Wave Energy Converter Power Take-Off Modeling and Validation From Experimental Bench Tests

This article describes the implementation of a new numerical model of the power take-off system installed in the Monterey Bay Aquarium Research Institute wave energy converter, a device developed to provide power to various oceanic research missions. The simultaneous presence of hydraulic, pneumatic, and electrical subsystems in the power take-off system represents a significant challenge in forging an accurate model able to replicate the main dynamic characteristics of the system. The validation of the new numerical model is addressed by comparing simulations with the measurements obtained during a series of bench tests. Data from the bench tests show good agreement with the numerical model. The validated model provides deeper insights into the complex nonlinear dynamics of the power take-off system and will support further performance improvements in the future.

Wave Energy Converter Power Take-Off Modeling and Validation From Experimental Bench Tests Simone Giorgi , Ryan G. Coe , Meagan M. Reasoner , Giorgio Bacelli , Dominic D. Forbush , Scott Jensen, François Cazenave , and Andrew Hamilton Abstract-This article describes the implementation of a new numerical model of the power take-off system installed in the Monterey Bay Aquarium Research Institute wave energy converter, a device developed to provide power to various oceanic research missions.The simultaneous presence of hydraulic, pneumatic, and electrical subsystems in the power take-off system represents a significant challenge in forging an accurate model able to replicate the main dynamic characteristics of the system.The validation of the new numerical model is addressed by comparing simulations with the measurements obtained during a series of bench tests.Data from the bench tests show good agreement with the numerical model.The validated model provides deeper insights into the complex nonlinear dynamics of the power take-off system and will support further performance improvements in the future.

I. INTRODUCTION
T HE MBARI-WEC is a small two-body point absorber wave energy converter (WEC) developed, constructed, and deployed by the Monterey Bay Aquarium Research Institute (MBARI) [1].This WEC has been in development by MBARI for about ten years, specifically with the objective of demonstrating this technology for the purposes of powering oceanographic instrumentation.MBARI is active in autonomous underwater vehicle (AUV) development and the ultimate goal of this WEC advancement is to power fleets of AUVs in remote locations for extended ocean process studies.
The MBARI-WEC is composed of a surface buoy and a submerged heave cone, which are mechanically connected by a power take-off (PTO) unit, as shown in Fig. 1.The PTO system the design to improve performance.The previous work regarding the MBARI-WEC [1] presented two different numerical models of the full MBARI-WEC system, accounting for hydrodynamic, PTO, and mooring dynamics, where the PTO is considered as a simplified black-box model.As part of this project, a new numerical PTO model to better describe the complex nonlinear dynamics of the PTO system is being developed.The contribution of this article is the presentation of the new nonlinear, time-domain numerical model, which has been developed by taking into consideration the real physical structures of the hydraulic, pneumatic, and electric PTO subsystems.The new gray-box model of the PTO has been finalized by identifying from some specifically designed bench tests, carried out in the MBARI laboratory, some of the model parameters.Furthermore, the article includes a validation of the new numerical model by comparing the simulations with measurements obtained during a series of bench tests.These tests have been performed for each subsystem independently, as well as for the complete PTO system.The implemented PTO model, described in this work, will contribute to a better understanding of the system and new knowledge for further performance improvements.
The rest of the article is organized as follows.First, in Section II, the hydraulic, pneumatic, and electrical subsystems of the PTO system are described.Section III outlines the numerically implemented MATLAB-Simulink-Simscape model.A comparison of the bench tests and simulations is shown in Section IV.Finally, Section V concludes this article.

II. PTO SYSTEM
The MBARI-WEC is a two-body system composed of a buoy at the surface and a subsurface heave cone located about 30 m below (see Fig. 1).Relative motion induced by the surface waves between the buoy and heave cone acts on the WECs PTO.The PTO system is a combination of electrical, hydraulic, and pneumatic subsystems (see Fig. 2).When in motion, the hydraulic piston drives oil through a fixed-displacement hydraulic motor, which is directly coupled to a three-phase brushless electric motor.As shown in Fig. 2, the hydraulic piston and the pneumatic air spring piston are rigidly connected together.Acting together, the hydraulic and pneumatic components can provide a damping and spring force to the system, respectively.The hydraulic, pneumatic air spring, and electrical subsystems are described in further detail in Sections II-A, II-B, and II-C, respectively.

