Development and Wave Tank Demonstration of a Fully Controlled Permanent Magnet Drive for a Heaving Wave Energy Converter

Abstract

One option for converting the energy in sea waves into renewable electricity is the development of floating wave energy converters coupled to electrical generators. For this to work, bespoke slow-speed electrical machines coupled to bidirectional power smoothing power electronic converters are required. This paper reports on the successful design and wave tank validation of an electric machine, power converter and fully controlled direct drive power take-off system coupled to two small scale heaving wave energy converters. The design, development and demonstration of linear generators and power converters is presented including some simulated and laboratory results. Demonstration of wave energy converters with pure electric drives, fully automated control, bidirectional power flow and active force management is almost unique and essential for future wave energy development. The results presented prove that direct-drive power take-off for wave energy devices is technically possible and can be used to implement an automated control system with bidirectional power flow in both resonant and non-resonant wave energy systems.

1. Introduction

Due to its large resource and high energy density, ocean wave energy is widely considered a potential source of renewable electricity generation. Many proposed devices produce slow-speed reciprocating motion, which poses a challenge to conventional electrical machines. A direct-drive power train would consist of a reciprocating electrical machine and power converter with no gearbox or other mechanical linkage. This paper gives details of the technology development of the electrical machine topology, power converter design and control implementation of a direct-drive power train for buoyant wave energy converters.

Numerous wave energy devices have been proposed over the preceding decades, with varying technical [1] and commercial success [2]. One of the many technical challenges relates to the discrepancy between the natural motion of ocean waves (slow speed and reciprocating) compared to the conventional motion of electrical machines (high speed, unidirectional, rotary). Wave energy devices, therefore, either need some form of mechanical linkage between the moving part and the electrical machine, or developers need to move away from using conventional electrical machines.

Only a small number of groups have proposed the development of bespoke linear generators for this application, such as [3,4,5,6], offering a direct-drive power take-off with all the implied mechanical simplifications. This approach poses challenges in terms of designing the electrical machine and associated power electronic converter, many of which are discussed in this paper.

Many more groups have proposed ‘non-direct drive’ Wave Energy Converters (WECs) and numerous examples have been demonstrated at sea, e.g., Corpower [7], Pelamis [8], Aquamarine, Ceto [9], Penguin [10]. Wave device developers tend to focus their power take-off engineering design effort on the mechanical speed enhancer, and use conventional electrical generators and power converters. Corpower has developed a linear to rotary mechanical gearbox interface; Carnegie, like Pelamis [11] and Aquamarine utilise hydraulics. For both Pelamis and Aquamarine, the hydraulic power take-off proved the limiting factor in the WEC design, and in Aquamarine’s case the power take-off failed completely. Commercially, it is widely accepted that the mechanical speed enhancer is the weak link between the wave device and the power take-off. The electrical generator used in speed enhanced power trains is either a high-speed induction machine or a field-wound synchronous machine, e.g., [8,12], whereas, for direct-drive systems, permanent magnet linear and rotary generators have been used.

Academic groups who research wave energy power take-off systems tend to focus on the development of novel low-speed direct-drive electrical generator topologies with inherent magnetic gearing, such as [13,14], or on optimising the electrical machine design [3]. A number of researchers have looked at linear machine topologies specifically for wave energy, most relying on rare earth permanent magnets (PMs) such as transverse flux [15], flux reversal [6], switched reluctance [16], tubular permanent magnet [17] or conventional ironed/ironless synchronous generators [18].

Of the numerous paper design studies that have been undertaken by these international universities in direct-drive wave energy, comparatively few include large scale, wave tank or at sea results except [19,20,21]. Examples of direct-drive systems implementing an automated control algorithm are fewer still. In this paper, we couple a fully direct-drive power take-off to two wave energy converter prototypes, which between them cover the characteristics of a wide range of wave energy converters.

Much work has been done on implementing a controller to maximise energy yield from a hydrodynamic perspective, see for example [22]. From the electrical development side, it is usually assumed that damping is controlled and fixed for a given wave state e.g., [23]. In this paper we implement an automated control system based on variable damping and maximum power point tracking.

