Modular Multilevel Converter for a Linear Generator for Wave Energy Converter

Abstract

In this paper we propose a modular multilevel converter for a linear generator for a wave energy converter. The coils of this generator are individually controlled to improve energy harvesting performance. This topology involves two stages. The first stage uses a full-bridge to control the harvested current with a reference generated by means of an MPPT method. The second stage uses a half-bridge to control the voltage of the DC-link and the output current. Furthermore, multilevel modular converters allow the generation of a medium DC voltagethat reduces the losses in energy transmission lines from offshore to a coupling common point on the shore.

1. Introduction

Concerns about global warming mitigation have increased the development of alternative renewable energy sources, such as solar (photovoltaic panels, PVs), wind (turbines), and ocean energy (wave energy converters). During the last ten years (2011–2021), renewable energies have matured and currently represent 37% of the total operating installed capacity of electrical energy, as shown in Figure 1. Wind power generation is continually improving and increased its power capacity by 93 GW in 2020. Furthermore, PV energy is the renewable energy with the most notable growth, for example, through new technologies such as organic solar cells or polymer solar cells [1,2,3,4,5].

Despite the increase in renewable energy generation, some sectors have a low percentage of final energy demand. Therefore, other renewable energy sources are required to meet the targets of gas emissions reductions, and for this purpose ocean energy is an excellent choice. This alternative energy has great potential due to its relatively high power density, but it is the largest untapped energy source [7].

Regarding ocean energy, there are six distinct sources (ocean waves, tidal range, tidal streams, ocean currents, temperature gradients, and salinity gradients), each with different technologies for their conversion. The development of ocean energy technologies has focused on tidal and wave energy, and the installed capacity of ocean energy rose to approximately 527 MW in late 2020 [6,7,8], as shown in Figure 1.

Ocean waves are generated through the kinetic energy exchange between the wind and the upper surface of the ocean into power. In this paper, we focus on an application using ocean waves to harvest energy. This is a valuable option because the estimated total wave power on the coastlines can reach up to $2TW$, which has a great potential to generate electrical energy [9].

Wave energy converters (WECs) are devices that capture wave energy mechanically and convert it into electrical energy. The development of WECs is still more complex than other types of renewable energies for two reasons. The first corresponds to the diversity of the wave resources in offshore and nearshore locations, which means that the design process involves difficulties in obtaining a high efficiency over the entire range of operations. The other reason involves the instability of energy intensity when capturing maximum energy [10,11,12,13].

There are plenty of WECs with different working principles and designs; point absorbers stand out among others because their size is smaller than the wavelengths of ocean waves. These systems harvest energy from the motion of bodies, which are submerged or floating on the surface wave [10,14].

Linear generators are a more efficient option to be used on a point absorber, for example, the Islandberg Project (Sweden), Seacap (France), PowerPod, and PowerPod II (UK), among others [15]. Furthermore, linear generator technology offers a simple and robust structure. In addition, linear generator design presents different properties, such as the use of a planar shape, reducing the detent force through its structure and tubular shape, producing a high energy density [14,16].

A tubular permanent magnet linear generator (TPMLG) can be designed with different topologies and shapes used for the permanent magnets (PMs) to obtain better flux linkage, cogging force reduction, and higher efficiency in WECs [17,18,19]. The most common topologies are radial, axial, and Halbach configurations for the implementation of TPMLGs [18,20]. Furthermore, topologies have also been developed to avoid demagnetization by modifying the air gap [21].

The development of high-voltage direct current (HVDC) systems based on the use of a voltage source converter (VSC) has increased significantly with the introduction of new semiconductor technologies. HVDC technologies are used in the long-line transmission of high power, the interconnection of different AC systems, the integration of renewable energy sources on a large scale, underground applications, and submarine applications [22,23].

MMC topologies use the same submodules to provide a step in a multilevel waveform and are easily adaptable to high voltage levels. The topology of high-voltage DC/AC conversion is based on the use of MMC converters on cascade submodules [23].

There are two basic submodules (SMs) in MMC topologies—the half-bridge, and the full-bridge submodule. The half-bridge has a voltage output equal to its capacitor voltage ($+v_C$) or zero (0). The full-bridge has a voltage output equal to its positive voltage ($+v_C$), its negative voltage ($-v_C$), or zero. The latter submodule has a higher cost because the number of semiconductors is doubled; however, the full-bridge has the advantage of providing a negative voltage level [24,25].