A. PTO Hydraulic Subsystem
A fixed-volume hydraulic system serves to transmit mechanical power between the piston rod and electrical motor/generator.The main components of this hydraulic subsystem are the hydraulic piston, relief valve, and hydraulic motor (see Fig. 2).The double-acting hydraulic ram has a piston rod on both sides, which results in an equal annular area of 8.87e-4 m 2 on both sides of the piston.The total piston stroke is equal to 2.03 m (see Table I).In this article, these two hydraulic piston chambers are called 1 and 2 (see Fig. 2).Fig. 2. MBARI-WEC PTO system divided into hydraulic, pneumatic air spring, and electrical subsystems.

TABLE I HYDRAULIC AND PNEUMATIC CYLINDER CHARACTERISTICS
The piston converts the heave motion of the buoy into a pressurized, bidirectional fluid flow, and the hydraulic motor converts the hydraulic fluid flow and pressure into rotational motion and torque.The shaft of the hydraulic motor is connected directly (no gearboxes are utilized) to the electric motor shaft, causing it to spin and generate electricity.Since there is no hydraulic rectification, the hydraulic and electric motors reverse rotary direction with a change in piston direction of motion.The hydraulic motor is a Parker F11-5 model, which is a bent axis (40 • angle between shaft and cylinder barrel) fixed displacement motor (see key parameters in Table II).The mechanical, volumetric, and total efficiency values are provided by the manufacturer in tabulated form and are shown in Fig. 3.The pressure relief valve shown in Fig. 2 is a critical safety component in the system that prevents an overspeed condition of the hydraulic and electric motors.To achieve this, the control electronics exert

B. Pneumatic Air Spring Subsystem
An air spring assembled in line with the hydraulic piston (see Fig. 4) is employed to accommodate the weight of the heave cone and maintain a taught connection between the buoy and heave cone.The air spring device is a pneumatic piston containing nitrogen gas in both piston chambers, having the rod only on one side of the piston head, which results in two different effective areas for chambers 3 and 4 (see Fig. 2 and Table I).The hydraulic and pneumatic pistons are rigidly connected together (the hydraulic and air spring subsystems have the same rod with a total moving mass of 48 kg).On both sides of the piston, a dead volume is present (see Table I).Apart from leakage across the piston that occurs at a very slow rate (days to weeks for a measurable change to occur), no mass flow occurs between chambers 3 and 4. As a first-order approximation, the restoring force provided by the air spring is about 8.9 kN when the rod extension is 1 m.