This paper therefore develops and demonstrates an electric power train specifically for direct-drive wave energy converters, consisting of slow-speed electrical machines, power converters and a control algorithm. A range of bespoke linear flux reversal type machines is discussed, demonstrated and compared in terms of operating power factor and efficiency. Voltage source and current source converters are introduced and compared as a means of power smoothing and implementing control of the wave energy device. Results from tank testing of a representative scale model of two wave energy devices coupled to the linear drive developed are then used to prove that the power train is capable of real-time automated control and power conversion in this application. Although the scale is small, this work represents a significant addition to the development and demonstration of direct-drive wave energy and builds on initial system development [24] and control results [25] previously presented by the authors.

2. Wave Energy

Figure 1 shows two conceptually simple direct-drive wave energy devices, consisting of a submerged generator coupled to a surface piercing or fully submerged buoy. Example devices of the surface piercing type have been demonstrated by Oregon [26] and Uppsala universities [19,20], whereas the fully submerged version is representative of Carnegie’s CETO device [9], for example. The design challenges associated with this device are shared by other similar concepts, of which there are many examples [7,9,21,27].

2.1. Model of the Wave Energy Converter

A common approach to simulating a wave energy device of the type shown in Figure 1 is to use simplifying assumptions relating to the sea-state and device reaction: sinusoidal monochromatic waves in deep water and limiting the degrees of motion to one axis only, i.e., heave in this example. The device dimensions and motion are relatively small compared to the wavelength and wave height. The mass or the force applied to the linear machine’s translator is sufficient to keep the tether between the buoy and translator taut at all times, such that it can be assumed the buoy and generator are rigidly coupled together. Under these conditions the equation of motion can be represented by a linear differential equation:

$$m\ddot{z}=A\left(\ddot{y}-\ddot{z}\right)+B\left(\dot{y}-\dot{z}\right)+C\left(y-z\right)$$

(1)

where m is the buoy mass, A, B and C are the coefficients expressing the added mass, damping and hydrostatic stiffness respectively, y is the vertical motion of the water surface and z is the vertical motion of the buoy.

Optimal reactive control of an oscillating device is achieved by applying the complex conjugate, or impedance matching, controlled force through the linear translator [27,28]. In essence, the control strategy is to tune the resonant frequency of the system to match the frequency of the incoming waves by means of the controlled force exerted on the buoy. There are significant limitations with the practical implementation of this analysis, especially when the device is operating near resonance in large or confused sea states when the equations of motion become highly non-linear, but it remains a good starting point for developing the controller.

The natural frequency for a floating body is given by:

$${\omega }_0=\sqrt{\frac{C}{m+A\left({\omega }_0\right)}}$$

(2)

where A(ω₀) is the added mass at the resonant frequency. By inspection, we can see that altering the mass, hydrostatic stiffness or added mass all affect the resonant frequency, ω₀, of the system.

Optimal control values for the power take-off can be deduced for a range of wave frequencies [28]:

$$B_{ePTO}\left(\omega \right)=B\left(\omega \right), K_{ePTO}\left(\omega \right)={\omega }^2\left(m+A\left(\omega \right)\right)-C$$

(3)

where B_ePTO (ω) is the damping coefficient and K_ePTO (ω) is the stiffness coefficient of the power train. Assuming linear behaviour, where the translator force is proportional to the instantaneous position and velocity of the buoy, the applied load impedance, Z_ePTO (ω), is:

$${\widehat{Z}}_{ePTO}\left(\omega \right)=B\left(\omega \right)-j\frac{K_{ePTO}\left(\omega \right)}{\omega }$$

(4)

In some devices, the value of K_ePTO (ω) < 0 for certain wave frequencies, the implication being that negative stiffness is required to tune the device over the full frequency range of incoming waves, i.e., reactive power flow must be provided. This requires a converter capable of four-quadrant operation with energy storage for the mechanical reactive power. See [27] for a discussion of this and more details on modelling of this type of wave energy device in general.

