Hardware-in-the-loop simulation of wave energy converters based on dielectric elastomer generators

A nonlinear lumped-parameter hydrodynamic model is used for describing the OWC oscillations in the time domain. This modelling approach is widely used in WEC modelling, as suggested, for instance, in [18] and [19]. The model describes the water column oscillations via the unsteady Bernoulli equation. This approach is applied under the assumptions that the OWC collector is fixed and the water column width and breadth are small compared to the wavelength, so that the water column displacement is approximated by the oscillation of a rigid vertical piston. We further assume that the total head at the OWC inlet is given by the wave pressure head, thus neglecting other contributions. The wave field in front of the structure is described by the potential flow theory. Specifically, the calculation of the wave pressure head is accomplished in the framework of the linear potential wave theory [20]. The inlet wave pressure at the OWC collector aperture is calculated via the so-called Cummins equation [21], which accounts for the effects of the hydrodynamic memory and the hydrodynamic added mass. The equation governing the water column dynamics [19] is: where \({p}_{S}\) is the relative pressure in the air chamber; \(A\left(\infty \right)\) is the infinite-frequency added mass; \(K\left(t\right)\) is called the retardation function; \(M\left({z}_{S}\right)\) is the time-varying inertia of the water column within the collector, which includes the effect of mass transfer between the water column and the wave field: it depends on the instantaneous water column level \({z}_{S}\) (i.e. the upward elevation with respect to the mean water level) and the OWC collector geometry; finally, \(C\left({z}_{S},{\dot{z}}_{S}\right)\) is a damping coefficient associated with the kinetic head of the water column, which depends on the water column position and velocity and on the system geometry [19]. Commonly, the calculation of the hydrodynamic parameters (\(A\left(\infty \right)\), \(K\left(t\right)\)) and the excitation is pursued via boundary-element codes [22], although analytical solutions are available for some specific geometrical configurations [19].

The system excitation is rendered by the wave pressure \(\Delta {p}^{\left(D\right)}\) calculated at the OWC inlet under the assumption of a diffracted wave field. The wave pressure \(\Delta {p}^{\left(D\right)}\) is calculated as a function of the input wave parameters: a sinusoidal wave pressure is used in the case of regular waves, whereas for irregular waves \(\Delta {p}^{\left(D\right)}\) is expressed as a finite sum of harmonic components whose amplitudes follow a prescribed spectral distribution [23].

A general case is studied in which the HIL setup is used to simulate OWC WECs that hold DEG PTOs which are similar to the physical prototype but present differences in several dimensional and layout features. Specifically, although the prototype system’s general architecture is equivalent to the simulated system, i.e. OWC combined with DEG-PTOs made with the same materials as the hardware prototype, the following differences can be introduced: (1) the simulated OWC can feature multiple CD-DEGs connected to the same air chamber (as opposed to a single physical CD-DEG sample); (2) the simulated CD-DEGs can have larger dimensions than those of the physical samples, which are bounded by hardware constraints; (3) the simulated CD-DEGs can feature a multi-layer architecture with a different number of layers compared to the hardware setup; (4) the simulated OWC can be larger than the physical setup (e.g., it can have a larger cross section).

A consistent dynamic equivalence between the simulated and the hardware systems can be obtained by introducing a coupling block (see Fig. 2a) in the HIL scheme, which applies suitable corrections: 1) to the measured air pressure fed into the software solver and, 2) to the commanded piston position calculated based on the hydrodynamic model solution. These corrections are calculated through convenient functions of the simulation variables that are presented in this section. In particular, we propose a procedure which guarantees that the physical CD-DEG sample is subject to the same strain and electric field time history as the DEGs in the simulated scenario, assuming same time scale (e.g., operating frequencies) for the simulation and the physical experiment. In order to distinguish the variables of the simulated scenario and of the hardware setup we hereby denote the first with subscript S and the latter with subscript H.