C. PTO Electrical Subsystem
As shown in Fig. 2, a Kollmorgen AKM63S brushless motor serves as the motor/generator in the system.This is a threephase permanent magnet motor with a maximum continuous current rating of 35 A (see motor characteristics in Table III).Electrical machines such as this are characterized by a linear relationship between the winding current I where k t is the torque constant, having a value (provided by the manufacturer) shown in Table III.
The three-phase motor windings are connected to a 10-kW (peak) 4-quadrant custom motor drive, which controls the motor winding current (and, therefore, indirectly the generated torque) to a set value by using field-oriented control techniques [2].While these are standard techniques in the permanent magnet servo motor industry, the power electronics that implement these behaviors in the MBARI-WEC have been custom developed and are implemented on a Microchip dsPIC33F microcontroller and DSP combination.The work to develop this custom motor driver was undertaken because the wave energy application at this scale has some very specific requirements, which are hard to meet using commercially available power electronics.In particular, the efficiency requirements of the system are important, especially at low speed because the rotor stops and changes direction with each wave.In this type of motor drive, switching losses in the transistors are a significant component of the total lost power, since these losses do not diminish at low speeds and low winding currents (such as other resistive losses in the system).For this reason, the MBARI motor driver turns OFF the transistor bridge at low speeds and torques to eliminate switching losses at these operating points.
Additional specialty features include the ability to bootstrap the system from a zero-speed and power-down condition.Because a WEC cannot be left unloaded without damage, the system needs to begin rotating and applying appropriate torques, even if the battery power to the system is not present due to depleted or disconnected batteries.Similarly, overspeed and overcurrent conditions must be safely handled should they occur, and the system needs to restart to normal operation on the next wave cycle without user intervention.All of these features and characteristics are somewhat specific to wave energy conversion and hard to achieve in commercially available motor-drive systems, which are often optimized for factory applications.
This system is a four-quadrant device that can act as a generator and convert mechanical energy to electrical energy, or as a motor in which electrical energy is converted to mechanical energy.By default, the system remains in generator mode but can be programmatically commanded to deliver any desired torque to the rotor, regardless of which torque-speed quadrant results.
When the prescribed electromagnetic torque opposes the motion of the motor, the system acts as a generator and boosts the dc bus voltage to 325 V for power transmission to the battery.In this default generation mode, the system is controlled to behave like a linear damper, in which the motor exerts a torque that is linearly proportional to the motor's rotational speed. ( Here, k d is the damping coefficient.In practice, the system implements a default damping coefficient that can be further adjusted by a scaling factor between 0.5 and 1.4.As per (1), I (rms) w is also linearly proportional to ω.When operating in generation mode, if the battery cannot accept the produced power (due to being full or disconnected), the energy is directed instead to a submerged load-dump, which is part of the PTO device.This load-dump resistor has a resistance of 7.2 Ω, and the current flowing to the load dump is controlled by a pulsewidth modulation technique.
Without programmatic intervention, the system will operate in the above-described generator mode; however, it is possible to override this in the system with an arbitrary torque command that may cause the system to operate in motor mode.In the case that the commanded torque and motor rotational speed are in the same direction, the system acts as a motor and drives the shaft with a torque proportional to the winding current, drawing energy out of the battery (up to a specified current limit).In all cases, the driver controller attempts to achieve the requested torque but will self-limit in some situations for device safety considerations.

III. SYSTEM MODELING AND IDENTIFICATION
The core of the PTO numerical model is an expression of Newton's second law applied to the moving piston rod.
Here, M is the total rod mass equal to 48 kg, x the rod displacement, f rod the external force applied on the rod (e.g., the force applied by the WEC float/heave-cone combination or by the laboratory hydraulic test piston during the bench tests), f as the air spring force, f hy the hydraulic force, and f fric the friction force.In (3), the positive direction corresponds to the extension direction of the piston rod (see Fig. 2).Since the piston is oriented horizontally in the laboratory, the gravitational force need not be considered.Note that when deployed at sea, the PTO is oriented vertically, therefore, necessitating the addition of a gravitational force in (3).This does not invalidate the lab tests as the performance of the device depends on the piston position and velocity only.
The hydraulic force f hy can be described as a function of p 1 and p 2 , respectively, the oil pressure in chambers 1 and 2 where p 1 and p 2 are numerically calculated from the interaction of the hydraulic and electrical subsystems (as shown in Fig. 2), which are described in Sections III-A and III-D, respectively.See Table I for numerical values of A 1 and A 2 .
Similarly, the pneumatic force f as can be determined from p 3 and p 4 , the nitrogen pressure in chambers 3 and 4 where p 3 and p 4 are calculated from the thermodynamic model of the air spring, explained in Section III-B (see Table I for numerical values of A 3 and A 4 ).The friction force f fric is evaluated from the hydraulicpneumatic rod friction model, described in Section III-C.The model described by (3) utilizes the rod displacement and velocity (x and ẋ) along with a specified electric-motor torque as inputs, and the total force applied to the rod f rod is the output.This arrangement allows straightforward integration into a simulation that includes the buoy and heave-cone dynamics and interactions with the environment, although that simulation is not included in this article.
The numerical model has been implemented using MATLAB-Simulink-Simscape software package and libraries.In particular, the Simscape software provides block libraries and special simulation tools for modeling physical systems.The model developed here relies on calibration from lab tests performed on the deployed hardware to determine various coefficients that are used to characterize the physical behavior of the systems.Sections III-A-III-D describe each subsystem model and the tests used to characterize these models.