2.2. Electric Power Take-Off Specification

A challenge for the electric power take-off designer is to rate the control system appropriately. For reactive power control of a wave energy converter (WEC), the converter must operate in all four quadrants and incorporate significant energy storage. The instantaneous force applied by the generator on the WEC is a combination of the time-varying real and reactive mechanical vectors derived from (4) with a fundamental frequency << 1 Hz. The relatively short time constants of the electrical power conversion system switches compared with those of the WEC indicate that the converter kVA requirements are defined by:

$${\mathrm{kVA}}_{converter}=\frac{\widehat{\sqrt{P{m}_{real}{}^2+P{m}_{reactive}{}^2}}}{{\eta }_g\ast cos{\varphi }_g}$$

(5)

where Pm_real and Pm_reactive are the combinations of real and reactive components resulting in maximum mechanical power demand to or from the WEC, η_g is the generator efficiency and cosϕ_g is the generator power factor. The maximum generator voltage, frequency and current are governed by the peak force and velocity applied to the translator and are WEC specific and, of course, the specific generator design.

A modular approach has been proposed whereby multiple generator sections are mechanically coupled and connected to multiple power train modules. The advantage of this approach is improved fault tolerance. A single module of the electrical power conversion system consists of a generator interface converter, an Energy Storage System (ESS) with its own DC-DC converter and a grid interface converter. Ideally, the ESS is dimensioned to absorb and supply the desired mechanical reactive power and also to level the peaks in the real power flow from the WEC.

Optimising the dimensions and location of the ESS in a wave farm is the subject of some research interest, particularly in relation to reducing flicker and cable losses at the grid interface [29]. In this application, however, we are interested in demonstrating an all-electric power train capable of locally providing reactive control of the WEC, hence the local ESS. Depending on the accepted design criteria, it is possible that the fluctuation in the mechanical reactive energy can significantly exceed the fluctuation in the real energy flows throughout a wave cycle with optimal reactive control [27].

3. Generator Topologies

3.1. Alternative Topologies

A brief comparison of the key features of alternative electrical machine topologies which are likely to be suitable for direct-drive wave energy is given in Figure 2. Relative force density has been estimated based on experience assuming low current density, i.e., for systems with no active cooling.

As the predicted motion of wave energy converters is reciprocating with a low peak velocity and likely to have a large peak-to-peak amplitude, flux reversal type permanent magnet machines, where all the active components are on the stator, have been the main focus of this work. The particular topology selected is known as a Vernier Hybrid Machine (VHM)—where the term hybrid relates to its mix between a flux reversal and a vernier machine. Figure 2 showed this to be towards the higher end of achievable force density machines for machines with no specific cooling. As the magnets and conductors are both mounted on the stator, the translator can be formed of a purely iron structure—advantageous for machines with a large translator overhang. Flux reversal machines, which share this property, tend to have a lower force density. Linear switched reluctance machines share this translator configuration and superconducting versions can, in theory, be developed to deliver previously undemonstrated force density, e.g., [30,31], but only assuming this technology matures.

3.2. The Vernier Hybrid Machine

The vernier hybrid machine (VHM) [32] is a member of the variable reluctance PM machine family and exhibits high force density due to the reluctance variation and flux reversal characteristics of the slotted translator. The translator consists of approximately equal slots and teeth fabricated from laminated steel, while multiple magnet poles and armature coils are mounted on the stators. Figure 3 shows a double sided three-phase linear VHM with six magnets mounted on each stator face, as developed in [33]. Alignment and un-alignment between stator tooth mounted magnet poles and translator teeth produce maximum and zero flux linkage respectively. Moving the teeth to adjacent magnets produces a reverse flux flow around the machine. Physically displacing the translator by a short distance, the coils see a large change in magnetic flux with a correspondingly high back emf and force production. This phenomenon is called magnetic gearing which is a non-contact method that enables direct-drive machines to reach higher force density without the use of a mechanical gearbox.

A design and performance analysis of a linear version of a VHM was presented in [34]. A fixed current density was applied to the coils and the geometry of the teeth and magnets was altered to achieve a maximum peak force. This is a sensible design strategy for minimising the use of the rare earth material, but the approach was found to lead to machines with an operating power factor of less than 0.2—leading to the need for a drastically overrated power converter.

Figure 3 shows surface-mounted magnet versions of this machine, but other configurations are possible. A number of alternative topologies have been investigated by the authors [25,35,36,37], driven primarily by the desire for a better power factor and a reduced mass of rare earth magnets. During the design studies, it was assumed that the VHM was connected to a fully controlled power converter to ensure that the current phase in the machine was controlled to be fully in the q-axis by controlling I as shown in Figure 4, a reasonable assertion as saliency in these machines was found to be minimal.