Since the deformed CD-DEG behaves as a thin elastomeric shell, the relative air pressure in the OWC air chamber is proportional to the CD-DEG initial thickness and the inverse of the base radius, with the proportionality factor (depending on the strain and the elastic properties) being the same in the scenario and the experiment. The following relation between the scenario and the measured air chamber pressure thus holds:where \({t}_{0}\) and \(e\) (with the appropriate subscripts) indicate the unstretched thickness and the CD-DEG’s base radius (see Fig. 1b). The right hand side of Eq. (2) provides a correction factor to be applied to the measured pressure \({p}_{H}\) so as to obtain the pressure \({p}_{S}\) to be fed in the software model (i.e., into Eq. (1)).

Since the CD-DEGs strain in the experiment and in the scenario are the same, the axial displacement of the CD-DEG centre in the experiment (\({h}_{H}\)) and in the scenario (\({h}_{S}\)) simply relate as follows:

A relationship between the water column displacement in the scenario \({z}_{S}\) and the commanded piston position \({z}_{H}\) is then identified, assuming that the air in the OWC chamber undergoes adiabatic transformations. Since the chamber volume variations are small compared to the initial volume, the following linearized models for the air chamber response are assumed: where \({p}_{atm}\) is the absolute atmospheric pressure; \(\gamma\) is the air’s adiabatic exponent; \({V}_{S}\) and \({V}_{H}\) are the initial air chamber volumes; \({S}_{S}\) is the simulated water column cross section, whereas \({S}_{H}\) is the hardware piston cross-section; \({N}_{S}\) is the number of CD-DEGs coupled with the air chamber that are devised for the simulated scenario. \({\Omega }_{\mathrm{S}}\) and \({\Omega }_{\mathrm{H}}\) are the volumes subtended by a CD-DEG in the scenario and in the experiment (positive for outward expansions), and they depend on the DEG strain and radius [6] as follows:

Using Eqs. (2, 3, 4, 5) leads to the following relationship for the commanded piston displacement:

According to Eq. (6), the commanded displacement \({z}_{H}\) is a sum of two terms: a kinematic term proportional to the simulated water column displacement \({z}_{S}\), and a term which accounts for the finite air chamber compressibility [24].

Since the power density (per unit dielectric material volume) converted by a CD-DEG in a cycle depends on the strain and the applied electric field [6], we require that the DEGs in the setup and in the simulated scenario are subject to the same electric field time-history and, hence, they generate the same electrical power density. In general, the CD-DEGs in the scenario and in the setup can have a different number of layers (\({n}_{{L}_{S}}\) and \({n}_{{L}_{H}}\) respectively), therefore, the voltage \({V}_{H}\) used in the tests differs from that envisaged in the scenario (\({V}_{S}\)) according to the following relation, which guarantees that the electric field is an invariant:

Equation (7) allows selecting a suitable number of layers \({n}_{{L}_{S}}\) for the scenario, so as to limit the output voltage \({V}_{S}\) consistently with technical constraints.

The total power output \({P}_{S}\) in the simulation scenario is obtained by multiplying the measured power, \({P}_{H}\), by the ratio of the total dielectric material volumes in the scenario and in the physical prototype, namely:

Equation (8) relies on the assumption that the CD-DEG efficiency is the same in the physical prototype and in the scenario, thus neglecting scale-dependent effects. This is, in general, a conservative assumption. It is indeed expected that CD-DEGs practically applied in relevant scenarios are the result of optimised manufacturing and assembly processes and potentially feature better efficiency and time-stable performance compared to the physical samples used in HIL tests.

In principle, the proposed rules allow for the simulation of DEG-based OWCs with arbitrary scale and dimensions, including the case of full-scale WECs. Compared to scaling rules for tank testing of DEG-OWCs (based on Froude scaling) [10], the proposed coupling block allows for the employment of DEG prototypes in which diameter and thickness scale factors (with respect to the scenario) are chosen independently from each other, hence granting better flexibility in terms of manufacturing. In fact, it is expected that the HIL simulator’s prediction would be more reliable when the dimensions of the tested DEG prototypes are close to those in the considered scenario. HIL simulations of large-scale systems with small-scale DEGs would, in turn, suffer from errors due to an imperfect rendering of scale-sensitive effects (e.g., the DEG inertia, the electrodes resistance) which are not accounted for by the coupling block.