In this paper, we propose a TPMLG with a radial configuration with a single stator and translator. Furthermore, we simulated this system with five individual coils which harvest energy, each with a full-bridge submodule. Finally, the power converter uses a full-bridge submodule to harvest energy and a half-bridge submodule at the output of the MMC.

2. System Model

A WEC system uses three energy conversion stages to feed the grid. The point absorbers have the advantage of reducing the two first stages in a direct stage. This design uses the action of the incident sea waves to move the translator of a linear generator without additional mechanical systems [26]. This device has a buoy on the surface of the waves and it is connected to an electrical generator system underwater, as shown in Figure 2. The last stage is the interface between the WECs and a coupling common point through a static transmission cable, and this provides a power signal to inject into the grid, as shown in Figure 2. The focus of this paper is on the direct stage.

2.1. Wave–Buoy Interaction

The wave elevation defines the potential energy ($E_p$) and the water fluid motion defines the kinetic energy ($E_k$). These energies are stored in the waves, as shown in Equation (1) [27,28].

$$E=E_p+E_k=\rho g{\int }_0^{\infty }S\left(f\right)df=\frac{\rho g{H}_s^2}{16}$$

(1)

where $\rho$ is the fluid density (approximately 1025 $\frac{kg}{m^3}$), g refers to the gravitational acceleration (9.8 $\frac{m}{s^2}$), E is total stored energy, and $S\left(f\right)$ is the spectrum distribution of the wave energy of a given location as a wave frequency function (f). The wave spectrum defines the average height ($H_s$), as shown in Equation (2); this is used to determine the wave power [27,29].

$$\frac{H_s^2}{16}={\int }_0^{\infty }S\left(f\right)df$$

(2)

The integral of the time-average energy transport per unit time and unit area in the direction of wave propagation (the x direction) is called the water-power level J or wave-energy transport, and its expression is shown below in Equation (3) using $v_g=g/2\omega$ for irregular plane waves on deep water [27,28].

$$J=\rho g{\int }_0^{\infty }S\left(f\right)v_g\left(f\right)df=\frac{\rho g^2{T}_J{H}_s^2}{64\pi }$$

(3)

where $T_j$ is the energy period. For example, an irregular wave of a buoy located in Valparaíso, Chile, where the depth is 4515 m, with energy period $T_J$ = 10.3 s and significant wave height $H_s$ = 2.3 m, the energy transport is J = 26.6 kW/m.

2.2. WEC Equation of Motion

The total force acting on a buoy can be decomposed as shown in Figure 3. The excitation force $F_e$ is produced by incident waves and the radiation force $F_r$ is produced by buoy oscillation. The buoyancy force has a part constant $F_b$ when the buoy stands still on the water surface $z_b=0$; this represents the weight of the buoy and translator, and it also has a variable part $F_h=\rho gS{z}_b$ related to the displacement of the buoy and the balanced position. There is a gravity force $M_b.g$ and a force line $F_{line}$. These forces are written in the frequency domain, as shown in Equation (4).

$$-w^2.M_b.z_b\left(w\right)=F_e+F_b-F_r-F_h-F_{line}-M_b.g$$

(4)

The force of the translator is the gravity force $M_t.g$, the line force is $F_{line}$, and the end-stop force is $F_{end}$. The latter force is generated when the translator hits the upper-end stop spring or the rubber damper hits the generator bottom. The $F_{PTO}$ is the electromagnetic damping force from the power take-off.

$$-w^2.M_t.z_t\left(w\right)=F_{line}-F_{end}-F_{PTO}-M_t.g$$

(5)

2.3. Power Take-Off

The PTO uses a linear generator with a tubular shape, divided into two structures. The translator has ring-shaped PMs or PMs arrays with a similar shape, which have an axial configuration (north orientation to outer radial); furthermore, PMs arrays are attached to the PMs’ support and a mobile vertical shaft, as shown in Figure 4. The stator has a tubular shape and supports five coils arranged around it.