For a fixed terminal voltage (V), the power factor (cosϕ) can be improved by flux concentration or reluctance minimisation of the magnetic circuit seen by the magnets, thus increasing the back emf. Alternatively, reducing the average inductance (X) or reducing the electric loading (I) will have the same effect. Several topologies are described in the following section, including cylindrical versions of the Figure 3 machine in the search for a higher performance machine which would integrate well into a wave energy device.

3.3. Stator Pole Topologies

The three concepts of flux concentration, consequent poles and Halbach arrays, which are commonly applied to other PM machines, have been applied to the linear VHM and are discussed below. A selection of photographs and figures of built and investigated pole faces are shown in Figure 5. The basic premise of this body of work was to improve the effective use of the PM material. For a fixed peak force output between designs, the improved performance was used to reduce the electric loading and hence improve the operating power factor and efficiency.

3.3.1. Consequent Poles

An improvement in magnet utilisation can often be achieved by switching to a consequent pole arrangement. Alternate magnets are replaced with soft magnetic poles (iron teeth) as shown in Figure 5a. This can result in a reduction in leakage flux between adjacent poles, discussed mathematically for rotary machines in [38] and demonstrated via simulation for linear machines in [36].

Detailed design, such as tapering the consequent pole and reducing the magnet pitch, has been used to further improve the performance [35]. A flux plot of a three-phase consequent pole VHM is shown in Figure 6. The left-hand phase is fully aligned (+1 unit flux linkage) and the other two phases are 120 and 240 degrees from alignment (−0.5 flux linking each).

3.3.2. Flux Concentration

Individual magnets can be split into two components in a V-shape configuration encompassing a soft magnetic triangular pole piece. The flux density at the translator surface of the pole piece can be concentrated to be higher than the remnant flux density, hence the interchangeable names of flux concentrated or V-shape. This is regularly used in automotive applications, for example. In machines with a high circumferential force, the pole piece must be held in by a rib which presents a leakage path. Alternatively, in machines with lower forces, pole pieces can be glued in place. Figure 5b shows the concept applied to the linear VHM and the results are given in [36].

3.3.3. Flux Concentration with Consequent Poles

Figure 5c is an attempt to capture the advantages of both consequent poles and flux concentration by the use of a stator with both concepts and results were presented in [36].

3.3.4. Halbach Array with Consequent Poles

A Halbach Array is where magnets of different orientations are used to shape the field and increase air gap flux density. In [39] a Halbach version of the consequent pole VHM was investigated, as shown in Figure 5d.

3.3.5. Topology Comparison

The four topologies introduced above have been extensively simulated and tested in the laboratory to validate findings.

Efficiency and operating power factor for the consequent pole, V-shape and Halbach consequent pole variants with a fixed magnet mass were compared using simulated data in [35]. The current density was varied in each variant to fix the force reacted—hence each configuration had a fixed power output and a fixed magnet mass. From this analysis, the V-shape topology required the lowest current and so offered the highest efficiency and highest operating power factor. The modelling was validated against basic experimental tests [35,36], where experience gained during prototype development highlighted the relative mechanical complexity of these concepts.

Table 1. Component count of stator variants of the VHM.

Topology	Component per Pole Pair
	Magnet	Pole piece	total
Surface Mount	2	0	2
Consequent Pole (CP)	1	0	1
V Shape	4	2	6
V-Shape CP	2	1	3
Halbach CP	3	0	3

shows the component count per pair of stator poles for the four concepts of Figure 5 and the original surface mounted version. The V-shape consequent pole machine shares many of the advantages of the V shape, but with half the component count is therefore the preferred topology for this study.

3.4. Cylindrical Topologies

As shown in Figure 1, for many wave energy converters the power take-off may be submerged and may also be subjected to torsional load from the capture element. A cylindrical axisymmetric cross section will allow for the use of conventional seals, reduce air gap closing forces and allow the translator to twist in reaction to torsional forces. In general terms, it has also been shown that cylindrical versions could give mass savings compared to flat machines with a short axial length [39].