The mechanical driving system is based on a vertically mounted pneumatic cylinder that has been purposely designed and built. The system, shown in Fig. 2b, is made of a main driving cylinder-piston system with a nominal diameter of 300 mm (that provides \({S}_{H}\)=707 cm²) and stroke of 500 mm. The linear motion of the piston is driven by a ball-screw stage by Festo with pitch of 20 mm and maximum force of 4500 N, that is actuated by an AKM52L Kollmorgen brushless motor that features rated continuous power of 2.4 kW, continuous torque (at stall) of 8.67 Nm, and maximum speed of 4590 rpm. The pneumatic cylinder holds a polycarbonate cylinder at its top, on which the CD-DEG assembly can be secured through a set of flanges.

Based on the volume swept by the piston and the maximum useful force (after friction losses in the O-rings), the setup can hold CD-DEGs with diameter of up to 500 mm and thickness of up to 0.2 mm after pre-stretch (i.e., 40 cm³ of dielectric material volume). Assuming convertible energy densities of 0.1–0.2 J/cm³ per unit of dielectric material volume [25], noticing that a CD-DEG performs two conversion cycles during a full-oscillation of the piston (one for upward and one for downward deformation), the considered samples can produce power outputs on the order of 10⁰–10¹ W if the piston moves at a frequency of 0.2–1 Hz. This frequency range is similar to that adopted in wave tanks for the testing of WEC prototypes at a scale between 1:30 and 1:10 (compared to a hypothetical full-scale plant installed at sea) [26]; that is, the HIL setup can be used to simulate prototypal devices over a wide dimensional range.

The control logics of DEG/OWCs is based on a prediction-free strategy, which only relies on the instantaneous measurement of the CD-DEG deformation. The control logics and power electronics used on the HIL simulator are the same as those used in [10]. A simple circuit (Fig. 3) is used, which relies on a HV amplifier (10/10B-HS by TREK), the CD-DEG sample, three HV switches (HM12-1A69-150 by MEDER Electronics), a constant capacitor \({C}_{a}\), and a draining resistor \(R\). The circuit allows for the implementation of the following four-phase control cycle corresponding to a full oscillation of the DEG between the flat and the maximally expanded configuration:

A schematic representation of the control cycle in terms of the CD-DEG’s electrical state variables (charge–voltage) is provided in Fig. 3: during phases (2) and (4) (quick charging/discharging transients) the CD-DEG position (and, hence, capacitance) holds approximately constant, whereas during phase (3) the state variables follow a trajectory which depends on the charge balance for the parallel of the CD-DEG and \({C}_{a}\). In practice, the capacitance might be subject to variations during phases (2) and (4), both because these phases are not instantaneous, and because the large voltage variations associated to charging/discharging might trigger DEG shape variations (in the same fashion as in DE actuators [27]). These variations have been shown to be negligible at the scale of the experimental CD-DEG prototypes that the HIL setup can hold, but they might become significant in case larger-scale systems are considered [10].

The use of in-parallel capacitance \({C}_{a}\), proposed in [25], allows for an increase of the cyclic convertible energy while limiting the voltage rise on the CD-DEG during generation phase (3). Although the proposed circuit does not enable energy storage, it makes it possible to easily estimate the net electrical energy generated in a cycle, which reads as follows:

The first term in Eq. (9) is the energy recovered from the DEG during phase (4), the second term is the energy spent to prime the DEG, while the third term is the energy transferred to \({C}_{a}\) during phase (3). The maximum capacitance \({C}_{max}\) reached by the CD-DEG in a cycle depends on the maximum deformation, and it can be estimated from the voltage drop on \({C}_{a}\) during the priming phase, namely:

The minimum capacitance \({C}_{min}\) is, in contrast, constant and it is measured in the flat CD-DEG configuration. This led the voltage \({V}_{2}\) to be also nearly constant (apart from minor fluctuations due to the effect of charge leaks) throughout the different cycles, with a value that is always lower than the charging voltage \({V}_{0}\) (see Fig. 3).

Switching from a phase to another of the control cycle is triggered based on air chamber pressure measurements. Although alternative measured or simulated state variables can be used to switch from a phase to another, pressure is used here as it is roughly proportional to DEG deformation and very easy to acquire. In particular, switching from phase (1) to phase (2) is performed based on the maxima/minima of the pressure profile; switching from phase (3) to phase (4) is triggered by the zero-crossings of the pressure; the duration of priming and discharging phases ((2), (4)) depends on characteristic charging/discharging times set by the circuit electrical dynamics.