The magnetic circuit can be analyzed by means of a lumped parameter method related to an electric circuit because they have the same behavior in a stable state [14]. The magnetic circuit is expressed in Equation (6), and it has a net reluctance (${\mathfrak{R}}_{xx}$) given by the path of magnetic flux shown in Figure 4, the magnetomotive force (${\mathcal{F}}_m$) produced by the PMs, and where $\mathsf{\Phi }$ indicates the magnetic flux.

$${\mathcal{F}}_m=\mathsf{\Phi }{\mathfrak{R}}_n$$

(6)

The reluctance (${\mathfrak{R}}_{xx}$) is defined by the flux path length ($l_x$), vacuum permeability (${\mu }_o$), the relative permeability of each material (${\mu }_{rx}$), and the cross area ($A_x$), [30], as follows:

$${\mathfrak{R}}_{xx}=\frac{l_x}{{\mu }_o{\mu }_{rx}{A}_x}$$

(7)

Figure 4. Schematic design of the linear generator. Figure based on [31].

Figure 4. Schematic design of the linear generator. Figure based on [31].

The partial reluctance of the structure describes the net reluctance, using Kirchhoff’s circuit laws [32], as shown below.

$$\begin{array}{cc}\hfill {\mathfrak{R}}_n=& {\mathfrak{R}}_{PM}+{\mathfrak{R}}_{oPMs}+{\mathfrak{R}}_{gap}+{\mathfrak{R}}_{cs}+{\mathfrak{R}}_c+{\mathfrak{R}}_l\hfill \\ \hfill & +{\mathfrak{R}}_{st}+{\mathfrak{R}}_{cs2}+{\mathfrak{R}}_{gap2}+{\mathfrak{R}}_{lPMs}+{\mathfrak{R}}_{iPMs}\hfill \end{array}$$

(8)

Equation (9) defines the magnetomotive force (${\mathcal{F}}_m$) generated by PMs. This is the relation between the product of the remanent magnetization of the PM ($B_r$) with the width of the PM ($w_{p}m$) and the product of the relative permeability of the PM ${\mu }_{rpm}$ with the vacuum permeability (${\mu }_o$).

$${\mathcal{F}}_m=\frac{B_r{w}_{p}m}{{\mu }_o{\mu }_{rpm}}$$

(9)

The flux linkage is considered a sinusoidal function of the vertical displacement (z) to define the magnetic flux for each coil, as shown below.

$$\varphi ={\mathsf{\Phi }}_{max}sin\left(\frac{\pi }{4{\tau }_m}z\right)=\frac{B_r{w}_m}{{\mu }_o{\mu }_{rm}{\mathfrak{R}}_n}sin\left(\frac{\pi }{4{\tau }_m}z\right)$$

(10)

Equation (10) is used to describe the electromotive force ($\epsilon$) in a coil with many turns (N) [33]; thus,

$$\begin{array}{c}\hfill \epsilon =-N\frac{d\varphi }{dh}\frac{dz}{dt}=-N\frac{B_r{w}_m}{{\mu }_o{\mu }_{rm}{\mathfrak{R}}_n}\frac{\pi }{4{\tau }_m}cos\left(\frac{\pi }{4{\tau }_m}z\right)\frac{dz}{dt}\end{array}$$

(11)

3. MMC Description

An MMC topology was used to harvest energy from each coil of the TPMLG, to control the current harvesting process and generate a medium voltage for transmission. Therefore, each coil was connected to a power converter submodule, and the power converter submodule outputs were connected in series to generate a medium voltage, as shown in Figure 5.

Each submodule was composed of two parts interconnected with a coupled capacitor, as illustrated in Figure 5: an AC-DC converter from a generator coil and a DC-DC converter connected in series to other converters. In the first stage, a full-bridge submodule was used to control the current harvesting with reference to the MPPT, which changed the direction of the torque drive. In the second stage, a half-bridge submodule was used because two states were required to control the capacitor voltage. The MMC connection could require a filter inductor if the transmission line inductance is insufficient to filter the switching harmonics.

For the converters shown in Figure 6, we used a controlled decoupled design. The DC-MMC part used a half-bridge submodule to control the voltage of the capacitor.

Through this configuration, one could add the submodules needed as coils have the generator. At the same time, this decreases the frequencies generated by power.

Previous studies on PV systems have presented the same topology to interconnect low-voltage systems (PV systems) to medium- or high-voltage DC systems through one or two voltage transformer steps. These works show the viability of implementing this topology to large-scale connected grids [34].

4. Control Strategy

The control strategy proposed here involves two decoupled control stages, one for an AC-DC converter with the MPPT and one for a DC-DC converter that assembles the MMC, as shown in Figure 6. The first control stage uses the result of the MPPT method as a reference to control the harvesting current of a coil of the generator. The second control stage aims to control each DC-link voltage and the current through all the submodules of the MMC.