To this end, two cylindrical versions of the surface-mounted E-core VHM, derived from the single-sided and double-sided flat variants were designed and tested. They consisted of a three-tooth [37] and a six-tooth [40] stator respectively. The two prototype stators are shown in Figure 7. No attempt was made to make a cylindrical version of the flux concentrated variants of this machine as it was hard to envisage a cylindrical-shaped flux pole piece. The stators were made of a single component of pressed soft magnetic composite (SMC). Although this worked well at a small scale, it is hard to imagine this manufacturing method being scaled up without the design becoming segmented. The cylindrical version is therefore not selected for further development at this stage.

3.5. Topology Selection and Build

All variants of these machines rely on the preservation of a small air gap—as the air gap grows with respect to the pole width the leakage path will increase. In original work on the VHM [32], the theoretical maximum force achievable was shown to be proportional to the physical clearance subtracted from the total magnetic air gap, as defined in Figure 3. It was suggested that around 80% of the magnetic air gap should be a magnet, so a 4 mm thick magnet requires an air gap of 1 mm. This is a real challenge in both flat and cylindrical machines with a large radius. To preserve the flux flow, the authors recommend the air gap should be around an order of magnitude less than the magnetic pole width for this machine to operate correctly. Even for large machines, the pole width must remain small due to the low velocity—and all the machines presented here have magnet widths of 12 mm and a target air gap of 1 mm. This represents a significant challenge to mechanical designers hoping to capture energy from waves with amplitudes of 2–3 m. There is more work to be done on the structural mass implications of making sufficiently stiff structures. A number of authors have used this structural challenge as a reason to pursue alternative electromagnetic topologies with lower inherent air gap closing force, such as ironless air-cored machines. Comparable performance can only be achieved by increased current density, and associated reducing in efficiency or more commonly a higher use of permanent magnet material, as discussed in [41] and developed in [42] for example.

The flux concentration effect in the V-shape stators gave a better tolerance to larger air gaps. In the present study, the flat cross section V-shape consequent pole machine was selected for further development in the wave energy system as the topology gives a good performance with half the component count of the purely V-shape topology and the cylindrical cross section variant appeared too challenging to implement at scale.

4. Power Converter

4.1. Options

The power train of a wave energy device is essentially a variable-speed drive, consisting of an electrical machine, an AC/DC machine side converter, a DC bus potentially including energy storage and a DC/AC grid side converter, as discussed in [25]. The generator charges the DC-link capacitor via a voltage source converter (VSC) or rectifier and then a voltage source inverter (VSI) is used to convert the DC-link voltage and current into grid frequency AC voltages and currents. For a constant speed application, this arrangement works well and the various components can be optimally selected. However, in wave energy, there are a number of additional requirements which influence the drive topology. For example, the generator is typically accelerating from zero to full speed and back to zero every few seconds. There is also an increased emphasis on reliability and fault tolerance, as servicing offshore platforms is weather dependent and loss of the drive means a loss of revenue. Furthermore, if it is required to control and tune the wave energy device to maximise power output, four-quadrant operation is required. Finally, local energy storage to smooth fluctuating real power and also implement the active control associated with tuning the wave energy converter requires significant reciprocating power flows over and above the average power being fed into the grid [27].

To address these requirements, a modular approach is proposed for the power take-off whereby multiple electrical generators are mechanically coupled and connected to multiple power converter modules. A single module of the drive is shown in Figure 8 for both a current-source and voltage-source topology, including a localised energy storage system (ESS).

The important elements of each module are the generator converter, the energy storage interface and the grid inverter. The ESS, using super capacitors, permits the dynamic exchange of mechanical energy during reactive power control of the wave energy converter and also to smooth the fluctuating real power through the grid interface.

4.2. Selection and Build

In [43] a discussion of the merits of a current source converter (CSC) based topology was compared with a voltage source converter (VSC) based topology for a direct-drive WEC application. The basis of this comparison was to evaluate the switching and conduction losses in the main switching and passive elements when operating at two representative states of the associated WEC and generator. The study concluded that for lower switching frequencies, i.e., <30–40 kHz, the VSC topology is more efficient but at higher switching frequencies, i.e., >40–65 kHz the CSC is more efficient. The motivation for a higher switching frequency is the reduction in passive component dimensions and costs for either topology. It is therefore implied that a CSC may lead to a more efficient smaller converter in this application.