The power electronics layout used here was selected based on its simplicity and capability to provide a straightforward estimate of the generated electrical energy. In principle, however, this power electronics can be replaced with other advanced circuits capable, e.g., to store the generated energy rather than draining it on a resistor.

The software environment for data acquisition, driving of the mechanical setup, and electrical control of the CD-DEG is implemented via Matlab and Simulink scripts, running the Simulink Real-Time software environment on a real-time target machine by Speedgoat at a sampling frequency of 1 kHz. Communication between the target control machine and the motor driver is handled via the EtherCAT protocol.

The voltages across the CD-DEG and \({C}_{a}\) are measured with two high voltage (HV) probes. The CD-DEG deformation is measured through the post-processing of the acquired frames of a high-speed camera (Point Grey GS3-U3-23S6M-C with lens 250F6C). Specifically, the time history of the axial displacement of the central point of the membrane is obtained by following the procedure described in [8]. The piston position is monitored through the motor’s integrated encoder. A pressure sensor (MPX12 by Freescale Semiconductor with custom conditioning circuit) is used to measure the pressure in the pneumatic chamber.

The U-OWC oscillation \({z}_{S}\) is governed by Eq. (1), with the mass \(M(z)\) and the damping \(C\left(z,\dot{z}\right)\) terms taking the following specific forms [19]:

Compared to the model in Sect. 2.1, additional terms are here considered which account for hydraulic head losses according to the so-called instantaneous acceleration based model [29]. \({C}_{in}=0.13\) and \({C}_{dg}=0.71\) in Eq. (11), are empirical coefficients [29]. R_h1 and R_h2 are called hydraulic radii of the vertical duct and of the chamber respectively [29], and they are defined as follows:

The hydrodynamic parameters (\(A\left(\infty \right)\), \(K(t)\)) and the excitation pressure as a function of the wave parameters are computed based on the analytical model presented in [19].

As regards coupling of the simulation and the physical variables, Eq. (6) requires the piston position \({z}_{H}\) to be commanded as a function of the simulated water column displacement \({z}_{S}\) and the measured pressure, \({p}_{H}\), which is naturally affected by noise. The presence of a moderate noise on \({p}_{H}\) does not affect the calculation of \({z}_{S}\), since the solution of Eq. (1) involves a double integration operation, which results in a smooth profile for \({z}_{S}\). In contrast to that, an additional term proportional to \({p}_{H}\) in the expression of \({z}_{H}\) (Eq. (6)) can lead to vibrations (which are especially critical because of the friction between piston and cylinder) or to close-loop instability in case signal filtering is applied. To overcome this issue and for the sake of simplicity, we chose to replace the measured pressure in Eq. (6) with a static averaged experimental trend of the air chamber pressure, \({p}_{H}={\stackrel{\sim }{p}}_{H}({z}_{H})\), hence expressing \({z}_{H}\) solely as a function of \({z}_{S}\). This choice granted the achievement of a smooth motion of the piston (regardless of pressure signal filtering quality), while introducing a limited error.

The CD-DEG has been built using the procedure introduced in [10]. The sample enjoys a multi-layer structure (see Fig. 1.b) with two in-parallel dielectric layers separated by three carbon grease (MG-Chemicals 846) electrodes: two ground electrodes on the outer faces of the stack, and a HV electrode in the middle. Each dielectric layer (1.5 mm thick before pre-stretch) has been obtained by bonding three VHB 4905 layers together.

The CD-DEG sample’s measured capacitance \({C}_{min}\) in the flat equilibrium configuration was 76.5 nF. The DEG was driven via the circuit and control logics described in Sect. 2.4, using an in-parallel capacitance \({C}_{a}=300\) nF, and draining resistance \(R=100\) kΩ. A voltage \({V}_{0}=9\) kV was used to prime \({C}_{a}\), leading to maximum electric fields on the CD-DEG in the order of 130 kV/mm, similar to [10].