A. Hydraulic PTO Subsystem Model
The hydraulic subsystem is modeled assuming the hydraulic oil is incompressible and isothermal; therefore, no fluid bulk modulus or thermodynamic effects are taken into account in this  model.The double-acting hydraulic cylinder model is based on (4) and the following equation [3]: where the subscript i assumes the value 1 or 2 (meaning chamber 1 or 2), q i is the chamber flow rate with direction convention shown in Fig. 5 and the numerical values of A 1 and A 2 are shown in Table I.The model assumes that the mechanical components are rigid with no deformations and that leakage flows are negligible.Furthermore, the friction between moving parts is not taken into account in the model of the double-acting hydraulic cylinder.Instead, the friction effect is evaluated specifically by the friction model described in Section III-C.The hydraulic motor is modeled as a constant volumetric displacement motor, able to operate in both forward and reverse directions depending on the rotation of the shaft.This component is able to extract or provide energy to the oil (working either as a motor or a pump), depending on the differential pressure applied across the device ports Δp and on the shaft rotational speed ω.Fig. 6 shows the four quadrant operations, where in quadrants I and III, the component works as a motor (Δp and ω have the same sign), and in quadrants II and IV , the component works as a pump (Δp and ω have opposite sign).Fig. 7 shows the model structure utilized for the hydraulic motor.For an ideal hydraulic motor, the volumetric flow q id and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the shaft torque τ id are given by q id = Dω (7a) where ω is the motor shaft speed, D is the cubic displacement per radian of shaft rotation, and Δp is the differential pressure applied across the motor ports.
Energy losses due to leakage flow and friction torque, present in any real hydraulic motor, are introduced into the model through the internal leakage volumetric flow rate q leak and the hydraulic motor friction torque τ (hm) fr q mot = q id + q leak (8a) where q mot is the net volumetric flow rate and τ mot is the net hydraulic motor torque.In particular, q leak and τ (hm) fr are computed by using tabulated values of the volumetric efficiency η v and mechanical efficiency η m , which are defined as [4], [5], [6] Both η v and η m vary with Δp and ω, as evident in the tabulated values provided by the motor manufacturer and shown in Fig. 3.
The relief valve has been modeled as an ideal component, which remains closed when the differential pressure between inlet and outlet ports is less than the preset pressure (equal to 200 bar in this system).When the differential pressure reaches the preset pressure, the valve opens and the differential pressure between inlet and outlet ports is stabilized at 200 bar.Note that in the simulations presented in this article, the relief valve never opens, since the differential pressure is persistently less than 200 bar.