For the VSC topology, a three-phase converter is employed with an AC low pass filter to mitigate the impact of high dv/dt on the generator cable and machine windings. It is assumed the DC link is maintained at a constant value based on the generator voltage specification. The system requires a bi-directional DC-DC converter to achieve the desired voltage control of the ESS. For the CSC topology, the commutation capacitors already provide a high degree of dv/dt reduction to the generator and cabling. The current to voltage DC-DC conversion for the ESS is a fundamental part of the chosen CSC topology.

As part of this work, a voltage source (Figure 9) and current source (Figure 10) converter have been built for comparison. Details of the current source design can be found in [44].

5. Wave Tank Testing

5.1. Test Setup

The electrical drive developed, consisting of the V-shape consequent pole VHM and the voltage source converter, is suitable for installation in a number of alternative wave energy device concepts. Two heaving buoys are selected here as case studies: one fully submerged and one semi-submerged. Figure 11 shows photographs of the buoys in and out of a wave tank, corresponding to the concepts presented in Figure 1.

For both buoys, to overcome the buoyancy force and keep the buoy submerged, the tether must be kept under tension. For an electrical system, this force offset would be a constant power drain and so to relieve the power train, a mechanical spring was used to provide this force. To remove the natural linear variation of force with the position of a mechanical spring, a mechanical linkage was devised to give a constant torque by altering pivot distance with spring extension, Figure 12. A ‘pretension actuator’ was used to tune the spring force to exactly counteract the buoyancy force. Equation (1) is therefore still valid for this system.

5.2. Wave Tank Testing

To demonstrate the system operation, the V-shape VHM and VSC were selected and joined to an existing scale model of a wave energy converter [9]. The buoy, machine and power converter were tested in a range of sea states and control regimes at FloWave energy research facility [45]. At full scale, it is envisaged that the submerged heaving buoy would have the power take-off situated below the buoy, similar to that shown earlier in Figure 1. For testing purposes, the power take-off unit, including the spring linkage, was located above the water line and coupled to the buoy via a pulley system, as shown schematically in Figure 13 and photographed in operation in Figure 14.

5.3. Steady State Operation

Typical results of steady state operation over a single mechanical cycle are shown in Figure 15. Translator velocity is derived from the position, which is measured from an optical encoder mounted to the translator. A Finite Impulse Response filter is used to smooth the results as the velocity in particular had a high noise content. Buoy force is measured by a load cell mounted on the join between the tether and the buoy, so measurements include the generator force, friction in the rig, and the constant tethering force provided by the spring. Mechanical power is calculated as the product of buoy force and velocity. It is hence the total mechanical power extracted from the oscillating body, including friction and electrical machine losses. As the wave period and amplitude are fixed and the controller is set to give a constant coefficient of damping from the linear machine (B_ePTO), after a few waves the results are settled and cyclical (Figure 16). The effect of any friction and cogging force in the rig and machine influences the oscillation to be non-sinusoidal, as visible most clearly in the velocity signal.

At this frequency (0.5 Hz), the buoy is seen to oscillate with a peak-to-peak height of 100 mm, whereas the wave peak-to-peak height (H) is 200 mm. The peak mechanical power is about 60 W, giving an average of 27 W. For reference, the power incident on a buoy with a 1 m diameter (D = 1) in these waves is estimated from (6) to be 78 W. As an energy converter, the buoy itself is therefore 35% efficient.

$$P=\frac{\rho H^2{g}^{2}T}{32\pi }$$

(6)

In (6), P is the power per m wave front of a pure sine wave, T is the period and H is the peak-to-peak wave height, as given in [46], for example.

Electrical power from the generator is calculated as the instantaneous values of voltage and current at the generator terminals. Figure 16 shows typical electrical values over a single mechanical cycle. The peak electrical power is about 40 W and the average is 11 W. The overall power take-off efficiency is therefore 45%. At this small scale, efficiency is dominated by the friction in the mechanical linkage and the translator bearings, rather than losses in the electrical machine.