We performed a set of tests to validate the HIL setup and assess its robustness against possible latencies, delays in the control loop, and simplifications due to the coupling criteria presented in Sect. 2.2. Similarly to [12], we assessed the ability of the HIL framework and setup to reproduce the results of a reference hydro-elastic model of the U-OWC and the DEG PTO. The target model, presented and validated in [11] via sea tests, relies on the same hydrodynamic model (validated, e.g., in [29]) used in the present HIL formulation and on a numerical model of the CD-DEG. Compared to [11], in the reference model we slightly updated the CD-DEG elastic parameters in order to better capture the pressure-deformation response of the sample under investigation, so as to mitigate the effect of the elastic parameters uncertainty. We compared the oscillation amplitudes of some relevant variables for the target model and the HIL setup in a set of regular wave tests with periods between 1.5 s and 3.5 s spaced apart by 0.5 s. Figure 5 compares the oscillation amplitudes (HIL simulations vs. reference model) of the water column displacement (Fig. 5a) and the relative air pressure (Fig. 5b) for a wave height of 50 mm. HIL data in the plots are projected to the scenario scale using the rules defined in Sect. 2.2. The plots show that the HIL framework faithfully replicates the trend and the amplitudes of the model, hence proving the robustness and stability of the closed-loop coupling of the software model and the physical setup. Mismatches between the model and the HIL data in terms of the free surface elevation data are larger than those in the air pressure data (e.g., at 0.5 Hz, the model overestimates the free surface elevation, while reasonably capturing the pressure). This is ascribable to inaccuracies in the CD-DEG model, which affects the relationship between water surface and pressure. In fact, the CD-DEG model relies on strong simplifications, and is thus prone to errors, which are not presented in the HIL simulator, where the CD-DEG model is replaced with a physical prototype [30].

Interestingly, the mismatch between the reference model and the HIL results higher at wave frequencies of 0.4–0.5 Hz, where the oscillation amplitudes are at their maximum (i.e., the system operates at the mechanical resonance). This is most probably due to the CD-DEG viscous dissipations, which are not accounted for in the target model and can influence the system response in the presence of large water column oscillations and velocities typical of the resonance condition. This means that the accurate description of the response of the system at resonance via a mathematical model is rather difficult. At the same time, the behaviour at resonance is of high interest, since this is the condition at which the best power/efficiency performance is recorded. This, hence, offers a strong motivation towards the employment of a HIL approach as opposed to purely numerical models.

The HIL setup and framework has been used to characterise the response of the reference U-OWC/DEG system in different wave conditions. Specifically, two datasets illustrating the functionality of the HIL test-bench are shown in Fig. 6. The plots show portions of time-series of the measured variables (piston position \({z}_{H}\), air chamber pressure \({p}_{H}\), membrane tip displacement \({h}_{H}\), and voltage \({V}_{H}\)) in two tests in which the simulated U-OWC was subject to incident irregular waves compatible with a JONSWAP frequency spectrum [23], similar to the actual sea states experimentally observed at NOEL [31]. The plot refers to a sea state with significant wave height \({H}_{S}=0.15\) m, peak period \({T}_{p}=1.65\) s and peak enhancement factor of 3.3, which is representative of a mild sea condition at NOEL. The figure compares the results of two test runs: 1) a passive test with no voltage applied on the DEG; and 2) a power generation test, with electrically controlled DEG. The two datasets present an initial overlap, as no voltage is initially applied on the DEG in both tests. Following that, the application of a voltage on the DEG visibly affects the system dynamics, leading to a reduction in the average amplitude of the oscillations, mainly due to the CD-DEG PTO damping. As shown by the voltage time-series, the voltage \({V}_{1}\) reached by the CD-DEG after priming changes at each cycle, owing to the different deformation amplitudes reached, whereas the maximum voltage \({V}_{2}\) is approximately the same (i.e., around 7 kV) for the different cycles (excluding the initial transient), as observed in Sect. 2.4. The controller was programmed to prevent the activation of the CD-DEG during certain cycles in which the deformation was small (namely, the measured pressure module is below 200 Pa), since in those situations the electrical losses might exceed the generated energy. In the power generation test, the average generated power in this test was calculated as the sum of the energies over the different cycles (Eq. (9)) divided by the total duration of the test. The average generated power (over a time window of 120 s) was 0.5 W, with peaks of up to 1.2 W (calculated from the generated energy and the duration of the cycles with maximum deformation). Based on Eq. (8), this renders a power of 3.2 W (with peaks of 7.7 W) in the considered scenario, i.e. a significant fraction (over 15% in average) of the power available in the considered mild sea state.

