms of the machine’s physical size and mass. If the
amplitude of the air gap flux density were 1T, then the
normal stress is about 200 kN/m2 while the shear stress is
in the order of 40 kN/m2. For a 100kW direct drive
machine running at 1m/s, the tangential force required
would be 100kN, which would require an air gap surface
area in the region of 2.5 m2. Hence the normal magnetic
attraction force would be of the order of 500 kN, which the
machine structure and bearing system would have to
overcome in order to maintain the air gap. In the AWS, the
maximum shear force is 1 MN, but to be able to produce
this force, an active surface area in the order of 20 m2 is
necessary. If amplitude of the flux density is 1T, then the
normal stress is 200kN/m2 giving an attraction force of 4
MN, which the bearings and support structure would have
All iron cored machines, that is those in which one iron
surface moves with respect to another iron surface, will
suffer from this large magnetic attraction force problem,
which as can be seen from the simple example above
becomes significant for low speed high force machines.
Figure 5 shows how these two force densities vary with
Figure 4: Electromagnetic forces in an electrical
Stress vs Flux Density
σ σ (kN/m2)
Normal stressShear stress
Figure 5: Variation in Normal and Shear stress.
4.3 Linear generators are expensive
The power available from linear motion is given by
F is the force, and
v is the speed.
The size of the generator is mainly determined by the
force is has to make. In wave energy, the speeds are mostly
rather low. If an amount of power has to be created at a low
speed, the force has to be high. This generally leads to large
and expensive machines.
There are a number of ways to deal with this problem.
If the power level would be kept the same while the
speed could be increased, the force of the generator
would decrease and therefore, the cost would also
decrease. Again it appears to be interesting to
investigate if the speed of the linear motion of the
linear generator could be increased.
Another interesting research issue is the question if
there are machine types where this limited force
density does not play such a dominant role. Examples
of these machines are variable reluctance generators,
transverse flux permanent magnet machines and
Vernier hybrid machines [3,17,19,20]. In literature,
these machines are known for their high force
densities, but they also have their drawbacks, such as a
complicated construction, a low power factor and
complicated iron losses. More research is necessary to
find out if these machines are suitable or not.
Another interesting research issue is the question if it
is possible to reduce the cost of the generator by using
cheaper windings. In the machine of figure 1,
distributed windings are used. In the machine of figure
6, concentrated coils are used instead . These
concentrated coils can be produced at a much lower
cost, because a much larger part of the winding
process can be done with machines instead of
manually. Also the end windings can be much shorter.
In machines with concentrated coils, different
combinations of numbers of poles and numbers of
slots are possible. The main drawback of machines
with concentrated windings is the increase in eddy-
current losses in the magnets and the back-iron. It
needs to be investigated further if this increase is
acceptable, and for which combination of numbers of
poles and numbers of slots this is acceptable.
Clever ways of constructing the generator, for
example by making it double-sided or cylindrical
could help to reduce the cost.
4.4 Heavily loaded bearings and maintenance
There are three main types of bearings and each type
may be assembled to allow either rotational or linear
motion between two elements (all but one degrees of
freedom are usually blocked):
Mechanical bearings (ball bearings, roller bearings,
etc.) represent the most common solution in a large
variety of applications, being rugged, reliable and cost
effective. Much of bearings design is about failure
analysis. Abrasion, fatigue and pressure-induced
weldings limit the lifetime and the load capacity of the
bearings. In more demanding applications, only
maintenance can keep them operating properly. As the
level of performance increases, in terms of precision,
speed, lifetime and load
technology can offer alternatives such as fluid or
Fluid bearings rely on a thin layer of liquid or gas to
support the load, separate and avoid direct contact
between the moving parts. According to the operating
principle and the fluid used, they may be broadly
classifies as hydrodynamic (which require continuous
motion), hydrostatic (require a pump) or gas bearings.
If compared with common bearings, fluid bearings are
highly versatile and almost maintenance-free. They
can be used in applications in which requirements for
load, speed or precision are too severe for ordinary
bearings. Besides seals and -if present- pumps, a
source of losses is fluid viscosity. Overall behaviour
in terms of losses may be far better than mechanical
bearings and, if the level of performance requested is
significantly high, the cost can be lower.
Magnetic bearings are bearings which support a load
using magnetic levitation. There is no contact between
the moving parts and thus friction is absent.
According to Earnshaw's theorem, permanent magnets
alone cannot provide stable levitation. Electromagnets
with continuous power input and active control
system are required. Safety bearings should be added
to avoid system damage in the case of either control or
power supply failure.
In electromechanic applications, linear bearings are used
primarily with linear motors, where some load has to be
moved along a prescribed straight path with a certain
accuracy. In other words, loads need to only translate in
one direction, and possibly move back to starting position
with high repeatability. The robotic uses of linear bearings
have opened up a promising market for the devices
operating with low thrust loads and high speed/precision.