During the testing, the generator phase current is forced to be in phase with the back emf, as shown in Figure 4 earlier. The generator operates at a power factor of around 0.7, meaning reactive power must be supplied to the machine by the converter. This manifests itself as the positive power per phase shown in the top left plot of Figure 16. The overall power across the three phases is shown in the top right of Figure 16 to be wholly negative, i.e., purely generation. Bidirectional power flow thorough the converter is therefore evident.

5.4. Case Study 1: Damping Variation in a Non-Resonant System

A plot of power output from the generator over a range of fixed amplitude varying frequency sine waves is shown in Figure 17. At each wave frequency, the generator reaction force is controlled to act as a damper with a variable coefficient (B_ePTO). As might be expected, the plot shows the familiar power capture curve implying there is an optimum damping coefficient for each wave state. This plot demonstrates both that the generator and power converter can successfully extract power from the sea, and also that the power converter can be used to implement control of the wave energy device itself. The earlier plots in Figure 14 and Figure 15 correspond to the system operating at the peak power point shown in Figure 17.

There is a slight shift with frequency in the value of B_ePTO which gives maximum output power. A controller would be required to tune the power take-off system for the predominant wave state. Overall, this plot confirms that the damping force of a generator can be used to control the power output of this non-resonant wave energy converter.

Efficiency at this small scale and using the arrangement of Figure 12 is dominated by friction. However, Figure 18 shows that a higher coefficient of damping decreases efficiency, as might be expected as a higher damping force requires a higher generator current. For some sea states, peak efficiency in this system therefore occurs when it is slightly underdamped compared to the maximum power point in Figure 17. In energy scavenging type applications, maximum output power is clearly of more importance than system efficiency.

5.5. Case Study 2: Resonant Systems and Spring Force Variation

Many wave energy converters sit on the sea surface, and so the buoyancy force varies with the relative submersion of the device, for example, the buoy shown earlier in Figure 1a. The resultant spring force means the device has a natural frequency of oscillation. If the predominant wave frequency matches the natural frequency (f_n) of the device, it will resonate and oscillate with a larger amplitude. Many wave energy developers aim for resonance in order to amplify oscillation velocity and increase the efficiency of power capture. In a direct-drive system, it is possible to introduce a generator force proportional to position, and so emulate an additional spring force on the system. This will alter the frequency response of the overall wave energy converter. To validate direct-drive power take-off in multiple types of wave energy converter, therefore, it has also been necessary to perform experiments with a semi-submerged surface piercing resonating buoy.

In any mechanical system, the angular velocity of resonance is equal to the square root of spring force divided by mass. For a surface piercing buoy constrained to oscillate in heave (vertically), the resonant frequency (f_n) is therefore defined by:

$$f_n=\frac{1}{2\pi }\sqrt{\frac{\rho g{A}_{wp}+k_g}{m+m_a}}$$

(7)

A_wp is the water plane area of the buoy, m is the mass and $\rho g{A}_{wp}$ corresponds to the spring force due to buoyancy variation with depth. k_g is a controllable spring force from the generator and m_a is the added mass of surrounding water that is effectively coupled to the buoy displacement. For an effective wave power device, the natural frequency, f_n, ideally coincides with the incoming wave frequency. Selecting the appropriate dimensions and mass of the buoy allows for some control of the natural frequency. Control of k_g allows the device to be tuned to a specific frequency.

Electrical machines do not scale down well, as friction and magnetic leakage become proportionally more significant. The size of buoy required to give meaningful power output at the smallest practical scale fixes the diameter and hence A_wp. The mass is defined by the submerged depth, which itself is limited by the wave tank. It was found that these constraints resulted in a system with a higher natural frequency than the capability of the wave tank being used, which required a natural resonant frequency in the heave of approximately 0.5 Hz. To manipulate the resonant frequency, a two-body system was devised, as proposed in [47] and shown in Figure 11b. The upper body (the buoy) interacts with the waves and the lower body increases the inertial mass. Ideally, the second body is coupled directly to the first at a sufficient depth to avoid interaction with the passing waves. The second body should be neutrally buoyant to ensure it acts to increase the inertia components of Equation (7).