B. Pneumatic (Air Spring) Subsystem Model
The air spring model is intended to replicate the main characteristics of the real system with minimum complexity and computation costs.In particular, the model must capture the presence of a hysteresis effect between the rod displacement/velocity and pneumatic force.This hysteresis can be attributed to the presence of heat exchange between the piston nitrogen chambers and the surroundings [7] and, therefore, depends upon the speed at which the piston moves.
The complex geometry of the air spring subsystem (see Section II-B, Fig. 4) is modeled as a simplified double-acting piston (see Fig. 8), having physical parameters shown in Table I.It is assumed that the piston casing is perfectly rigid and the moving interface is perfectly sealed (i.e., each chamber is a closed system with no material exchange with the surroundings).Furthermore, the gas pressure and temperature are assumed homogeneous in each chamber, and no friction between the rod and the cylinder is introduced into the air spring model (a separate friction model accounts for the air spring friction, as explained in Section III-C).The nitrogen is modeled as an ideal gas, obeying the equation of state [8], [9] pV = nRT (10) where p is the pressure, T the absolute temperature, V the volume, and R = 0.2968 kJ/(kg K) the nitrogen gas constant [8].
In each chamber, the gas pressure, volume, and temperature evolve based on (10) and on the energy balance equation (first law of thermodynamics) for a closed system [8], [9] Here, ΔU is the variation of the internal energy of a closed system (for an ideal gas, the internal energy is only a function of temperature), Q the amount of energy supplied to the system in the form of heat, and W the work done by the system on its Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.surroundings.Equation (11) shows that, in a closed system, the only way for the inner gas to exchange energy with the external environment is through the heat flow across the gas boundary and mechanical work (as the pressure-volume work done by the gas on the moving interface).
In the proposed simplified heat transfer model, it is assumed that each gas chamber transfers a total heat according to [8], [10] where the subscript i indicates chamber 3 or 4, Qi is the rate of heat transfer, T amb is the ambient temperature (assumed equal to 10 • C = 283.15K), T i is the chamber temperature, and K i is the heat transfer coefficient that characterizes the model.To complete the model, the coefficients K 3 and K 4 have been identified by fitting the model to experimental data collected during lab tests, which are characterized by a sequence of constant velocity motions (see Fig. 9).Figs. 10 and 11 show the comparison of the experimental data with the simulation results after the identification of K 3 and K 4 , where a good reproduction of the hysteresis effect is observed.

C. Hydraulic-Pneumatic Rod Friction Model
As described in Section II, the hydraulic piston and the pneumatic air spring are rigidly connected together by sharing the same rod (see Fig. 2).The rod and pistons are subjected to friction forces due to relative movement between surfaces in contact; in particular, the cylinder seals are the primary contributors.In some cases, the seals in this system are asymmetric, sealing pressure in only one direction.This causes a dependence on the direction of travel that can be seen in the test data.The hydraulic-pneumatic rod friction model is based on an assumption of a nonlinear algebraic relationship between rod velocity and friction force.In particular, the total friction force is modeled as the superposition of a Coulomb friction and a linear damping [11], [12], [13].The Coulomb friction is a constant friction contribution only dependent on the sign of the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.velocity (not the magnitude), as shown in Fig. 12(a).The viscous damping is assumed to be linearly proportional to the velocity [see Fig. 12(b)].The superposition of the Coulomb friction and linear damping results in a discontinuity around the origin [see Fig. 12(c)], which represents a challenge for the numerical solutions.Therefore, the discontinuity has been removed by adding a linear connection passing through the origin, as shown in Fig. 12(d).
From (3) to ( 5), the friction force can be expressed as which is utilized to provide an indirect measurement of the friction force from bench tests, since the time-evolution values of p 1 , p 2 , p 3 , p 4 , x, and f rod are measured.In particular, the experimental data from the bench tests with constant velocity motions are used.For different velocity magnitudes and directions, it is possible to extract the constant speed value S i and the constant friction force value F i .The S i and F i values can be used to fit two distinct linear models (one model for positive speed and one model for negative speed) utilizing a least-squares methodology [14].By carrying out a matrix of tests with different velocities and damping coefficients applied by the electric motor, five different friction models have been identified, showing a dependence of the friction model on the hydraulic loading of the system (see Fig. 13).The models shown in Fig. 13 have been averaged to obtain a single friction model applicable for all conditions.