Figure 7 shows the simulated voltage of a TPMLG with five coils to show its behavior. Furthermore, the TPMLG has five PM arrays, which are moving at 0.1 Hz.

4.1. Harvesting Control

The MPPT algorithm determines the current reference for the AC-DC stage to maximize the power of each coil. In this paper, we present an algorithm for the MPPT based on the nature of the frequency of the waves and the electrical system measurements. A PI regulator controls the error generated by the MPPT reference and the generated current. Equation (12) shows the transfer function used to design the PI for the modulation index of the full-bridge submodule.

$$\frac{i_{gk}}{m_k}=-\frac{v_{ck}}{R_e+s{L}_e}$$

(12)

where $i_{gk}$ is the input current (coil current), $m_k$ is he modulation index for the converter, $R_e$ is the coil resistance, and $L_e$ is coil inductance. $v_{ck}$ is the capacitor voltage and it is considered a constant parameter with a value of 100 V for the simulation. This control scheme is shown in Figure 6.

Figure 8 shows the input signal and output signal of the PI regulator from the first submodule. It shows the switching signals at the PMW modulator’s bipolar output used to control each semiconductor.

The ideal MPPT reference would be the exact behavior of the vertical position of the buoy in relation to the wave, but an absolute reference is estimated to have the following expression:

$$m_{tot}\frac{dv}{dt}+k_{s}z=0$$

(13)

$${\beta }_g={\beta }_w$$

(14)

where $m_{tot}$ is a sum of the floater mass and the added mass, $k_s$ is the spring constant point absorber, ${\beta }_g$ is the damping coefficient provided by the generator, and ${\beta }_w$ hydrodynamic damping coefficient of the point absorber.

4.2. DC-Link Voltage Control

This control scheme allows one to control the DC-link voltage. The submodules operate at different power levels due to the differing translator positions, and each uses a PI regulator. The PI is designed with the transfer function shown in Equation (15) to generate a modulation index for the half-bridge submodule.

$$\frac{v_{ck}}{m_k^{\prime }}=-\frac{i_o}{s_c}$$

(15)

where $v_{ck}$ is the DC-link voltage, $m_k^{\prime }$ is the modulation index for the converter, and C is the capacitance of the DC-link. $i_o$ is the output current and is considered a constant parameter. The capacitor current $i_c$ is considered a disturbance of the system, so it does not appear in the control law.

Figure 9 shows the input signal and output signal of the PI regulator to the first submodule. It shows the switching signals at the output of the modulation stage used to control two semiconductors.

4.3. Current Control

Loop current control injects a current into the DC output. The topology shown in Figure 10 represents a simplified model of the power circuit. The total current $i_T$ flows through the MMC submodules connected in series, and the total converter voltage $v_o$ is the sum of all submodule voltages, as shown in Figure 5.

$$v_o=\sum _{k=1}^N{v}_k$$

(16)

Equation (16) shows the output voltage of module k ($v_k$) and the total number of DC submodules (N) in the converter (N=5 in this case). The following equation uses Kirchhoff’s voltage law in the topology equivalent of Figure 11.

$$v_T=v_o+i_o{R}_T+L_T\frac{d{i}_o}{dt}$$

(17)

where $v_T$ represent the voltage grid, $i_o$ is the output current, and $R_s$, $L_s$ are the line parameters. Therefore, an analysis of the power and the sum of all the individual DC-link voltages involves the use of the transfer function to design a PI control to generate the average modulation index and control the grid current, as shown below.

$$\frac{i_o^2}{m_{i}k}=\frac{N{V}_{ck}}{\frac{L_L}{2}+R_L}$$

(18)

where $v_{ck}$ is the DC-link average of the submodules, N is the number of submodules, and $m_i$ is the modulation index average.

5. Conclusions

Wave energy is an under-exploited resource with great potential. Direct absorbers with linear generators present better results because they do not have mechanical connections; they also have the advantage of being placed offshore, avoiding visual contamination and problems with inhabitants near the coast.

In this study, we proposed to improve the energy harvesting performance of WECs by means of a modular multilevel converter to drive a linear generator with independent coils by controlling the magnetic flux and torque depending on the translator position and increasing the efficiency of power transmission. The connection of the windings in series can be undertaken to generate a higher voltage and increase the efficiency of the transmission of power to the coast.