Following that, we performed a systematic study of the coupled U-OWC/DEG dynamic response through regular wave tests. These tests are an effective tool to investigate the system frequency response [10], and they provide useful information that would be unavailable via sea tests with purely stochastic excitation. Consistently with the hardware setup limitations, we ran regular wave tests using a constant wave height \(H=50\) mm and wave periods within the typical frequency range at NOEL. Each regular wave test comprised a first phase in which the electrical control was not active (namely, no voltage was present on the CD-DEG sample), and a second phase in which the DEG was actively controlled. This allowed for a characterisation of the system steady-state mechanical passive response and the electrically active response.

Some relevant experimental time-series relative to two regular wave tests are shown in Fig. 7. In particular, the dataset in Fig. 7.a refers to a wave period \(T=2.5\) s (frequency \(f=0.4\) Hz), i.e., close to the system natural frequency (as observed in Sect. 3.3), whereas the dataset in Fig. 7.b refers to a wave period \(T=3.5\) s (frequency \(f=0.29\) Hz), i.e. below the system natural frequency. Applying electrical activation on the U-OWC/DEG has the double effect of causing a decrease in the CD-DEG stiffness and damping the system oscillations. As a consequence:

The air chamber pressure time-series further show visible drops in correspondence of the charging transients, as a result of the CD-DEG stiffness decrease due to electrical activation. The oscillatory behavior of the pressure in correspondence of the zero crossings is ascribable to a partial loss of tension in the membrane, possibly due to viscoelasticity.

An overview of regular wave test results is presented in Fig. 8, which shows the oscillation amplitudes of water column position, air pressure, and CD-DEG tip position (at the scale of the scenario) for the passive mechanical case and the electrically active case. The natural frequency in the presence of electrical activation slightly decreases (although in Fig. 8 the maxima stay at 0.4 Hz, because of the coarse resolution of the chosen frequency grid) and the oscillation amplitudes increase (decrease) at low (high) frequencies upon control application, because of electrically-induced variations in the system natural frequency. The peak values in the water column motion are lower in the presence of electrical activation, as a result of the PTO damping.

The power \({P}_{H}\) generated by the CD-DEG sample and the corresponding power output \({P}_{S}\) of the U-OWC plant are shown in Fig. 9. The generated power \({P}_{H}\) is estimated from the average energy generated during the different cycles (Eq. (9)), whereas \({P}_{S}\) is calculated using Eq. (8). The CD-DEG samples generated powers over 1 W, corresponding to a few Watts at the scenario scale. This is a rather small power considering the dimensions of the plant, but it represents a significant fraction of the input power of the mild incident waves considered in these tests. The present results hence confirm the effectiveness of the proposed control strategy for the CD-DEG and effectiveness of the DEG PTO in combination with a dynamical system and broadband mechanical power sources. Larger powers could be obtained (both in the scenario and with the HIL test-bench) by selecting DEGs with a larger thickness (subject to more energetic sea states), at the expense of a greater experimental burden.

The present results are encouraging in view of future sea tests implementing the scenario simulated in this article. Such tests are indeed feasible with the simplified manufacturing technology used here and in previous works [10] and with simple laboratory-scale electronics. These results suggest that, by selecting a suitable window of mild sea conditions, it would be possible to achieve power generation from the waves up to nearly 10 W. Despite representing a modest energy income in relation to the target U-OWC WEC, this would represent the largest power generation test ever realized with DEGs, and would pave the way to further research on DEG PTOs scaling-up via improved electronics, materials and manufacturing strategies.

In addition to that, the HIL setup might be employed in the simulation of larger scale systems, e.g., the full-scale DEG-based U-OWC design envisaged in [11]. In principle, this full-scale WEC can be simulated with the presented HIL setup, by simply relying on the coupling laws discussed in Sect. 2.2. In practice, this would require the use of a different DE material compared to the leaky and unreliable acrylic used in this study.