D. Electrical Subsystem Model
The three-phase brushless motor is characterized by a linear relationship between the winding current and the generated Fig. 13.Least-squares fitting to the experimental data obtained from constant speed motions.The data points used correspond only to the constant-speed portions of the tests and each curve corresponds to a different amount of hydraulic pressure during the motions.As expected, a weak dependence on hydraulic pressure is exhibited.electromagnetic torque, as shown in (1).The friction torque present in the motor τ (em) fr can be modeled as the superposition of the Coulomb and viscous damping (a similar approach has been used for the rod friction model in Section III-C) [2], [13]: where k th determines how fast the tanh function changes from -1 to +1, τ c is the Coulomb friction torque, and k v the viscous coefficient (see Table III).Other models use sign(ω) instead of tanh(ω/k tanh ) but the model given by ( 14) exhibits a higher numerical stability [13].Fig. 14 shows the algebraic relationship between τ (em) fr and ω, described by (14).The core of the implemented electric PTO subsystem model is represented by a Simscape-Electrical motor and driver block, representing a generic motor and driver system, which abstracts the motor, driver, and control details, allowing for fast simulation at a system level (since the electrical dynamics are much faster than those of the rest of the system).
The real motor and driver system exhibits a torque saturation effect, visible on the plane speed-torque (see Fig. 15).This effect is due to the motor-driver actively limiting the amount of current that can be drawn from the battery system.This saturation effect parameterization is inserted into the Simscape block as a set of speed data points and corresponding maximum torque values.Another fundamental characteristic of the real motor and driver system is the internal total power losses, P tot loss , which is the summation of the mechanical losses P mec loss due to the electric motor friction force, and the electrical losses P elec loss due to the generated heat by the electrical components P tot loss = P mec loss + P elec loss (15) where the mechanical power loss due to friction is Note the presence of the minus sign in ( 16), because P mec loss is considered positive when leaving the system.The Simscape block is parameterized by providing tabulated values of P elec loss versus motor speed and torque, which can be identified from the experimental data.Indeed, P tot loss is the difference of the incoming mechanical power P mec and the outgoing electric power P elec (see Fig. 16)     Electric batteries are components based on distributed electrochemical reactions, occurring in the electrodes and electrolyte.The complex inner mechanisms of a battery exhibit an interdependency between variables, such as temperature distribution, electric charge concentration, voltage, electric current, and state of charge.Different battery models exist, characterized by various levels of complexity, accuracy, and computational cost [15], [16].In this effort, the battery and damping load (see Section II-C) have been significantly simplified, by modeling their combined behavior as a 325-V ideal constant voltage source, where voltage dependency to load variation, SoC, temperature, internal parameters, or any other transient phenomenon is neglected.

IV. COMPLETE PTO MODEL VALIDATION
Validation tests were conducted by using the experimentally measured rod speed as an input signal to the simulations together with the target torque for the PTO electric motor.A series of five sinusoidal rod oscillations were tested.In addition, the measured hydraulic and pneumatic chamber pressures p 1 , p 2 , p 3 , and p 4 are utilized as initial conditions in the simulations.The complete results are plotted in Fig. 18.In addition, a shorter time period (510 ≤ t ≤ 560) is plotted in Fig. 19 to provide a more detailed comparison between the measured and simulated results.
The matching between both the individual pneumatic chamber pressures and the overall pneumatic force shows very good agreement.The simulation's prediction for the hydraulic pressures shows that less accuracy has been achieved here compared to the pneumatic pressures.However, the discrepancy is less severe when the force due to the hydraulic pressures is considered.This indicates some offsetting errors in the simulation or system identification process but the resulting hydraulic force is the important quantity for the purpose of this simulator.
Similarly, the predicted friction force from the rod is fairly accurate, although there is some low amplitude high-frequency noise in the simulated friction force during periods when the rod velocity is zero.This is due to small velocities measured experimentally and applied to the simulation, which are amplified into greater friction force fluctuations by the large algebraic curve slope around the origin of the friction model (see Fig. 12-Pane 4).Since these high-frequency fluctuations are significant only when the rod piston velocity is small, their presence is not considered particularly problematic when evaluating the PTO performance.
The ability of the numerical model to accurately predict the electrical power is considered essential in evaluating the suitability of the numerical model.The simulated electric power, which is calculated as the product of V bus and I bus , has very good agreement with the experimental measurements.Note that the total electrical power is shown as the sum of power being delivered to the battery and power being dissipated by the load dump.The battery voltages and currents are not shown in this comparison because the specific battery behavior has not been included in this modeling effort.Instead, the battery is modeled as a constant voltage source and without internal resistance.This eliminates the challenge of matching the simulations to the exact state of charge of the batteries during the testing, at the expense of accuracy because the converter efficiency depends weakly on the bus voltage of the system.In practice, much of the electrical power generated during the testing is dissipated by the load-dump, so these shortcomings of the model do not appear in the electrical power comparison.A planned upgrade to the system is a change from the currently used lead-acid battery to a lithium iron phosphate system.This new battery has a significantly lower internal resistance so the swing in bus voltage with power will be considerably reduced.