Figure 19 shows results from the two-body buoy of Figure 11b being excited by waves of 0.6 Hz, close to its natural frequency. In this plot, generator force is controlled to be a function of both position and velocity—i.e., a spring (k_g) and damping (B_ePTO) force. Results at low damping can be seen to have two peaks: corresponding to two modes of resonance. It proves that a direct-drive power take-off with an effective spring force can be used to control resonance and maximise output power in resonant wave energy converter. The two peaks correspond to two modes of oscillation: the planned heave and a parasitic pitching mode. Figure 19 also proves, therefore, that the electric machine of a direct-drive wave energy converter can be used to control the mode of oscillation.

5.6. Automated Control and Transient Results

When deployed at sea, the dominant frequency and amplitude of the waves will change. For optimal performance, the frequency response of the wave energy converter will need to alter, perhaps by changing the spring or damping components of power take-off force. Optimal values can be selected by advanced knowledge of the sea state and an accurate hydrodynamic model of the device. Alternatively, a simple representative control algorithm based on a ‘cycling’ maximum power point tracking algorithm (MPPT) [48,49] has been designed and implemented [25].

MPPT algorithms provide a functionally robust and pragmatic solution to the optimisation problem. As a non-model-based algorithm, they can be rapidly deployed and quickly tuned online, avoiding many of the difficulties associated with model-based approaches such as accounting for complicated multidimensional dynamics [50]. Under simple conditions, such as damping control which yields a convex mapping between the damping factor and output power (see Figure 17), MPPT based approaches are particularly well suited, as the gradient ascent type algorithm is able to easily assert optimal points [48,50,51]. However, under more complex conditions, such as spring-damping control, the performance may be compromised due to the presence of multiple local optimal points associated with the various resonances in each mode of oscillation, such as that shown in Figure 19. As with any locally minimising algorithm, the found solution depends on the initialisation.

The ‘cycling’ MPPT algorithm that was used is an application of a recent technique developed in [48]. This algorithm in particular improves on the deficiencies of standard ‘perturb and observe’ type MPPT algorithms in irregular seas. In brief, the algorithm operates by constantly perturbing the control variable (e.g., damping factor) over a number of sea-wave cycles using a sinusoidal waveform. Over a prescribed interval (e.g., taking a few cycles of the perturbation waveform), the statistical covariance between this waveform and the output power is established. Utilising the resulting covariance, the control variable can hence be updated proportionally to the ‘strength’ of the relationship. The algorithm can be tuned by varying the magnitude of the perturbation waveform to increase the sensitivity/effective region, and, passing the covariance result through an additional gain parameter to modify its effect on the control variable; these can be tuned online with relative ease. A useful trait of the algorithm is that parallel instances using different perturbation waveform periods can be run to control multiple variables. Further details and examples of the cycling MPPT algorithm can be found in [48,49].

In practice, the cycling MPPT algorithm was able to effectively optimise the test platform under damping control action. Figure 20 illustrates the experimental results of the algorithm under a 0.75 Hz regular sea with a 0.25 m wave-height run for seven minutes [25]. MPPT update intervals were one minute. The results illustrate the algorithm actively adjusts the damping coefficient to optimise the output power response, increasing the average output by approximately 30%, and nearly doubling the peak power. The effect of the control solution is particularly evident in the first 200 s of run-time (extending past the initial start-up transient interval which finishes at approximately 30 s) where the average and peak power outputs drastically improve. These results hence show the automatic implementation of a control algorithm in a direct-drive wave energy converter.

6. Conclusions

Many wave energy devices are anticipated to produce slow-speed reciprocating motion. This paper has discussed the development of electrical drives suitable for direct-drive electrical power take-off. Several topologies of linear permanent magnet machine have been presented. Of these, it appears the flux concentrated ‘V-Shape’ version of the vernier hybrid machine can convert the power at the most favourable power factor and with the highest efficiency. A consequent pole version of this topology was developed further, as it offered similar advantages with just half the component count per stator pole. Cylindrical versions of some of the machines are possible, although for the V-shape machine it will be hard to manufacture at scale. A flat cross section version was therefore built and tested. For the power converter, the requirements and design of two configurations has been presented, with results from the voltage source version presented.

Sample results of the whole system tested in a scaled wave tank and coupled to two buoys prove this technology is suitable for use in a controlled wave energy converter. The electric drive train is used to manipulate the power output, system efficiency and mode of oscillation of the two wave energy converters. Finally, a maximum power point tracker technique has been employed to prove that automated control is possible on regular seas.