On the other hand, roller bearings for overhead cranes
represent an example of an application in which accuracy is
less important than loading capability. In all these
applications, adopted bearings are mostly mechanical and
thus require either ordinary maintenance or replacement.
In a conventional linear machine, the attractive magnetic
forces between stator and translator are usually much
higher than the propulsive force. Therefore, as the size and
the power level of the machine increase, it may not be easy
to design bearings that can deal with the resulting forces
without regular maintenance.
Figure 6: Linear permanent-magnet machine with
When dealing with small off-shore wave energy devices
implementing linear generators as power take-off system,
such as floating power buoys rated up to a few tens of kW,
maintenance is not expected to represent a critical issue.
Things change considerably with a 2MW submerged power
plant of the class of the AWS. The first full-size prototype
had a weight of 7000 tons while the weight of the floater
alone was about 400 tons. Most of the weight was due to
the pontoon and ballasts, designed to transport and keep the
device safe in place. Even assuming a future version of the
AWS free of ballasts and pontoon, its huge dimensions and
deployment in rough ocean sites would encourage neither
frequent ashore recovery nor on-site long operations for
The bearing design is crucial in maintaining a physical
air gap between stationary an moving parts. The
coefficients of friction for various plain bearings are listed
Plain bearing, Teflon
The power lost due to friction is given by
v is the velocity,
F is the load force,
f is the friction coefficient.
For the same load force, F, hydrostatic bearings offer the
best performance in terms of power lost. With such a
bearing the working fluid could be in the air gap. Seawater
would be the obvious fluid to use but there are then design
issues to be overcome such as corrosion and the operation
of windings in water. The issue of corrosion was discussed
in  with respect to permanent magnets. Figures 7 and 8
show the effect of seawater on magnets using currently
available magnet coatings.
The attractive magnetic forces between stator and
rotor are rather high, typically 200 kN/m2, resulting in high
values of F in (6). As illustrated above there are challenges
to overcome to design bearings that can deal with these
forces without maintenance. However, there are a few
ways of reducing bearing loads.
If the generator is constructed double-sided (as has
been done in the AWS) this results in a significant
reduction in bearing loads. However, because of
manufacturing inaccuracies of the huge construction,
the bearing loads remained considerable. Irregular
bearing surface and heavy loads may quickly cause
failure of roller bearings.
In a double-sided machine, by means of a back-to-
back voltage source inverter, the phases of stator
currents could be controlled in such a way that the
attractive forces between stator and rotor are balanced
with limited (below 5%) additional copper losses .
Because of the limited speed of the machine, attractive
and propulsive force are practically independent which
means that they can be controlled without additional
sets of coils and without affecting the process of wave
energy absorption . It is however not clear how to
0.12 – 0.14
10-2 – 10-3
10-2 – 10-3
10-3 – 10-6
evaluate exactly the attractive forces during operation,
since they depend upon position and structure
deformation due to stresses and temperature
As a next step, the bearings could be made completely
magnetic in all degrees of freedom. This would result
in a bearing system that is in principle maintenance-
free. However, the complexity of the system and its
control (considering that air gap length is about 5 mm)
would increase significantly. Instead of double-sided
machines, multi-sided or even cylindrical machines
could be considered, divided into a number of
independent sub-machines to provide proper control.
Extensive use of power electronics implies that the
electrical losses in the system would increase
significantly. Also copper losses may be higher. It
could be investigated whether this is acceptable or not.
Elimination of the large attraction force will
significantly reduce the bearing load due to
electromagnetic forces. This can be achieved using
air-cored permanent magnet machines as discussed in
Using a single set of magnetic bearings for the generator
and the floater does not seem realistic, because of the small
air gap and the large hydrodynamic forces acting on the
floater, unless superconductivity is considered. The floater
may instead use hydrostatic bearings. (This alternative is
also valid for the generator.) From the point of view of
maintenance, the most critical element would be
represented by the pump which is needed to operate this
type of fluid bearings. (Hydrodynamic bearings are
probably not viable because the moving parts stop twice
per cycle and the overall speed is limited.) The pump may
be a removable module located on the topmost part of the
floater, a few meters underwater, and thus it could be easily
Figure 7: Standard coating for magnets (A) New, (B)
after 6 weeks and (C) after 2 years submersion in
Figure 8: Alternative coatings for magnets (i) as
new, (ii) after years submerged in sea water
5 Summary of interesting research work
Sensible directions for further work are the following.
It makes sense to investigate if the speed of the linear
motion of the wave energy converter be increased.
It makes sense to investigate other generator types
with higher force densities and possibly better
efficiencies, such as transverse flux permanent magnet
As well as high force density machines it makes sense
to further investigate air-cored machines in terms of
their potential for a highly integrated electrical-
mechanical-structural design solution.
It makes sense to investigate generator constructions
that result in a cheaper generator, for example using a
cylindrical generator or a generator with concentrated
coils instead of distributed coils.
This work was supported in part by a Marie Curie Early
Stage Research Training Fellowship of the European
Community’s Sixth Framework Programme under contract
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