V. CONCLUSION
This article describes the implementation of a new MATLAB-Simulink-Simscape model for the MBARI-WEC PTO, a system composed of hydraulic, pneumatic, and electrical subsystems, developed to provide power to various oceanic research missions.The model has been created by utilizing both manufacturer component datasheet values and information extracted from bench tests carried out on specific subsystems in the MBARI laboratory.The model is the result of the compromise between the conflicting requirements of having an accurate model without introducing excessive complexity in the model structure, which necessarily leads to the adoption of simplifying assumptions, in exchange for better numerical efficiency.
The validation of the model is carried out by comparing the simulation results with harmonically driven experimental tests.In general, the PTO model shows good performance in replicating the main dynamics of the real system.The implemented thermodynamic air spring model exhibits an excellent ability to predict the pneumatic force originated by the nitrogen chamber pressures and in repeating the hysteresis effect measured in the bench tests.The hydraulic subsystem shows the ability to model the frequency and phase of the measured hydraulic force, even if the predicted oscillation amplitude is persistently smaller than the measured one.
In general, the quantities predicted by the electrical subsystem model have a good resemblance to the experimental data.Indeed, the electric motor rotational speed is predicted with excellent agreement and the bus current shows a good similarity; furthermore, despite the fact the battery is modeled as an ideal constant voltage source, the electric power exported to the electric bus shows good agreement.
A number of areas of possible improvement of the model have been identified, which could lead to a higher prediction accuracy, which is as follows: 1) addition of some extra hydraulic loads (such as pipe restrictions) to increase the oil pressure and consequently the force generated by the hydraulic piston; 2) more complex friction model and/or filter to address highfrequency noise present near zero speed; 3) inclusion of internal resistance and state-of-charge modeling for the battery system.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 11 .
Fig. 11.Comparison of pneumatic spring simulation and test data illustrating the hysteresis effect represented on the velocity-force plane.

Fig. 12 .
Fig. 12. Hydraulic-pneumatic rod friction model.(a) Coulomb friction force.(b) Linear viscous friction force.(c) Superposition of Coulomb and linear viscous friction force, with discontinuity around the origin.(d) Utilized friction model obtained by removing the discontinuity around the origin.

Fig. 15 .
Fig.15.Electric motor test points (note operational envelope is limited by power saturation effect).

Fig. 16 .
Fig. 16.Block diagram of the PTO motor and driver electrical losses.

Fig. 17 .
Fig. 17.Power losses in the PTO motor and driver (from measurements).(a) Total losses.(b) Mechanical losses.(c) Electrical losses utilized to parameterize the Simscape-Electrical motor and driver block.Surface colors correlate with vertical axes.
(c) illustrates the electrical losses obtained from (18), which have been utilized to parameterize the Simscape-Electrical motor and driver block.

Fig. 18 .
Fig. 18.Validation results with measured rod velocity used to drive numerical simulation.

Fig. 19 .
Fig. 19.Validation results with measured rod velocity used to drive numerical simulation shown for a reduced time period to enhance detail.

Simone
Giorgi received the master's degree in electronics from the Università degli Studi di Ferrara, Ferrara, Italy, in 2002, and the master's and Ph.D. degrees in renewable energy systems from Maynooth University, Maynooth, Ireland, in 2012 and 2017, respectively.Since 2017, he has been a Renewable Energy Engineer and a Consultant with Ocean Wave Venture.His research interest focuses on the numerical modeling of wave energy devices, resource assessments, and techno-economic analyses.